On the variety generated by all nilpotent lattice-ordered groups
V.
V.
Bludov;
A.
M. W.
Glass
5179-5192
Abstract: In 1974, J. Martinez introduced the variety ${\mathcal W}$ of weakly Abelian lattice-ordered groups; it is defined by the identity $\displaystyle x^{-1}(y\vee 1)x\vee (y\vee 1)^2=(y\vee 1)^2.$
Bumpy metrics and closed parametrized minimal surfaces in Riemannian manifolds
John
Douglas
Moore
5193-5256
Abstract: The purpose of this article is to study conformal harmonic maps $f:\Sigma \rightarrow M$, where $\Sigma$ is a closed Riemann surface and $M$ is a compact Riemannian manifold of dimension at least four. Such maps define parametrized minimal surfaces, possibly with branch points. We show that when the ambient manifold $M$ is given a generic metric, all prime closed parametrized minimal surfaces are free of branch points, and are as Morse nondegenerate as allowed by the group of automorphisms of $\Sigma$. They are Morse nondegenerate in the usual sense if $\Sigma$ has genus at least two, lie on two-dimensional nondegenerate critical submanifolds if $\Sigma$ has genus one, and on six-dimensional nondegenerate critical submanifolds if $\Sigma$ has genus zero.
The 3-manifold recognition problem
Robert
J.
Daverman;
Thomas
L.
Thickstun
5257-5270
Abstract: We introduce a natural Relative Simplicial Approximation Property for maps from a 2-cell to a generalized 3-manifold and prove that, modulo the Poincaré Conjecture, 3-manifolds are precisely the generalized 3-manifolds satisfying this approximation property. The central technical result establishes that every generalized 3-manifold with this Relative Simplicial Approximation Property is the cell-like image of some generalized 3-manifold having just a 0-dimensional set of nonmanifold singularities.
Low-pass filters and representations of the Baumslag Solitar group
Dorin
Ervin
Dutkay
5271-5291
Abstract: We analyze representations of the Baumslag Solitar group $\displaystyle BS(1,N)=\langle u,t\,\vert\,utu^{-1}=t^N\rangle$ that admit wavelets and show how such representations can be constructed from a given low-pass filter. We describe the direct integral decomposition for some examples and derive from it a general criterion for the existence of solutions for scaling equations. As another application, we construct a Fourier transform for some Hausdorff measures.
Sign-changing critical points from linking type theorems
M.
Schechter;
W.
Zou
5293-5318
Abstract: In this paper, the relationships between sign-changing critical point theorems and the linking type theorems of M. Schechter and the saddle point theorems of P. Rabinowitz are established. The abstract results are applied to the study of the existence of sign-changing solutions for the nonlinear Schrödinger equation $-\Delta u +V(x)u = f(x, u), u \in H^1({\mathbf{R}}^N),$ where $f(x, u)$ is a Carathéodory function. Problems of jumping or oscillating nonlinearities and of double resonance are considered.
Scattering theory for the elastic wave equation in perturbed half-spaces
Mishio
Kawashita;
Wakako
Kawashita;
Hideo
Soga
5319-5350
Abstract: In this paper we consider the linear elastic wave equation with the free boundary condition (the Neumann condition), and formulate a scattering theory of the Lax and Phillips type and a representation of the scattering kernel. We are interested in surface waves (the Rayleigh wave, etc.) connected closely with situations of boundaries, and make the formulations intending to extract this connection. The half-space is selected as the free space, and making dents on the boundary is considered as a perturbation from the flat one. Since the lacuna property for the solutions in the outgoing and incoming spaces does not hold because of the existence of the surface waves, instead of it, certain decay estimates for the free space solutions and a weak version of the Morawetz arguments are used to formulate the scattering theory. We construct the representation of the scattering kernel with outgoing scattered plane waves. In this step, again because of the existence of the surface waves, we need to introduce new outgoing and incoming conditions for the time dependent solutions to ensure uniqueness of the solutions. This introduction is essential to show the representation by reasoning similar to the case of the reduced wave equation.
Boundary relations and their Weyl families
Vladimir
Derkach;
Seppo
Hassi;
Mark
Malamud;
Henk
de Snoo
5351-5400
Abstract: The concepts of boundary relations and the corresponding Weyl families are introduced. Let $S$ be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space $\mathfrak{H}$, let $\mathcal{H}$ be an auxiliary Hilbert space, let $\displaystyle J_\mathfrak{H}=\begin{pmatrix}0&-iI_\mathfrak{H} iI_\mathfrak{H} & 0\end{pmatrix},$ and let $J_\mathcal{H}$ be defined analogously. A unitary relation $\Gamma$ from the Krein space $(\mathfrak{H}^2,J_\mathfrak{H})$ to the Krein space $(\mathcal{H}^2,J_\mathcal{H})$ is called a boundary relation for the adjoint $S^*$ if $\ker \Gamma=S$. The corresponding Weyl family $M(\lambda)$ is defined as the family of images of the defect subspaces $\widehat{\mathfrak{N}}_\lambda$, $\lambda\in \mathbb{C}\setminus\mathbb{R}$, under $\Gamma$. Here $\Gamma$ need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space $\mathcal{H}$ and the class of unitary relations $\Gamma:(\mathfrak{H}^2,J_\mathfrak{H})\to(\mathcal{H}^2,J_\mathcal{H})$, it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every $\mathcal{H}$-valued maximal dissipative (for $\lambda\in\mathbb{C}_+$) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.
Hilbert functions of points on Schubert varieties in the symplectic Grassmannian
Sudhir
R.
Ghorpade;
K.
N.
Raghavan
5401-5423
Abstract: We give an explicit combinatorial description of the multiplicity as well as the Hilbert function of the tangent cone at any point on a Schubert variety in the symplectic Grassmannian.
The braid index is not additive for the connected sum of 2-knots
Seiichi
Kamada;
Shin
Satoh;
Manabu
Takabayashi
5425-5439
Abstract: Any $2$-dimensional knot $K$ can be presented in a braid form, and its braid index, ${Braid}(K)$, is defined. For the connected sum $K_1\char93 K_2$ of $2$-knots $K_1$ and $K_2$, it is easily seen that ${Braid}(K_1\char93 K_2)\leq {B}(K_1) + {B}(K_2) -1$ holds. Birman and Menasco proved that the braid index (minus one) is additive for the connected sum of $1$-dimensional knots; the equality holds for $1$-knots. We prove that the equality does not hold for $2$-knots unless $K_1$ or $K_2$ is a trivial $2$-knot. We also prove that the $2$-knot obtained from a granny knot by Artin's spinning is of braid index $4$, and there are infinitely many $2$-knots of braid index $4$.
Deformation theory of abelian categories
Wendy
Lowen;
Michel
Van den Bergh
5441-5483
Abstract: In this paper we develop the basic infinitesimal deformation theory of abelian categories. This theory yields a natural generalization of the well-known deformation theory of algebras developed by Gerstenhaber. As part of our deformation theory we define a notion of flatness for abelian categories. We show that various basic properties are preserved under flat deformations, and we construct several equivalences between deformation problems.
On a class of special linear systems of $\mathbb{P}^3$
Antonio
Laface;
Luca
Ugaglia
5485-5500
Abstract: In this paper we deal with linear systems of $\mathbb{P}^3$ through fat points. We consider the behavior of these systems under a cubic Cremona transformation that allows us to produce a class of special systems which we conjecture to be the only ones.
Fourier expansions of functions with bounded variation of several variables
Leonardo
Colzani
5501-5521
Abstract: In the first part of the paper we establish the pointwise convergence as $t\rightarrow +\infty$ for convolution operators $\int_{\mathbb{R}^{d}}t^{d}K\left( ty\right) \varphi (x-y)dy$ under the assumptions that $\varphi (y)$ has integrable derivatives up to an order $\alpha$ and that $\left\vert K(y)\right\vert \leq c\left( 1+\left\vert y\right\vert \right) ^{-\beta }$ with $\alpha +\beta >d$. We also estimate the Hausdorff dimension of the set where divergence may occur. In particular, when the kernel is the Fourier transform of a bounded set in the plane, we recover a two-dimensional analog of the Dirichlet theorem on the convergence of Fourier series of functions with bounded variation. In the second part of the paper we prove an equiconvergence result between Fourier integrals on euclidean spaces and expansions in eigenfunctions of elliptic operators on manifolds, which allows us to transfer some of the results proved for Fourier integrals to eigenfunction expansions. Finally, we present some examples of different behaviors between Fourier integrals, Fourier series and spherical harmonic expansions.
Singularities of linear systems and the Waring problem
Massimiliano
Mella
5523-5538
Abstract: The Waring problem for homogeneous forms asks for additive decomposition of a form $f$ into powers of linear forms. A classical problem is to determine when such a decomposition is unique. In this paper we answer this question when the degree of $f$ is greater than the number of variables. To do this we translate the algebraic statement into a geometric one concerning the singularities of linear systems of $\mathbb{P}^n$ with assigned singularities.
Blaschke- and Minkowski-endomorphisms of convex bodies
Markus
Kiderlen
5539-5564
Abstract: We consider maps of the family of convex bodies in Euclidean $d$-dimensional space into itself that are compatible with certain structures on this family: A Minkowski-endomorphism is a continuous, Minkowski-additive map that commutes with rotations. For $d\ge 3$, a representation theorem for such maps is given, showing that they are mixtures of certain prototypes. These prototypes are obtained by applying the generalized spherical Radon transform to support functions. We give a complete characterization of weakly monotonic Minkowski-endomorphisms. A corresponding theory is developed for Blaschke-endomorphisms, where additivity is now understood with respect to Blaschke-addition. Using a special mixed volume, an adjoining operator can be introduced. This operator allows one to identify the class of Blaschke-endomorphisms with the class of weakly monotonic, non-degenerate and translation-covariant Minkowski-endomorphisms. The following application is also shown: If a (weakly monotonic and) non-trivial endomorphism maps a convex body to a homothet of itself, then this body must be a ball.
Varieties with small discriminant variety
Antonio
Lanteri;
Roberto
Muñoz
5565-5585
Abstract: Let $X$ be a smooth complex projective variety, let $L$ be an ample and spanned line bundle on $X$, $V\subseteq H^{0}(X,L)$ defining a morphism $\phi _{V}:X \to \mathbb{P}^{N}$ and let $\mathcal{D}(X,V)$ be its discriminant locus, the variety parameterizing the singular elements of $\vert V\vert$. We present two bounds on the dimension of $\mathcal{D}(X,V)$ and its main component relying on the geometry of $\phi _{V}(X) \subset \mathbb{P}^{N}$. Classification results for triplets $(X,L,V)$ reaching the bounds as well as significant examples are provided.
Asymptotic compactness and absorbing sets for 2D stochastic Navier-Stokes equations on some unbounded domains
Zdzislaw
Brzezniak;
Yuhong
Li
5587-5629
Abstract: We introduce a notion of an asymptotically compact (AC) random dynamical system (RDS). We prove that for an AC RDS the $\Omega$-limit set $\Omega_B(\omega)$ of any bounded set $B$ is nonempty, compact, strictly invariant and attracts the set $B$. We establish that the $2$D Navier Stokes Equations (NSEs) in a domain satisfying the Poincaré inequality perturbed by an additive irregular noise generate an AC RDS in the energy space $\mathrm{H}$. As a consequence we deduce existence of an invariant measure for such NSEs. Our study generalizes on the one hand the earlier results by Flandoli-Crauel (1994) and Schmalfuss (1992) obtained in the case of bounded domains and regular noise, and on the other hand the results by Rosa (1998) for the deterministic NSEs.
Corrigendum to ``West's problem on equivariant hyperspaces and Banach-Mazur compacta''
Sergey
Antonyan
5631-5633
Corrections to ``Involutions fixing $\mathbb{RP}^{\text{odd}}\sqcup P(h,i)$, II''
Bo
Chen;
Zhi
Lü
5635-5638
Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups
Mara
D.
Neusel
4689-4720
Abstract: We consider purely inseparable extensions $\textrm{H}\hookrightarrow \sqrt[\mathscr{P}^*]{\textrm{H}}$ of unstable Noetherian integral domains over the Steenrod algebra. It turns out that there exists a finite group $G\le\textrm{GL}(V)$ and a vector space decomposition $V=W_0\oplus W_1\oplus\dotsb\oplus W_e$ such that $\overline{\textrm{H}}=(\mathbb{F}[W_0] \otimes\mathbb{F}[W_1]^p\otimes\dotsb\otimes\mathbb{F}[W_e]^{p^e})^G$ and $\overline{\sqrt[\mathscr{P}^*]{\textrm{H}}}=\mathbb{F}[V]^G$, where $\overline{(-)}$ denotes the integral closure. Moreover, $\textrm{H}$ is Cohen-Macaulay if and only if $\sqrt[\mathscr{P}^*]{\textrm{H}}$ is Cohen-Macaulay. Furthermore, $\overline{\textrm{H}}$ is polynomial if and only if $\sqrt[\mathscr{P}^*]{\textrm{H}}$ is polynomial, and $\sqrt[\mathscr{P}^*]{\textrm{H}}=\mathbb{F}[h_1,\dotsc,h_n]$ if and only if $\displaystyle \textrm{H}=\mathbb{F}[h_1,\dotsc,h_{n_0},h_{n_0+1}^p,\dotsc,h_{n_1}^p, h_{n_1+1}^{p^2},\dotsc,h_{n_e}^{p^e}],$ where $n_e=n$ and $n_i=\dim_{\mathbb{F}}(W_0\oplus\dotsb\oplus W_i)$.
Groupoid cohomology and extensions
Jean-Louis
Tu
4721-4747
Abstract: We show that Haefliger's cohomology for étale groupoids, Moore's cohomology for locally compact groups and the Brauer group of a locally compact groupoid are all particular cases of sheaf (or Cech) cohomology for topological simplicial spaces.
A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory
G.
C.
Bell;
A.
N.
Dranishnikov
4749-4764
Abstract: We prove an asymptotic analog of the classical Hurewicz theorem on mappings that lower dimension. This theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite-dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension. As applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their Hirsch length. We are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups, amalgamated free products and the hyperbolization of metric spaces possessing the Higson property.
On neoclassical Schottky groups
Rubén
Hidalgo;
Bernard
Maskit
4765-4792
Abstract: The goal of this paper is to describe a theoretical construction of an infinite collection of non-classical Schottky groups. We first show that there are infinitely many non-classical noded Schottky groups on the boundary of Schottky space, and we show that infinitely many of these are ``sufficiently complicated''. We then show that every Schottky group in an appropriately defined relative conical neighborhood of any sufficiently complicated noded Schottky group is necessarily non-classical. Finally, we construct two examples; the first is a noded Riemann surface of genus $3$ that cannot be uniformized by any neoclassical Schottky group (i.e., classical noded Schottky group); the second is an explicit example of a sufficiently complicated noded Schottky group in genus $3$.
On the characterization of the kernel of the geodesic X-ray transform
Eduardo
Chappa
4793-4807
Abstract: Let $\overline{\Omega}$ be a compact manifold with boundary. We consider covariant symmetric tensor fields of order two that belong to a Sobolev space $H^{k}(\overline{\Omega}), k \geq 1$. We prove, under the assumption that the metric is simple, that solenoidal tensor fields that belong to the kernel of the geodesic X-ray transform are smooth up to the boundary. As a corollary we obtain that they form a finite-dimensional set in $H^{k}$.
On $C^\infty$ and Gevrey regularity of sublaplacians
A.
Alexandrou
Himonas;
Gerson
Petronilho
4809-4820
Abstract: In this paper we consider zero order perturbations of a class of sublaplacians on the two-dimensional torus and give sufficient conditions for global $C^\infty$ regularity to persist. In the case of analytic coefficients, we prove Gevrey regularity for a general class of sublaplacians when the finite type condition holds.
Stationary isothermic surfaces and uniformly dense domains
R.
Magnanini;
J.
Prajapat;
S.
Sakaguchi
4821-4841
Abstract: We establish a relationship between stationary isothermic surfaces and uniformly dense domains. A stationary isothermic surface is a level surface of temperature which does not evolve with time. A domain $\Omega$ in the $N$-dimensional Euclidean space $\mathbb{R}^N$ is said to be uniformly dense in a surface $\Gamma\subset\mathbb{R}^N$ of codimension $1$ if, for every small $r>0,$ the volume of the intersection of $\Omega$ with a ball of radius $r$ and center $x$ does not depend on $x$ for $x\in\Gamma.$ We prove that the boundary of every uniformly dense domain which is bounded (or whose complement is bounded) must be a sphere. We then examine a uniformly dense domain with unbounded boundary $\partial\Omega$, and we show that the principal curvatures of $\partial\Omega$ satisfy certain identities. The case in which the surface $\Gamma$ coincides with $\partial\Omega$ is particularly interesting. In fact, we show that, if the boundary of a uniformly dense domain is connected, then (i) if $N=2$, it must be either a circle or a straight line and (ii) if $N=3,$ it must be either a sphere, a spherical cylinder or a minimal surface. We conclude with a discussion on uniformly dense domains whose boundary is a minimal surface.
$W^{2,p}$--estimates for the linearized Monge--Ampère equation
Cristian
E.
Gutiérrez;
Federico
Tournier
4843-4872
Abstract: Let $\Omega \subseteq \mathbb{R}^n$ be a strictly convex domain and let $\phi \in C^2(\Omega)$ be a convex function such that $\lambda \leq$ det$D^2\phi \leq\Lambda$ in $\Omega$. The linearized Monge-Ampère equation is $\displaystyle L_{\Phi}u=\textrm{trace}(\Phi D^2u)=f,$ where $\Phi = ($det$D^2\phi)(D^2\phi)^{-1}$ is the matrix of cofactors of $D^2\phi$. We prove that there exist $p>0$ and $C>0$ depending only on $n,\lambda,\Lambda$, and $\textrm{dist}(\Omega^\prime,\Omega)$ such that $\displaystyle \Vert D^2u\Vert _{L^p(\Omega^\prime)}\leq C(\Vert u\Vert _{L^\infty(\Omega)}+\Vert f\Vert _{L^n(\Omega)})$ for all solutions $u\in C^2(\Omega)$ to the equation $L_{\Phi}u=f$.
Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators
Andrea
Pascucci;
Sergio
Polidoro
4873-4893
Abstract: We prove a global Harnack inequality for a class of degenerate evolution operators by repeatedly using an invariant local Harnack inequality. As a consequence we obtain an accurate Gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators.
The intersection of a matroid and a simplicial complex
Ron
Aharoni;
Eli
Berger
4895-4917
Abstract: A classical theorem of Edmonds provides a min-max formula relating the maximal size of a set in the intersection of two matroids to a ``covering" parameter. We generalize this theorem, replacing one of the matroids by a general simplicial complex. One application is a solution of the case $r=3$ of a matroidal version of Ryser's conjecture. Another is an upper bound on the minimal number of sets belonging to the intersection of two matroids, needed to cover their common ground set. This, in turn, is used to derive a weakened version of a conjecture of Rota. Bounds are also found on the dual parameter--the maximal number of disjoint sets, all spanning in each of two given matroids. We study in detail the case in which the complex is the complex of independent sets of a graph, and prove generalizations of known results on ``independent systems of representatives" (which are the special case in which the matroid is a partition matroid). In particular, we define a notion of $k$-matroidal colorability of a graph, and prove a fractional version of a conjecture, that every graph $G$ is $2\Delta(G)$-matroidally colorable. The methods used are mostly topological.
Semifree symplectic circle actions on $4$-orbifolds
L.
Godinho
4919-4933
Abstract: A theorem of Tolman and Weitsman states that all symplectic semifree circle actions with isolated fixed points on compact symplectic manifolds must be Hamiltonian and have the same equivariant cohomology and Chern classes of $(\mathbb{C}P^1)^n$ equipped with the standard diagonal circle action. In this paper, we show that the situation is much different when we consider compact symplectic orbifolds. Focusing on $4$-orbifolds with isolated cone singularities, we show that such actions, besides being Hamiltonian, can now be obtained from either $S^2\times S^2$ or a weighted projective space, or a quotient of one of these spaces by a finite cyclic group, by a sequence of special weighted blow-ups at fixed points. In particular, they can have any number of fixed points.
The invariant factors of the incidence matrices of points and subspaces in $\operatorname{PG}(n,q)$ and $\operatorname{AG}(n,q)$
David
B.
Chandler;
Peter
Sin;
Qing
Xiang
4935-4957
Abstract: We determine the Smith normal forms of the incidence matrices of points and projective $(r-1)$-dimensional subspaces of $\operatorname{PG}(n,q)$ and of the incidence matrices of points and $r$-dimensional affine subspaces of $\operatorname{AG}(n,q)$ for all $n$, $r$, and arbitrary prime power $q$.
Open loci of graded modules
Christel
Rotthaus;
Liana
M.
Sega
4959-4980
Abstract: Let $A=\bigoplus_{i\in \mathbb{N}}A_i$ be an excellent homogeneous Noetherian graded ring and let $M=\bigoplus_{n\in \mathbb{Z}}M_n$ be a finitely generated graded $A$-module. We consider $M$ as a module over $A_0$ and show that the $(S_k)$-loci of $M$ are open in $\operatorname{Spec}(A_0)$. In particular, the Cohen-Macaulay locus $U^0_{CM}=\{{\mathfrak{p}}\in \operatorname{Spec}(A_0) \mid M_\mathfrak{p} \;$ is Cohen-Macaulay$\}$ is an open subset of $\operatorname{Spec}(A_0)$. We also show that the $(S_k)$-loci on the homogeneous parts $M_n$ of $M$ are eventually stable. As an application we obtain that for a finitely generated Cohen-Macaulay module $M$ over an excellent ring $A$ and for an ideal $I\subseteq A$ which is not contained in any minimal prime of $M$, the $(S_k)$-loci for the modules $M/I^nM$ are eventually stable.
Partitioning $\alpha$--large sets: Some lower bounds
Teresa
Bigorajska;
Henryk
Kotlarski
4981-5001
Abstract: Let $\alpha\rightarrow(\beta)_m^r$ denote the property: if $A$ is an $\alpha$-large set of natural numbers and $[A]^r$ is partitioned into $m$ parts, then there exists a $\beta$-large subset of $A$ which is homogeneous for this partition. Here the notion of largeness is in the sense of the so-called Hardy hierarchy. We give a lower bound for $\alpha$ in terms of $\beta,m,r$ for some specific $\beta$.
On almost one-to-one maps
Alexander
Blokh;
Lex
Oversteegen;
E.
D.
Tymchatyn
5003-5014
Abstract: A continuous map $f:X\to Y$ of topological spaces $X, Y$ is said to be almost $1$-to-$1$ if the set of the points $x\in X$ such that $f^{-1}(f(x))=\{x\}$ is dense in $X$; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and $\sigma$-compact spaces (e.g., $n$-manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if $f$ is a minimal self-mapping of a 2-manifold $M$, then point preimages under $f$ are tree-like continua and either $M$ is a union of 2-tori, or $M$ is a union of Klein bottles permuted by $f$.
Multiplier ideals of hyperplane arrangements
Mircea
Mustata
5015-5023
Abstract: In this note we compute multiplier ideals of hyperplane arrangements. This is done using the interpretation of multiplier ideals in terms of spaces of arcs by Ein, Lazarsfeld, and Mustata (2004).
Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes
Tadeusz
Kulczycki;
Bartlomiej
Siudeja
5025-5057
Abstract: Let $X_t$ be the relativistic $\alpha$-stable process in $\mathbf{R}^d$, $\alpha \in (0,2)$, $d > \alpha$, with infinitesimal generator $H_0^{(\alpha)}= - ((-\Delta +m^{2/\alpha})^{\alpha/2}-m)$. We study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup $T_t$ for this process with generator $H_0^{(\alpha)} - V$, $V \ge 0$, $V$ locally bounded. We prove that if $\lim_{\vert x\vert \to \infty} V(x) = \infty$, then for every $t >0$ the operator $T_t$ is compact. We consider the class $\mathcal{V}$ of potentials $V$ such that $V \ge 0$, $\lim_{\vert x\vert \to \infty} V(x) = \infty$ and $V$ is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For $V$ in the class $\mathcal{V}$ we show that the semigroup $T_t$ is IU if and only if $\lim_{\vert x\vert \to \infty} V(x)/\vert x\vert = \infty$. If this condition is satisfied we also obtain sharp estimates of the first eigenfunction $\phi_1$ for $T_t$. In particular, when $V(x) = \vert x\vert^{\beta}$, $\beta > 0$, then the semigroup $T_t$ is IU if and only if $\beta >1$. For $\beta >1$ the first eigenfunction $\phi_1(x)$ is comparable to $\displaystyle \exp(-m^{1/{\alpha}}\vert x\vert) \, (\vert x\vert + 1)^{(-d - \alpha - 2 \beta -1 )/2}.$
Ratio limit theorem for parabolic horn-shaped domains
Pierre
Collet;
Servet
Martinez;
Jaime
San Martin
5059-5082
Abstract: We prove that for horn-shaped domains of parabolic type, the ratio of the heat kernel at different fixed points has a limit when the time tends to infinity. We also give an explicit formula for the limit in terms of the harmonic functions.
Frequently hypercyclic operators
Frédéric
Bayart;
Sophie
Grivaux
5083-5117
Abstract: We investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators $T$ on separable complex $\mathcal{F}$-spaces: $T$ is frequently hypercyclic if there exists a vector $x$ such that for every nonempty open subset $U$ of $X$, the set of integers $n$ such that $T^{n}x$ belongs to $U$ has positive lower density. We give several criteria for frequent hypercyclicity, and this leads us in particular to study linear transformations from the point of view of ergodic theory. Several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting.
Partial hyperbolicity or dense elliptic periodic points for $C^1$-generic symplectic diffeomorphisms
Radu
Saghin;
Zhihong
Xia
5119-5138
Abstract: We prove that if a symplectic diffeomorphism is not partially hyperbolic, then with an arbitrarily small $C^1$ perturbation we can create a totally elliptic periodic point inside any given open set. As a consequence, a $C^1$-generic symplectic diffeomorphism is either partially hyperbolic or it has dense elliptic periodic points. This extends the similar results of S. Newhouse in dimension 2 and M.-C. Arnaud in dimension 4. Another interesting consequence is that stably ergodic symplectic diffeomorphisms must be partially hyperbolic, a converse to Shub-Pugh's stable ergodicity conjecture for the symplectic case.
Jointly hyponormal pairs of commuting subnormal operators need not be jointly subnormal
Raúl
E.
Curto;
Jasang
Yoon
5139-5159
Abstract: We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of R. Curto, P. Muhly and J. Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality.
Inverse scattering with fixed energy and an inverse eigenvalue problem on the half-line
Miklós
Horváth
5161-5177
Abstract: Recently A. G. Ramm (1999) has shown that a subset of phase shifts $\delta_l$, $l=0,1,\ldots$, determines the potential if the indices of the known shifts satisfy the Müntz condition $\sum_{l\neq0,l\in L}\frac{1}{l}=\infty$. We prove the necessity of this condition in some classes of potentials. The problem is reduced to an inverse eigenvalue problem for the half-line Schrödinger operators.
Paley--Wiener theorems for the Dunkl transform
Marcel
de Jeu
4225-4250
Abstract: We conjecture a geometrical form of the Paley-Wiener theorem for the Dunkl transform and prove three instances thereof, by using a reduction to the one-dimensional even case, shift operators, and a limit transition from Opdam's results for the graded Hecke algebra, respectively. These Paley-Wiener theorems are used to extend Dunkl's intertwining operator to arbitrary smooth functions. Furthermore, the connection between Dunkl operators and the Cartan motion group is established. It is shown how the algebra of radial parts of invariant differential operators can be described explicitly in terms of Dunkl operators. This description implies that the generalized Bessel functions coincide with the spherical functions. In this context of the Cartan motion group, the restriction of Dunkl's intertwining operator to the invariants can be interpreted in terms of the Abel transform. We also show that, for certain values of the multiplicities of the restricted roots, the Abel transform is essentially inverted by a differential operator.
Whitney towers and gropes in 4--manifolds
Rob
Schneiderman
4251-4278
Abstract: Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2-complexes called gropes. This paper describes a precise correspondence between embedded gropes in 4-manifolds and the failure of the Whitney move in terms of iterated `towers' of Whitney disks. The `flexibility' of these Whitney towers is used to demonstrate some geometric consequences for knot and link concordance connected to $n$-solvability, $k$-cobordism and grope concordance. The key observation is that the essential structure of gropes and Whitney towers can be described by embedded unitrivalent trees which can be controlled during surgeries and Whitney moves. It is shown that a Whitney move in a Whitney tower induces an IHX (Jacobi) relation on the embedded trees.
Root invariants in the Adams spectral sequence
Mark
Behrens
4279-4341
Abstract: Let $E$ be a ring spectrum for which the $E$-Adams spectral sequence converges. We define a variant of Mahowald's root invariant called the `filtered root invariant' which takes values in the $E_1$ term of the $E$-Adams spectral sequence. The main theorems of this paper are concerned with when these filtered root invariants detect the actual root invariant, and explain a relationship between filtered root invariants and differentials and compositions in the $E$-Adams spectral sequence. These theorems are compared to some known computations of root invariants at the prime $2$. We use the filtered root invariants to compute some low-dimensional root invariants of $v_1$-periodic elements at the prime $3$. We also compute the root invariants of some infinite $v_1$-periodic families of elements at the prime $3$.
Functional distribution of $L(s, \chi_d)$ with real characters and denseness of quadratic class numbers
Hidehiko
Mishou;
Hirofumi
Nagoshi
4343-4366
Abstract: We investigate the functional distribution of $L$-functions $L(s, \chi_d)$ with real primitive characters $\chi_d$ on the region $1/2 < \operatorname{Re} s <1$ as $d$ varies over fundamental discriminants. Actually we establish the so-called universality theorem for $L(s, \chi_d)$ in the $d$-aspect. From this theorem we can, of course, deduce some results concerning the value distribution and the non-vanishing. As another corollary, it follows that for any fixed $a, b$ with $1/2< a< b<1$ and positive integers $r', m$, there exist infinitely many $d$ such that for every $L^{(r)} (s, \chi_d)$ has at least $m$ zeros on the interval $[a, b]$ in the real axis. We also study the value distribution of $L(s, \chi_d)$ for fixed $s$ with $\operatorname{Re} s =1$ and variable $d$, and obtain the denseness result concerning class numbers of quadratic fields.
Frankel's theorem in the symplectic category
Min
Kyu
Kim
4367-4377
Abstract: We prove that if an $(n-1)$-dimensional torus acts symplectically on a $2n$-dimensional symplectic manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regard it as a symplectic version of Frankel's theorem which says that a Kähler circle action has a fixed point if and only if it is Hamiltonian. The case of $n=2$ is the well-known theorem by McDuff.
$\mathbf{h}$-principles for hypersurfaces with prescribed principle curvatures and directions
Mohammad
Ghomi;
Marek
Kossowski
4379-4393
Abstract: We prove that any compact orientable hypersurface with boundary immersed (resp. embedded) in Euclidean space is regularly homotopic (resp. isotopic) to a hypersurface with principal directions which may have any prescribed homotopy type, and principal curvatures each of which may be prescribed to within an arbitrary small error of any constant. Further we construct regular homotopies (resp. isotopies) which control the principal curvatures and directions of hypersurfaces in a variety of ways. These results, which we prove by holonomic approximation, establish some h-principles in the sense of Gromov, and generalize theorems of Gluck and Pan on embedding and knotting of positively curved surfaces in 3-space.
A preparation theorem for Weierstrass systems
Daniel
J.
Miller
4395-4439
Abstract: It is shown that Lion and Rolin's preparation theorem for globally subanalytic functions holds for the collection of definable functions in any expansion of the real ordered field by a Weierstrass system.
Transplantation and multiplier theorems for Fourier-Bessel expansions
Óscar
Ciaurri;
Krzysztof
Stempak
4441-4465
Abstract: Proved are weighted transplantation inequalities for Fourier-Bessel expansions. These extend known results on this subject by considering the largest possible range of parameters, allowing more weights and admitting a shift. The results are then used to produce a fairly general multiplier theorem with power weights for considered expansions. Also fractional integral results and conjugate function norm inequalities for these expansions are proved.
Approximation and regularization of Lipschitz functions: Convergence of the gradients
Marc-Olivier
Czarnecki;
Ludovic
Rifford
4467-4520
Abstract: We examine the possible extensions to the Lipschitzian setting of the classical result on $C^1$-convergence: first (approximation), if a sequence $(f_n)$ of functions of class $C^1$ from $\mathbb{R}^N$ to $\mathbb{R}$ converges uniformly to a function $f$ of class $C^1$, then the gradient of $f$ is a limit of gradients of $f_n$ in the sense that $\operatorname{graph}(\nabla f)\subset \liminf_{n\to +\infty} \operatorname{graph}(\nabla f_n)$; second (regularization), the functions $(f_n)$ can be chosen to be of class $C^{\infty}$ and $C^1$-converging to $f$ in the sense that $\lim_{n\to +\infty} \Vert f_n-f\Vert _{\infty}+ \Vert\nabla f_n-\nabla f\Vert _{\infty}=0$. In other words, the space of $C^{\infty}$ functions is dense in the space of $C^1$ functions endowed with the $C^1$ pseudo-norm. We first deepen the properties of Warga's counterexample (1981) for the extension of the approximation part to the Lipschitzian setting. This part cannot be extended, even if one restricts the approximation schemes to the classical convolution and the Lasry-Lions regularization. We thus make more precise various results in the literature on the convergence of subdifferentials. We then show that the regularization part can be extended to the Lipschitzian setting, namely if $f:\mathbb{R}^N \rightarrow {\mathbb{R}}$ is a locally Lipschitz function, we build a sequence of smooth functions $(f_n)_{n \in \mathbb{N}}$ such that $\displaystyle \lim_{n\to +\infty} \Vert f_n-f\Vert _{\infty}=0,$ $\displaystyle \lim_{n\to +\infty} d_{Haus}(\operatorname{graph}(\nabla f_n), \operatorname{graph}(\partial f))=0.$ In other words, the space of $C^{\infty}$ functions is dense in the space of locally Lipschitz functions endowed with an appropriate Lipschitz pseudo-distance. Up to now, Rockafellar and Wets (1998) have shown that the convolution procedure permits us to have the equality $\limsup_{n\to +\infty} \operatorname{graph}(\nabla f_n) =\operatorname{graph}(\partial f)$, which cannot provide the exactness of our result. As a consequence, we obtain a similar result on the regularization of epi-Lipschitz sets. With both functional and set parts, we improve previous results in the literature on the regularization of functions and sets.
Generic systems of co-rank one vector distributions
Howard
Jacobowitz
4521-4531
Abstract: This paper studies a generic class of sub-bundles of the complexified tangent bundle. Involutive, generic structures always exist and have Levi forms with only simple zeroes. For a compact, orientable three-manifold the Chern class of the sub-bundle is mod $2$ equivalent to the Poincaré dual of the characteristic set of the associated system of linear partial differential equations.
Martingales and character ratios
Jason
Fulman
4533-4552
Abstract: Some general connections between martingales and character ratios of finite groups are developed. As an application we sharpen the convergence rate in a central limit theorem for the character ratio of a random representation of the symmetric group on transpositions. A generalization of these results is given for Jack measure on partitions. We also give a probabilistic proof of a result of Burnside and Brauer on the decomposition of tensor products.
Spectra of quantized hyperalgebras
William
Chin;
Leonid
Krop
4553-4567
Abstract: We describe the prime and primitive spectra for quantized enveloping algebras at roots of 1 in characteristic zero in terms of the prime spectrum of the underlying enveloping algebra. Our methods come from the theory of Hopf algebra crossed products. For primitive ideals we obtain an analogue of Duflo's Theorem, which says that every primitive ideal is the annihilator of a simple highest weight module. This depends on an extension of Lusztig's tensor product theorem.
The spectrum of twisted Dirac operators on compact flat manifolds
Roberto
J.
Miatello;
Ricardo
A.
Podestá
4569-4603
Abstract: Let $M$ be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of $M$, and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group $\mathbb{Z}_2^k$, we give a very simple expression for the multiplicities of eigenvalues that allows us to compute explicitly the $\eta$-series, in terms of values of Hurwitz zeta functions, and the $\eta$-invariant. We give the dimension of the space of harmonic spinors and characterize all $\mathbb{Z}_2^k$-manifolds having asymmetric Dirac spectrum. Furthermore, we exhibit many examples of Dirac isospectral pairs of $\mathbb{Z}_2^k$-manifolds which do not satisfy other types of isospectrality. In one of the main examples, we construct a large family of Dirac isospectral compact flat $n$-manifolds, pairwise nonhomeomorphic to each other of the order of $a^n$.
Finite edge-transitive Cayley graphs and rotary Cayley maps
Cai
Heng
Li
4605-4635
Abstract: This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and core-free), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various combinatorial objects, such as half-transitive graphs, (orientable and non-orientable) regular Cayley maps, vertex-transitive non-Cayley graphs, and permutation groups containing certain regular subgroups. In particular, a characterisation is given of locally primitive holomorph Cayley graphs, and a classification is given of rotary Cayley maps of simple groups. Also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup.
Vanishing and non-vanishing of traces of Hecke operators
Jeremy
Rouse
4637-4651
Abstract: Using a reformulation of the Eichler-Selberg trace formula, due to Frechette, Ono and Papanikolas, we consider the problem of the vanishing (resp. non-vanishing) of traces of Hecke operators on spaces of even weight cusp forms with trivial Nebentypus character. For example, we show that for a fixed operator and weight, the set of levels for which the trace vanishes is effectively computable. Also, for a fixed operator the set of weights for which the trace vanishes (for any level) is finite. These results motivate the ``generalized Lehmer conjecture'', that the trace does not vanish for even weights $2k \geq 16$ or $2k = 12$.
On the non-unitary unramified dual for classical $p$--adic groups
Goran
Muic
4653-4687
Abstract: In this paper we give a Zelevinsky type classification of unramified irreducible representations of split classical groups.
Dimension of hyperbolic measures of random diffeomorphisms
Pei-Dong
Liu;
Jian-Sheng
Xie
3751-3780
Abstract: We consider dynamics of compositions of stationary random $C^2$ diffeomorphisms. We will prove that the sample measures of an ergodic hyperbolic invariant measure of the system are exact dimensional. This is an extension to random diffeomorphisms of the main result of Barreira, Pesin and Schmeling (1999), which proves the Eckmann-Ruelle dimension conjecture for a deterministic diffeomorphism.
$3$-manifolds with planar presentations and the width of satellite knots
Martin
Scharlemann;
Jennifer
Schultens
3781-3805
Abstract: We consider compact $3$-manifolds $M$ having a submersion $h$ to $R$ in which each generic point inverse is a planar surface. The standard height function on a submanifold of $S^{3}$ is a motivating example. To $(M, h)$ we associate a connectivity graph $\Gamma$. For $M \subset S^{3}$, $\Gamma$ is a tree if and only if there is a Fox reimbedding of $M$ which carries horizontal circles to a complete collection of complementary meridian circles. On the other hand, if the connectivity graph of $S^{3} - M$ is a tree, then there is a level-preserving reimbedding of $M$ so that $S^{3} - M$ is a connected sum of handlebodies. Corollary. $\bullet$ The width of a satellite knot is no less than the width of its pattern knot and so $\bullet$ $w(K_{1} \char93 K_{2}) \geq max(w(K_{1}), w(K_{2}))$.
Complete nonorientable minimal surfaces in a ball of $\mathbb{R}^3$
F.
J.
López;
Francisco
Martin;
Santiago
Morales
3807-3820
Abstract: The existence of complete minimal surfaces in a ball was proved by N. Nadirashvili in 1996. However, the construction of such surfaces with nontrivial topology remained open. In 2002, the authors showed examples of complete orientable minimal surfaces with arbitrary genus and one end. In this paper we construct complete bounded nonorientable minimal surfaces in $\mathbb{R}^3$ with arbitrary finite topology. The method we present here can also be used to construct orientable complete minimal surfaces with arbitrary genus and number of ends.
The computational complexity of knot genus and spanning area
Ian
Agol;
Joel
Hass;
William
Thurston
3821-3850
Abstract: We show that the problem of deciding whether a polygonal knot in a closed three-dimensional manifold bounds a surface of genus at most $g$ is NP-complete. We also show that the problem of deciding whether a curve in a PL manifold bounds a surface of area less than a given constant $C$ is NP-hard.
On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces
Athanassios
G.
Kartsatos;
Igor
V.
Skrypnik
3851-3881
Abstract: Let $X$ be a real reflexive Banach space with dual $X^{*}$ and $G\subset X$open and bounded and such that $0\in G.$ Let $T:X\supset D(T)\to 2^{X^{*}}$be maximal monotone with $0\in D(T)$ and $0\in T(0),$ and $C:X\supset D(C)\to X^{*}$ with $0\in D(C)$ and $C(0)\neq 0.$ A general and more unified eigenvalue theory is developed for the pair of operators $(T,C).$ Further conditions are given for the existence of a pair $(\lambda ,x) \in (0,\infty )\times (D(T+C)\cap \partial G)$ such that \begin{displaymath}(**)\quad\qquad\qquad\qquad\qquad\qquad\qquad Tx+\lambda Cx\owns 0.\quad\qquad\qquad\qquad\qquad\qquad\qquad\end{displaymath} The ``implicit" eigenvalue problem, with $C(\lambda ,x)$ in place of $\lambda Cx,$ is also considered. The existence of continuous branches of eigenvectors of infinite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators $T,~C.$ No compactness assumptions have been made in most of the results. The degree theories of Browder and Skrypnik are used, as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators. Applications to nonlinear partial differential equations are included.
Calculus on the Sierpinski gasket II: Point singularities, eigenfunctions, and normal derivatives of the heat kernel
Nitsan
Ben-Gal;
Abby
Shaw-Krauss;
Robert
S.
Strichartz;
Clint
Young
3883-3936
Abstract: This paper continues the study of fundamental properties of elementary functions on the Sierpinski gasket (SG) related to the Laplacian defined by Kigami: harmonic functions, multiharmonic functions, and eigenfunctions of the Laplacian. We describe the possible point singularities of such functions, and we use the description at certain periodic points to motivate the definition of local derivatives at these points. We study the global behavior of eigenfunctions on all generic infinite blow-ups of SG, and construct eigenfunctions that decay at infinity in certain directions. We study the asymptotic behavior of normal derivatives of Dirichlet eigenfunctions at boundary points, and give experimental evidence for the behavior of the normal derivatives of the heat kernel at boundary points.
Topological obstructions to certain Lie group actions on manifolds
Pisheng
Ding
3937-3967
Abstract: Given a smooth closed $S^{1}$-manifold $M$, this article studies the extent to which certain numbers of the form $\left( f^{\ast}\left( x\right) \cdot P\cdot C\right) \left[ M\right]$ are determined by the fixed-point set $M^{S^{1}}$, where $f:M\rightarrow K\left( \pi_{1}\left( M\right), 1\right)$ classifies the universal cover of $M$, $x\in H^{\ast}\left( \pi_{1}\left( M\right) ;\mathbb{Q}\right)$, $P$ is a polynomial in the Pontrjagin classes of $M$, and $C$ is in the subalgebra of $H^{\ast}\left( M;\mathbb{Q}\right)$ generated by $H^{2}\left( M;\mathbb{Q}\right)$. When $M^{S^{1}}=\varnothing$, various vanishing theorems follow, giving obstructions to certain fixed-point-free actions. For example, if a fixed-point-free $S^{1}$-action extends to an action by some semisimple compact Lie group $G$, then $\left( f^{\ast}(x)\cdot P\cdot C\right) [M]=0$. Similar vanishing results are obtained for spin manifolds admitting certain $S^{1}$-actions.
Toroidal orbifolds, gerbes and group cohomology
Alejandro
Adem;
Jianzhong
Pan
3969-3983
Abstract: In this paper we compute the integral cohomology of certain semi-direct products of the form $\mathbb{Z}^n\rtimes G$, arising from a linear $G$ action on the $n$-torus, where $G$ is a finite group. The main application is the complete calculation of torsion gerbes for six-dimensional examples arising in string theory.
An alternative approach to homotopy operations
Marcel
Bökstedt;
Iver
Ottosen
3985-3995
Abstract: We give a particular choice of the higher Eilenberg-Mac Lane maps by a recursive formula. This choice leads to a simple description of the homotopy operations for simplicial ${\bf Z}/2$-algebras.
Manifolds with an $SU(2)$-action on the tangent bundle
Roger
Bielawski
3997-4019
Abstract: We study manifolds arising as spaces of sections of complex manifolds fibering over ${\mathbb C}P^1$ with the normal bundle of each section isomorphic to $\mathcal{O}(k)\otimes {\mathbb C}^n$.
On the Cauchy problem of degenerate hyperbolic equations
Qing
Han;
Jia-Xing
Hong;
Chang-Shou
Lin
4021-4044
Abstract: In this paper, we study a class of degenerate hyperbolic equations and prove the existence of smooth solutions for Cauchy problems. The existence result is based on a priori estimates of Sobolev norms of solutions. Such estimates illustrate a loss of derivatives because of the degeneracy.
Seifert-fibered surgeries which do not arise from primitive/Seifert-fibered constructions
Thomas
Mattman;
Katura
Miyazaki;
Kimihiko
Motegi
4045-4055
Abstract: We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert-fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The $(-3,3,5)$-pretzel knot belongs to both of the infinite families.
Scott's rigidity theorem for Seifert fibered spaces; revisited
Teruhiko
Soma
4057-4070
Abstract: We will present a new proof of the rigidity theorem for Seifert fibered spaces of infinite $\pi_1$ by Scott (1983) in the case when the base of the fibration is a hyperbolic triangle 2-orbifold. Our proof is based on arguments in the rigidity theorem for hyperbolic 3-manifolds by Gabai (1997).
A moment problem and a family of integral evaluations
Jacob
S.
Christiansen;
Mourad
E. H.
Ismail
4071-4097
Abstract: We study the Al-Salam-Chihara polynomials when $q>1$. Several solutions of the associated moment problem are found, and the orthogonality relations lead to explicit evaluations of several integrals. The polynomials are shown to have raising and lowering operators and a second order operator equation of Sturm-Liouville type whose eigenvalues are found explicitly. We also derive new measures with respect to which the Ismail-Masson system of rational functions is biorthogonal. An integral representation of the right inverse of a divided difference operator is also obtained.
Maximal theorems for the directional Hilbert transform on the plane
Michael
T.
Lacey;
Xiaochun
Li
4099-4117
Abstract: For a Schwartz function $f$ on the plane and a non-zero $v\in\mathbb{R}^2$ define the Hilbert transform of $f$ in the direction $v$ to be $\displaystyle \operatorname H_vf(x)=$p.v.$\displaystyle \int_{\mathbb{R}} f(x-vy)\; \frac{dy}y.$ Let $\zeta$ be a Schwartz function with frequency support in the annulus $1\le\vert\xi\vert\le2$, and ${\boldsymbol \zeta}f=\zeta*f$. We prove that the maximal operator $\sup_{\vert v\vert=1}\vert\operatorname H_v{\boldsymbol \zeta} f\vert$ maps $L^2$ into weak $L^2$, and $L^p$ into $L^p$ for $p>2$. The $L^2$ estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series. Indeed, our main theorem implies this result on Fourier series.
The monomial ideal of a finite meet-semilattice
Jürgen
Herzog;
Takayuki
Hibi;
Xinxian
Zheng
4119-4134
Abstract: Squarefree monomial ideals arising from finite meet-semilattices and their free resolutions are studied. For the squarefree monomial ideals corresponding to poset ideals in a distributive lattice, the Alexander dual is computed.
Geometric characterization of strongly normal extensions
Jerald
J.
Kovacic
4135-4157
Abstract: This paper continues previous work in which we developed the Galois theory of strongly normal extensions using differential schemes. In the present paper we derive two main results. First, we show that an extension is strongly normal if and only if a certain differential scheme splits, i.e. is obtained by base extension of a scheme over constants. This gives a geometric characterization to the notion of strongly normal. Second, we show that Picard-Vessiot extensions are characterized by their Galois group being affine. Our proofs are elementary and do not use ``group chunks'' or cohomology. We end by recalling some important results about strongly normal extensions with the hope of spurring future research.
Homomorphisms between Weyl modules for $\operatorname{SL}_3(k)$
Anton
Cox;
Alison
Parker
4159-4207
Abstract: We classify all homomorphisms between Weyl modules for $\operatorname{SL}_3(k)$ when $k$ is an algebraically closed field of characteristic at least three, and show that the $\operatorname{Hom}$-spaces are all at most one dimensional. As a corollary we obtain all homomorphisms between Specht modules for the symmetric group when the labelling partitions have at most three parts and the prime is at least three. We conclude by showing how a result of Fayers and Lyle on Hom-spaces for Specht modules is related to earlier work of Donkin for algebraic groups.
Busemann points of infinite graphs
Corran
Webster;
Adam
Winchester
4209-4224
Abstract: We provide a geometric condition which determines whether or not every point on the metric boundary of a graph with the standard path metric is a Busemann point, that is, it is the limit point of a geodesic ray. We apply this and a related condition to investigate the structure of the metric boundary of Cayley graphs. We show that groups such as the braid group and the discrete Heisenberg group have boundary points of the Cayley graph which are not Busemann points when equipped with their usual generators.
Lower and upper Loeb-integrals
D.
Landers;
L.
Rogge
3263-3283
Abstract: We introduce the concepts of lower and upper Loeb-integrals for an internal integration structure. These are concepts which are similarly useful for Loebs internal integration theory as the concepts of inner and outer Loeb-measures for Loebs measure theory.
Prime geodesic theorem for higher-dimensional hyperbolic manifold
Maki
Nakasuji
3285-3303
Abstract: For a $(d+1)$-dimensional hyperbolic manifold $\mathcal{M}$, we consider an estimate of the error term of the prime geodesic theorem. Put the fundamental group $\Gamma$ of $\mathcal{M}$ to be a discrete subgroup of $SO_e(d+1, 1)$ with cofinite volume. When the contribution of the discrete spectrum of the Laplace-Beltrami operator is larger than that of the continuous spectrum in Weyl's law, we obtained a lower estimate $\Omega_{\pm}(\tfrac{x^{d/2}(\log\log x)^{1/(d+1)}}{\log x})$ as $x$ goes to $\infty$.
On decompositions in homotopy theory
Brayton
Gray
3305-3328
Abstract: We first describe Krull-Schmidt theorems decomposing $H$ spaces and simply-connected co-$H$ spaces into atomic factors in the category of pointed nilpotent $p$-complete spaces of finite type. We use this to construct a 1-1 correspondence between homotopy types of atomic $H$ spaces and homotopy types of atomic co-$H$ spaces, and construct a split fibration which connects them and illuminates the decomposition. Various properties of these constructions are analyzed.
Kirwan-Novikov inequalities on a manifold with boundary
Maxim
Braverman;
Valentin
Silantyev
3329-3361
Abstract: We extend the Novikov Morse-type inequalities for closed 1-forms in 2 directions. First, we consider manifolds with boundary. Second, we allow a very degenerate structure of the critical set of the form, assuming only that the form is non-degenerated in the sense of Kirwan. In particular, we obtain a generalization of a result of Floer about the usual Morse inequalities on a manifold with boundary. We also obtain an equivariant version of our inequalities. Our proof is based on an application of the Witten deformation technique. The main novelty here is that we consider the neighborhood of the critical set as a manifold with a cylindrical end. This leads to a considerable simplification of the local analysis. In particular, we obtain a new analytic proof of the Morse-Bott inequalities on a closed manifold.
Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients
Günther
Hörmann;
Michael
Oberguggenberger;
Stevan
Pilipovic
3363-3383
Abstract: We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols. Methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non-smooth lower order terms. We include simplified conditions which are applicable in special cases of interest.
Curvilinear base points, local complete intersection and Koszul syzygies in biprojective spaces
J.
William
Hoffman;
Hao
Hao
Wang
3385-3398
Abstract: Let $I = \langle f_1 , f_2 , f_3\rangle$ be a bigraded ideal in the bigraded polynomial ring $k[s, u; t, v]$. Assume that $I$ has codimension 2. Then $Z = \mathbb{V}(I) \subset \mathbf{P}^{1} \times \mathbf{P}^{1}$ is a finite set of points. We prove that if $Z$ is a local complete intersection, then any syzygy of the $f_i$ vanishing at $Z$, and in a certain degree range, is in the module of Koszul syzygies. This is an analog of a recent result of Cox and Schenck (2003).
Signature invariants of covering links
Jae
Choon
Cha;
Ki
Hyoung
Ko
3399-3412
Abstract: We apply the theory of signature invariants of links in rational homology spheres to covering links of homology boundary links. From patterns and Seifert matrices of homology boundary links, we derive an explicit formula to compute signature invariants of their covering links. Using the formula, we produce fused boundary links that are positive mutants of ribbon links but are not concordant to boundary links. We also show that for any finite collection of patterns, there are homology boundary links that are not concordant to any homology boundary links admitting a pattern in the collection.
A general theory of almost convex functions
S.
J.
Dilworth;
Ralph
Howard;
James
W.
Roberts
3413-3445
Abstract: Let $\Delta_m=\{(t_0,\dots, t_m)\in \mathbf{R}^{m+1}: t_i\ge 0, \sum_{i=0}^mt_i=1\}$ be the standard $m$-dimensional simplex and let $\varnothing\ne S\subset \bigcup_{m=1}^\infty\Delta_m$. Then a function $h\colon C\to \mathbf{R}$ with domain a convex set in a real vector space is $S$-almost convex iff for all $(t_0,\dots, t_m)\in S$ and $x_0,\dots, x_m\in C$ the inequality $\displaystyle h(t_0x_0+\dots+t_mx_m)\le 1+ t_0h(x_0)+\cdots+t_mh(x_m)$ holds. A detailed study of the properties of $S$-almost convex functions is made. If $S$ contains at least one point that is not a vertex, then an extremal $S$-almost convex function $E_S\colon \Delta_n\to \mathbf{R}$ is constructed with the properties that it vanishes on the vertices of $\Delta_n$ and if $h\colon \Delta_n\to \mathbf{R}$ is any bounded $S$-almost convex function with $h(e_k)\le 0$ on the vertices of $\Delta_n$, then $h(x)\le E_S(x)$ for all $x\in \Delta_n$. In the special case $S=\{(1/(m+1),\dotsc, 1/(m+1))\}$, the barycenter of $\Delta_m$, very explicit formulas are given for $E_S$ and $\kappa_S(n)=\sup_{x\in\Delta_n}E_S(x)$. These are of interest, as $E_S$ and $\kappa_S(n)$ are extremal in various geometric and analytic inequalities and theorems.
Explicit bounds for the finite jet determination problem
Bernhard
Lamel
3447-3457
Abstract: We introduce biholomorphic invariants for (germs of) rigid holomorphically nondegenerate real hypersurfaces in complex space and show how they can be used to compute explicit bounds on the order of jets for which biholomorphisms of the hypersurface are determined uniquely by their jets. The main result which allows us to derive these bounds is a theorem which shows that solutions of certain singular analytic equations are uniquely determined by their $1$-jet.
A geometric characterization of interpolation in $\hat{\mathcal{E}}^\prime(\mathbb{R})$
Xavier
Massaneda;
Joaquim
Ortega-Cerdà;
Myriam
Ounaïes
3459-3472
Abstract: We give a geometric description of the interpolating varieties for the algebra of Fourier transforms of distributions (or Beurling ultradistributions) with compact support on the real line.
Dynamical forcing of circular groups
Danny
Calegari
3473-3491
Abstract: In this paper we introduce and study the notion of dynamical forcing. Basically, we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties. As an application, we show that the set $X \subset \mathbb{R} /\mathbb{Z}$ consisting of rotation numbers $\theta$ which can be forced by finitely presented groups is an infinitely generated $\mathbb{Q}$-module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number $\theta$ is forced by a pair $(G_\theta,\alpha)$, where $G_\theta$ is a finitely presented group $G_\theta$ and $\alpha \in G_\theta$ is some element, if the set of rotation numbers of $\rho(\alpha)$ as $\rho$varies over $\rho \in \operatorname{Hom}(G_\theta,\operatorname{Homeo}^+(S^1))$ is precisely the set $\lbrace 0, \pm \theta \rbrace$. We show that the set of subsets of $\mathbb{R} /\mathbb{Z}$ which are of the form \begin{displaymath}\operatorname{rot}(X(G,\alpha)) = \lbrace r \in \mathbb{R} /... ... \in \operatorname{Hom}(G,\operatorname{Homeo}^+(S^1)) \rbrace,\end{displaymath} where $G$ varies over countable groups, are exactly the set of closed subsets which contain $0$ and are invariant under $x \to -x$. Moreover, we show that every such subset can be approximated from above by $\operatorname{rot}(X(G_i,\alpha_i))$ for finitely presented $G_i$. As another application, we construct a finitely generated group $\Gamma$ which acts faithfully on the circle, but which does not admit any faithful $C^1$action, thus answering in the negative a question of John Franks.
Exponents for $B$-stable ideals
Eric
Sommers;
Julianna
Tymoczko
3493-3509
Abstract: Let $G$ be a simple algebraic group over the complex numbers containing a Borel subgroup $B$. Given a $B$-stable ideal $I$ in the nilradical of the Lie algebra of $B$, we define natural numbers $m_1, m_2, \dots, m_k$ which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types $A_n, B_n, C_n$ and some other types. When $I = 0$, we recover the usual exponents of $G$ by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincaré polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.
Bilinear restriction estimates for surfaces with curvatures of different signs
Sanghyuk
Lee
3511-3533
Abstract: Recently, the sharp $L^2$-bilinear (adjoint) restriction estimates for the cone and the paraboloid were established by Wolff and Tao, respectively. Their results rely on the fact that for the cone and the paraboloid, the nonzero principal curvatures have the same sign. We generalize those bilinear restriction estimates to surfaces with curvatures of different signs.
Nonnegative solvability of linear equations in certain ordered rings
Philip
Scowcroft
3535-3570
Abstract: In the integers and in certain densely ordered rings that are not fields, projections of the solution set of finitely many homogeneous weak linear inequalities may be defined by finitely many congruence inequalities, where a congruence inequality combines a weak inequality with a system of congruences. These results extend well-known facts about systems of weak linear inequalities over ordered fields and imply corresponding analogues of Farkas' Lemma on nonnegative solvability of systems of linear equations.
Extended degree functions and monomial modules
Uwe
Nagel;
Tim
Römer
3571-3589
Abstract: The arithmetic degree, the smallest extended degree, and the homological degree are invariants that have been proposed as alternatives of the degree of a module if this module is not Cohen-Macaulay. We compare these degree functions and study their behavior when passing to the generic initial or the lexicographic submodule. This leads to various bounds and to counterexamples to a conjecture of Gunston and Vasconcelos, respectively. Particular attention is given to the class of sequentially Cohen-Macaulay modules. The results in this case lead to an algorithm that computes the smallest extended degree.
Frobenius morphisms and representations of algebras
Bangming
Deng;
Jie
Du
3591-3622
Abstract: By introducing Frobenius morphisms $F$ on algebras $A$ and their modules over the algebraic closure ${\overline {\mathbb{F}}}_q$ of the finite field ${\mathbb{F}}_q$ of $q$ elements, we establish a relation between the representation theory of $A$ over $\overline {\mathbb{F}}_q$ and that of the $F$-fixed point algebra $A^F$ over ${\mathbb{F}}_q$. More precisely, we prove that the category mod-$A^F$ of finite-dimensional $A^F$-modules is equivalent to the subcategory of finite-dimensional $F$-stable $A$-modules, and, when $A$ is finite dimensional, we establish a bijection between the isoclasses of indecomposable $A^F$-modules and the $F$-orbits of the isoclasses of indecomposable $A$-modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over ${\mathbb{F}}_q$ can be interpreted as $F$-stable representations of the corresponding quiver over $\overline {\mathbb{F}}_q$. We further prove that every finite-dimensional hereditary algebra over ${\mathbb{F}}_q$ is Morita equivalent to some $A^F$, where $A$ is the path algebra of a quiver $Q$ over $\overline {\mathbb{F}}_q$ and $F$ is induced from a certain automorphism of $Q$. A close relation between the Auslander-Reiten theories for $A$ and $A^F$ is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of $A^F$ is obtained by ``folding" the Auslander-Reiten quiver of $A$. Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver over ${\mathbb{F}}_q$ with a given dimension vector and to generalize Kac's theorem for all modulated quivers and their associated Kac-Moody algebras defined by symmetrizable generalized Cartan matrices.
On Cartan matrices and lower defect groups for covering groups of symmetric groups
Christine
Bessenrodt;
Jørn
B.
Olsson
3623-3635
Abstract: We determine the elementary divisors of the Cartan matrices of spin $p$-blocks of the covering groups of the symmetric groups when $p$ is an odd prime. As a consequence, we also compute the determinants of these Cartan matrices, and in particular we confirm a conjecture by Brundan and Kleshchev that these determinants depend only on the weight but not on the sign of the block.
Unital bimodules over the simple Jordan superalgebra $D(t)$
Consuelo
Martínez;
Efim
Zelmanov
3637-3649
Abstract: We classify indecomposable finite dimensional bimodules over Jordan superalgebras $D(t)$, $t \neq -1,0,1$.
Geometric structures as deformed infinitesimal symmetries
Anthony
D.
Blaom
3651-3671
Abstract: A general model for geometric structures on differentiable manifolds is obtained by deforming infinitesimal symmetries. Specifically, this model consists of a Lie algebroid, equipped with an affine connection compatible with the Lie algebroid structure. The curvature of this connection vanishes precisely when the structure is locally symmetric. This model generalizes Cartan geometries, a substantial class, to the intransitive case. Simple examples are surveyed and corresponding local obstructions to symmetry are identified. These examples include foliations, Riemannian structures, infinitesimal $G$-structures, symplectic and Poisson structures.
On the Eshelby-Kostrov property for the wave equation in the plane
M.
A.
Herrero;
G.
E.
Oleaga;
J.
J. L.
Velázquez
3673-3695
Abstract: This work deals with the linear wave equation considered in the whole plane $\mathbb{R}^{2}$ except for a rectilinear moving slit, represented by a curve $\Gamma\left( t\right) =\left\{ \left( x_{1},0\right) :-\infty<x_{1}<\lambda\left( t\right) \right\}$ with $t\geq0.$ Along $\Gamma\left( t\right) ,$ either homogeneous Dirichlet or Neumann boundary conditions are imposed. We discuss existence and uniqueness for these problems, and derive explicit representation formulae for solutions. The latter have a simple geometrical interpretation, and in particular allow us to derive precise asymptotic expansions for solutions near the tip of the curve. In the Neumann case, we thus recover a classical result in fracture dynamics, namely the form of the stress intensity factor in crack propagation under antiplane shear conditions.
On linear transformations preserving the Pólya frequency property
Petter
Brändén
3697-3716
Abstract: We prove that certain linear operators preserve the Pólya frequency property and real-rootedness, and apply our results to settle some conjectures and open problems in combinatorics proposed by Bóna, Brenti and Reiner-Welker.
Length, multiplicity, and multiplier ideals
Tommaso
de Fernex
3717-3731
Abstract: Let $(R,\mathfrak{m})$ be an $n$-dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an $\mathfrak{m}$-primary ideal $\mathfrak{a}$ of $R$, the relationship between the singularities of the scheme defined by $\mathfrak{a}$ and those defined by the multiplier ideals $\mathcal{J}(\mathfrak{a}^c)$, with $c$ varying in $\mathbb{Q}_+$, are quantified in this paper by showing that the Samuel multiplicity of $\mathfrak{a}$ satisfies $e(\mathfrak{a}) \ge (n+k)^n/c^n$ whenever $\mathcal{J}(\mathfrak{a}^c) \subseteq \mathfrak{m}^{k+1}$. This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustata and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.
On higher syzygies of ruled surfaces
Euisung
Park
3733-3749
Abstract: We study higher syzygies of a ruled surface $X$ over a curve of genus $g$ with the numerical invariant $e$. Let $L \in$ Pic$X$ be a line bundle in the numerical class of $aC_0 +bf$. We prove that for $0 \leq e \leq g-3$, $L$ satisfies property $N_p$ if $a \geq p+2$ and $b-ae \geq 3g-1-e+p$, and for $e \geq g-2$, $L$ satisfies property $N_p$ if $a \geq p+2$ and $b-ae\geq 2g+1+p$. By using these facts, we obtain Mukai-type results. For ample line bundles $A_i$, we show that $K_X + A_1 + \cdots + A_q$ satisfies property $N_p$ when $0 \leq e < \frac{g-3}{2}$ and $q \geq g-2e+1 +p$ or when $e \geq \frac{g-3}{2}$ and $q \geq p+4$. Therefore we prove Mukai's conjecture for ruled surface with $e \geq \frac{g-3}{2}$. We also prove that when $X$ is an elliptic ruled surface with $e \geq 0$, $L$ satisfies property $N_p$ if and only if $a \geq 1$ and $b-ae\geq 3+p$.
On the $p$-compact groups corresponding to the $p$-adic reflection groups $G(q,r,n)$
Natàlia
Castellana
2799-2819
Abstract: There exists an infinite family of $p$-compact groups whose Weyl groups correspond to the finite $p$-adic pseudoreflection groups $G(q,r,n)$ of family 2a in the Clark-Ewing list. In this paper we study these $p$-compact groups. In particular, we construct an analog of the classical Whitney sum map, a family of monomorphisms and a spherical fibration which produces an analog of the classical $J$-homomorphism. Finally, we also describe a faithful complexification homomorphism from these $p$-compact groups to the $p$-completion of unitary compact Lie groups.
Entire majorants via Euler--Maclaurin summation
Friedrich
Littmann
2821-2836
Abstract: It is the aim of this article to give extremal majorants of type $2\pi\delta$ for the class of functions $f_n(x)=$sgn$(x)x^n$, where $n\in\mathbb{N}$. As applications we obtain positive definite extensions to $\mathbb{R}$ of $\pm(it)^{-m}$ defined on $\mathbb{R}\backslash[-1,1]$, where $m\in\mathbb{N}$, optimal bounds in Hilbert-type inequalities for the class of functions $(it)^{-m}$, and majorants of type $2\pi$ for functions whose graphs are trapezoids.
Unique continuation for the two-dimensional anisotropic elasticity system and its applications to inverse problems
Gen
Nakamura;
Jenn-Nan
Wang
2837-2853
Abstract: Under some generic assumptions we prove the unique continuation property for the two-dimensional inhomogeneous anisotropic elasticity system. Having established the unique continuation property, we then investigate the inverse problem of reconstructing the inclusion or cavity embedded in a plane elastic body with inhomogeneous anisotropic medium by infinitely many localized boundary measurements.
The flat model structure on complexes of sheaves
James
Gillespie
2855-2874
Abstract: Let $\mathbf{Ch}(\mathcal{O})$ be the category of chain complexes of $\mathcal{O}$-modules on a topological space $T$ (where $\mathcal{O}$ is a sheaf of rings on $T$). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on $\mathbf{Ch}(\mathcal{O})$. As a corollary, we have a general framework for doing homological algebra in the category $\mathbf{Sh}(\mathcal{O})$ of $\mathcal{O}$-modules. I.e., we have a natural way to define the functors $\operatorname{Ext}$ and $\operatorname{Tor}$ in $\mathbf{Sh}(\mathcal{O})$.
The parameterized Steiner problem and the singular Plateau problem via energy
Chikako
Mese;
Sumio
Yamada
2875-2895
Abstract: The Steiner problem is the problem of finding the shortest network connecting a given set of points. By the singular Plateau Problem, we will mean the problem of finding an area-minimizing surface (or a set of surfaces adjoined so that it is homeomorphic to a 2-complex) spanning a graph. In this paper, we study the parametric versions of the Steiner problem and the singular Plateau problem by a variational method using a modified energy functional for maps. The main results are that the solutions of our one- and two-dimensional variational problems yield length and area minimizing maps respectively, i.e. we provide new methods to solve the Steiner and singular Plateau problems by the use of energy functionals. Furthermore, we show that these solutions satisfy a natural balancing condition along its singular sets. The key issue involved in the two-dimensional problem is the understanding of the moduli space of conformal structures on a 2-complex.
Quillen stratification for Hochschild cohomology of blocks
Jonathan
Pakianathan;
Sarah
Witherspoon;
and with an appendix by
Stephen
F.
Siegel
2897-2916
Abstract: We decompose the maximal ideal spectrum of the Hochschild cohomology ring of a block of a finite group into a disjoint union of subvarieties corresponding to elementary abelian $p$-subgroups of a defect group. These subvarieties are described in terms of group cohomological varieties and the Alperin-Broué correspondence on blocks. Our description leads in particular to a homeomorphism between the Hochschild variety of the principal block and the group cohomological variety. The proofs require a result of Stephen F. Siegel, given in the Appendix, which states that nilpotency in Hochschild cohomology is detected on elementary abelian $p$-subgroups.
On Hölder continuous Riemannian and Finsler metrics
Alexander
Lytchak;
Asli
Yaman
2917-2926
Abstract: We discuss smoothness of geodesics in Riemannian and Finsler metrics.
Quantum cohomology and $S^1$-actions with isolated fixed points
Eduardo
Gonzalez
2927-2948
Abstract: This paper studies symplectic manifolds that admit semi-free circle actions with isolated fixed points. We prove, using results on the Seidel element, that the (small) quantum cohomology of a $2n$-dimensional manifold of this type is isomorphic to the (small) quantum cohomology of a product of $n$ copies of $\mathbb{P}^1$. This generalizes a result due to Tolman and Weitsman.
Alexander polynomials of equivariant slice and ribbon knots in $S^3$
James
F.
Davis;
Swatee
Naik
2949-2964
Abstract: This paper gives an algebraic characterization of Alexander polynomials of equivariant ribbon knots and a factorization condition satisfied by Alexander polynomials of equivariant slice knots.
Besov spaces with non-doubling measures
Donggao
Deng;
Yongsheng
Han;
Dachun
Yang
2965-3001
Abstract: Suppose that $\mu$ is a Radon measure on ${\mathbb R}^d,$ which may be non-doubling. The only condition on $\mu$ is the growth condition, namely, there is a constant $C_0>0$ such that for all $x\in {\rm {\,supp\,}}(\mu)$ and $r>0,$ \begin{displaymath}\mu (B(x, r))\le C_0r^n,\end{displaymath} where $0<n\leq d.$ In this paper, the authors establish a theory of Besov spaces $\dot B^s_{pq}(\mu)$ for $1\le p, q\le\infty$ and $\vert s\vert<\theta$, where $\theta>0$ is a real number which depends on the non-doubling measure $\mu$, $C_0$, $n$ and $d$. The method used to define these spaces is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces are obtained.
Coisotropic and polar actions on compact irreducible Hermitian symmetric spaces
Leonardo
Biliotti
3003-3022
Abstract: We obtain the full classification of coisotropic and polar isometric actions of compact Lie groups on irreducible Hermitian symmetric spaces.
Invariance in $\boldsymbol{\mathcal{E}^*}$ and $\boldsymbol{\mathcal{E}_\Pi}$
Rebecca
Weber
3023-3059
Abstract: We define $G$, a substructure of $\mathcal{E}_\Pi$ (the lattice of $\Pi^0_1$ classes), and show that a quotient structure of $G$, $G^\diamondsuit$, is isomorphic to $\mathcal{E}^*$. The result builds on the $\Delta^0_3$ isomorphism machinery, and allows us to transfer invariant classes from $\mathcal{E}^*$ to $\mathcal{E}_\Pi$, though not, in general, orbits. Further properties of $G^\diamondsuit$ and ramifications of the isomorphism are explored, including degrees of equivalence classes and degree invariance.
Finite Bruck loops
Michael
Aschbacher;
Michael
K.
Kinyon;
J.
D.
Phillips
3061-3075
Abstract: Bruck loops are Bol loops satisfying the automorphic inverse property. We prove a structure theorem for finite Bruck loops $X$, showing that $X$ is essentially the direct product of a Bruck loop of odd order with a $2$-element Bruck loop. The former class of loops is well understood. We identify the minimal obstructions to the conjecture that all finite $2$-element Bruck loops are $2$-loops, leaving open the question of whether such obstructions actually exist.
Projective Fraïssé limits and the pseudo-arc
Trevor
Irwin;
Slawomir
Solecki
3077-3096
Abstract: The aim of the present work is to develop a dualization of the Fraïssé limit construction from model theory and to indicate its surprising connections with the pseudo-arc. As corollaries of general results on the dual Fraïssé limits, we obtain Mioduszewski's theorem on surjective universality of the pseudo-arc among chainable continua and a theorem on projective homogeneity of the pseudo-arc (which generalizes a result of Lewis and Smith on density of homeomorphisms of the pseudo-arc among surjective continuous maps from the pseudo-arc to itself). We also get a new characterization of the pseudo-arc via the projective homogeneity property.
A structure theorem for the elementary unimodular vector group
Selby
Jose;
Ravi
A.
Rao
3097-3112
Abstract: Given a pair of vectors $v,w\in R^{r+1}$ with $\langle v,w\rangle=v\cdot w^T=1$, A. Suslin constructed a matrix $S_r(v,w)\in Sl_{2^r}(R)$. We study the subgroup $SUm_r(R)$ generated by these matrices, and its (elementary) subgroup $EUm_r(R)$ generated by the matrices $S_r(e_1\varepsilon,e_1\varepsilon^{T^{-1}})$, for $\varepsilon\in E_{r+1}(R)$. The basic calculus for $EUm_r(R)$ is developed via a key lemma, and a fundamental property of Suslin matrices is derived.
Commutative ideal theory without finiteness conditions: Completely irreducible ideals
Laszlo
Fuchs;
William
Heinzer;
Bruce
Olberding
3113-3131
Abstract: An ideal of a ring is completely irreducible if it is not the intersection of any set of proper overideals. We investigate the structure of completely irrreducible ideals in a commutative ring without finiteness conditions. It is known that every ideal of a ring is an intersection of completely irreducible ideals. We characterize in several ways those ideals that admit a representation as an irredundant intersection of completely irreducible ideals, and we study the question of uniqueness of such representations. We characterize those commutative rings in which every ideal is an irredundant intersection of completely irreducible ideals.
Maximal families of Gorenstein algebras
Jan
O.
Kleppe
3133-3167
Abstract: The purpose of this paper is to study maximal irreducible families of Gorenstein quotients of a polynomial ring $R$. Let $\operatorname{GradAlg}^H(R)$ be the scheme parametrizing graded quotients of $R$ with Hilbert function $H$. We prove there is a close relationship between the irreducible components of $\operatorname{GradAlg}^H(R)$, whose general member is a Gorenstein codimension $(c+1)$ quotient, and the irreducible components of $\operatorname{GradAlg}^{H'}(R)$ to ``Gorenstein'' components of $\operatorname{GradAlg}^{H}(R)$, in which generically smooth components correspond. Moreover the dimension of the ``Gorenstein'' components is computed in terms of the dimension of the corresponding ``Cohen-Macaulay'' component and a sum of two invariants of $B$. Using linkage by a complete intersection we show how to compute these invariants. Linkage also turns out to be quite effective in verifying the assumptions which appear in a generalization of the main theorem.
Layers and spikes in non-homogeneous bistable reaction-diffusion equations
Shangbing
Ai;
Xinfu
Chen;
Stuart
P.
Hastings
3169-3206
Abstract: We study $\varepsilon^2\ddot{u}=f(u,x)=A\, u\, (1-u)\,(\phi-u)$, where $A=A(u,x)>0$, $\phi=\phi(x)\in(0,1)$, and $\varepsilon>0$ is sufficiently small, on an interval $[0,L]$ with boundary conditions $\dot{u}=0$ at $x=0,L$. All solutions with an $\varepsilon$ independent number of oscillations are analyzed. Existence of complicated patterns of layers and spikes is proved, and their Morse index is determined. It is observed that the results extend to $f=A(u,x)\; (u-\phi_-)\,(u-\phi)\,(u-\phi_+)$ with $\phi_-(x)<\phi(x)<\phi_+(x)$ and also to an infinite interval.
Moduli of curves and spin structures via algebraic geometry
Gilberto
Bini;
Claudio
Fontanari
3207-3217
Abstract: Here we investigate some birational properties of two collections of moduli spaces, namely moduli spaces of (pointed) stable curves and of (pointed) spin curves. In particular, we focus on vanishings of Hodge numbers of type $(p,0)$ and on computations of Kodaira dimension. Our methods are purely algebro-geometric and rely on an induction argument on the number of marked points and the genus of the curves.
Effective cones of quotients of moduli spaces of stable $n$-pointed curves of genus zero
William
F.
Rulla
3219-3237
Abstract: Let $X_n := \overline{M}_{0,n}$, the moduli space of $n$-pointed stable genus zero curves, and let $X_{n,m}$ be the quotient of $X_n$ by the action of $\mathcal{S}_{n-m}$ on the last $n-m$ marked points. The cones of effective divisors $\overline{NE}^1(X_{n,m})$, $m = 0,1,2$, are calculated. Using this, upper bounds for the cones $Mov(X_{n,m})$ generated by divisors with moving linear systems are calculated, $m = 0,1$, along with the induced bounds on the cones of ample divisors of $\overline{M}_g$ and $\overline{M}_{g,1}$. As an application, the cone $\overline{NE}^1(\overline{M}_{2,1})$ is analyzed in detail.
An infinitary extension of the Graham--Rothschild Parameter Sets Theorem
Timothy
J.
Carlson;
Neil
Hindman;
Dona
Strauss
3239-3262
Abstract: The Graham-Rothschild Parameter Sets Theorem is one of the most powerful results of Ramsey Theory. (The Hales-Jewett Theorem is its most trivial instance.) Using the algebra of $\beta S$, the Stone-Cech compactification of a discrete semigroup, we derive an infinitary extension of the Graham-Rothschild Parameter Sets Theorem. Even the simplest finite instance of this extension is a significant extension of the original. The original theorem says that whenever $k<m$ in $\mathbb{N}$ and the $k$-parameter words are colored with finitely many colors, there exist a color and an $m$-parameter word $w$ with the property that whenever a $k$-parameter word of length $m$ is substituted in $w$, the result is in the specified color. The ``simplest finite instance'' referred to above is that, given finite colorings of the $k$-parameter words for each $k<m$, there is one $m$-parameter word which works for each $k$. Some additional Ramsey Theoretic consequences are derived. We also observe that, unlike any other Ramsey Theoretic result of which we are aware, central sets are not necessarily good enough for even the $k=1$ and $m=2$ version of the Graham-Rothschild Parameter Sets Theorem.
$\boldsymbol{\pi_*}$-kernels of Lie groups
Ken-ichi
Maruyama
2335-2351
Abstract: We study a filtration on the group of homotopy classes of self maps of a compact Lie group associated with homotopy groups. We determine these filtrations of $SU(3)$ and $Sp(2)$ completely. We introduce two natural invariants $lz_p(X)$ and $sz_p(X)$ defined by the filtration, where $p$ is a prime number, and compute the invariants for simple Lie groups in the cases where Lie groups are $p$-regular or quasi $p$-regular. We apply our results to the groups of self homotopy equivalences.
Associahedra, cellular $W$-construction and products of $A_\infty$-algebras
Martin
Markl;
Steve
Shnider
2353-2372
Abstract: The aim of this paper is to construct a functorial tensor product of $A_\infty$-algebras or, equivalently, an explicit diagonal for the operad of cellular chains, over the integers, of the Stasheff associahedron. These constructions in fact already appeared (Saneblidze and Umble, 2000 and 2002); we will try to give a more conceptual presentation. We also prove that there does not exist a coassociative diagonal.
Koszul duality and equivalences of categories
Gunnar
Fløystad
2373-2398
Abstract: Let $A$ and $A^{!}$ be dual Koszul algebras. By Positselski a filtered algebra $U$ with gr$\,U = A$ is Koszul dual to a differential graded algebra $(A^{!},d)$. We relate the module categories of this dual pair by a $\otimes-$Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg, and Soergel.
On the contact geometry of nodal sets
R.
Komendarczyk
2399-2413
Abstract: In the 3-dimensional Riemannian geometry, contact structures equipped with an adapted Riemannian metric are divergence-free, nondegenerate eigenforms of the Laplace-Beltrami operator. We trace out a two-dimensional consequence of this fact: there is a close relationship between the topology of the contact structure on a convex surface in the 3-manifold (the dividing curves) and the nodal curves of Laplacian eigenfunctions on that surface. Motivated by this relationship, we consider a topological version of Payne's conjecture for the free membrane problem. We construct counterexamples to Payne's conjecture for closed Riemannian surfaces. In light of the correspondence between the nodal lines and dividing curves, we interpret the conjecture in terms of the tight versus overtwisted dichotomy for contact structures.
Existence and regularity of isometries
Michael
Taylor
2415-2423
Abstract: We use local harmonic coordinates to establish sharp results on the regularity of isometric maps between Riemannian manifolds whose metric tensors have limited regularity (e.g., are Hölder continuous). We also discuss the issue of local flatness and of local isometric embedding with given first and second fundamental form, in the context of limited smoothness.
On the shape of the moduli of spherical minimal immersions
Gabor
Toth
2425-2446
Abstract: The DoCarmo-Wallach moduli space parametrizing spherical minimal immersions of a Riemannian manifold $M$ is a compact convex body in a linear space of tracefree symmetric endomorphisms of an eigenspace of $M$. In this paper we define and study a sequence of metric invariants $\sigma_m$, $m\geq 1$, associated to a compact convex body $\mathcal{L}$ with base point $\mathcal{O}$ in the interior of $\mathcal{L}$. The invariant $\sigma_m$ measures how lopsided $\mathcal{L}$ is in dimension $m$ with respect to $\mathcal{O}$. The results are then appplied to the DoCarmo-Wallach moduli space. We also give an efficient algorithm to calculate $\sigma_m$ for convex polytopes.
Global well-posedness in the energy space for a modified KP II equation via the Miura transform
Carlos
E.
Kenig;
Yvan
Martel
2447-2488
Abstract: We prove global well-posedness of the initial value problem for a modified Kadomtsev-Petviashvili II (mKP II) equation in the energy space. The proof proceeds in three main steps and involves several different techniques. In the first step, we make use of several linear estimates to solve a fourth-order parabolic regularization of the mKP II equation by a fixed point argument, for regular initial data (one estimate is similar to the sharp Kato smoothing effect proved for the KdV equation by Kenig, Ponce, and Vega, 1991). Then, compactness arguments (the energy method performed through the Miura transform) give the existence of a local solution of the mKP II equation for regular data. Finally, we approximate any data in the energy space by a sequence of smooth initial data. Using Bourgain's result concerning the global well-posedness of the KP II equation in $L^2$ and the Miura transformation, we obtain convergence of the sequence of smooth solutions to a solution of mKP II in the energy space.
A $(p,q)$ version of Bourgain's theorem
John
J.
Benedetto;
Alexander
M.
Powell
2489-2505
Abstract: Let $1<p,q<\infty$ satisfy $\frac{1}{p} + \frac{1}{q} =1.$ We construct an orthonormal basis $\{ b_n \}$ for $L^2 (\mathbb{R})$ such that $\Delta_p ( b_n )$ and $\Delta_q (\widehat{b_n})$ are both uniformly bounded in $n$. Here $\Delta_{\lambda} (f) \equiv {\rm inf}_{a \in \mathbb{R}} \left( \int \vert x - a\vert^{\lambda} \vert f(x)\vert^2 dx \right)^{\frac{1}{2}}$. This generalizes a theorem of Bourgain and is closely related to recent results on the Balian-Low theorem.
Resolutions for metrizable compacta in extension theory
Leonard
R.
Rubin;
Philip
J.
Schapiro
2507-2536
Abstract: We prove a $K$-resolution theorem for simply connected CW- complexes $K$ in extension theory in the class of metrizable compacta $X$. This means that if $K$ is a connected CW-complex, $G$ is an abelian group, $n\in \mathbb N _{\geq 2}$, $G=\pi _{n}(K)$, $\pi _{k}(K)=0$ for $0\leq k<n$, and $\operatorname{extdim} X\leq K$ (in the sense of extension theory, that is, $K$ is an absolute extensor for $X$), then there exists a metrizable compactum $Z$ and a surjective map $\pi :Z\rightarrow X$ such that: (a) $\pi$ is $G$-acyclic, (b) $\dim Z\leq n+1$, and (c) $\operatorname{extdim} Z\leq K$. This implies the $G$-resolution theorem for arbitrary abelian groups $G$ for cohomological dimension $\dim _{G} X\leq n$ when $n\in \mathbb N_{\geq 2}$. Thus, in case $K$ is an Eilenberg-MacLane complex of type $K(G,n)$, then (c) becomes $\dim _{G} Z\leq n$. If in addition $\pi _{n+1}(K)=0$, then (a) can be replaced by the stronger statement, (aa) $\pi$ is $K$-acyclic. To say that a map $\pi$ is $K$-acyclic means that for each $x\in X$, every map of the fiber $\pi ^{-1}(x)$ to $K$ is nullhomotopic.
Construction of stable equivalences of Morita type for finite-dimensional algebras I
Yuming
Liu;
Changchang
Xi
2537-2560
Abstract: In the representation theory of finite groups, the stable equivalence of Morita type plays an important role. For general finite-dimensional algebras, this notion is still of particular interest. However, except for the class of self-injective algebras, one does not know much on the existence of such equivalences between two finite-dimensional algebras; in fact, even a non-trivial example is not known. In this paper, we provide two methods to produce new stable equivalences of Morita type from given ones. The main results are Corollary 1.2 and Theorem 1.3. Here the algebras considered are not necessarily self-injective. As a consequence of our constructions, we give an example of a stable equivalence of Morita type between two algebras of global dimension $4$, such that one of them is quasi-hereditary and the other is not. This shows that stable equivalences of Morita type do not preserve the quasi-heredity of algebras. As another by-product, we construct a Morita equivalence inside each given stable equivalence of Morita type between algebras $A$ and $B$. This leads not only to a general formulation of a result by Linckelmann (1996), but also to a nice correspondence of some torsion pairs in $A$-mod with those in $B$-mod if both $A$ and $B$are symmetric algebras. Moreover, under the assumption of symmetric algebras we can get a new stable equivalence of Morita type. Finally, we point out that stable equivalences of Morita type are preserved under separable extensions of ground fields.
A generalization of Marshall's equivalence relation
Ido
Efrat
2561-2577
Abstract: For $p$ prime and for a field $F$ containing a root of unity of order $p$, we generalize Marshall's equivalence relation on orderings to arbitrary subgroups of $F^{\times }$ of index $p$. The equivalence classes then correspond to free pro-$p$ factors of the maximal pro-$p$ Galois group of $F$. We generalize to this setting results of Jacob on the maximal pro-$2$ Galois group of a Pythagorean field.
Bicyclic algebras of prime exponent over function fields
Boris
È.
Kunyavskii;
Louis
H.
Rowen;
Sergey
V.
Tikhonov;
Vyacheslav
I.
Yanchevskii
2579-2610
Abstract: We examine some properties of bicyclic algebras, i.e. the tensor product of two cyclic algebras, defined over a purely transcendental function field in one variable. We focus on the following problem: When does the set of local invariants of such an algebra coincide with the set of local invariants of some cyclic algebra? Although we show this is not always the case, we determine when it happens for the case where all degeneration points are defined over the ground field. Our main tool is Faddeev's theory. We also study a geometric counterpart of this problem (pencils of Severi-Brauer varieties with prescribed degeneration data).
The degree of the variety of rational ruled surfaces and Gromov-Witten invariants
Cristina
Martínez
2611-2624
Abstract: We compute the degree of the variety parametrizing rational ruled surfaces of degree $d$ in $\mathbb{P} ^{3}$ by relating the problem to Gromov-Witten invariants and Quantum cohomology.
Inequalities for eigenvalues of a clamped plate problem
Qing-Ming
Cheng;
Hongcang
Yang
2625-2635
Abstract: Let $D$ be a connected bounded domain in an $n$-dimensional Euclidean space $\mathbb{R}^n$. Assume that $\displaystyle 0 < \lambda_1 <\lambda_2 \le \cdots \le \lambda_k \le \cdots$ are eigenvalues of a clamped plate problem or an eigenvalue problem for the Dirichlet biharmonic operator: $\displaystyle \left \{ \aligned&\Delta^2 u =\lambda u, \text{ in$D$,} &u\ve... ...rac {\partial u}{\partial n}\right \vert _{\partial D}=0. \endaligned \right .$ Then, we give an upper bound of the $(k+1)$-th eigenvalue $\lambda_{k+1}$ in terms of the first $k$ eigenvalues, which is independent of the domain $D$, that is, we prove the following: $\displaystyle \lambda_{k+1} \le \frac 1k\sum_{i=1}^k \lambda_i +\left [\frac {8... ...ac 1k\sum_{i=1}^k \biggl[ \lambda_i(\lambda_{k+1} -\lambda_i) \biggl ]^{1/2}.$ Further, a more explicit inequality of eigenvalues is also obtained.
Teichmüller mapping class group of the universal hyperbolic solenoid
Vladimir
Markovic;
Dragomir
Saric
2637-2650
Abstract: We show that the homotopy class of a quasiconformal self-map of the universal hyperbolic solenoid $H_\infty$ is the same as its isotopy class and that the uniform convergence of quasiconformal self-maps of $H_\infty$ to the identity forces them to be homotopic to conformal maps. We identify a dense subset of $\mathcal{T}(H_\infty )$ such that the orbit under the base leaf preserving mapping class group $MCG_{BLP}(H_\infty)$ of any point in this subset has accumulation points in the Teichmüller space $\mathcal{T}(H_\infty )$. Moreover, we show that finite subgroups of $MCG_{BLP}(H_\infty )$ are necessarily cyclic and that each point of $\mathcal{T}(H_\infty)$ has an infinite isotropy subgroup in $MCG_{BLP}(H_\infty )$.
Bounded Hochschild cohomology of Banach algebras with a matrix-like structure
Niels
Grønbæk
2651-2662
Abstract: Let $\mathfrak{B}$ be a unital Banach algebra. A projection in $\mathfrak{B}$ which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal $\mathfrak{A}$ in $\mathfrak{B}$. In this set-up we prove a theorem to the effect that the bounded cohomology $\mathcal{H}^{n}(\mathfrak{A}, \mathfrak{A}^{*})$ vanishes for all $n\geq 1$. The hypotheses of this theorem involve (i) strong H-unitality of $\mathfrak{A}$, (ii) a growth condition on diagonal matrices in $\mathfrak{A}$, and (iii) an extension of $\mathfrak{A}$ in $\mathfrak{B}$ by an amenable Banach algebra. As a corollary we show that if $X$ is an infinite dimensional Banach space with the bounded approximation property, $L_{1}(\mu ,\Omega )$ is an infinite dimensional $L_{1}$-space, and $\mathfrak{A}$ is the Banach algebra of approximable operators on $L_{p}(X,\mu ,\Omega )\;(1\leq p<\infty )$, then $\mathcal{H}^{n}(\mathfrak{A},\mathfrak{A}^{*})=(0)$ for all $n\geq 0$.
Uniform asymptotics for Jacobi polynomials with varying large negative parameters--- a Riemann-Hilbert approach
R.
Wong;
Wenjun
Zhang
2663-2694
Abstract: An asymptotic expansion is derived for the Jacobi polynomials $P_{n}^{(\alpha_{n},\beta_{n})}(z)$ with varying parameters $\alpha_{n}=-nA+a$ and $\beta_n=-nB+b$, where $A>1, B>1$ and $a,b$ are constants. Our expansion is uniformly valid in the upper half-plane $\overline{\mathbb{C}}^+=\{z:{Im}\; z \geq 0\}$. A corresponding expansion is also given for the lower half-plane $\overline{\mathbb{C}}^-=\{z:{Im}\; z \leq 0\}$. Our approach is based on the steepest-descent method for Riemann-Hilbert problems introduced by Deift and Zhou (1993). The two asymptotic expansions hold, in particular, in regions containing the curve $L$, which is the support of the equilibrium measure associated with these polynomials. Furthermore, it is shown that the zeros of these polynomials all lie on one side of $L$, and tend to $L$ as $n \to \infty$.
Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices
Karlheinz
Gröchenig;
Michael
Leinert
2695-2711
Abstract: We investigate the symbolic calculus for a large class of matrix algebras that are defined by the off-diagonal decay of infinite matrices. Applications are given to the symmetry of some highly non-commutative Banach algebras, to the analysis of twisted convolution, and to the theory of localized frames.
Theta lifting of nilpotent orbits for symmetric pairs
Kyo
Nishiyama;
Hiroyuki
Ochiai;
Chen-bo
Zhu
2713-2734
Abstract: We consider a reductive dual pair $(G, G')$ in the stable range with $G'$ the smaller member and of Hermitian symmetric type. We study the theta lifting of nilpotent $K_{\mathbb{C}}$-module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair $( G, K)$. As an application, we prove sphericality and normality of the closure of certain nilpotent $K_{\mathbb{C}}$-orbits obtained in this way. We also give integral formulas for their degrees.
Construction and properties of quasi-linear functionals
Ørjan
Johansen;
Alf
B.
Rustad
2735-2758
Abstract: Quasi-linear functionals are shown to be uniformly continuous and decomposable into a difference of two quasi-integrals. A predual space for the quasi-linear functionals inducing the weak*-topology is given. General constructions of quasi-linear functionals by solid set-functions and q-functions are given.
Block combinatorics
V.
Farmaki;
S.
Negrepontis
2759-2779
Abstract: In this paper we extend the block combinatorics partition theorems of Hindman and Milliken-Taylor in the setting of the recursive system of the block Schreier families $(\mathcal{B}^\xi)$, consisting of families defined for every countable ordinal $\xi$. Results contain (a) a block partition Ramsey theorem for every countable ordinal $\xi$ (Hindman's Theorem corresponding to $\xi=1$, and the Milliken-Taylor Theorem to $\xi$ a finite ordinal), (b) a countable ordinal form of the block Nash-Williams partition theorem, and (c) a countable ordinal block partition theorem for sets closed in the infinite block analogue of Ellentuck's topology.
Twist points of planar domains
Nicola
Arcozzi;
Enrico
Casadio Tarabusi;
Fausto
Di Biase;
Massimo
A.
Picardello
2781-2798
Abstract: We establish a potential theoretic approach to the study of twist points in the boundary of simply connected planar domains.
Algebraic Goodwillie calculus and a cotriple model for the remainder
Andrew
Mauer-Oats
1869-1895
Abstract: Goodwillie has defined a tower of approximations for a functor from spaces to spaces that is analogous to the Taylor series of a function. His $n^{\text{th}}$ order approximation $P_n F$ at a space $X$ depends on the values of $F$ on coproducts of large suspensions of the space: $F(\vee \Sigma^M X)$. We define an ``algebraic'' version of the Goodwillie tower, $P_n^{\text{alg}} F(X)$, that depends only on the behavior of $F$ on coproducts of $X$. When $F$ is a functor to connected spaces or grouplike $H$-spaces, the functor $P_n^{\text{alg}} F$ is the base of a fibration $\displaystyle \vert{\bot^{*+1} F}\vert \rightarrow F \rightarrow P_n^{\text{alg}} F,$ whose fiber is the simplicial space associated to a cotriple $\bot$ built from the $(n+1)^{\text{st}}$ cross effect of the functor $F$. In a range in which $F$ commutes with realizations (for instance, when $F$ is the identity functor of spaces), the algebraic Goodwillie tower agrees with the ordinary (topological) Goodwillie tower, so this theory gives a way of studying the Goodwillie approximation to a functor $F$ in many interesting cases.
Non-Moishezon twistor spaces of $4{\mathbf{CP}}^2$ with non-trivial automorphism group
Nobuhiro
Honda
1897-1920
Abstract: We show that a twistor space of a self-dual metric on $4{\mathbf{CP}}^2$ with $U(1)$-isometry is not Moishezon iff there is a $\mathbf{C}^*$-orbit biholomorphic to a smooth elliptic curve, where the $\mathbf C^*$-action is the complexification of the $U(1)$-action on the twistor space. It follows that the $U(1)$-isometry has a two-sphere whose isotropy group is $\mathbf Z_2$. We also prove the existence of such twistor spaces in a strong form to show that a problem of Campana and Kreußler is affirmative even though a twistor space is required to have a non-trivial automorphism group.
On topological invariants of stratified maps with non-Witt target
Markus
Banagl
1921-1935
Abstract: The Cappell-Shaneson decomposition theorem for self-dual sheaves asserts that on a space with only even-codimensional strata any self-dual sheaf is cobordant to an orthogonal sum of twisted intersection chain sheaves associated to the various strata. In sharp contrast to this result, we prove that on a space with only odd-codimensional strata (not necessarily Witt), any self-dual sheaf is cobordant to an intersection chain sheaf associated to the top stratum: the strata of odd codimension do not contribute terms. As a consequence, we obtain formulae for the pushforward of characteristic classes under a stratified map whose target need not satisfy the Witt space condition. To prove these results, we introduce a new category of superperverse sheaves, which we show to be abelian. Finally, we apply the results to the study of desingularization of non-Witt spaces and exhibit a singular space which admits a PL resolution in the sense of M. Kato, but no resolution by a stratified map.
Limiting weak--type behavior for singular integral and maximal operators
Prabhu
Janakiraman
1937-1952
Abstract: The following limit result holds for the weak-type (1,1) constant of dilation-commuting singular integral operator $T$ in $\mathbb{R}^n$: for $f\in L^1(\mathbb{R}^n)$, $f\geq 0$, $\displaystyle \lim_{\lambda\rightarrow 0} \lambda\hspace{1mm}m\{x\in\mathbb{R}^... ...a\} = \frac{1}{n} \int_{S^{n-1}}\vert\Omega(x)\vert d\sigma(x)\Vert f\Vert _1.$ For the maximal operator $M$, the corresponding result is $\displaystyle \lim_{\lambda\rightarrow 0} \lambda\hspace{1mm}m\{x\in\mathbb{R}^n: \vert Mf(x)\vert>\lambda\} = \Vert f\Vert _1.$
Symplectic forms invariant under free circle actions on 4-manifolds
Boguslaw
Hajduk;
Rafal
Walczak
1953-1970
Abstract: Let $M$ be a smooth closed 4-manifold with a free circle action generated by a vector field $X.$ Then for any invariant symplectic form $\omega$ on $M$ the contracted form $\iota_X\omega$ is non-vanishing. Using the map $\omega \mapsto \iota_X\omega$ and the related map to $H^1(M \slash S^1,\mathbb{R})$ we study the topology of the space $S_{inv}(M)$ of invariant symplectic forms on $M.$ For example, the first map is proved to be a homotopy equivalence. This reduces examination of homotopy properties of $S_{inv}$ to that of the space $\mathcal{N}_L$ of non-vanishing closed 1-forms satisfying certain cohomology conditions. In particular we give a description of $\pi_0S_{inv}(M)$ in terms of the unit ball of Thurston's norm and calculate higher homotopy groups in some cases. Our calculations show that the homotopy type of the space of non-vanishing 1-forms representing a fixed cohomology class can be non-trivial for some torus bundles over the circle. This provides a counterexample to an open problem related to the Blank-Laudenbach theorem (which says that such spaces are connected for any closed 3-manifold). Finally, we prove some theorems on lifting almost complex structures to symplectic forms in the invariant case.
Torus actions on weakly pseudoconvex spaces
Stefano
Trapani
1971-1981
Abstract: We show that the univalent local actions of the complexification of a compact connected Lie group $K$ on a weakly pseudoconvex space where $K$ is acting holomorphically have a universal orbit convex weakly pseudoconvex complexification. We also show that if $K$ is a torus, then every holomorphic action of $K$ on a weakly pseudoconvex space extends to a univalent local action of $K^{\mathbf{C}}.$
Filtrations in semisimple Lie algebras, I
Y.
Barnea;
D.
S.
Passman
1983-2010
Abstract: In this paper, we study the maximal bounded $\mathbb{Z}$-filtrations of a complex semisimple Lie algebra $L$. Specifically, we show that if $L$ is simple of classical type $A_n$, $B_n$, $C_n$ or $D_n$, then these filtrations correspond uniquely to a precise set of linear functionals on its root space. We obtain partial, but not definitive, results in this direction for the remaining exceptional algebras. Maximal bounded filtrations were first introduced in the context of classifying the maximal graded subalgebras of affine Kac-Moody algebras, and the maximal graded subalgebras of loop toroidal Lie algebras. Indeed, our main results complete this classification in most cases. Finally, we briefly discuss the analogous question for bounded filtrations with respect to other Archimedean ordered groups.
New properties of convex functions in the Heisenberg group
Nicola
Garofalo;
Federico
Tournier
2011-2055
Abstract: We prove some new properties of the weakly $H$-convex functions recently introduced by Danielli, Garofalo and Nhieu. As an interesting application of our results we prove a theorem of Busemann-Feller-Alexandrov type in the Heisenberg groups $\mathbb{H}^n$, $n=1,2$.
Almost complex manifolds and Cartan's uniqueness theorem
Kang-Hyurk
Lee
2057-2069
Abstract: We present a generalization of Cartan's uniqueness theorem to the almost complex manifolds.
Resonances for steplike potentials: Forward and inverse results
T.
Christiansen
2071-2089
Abstract: We consider resonances associated to the one dimensional Schrödinger operator $-\frac{d^2}{dx^2}+V(x)$, where $V(x)=V_+$ if $x>x_M$ and $V(x)=V_-$ if $x<-x_M$, with $V_+\not = V_-$. We obtain asymptotics of the resonance-counting function for several regions. Moreover, we show that in several situations, the resonances, $V_+$, and $V_-$ determine $V$ uniquely up to translation.
Stable mapping class groups of $4$-manifolds with boundary
Osamu
Saeki
2091-2104
Abstract: We give a complete algebraic description of the mapping class groups of compact simply connected 4-manifolds with boundary up to connected sum with copies of $S^2 \times S^2$.
Instability of standing waves of the Schrödinger equation with inhomogeneous nonlinearity
Yue
Liu;
Xiao-Ping
Wang;
Ke
Wang
2105-2122
Abstract: This paper is concerned with the inhomogeneous nonlinear Shrödinger equation (INLS-equation) \begin{displaymath}i u_t + \Delta u + V(\epsilon x) \vert u\vert^p u = 0, \; x \in {\mathbf R}^N. \end{displaymath} In the critical and supercritical cases $p \ge 4/N,$ with $N \ge 2,$ it is shown here that standing-wave solutions of (INLS-equation) on $H^1({\mathbf R}^N)$ perturbation are nonlinearly unstable or unstable by blow-up under certain conditions on the potential term V with a small $\epsilon > 0.$
Second-order hyperbolic s.p.d.e.'s driven by homogeneous Gaussian noise on a hyperplane
Robert
C.
Dalang;
Olivier
Lévêque
2123-2159
Abstract: We study a class of hyperbolic stochastic partial differential equations in Euclidean space, that includes the wave equation and the telegraph equation, driven by Gaussian noise concentrated on a hyperplane. The noise is assumed to be white in time but spatially homogeneous within the hyperplane. Two natural notions of solutions are function-valued solutions and random field solutions. For the linear form of the equations, we identify the necessary and sufficient condition on the spectral measure of the spatial covariance for existence of each type of solution, and it turns out that the conditions differ. In spatial dimensions 2 and 3, under the condition for existence of a random field solution to the linear form of the equation, we prove existence and uniqueness of a random field solution to non-linear forms of the equation.
Dual decompositions of 4-manifolds II: Linear invariants
Frank
Quinn
2161-2181
Abstract: This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index $\leq 2$. Part I (Trans. Amer. Math. Soc. 354 (2002), 1373-1392) gave existence results in terms of spines and chain complexes over the fundamental group of the ambient manifold. Here we assume that one side of a decomposition has larger fundamental group, and use this to define algebraic-topological invariants. These reveal a basic asymmetry in these decompositions: subtle changes on one side can force algebraic-topologically detectable changes on the other. A solvable iteration of the basic invariant gives an ``obstruction theory'' using lower commutator quotients. By thinking of a 2-handlebody as essentially determined by the links used as attaching maps for its 2-handles, this theory can be thought of as giving ``ambient'' link invariants. The moves used are related to the grope cobordism of links developed by Conant-Teichner, and the Cochran-Orr-Teichner filtration of the link concordance groups. The invariants give algebraically sophisticated ``finite type'' invariants in the sense of Vassilaev.
Cohomology theories based on Gorenstein injective modules
Javad
Asadollahi;
Shokrollah
Salarian
2183-2203
Abstract: In this paper we study relative and Tate cohomology of modules of finite Gorenstein injective dimension. Using these cohomology theories, we present variations of Grothendieck local cohomology modules, namely Gorenstein and Tate local cohomology modules. By applying a sort of Avramov-Martsinkovsky exact sequence, we show that these two variations of local cohomology are tightly connected to the generalized local cohomology modules introduced by J. Herzog. We discuss some properties of these modules and give some results concerning their vanishing and non-vanishing.
Relative entropy functions for factor maps between subshifts
Sujin
Shin
2205-2216
Abstract: Let $(X, S)$ and $(Y, T)$ be topological dynamical systems and $\pi : X \rightarrow Y$ a factor map. A function $F \in C (X)$ is a compensation function for $\pi$ if $P (F + \phi \circ \pi ) = P (\phi)$ for all $\phi \in C(Y)$. For a factor code between subshifts of finite type, we analyze the associated relative entropy function and give a necessary condition for the existence of saturated compensation functions. Necessary and sufficient conditions for a map to be a saturated compensation function will be provided.
Equivalence of domains arising from duality of orbits on flag manifolds
Toshihiko
Matsuki
2217-2245
Abstract: S. Gindikin and the author defined a $G_{\mathbb R}$- $K_{\mathbb C}$ invariant subset $C(S)$ of $G_{\mathbb C}$ for each $K_{\mathbb C}$-orbit $S$ on every flag manifold $G_{\mathbb C}/P$ and conjectured that the connected component $C(S)_0$ of the identity would be equal to the Akhiezer-Gindikin domain $D$ if $S$ is of non-holomorphic type by computing many examples. In this paper, we first prove this conjecture for the open $K_{\mathbb C}$-orbit $S$ on an ``arbitrary'' flag manifold generalizing the result of Barchini. This conjecture for closed $S$ was solved by J. A. Wolf and R. Zierau for Hermitian cases and by G. Fels and A. Huckleberry for non-Hermitian cases. We also deduce an alternative proof of this result for non-Hermitian cases.
Intersecting curves and algebraic subgroups: Conjectures and more results
E.
Bombieri;
D.
Masser;
U.
Zannier
2247-2257
Abstract: This paper solves in the affirmative, up to dimension $n=5$, a question raised in an earlier paper by the authors. The equivalence of the problem with a conjecture of Shou-Wu Zhang is proved in the Appendix.
The fourth power moment of automorphic $L$-functions for $GL(2)$ over a short interval
Yangbo
Ye
2259-2268
Abstract: In this paper we will prove bounds for the fourth power moment in the $t$ aspect over a short interval of automorphic $L$-functions $L(s,g)$ for $GL(2)$ on the central critical line Re$s=1/2$. Here $g$ is a fixed holomorphic or Maass Hecke eigenform for the modular group $SL_{2}(\mathbb{Z})$, or in certain cases, for the Hecke congruence subgroup $\Gamma _{0}({\mathcal{N}})$ with $\mathcal{N}>1$. The short interval is from a large $K$ to $K+K^{103/135+\varepsilon }$. The proof is based on an estimate in the proof of subconvexity bounds for Rankin-Selberg $L$-function for Maass forms by Jianya Liu and Yangbo Ye (2002) and Yuk-Kam Lau, Jianya Liu, and Yangbo Ye (2004), which in turn relies on the Kuznetsov formula (1981) and bounds for shifted convolution sums of Fourier coefficients of a cusp form proved by Sarnak (2001) and by Lau, Liu, and Ye (2004).
Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence
Henry
K.
Schenck;
Alexander
I.
Suciu
2269-2289
Abstract: If $\mathcal A$ is a complex hyperplane arrangement, with complement $X$, we show that the Chen ranks of $G=\pi_1(X)$ are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring $A=H^*(X,\Bbbk)$, viewed as a module over the exterior algebra $E$ on $\mathcal A$: $\displaystyle \theta_k(G) = \dim_{\Bbbk}\operatorname{Tor}^E_{k-1}(A,\Bbbk)_k,$ for $k\ge 2$$\displaystyle ,$ where $\Bbbk$ is a field of characteristic 0. The Chen ranks conjecture asserts that, for $k$ sufficiently large, $\theta_k(G) =(k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}$, where $h_r$ is the number of $r$-dimensional components of the projective resonance variety $\mathcal R^{1}(\mathcal A)$. Our earlier work on the resolution of $A$ over $E$ and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of $\mathcal R ^{1}(\mathcal A)$ and a localization argument, we establish the inequality $\displaystyle \theta_k(G) \ge (k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k},$ for $k\gg 0$$\displaystyle ,$ for arbitrary $\mathcal A$. Finally, we show that there is a polynomial $\mathrm{P}(t)$ of degree equal to the dimension of $\mathcal R^1(\mathcal A)$, such that $\theta_k(G) = \mathrm{P}(k)$, for all $k\gg 0$.
Canard solutions at non-generic turning points
Peter
De Maesschalck;
Freddy
Dumortier
2291-2334
Abstract: This paper deals with singular perturbation problems for vector fields on $2$-dimensional manifolds. ``Canard solutions'' are solutions that, starting near an attracting normally hyperbolic branch of the singular curve, cross a ``turning point'' and follow for a while a normally repelling branch of the singular curve. Following the geometric ideas developed by Dumortier and Roussarie in 1996 for the study of canard solutions near a generic turning point, we study canard solutions near non-generic turning points. Characterization of manifolds of canard solutions is given in terms of boundary conditions, their regularity properties are studied and the relation is described with the more traditional asymptotic approach. It reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work. Since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve, we also study their contact and its relation to the ``control curve''.
A moment approach to analyze zeros of triangular polynomial sets
Jean
B.
Lasserre
1403-1420
Abstract: Let $I=\langle g_1,\ldots, g_n\rangle$ be a zero-dimensional ideal of $\mathbb{R}[x_1,\ldots ,x_n]$ such that its associated set $\mathbb{G}$ of polynomial equations $g_i(x)=0$ for all $i=1,\ldots ,n$ is in triangular form. By introducing multivariate Newton sums we provide a numerical characterization of polynomials in $\sqrt{I}$. We also provide a necessary and sufficient (numerical) condition for all the zeros of $\mathbb{G}$ to be in a given set $\mathbb{K}\subset \mathbb{C}^n$, without explicitly computing the zeros. In addition, we also provide a necessary and sufficient condition on the coefficients of the $g_i$'s for $\mathbb{G}$ to have (a) only real zeros, (b) to have only real zeros, all contained in a given semi-algebraic set $\mathbb{K}\subset\mathbb{R}^n$. In the proof technique, we use a deep result of Curto and Fialkow (2000) on the $\mathbb{K}$-moment problem, and the conditions we provide are given in terms of positive definiteness of some related moment and localizing matrices depending on the $g_i$'s via the Newton sums of $\mathbb{G}$. In addition, the number of distinct real zeros is shown to be the maximal rank of a related moment matrix.
Covering a compact set in a Banach space by an operator range of a Banach space with basis
V.
P.
Fonf;
W.
B.
Johnson;
A.
M.
Plichko;
V.
V.
Shevchyk
1421-1434
Abstract: A Banach space $X$ has the approximation property if and only if every compact set in $X$ is in the range of a one-to-one bounded linear operator from a space that has a Schauder basis. Characterizations are given for $\mathcal{L}_p$spaces and quotients of $\mathcal{L}_p$ spaces in terms of covering compact sets in $X$ by operator ranges from $\mathcal{L}_p$ spaces. A Banach space $X$is a $\mathcal{L}_1$ space if and only if every compact set in $X$ is contained in the closed convex symmetric hull of a basic sequence which converges to zero.
Sharp dimension estimates of holomorphic functions and rigidity
Bing-Long
Chen;
Xiao-Yong
Fu;
Le
Yin;
Xi-Ping
Zhu
1435-1454
Abstract: Let $M^n$ be a complete noncompact Kähler manifold of complex dimension $n$ with nonnegative holomorphic bisectional curvature. Denote by $\mathcal{O}_d(M^n)$ the space of holomorphic functions of polynomial growth of degree at most $d$ on $M^n$. In this paper we prove that $\displaystyle dim_{\mathbb{C}}{\mathcal{O}}_d(M^n)\leq dim_{\mathbb{C}}{\mathcal{O}}_{[d]}(\mathbb{C}^n),$ for all $d>0$, with equality for some positive integer $d$ if and only if $M^n$ is holomorphically isometric to $\mathbb{C}^n$. We also obtain sharp improved dimension estimates when its volume growth is not maximal or its Ricci curvature is positive somewhere.
Low-dimensional homogeneous Einstein manifolds
Christoph
Böhm;
Megan
M.
Kerr
1455-1468
Abstract: We show that compact, simply connected homogeneous spaces up to dimension $11$ admit homogeneous Einstein metrics.
Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces
Marcin
Bownik;
Kwok-Pun
Ho
1469-1510
Abstract: Weighted anisotropic Triebel-Lizorkin spaces are introduced and studied with the use of discrete wavelet transforms. This study extends the isotropic methods of dyadic $\varphi$-transforms of Frazier and Jawerth (1985, 1989) to non-isotropic settings associated with general expansive matrix dilations and $A_\infty$ weights. In close analogy with the isotropic theory, we show that weighted anisotropic Triebel-Lizorkin spaces are characterized by the magnitude of the $\varphi$-transforms in appropriate sequence spaces. We also introduce non-isotropic analogues of the class of almost diagonal operators and we obtain atomic and molecular decompositions of these spaces, thus extending isotropic results of Frazier and Jawerth.
Local theta correspondence for small unitary groups
Shu-Yen
Pan
1511-1535
Abstract: In this paper we give an explicit parameterization of the local theta correspondence of supercuspidal representations for the reductive dual pairs $({\rm U}_1(F),{\rm U}_1(F))$, $({\rm U}_1(F),{\rm U}_{1,1}(F))$, $({\rm U}_1(F),{\rm U}_{2}(F))$, and $({\rm U}_1(F),{\rm U}_{1,2}(F))$ of unitary groups over a nonarchimedean local field $F$ of odd residue characteristic.
Algebra of dimension theory
Jerzy
Dydak
1537-1561
Abstract: The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory, this algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes $\dim(A)$ of graded groups $A$. There are two geometric interpretations of these equivalence classes: 1) For pointed CW complexes $K$ and $L$, $\dim(H_\ast(K))=\dim(H_\ast(L))$ if and only if the infinite symmetric products $SP(K)$ and $SP(L)$ are of the same extension type (i.e., $SP(K)\in AE(X)$ iff $SP(L)\in AE(X)$ for all compact $X$). 2) For pointed compact spaces $X$ and $Y$, $\dim(\mathcal{H}^{-\ast}(X))=\dim(\mathcal{H}^{-\ast}(Y))$if and only if $X$ and $Y$ are of the same dimension type (i.e., $\dim_G(X)=\dim_G(Y)$ for all Abelian groups $G$). Dranishnikov's version of the Hurewicz Theorem in extension theory becomes $\dim(\pi_\ast(K))=\dim(H_\ast(K))$ for all simply connected $K$. The concept of cohomological dimension $\dim_A(X)$of a pointed compact space $X$ with respect to a graded group $A$ is introduced. It turns out $\dim_A(X) \leq 0$ iff $\dim_{A(n)}(X)\leq n$for all $n\in\mathbf{Z}$. If $A$ and $B$ are two positive graded groups, then $\dim(A)=\dim(B)$ if and only if $\dim_A(X)=\dim_B(X)$for all compact $X$.
Average size of $2$-Selmer groups of elliptic curves, I
Gang
Yu
1563-1584
Abstract: In this paper, we study a class of elliptic curves over $\mathbb{Q}$ with $\mathbb{Q}$-torsion group ${\mathbb{Z}}_{2}\times\mathbb{Z}_{2}$, and prove that the average order of the $2$-Selmer groups is bounded.
Stable geometric dimension of vector bundles over even-dimensional real projective spaces
Martin
Bendersky;
Donald
M.
Davis;
Mark
Mahowald
1585-1603
Abstract: In 1981, Davis, Gitler, and Mahowald determined the geometric dimension of stable vector bundles of order $2^e$ over $RP^{n}$ if $n$ is even and sufficiently large and $e\ge75$. In this paper, we use the Bendersky-Davis computation of $v_1^{-1}\pi _*(SO(m))$ to show that the 1981 result extends to all $e\ge5$ (still provided that $n$ is sufficiently large). If $e\le4$, the result is often different due to anomalies in the formula for $v_1^{-1}\pi_*(SO(m))$ when $m\le8$, but we also determine the stable geometric dimension in these cases.
On the Cohen-Macaulay property of multiplicative invariants
Martin
Lorenz
1605-1617
Abstract: We investigate the Cohen-Macaulay property for rings of invariants under multiplicative actions of a finite group $\mathcal{G}$. By definition, these are $\mathcal{G}$-actions on Laurent polynomial algebras $\Bbbk[x_1^{\pm 1},\dots,x_n^{\pm 1}]$that stabilize the multiplicative group consisting of all monomials in the variables $x_i$. For the most part, we concentrate on the case where the base ring $\Bbbk$ is $\mathbb{Z}$. Our main result states that if $\mathcal{G}$ acts non-trivially and the invariant ring $\mathbb{Z} [x_1^{\pm 1},\dots,x_n^{\pm 1}]^\mathcal{G}$ is Cohen-Macaulay, then the abelianized isotropy groups ${\mathcal{G}}_m^{{ab}}$ of all monomials $m$ are generated by the bireflections in $\mathcal{G}_m$ and at least one ${\mathcal{G}}_m^{{ab}}$ is non-trivial. As an application, we prove the multiplicative version of Kemper's $3$-copies conjecture.
Innately transitive subgroups of wreath products in product action
Robert
W.
Baddeley;
Cheryl
E.
Praeger;
Csaba
Schneider
1619-1641
Abstract: A permutation group is innately transitive if it has a transitive minimal normal subgroup, which is referred to as a plinth. We study the class of finite, innately transitive permutation groups that can be embedded into wreath products in product action. This investigation is carried out by observing that such a wreath product preserves a natural Cartesian decomposition of the underlying set. Previously we classified the possible embeddings in the case where the plinth is simple. Here we extend that classification and identify several different types of Cartesian decompositions that can be preserved by an innately transitive group with a non-abelian plinth. These different types of decompositions lead to different types of embeddings of the acting group into wreath products in product action. We also obtain a full characterisation of embeddings of innately transitive groups with diagonal type into such wreath products.
The geometry of symplectic pairs
G.
Bande;
D.
Kotschick
1643-1655
Abstract: We study the geometry of manifolds carrying symplectic pairs consisting of two closed $2$-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.
Height uniformity for integral points on elliptic curves
Su-ion
Ih
1657-1675
Abstract: We recall the result of D. Abramovich and its generalization by P. Pacelli on the uniformity for stably integral points on elliptic curves. It says that the Lang-Vojta conjecture on the distribution of integral points on a variety of logarithmic general type implies the uniformity for the numbers of stably integral points on elliptic curves. In this paper we will investigate its analogue for their heights under the assumption of the Vojta conjecture. Basically, we will show that the Vojta conjecture gives a naturally expected simple uniformity for their heights.
Parallel focal structure and singular Riemannian foliations
Dirk
Töben
1677-1704
Abstract: We give a necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds. We show that this is a singular Riemannian foliation with complete orthogonal transversals. For this object we construct an action on the transversals that generalizes the Weyl group action for polar actions.
Poisson structures on complex flag manifolds associated with real forms
Philip
Foth;
Jiang-Hua
Lu
1705-1714
Abstract: For a complex semisimple Lie group $G$ and a real form $G_0$ we define a Poisson structure on the variety of Borel subgroups of $G$ with the property that all $G_0$-orbits in $X$ as well as all Bruhat cells (for a suitable choice of a Borel subgroup of $G$) are Poisson submanifolds. In particular, we show that every non-empty intersection of a $G_0$-orbit and a Bruhat cell is a regular Poisson manifold, and we compute the dimension of its symplectic leaves.
Multiple homoclinic orbits in conservative and reversible systems
Ale
Jan
Homburg;
Jürgen
Knobloch
1715-1740
Abstract: We study dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows. We consider the existence of two homoclinic orbits in the bellows configuration, where the homoclinic orbits approach the equilibrium along the same direction for positive and negative times. In conservative systems one finds one parameter families of suspended horseshoes, parameterized by the level of the first integral. A somewhat similar picture occurs in reversible systems, with two homoclinic orbits that are both symmetric. The lack of a first integral implies that complete horseshoes do not exist. We provide a description of orbits that necessarily do exist. A second possible configuration in reversible systems occurs if a non-symmetric homoclinic orbit exists and forms a bellows together with its symmetric image. We describe the nonwandering set in an unfolding. The nonwandering set is shown to simultaneously contain one-parameter families of periodic orbits, hyperbolic periodic orbits of different index, and heteroclinic cycles between these periodic orbits.
On polynomial-factorial diophantine equations
Daniel
Berend;
Jørgen
E.
Harmse
1741-1779
Abstract: We study equations of the form $P(x)=n!$ and show that for some classes of polynomials $P$ the equation has only finitely many solutions. This is the case, say, if $P$ is irreducible (of degree greater than 1) or has an irreducible factor of ``relatively large" degree. This is also the case if the factorization of $P$ contains some ``large" power(s) of irreducible(s). For example, we can show that the equation $x^{r}(x+1)=n!$ has only finitely many solutions for $r\ge 4$, but not that this is the case for $1\le r\le 3$ (although it undoubtedly should be). We also study the equation $P(x)=H_{n}$, where $(H_{n})$ is one of several other ``highly divisible" sequences, proving again that for various classes of polynomials these equations have only finitely many solutions.
Automorphisms of Coxeter groups
Patrick
Bahls
1781-1796
Abstract: We compute ${\rm Aut}(W)$ for any even Coxeter group whose Coxeter diagram is connected, contains no edges labeled 2, and cannot be separated into more than 2 connected components by removing a single vertex. The description is given explicitly in terms of the given presentation for the Coxeter group and admits an easy characterization of those groups $W$ for which ${\rm Out}(W)$ is finite.
On the correlations of directions in the Euclidean plane
Florin
P.
Boca;
Alexandru
Zaharescu
1797-1825
Abstract: Let ${\mathcal{R}}^{(\nu )}_{(x,y),Q}$ denote the repartition of the $\nu$-level correlation measure of the finite set of directions $P_{(x,y)}P$, where $P_{(x,y)}$ is the fixed point $(x,y)\in [0,1)^{2}$ and $P$ is an integer lattice point in the square $[-Q,Q]^{2}$. We show that the average of the pair correlation repartition ${\mathcal{R}}^{(2)}_{(x,y),Q}$ over $(x,y)$ in a fixed disc ${\mathbb{D}}_{0}$ converges as $Q\rightarrow \infty$. More precisely we prove, for every $\lambda \in {\mathbb{R}}_{+}$ and $0<\delta <\frac{1}{10}$, the estimate \begin{displaymath}\frac{1}{\operatorname{Area} ({\mathbb{D}}_{0})} \iint \limi... ...1}{10}+\delta }) \qquad \text{\rm as$Q\rightarrow \infty$.} \end{displaymath} We also prove that for each individual point $(x,y)\in [0,1)^{2}$, the $6$-level correlation ${\mathcal{R}}^{(6)}_{(x,y),Q}(\lambda )$diverges at any point $\lambda \in {\mathbb{R}}^{5}_{+}$ as $Q\rightarrow \infty$, and we give an explicit lower bound for the rate of divergence.
On the hyperbolicity of the period-doubling fixed point
Daniel
Smania
1827-1846
Abstract: We give a new proof of the hyperbolicity of the fixed point for the period-doubling renormalization operator using the local dynamics near a semi-attractive fixed point (in a Banach space) and the theory of holomorphic motions. We also give a new proof of the exponential contraction of the Feigenbaum renormalization operator in the hybrid class of the period-doubling fixed point: our proof uses the non-existence of invariant line fields in the period-doubling tower (C. McMullen), the topological convergence (D. Sullivan), and a new infinitesimal argument.
Crystals of type $D_n^{(1)}$ and Young walls
Hyeonmi
Lee
1847-1867
Abstract: We give a new realization of arbitrary level perfect crystals and arbitrary level irreducible highest weight crystals of type $D_n^{(1)}$, in the language of Young walls. We refine the notions of splitting of blocks and slices that have appeared in our previous works, and these play crucial roles in the construction of crystals. The perfect crystals are realized as the set of equivalence classes of slices, and the irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls which, in turn, are concatenations of slices.
$L^1$--framework for continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations
Gui-Qiang
Chen;
Kenneth
H.
Karlsen
937-963
Abstract: We develop a general $L^1$-framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the Chen-Perthame kinetic approach. We apply our $L^1$-framework to establish an explicit estimate for continuous dependence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method, without imposition of bounded variation of the approximate solutions. Finally, as an example of a direct application of this framework to numerical methods, we focus on a linear convection-diffusion model equation and derive an $L^1$ error estimate for an upwind-central finite difference scheme.
Generalized interpolation in $H^\infty$ with a complexity constraint
Christopher
I.
Byrnes;
Tryphon
T.
Georgiou;
Anders
Lindquist;
Alexander
Megretski
965-987
Abstract: In a seminal paper, Sarason generalized some classical interpolation problems for $H^\infty$ functions on the unit disc to problems concerning lifting onto $H^2$ of an operator $T$ that is defined on $\EuScript{K} =H^2\ominus\phi H^2$ ($\phi$ is an inner function) and commutes with the (compressed) shift $S$. In particular, he showed that interpolants (i.e., $f\in H^\infty$ such that $f(S)=T$) having norm equal to $\Vert T\Vert$ exist, and that in certain cases such an $f$ is unique and can be expressed as a fraction $f=b/a$ with $a,b\in\EuScript{K}$. In this paper, we study interpolants that are such fractions of $\EuScript{K}$ functions and are bounded in norm by $1$ (assuming that $\Vert T\Vert<1$, in which case they always exist). We parameterize the collection of all such pairs $(a,b)\in\EuScript{K}\times\EuScript{K}$ and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where $\phi$ is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.
Invariants, Boolean algebras and ACA$_{0}^{+}$
Richard
A.
Shore
989-1014
Abstract: The sentences asserting the existence of invariants for mathematical structures are usually third order ones. We develop a general approach to analyzing the strength of such statements in second order arithmetic in the spirit of reverse mathematics. We discuss a number of simple examples that are equivalent to ACA$_{0}$. Our major results are that the existence of elementary equivalence invariants for Boolean algebras and isomorphism invariants for dense Boolean algebras are both of the same strength as ACA$_{0}^{+}$. This system corresponds to the assertion that $X^{(\omega)}$(the arithmetic jump of $X$) exists for every set $X$. These are essentially the first theorems known to be of this proof theoretic strength. The proof begins with an analogous result about these invariants on recursive (dense) Boolean algebras coding $0^{(\omega)}$.
Horrocks theory and the Bernstein-Gel'fand-Gel'fand correspondence
I.
Coanda;
G.
Trautmann
1015-1031
Abstract: We construct an explicit equivalence between a category of complexes over the exterior algebra, which we call HT-complexes, and the stable category of vector bundles on the corresponding projective space, essentially translating into more fancy terms the results of Trautmann (1978) which, in turn, were influenced by ideas of Horrocks (1964), (1980). However, the result expressed by Theorem 5.1 and its corollary, which establishes a relation between the Tate resolutions over the exterior algebra (described in a paper by Eisenbud, Fløystad, and Schreyer) and HT-complexes, might be new, although, perhaps, not a surprise to experts.
The integral cohomology of the Bianchi groups
Ethan
Berkove
1033-1049
Abstract: We calculate the integral cohomology ring structure for various members of the Bianchi group family. The main tools we use are the Bockstein spectral sequence and a long exact sequence derived from Bass-Serre theory.
Distance between toroidal surgeries on hyperbolic knots in the $3$-sphere
Masakazu
Teragaito
1051-1075
Abstract: For a hyperbolic knot in the $3$-sphere, at most finitely many Dehn surgeries yield non-hyperbolic $3$-manifolds. As a typical case of such an exceptional surgery, a toroidal surgery is one that yields a closed $3$-manifold containing an incompressible torus. The slope corresponding to a toroidal surgery, called a toroidal slope, is known to be integral or half-integral. We show that the distance between two integral toroidal slopes for a hyperbolic knot, except the figure-eight knot, is at most four.
Mayer brackets and solvability of PDEs -- II
Boris
Kruglikov;
Valentin
Lychagin
1077-1103
Abstract: For the Spencer $\delta$-cohomologies of a symbolic system we construct a spectral sequence associated with a subspace. We calculate the sequence for the systems of Cohen-Macaulay type and obtain a reduction theorem, which facilitates computation of $\delta$-cohomologies by reducing dimension of the system. Using this algebraic result we prove an efficient compatibility criterion for a system of two scalar non-linear PDEs on a manifold of any dimension in terms of (generalized) Mayer brackets.
Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces
Gil
Solanes
1105-1115
Abstract: We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.
Duality for Hopf orders
Robert
G.
Underwood;
Lindsay
N.
Childs
1117-1163
Abstract: In this paper we use duality to construct new classes of Hopf orders in the group algebra $KC_{p^3}$, where $K$ is a finite extension of $\mathbb{Q} _p$ and $C_{p^3}$ denotes the cyclic group of order $p^3$. Included in this collection is a subcollection of Hopf orders which are realizable as Galois groups.
Damped wave equation with a critical nonlinearity
Nakao
Hayashi;
Elena
I.
Kaikina;
Pavel
I.
Naumkin
1165-1185
Abstract: We study large time asymptotics of small solutions to the Cauchy problem for nonlinear damped wave equations with a critical nonlinearity $\begin{displaymath}\left\{ \begin{array}{c} \partial _{t}^{2}u+\partial _{t}u-\... ...u_{1}\left( x\right) ,x\in \mathbf{R}^{n}, \end{array}\right. \end{displaymath}$ where $\varepsilon >0,$ and space dimensions $n=1,2,3$. Assume that the initial data \begin{displaymath}u_{0}\in \mathbf{H}^{\delta ,0}\cap \mathbf{H}^{0,\delta },\t... ..._{1}\in \mathbf{H}^{\delta -1,0}\cap \mathbf{H}^{-1,\delta }, \end{displaymath} where $\delta >\frac{n}{2},$ weighted Sobolev spaces are $\begin{displaymath}\mathbf{H}^{l,m}=\left\{ \phi \in \mathbf{L}^{2};\left\Vert \... ...left( x\right) \right\Vert _{\mathbf{L}^{2}}<\infty \right\} , \end{displaymath}$ $\left\langle x\right\rangle =\sqrt{1+x^{2}}.$ Also we suppose that $\begin{displaymath}\lambda \theta ^{\frac{2}{n}}>0,\int u_{0}\left( x\right) dx>0, \end{displaymath}$ where $\begin{displaymath}\text{ }\theta =\int \left( u_{0}\left( x\right) +u_{1}\left( x\right) \right) dx\text{.} \end{displaymath}$ Then we prove that there exists a positive $\varepsilon _{0}$ such that the Cauchy problem above has a unique global solution $u\in \mathbf{C}\left( \left[ 0,\infty \right) ;\mathbf{H}^{\delta ,0}\right)$ satisfying the time decay property \begin{displaymath}\left\Vert u\left( t\right) -\varepsilon \theta G\left( t,x\r... ...gle t\right\rangle ^{-\frac{n}{2}\left( 1-\frac{1}{p}\right) } \end{displaymath} for all $t>0,$ $1\leq p\leq \infty ,$ where $\varepsilon \in \left( 0,\varepsilon _{0}\right] .$
Automorphisms of fiber surfaces of genus $2$, inducing the identity in cohomology
Jin-Xing
Cai
1187-1201
Abstract: Let $S$ be a complex non-singular projective surface of general type with a genus $2$ fibration and $\chi (\mathcal O_S)\geq 5$. Let $G \subset\operatorname{Aut}S$ be a non-trivial subgroup of automorphisms of $S$, inducing trivial actions on $H^i(S,\mathbb{Q})$ for all $i$. Then $\vert G\vert=2$, $K_S^2=4\chi (\mathcal O_S)$ and $q(S)=1$. Examples of such surfaces are given.
Constructive recognition of $\mathrm{PSL}(2, q)$
M.
D. E.
Conder;
C.
R.
Leedham-Green;
E.
A.
O'Brien
1203-1221
Abstract: Existing black box and other algorithms for explicitly recognising groups of Lie type over $\mathrm{GF}(q)$ have asymptotic running times which are polynomial in $q$, whereas the input size involves only $\log q$. This has represented a serious obstruction to the efficient recognition of such groups. Recently, Brooksbank and Kantor devised new explicit recognition algorithms for classical groups; these run in time that is polynomial in the size of the input, given an oracle that recognises $\mathrm{PSL}(2,q)$ explicitly. The present paper, in conjunction with an earlier paper by the first two authors, provides such an oracle. The earlier paper produced an algorithm for explicitly recognising $\mathrm{SL}(2,q)$ in its natural representation in polynomial time, given a discrete logarithm oracle for $\mathrm{GF}(q)$. The algorithm presented here takes as input a generating set for a subgroup $G$ of $\mathrm{GL}(d,F)$ that is isomorphic modulo scalars to $\mathrm{PSL}(2,q)$, where $F$ is a finite field of the same characteristic as $\mathrm{GF}(q)$; it returns the natural representation of $G$ modulo scalars. Since a faithful projective representation of $\mathrm{PSL}(2,q)$ in cross characteristic, or a faithful permutation representation of this group, is necessarily of size that is polynomial in $q$ rather than in $\log q$, elementary algorithms will recognise $\mathrm{PSL} (2,q)$ explicitly in polynomial time in these cases. Given a discrete logarithm oracle for $\mathrm{GF}(q)$, our algorithm thus provides the required polynomial time oracle for recognising $\mathrm{PSL}(2,q)$ explicitly in the remaining case, namely for representations in the natural characteristic. This leads to a partial solution of a question posed by Babai and Shalev: if $G$ is a matrix group in characteristic $p$, determine in polynomial time whether or not $O_p(G)$ is trivial.
Brownian intersection local times: Exponential moments and law of large masses
Wolfgang
König;
Peter
Mörters
1223-1255
Abstract: Consider $p$ independent Brownian motions in $\mathbb{R} ^d$, each running up to its first exit time from an open domain $B$, and their intersection local time $\ell$ as a measure on $B$. We give a sharp criterion for the finiteness of exponential moments, \begin{displaymath}\mathbb{E}\Big[\exp\Big(\sum_{i=1}^n \langle\varphi_i, \ell \rangle^{1/p}\Big) \Big],\end{displaymath} where $\varphi_1, \dots, \varphi_n$ are nonnegative, bounded functions with compact support in $B$. We also derive a law of large numbers for intersection local time conditioned to have large total mass.
Gorenstein projective dimension for complexes
Oana
Veliche
1257-1283
Abstract: We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.
Complex symmetric operators and applications
Stephan
Ramon
Garcia;
Mihai
Putinar
1285-1315
Abstract: We study a few classes of Hilbert space operators whose matrix representations are complex symmetric with respect to a preferred orthonormal basis. The existence of this additional symmetry has notable implications and, in particular, it explains from a unifying point of view some classical results. We explore applications of this symmetry to Jordan canonical models, self-adjoint extensions of symmetric operators, rank-one unitary perturbations of the compressed shift, Darlington synthesis and matrix-valued inner functions, and free bounded analytic interpolation in the disk.
Invariant pre-foliations for non-resonant non-uniformly hyperbolic systems
Ernest
Fontich;
Rafael
de la Llave;
Pau
Martín
1317-1345
Abstract: Given an orbit whose linearization has invariant subspaces satisfying some non-resonance conditions in the exponential rates of growth, we prove existence of invariant manifolds tangent to these subspaces. The exponential rates of growth can be understood either in the sense of Lyapunov exponents or in the sense of exponential dichotomies. These manifolds can correspond to ``slow manifolds'', which characterize the asymptotic convergence. Let $\{x_i\}_{i \in \mathbb{N} }$ be a regular orbit of a $C^2$ dynamical system $f$. Let $S$ be a subset of its Lyapunov exponents. Assume that all the Lyapunov exponents in $S$ are negative and that the sums of Lyapunov exponents in $S$ do not agree with any Lyapunov exponent in the complement of $S.$ Denote by $E^S_{x_i}$ the linear spaces spanned by the spaces associated to the Lyapunov exponents in $S.$ We show that there are smooth manifolds $W^S_{x_i}$ such that $f(W^S_{x_i}) \subset W^S_{x_{i+1}}$ and $T_{x_i} W^S_{x_i} = E^S_{x_i}$. We establish the same results for orbits satisfying dichotomies and whose rates of growth satisfy similar non-resonance conditions. These systems of invariant manifolds are not, in general, a foliation.
Quivers with relations arising from clusters $(A_n$ case)
P.
Caldero;
F.
Chapoton;
R.
Schiffler
1347-1364
Abstract: Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let $U$ be a cluster algebra of type $A_n$. We associate to each cluster $C$ of $U$ an abelian category $\mathcal{C}_C$ such that the indecomposable objects of $\mathcal{C}_C$ are in natural correspondence with the cluster variables of $U$ which are not in $C$. We give an algebraic realization and a geometric realization of $\mathcal{C}_C$. Then, we generalize the ``denominator theorem'' of Fomin and Zelevinsky to any cluster.
Contact reduction and groupoid actions
Marco
Zambon;
Chenchang
Zhu
1365-1401
Abstract: We introduce a new method to perform reduction of contact manifolds that extends Willett's and Albert's results. To carry out our reduction procedure all we need is a complete Jacobi map $J:M \rightarrow \Gamma_0$ from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of $\Gamma_0$ on $M$, and we show that the quotients of fibers $J^{-1}(x)$ by suitable Lie subgroups $\Gamma_x$ are either contact or locally conformal symplectic manifolds with structures induced by the one on $M$. We show that Willett's reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett's reduction can be performed only at distinguished points. As an application we obtain Kostant's prequantizations of coadjoint orbits. Finally we present several examples where we obtain classical contact manifolds as reduced spaces.
On meromorphic functions with finite logarithmic order
Peter
Tien-Yu
Chern
473-489
Abstract: By using a slow growth scale, the logarithmic order, with which to measure the growth of functions, we obtain basic results on the value distribution of a class of meromorphic functions of zero order.
Constant mean curvature surfaces in $M^2\times \mathbf{R}$
David
Hoffman;
Jorge
H. S.
de Lira;
Harold
Rosenberg
491-507
Abstract: The subject of this paper is properly embedded $H-$surfaces in Riemannian three manifolds of the form $M^2\times \mathbf{R}$, where $M^2$ is a complete Riemannian surface. When $M^2={\mathbf R}^2$, we are in the classical domain of $H-$surfaces in ${\mathbf R}^3$. In general, we will make some assumptions about $M^2$ in order to prove stronger results, or to show the effects of curvature bounds in $M^2$ on the behavior of $H-$surfaces in $M^2\times \mathbf{R}$.
Steinberg symbols modulo the trace class, holonomy, and limit theorems for Toeplitz determinants
Richard
W.
Carey;
Joel
D.
Pincus
509-551
Abstract: Suppose that $\phi=\psi z^\gamma$ where $\gamma\in Z_+$ and $\psi \in \text{\rm Lip}_\beta,\,{1\over 2}<\beta<1$, and the Toeplitz operator $T_\psi$ is invertible. Let $D_n(T_\phi)$ be the determinant of the Toeplitz matrix $((\hat\phi _{i,j}))=((\hat\phi _{i-j})),\quad 0\leq i,j\leq n ,$ where $\hat \phi_k={1\over 2\pi}\int_0^{2\pi} \phi(\theta)e^{-ik\theta}\, d\theta$. Let $P_n$ be the orthogonal projection onto $\ker {S^*}^{n+1}=\bigvee\{1,e^{i\theta}, e^{2i\theta},\ldots, e^{in\theta}\},$where $S=T_z$; set $Q_n=1-P_n$, let $H_\omega$ denote the Hankel operator associated to $\omega$, and set $\tilde\omega(t)=\omega({1\over t})$ for $t\in \mathbb{T}$. For the Wiener-Hopf factorization $\psi=f\bar g$ where $f, g$ and ${1\over f },{1\over g}\in \text{\rm Lip}_\beta\cap H^\infty(\mathbb{T} ), {1\over 2}<\beta<1$, put $E(\psi)=\exp\sum_{k=1}^\infty k(\log f)_k(\log \bar g)_{-k}$, $G(\psi)=\exp(\log\psi)_0.$ Theorem A. $D_n(T_\phi)=(-1)^{(n+1)\gamma} G(\psi)^{n+1}E(\psi) G({\bar g\over f})^\gamma$ $\cdot \det\bigg((T_{{f\over \bar g}z^{n+1}}\cdot [1-H_{\bar g\over f} Q_{n-\gam... ...^{\alpha-1},z^{\tau-1})\bigg)_{\gamma \times \gamma} \cdot [1+O(n^{1-2\beta})].$ Let $H^2(\mathbb{T} )= {\mathcal X}\dotplus {\mathcal Y}$ be a decomposition into $T_\phi T_{\phi^{-1}}$invariant subspaces, ${\mathcal X}= \bigcap_{n=1}^\infty\operatorname{ran} (T_\phi T_{\phi^{-1}})^n$and ${\mathcal Y}=\bigcup _{n=1}^\infty\ker (T_\phi T_{\phi^{-1}})^n$, so that $T_\phi T_{\phi^{-1}}$ restricted to ${\mathcal X}$ is invertible, ${\mathcal Y}$ is finite dimensional, and $T_\phi T_{\phi^{-1}}$ restricted to ${\mathcal Y}$ is nilpotent. Let $\{w_\alpha\}_1^\gamma$ be the basis $\{T_f z^\alpha\}_0^{\gamma-1}$ for the null space of $T_\phi T_{\phi^{-1}}$, and let $u_\alpha$ be the top vector in a Jordan root vector chain of length $m_\alpha+1$ lying over $(-1)^{m_\alpha}w_\alpha$, i.e., $(T_\phi T_{\phi^{-1}})^{m_\alpha}u_\alpha =(-1)^{m_\alpha}w_\alpha$where $m_\alpha=\max\{m\in Z_+:\exists x\,\text{\rm so that} (T_\phi T_{\phi^{-1}})^mx=w_\alpha\}^{-1}$. Theorem B. $E( \psi) G({\bar g\over f})^\gamma=$ $ {\prod_{\lambda\in\sigma(T_{\phi} T_{\phi^{-1}})\setminus \{0\}}\,\lambda}\over \det( u_\alpha,T_{1\over g}z^{\tau-1})$ $=\left (\bar g\cup f\times {\bar g\over f}\cup z^\gamma\right )(\mathbb{T} )$, the holonomy of a Deligne bundle with connection defined by the factorization $\phi= f\bar gz^\gamma$. Note that the generalizations of the Szegö limit theorem for $D_n(T_\phi)$which have appeared in the literature with $1$ instead of $[1-H_{\bar g\over f} Q_{n-\gamma} H_{({f\over \bar g})^{\tilde{}}}]^{-1}$ have the defect that the limit of ${D_n(T_\phi)\over (-1)^{(n+1)\gamma} G(\psi)^{n+1} \det(T_{{f\over \bar g}z^{n+1}}z^{\alpha-1},z^{\tau-1})}$ does not exist in general. An example is given with $D_n(T_\phi)\neq 0$yet $D_{\gamma-1}(T_{{f\over \bar g}z^{n+1}})=0$ for infinitely many $n$.
Norms and essential norms of linear combinations of endomorphisms
Pamela
Gorkin;
Raymond
Mortini
553-571
Abstract: We compute norms and essential norms of linear combinations of endomorphisms on uniform algebras.
MacNeille completions and canonical extensions
Mai
Gehrke;
John
Harding;
Yde
Venema
573-590
Abstract: Let $V$ be a variety of monotone bounded lattice expansions, that is, bounded lattices endowed with additional operations, each of which is order preserving or reversing in each coordinate. We prove that if $V$ is closed under MacNeille completions, then it is also closed under canonical extensions. As a corollary we show that in the case of Boolean algebras with operators, any such variety $V$ is generated by an elementary class of relational structures. Our main technical construction reveals that the canonical extension of a monotone bounded lattice expansion can be embedded in the MacNeille completion of any sufficiently saturated elementary extension of the original structure.
Multi-scale Young measures
Pablo
Pedregal
591-602
Abstract: We introduce multi-scale Young measures to deal with problems where multi-scale phenomena are relevant. We prove some interesting representation results that allow the use of these families of measures in practice, and illustrate its applicability by treating, from this perspective, multi-scale convergence and homogenization of multiple integrals.
An explicit characterization of Calogero--Moser systems
Fritz
Gesztesy;
Karl
Unterkofler;
Rudi
Weikard
603-656
Abstract: Combining theorems of Halphen, Floquet, and Picard and a Frobenius type analysis, we characterize rational, meromorphic simply periodic, and elliptic KdV potentials. In particular, we explicitly describe the proper extension of the Airault-McKean-Moser locus associated with these three classes of algebro-geometric solutions of the KdV hierarchy with special emphasis on the case of multiple collisions between the poles of solutions. This solves a problem left open since the mid-1970s.
Newton polygons and local integrability of negative powers of smooth functions in the plane
Michael
Greenblatt
657-670
Abstract: Let $f(x,y)$ be any smooth real-valued function with $f(0,0)=0$. For a sufficiently small neighborhood $U$ of the origin, we study the number \begin{displaymath}\sup\left\{\epsilon:\int_U \vert f(x,y)\vert^{-\epsilon}<\infty\right\}. \end{displaymath} It is known that sometimes this number can be expressed in a natural way using the Newton polygon of $f$. We provide necessary and sufficient conditions for this Newton polygon characterization to hold. The behavior of the integral at the supremal $\epsilon$ is also analyzed.
Some quotient Hopf algebras of the dual Steenrod algebra
J.
H.
Palmieri
671-685
Abstract: Fix a prime $p$, and let $A$ be the polynomial part of the dual Steenrod algebra. The Frobenius map on $A$ induces the Steenrod operation $\widetilde{\mathscr{P}}^{0}$on cohomology, and in this paper, we investigate this operation. We point out that if $p=2$, then for any element in the cohomology of $A$, if one applies $\widetilde{\mathscr{P}}^{0}$ enough times, the resulting element is nilpotent. We conjecture that the same is true at odd primes, and that ``enough times'' should be ``once.'' The bulk of the paper is a study of some quotients of $A$ in which the Frobenius is an isomorphism of order $n$. We show that these quotients are dual to group algebras, the resulting groups are torsion-free, and hence every element in Ext over these quotients is nilpotent. We also try to relate these results to the questions about $\widetilde{\mathscr{P}}^{0}$. The dual complete Steenrod algebra makes an appearance.
Equivariant Gysin maps and pulling back fixed points
Bernhard
Hanke;
Volker
Puppe
687-702
Abstract: We develop a new approach to the pulling back fixed points theorem of W. Browder and use it in order to prove various generalizations of this result.
On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo $5$ and generalizations
Alexander
Berkovich;
Frank
G.
Garvan
703-726
Abstract: In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic ${\mathcal O}(\pi)$ denotes the number of odd parts of the partition $\pi$and $\pi'$ is the conjugate of $\pi$. In a forthcoming paper, Andrews proved the following refinement of Ramanujan's partition congruence mod $5$: \begin{align*}p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod{5}, p(n) &= p_0(n) + p_2(n), \end{align*} where $p_i(n)$ ($i=0,2$) denotes the number of partitions of $n$ with $\mathrm{srank}\equiv i\pmod{4}$ and $p(n)$ is the number of unrestricted partitions of $n$. Andrews asked for a partition statistic that would divide the partitions enumerated by $p_i(5n+4)$ ($i=0,2$) into five equinumerous classes. In this paper we discuss three such statistics: the ST-crank, the $2$-quotient-rank and the $5$-core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the $2$-quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan's congruence mod $5$. This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo $5$. Finally, we discuss some new formulas for partitions that are $5$-cores and discuss an intriguing relation between $3$-cores and the Andrews-Garvan crank.
Cusp size bounds from singular surfaces in hyperbolic 3-manifolds
C.
Adams;
A.
Colestock;
J.
Fowler;
W.
Gillam;
E.
Katerman
727-741
Abstract: Singular maps of surfaces into a hyperbolic 3-manifold are utilized to find upper bounds on meridian length, $\ell$-curve length and maximal cusp volume for the manifold. This allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3-manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9. We also find particular upper bounds on meridian length, $\ell$-curve length and maximal cusp volume for hyperbolic knots in $\mathbb{S} ^3$ depending on crossing number. Particular improved bounds are obtained for alternating knots.
Isovariant Borsuk-Ulam results for pseudofree circle actions and their converse
Ikumitsu
Nagasaki
743-757
Abstract: In this paper we shall study the existence of an $S^1$-isovariant map from a rational homology sphere $M$ with pseudofree action to a representation sphere $SW$. We first show some isovariant Borsuk-Ulam type results. Next we shall consider the converse of those results and show that there exists an $S^1$-isovariant map from $M$ to $SW$ under suitable conditions.
Surfaces of general type with $p_g=q=1, K^2=8$
and bicanonical map of degree $2$
Francesco
Polizzi
759-798
Abstract: We classify the minimal algebraic surfaces of general type with $p_g=q=1, \; K^2=8$ and bicanonical map of degree $2$. It will turn out that they are isogenous to a product of curves, i.e. if $S$ is such a surface, then there exist two smooth curves $C, \; F$ and a finite group $G$ acting freely on $C \times F$ such that $S = (C \times F)/G$. We describe the $C, \; F$ and $G$that occur. In particular the curve $C$ is a hyperelliptic-bielliptic curve of genus $3$, and the bicanonical map $\phi$ of $S$ is composed with the involution $\sigma$ induced on $S$ by $\tau \times id: C \times F \longrightarrow C \times F$, where $\tau$ is the hyperelliptic involution of $C$. In this way we obtain three families of surfaces with $p_g=q=1, \; K^2=8$which yield the first-known examples of surfaces with these invariants. We compute their dimension and we show that they are three generically smooth, irreducible components of the moduli space $\mathcal{M}$ of surfaces with $p_g=q=1, \; K^2=8$. Moreover, we give an alternative description of these surfaces as double covers of the plane, recovering a construction proposed by Du Val.
Lagrangian submanifolds and moment convexity
Bernhard
Krötz;
Michael
Otto
799-818
Abstract: We consider a Hamiltonian torus action $T\times M \rightarrow M$ on a compact connected symplectic manifold $M$ and its associated momentum map $\Phi$. For certain Lagrangian submanifolds $Q\subseteq M$ we show that $\Phi (Q)$ is convex. The submanifolds $Q$ arise as the fixed point set of an involutive diffeomorphism $\tau :M\rightarrow M$ which satisfies several compatibility conditions with the torus action, but which is in general not anti-symplectic. As an application we complete a symplectic proof of Kostant's non-linear convexity theorem.
The general hyperplane section of a curve
Elisa
Gorla
819-869
Abstract: In this paper, we discuss some necessary and sufficient conditions for a curve to be arithmetically Cohen-Macaulay, in terms of its general hyperplane section. We obtain a characterization of the degree matrices that can occur for points in the plane that are the general plane section of a non-arithmetically Cohen-Macaulay curve of $\mathbf{P}^3$. We prove that almost all the degree matrices with positive subdiagonal that occur for the general plane section of a non-arithmetically Cohen-Macaulay curve of $\mathbf{P}^3$, arise also as degree matrices of some smooth, integral, non-arithmetically Cohen-Macaulay curve, and we characterize the exceptions. We give a necessary condition on the graded Betti numbers of the general plane section of an arithmetically Buchsbaum (non-arithmetically Cohen-Macaulay) curve in $\mathbf{P}^n$. For curves in $\mathbf{P}^3$, we show that any set of Betti numbers that satisfies that condition can be realized as the Betti numbers of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay curve. We also show that the matrices that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, integral, (smooth) non-arithmetically Cohen-Macaulay space curve are exactly those that arise as a degree matrix of the general plane section of an arithmetically Buchsbaum, non-arithmetically Cohen-Macaulay space curve and have positive subdiagonal. We also prove some bounds on the dimension of the deficiency module of an arithmetically Buchsbaum space curve in terms of the degree matrix of the general plane section of the curve, and we prove that they are sharp.
Nondegenerate $q$-biresolving textile systems and expansive automorphisms of onesided full shifts
Masakazu
Nasu
871-891
Abstract: We study nondegenerate, $q$-biresolving textile systems and using properties of them, we prove a conjecture of Boyle and Maass on arithmetic constraints for expansive automorphisms of onesided full shifts and positively expansive endomorphisms of mixing topological Markov shifts. A similar result is also obtained for expansive leftmost-permutive endomorphisms of onesided full shifts.
Representation formulae and inequalities for solutions of a class of second order partial differential equations
Lorenzo
D'Ambrosio;
Enzo
Mitidieri;
Stanislav
I.
Pohozaev
893-910
Abstract: Let $L$ be a possibly degenerate second order differential operator and let $\Gamma_\eta=d^{2-Q}$ be its fundamental solution at $\eta$; here $d$ is a suitable distance. In this paper we study necessary and sufficient conditions for the weak solutions of $-Lu\ge f(\xi,u)\ge 0$ on ${\mathbb{R}}^N$ to satisfy the representation formula \begin{displaymath}(\mbox R)\qquad\qquad\qquad\qquad\qquad u(\eta)\ge\int_{\mat... ...amma_\eta f(\xi,u) \,d\xi.\qquad\qquad\qquad\qquad\qquad\qquad \end{displaymath} We prove that (R) holds provided $f(\xi,\cdot)$ is superlinear, without any assumption on the behavior of $u$ at infinity. On the other hand, if $u$ satisfies the condition \begin{displaymath}\liminf_{R\rightarrow\infty} {-\int}_{R\le d(\xi)\le 2R}\vert u(\xi)\vert d\xi =0,\end{displaymath} then (R) holds with no growth assumptions on $f(\xi,\cdot)$.
Uniform bounds under increment conditions
Michel
Weber
911-936
Abstract: We apply a majorizing measure theorem of Talagrand to obtain uniform bounds for sums of random variables satisfying increment conditions of the type considered in Gál-Koksma Theorems. We give some applications.
Homotopical variations and high-dimensional Zariski-van Kampen theorems
D.
Chéniot;
C.
Eyral
1-10
Abstract: We give a new definition of the homotopical variation operators occurring in a recent high-dimensional Zariski-van Kampen theorem, a definition which opens the way to further generalizations of theorems of this kind.
Polar sets on metric spaces
Juha
Kinnunen;
Nageswari
Shanmugalingam
11-37
Abstract: We show that if $X$ is a proper metric measure space equipped with a doubling measure supporting a Poincaré inequality, then subsets of $X$ with zero $p$-capacity are precisely the $p$-polar sets; that is, a relatively compact subset of a domain in $X$ is of zero $p$-capacity if and only if there exists a $p$-superharmonic function whose set of singularities contains the given set. In addition, we prove that if $X$ is a $p$-hyperbolic metric space, then the $p$-superharmonic function can be required to be $p$-superharmonic on the entire space $X$. We also study the the following question: If a set is of zero $p$-capacity, does there exist a $p$-superharmonic function whose set of singularities is precisely the given set?
A generalization of Euler's hypergeometric transformation
Robert
S.
Maier
39-57
Abstract: Euler's transformation formula for the Gauss hypergeometric function ${}_2F_1$ is extended to hypergeometric functions of higher order. Unusually, the generalized transformation constrains the hypergeometric function parameters algebraically but not linearly. Its consequences for hypergeometric summation are explored. It has as a corollary a summation formula of Slater. From this formula new one-term evaluations of ${}_2F_1(-1)$ and ${}_3F_2(1)$ are derived by applying transformations in the Thomae group. Their parameters are also constrained nonlinearly. Several new one-term evaluations of ${}_2F_1(-1)$ with linearly constrained parameters are derived as well.
The complexity of recursion theoretic games
Martin
Kummer
59-86
Abstract: We show that some natural games introduced by Lachlan in 1970 as a model of recursion theoretic constructions are undecidable, contrary to what was previously conjectured. Several consequences are pointed out; for instance, the set of all $\Pi_2$-sentences that are uniformly valid in the lattice of recursively enumerable sets is undecidable. Furthermore we show that these games are equivalent to natural subclasses of effectively presented Borel games.
Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds
Yasushi
Homma
87-114
Abstract: Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give formulas in the enveloping algebra that induce not only identities for higher Casimir elements but also all Bochner-Weitzenböck formulas for gradients. As applications, we give some vanishing theorems.
Morse theory from an algebraic viewpoint
Emil
Sköldberg
115-129
Abstract: Forman's discrete Morse theory is studied from an algebraic viewpoint, and we show how this theory can be extended to chain complexes of modules over arbitrary rings. As applications we compute the homologies of a certain family of nilpotent Lie algebras, and show how the algebraic Morse theory can be used to derive the classical Anick resolution as well as a new two-sided Anick resolution.
On the $K$-theory and topological cyclic homology of smooth schemes over a discrete valuation ring
Thomas
Geisser;
Lars
Hesselholt
131-145
Abstract: We show that for a smooth and proper scheme over a henselian discrete valuation ring of mixed characteristic $(0,p)$, the $p$-adic étale $K$-theory and $p$-adic topological cyclic homology agree.
The limiting absorption principle for the two-dimensional inhomogeneous anisotropic elasticity system
Gen
Nakamura;
Jenn-Nan
Wang
147-165
Abstract: In this work we establish the limiting absorption principle for the two-dimensional steady-state elasticity system in an inhomogeneous aniso- tropic medium. We then use the limiting absorption principle to prove the existence of a radiation solution to the exterior Dirichlet or Neumann boundary value problems for such a system. In order to define the radiation solution, we need to impose certain appropriate radiation conditions at infinity. It should be remarked that even though in this paper we assume that the medium is homogeneous outside of a large domain, it still preserves anisotropy. Thus the classical Kupradze's radiation conditions for the isotropic system are not suitable in our problem and new radiation conditions are required. The uniqueness of the radiation solution plays a key role in establishing the limiting absorption principle. To prove the uniqueness of the radiation solution, we make use of the unique continuation property, which was recently obtained by the authors. The study of this work is motivated by related inverse problems in the anisotropic elasticity system. The existence and uniqueness of the radiation solution are fundamental questions in the investigation of inverse problems.
Quantifier elimination for algebraic $D$-groups
Piotr
Kowalski;
Anand
Pillay
167-181
Abstract: We prove that if $G$ is an algebraic $D$-group (in the sense of Buium over a differentially closed field $(K,\partial)$ of characteristic $0$, then the first order structure consisting of $G$ together with the algebraic $D$-subvarieties of $G, G\times G,\dots$, has quantifier-elimination. In other words, the projection on $G^{n}$ of a $D$-constructible subset of $G^{n+1}$ is $D$-constructible. Among the consequences is that any finite-dimensional differential algebraic group is interpretable in an algebraically closed field.
A new approach to the theory of classical hypergeometric polynomials
José
Manuel
Marco;
Javier
Parcet
183-214
Abstract: In this paper we present a unified approach to the spectral analysis of a hypergeometric type operator whose eigenfunctions include the classical orthogonal polynomials. We write the eigenfunctions of this operator by means of a new Taylor formula for operators of Askey-Wilson type. This gives rise to some expressions for the eigenfunctions, which are unknown in such a general setting. Our methods also give a general Rodrigues formula from which several well-known formulas of Rodrigues-type can be obtained directly. Moreover, other new Rodrigues-type formulas come out when seeking for regular solutions of the associated functional equations. The main difference here is that, in contrast with the formulas appearing in the literature, we get non-ramified solutions which are useful for applications in combinatorics. Another fact, that becomes clear in this paper, is the role played by the theory of elliptic functions in the connection between ramified and non-ramified solutions.
Symmetric functions in noncommuting variables
Mercedes
H.
Rosas;
Bruce
E.
Sagan
215-232
Abstract: Consider the algebra $\mathbb{Q}\langle \langle x_1,x_2,\ldots\rangle \rangle$ of formal power series in countably many noncommuting variables over the rationals. The subalgebra $\Pi(x_1,x_2,\ldots)$of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur symmetric functions as well as investigating their properties.
Real and complex earthquakes
Dragomir
Saric
233-249
Abstract: We consider (real) earthquakes and, by their extensions, complex earthquakes of the hyperbolic plane $\mathbb{H} ^2$. We show that an earthquake restricted to the boundary $S^1$ of $\mathbb{H} ^2$ is a quasisymmetric map if and only if its earthquake measure is bounded. Multiplying an earthquake measure by a positive parameter we obtain an earthquake path. Consequently, an earthquake path with a bounded measure is a path in the universal Teichmüller space. We extend the real parameter for a bounded earthquake into the complex parameter with small imaginary part. Such obtained complex earthquake (or bending) is holomorphic in the parameter. Moreover, the restrictions to $S^1$ of a bending with complex parameter of small imaginary part is a holomorphic motion of $S^1$in the complex plane. In particular, a real earthquake path with bounded earthquake measure is analytic in its parameter.
Some counterexamples to a generalized Saari's conjecture
Gareth
E.
Roberts
251-265
Abstract: For the Newtonian $n$-body problem, Saari's conjecture states that the only solutions with a constant moment of inertia are relative equilibria, solutions rigidly rotating about their center of mass. We consider the same conjecture applied to Hamiltonian systems with power-law potential functions. A family of counterexamples is given in the five-body problem (including the Newtonian case) where one of the masses is taken to be negative. The conjecture is also shown to be false in the case of the inverse square potential and two kinds of counterexamples are presented. One type includes solutions with collisions, derived analytically, while the other consists of periodic solutions shown to exist using standard variational methods.
On the Castelnuovo-Mumford regularity of connected curves
Daniel
Giaimo
267-284
Abstract: In this paper we prove that the regularity of a connected curve is bounded by its degree minus its codimension plus 1. We also investigate the structure of connected curves for which this bound is optimal. In particular, we construct connected curves of arbitrarily high degree in $\mathbb{P} ^4$ having maximal regularity, but no extremal secants. We also show that any connected curve in $\mathbb{P} ^3$ of degree at least 5 with maximal regularity and no linear components has an extremal secant.
Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics
B.
M.
Hambly;
Michel
L.
Lapidus
285-314
Abstract: In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that by using a random recursive self-similar construction, it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary.
An invariant of tangle cobordisms
Mikhail
Khovanov
315-327
Abstract: We construct a new invariant of tangle cobordisms. The invariant of a tangle is a complex of bimodules over certain rings, well-defined up to chain homotopy equivalence. The invariant of a tangle cobordism is a homomorphism between complexes of bimodules assigned to boundaries of the cobordism.
The automorphism tower of groups acting on rooted trees
Laurent
Bartholdi;
Said
N.
Sidki
329-358
Abstract: The group of isometries $\operatorname{Aut}(\mathcal{T}_n)$ of a rooted $n$-ary tree, and many of its subgroups with branching structure, have groups of automorphisms induced by conjugation in $\operatorname{Aut}(\mathcal{T}_n)$. This fact has stimulated the computation of the group of automorphisms of such well-known examples as the group $\mathfrak{G}$ studied by R. Grigorchuk, and the group $\ddot\Gamma$ studied by N. Gupta and the second author. In this paper, we pursue the larger theme of towers of automorphisms of groups of tree isometries such as $\mathfrak{G}$ and $\ddot\Gamma$. We describe this tower for all subgroups of $\operatorname{Aut}(\mathcal{T}_2)$ which decompose as infinitely iterated wreath products. Furthermore, we fully describe the towers of $\mathfrak{G}$ and $\ddot\Gamma$. More precisely, the tower of $\mathfrak{G}$ is infinite countable, and the terms of the tower are $2$-groups. Quotients of successive terms are infinite elementary abelian $2$-groups. In contrast, the tower of $\ddot\Gamma$ has length $2$, and its terms are $\{2,3\}$-groups. We show that $\operatorname{Aut}^2(\ddot\Gamma) /\operatorname{Aut}(\ddot\Gamma)$ is an elementary abelian $3$-group of countably infinite rank, while $\operatorname{Aut}^3(\ddot\Gamma)=\operatorname{Aut}^2(\ddot\Gamma)$.
Unramified cohomology of classifying varieties for exceptional simply connected groups
Skip
Garibaldi
359-371
Abstract: Let $BG$ be a classifying variety for an exceptional simple simply connected algebraic group $G$. We compute the degree 3 unramified Galois cohomology of $BG$ with values in
Polygonal invariant curves for a planar piecewise isometry
Peter
Ashwin;
Arek
Goetz
373-390
Abstract: We investigate a remarkable new planar piecewise isometry whose generating map is a permutation of four cones. For this system we prove the coexistence of an infinite number of periodic components and an uncountable number of transitive components. The union of all periodic components is an invariant pentagon with unequal sides. Transitive components are invariant curves on which the dynamics are conjugate to a transitive interval exchange. The restriction of the map to the invariant pentagonal region is the first known piecewise isometric system for which there exist an infinite number of periodic components but the only aperiodic points are on the boundary of the region. The proofs are based on exact calculations in a rational cyclotomic field. We use the system to shed some light on a conjecture that PWIs can possess transitive invariant curves that are not smooth.
A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal
Volker
Runde
391-402
Abstract: Let $G$ be a locally compact group, and let $\mathcal{WAP}(G)$ denote the space of weakly almost periodic functions on $G$. We show that, if $G$ is a $[\operatorname{SIN}]$-group, but not compact, then the dual Banach algebra $\mathcal{WAP}(G)^\ast$ does not have a normal, virtual diagonal. Consequently, whenever $G$ is an amenable, non-compact $[\operatorname{SIN}]$-group, $\mathcal{WAP}(G)^\ast$ is an example of a Connes-amenable, dual Banach algebra without a normal, virtual diagonal. On the other hand, there are amenable, non-compact, locally compact groups $G$ such that $\mathcal{WAP}(G)^\ast$ does have a normal, virtual diagonal.
Quasi-finite modules for Lie superalgebras of infinite rank
Ngau
Lam;
R.
B.
Zhang
403-439
Abstract: We classify the quasi-finite irreducible highest weight modules over the infinite rank Lie superalgebras $\widehat{\rm gl}_{\infty\vert\infty}$, $\widehat{\mathcal{C}}$and $\widehat{\mathcal{ D}}$, and determine the necessary and sufficient conditions for such modules to be unitarizable. The unitarizable irreducible modules are constructed in terms of Fock spaces of free quantum fields, and explicit formulae for their formal characters are also obtained by investigating Howe dualities between the infinite rank Lie superalgebras and classical Lie groups.
A density theorem on automorphic $L$-functions and some applications
Yuk-Kam
Lau;
Jie
Wu
441-472
Abstract: We establish a density theorem on automorphic $L$-functions and give some applications on the extreme values of these $L$-functions at $s=1$ and the distribution of the Hecke eigenvalue of holomorphic cusp forms.
Quantum groups, differential calculi and the eigenvalues of the Laplacian
J.
Kustermans;
G.
J.
Murphy;
L.
Tuset
4681-4717
Abstract: We study $*$-differential calculi over compact quantum groups in the sense of S.L. Woronowicz. Our principal results are the construction of a Hodge operator commuting with the Laplacian, the derivation of a corresponding Hodge decomposition of the calculus of forms, and, for Woronowicz' first calculus, the calculation of the eigenvalues of the Laplacian.
$k$-hyponormality of finite rank perturbations of unilateral weighted shifts
Raúl
E.
Curto;
Woo
Young
Lee
4719-4737
Abstract: In this paper we explore finite rank perturbations of unilateral weighted shifts $W_{\alpha }$. First, we prove that the subnormality of $W_{\alpha }$ is never stable under nonzero finite rank perturbations unless the perturbation occurs at the zeroth weight. Second, we establish that 2-hyponormality implies positive quadratic hyponormality, in the sense that the Maclaurin coefficients of $D_{n}(s):=\text{det}\,P_{n}\,[(W_{\alpha }+sW_{\alpha }^{2})^{*},\, W_{\alpha }+s W_{\alpha }^{2}]\,P_{n}$are nonnegative, for every $n\ge 0$, where $P_{n}$ denotes the orthogonal projection onto the basis vectors $\{e_{0},\cdots ,e_{n}\}$. Finally, for $\alpha$ strictly increasing and $W_{\alpha }$ 2-hyponormal, we show that for a small finite-rank perturbation $\alpha ^{\prime }$ of $\alpha$, the shift $W_{\alpha ^{\prime }}$ remains quadratically hyponormal.
Canonical forms of Borel functions on the Milliken space
Olaf
Klein;
Otmar
Spinas
4739-4769
Abstract: The goal of this paper is to canonize Borel measurable mappings $\Delta\colon\Omega^\omega\to\mathbb{R}$, where $\Omega^\omega$ is the Milliken space, i.e., the space of all increasing infinite sequences of pairwise disjoint nonempty finite sets of $\omega$. This main result is a common generalization of a theorem of Taylor and a theorem of Prömel and Voigt.
A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior
Xinfu
Chen;
Shangbin
Cui;
Avner
Friedman
4771-4804
Abstract: In this paper we study a free boundary problem modeling the growth of radially symmetric tumors with two populations of cells: proliferating cells and quiescent cells. The densities of these cells satisfy a system of nonlinear first order hyperbolic equations in the tumor, and the tumor's surface is a free boundary $r=R(t)$. The nutrient concentration satisfies a diffusion equation, and $R(t)$ satisfies an integro-differential equation. It is known that this problem has a unique stationary solution with $R(t)\equiv R_s$. We prove that (i) if $\lim _{T\to \infty} \int^{T+1}_T \vert\dot R(t)\vert\,dt=0$, then $\lim_{t\to \infty}R(t)=R_s$, and (ii) the stationary solution is linearly asymptotically stable.
A simple algorithm for principalization of monomial ideals
Russell
A.
Goward Jr.
4805-4812
Abstract: In this paper, we give a simple constructive proof of principalization of monomial ideals and the global analog. This also gives an algorithm for principalization.
Cofinality of the nonstationary ideal
Pierre
Matet;
Andrzej
Roslanowski;
Saharon
Shelah
4813-4837
Abstract: We show that the reduced cofinality of the nonstationary ideal ${\mathcal N S}_\kappa$ on a regular uncountable cardinal $\kappa$ may be less than its cofinality, where the reduced cofinality of ${\mathcal N S}_\kappa$ is the least cardinality of any family ${\mathcal F}$ of nonstationary subsets of $\kappa$ such that every nonstationary subset of $\kappa$ can be covered by less than $\kappa$ many members of ${\mathcal F}$. For this we investigate connections of the various cofinalities of ${\mathcal N S}_\kappa$ with other cardinal characteristics of ${}^{\textstyle\kappa}\kappa$ and we also give a property of forcing notions (called manageability) which is preserved in ${<}\kappa$-support iterations and which implies that the forcing notion preserves non-meagerness of subsets of ${}^{\textstyle\kappa}\kappa$ (and does not collapse cardinals nor changes cofinalities).
Complete analytic equivalence relations
Alain
Louveau;
Christian
Rosendal
4839-4866
Abstract: We prove that various concrete analytic equivalence relations arising in model theory or analysis are complete, i.e. maximum in the Borel reducibility ordering. The proofs use some general results concerning the wider class of analytic quasi-orders.
Affine pseudo-planes and cancellation problem
Kayo
Masuda;
Masayoshi
Miyanishi
4867-4883
Abstract: We define affine pseudo-planes as one class of $\mathbb{Q}$-homology planes. It is shown that there exists an infinite-dimensional family of non-isomorphic affine pseudo-planes which become isomorphic to each other by taking products with the affine line $\mathbb{A} ^1$. Moreover, we show that there exists an infinite-dimensional family of the universal coverings of affine pseudo-planes with a cyclic group acting as the Galois group, which have the equivariant non-cancellation property. Our family contains the surfaces without the cancellation property, due to Danielewski-Fieseler and tom Dieck.
Extended Hardy-Littlewood inequalities and some applications
Hichem
Hajaiej
4885-4896
Abstract: We establish conditions under which the extended Hardy-Little- wood inequality \begin{displaymath}\int \limits_{\mathbb{R} ^N} H\big{(}\vert x\vert,\, u_1(x), ... ...ig{(}\vert x\vert,\, u_1^*(x), \ldots , u_m^*(x)\big{)}\, dx, \end{displaymath} where each $u_i$ is non-negative and $u_i^*$ denotes its Schwarz symmetrization, holds. We also determine appropriate monotonicity assumptions on $H$ such that equality occurs in the above inequality if and only if each $u_i$ is Schwarz symmetric. We end this paper with some applications of our results in the calculus of variations and partial differential equations.
A Gieseker type degeneration of moduli stacks of vector bundles on curves
Ivan
Kausz
4897-4955
Abstract: We construct a new degeneration of the moduli stack of vector bundles over a smooth curve when the curve degenerates to a singular curve which is irreducible with one double point. We prove that the total space of the degeneration is smooth and its special fibre is a divisor with normal crossings. Furthermore, we give a precise description of how the normalization of the special fibre of the degeneration is related to the moduli space of vector bundles over the desingularized curve.
Regulating flows, topology of foliations and rigidity
Sérgio
R.
Fenley
4957-5000
Abstract: A flow transverse to a foliation is regulating if, in the universal cover, an arbitrary orbit of the flow intersects every leaf of the lifted foliation. This implies that the foliation is $\mathbf{R}$-covered, that is, its leaf space in the universal cover is homeomorphic to the reals. We analyse the converse of this implication to study the topology of the leaf space of certain foliations. We prove that if a pseudo-Anosov flow is transverse to an $\mathbf{R}$-covered foliation and the flow is not an $\mathbf{R}$-covered Anosov flow, then the flow is regulating for the foliation. Using this we show that several interesting classes of foliations are not $\mathbf{R}$-covered. Finally we show a rigidity result: if an $\mathbf{R}$-covered Anosov flow is transverse to a foliation but is not regulating, then the foliation blows down to one topologically conjugate to the stable or unstable foliations of the transverse flow.
Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces
Siva
R.
Athreya;
Richard
F.
Bass;
Edwin
A.
Perkins
5001-5029
Abstract: We introduce a new method for proving the estimate \begin{displaymath}\left\Vert\frac{\partial^2 u}{\partial x_i \partial x_j} \right\Vert_{C^\alpha}\leq c\Vert f\Vert _{C^\alpha},\end{displaymath} where $u$ solves the equation $\Delta u-\lambda u=f$. The method can be applied to the Laplacian on $\mathbb{R}^\infty$. It also allows us to obtain similar estimates when we replace the Laplacian by an infinite-dimensional Ornstein-Uhlenbeck operator or other elliptic operators. These operators arise naturally in martingale problems arising from measure-valued branching diffusions and from stochastic partial differential equations.
An algebraic approach to multiresolution analysis
Richard
Foote
5031-5050
Abstract: The notion of a weak multiresolution analysis is defined over an arbitrary field in terms of cyclic modules for a certain affine group ring. In this setting the basic properties of weak multiresolution analyses are established, including characterizations of their submodules and quotient modules, the existence and uniqueness of reduced scaling equations, and the existence of wavelet bases. These results yield some standard facts on classical multiresolution analyses over the reals as special cases, but provide a different perspective by not relying on orthogonality or topology. Connections with other areas of algebra and possible further directions are mentioned.
Filtrations in semisimple rings
D.
S.
Passman
5051-5066
Abstract: In this paper, we describe the maximal bounded $\mathbb{Z}$-filtrations of Artinian semisimple rings. These turn out to be the filtrations associated to finite $\mathbb{Z}$-gradings. We also consider simple Artinian rings with involution, in characteristic $\neq 2$, and we determine those bounded $\mathbb{Z}$-filtrations that are maximal subject to being stable under the action of the involution. Finally, we briefly discuss the analogous questions for filtrations with respect to other Archimedean ordered groups.
Minimal invariant tori in the resonant regions for nearly integrable Hamiltonian systems
Chong-Qing
Cheng
5067-5095
Abstract: Consider a real analytical Hamiltonian system of KAM type $H(p,q)$ $=N(p)+P(p,q)$ that has $n$ degrees of freedom ($n>2$) and is positive definite in $p$. Let $\Omega =\{\omega\in \mathbb R^n \vert\langle \bar k,\omega\rangle =0, \bar k\in\mathbb Z^n\}$. In this paper we show that for most rotation vectors in $\Omega$, in the sense of ($n-1$)-dimensional Lebesgue measure, there is at least one ($n-1$)-dimensional invariant torus. These tori are the support of corresponding minimal measures. The Lebesgue measure estimate on this set is uniformly valid for any perturbation.
The cyclic and simplicial cohomology of $l^1(\mathbf{N})$
Frédéric
Gourdeau;
B.
E.
Johnson;
Michael
C.
White
5097-5113
Abstract: Let $\mathcal{A}=l^1(\mathbf Z_+)$ be the unital semigroup algebra of $\mathbf N$. We show that the cyclic cohomology groups $\mathcal{H}^n(\mathcal{A},\mathcal{A}')$ vanish for $n\ge 2$. The results obtained are extended to unital algebras $l^1(S)$ for some other semigroups of $\mathbf{R}$.
Hausdorff measures, dimensions and mutual singularity
Manav
Das
4249-4268
Abstract: Let $(X,d)$ be a metric space. For a probability measure $\mu$ on a subset $E$of $X$ and a Vitali cover $V$ of $E$, we introduce the notion of a $b_{\mu}$-Vitali subcover $V_{\mu}$, and compare the Hausdorff measures of $E$with respect to these two collections. As an application, we consider graph directed self-similar measures $\mu$ and $\nu$ in $\mathbb{R}^{d}$ satisfying the open set condition. Using the notion of pointwise local dimension of $\mu$with respect to $\nu$, we show how the Hausdorff dimension of some general multifractal sets may be computed using an appropriate stochastic process. As another application, we show that Olsen's multifractal Hausdorff measures are mutually singular.
First countable, countably compact spaces and the continuum hypothesis
Todd
Eisworth;
Peter
Nyikos
4269-4299
Abstract: We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy of $\omega_1$ with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize Saharon Shelah's iteration theorems appearing in Chapters V and VIII of Proper and improper forcing (1998), as well as Eisworth and Roitman's (1999) iteration theorem. We close the paper with a ZFC example (constructed using Shelah's club-guessing sequences) that shows similar results do not hold for closed pre-images of $\omega_2$.
Classification problems in continuum theory
Riccardo
Camerlo;
Udayan
B.
Darji;
Alberto
Marcone
4301-4328
Abstract: We study several natural classes and relations occurring in continuum theory from the viewpoint of descriptive set theory and infinite combinatorics. We provide useful characterizations for the relation of likeness among dendrites and show that it is a bqo with countably many equivalence classes. For dendrites with finitely many branch points the homeomorphism and quasi-homeomorphism classes coincide, and the minimal quasi-homeomorphism classes among dendrites with infinitely many branch points are identified. In contrast, we prove that the homeomorphism relation between dendrites is $S_\infty$-universal. It is shown that the classes of trees and graphs are both $\mathrm{D}_{2}({{\boldsymbol \Sigma_{3}^{0}}})$-complete, the class of dendrites is ${{\boldsymbol\Pi_{3}^{0}}}$-complete, and the class of all continua homeomorphic to a graph or dendrite with finitely many branch points is ${{\boldsymbol\Pi_{3}^{0}}}$-complete. We also show that if $G$ is a nondegenerate finitely triangulable continuum, then the class of $G$-like continua is ${\boldsymbol\Pi_{2}^{0}}$-complete.
Towers of 2-covers of hyperelliptic curves
Nils
Bruin;
E.
Victor
Flynn
4329-4347
Abstract: In this article, we give a way of constructing an unramified Galois-cover of a hyperelliptic curve. The geometric Galois-group is an elementary abelian $2$-group. The construction does not make use of the embedding of the curve in its Jacobian, and it readily displays all subcovers. We show that the cover we construct is isomorphic to the pullback along the multiplication-by-$2$ map of an embedding of the curve in its Jacobian. We show that the constructed cover has an abundance of elliptic and hyperelliptic subcovers. This makes this cover especially suited for covering techniques employed for determining the rational points on curves. In particular the hyperelliptic subcovers give a chance for applying the method iteratively, thus creating towers of elementary abelian 2-covers of hyperelliptic curves. As an application, we determine the rational points on the genus $2$ curve arising from the question of whether the sum of the first $n$ fourth powers can ever be a square. For this curve, a simple covering step fails, but a second step succeeds.
Uniform properties of rigid subanalytic sets
Leonard
Lipshitz;
Zachary
Robinson
4349-4377
Abstract: In the context of rigid analytic spaces over a non-Archimedean valued field, a rigid subanalytic set is a Boolean combination of images of rigid analytic maps. We give an analytic quantifier elimination theorem for (complete) algebraically closed valued fields that is independent of the field; in particular, the analytic quantifier elimination is independent of the valued field's characteristic, residue field and value group, in close analogy to the algebraic case. This provides uniformity results about rigid subanalytic sets. We obtain uniform versions of smooth stratification for subanalytic sets and the \Lojasiewicz inequalities, as well as a unfiorm description of the closure of a rigid semianalytic set.
On the power series coefficients of certain quotients of Eisenstein series
Bruce
C.
Berndt;
Paul
R.
Bialek
4379-4412
Abstract: In their last joint paper, Hardy and Ramanujan examined the coefficients of modular forms with a simple pole in a fundamental region. In particular, they focused on the reciprocal of the Eisenstein series $E_6(\tau)$. In letters written to Hardy from nursing homes, Ramanujan stated without proof several more results of this sort. The purpose of this paper is to prove most of these claims.
Functional equations and their related operads
Vahagn
Minasian
4413-4443
Abstract: Using functional equations, we define functors that generalize standard examples from calculus of one variable. Examples of such functors are discussed, and their Taylor towers are computed. We also show that these functors factor through objects enriched over the homology of little $n$-cubes operads and discuss the relationship between functors defined via functional equations and operads. In addition, we compute the differentials of the forgetful functor from the category of $n$-Poisson algebras in terms of the homology of configuration spaces.
Riemannian nilmanifolds and the trace formula
Ruth
Gornet
4445-4479
Abstract: This paper examines the clean intersection hypothesis required for the expression of the wave invariants, computed from the asymptotic expansion of the classical wave trace by Duistermaat and Guillemin. The main result of this paper is the calculation of a necessary and sufficient condition for an arbitrary Riemannian two-step nilmanifold to satisfy the clean intersection hypothesis. This condition is stated in terms of metric Lie algebra data. We use the calculation to show that generic two-step nilmanifolds satisfy the clean intersection hypothesis. In contrast, we also show that the family of two-step nilmanifolds that fail the clean intersection hypothesis are dense in the family of two-step nilmanifolds. Finally, we give examples of nilmanifolds that fail the clean intersection hypothesis.
On the absolutely continuous spectrum of one-dimensional quasi-periodic Schrödinger operators in the adiabatic limit
Alexander
Fedotov;
Frédéric
Klopp
4481-4516
Abstract: In this paper we study the spectral properties of families of quasi-periodic Schrödinger operators on the real line in the adiabatic limit in the case when the adiabatic iso-energetic curves are extended along the position direction. We prove that, in energy intervals where this is the case, most of the spectrum is purely absolutely continuous in the adiabatic limit, and that the associated generalized eigenfunctions are Bloch-Floquet solutions. RÉSUMÉ. Cet article est consacré à l'étude du spectre de certaines familles d'équations de Schrödinger quasi-périodiques sur l'axe réel lorsque les variétés iso-énergetiques adiabatiques sont étendues dans la direction des positions. Nous démontrons que, dans un intervalle d'énergie où ceci est le cas, le spectre est dans sa majeure partie purement absolument continu et que les fonctions propres généralisées correspondantes sont des fonctions de Bloch-Floquet.
On the mod $p$ cohomology of $BPU(p)$
Ales
Vavpetic;
Antonio
Viruel
4517-4532
Abstract: We study the mod $p$ cohomology of the classifying space of the projective unitary group $PU(p)$. We first prove that conjectures due to J.F. Adams and Kono and Yagita (1993) about the structure of the mod $p$ cohomology of the classifying space of connected compact Lie groups hold in the case of $PU(p)$. Finally, we prove that the classifying space of the projective unitary group $PU(p)$ is determined by its mod $p$ cohomology as an unstable algebra over the Steenrod algebra for $p>3$, completing previous work by Dwyer, Miller and Wilkerson (1992) and Broto and Viruel (1998) for the cases $p=2,3$.
Free and semi-inert cell attachments
Peter
Bubenik
4533-4553
Abstract: Let $Y$ be the space obtained by attaching a finite-type wedge of cells to a simply-connected, finite-type CW-complex. We introduce the free and semi-inert conditions on the attaching map which broadly generalize the previously-studied inert condition. Under these conditions we determine $H_*(\Omega Y;R)$ as an $R$-module and as an $R$-algebra, respectively. Under a further condition we show that $H_*(\Omega Y;R)$ is generated by Hurewicz images. As an example we study an infinite family of spaces constructed using only semi-inert cell attachments.
Aleksandrov surfaces and hyperbolicity
Byung-Geun
Oh
4555-4577
Abstract: Aleksandrov surfaces are a generalization of two-dimensional Riemannian manifolds, and it is known that every open simply-connected Aleksandrov surface is conformally equivalent either to the unit disc (hyperbolic case) or to the plane (parabolic case). We prove a criterion for hyperbolicity of Aleksandrov surfaces which have nice tilings and where negative curvature dominates. We then apply this to generalize a result of Nevanlinna and give a partial answer for his conjecture about line complexes.
Canonical varieties with no canonical axiomatisation
Ian
Hodkinson;
Yde
Venema
4579-4605
Abstract: We give a simple example of a variety $\mathbf{V}$ of modal algebras that is canonical but cannot be axiomatised by canonical equations or first-order sentences. We then show that the variety $\mathbf{RRA}$ of representable relation algebras, although canonical, has no canonical axiomatisation. Indeed, we show that every axiomatisation of these varieties involves infinitely many non-canonical sentences. Using probabilistic methods of Erdos, we construct an infinite sequence $G_0,G_1,\ldots$ of finite graphs with arbitrarily large chromatic number, such that each $G_n$ is a bounded morphic image of $G_{n+1}$ and has no odd cycles of length at most $n$. The inverse limit of the sequence is a graph with no odd cycles, and hence is 2-colourable. It follows that a modal algebra (respectively, a relation algebra) obtained from the $G_n$ satisfies arbitrarily many axioms from a certain axiomatisation of $\mathbf{V} (\mathbf{RRA})$, while its canonical extension satisfies only a bounded number of them. First-order compactness will now establish that $\mathbf{V} (\mathbf{RRA})$ has no canonical axiomatisation. A variant of this argument shows that all axiomatisations of these classes have infinitely many non-canonical sentences.
Nonautonomous Kato classes of measures and Feynman-Kac propagators
Archil
Gulisashvili
4607-4632
Abstract: The behavior of the Feynman-Kac propagator corresponding to a time-dependent measure on $R^n$ is studied. We prove the boundedness of the propagator in various function spaces on $R^n$, and obtain a uniqueness theorem for an exponentially bounded distributional solution to a nonautonomous heat equation.
Irregular hypergeometric systems associated with a singular monomial curve
María
Isabel
Hartillo-Hermoso
4633-4646
Abstract: In this paper we study irregular hypergeometric systems defined by one row. Specifically, we calculate slopes of such systems. In the case of reduced semigroups, we generalize the case studied by Castro and Takayama. In all the cases we find that there always exists a slope with respect to a hyperplane of this system. Only in the case of an irregular system defined by a $1\times 2$integer matrix we might need a change of coordinates to study slopes at infinity. In the other cases slopes are always at the origin, defined with respect to a hyperplane. We also compute all the $L$-characteristic varieties of the system, so we have a section of the Gröbner fan of the module defined by the hypergeometric system.
Unipotent flat bundles and Higgs bundles over compact Kähler manifolds
Silke
Lekaus
4647-4659
Abstract: We characterize those unipotent representations of the fundamental group $\pi_1(X,x)$ of a compact Kähler manifold $X$, which correspond to a Higgs bundle whose underlying Higgs field is equal to zero. The characterization is parallel to the one that R. Hain gave of those unipotent representations of $\pi_1(X,x)$ that can be realized as the monodromy of a flat connection on the holomorphically trivial vector bundle. We see that Hain's result and ours follow from a careful study of Simpson's correspondence between Higgs bundles and local systems.
Outer factorizations in one and several variables
Michael
A.
Dritschel;
Hugo
J.
Woerdeman
4661-4679
Abstract: A multivariate version of Rosenblum's Fejér-Riesz theorem on outer factorization of trigonometric polynomials with operator coefficients is considered. Due to a simplification of the proof of the single variable case, new necessary and sufficient conditions for the multivariable outer factorization problem are formulated and proved.
Strong CHIP, normality, and linear regularity of convex sets
Andrew
Bakan;
Frank
Deutsch;
Wu
Li
3831-3863
Abstract: We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a finite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed convex cones, the dual normal property is the property (G) introduced by Jameson. Normality of convex sets provides a new perspective on the relationship between the strong conical hull intersection property (strong CHIP) and various regularity properties. In particular, we prove that the weak normal property is a dual characterization of the strong CHIP, and the uniform normal property is a characterization of the linear regularity. Moreover, the linear regularity is equivalent to the fact that the normality constant for feasible direction cones of the convex sets at $x$ is bounded away from 0 uniformly over all points in the intersection of these convex sets.
Ramsey families of subtrees of the dyadic tree
Vassilis
Kanellopoulos
3865-3886
Abstract: We show that for every rooted, finitely branching, pruned tree $T$of height $\omega$ there exists a family $\mathcal{F}$ which consists of order isomorphic to $T$ subtrees of the dyadic tree $C=\{0,1\}^{<\mathbb{N} }$ with the following properties: (i) the family $\mathcal{F}$ is a $G_\delta$ subset of $2^C$; (ii) every perfect subtree of $C$ contains a member of $\mathcal{F}$; (iii) if $K$ is an analytic subset of $\mathcal{F}$, then for every perfect subtree $S$ of $C$ there exists a perfect subtree $S'$ of $S$ such that the set
$L^p$ improving estimates for some classes of Radon transforms
Chan
Woo
Yang
3887-3903
Abstract: In this paper, we give $L^p-L^q$ estimates and the $L^p$ regularizing estimate of Radon transforms associated to real analytic functions, and we also give estimates of the decay rate of the $L^p$ operator norm of corresponding oscillatory integral operators. For $L^p-L^q$estimates and estimates of the decay rate of the $L^p$ operator norm we obtain sharp results except for extreme points; however, for $L^p$regularity we allow some restrictions on the phase function.
The Poincaré metric and isoperimetric inequalities for hyperbolic polygons
Roger
W.
Barnard;
Petros
Hadjicostas;
Alexander
Yu.
Solynin
3905-3932
Abstract: We prove several isoperimetric inequalities for the conformal radius (or equivalently for the Poincaré density) of polygons on the hyperbolic plane. Our results include, as limit cases, the isoperimetric inequality for the conformal radius of Euclidean $n$-gons conjectured by G. Pólya and G. Szegö in 1951 and a similar inequality for the hyperbolic $n$-gons of the maximal hyperbolic area conjectured by J. Hersch. Both conjectures have been proved in previous papers by the third author. Our approach uses the method based on a special triangulation of polygons and weighted inequalities for the reduced modules of trilaterals developed by A. Yu. Solynin. We also employ the dissymmetrization transformation of V. N. Dubinin. As an important part of our proofs, we obtain monotonicity and convexity results for special combinations of the Euler gamma functions, which appear to have a significant interest in their own right.
The uniform companion for large differential fields of characteristic 0
Marcus
Tressl
3933-3951
Abstract: We show that there is a theory UC of differential fields (in several commuting derivatives) of characteristic $0$, which serves as a model companion for every theory of large and differential fields extending a model complete theory of pure fields. As an application, we introduce differentially closed ordered fields, differentially closed p-adic fields and differentially closed pseudo-finite fields.
Turing patterns in the Lengyel-Epstein system for the CIMA reaction
Wei-Ming
Ni;
Moxun
Tang
3953-3969
Abstract: The first experimental evidence of the Turing pattern was observed by De Kepper and her associates (1990) on the CIMA reaction in an open unstirred gel reactor, almost 40 years after Turing's prediction. Lengyel and Epstein characterized this famous experiment using a system of reaction-diffusion equations. In this paper we report some fundamental analytic properties of the Lengyel-Epstein system. Our result also indicates that if either of the initial concentrations of the reactants, the size of the reactor, or the effective diffusion rate, are not large enough, then the system does not admit nonconstant steady states. A priori estimates are fundamental to our approach for this nonexistence result. The degree theory was combined with the a priori estimates to derive existence of nonconstant steady states.
Double forms, curvature structures and the $(p,q)$-curvatures
M.-L.
Labbi
3971-3992
Abstract: We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the $(p,q)$-curvatures. They are a generalization of the $p$-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for $p=0$, the $(0,q)$-curvatures coincide with the H. Weyl curvature invariants, for $p=1$ the $(1,q)$-curvatures are the curvatures of generalized Einstein tensors, and for $q=1$ the $(p,1)$-curvatures coincide with the $p$-curvatures. Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension $n\geq 4$, and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.
Coloring-flow duality of embedded graphs
Matt
DeVos;
Luis
Goddyn;
Bojan
Mohar;
Dirk
Vertigan;
Xuding
Zhu
3993-4016
Abstract: Let $G$ be a directed graph embedded in a surface. A map $\phi : E(G) \rightarrow \mathbb{R}$ is a tension if for every circuit $C \subseteq G$, the sum of $\phi$ on the forward edges of $C$ is equal to the sum of $\phi$ on the backward edges of $C$. If this condition is satisfied for every circuit of $G$ which is a contractible curve in the surface, then $\phi$ is a local tension. If $1 \le \vert\phi(e)\vert \le \alpha-1$ holds for every $e \in E(G)$, we say that $\phi$ is a (local) $\alpha$-tension. We define the circular chromatic number and the local circular chromatic number of $G$ by $\chi_{\rm c}(G) =\inf \{\alpha \in \mathbb{R}\mid \mbox{$G$has an$\alpha$-tension} \}$and $\chi_{\rm loc}(G) = \inf \{ \alpha \in \mathbb{R}\mid \mbox{$G$\space has a local$\alpha$-tension} \}$, respectively. The invariant $\chi_{\rm c}$ is a refinement of the usual chromatic number, whereas $\chi_{\rm loc}$ is closely related to Tutte's flow index and Bouchet's biflow index of the surface dual $G^*$. From the definitions we have $\chi_{\rm loc}(G) \le \chi_{\rm c}(G)$. The main result of this paper is a far-reaching generalization of Tutte's coloring-flow duality in planar graphs. It is proved that for every surface $\mathbb{X}$ and every $\varepsilon > 0$, there exists an integer $M$ so that $\chi_{\rm c}(G) \le \chi_{\rm loc}(G)+\varepsilon$ holds for every graph embedded in $\mathbb{X}$ with edge-width at least $M$, where the edge-width is the length of a shortest noncontractible circuit in $G$. In 1996, Youngs discovered that every quadrangulation of the projective plane has chromatic number 2 or 4, but never 3. As an application of the main result we show that such `bimodal' behavior can be observed in $\chi_{\rm loc}$, and thus in $\chi_{\rm c}$for two generic classes of embedded graphs: those that are triangulations and those whose face boundaries all have even length. In particular, if $G$ is embedded in some surface with large edge-width and all its faces have even length $\le 2r$, then $\chi_{\rm c}(G)\in [2,2+\varepsilon] \cup [\frac{2r}{r-1},4]$. Similarly, if $G$ is a triangulation with large edge-width, then $\chi_{\rm c}(G)\in [3,3+\varepsilon] \cup [4,5]$. It is also shown that there exist Eulerian triangulations of arbitrarily large edge-width on nonorientable surfaces whose circular chromatic number is equal to 5.
Inequalities for finite group permutation modules
Daniel
Goldstein;
Robert
M.
Guralnick;
I.
M.
Isaacs
4017-4042
Abstract: If $f$ is a nonzero complex-valued function defined on a finite abelian group $A$ and $\hat f$ is its Fourier transform, then $\vert\operatorname{supp}(f)\vert\vert\operatorname{supp}({\hat f})\vert \ge \vert A\vert$, where $\operatorname{supp}(f)$ and $\operatorname{supp}({\hat f})$ are the supports of $f$ and $\hat f$. In this paper we generalize this known result in several directions. In particular, we prove an analogous inequality where the abelian group $A$ is replaced by a transitive right $G$-set, where $G$ is an arbitrary finite group. We obtain stronger inequalities when the $G$-set is primitive, and we determine the primitive groups for which equality holds. We also explore connections between inequalities of this type and a result of Chebotarëv on complex roots of unity, and we thereby obtain a new proof of Chebotarëv's theorem.
Inverse spectral problem for normal matrices and the Gauss-Lucas theorem
S.
M.
Malamud
4043-4064
Abstract: We establish an analog of the Cauchy-Poincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem. Using this solution we generalize and extend the Gauss-Lucas theorem and prove the old conjecture of de Bruijn-Springer on the location of the roots of a complex polynomial and its derivative and an analog of Rolle's theorem, conjectured by Schoenberg.
The cohomology of the Steenrod algebra and representations of the general linear groups
Nguyên
H. V.
Hung
4065 - 4089
Abstract: Let $Tr_k$ be the algebraic transfer that maps from the coinvariants of certain $GL_k$-representations to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer $tr_k: \pi_*^S((B\mathbb{V} _k)_+) \to \pi_*^S(S^0)$. It has been shown that the algebraic transfer is highly nontrivial, more precisely, that $Tr_k$ is an isomorphism for $k=1, 2, 3$ and that $Tr= \bigoplus_k Tr_k$ is a homomorphism of algebras. In this paper, we first recognize the phenomenon that if we start from any degree $d$ and apply $Sq^0$ repeatedly at most $(k-2)$ times, then we get into the region in which all the iterated squaring operations are isomorphisms on the coinvariants of the $GL_k$-representations. As a consequence, every finite $Sq^0$-family in the coinvariants has at most $(k-2)$ nonzero elements. Two applications are exploited. The first main theorem is that $Tr_k$ is not an isomorphism for $k\geq 5$. Furthermore, for every $k>5$, there are infinitely many degrees in which $Tr_k$ is not an isomorphism. We also show that if $Tr_{\ell}$ detects a nonzero element in certain degrees of $\text{Ker}(Sq^0)$, then it is not a monomorphism and further, for each $k>\ell$, $Tr_k$ is not a monomorphism in infinitely many degrees. The second main theorem is that the elements of any $Sq^0$-family in the cohomology of the Steenrod algebra, except at most its first $(k-2)$ elements, are either all detected or all not detected by $Tr_k$, for every $k$. Applications of this study to the cases $k=4$ and $5$ show that $Tr_4$ does not detect the three families $g$, $D_3$ and $p'$, and that $Tr_5$ does not detect the family $\{h_{n+1}g_n \vert\; n\geq 1\}$.
Poincaré-Hopf inequalities
M.
A.
Bertolim;
M.
P.
Mello;
K.
A.
de Rezende
4091-4129
Abstract: In this article the main theorem establishes the necessity and sufficiency of the Poincaré-Hopf inequalities in order for the Morse inequalities to hold. The convex hull of the collection of all Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data determines a Morse polytope defined on the nonnegative orthant. Using results from network flow theory, a scheme is provided for constructing all possible Betti number vectors which satisfy the Morse inequalities for a pre-assigned index data. Geometrical properties of this polytope are described.
Nonuniform hyperbolicity for singular hyperbolic attractors
Wilmer
J.
Colmenárez Rodriguez
4131-4140
Abstract: In this paper we show nonuniform hyperbolicity for a class of attractors of $C^2$ flows in dimension three. These attractors are partially hyperbolic with central direction being volume expanding, contain dense periodic orbits and hyperbolic singularities of the associated vector field. Classical expanding Lorenz attractors are the main examples in this class.
On the finite embeddability property for residuated ordered groupoids
W.
J.
Blok;
C.
J.
van Alten
4141-4157
Abstract: The finite embeddability property (FEP) for integral, commutative residuated ordered monoids was established by W. J. Blok and C. J. van Alten in 2002. Using Higman's finite basis theorem for divisibility orders we prove that the assumptions of commutativity and associativity are not required: the classes of integral residuated ordered monoids and integral residuated ordered groupoids have the FEP as well. The same holds for their respective subclasses of (bounded) (semi-)lattice ordered structures. The assumption of integrality cannot be dropped in general--the class of commutative, residuated, lattice ordered monoids does not have the FEP--but the class of $n$-potent commutative residuated lattice ordered monoids does have the FEP, for any $n < \omega$.
Prescribing analytic singularities for solutions of a class of vector fields on the torus
Adalberto
P.
Bergamasco;
Sérgio
Luís
Zani
4159-4174
Abstract: We consider the operator $L=\partial_t+(a(t)+ib(t))\partial_x$ acting on distributions on the two-torus $\mathbb T^2,$ where $a$ and $b$ are real-valued, real analytic functions defined on the unit circle $\mathbb T^1.$We prove, among other things, that when $b$ changes sign, given any subset $\Sigma$ of the set of the local extrema of the local primitives of $b,$ there exists a singular solution of $L$ such that the $t-$projection of its analytic singular support is $\Sigma;$ furthermore, for any $\tau\in\Sigma$ and any closed subset $F$ of $\mathbb T^1_x$ there exists $Lu\in C^\omega(\mathbb T^2)$ and $\operatorname{sing\, supp_A}(u)=\{\tau\}\times F.$ We also provide a microlocal result concerning the trace of $u$ at $t=\tau.$
Classification of regular maps of negative prime Euler characteristic
Antonio
Breda
d'Azevedo;
Roman
Nedela;
Jozef
Sirán
4175-4190
Abstract: We give a classification of all regular maps on nonorientable surfaces with a negative odd prime Euler characteristic (equivalently, on nonorientable surfaces of genus $p+2$ where $p$is an odd prime). A consequence of our classification is that there are no regular maps on nonorientable surfaces of genus $p+2$where $p$ is a prime such that $p\equiv 1$ (mod $12$) and $p\ne 13$.
Minkowski valuations
Monika
Ludwig
4191-4213
Abstract: Centroid and difference bodies define $\operatorname{SL}(n)$ equivariant operators on convex bodies and these operators are valuations with respect to Minkowski addition. We derive a classification of $\operatorname{SL}(n)$equivariant Minkowski valuations and give a characterization of these operators. We also derive a classification of $\operatorname{SL}(n)$contravariant Minkowski valuations and of $L_p$-Minkowski valuations.
Saari's conjecture for the collinear $n$-body problem
Florin
Diacu;
Ernesto
Pérez-Chavela;
Manuele
Santoprete
4215-4223
Abstract: In 1970 Don Saari conjectured that the only solutions of the Newtonian $n$-body problem that have constant moment of inertia are the relative equilibria. We prove this conjecture in the collinear case for any potential that involves only the mutual distances. Furthermore, in the case of homogeneous potentials, we show that the only collinear and non-zero angular momentum solutions are homographic motions with central configurations.
Elliptic planar vector fields with degeneracies
Abdelhamid
Meziani
4225-4248
Abstract: This paper deals with the normalization of elliptic vector fields in the plane that degenerate along a simple and closed curve. The associated homogeneous equation $Lu=0$ is studied and an application to a degenerate Beltrami equation is given.
Harmonic calculus on fractals---A measure geometric approach II
M.
Zähle
3407-3423
Abstract: Riesz potentials of fractal measures $\mu$ in metric spaces and their inverses are introduced. They define self-adjoint operators in the Hilbert space $L_2(\mu)$ and the former are shown to be compact. In the Euclidean case the corresponding spectral asymptotics are derived with Besov space methods. The inverses of the Riesz potentials are fractal pseudodifferential operators. For the order two operator the spectral dimension coincides with the Hausdorff dimension of the underlying fractal.
Notes on limits of Sobolev spaces and the continuity of interpolation scales
Mario
Milman
3425-3442
Abstract: We extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz'ya-Shaposhnikova (2002), on limits of Sobolev spaces, to the setting of interpolation scales. This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points. A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained. We also give a new approach to the problem of how to recognize constant functions via Sobolev conditions.
Rhombic embeddings of planar quad-graphs
Richard
Kenyon;
Jean-Marc
Schlenker
3443-3458
Abstract: Given a finite or infinite planar graph all of whose faces have degree $4$, we study embeddings in the plane in which all edges have length $1$, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the existence of such an embedding, as well as a description of the set of all such embeddings. RÉSUMÉ. Etant donné un graphe planaire, fini ou infini, dont toutes les faces sont de degré $4$, on étudie ses plongements dans le plan dont toutes les arêtes sont de longueur $1$, c'est à dire dont toutes les faces sont des losanges. On donne une condition nécessaire et suffisante pour l'existence d'un tel plongement, et on décrit l'ensemble de ces plongements.
How to obtain transience from bounded radial mean curvature
Steen
Markvorsen;
Vicente
Palmer
3459-3479
Abstract: We show that Brownian motion on any unbounded submanifold $P$ in an ambient manifold $N$ with a pole $p$ is transient if the following conditions are satisfied: The $p$-radial mean curvatures of $P$ are sufficiently small outside a compact set and the $p$-radial sectional curvatures of $N$ are sufficiently negative. The `sufficiency' conditions are obtained via comparison with explicit transience criteria for radially drifted Brownian motion in warped product model spaces.
Cremer fixed points and small cycles
Lia
Petracovici
3481-3491
Abstract: Let $\lambda= e^{2\pi i \alpha}$, $\alpha \in \mathbb{R}\setminus \mathbb{Q}$, and let $(p_n/q_n)$ denote the sequence of convergents to the regular continued fraction of $\alpha$. Let $f$ be a function holomorphic at the origin, with a power series of the form $f(z)= \lambda z+\sum _{n=2}^{\infty}a_nz^n$. We assume that for infinitely many $n$ we simultaneously have (i) $\log \log q_{n+1} \geq 3\log q_n$, (ii) the coefficients $a_{1+q_n}$ stay outside two small disks, and (iii) the series $f(z)$ is lacunary, with $a_j=0$ for $2+q_n\leq j \leq q_n^{1+q_n}-1$. We then prove that $f(z)$ has infinitely many periodic orbits in every neighborhood of the origin.
Fixed point index in symmetric products
José
M.
Salazar
3493-3508
Abstract: Let $U$ be an open subset of a locally compact metric ANR $X$ and let $f:U \rightarrow X$ be a continuous map. In this paper we study the fixed point index of the map that $f$ induces in the $n$-symmetric product of $X$, $F_{n}(X)$. This index can detect the existence of periodic orbits of period $\leq n$ of $f$, and it can be used to obtain the Euler characteristic of the $n$-symmetric product of a manifold $X$, $\chi(F_{n}(X))$. We compute $\chi(F_{n}(X))$ for all orientable compact surfaces without boundary.
Tangent algebraic subvarieties of vector fields
Juan
B.
Sancho de Salas
3509-3523
Abstract: An algebraic commutative group $G$ is associated to any vector field $D$ on a complete algebraic variety $X$. The group $G$ acts on $X$ and its orbits are the minimal subvarieties of $X$ which are tangent to $D$. This group is computed in the case of a vector field on $\mathbb{P}_n$.
Clones from creatures
Martin
Goldstern;
Saharon
Shelah
3525-3551
Abstract: We show that (consistently) there is a clone $\mathcal{C}$ on a countable set such that the interval of clones above $\mathcal{C}$ is linearly ordered and has no coatoms.
Smooth projective varieties with extremal or next to extremal curvilinear secant subspaces
Sijong
Kwak
3553-3566
Abstract: We intend to give a classification of smooth nondegenerate projective varieties admitting extremal or next to extremal curvilinear secant subspaces. Gruson, Lazarsfeld and Peskine classified all projective integral curves with extremal secant lines. On the other hand, if a locally Cohen-Macaulay variety $X^{n}\subset \mathbb{P}^{n+e}$ of degree $d$meets with a linear subspace $L$ of dimension $\beta$ at finite points, then $\operatorname{length} {(X\cap L)}\le d-e+\beta$ as a finite scheme. A linear subspace $L$ for which the above length attains maximal possible value is called an extremal secant subspace and such $L$ for which $\operatorname{length}{(X\cap L)}= d-e+\beta -1$ is called a next to extremal secant subspace. In this paper, we show that if a smooth variety $X$ of degree $d\ge 6$ has extremal or next to extremal curvilinear secant subspaces, then it is either Del Pezzo or a scroll over a curve of genus $g\le 1$. This generalizes the results of Gruson, Lazarsfeld and Peskine (1983) for curves and the work of M-A. Bertin (2002) who classified smooth higher dimensional varieties with extremal secant lines. This is also motivated and closely related to establishing an upper bound for the Castelnuovo-Mumford regularity and giving a classification of the varieties on the boundary.
Homological and finiteness properties of picture groups
Daniel
S.
Farley
3567-3584
Abstract: Picture groups are a class of groups introduced by Guba and Sapir. Known examples include Thompson's groups $F$, $T$, and $V$. In this paper, a large class of picture groups is proved to be of type $F_{\infty}$. A Morse-theoretic argument shows that, for a given picture group, the rational homology vanishes in almost all dimensions.
Comparing Castelnuovo-Mumford regularity and extended degree: The borderline cases
Uwe
Nagel
3585-3603
Abstract: Castelnuovo-Mumford regularity and any extended degree function can be thought of as complexity measures for the structure of finitely generated graded modules. A recent result of Doering, Gunston, and Vasconcelos shows that both can be compared in the case of a graded algebra. We extend this result to modules and analyze when the estimate is in fact an equality. A complete classification is obtained if we choose as extended degree the homological or the smallest extended degree. The corresponding algebras are characterized in three ways: by relations among the algebra generators, by using generic initial ideals, and by their Hilbert series.
Depth and cohomological connectivity in modular invariant theory
Peter
Fleischmann;
Gregor
Kemper;
R.
James
Shank
3605-3621
Abstract: Let $G$ be a finite group acting linearly on a finite-dimensional vector space $V$ over a field $K$ of characteristic $p$. Assume that $p$ divides the order of $G$ so that $V$ is a modular representation and let $P$ be a Sylow $p$-subgroup for $G$. Define the cohomological connectivity of the symmetric algebra $S(V^*)$ to be the smallest positive integer $m$ such that $H^m(G,S(V^*))\not=0$. We show that $\min\left\{\dim_K(V^P) + m + 1,\dim_K(V)\right\}$is a lower bound for the depth of $S(V^*)^G$. We characterize those representations for which the lower bound is sharp and give several examples of representations satisfying the criterion. In particular, we show that if $G$ is $p$-nilpotent and $P$ is cyclic, then, for any modular representation, the depth of $S(V^*)^G$is $\min\left\{\dim_K(V^P) + 2,\dim_K(V)\right\}$.
Suspensions of crossed and quadratic complexes, co-H-structures and applications
Fernando
Muro
3623-3653
Abstract: Crossed and quadratic modules are algebraic models of the 2-type and the 3-type of a space, respectively. In this paper we compute a purely algebraic suspension functor from crossed to quadratic modules which sends a 2-type to the 3-type of its suspension. We also give some applications in homotopy theory and group theory.
Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups
Huy
Tài
Hà;
Ngô
Viêt
Trung
3655-3672
Abstract: This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a surprising complete answer to a question poised by Cutkosky and Herzog. Let $Y$ be the blow-up of a projective scheme $X = \operatorname{Proj} R$ along the ideal sheaf of $I \subset R$. It is known that there are embeddings $Y \cong \operatorname{Proj} k[(I^e)_c]$for $c \ge d(I)e + 1$, where $d(I)$ denotes the maximal generating degree of $I$, and that there exists a Cohen-Macaulay ring of the form $k[(I^e)_c]$(which gives an arithmetic Macaulayfication of $X$) if and only if $H^0(Y,\mathcal{O}_Y) = k$, $H^i(Y,\mathcal{O}_Y) = 0$ for $i = 1,..., \dim Y-1$, and $Y$ is equidimensional and Cohen-Macaulay. We show that under these conditions, there are well-determined invariants $\varepsilon$ and $e_0$ such that $k[(I^e)_c]$ is Cohen-Macaulay for all $c > d(I)e + \varepsilon$ and $e > e_0$, and that these bounds are the best possible. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form $R[(I^e)_ct]$. If $R$ has negative $a^*$-invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if $\pi_*\mathcal{O}_Y = \mathcal{O}_X$, $R^i\pi_*\mathcal{O}_Y = 0$ for $i > 0$, and $Y$ is equidimensional and Cohen-Macaulay. Moreover, these conditions imply the Cohen-Macaulayness of $R[(I^e)_ct]$ for all $c > d(I)e + \varepsilon$ and $e> e_0$.
Degeneration of linear systems through fat points on $K3$ surfaces
Cindy
De Volder;
Antonio
Laface
3673-3682
Abstract: In this paper we introduce a technique to degenerate $K3$surfaces and linear systems through fat points in general position on $K3$surfaces. Using this degeneration we show that on generic $K3$ surfaces it is enough to prove that linear systems with one fat point are non-special in order to obtain the non-speciality of homogeneous linear systems through $n = 4^u9^w$fat points in general position. Moreover, we use this degeneration to obtain a result for homogeneous linear systems through $n = 4^u9^w$ fat points in general position on a general quartic surface in $\mathbb{P}^3$.
Asymptotic properties of convolution operators and limits of triangular arrays on locally compact groups
Yves
Guivarc'h;
Riddhi
Shah
3683-3723
Abstract: We consider a locally compact group $G$ and a limiting measure $\mu$of a commutative infinitesimal triangular system (c.i.t.s.) $\Delta$of probability measures on $G$. We show, under some restrictions on $G$, $\mu$ or $\Delta$, that $\mu$ belongs to a continuous one-parameter convolution semigroup. In particular, this result is valid for symmetric c.i.t.s. $\Delta$ on any locally compact group $G$. It is also valid for a limiting measure $\mu$ which has `full' support on a Zariski connected $\mathbb{F}$-algebraic group $G$, where $\mathbb{F}$ is a local field, and any one of the following conditions is satisfied: (1) $G$ is a compact extension of a closed solvable normal subgroup, in particular, $G$ is amenable, (2) $\mu$ has finite one-moment or (3) $\mu$ has density and in case the characteristic of $\mathbb{F}$ is positive, the radical of $G$ is $\mathbb{F}$-defined. We also discuss the spectral radius of the convolution operator of a probability measure on a locally compact group $G$, we show that it is always positive for any probability measure on $G$, and it is also multiplicative in case of symmetric commuting measures.
Complex immersions in Kähler manifolds of positive holomorphic $k$-Ricci curvature
Fuquan
Fang;
Sérgio
Mendonça
3725-3738
Abstract: The main purpose of this paper is to prove several connectedness theorems for complex immersions of closed manifolds in Kähler manifolds with positive holomorphic $k$-Ricci curvature. In particular this generalizes the classical Lefschetz hyperplane section theorem for projective varieties. As an immediate geometric application we prove that a complex immersion of an $n$-dimensional closed manifold in a simply connected closed Kähler $m$-manifold $M$ with positive holomorphic $k$-Ricci curvature is an embedding, provided that $2n\ge m+k$. This assertion for $k=1$ follows from the Fulton-Hansen theorem (1979).
Applications of the Wold decomposition to the study of row contractions associated with directed graphs
Elias
Katsoulis;
David
W.
Kribs
3739-3755
Abstract: Based on a Wold decomposition for families of partial isometries and projections of Cuntz-Krieger-Toeplitz-type, we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks. We prove a Szego-type factorization theorem for CKT families, which leads to information on the structure of the unit ball in free semigroupoid algebras, and show that joint similarity implies joint unitary equivalence for such families. For each graph we prove a generalization of von Neumann's inequality which applies to row contractions of operators on Hilbert space which are related to the graph in a natural way. This yields a functional calculus determined by quiver algebras and free semigroupoid algebras. We establish a generalization of Coburn's theorem for the $\mathrm{C}^*$-algebra of a CKT family, and prove a universality theorem for $\mathrm{C}^*$-algebras generated by these families. In both cases, the $\mathrm{C}^*$-algebras generated by quiver algebras play the universal role.
Lagrangian tori in homotopy elliptic surfaces
Tolga
Etgü;
David
McKinnon;
B.
Doug
Park
3757-3774
Abstract: Let $E(1)_K$ denote the symplectic four-manifold, homotopy equivalent to the rational elliptic surface, corresponding to a fibred knot $K$ in $S^3$ constructed by R. Fintushel and R. J. Stern in 1998. We construct a family of nullhomologous Lagrangian tori in $E(1)_K$ and prove that infinitely many of these tori have complements with mutually non-isomorphic fundamental groups if the Alexander polynomial of $K$ has some irreducible factor which does not divide $t^n-1$ for any positive integer $n$. We also show how these tori can be non-isotopically embedded as nullhomologous Lagrangian submanifolds in other symplectic $4$-manifolds.
Renorming James tree space
Petr
Hájek;
Jan
Rychtár
3775-3788
Abstract: We show that the James tree space $JT$ can be renormed to be Lipschitz separated. This negatively answers the question of J. Borwein, J. Giles and J. Vanderwerff as to whether every Lipschitz separated Banach space is an Asplund space.
Differentiation evens out zero spacings
David
W.
Farmer;
Robert
C.
Rhoades
3789-3811
Abstract: If $f$ is a polynomial with all of its roots on the real line, then the roots of the derivative $f'$ are more evenly spaced than the roots of $f$. The same holds for a real entire function of order 1 with all its zeros on a line. In particular, we show that if $f$ is entire of order 1 and has sufficient regularity in its zero spacing, then under repeated differentiation the function approaches, after normalization, the cosine function. We also study polynomials with all their zeros on a circle, and we find a close analogy between the two situations. This sheds light on the spacing between zeros of the Riemann zeta-function and its connection to random matrix polynomials.
On Ore's conjecture and its developments
Ilaria
Del Corso;
Roberto
Dvornicich
3813-3829
Abstract: The $p$-component of the index of a number field $K$, ${ \rm ind}_p(K)$, depends only on the completions of $K$ at the primes over $p$. More precisely, ${\rm ind}_p(K)$ equals the index of the $\mathbb{Q} _p$-algebra $K\otimes\mathbb{Q} _p$. If $K$ is normal, then $K\otimes\mathbb{Q} _p\cong L^n$ for some $L$ normal over $\mathbb{Q} _p$ and some $n$, and we write $I_p(nL)$ for its index. In this paper we describe an effective procedure to compute $I_p(nL)$ for all $n$ and all normal and tamely ramified extensions $L$ of $\mathbb{Q} _p$, hence to determine ${\rm ind}_p(K)$ for all Galois number fields that are tamely ramified at $p$. Using our procedure, we are able to exhibit a counterexample to a conjecture of Nart (1985) on the behaviour of $I_p(nL)$.
Regular domains in homogeneous groups
Roberto
Monti;
Daniele
Morbidelli
2975-3011
Abstract: We study John, uniform and non-tangentially accessible domains in homogeneous groups of steps 2 and 3. We show that $C^{1,1}$ domains in groups of step 2 are non-tangentially accessible and we give an explicit condition which ensures the John property in groups of step 3.
Mixing times of the biased card shuffling and the asymmetric exclusion process
Itai
Benjamini;
Noam
Berger;
Christopher
Hoffman;
Elchanan
Mossel
3013-3029
Abstract: Consider the following method of card shuffling. Start with a deck of $N$cards numbered 1 through $N$. Fix a parameter $p$ between 0 and 1. In this model a ``shuffle'' consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability $p$. If the coin comes up heads, then we arrange the two cards so that the lower-numbered card comes before the higher-numbered card. If the coin comes up tails, then we arrange the cards with the higher-numbered card first. In this paper we prove that for all $p\ne 1/2$, the mixing time of this card shuffling is $O(N^2)$, as conjectured by Diaconis and Ram (2000). Our result is a rare case of an exact estimate for the convergence rate of the Metropolis algorithm. A novel feature of our proof is that the analysis of an infinite (asymmetric exclusion) process plays an essential role in bounding the mixing time of a finite process.
Gauss-Manin connections for arrangements, III Formal connections
Daniel
C.
Cohen;
Peter
Orlik
3031-3050
Abstract: We study the Gauss-Manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a complex rank one local system. We define formal Gauss-Manin connection matrices in the Aomoto complex and prove that, for all arrangements and all local systems, these formal connection matrices specialize to Gauss-Manin connection matrices.
Descent representations and multivariate statistics
Ron
M.
Adin;
Francesco
Brenti;
Yuval
Roichman
3051-3082
Abstract: Combinatorial identities on Weyl groups of types $A$ and $B$ are derived from special bases of the corresponding coinvariant algebras. Using the Garsia-Stanton descent basis of the coinvariant algebra of type $A$ we give a new construction of the Solomon descent representations. An extension of the descent basis to type $B$, using new multivariate statistics on the group, yields a refinement of the descent representations. These constructions are then applied to refine well-known decomposition rules of the coinvariant algebra and to generalize various identities.
A converse to Dye's theorem
Greg
Hjorth
3083-3103
Abstract: Every non-amenable countable group induces orbit inequivalent ergodic equivalence relations on standard Borel probability spaces. Not every free, ergodic, measure preserving action of $\mathbb{F} _2$ on a standard Borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces. There is a treeable non-hyperfinite Borel equivalence relation which is not universal for treeable in the $\leq_B$ ordering.
A computer-assisted proof of Saari's conjecture for the planar three-body problem
Richard
Moeckel
3105-3117
Abstract: The five relative equilibria of the three-body problem give rise to solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that these rigid motions are the only solutions with constant moment of inertia. This result will be proved here for the planar problem with three nonzero masses with the help of some computational algebra and geometry.
A countable Teichmüller modular group
Katsuhiko
Matsuzaki
3119-3131
Abstract: We construct an example of a Riemann surface of infinite topological type for which the Teichmüller modular group consists of only a countable number of elements. We also consider distinguished properties which the Teichmüller space of this Riemann surface possesses.
Valence of complex-valued planar harmonic functions
Genevra
Neumann
3133-3167
Abstract: The valence of a function $f$ at a point $w$is the number of distinct, finite solutions to $f(z) = w$. Let $f$ be a complex-valued harmonic function in an open set $R \subseteq \mathbb{C}$. Let $S$ denote the critical set of $f$and $C(f)$ the global cluster set of $f$. We show that $f(S) \cup C(f)$ partitions the complex plane into regions of constant valence. We give some conditions such that $f(S) \cup C(f)$has empty interior. We also show that a component $R_0 \subseteq R \backslash f^{-1} (f(S) \cup C(f))$ is an $n_0$-fold covering of some component $\Omega_0 \subseteq \mathbb{C}\backslash (f(S) \cup C(f))$. If $\Omega_0$ is simply connected, then $f$ is univalent on $R_0$. We explore conditions for combining adjacent components to form a larger region of univalence. Those results which hold for $C^1$ functions on open sets in $\mathbb{R} ^2$ are first stated in that form and then applied to the case of planar harmonic functions. If $f$ is a light, harmonic function in the complex plane, we apply a structure theorem of Lyzzaik to gain information about the difference in valence between components of $\mathbb{C}\backslash (f(S) \cup C(f))$sharing a common boundary arc in $f(S) \backslash C(f)$.
Entire solutions of certain partial differential equations and factorization of partial derivatives
Bao
Qin
Li
3169-3177
Abstract: We show that the problem of characterizing entire solutions of certain partial differential equations and the problem of characterizing common right factors of partial derivatives of meromorphic functions in $\mathbf{C}^{2}$ are closely related, and characterizations will be given using their relations.
Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations
Hisaaki
Endo;
Seiji
Nagami
3179-3199
Abstract: We introduce the notion of signature for relations in mapping class groups and show that the signature of a Lefschetz fibration over the 2-sphere is the sum of the signatures for basic relations contained in its monodromy. Combining explicit calculations of the signature cocycle with a technique of substituting positive relations, we give some new examples of non-holomorphic Lefschetz fibrations of genus $3, 4$ and $5$ which violate slope bounds for non-hyperelliptic fibrations on algebraic surfaces of general type.
Abelian categories, almost split sequences, and comodules
Mark
Kleiner;
Idun
Reiten
3201-3214
Abstract: The following are equivalent for a skeletally small abelian Hom-finite category over a field with enough injectives and each simple object being an epimorphic image of a projective object of finite length. (a) Each indecomposable injective has a simple subobject. (b) The category is equivalent to the category of socle-finitely copresented right comodules over a right semiperfect and right cocoherent coalgebra such that each simple right comodule is socle-finitely copresented. (c) The category has left almost split sequences.
Groups of units of integral group rings of Kleinian type
Antonio
Pita;
Ángel
del Río;
Manuel
Ruiz
3215-3237
Abstract: We explore a method to obtain presentations of the group of units of an integral group ring of some finite groups by using methods on Kleinian groups. We classify the nilpotent finite groups with central commutator for which the method works and apply the method for two concrete groups of order 16.
The smoothing property for a class of doubly nonlinear parabolic equations
Carsten
Ebmeyer;
José
Miguel
Urbano
3239-3253
Abstract: We consider a class of doubly nonlinear parabolic equations used in modeling free boundaries with a finite speed of propagation. We prove that nonnegative weak solutions satisfy a smoothing property; this is a well-known feature in some particular cases such as the porous medium equation or the parabolic $p$-Laplace equation. The result is obtained via regularization and a comparison theorem.
Representation dimension: An invariant under stable equivalence
Xiangqian
Guo
3255-3263
Abstract: In this paper, we prove that the representation dimension is an invariant under stable equivalence.
Spike-layered solutions for an elliptic system with Neumann boundary conditions
Miguel
Ramos;
Jianfu
Yang
3265-3284
Abstract: We prove the existence of nonconstant positive solutions for a system of the form $-\varepsilon^2\Delta u + u = g(v)$, $-\varepsilon^2\Delta v + v = f(u)$ in $\Omega$, with Neumann boundary conditions on $\partial \Omega$, where $\Omega$ is a smooth bounded domain and $f$, $g$are power-type nonlinearities having superlinear and subcritical growth at infinity. For small values of $\varepsilon$, the corresponding solutions $u_{\varepsilon}$ and $v_{\varepsilon}$ admit a unique maximum point which is located at the boundary of $\Omega$.
Seshadri constants at very general points
Michael
Nakamaye
3285-3297
Abstract: We study the local positivity of an ample line bundle at a very general point of a smooth projective variety. We obtain a slight improvement of the result of Ein, Küchle, and Lazarsfeld.
Generating the surface mapping class group by two elements
Mustafa
Korkmaz
3299-3310
Abstract: Wajnryb proved in 1996 that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two elements, again one of which is a Dehn twist. Another result we prove is that the mapping class groups are also generated by two elements of finite order.
Thurston's weak metric on the Teichmüller space of the torus
Abdelhadi
Belkhirat;
Athanase
Papadopoulos;
Marc
Troyanov
3311-3324
Abstract: We define and study a natural weak metric on the Teichmüller space of the torus. A similar metric has been defined by W. Thurston on the Teichmüller space of higher genus surfaces and our definition is motivated by Thurston's definition. However, we shall see that in the case of the torus, this metric has a different behaviour than on higher genus surfaces.
Torsion subgroups of elliptic curves in short Weierstrass form
Michael
A.
Bennett;
Patrick
Ingram
3325-3337
Abstract: In a recent paper by M. Wieczorek, a claim is made regarding the possible rational torsion subgroups of elliptic curves $E/\mathbb{Q}$ in short Weierstrass form, subject to certain inequalities for their coefficients. We provide a series of counterexamples to this claim and explore a number of related results. In particular, we show that, for any $\varepsilon>0$, all but finitely many curves \begin{displaymath}E_{A,B} \; : \; y^2 = x^3 + A x + B, \end{displaymath} where $A$ and $B$ are integers satisfying $A>\vert B\vert^{1+\varepsilon}>0$, have rational torsion subgroups of order either one or three. If we modify our demands upon the coefficients to $\vert A\vert>\vert B\vert^{2+\varepsilon}>0$, then the $E_{A,B}$ now have trivial rational torsion, with at most finitely many exceptions, at least under the assumption of the abc-conjecture of Masser and Oesterlé.
Telescoping, rational-valued series, and zeta functions
J.
Marshall
Ash;
Stefan
Catoiu
3339-3358
Abstract: We give an effective procedure for determining whether or not a series $\sum_{n=M}^{N}r\left( n\right)$ telescopes when $r\left( n\right)$ is a rational function with complex coefficients. We give new examples of series $\left( \ast\right) \sum_{n=1}^{\infty}r\left( n\right)$, where $r\left( n\right)$ is a rational function with integer coefficients, that add up to a rational number. Generalizations of the Euler phi function and the Riemann zeta function are involved. We give an effective procedure for determining which numbers of the form $\left( \ast\right)$ are rational. This procedure is conditional on 3 conjectures, which are shown to be equivalent to conjectures involving the linear independence over the rationals of certain sets of real numbers. For example, one of the conjectures is shown to be equivalent to the well-known conjecture that the set $\left\{ \zeta\left( s\right) :s=2,3,4,\dots\right\}$ is linearly independent, where $\zeta\left( s\right) =\sum n^{-s}$ is the Riemann zeta function. Some series of the form $\sum_{n}s\left( \sqrt[r]{n},\sqrt[r]{n+1} ,\cdots,\sqrt[r]{n+k}\right)$, where $s$ is a quotient of symmetric polynomials, are shown to be telescoping, as is $\sum1/(n!+\left( n-1\right) !)$. Quantum versions of these examples are also given.
On compact symplectic manifolds with Lie group symmetries
Daniel
Guan
3359-3373
Abstract: In this note we give a structure theorem for a finite-dimensional subgroup of the automorphism group of a compact symplectic manifold. An application of this result is a simpler and more transparent proof of the classification of compact homogeneous spaces with invariant symplectic structures. We also give another proof of the classification from the general theory of compact homogeneous spaces which leads us to a splitting conjecture on compact homogeneous spaces with symplectic structures (which are not necessary invariant under the group action) that makes the classification of this kind of manifold possible.
An approximate universal coefficient theorem
Huaxin
Lin
3375-3405
Abstract: An approximate Universal Coefficient Theorem (AUCT) for certain $C^*$-algebras is established. We present a proof that Kirchberg-Phillips's classification theorem for separable nuclear purely infinite simple $C^*$-algebras is valid for $C^*$-algebras satisfying the AUCT instead of the UCT. It is proved that two versions of AUCT are in fact the same. We also show that $C^*$-algebras that are locally approximated by $C^*$-algebras satisfying the AUCT satisfy the AUCT. As an application, we prove that certain simple $C^*$-algebras which are locally type I are in fact isomorphic to simple AH-algebras. As another application, we show that a sequence of residually finite-dimensional $C^*$-algebras which are asymptotically nuclear and which asymptotically satisfies the AUCT can be embedded into the same simple AF-algebra.
A rigid subspace of the real line whose square is a homogeneous subspace of the plane
L.
Brian
Lawrence
2535-2556
Abstract: Working in ZFC, we give an example as indicated in the title.
Cycles on curves over global fields of positive characteristic
Reza
Akhtar
2557-2569
Abstract: Let $k$ be a global field of positive characteristic, and let $\sigma: X \longrightarrow \operatorname{Spec} k$ be a smooth projective curve. We study the zero-dimensional cycle group $V(X) =\operatorname{Ker}(\sigma_*: SK_1(X) \rightarrow K_1(k))$ and the one-dimensional cycle group $W(X) =\operatorname{coker}(\sigma^*: K_2(k) \rightarrow H^0_{Zar}(X, \mathcal{K}_2))$, addressing the conjecture that $V(X)$ is torsion and $W(X)$ is finitely generated. The main idea is to use Abhyankar's Theorem on resolution of singularities to relate the study of these cycle groups to that of the $K$-groups of a certain smooth projective surface over a finite field.
On nonlinear wave equations with degenerate damping and source terms
Viorel
Barbu;
Irena
Lasiecka;
Mohammad
A.
Rammaha
2571-2611
Abstract: In this article we focus on the global well-posedness of the differential equation $u_{tt}- \Delta u + \vert u\vert^k\partial j(u_t) = \vert u\vert^{ p-1}u \, \text{ in } \Omega \times (0,T)$, where $\partial j$ is a sub-differential of a continuous convex function $j$. Under some conditions on $j$ and the parameters in the equations, we obtain several results on the existence of global solutions, uniqueness, nonexistence and propagation of regularity. Under nominal assumptions on the parameters we establish the existence of global generalized solutions. With further restrictions on the parameters we prove the existence and uniqueness of a global weak solution. In addition, we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent $p$ is greater than the critical value $k+m$, and the initial energy is negative. We also address the issue of propagation of regularity. Specifically, under some restriction on the parameters, we prove that solutions that correspond to any regular initial data such that $u_0\in H^2(\Omega)\cap H^1_0(\Omega)$, $u_1 \in H^1_0(\Omega)$ are indeed strong solutions.
The Bergman metric and the pluricomplex Green function
Zbigniew
Blocki
2613-2625
Abstract: We improve a lower bound for the Bergman distance in smooth pseudoconvex domains due to Diederich and Ohsawa. As the main tool we use the pluricomplex Green function and an $L^2$-estimate for the $\overline\partial$-operator of Donnelly and Fefferman.
Estimates of the derivatives for parabolic operators with unbounded coefficients
Marcello
Bertoldi;
Luca
Lorenzi
2627-2664
Abstract: We consider a class of second-order uniformly elliptic operators $\mathcal{A}$ with unbounded coefficients in $\mathbb{R}^N$. Using a Bernstein approach we provide several uniform estimates for the semigroup $T(t)$ generated by the realization of the operator $\mathcal{A}$ in the space of all bounded and continuous or Hölder continuous functions in $\mathbb{R}^N$. As a consequence, we obtain optimal Schauder estimates for the solution to both the elliptic equation $\lambda u-\mathcal{A}u=f$ ($\lambda>0$) and the nonhomogeneous Dirichlet Cauchy problem $D_tu=\mathcal{A}u+g$. Then, we prove two different kinds of pointwise estimates of $T(t)$ that can be used to prove a Liouville-type theorem. Finally, we provide sharp estimates of the semigroup $T(t)$ in weighted $L^p$-spaces related to the invariant measure associated with the semigroup.
On homeomorphism groups of Menger continua
Jan
J.
Dijkstra
2665-2679
Abstract: It is shown that the homeomorphism groups of the (generalized) Sierpinski carpet and the universal Menger continua are not zero-dimensional. These results were corollaries to a 1966 theorem of Brechner. New proofs were needed because we also show that Brechner's proof is inadequate. The method by which we obtain our results, the construction of closed imbeddings of complete Erdos space in the homeomorphism groups, is of independent interest.
Good measures on Cantor space
Ethan
Akin
2681-2722
Abstract: While there is, up to homeomorphism, only one Cantor space, i.e. one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor space which are not topologically equivalent. The clopen values set for a full, nonatomic measure $\mu$ is the countable dense subset $\{ \mu(U) : U$ is clopen$\}$ of the unit interval. It is a topological invariant for the measure. For the class of good measures it is a complete invariant. A full, nonatomic measure $\mu$ is good if whenever $U, V$ are clopen sets with $\mu(U) < \mu(V)$, there exists $W$ a clopen subset of $V$ such that $\mu(W) = \mu(U)$. These measures have interesting dynamical properties. They are exactly the measures which arise from uniquely ergodic minimal systems on Cantor space. For some of them there is a unique generic measure-preserving homeomorphism. That is, within the Polish group of such homeomorphisms there is a dense, $G_{\delta}$ conjugacy class.
Brauer groups of genus zero extensions of number fields
Jack
Sonn;
John
Swallow
2723-2738
Abstract: We determine the isomorphism class of the Brauer groups of certain nonrational genus zero extensions of number fields. In particular, for all genus zero extensions $E$ of the rational numbers $\mathbb{Q}$ that are split by $\mathbb{Q} (\sqrt{2})$, $\operatorname{Br}(E)\cong \operatorname{Br}(\mathbb{Q} (t))$.
Finite dimensional representations of invariant differential operators
Ian
M.
Musson;
Sonia
L.
Rueda
2739-2752
Abstract: Let $k$ be an algebraically closed field of characteristic $0$, $Y=k^{r}\times {(k^{\times})}^{s}$, and let $G$ be an algebraic torus acting diagonally on the ring of algebraic differential operators $\mathcal{D} (Y)$. We give necessary and sufficient conditions for $\mathcal{D}(Y)^G$ to have enough simple finite dimensional representations, in the sense that the intersection of the kernels of all the simple finite dimensional representations is zero. As an application we show that if $K\longrightarrow GL(V)$ is a representation of a reductive group $K$ and if zero is not a weight of a maximal torus of $K$ on $V$, then $\mathcal{D} (V)^K$ has enough finite dimensional representations. We also construct examples of FCR-algebras with any integer GK dimension $\geq 3$.
Homotopical localizations of module spectra
Carles
Casacuberta;
Javier
J.
Gutiérrez
2753-2770
Abstract: We prove that stable $f$-localizations (where $f$ is any map of spectra) preserve ring spectrum structures and module spectrum structures, under suitable hypotheses, and we use this fact to describe all possible localizations of the integral Eilenberg-MacLane spectrum $H{\mathbb{Z} }$. As a consequence of this study, we infer that localizations of stable GEMs are stable GEMs, and it also follows that there is a proper class of nonequivalent stable localizations.
Commutative ideal theory without finiteness conditions: Primal ideals
Laszlo
Fuchs;
William
Heinzer;
Bruce
Olberding
2771-2798
Abstract: Our goal is to establish an efficient decomposition of an ideal $A$ of a commutative ring $R$ as an intersection of primal ideals. We prove the existence of a canonical primal decomposition: $A = \bigcap_{P \in \mathcal{X}_A}A_{(P)}$, where the $A_{(P)}$ are isolated components of $A$ that are primal ideals having distinct and incomparable adjoint primes $P$. For this purpose we define the set $\operatorname{Ass}(A)$ of associated primes of the ideal $A$ to be those defined and studied by Krull. We determine conditions for the canonical primal decomposition to be irredundant, or residually maximal, or the unique representation of $A$ as an irredundant intersection of isolated components of $A$. Using our canonical primal decomposition, we obtain an affirmative answer to a question raised by Fuchs, and also prove for $P \in \operatorname{Spec}R$ that an ideal $A \subseteq P$ is an intersection of $P$-primal ideals if and only if the elements of $R \setminus P$ are prime to $A$. We prove that the following conditions are equivalent: (i) the ring $R$ is arithmetical, (ii) every primal ideal of $R$ is irreducible, (iii) each proper ideal of $R$ is an intersection of its irreducible isolated components. We classify the rings for which the canonical primal decomposition of each proper ideal is an irredundant decomposition of irreducible ideals as precisely the arithmetical rings with Noetherian maximal spectrum. In particular, the integral domains having these equivalent properties are the Prüfer domains possessing a certain property.
Poisson brackets associated to the conformal geometry of curves
G.
Marí Beffa
2799-2827
Abstract: In this paper we present an invariant moving frame, in the group theoretical sense, along curves in the Möbius sphere. This moving frame will describe the relationship between all conformal differential invariants for curves that appear in the literature. Using this frame we first show that the Kac-Moody Poisson bracket on $Lo(n+1,1)^\ast$ can be Poisson reduced to the space of conformal differential invariants of curves. The resulting bracket will be the conformal analogue of the Adler-Gel'fand-Dikii bracket. Secondly, a conformally invariant flow of curves induces naturally an evolution on the differential invariants of the flow. We give the conditions on the invariant flow ensuring that the induced evolution is Hamiltonian with respect to the reduced Poisson bracket. Because of a certain parallelism with the Euclidean case we study what we call Frenet and natural cases. We comment on the implications for completely integrable systems, and describe conformal analogues of the Hasimoto transformation.
Saddle surfaces in singular spaces
Dimitrios
E.
Kalikakis
2829-2841
Abstract: The notion of a saddle surface is well known in Euclidean space. In this work we extend the idea of a saddle surface to geodesically connected metric spaces. We prove that any solution of the Dirichlet problem for the Sobolev energy in a nonpositively curved space is a saddle surface. Further, we show that the space of saddle surfaces in a nonpositively curved space is a complete space in the Fréchet distance. We also prove a compactness theorem for saddle surfaces in spaces of curvature bounded from above; in spaces of constant curvature we obtain a stronger result based on an isoperimetric inequality for a saddle surface. These results generalize difficult theorems of S.Z. Shefel' on compactness of saddle surfaces in a Euclidean space.
Weighted estimates in $L^{2}$ for Laplace's equation on Lipschitz domains
Zhongwei
Shen
2843-2870
Abstract: Let $\Omega \subset \mathbb{R}^{d}$, $d\ge 3$, be a bounded Lipschitz domain. For Laplace's equation $\Delta u=0$ in $\Omega$, we study the Dirichlet and Neumann problems with boundary data in the weighted space $L^{2}(\partial \Omega ,\omega _{\alpha }d\sigma )$, where $\omega _{\alpha }(Q) =\vert Q-Q_{0}\vert^{\alpha }$, $Q_{0}$ is a fixed point on $\partial \Omega$, and $d\sigma$ denotes the surface measure on $\partial \Omega$. We prove that there exists $\varepsilon =\varepsilon (\Omega )\in (0,2]$ such that the Dirichlet problem is uniquely solvable if $1-d<\alpha <d-3+\varepsilon$, and the Neumann problem is uniquely solvable if $3-d-\varepsilon <\alpha <d-1$. If $\Omega$ is a $C^{1}$ domain, one may take $\varepsilon =2$. The regularity for the Dirichlet problem with data in the weighted Sobolev space $L^{2}_{1}(\partial \Omega ,\omega _{\alpha }d\sigma )$ is also considered. Finally we establish the weighted $L^{2}$ estimates with general $A_{p}$weights for the Dirichlet and regularity problems.
Dimension of families of determinantal schemes
Jan
O.
Kleppe;
Rosa
M.
Miró-Roig
2871-2907
Abstract: A scheme $X\subset \mathbb{P} ^{n+c}$ of codimension $c$ is called standard determinantal if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous $t \times (t+c-1)$ matrix and $X$ is said to be good determinantal if it is standard determinantal and a generic complete intersection. Given integers $a_0,a_1,...,a_{t+c-2}$ and $b_1,...,b_t$ we denote by $W(\underline{b};\underline{a})\subset \operatorname{Hilb} ^p(\mathbb{P} ^{n+c})$(resp. $W_s(\underline{b};\underline{a})$) the locus of good (resp. standard) determinantal schemes $X\subset \mathbb{P} ^{n+c}$ of codimension $c$ defined by the maximal minors of a $t\times (t+c-1)$ matrix $(f_{ij})^{i=1,...,t}_{j=0,...,t+c-2}$ where $f_{ij}\in k[x_0,x_1,...,x_{n+c}]$ is a homogeneous polynomial of degree $a_j-b_i$. In this paper we address the following three fundamental problems: To determine (1) the dimension of $W(\underline{b};\underline{a})$ (resp. $W_s(\underline{b};\underline{a})$) in terms of $a_j$ and $b_i$, (2) whether the closure of $W(\underline{b};\underline{a})$ is an irreducible component of $\operatorname{Hilb} ^p(\mathbb{P} ^{n+c})$, and (3) when $\operatorname{Hilb} ^p(\mathbb{P} ^{n+c})$ is generically smooth along $W(\underline{b};\underline{a})$. Concerning question (1) we give an upper bound for the dimension of $W(\underline{b};\underline{a})$ (resp. $W_s(\underline{b};\underline{a})$) which works for all integers $a_0,a_1,...,a_{t+c-2}$ and $b_1,...,b_t$, and we conjecture that this bound is sharp. The conjecture is proved for $2\le c\le 5$, and for $c\ge 6$ under some restriction on $a_0,a_1,...,a_{t+c-2}$and $b_1,...,b_t$. For questions (2) and (3) we have an affirmative answer for $2\le c \le 4$ and $n\ge 2$, and for $c\ge 5$ under certain numerical assumptions.
Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities
Didier
Smets
2909-2938
Abstract: We study a time-independent nonlinear Schrödinger equation with an attractive inverse square potential and a nonautonomous nonlinearity whose power is the critical Sobolev exponent. The problem shares a strong resemblance with the prescribed scalar curvature problem on the standard sphere. Particular attention is paid to the blow-up possibilities, i.e. the critical points at infinity of the corresponding variational problem. Due to the strong singularity in the potential, some new phenomenon appear. A complete existence result is obtained in dimension 4 using a detailed analysis of the gradient flow lines.
Translation and shuffling of projectively presentable modules and a categorification of a parabolic Hecke module
Volodymyr
Mazorchuk;
Catharina
Stroppel
2939-2973
Abstract: We investigate certain singular categories of Harish-Chandra bimodules realized as the category of $\mathfrak{p}$-presentable modules in the principal block of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$. This category is equivalent to the module category of a properly stratified algebra. We describe the socles and endomorphism rings of standard objects in this category. Further, we consider translation and shuffling functors and their action on the standard modules. Finally, we study a graded version of this category; in particular, we give a graded version of the properly stratified structure, and use graded versions of translation functors to categorify a parabolic Hecke module.
The limit sets of Schottky quasiconformal groups are uniformly perfect
Xiaosheng
Li
2119-2132
Abstract: In this paper we study Schottky quasiconformal groups. We show that the limit sets of Schottky quasiconformal groups are uniformly perfect, and that the limit set of a given discrete non-elementary quasiconformal group has positive Hausdorff dimension.
Analysis of a coupled system of kinetic equations and conservation laws: Rigorous derivation and existence theory via defect measures
M.
Tidriri
2133-2164
Abstract: In this paper we introduce a coupled system of kinetic equations of B.G.K. type and then study its hydrodynamic limit. We obtain as a consequence the rigorous derivation and existence theory for a coupled system of kinetic equations and their hydrodynamic (conservation laws) limit. The latter is a particular case of the coupled system of Boltzmann and Euler equations. A fundamental element in this study is the rigorous derivation and justification of the interface conditions between the kinetic model and its hydrodynamic conservation laws limit, which is obtained using a new regularity theory introduced herein.
Stable rank and real rank for some classes of group $C^\ast$-algebras
Robert
J.
Archbold;
Eberhard
Kaniuth
2165-2186
Abstract: We investigate the real and stable rank of the $C^\ast$-algebras of locally compact groups with relatively compact conjugacy classes or finite-dimensional irreducible representations. Estimates and formulae are given in terms of the group-theoretic rank.
Duality and uniqueness of convex solutions to stationary Hamilton-Jacobi equations
Rafal
Goebel
2187-2203
Abstract: Value functions for convex optimal control problems on infinite time intervals are studied in the framework of duality. Hamilton-Jacobi characterizations and the conjugacy of primal and dual value functions are of main interest. Close ties between the uniqueness of convex solutions to a Hamilton-Jacobi equation, the uniqueness of such solutions to a dual Hamilton-Jacobi equation, and the conjugacy of primal and dual value functions are displayed. Simultaneous approximation of primal and dual infinite horizon problems with a pair of dual problems on finite horizon, for which the value functions are conjugate, leads to sufficient conditions on the conjugacy of the infinite time horizon value functions. Consequently, uniqueness results for the Hamilton-Jacobi equation are established. Little regularity is assumed on the cost functions in the control problems, correspondingly, the Hamiltonians need not display any strict convexity and may have several saddle points.
Orbifolds and analytic torsions
Xiaonan
Ma
2205-2233
Abstract: In this paper, we calculate the behavior of the Quillen metric by orbifold immersions. We thus extend a formula of Bismut-Lebeau to the orbifold case. R´ESUMÉ. Dans cet article, on calcule le comportement de métrique de Quillen par immersions d'orbifold. On étend ainsi une formule de Bismut-Lebeau au cas d'orbifold.
Anosov automorphisms on compact nilmanifolds associated with graphs
S.
G.
Dani;
Meera
G.
Mainkar
2235-2251
Abstract: We associate with each graph $(S,E)$ a $2$-step simply connected nilpotent Lie group $N$ and a lattice $\Gamma$ in $N$. We determine the group of Lie automorphisms of $N$ and apply the result to describe a necessary and sufficient condition, in terms of the graph, for the compact nilmanifold $N/\Gamma$ to admit an Anosov automorphism. Using the criterion we obtain new examples of compact nilmanifolds admitting Anosov automorphisms, and conclude that for every $n\geq 17$ there exist a $n$-dimensional $2$-step simply connected nilpotent Lie group $N$ which is indecomposable (not a direct product of lower dimensional nilpotent Lie groups), and a lattice $\Gamma$ in $N$ such that $N/\Gamma$ admits an Anosov automorphism; we give also a lower bound on the number of mutually nonisomorphic Lie groups $N$ of a given dimension, satisfying the condition. Necessary and sufficient conditions are also described for a compact nilmanifold as above to admit ergodic automorphisms.
Homological algebra for the representation Green functor for abelian groups
Joana
Ventura
2253-2289
Abstract: In this paper we compute some derived functors ${Ext}$ of the internal homomorphism functor in the category of modules over the representation Green functor. This internal homomorphism functor is the left adjoint of the box product. When the group is a cyclic $2$-group, we construct a projective resolution of the module fixed point functor, and that allows a direct computation of the graded Green functor ${Ext}$. When the group is $G=\mathbb{Z} /2\times\mathbb{Z} /2$, we can still build a projective resolution, but we do not have explicit formulas for the differentials. The resolution is built from long exact sequences of projective modules over the representation functor for the subgroups of $G$ by using exact functors between these categories of modules. This induces a filtration which gives a spectral sequence which converges to the desired ${Ext}$ functors.
Positivity preserving transformations for $q$-binomial coefficients
Alexander
Berkovich;
S.
Ole
Warnaar
2291-2351
Abstract: Several new transformations for $q$-binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving transformations, as well as their connection with the Bailey lemma, many new summation and transformation formulas for basic hypergeometric series are found. The new $q$-binomial transformations are also applied to obtain multisum Rogers-Ramanujan identities, to find new representations for the Rogers-Szegö polynomials, and to make some progress on Bressoud's generalized Borwein conjecture. For the original Borwein conjecture we formulate a refinement based on new triple sum representations of the Borwein polynomials.
Symmetric functions and the phase problem in crystallography
J.
Buhler;
Z.
Reichstein
2353-2377
Abstract: The calculation of crystal structure from X-ray diffraction data requires that the phases of the ``structure factors'' (Fourier coefficients) determined by scattering be deduced from the absolute values of those structure factors. Motivated by a question of Herbert Hauptman, we consider the problem of determining phases by direct algebraic means in the case of crystal structures with $n$ equal atoms in the unit cell, with $n$ small. We rephrase the problem as a question about multiplicative invariants for a particular finite group action. We show that the absolute values form a generating set for the field of invariants of this action, and consider the problem of making this theorem constructive and practical; the most promising approach for deriving explicit formulas uses SAGBI bases.
On degrees of irreducible Brauer characters
W.
Willems
2379-2387
Abstract: Based on a large amount of examples, which we have checked so far, we conjecture that
Characteristic subsurfaces and Dehn filling
Steve
Boyer;
Marc
Culler;
Peter
B.
Shalen;
Xingru
Zhang
2389-2444
Abstract: Let $M$ be a simple knot manifold. Using the characteristic submanifold theory and the combinatorics of graphs in surfaces, we develop a method for bounding the distance between the boundary slope of an essential surface in $M$ which is not a fiber or a semi-fiber, and the boundary slope of a certain type of singular surface. Applications include bounds on the distances between exceptional Dehn surgery slopes. It is shown that if the fundamental group of $M(\alpha)$ has no non-abelian free subgroup, and if $M(\beta)$ is a reducible manifold which is not homeomorphic to $S^1 \times S^2$ or $P^3 \char93 P^3$, then $\Delta(\alpha, \beta)\le 5$. Under the same condition on $M(\beta)$, it is shown that if $M(\alpha)$ is Seifert fibered, then $\Delta(\alpha, \beta)\le 6$. Moreover, in the latter situation, character variety techniques are used to characterize the topological types of $M(\alpha)$ and $M(\beta)$ in case the bound of $6$ is attained.
Weighted rearrangement inequalities for local sharp maximal functions
Andrei
K.
Lerner
2445-2465
Abstract: Several weighted rearrangement inequalities for uncentered and centered local sharp functions are proved. These results are applied to obtain new weighted weak-type and strong-type estimates for singular integrals. A self-improving property of sharp function inequalities is established.
Smoothness of equisingular families of curves
Thomas
Keilen
2467-2481
Abstract: Francesco Severi (1921) showed that equisingular families of plane nodal curves are T-smooth, i.e. smooth of the expected dimension, whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a divisor $D$ on a smooth projective surface $\Sigma$ it thus makes sense to look for conditions which ensure that the family $V_{\vert D\vert}^{irr}\big(\mathcal{S}_1,\ldots,\mathcal{S}_r\big)$ of irreducible curves in the linear system $\vert D\vert _l$ with precisely $r$ singular points of types $\mathcal{S}_1,\ldots,\mathcal{S}_r$ is T-smooth. Considering different surfaces including the projective plane, general surfaces in $\mathbb{P} _{\mathbb{C} }^3$, products of curves and geometrically ruled surfaces, we produce a sufficient condition of the type \begin{displaymath}\sum\limits_{i=1}^r\gamma_\alpha(\mathcal{S}_i) < \gamma\cdot (D- K_\Sigma)^2, \end{displaymath} where $\gamma_\alpha$ is some invariant of the singularity type and $\gamma$ is some constant. This generalises the results of Greuel, Lossen, and Shustin (2001) for the plane case, combining their methods and the method of Bogomolov instability. For many singularity types the $\gamma_\alpha$-invariant leads to essentially better conditions than the invariants used by Greuel, Lossen, and Shustin (1997), and for most classes of geometrically ruled surfaces our results are the first known for T-smoothness at all.
Clustering in coagulation-fragmentation processes, random combinatorial structures and additive number systems: Asymptotic formulae and limiting laws
Gregory
A.
Freiman;
Boris
L.
Granovsky
2483-2507
Abstract: We develop a unified approach to the problem of clustering in the three different fields of applications indicated in the title of the paper, in the case when the parametric function of the models is regularly varying with positive exponent. The approach is based on Khintchine's probabilistic method that grew out of the Darwin-Fowler method in statistical physics. Our main result is the derivation of asymptotic formulae for the distribution of the largest and the smallest clusters (= components), as the total size of a structure (= number of particles) goes to infinity. We discover that $n^{\frac{1}{l+1}}$ is the threshold for the limiting distribution of the largest cluster. As a by-product of our study, we prove the independence of the numbers of groups of fixed sizes, as $n\to \infty.$ This is in accordance with the general principle of asymptotic independence of sites in mean-field models. The latter principle is commonly accepted in statistical physics, but not rigorously proved.
On some constants in the supercuspidal characters of $\operatorname{GL}_l$, $l$ a prime $\neq p$
Tetsuya
Takahashi
2509-2526
Abstract: The article gives explicit values of some constants which appear in the character formula for the irreducible supercuspidal representation of $\operatorname{GL}_l(F)$ for $F$ a local field of the residual characteristic $p\neq l$.
Upper bounds for the number of solutions of a Diophantine equation
M.
Z.
Garaev
2527-2534
Abstract: We give upper bound estimates for the number of solutions of a certain diophantine equation. Our results can be applied to obtain new lower bound estimates for the $L_1$-norm of certain exponential sums.
A new Löwenheim-Skolem theorem
Matthew
Foreman;
Stevo
Todorcevic
1693-1715
Abstract: This paper establishes a refinement of the classical Löwenheim-Skolem theorem. The main result shows that any first order structure has a countable elementary substructure with strong second order properties. Several consequences for Singular Cardinals Combinatorics are deduced from this.
Compactness of isospectral potentials
Harold
Donnelly
1717-1730
Abstract: The Schrödinger operator $-\Delta+V$, of a compact Riemannian manifold $M$, has pure point spectrum. Suppose that $V_0$ is a smooth reference potential. Various criteria are given which guarantee the compactness of all $V$satisfying $\operatorname{spec}(-\Delta+V)=\operatorname{spec}(-\Delta+V_0)$. In particular, compactness is proved assuming an a priori bound on the $W_{s,2}(M)$ norm of $V$, where $s>n/2-2$ and $n=\dim M$. This improves earlier work of Brüning. An example involving singular potentials suggests that the condition $s>n/2-2$ is appropriate. Compactness is also proved for non-negative isospectral potentials in dimensions $n\le 9$.
Coisotropic and polar actions on complex Grassmannians
Leonardo
Biliotti;
Anna
Gori
1731-1751
Abstract: The main result of the paper is the complete classification of the compact connected Lie groups acting coisotropically on complex Grassmannians. This is used to determine the polar actions on the same manifolds.
Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions
Wen-Xiu
Ma;
Yuncheng
You
1753-1778
Abstract: A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.
From \boldmath$\Gamma$-spaces to algebraic theories
Bernard
Badzioch
1779-1799
Abstract: The paper examines semi-theories, that is, formalisms of the type of the $\Gamma$-spaces of Segal which describe homotopy structures on topological spaces. It is shown that for any semi-theory one can find an algebraic theory describing the same structure on spaces as the original semi-theory. As a consequence one obtains a criterion for establishing when two semi-theories describe equivalent homotopy structures.
Existence and asymptotic behavior for a singular parabolic equation
Juan
Dávila;
Marcelo
Montenegro
1801-1828
Abstract: We prove global existence of nonnegative solutions to the singular parabolic equation $u_t -\Delta u + \raise 1.5pt\hbox{$\chi$}_{ \{ u>0 \} } ( -u^{-\beta} + \lambda f(u) )=0$ in a smooth bounded domain $\Omega\subset\mathbb{R} ^N$ with zero Dirichlet boundary condition and initial condition $u_0 \in C(\Omega)$, $u_0 \geq 0$. In some cases we are also able to treat $u_0 \in L^\infty(\Omega)$. Then we show that if the stationary problem admits no solution which is positive a.e., then the solutions of the parabolic problem must vanish in finite time, a phenomenon called ``quenching''. We also establish a converse of this fact and study the solutions with a positive initial condition that leads to uniqueness on an appropriate class of functions.
The baseleaf preserving mapping class group of the universal hyperbolic solenoid
Chris
Odden
1829-1858
Abstract: Given a closed surface $X$, the covering solenoid $\mathbf{X}_\infty$ is by definition the inverse limit of all finite covering surfaces over $X$. If the genus of $X$ is greater than one, then there is only one homeomorphism type of covering solenoid; it is called the universal hyperbolic solenoid. In this paper we describe the topology of $\Gamma(\mathbf{X}_\infty)$, the mapping class group of the universal hyperbolic solenoid. Central to this description is the notion of a virtual automorphism group. The main result is that there is a natural isomorphism of the baseleaf preserving mapping class group of $\mathbf{X}_\infty$ onto the virtual automorphism group of $\pi_1(X,*)$. This may be regarded as a genus independent generalization of the theorem of Dehn, Nielsen, Baer, and Epstein that the pointed mapping class group $\Gamma(X,*)$ is isomorphic to the automorphism group of $\pi_1(X,*)$.
Graphs of zeros of analytic families
Alexander
Brudnyi
1859-1875
Abstract: Let $\mathcal{F}:=\{f_{\lambda}\}$ be a family of holomorphic functions in a domain $D\subset\mathbb{C}$ depending holomorphically on $\lambda\in U\subset\mathbb{C}^{n}$. We study the distribution of zeros of $\{f_{\lambda}\}$ in a subdomain $R\subset\subset D$ whose boundary is a closed non-singular analytic curve. As an application, we obtain several results about distributions of zeros of families of generalized exponential polynomials and displacement maps related to certain ODE's.
Poset fiber theorems
Anders
Björner;
Michelle
L.
Wachs;
Volkmar
Welker
1877-1899
Abstract: Suppose that $f:P \to Q$ is a poset map whose fibers $f^{-1}(Q_{\le q})$ are sufficiently well connected. Our main result is a formula expressing the homotopy type of $P$ in terms of $Q$ and the fibers. Several fiber theorems from the literature (due to Babson, Baclawski and Quillen) are obtained as consequences or special cases. Homology, Cohen-Macaulay, and equivariant versions are given, and some applications are discussed.
Plane Cremona maps, exceptional curves and roots
Maria
Alberich-Carramiñana
1901-1914
Abstract: We address three different questions concerning exceptional and root divisors (of arithmetic genus zero and of self-intersection $-1$ and $-2$, respectively) on a smooth complex projective surface $S$ which admits a birational morphism $\pi$ to $\mathbb{P} ^{2}$. The first one is to find criteria for the properness of these divisors, that is, to characterize when the class of $C$ is in the $W$-orbit of the class of the total transform of some point blown up by $\pi$ if $C$ is exceptional, or in the $W$-orbit of a simple root if $C$ is root, where $W$ is the Weyl group acting on $\operatorname{Pic}S$; we give an arithmetical criterion, which adapts an analogous criterion suggested by Hudson for homaloidal divisors, and a geometrical one. Secondly, we prove that the irreducibility of the exceptional or root divisor $C$ is a necessary and sufficient condition in order that $\pi_{\ast} (C)$ could be transformed into a line by some plane Cremona map, and in most cases for its contractibility. Finally, we provide irreducibility criteria for proper homaloidal, exceptional and effective root divisors.
Blow-up examples for second order elliptic PDEs of critical Sobolev growth
Olivier
Druet;
Emmanuel
Hebey
1915-1929
Abstract: Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$, and $\Delta_g = -div_g\nabla$ be the Laplace-Beltrami operator. Let also $2^\star$ be the critical Sobolev exponent for the embedding of the Sobolev space $H_1^2(M)$ into Lebesgue's spaces, and $h$ be a smooth function on $M$. Elliptic equations of critical Sobolev growth such as \begin{displaymath}(E)\qquad\qquad\qquad\qquad\qquad\qquad\Delta_gu + hu = u^{2^\star-1} \qquad\qquad\qquad\qquad\qquad\qquad\end{displaymath} have been the target of investigation for decades. A very nice $H_1^2$-theory for the asymptotic behaviour of solutions of such equations has been available since the 1980's. The $C^0$-theory was recently developed by Druet-Hebey-Robert. Such a theory provides sharp pointwise estimates for the asymptotic behaviour of solutions of $(E)$. It was used as a key point by Druet to prove compactness results for equations such as $(E)$. An important issue in the field of blow-up analysis, in particular with respect to previous work by Druet and Druet-Hebey-Robert, is to get explicit nontrivial examples of blowing-up sequences of solutions of $(E)$. We present such examples in this article.
Associativity of crossed products by partial actions, enveloping actions and partial representations
M.
Dokuchaev;
R.
Exel
1931-1952
Abstract: Given a partial action $\alpha$ of a group $G$ on an associative algebra $\mathcal{A}$, we consider the crossed product $\mathcal{A}\rtimes _\alpha G$. Using the algebras of multipliers, we generalize a result of Exel (1997) on the associativity of $\mathcal{A}\rtimes_\alpha G$ obtained in the context of $C^*$-algebras. In particular, we prove that $\mathcal{A} \rtimes_{\alpha} G$ is associative, provided that $\mathcal{A}$ is semiprime. We also give a criterion for the existence of a global extension of a given partial action on an algebra, and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with a crossed product structure.
Extension-orthogonal components of preprojective varieties
Christof
Geiß;
Jan
Schröer
1953-1962
Abstract: Let $Q$ be a Dynkin quiver, and let $\Lambda$ be the corresponding preprojective algebra. Let ${\mathcal C} = \{ C_i \mid i \in I \}$ be a set of pairwise different indecomposable irreducible components of varieties of $\Lambda$-modules such that generically there are no extensions between $C_i$ and $C_j$ for all $i,j$. We show that the number of elements in ${\mathcal C}$ is at most the number of positive roots of $Q$. Furthermore, we give a module-theoretic interpretation of Leclerc's counterexample to a conjecture of Berenstein and Zelevinsky.
Toric residue and combinatorial degree
Ivan
Soprounov
1963-1975
Abstract: Consider an $n$-dimensional projective toric variety $X$defined by a convex lattice polytope $P$. David Cox introduced the toric residue map given by a collection of $n+1$ divisors $(Z_0,\dots,Z_n)$ on $X$. In the case when the $Z_i$ are $\mathbb{T}$-invariant divisors whose sum is $X\setminus\mathbb{T}$, the toric residue map is the multiplication by an integer number. We show that this number is the degree of a certain map from the boundary of the polytope $P$ to the boundary of a simplex. This degree can be computed combinatorially. We also study radical monomial ideals $I$ of the homogeneous coordinate ring of $X$. We give a necessary and sufficient condition for a homogeneous polynomial of semiample degree to belong to $I$in terms of geometry of toric varieties and combinatorics of fans. Both results have applications to the problem of constructing an element of residue one for semiample degrees.
Threefolds with vanishing Hodge cohomology
Jing
Zhang
1977-1994
Abstract: We consider algebraic manifolds $Y$ of dimension 3 over $\mathbb{C}$ with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$ and $i>0$. Let $X$ be a smooth completion of $Y$ with $D=X-Y$, an effective divisor on $X$ with normal crossings. If the $D$-dimension of $X$ is not zero, then $Y$ is a fibre space over a smooth affine curve $C$ (i.e., we have a surjective morphism from $Y$to $C$ such that the general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of $X$ is $-\infty$ and the $D$-dimension of $X$ is 1. We also discuss sufficient conditions from the behavior of fibres or higher direct images to guarantee the global vanishing of Hodge cohomology and the affineness of $Y$.
Finite quotients of rings and applications to subgroup separability of linear groups
Emily
Hamilton
1995-2006
Abstract: In this paper we apply results from algebraic number theory to subgroup separability of linear groups. We then state applications to subgroup separability of free products with amalgamation of hyperbolic $3$-manifold groups.
Cut numbers of $3$-manifolds
Adam
S.
Sikora
2007-2020
Abstract: We investigate the relations between the cut number, $c(M),$ and the first Betti number, $b_1(M),$ of $3$-manifolds $M.$ We prove that the cut number of a ``generic'' $3$-manifold $M$ is at most $2.$ This is a rather unexpected result since specific examples of $3$-manifolds with large $b_1(M)$ and $c(M)\leq 2$ are hard to construct. We also prove that for any complex semisimple Lie algebra $\mathfrak g$ there exists a $3$-manifold $M$ with $b_1(M)=dim\, \mathfrak g$ and $c(M)\leq rank\, \mathfrak g.$ Such manifolds can be explicitly constructed.
Mansfield's imprimitivity theorem for full crossed products
S.
Kaliszewski;
John
Quigg
2021-2042
Abstract: For any maximal coaction $(A,G,\delta)$ and any closed normal subgroup $N$ of $G$, there exists an imprimitivity bimodule $Y_{G/N}^G(A)$ between the full crossed product $A\times_\delta G\times_{\widehat\delta\vert}N$ and $A\times_{\delta\vert}G/N$, together with $\operatorname{Inf}\widehat{\widehat\delta\vert}-\delta^{\text{dec}}$ compatible coaction $\delta_Y$ of $G$. The assignment $(A,\delta)\mapsto (Y_{G/N}^G(A),\delta_Y)$implements a natural equivalence between the crossed-product functors `` ${}\times G\times N$'' and `` ${}\times G/N$'', in the category whose objects are maximal coactions of $G$ and whose morphisms are isomorphism classes of right-Hilbert bimodule coactions of $G$.
On the theory of elliptic functions based on ${}_{2}F_{1}(\frac{1}{3},\frac{2}{3};\frac{1}{2};z)$
Li-Chien
Shen
2043-2058
Abstract: Based on properties of the hypergeometric series ${}_{2}F_{1}(\frac{1}{3},\frac{2}{3};\frac{1}{2};z)$, we develop a theory of elliptic functions that shares many striking similarities with the classical Jacobian elliptic functions.
Small deviations of weighted fractional processes and average non--linear approximation
Mikhail
A.
Lifshits;
Werner
Linde
2059-2079
Abstract: We investigate the small deviation problem for weighted fractional Brownian motions in $L_q$-norm, $1\le q\le\infty$. Let $B^H$ be a fractional Brownian motion with Hurst index $0<H<1$. If $1/r:=H+1/q$, then our main result asserts \begin{displaymath}\lim_{\varepsilon\to 0} \varepsilon^{1/H}\log \mathbb{P}\left... ...c(H,q)\cdot\left\Vert{\rho}\right\Vert _{L_r(0,\infty)}^{1/H}, \end{displaymath} provided the weight function $\rho$satisfies a condition slightly stronger than the $r$-integrability. Thus we extend earlier results for Brownian motion, i.e. $H=1/2$, to the fractional case. Our basic tools are entropy estimates for fractional integration operators, a non-linear approximation technique for Gaussian processes as well as sharp entropy estimates for $l_q$-sums of linear operators defined on a Hilbert space.
Generalized spherical functions on reductive $p$-adic groups
Jing-Song
Huang;
Marko
Tadic
2081-2117
Abstract: Let $G$ be the group of rational points of a connected reductive $p$-adic group and let $K$ be a maximal compact subgroup satisfying conditions of Theorem 5 from Harish-Chandra (1970). Generalized spherical functions on $G$ are eigenfunctions for the action of the Bernstein center, which satisfy a transformation property for the action of $K$. In this paper we show that spaces of generalized spherical functions are finite dimensional. We compute dimensions of spaces of generalized spherical functions on a Zariski open dense set of infinitesimal characters. As a consequence, we get that on that Zariski open dense set of infinitesimal characters, the dimension of the space of generalized spherical functions is constant on each connected component of infinitesimal characters. We also obtain the formula for the generalized spherical functions by integrals of Eisenstein type. On the Zariski open dense set of infinitesimal characters that we have mentioned above, these integrals then give the formula for all the generalized spherical functions. At the end, let as mention that among others we prove that there exists a Zariski open dense subset of infinitesimal characters such that the category of smooth representations of $G$ with fixed infinitesimal character belonging to this subset is semi-simple.
Subsmooth sets: Functional characterizations and related concepts
D.
Aussel;
A.
Daniilidis;
L.
Thibault
1275-1301
Abstract: Prox-regularity of a set (Poliquin-Rockafellar-Thibault, 2000), or its global version, proximal smoothness (Clarke-Stern-Wolenski, 1995) plays an important role in variational analysis, not only because it is associated with some fundamental properties as the local continuous differentiability of the function $\mbox{dist}\,(C;\cdot)$, or the local uniqueness of the projection mapping, but also because in the case where $C$is the epigraph of a locally Lipschitz function, it is equivalent to the weak convexity (lower-C$^{2}$ property) of the function. In this paper we provide an adapted geometrical concept, called subsmoothness, which permits an epigraphic characterization of the approximate convex functions (or lower-C$^{1}$ property). Subsmooth sets turn out to be naturally situated between the classes of prox-regular and of nearly radial sets. This latter class has been recently introduced by Lewis in 2002. We hereby relate it to the Mifflin semismooth functions.
An unusual self-adjoint linear partial differential operator
W.
N.
Everitt;
L.
Markus;
M.
Plum
1303-1324
Abstract: In an American Mathematical Society Memoir, published in 2003, the authors Everitt and Markus apply their prior theory of symplectic algebra to the study of symmetric linear partial differential expressions, and the generation of self-adjoint differential operators in Sobolev Hilbert spaces. In the case when the differential expression has smooth coefficients on the closure of a bounded open region, in Euclidean space, and when the region has a smooth boundary, this theory leads to the construction of certain self-adjoint partial differential operators which cannot be defined by applying classical or generalized conditions on the boundary of the open region. This present paper concerns the spectral properties of one of these unusual self-adjoint operators, sometimes called the ``Harmonic'' operator. The boundary value problems considered in the Memoir (see above) and in this paper are called regular in that the cofficients of the differential expression do not have singularities within or on the boundary of the region; also the region is bounded and has a smooth boundary. Under these and some additional technical conditions it is shown in the Memoir, and emphasized in this present paper, that all the self-adjoint operators considered are explicitly determined on their domains by the partial differential expression; this property makes a remarkable comparison with the case of symmetric ordinary differential expressions. In the regular ordinary case the spectrum of all the self-adjoint operators is discrete in that it consists of a countable number of eigenvalues with no finite point of accumulation, and each eigenvalue is of finite multiplicity. Thus the essential spectrum of all these operators is empty. This spectral property extends to the present partial differential case for the classical Dirichlet and Neumann operators but not to the Harmonic operator. It is shown in this paper that the Harmonic operator has an eigenvalue of infinite multiplicity at the origin of the complex spectral plane; thus the essential spectrum of this operator is not empty. Both the weak and strong formulations of the Harmonic boundary value problem are considered; these two formulations are shown to be equivalent. In the final section of the paper examples are considered which show that the Harmonic operator, defined by the methods of symplectic algebra, has a domain that cannot be determined by applying either classical or generalized local conditions on the boundary of the region.
Parameter-shifted shadowing property for geometric Lorenz attractors
Shin
Kiriki;
Teruhiko
Soma
1325-1339
Abstract: In this paper, we will show that any geometric Lorenz flow in a definite class satisfies the parameter-shifted shadowing property.
On adic genus and lambda-rings
Donald
Yau
1341-1348
Abstract: Sufficient conditions on a space are given which guarantee that the $K$-theory ring is an invariant of the adic genus. An immediate consequence of this result about adic genus is that for any positive integer $n$, the power series ring $\mathbf{Z} \lbrack \lbrack x_1, \ldots , x_n \rbrack \rbrack$ admits uncountably many pairwise non-isomorphic $\lambda$-ring structures.
Serre duality for non-commutative ${\mathbb{P}}^{1}$-bundles
Adam
Nyman
1349-1416
Abstract: Let $X$ be a smooth scheme of finite type over a field $K$, let $\mathcal{E}$ be a locally free $\mathcal{O}_{X}$-bimodule of rank $n$, and let $\mathcal{A}$ be the non-commutative symmetric algebra generated by $\mathcal{E}$. We construct an internal $\operatorname{Hom}$ functor, ${\underline{{\mathcal{H}}\textit{om}}_{\mathsf{Gr} \mathcal{A}}} (-,-)$, on the category of graded right $\mathcal{A}$-modules. When $\mathcal{E}$ has rank 2, we prove that $\mathcal{A}$ is Gorenstein by computing the right derived functors of ${\underline{{\mathcal{H}}\textit{om}}_{\mathsf{Gr} \mathcal{A}}} (\mathcal{O}_{X},-)$. When $X$ is a smooth projective variety, we prove a version of Serre Duality for ${\mathsf{Proj}} \mathcal{A}$ using the right derived functors of $\underset{n \to \infty}{\lim} \underline{\mathcal{H}\textit{om}}_{\mathsf{Gr} \mathcal{A}} (\mathcal{A}/\mathcal{A}_{\geq n}, -)$.
An iterative construction of Gorenstein ideals
C.
Bocci;
G.
Dalzotto;
R.
Notari;
M.
L.
Spreafico
1417-1444
Abstract: In this paper, we present a method to inductively construct Gorenstein ideals of any codimension $c.$ We start from a Gorenstein ideal $I$ of codimension $c$ contained in a complete intersection ideal $J$ of the same codimension, and we prove that under suitable hypotheses there exists a new Gorenstein ideal contained in the residual ideal $I : J.$ We compare some numerical data of the starting and the resulting Gorenstein ideals of the construction. We compare also the Buchsbaum-Eisenbud matrices of the two ideals, in the codimension three case. Furthermore, we show that this construction is independent from the other known geometrical constructions of Gorenstein ideals, providing examples.
Hyperpolygon spaces and their cores
Megumi
Harada;
Nicholas
Proudfoot
1445-1467
Abstract: Given an $n$-tuple of positive real numbers $(\alpha_1,\ldots,\alpha_n)$, Konno (2000) defines the hyperpolygon space $X(\alpha)$, a hyperkähler analogue of the Kähler variety $M(\alpha)$ parametrizing polygons in $\mathbb{R} ^3$with edge lengths $(\alpha_1,\ldots,\alpha_n)$. The polygon space $M(\alpha)$can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, $X(\alpha)$ is the hyperkähler quiver variety defined by Nakajima. A quiver variety admits a natural $\mathbb{C} ^*$-action, and the union of the precompact orbits is called the core. We study the components of the core of $X(\alpha)$, interpreting each one as a moduli space of pairs of polygons in $\mathbb{R} ^3$with certain properties. Konno gives a presentation of the cohomology ring of $X(\alpha)$; we extend this result by computing the $\mathbb{C} ^*$-equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.
Hardy space of exact forms on $\mathbb{R}^N$
Zengjian
Lou;
Alan
McIntosh
1469-1496
Abstract: We show that the Hardy space of divergence-free vector fields on $\mathbb{R}^{3}$ has a divergence-free atomic decomposition, and thus we characterize its dual as a variant of $BMO$. Using the duality result we prove a ``div-curl" type theorem: for $b$ in $L^{2}_{loc}(\mathbb{R}^{3}, \mathbb{R}^{3})$, $\sup \int b\cdot (\nabla u\times \nabla v) dx$ is equivalent to a $BMO$-type norm of $b$, where the supremum is taken over all $u, v\in W^{1,2}(\mathbb{R}^{3})$ with $\Vert\nabla u\Vert _{L^{2}}, \Vert\nabla v\Vert _{L^{2}}\le 1.$ This theorem is used to obtain some coercivity results for quadratic forms which arise in the linearization of polyconvex variational integrals studied in nonlinear elasticity. In addition, we introduce Hardy spaces of exact forms on $\mathbb{R}^N$, study their atomic decompositions and dual spaces, and establish ``div-curl" type theorems on $\mathbb{R}^N$.
Measurable Kac cohomology for bicrossed products
Saad
Baaj;
Georges
Skandalis;
Stefaan
Vaes
1497-1524
Abstract: We study the Kac cohomology for matched pairs of locally compact groups. This cohomology theory arises from the extension theory of locally compact quantum groups. We prove a measurable version of the Kac exact sequence and provide methods to compute the cohomology. We give explicit calculations in several examples using results of Moore and Wigner.
On the singular spectrum of Schrödinger operators with decaying potential
S.
Denisov;
S.
Kupin
1525-1544
Abstract: The relation between Hausdorff dimension of the singular spectrum of a Schrödinger operator and the decay of its potential has been extensively studied in many papers. In this work, we address similar questions from a different point of view. Our approach relies on the study of the so-called Krein systems. For Schrödinger operators, we show that some bounds on the singular spectrum, obtained recently by Remling and Christ-Kiselev, are optimal.
A sharp weak type $(p,p)$ inequality $(p>2)$ for martingale transforms and other subordinate martingales
Jiyeon
Suh
1545-1564
Abstract: If $(d_{n})_{n\geq 0}$ is a martingale difference sequence, $(\varepsilon_{n})_{n\geq 0}$ a sequence of numbers in $\{ 1,-1\}$, and $n$ a positive integer, then \begin{displaymath}P(\max _{0\leq m\leq n}\vert \sum_{k=0}^{m} \varepsilon_{k}d_... ...geq 1) \leq \alpha_{p}\Vert\sum_{k=0}^{n} d_{k}\Vert_{p}^{p}. \end{displaymath} Here $\alpha_{p}$ denotes the best constant. If $1\leq p\leq 2$, then $\alpha_{p}= 2/\Gamma(p+1)$ as was shown by Burkholder. We show here that $\alpha_p=p^{p-1}/2$ for the case $p > 2$, and that $p^{p-1}/2$ is also the best constant in the analogous inequality for two martingales $M$ and $N$ indexed by $[0,\infty)$, right continuous with limits from the left, adapted to the same filtration, and such that $[M,M]_t-[N,N]_t$ is nonnegative and nondecreasing in $t$. In Section 7, we prove a similar inequality for harmonic functions.
Persistence of lower dimensional tori of general types in Hamiltonian systems
Yong
Li;
Yingfei
Yi
1565-1600
Abstract: This work is a generalization to a result of J. You (1999). We study the persistence of lower dimensional tori of general type in Hamiltonian systems of general normal forms. By introducing a modified linear KAM iterative scheme to deal with small divisors, we shall prove a persistence result, under a Melnikov type of non-resonance condition, which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori.
Stable branching rules for classical symmetric pairs
Roger
Howe;
Eng-Chye
Tan;
Jeb
F.
Willenbring
1601-1626
Abstract: We approach the problem of obtaining branching rules from the point of view of dual reductive pairs. Specifically, we obtain a stable branching rule for each of $10$ classical families of symmetric pairs. In each case, the branching multiplicities are expressed in terms of Littlewood-Richardson coefficients. Some of the formulas are classical and include, for example, Littlewood's restriction rule as a special case.
The Tits boundary of a $\text{CAT}(0)$ 2-complex
Xiangdong
Xie
1627-1661
Abstract: We investigate the Tits boundary of $\text{CAT}(0)$ $2$-complexes that have only a finite number of isometry types of cells. In particular, we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary to be the endpoints of a geodesic in the $2$-complex and for a group generated by two hyperbolic isometries to contain a free group. We also show that if two $\text{CAT}(0)$ $2$-complexes are quasi-isometric, then the cores of their Tits boundaries are bi-Lipschitz.
On Bombieri's asymptotic sieve
Kevin
Ford
1663-1674
Abstract: If a sequence $(a_n)$ of non-negative real numbers has ``best possible'' distribution in arithmetic progressions, Bombieri showed that one can deduce an asymptotic formula for the sum $\sum_{n\le x} a_n \Lambda_k(n)$ for $k\ge 2$. By constructing appropriate sequences, we show that any weakening of the well-distribution property is not sufficient to deduce the same conclusion.
Powers in recurrence sequences: Pell equations
Michael
A.
Bennett
1675-1691
Abstract: In this paper, we present a new technique for determining all perfect powers in so-called Pell sequences. To be precise, given a positive nonsquare integer $D$, we show how to (practically) solve Diophantine equations of the form \begin{displaymath}x^2 - Dy^{2n} =1 \end{displaymath} in integers $x, y$ and $n \geq 2$. Our method relies upon Frey curves and corresponding Galois representations and eschews lower bounds for linear forms in logarithms. Along the way, we sharpen and generalize work of Cao, Af Ekenstam, Ljunggren and Tartakowsky on these and related questions.
A quadratic approximation to the Sendov radius near the unit circle
Michael
J.
Miller
851-873
Abstract: Define $S(n,\beta)$ to be the set of complex polynomials of degree $n\ge2$ with all roots in the unit disk and at least one root at $\beta$. For a polynomial $P$, define $\vert P\vert _\beta$ to be the distance between $\beta$ and the closest root of the derivative $P'$. Finally, define $r_n(\beta)=\sup \{ \vert P\vert _\beta : P \in S(n,\beta) \}$. In this notation, a conjecture of Bl. Sendov claims that $r_n(\beta)\le1$. In this paper we investigate Sendov's conjecture near the unit circle, by computing constants $C_1$ and $C_2$ (depending only on $n$) such that $r_n(\beta)\sim1+C_1(1-\vert\beta\vert)+C_2(1-\vert\beta\vert)^2$ for $\vert\beta\vert$near $1$. We also consider some consequences of this approximation, including a hint of where one might look for a counterexample to Sendov's conjecture.
On the degenerate Beltrami equation
V.
Gutlyanskii;
O.
Martio;
T.
Sugawa;
M.
Vuorinen
875-900
Abstract: We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient $\mu(z)$ has the norm $\Vert\mu\Vert _\infty=1.$Sufficient conditions for the existence of a homeomorphic solution to the Beltrami equation on the Riemann sphere are given in terms of the directional dilatation coefficients of $\mu.$A uniqueness theorem is also proved when the singular set $\operatorname{Sing} (\mu)$ of $\mu$is contained in a totally disconnected compact set with an additional thinness condition on $\operatorname{Sing}(\mu).$
Geometry of Fermat adeles
Alexandru
Buium
901-964
Abstract: If $L(a,s):=\sum_n c(n,a)n^{-s}$ is a family of ``geometric'' $L-$functions depending on a parameter $a$, then the function $(p,a)\mapsto c(p,a)$, where $p$ runs through the set of prime integers, is not a rational function and hence is not a function belonging to algebraic geometry. The aim of the paper is to show that if one enlarges algebraic geometry by ``adjoining a Fermat quotient operation'', then the functions $c(p,a)$ become functions in the enlarged geometry at least for $L-$functions of curves and Abelian varieties.
Resultants and discriminants of Chebyshev and related polynomials
Karl
Dilcher;
Kenneth
B.
Stolarsky
965-981
Abstract: We show that the resultants with respect to $x$ of certain linear forms in Chebyshev polynomials with argument $x$ are again linear forms in Chebyshev polynomials. Their coefficients and arguments are certain rational functions of the coefficients of the original forms. We apply this to establish several related results involving resultants and discriminants of polynomials, including certain self-reciprocal quadrinomials.
On the construction of certain 6-dimensional symplectic manifolds with Hamiltonian circle actions
Hui
Li
983-998
Abstract: Let $(M, \omega)$ be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian $S^1$ action such that the fixed point set consists of isolated points or surfaces. Assume dim $H^2(M)<3$. In an earlier paper, we defined a certain invariant of such spaces which consists of fixed point data and twist type, and we divided the possible values of these invariants into six ``types''. In this paper, we construct such manifolds with these ``types''. As a consequence, we have a precise list of the values of these invariants.
The number of certain integral polynomials and nonrecursive sets of integers, Part 1
Tamás
Erdélyi;
Harvey
Friedman
999-1011
Abstract: Given $r > 2$, we establish a good upper bound for the number of multivariate polynomials (with as many variables and with as large degree as we wish) with integer coefficients mapping the ``cube'' with real coordinates from $[-r,r]$ into $[-t,t]$. This directly translates to a nice statement in logic (more specifically recursion theory) with a corresponding phase transition case of 2 being open. We think this situation will be of real interest to logicians. Other related questions are also considered. In most of these problems our main idea is to write the multivariate polynomials as a linear combination of products of scaled Chebyshev polynomials of one variable.
The number of certain integral polynomials and nonrecursive sets of integers, Part 2
Harvey
M.
Friedman
1013-1023
Abstract: We present some examples of mathematically natural nonrecursive sets of integers and relations on integers by combining results from Part 1, from recursion theory, and from the negative solution to Hilbert's 10th Problem.
Elliptic equations with BMO coefficients in Lipschitz domains
Sun-Sig
Byun
1025-1046
Abstract: In this paper, we study inhomogeneous Dirichlet problems for elliptic equations in divergence form. Optimal regularity requirements on the coefficients and domains for the $W^{1,p} (1<p<\infty)$ estimates are obtained. The principal coefficients are supposed to be in the John-Nirenberg space with small BMO semi-norms. The domain is supposed to have Lipschitz boundary with small Lipschitz constant. These conditions for the $W^{1,p}$ theory do not just weaken the requirements on the coefficients; they also lead to a more general geometric condition on the domain.
On the $p^e$-torsion of elliptic curves and elliptic surfaces in characteristic $p$
Andreas
Schweizer
1047-1059
Abstract: We study the extension generated by the $x$-coordinates of the $p^e$-torsion points of an elliptic curve over a function field of characteristic $p$. If $S\to C$ is a non-isotrivial elliptic surface in characteristic $p$ with a $p^e$-torsion section, then for $p^e>11$ our results imply restrictions on the genus, the gonality, and the $p$-rank of the base curve $C$, whereas for $p^e\le 11$ such a surface can be constructed over any base curve $C$. We also describe explicitly all occurring $p^e$ in the cases where the surface $S$ is rational or $K3$ or the base curve $C$ is rational, elliptic or hyperelliptic.
Parametric Bäcklund transformations I: Phenomenology
Jeanne
N.
Clelland;
Thomas
A.
Ivey
1061-1093
Abstract: We begin an exploration of parametric Bäcklund transformations for hyperbolic Monge-Ampère systems. (The appearance of an arbitrary parameter in the transformation is a feature of several well-known completely integrable PDEs.) We compute invariants for such transformations and explore the behavior of four examples, two of which are new, in terms of their invariants, symmetries, and conservation laws. We prove some preliminary results and indicate directions for further research.
Gröbner bases of associative algebras and the Hochschild cohomology
Yuji
Kobayashi
1095-1124
Abstract: We give an algorithmic way to construct a free bimodule resolution of an algebra admitting a Gröbner base. It enables us to compute the Hochschild (co)homology of the algebra. Let $A$ be a finitely generated algebra over a commutative ring $K$ with a (possibly infinite) Gröbner base $G$ on a free algebra $F$, that is, $A$ is the quotient $F/I(G)$ with the ideal $I(G)$ of $F$ generated by $G$. Given a Gröbner base $H$ for an $A$-subbimodule $L$ of the free $A$-bimodule $A \cdot X \cdot A = A_K \otimes K \cdot X \otimes_KA$ generated by a set $X$, we have a morphism $\partial$ of $A$-bimodules from the free $A$-bimodule $A \cdot H \cdot A$ generated by $H$ to $A \cdot X \cdot A$ sending the generator $[h]$to the element $h \in H$. We construct a Gröbner base $C$ on $F \cdot H \cdot F$ for the $A$-subbimodule Ker($\partial$) of $A \cdot H \cdot A$, and with this $C$ we have the free $A$-bimodule $A \cdot C \cdot A$ generated by $C$ and an exact sequence $A \cdot C \cdot A \rightarrow A \cdot H \cdot A \rightarrow A \cdot X \cdot A$. Applying this construction inductively to the $A$-bimodule $A$ itself, we have a free $A$-bimodule resolution of $A$.
Weakly compact approximation in Banach spaces
Edward
Odell;
Hans-Olav
Tylli
1125-1159
Abstract: The Banach space $E$ has the weakly compact approximation property (W.A.P. for short) if there is a constant $C < \infty$so that for any weakly compact set $D \subset E$ and $\varepsilon > 0$there is a weakly compact operator $V: E \to E$ satisfying $\sup _{x\in D} \Vert x - Vx \Vert < \varepsilon$ and $\Vert V\Vert \leq C$. We give several examples of Banach spaces both with and without this approximation property. Our main results demonstrate that the James-type spaces from a general class of quasi-reflexive spaces (which contains the classical James' space $J$) have the W.A.P, but that James' tree space $JT$ fails to have the W.A.P. It is also shown that the dual $J^{*}$ has the W.A.P. It follows that the Banach algebras $W(J)$ and $W(J^{*})$, consisting of the weakly compact operators, have bounded left approximate identities. Among the other results we obtain a concrete Banach space $Y$ so that $Y$ fails to have the W.A.P., but $Y$ has this approximation property without the uniform bound $C$.
Long-time behavior for a nonlinear fourth-order parabolic equation
María
J.
Cáceres;
J.
A.
Carrillo;
G.
Toscani
1161-1175
Abstract: We study the asymptotic behavior of solutions of the initial- boundary value problem, with periodic boundary conditions, for a fourth-order nonlinear degenerate diffusion equation with a logarithmic nonlinearity. For strictly positive and suitably small initial data we show that a positive solution exponentially approaches its mean as time tends to infinity. These results are derived by analyzing the equation verified by the logarithm of the solution.
Harmonic maps $\mathbf{M^3 \rightarrow S^1}$ and 2-cycles, realizing the Thurston norm
Gabriel
Katz
1177-1224
Abstract: Let $M^3$ be an oriented 3-manifold. We investigate when one of the fibers or a combination of fiber components, $F_{best}$, of a harmonic map $f: M^3 \rightarrow S^1$ with Morse-type singularities delivers the Thurston norm $\chi_-([F_{best}])$ of its homology class $[F_{best}] \in H_2(M^3; \mathbb{Z} )$. In particular, for a map $f$ with connected fibers and any well-positioned oriented surface $\Sigma \subset M$ in the homology class of a fiber, we show that the Thurston number $\chi_-(\Sigma)$ satisfies an inequality \begin{displaymath}\chi_-(\Sigma) \geq \chi_-(F_{best}) - \rho^\circ(\Sigma, f)\cdot Var_{\chi_-}(f).\end{displaymath} Here the variation $Var_{\chi_-}(f)$ is can be expressed in terms of the $\chi_-$-invariants of the fiber components, and the twist $\rho^\circ(\Sigma, f)$ measures the complexity of the intersection of $\Sigma$ with a particular set $F_R$ of ``bad" fiber components. This complexity is tightly linked with the optimal ``$\tilde f$-height" of $\Sigma$, being lifted to the $f$-induced cyclic cover $\tilde M^3 \rightarrow M^3$. Based on these invariants, for any Morse map $f$, we introduce the notion of its twist $\rho_{\chi_-}(f)$. We prove that, for a harmonic $f$, $\chi_-([F_{best}]) = \, \chi_-(F_{best})$ if and only if $\rho_{\chi_-}(f) = 0$.
Stable and finite Morse index solutions on $\mathbf{R}^n$ or on bounded domains with small diffusion
E.
N.
Dancer
1225-1243
Abstract: In this paper, we study bounded solutions of $- \Delta u = f (u)$ on $\mathbf{R}^n$ (where $n = 2$ and sometimes $n = 3$) and show that, for most $f$'s, the weakly stable and finite Morse index solutions are quite simple. We then use this to obtain a very good understanding of the stable and bounded Morse index solutions of $- \epsilon^2 \Delta u = f (u)$ on $\Omega$ with Dirichlet or Neumann boundary conditions for small $\epsilon$.
Brownian motion in twisted domains
Dante
DeBlassie;
Robert
Smits
1245-1274
Abstract: The tail behavior of a Brownian motion's exit time from an unbounded domain depends upon the growth of the ``inner radius'' of the domain. In this article we quantify this idea by introducing the notion of a twisted domain in the plane. Roughly speaking, such a domain is generated by a planar curve as follows. As a traveler proceeds out along the curve, the boundary curves of the domain are obtained by moving out $\pm g(r)$ units along the unit normal to the curve when the traveler is $r$ units away from the origin. The function $g$ is called the growth radius. Such domains can be highly nonconvex and asymmetric. We give a detailed account of the case $g(r) = \gamma r^p$, $0<p\le 1$. When $p=1$, a twisted domain can reasonably be interpreted as a ``twisted cone.''
On the complexity of the integral closure
Bernd
Ulrich;
Wolmer
V.
Vasconcelos
425-442
Abstract: The computation of the integral closure of an affine ring has been the focus of several modern algorithms. We will treat here one related problem: the number of generators the integral closure of an affine ring may require. This number, and the degrees of the generators in the graded case, are major measures of cost of the computation. We prove several polynomial type bounds for various kinds of algebras, and establish in characteristic zero an exponential type bound for homogeneous algebras with a small singular locus.
Quantum cohomology of partial flag manifolds
Anders
Skovsted
Buch
443-458
Abstract: We give elementary geometric proofs of the structure theorems for the (small) quantum cohomology of partial flag varieties $\operatorname{SL}(n)/P$, including the quantum Pieri and quantum Giambelli formulas and the presentation.
A tracial quantum central limit theorem
Greg
Kuperberg
459-471
Abstract: We prove a central limit theorem for non-commutative random variables in a von Neumann algebra with a tracial state: Any non-commutative polynomial of averages of i.i.d. samples converges to a classical limit. The proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the Baker-Campbell-Hausdorff expansion. The result can be considered a generalization of Johansson's theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity.
On the behavior of the algebraic transfer
Robert
R.
Bruner;
Lê
M.
Hà;
Nguyên
H. V.
Hung
473-487
Abstract: Let $Tr_k:\mathbb{F}_2\underset{GL_k}{\otimes} PH_i(B\mathbb{V}_k)\to Ext_{\mathcal{A}}^{k,k+i}(\mathbb{F}_2, \mathbb{F}_2)$ be the algebraic transfer, which is defined by W. Singer as an algebraic version of the geometrical transfer $tr_k: \pi_*^S((B\mathbb{V} _k)_+) \to \pi_*^S(S^0)$. It has been shown that the algebraic transfer is highly nontrivial and, more precisely, that $Tr_k$ is an isomorphism for $k=1, 2, 3$. However, Singer showed that $Tr_5$ is not an epimorphism. In this paper, we prove that $Tr_4$does not detect the nonzero element $g_s\in Ext_{\mathcal{A}}^{4,12\cdot 2^s}(\mathbb{F}_2, \mathbb{F}_2)$ for every $s\geq 1$. As a consequence, the localized $(Sq^0)^{-1}Tr_4$ given by inverting the squaring operation $Sq^0$ is not an epimorphism. This gives a negative answer to a prediction by Minami.
Moduli of suspension spectra
John
R.
Klein
489-507
Abstract: For a $1$-connected spectrum $E$, we study the moduli space of suspension spectra which come equipped with a weak equivalence to $E$. We construct a spectral sequence converging to the homotopy of the moduli space in positive degrees. In the metastable range, we get a complete homotopical classification of the path components of the moduli space. Our main tool is Goodwillie's calculus of homotopy functors.
Discrete Morse functions from lexicographic orders
Eric
Babson;
Patricia
Hersh
509-534
Abstract: This paper shows how to construct a discrete Morse function with a relatively small number of critical cells for the order complex of any finite poset with $\hat{0}$ and $\hat{1}$ from any lexicographic order on its maximal chains. Specifically, if we attach facets according to the lexicographic order on maximal chains, then each facet contributes at most one new face which is critical, and at most one Betti number changes; facets which do not change the homotopy type also do not contribute any critical faces. Dimensions of critical faces as well as a description of which facet attachments change the homotopy type are provided in terms of interval systems associated to the facets. As one application, the Möbius function may be computed as the alternating sum of Morse numbers. The above construction enables us to prove that the poset $\Pi_n/S_{\lambda }$ of partitions of a set $\{ 1^{\lambda_1 },\dots ,k^{\lambda_k }\}$ with repetition is homotopy equivalent to a wedge of spheres of top dimension when $\lambda$ is a hook-shaped partition; it is likely that the proof may be extended to a larger class of $\lambda$ and perhaps to all $\lambda$, despite a result of Ziegler (1986) which shows that $\Pi_n/S_{\lambda }$ is not always Cohen-Macaulay.
Knot theory for self-indexed graphs
Matías
Graña;
Vladimir
Turaev
535-553
Abstract: We introduce and study so-called self-indexed graphs. These are (oriented) finite graphs endowed with a map from the set of edges to the set of vertices. Such graphs naturally arise from classical knot and link diagrams. In fact, the graphs resulting from link diagrams have an additional structure, an integral flow. We call a self-indexed graph with integral flow a comte. The analogy with links allows us to define transformations of comtes generalizing the Reidemeister moves on link diagrams. We show that many invariants of links can be generalized to comtes, most notably the linking number, the Alexander polynomials, the link group, etc. We also discuss finite type invariants and quandle cocycle invariants of comtes.
Stein's method and Plancherel measure of the symmetric group
Jason
Fulman
555-570
Abstract: We initiate a Stein's method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov's central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein's method arises from the theory of harmonic functions on Bratelli diagrams. We also find the spectrum of the Markov chain on partitions underlying the construction of the exchangeable pair. This yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group.
Analysis on products of fractals
Robert
S.
Strichartz
571-615
Abstract: For a class of post-critically finite (p.c.f.) fractals, which includes the Sierpinski gasket (SG), there is a satisfactory theory of analysis due to Kigami, including energy, harmonic functions and Laplacians. In particular, the Laplacian coincides with the generator of a stochastic process constructed independently by probabilistic methods. The probabilistic method is also available for non-p.c.f. fractals such as the Sierpinski carpet. In this paper we show how to extend Kigami's construction to products of p.c.f. fractals. Since the products are not themselves p.c.f., this gives the first glimpse of what the analytic theory could accomplish in the non-p.c.f. setting. There are some important differences that arise in this setting. It is no longer true that points have positive capacity, so functions of finite energy are not necessarily continuous. Also the boundary of the fractal is no longer finite, so boundary conditions need to be dealt with in a more involved manner. All in all, the theory resembles PDE theory while in the p.c.f. case it is much closer to ODE theory.
On structurally stable diffeomorphisms with codimension one expanding attractors
V.
Grines;
E.
Zhuzhoma
617-667
Abstract: We show that if a closed $n$-manifold $M^n$ $(n\ge 3)$ admits a structurally stable diffeomorphism $f$ with an orientable expanding attractor $\Omega$ of codimension one, then $M^n$ is homotopy equivalent to the $n$-torus $T^n$ and is homeomorphic to $T^n$ for $n\ne 4$. Moreover, there are no nontrivial basic sets of $f$ different from $\Omega$. This allows us to classify, up to conjugacy, structurally stable diffeomorphisms having codimension one orientable expanding attractors and contracting repellers on $T^n$, $n\ge 3$.
Dynamical systems disjoint from any minimal system
Wen
Huang;
Xiangdong
Ye
669-694
Abstract: Furstenberg showed that if two topological systems $(X,T)$ and $(Y,S)$ are disjoint, then one of them, say $(Y,S)$, is minimal. When $(Y,S)$ is nontrivial, we prove that $(X,T)$ must have dense recurrent points, and there are countably many maximal transitive subsystems of $(X,T)$ such that their union is dense and each of them is disjoint from $(Y,S)$. Showing that a weakly mixing system with dense periodic points is in ${\mathcal{M}}^{\perp }$, the collection of all systems disjoint from any minimal system, Furstenberg asked the question to characterize the systems in ${\mathcal{M}}^{\perp }$. We show that a weakly mixing system with dense regular minimal points is in ${\mathcal{M}}^{\perp }$, and each system in ${\mathcal{M}}^{\perp }$ has dense minimal points and it is weakly mixing if it is transitive. Transitive systems in ${\mathcal{M}}^{\perp }$ and having no periodic points are constructed. Moreover, we show that there is a distal system in ${\mathcal{M}}^{\perp }$. Recently, Weiss showed that a system is weakly disjoint from all weakly mixing systems iff it is topologically ergodic. We construct an example which is weakly disjoint from all topologically ergodic systems and is not weakly mixing.
Finite time blow-up for a dyadic model of the Euler equations
Nets
Hawk
Katz;
Natasa
Pavlovic
695-708
Abstract: We introduce a dyadic model for the Euler equations and the Navier-Stokes equations with hyper-dissipation in three dimensions. For the dyadic Euler equations we prove finite time blow-up. In the context of the dyadic Navier-Stokes equations with hyper-dissipation we prove finite time blow-up in the case when the dissipation degree is sufficiently small.
The relationship between homological properties and representation theoretic realization of artin algebras
Osamu
Iyama
709-734
Abstract: We will study the relationship of quite different objects in the theory of artin algebras, namely Auslander-regular rings of global dimension two, torsion theories, $\tau$-categories and almost abelian categories. We will apply our results to characterization problems of Auslander-Reiten quivers.
On the Cohen-Macaulay modules of graded subrings
Douglas
Hanes
735-756
Abstract: We give several characterizations for the linearity property for a maximal Cohen-Macaulay module over a local or graded ring, as well as proofs of existence in some new cases. In particular, we prove that the existence of such modules is preserved when taking Segre products, as well as when passing to Veronese subrings in low dimensions. The former result even yields new results on the existence of finitely generated maximal Cohen-Macaulay modules over non-Cohen-Macaulay rings.
The Dirichlet problem for harmonic maps from Riemannian polyhedra to spaces of upper bounded curvature
Bent
Fuglede
757-792
Abstract: This is a continuation of the Cambridge Tract ``Harmonic maps between Riemannian polyhedra'', by J. Eells and the present author. The variational solution to the Dirichlet problem for harmonic maps with countinuous boundary data is shown to be continuous up to the boundary, and thereby uniquely determined. The domain space is a compact admissible Riemannian polyhedron with boundary, while the target can be, for example, a simply connected complete geodesic space of nonpositive Alexandrov curvature; alternatively, the target may have upper bounded curvature provided that the maps have a suitably small range. Essentially in the former setting it is further shown that a harmonic map pulls convex functions in the target back to subharmonic functions in the domain.
A note on the hyperbolic 4--orbifold of minimal volume
Ruth
Kellerhals
793-793
Abstract: Paper withdrawn by author after original posting date of July 16, 2004 and prior to preparation of the printed issue.
The Aronsson-Euler equation for absolutely minimizing Lipschitz extensions with respect to Carnot-Carathéodory metrics
Thomas
Bieske;
Luca
Capogna
795-823
Abstract: We derive the Euler-Lagrange equation (also known in this setting as the Aronsson-Euler equation) for absolute minimizers of the $L^{\infty}$variational problem \begin{displaymath}\begin{cases} \inf \vert\vert\nabla_0 u\vert\vert _{L^{\infty... ...g\in Lip(\partial\Omega) \text{ on }\partial\Omega, \end{cases}\end{displaymath} where $\Omega\subset \mathbf{G}$ is an open subset of a Carnot group, $\nabla_0 u$ denotes the horizontal gradient of $u:\Omega\to \mathbb{R}$, and the Lipschitz class is defined in relation to the Carnot-Carathéodory metric. In particular, we show that absolute minimizers are infinite harmonic in the viscosity sense. As a corollary we obtain the uniqueness of absolute minimizers in a large class of groups. This result extends previous work of Jensen and of Crandall, Evans and Gariepy. We also derive the Aronsson-Euler equation for more ``regular" absolutely minimizing Lipschitz extensions corresponding to those Carnot-Carathéodory metrics which are associated to ``free" systems of vector fields.
A theta function identity and its implications
Zhi-Guo
Liu
825-835
Abstract: In this paper we prove a general theta function identity with four parameters by employing the complex variable theory of elliptic functions. This identity plays a central role for the cubic theta function identities. We use this identity to re-derive some important identities of Hirschhorn, Garvan and Borwein about cubic theta functions. We also prove some other cubic theta function identities. A new representation for $\prod_{n=1}^\infty(1-q^n)^{10}$is given. The proofs are self-contained and elementary.
Harnack inequalities for non-local operators of variable order
Richard
F.
Bass;
Moritz
Kassmann
837-850
Abstract: We consider harmonic functions with respect to the operator \begin{displaymath}\mathcal{L} u(x)=\int [u(x+h)-u(x)-1_{(\vert h\vert\leq 1)} h\cdot \nabla u(x)] n(x,h) \, dh. \end{displaymath} Under suitable conditions on $n(x,h)$ we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. The operator $\mathcal{L}$ is allowed to be anisotropic and of variable order.
Operators on $C(K)$ spaces preserving copies of Schreier spaces
Ioannis
Gasparis
1-30
Abstract: It is proved that an operator \(T \colon C(K) \to X\), \(K\)compact metrizable, \(X\) a separable Banach space, for which the \(\epsilon\)-Szlenk index of \(T^*(B_{X^*})\) is greater than or equal to \(\omega^\xi\), \(\xi < \omega_1\), is an isomorphism on a subspace of \(C(K)\) isomorphic to \(X_\xi\), the Schreier space of order \(\xi\). As a corollary, one obtains that a complemented subspace of \(C(K)\) with Szlenk index equal to \(\omega^{\xi + 1}\) contains a subspace isomorphic to \(X_\xi\).
Symmetrically approximately continuous functions, consistent density theorems, and Fubini type inequalities
P.
D.
Humke;
M.
Laczkovich
31-44
Abstract: Using the continuum hypothesis, Sierpinski constructed a nonmeasurable function $f$ such that $\{ h: f(x+h)\ne f(x-h)\}$ is countable for every $x.$ Clearly, such a function is symmetrically approximately continuous everywhere. Here we to show that Sierpinski's example cannot be constructed in ZFC. Moreover, we show it is consistent with ZFC that if a function is symmetrically approximately continuous almost everywhere, then it is measurable.
The $\alpha$-invariant on certain surfaces with symmetry groups
Jian
Song
45-57
Abstract: The global holomorphic $\alpha$-invariant introduced by Tian is closely related to the existence of Kähler-Einstein metrics. We apply the result of Tian, Yau and Zelditch on polarized Kähler metrics to approximate plurisubharmonic functions and compute the $\alpha$-invariant on $CP^2\char93 n\overline{CP^2}$ for $n=1,2,3$.
Local zeta function for curves, non-degeneracy conditions and Newton polygons
M.
J.
Saia;
W.
A.
Zuniga-Galindo
59-88
Abstract: This paper is dedicated to a description of the poles of the Igusa local zeta function $Z(s,f,v)$ when $f(x,y)$ satisfies a new non-degeneracy condition called arithmetic non-degeneracy. More precisely, we attach to each polynomial $f(x,y)$ a collection of convex sets $\Gamma ^{A}(f)=\left\{ \Gamma _{f,1},\dots ,\Gamma _{f,l_{0}}\right\}$called the arithmetic Newton polygon of $f(x,y)$, and introduce the notion of arithmetic non-degeneracy with respect to $\Gamma ^{A}(f)$. If $L_{v}$ is a $p$-adic field, and $f(x,y)\in L_{v}\left[ x,y \right]$ is arithmetically non-degenerate, then the poles of $Z(s,f,v)$ can be described explicitly in terms of the equations of the straight segments that form the boundaries of the convex sets $\Gamma _{f,1},\dots , \Gamma _{f,l_{0}}$. Moreover, the proof of the main result gives an effective procedure for computing $Z(s,f,v)$.
Some logical metatheorems with applications in functional analysis
Ulrich
Kohlenbach
89-128
Abstract: In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. Here `uniform' means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, CAT(0)-spaces, normed linear spaces, uniformly convex spaces, as well as inner product spaces.
A Lyndon-Hochschild-Serre spectral sequence for certain homotopy fixed point spectra
Ethan
S.
Devinatz
129-150
Abstract: Let $H$ and $K$ be closed subgroups of the extended Morava stabilizer group $G_n$ and suppose that $H$ is normal in $K$. We construct a strongly convergent spectral sequence \begin{displaymath}H^\ast_c(K/H, (E^{hH}_n)^\ast X) \Rightarrow (E^{hK}_n)^\ast X, \end{displaymath} where $E^{hH}_n$ and $E^{hK}_n$ are the continuous homotopy fixed point spectra of Devinatz and Hopkins. This spectral sequence turns out to be an Adams spectral sequence in the category of $K(n)_\ast$-local $E^{hK}_n$-modules.
Boundary Hölder and $L^p$ estimates for local solutions of the tangential Cauchy-Riemann equation
Christine
Laurent-Thiébaut;
Mei-Chi
Shaw
151-177
Abstract: We study the local solvability of the tangential Cauchy-Riemann equation on an open neighborhood $\omega$ of a point $z_0\in M$ when $M$ is a generic $q$-concave $CR$ manifold of real codimension $k$ in $\mathbb{C} ^n$, where $1\le k\le n-1$. Our method is to first derive a homotopy formula for $\overline\partial_b$ in $\omega$ when $\omega$ is the intersection of $M$ with a strongly pseudoconvex domain. The homotopy formula gives a local solution operator for any $\overline\partial_b$-closed form on $\omega$ without shrinking. We obtain Hölder and $L^p$ estimates up to the boundary for the solution operator. RÉSUMÉ. Nous étudions la résolubilité locale de l'opérateur de Cauchy- Riemann tangentiel sur un voisinage $\omega$ d'un point $z_0$d'une sous-variété $CR$ générique $q$-concave $M$ de codimension quelconque de $\mathbb C^n$. Nous construisons une formule d'homotopie pour le $\overline\partial_b$ sur $\omega$, lorsque $\omega$ est l'intersection de $M$ et d'un domaine strictement pseudoconvexe. Nous obtenons ainsi un opérateur de résolution pour toute forme $\overline\partial_b$-fermée sur $\omega$. Nous en déduisons des estimations $L^p$ et des estimations hölderiennes jusqu'au bord pour la solution de l'équation de Cauchy-Riemann tangentielle sur $\omega$.
On a refinement of the generalized Catalan numbers for Weyl groups
Christos
A.
Athanasiadis
179-196
Abstract: Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$, spanning a Euclidean space $V$, and let $m$ be a positive integer. It is known that the set of regions into which the fundamental chamber of $W$ is dissected by the hyperplanes in $V$ of the form $(\alpha, x) = k$ for $\alpha \in \Phi$ and $k = 1, 2,\dots,m$ is equinumerous to the set of orbits of the action of $W$ on the quotient $\check{Q} / \, (mh+1) \, \check{Q}$. A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of $\Phi$, are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case $m=1$ and $\Phi = A_{n-1}$.
One-dimensional dynamical systems and Benford's law
Arno
Berger;
Leonid
A.
Bunimovich;
Theodore
P.
Hill
197-219
Abstract: Near a stable fixed point at 0 or $\infty$, many real-valued dynamical systems follow Benford's law: under iteration of a map $T$ the proportion of values in $\{x, T(x), T^2(x),\dots, T^n(x)\}$ with mantissa (base $b$) less than $t$ tends to $\log_bt$ for all $t$ in $[1,b)$ as $n\to\infty$, for all integer bases $b>1$. In particular, the orbits under most power, exponential, and rational functions (or any successive combination thereof), follow Benford's law for almost all sufficiently large initial values. For linearly-dominated systems, convergence to Benford's distribution occurs for every $x$, but for essentially nonlinear systems, exceptional sets may exist. Extensions to nonautonomous dynamical systems are given, and the results are applied to show that many differential equations such as $\dot x=F(x)$, where $F$ is $C^2$ with
Elements of specified order in simple algebraic groups
R.
Lawther
221-245
Abstract: In this paper we let $G$ be a simple algebraic group and $r$ be a natural number, and consider the codimension in $G$ of the variety of elements $g\in G$ satisfying $g^r=1$. We shall obtain a lower bound for this codimension which is independent of characteristic, and show that it is attained if $G$ is of adjoint type.
Well-posedness of the Dirichlet problem for the non-linear diffusion equation in non-smooth domains
Ugur
G.
Abdulla
247-265
Abstract: We investigate the Dirichlet problem for the parablic equation \begin{displaymath}u_t = \Delta u^m, m > 0, \end{displaymath} in a non-smooth domain $\Omega \subset \mathbb{R}^{N+1}, N \geq 2$. In a recent paper [U.G. Abdulla, J. Math. Anal. Appl., 260, 2 (2001), 384-403] existence and boundary regularity results were established. In this paper we present uniqueness and comparison theorems and results on the continuous dependence of the solution on the initial-boundary data. In particular, we prove $L_1$-contraction estimation in general non-smooth domains.
Dupin indicatrices and families of curve congruences
J.
W.
Bruce;
F.
Tari
267-285
Abstract: We study a number of natural families of binary differential equations (BDE's) on a smooth surface $M$ in ${\mathbb{R}}^3$. One, introduced by G. J. Fletcher in 1996, interpolates between the asymptotic and principal BDE's, another between the characteristic and principal BDE's. The locus of singular points of the members of these families determine curves on the surface. In these two cases they are the tangency points of the discriminant sets (given by a fixed ratio of principle curvatures) with the characteristic (resp. asymptotic) BDE. More generally, we consider a natural class of BDE's on such a surface $M$, and show how the pencil of BDE's joining certain pairs are related to a third BDE of the given class, the so-called polar BDE. This explains, in particular, why the principal, asymptotic and characteristic BDE's are intimately related.
Stability of transonic shock fronts in two-dimensional Euler systems
Shuxing
Chen
287-308
Abstract: We study the stability of stationary transonic shock fronts under two-dimensional perturbation in gas dynamics. The motion of the gas is described by the full Euler system. The system is hyperbolic ahead of the shock front, and is a hyperbolic-elliptic composed system behind the shock front. The stability of the shock front and the downstream flow under two-dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic-elliptic composed system. We develop a method to deal with boundary value problems for such systems. The crucial point is to decompose the system to a canonical form, in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients. By several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem. Our result indicates the stability of the transonic shock front and the flow field behind the shock.
Glauberman-Watanabe corresponding $p$-blocks of finite groups with normal defect groups are Morita equivalent
Morton
E.
Harris
309-335
Abstract: Let $G$ be a finite group and let $A$ be a solvable finite group that acts on $G$ such that the orders of $G$ and $A$are relatively prime. Let $b$ be a $p$-block of $G$ with normal defect group $D$ such that $A$ stabilizes $b$ and $D\leq C_{G}(A)$. Then there is a Morita equivalence between the block $b$ and its Watanabe correspondent block $W(b)$ of $C_{G}(A)$ given by a bimodule $M$ with vertex $\Delta D$ and trivial source that on the character level induces the Glauberman correspondence (and which is an isotypy by a theorem of Watanabe).
Estimations $L^p$ des fonctions du Laplacien sur les variétés cuspidales
Hong-Quan
Li
337-354
Abstract: Le but de cet article est d'étudier la continuité $L^p$ des fonctions du Laplacien sur les variétés cuspidales.
Maximal holonomy of infra-nilmanifolds with $2$-dimensional quaternionic Heisenberg geometry
Ku
Yong
Ha;
Jong
Bum
Lee;
Kyung
Bai
Lee
355-383
Abstract: Let $\mathbf{H}_{4n-1}(\mathbb{H} )$ be the quaternionic Heisenberg group of real dimension $4n-1$ and let $I_{n}$ denote the maximal order of the holonomy groups of all infra-nilmanifolds with $\mathbf{H}_{4n-1}(\mathbb{H} )$-geometry. We prove that $I_2=48$. As an application, by applying Kim and Parker's result, we obtain that the minimum volume of a $2$-dimensional quaternionic hyperbolic manifold with $k$ cusps is at least $\frac{\sqrt{2}k}{720}.$
Lack of natural weighted estimates for some singular integral operators
José
María
Martell;
Carlos
Pérez;
Rodrigo
Trujillo-González
385-396
Abstract: We show that the classical Hörmander condition, or analogously the $L^r$-Hörmander condition, for singular integral operators $T$ is not sufficient to derive Coifman's inequality \begin{displaymath}\int_{\mathbb{R} ^n} \vert Tf(x)\vert^p\, w(x)\, dx \le C\,\int_{\mathbb{R} ^n} M f(x)^p\, w(x)\,dx, \end{displaymath} where $0<p<\infty$, $M$ is the Hardy-Littlewood maximal operator, $w$ is any $A_{\infty}$ weight and $C$ is a constant depending upon $p$ and the $A_{\infty}$ constant of $w$. This estimate is well known to hold when $T$ is a Calderón-Zygmund operator. As a consequence we deduce that the following estimate does not hold: \begin{displaymath}\int_{\mathbb{R} ^n} \vert Tf(x)\vert^p\, w(x)\, dx \le C\,\int_{\mathbb{R} ^n} Mf(x)^p\, Mw(x)\,dx, \end{displaymath} where $0<p\le 1$ and where $w$ is an arbitrary weight. However, by a recent result due to A. Lerner, this inequality is satisfied whenever $T$ is a Calderón-Zygmund operator. One of the main ingredients of the proof is a very general extrapolation theorem for $A_\infty$ weights.
Convergence of double Fourier series and $W$-classes
M.
I.
Dyachenko;
D.
Waterman
397-407
Abstract: The double Fourier series of functions of the generalized bounded variation class $\{n/\ln (n+1)\}^{\ast }BV$ are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class $\Lambda ^{\ast }BV,$ defined here, have quadrant limits at every point and, for $f\in \Lambda ^{\ast }BV,$ there exist at most countable sets $P$ and $Q$ such that, for $x\notin P$ and $y\notin Q,$ $f$is continuous at $(x,y)$. It is shown that the previously studied class $\Lambda BV$ contains essentially discontinuous functions unless the sequence $\Lambda$ satisfies a strong condition.
A novel dual approach to nonlinear semigroups of Lipschitz operators
Jigen
Peng;
Zongben
Xu
409-424
Abstract: Lipschitzian semigroup refers to a one-parameter semigroup of Lipschitz operators that is strongly continuous in the parameter. It contains $C_{0}$-semigroup, nonlinear semigroup of contractions and uniformly $k$-Lipschitzian semigroup as special cases. In this paper, through developing a series of Lipschitz dual notions, we establish an analysis approach to Lipschitzian semigroup. It is mainly proved that a (nonlinear) Lipschitzian semigroup can be isometrically embedded into a certain $C_{0}$-semigroup. As application results, two representation formulas of Lipschitzian semigroup are established, and many asymptotic properties of $C_{0}$-semigroup are generalized to Lipschitzian semigroup.
Expansiveness of algebraic actions on connected groups
Siddhartha
Bhattacharya
4687-4700
Abstract: We study endomorphism actions of a discrete semigroup $\Gamma$ on a connected group $G$. We give a necessary and sufficient condition for expansiveness of such actions provided $G$ is either a Lie group or a solenoid.
Quaternionic algebraic cycles and reality
Pedro
F.
dos Santos;
Paulo
Lima-Filho
4701-4736
Abstract: In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real Brauer-Severi varieties, under the action of the Galois group $Gal({\mathbb C} / {\mathbb R})$. Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymour's quaternionic $K$-theory, and the other one classifies an equivariant cohomology theory ${\mathfrak Z}^*(-)$ which is a natural recipient of characteristic classes $KH^*(X) \to {\mathfrak Z}^*(X)$ for quaternionic bundles over Real spaces $X$.
Complete second order linear differential operator equations in Hilbert space and applications in hydrodynamics
N.
D.
Kopachevsky;
R.
Mennicken;
Ju.
S.
Pashkova;
C.
Tretter
4737-4766
Abstract: We study the Cauchy problem for a complete second order linear differential operator equation in a Hilbert space ${\mathcal H}$ of the form \begin{displaymath}{\mathcal D}(F)\subset{\mathcal D}(B),\quad {\mathcal D}(F)\subset {\mathcal D}(K). \end{displaymath} We also suppose that $F$ and $B$ are bounded from below, but the operator coefficients are not assumed to commute. The main results concern the existence of strong solutions to the stated Cauchy problem and applications of these results to the Cauchy problem associated with small motions of some hydrodynamical systems.
Geometric aspects of frame representations of abelian groups
Akram
Aldroubi;
David
Larson;
Wai-Shing
Tang;
Eric
Weber
4767-4786
Abstract: We consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.
Second order parabolic equations in Banach spaces with dynamic boundary conditions
Ti-Jun
Xiao;
Jin
Liang
4787-4809
Abstract: In this paper, we exhibit a unified treatment of the mixed initial boundary value problem for second order (in time) parabolic linear differential equations in Banach spaces, whose boundary conditions are of a dynamical nature. Results regarding existence, uniqueness, continuous dependence (on initial data) and regularity of classical and strict solutions are established. Moreover, several examples are given as samples for possible applications.
On peak-interpolation manifolds for $\boldsymbol{A}\boldsymbol{(}\boldsymbol{\Omega}\boldsymbol{)}$ for convex domains in $\boldsymbol{\mathbb{C}}^{\boldsymbol{n}}$
Gautam
Bharali
4811-4827
Abstract: Let $\Omega$ be a bounded, weakly convex domain in ${\mathbb{C} }^n$, $n\geq 2$, having real-analytic boundary. $A(\Omega)$ is the algebra of all functions holomorphic in $\Omega$ and continuous up to the boundary. A submanifold $\boldsymbol{M}\subset \partial \Omega$ is said to be complex-tangential if $T_p(\boldsymbol{M})$ lies in the maximal complex subspace of $T_p(\partial \Omega)$ for each $p\in\boldsymbol{M}$. We show that for real-analytic submanifolds $\boldsymbol{M}\subset\partial \Omega$, if $\boldsymbol{M}$ is complex-tangential, then every compact subset of $\boldsymbol{M}$ is a peak-interpolation set for $A(\Omega)$.
An analogue of continued fractions in number theory for Nevanlinna theory
Zhuan
Ye
4829-4838
Abstract: We show an analogue of continued fractions in approximation to irrational numbers by rationals for Nevanlinna theory. The analogue is a sequence of points in the complex plane which approaches a given finite set of points and at a given rate in the sense of Nevanlinna theory.
Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle
Jason
L.
Metcalfe
4839-4855
Abstract: In this paper, we show that certain local Strichartz estimates for solutions of the wave equation exterior to a convex obstacle can be extended to estimates that are global in both space and time. This extends the work that was done previously by H. Smith and C. Sogge in odd spatial dimensions. In order to prove the global estimates, we explore weighted Strichartz estimates for solutions of the wave equation when the Cauchy data and forcing term are compactly supported.
On orbital partitions and exceptionality of primitive permutation groups
R.
M.
Guralnick;
Cai
Heng
Li;
Cheryl
E.
Praeger;
J.
Saxl
4857-4872
Abstract: Let $G$ and $X$ be transitive permutation groups on a set $\Omega$ such that $G$ is a normal subgroup of $X$. The overgroup $X$ induces a natural action on the set $\operatorname{Orbl}(G,\Omega)$ of non-trivial orbitals of $G$ on $\Omega$. In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples $(G,X,\Omega)$ where $X$fixes no elements of $\operatorname{Orbl}(G,\Omega)$; such triples are called exceptional. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples $(G,X,\Omega,\mathcal{P})$ where $\mathcal{P}$ is a partition of $\operatorname{Orbl}(G,\Omega)$ such that $X$ is transitive on $\mathcal{P}$; such a quadruple is called a TOD (transitive orbital decomposition). It follows easily that the triple $(G,X,\Omega)$ in a TOD $(G,X,\Omega,\mathcal{P})$is exceptional; conversely if an exceptional triple $(G,X,\Omega)$ is such that $X/G$ is cyclic of prime-power order, then there exists a partition $\mathcal{P}$ of $\operatorname{Orbl}(G,\Omega)$ such that $(G,X,\Omega,\mathcal{P})$ is a TOD. This paper characterizes TODs $(G,X,\Omega,\mathcal{P})$ such that $X^\Omega$ is primitive and $X/G$ is cyclic of prime-power order. An application is given to the classification of self-complementary vertex-transitive graphs.
Change of rings in deformation theory of modules
Runar
Ile
4873-4896
Abstract: Given a $B$-module $M$ and any presentation $B=A/J$, the obstruction theory of $M$ as a $B$-module is determined by the usual obstruction class $\mathrm{o}_{ \scriptscriptstyle{A}}^{\scriptscriptstyle{}}$ for deforming $M$ as an $A$-module and a new obstruction class $\mathrm{o}_{ \scriptscriptstyle{J}}^{\scriptscriptstyle{}}$. These two classes give the tool for constructing two obstruction maps which depend on each other and which characterise the hull of the deformation functor. We obtain relations between the obstruction classes by studying a change of rings spectral sequence and by representing certain classes as elements in the Yoneda complex. Calculation of the deformation functor of $M$ as a $B$-module, including the (generalised) Massey products, is thus possible within any $A$-free $2$-presentation of $M$.
A local limit theorem for closed geodesics and homology
Richard
Sharp
4897-4908
Abstract: In this paper, we study the distribution of closed geodesics on a compact negatively curved manifold. We concentrate on geodesics lying in a prescribed homology class and, under certain conditions, obtain a local limit theorem to describe the asymptotic behaviour of the associated counting function as the homology class varies.
Characterizations of regular almost periodicity in compact minimal abelian flows
Alica
Miller;
Joseph
Rosenblatt
4909-4929
Abstract: Regular almost periodicity in compact minimal abelian flows was characterized for the case of discrete acting group by W. Gottschalk and G. Hedlund and for the case of $0$-dimensional phase space by W. Gottschalk a few decades ago. In 1995 J. Egawa gave characterizations for the case when the acting group is $\mathbb{R}$. We extend Egawa's results to the case of an arbitrary abelian acting group and a not necessarily metrizable phase space. We then show how our statements imply previously known characterizations in each of the three special cases and give various other applications (characterization of regularly almost periodic functions on arbitrary abelian topological groups, classification of uniformly regularly almost periodic compact minimal $\mathbb{Z}$- and $\mathbb{R}$-flows, conditions equivalent with uniform regular almost periodicity, etc.).
The Perron-Frobenius theorem for homogeneous, monotone functions
Stéphane
Gaubert;
Jeremy
Gunawardena
4931-4950
Abstract: If $A$ is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that $A$ has an eigenvector in the positive cone, $(\mathbb R^{+})^n$. We associate a directed graph to any homogeneous, monotone function, $f: (\mathbb R^{+})^n \rightarrow (\mathbb R^{+})^n$, and show that if the graph is strongly connected, then $f$ has a (nonlinear) eigenvector in $(\mathbb R^{+})^n$. Several results in the literature emerge as corollaries. Our methods show that the Perron-Frobenius theorem is ``really'' about the boundedness of invariant subsets in the Hilbert projective metric. They lead to further existence results and open problems.
Spectral properties and dynamics of quantized Henon maps
Brendan
Weickert
4951-4968
Abstract: We study a generalization of the Airy function, and use its properties to investigate the dynamics and spectral properties of the unitary operators on $L^2(\mathbf{R})$ of the form $U_c:=Fe^{i(q(x)+cx)}$, where $q$ is a real polynomial of odd degree, $c$ is a real number, and $F$ is the Fourier transform. We show that $U_c$ is a quantization of the classical Henon map \begin{align*}f_\lambda:\mathbf{R}^2 &\to \mathbf{R}^2 , (x,y) &\mapsto (y+q'(x)+c,-x), \end{align*} and show that for $c>0$ sufficiently large, $U_c$ has purely continuous spectrum. This fact has implications for the dynamics of $U_c$, which are shown to correspond when the condition is satisfied to the dynamics of its classical counterpart on $\mathbf{R}^2$.
Cross characteristic representations of even characteristic symplectic groups
Robert
M.
Guralnick;
Pham
Huu
Tiep
4969-5023
Abstract: We classify the small irreducible representations of $Sp_{2n}(q)$ with $q$ even in odd characteristic. This improves even the known results for complex representations. The smallest representation for this group is much larger than in the case when $q$ is odd. This makes the problem much more difficult.
Blakers-Massey elements and exotic diffeomorphisms of $S^6$ and $S^{14}$ via geodesics
C.
E.
Durán;
A.
Mendoza;
A.
Rigas
5025-5043
Abstract: We use the geometry of the geodesics of a certain left-invariant metric on the Lie group $Sp(2)$ to find explicit related formulas for two topological objects: the Blakers-Massey element (a generator of $\pi_6(S^3)$) and an exotic (i.e. not isotopic to the identity) diffeomorphism of $S^6$ (C. E. Durán, 2001). These formulas depend on two quaternions and their conjugates and we produce their extensions to the octonions through formulas for a generator of $\pi _{14}(S^{7})$ and exotic diffeomorphisms of $S^{14}$, thus giving explicit gluing maps for half of the 15-dimensional exotic spheres expressed as the union of two 15-disks.
Radon transforms on affine Grassmannians
Boris
Rubin
5045-5070
Abstract: We develop an analytic approach to the Radon transform $\hat f (\zeta)=\int_{\tau\subset \zeta} f (\tau)$, where $f(\tau)$ is a function on the affine Grassmann manifold $G(n,k)$ of $k$-dimensional planes in $\mathbb{R}^n$, and $\zeta$ is a $k'$-dimensional plane in the similar manifold $G(n,k'), \; k'>k$. For $f \in L^p (G(n,k))$, we prove that this transform is finite almost everywhere on $G(n,k')$ if and only if $\mathbb{R}^{n+1}$. It is proved that the dual Radon transform can be explicitly inverted for
On the representation of integers as linear combinations of consecutive values of a polynomial
Jacques
Boulanger;
Jean-Luc
Chabert
5071-5088
Abstract: Let $K$ be a cyclotomic field with ring of integers $\mathcal{O}_{K}$ and let $f$ be a polynomial whose values on $\mathbb{Z}$ belong to $\mathcal{O}_{K}$. If the ideal of $\mathcal{O}_{K}$ generated by the values of $f$ on $\mathbb{Z}$ is $\mathcal{O}_{K}$ itself, then every algebraic integer $N$ of $K$ may be written in the following form: \begin{displaymath}N=\sum_{k=1}^l\;\varepsilon_{k}f(k)\end{displaymath} for some integer $l$, where the $\varepsilon_{k}$'s are roots of unity of $K$. Moreover, there are two effective constants $A$ and $B$ such that the least integer $l$ (for a fixed $N$) is less than $A\,\widetilde{N}+B$, where \begin{displaymath}\widetilde{N}= \max_{\sigma\in Gal(K/\mathbb{Q} )} \; \vert \sigma (N) \vert.\end{displaymath}
Character sums and congruences with $n!$
Moubariz
Z.
Garaev;
Florian
Luca;
Igor
E.
Shparlinski
5089-5102
Abstract: We estimate character sums with $n!$, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime $p$ and obtain new information about the spacings between quadratic nonresidues modulo $p$. In particular, we show that there exists a positive integer $n\ll p^{1/2+\varepsilon}$ such that $n!$ is a primitive root modulo $p$. We also show that every nonzero congruence class $a \not \equiv 0 \pmod p$can be represented as a product of 7 factorials, $a \equiv n_1! \ldots n_7! \pmod p$, where $\max \{n_i \vert i=1,\ldots, 7\}=O(p^{11/12+\varepsilon})$, and we find the asymptotic formula for the number of such representations. Finally, we show that products of 4 factorials $n_1!n_2!n_3!n_4!,$ with $\max\{n_1, n_2, n_3, n_4\}=O(p^{6/7+\varepsilon})$ represent ``almost all'' residue classes modulo p, and that products of 3 factorials $n_1!n_2!n_3!$ with $\max\{n_1, n_2, n_3\}=O(p^{5/6+\varepsilon})$ are uniformly distributed modulo $p$.
Morse index and uniqueness for positive solutions of radial $p$-Laplace equations
Amandine
Aftalion;
Filomena
Pacella
4255-4272
Abstract: We study the positive radial solutions of the Dirichlet problem $\Delta_p u+f(u)=0$ in $B$, $u>0$ in $B$, $u=0$ on $\partial B$, where $\Delta_p u=\operatorname{div}(\vert\nabla u\vert^{p-2}\nabla u)$, $p>1$, is the $p$-Laplace operator, $B$ is the unit ball in $\mathbb{R} ^n$ centered at the origin and $f$ is a $C^1$ function. We are able to get results on the spectrum of the linearized operator in a suitable weighted space of radial functions and derive from this information on the Morse index. In particular, we show that positive radial solutions of Mountain Pass type have Morse index 1 in the subspace of radial functions of $W_0^{1,p}(B)$. We use this to prove uniqueness and nondegeneracy of positive radial solutions when $f$ is of the type $u^s+u^q$ and $p\geq 2$.
Localization for a porous medium type equation in high dimensions
Changfeng
Gui;
Xiaosong
Kang
4273-4285
Abstract: We prove the strict localization for a porous medium type equation with a source term, $u_{t}= \nabla(u^ {\sigma} \nabla u)+u^ \beta$, $x \in \mathbf{R}^ N$, $N>1$, $\beta >\sigma +1$, $\sigma>0,$ in the case of arbitrary compactly supported initial functions $u_0$. We also otain an estimate of the size of the localization in terms of the support of the initial data $\operatorname{supp}u_0$ and the blow-up time $T$. Our results extend the well-known one dimensional result of Galaktionov and solve an open question regarding high dimensions.
A class of \boldmath{$C^*$}-algebras generalizing both graph algebras and homeomorphism \boldmath{$C^*$}-algebras I, fundamental results
Takeshi
Katsura
4287-4322
Abstract: We introduce a new class of $C^*$-algebras, which is a generalization of both graph algebras and homeomorphism $C^*$-algebras. This class is very large and also very tractable. We prove the so-called gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem, and compute the $K$-groups of our algebras.
Multi-point Taylor expansions of analytic functions
José
L.
López;
Nico
M.
Temme
4323-4342
Abstract: Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.
Nonexistence of abelian difference sets: Lander's conjecture for prime power orders
Ka
Hin
Leung;
Siu Lun
Ma;
Bernhard
Schmidt
4343-4358
Abstract: In 1963 Ryser conjectured that there are no circulant Hadamard matrices of order $>4$ and no cyclic difference sets whose order is not coprime to the group order. These conjectures are special cases of Lander's conjecture which asserts that there is no abelian group with a cyclic Sylow $p$-subgroup containing a difference set of order divisible by $p$. We verify Lander's conjecture for all difference sets whose order is a power of a prime greater than 3.
On the $L_{p}$-Minkowski problem
Erwin
Lutwak;
Deane
Yang;
Gaoyong
Zhang
4359-4370
Abstract: A volume-normalized formulation of the $L_{p}$-Minkowski problem is presented. This formulation has the advantage that a solution is possible for all $p\ge 1$, including the degenerate case where the index $p$ is equal to the dimension of the ambient space. A new approach to the $L_{p}$-Minkowski problem is presented, which solves the volume-normalized formulation for even data and all $p\ge 1$.
3-manifolds that admit knotted solenoids as attractors
Boju
Jiang;
Yi
Ni;
Shicheng
Wang
4371-4382
Abstract: Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? The result presents some intrinsic symmetries for a class of 3-manifolds.
On the Harnack inequality for a class of hypoelliptic evolution equations
Andrea
Pascucci;
Sergio
Polidoro
4383-4394
Abstract: We give a direct proof of the Harnack inequality for a class of degenerate evolution operators which contains the linearized prototypes of the Kolmogorov and Fokker-Planck operators. We also improve the known results in that we find explicitly the optimal constant of the inequality.
Isolating blocks near the collinear relative equilibria of the three-body problem
Richard
Moeckel
4395-4425
Abstract: The collinear relative equilibrium solutions are among the few explicitly known periodic solutions of the Newtonian three-body problem. When the energy and angular momentum constants are varied slightly, these unstable periodic orbits become normally hyperbolic invariant spheres whose stable and unstable manifolds form separatrices in the integral manifolds. The goal of this paper is to construct simple isolating blocks for these invariant spheres analogous to those introduced by Conley in the restricted three-body problem. This allows continuation of the invariant set and the separatrices to energies and angular momenta far from those of the relative equilibrium.
The loss of tightness of time distributions for homeomorphisms of the circle
Zaqueu
Coelho
4427-4445
Abstract: For a minimal circle homeomorphism $f$ we study convergence in law of rescaled hitting time point process of an interval of length $\varepsilon>0$. Although the point process in the natural time scale never converges in law, we study all possible limits under a subsequence. The new feature is the fact that, for rotation numbers of unbounded type, there is a sequence $\varepsilon_{n}$ going to zero exhibiting coexistence of two non-trivial asymptotic limit point processes depending on the choice of time scales used when rescaling the point process. The phenomenon of loss of tightness of the first hitting time distribution is an indication of this coexistence behaviour. Moreover, tightness occurs if and only if the rotation number is of bounded type. Therefore tightness of time distributions is an intrinsic property of badly approximable irrational rotation numbers.
Oppenheim conjecture for pairs consisting of a linear form and a quadratic form
Alexander
Gorodnik
4447-4463
Abstract: Let $Q$ be a nondegenerate quadratic form and $L$ a nonzero linear form of dimension $d>3$. As a generalization of the Oppenheim conjecture, we prove that the set $\{(Q(x),L(x)):x\in\mathbb{Z} ^d\}$ is dense in $\mathbb{R} ^2$ provided that $Q$ and $L$ satisfy some natural conditions. The proof uses dynamics on homogeneous spaces of Lie groups.
Spécialisation de la $R$-équivalence pour les groupes réductifs
Philippe
Gille
4465-4474
Abstract: Soit $G/k$ un groupe réductif défini sur un corps $k$ de caractéristique distincte de $2$. On montre que le groupes des classes de $R$-équivalence de $G(k)$ne change pas lorsque l'on passe de $k$ au corps des séries de Laurent $k((t))$, c'est-à-dire que l'on a un isomorphisme naturel $G(k)/R \buildrel\sim\over\longrightarrow G\bigl( k((t)) \bigr)/R$. ABSTRACT. Let $G/k$ be a reductive group defined over a field of characteristic $\not =2$. We show that the group of $R$-equivalence for $G(k)$ is invariant by the change of fields $k((t))/k$ given by the Laurent series. In other words, there is a natural isomorphism $G(k)/R \buildrel\sim\over\longrightarrow G\bigl( k((t)) \bigr)/R$.
Radon's inversion formulas
W.
R.
Madych
4475-4491
Abstract: Radon showed the pointwise validity of his celebrated inversion formulas for the Radon transform of a function $f$ of two real variables (formulas (III) and (III$'$) in J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Nat. kl. 69 (1917), 262-277) under the assumption that $f$ is continuous and satisfies two other technical conditions. In this work, using the methods of modern analysis, we show that these technical conditions can be relaxed. For example, the assumption that $f$ be in $L^p(\mathbb{R} ^2)$for some $p$ satisfying $4/3<p<2$ suffices to guarantee the almost everywhere existence of its Radon transform and the almost everywhere validity of Radon's inversion formulas.
Variation inequalities for the Fejér and Poisson kernels
Roger
L.
Jones;
Gang
Wang
4493-4518
Abstract: In this paper we show that the $\varrho$-th order variation operator, for both the Fejér and Poisson kernels, are bounded from $L^p$ to $L^p$, $1<p<\infty$, when $\varrho >2$. Counterexamples are given if $\varrho =2$.
How to make a triangulation of $S^3$ polytopal
Simon
A.
King
4519-4542
Abstract: We introduce a numerical isomorphism invariant $p(\mathcal{T})$ for any triangulation $\mathcal{T}$ of $S^3$. Although its definition is purely topological (inspired by the bridge number of knots), $p(\mathcal{T})$ reflects the geometric properties of $\mathcal{T}$. Specifically, if $\mathcal{T}$ is polytopal or shellable, then $p(\mathcal{T})$is ``small'' in the sense that we obtain a linear upper bound for $p(\mathcal{T})$ in the number $n=n(\mathcal{T})$ of tetrahedra of $\mathcal{T}$. Conversely, if $p(\mathcal{T})$ is ``small'', then $\mathcal{T}$is ``almost'' polytopal, since we show how to transform $\mathcal{T}$ into a polytopal triangulation by $O((p(\mathcal{T}))^2)$ local subdivisions. The minimal number of local subdivisions needed to transform $\mathcal{T}$ into a polytopal triangulation is at least $\frac{p(\mathcal{T})}{3n}-n-2$. Using our previous results [The size of triangulations supporting a given link, Geometry & Topology 5 (2001), 369-398], we obtain a general upper bound for $p(\mathcal{T})$ exponential in $n^2$. We prove here by explicit constructions that there is no general subexponential upper bound for $p(\mathcal{T})$ in $n$. Thus, we obtain triangulations that are ``very far'' from being polytopal. Our results yield a recognition algorithm for $S^3$ that is conceptually simpler, although somewhat slower, than the famous Rubinstein-Thompson algorithm.
Subgroups of $\operatorname{Diff}^{\infty}_+ (\mathbb S^1)$ acting transitively on $4$-tuples
Julio
C.
Rebelo
4543-4557
Abstract: We consider subgroups of $C^{\infty}$-diffeomorphisms of the circle $\mathbb S^1$which act transitively on $4$-tuples of points. We show, in particular, that these subgroups are dense in the group of homeomorphisms of $\mathbb S^1$. A stronger result concerning $C^{\infty}$-approximations is obtained as well. The techniques employed in this paper rely on Lie algebra ideas and they also provide partial generalizations to the differentiable case of some results previously established in the analytic category.
Value groups, residue fields, and bad places of rational function fields
Franz-Viktor
Kuhlmann
4559-4600
Abstract: We classify all possible extensions of a valuation from a ground field $K$ to a rational function field in one or several variables over $K$. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field $K(x)$ in one variable, we consider the relative algebraic closure of $K$ in the henselization of $K(x)$ with respect to the given extension, and we show that this can be any countably generated separable-algebraic extension of $K$. In the ``tame case'', we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the $p$-adics.
Existence and characterization of regions minimizing perimeter under a volume constraint inside Euclidean cones
Manuel
Ritoré;
César
Rosales
4601-4622
Abstract: We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some criteria ensuring existence of isoperimetric regions: for instance, local convexity of the cone at some boundary point. We also characterize which are the stable regions in a convex cone, i.e., second order minima of perimeter under a volume constraint. From this it follows that the isoperimetric regions in a convex cone are the euclidean balls centered at the vertex intersected with the cone.
Real loci of symplectic reductions
R.
F.
Goldin;
T.
S.
Holm
4623-4642
Abstract: Let $M$ be a compact, connected symplectic manifold with a Hamiltonian action of a compact $n$-dimensional torus $T$. Suppose that $M$ is equipped with an anti-symplectic involution $\sigma$ compatible with the $T$-action. The real locus of $M$ is the fixed point set $M^\sigma$ of $\sigma$. Duistermaat introduced real loci, and extended several theorems of symplectic geometry to real loci. In this paper, we extend another classical result of symplectic geometry to real loci: the Kirwan surjectivity theorem. In addition, we compute the kernel of the real Kirwan map. These results are direct consequences of techniques introduced by Tolman and Weitsman. In some examples, these results allow us to show that a symplectic reduction $M/ /T$ has the same ordinary cohomology as its real locus $(M/ /T)^{\sigma_{red}}$, with degrees halved. This extends Duistermaat's original result on real loci to a case in which there is not a natural Hamiltonian torus action.
Viscosity solutions, almost everywhere solutions and explicit formulas
Bernard
Dacorogna;
Paolo
Marcellini
4643-4653
Abstract: Consider the differential inclusion $Du\in E$ in $\mathbb{R} ^{n}$. We exhibit an explicit solution that we call fundamental. It also turns out to be a viscosity solution when properly defining this notion. Finally, we consider a Dirichlet problem associated to the differential inclusion and we give an iterative procedure for finding a solution.
Convolution roots of radial positive definite functions with compact support
Werner
Ehm;
Tilmann
Gneiting;
Donald
Richards
4655-4685
Abstract: A classical theorem of Boas, Kac, and Krein states that a characteristic function $\varphi$ with $\varphi(x) = 0$ for $\vert x\vert \geq \tau$ admits a representation of the form \begin{displaymath}\varphi(x) = \int u(y) \hspace{0.2mm} \overline{u(y+x)} \, \mathrm{d}y, \qquad x \in \mathbb{R}, \end{displaymath} where the convolution root $u \in L^2(\mathbb{R})$ is complex-valued with $u(x) = 0$ for $\vert x\vert \geq \tau/2$. The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examines the Boas-Kac representation under additional constraints: If $\varphi$ is real-valued and even, can the convolution root $u$ be chosen as a real-valued and/or even function? A complete answer in terms of the zeros of the Fourier transform of $\varphi$ is obtained. Furthermore, the analogous problem for radially symmetric functions defined on $\mathbb{R}^d$ is solved. Perhaps surprisingly, there are compactly supported, radial positive definite functions that do not admit a convolution root with half-support. However, under the additional assumption of nonnegativity, radially symmetric convolution roots with half-support exist. Further results in this paper include a characterization of extreme points, pointwise and integral bounds (Turán's problem), and a unified solution to a minimization problem for compactly supported positive definite functions. Specifically, if $f$ is a probability density on $\mathbb{R}^d$ whose characteristic function $\varphi$ vanishes outside the unit ball, then \begin{displaymath}\int \vert x\vert^2 f(x) \, \mathrm{d}x = - \Delta \varphi(0) \geq 4 \, j_{(d-2)/2}^2 \end{displaymath} where $j_\nu$ denotes the first positive zero of the Bessel function $J_\nu$, and the estimate is sharp. Applications to spatial moving average processes, geostatistical simulation, crystallography, optics, and phase retrieval are noted. In particular, a real-valued half-support convolution root of the spherical correlation function in $\mathbb{R}^2$ does not exist.
Higher homotopy commutativity of $H$-spaces and the permuto-associahedra
Yutaka
Hemmi;
Yusuke
Kawamoto
3823-3839
Abstract: In this paper, we give a combinatorial definition of a higher homotopy commutativity of the multiplication for an $A_n$-space. To give the definition, we use polyhedra called the permuto-associahedra which are constructed by Kapranov. We also show that if a connected $A_p$-space has the finitely generated mod $p$ cohomology for a prime $p$ and the multiplication of it is homotopy commutative of the $p$-th order, then it has the mod $p$ homotopy type of a finite product of Eilenberg-Mac Lane spaces $K(\mathbb{Z},1)$s, $K(\mathbb{Z},2)$s and $K(\mathbb{Z}/p^i,1)$s for $i\ge 1$.
Latroids and their representation by codes over modules
Dirk
Vertigan
3841-3868
Abstract: It has been known for some time that there is a connection between linear codes over fields and matroids represented over fields. In fact a generator matrix for a linear code over a field is also a representation of a matroid over that field. There are intimately related operations of deletion, contraction, minors and duality on both the code and the matroid. The weight enumerator of the code is an evaluation of the Tutte polynomial of the matroid, and a standard identity relating the Tutte polynomials of dual matroids gives rise to a MacWilliams identity relating the weight enumerators of dual codes. More recently, codes over rings and modules have been considered, and MacWilliams type identities have been found in certain cases. In this paper we consider codes over rings and modules with code duality based on a Morita duality of categories of modules. To these we associate latroids, defined here. We generalize notions of deletion, contraction, minors and duality, on both codes and latroids, and examine all natural relations among these. We define generating functions associated with codes and latroids, and prove identities relating them, generalizing above-mentioned generating functions and identities.
Quantum deformations of fundamental groups of oriented $3$-manifolds
Uwe
Kaiser
3869-3880
Abstract: We compute two-term skein modules of framed oriented links in oriented $3$-manifolds. They contain the self-writhe and total linking number invariants of framed oriented links in a universal way. The relations in a natural presentation of the skein module are interpreted as monodromies in the space of immersions of circles into the $3$-manifold.
The Deligne complex for the four-strand braid group
Ruth
Charney
3881-3897
Abstract: This paper concerns the homotopy type of hyperplane arrangements associated to infinite Coxeter groups acting as reflection groups on $\mathbb C^n$. A long-standing conjecture states that the complement of such an arrangement should be aspherical. Some partial results on this conjecture were previously obtained by the author and M. Davis. In this paper, we extend those results to another class of Coxeter groups. The key technical result is that the spherical Deligne complex for the 4-strand braid group is CAT(1).
Surface superconductivity in $3$ dimensions
Xing-Bin
Pan
3899-3937
Abstract: We study the Ginzburg-Landau system for a superconductor occupying a $3$-dimensional bounded domain, and improve the estimate of the upper critical field $H_{C_{3}}$ obtained by K. Lu and X. Pan in J. Diff. Eqns., 168 (2000), 386-452. We also analyze the behavior of the order parameters. We show that, under an applied magnetic field lying below and not far from $H_{C_{3}}$, order parameters concentrate in a vicinity of a sheath of the surface that is tangential to the applied field, and exponentially decay both in the normal and tangential directions away from the sheath in the $L^{2}$sense. As the applied field decreases further but keeps in between and away from $H_{C_{2}}$ and $H_{C_{3}}$, the superconducting sheath expands but does not cover the entire surface, and superconductivity at the surface portion orthogonal to the applied field is always very weak. This phenomenon is significantly different to the surface superconductivity on a cylinder of infinite height studied by X. Pan in Comm. Math. Phys., 228 (2002), 327-370, where under an axial applied field lying in-between $H_{C_{2}}$ and $H_{C_{3}}$ the entire surface is in the superconducting state.
Identities of graded algebras and codimension growth
Yu.
A.
Bahturin;
M.
V.
Zaicev
3939-3950
Abstract: Let $A=\oplus_{g\in G}A_g$ be a $G$-graded associative algebra over a field of characteristic zero. In this paper we develop a conjecture that relates the exponent of the growth of polynomial identities of the identity component $A_e$ to that of the whole of $A$, in the case where the support of the grading is finite. We prove the conjecture in several natural cases, one of them being the case where $A$ is finite dimensional and $A_e$ has polynomial growth.
When does the subadditivity theorem for multiplier ideals hold?
Shunsuke
Takagi;
Kei-ichi
Watanabe
3951-3961
Abstract: Demailly, Ein and Lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties, which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals. We prove that, in the two-dimensional case, the subadditivity theorem holds on log terminal singularities. However, in the higher dimensional case, we have several counterexamples. We consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three-dimensional toric ring.
A modified Brauer algebra as centralizer algebra of the unitary group
Alberto
Elduque
3963-3983
Abstract: The centralizer algebra of the action of $U(n)$ on the real tensor powers $\otimes_\mathbb{R}^r V$ of its natural module, $V=\mathbb{C}^n$, is described by means of a modification in the multiplication of the signed Brauer algebras. The relationships of this algebra with the invariants for $U(n)$ and with the decomposition of $\otimes_\mathbb{R}^r V$ into irreducible submodules is considered.
On the asymptotic behavior of a complete bounded minimal surface in $\mathbb{R}^3$
Francisco
Martín;
Santiago
Morales
3985-3994
Abstract: In this paper we construct an example of a complete minimal disk which is properly immersed in a ball of $\mathbb{R} ^3$.
Flat holomorphic connections on principal bundles over a projective manifold
Indranil
Biswas;
S.
Subramanian
3995-4018
Abstract: Let $G$ be a connected complex linear algebraic group and $R_u(G)$ its unipotent radical. A principal $G$-bundle $E_G$ over a projective manifold $M$ will be called polystable if the associated principal $G/R_u(G)$-bundle is so. A $G$-bundle $E_G$ over $M$ is polystable with vanishing characteristic classes of degrees one and two if and only if $E_G$ admits a flat holomorphic connection with the property that the image in $G/R_u(G)$ of the monodromy of the connection is contained in a maximal compact subgroup of $G/R_u(G)$.
Quadratic forms and Pfister neighbors in characteristic 2
Detlev
W.
Hoffmann;
Ahmed
Laghribi
4019-4053
Abstract: We study Pfister neighbors and their characterization over fields of characteristic $2$, where we include the case of singular forms. We give a somewhat simplified proof of a theorem of Fitzgerald which provides a criterion for when a nonsingular quadratic form $q$ is similar to a Pfister form in terms of the hyperbolicity of this form over the function field of a form $\varphi$ which is dominated by $q$. From this, we derive an analogue in characteristic $2$ of a result by Knebusch saying that, in characteristic $\neq 2$, a form is a Pfister neighbor if its anisotropic part over its own function field is defined over the base field. Our result includes certain cases of singular forms, but we also give examples which show that Knebusch's result generally fails in characteristic $2$ for singular forms. As an application, we characterize certain forms of height $1$ in the sense of Knebusch whose quasi-linear parts are of small dimension. We also develop some of the basics of a theory of totally singular quadratic forms. This is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper.
Conservation laws for a class of third order evolutionary differential systems
Sung Ho
Wang
4055-4073
Abstract: Conservation laws of third order quasi-linear scalar evolution equations are studied via exterior differential system and characteristic cohomology. We find a subspace of 2-forms in the infinite prolongation space in which every conservation law has a unique representative. Analysis of the structure of this subspace based upon the symbol of the differential equation leads to a universal integrability condition for an evolution equation to admit any higher order (weight) conservation laws. As an example, we give a complete classification of a class of evolution equations which admit conservation laws of the first three consecutive weights $-1$, $1$, $3$. The differential system describing the flow of a curve in the plane by the derivative of its curvature with respect to the arc length is also shown to exhibit the KdV property, i.e., an infinite sequence of conservation laws of distinct weights.
Homotopy groups of $K$-contact toric manifolds
Eugene
Lerman
4075-4083
Abstract: Contact toric manifolds of Reeb type are a subclass of contact toric manifolds which have the property that they are classified by the images of the associated moment maps. We compute their first and second homotopy group terms of the images of the moment map. We also explain why they are $K$-contact.
Green's functions for elliptic and parabolic equations with random coefficients II
Joseph
G.
Conlon
4085-4142
Abstract: This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Green's functions for the equations are then random variables. Regularity properties for expectation values of Green's functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green's function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.
On the classification of full factors of type III
Dimitri
Shlyakhtenko
4143-4159
Abstract: We introduce a new invariant $\mathscr{S}(M)$ for type III factors $M$ with no almost-periodic weights. We compute this invariant for certain free Araki-Woods factors. We show that Connes' invariant $\tau$cannot distinguish all isomorphism classes of free Araki-Woods factors. We show that there exists a continuum of mutually non-isomorphic free Araki-Woods factors, each without almost-periodic weights.
Dual Radon transforms on affine Grassmann manifolds
Fulton
B.
Gonzalez;
Tomoyuki
Kakehi
4161-4180
Abstract: Fix $0 \leq p < q \leq n-1$, and let $G(p,n)$ and $G(q,n)$denote the affine Grassmann manifolds of $p$- and $q$-planes in $\mathbb{R} ^n$. We investigate the Radon transform $\mathcal{R}^{(q,p)} : C^{\infty} (G(q,n)) \to C^{\infty} (G(p,n))$associated with the inclusion incidence relation. For the generic case $\dim G(q,n) < \dim G(p,n)$ and $p+q > n$, we will show that the range of this transform is given by smooth functions on $G(p,n)$ annihilated by a system of Pfaffian type differential operators. We also study aspects of the exceptional case $p+q =n$.
Varieties of tori and Cartan subalgebras of restricted Lie algebras
Rolf
Farnsteiner
4181-4236
Abstract: This paper investigates varieties of tori and Cartan subalgebras of a finite-dimensional restricted Lie algebra $(\mathfrak{g},[p])$, defined over an algebraically closed field $k$ of positive characteristic $p$. We begin by showing that schemes of tori may be used as a tool to retrieve results by A. Premet on regular Cartan subalgebras. Moreover, they give rise to principal fibre bundles, whose structure groups coincide with the Weyl groups in case $\mathfrak{g}= \operatorname{Lie}(\mathcal{G})$ is the Lie algebra of a smooth group $\mathcal{G}$. For solvable Lie algebras, varieties of tori are full affine spaces, while simple Lie algebras of classical or Cartan type cannot have varieties of this type. In the final sections the quasi-projective variety of Cartan subalgebras of minimal dimension ${\rm rk}(\mathfrak{g})$ is shown to be irreducible of dimension $\dim_k\mathfrak{g}-{\rm rk}(\mathfrak{g})$, with Premet's regular Cartan subalgebras belonging to the regular locus.
Elliptic Apostol sums and their reciprocity laws
Shinji
Fukuhara;
Noriko
Yui
4237-4254
Abstract: We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter $\tau$ having positive imaginary part. When $\tau\to i\infty$, these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable $\tau$. We obtain a reciprocity law for these sums, which gives rise to new relations between certain modular forms (of one variable).
Complete linear Weingarten surfaces of Bryant type. A Plateau problem at infinity
José
Antonio
Gálvez;
Antonio
Martínez;
Francisco
Milán
3405-3428
Abstract: In this paper we study a large class of Weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3-space. We show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the Euclidean 3-space and we use it to study their completeness. We also establish some existence and uniqueness theorems by studing the Plateau problem at infinity: when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family? and, how many embedded solutions are there?
$L^p\to L^q$ regularity of Fourier integral operators with caustics
Andrew
Comech
3429-3454
Abstract: The caustics of Fourier integral operators are defined as caustics of the corresponding Schwartz kernels (Lagrangian distributions on $X\times Y$). The caustic set $\Sigma(\matheurb{ C})$of the canonical relation is characterized as the set of points where the rank of the projection $\pi:\matheurb{ C}\to X\times Y$is smaller than its maximal value, $\dim(X\times Y)-1$. We derive the $L^ p(Y)\to L^ q(X)$ estimates on Fourier integral operators with caustics of corank $1$(such as caustics of type $A_{m+1}$, $m\in{\mathbb N}$). For the values of $p$ and $q$outside of a certain neighborhood of the line of duality, $q=p'$, the $L^ p\to L^ q$ estimates are proved to be caustics-insensitive. We apply our results to the analysis of the blow-up of the estimates on the half-wave operator just before the geodesic flow forms caustics.
Jack polynomials and some identities for partitions
Michel
Lassalle
3455-3476
Abstract: We prove an identity about partitions involving new combinatorial coefficients. The proof given is using a generating function. As an application we obtain the explicit expression of two shifted symmetric functions, related with Jack polynomials. These quantities are the moments of the ``$\alpha$-content'' random variable with respect to some transition probability distributions.
The best constant of the Moser-Trudinger inequality on $\textbf{S}^2$
Yuji
Sano
3477-3482
Abstract: We consider the best constant of the Moser-Trudinger inequality on $\textbf{S}^2$ under a certain orthogonality condition. Applying Moser's calculation, we construct a counterexample to the sharper inequality with the condition.
Definability in the lattice of equational theories of commutative semigroups
Andrzej
Kisielewicz
3483-3504
Abstract: In this paper we study first-order definability in the lattice of equational theories of commutative semigroups. In a series of papers, J. Jezek, solving problems posed by A. Tarski and R. McKenzie, has proved, in particular, that each equational theory is first-order definable in the lattice of equational theories of a given type, up to automorphism, and that such lattices have no automorphisms besides the obvious syntactically defined ones (with exceptions for special unary types). He has proved also that the most important classes of theories of a given type are so definable. In a later paper, Jezek and McKenzie have ``almost proved" the same facts for the lattice of equational theories of semigroups. There were good reasons to believe that the same can be proved for the lattice of equational theories of commutative semigroups. In this paper, however, we show that the case of commutative semigroups is different.
An extended urn model with application to approximation
Fengxin
Chen
3505-3515
Abstract: Pólya's urn model from probability theory is extended to obtain a class of approximation operators for which the Weierstrass Approximation Theorem holds.
Parabolic evolution equations with asymptotically autonomous delay
Roland
Schnaubelt
3517-3543
Abstract: We study retarded parabolic non-autonomous evolution equations whose coefficients converge as $t\to\infty$, such that the autonomous problem in the limit has an exponential dichotomy. Then the non-autonomous problem inherits the exponential dichotomy, and the solution of the inhomogeneous equation tends to the stationary solution at infinity. We use a generalized characteristic equation to deduce the exponential dichotomy and new representation formulas for the solution of the inhomogeneous equation.
Nonlinearizable actions of dihedral groups on affine space
Kayo
Masuda
3545-3556
Abstract: Let $G$ be a reductive, non-abelian, algebraic group defined over $\mathbb{C}$. We investigate algebraic $G$-actions on the total spaces of non-trivial algebraic $G$-vector bundles over $G$-modules with great interest in the case that $G$ is a dihedral group. We construct a map classifying such actions of a dihedral group in some cases and describe the spaces of those non-linearizable actions in some examples.
On the structure of a sofic shift space
Klaus
Thomsen
3557-3619
Abstract: The structure of a sofic shift space is investigated, and Krieger's embedding theorem and Boyle's factor theorem are generalized to a large class of sofic shifts.
Realizability of modules over Tate cohomology
David
Benson;
Henning
Krause;
Stefan
Schwede
3621-3668
Abstract: Let $k$ be a field and let $G$ be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology $\gamma_G\in HH^{3,-1}\hat H^*(G,k)$ with the following property. Given a graded $\hat H^*(G,k)$-module $X$, the image of $\gamma_G$in ${\text{\rm Ext}}^{3,-1}_{\hat H^*(G,k)}(X,X)$ vanishes if and only if $X$ is isomorphic to a direct summand of $\hat H^*(G,M)$ for some $kG$-module $M$. The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a differential graded algebra $A$, there is also a canonical element of Hochschild cohomology $HH^{3,-1}H^*(A)$ which is a predecessor for these obstructions.
Linking numbers in rational homology $3$-spheres, cyclic branched covers and infinite cyclic covers
Józef
H.
Przytycki;
Akira
Yasuhara
3669-3685
Abstract: We study the linking numbers in a rational homology $3$-sphere and in the infinite cyclic cover of the complement of a knot. They take values in $\mathbb{Q}$ and in ${Q}(\mathbb{Z}[t,t^{-1}])$, respectively, where ${Q}(\mathbb{Z}[t,t^{-1}])$ denotes the quotient field of $\mathbb{Z}[t,t^{-1}]$. It is known that the modulo- $\mathbb{Z}$ linking number in the rational homology $3$-sphere is determined by the linking matrix of the framed link and that the modulo- $\mathbb{Z}[t,t^{-1}]$ linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate `modulo $\mathbb{Z}$' and `modulo $\mathbb{Z}[t,t^{-1}]$'. When the finite cyclic cover of the $3$-sphere branched over a knot is a rational homology $3$-sphere, the linking number of a pair in the preimage of a link in the $3$-sphere is determined by the Goeritz/Seifert matrix of the knot.
Ideals in a perfect closure, linear growth of primary decompositions, and tight closure
Rodney
Y.
Sharp;
Nicole
Nossem
3687-3720
Abstract: This paper is concerned with tight closure in a commutative Noetherian ring $R$ of prime characteristic $p$, and is motivated by an argument of K. E. Smith and I. Swanson that shows that, if the sequence of Frobenius powers of a proper ideal ${\mathfrak{a}}$ of $R$has linear growth of primary decompositions, then tight closure (of ${\mathfrak{a}}$) `commutes with localization at the powers of a single element'. It is shown in this paper that, provided $R$ has a weak test element, linear growth of primary decompositions for other sequences of ideals of $R$ that approximate, in a certain sense, the sequence of Frobenius powers of ${\mathfrak{a}}$ would not only be just as good in this context, but, in the presence of a certain additional finiteness property, would actually imply that tight closure (of ${\mathfrak{a}}$) commutes with localization at an arbitrary multiplicatively closed subset of $R$. Work of M. Katzman on the localization problem for tight closure raised the question as to whether the union of the associated primes of the tight closures of the Frobenius powers of ${\mathfrak{a}}$has only finitely many maximal members. This paper develops, through a careful analysis of the ideal theory of the perfect closure of $R$, strategies for showing that tight closure (of a specified ideal ${\mathfrak{a}}$ of $R$) commutes with localization at an arbitrary multiplicatively closed subset of $R$and for showing that the union of the associated primes of the tight closures of the Frobenius powers of ${\mathfrak{a}}$is actually a finite set. Several applications of the strategies are presented; in most of them it was already known that tight closure commutes with localization, but the resulting affirmative answers to Katzman's question in the various situations considered are believed to be new.
A positivstellensatz for non-commutative polynomials
J.
William
Helton;
Scott
A.
McCullough
3721-3737
Abstract: A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the commutative case. A broader issue is, to what extent does real semi-algebraic geometry extend to non-commutative polynomials? Our ``strict" Positivstellensatz is positive news, on the opposite extreme from strict positivity would be a Real Nullstellensatz. We give an example which shows that there is no non-commutative Real Nullstellensatz along certain lines. However, we include a successful type of non-commutative Nullstellensatz proved by George Bergman.
Non-isotopic symplectic tori in the same homology class
Tolga
Etgü;
B.
Doug
Park
3739-3750
Abstract: For any pair of integers $n\geq 1$ and $q\geq 2$, we construct an infinite family of mutually non-isotopic symplectic tori representing the homology class $q[F]$ of an elliptic surface $E(n)$, where $[F]$ is the homology class of the fiber. We also show how such families can be non-isotopically and symplectically embedded into a more general class of symplectic $4$-manifolds.
A class of processes on the path space over a compact Riemannian manifold with unbounded diffusion
Jörg-Uwe
Löbus
3751-3767
Abstract: A class of diffusion processes on the path space over a compact Riemannian manifold is constructed. The diffusion of such a process is governed by an unbounded operator. A representation of the associated generator is derived and the existence of a certain local second moment is shown.
The double bubble problem on the flat two-torus
Joseph
Corneli;
Paul
Holt;
George
Lee;
Nicholas
Leger;
Eric
Schoenfeld;
Benjamin
Steinhurst
3769-3820
Abstract: We characterize the perimeter-minimizing double bubbles on all flat two-tori and, as corollaries, on the flat infinite cylinder and the flat infinite strip with free boundary. Specifically, we show that there are five distinct types of minimizers on flat two-tori, depending on the areas to be enclosed.
Addendum to ``Symmetrization, symmetric stable processes, and Riesz capacities''
Dimitrios
Betsakos
3821-3821
Eigenfunctions of the Laplacian acting on degree zero bundles over special Riemann surfaces
Marco
Matone
2989-3004
Abstract: We find an infinite set of eigenfunctions for the Laplacian with respect to a flat metric with conical singularities and acting on degree zero bundles over special Riemann surfaces of genus greater than one. These special surfaces correspond to Riemann period matrices satisfying a set of equations which lead to a number theoretical problem. It turns out that these surfaces precisely correspond to branched covering of the torus. This reflects in a Jacobian with a particular kind of complex multiplication.
A new variational characterization of $n$-dimensional space forms
Zejun
Hu;
Haizhong
Li
3005-3023
Abstract: A Riemannian manifold $(M^n,g)$ is associated with a Schouten $(0,2)$-tensor $C_g$ which is a naturally defined Codazzi tensor in case $(M^n,g)$ is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional $\mathcal{F}_k[g]=\int_M\sigma_k(C_g)dvol_g$ defined on $\mathcal{M}_1=\{g\in\mathcal{M}\vert Vol(g)=1\}$, where $\mathcal{M}$ is the space of smooth Riemannian metrics on a compact smooth manifold $M$ and $\{\sigma_k(C_g), 1\leq k\leq n\}$ is the elementary symmetric functions of the eigenvalues of $C_g$ with respect to $g$. We prove that if $n\geq 5$ and a conformally flat metric $g$ is a critical point of $\mathcal{F}_2\vert _{\mathcal{M}_1}$ with $\mathcal{F}_2[g]\geq0$, then $g$ must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of $\mathcal{F}_2\vert _{\mathcal{M}_1}$ with $\mathcal{F}_2[g]\geq0$ characterized the three-dimensional space forms.
The $\forall\exists$-theory of $\mathcal{R}(\leq,\vee,\wedge)$ is undecidable
Russell
G.
Miller;
Andre
O.
Nies;
Richard
A.
Shore
3025-3067
Abstract: The three quantifier theory of $(\mathcal{R},\leq_{T})$, the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of $\mathcal{R}$ that lies between the two and three quantifier theories with $\leq_{T}$ but includes function symbols. Theorem. The two quantifier theory of $(\mathcal{R},\leq ,\vee,\wedge)$, the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on $\mathcal{R}$) is undecidable. The same result holds for various lattices of ideals of $\mathcal{R}$ which are natural extensions of $\mathcal{R}$ preserving join and infimum when it exits.
Szegö kernels and finite group actions
Roberto
Paoletti
3069-3076
Abstract: In the context of almost complex quantization, a natural generalization of algebro-geometric linear series on a compact symplectic manifold has been proposed. Here we suppose given a compatible action of a finite group and consider the linear subseries associated to the irreducible representations of $G$, give conditions under which these are base-point-free and study properties of the associated projective morphisms. The results obtained are new even in the complex projective case.
Homotopy equivalence of two families of complexes
Giandomenico
Boffi;
David
A.
Buchsbaum
3077-3107
Abstract: An explicit homotopy equivalence is established between two families of complexes, both of which generalize the classical Koszul complex.
Trees, parking functions, syzygies, and deformations of monomial ideals
Alexander
Postnikov;
Boris
Shapiro
3109-3142
Abstract: For a graph $G$, we construct two algebras whose dimensions are both equal to the number of spanning trees of $G$. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to $G$-parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.
Core versus graded core, and global sections of line bundles
Eero
Hyry;
Karen
E.
Smith
3143-3166
Abstract: We find formulas for the graded core of certain $\mathfrak{m}$-primary ideals in a graded ring. In particular, if $S$ is the section ring of an ample line bundle on a Cohen-Macaulay complex projective variety, we show that under a suitable hypothesis, the core and graded core of the ideal of $S$ generated by all elements of degrees at least $N$ (for some, equivalently every, large $N$) are equal if and only if the line bundle admits a non-zero global section. We also prove a formula for the graded core of the powers of the unique homogeneous maximal ideal in a standard graded Cohen-Macaulay ring of arbitrary characteristic. Several open problems are posed whose solutions would lead to progress on a non-vanishing conjecture of Kawamata.
Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc
Pekka
J.
Nieminen;
Eero
Saksman
3167-3187
Abstract: Let $\phi$ be a holomorphic self-map of the unit disc $\mathbb{D}$. For every $\alpha \in \partial\mathbb{D}$, there is a measure $\tau_\alpha$ on $\partial\mathbb{D}$ (sometimes called Aleksandrov measure) defined by the Poisson representation $\operatorname{Re}(\alpha+\phi(z))/(\alpha-\phi(z)) = \int P(z,\zeta) \,d\tau_\alpha(\zeta)$. Its singular part $\sigma_\alpha$ measures in a natural way the ``affinity'' of $\phi$ for the boundary value $\alpha$. The affinity for values $w$ inside $\mathbb{D}$ is provided by the Nevanlinna counting function $N(w)$ of $\phi$. We introduce a natural measure-valued refinement $M_w$ of $N(w)$ and establish that the measures $\{\sigma_\alpha\}_{\alpha\in\partial\mathbb{D}}$are obtained as boundary values of the refined Nevanlinna counting function $M$. More precisely, we prove that $\sigma_\alpha$ is the weak$^*$ limit of $M_w$ whenever $w$ converges to $\alpha$non-tangentially outside a small exceptional set $E$. We obtain a sharp estimate for the size of $E$ in the sense of capacity.
On Diophantine definability and decidability in some infinite totally real extensions of $\mathbb Q$
Alexandra
Shlapentokh
3189-3207
Abstract: Let $M$ be a number field, and $W_M$ a set of its non-Archimedean primes. Then let $O_{M,W_M} = \{x \in M\vert \operatorname{ord}_{\mathfrak{t}}x \geq 0, \, \forall \mathfrak{t} \, \not \in W_M\}$. Let $\{p_1,\ldots,p_r\}$ be a finite set of prime numbers. Let $F_{inf}$ be the field generated by all the $p_i^{j}$-th roots of unity as $j \rightarrow \infty$ and $i=1,\ldots,r$. Let $K_{inf}$ be the largest totally real subfield of $F_{inf}$. Then for any $\varepsilon > 0$, there exist a number field $M \subset K_{inf}$, and a set $W_M$ of non-Archimedean primes of $M$ such that $W_M$ has density greater than $1-\varepsilon$, and $\mathbb{Z}$ has a Diophantine definition over the integral closure of $O_{M,W_M}$ in $K_{inf}$.
Uncorrelatedness and orthogonality for vector-valued processes
Peter
A.
Loeb;
Horst
Osswald;
Yeneng
Sun;
Zhixiang
Zhang
3209-3225
Abstract: For a square integrable vector-valued process $f$ on the Loeb product space, it is shown that vector orthogonality is almost equivalent to componentwise scalar orthogonality. Various characterizations of almost sure uncorrelatedness for $f$ are presented. The process $f$ is also related to multilinear forms on the target Hilbert space. Finally, a general structure result for $f$ involving the biorthogonal representation for the conditional expectation of $f$ with respect to the usual product $\sigma$-algebra is presented.
Existence of local sufficiently smooth solutions to the complex Monge-Ampère equation
Saoussen
Kallel-Jallouli
3227-3242
Abstract: We prove the $C^{\infty }$ local solvability of the $n$-dimensional complex Monge-Ampère equation $\det \left( u_{i\overline{j}}\right) =f\left( z,u,\nabla u\right)$, $f\geq 0$, in a neighborhood of any point $z_{0}$where $f\left( z_{0}\right) =0$.
Construction and recognition of hyperbolic 3-manifolds with geodesic boundary
Roberto
Frigerio;
Carlo
Petronio
3243-3282
Abstract: We extend to the context of hyperbolic 3-manifolds with geodesic boundary Thurston's approach to hyperbolization by means of geometric triangulations. In particular, we introduce moduli for (partially) truncated hyperbolic tetrahedra, and we discuss consistency and completeness equations. Moreover, building on previous work of Ushijima, we extend Weeks' tilt formula algorithm, which computes the Epstein-Penner canonical decomposition, to an algorithm that computes the Kojima decomposition. Our theory has been exploited to classify all the orientable finite-volume hyperbolic $3$-manifolds with non-empty compact geodesic boundary admitting an ideal triangulation with at most four tetrahedra. The theory is particularly interesting in the case of complete finite-volume manifolds with geodesic boundary in which the boundary is non-compact. We include this case using a suitable adjustment of the notion of ideal triangulation, and we show how this case arises within the theory of knots and links.
Infinitely many solutions to fourth order superlinear periodic problems
Monica
Conti;
Susanna
Terracini;
Gianmaria
Verzini
3283-3300
Abstract: We present a new min-max approach to the search of multiple $T$-periodic solutions to a class of fourth order equations \begin{displaymath}u^{iv}(t)-c u''(t)=f(t,u(t)),\hspace{5mm}t\in[0,T],\end{displaymath} where $f(t,u)$ is continuous, $T$-periodic in $t$ and satisfies a superlinearity assumption when $\vert u\vert\to\infty$. For every $n\in\mathbb{N}$, we prove the existence of a $T$-periodic solution having exactly $2n$ zeroes in $(0,T]$.
The cohomology of certain Hopf algebras associated with $p$-groups
Justin
M.
Mauger
3301-3323
Abstract: We study the cohomology $H^*(A)=\operatorname{Ext}_A^*(k,k)$ of a locally finite, connected, cocommutative Hopf algebra $A$ over $k=\mathbb{F} _p$. Specifically, we are interested in those algebras $A$ for which $H^*(A)$ is generated as an algebra by $H^1(A)$ and $H^2(A)$. We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras $F\rightarrow A\rightarrow B$ with $F$ monogenic and $B$ semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for $A$ to be semi-Koszul. Special attention is given to the case in which $A$ is the restricted universal enveloping algebra of the Lie algebra obtained from the mod-$p$ lower central series of a $p$-group. We show that the algebras arising in this way from extensions by $\mathbb{Z} /(p)$ of an abelian $p$-group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 $p$-groups, and it is shown that these are all semi-Koszul for $p\geq 5$.
On the divergence of the Rogers-Ramanujan continued fraction on the unit circle
Douglas
Bowman;
James
Mc Laughlin
3325-3347
Abstract: This paper studies ordinary and general convergence of the Rogers-Ramanujan continued fraction. Let the continued fraction expansion of any irrational number $t \in (0,1)$be denoted by $[0,e_{1}(t),e_{2}(t),\cdots]$ and let the $i$-th convergent of this continued fraction expansion be denoted by $c_{i}(t)/d_{i}(t)$. Let \begin{displaymath}S=\{t \in (0,1): e_{i+1}(t) \geq \phi^{d_{i}(t)} \text{ infinitely often}\}, \end{displaymath} where $\phi = (\sqrt{5}+1)/2$. Let $Y_{S} =\{\exp(2 \pi i t): t \in S \}$. It is shown that if $y \in Y_{S}$, then the Rogers-Ramanujan continued fraction $R(y)$ diverges at $y$. $S$ is an uncountable set of measure zero. It is also shown that there is an uncountable set of points $G \subset Y_{S}$such that if $y \in G$, then $R(y)$ does not converge generally. It is further shown that $R(y)$ does not converge generally for $\vert y\vert > 1$. However we show that $R(y)$ does converge generally if $y$ is a primitive $5m$-th root of unity, for some $m \in \mathbb{N}$. Combining this result with a theorem of I. Schur then gives that the continued fraction converges generally at all roots of unity.
Integrals, partitions, and cellular automata
Alexander
E.
Holroyd;
Thomas
M.
Liggett;
Dan
Romik
3349-3368
Abstract: We prove that \begin{displaymath}\int_0^1\frac{-\log f(x)}xdx=\frac{\pi^2}{3ab},\end{displaymath} where $f(x)$ is the decreasing function that satisfies $f^a-f^b=x^a-x^b$, for $0<a<b$. When $a$ is an integer and $b=a+1$ we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having $a$ consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.
The flat model structure on $\mathbf{Ch}(R)$
James
Gillespie
3369-3390
Abstract: Given a cotorsion pair $(\mathcal{A},\mathcal{B})$ in an abelian category $\mathcal{C}$ with enough $\mathcal{A}$ objects and enough $\mathcal{B}$ objects, we define two cotorsion pairs in the category $\mathbf{Ch(\mathcal{C})}$ of unbounded chain complexes. We see that these two cotorsion pairs are related in a nice way when $(\mathcal{A},\mathcal{B})$ is hereditary. We then show that both of these induced cotorsion pairs are complete when $(\mathcal{A},\mathcal{B})$ is the ``flat'' cotorsion pair of $R$-modules. This proves the flat cover conjecture for (possibly unbounded) chain complexes and also gives us a new ``flat'' model category structure on $\mathbf{Ch}(R)$. In the last section we use the theory of model categories to show that we can define $\operatorname{Ext}^n_R(M,N)$using a flat resolution of $M$ and a cotorsion coresolution of $N$.
Rationality, regularity, and $C_2$-cofiniteness
Toshiyuki
Abe;
Geoffrey
Buhl;
Chongying
Dong
3391-3402
Abstract: We demonstrate that, for vertex operator algebras of CFT type, $C_2$-cofiniteness and rationality is equivalent to regularity. For $C_2$-cofinite vertex operator algebras, we show that irreducible weak modules are ordinary modules and $C_2$-cofinite, $V_L^+$ is $C_2$-cofinite, and the fusion rules are finite.
Errata to ``On the structure of weight modules"
Ivan
Dimitrov;
Olivier
Mathieu;
Ivan
Penkov
3403-3404
A join theorem for the computably enumerable degrees
Carl
G.
Jockusch Jr.;
Angsheng
Li;
Yue
Yang
2557-2568
Abstract: It is shown that for any computably enumerable (c.e.) degree $\mathbf{w}$, if $\mathbf{w\not=0}$, then there is a c.e. degree $\mathbf{a}$ such that $\mathbf{a}$ is low$_2$and $\mathbf{a\lor w}$ is high). It follows from this and previous work of P. Cholak, M. Groszek and T. Slaman that the low and low$_2$ c.e. degrees are not elementarily equivalent as partial orderings.
Thomason's theorem for varieties over algebraically closed fields
Mark
E.
Walker
2569-2648
Abstract: We present a novel proof of Thomason's theorem relating Bott inverted algebraic $K$-theory with finite coefficients and étale cohomology for smooth varieties over algebraically closed ground fields. Our proof involves first introducing a new theory, which we term algebraic $K$-homology, and proving it satisfies étale descent (with finite coefficients) on the category of normal, Cohen-Macaulay varieties. Then, we prove algebraic $K$-homology and algebraic $K$-theory (each taken with finite coefficients) coincide on smooth varieties upon inverting the Bott element.
Subvarieties of general type on a general projective hypersurface
Gianluca
Pacienza
2649-2661
Abstract: We study subvarieties of a general projective degree $d$ hypersurface $X_d\subset \mathbf{P}^n$. Our main theorem, which improves previous results of L. Ein and C. Voisin, implies in particular the following sharp corollary: any subvariety of a general hypersurface $X_{d}\subset {\mathbf P}^n$, for $n\geq 6$ and $d\geq 2n-2$, is of general type.
Sums of squares in real rings
José
F.
Fernando;
Jesús
M.
Ruiz;
Claus
Scheiderer
2663-2684
Abstract: Let $A$ be an excellent ring. We show that if the real dimension of $A$ is at least three then $A$ has infinite Pythagoras number, and there exists a positive semidefinite element in $A$ which is not a sum of squares in $A$.
$L^{2}$-metrics, projective flatness and families of polarized abelian varieties
Wing-Keung
To;
Lin
Weng
2685-2707
Abstract: We compute the curvature of the $L^{2}$-metric on the direct image of a family of Hermitian holomorphic vector bundles over a family of compact Kähler manifolds. As an application, we show that the $L^{2}$-metric on the direct image of a family of ample line bundles over a family of abelian varieties and equipped with a family of canonical Hermitian metrics is always projectively flat. When the parameter space is a compact Kähler manifold, this leads to the poly-stability of the direct image with respect to any Kähler form on the parameter space.
Fundamental solutions for non-divergence form operators on stratified groups
Andrea
Bonfiglioli;
Ermanno
Lanconelli;
Francesco
Uguzzoni
2709-2737
Abstract: We construct the fundamental solutions $\Gamma$ and $\gamma$for the non-divergence form operators $\,{\textstyle\sum_{i,\,j}\,} a_{i,\,j}(x,t)\,X_iX_j\,-\,\partial_t\,$ and ${\,\textstyle\sum_{i,\,j}}\,a_{i,\,j}(x)\,X_iX_j$, where the $X_i$'s are Hörmander vector fields generating a stratified group $\mathbb{G}$ and $(a_{i,j})_{i,j}$ is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates of $\Gamma$ and its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates of $\Gamma$ allow us to construct $\gamma$ using both $t$-saturation and approximation arguments.
Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation
Víctor
Padrón
2739-2756
Abstract: In this paper we study the equation \begin{displaymath}u_t=\Delta(\phi(u) - \lambda f(u) + \lambda u_t) + f(u) \end{displaymath} in a bounded domain of $\mathbb{R} ^d$, $d\ge1$, with homogeneous boundary conditions of the Neumann type, as a model of aggregating population with a migration rate determined by $\phi$, and total birth and mortality rates characterized by $f$. We will show that the aggregating mechanism induced by $\phi(u)$ allows the survival of a species in danger of extinction. Numerical simulations suggest that the solutions stabilize asymptotically in time to a not necessarily homogeneous stationary solution. This is shown to be the case for a particular version of the function $\phi(u)$.
Dynamical approach to some problems in integral geometry
Boris
Paneah
2757-2780
Abstract: As is well known, the main problem in integral geometry is to reconstruct a function in a given domain $D$, where its integrals over a family of subdomains in $D$ are known. Such a problem is interesting not only as an object of pure analysis, but also in connection with various applications in practical disciplines. The most remarkable example of such a connection is the Radon problem and tomography. In this paper we solve one of these problems when $D$ is a bounded domain in ${\mathbb{R}}^2$ with a piecewise smooth boundary. Some intermediate results related to dynamical systems with two generators and to some functional-integral equations are new and interesting per se. As an application of the results obtained we briefly study a boundary problem for a general third order hyperbolic partial differential equation in a bounded domain $D\subset {\mathbb{R}}^2$ with data on the whole boundary $\partial D$.
The peak algebra and the descent algebras of types B and D
Marcelo
Aguiar;
Nantel
Bergeron;
Kathryn
Nyman
2781-2824
Abstract: We show the existence of a unital subalgebra $\mathfrak{P}_n$ of the symmetric group algebra linearly spanned by sums of permutations with a common peak set, which we call the peak algebra. We show that $\mathfrak{P}_n$ is the image of the descent algebra of type B under the map to the descent algebra of type A which forgets the signs, and also the image of the descent algebra of type D. The algebra $\mathfrak{P}_n$ contains a two-sided ideal $\overset{\circ}{\mathfrak{P}}_n$ which is defined in terms of interior peaks. This object was introduced in previous work by Nyman (2003); we find that it is the image of certain ideals of the descent algebras of types B and D. We derive an exact sequence of the form $0\to\overset{\circ}{\mathfrak{P}}_n \to\mathfrak{P}_n\to\mathfrak{P}_{n-2}\to 0$. We obtain this and many other properties of the peak algebra and its peak ideal by first establishing analogous results for signed permutations and then forgetting the signs. In particular, we construct two new commutative semisimple subalgebras of the descent algebra (of dimensions $n$ and $\lfloor\frac{n}{2}\rfloor+1)$ by grouping permutations according to their number of peaks or interior peaks. We discuss the Hopf algebraic structures that exist on the direct sums of the spaces $\mathfrak{P}_n$ and $\overset{\circ}{\mathfrak{P}}_n$ over $n\geq 0$ and explain the connection with previous work of Stembridge (1997); we also obtain new properties of his descents-to-peaks map and construct a type B analog.
Combinatorial properties of Thompson's group $F$
Sean
Cleary;
Jennifer
Taback
2825-2849
Abstract: We study some combinatorial consequences of Blake Fordham's theorems on the word metric of Thompson's group $F$ in the standard two generator presentation. We explore connections between the tree pair diagram representing an element $w$ of $F$, its normal form in the infinite presentation, its word length, and minimal length representatives of it. We estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates. In addition we explore some properties of the Cayley graph of $F$ with respect to the two generator finite presentation. Namely, we exhibit the form of ``dead end'' elements in this Cayley graph, and show that it has no ``deep pockets''. Finally, we discuss a simple method for constructing minimal length representatives for strictly positive or negative words.
Metrical diophantine approximation for continued fraction like maps of the interval
Andrew
Haas;
David
Molnar
2851-2870
Abstract: We study the metrical properties of a class of continued fraction-like mappings of the unit interval, each of which is defined as the fractional part of a Möbius transformation taking the endpoints of the interval to zero and infinity.
The ABC theorem for higher-dimensional function fields
Liang-Chung
Hsia;
Julie
Tzu-Yueh
Wang
2871-2887
Abstract: We generalize the ABC theorems to the function field of a variety over an algebraically closed field of arbitrary characteristic which is non-singular in codimension one. We also obtain an upper bound for the minimal order sequence of Wronskians over such function fields of positive characteristic.
A separable Brown-Douglas-Fillmore theorem and weak stability
Huaxin
Lin
2889-2925
Abstract: We give a separable Brown-Douglas-Fillmore theorem. Let $A$ be a separable amenable $C^*$-algebra which satisfies the approximate UCT, $B$ be a unital separable amenable purely infinite simple $C^*$-algebra and $h_1, \, h_2: A\to B$ be two monomorphisms. We show that $h_1$ and $h_2$ are approximately unitarily equivalent if and only if $[h_1]=[h_2]\,\,\,\,{\rm in}\,\,\, KL(A,B).$ We prove that, for any $\varepsilon>0$ and any finite subset $\mathcal{F}\subset A$, there exist $\delta>0$ and a finite subset $\mathcal{G}\subset A$ satisfying the following: for any amenable purely infinite simple $C^*$-algebra $B$ and for any contractive positive linear map $L: A\to B$ such that \begin{displaymath}\Vert L(ab)-L(a)L(b)\Vert<\delta\quad{and}\quad \Vert L(a)\Vert\ge (1/2)\Vert a\Vert \end{displaymath} for all $a\in \mathcal{G},$ there exists a homomorphism $h: A\to B$such that \begin{displaymath}\Vert h(a)-L(a)\Vert<\varepsilon\,\,\,\,\,{\rm for\,\,\,all}\,\,\, a\in \mathcal{F} \end{displaymath} provided, in addition, that $K_i(A)$ are finitely generated. We also show that every separable amenable simple $C^*$-algebra $A$ with finitely generated $K$-theory which is in the so-called bootstrap class is weakly stable with respect to the class of amenable purely infinite simple $C^*$-algebras. As an application, related to perturbations in the rotation $C^*$-algebras studied by U. Haagerup and M. Rørdam, we show that for any irrational number $\theta$ and any $\varepsilon>0$ there is $\delta>0$ such that in any unital amenable purely infinite simple $C^*$-algebra $B$ if \begin{displaymath}\Vert uv-e^{i\theta\pi}vu\Vert<\delta \end{displaymath} for a pair of unitaries, then there exists a pair of unitaries $u_1$ and $v_1$ in $B$ such that \begin{displaymath}u_1v_1=e^{i\theta\pi}v_1u_1,\,\,\,\,\,\Vert u_1-u\Vert<\varepsilon\quad\text{and} \quad\Vert v_1-v\Vert<\varepsilon. \end{displaymath}
Maps between non-commutative spaces
S.
Paul
Smith
2927-2944
Abstract: Let $J$ be a graded ideal in a not necessarily commutative graded $k$-algebra $A=A_0 \oplus A_1 \oplus \cdots$ in which $\dim_k A_i < \infty$ for all $i$. We show that the map $A \to A/J$ induces a closed immersion $i:\operatorname{Proj}_{nc} A/J \to \operatorname{Proj}_{nc}A$ between the non-commutative projective spaces with homogeneous coordinate rings $A$ and $A/J$. We also examine two other kinds of maps between non-commutative spaces. First, a homomorphism $\phi:A \to B$ between not necessarily commutative $\mathbb{N}$-graded rings induces an affine map $\operatorname{Proj}_{nc} B \supset U \to \operatorname{Proj}_{nc} A$from a non-empty open subspace $U \subset \operatorname{Proj}_{nc} B$. Second, if $A$ is a right noetherian connected graded algebra (not necessarily generated in degree one), and $A^{(n)}$ is a Veronese subalgebra of $A$, there is a map $\operatorname{Proj}_{nc} A \to \operatorname{Proj}_{nc} A^{(n)}$; we identify open subspaces on which this map is an isomorphism. Applying these general results when $A$ is (a quotient of) a weighted polynomial ring produces a non-commutative resolution of (a closed subscheme of) a weighted projective space.
Koszul homology and extremal properties of Gin and Lex
Aldo
Conca
2945-2961
Abstract: For every homogeneous ideal $I$ in a polynomial ring $R$ and for every $p\leq\dim R$ we consider the Koszul homology $H_i(p,R/I)$ with respect to a sequence of $p$ of generic linear forms. The Koszul-Betti number $\beta_{ijp}(R/I)$ is, by definition, the dimension of the degree $j$ part of $H_i(p,R/I)$. In characteristic $0$, we show that the Koszul-Betti numbers of any ideal $I$ are bounded above by those of the gin-revlex $\mathrm{Gin}(I)$ of $I$ and also by those of the Lex-segment $\mathrm{Lex}(I)$ of $I$. We show that $\beta_{ijp}(R/I)=\beta_{ijp}(R/\mathrm{Gin}(I))$ iff $I$ is componentwise linear and that and $\beta_{ijp}(R/I)=\beta_{ijp}(R/\mathrm{Lex}(I))$ iff $I$is Gotzmann. We also investigate the set $\mathrm{Gins}(I)$ of all the gin of $I$ and show that the Koszul-Betti numbers of any ideal in $\mathrm{Gins}(I)$are bounded below by those of the gin-revlex of $I$. On the other hand, we present examples showing that in general there is no $J$ is $\mathrm{Gins}(I)$such that the Koszul-Betti numbers of any ideal in $\mathrm{Gins}(I)$ are bounded above by those of $J$.
Hermitian metrics inducing the Poincaré metric, in the leaves of a singular holomorphic foliation by curves
A.
Lins
Neto;
J.
C. Canille
Martins
2963-2988
Abstract: In this paper we consider the problem of uniformization of the leaves of a holomorphic foliation by curves in a complex manifold $M$. We consider the following problems: 1. When is the uniformization function $\lambda _{g}$, with respect to some metric $g$, continuous? It is known that the metric $\frac{g}{4\lambda _{g}}$ induces the Poincaré metric on the leaves. 2. When is the metric $\frac{g}{4\lambda _{g}}$ complete? We extend the concept of ultra-hyperbolic metric, introduced by Ahlfors in 1938, for singular foliations by curves, and we prove that if there exists a complete ultra-hyperbolic metric $g$, then $\lambda _{g}$ is continuous and $\frac{g}{4\lambda _{g}}$ is complete. In some local cases we construct such metrics, including the saddle-node (Theorem 1) and singularities given by vector fields with the first non-zero jet isolated (Theorem 2). We also give an example where for any metric $g$, $\frac{g}{4\,\lambda _{g}}$ is not complete (§3.2).
Sur les transformées de Riesz dans le cas du Laplacien avec drift
Noël
Lohoué;
Sami
Mustapha
2139-2147
Abstract: We prove $L^p$ estimates for Riesz transforms with drift.
Hardy inequalities with optimal constants and remainder terms
Filippo
Gazzola;
Hans-Christoph
Grunau;
Enzo
Mitidieri
2149-2168
Abstract: We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces $W_0^{1,p}$ and in higher-order Sobolev spaces on a bounded domain $\Omega\subset\mathbb{R} ^n$ can be refined by adding remainder terms which involve $L^p$ norms. In the higher-order case further $L^p$ norms with lower-order singular weights arise. The case $1<p<2$ being more involved requires a different technique and is developed only in the space $W_0^{1,p}$.
A unified approach to improved $L^p$ Hardy inequalities with best constants
G.
Barbatis;
S.
Filippas;
A.
Tertikas
2169-2196
Abstract: We present a unified approach to improved $L^p$ Hardy inequalities in $\mathbf{R}^N$. We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where the distance is taken from a surface of codimension $1<k<N$. In our main result, we add to the right hand side of the classical Hardy inequality a weighted $L^p$ norm with optimal weight and best constant. We also prove nonhomogeneous improved Hardy inequalities, where the right hand side involves weighted $L^q$ norms, $q \neq p$.
Luzin gaps
Ilijas
Farah
2197-2239
Abstract: We isolate a class of $F_{\sigma\delta}$ ideals on $\mathbb{N}$ that includes all analytic P-ideals and all $F_\sigma$ ideals, and introduce `Luzin gaps' in their quotients. A dichotomy for Luzin gaps allows us to freeze gaps, and prove some gap preservation results. Most importantly, under PFA all isomorphisms between quotient algebras over these ideals have continuous liftings. This gives a partial confirmation to the author's rigidity conjecture for quotients $\mathcal{P}(\mathbb{N} )/\mathcal{I}$. We also prove that the ideals $\operatorname{NWD}(\mathbb{Q} )$ and $\operatorname{NULL}(\mathbb{Q} )$have the Radon-Nikodým property, and (using OCA$_\infty$) a uniformization result for $\mathcal{K}$-coherent families of continuous partial functions.
Generic integral manifolds for weight two period domains
James
A.
Carlson;
Domingo
Toledo
2241-2249
Abstract: We define the notion of a generic integral element for the Griffiths distribution on a weight two period domain, draw the analogy with the classical contact distribution, and then show how to explicitly construct an infinite-dimensional family of integral manifolds tangent to a given element.
Maximum norms of random sums and transient pattern formation
Thomas
Wanner
2251-2279
Abstract: Many interesting and complicated patterns in the applied sciences are formed through transient pattern formation processes. In this paper we concentrate on the phenomenon of spinodal decomposition in metal alloys as described by the Cahn-Hilliard equation. This model depends on a small parameter, and one is generally interested in establishing sharp lower bounds on the amplitudes of the patterns as the parameter approaches zero. Recent results on spinodal decomposition have produced such lower bounds. Unfortunately, for higher-dimensional base domains these bounds are orders of magnitude smaller than what one would expect from simulations and experiments. The bounds exhibit a dependence on the dimension of the domain, which from a theoretical point of view seemed unavoidable, but which could not be observed in practice. In this paper we resolve this apparent paradox. By employing probabilistic methods, we can improve the lower bounds for certain domains and remove the dimension dependence. We thereby obtain optimal results which close the gap between analytical methods and numerical observations, and provide more insight into the nature of the decomposition process. We also indicate how our results can be adapted to other situations.
Parametrized $\diamondsuit$ principles
Justin
Tatch
Moore;
Michael
Hrusák;
Mirna
Dzamonja
2281-2306
Abstract: We will present a collection of guessing principles which have a similar relationship to $\diamondsuit$ as cardinal invariants of the continuum have to ${CH}$. The purpose is to provide a means for systematically analyzing $\diamondsuit$ and its consequences. It also provides for a unified approach for understanding the status of a number of consequences of ${CH}$ and $\diamondsuit$in models such as those of Laver, Miller, and Sacks.
On the adjunction mapping of very ample vector bundles of corank one
Antonio
Lanteri;
Marino
Palleschi;
Andrew
J.
Sommese
2307-2324
Abstract: Let $\mathcal{E}$ be a very ample vector bundle of rank $n-1$ over a smooth complex projective variety $X$ of dimension $n\geq 3$. The structure of $(X,\mathcal{E})$ being known when $\kappa (K_{X} + \det \mathcal{E}) \leq 0$, we investigate the structure of the adjunction mapping when $0 < \kappa (K_{X} + \det \mathcal{E}) < n$.
Units in some families of algebraic number fields
L.
Ya.
Vulakh
2325-2348
Abstract: Multi-dimensional continued fractions associated with $GL_n({\mathbf Z})$ are introduced and applied to find systems of fundamental units in some families of totally real fields and fields with signature (2,1).
Young wall realization of crystal bases for classical Lie algebras
Seok-Jin
Kang;
Jeong-Ah
Kim;
Hyeonmi
Lee;
Dong-Uy
Shin
2349-2378
Abstract: In this paper, we give a new realization of crystal bases for finite-dimensional irreducible modules over classical Lie algebras. The basis vectors are parameterized by certain Young walls lying between highest weight and lowest weight vectors.
Phase transitions in phylogeny
Elchanan
Mossel
2379-2404
Abstract: We apply the theory of Markov random fields on trees to derive a phase transition in the number of samples needed in order to reconstruct phylogenies. We consider the Cavender-Farris-Neyman model of evolution on trees, where all the inner nodes have degree at least $3$, and the net transition on each edge is bounded by $\epsilon$. Motivated by a conjecture by M. Steel, we show that if $2 (1 - 2 \epsilon)^2 > 1$, then for balanced trees, the topology of the underlying tree, having $n$ leaves, can be reconstructed from $O(\log n)$ samples (characters) at the leaves. On the other hand, we show that if $2 (1 - 2 \epsilon)^2 < 1$, then there exist topologies which require at least $n^{\Omega(1)}$ samples for reconstruction. Our results are the first rigorous results to establish the role of phase transitions for Markov random fields on trees, as studied in probability, statistical physics and information theory, for the study of phylogenies in mathematical biology.
Stable representatives for symmetric automorphisms of groups and the general form of the Scott conjecture
Mihalis
Sykiotis
2405-2441
Abstract: Let $G$ be a group acting on a tree $X$ such that all edge stabilizers are finite. We extend Bestvina-Handel's theory of train tracks for automorphisms of free groups to automorphisms of $G$ which permute vertex stabilizers. Using this extension we show that there is an upper bound depending only on $G$ for the complexity of the graph of groups decomposition of the fixed subgroups of such automorphisms of $G$.
Dialgebra cohomology as a G-algebra
Anita
Majumdar;
Goutam
Mukherjee
2443-2457
Abstract: It is well known that the Hochschild cohomology $H^*(A,A)$ of an associative algebra $A$ admits a G-algebra structure. In this paper we show that the dialgebra cohomology $HY^*(D,D)$ of an associative dialgebra $D$ has a similar structure, which is induced from a homotopy G-algebra structure on the dialgebra cochain complex $CY^*(D,D)$.
Norms of linear-fractional composition operators
P.
S.
Bourdon;
E.
E.
Fry;
C.
Hammond;
C.
H.
Spofford
2459-2480
Abstract: We obtain a representation for the norm of the composition operator $C_\phi$ on the Hardy space $H^2$ whenever $\phi$ is a linear-fractional mapping of the form $\phi(z) = b/(cz +d)$. The representation shows that, for such mappings $\phi$, the norm of $C_\phi$ always exceeds the essential norm of $C_\phi$. Moreover, it shows that a formula obtained by Cowen for the norms of composition operators induced by mappings of the form $\phi(z) = sz +t$ has no natural generalization that would yield the norms of all linear-fractional composition operators. For rational numbers $s$ and $t$, Cowen's formula yields an algebraic number as the norm; we show, e.g., that the norm of $C_{1/(2-z)}$ is a transcendental number. Our principal results are based on a process that allows us to associate with each non-compact linear-fractional composition operator $C_\phi$, for which $\Vert C_\phi\Vert> \Vert C_\phi\Vert _e$, an equation whose maximum (real) solution is $\Vert C_\phi\Vert^2$. Our work answers a number of questions in the literature; for example, we settle an issue raised by Cowen and MacCluer concerning co-hyponormality of a certain family of composition operators.
Presentations of noneffective orbifolds
Andre
Henriques;
David
S.
Metzler
2481-2499
Abstract: It is well known that an effective orbifold $M$ (one for which the local stabilizer groups act effectively) can be presented as a quotient of a smooth manifold $P$ by a locally free action of a compact Lie group $K$. We use the language of groupoids to provide a partial answer to the question of whether a noneffective orbifold can be so presented. We also note some connections to stacks and gerbes.
The length of harmonic forms on a compact Riemannian manifold
Paul-Andi
Nagy;
Constantin
Vernicos
2501-2513
Abstract: We study $(n+1)$-dimensional Riemannian manifolds with harmonic forms of constant length and first Betti number equal to $n$ showing that they are $2$-step nilmanifolds with some special metrics. We also characterize, in terms of properties on the product of harmonic forms, the left-invariant metrics among them. This allows us to clarify the case of equality in the stable isosytolic inequalities in that setting. We also discuss other values of the Betti number.
Automorphisms of subfactors from commuting squares
Anne
Louise
Svendsen
2515-2543
Abstract: We study an infinite series of irreducible, hyperfinite subfactors, which are obtained from an initial commuting square by iterating Jones' basic construction. They were constructed by Haagerup and Schou and have $A_{\infty}$as principal graphs, which means that their standard invariant is ``trivial''. We use certain symmetries of the initial commuting squares to construct explicitly non-trivial outer automorphisms of these subfactors. These automorphisms capture information about the subfactors which is not contained in the standard invariant.
Poincaré's closed geodesic on a convex surface
Wilhelm
P. A.
Klingenberg
2545-2556
Abstract: We present a new proof for the existence of a simple closed geodesic on a convex surface $M$. This result is due originally to Poincaré. The proof uses the ${2k}$-dimensional Riemannian manifold ${_{k}\Lambda M} = \hbox {(briefly)} \Lambda$ of piecewise geodesic closed curves on $M$ with a fixed number $k$ of corners, $k$ chosen sufficiently large. In $\Lambda$ we consider a submanifold $\overset{\approx }{\Lambda }_{0}$ formed by those elements of $\Lambda$ which are simple regular and divide $M$ into two parts of equal total curvature $2\pi$. The main burden of the proof is to show that the energy integral $E$, restricted to $\overset{\approx }{\Lambda }_{0}$, assumes its infimum. At the end we give some indications of how our methods yield a new proof also for the existence of three simple closed geodesics on $M$.
Coordinates in two variables over a $\mathbb{Q}$-algebra
Arno
van den Essen;
Peter
van Rossum
1691-1703
Abstract: This paper studies coordinates in two variables over a $\mathbb{Q}$-algebra. It gives several ways to characterize such coordinates. Also, various results about coordinates in two variables that were previously only known for fields, are extended to arbitrary $\mathbb{Q}$-algebras.
Root vectors for geometrically simple two-parameter eigenvalues
Paul
Binding;
Tomaz
Kosir
1705-1726
Abstract: A class of two-parameter eigenvalue problems involving generally nonselfadjoint and unbounded operators is studied. A basis for the root subspace at a geometrically simple eigenvalue of Fredholm type is computed in terms of the underlying two-parameter system. Comparison with Faierman's work on two-parameter boundary value problems of Sturm-Liouville type is given as an application.
Commuting Toeplitz operators on the polydisk
Boo
Rim
Choe;
Hyungwoon
Koo;
Young
Joo
Lee
1727-1749
Abstract: We obtain characterizations of (essentially) commuting Toeplitz operators with pluriharmonic symbols on the Bergman space of the polydisk. We show that commuting and essential commuting properties are the same for dimensions bigger than 2, while they are not for dimensions less than or equal to 2. Also, the corresponding results for semi-commutators are obtained.
Newton polyhedra, unstable faces and the poles of Igusa's local zeta function
Kathleen
Hoornaert
1751-1779
Abstract: In this paper we examine when the order of a pole of Igusa's local zeta function associated to a polynomial $f$ is smaller than ``expected''. We carry out this study in the case that $f$ is sufficiently non-degenerate with respect to its Newton polyhedron $\Gamma(f)$, and the main result of this paper is a proof of one of the conjectures of Denef and Sargos. Our technique consists in reducing our question about the polynomial $f$ to the same question about polynomials $f_\mu$, where $\mu$ are faces of $\Gamma(f)$ depending on the examined pole and $f_\mu$ is obtained from $f$ by throwing away all monomials of $f$ whose exponents do not belong to $\mu$. Secondly, we obtain a formula for Igusa's local zeta function associated to a polynomial $f_\mu$, with $\mu$ unstable, which shows that, in this case, the upperbound for the order of the examined pole is obviously smaller than ``expected''.
A nonstandard Riemann existence theorem
Rahim
Moosa
1781-1797
Abstract: We study elementary extensions of compact complex spaces and deduce that every complete type of dimension $1$ is internal to projective space. This amounts to a nonstandard version of the Riemann Existence Theorem, and answers a question posed by Anand Pillay.
Algebraic $\mathbb{Z}^d$-actions of entropy rank one
Manfred
Einsiedler;
Douglas
Lind
1799-1831
Abstract: We investigate algebraic $\mathbb Z^d$-actions of entropy rank one, namely those for which each element has finite entropy. Such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery. This leads to numerous alternative characterizations of entropy rank one, both geometric and algebraic. We then compute the measure entropy of a class of skew products, where the fiber maps are elements from an algebraic $\mathbb Z^d$-action of entropy rank one. This leads, via the relative variational principle, to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures. We use this to settle a conjecture concerning the relational entropy of commuting toral automorphisms.
Poincaré series of resolutions of surface singularities
Steven
Dale
Cutkosky;
Jürgen
Herzog;
Ana
Reguera
1833-1874
Abstract: Let $X\rightarrow\mathrm{spec}(R)$ be a resolution of singularities of a normal surface singularity $\mathrm{spec}(R)$, with integral exceptional divisors $E_1,\dotsc,E_r$. We consider the Poincaré series \begin{displaymath}g= \sum_{\underline{n}\in\mathbf{N}^r} h(\underline{n})t^{\underline{n}}, \end{displaymath} where \begin{displaymath}h(\underline{n})=\ell(R/\Gamma(X,\mathcal{O}_X(-n_1E-1-\cdots-n_rE_r)). \end{displaymath} We show that if $R/m$ has characteristic zero and $\mathrm{Pic}^0(X)$ is a semi-abelian variety, then the Poincaré series $g$ is rational. However, we give examples to show that this series can be irrational if either of these conditions fails.
Uniqueness of varieties of minimal degree containing a given scheme
M.
Casanellas
1875-1888
Abstract: We prove that if $X \subset \mathbb{P} ^N$ has dimension $k$ and it is $r$-Buchsbaum with $r>\max{(\operatorname{codim}{X}-k,0)}$, then $X$ is contained in at most one variety of minimal degree and dimension $k+1$.
Asymptotic behavior of the solutions of linear and quasilinear elliptic equations on $\mathbb{R}^{N}$
Patrick
J.
Rabier
1889-1907
Abstract: We investigate the relationship between the decay at infinity of the right-hand side $f$ and solutions $u$ of an equation $Lu=f$ when $L$ is a second order elliptic operator on $\mathbb{R} ^{N}.$ It is shown that when $L$is Fredholm, $u$ inherits the type of decay of $f$ (for instance, exponential, or power-like). In particular, the generalized eigenfunctions associated with all the Fredholm eigenvalues of $L,$ isolated or not, decay exponentially. No use is made of spectral theory. The result is next extended when $L$ is replaced by a Fredholm quasilinear operator. Various generalizations to other unbounded domains, higher order operators or elliptic systems are possible and briefly alluded to, but not discussed in detail.
Automorphic forms and differentiability properties
Fernando
Chamizo
1909-1935
Abstract: We consider Fourier series given by a type of fractional integral of automorphic forms, and we study their local and global properties, especially differentiability and fractal dimension of the graph of their real and imaginary parts. In this way we can construct fractal objects and continuous non-differentiable functions associated with elliptic curves and theta functions.
Simple Bratteli diagrams with a Gödel-incomplete C*-equivalence problem
Daniele
Mundici
1937-1955
Abstract: An abstract simplicial complex is a finite family of subsets of a finite set, closed under subsets. Every abstract simplicial complex $\mathcal{C}$ naturally determines a Bratteli diagram and a stable AF-algebra $A(\mathcal{C})$. Consider the following problem: INPUT: a pair of abstract simplicial complexes $\mathcal{C}$ and $A(\mathcal{C})$ isomorphic to
Semilattices of finitely generated ideals of exchange rings with finite stable rank
F.
Wehrung
1957-1970
Abstract: We find a distributive \ensuremath{(\vee,0,1)}-semilattice $S_{\omega_1}$ of size $\aleph_1$ that is not isomorphic to the maximal semilattice quotient of any Riesz monoid endowed with an order-unit of finite stable rank. We thus obtain solutions to various open problems in ring theory and in lattice theory. In particular: -- There is no exchange ring (thus, no von Neumann regular ring and no C*-algebra of real rank zero) with finite stable rank whose semilattice of finitely generated, idempotent-generated two-sided ideals is isomorphic to $S_{\omega_1}$. -- There is no locally finite, modular lattice whose semilattice of finitely generated congruences is isomorphic to $S_{\omega_1}$. These results are established by constructing an infinitary statement, denoted here by $\mathrm{URP_{sr}}$, that holds in the maximal semilattice quotient of every Riesz monoid endowed with an order-unit of finite stable rank, but not in the semilattice $S_{\omega_1}$.
On restrictions of modular spin representations of symmetric and alternating groups
Alexander
S.
Kleshchev;
Pham
Huu
Tiep
1971-1999
Abstract: Let $\mathbb F$ be an algebraically closed field of characteristic $p$ and $H$ be an almost simple group or a central extension of an almost simple group. An important problem in representation theory is to classify the subgroups $G$ of $H$ and $\mathbb F H$-modules $V$ such that the restriction $V{\downarrow}_G$ is irreducible. For example, this problem is a natural part of the program of describing maximal subgroups in finite classical groups. In this paper we investigate the case of the problem where $H$ is the Schur's double cover $\hat A_n$ or $\hat S_n$.
Varying the time-frequency lattice of Gabor frames
Hans
G.
Feichtinger;
Norbert
Kaiblinger
2001-2023
Abstract: A Gabor or Weyl-Heisenberg frame for $L^2(\mathbb{R}^d)$is generated by time-frequency shifts of a square-integrable function, the Gabor atom, along a time-frequency lattice. The dual frame is again a Gabor frame, generated by the dual atom. In general, Gabor frames are not stable under a perturbation of the lattice constants; that is, even for arbitrarily small changes of the parameters the frame property can be lost. In contrast, as a main result we show that this kind of stability does hold for Gabor frames generated by a Gabor atom from the modulation space $M^1(\mathbb{R}^d)$, which is a dense subspace of $L^2(\mathbb{R}^d)$. Moreover, in this case the dual atom depends continuously on the lattice constants. In fact, we prove these results for more general weighted modulation spaces. As a consequence, we obtain for Gabor atoms from the Schwartz class that the continuous dependence of the dual atom holds even in the Schwartz topology. Also, we complement these main results by corresponding statements for Gabor Riesz sequences and their biorthogonal system.
The periodic isoperimetric problem
Laurent
Hauswirth;
Joaquín
Pérez;
Pascal
Romon;
Antonio
Ros
2025-2047
Abstract: Given a discrete group $G$ of isometries of $\mathbb{R} ^3$, we study the $G$-isoperimetric problem, which consists of minimizing area (modulo $G$) among surfaces in $\mathbb{R} ^3$ which enclose a $G$-invariant region with a prescribed volume fraction. If $G$ is a line group, we prove that solutions are either families of round spheres or right cylinders. In the doubly periodic case we prove that for most rank two lattices, solutions must be spheres, cylinders or planes. For the remaining rank two lattices we show, among other results, an isoperimetric inequality in terms of the topology of the isoperimetric surfaces. Finally, we study the case where $G=Pm\overline{3}m$ (the group of symmetries of the integer rank three lattice $\mathbb{Z} ^3$) and other crystallographic groups of cubic type. We prove that isoperimetric solutions must be spheres if the prescribed volume fraction is less than $1/6$, and we give an isoperimetric inequality for $G$-invariant regions that, for instance, implies that the area (modulo $\mathbb{Z} ^3$) of a surface dividing the three space in two $G$-invariant regions with equal volume fractions, is at least $2.19$ (the conjectured solution is the classical $P$ Schwarz triply periodic minimal surface whose area is $\sim 2.34$). Another consequence of this isoperimetric inequality is that $Pm\overline{3}m$-symmetric surfaces (other than families of spheres) cannot be isoperimetric for the lattice group $\mathbb{Z} ^3$.
Spinors as automorphisms of the tangent bundle
Alexandru
Scorpan
2049-2066
Abstract: We show that, on a $4$-manifold $M$ endowed with a $\operatorname{spin}^{\mathbb{C} }$-structure induced by an almost-complex structure, a self-dual (positive) spinor field $\phi\in\Gamma(W^+)$ is the same as a bundle morphism $\phi:T_M\to T_M$ acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of $\phi$ on tangent vectors, and that the squaring map $\sigma:\mathcal{W}^+\to\Lambda^+$ acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kähler and symplectic structures.
Gromov translation algebras over discrete trees are exchange rings
P.
Ara;
K.
C.
O'Meara;
F.
Perera
2067-2079
Abstract: It is shown that the Gromov translation ring of a discrete tree over a von Neumann regular ring is an exchange ring. This provides a new source of exchange rings, including, for example, the algebras $G(0)$ of $\omega\times\omega$ matrices (over a field) of constant bandwidth. An extension of these ideas shows that for all real numbers $r$ in the unit interval $[0,1]$, the growth algebras $G(r)$(introduced by Hannah and O'Meara in 1993) are exchange rings. Consequently, over a countable field, countable-dimensional exchange algebras can take any prescribed bandwidth dimension $r$in $[0,1]$.
Positive laws in fixed points
Pavel
Shumyatsky
2081-2091
Abstract: Let $A$ be an elementary abelian group of order at least $q^3$ acting on a finite $q'$-group $G$in such a manner that $C_G(a)$ satisfies a positive law of degree $n$ for any $a\in A^\char93$. It is proved that the entire group $G$ satisfies a positive law of degree bounded by a function of $q$ and $n$ only.
Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems
Donatella
Donatelli;
Pierangelo
Marcati
2093-2121
Abstract: In this paper we investigate the diffusive zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form: \begin{displaymath}W_{t}(x,t) + \frac{1}{\varepsilon}A(x,D)W(x,t)= \frac{1}{\varepsilon ^2} B(x,W(x,t))+\frac{1}{\varepsilon} D(W(x,t))+E(W(x,t)).\end{displaymath} We analyze the singular convergence, as $\varepsilon \downarrow 0$, in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps: (i) We single out algebraic ``structure conditions'' on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories. (ii) We deduce ``energy estimates '', uniformly in $\varepsilon$, by assuming the existence of a symmetrizer having the so-called block structure and by assuming ``dissipativity conditions'' on $B$. (iii) We assume a Kawashima type condition and perform the convergence analysis by using generalizations of compensated compactness due to Tartar and Gérard. Finally, we include examples that show how to use our theory to approximate any quasilinear parabolic systems, satisfying the Petrowski parabolicity condition, or general reaction diffusion systems, including Chemotaxis and Brusselator type systems.
The steepest point of the boundary layers of singularly perturbed semilinear elliptic problems
T.
Shibata
2123-2135
Abstract: We consider the nonlinear singularly perturbed problem \begin{displaymath}-\epsilon^2\Delta u = f(u), \enskip u > 0 \quad \mbox{in} \enskip \Omega, u = 0 \quad \mbox{on} \enskip \partial\Omega, \end{displaymath} where $\Omega \subset {\mathbf{R}}^N$ ($N \ge 2$) is an appropriately smooth bounded domain and $\epsilon > 0$ is a small parameter. It is known that under some conditions on $f$, the solution $u_\epsilon$ corresponding to $\epsilon$ develops boundary layers when $\epsilon \to 0$. We determine the steepest point of the boundary layers on the boundary by establishing an asymptotic formula for the slope of the boundary layers with exact second term.
Erratum to ``The central limit problem for convex bodies''
Milla
Anttila;
Keith
Ball;
Irini
Perissinaki
2137-2137
Splitting criteria for homotopy functors of spectra
Phichet
Chaoha
1271-1280
Abstract: We explore the interaction between the Taylor tower and cotower, as defined in deriving calculus with cotriples and dual calculus for functors to spectra of functors of spectra. This leads to new splitting criteria which generalize the results in dual calculus for functors to spectra.
Random gaps under CH
James
Hirschorn
1281-1290
Abstract: It is proved that if the Continuum Hypothesis is true, then one random real always produces a destructible $(\omega_1,\omega_1)$ gap.
Involutions fixing ${\mathbb{RP}}^{\text{odd}} \sqcup P(h,i)$, II
Zhi
Lü
1291-1314
Abstract: This paper studies the equivariant cobordism classification of all involutions fixing a disjoint union of an odd-dimensional real projective space ${\mathbb{RP}}^j$ with its normal bundle nonbounding and a Dold manifold $P(h,i)$ with $h$ a positive even and $i>0$. The complete analysis of the equivariant cobordism classes of such involutions is given except that the upper and lower bounds on the codimension of $P(h,i)$ may not be best possible. In particular, we find that there exist such involutions with nonstandard normal bundle to $P(h,i)$. Together with the results of part I of this title (Trans. Amer. Math. Soc. 354 (2002), 4539-4570), the argument for involutions fixing ${\mathbb{RP}}^{\text{odd}}\sqcup P(h,i)$ is finished.
The geometry of analytic varieties satisfying the local Phragmén-Lindelöf condition and a geometric characterization of the partial differential operators that are surjective on $\mathcal{A}(\mathbb{R}^4)$
Rüdiger
W.
Braun;
Reinhold
Meise;
B.
A.
Taylor
1315-1383
Abstract: The local Phragmén-Lindelöf condition for analytic subvarieties of $\mathbb{C} ^n$ at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators, as Hörmander has shown. Here, necessary geometric conditions for this Phragmén-Lindelöf condition are derived. They are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in $\mathbb{C} ^3$. The latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on $\mathbb{R} ^4$.
There are no unexpected tunnel number one knots of genus one
Martin
Scharlemann
1385-1442
Abstract: We show that the only knots that are tunnel number one and genus one are those that are already known: $2$-bridge knots obtained by plumbing together two unknotted annuli and the satellite examples classified by Eudave-Muñoz and by Morimoto and Sakuma. This confirms a conjecture first made by Goda and Teragaito.
Extension of CR-functions into weighted wedges through families of nonsmooth analytic discs
Dmitri
Zaitsev;
Giuseppe
Zampieri
1443-1462
Abstract: The goal of this paper is to develop a theory of nonsmooth analytic discs attached to domains with Lipschitz boundary in real submanifolds of $\mathbb{C} ^{n}$. We then apply this technique to establish a propagation principle for wedge extendibility of CR-functions on these domains along CR-curves and along boundaries of attached analytic discs. The technique from this paper has been also extensively used by the authors recently to obtain sharp results on wedge extension of CR-functions on wedges in prescribed directions extending results of BOGGESS-POLKING and EASTWOOD-GRAHAM.
LS-category of compact Hausdorff foliations
Hellen
Colman;
Steven
Hurder
1463-1487
Abstract: The transverse (saturated) Lusternik-Schnirelmann category of foliations, introduced by the first author, is an invariant of foliated homotopy type with values in $\{1,2, \ldots, \infty\}$. A foliation with all leaves compact and Hausdorff leaf space $M/\mathcal{F}$ is called compact Hausdorff. The transverse saturated category $\operatorname{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}M$ of a compact Hausdorff foliation is always finite. In this paper we study the transverse category of compact Hausdorff foliations. Our main result provides upper and lower bounds on the transverse category $\operatorname{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}(M)$ in terms of the geometry of $\mathcal{F}$ and the Epstein filtration of the exceptional set $\mathcal{E}$. The exceptional set is the closed saturated foliated space which is the union of the leaves with non-trivial holonomy. We prove that \begin{displaymath}\max \{\operatorname{cat}(M/{\mathcal{F}}), \operatorname{ca... ...me{cat}_{\mathbin{\cap{\mkern-9mu}\mid}\,\,}(\mathcal{E}) + q.\end{displaymath} We give examples to show that both the upper and lower bounds are realized, so the estimate is sharp. We also construct a family of examples for which the transverse category for a compact Hausdorff foliation can be arbitrarily large, though the category of the leaf spaces is constant.
Analytic $p$-adic cell decomposition and integrals
Raf
Cluckers
1489-1499
Abstract: We prove a conjecture of Denef on parameterized $p$-adic analytic integrals using an analytic cell decomposition theorem, which we also prove in this paper. This cell decomposition theorem describes piecewise the valuation of analytic functions (and more generally of subanalytic functions), the pieces being geometrically simple sets, called cells. We also classify subanalytic sets up to subanalytic bijection.
Stability of parabolic Harnack inequalities
Martin
T.
Barlow;
Richard
F.
Bass
1501-1533
Abstract: Let $(G,E)$ be a graph with weights $\{a_{xy}\}$ for which a parabolic Harnack inequality holds with space-time scaling exponent $\beta \ge 2$. Suppose $\{a_{xy}\}$. We prove that this parabolic Harnack inequality also holds for $(G,E)$ with the weights
Modular Shimura varieties and forgetful maps
Victor
Rotger
1535-1550
Abstract: In this note we consider several maps that occur naturally between modular Shimura varieties, Hilbert-Blumenthal varieties and the moduli spaces of polarized abelian varieties when forgetting certain endomorphism structures. We prove that, up to birational equivalences, these forgetful maps coincide with the natural projection by suitable abelian groups of Atkin-Lehner involutions.
Standard noncommuting and commuting dilations of commuting tuples
B.
V. Rajarama
Bhat;
Tirthankar
Bhattacharyya;
Santanu
Dey
1551-1568
Abstract: We introduce a notion called `maximal commuting piece' for tuples of Hilbert space operators. Given a commuting tuple of operators forming a row contraction, there are two commonly used dilations in multivariable operator theory. First there is the minimal isometric dilation consisting of isometries with orthogonal ranges, and hence it is a noncommuting tuple. There is also a commuting dilation related with a standard commuting tuple on boson Fock space. We show that this commuting dilation is the maximal commuting piece of the minimal isometric dilation. We use this result to classify all representations of the Cuntz algebra $\mathcal{O}_n$ coming from dilations of commuting tuples.
Cohomology operations for Lie algebras
Grant
Cairns;
Barry
Jessup
1569-1583
Abstract: If $L$ is a Lie algebra over $\mathbb{R}$ and $Z$its centre, the natural inclusion $Z\hookrightarrow (L^{*})^{*}$ extends to a representation $i^{*}\,:\,\Lambda Z\to \operatorname{End} H^{*}(L,\mathbb{R})$ of the exterior algebra of $Z$ in the cohomology of $L$. We begin a study of this representation by examining its Poincaré duality properties, its associated higher cohomology operations and its relevance to the toral rank conjecture. In particular, by using harmonic forms we show that the higher operations presented by Goresky, Kottwitz and MacPherson (1998) form a subalgebra of $\operatorname{End} H^{*}(L,\mathbb{R})$, and that they can be assembled to yield an explicit Hirsch-Brown model for the Borel construction associated to $0\to Z\to L\to L/Z\to 0$.
Copolarity of isometric actions
Claudio
Gorodski;
Carlos
Olmos;
Ruy
Tojeiro
1585-1608
Abstract: We introduce a new integral invariant for isometric actions of compact Lie groups, the copolarity. Roughly speaking, it measures how far from being polar the action is. We generalize some results about polar actions in this context. In particular, we develop some of the structural theory of copolarity $k$ representations, we classify the irreducible representations of copolarity one, and we relate the copolarity of an isometric action to the concept of variational completeness in the sense of Bott and Samelson.
Lattice invariants and the center of the generic division ring
Esther
Beneish
1609-1622
Abstract: Let $G$ be a finite group, let $M$ be a $ZG$-lattice, and let $F$ be a field of characteristic zero containing primitive $p^{{th}}$ roots of 1. Let $F(M)$ be the quotient field of the group algebra of the abelian group $M$. It is well known that if $M$ is quasi-permutation and $G$-faithful, then $F(M)^G$ is stably equivalent to $F(ZG)^G$. Let $C_n$ be the center of the division ring of $n\times n$ generic matrices over $F$. Let $S_n$ be the symmetric group on $n$symbols. Let $p$ be a prime. We show that there exist a split group extension $G'$of $S_p$ by a $p$-elementary group, a $G'$-faithful quasi-permutation $ZG'$-lattice $M$, and a one-cocycle $\alpha$ in
Overpartitions
Sylvie
Corteel;
Jeremy
Lovejoy
1623-1635
Abstract: We discuss a generalization of partitions, called overpartitions, which have proven useful in several combinatorial studies of basic hypergeometric series. After showing how a number of finite products occurring in $q$-series have natural interpretations in terms of overpartitions, we present an introduction to their rich structure as revealed by $q$-series identities.
Limit theorems for partially hyperbolic systems
Dmitry
Dolgopyat
1637-1689
Abstract: We consider a large class of partially hyperbolic systems containing, among others, affine maps, frame flows on negatively curved manifolds, and mostly contracting diffeomorphisms. If the rate of mixing is sufficiently high, the system satisfies many classical limit theorems of probability theory.
On the Weyl tensor of a self-dual complex 4-manifold
Florin
Alexandru
Belgun
853-880
Abstract: We study complex 4-manifolds with holomorphic self-dual conformal structures, and we obtain an interpretation of the Weyl tensor of such a manifold as the projective curvature of a field of cones on the ambitwistor space. In particular, its vanishing is implied by the existence of some compact, simply-connected, null-geodesics. We also show that the projective structure of the $\beta$-surfaces of a self-dual manifold is flat. All these results are illustrated in detail in the case of the complexification of $\mathbb{CP} ^2$.
Hilbert spaces of Dirichlet series and their multipliers
John
E.
McCarthy
881-893
Abstract: We consider various Hilbert spaces of Dirichlet series whose norms are given by weighted $\ell^2$ norms of the Dirichlet coefficients. We describe the multiplier algebras of these spaces. The functions in the multiplier algebra may or may not extend to be analytic on a larger half-plane than the functions in the Hilbert space.
Symplectic semifield planes and ${\mathbb Z}_4$--linear codes
William
M.
Kantor;
Michael
E.
Williams
895-938
Abstract: There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and $\mathbb{Z}_4$-linear Kerdock and Preparata codes on the other. These inter-relationships lead to the construction of large numbers of objects of each type. In the geometric context we construct and study large numbers of nonisomorphic affine planes coordinatized by semifields; or, equivalently, large numbers of non-isotopic semifields: their numbers are not bounded above by any polynomial in the order of the plane. In the coding theory context we construct and study large numbers of $\mathbb{Z}_4$-linear Kerdock and Preparata codes. All of these are obtained using large numbers of orthogonal spreads of orthogonal spaces of maximal Witt index over finite fields of characteristic 2. We also obtain large numbers of ``boring'' affine planes in the sense that the full collineation group fixes the line at infinity pointwise, as well as large numbers of Kerdock codes ``boring'' in the sense that each has as small an automorphism group as possible. The connection with affine planes is a crucial tool used to prove inequivalence theorems concerning the orthogonal spreads and associated codes, and also to determine their full automorphism groups.
Low-degree points on Hurwitz-Klein curves
Pavlos
Tzermias
939-951
Abstract: We investigate low-degree points on the Fermat curve of degree 13, the Snyder quintic curve and the Klein quartic curve. We compute all quadratic points on these curves and use Coleman's effective Chabauty method to obtain bounds for the number of cubic points on each of the former two curves.
Analytic order of singular and critical points
Eugenii
Shustin
953-985
Abstract: We deal with the following closely related problems: (i) For a germ of a reduced plane analytic curve, what is the minimal degree of an algebraic curve with a singular point analytically equivalent (isomorphic) to the given one? (ii) For a germ of a holomorphic function in two variables with an isolated critical point, what is the minimal degree of a polynomial, equivalent to the given function up to a local holomorphic coordinate change? Classically known estimates for such a degree $d$ in these questions are $\sqrt{\mu}+1\le d\le \mu+1$, where $\mu$ is the Milnor number. Our result in both the problems is $d\le a\sqrt{\mu}$ with an absolute constant $a$. As a corollary, we obtain asymptotically proper sufficient conditions for the existence of algebraic curves with prescribed singularities on smooth algebraic surfaces.
Smooth solutions to a class of free boundary parabolic problems
Olivier
Baconneau;
Alessandra
Lunardi
987-1005
Abstract: We establish existence, uniqueness, and regularity results for solutions to a class of free boundary parabolic problems, including the free boundary heat equation which arises in the so-called ``focusing problem'' in the mathematical theory of combustion. Such solutions are proved to be smooth with respect to time for positive $t$, if the data are smooth.
The 2-twist-spun trefoil has the triple point number four
Shin
Satoh;
Akiko
Shima
1007-1024
Abstract: The triple point number of an embedded surface in 4-space is the minimal number of the triple points on all the projection images into 3-space. We show that the 2-twist-spun trefoil has the triple point number four.
Extensions contained in ideals
Dan
Kucerovsky
1025-1043
Abstract: We prove a Weyl-von Neumann type absorption theorem for extensions which are not full, and give a condition for constructing infinite repeats contained in an ideal. We also clear up some questions associated with the purely large criterion for full extensions to be absorbing.
Semilinear parabolic equations involving measures and low regularity data
H.
Amann;
P.
Quittner
1045-1119
Abstract: A detailed study of abstract semilinear evolution equations of the form $\dot u+Au=\mu(u)$ is undertaken, where $-A$ generates an analytic semigroup and $\mu(u)$ is a Banach space valued measure depending on the solution. Then it is shown that the general theorems apply to a variety of semilinear parabolic boundary value problems involving measures in the interior and on the boundary of the domain. These results extend far beyond the known results in this field. A particularly new feature is the fact that the measures may depend nonlinearly and possibly nonlocally on the solution.
A Capelli Harish-Chandra homomorphism
Tomasz
Przebinda
1121-1154
Abstract: For a real reductive dual pair the Capelli identities define a homomorphism $\mathcal{C}$ from the center of the universal enveloping algebra of the larger group to the center of the universal enveloping algebra of the smaller group. In terms of the Harish-Chandra isomorphism, this map involves a $\rho$-shift. We view a dual pair as a Lie supergroup and offer a construction of the homomorphism $\mathcal{C}$ based solely on the Harish-Chandra's radial component maps. Thus we provide a geometric interpretation of the $\rho$-shift.
Character degree graphs and normal subgroups
I.
M.
Isaacs
1155-1183
Abstract: We consider the degrees of those irreducible characters of a group $G$whose kernels do not contain a given normal subgroup $N$. We show that if
Complete hyperelliptic integrals of the first kind and their non-oscillation
Lubomir
Gavrilov;
Iliya
D.
Iliev
1185-1207
Abstract: Let $P(x)$ be a real polynomial of degree $2g+1$, $H=y^2+P(x)$ and $\delta(h)$ be an oval contained in the level set $\{H=h\}$. We study complete Abelian integrals of the form \begin{displaymath}I(h)=\int_{\delta(h)} \frac{(\alpha_0+\alpha_1 x+\ldots + \alpha_{g-1}x^{g-1})dx}{y}, \;\;h\in \Sigma, \end{displaymath} where $\alpha_i$ are real and $\Sigma\subset \mathbb{R}$ is a maximal open interval on which a continuous family of ovals $\{\delta(h)\}$ exists. We show that the $g$-dimensional real vector space of these integrals is not Chebyshev in general: for any $g>1$, there are hyperelliptic Hamiltonians $H$ and continuous families of ovals $\delta(h)\subset\{H=h\}$, $h\in\Sigma$, such that the Abelian integral $I(h)$ can have at least $[\frac32g]-1$ zeros in $\Sigma$. Our main result is Theorem 1 in which we show that when $g=2$, exceptional families of ovals $\{\delta(h)\}$ exist, such that the corresponding vector space is still Chebyshev.
How to do a $p$-descent on an elliptic curve
Edward
F.
Schaefer;
Michael
Stoll
1209-1231
Abstract: In this paper, we describe an algorithm that reduces the computation of the (full) $p$-Selmer group of an elliptic curve $E$ over a number field to standard number field computations such as determining the ($p$-torsion of) the $S$-class group and a basis of the $S$-units modulo $p$th powers for a suitable set $S$ of primes. In particular, we give a result reducing this set $S$ of `bad primes' to a very small set, which in many cases only contains the primes above $p$. As of today, this provides a feasible algorithm for performing a full $3$-descent on an elliptic curve over $\mathbb Q$, but the range of our algorithm will certainly be enlarged by future improvements in computational algebraic number theory. When the Galois module structure of $E[p]$ is favorable, simplifications are possible and $p$-descents for larger $p$ are accessible even today. To demonstrate how the method works, several worked examples are included.
Universal covers for Hausdorff limits of noncompact spaces
Christina
Sormani;
Guofang
Wei
1233-1270
Abstract: We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of complete manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature, then $Y$ has a universal cover.
Asymptotic relations among Fourier coefficients of automorphic eigenfunctions
Scott
A.
Wolpert
427-456
Abstract: A detailed stationary phase analysis is presented for noncompact parameter ranges of the family of elementary eigenfunctions on the hyperbolic plane $\mathcal{K}(z)=y^{1/2}K_{ir}(2\pi my)e^{2\pi im x}$, $z=x+iy$, $\lambda=\frac14+r^2$ the eigenvalue, $s=2\pi m\lambda^{-1/2}$ and $K_{ir}$ the Macdonald-Bessel function. The phase velocity of $\mathcal{K}$ on $\{\vert s\vert Im z\le1\}$ is a double-valued vector field, the tangent field to the pencil of geodesics $\mathcal{G}$ tangent to the horocycle $\{\vert s\vert Im z =1 \}$. For $A\in SL(2;\mathbb{R} )$ a multi-term stationary phase expansion is presented in $\lambda$ for $\mathcal{K}(Az)e^{2\pi in\,Re z}$ uniform in parameters. An application is made to find an asymptotic relation for the Fourier coefficients of a family of automorphic eigenfunctions. In particular, for $\psi$automorphic with coefficients $\{a_n\}$ and eigenvalue $\lambda$ it is shown for the special range $n\sim \lambda^{1/2}$ that $a_n$ is $O(\lambda^{1/4}\,e^{\pi\lambda^{1/2}/2})$ for $\lambda$ large, improving by an order of magnitude for this special range upon the estimate from the general Hecke bound $O(\vert n\vert^{1/2}\lambda^{1/4}\,e^{\pi\lambda^{1/2}/2})$. An exposition of the WKB asymptotics of the Macdonald-Bessel functions is presented.
Symmetries of flat rank two distributions and sub-Riemannian structures
Yuri
L.
Sachkov
457-494
Abstract: Flat sub-Riemannian structures are local approximations -- nilpotentizations -- of sub-Riemannian structures at regular points. Lie algebras of symmetries of flat maximal growth distributions and sub-Riemannian structures of rank two are computed in dimensions 3, 4, and 5.
A version of Gordon's theorem for multi-dimensional Schrödinger operators
David
Damanik
495-507
Abstract: We consider discrete Schrödinger operators $H = \Delta + V$ in $\ell^2(\mathbb{Z} ^d)$with $d \ge 1$, and study the eigenvalue problem for these operators. It is shown that the point spectrum is empty if the potential $V$ is sufficiently well approximated by periodic potentials. This criterion is applied to quasiperiodic $V$ and to so-called Fibonacci-type superlattices.
Simple birational extensions of the polynomial algebra $\mathbb{C}^{[3]}$
Shulim
Kaliman;
Stéphane
Vénéreau;
Mikhail
Zaidenberg
509-555
Abstract: The Abhyankar-Sathaye Problem asks whether any biregular embedding $\varphi:\mathbb{C}^k\hookrightarrow\mathbb{C}^n$ can be rectified, that is, whether there exists an automorphism $\alpha\in{\operatorname{Aut}}\,\mathbb{C}^n$ such that $\alpha\circ\varphi$ is a linear embedding. Here we study this problem for the embeddings $\varphi:\mathbb{C}^3\hookrightarrow \mathbb{C}^4$ whose image $X=\varphi(\mathbb{C}^3)$ is given in $\mathbb{C}^4$ by an equation $p=f(x,y)u+g(x,y,z)=0$, where $f\in\mathbb{C}[x,y]\backslash\{0\}$ and $g\in\mathbb{C}[x,y,z]$. Under certain additional assumptions we show that, indeed, the polynomial $p$ is a variable of the polynomial ring $\mathbb{C}^{[4]}=\mathbb{C}[x,y,z,u]$ (i.e., a coordinate of a polynomial automorphism of $\mathbb{C}^4$). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings $\mathbb{C}^2\hookrightarrow\mathbb{C}^3$. Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial $p$ as above, a criterion for when $X=p^{-1}(0)\simeq\mathbb{C}^3$.
Truncated second main theorem with moving targets
Min
Ru;
Julie
Tzu-Yueh
Wang
557-571
Abstract: We prove a truncated Second Main Theorem for holomorphic curves intersecting a finite set of moving or fixed hyperplanes. The set of hyperplanes is assumed to be non-degenerate. Previously only general position or subgeneral position was considered.
The two-by-two spectral Nevanlinna-Pick problem
Jim
Agler;
N.
J.
Young
573-585
Abstract: We give a criterion for the existence of an analytic $2 \times 2$matrix-valued function on the disc satisfying a finite set of interpolation conditions and having spectral radius bounded by $1$. We also give a realization theorem for analytic functions from the disc to the symmetrised bidisc.
Chern numbers of ample vector bundles on toric surfaces
Sandra
Di Rocco;
Andrew
J.
Sommese
587-598
Abstract: This article shows a number of strong inequalities that hold for the Chern numbers $c_1^2$, $c_2$ of any ample vector bundle $\mathcal{E}$ of rank $r$ on a smooth toric projective surface, $S$, whose topological Euler characteristic is $e(S)$. One general lower bound for $c_1^2$ proven in this article has leading term $(4r+2)e(S)\ln_2\left(\tfrac{e(S)}{12}\right)$. Using Bogomolov instability, strong lower bounds for $c_2$ are also given. Using the new inequalities, the exceptions to the lower bounds $c_1^2> 4e(S)$ and $c_2>e(S)$ are classified.
The distribution of prime ideals of imaginary quadratic fields
G.
Harman;
A.
Kumchev;
P.
A.
Lewis
599-620
Abstract: Let $Q(x, y)$ be a primitive positive definite quadratic form with integer coefficients. Then, for all $(s, t)\in \mathbb R^2$ there exist $(m, n) \in \mathbb Z^2$ such that $Q(m, n)$ is prime and \begin{displaymath}Q(m - s, n - t) \ll Q(s, t)^{0.53} + 1. \end{displaymath} This is deduced from another result giving an estimate for the number of prime ideals in an ideal class of an imaginary quadratic number field that fall in a given sector and whose norm lies in a short interval.
Examples of pleating varieties for twice punctured tori
Raquel
Díaz;
Caroline
Series
621-658
Abstract: We give an explicit description of some pleating varieties (sets with a fixed set of bending lines in the convex hull boundary) in the quasi-Fuchsian space of the twice punctured torus. In accordance with a conjecture of the second author, we show that their closures intersect Fuchsian space in the simplices of minima introduced by Kerckhoff. All computations are done using complex Fenchel-Nielsen coordinates for quasi-Fuchsian space referred to a maximal system of curves.
Variational principles for circle patterns and Koebe's theorem
Alexander
I.
Bobenko;
Boris
A.
Springborn
659-689
Abstract: We prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces. Our method is based on two new functionals--one for the Euclidean and one for the hyperbolic case. We show how Colin de Verdière's, Brägger's and Rivin's functionals can be derived from ours.
On a conjecture of Whittaker concerning uniformization of hyperelliptic curves
Ernesto
Girondo;
Gabino
González-Diez
691-702
Abstract: This article concerns an old conjecture due to E. T. Whittaker, aiming to describe the group uniformizing an arbitrary hyperelliptic Riemann surface $y^2=\prod_{i=1}^{2g+2}(x-a_i)$ as an index two subgroup of the monodromy group of an explicit second order linear differential equation with singularities at the values $a_i$. Whittaker and collaborators in the thirties, and R. Rankin some twenty years later, were able to prove the conjecture for several families of hyperelliptic surfaces, characterized by the fact that they admit a large group of symmetries. However, general results of the analytic theory of moduli of Riemann surfaces, developed later, imply that Whittaker's conjecture cannot be true in its full generality. Recently, numerical computations have shown that Whittaker's prediction is incorrect for random surfaces, and in fact it has been conjectured that it only holds for the known cases of surfaces with a large group of automorphisms. The main goal of this paper is to prove that having many automorphisms is not a necessary condition for a surface to satisfy Whittaker's conjecture.
Some Picard theorems for minimal surfaces
Francisco
J.
López
703-733
Abstract: This paper deals with the study of those closed subsets $F \subset \mathbb{R} ^3$ for which the following statement holds: If $S$ is a properly immersed minimal surface in $\mathbb{R} ^3$ of finite topology that is eventually disjoint from $F,$ then $S$ has finite total curvature. The same question is also considered when the conclusion is finite type or parabolicity.
Symmetrization, symmetric stable processes, and Riesz capacities
Dimitrios
Betsakos
735-755
Abstract: Let $\texttt{X}_t$ be a symmetric $\alpha$-stable process killed on exiting an open subset $D$ of $\mathbb R^n$. We prove a theorem that describes the behavior of its transition probabilities under polarization. We show that this result implies that the probability of hitting a given set $B$ in the complement of $D$ in the first exit moment from $D$ increases when $D$ and $B$ are polarized. It can also lead to symmetrization theorems for hitting probabilities, Green functions, and Riesz capacities. One such theorem is the following: Among all compact sets $K$ in $\mathbb R^n$with given volume, the balls have the least $\alpha$-capacity ( $0<\alpha<2$).
Deriving calculus with cotriples
B.
Johnson;
R.
McCarthy
757-803
Abstract: We construct a Taylor tower for functors from pointed categories to abelian categories via cotriples associated to cross effect functors. The tower was inspired by Goodwillie's Taylor tower for functors of spaces, and is related to Dold and Puppe's stable derived functors and Mac Lane's $Q$-construction. We study the layers, $D_{n}F=\text{\rm fiber}(P_{n}F\rightarrow P_{n-1}F)$, and the limit of the tower. For the latter we determine a condition on the cross effects that guarantees convergence. We define differentials for functors, and establish chain and product rules for them. We conclude by studying exponential functors in this setting and describing their Taylor towers.
The geometry of profinite graphs with applications to free groups and finite monoids
K.
Auinger;
B.
Steinberg
805-851
Abstract: We initiate the study of the class of profinite graphs $\Gamma$ defined by the following geometric property: for any two vertices $v$ and $w$ of $\Gamma$, there is a (unique) smallest connected profinite subgraph of $\Gamma$ containing them; such graphs are called tree-like. Profinite trees in the sense of Gildenhuys and Ribes are tree-like, but the converse is not true. A profinite group is then said to be dendral if it has a tree-like Cayley graph with respect to some generating set; a Bass-Serre type characterization of dendral groups is provided. Also, such groups (including free profinite groups) are shown to enjoy a certain small cancellation condition. We define a pseudovariety of groups $\mathbf{H}$ to be arboreous if all finitely generated free pro- $\mathbf{H}$ groups are dendral (with respect to a free generating set). Our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids. We prove, for arboreous pseudovarieties $\mathbf{H}$, a pro- $\mathbf{H}$ analog of the Ribes and Zalesski{\u{\i}}\kern.15em product theorem for the profinite topology on a free group. Also, arboreous pseudovarieties are characterized as precisely the solutions $\mathbf{H}$ to the much studied pseudovariety equation $\mathbf{J}\ast\mathbf{H}= \mathbf{J}\mathrel{{\mbox{\textcircled{\petite m}}}}\mathbf{H}$.
Cartan-decomposition subgroups of $\operatorname{SO}(2,n)$
Hee
Oh;
Dave
Witte
Morris
1-38
Abstract: For $G = \operatorname{SL} (3,\mathord{\mathbb{R} })$ and $G = \operatorname{SO}(2,n)$, we give explicit, practical conditions that determine whether or not a closed, connected subgroup $H$of $G$ has the property that there exists a compact subset $C$ of $G$with $CHC = G$. To do this, we fix a Cartan decomposition $G = K A^+ K$of $G$, and then carry out an approximate calculation of $(KHK) \cap A^+$for each closed, connected subgroup $H$ of $G$.
Cuntz-Krieger algebras of infinite graphs and matrices
Iain
Raeburn;
Wojciech
Szymanski
39-59
Abstract: We show that the Cuntz-Krieger algebras of infinite graphs and infinite $\{0,1\}$-matrices can be approximated by those of finite graphs. We then use these approximations to deduce the main uniqueness theorems for Cuntz-Krieger algebras and to compute their $K$-theory. Since the finite approximating graphs have sinks, we have to calculate the $K$-theory of Cuntz-Krieger algebras of graphs with sinks, and the direct methods we use to do this should be of independent interest.
Semi-linear homology $G$-spheres and their equivariant inertia groups
Zhi
Lü
61-71
Abstract: This paper introduces an abelian group $H\Theta_V^G$ for all semi-linear homology $G$-spheres, which corresponds to a known abelian group $\Theta_V^G$ for all semi-linear homotopy $G$-spheres, where $G$ is a compact Lie group and $V$ is a $G$-representation with $\dim V^G>0$. Then using equivariant surgery techniques, we study the relation between both $H\Theta_V^G$ and $\Theta_V^G$ when $G$ is finite. The main result is that under the conditions that $G$-action is semi-free and $\dim V-\dim V^G\geq 3$ with $\dim V^G >0$, the homomorphism $T: \Theta_V^G\longrightarrow H\Theta_V^G$defined by $T([\Sigma]_G)=\langle \Sigma\rangle_G$ is an isomorphism if $\dim V^G\not=3,4$, and a monomorphism if $\dim V^G=4$. This is an equivariant analog of a well-known result in differential topology. Such a result is also applied to the equivariant inertia groups of semi-linear homology $G$-spheres.
Compact covering mappings between Borel sets and the size of constructible reals
Gabriel
Debs;
Jean
Saint
Raymond
73-117
Abstract: We prove that the topological statement: ``Any compact covering mapping between two Borel sets is inductively perfect" is equivalent to the set-theoretical statement: $\lq\lq \,\forall\alpha\in \omega^\omega,\; \aleph_1^{L(\alpha)}<\aleph_1$".
Backward stability for polynomial maps with locally connected Julia sets
Alexander
Blokh;
Lex
Oversteegen
119-133
Abstract: We study topological dynamics on unshielded planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability. Then we deduce from this that a polynomial $f$ with a locally connected Julia set is backward stable outside any neighborhood of its attracting and neutral cycles. For a conformal measure $\mu$ this easily implies that one of the following holds: 1. for $\mu$-a.e. $x\in J(f)$, $\omega(x)=J(f)$; 2. for $\mu$-a.e. $x\in J(f)$, $\omega(x)=\omega(c(x))$ for a critical point $c(x)$depending on $x$.
Geometric aspects of Sturm-Liouville problems II. Space of boundary conditions for left-definiteness
Kevin
Haertzen;
Qingkai
Kong;
Hongyou
Wu;
Anton
Zettl
135-157
Abstract: For a given regular Sturm-Liouville equation with an indefinite weight function, we explicitly describe the space of left-definite selfadjoint boundary conditions. The description only uses one value of a fundamental solution of the matrix form of the equation. As a consequence we show that this space has the shape of a solid consisting of two cones sharing a common base.
Closed product formulas for extensions of generalized Verma modules
Riccardo
Biagioli
159-184
Abstract: We give explicit combinatorial product formulas for the polynomials encoding the dimensions of the spaces of extensions of $(g,p)$-generalized Verma modules, in the cases when $(g,p)$corresponds to an indecomposable classic Hermitian symmetric pair. The formulas imply that these dimensions are combinatorial invariants. We also discuss how these polynomials, defined by Shelton, are related to the parabolic $R$-polynomials introduced by Deodhar.
An index for gauge-invariant operators and the Dixmier-Douady invariant
Victor
Nistor;
Evgenij
Troitsky
185-218
Abstract: Let $\mathcal{G}\to B$ be a bundle of compact Lie groups acting on a fiber bundle $Y \to B$. In this paper we introduce and study gauge-equivariant $K$-theory groups $K_\mathcal{G}^i(Y)$. These groups satisfy the usual properties of the equivariant $K$-theory groups, but also some new phenomena arise due to the topological non-triviality of the bundle $\mathcal{G}\to B$. As an application, we define a gauge-equivariant index for a family of elliptic operators $(P_b)_{b \in B}$ invariant with respect to the action of $\mathcal{G}\to B$, which, in this approach, is an element of $K_\mathcal{G}^0(B)$. We then give another definition of the gauge-equivariant index as an element of $K_0(C^*(\mathcal{G}))$, the $K$-theory group of the Banach algebra $C^*(\mathcal{G})$. We prove that $K_0(C^*(\mathcal{G})) \simeq K^0_\mathcal{G}(\mathcal{G})$ and that the two definitions of the gauge-equivariant index are equivalent. The algebra $C^*(\mathcal{G})$ is the algebra of continuous sections of a certain field of $C^*$-algebras with non-trivial Dixmier-Douady invariant. The gauge-equivariant $K$-theory groups are thus examples of twisted $K$-theory groups, which have recently turned out to be useful in the study of Ramond-Ramond fields.
Vassiliev invariants for braids on surfaces
Juan
González-Meneses;
Luis
Paris
219-243
Abstract: We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit a universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the surface.
Ideals of the cohomology rings of Hilbert schemes and their applications
Wei-Ping
Li;
Zhenbo
Qin;
Weiqiang
Wang
245-265
Abstract: We study the ideals of the rational cohomology ring of the Hilbert scheme $X^{[n]}$ of $n$ points on a smooth projective surface $X$. As an application, for a large class of smooth quasi-projective surfaces $X$, we show that every cup product structure constant of $H^*(X^{[n]})$ is independent of $n$; moreover, we obtain two sets of ring generators for the cohomology ring $H^*(X^{[n]})$. Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between $H^*(X^{[n]}; \mathbb{C} )$ and $H^*_{\rm orb}(X^n/S_n; \mathbb{C} )$ for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.
Framings of knots satisfying differential relations
James
J.
Hebda;
Chichen
M.
Tsau
267-281
Abstract: This paper introduces the notion of a differential framing relation for knots in a three-dimensional manifold. There is a canonical map from the space of knots that satisfy a framing relation into the space of framed knots. Under reasonable assumptions this canonical map is a weak homotopy equivalence.
Embedded minimal disks: Proper versus nonproper---global versus local
Tobias
H.
Colding;
William
P.
Minicozzi II
283-289
Abstract: We construct a sequence of compact embedded minimal disks in a ball in $\mathbf{R}^3$ with boundaries in the boundary of the ball and where the curvatures blow up only at the center. The sequence converges to a limit which is not smooth and not proper. If instead the sequence of embedded disks had boundaries in a sequence of balls with radii tending to infinity, then we have shown previously that any limit must be smooth and proper.
Analysing finite locally $s$-arc transitive graphs
Michael
Giudici;
Cai
Heng
Li;
Cheryl
E.
Praeger
291-317
Abstract: We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms $G$ and are either locally $(G,s)$-arc transitive for $s \geq 2$ or $G$-locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of $G$. Given a normal subgroup $N$ which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of $N$ preserves both local primitivity and local $s$-arc transitivity and leads us to study graphs where $G$ acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for $G$ in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section.
Eigenvalue and gap estimates for the Laplacian acting on $p$-forms
Pierre
Guerini;
Alessandro
Savo
319-344
Abstract: We study the gap of the first eigenvalue of the Hodge Laplacian acting on $p$-differential forms of a manifold with boundary, for consecutive values of the degree $p$. We first show that the gap may assume any sign. Then we give sufficient conditions on the intrinsic and extrinsic geometry to control it. Finally, we estimate the first Hodge eigenvalue of manifolds whose boundaries have some degree of convexity.
Exponential sums on $\mathbf{A}^n$, II
Alan
Adolphson;
Steven
Sperber
345-369
Abstract: We prove a vanishing theorem for the $p$-adic cohomology of exponential sums on $\mathbf{A}^n$. In particular, we obtain new classes of exponential sums on $\mathbf{A}^n$ that have a single nonvanishing $p$-adic cohomology group. The dimension of this cohomology group equals a sum of Milnor numbers.
Slopes of vector bundles on projective curves and applications to tight closure problems
Holger
Brenner
371-392
Abstract: We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from below for the tight closure of a homogeneous $R_+$-primary ideal in a two-dimensional normal standard-graded algebra $R$ in terms of the minimal and the maximal slope of the sheaf of relations for some ideal generators. If moreover this sheaf of relations is semistable, then both degree estimates coincide and we get a vanishing type theorem.
Asymptotics of the transition probabilities of the simple random walk on self-similar graphs
Bernhard
Krön;
Elmar
Teufl
393-414
Abstract: It is shown explicitly how self-similar graphs can be obtained as `blow-up' constructions of finite cell graphs $\hat C$. This yields a larger family of graphs than the graphs obtained by discretising continuous self-similar fractals. For a class of symmetrically self-similar graphs we study the simple random walk on a cell graph $\hat C$, starting at a vertex $v$ of the boundary of $\hat C$. It is proved that the expected number of returns to $v$before hitting another vertex in the boundary coincides with the resistance scaling factor. Using techniques from complex rational iteration and singularity analysis for Green functions, we compute the asymptotic behaviour of the $n$-step transition probabilities of the simple random walk on the whole graph. The results of Grabner and Woess for the Sierpinski graph are generalised to the class of symmetrically self-similar graphs, and at the same time the error term of the asymptotic expression is improved. Finally, we present a criterion for the occurrence of oscillating phenomena of the $n$-step transition probabilities.
The classical problem of the calculus of variations in the autonomous case: Relaxation and Lipschitzianity of solutions
Arrigo
Cellina
415-426
Abstract: We consider the problem of minimizing \begin{displaymath}\int _{a}^{b} L(x(t),x^{\prime }(t)) \, dt, \qquad x(a)=A, x(b)=B.\end{displaymath} Under the assumption that the Lagrangian $L$is continuous and satisfies a growth assumption that does not imply superlinear growth, we provide a result on the relaxation of the functional and show that a solution to the minimum problem is Lipschitzian.
Causal compactification of compactly causal spaces
Frank
Betten
4699-4721
Abstract: We give a classification of causal compactifications of compactly causal spaces. Introduced by Ólafsson and Ørsted, for a compactly causal space $G/H$, these compactifications are given by $G$-orbits in the Bergman-Silov boundary of $G_1/K_1$, with $G \subset G_1$ and $(G_1, K_1, \theta)$ a Hermitian symmetric space of tube type. For the classical spaces an explicit construction is presented.
The central limit problem for convex bodies
Milla
Anttila;
Keith
Ball;
Irini
Perissinaki
4723-4735
Abstract: It is shown that every symmetric convex body which satisfies a kind of weak law of large numbers has the property that almost all its marginal distributions are approximately Gaussian. Several quite broad classes of bodies are shown to satisfy the condition.
A compactification of open varieties
Yi
Hu
4737-4753
Abstract: In this paper we prove a general method to compactify certain open varieties by adding normal crossing divisors. This is done by showing that blowing up along an arrangement of subvarieties can be carried out. Important examples such as Ulyanov's configuration spaces and complements of arrangements of linear subspaces in projective spaces, etc., are covered. Intersection ring and (nonrecursive) Hodge polynomials are computed. Furthermore, some general structures arising from the blowup process are also described and studied.
A Baire's category method for the Dirichlet problem of quasiregular mappings
Baisheng
Yan
4755-4765
Abstract: We adopt the idea of Baire's category method as presented in a series of papers by Dacorogna and Marcellini to study the boundary value problem for quasiregular mappings in space. Our main result is to prove that for any $\epsilon>0$ and any piece-wise affine map $\varphi\in W^{1,n}(\Omega;\mathbf{R}^n)$ with $\vert D\varphi(x)\vert^n\le L\det D\varphi(x)$ for almost every $x\in\Omega$ there exists a map $u\in W^{1,n}(\Omega;\mathbf{R}^n)$ such that \begin{displaymath}\begin{cases} \vert Du(x)\vert^n=L\det Du(x)\quad\text{a.e.} ... ...,\quad\Vert u-\varphi\Vert _{L^n(\Omega)}<\epsilon. \end{cases}\end{displaymath} The theorems of Dacorogna and Marcellini do not directly apply to our result since the involved sets are unbounded. Our proof is elementary and does not require any notion of polyconvexity, quasiconvexity or rank-one convexity in the vectorial calculus of variations, as required in the papers by the quoted authors.
Non-trivial quadratic approximations to zero of a family of cubic Pisot numbers
Peter
Borwein;
Kevin
G.
Hare
4767-4779
Abstract: This paper gives exact rates of quadratic approximations to an infinite class of cubic Pisot numbers. We show that for any cubic Pisot number $q$, with minimal polynomial $p$, such that $p(0) = -1$, and where $p$ has only one real root, then there exists a $C(q)$, explicitly given here, such that: (1) For all $\epsilon > 0$, all but finitely many integer quadratics $P$ satisfy \begin{displaymath}\vert P(q)\vert \geq \frac{C(q) - \epsilon}{H(P)^2}\end{displaymath} where $H$ is the height function. (2) For all $\epsilon > 0$ there exists a sequence of integer quadratics $P_n(q)$ such that \begin{displaymath}\vert P_n(q)\vert \leq \frac{C(q) + \epsilon}{H(P_n)^2}.\end{displaymath} Furthermore, $C(q) < 1$ for all $q$ in this class of cubic Pisot numbers. What is surprising about this result is how precise it is, giving exact upper and lower bounds for these approximations.
Uniqueness of the density in an inverse problem for isotropic elastodynamics
Lizabeth
V.
Rachele
4781-4806
Abstract: We consider the unique determination of the density of a nonhomogeneous, isotropic elastic object from measurements made at the surface. We model the behavior of the bounded, 3-dimensional object by the linear, hyperbolic system of operators for isotropic elastodynamics. The material properties of the object (its density and elastic properties) correspond to the smooth coefficients of these differential operators. The data for this inverse problem, in the form of the correspondence between applied surface tractions and resulting surface displacements, is modeled by the dynamic Dirichlet-to-Neumann map on a finite time interval. In an earlier paper we show that the speeds $c_{p/s}$ of (compressional and sheer) wave propagation through the object are uniquely determined by the Dirichlet-to-Neumann map. Here we extend that result by showing that the density is also determined in the interior by the Dirichlet-to-Neumann map in the case, for example, that $c_p = 2 c_s$ at only isolated points in the object. We use techniques from microlocal analysis and integral geometry to solve this fully three-dimensional problem.
A local characterization of simply-laced crystals
John
R.
Stembridge
4807-4823
Abstract: We provide a simple list of axioms that characterize the crystal graphs of integrable highest weight modules for simply-laced quantum Kac-Moody algebras.
A geometric characterization of Vassiliev invariants
Michael
Eisermann
4825-4846
Abstract: It is a well-known paradigm to consider Vassiliev invariants as polynomials on the set of knots. We prove the following characterization: a rational knot invariant is a Vassiliev invariant of degree $\le m$ if and only if it is a polynomial of degree $\le m$ on every geometric sequence of knots. Here a sequence $K_z$ with $z\in\mathbb{Z}$ is called geometric if the knots $K_z$ coincide outside a ball $B$, inside of which they satisfy $K_z \cap B = \tau^z$ for all $z$ and some pure braid $\tau$. As an application we show that the torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: there exist knots in $\mathbb{S} ^1\times\mathbb{S} ^2$that can be distinguished by $\mathbb{Z} {/}{2}$-invariants of finite type but not by rational invariants of finite type. In order to obtain such torsion invariants we construct over $\mathbb{Z}$ a universal Vassiliev invariant of degree $1$ for knots in $\mathbb{S} ^1\times\mathbb{S} ^2$.
Tight surfaces in three-dimensional compact Euclidean space forms
Marc-Oliver
Otto
4847-4863
Abstract: In this paper we define and discuss tight surfaces -- smooth or polyhedral -- in three-dimensional compact Euclidean space forms and prove existence and non-existence results. It will be shown that all orientable and most of the non-orientable surfaces can be tightly immersed in any of these space forms.
The limiting curve of Jarník's polygons
Greg
Martin
4865-4880
Abstract: In 1925, Jarník defined a sequence of convex polygons for use in constructing curves containing many lattice points relative to their curvatures. Properly scaled, these polygons converge to a certain limiting curve. In this paper we identify this limiting curve precisely, showing that it consists piecewise of arcs of parabolas, and we discuss the analogous problem for sequences of polygons arising from generalizations of Jarník's construction.
Stratified transversality by isotopy
C.
Murolo;
D.
J. A.
Trotman;
A.
A.
Du Plessis
4881-4900
Abstract: For $\mathcal{X}$ an abstract stratified set or a $(w)$-regular stratification, hence for any $(b)$-, $(c)$- or $(L)$-regular stratification, we prove that after stratified isotopy of $\mathcal{X}$, a stratified subspace $\mathcal{W}$ of $\mathcal{X}$, or a stratified map $h : \mathcal{Z} \to \mathcal{X}$, can be made transverse to a fixed stratified map $g: \mathcal{Y} \to \mathcal{X}$.
Families of nodal curves on projective threefolds and their regularity via postulation of nodes
Flaminio
Flamini
4901-4932
Abstract: The main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds. Precisely, given a smooth projective threefold $X$, a rank-two vector bundle $\mathcal{E}$ on $X$, and integers $k\geq 0$, $\delta >0$, denote by ${\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ the subscheme of ${\mathbb{P}}(H^0({\mathcal{E}}(k)))$ parametrizing global sections of ${\mathcal{E}}(k)$ whose zero-loci are irreducible $\delta$-nodal curves on $X$. We present a new cohomological description of the tangent space $T_{[s]}({\mathcal{V}}_{\delta} ({\mathcal{E}} (k)))$ at a point $[s]\in {\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$. This description enables us to determine effective and uniform upper bounds for $\delta$, which are linear polynomials in $k$, such that the family ${\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ is smooth and of the expected dimension (regular, for short). The almost sharpness of our bounds is shown by some interesting examples. Furthermore, when $X$ is assumed to be a Fano or a Calabi-Yau threefold, we study in detail the regularity property of a point $[s] \in {\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$related to the postulation of the nodes of its zero-locus $C = V(s) \subset X$. Roughly speaking, when the nodes of $C$ are assumed to be in general position either on $X$, or on an irreducible divisor of $X$ having at worst log-terminal singularities or to lie on a l.c.i. and subcanonical curve in $X$, we find upper bounds on $\delta$ which are, respectively, cubic, quadratic and linear polynomials in $k$ ensuring the regularity of ${\mathcal{V}}_{\delta} ({\mathcal{E}} (k))$ at $[s]$. Finally, when $X= \mathbb{P}^3$, we also discuss some interesting geometric properties of the curves given by sections parametrized by ${\mathcal{V}}_{\delta} ({\mathcal{E}} \otimes \mathcal{O}_X(k))$.
Spines and topology of thin Riemannian manifolds
Stephanie
B.
Alexander;
Richard
L.
Bishop
4933-4954
Abstract: Consider Riemannian manifolds $M$ for which the sectional curvature of $M$ and second fundamental form of the boundary $B$ are bounded above by one in absolute value. Previously we proved that if $M$ has sufficiently small inradius (i.e. all points are sufficiently close to the boundary), then the cut locus of $B$ exhibits canonical branching behavior of arbitrarily low branching number. In particular, if $M$is thin in the sense that its inradius is less than a certain universal constant (known to lie between $.108$ and $.203$), then $M$collapses to a triply branched simple polyhedral spine. We use a graphical representation of the stratification structure of such a collapse, and relate numerical invariants of the graph to topological invariants of $M$ when $B$ is simply connected. In particular, the number of connected strata of the cut locus is a topological invariant. When $M$ is $3$-dimensional and compact, $M$ has complexity $0$ in the sense of Matveev, and is a connected sum of $p$ copies of the real projective space $P^3$, $t$ copies chosen from the lens spaces $L(3,\pm1)$, and $\ell$ handles chosen from $S^2\times S^1$ or $S^2\tilde\times S^1$, with $\beta$ 3-balls removed, where $p+t+\ell +\beta \ge 2$. Moreover, we construct a thin metric for every graph, and hence for every homeomorphism type on the list.
On the equations defining toric l.c.i.-singularities
Dimitrios
I.
Dais;
Martin
Henk
4955-4984
Abstract: Based on Nakajima's Classification Theorem we describe the precise form of the binomial equations which determine toric locally complete intersection (``l.c.i.'') singularities.
A pair of difference differential equations of Euler-Cauchy type
David
M.
Bradley
4985-5002
Abstract: We study two classes of linear difference differential equations analogous to Euler-Cauchy ordinary differential equations, but in which multiple arguments are shifted forward or backward by fixed amounts. Special cases of these equations have arisen in diverse branches of number theory and combinatorics. They are also of use in linear control theory. Here, we study these equations in a general setting. Building on previous work going back to de Bruijn, we show how adjoint equations arise naturally in the problem of uniqueness of solutions. Exploiting the adjoint relationship in a new way leads to a significant strengthening of previous uniqueness results. Specifically, we prove here that the general Euler-Cauchy difference differential equation with advanced arguments has a unique solution (up to a multiplicative constant) in the class of functions bounded by an exponential function on the positive real line. For the closely related class of equations with retarded arguments, we focus on a corresponding class of solutions, locating and classifying the points of discontinuity. We also provide an explicit asymptotic expansion at infinity.
Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations
Rabi
N.
Bhattacharya;
Larry
Chen;
Scott
Dobson;
Ronald
B.
Guenther;
Chris
Orum;
Mina
Ossiander;
Enrique
Thomann;
Edward
C.
Waymire
5003-5040
Abstract: A general method is developed to obtain conditions on initial data and forcing terms for the global existence of unique regular solutions to incompressible 3d Navier-Stokes equations. The basic idea generalizes a probabilistic approach introduced by LeJan and Sznitman (1997) to obtain weak solutions whose Fourier transform may be represented by an expected value of a stochastic cascade. A functional analytic framework is also developed which partially connects stochastic iterations and certain Picard iterates. Some local existence and uniqueness results are also obtained by contractive mapping conditions on the Picard iteration.
Regularity of isoperimetric hypersurfaces in Riemannian manifolds
Frank
Morgan
5041-5052
Abstract: We add to the literature the well-known fact that an isoperimetric hypersurface $S$ of dimension at most six in a smooth Riemannian manifold $M$ is a smooth submanifold. If the metric is merely Lipschitz, then $S$ is still Hölder differentiable.
The free entropy dimension of hyperfinite von Neumann algebras
Kenley
Jung
5053-5089
Abstract: Suppose $M$ is a hyperfinite von Neumann algebra with a normal, tracial state $\varphi$ and $\{a_1,\ldots,a_n\}$ is a set of selfadjoint generators for $M$. We calculate $\delta_0(a_1,\ldots,a_n)$, the modified free entropy dimension of $\{a_1,\ldots,a_n\}$. Moreover, we show that $\delta_0(a_1,\ldots,a_n)$ depends only on $M$ and $\varphi$. Consequently, $\delta_0(a_1,\ldots,a_n)$ is independent of the choice of generators for $M$. In the course of the argument we show that if $\{b_1,\ldots,b_n\}$ is a set of selfadjoint generators for a von Neumann algebra $\mathcal R$ with a normal, tracial state and $\{b_1,\ldots,b_n\}$has finite-dimensional approximants, then $\delta_0(N) \leq \delta_0(b_1,\ldots,b_n)$ for any hyperfinite von Neumann subalgebra $N$of $\mathcal R.$ Combined with a result by Voiculescu, this implies that if $\mathcal R$ has a regular diffuse hyperfinite von Neumann subalgebra, then $\delta_0(b_1,\ldots,b_n)=1$.
Codimension growth and minimal superalgebras
A.
Giambruno;
M.
Zaicev
5091-5117
Abstract: A celebrated theorem of Kemer (1978) states that any algebra satisfying a polynomial identity over a field of characteristic zero is PI-equivalent to the Grassmann envelope $G(A)$ of a finite dimensional superalgebra $A$. In this paper, by exploiting the basic properties of the exponent of a PI-algebra proved by Giambruno and Zaicev (1999), we define and classify the minimal superalgebras of a given exponent over a field of characteristic zero. In particular we prove that these algebras can be realized as block-triangular matrix algebras over the base field. The importance of such algebras is readily proved: $A$ is a minimal superalgebra if and only if the ideal of identities of $G(A)$ is a product of verbally prime T-ideals. Also, such superalgebras allow us to classify all minimal varieties of a given exponent i.e., varieties $\mathcal{V}$ such that $\exp({\mathcal{V}})=d\ge 2$ and $\exp(\mathcal{U})<d$ for all proper subvarieties ${\mathcal{U}}$ of ${\mathcal{V}}$. This proves in the positive a conjecture of Drensky (1988). As a corollary we obtain that there is only a finite number of minimal varieties for any given exponent. A classification of minimal varieties of finite basic rank was proved by the authors (2003). As an application we give an effective way for computing the exponent of a T-ideal given by generators and we discuss the problem of what functions can appear as growth functions of varieties of algebras.
Erratum to ``A Berger-Green type inequality for compact Lorentzian manifolds"
Manuel
Gutiérrez;
Francisco
J.
Palomo;
Alfonso
Romero
5119-5120
On model complete differential fields
E.
Hrushovski;
M.
Itai
4267-4296
Abstract: We develop a geometric approach to definable sets in differentially closed fields, with emphasis on the question of orthogonality to a given strongly minimal set. Equivalently, within a family of ordinary differential equations, we consider those equations that can be transformed, by differential-algebraic transformations, so as to yield solutions of a given fixed first-order ODE $X$. We show that this sub-family is usually definable (in particular if $X$ lives on a curve of positive genus). As a corollary, we show the existence of many model-complete, superstable theories of differential fields.
On cubic lacunary Fourier series
Joseph
L.
Gerver
4297-4347
Abstract: For $2<\beta <4$, we analyze the behavior, near the rational points $x=p\pi /q$, of $\sum^\infty_{n=1}n^{-\beta }\exp (ixn^{3})$, considered as a function of $x$. We expand this series into a constant term, a term on the order of $(x-p\pi /q)^{(\beta -1)/3}$, a term linear in $x-p\pi /q$, a ``chirp" term on the order of $(x-p\pi /q)^{(2\beta -1)/4}$, and an error term on the order of $(x-p\pi /q)^{\beta /2}$. At every such rational point, the left and right derivatives are either both finite (and equal) or both infinite, in contrast with the quadratic series, where the derivative is often finite on one side and infinite on the other. However, in the cubic series, again in contrast with the quadratic case, the chirp term generally has a different set of frequencies and amplitudes on the right and left sides. Finally, we show that almost every irrational point can be closely approximated, in a suitable Diophantine sense, by rational points where the cubic series has an infinite derivative. This implies that when $\beta \le (\sqrt {97}-1)/4=2.212\dots$, both the real and imaginary parts of the cubic series are differentiable almost nowhere.
Rigidity in holomorphic and quasiregular dynamics
Gaven
J.
Martin;
Volker
Mayer
4349-4363
Abstract: We consider rigidity phenomena for holomorphic functions and then more generally for uniformly quasiregular maps.
Hyperplane arrangement cohomology and monomials in the exterior algebra
David
Eisenbud;
Sorin
Popescu;
Sergey
Yuzvinsky
4365-4383
Abstract: We show that if $X$ is the complement of a complex hyperplane arrangement, then the homology of $X$ has linear free resolution as a module over the exterior algebra on the first cohomology of $X$. We study invariants of $X$ that can be deduced from this resolution. A key ingredient is a result of Aramova, Avramov, and Herzog (2000) on resolutions of monomial ideals in the exterior algebra. We give a new conceptual proof of this result.
Group actions on one-manifolds, II: Extensions of Hölder's Theorem
Benson
Farb;
John
Franks
4385-4396
Abstract: This self-contained paper is part of a series seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. In this paper we consider groups which act on $\mathbf R$ with restrictions on the fixed point set of each element. One result is a topological characterization of affine groups in $\mathrm{Diff}^2(\mathbf R)$ as those groups whose elements have at most one fixed point.
Sheaf cohomology and free resolutions over exterior algebras
David
Eisenbud;
Gunnar
Fløystad;
Frank-Olaf
Schreyer
4397-4426
Abstract: We derive an explicit version of the Bernstein-Gel'fand-Gel'fand (BGG) correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free resolutions over its ``Koszul dual'' exterior algebra. Among the facts about the BGG correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the ``linear part'' of a resolution over the exterior algebra. We explore the structure of free resolutions over an exterior algebra. For example, we show that such resolutions are eventually dominated by their ``linear parts" in the sense that erasing all terms of degree $>1$ in the complex yields a new complex which is eventually exact. As applications we give a construction of the Beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms. The explicitness of our version allows us to prove two conjectures about the morphisms in the monad, and we get an efficient method for machine computation of the cohomology of sheaves. We also construct all the monads for a sheaf that can be built from sums of line bundles, and show that they are often characterized by numerical data.
Examples for the mod $p$ motivic cohomology of classifying spaces
Nobuaki
Yagita
4427-4450
Abstract: Let $BG$ be the classifying space of a compact Lie group $G$. Some examples of computations of the motivic cohomology $H^{*,*}(BG;\mathbb{Z}/p)$ are given, by comparing with $H^*(BG;\mathbb{Z}/p)$, $CH^*(BG)$ and $BP^*(BG)$.
Fitting's Lemma for $\mathbb{Z}/2$-graded modules
David
Eisenbud;
Jerzy
Weyman
4451-4473
Abstract: Let $\phi :\; R^{m}\to R^{d}$be a map of free modules over a commutative ring $R$. Fitting's Lemma shows that the ``Fitting ideal,'' the ideal of $d\times d$ minors of $\phi$, annihilates the cokernel of $\phi$ and is a good approximation to the whole annihilator in a certain sense. In characteristic 0 we define a Fitting ideal in the more general case of a map of graded free modules over a $\mathbb{Z}/2$-graded skew-commutative algebra and prove corresponding theorems about the annihilator; for example, the Fitting ideal and the annihilator of the cokernel are equal in the generic case. Our results generalize the classical Fitting Lemma in the commutative case and extend a key result of Green (1999) in the exterior algebra case. They depend on the Berele-Regev theory of representations of general linear Lie superalgebras. In the purely even and purely odd cases we also offer a standard basis approach to the module $\operatorname{coker}\phi$ when $\phi$ is a generic matrix.
The differential Galois theory of strongly normal extensions
Jerald
J.
Kovacic
4475-4522
Abstract: Differential Galois theory, the theory of strongly normal extensions, has unfortunately languished. This may be due to its reliance on Kolchin's elegant, but not widely adopted, axiomatization of the theory of algebraic groups. This paper attempts to revive the theory using a differential scheme in place of those axioms. We also avoid using a universal differential field, instead relying on a certain tensor product. We identify automorphisms of a strongly normal extension with maximal differential ideals of this tensor product, thus identifying the Galois group with the closed points of an affine differential scheme. Moreover, the tensor product has a natural coring structure which translates into the Galois group operation: composition of automorphisms. This affine differential scheme splits, i.e. is obtained by base extension from a (not differential, not necessarily affine) group scheme. As a consequence, the Galois group is canonically isomorphic to the closed, or rational, points of a group scheme defined over constants. We obtain the fundamental theorem of differential Galois theory, giving a bijective correspondence between subgroup schemes and intermediate differential fields. On the way to this result we study certain aspects of differential algebraic geometry, e.g. closed immersions, products, local ringed space of constants, and split differential schemes.
Twisted sums with $C(K)$ spaces
F.
Cabello
Sánchez;
J.
M. F.
Castillo;
N.
J.
Kalton;
D.
T.
Yost
4523-4541
Abstract: If $X$ is a separable Banach space, we consider the existence of non-trivial twisted sums $0\to C(K)\to Y\to X\to 0$, where $K=[0,1]$ or $\omega^{\omega}.$For the case $K=[0,1]$ we show that there exists a twisted sum whose quotient map is strictly singular if and only if $X$ contains no copy of $\ell_1$. If $K=\omega^{\omega}$ we prove an analogue of a theorem of Johnson and Zippin (for $K=[0,1]$) by showing that all such twisted sums are trivial if $X$ is the dual of a space with summable Szlenk index (e.g., $X$ could be Tsirelson's space); a converse is established under the assumption that $X$ has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with $C(\omega^{\omega})$ with strictly singular quotient map.
Semi-free Hamiltonian circle actions on 6-dimensional symplectic manifolds
Hui
Li
4543-4568
Abstract: Assume $(M, \omega)$ is a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action, such that the fixed point set consists of isolated points or compact orientable surfaces. We restrict attention to the case $\dim H^2(M)<3$. We give a complete list of the possible manifolds, and determine their equivariant cohomology rings and equivariant Chern classes. Some of these manifolds are classified up to diffeomorphism. We also show the existence for a few cases.
A constructive Schwarz reflection principle
Jeremy
Clark
4569-4579
Abstract: We prove a constructive version of the Schwarz reflection principle. Our proof techniques are in line with Bishop's development of constructive analysis. The principle we prove enables us to reflect analytic functions in the real line, given that the imaginary part of the function converges to zero near the real line in a uniform fashion. This form of convergence to zero is classically equivalent to pointwise convergence, but may be a stronger condition from the constructivist point of view.
Maximal complexifications of certain homogeneous Riemannian manifolds
S.
Halverscheid;
A.
Iannuzzi
4581-4594
Abstract: Let $M=G/K$ be a homogeneous Riemannian manifold with $\dim_{\mathbb{C}} G^{\mathbb{C}} = \dim_{\mathbb{R}} G$, where $G^{\mathbb{C}}$ denotes the universal complexification of $G$. Under certain extensibility assumptions on the geodesic flow of $M$, we give a characterization of the maximal domain of definition in $TM$ for the adapted complex structure and show that it is unique. For instance, this can be done for generalized Heisenberg groups and naturally reductive homogeneous Riemannian spaces. As an application it is shown that the case of generalized Heisenberg groups yields examples of maximal domains of definition for the adapted complex structure which are neither holomorphically separable nor holomorphically convex.
Open 3-manifolds whose fundamental groups have infinite center, and a torus theorem for 3-orbifolds
Sylvain
Maillot
4595-4638
Abstract: Our main result is a characterization of open Seifert fibered $3$-manifolds in terms of the fundamental group and large-scale geometric properties of a triangulation. As an application, we extend the Seifert Fiber Space Theorem and the Torus Theorem to a class of $3$-orbifolds.
Baxter algebras and Hopf algebras
George
E.
Andrews;
Li
Guo;
William
Keigher;
Ken
Ono
4639-4656
Abstract: By applying a recent construction of free Baxter algebras, we obtain a new class of Hopf algebras that generalizes the classical divided power Hopf algebra. We also study conditions under which these Hopf algebras are isomorphic.
On the Diophantine equation $G_n(x)=G_m(P(x))$: Higher-order recurrences
Clemens
Fuchs;
Attila
Petho;
Robert
F.
Tichy
4657-4681
Abstract: Let $\mathbf{K}$ be a field of characteristic $0$ and let $(G_{n}(x))_{n=0}^{\infty}$ be a linear recurring sequence of degree $d$ in $\mathbf{K}[x]$ defined by the initial terms $G_0,\ldots,G_{d-1}\in\mathbf{K}[x]$ and by the difference equation \begin{displaymath}G_{n+d}(x)=A_{d-1}(x)G_{n+d-1}(x)+\cdots+A_0(x)G_{n}(x), \quad \mbox{for} \,\, n\geq 0,\end{displaymath} with $A_0,\ldots,A_{d-1}\in\mathbf{K}[x]$. Finally, let $P(x)$ be an element of $\mathbf{K}[x]$. In this paper we are giving fairly general conditions depending only on $G_0,\ldots,G_{d-1},$ on $P$, and on $A_0,\ldots,A_{d-1}$ under which the Diophantine equation \begin{displaymath}G_{n}(x)=G_{m}(P(x))\end{displaymath} has only finitely many solutions $(n,m)\in \mathbb{Z}^{2},n,m\geq 0$. Moreover, we are giving an upper bound for the number of solutions, which depends only on $d$. This paper is a continuation of the work of the authors on this equation in the case of second-order linear recurring sequences.
Compact composition operators on Besov spaces
Maria
Tjani
4683-4698
Abstract: We give a Carleson measure characterization of the compact composition operators on Besov spaces. We use this characterization to show that every compact composition operator on a Besov space is compact on the Bloch space. Finally we give conditions that guarantee that the converse holds.
Meromorphic groups
Anand
Pillay;
Thomas
Scanlon
3843-3859
Abstract: We show that a connected group interpretable in a compact complex manifold (a meromorphic group) is definably an extension of a complex torus by a linear algebraic group, generalizing results of Fujiki. A special case of this result, as well as one of the ingredients in the proof, is that a strongly minimal modular meromorphic group is a complex torus, answering a question of Hrushovski. As a consequence, we show that a simple compact complex manifold has algebraic and Kummer dimension zero if and only if its generic type is trivial.
Poset block equivalence of integral matrices
Mike
Boyle;
Danrun
Huang
3861-3886
Abstract: Given square matrices $B$ and $B'$ with a poset-indexed block structure (for which an $ij$ block is zero unless $i\preceq j$), when are there invertible matrices $U$ and $V$ with this required-zero-block structure such that $UBV = B'$? We give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain $\mathcal R$. As one application, when $\mathcal R$ is a field we classify such matrices up to similarity by matrices respecting the block structure. We also give complete invariants for equivalence under the additional requirement that the diagonal blocks of $U$ and $V$ have determinant $1$. The invariants involve an associated diagram (the ``$K$-web'') of $\mathcal R$-module homomorphisms. The study is motivated by applications to symbolic dynamics and $C^*$-algebras.
Multiple orthogonal polynomials for classical weights
A.
I.
Aptekarev;
A.
Branquinho;
W.
Van Assche
3887-3914
Abstract: A new set of special functions, which has a wide range of applications from number theory to integrability of nonlinear dynamical systems, is described. We study multiple orthogonal polynomials with respect to $p > 1$ weights satisfying Pearson's equation. In particular, we give a classification of multiple orthogonal polynomials with respect to classical weights, which is based on properties of the corresponding Rodrigues operators. We show that the multiple orthogonal polynomials in our classification satisfy a linear differential equation of order $p+1$. We also obtain explicit formulas and recurrence relations for these polynomials.
Maximal singular loci of Schubert varieties in $SL(n)/B$
Sara
C.
Billey;
Gregory
S.
Warrington
3915-3945
Abstract: Schubert varieties in the flag manifold $SL(n)/B$ play a key role in our understanding of projective varieties. One important problem is to determine the locus of singular points in a variety. In 1990, Lakshmibai and Sandhya showed that the Schubert variety $X_w$ is nonsingular if and only if $w$ avoids the patterns $4231$ and $3412$. They also gave a conjectural description of the singular locus of $X_w$. In 1999, Gasharov proved one direction of their conjecture. In this paper we give an explicit combinatorial description of the irreducible components of the singular locus of the Schubert variety $X_w$ for any element $w\in \mathfrak{S}_n$. In doing so, we prove both directions of the Lakshmibai-Sandhya conjecture. These irreducible components are indexed by permutations which differ from $w$ by a cycle depending naturally on a $4231$ or $3412$ pattern in $w$. Our description of the irreducible components is computationally more efficient ($O(n^6)$) than the previously best known algorithms, which were all exponential in time. Furthermore, we give simple formulas for calculating the Kazhdan-Lusztig polynomials at the maximum singular points.
Quandle cohomology and state-sum invariants of knotted curves and surfaces
J.
Scott
Carter;
Daniel
Jelsovsky;
Seiichi
Kamada;
Laurel
Langford;
Masahico
Saito
3947-3989
Abstract: The 2-twist spun trefoil is an example of a sphere that is knotted in 4-dimensional space. A proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed. Our proof is based on a state-sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles (also known as distributive groupoids). A quandle is a set with a binary operation -- the axioms of which model the Reidemeister moves in classical knot theory. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define state-sum invariants for knotted circles in $3$-space and knotted surfaces in $4$-space. Cohomology groups of various quandles are computed herein and applied to the study of the state-sum invariants. Non-triviality of the invariants is proved for a variety of knots and links, and conversely, knot invariants are used to prove non-triviality of cohomology for a variety of quandles.
Spin Borromean surgeries
Gwénaël
Massuyeau
3991-4017
Abstract: In 1986, Matveev defined the notion of Borromean surgery for closed oriented $3$-manifolds and showed that the equivalence relation generated by this move is characterized by the pair (first Betti number, linking form up to isomorphism). We explain how this extends for $3$-manifolds with spin structure if we replace the linking form by the quadratic form defined by the spin structure. We then show that the equivalence relation among closed spin $3$-manifolds generated by spin Borromean surgeries is characterized by the triple (first Betti number, linking form up to isomorphism, Rochlin invariant modulo $8$).
Fractafolds based on the Sierpinski gasket and their spectra
Robert
S.
Strichartz
4019-4043
Abstract: We introduce the notion of ``fractafold'', which is to a fractal what a manifold is to a Euclidean half-space. We specialize to the case when the fractal is the Sierpinski gasket SG. We show that each such compact fractafold can be given by a cellular construction based on a finite cell graph $G$, which is $3$-regular in the case that the fractafold has no boundary. We show explicitly how to obtain the spectrum of the fractafold from the spectrum of the graph, using the spectral decimation method of Fukushima and Shima. This enables us to obtain isospectral pairs of nonhomeomorphic fractafolds. We also show that although SG is topologically rigid, there are fractafolds based on SG that are not topologically rigid.
Generalized hyperelliptic surfaces
Francesco
Zucconi
4045-4059
Abstract: This article presents some results on the surfaces of general type whose Albanese morphism is a holomorphic fibre bundle.
Tribasic integrals and identities of Rogers-Ramanujan type
M.
E. H.
Ismail;
D.
Stanton
4061-4091
Abstract: Some integrals involving three bases are evaluated as infinite products using complex analysis. Many special cases of these integrals may be evaluated in another way to find infinite sum representations for these infinite products. The resulting identities are identities of Rogers-Ramanujan type. Some integer partition interpretations of these identities are given. Generalizations of the Rogers-Ramanujan type identities involving polynomials are given again as corollaries of integral evaluations.
Burghelea-Friedlander-Kappeler's gluing formula for the zeta-determinant and its applications to the adiabatic decompositions of the zeta-determinant and the analytic torsion
Yoonweon
Lee
4093-4110
Abstract: The gluing formula of the zeta-determinant of a Laplacian given by Burghelea, Friedlander and Kappeler contains an unknown constant. In this paper we compute this constant to complete the formula under an assumption that the product structure is given near the boundary. As applications of this result, we prove the adiabatic decomposition theorems of the zeta-determinant of a Laplacian with respect to the Dirichlet and Neumann boundary conditions and of the analytic torsion with respect to the absolute and relative boundary conditions.
Inverse functions of polynomials and orthogonal polynomials as operator monotone functions
Mitsuru
Uchiyama
4111-4123
Abstract: We study the operator monotonicity of the inverse of every polynomial with a positive leading coefficient. Let $\{p_n\}_{n=0}^{\infty}$ be a sequence of orthonormal polynomials and $p_{n+}$ the restriction of $p_n$ to $[a_n, \infty)$, where $a_n$ is the maximum zero of $p_n$. Then $p_{n+}^{-1}$ and the composite $p_{n-1}\circ p_{n+}^{-1}$ are operator monotone on $[0, \infty)$. Furthermore, for every polynomial $p$ with a positive leading coefficient there is a real number $a$ so that the inverse function of $p(t+a)-p(a)$ defined on $[0,\infty)$is semi-operator monotone, that is, for matrices $A,B \geq 0$, $(p(A+a)-p(a))^2 \leq ((p(B+a)-p(a))^{2}$ implies $A^2\leq B^2.$
The structure of equicontinuous maps
Jie-Hua
Mai
4125-4136
Abstract: Let $(X,d)$ be a metric space, and $f:X\rightarrow X$ be a continuous map. In this paper we prove that if $R(f)$ is compact, and $\omega (x,f)\not =\emptyset$ for all $x\in X$, then $f$ is equicontinuous if and only if there exist a pointwise recurrent isometric homeomorphism $h$ and a non-expanding map $g$ that is pointwise convergent to a fixed point $v_{0}$ such that $f$ is uniformly conjugate to a subsystem $(h\times g)\vert S$ of the product map $h\times g$. In addition, we give some still simpler necessary and sufficient conditions of equicontinuous graph maps.
Heegner zeros of theta functions
Jorge
Jimenez-Urroz;
Tonghai
Yang
4137-4149
Abstract: Heegner divisors play an important role in number theory. However, little is known on whether a modular form has Heegner zeros. In this paper, we start to study this question for a family of classical theta functions, and prove a quantitative result, which roughly says that many of these theta functions have a Heegner zero of discriminant $-7$. This leads to some interesting questions on the arithmetic of certain elliptic curves, which we also address here.
Geometry of graph varieties
Jeremy
L.
Martin
4151-4169
Abstract: A picture $\mathbf{P}$ of a graph $G=(V,E)$ consists of a point $\mathbf{P}(v)$ for each vertex $v \in V$ and a line $\mathbf{P}(e)$ for each edge $e \in E$, all lying in the projective plane over a field $\mathbf k$ and subject to containment conditions corresponding to incidence in $G$. A graph variety is an algebraic set whose points parametrize pictures of $G$. We consider three kinds of graph varieties: the picture space $\mathcal{X}(G)$ of all pictures; the picture variety $\mathcal{V}(G)$, an irreducible component of $\mathcal{X}(G)$ of dimension $2\vert V\vert$, defined as the closure of the set of pictures on which all the $\mathbf{P}(v)$ are distinct; and the slope variety $\mathcal{S}(G)$, obtained by forgetting all data except the slopes of the lines $\mathbf{P}(e)$. We use combinatorial techniques (in particular, the theory of combinatorial rigidity) to obtain the following geometric and algebraic information on these varieties: (1) a description and combinatorial interpretation of equations defining each variety set-theoretically; (2) a description of the irreducible components of $\mathcal{X}(G)$; (3) a proof that $\mathcal{V}(G)$ and $\mathcal{S}(G)$ are Cohen-Macaulay when $G$ satisfies a sparsity condition, rigidity independence. In addition, our techniques yield a new proof of the equality of two matroids studied in rigidity theory.
Generalized associahedra via quiver representations
Robert
Marsh;
Markus
Reineke;
Andrei
Zelevinsky
4171-4186
Abstract: We provide a quiver-theoretic interpretation of certain smooth complete simplicial fans associated to arbitrary finite root systems in recent work of S. Fomin and A. Zelevinsky. The main properties of these fans then become easy consequences of the known facts about tilting modules due to K. Bongartz, D. Happel and C. M. Ringel.
Fibred knots and twisted Alexander invariants
Jae
Choon
Cha
4187-4200
Abstract: We study the twisted Alexander invariants of fibred knots. We establish necessary conditions on the twisted Alexander invariants for a knot to be fibred, and develop a practical method to compute the twisted Alexander invariants from the homotopy type of a monodromy. It is illustrated that the twisted Alexander invariants carry more information on fibredness than the classical Alexander invariants, even for knots with trivial Alexander polynomials.
Sub-bundles of the complexified tangent bundle
Howard
Jacobowitz;
Gerardo
Mendoza
4201-4222
Abstract: We study embeddings of complex vector bundles, especially line bundles, in the complexification of the tangent bundle of a manifold. The aim is to understand implications of properties of interest in partial differential equations.
The autohomeomorphism group of the Cech-Stone compactification of the integers
Juris
Steprans
4223-4240
Abstract: It is shown to be consistent that there is a nontrivial autohomeomorphism of $\beta{\mathbb N} \setminus {\mathbb N}$, yet all such autohomeomorphisms are trivial on a dense $P$-ideal. Furthermore, the cardinality of the autohomeomorphism group of $\beta{\mathbb N} \setminus {\mathbb N}$ can be any regular cardinal between $2^{\aleph_0}$ and $2^{2^{\aleph_0}}$. The model used is one due to Velickovic in which, coincidentally, Martin's Axiom also holds.
The geometry of 1-based minimal types
Tristram
de Piro;
Byunghan
Kim
4241-4263
Abstract: In this paper, we study the geometry of a (nontrivial) 1-based $SU$ rank-1 complete type. We show that if the (localized, resp.) geometry of the type is modular, then the (localized, resp.) geometry is projective over a division ring. However, unlike the stable case, we construct a locally modular type that is not affine. For the general 1-based case, we prove that even if the geometry of the type itself is not projective over a division ring, it is when we consider a 2-fold or 3-fold of the geometry altogether. In particular, it follows that in any $\omega$-categorical, nontrivial, 1-based theory, a vector space over a finite field is interpretable.
Errata to ``Metric character of Hamilton--Jacobi equations''
Antonio
Siconolfi
4265-4265
Accelerating the convergence of the method of alternating projections
Heinz
H.
Bauschke;
Frank
Deutsch;
Hein
Hundal;
Sung-Ho
Park
3433-3461
Abstract: The powerful von Neumann-Halperin method of alternating projections (MAP) is an algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces. It achieves this by reducing the problem to an iterative scheme which involves only computing best approximations from the individual subspaces which make up the intersection. The main practical drawback of this algorithm, at least for some applications, is that the method is slowly convergent. In this paper, we consider a general class of iterative methods which includes the MAP as a special case. For such methods, we study an ``accelerated'' version of this algorithm that was considered earlier by Gubin, Polyak, and Raik (1967) and by Gearhart and Koshy (1989). We show that the accelerated algorithm converges faster than the MAP in the case of two subspaces, but is, in general, not faster than the MAP for more than two subspaces! However, for a ``symmetric'' version of the MAP, the accelerated algorithm always converges faster for any number of subspaces. Our proof seems to require the use of the Spectral Theorem for selfadjoint mappings.
Anderson's double complex and gamma monomials for rational function fields
Sunghan
Bae;
Ernst-Ulrich
Gekeler;
Pyung-Lyun
Kang;
Linsheng
Yin
3463-3474
Abstract: We investigate algebraic $\Gamma$-monomials of Thakur's positive characteristic $\Gamma$-function, by using Anderson and Das' double complex method of computing the sign cohomology of the universal ordinary distribution. We prove that the $\Gamma$-monomial associated to an element of the second sign cohomology of the universal ordinary distribution of $\mathbb{F} _{q}(T)$generates a Kummer extension of some Carlitz cyclotomic function field, which is also a Galois extension of the base field $\mathbb{F} _{q}(T)$. These results are characteristic-$p$ analogues of those of Deligne on classical $\Gamma$-monomials, proofs of which were given by Das using the double complex method. In this paper, we also obtain some results on $e$-monomials of Carlitz's exponential function.
Remarks about uniform boundedness of rational points over function fields
Lucia
Caporaso
3475-3484
Abstract: We prove certain uniform versions of the Mordell Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points.
Irreducibility of equisingular families of curves
Thomas
Keilen
3485-3512
Abstract: In 1985 Joe Harris proved the long-standing claim of Severi that equisingular families of plane nodal curves are irreducible whenever they are nonempty. For families with more complicated singularities this is no longer true. Given a divisor $D$ on a smooth projective surface $\Sigma$ it thus makes sense to look for conditions which ensure that the family $V_{\vert D\vert}^{irr}\big(\mathcal{S}_1,\ldots,\mathcal{S}_r\big)$ of irreducible curves in the linear system $\vert D\vert _l$ with precisely $r$ singular points of types $\mathcal{S}_1,\ldots,\mathcal{S}_r$ is irreducible. Considering different surfaces, including general surfaces in $\mathbb P_{\mathbb C}^3$ and products of curves, we produce a sufficient condition of the type \begin{displaymath}\sum\limits_{i=1}^r\deg\big(X(\mathcal{S}_i)\big)^2 < \gamma\cdot (D- K_\Sigma)^2, \end{displaymath} where $\gamma$ is some constant and $X(\mathcal{S}_i)$ some zero-dimensional scheme associated to the singularity type. Our results carry the same asymptotics as the best known results in this direction in the plane case, even though the coefficient is worse. For most of the surfaces considered these are the only known results in that direction.
Planar convex bodies, Fourier transform, lattice points, and irregularities of distribution
L.
Brandolini;
A.
Iosevich;
G.
Travaglini
3513-3535
Abstract: Let $B$ be a convex body in the plane. The purpose of this paper is a systematic study of the geometric properties of the boundary of $B$, and the consequences of these properties for the distribution of lattice points in rotated and translated copies of $\rho B$ ($\rho$ being a large positive number), irregularities of distribution, and the spherical average decay of the Fourier transform of the characteristic function of $B$. The analysis makes use of two notions of ``dimension'' of a convex set. The first notion is defined in terms of the number of sides required to approximate a convex set by a polygon up to a certain degree of accuracy. The second is the fractal dimension of the image of the Gauss map of $B$. The results stated in terms of these quantities are essentially sharp and lead to a nearly complete description of the problems in question.
A free boundary problem for a singular system of differential equations: An application to a model of tumor growth
Shangbin
Cui;
Avner
Friedman
3537-3590
Abstract: In this paper we consider a free boundary problem for a nonlinear system of two ordinary differential equations, one of which is singular at some points, including the initial point $r=0$. Because of the singularity at $r=0$, the initial value problem has a one-parameter family of solutions. We prove that there exists a unique solution to the free boundary problem. The proof of existence employs two ``shooting'' parameters. Analysis of the profiles of solutions of the initial value problem and tools such as comparison theorems and weak limits of solutions play an important role in the proof. The system considered here is motivated by a model in tumor growth, but the methods developed should be applicable to more general systems.
Sharp Fourier type and cotype with respect to compact semisimple Lie groups
José
García-Cuerva;
José
Manuel
Marco;
Javier
Parcet
3591-3609
Abstract: Sharp Fourier type and cotype of Lebesgue spaces and Schatten classes with respect to an arbitrary compact semisimple Lie group are investigated. In the process, a local variant of the Hausdorff-Young inequality on such groups is given.
Left-determined model categories and universal homotopy theories
J.
Rosicky;
W.
Tholen
3611-3623
Abstract: We say that a model category is left-determined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not left-determined, we show that its non-oriented variant, the category of symmetric simplicial sets (in the sense of Lawvere and Grandis) carries a natural left-determined model category structure. This is used to give another and, as we believe simpler, proof of a recent result of D. Dugger about universal homotopy theories.
The combinatorial rigidity conjecture is false for cubic polynomials
Christian
Henriksen
3625-3639
Abstract: We show that there exist two cubic polynomials with connected Julia sets which are combinatorially equivalent but not topologically conjugate on their Julia sets. This disproves a conjecture by McMullen from 1995.
Zero entropy, non-integrable geodesic flows and a non-commutative rotation vector
Leo
T.
Butler
3641-3650
Abstract: Let $\mathfrak g$ be a $2$-step nilpotent Lie algebra; we say $\mathfrak g$ is non-integrable if, for a generic pair of points $\mathfrak g ={\mathrm {Lie}}(G)$ is non-integrable, $D < G$ is a cocompact subgroup, and ${\mathbf g}$ is a left-invariant Riemannian metric, then the geodesic flow of ${\mathbf g}$ on $T^*(D \backslash G)$ is neither Liouville nor non-commutatively integrable with $C^0$ first integrals. The proof uses a generalization of the rotation vector pioneered by Benardete and Mitchell.
Complete homogeneous varieties: Structure and classification
Carlos
Sancho de Salas
3651-3667
Abstract: Homogeneous varieties are those whose group of automorphisms acts transitively on them. In this paper we prove that any complete homogeneous variety splits in a unique way as a product of an abelian variety and a parabolic variety. This is obtained by proving a rigidity theorem for the parabolic subgroups of a linear group. Finally, using the results of Wenzel on the classification of parabolic subgroups of a linear group and the results of Demazure on the automorphisms of a flag variety, we obtain the classification of the parabolic varieties (in characteristic different from $2,3$). This, together with the moduli of abelian varieties, concludes the classification of the complete homogeneous varieties.
A path-transformation for random walks and the Robinson-Schensted correspondence
Neil
O'Connell
3669-3697
Abstract: The author and Marc Yor recently introduced a path-transformation $G^{(k)}$ with the property that, for $X$ belonging to a certain class of random walks on $\mathbb{Z}_+^k$, the transformed walk $G^{(k)}(X)$has the same law as the original walk conditioned never to exit the Weyl chamber $\{x: x_1\le\cdots\le x_k\}$. In this paper, we show that $G^{(k)}$ is closely related to the Robinson-Schensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law of $X$ and $G^{(k)}(X)$. The corresponding results for the Brownian model are recovered by Donsker's theorem. These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory. The connection we make between the path-transformation $G^{(k)}$ and the Robinson-Schensted algorithm also provides a new formula and interpretation for the latter. This can be used to study properties of the Robinson-Schensted algorithm and, moreover, extends easily to a continuous setting.
On the Iwasawa $\lambda$-invariants of real abelian fields
Takae
Tsuji
3699-3714
Abstract: For a prime number $p$ and a number field $k$, let $A_\infty$ denote the projective limit of the $p$-parts of the ideal class groups of the intermediate fields of the cyclotomic $\mathbb{Z} _p$-extension over $k$. It is conjectured that $A_\infty$ is finite if $k$ is totally real. When $p$ is an odd prime and $k$ is a real abelian field, we give a criterion for the conjecture, which is a generalization of results of Ichimura and Sumida. Furthermore, in a special case where $p$ divides the degree of $k$, we also obtain a rather simple criterion.
Pseudo-holomorphic curves in complex Grassmann manifolds
Xiaoxiang
Jiao;
Jiagui
Peng
3715-3726
Abstract: It is proved that the Kähler angle of the pseudo-holomorphic sphere of constant curvature in complex Grassmannians is constant. At the same time we also prove several pinching theorems for the curvature and the Kähler angle of the pseudo-holomorphic spheres in complex Grassmannians with non-degenerate associated harmonic sequence.
The periodic Euler-Bernoulli equation
Vassilis
G.
Papanicolaou
3727-3759
Abstract: We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem \begin{displaymath}\left[ a(x)u^{\prime \prime }(x)\right] ^{\prime \prime }=\lambda \rho (x)u(x),\qquad -\infty <x<\infty , \end{displaymath} where the functions $a$ and $\rho$ are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by $a$ and $\rho$. Here we develop a theory analogous to the theory of the Hill operator $-(d/dx)^2+q(x)$. We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or $\psi$-spectrum. Our new analysis begins with a detailed study of the zeros of the function $F(\lambda ;k)$, for any given ``quasimomentum'' $k\in \mathbb{C}$, where $F(\lambda ;k)=0$ is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to $F(\lambda ;k)$ is $\Delta (\lambda )-2\cos (kb)$, where $\Delta (\lambda )$ is the discriminant and $b$ the period of $q$). We show that the multiplicity $m(\lambda ^{\ast })$ of any zero $\lambda ^{\ast }$ of $F(\lambda ;k)$ can be one or two and $m(\lambda ^{\ast })=2$ (for some $k$) if and only if $\lambda ^{\ast }$ is also a zero of another entire function $D(\lambda )$, independent of $k$. Furthermore, we show that $D(\lambda )$ has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each $\psi$-gap. If $\lambda ^{\ast }$ is a double zero of $F(\lambda ;k)$, it may happen that there is only one Floquet solution with quasimomentum $k$; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree. Next we show that if $(\alpha ,\beta )$ is an open $\psi$-gap of the pseudospectrum (i.e., $\alpha <\beta$), then the Floquet matrix $T(\lambda )$has a specific Jordan anomaly at $\lambda =\alpha$ and $\lambda =\beta$. We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by $\{\mu _n\}_{n\in \mathbb{Z}}$ the eigenvalues of this multipoint problem and show that $\{\mu _n\}_{n\in \mathbb{Z}}$ is also characterized as the set of values of $\lambda$ for which there is a proper Floquet solution $f(x;\lambda )$ such that $f(0;\lambda )=0$. We also show (Theorem 7) that each gap of the $L^{2}(\mathbb{R})$-spectrum contains exactly one $\mu _{n}$ and each $\psi$-gap of the pseudospectrum contains exactly two $\mu _{n}$'s, counting multiplicities. Here when we say ``gap'' or ``$\psi$-gap'' we also include the endpoints (so that when two consecutive bands or $\psi$-bands touch, the in-between collapsed gap, or $\psi$-gap, is a point). We believe that $\{\mu _{n}\}_{n\in \mathbb{Z}}$ can be used to formulate the associated inverse spectral problem. As an application of Theorem 7, we show that if $\nu ^{*}$ is a collapsed (``closed'') $\psi$-gap, then the Floquet matrix $T(\nu ^{*})$ is diagonalizable. Some of the above results were conjectured in our previous works. However, our conjecture that if all the $\psi$-gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.
Singularities of the hypergeometric system associated with a monomial curve
Francisco
Jesús
Castro-Jiménez;
Nobuki
Takayama
3761-3775
Abstract: We compute, using $\mathcal{D}$-module restrictions, the slopes of the irregular hypergeometric system associated with a monomial curve. We also study rational solutions and reducibility of such systems.
Asymptotics for logical limit laws: When the growth of the components is in an RT class
Jason
P.
Bell;
Stanley
N.
Burris
3777-3794
Abstract: Compton's method of proving monadic second-order limit laws is based on analyzing the generating function of a class of finite structures. For applications of his deeper results we previously relied on asymptotics obtained using Cauchy's integral formula. In this paper we develop elementary techniques, based on a Tauberian theorem of Schur, that significantly extend the classes of structures for which we know that Compton's theory can be applied.
Combinatorics of rooted trees and Hopf algebras
Michael
E.
Hoffman
3795-3811
Abstract: We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally associates a multiplicity to each pair of rooted trees. By using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the associated multiplicities. Now the symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators, as well as the inner product mentioned above, to Kreimer's Hopf algebra. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this leads to the Hopf algebra of Grossman and Larson. We show that the inner product on rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with the graded dual of Kreimer's Hopf algebra, correcting an earlier result of Panaite.
Connections with prescribed first Pontrjagin form
Mahuya
Datta
3813-3824
Abstract: Let $P$ be a principal $O(n)$ bundle over a $C^\infty$manifold $M$ of dimension $m$. If $n\geq 5m+4+4\binom{m+1}{4}$, then we prove that every differential 4-form representing the first Pontrjagin class of $P$ is the Pontrjagin form of some connection on $P$.
Self-intersection class for singularities and its application to fold maps
Toru
Ohmoto;
Osamu
Saeki;
Kazuhiro
Sakuma
3825-3838
Abstract: Let $f :M \to N$ be a generic smooth map with corank one singularities between manifolds, and let $S(f)$ be the singular point set of $f$. We define the self-intersection class $I(S(f)) \in H^*(M; \mathbf{Z})$ of $S(f)$using an incident class introduced by Rimányi but with twisted coefficients, and give a formula for $I(S(f))$ in terms of characteristic classes of the manifolds. We then apply the formula to the existence problem of fold maps.
Erratum to ``Arens regularity of the algebra $A \hat{\otimes} B$''
A.
Ülger
3839-3839
Erratum to ``Spherical classes and the algebraic transfer''
Nguyên
H. V.
Hung
3841-3842
Homology of pseudodifferential operators on manifolds with fibered cusps
Robert
Lauter;
Sergiu
Moroianu
3009-3046
Abstract: The Hochschild homology of the algebra of pseudodifferential operators on a manifold with fibered cusps, introduced by Mazzeo and Melrose, is studied and computed using the approach of Brylinski and Getzler. One of the main technical tools is a new convergence criterion for tri-filtered half-plane spectral sequences. Using trace-like functionals that generate the $0$-dimensional Hochschild cohomology groups, the index of a fully elliptic fibered cusp operator is expressed as the sum of a local contribution of Atiyah-Singer type and a global term on the boundary. We announce a result relating this boundary term to the adiabatic limit of the eta invariant in a particular case.
Moderate deviation principles for trajectories of sums of independent Banach space valued random variables
Yijun
Hu;
Tzong-Yow
Lee
3047-3064
Abstract: Let $\{X_n\}$ be a sequence of i.i.d. random vectors with values in a separable Banach space. Moderate deviation principles for trajectories of sums of $\{X_n\}$ are proved, which generalize related results of Borovkov and Mogulskii (1980) and Deshayes and Picard (1979). As an application, functional laws of the iterated logarithm are given. The paper also contains concluding remarks, with examples, on extending results for partial sums to corresponding ones for trajectory setting.
Weierstrass functions with random phases
Yanick
Heurteaux
3065-3077
Abstract: Consider the function \begin{displaymath}f_\theta(x)=\sum_{n=0}^{+\infty}b^{-n\alpha}g(b^nx+\theta_n),\end{displaymath} where $b>1$, $0<\alpha<1$, and $g$ is a non-constant 1-periodic Lipschitz function. The phases $\theta_n$ are chosen independently with respect to the uniform probability measure on $[0,1]$. We prove that with probability one, we can choose a sequence of scales $\delta_k\searrow 0$ such that for every interval $I$ of length $\vert I\vert=\delta_k$, the oscillation of $f_\theta$ satisfies $\operatorname{osc}(f_\theta,I)\geq C\vert I\vert^\alpha$. Moreover, the inequality $\operatorname{osc}(f_\theta,I)\geq C\vert I\vert^{\alpha+\varepsilon}$ is almost surely true at every scale. When $b$ is a transcendental number, these results can be improved: the minoration $\operatorname{osc}(f_\theta,I)\geq C\vert I\vert^\alpha$ is true for every choice of the phases $\theta_n$ and at every scale.
Explicit Lower bounds for residues at $s=1$ of Dedekind zeta functions and relative class numbers of CM-fields
Stéphane
Louboutin
3079-3098
Abstract: Let $S$ be a given set of positive rational primes. Assume that the value of the Dedekind zeta function $\zeta_K$ of a number field $K$ is less than or equal to zero at some real point $\beta$ in the range ${1\over 2} <\beta <1$. We give explicit lower bounds on the residue at $s=1$ of this Dedekind zeta function which depend on $\beta$, the absolute value $d_K$of the discriminant of $K$ and the behavior in $K$ of the rational primes $p\in S$. Now, let $k$ be a real abelian number field and let $\beta$ be any real zero of the zeta function of $k$. We give an upper bound on the residue at $s=1$ of $\zeta_k$which depends on $\beta$, $d_k$ and the behavior in $k$ of the rational primes $p\in S$. By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields $K$ which depend on the behavior in $K$ of the rational primes $p\in S$. We will then show that these new lower bounds for relative class numbers are of paramount importance for solving, for example, the exponent-two class group problem for the non-normal quartic CM-fields. Finally, we will prove Brauer-Siegel-like results about the asymptotic behavior of relative class numbers of CM-fields.
Primitive free cubics with specified norm and trace
Sophie
Huczynska;
Stephen
D.
Cohen
3099-3116
Abstract: The existence of a primitive free (normal) cubic $x^3-ax^2+cx-b$ over a finite field $F$ with arbitrary specified values of $a$ ($\neq 0$) and $b$ (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.
D-log and formal flow for analytic isomorphisms of n-space
David
Wright;
Wenhua
Zhao
3117-3141
Abstract: Given a formal map $F=(F_1,\ldots,F_n)$ of the form $z+\text{higher-order}$ terms, we give tree expansion formulas and associated algorithms for the D-Log of $F$ and the formal flow $F_t$. The coefficients that appear in these formulas can be viewed as certain generalizations of the Bernoulli numbers and the Bernoulli polynomials. Moreover, the coefficient polynomials in the formal flow formula coincide with the strict order polynomials in combinatorics for the partially ordered sets induced by trees. Applications of these formulas to the Jacobian Conjecture are discussed.
A generalization of tight closure and multiplier ideals
Nobuo
Hara;
Ken-ichi
Yoshida
3143-3174
Abstract: We introduce a new variant of tight closure associated to any fixed ideal $\mathfrak{a}$, which we call $\mathfrak{a}$-tight closure, and study various properties thereof. In our theory, the annihilator ideal $\tau (\mathfrak{a})$ of all $\mathfrak{a}$-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal $\tau (\mathfrak{a})$ and the multiplier ideal associated to $\mathfrak{a}$ (or, the adjoint of $\mathfrak{a}$ in Lipman's sense) in normal $\mathbb{Q}$-Gorenstein rings reduced from characteristic zero to characteristic $p \gg 0$. Also, in fixed prime characteristic, we establish some properties of $\tau (\mathfrak{a})$ similar to those of multiplier ideals (e.g., a Briançon-Skoda-type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal $\tau (\mathfrak{a})$ and the F-rationality of Rees algebras.
Seshadri constants on Jacobian of curves
Jian
Kong
3175-3180
Abstract: We compute the Seshadri constants on the Jacobian of hyperelliptic curves, as well as of curves with genus three and four. For higher genus curves we conclude that if the Seshadri constants of their Jacobian are less than 2, then the curves must be hyperelliptic.
On the Clifford algebra of a binary form
Rajesh
S.
Kulkarni
3181-3208
Abstract: The Clifford algebra $C_f$ of a binary form $f$ of degree $d$is the $k$-algebra $k\{x, y\}/I$, where $I$ is the ideal generated by $\{(\alpha x + \beta y)^d - f(\alpha, \beta) \mid \alpha, \beta \in k\}$. $C_f$ has a natural homomorphic image $A_f$ that is a rank $d^2$ Azumaya algebra over its center. We prove that the center is isomorphic to the coordinate ring of the complement of an explicit $\Theta$-divisor in $\ensuremath{{Pic}_{C/k}^{d + g - 1}}$, where $C$ is the curve $(w^d - f(u, v))$ and $g$is the genus of $C$.
Projective normality of abelian varieties
Jaya
N.
Iyer
3209-3216
Abstract: We show that ample line bundles $L$ on a $g$-dimensional simple abelian variety $A$, satisfying $h^0(A,L)>2^g\cdot g!$, give projective normal embeddings, for all $g\geq 1$.
Clustering of critical points in Lefschetz fibrations and the symplectic Szpiro inequality
V.
Braungardt;
D.
Kotschick
3217-3226
Abstract: We prove upper bounds for the number of critical points in semi- stable symplectic Lefschetz fibrations. We also obtain a new lower bound for the number of nonseparating vanishing cycles in Lefschetz pencils and reprove the known lower bounds for the commutator lengths of Dehn twists.
On measures of maximal and full dimension for polynomial automorphisms of $\mathbb{C}^2$
Christian
Wolf
3227-3239
Abstract: For a hyperbolic polynomial automorphism of $\mathbb{C} ^2$, we show the existence of a measure of maximal dimension and identify the conditions under which a measure of full dimension exists.
Hausdorff dimension and asymptotic cycles
Mark
Pollicott
3241-3252
Abstract: We carry out a multifractal analysis for the asymptotic cycles for compact Riemann surfaces of genus $g \geq 2$. This describes the set of unit tangent vectors for which the associated orbit has a given asymptotic cycle in homology.
Constructions preserving Hilbert space uniform embeddability of discrete groups
Marius
Dadarlat;
Erik
Guentner
3253-3275
Abstract: Uniform embeddability (in a Hilbert space), introduced by Gromov, is a geometric property of metric spaces. As applied to countable discrete groups, it has important consequences for the Novikov conjecture. Exactness, introduced and studied extensively by Kirchberg and Wassermann, is a functional analytic property of locally compact groups. Recently it has become apparent that, as properties of countable discrete groups, uniform embeddability and exactness are closely related. We further develop the parallel between these classes by proving that the class of uniformly embeddable groups shares a number of permanence properties with the class of exact groups. In particular, we prove that it is closed under direct and free products (with and without amalgam), inductive limits and certain extensions.
Vitali covering theorem in Hilbert space
Jaroslav
Tiser
3277-3289
Abstract: It is shown that the statement of the Vitali Covering Theorem does not hold for a certain class of measures in a Hilbert space. This class contains all infinite-dimensional Gaussian measures.
On idempotents in reduced enveloping algebras
George
B.
Seligman
3291-3300
Abstract: Explicit constructions are given for idempotents that generate all projective indecomposable modules for certain finite-dimensional quotients of the universal enveloping algebra of the Lie algebra $s\ell (2)$ in odd prime characteristic. The program is put in a general context, although constructions are only carried through in the case of $s\ell (2)$.
Stability of infinite-dimensional sampled-data systems
Hartmut
Logemann;
Richard
Rebarber;
Stuart
Townley
3301-3328
Abstract: Suppose that a static-state feedback stabilizes a continuous-time linear infinite-dimensional control system. We consider the following question: if we construct a sampled-data controller by applying an idealized sample-and-hold process to a continuous-time stabilizing feedback, will this sampled-data controller stabilize the system for all sufficiently small sampling times? Here the state space $X$ and the control space $U$ are Hilbert spaces, the system is of the form $\dot x(t) = Ax(t) + Bu(t)$, where $A$ is the generator of a strongly continuous semigroup on $X$, and the continuous time feedback is $u(t) = Fx(t)$. The answer to the above question is known to be ``yes'' if $X$ and $U$ are finite-dimensional spaces. In the infinite-dimensional case, if $F$ is not compact, then it is easy to find counterexamples. Therefore, we restrict attention to compact feedback. We show that the answer to the above question is ``yes'', if $B$ is a bounded operator from $U$ into $X$. Moreover, if $B$ is unbounded, we show that the answer ``yes'' remains correct, provided that the semigroup generated by $A$ is analytic. We use the theory developed for static-state feedback to obtain analogous results for dynamic-output feedback control.
Approximations for Gabor and wavelet frames
Deguang
Han
3329-3342
Abstract: Let $\psi$ be a frame vector under the action of a collection of unitary operators $\mathcal U$. Motivated by the recent work of Frank, Paulsen and Tiballi and some application aspects of Gabor and wavelet frames, we consider the existence and uniqueness of the best approximation by normalized tight frame vectors. We prove that for any frame induced by a projective unitary representation for a countable discrete group, the best normalized tight frame (NTF) approximation exists and is unique. Therefore it applies to Gabor frames (including Gabor frames for subspaces) and frames induced by translation groups. Similar results hold for semi-orthogonal wavelet frames.
Mean curvature flow, orbits, moment maps
Tommaso
Pacini
3343-3357
Abstract: Given a compact Riemannian manifold together with a group of isometries, we discuss MCF of the orbits and some applications: e.g., finding minimal orbits. We then specialize to Lagrangian orbits in Kaehler manifolds. In particular, in the Kaehler-Einstein case we find a relation between MCF and moment maps which, for example, proves that the minimal Lagrangian orbits are isolated.
Singular integrals on symmetric spaces, II
Alexandru
D.
Ionescu
3359-3378
Abstract: We extend some of our earlier results on boundedness of singular integrals on symmetric spaces of real rank one to arbitrary noncompact symmetric spaces. Our main theorem is a transference principle for operators defined by $\mathbb{K}$-bi-invariant kernels with certain large scale cancellation properties. As an application we prove $L^p$ boundedness of operators defined by Fourier multipliers that satisfy singular differential inequalities of the Hörmander-Michlin type.
West's problem on equivariant hyperspaces and Banach-Mazur compacta
Sergey
Antonyan
3379-3404
Abstract: Let $G$ be a compact Lie group, $X$ a metric $G$-space, and $\exp X$ the hyperspace of all nonempty compact subsets of $X$ endowed with the Hausdorff metric topology and with the induced action of $G$. We prove that the following three assertions are equivalent: (a) $X$ is locally continuum-connected (resp., connected and locally continuum-connected); (b) $\exp X$ is a $G$-ANR (resp., a $G$-AR); (c) $(\exp X)/G$ is an ANR (resp., an AR). This is applied to show that $(\exp G)/G$ is an ANR (resp., an AR) for each compact (resp., connected) Lie group $G$. If $G$ is a finite group, then $(\exp X)/G$ is a Hilbert cube whenever $X$ is a nondegenerate Peano continuum. Let $L(n)$ be the hyperspace of all centrally symmetric, compact, convex bodies $A\subset \mathbb{R}^n$, $n\ge 2$, for which the ordinary Euclidean unit ball is the ellipsoid of minimal volume containing $A$, and let $L_0(n)$ be the complement of the unique $O(n)$-fixed point in $L(n)$. We prove that: (1) for each closed subgroup $H\subset O(n)$, $L_0(n)/H$ is a Hilbert cube manifold; (2) for each closed subgroup $K\subset O(n)$ acting non-transitively on $S^{n-1}$, the $K$-orbit space $L(n)/K$ and the $K$-fixed point set $L(n)[K]$ are Hilbert cubes. As an application we establish new topological models for tha Banach-Mazur compacta $L(n)/O(n)$ and prove that $L_0(n)$ and $(\exp S^{n-1})\setminus\{S^{n-1}\}$ have the same $O(n)$-homotopy type.
Local solvability and hypoellipticity for semilinear anisotropic partial differential equations
Giuseppe
de Donno;
Alessandro
Oliaro
3405-3432
Abstract: We propose a unified approach, based on methods from microlocal analysis, for characterizing the local solvability and hypoellipticity in $C^\infty$ and Gevrey $G^\sigma$ classes of $2$-variable semilinear anisotropic partial differential operators with multiple characteristics. The conditions imposed on the lower-order terms of the linear part of the operator are optimal.
Algorithms for nonlinear piecewise polynomial approximation: Theoretical aspects
Borislav
Karaivanov;
Pencho
Petrushev;
Robert
C.
Sharpley
2585-2631
Abstract: In this article algorithms are developed for nonlinear $n$-term Courant element approximation of functions in $L_p$ ( $0 < p \le \infty$) on bounded polygonal domains in $\mathbb{R} ^2$. Redundant collections of Courant elements, which are generated by multilevel nested triangulations allowing arbitrarily sharp angles, are investigated. Scalable algorithms are derived for nonlinear approximation which both capture the rate of the best approximation and provide the basis for numerical implementation. Simple thresholding criteria enable approximation of a target function $f$ to optimally high asymptotic rates which are determined and automatically achieved by the inherent smoothness of $f$. The algorithms provide direct approximation estimates and permit utilization of the general Jackson-Bernstein machinery to characterize $n$-term Courant element approximation in terms of a scale of smoothness spaces ($B$-spaces) which govern the approximation rates.
The almost-disjointness number may have countable cofinality
Jörg
Brendle
2633-2649
Abstract: We show that it is consistent for the almost-disjointness number $\mathfrak{a}$ to have countable cofinality. For example, it may be equal to $\aleph_\omega$.
Cyclicity of CM elliptic curves modulo $p$
Alina
Carmen
Cojocaru
2651-2662
Abstract: Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and with complex multiplication. For a prime $p$ of good reduction, let $\overline{E}$ be the reduction of $E$ modulo $p.$ We find the density of the primes $p \leq x$ for which $\overline{E}(\mathbb{F} _p)$ is a cyclic group. An asymptotic formula for these primes had been obtained conditionally by J.-P. Serre in 1976, and unconditionally by Ram Murty in 1979. The aim of this paper is to give a new simpler unconditional proof of this asymptotic formula and also to provide explicit error terms in the formula.
Taylor expansion of an Eisenstein series
Tonghai
Yang
2663-2674
Abstract: In this paper, we give an explicit formula for the first two terms of the Taylor expansion of a classical Eisenstein series of weight $2k+1$ for $\Gamma_{0}(q)$. Both the first term and the second term have interesting arithmetic interpretations. We apply the result to compute the central derivative of some Hecke $L$-functions.
Systems of diagonal Diophantine inequalities
Eric
Freeman
2675-2713
Abstract: We treat systems of real diagonal forms $F_1({\mathbf x}), F_2({\mathbf x}), \ldots, F_R({\mathbf x})$ of degree $k$, in $s$ variables. We give a lower bound $s_0(R,k)$, which depends only on $R$ and $k$, such that if $s \geq s_0(R,k)$ holds, then, under certain conditions on the forms, and for any positive real number $\epsilon$, there is a nonzero integral simultaneous solution $\displaystyle{{\mathbf x}\in {\mathbb Z}^s}$ of the system of Diophantine inequalities $\vert F_i({\mathbf x})\vert < \epsilon$ for $1 \leq i \leq R$. In particular, our result is one of the first to treat systems of inequalities of even degree. The result is an extension of earlier work by the author on quadratic forms. Also, a restriction in that work is removed, which enables us to now treat combined systems of Diophantine equations and inequalities.
On the canonical rings of covers of surfaces of minimal degree
Francisco
Javier
Gallego;
Bangere
P.
Purnaprajna
2715-2732
Abstract: In one of the main results of this paper, we find the degrees of the generators of the canonical ring of a regular algebraic surface $X$ of general type defined over a field of characteristic $0$, under the hypothesis that the canonical divisor of $X$ determines a morphism $\varphi$ from $X$ to a surface of minimal degree $Y$. As a corollary of our results and results of Ciliberto and Green, we obtain a necessary and sufficient condition for the canonical ring of $X$ to be generated in degree less than or equal to $2$. We construct new examples of surfaces satisfying the hypothesis of our theorem and prove results which show that many a priori plausible examples cannot exist. Our methods are to exploit the $\mathcal{O}_{Y}$-algebra structure on $\varphi_{*}\mathcal{O}_{X}$. These methods have other applications, including those on Calabi-Yau threefolds. We prove new results on homogeneous rings associated to a polarized Calabi-Yau threefold and also prove some existence theorems for Calabi-Yau covers of threefolds of minimal degree. These have consequences towards constructing new examples of Calabi-Yau threefolds.
A classification and examples of rank one chain domains
H.
H.
Brungs;
N.
I.
Dubrovin
2733-2753
Abstract: A chain order of a skew field $D$ is a subring $R$ of $D$ so that $d\in D\backslash R$ implies $d^{-1}\in R.$ Such a ring $R$ has rank one if $J(R)$, the Jacobson radical of $R,$ is its only nonzero completely prime ideal. We show that a rank one chain order of $D$ is either invariant, in which case $R$ corresponds to a real-valued valuation of $D,$ or $R$ is nearly simple, in which case $R,$ $J(R)$ and $(0)$ are the only ideals of $R,$ or $R$ is exceptional in which case $R$ contains a prime ideal $Q$that is not completely prime. We use the group $\mathcal{M}(R)$ of divisorial $R{\text{-}ideals}$ of $D$ with the subgroup $\mathcal{H}(R)$ of principal $R{\text{-}ideals}$ to characterize these cases. The exceptional case subdivides further into infinitely many cases depending on the index $k$of $\mathcal{H}(R)$ in $\mathcal{M}(R).$Using the covering group $\mathbb{G}$ of $\operatorname{SL}(2,\mathbb{R} )$ and the result that the group ring $T\mathbb{G}$ is embeddable into a skew field for $T$ a skew field, examples of rank one chain orders are constructed for each possible exceptional case.
On the spectral sequence constructors of Guichardet and Stefan
Donald
W.
Barnes
2755-2769
Abstract: The concept of a spectral sequence constructor is generalised to Hopf Galois extensions. The spectral sequence constructions that are given by Guichardet for crossed product algebras are also generalised and shown to provide examples. It is shown that all spectral sequence constructors for Hopf Galois extensions construct the same spectral sequence.
Formality in an equivariant setting
Steven
Lillywhite
2771-2793
Abstract: We define and discuss $G$-formality for certain spaces endowed with an action by a compact Lie group. This concept is essentially formality of the Borel construction of the space in a category of commutative differential graded algebras over $R=H^\bullet(BG)$. These results may be applied in computing the equivariant cohomology of their loop spaces.
Large rectangular semigroups in Stone-Cech compactifications
Neil
Hindman;
Dona
Strauss;
Yevhen
Zelenyuk
2795-2812
Abstract: We show that large rectangular semigroups can be found in certain Stone-Cech compactifications. In particular, there are copies of the $2^{\mathfrak{c}}\times 2^{\mathfrak{c}}$rectangular semigroup in the smallest ideal of $(\beta\mathbb{N},+)$, and so, a semigroup consisting of idempotents can be embedded in the smallest ideal of $(\beta\mathbb{N},+)$ if and only if it is a subsemigroup of the $2^{\mathfrak{c}}\times 2^{\mathfrak{c}}$ rectangular semigroup. In fact, we show that for any ordinal $\lambda$ with cardinality at most $\mathfrak{c}$, $\beta{\mathbb{N}}$ contains a semigroup of idempotents whose rectangular components are all copies of the $2^{\mathfrak{c}}\times 2^{\mathfrak{c}}$ rectangular semigroup and form a decreasing chain indexed by $\lambda+1$, with the minimum component contained in the smallest ideal of $\beta\mathbb{N}$. As a fortuitous corollary we obtain the fact that there are $\leq_{L}$-chains of idempotents of length $\mathfrak{c}$ in $\beta \mathbb{N}$. We show also that there are copies of the direct product of the $2^{\mathfrak{c}}\times 2^{\mathfrak{c}}$ rectangular semigroup with the free group on $2^{\mathfrak{c}}$ generators contained in the smallest ideal of $\beta\mathbb{N}$.
Galois groups of quantum group actions and regularity of fixed-point algebras
Takehiko
Yamanouchi
2813-2828
Abstract: It is shown that, for a minimal and integrable action of a locally compact quantum group on a factor, the group of automorphisms of the factor leaving the fixed-point algebra pointwise invariant is identified with the intrinsic group of the dual quantum group. It is proven also that, for such an action, the regularity of the fixed-point algebra is equivalent to the cocommutativity of the quantum group.
Composition operators acting on holomorphic Sobolev spaces
Boo
Rim
Choe;
Hyungwoon
Koo;
Wayne
Smith
2829-2855
Abstract: We study the action of composition operators on Sobolev spaces of analytic functions having fractional derivatives in some weighted Bergman space or Hardy space on the unit disk. Criteria for when such operators are bounded or compact are given. In particular, we find the precise range of orders of fractional derivatives for which all composition operators are bounded on such spaces. Sharp results about boundedness and compactness of a composition operator are also given when the inducing map is polygonal.
Distributions of corank 1 and their characteristic vector fields
B.
Jakubczyk;
M.
Zhitomirskii
2857-2883
Abstract: We prove that any 1-parameter family of corank 1 distributions (or Pfaff equations) on a compact manifold $M^{n}$ is trivializable, i.e., transformable to a constant family by a family of diffeomorphisms, if all distributions of the family have the same characteristic line field. The characteristic line field is a field of tangent lines which is invariantly assigned to a corank one distribution. It is defined on $M^{n}$, if $n=2k$, or on a subset of $M^{n}$ called the Martinet hypersurface, if $n=2k+1$. Our second main result states that if two corank one distributions have the same characteristic line field and are close to each other, then they are equivalent via a diffeomorphism. This holds under a weak assumption on the singularities of the distributions. The second result implies that the abnormal curves of a distribution determine the equivalence class of the distribution, among distributions close to a given one.
When are the tangent sphere bundles of a Riemannian manifold reducible?
E.
Boeckx
2885-2903
Abstract: We determine all Riemannian manifolds for which the tangent sphere bundles, equipped with the Sasaki metric, are local or global Riemannian product manifolds.
Criteria for large deviations
Henri
Comman
2905-2923
Abstract: We give the general variational form of \begin{displaymath}\limsup(\int_X e^{h(x)/t_{\alpha}}\mu_{\alpha}(dx))^{t_{\alpha}}\end{displaymath} for any bounded above Borel measurable function $h$ on a topological space $X$, where $(\mu_{\alpha})$ is a net of Borel probability measures on $X$, and $(t_{\alpha})$ a net in $]0,\infty[$ converging to $0$. When $X$ is normal, we obtain a criterion in order to have a limit in the above expression for all $h$ continuous bounded, and deduce new criteria of a large deviation principle with not necessarily tight rate function; this allows us to remove the tightness hypothesis in various classical theorems.
Integration by parts formulas involving generalized Fourier-Feynman transforms on function space
Seung
Jun
Chang;
Jae
Gil
Choi;
David
Skoug
2925-2948
Abstract: In an upcoming paper, Chang and Skoug used a generalized Brownian motion process to define a generalized analytic Feynman integral and a generalized analytic Fourier-Feynman transform. In this paper we establish several integration by parts formulas involving generalized Feynman integrals, generalized Fourier-Feynman transforms, and the first variation of functionals of the form $F(x)=f(\langle {\alpha _{1} , x}\rangle, \dots , \langle {\alpha _{n} , x}\rangle )$ where $\langle {\alpha ,x}\rangle$ denotes the Paley-Wiener-Zygmund stochastic integral $\int _{0}^{T} \alpha (t) d x(t)$.
Thermodynamic formalism for countable to one Markov systems
Michiko
Yuri
2949-2971
Abstract: For countable to one transitive Markov systems we establish thermodynamic formalism for non-Hölder potentials in nonhyperbolic situations. We present a new method for the construction of conformal measures that satisfy the weak Gibbs property for potentials of weak bounded variation and show the existence of equilibrium states equivalent to the weak Gibbs measures. We see that certain periodic orbits cause a phase transition, non-Gibbsianness and force the decay of correlations to be slow. We apply our results to higher-dimensional maps with indifferent periodic points.
Strongly indefinite functionals and multiple solutions of elliptic systems
D.
G.
De Figueiredo;
Y.
H.
Ding
2973-2989
Abstract: We study existence and multiplicity of solutions of the elliptic system \begin{displaymath}\begin{cases}-\Delta u =H_u(x,u,v) & \text{in} \Omega, -\D... ...\quad u(x) =v(x)=0 \quad\text{on} \partial \Omega ,\end{cases}\end{displaymath} where $\Omega\subset\mathbb{R}^N, N\geq 3$, is a smooth bounded domain and $H\in \mathcal{C}^1(\bar{\Omega}\times\mathbb{R}^2, \mathbb{R})$. We assume that the nonlinear term \begin{displaymath}H(x,u,v)\sim \vert u\vert^p + \vert v\vert^q + R(x,u,v) \te... ...ert\to\infty}\frac{R(x,u,v)}{\vert u\vert^p+\vert v\vert^q}=0, \end{displaymath} where $p\in (1, 2^*)$, $2^*:=2N/(N-2)$, and $q\in (1, \infty)$. So some supercritical systems are included. Nontrivial solutions are obtained. When $H(x,u,v)$ is even in $(u,v)$, we show that the system possesses a sequence of solutions associated with a sequence of positive energies (resp. negative energies) going toward infinity (resp. zero) if $p>2$ (resp. $p<2$). All results are proved using variational methods. Some new critical point theorems for strongly indefinite functionals are proved.
Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems
F.
Rousset
2991-3008
Abstract: We consider an initial boundary value problem for a symmetrizable mixed hyperbolic-parabolic system of conservation laws with a small viscosity $\varepsilon$, $u^\varepsilon_t+F(u^\varepsilon)_x =\varepsilon(B(u^\varepsilon) u^\varepsilon_x )_x .$ When the boundary is noncharacteristic for both the viscous and the inviscid system, and the boundary condition dissipative, we show that $u^\varepsilon$ converges to a solution of the inviscid system before the formation of shocks if the amplitude of the boundary layer is sufficiently small. This generalizes previous results obtained for $B$ invertible and the linear study of Serre and Zumbrun obtained for a pure Dirichlet's boundary condition.
Central Kähler metrics
Gideon
Maschler
2161-2182
Abstract: The determinant of the Ricci endomorphism of a Kähler metric is called its central curvature, a notion well-defined even in the Riemannian context. This work investigates two types of Kähler metrics in which this curvature potential gives rise to a potential for a gradient holomorphic vector field. These metric types generalize the Kähler-Einstein notion as well as that of Bando and Mabuchi (1986). Whenever possible the central curvature is treated in analogy with the scalar curvature, and the metrics are compared with the extremal Kähler metrics of Calabi. An analog of the Futaki invariant is employed, both invariants belonging to a family described in the language of holomorphic equivariant cohomology. It is shown that one of the metric types realizes the minimum of an $L^2$ functional defined on the space of Kähler metrics in a given Kähler class. For metrics of constant central curvature, results are obtained regarding existence, uniqueness and a partial classification in complex dimension two. Consequently, on a manifold of Fano type, such metrics and Kähler-Einstein metrics can only exist concurrently. An existence result for the case of non-constant central curvature is stated, and proved in a sequel to this work.
Central Kähler metrics with non-constant central curvature
Andrew
D.
Hwang;
Gideon
Maschler
2183-2203
Abstract: The central curvature of a Riemannian metric is the determinant of its Ricci endomorphism, while the scalar curvature is its trace. A Kähler metric is called central if the gradient of its central curvature is a holomorphic vector field. Such metrics may be viewed as analogs of the extremal Kähler metrics defined by Calabi. In this work, central metrics of non-constant central curvature are constructed on various ruled surfaces, most notably the first Hirzebruch surface. This is achieved via the momentum construction of Hwang and Singer, a variant of an ansatz employed by Calabi (1979) and by Koiso and Sakane (1986). Non-existence, real-analyticity and positivity properties of central metrics arising in this ansatz are also established.
On a measure in Wiener space and applications
K.
S.
Ryu;
M.
K.
Im
2205-2222
Abstract: In this article, we consider a measure in Wiener space, induced by the sum of measures associated with an uncountable set of positive real numbers, and investigate the basic properties of this measure. We apply this measure to the various theories related to Wiener space. In particular, we can obtain a partial answer to Johnson and Skoug's open problems, raised in their 1979 paper. Moreover, we can improve and clarify some theories related to Wiener space.
Contractive projections and operator spaces
Matthew
Neal;
Bernard
Russo
2223-2262
Abstract: Parallel to the study of finite-dimensional Banach spaces, there is a growing interest in the corresponding local theory of operator spaces. We define a family of Hilbertian operator spaces $H_n^k$, $1\le k\le n$, generalizing the row and column Hilbert spaces $R_n$ and $C_n$, and we show that an atomic subspace $X\subset B(H)$ that is the range of a contractive projection on $B(H)$is isometrically completely contractive to an $\ell^\infty$-sum of the $H_n^k$ and Cartan factors of types 1 to 4. In particular, for finite-dimensional $X$, this answers a question posed by Oikhberg and Rosenthal. Explicit in the proof is a classification up to complete isometry of atomic w$^*$-closed $JW^*$-triples without an infinite-dimensional rank 1 w$^*$-closed ideal.
Non-crossing cumulants of type B
Philippe
Biane;
Frederick
Goodman;
Alexandru
Nica
2263-2303
Abstract: We establish connections between the lattices of non-crossing partitions of type B introduced by V. Reiner, and the framework of the free probability theory of D. Voiculescu. Lattices of non-crossing partitions (of type A, up to now) have played an important role in the combinatorics of free probability, primarily via the non-crossing cumulants of R. Speicher. Here we introduce the concept of non-crossing cumulant of type B; the inspiration for its definition is found by looking at an operation of ``restricted convolution of multiplicative functions'', studied in parallel for functions on symmetric groups (in type A) and on hyperoctahedral groups (in type B). The non-crossing cumulants of type B live in an appropriate framework of ``non-commutative probability space of type B'', and are closely related to a type B analogue for the R-transform of Voiculescu (which is the free probabilistic counterpart of the Fourier transform). By starting from a condition of ``vanishing of mixed cumulants of type B'', we obtain an analogue of type B for the concept of free independence for random variables in a non-commutative probability space.
Four-weight spin models and Jones pairs
Ada
Chan;
Chris
Godsil;
Akihiro
Munemasa
2305-2325
Abstract: We introduce and discuss Jones pairs. These provide a generalization and a new approach to the four-weight spin models of Bannai and Bannai. We show that each four-weight spin model determines a ``dual'' pair of association schemes.
An elementary invariant problem and general linear group cohomology restricted to the diagonal subgroup
Marian
F.
Anton
2327-2340
Abstract: Conjecturally, for $p$ an odd prime and $R$ a certain ring of $p$-integers, the stable general linear group $GL(R)$ and the étale model for its classifying space have isomorphic mod $p$ cohomology rings. In particular, these two cohomology rings should have the same image with respect to the restriction map to the diagonal subgroup. We show that a strong unstable version of this last property holds for any rank if $p$ is regular and certain homology classes for $SL_2(R)$ vanish. We check that this criterion is satisfied for $p=3$ as evidence for the conjecture.
Induction theorems of surgery obstruction groups
Masaharu
Morimoto
2341-2384
Abstract: Let $G$ be a finite group. It is well known that a Mackey functor $\{ H \mapsto M(H) \}$ is a module over the Burnside ring functor $\{ H \mapsto \Omega(H) \}$, where $H$ ranges over the set of all subgroups of $G$. For a fixed homomorphism $w : G \to \{ -1, 1 \}$, the Wall group functor $\{ H \mapsto L_n^h ({\mathbb Z}[H], w\vert _H) \}$ is not a Mackey functor if $w$ is nontrivial. In this paper, we show that the Wall group functor is a module over the Burnside ring functor as well as over the Grothendieck-Witt ring functor $\{ H \mapsto {\mathrm{GW}}_0 ({\mathbb Z}, H) \}$. In fact, we prove a more general result, that the functor assigning the equivariant surgery obstruction group on manifolds with middle-dimensional singular sets to each subgroup of $G$ is a module over the Burnside ring functor as well as over the special Grothendieck-Witt ring functor. As an application, we obtain a computable property of the functor described with an element in the Burnside ring.
Finiteness theorems for positive definite $n$-regular quadratic forms
Wai Kiu
Chan;
Byeong-Kweon
Oh
2385-2396
Abstract: An integral quadratic form $f$ of $m$ variables is said to be $n$-regular if $f$ globally represents all quadratic forms of $n$ variables that are represented by the genus of $f$. For any $n \geq 2$, it is shown that up to equivalence, there are only finitely many primitive positive definite integral quadratic forms of $n + 3$variables that are $n$-regular. We also investigate similar finiteness results for almost $n$-regular and spinor $n$-regular quadratic forms. It is shown that for any $n \geq 2$, there are only finitely many equivalence classes of primitive positive definite spinor or almost $n$-regular quadratic forms of $n + 2$ variables. These generalize the finiteness result for 2-regular quaternary quadratic forms proved by Earnest (1994).
On Ramanujan's continued fraction for $(q^2;q^3)_{\infty}/(q;q^3)_{\infty}$
George
E.
Andrews;
Bruce
C.
Berndt;
Jaebum
Sohn;
Ae Ja
Yee;
Alexandru
Zaharescu
2397-2411
Abstract: The continued fraction in the title is perhaps the deepest of Ramanujan's $q$-continued fractions. We give a new proof of this continued fraction, more elementary and shorter than the only known proof by Andrews, Berndt, Jacobsen, and Lamphere. On page 45 in his lost notebook, Ramanujan states an asymptotic formula for a continued fraction generalizing that in the title. The second main goal of this paper is to prove this asymptotic formula.
A positive radial product formula for the Dunkl kernel
Margit
Rösler
2413-2438
Abstract: It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial here means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.
Stationary sets for the wave equation in crystallographic domains
Mark
L.
Agranovsky;
Eric
Todd
Quinto
2439-2451
Abstract: Let $W$ be a crystallographic group in $\mathbb R^n$ generated by reflections and let $\Omega$ be the fundamental domain of $W.$ We characterize stationary sets for the wave equation in $\Omega$ when the initial data is supported in the interior of $\Omega.$ The stationary sets are the sets of time-invariant zeros of nontrivial solutions that are identically zero at $t=0$. We show that, for these initial data, the $(n-1)$-dimensional part of the stationary sets consists of hyperplanes that are mirrors of a crystallographic group $\tilde W$, $W<\tilde W.$ This part comes from a corresponding odd symmetry of the initial data. In physical language, the result is that if the initial source is localized strictly inside of the crystalline $\Omega$, then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one.
On $L^{p}$ continuity of singular Fourier integral operators
Andrew
Comech;
Scipio
Cuccagna
2453-2476
Abstract: We derive $L^{p}$ continuity of Fourier integral operators with one-sided fold singularities. The argument is based on interpolation of (asymptotics of) $L^{2}$ estimates and $\matheurm{H}^1\to L^1$ estimates. We derive the latter estimates elaborating arguments of Seeger, Sogge, and Stein's 1991 paper. We apply our results to the study of the $L^{p}$ regularity properties of the restrictions of solutions to hyperbolic equations onto timelike hypersurfaces and onto hypersurfaces with characteristic points.
Regularity of weak solutions to the Monge--Ampère equation
Cristian
E.
Gutiérrez;
David
Hartenstine
2477-2500
Abstract: We study the properties of generalized solutions to the Monge-Ampère equation $\det D^2 u = \nu$, where the Borel measure $\nu$ satisfies a condition, introduced by Jerison, that is weaker than the doubling property. When $\nu = f \, dx$, this condition, which we call $D_{\epsilon}$, admits the possibility of $f$ vanishing or becoming infinite. Our analysis extends the regularity theory (due to Caffarelli) available when $0 < \lambda \leq f \leq \Lambda < \infty$, which implies that $\nu = f \, dx$is doubling. The main difference between the $D_{\epsilon}$ case and the case when $f$ is bounded between two positive constants is the need to use a variant of the Aleksandrov maximum principle (due to Jerison) and some tools from convex geometry, in particular the Hausdorff metric.
On Ginzburg's bivariant Chern classes
Shoji
Yokura
2501-2521
Abstract: The convolution product is an important tool in geometric representation theory. Ginzburg constructed the ``bivariant" Chern class operation from a certain convolution algebra of Lagrangian cycles to the convolution algebra of Borel-Moore homology. In this paper we prove a ``constructible function version" of one of Ginzburg's results; motivated by its proof, we introduce another bivariant algebraic homology theory $s\mathbb{AH}$ on smooth morphisms of nonsingular varieties and show that the Ginzburg bivariant Chern class is the unique Grothendieck transformation from the Fulton-MacPherson bivariant theory of constructible functions to this new bivariant algebraic homology theory, modulo a reasonable conjecture. Furthermore, taking a hint from this conjecture, we introduce another bivariant theory $\mathbb{GF}$ of constructible functions, and we show that the Ginzburg bivariant Chern class is the unique Grothendieck transformation from $\mathbb{GF}$ to $s\mathbb{AH}$satisfying the ``normalization condition" and that it becomes the Chern-Schwartz-MacPherson class when restricted to the morphisms to a point.
Construction of $t$-structures and equivalences of derived categories
Leovigildo Alonso
Tarrío;
Ana Jeremías
López;
María José
Souto
Salorio
2523-2543
Abstract: We associate a $t$-structure to a family of objects in $\boldsymbol{\mathsf{D}}(\mathcal{A})$, the derived category of a Grothendieck category $\mathcal{A}$. Using general results on $t$-structures, we give a new proof of Rickard's theorem on equivalence of bounded derived categories of modules. Also, we extend this result to bounded derived categories of quasi-coherent sheaves on separated divisorial schemes obtaining, in particular, Be{\u{\i}}\kern.15emlinson's equivalences.
A $C^1$ function for which the $\omega$-limit points are not contained in the closure of the periodic points
Emma
D'Aniello;
T.
H.
Steele
2545-2556
Abstract: We develop a $C^1$ function $f: [- \frac{1}{6}, 1] \rightarrow [- \frac{1}{6}, 1]$ for which $\Lambda(f) \not= \overline{P(f)}$. This answers a query from Block and Coppel (1992).
Tame sets, dominating maps, and complex tori
Gregery
T.
Buzzard
2557-2568
Abstract: A discrete subset of $\mathbb C^n$ is said to be tame if there is an automorphism of $\mathbb C^n$ taking the given discrete subset to a subset of a complex line; such tame sets are known to allow interpolation by automorphisms. We give here a fairly general sufficient condition for a discrete set to be tame. In a related direction, we show that for certain discrete sets in $\mathbb C^n$ there is an injective holomorphic map from $\mathbb C^n$ into itself whose image avoids an $\epsilon$-neighborhood of the discrete set. Among other things, this is used to show that, given any complex $n$-torus and any finite set in this torus, there exist an open set containing the finite set and a locally biholomorphic map from $\mathbb C^n$ into the complement of this open set.
Fixed points of commuting holomorphic mappings other than the Wolff point
Filippo
Bracci
2569-2584
Abstract: Let $\Delta$ be the unit disc of $\mathbb C$ and let $f,g \in \mathrm{Hol}(\Delta,\Delta)$ be such that $f \circ g = g \circ f$. For $A>1$, let $\mathrm{Fix}_A (f)$. In particular, we prove that $g(\mathrm{Fix}_A (f))\subseteq \mathrm{Fix}_A (f)$. As a consequence, besides conditions for $\mathrm{Fix}_A(f) \cap \mathrm{Fix}_A(g) \neq \emptyset$, we prove a conjecture of C. Cowen in case $f$ and $g$ are univalent mappings.
Extender-based Radin forcing
Carmi
Merimovich
1729-1772
Abstract: We define extender sequences, generalizing measure sequences of Radin forcing. Using the extender sequences, we show how to combine the Gitik-Magidor forcing for adding many Prikry sequences with Radin forcing. We show that this forcing satisfies a Prikry-like condition, destroys no cardinals, and has a kind of properness. Depending on the large cardinals we start with, this forcing can blow the power of a cardinal together with changing its cofinality to a prescribed value. It can even blow the power of a cardinal while keeping it regular or measurable.
Castelnuovo-Mumford regularity and extended degree
Maria Evelina
Rossi;
Ngô
Viêt
Trung;
Giuseppe
Valla
1773-1786
Abstract: Our main result shows that the Castelnuovo-Mumford regularity of the tangent cone of a local ring $A$ is effectively bounded by the dimension and any extended degree of $A$. From this it follows that there are only a finite number of Hilbert-Samuel functions of local rings with given dimension and extended degree.
On a problem of W. J. LeVeque concerning metric diophantine approximation
Michael
Fuchs
1787-1801
Abstract: We consider the diophantine approximation problem \begin{displaymath}\left\vert x-\frac{p}{q}\right\vert\leq\frac{f(\log q)}{q^2} \end{displaymath} where $f$ is a fixed function satisfying suitable assumptions. Suppose that $x$ is randomly chosen in the unit interval. In a series of papers that appeared in earlier issues of this journal, LeVeque raised the question of whether or not the central limit theorem holds for the solution set of the above inequality (compare also with some work of Erdos). Here, we are going to extend and solve LeVeque's problem.
Cyclotomic units and Stickelberger ideals of global function fields
Jaehyun
Ahn;
Sunghan
Bae;
Hwanyup
Jung
1803-1818
Abstract: In this paper, we define the group of cyclotomic units and Stickelberger ideals in any subfield of the cyclotomic function field. We also calculate the index of the group of cyclotomic units in the total unit group in some special cases and the index of Stickelberger ideals in the integral group ring.
Humbert surfaces and the Kummer plane
Christina
Birkenhake;
Hannes
Wilhelm
1819-1841
Abstract: A Humbert surface is a hypersurface of the moduli space $\mathcal A_2$ of principally polarized abelian surfaces defined by an equation of the form $az_1+bz_2+cz_3+d(z_2^2-z_1z_3)+e=0$ with integers $a,\ldots,e$. We give geometric characterizations of such Humbert surfaces in terms of the presence of certain curves on the associated Kummer plane. Intriguingly this shows that a certain plane configuration of lines and curves already carries all information about principally polarized abelian surfaces admitting a symmetric endomorphism with given discriminant.
The stringy E-function of the moduli space of rank 2 bundles over a Riemann surface of genus 3
Young-Hoon
Kiem
1843-1856
Abstract: We compute the stringy E-function (or the motivic integral) of the moduli space of rank 2 bundles over a Riemann surface of genus 3. In doing so, we answer a question of Batyrev about the stringy E-functions of the GIT quotients of linear representations.
Hyperbolic $2$-spheres with conical singularities, accessory parameters and Kähler metrics on ${\mathcal{M}}_{0,n}$
Leon
Takhtajan;
Peter
Zograf
1857-1867
Abstract: We show that the real-valued function $S_\alpha$ on the moduli space ${\mathcal{M}}_{0,n}$ of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic $2$-sphere with $n\geq 3$ conical singularities of arbitrary orders $\alpha=\{\alpha_1,\dots, \alpha_n\}$, generates accessory parameters of the associated Fuchsian differential equation as their common antiderivative. We introduce a family of Kähler metrics on ${\mathcal{M}}_{0,n}$ parameterized by the set of orders $\alpha$, explicitly relate accessory parameters to these metrics, and prove that the functions $S_\alpha$ are their Kähler potentials.
Steenrod operations in Chow theory
Patrick
Brosnan
1869-1903
Abstract: An action of the Steenrod algebra is constructed on the mod $p$ Chow theory of varieties over a field of characteristic different from $p$, answering a question posed in Fulton's Intersection Theory. The action agrees with the action of the Steenrod algebra used by Voevodsky in his proof of the Milnor conjecture. However, the construction uses only basic functorial properties of equivariant intersection theory.
Mappings of finite distortion: The sharp modulus of continuity
Pekka
Koskela;
Jani
Onninen
1905-1920
Abstract: We establish an essentially sharp modulus of continuity for mappings of subexponentially integrable distortion.
Holomorphic extensions from open families of circles
Josip
Globevnik
1921-1931
Abstract: For a circle $\Gamma =\{ z\in \mathbb{C}\colon \vert z-c\vert=\rho \}$ write $\Lambda (\Gamma )=\{ (z,w)\colon (z-a)(w-\overline{a}) =\rho ^{2}, 0<\vert z-a\vert<\rho \}$. A continuous function $f$ on $\Gamma$ extends holomorphically from $\Gamma$(into the disc bounded by $\Gamma$) if and only if the function $F(z,\overline{z})=f(z)$ defined on $\{(z,\overline{z})\colon z\in \Gamma \}$ has a bounded holomorphic extension into $\Lambda (\Gamma )$. In the paper we consider open connected families of circles $\mathcal{C}$, write $U=\bigcup \{ \Gamma \colon \Gamma \in \mathcal{C}\}$, and assume that a continuous function on $U$ extends holomorphically from each $\Gamma \in \mathcal{C}$. We show that this happens if and only if the function $F(z, \overline{z})=f(z)$ defined on $\{ (z,\overline{z})\colon z\in U\}$ has a bounded holomorphic extension into the domain $\bigcup \{ \Lambda (\Gamma )\colon \Gamma \in \mathcal{Q}\}$ for each open family $\mathcal{Q}$ compactly contained in $\mathcal{C}$. This allows us to use known facts from several complex variables. In particular, we use the edge of the wedge theorem to prove a theorem on real analyticity of such functions.
Ricci flatness of asymptotically locally Euclidean metrics
Lei
Ni;
Yuguang
Shi;
Luen-Fai
Tam
1933-1959
Abstract: In this article we study the metric property and the function theory of asymptotically locally Euclidean (ALE) Kähler manifolds. In particular, we prove the Ricci flatness under the assumption that the Ricci curvature of such manifolds is either nonnegative or nonpositive. The result provides a generalization of previous gap type theorems established by Greene and Wu, Mok, Siu and Yau, etc. It can also be thought of as a general positive mass type result. The method also proves the Liouville properties of plurisubharmonic functions on such manifolds. We also give a characterization of Ricci flatness of an ALE Kähler manifold with nonnegative Ricci curvature in terms of the structure of its cone at infinity.
Potential theory on Lipschitz domains in Riemannian manifolds: The case of Dini metric tensors
Marius
Mitrea;
Michael
Taylor
1961-1985
Abstract: We study the applicability of the method of layer potentials in the treatment of boundary value problems for the Laplace-Beltrami operator on Lipschitz sub-domains of Riemannian manifolds, in the case when the metric tensor $g_{jk} dx_j\otimes dx_k$ has low regularity. Under the assumption that \begin{displaymath}\vert g_{jk}(x)-g_{jk}(y)\vert\leq C\,\omega(\vert x-y\vert),\end{displaymath} where the modulus of continuity $\omega$ satisfies a Dini-type condition, we prove the well-posedness of the classical Dirichlet and Neumann problems with $L^p$ boundary data, for sharp ranges of $p$'s and with optimal nontangential maximal function estimates.
Metric character of Hamilton--Jacobi equations
Antonio
Siconolfi
1987-2009
Abstract: We deal with the metrics related to Hamilton-Jacobi equations of eikonal type. If no convexity conditions are assumed on the Hamiltonian, these metrics are expressed by an $\inf$-$\sup$ formula involving certain level sets of the Hamiltonian. In the case where these level sets are star-shaped with respect to 0, we study the induced length metric and show that it coincides with the Finsler metric related to a suitable convexification of the equation.
Functorial Hodge identities and quantization
M.
J.
Slupinski
2011-2046
Abstract: By a uniform abstract procedure, we obtain integrated forms of the classical Hodge identities for Riemannian, Kähler and hyper-Kähler manifolds, as well as of the analogous identities for metrics of arbitrary signature. These identities depend only on the type of geometry and, for each of the three types of geometry, define a multiplicative functor from the corresponding category of real, graded, flat vector bundles to the category of infinite-dimensional $\mathbf{Z}_{2}$-projective representations of an algebraic structure. We define new multiplicative numerical invariants of closed Kähler and hyper-Kähler manifolds which are invariant under deformations of the metric.
Non-solvability for a class of left-invariant second-order differential operators on the Heisenberg group
Detlef
Müller;
Marco
M.
Peloso
2047-2064
Abstract: We study the question of local solvability for second-order, left-invariant differential operators on the Heisenberg group $\mathbb{H} _n$, of the form \begin{displaymath}\mathcal{P}_\Lambda= \sum_{i,j=1}^{n} \lambda_{ij}X_i Y_j={\,}^t X\Lambda Y, \end{displaymath} where $\Lambda=(\lambda_{ij})$ is a complex $n\times n$matrix. Such operators never satisfy a cone condition in the sense of Sjöstrand and Hörmander. We may assume that $\mathcal{P}_\Lambda$ cannot be viewed as a differential operator on a lower-dimensional Heisenberg group. Under the mild condition that $\operatorname{Re}\Lambda,$ $\operatorname{Im}\Lambda$ and their commutator are linearly independent, we show that $\mathcal{P}_\Lambda$ is not locally solvable, even in the presence of lower-order terms, provided that $n\ge7$. In the case $n=3$ we show that there are some operators of the form described above that are locally solvable. This result extends to the Heisenberg group $\mathbb{H} _3$ a phenomenon first observed by Karadzhov and Müller in the case of $\mathbb{H} _2.$ It is interesting to notice that the analysis of the exceptional operators for the case $n=3$turns out to be more elementary than in the case $n=2.$When $3\le n\le 6$ the analysis of these operators seems to become quite complex, from a technical point of view, and it remains open at this time.
Heat kernels on metric measure spaces and an application to semilinear elliptic equations
Alexander
Grigor'yan;
Jiaxin
Hu;
Ka-Sing
Lau
2065-2095
Abstract: We consider a metric measure space $(M,d,\mu )$ and a heat kernel $p_{t}(x,y)$ on $M$ satisfying certain upper and lower estimates, which depend on two parameters $\alpha$ and $\beta$. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space $(M,d,\mu )$. Namely, $\alpha$ is the Hausdorff dimension of this space, whereas $\beta$, called the walk dimension, is determined via the properties of the family of Besov spaces $W^{\sigma ,2}$ on $M$. Moreover, the parameters $\alpha$ and $\beta$ are related by the inequalities $2\leq \beta \leq \alpha +1$. We prove also the embedding theorems for the space $W^{\beta /2,2}$, and use them to obtain the existence results for weak solutions to semilinear elliptic equations on $M$ of the form \begin{displaymath}-\mathcal{L}u+f(x,u)=g(x), \end{displaymath} where $\mathcal{L}$ is the generator of the semigroup associated with $p_{t}$. The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpinski carpet in ${\mathbb{R}^{n}}$.
A black-box group algorithm for recognizing finite symmetric and alternating groups, I
Robert
Beals;
Charles
R.
Leedham-Green;
Alice
C.
Niemeyer;
Cheryl
E.
Praeger;
Ákos
Seress
2097-2113
Abstract: We present a Las Vegas algorithm which, for a given black-box group known to be isomorphic to a symmetric or alternating group, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm has applications in computations with matrix groups and permutation groups. In this paper, we handle the case when the degree $n$ of the standard permutation representation is part of the input. In a sequel, we shall treat the case when the value of $n$ is not known in advance. As an important ingredient in the theoretical basis for the algorithm, we prove the following result about the orders of elements of $S_n$: the conditional probability that a random element $\sigma \in S_n$is an $n$-cycle, given that $\sigma^n=1$, is at least $1/10$.
Oscillation and variation for singular integrals in higher dimensions
James
T.
Campbell;
Roger
L.
Jones;
Karin
Reinhold;
Máté
Wierdl
2115-2137
Abstract: In this paper we continue our investigations of square function inequalities in harmonic analysis. Here we investigate oscillation and variation inequalities for singular integral operators in dimensions $d \geq 1$. Our estimates give quantitative information on the speed of convergence of truncations of a singular integral operator, including upcrossing and $\lambda$ jump inequalities.
Local power series quotients of commutative Banach and Fréchet algebras
Marc
P.
Thomas
2139-2160
Abstract: We consider the relationship between derivations and local power series quotients for a locally multiplicatively convex Fréchet algebra. In §2 we derive necessary conditions for a commutative Fréchet algebra to have a local power series quotient. Our main result here is Proposition 2.6, which shows that if the generating element has finite closed descent, the algebra cannot be simply a radical algebra with identity adjoined--it must have nontrivial representation theory; if the generating element does not have finite closed descent, then the algebra cannot be a Banach algebra, and the generating element must be locally nilpotent (but non-nilpotent) in an associated quotient algebra. In §3 we consider a fundamental situation which leads to local power series quotients. Let $D$ be a derivation on a commutative radical Fréchet algebra ${\mathcal{R}}^{\sharp }$ with identity adjoined. We show in Theorem 3.10 that if the discontinuity of $D$ is not concentrated in the (Jacobson) radical, then ${\mathcal{R}}^{\sharp }$ has a local power series quotient. The question of whether such a derivation can have a separating ideal so large it actually contains the identity element has been recently settled in the affirmative by C. J. Read.
Besov-Morrey spaces: Function space theory and applications to non-linear PDE
Anna
L.
Mazzucato
1297-1364
Abstract: This paper is devoted to the analysis of function spaces modeled on Besov spaces and their applications to non-linear partial differential equations, with emphasis on the incompressible, isotropic Navier-Stokes system and semi-linear heat equations. Specifically, we consider the class, introduced by Hideo Kozono and Masao Yamazaki, of Besov spaces based on Morrey spaces, which we call Besov-Morrey or BM spaces. We obtain equivalent representations in terms of the Weierstrass semigroup and wavelets, and various embeddings in classical spaces. We then establish pseudo-differential and para-differential estimates. Our results cover non-regular and exotic symbols. Although the heat semigroup is not strongly continuous on Morrey spaces, we show that its action defines an equivalent norm. In particular, homogeneous BM spaces belong to a larger class constructed by Grzegorz Karch to analyze scaling in parabolic equations. We compare Karch's results with those of Kozono and Yamazaki and generalize them by obtaining short-time existence and uniqueness of solutions for arbitrary data with subcritical regularity. We exploit pseudo-differential calculus to extend the analysis to compact, smooth, boundaryless, Riemannian manifolds. BM spaces are defined by means of partitions of unity and coordinate patches, and intrinsically in terms of functions of the Laplace operator.
Existence and uniqueness for a semilinear elliptic problem on Lipschitz domains in Riemannian manifolds II
Martin
Dindos
1365-1399
Abstract: Extending our recent work for the semilinear elliptic equation on Lipschitz domains, we study a general second-order Dirichlet problem $Lu-F(x,u)=0$ in $\Omega$. We improve our previous results by studying more general nonlinear terms $F(x,u)$ with polynomial (and in some cases exponential) growth in the variable $u$. We also study the case of nonnegative solutions.
On one-dimensional self-similar tilings and $pq$-tiles
Ka-Sing
Lau;
Hui
Rao
1401-1414
Abstract: Let $b \geq 2$ be an integer base, $\mathcal{D} = \{ 0, d_1, \cdots , d_{b-1}\} \subset \mathbb{Z}$ a digit set and $T = T(b, \mathcal{D})$the set of radix expansions. It is well known that if $T$ has nonvoid interior, then $T$ can tile $\mathbb{R}$ with some translation set $\mathcal{J}$ ($T$ is called a tile and $\mathcal{D}$ a tile digit set). There are two fundamental questions studied in the literature: (i) describe the structure of $\mathcal{J}$; (ii) for a given $b$, characterize $\mathcal{D}$ so that $T$ is a tile. We show that for a given pair $(b,\mathcal{D})$, there is a unique self-replicating translation set $\mathcal{J} \subset \mathbb{Z}$, and it has period $b^m$ for some $m \in \mathbb{N}$. This completes some earlier work of Kenyon. Our main result for (ii) is to characterize the tile digit sets for $b = pq$ when $p,q$ are distinct primes. The only other known characterization is for $b = p^l$, due to Lagarias and Wang. The proof for the $pq$ case depends on the techniques of Kenyon and De Bruijn on the cyclotomic polynomials, and also on an extension of the product-form digit set of Odlyzko.
On the capacity of sets of divergence associated with the spherical partial integral operator
Emmanuel
Montini
1415-1441
Abstract: In this article, we study the pointwise convergence of the spherical partial integral operator $S_Rf(x)=\int_{B(0,R)} \hat{f} (y) e^{2\pi ix\cdot y}dy$ when it is applied to functions with a certain amount of smoothness. In particular, for $f\in \mathcal{L}_{\alpha}^p(\mathbb{R} ^n)$, $\tfrac{n-1}{2} <\alpha\leq\tfrac{n}{p}$, $2\leq p<\tfrac{2n}{n-1}$, we prove that $S_Rf(x)\to G_{\alpha} *g(x)$ $C_{\alpha,p}$-quasieverywhere on $\mathbb{R} ^n$, where $g\in L^p({\mathbb{R} }^n )$ is such that $f=G_{\alpha}*g$ almost everywhere. A weaker version of this result in the range $0<\alpha\leq\tfrac{n-1}{2}$ as well as some related localisation principles are also obtained. For $1\leq p<2-\tfrac{1}{n}$ and $0\leq\alpha <\tfrac{(2-p)n-1}{2p}$, we construct a function $f\in\mathcal{L}_\alpha^p(\mathbb{R} ^n)$ such that $S_Rf(x)$ diverges everywhere.
Square-integrability modulo a subgroup
G.
Cassinelli;
E.
De Vito
1443-1465
Abstract: We prove a weak form of the Frobenius reciprocity theorem for locally compact groups. As a consequence, we propose a definition of square-integrable representation modulo a subgroup that clarifies the relations between coherent states, wavelet transforms and covariant localisation observables. A self-contained proof of the imprimitivity theorem for covariant positive operator-valued measures is given.
Lebesgue type decomposition of subspaces of Fourier-Stieltjes algebras
E.
Kaniuth;
A.
T.
Lau;
G.
Schlichting
1467-1490
Abstract: Let $G$ be a locally compact group and let $A(G)$ and $B(G)$ be the Fourier algebra and the Fourier-Stieltjes algebra of $G$, respectively. For any unitary representation $\pi$ of $G$, let $B_\pi(G)$ denote the $w^\ast$-closed linear subspace of $B(G)$ generated by all coefficient functions of $\pi$, and $B_\pi^0(G)$ the closure of $B_\pi(G) \cap A_c(G)$, where $A_c(G)$ consists of all functions in $A(G)$ with compact support. In this paper we present descriptions of $B_\pi^0(G)$ and its orthogonal complement $B_\pi^s(G)$ in $B_\pi(G)$, generalizing a recent result of T. Miao. We show that for some classes of locally compact groups $G$, there is a dichotomy in the sense that for arbitrary $\pi$, either $B_\pi^0(G) = \{0\}$ or $B_\pi^0(G) = A(G)$. We also characterize functions in ${\mathcal B}_\pi^0(G) = A_c(G) + B_\pi^0(G)$and study the question of whether ${\mathcal B}_\pi^0(G) = A(G)$ implies that $\pi$ weakly contains the regular representation.
Some two-step and three-step nilpotent Lie groups with small automorphism groups
S.
G.
Dani
1491-1503
Abstract: We construct examples of two-step and three-step nilpotent Lie groups whose automorphism groups are ``small'' in the sense of either not having a dense orbit for the action on the Lie group, or being nilpotent (the latter being stronger). From the results we also get new examples of compact manifolds covered by two-step simply connected nilpotent Lie groups which do not admit Anosov automorphisms.
Quadratic iterations to ${\pi}$ associated with elliptic functions to the cubic and septic base
Heng
Huat
Chan;
Kok
Seng
Chua;
Patrick
Solé
1505-1520
Abstract: In this paper, properties of the functions $A_d(q)$, $B_d(q)$ and $C_d(q)$ are derived. Specializing at $d=1$ and $2$, we construct two new quadratic iterations to $\pi$. These are analogues of previous iterations discovered by the Borweins (1987), J. M. Borwein and F. G. Garvan (1997), and H. H. Chan (2002). Two new transformations of the hypergeometric series $_2F_1(1/3,1/6;1;z)$are also derived.
Higher Weierstrass points on $X_{0}(p)$
Scott
Ahlgren;
Matthew
Papanikolas
1521-1535
Abstract: We study the arithmetic properties of higher Weierstrass points on modular curves $X_{0}(p)$ for primes $p$. In particular, for $r\in \{2, 3, 4, 5\}$, we obtain a relationship between the reductions modulo $p$ of the collection of $r$-Weierstrass points on $X_{0}(p)$ and the supersingular locus in characteristic $p$.
On extendability of group actions on compact Riemann surfaces
Emilio
Bujalance;
F.
J.
Cirre;
Marston
Conder
1537-1557
Abstract: The question of whether a given group $G$ which acts faithfully on a compact Riemann surface $X$ of genus $g\ge 2$ is the full group of automorphisms of $X$ (or some other such surface of the same genus) is considered. Conditions are derived for the extendability of the action of the group $G$ in terms of a concrete partial presentation for $G$associated with the relevant branching data, using Singerman's list of signatures of Fuchsian groups that are not finitely maximal. By way of illustration, the results are applied to the special case where $G$ is a non-cyclic abelian group.
Local geometry of singular real analytic surfaces
Daniel
Grieser
1559-1577
Abstract: Let $V\subset\mathbb{R} ^N$ be a compact real analytic surface with isolated singularities, and assume its smooth part $V_0$ is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on $\mathbb{R} ^N$. We prove: 1. Each point of $V$ has a neighborhood which is quasi-isometric (naturally and ``almost isometrically'') to a union of metric cones and horns, glued at their tips. 2. A full asymptotic expansion, for any $p\in V$, of the length of $V\cap\{q:{\rm dist\,}(q,p)=r\}$ as $r\to0$. 3. A Gauss-Bonnet Theorem, saying that each singular point contributes $1-l/(2\pi)$, where $l$ is the coefficient of the linear term in the expansion of (2). 4. The $L^2$ Stokes Theorem, selfadjointness and discreteness of the Laplace-Beltrami operator on $V_0$, an estimate on the heat kernel, and a Gauss-Bonnet Theorem for the $L^2$ Euler characteristic. As a central tool we use resolution of singularities.
Approximation of plurisubharmonic functions by multipole Green functions
Evgeny
A.
Poletsky
1579-1591
Abstract: For a strongly hyperconvex domain $D\subset{{\mathbb{C}}}^n$ we prove that multipole pluricomplex Green functions are dense in the cone in $L^1(D)$ of negative plurisubharmonic functions with zero boundary values.
Monomial bases for $q$-Schur algebras
Jie
Du;
Brian
Parshall
1593-1620
Abstract: Using the Beilinson-Lusztig-MacPherson construction of the quantized enveloping algebra of $\mathfrak{gl}_n$ and its associated monomial basis, we investigate $q$-Schur algebras $\mathbf{S}_q(n,r)$ as ``little quantum groups". We give a presentation for $\mathbf{S}_q(n,r)$ and obtain a new basis for the integral $q$-Schur algebra $S_q(n,r)$, which consists of certain monomials in the original generators. Finally, when $n\geqslant r$, we interpret the Hecke algebra part of the monomial basis for $S_q(n,r)$ in terms of Kazhdan-Lusztig basis elements.
Logmodularity and isometries of operator algebras
David
P.
Blecher;
Louis
E.
Labuschagne
1621-1646
Abstract: We generalize some facts about function algebras to operator algebras, using the ``noncommutative Shilov boundary'' or ``$C^*$-envelope'' first considered by Arveson. In the first part we study and characterize complete isometries between operator algebras. In the second part we introduce and study a notion of logmodularity for operator algebras. We also give a result on conditional expectations. Many miscellaneous applications are provided.
The $D$--module structure of $R[F]$--modules
Manuel
Blickle
1647-1668
Abstract: Let $R$ be a regular ring, essentially of finite type over a perfect field $k$. An $R$-module $\mathcal{M}$ is called a unit $R[F]$-module if it comes equipped with an isomorphism $F^{e*} \mathcal{M} \xrightarrow{ \ }\mathcal{M}$, where $F$ denotes the Frobenius map on $\operatorname{Spec}R$, and $F^{e*}$ is the associated pullback functor. It is well known that $\mathcal{M}$ then carries a natural $D_R$-module structure. In this paper we investigate the relation between the unit $R[F]$-structure and the induced $D_R$-structure on $\mathcal{M}$. In particular, it is shown that if $k$ is algebraically closed and $\mathcal{M}$ is a simple finitely generated unit $R[F]$-module, then it is also simple as a $D_R$-module. An example showing the necessity of $k$ being algebraically closed is also given.
Seiberg-Witten invariants, orbifolds, and circle actions
Scott
Jeremy
Baldridge
1669-1697
Abstract: The main result of this paper is a formula for calculating the Seiberg-Witten invariants of 4-manifolds with fixed-point-free circle actions. This is done by showing under suitable conditions the existence of a diffeomorphism between the moduli space of the 4-manifold and the moduli space of the quotient 3-orbifold. Two corollaries include the fact that $b_+ {>} 1$ $4$-manifolds with fixed-point-free circle actions are simple type and a new proof of the equality $\mathcal{SW}_{Y^3\times S^1} = \mathcal{SW}_{Y^3}$. An infinite number of $4$-manifolds with $b_+=1$ whose Seiberg-Witten invariants are still diffeomorphism invariants is constructed and studied.
Couples contacto-symplectiques
Gianluca
Bande
1699-1711
Abstract: We introduce a new geometric structure on differentiable manifolds. A contact-symplectic pair on a manifold $M$ is a pair $\left( \alpha ,\eta \right)$ where $\alpha$ is a Pfaffian form of constant class $2k+1$ and $\eta$ a $2$-form of constant class$2h$ such that $\alpha \wedge d\alpha ^{k}\wedge \eta ^{h}$ is a volume form. Each form has a characteristic foliation whose leaves are symplectic and contact manifolds respectively. These foliations are transverse and complementary. Some other differential objects are associated to it. We give a local model and several existence theorems on nilpotent Lie groups, nilmanifolds and principal torus bundles. As a deep application of this theory, we give a negative answer to the famous Reeb's problem which asks if every vector field without closed 1-codimensional transversal on a manifold having contact forms is the Reeb vector field of a contact form.
Projectively flat Finsler metrics of constant flag curvature
Zhongmin
Shen
1713-1728
Abstract: Finsler metrics on an open subset in ${R}^n$ with straight geodesics are said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). In this paper, we discuss the classification problem on projective Finsler metrics of constant flag curvature. We express them by a Taylor expansion or an algebraic formula. Many examples constructed in this paper can be used as models in Finsler geometry.
Asymptotics for the nonlinear dissipative wave equation
Tokio
Matsuyama
865-899
Abstract: We are interested in the asymptotic behaviour of global classical solutions to the initial-boundary value problem for the nonlinear dissipative wave equation in the whole space or the exterior domain outside a star-shaped obstacle. We shall treat the nonlinear dissipative term like $a_1 (1+\vert x \vert)^{-\delta} \vert u_t \vert^{\beta} u_t$ $(a_1$, $\beta$, $\delta>0)$ and prove that the energy does not in general decay. Further, we can deduce that the classical solution is asymptotically free and the local energy decays at a certain rate as the time goes to infinity.
Hölder regularity for a Kolmogorov equation
Andrea
Pascucci
901-924
Abstract: We study the interior regularity properties of the solutions to the degenerate parabolic equation, \begin{displaymath}\Delta_{x}u+b\partial_{y}u-\partial_{t}u=f, \qquad (x,y,t)\in \mathbb{R} ^{N}\times \mathbb{R}\times\mathbb{R} ,\end{displaymath} which arises in mathematical finance and in the theory of diffusion processes.
Some properties of the Schouten tensor and applications to conformal geometry
Pengfei
Guan;
Jeff
Viaclovsky;
Guofang
Wang
925-933
Abstract: The Riemannian curvature tensor decomposes into a conformally invariant part, the Weyl tensor, and a non-conformally invariant part, the Schouten tensor. A study of the $k$th elementary symmetric function of the eigenvalues of the Schouten tensor was initiated in an earlier paper by the second author, and a natural condition to impose is that the eigenvalues of the Schouten tensor are in a certain cone, $\Gamma_k^+$. We prove that this eigenvalue condition for $k \geq n/2$ implies that the Ricci curvature is positive. We then consider some applications to the locally conformally flat case, in particular, to extremal metrics of $\sigma_k$-curvature functionals and conformal quermassintegral inequalities, using the results of the first and third authors.
The double of a hyperbolic manifold and non-positively curved exotic $PL$ structures
Pedro
Ontaneda
935-965
Abstract: We give examples of non-compact finite volume real hyperbolic manifolds of dimension greater than five, such that their doubles admit at least three non-equivalent smoothable $PL$ structures, two of which admit a Riemannian metric of non-positive curvature while the third does not. We also prove that the doubles of non-compact finite volume real hyperbolic manifolds of dimension greater than four are differentiably rigid.
Non-independence of excursions of the Brownian sheet and of additive Brownian motion
Robert
C.
Dalang;
T.
Mountford
967-985
Abstract: A classical and important property of Brownian motion is that given its zero set, distinct excursions away from zero are independent. In this paper, we examine the analogous question for the Brownian sheet, and also for additive Brownian motion. Our main result is that given the level set of the Brownian sheet at level zero, distinct excursions of the sheet away from zero are not independent. In fact, given the zero set of the Brownian sheet in the entire non-negative quadrant, and the sign of all but a finite number of excursions away from zero, the signs of the remaining excursions are determined. For additive Brownian motion, we prove the following definitive result: given the zero set of additive Brownian motion and the sign of a single excursion, the signs of all other excursions are determined. In an appendix by John B. Walsh, it is shown that given the absolute value of the sheet in the entire quadrant and, in addition, the sign of the sheet at a fixed, non-random time point, then the whole sheet can be recovered.
Supercongruences between truncated $_{2}F_{1}$ hypergeometric functions and their Gaussian analogs
Eric
Mortenson
987-1007
Abstract: Fernando Rodriguez-Villegas has conjectured a number of supercongruences for hypergeometric Calabi-Yau manifolds of dimension $d\le 3$. For manifolds of dimension $d=1$, he observed four potential supercongruences. Later the author proved one of the four. Motivated by Rodriguez-Villegas's work, in the present paper we prove a general result on supercongruences between values of truncated $_{2}F_{1}$hypergeometric functions and Gaussian hypergeometric functions. As a corollary to that result, we prove the three remaining supercongruences.
Cyclic covers of rings with rational singularities
Anurag
K.
Singh
1009-1024
Abstract: We examine some recent work of Phillip Griffith on étale covers and fibered products from the point of view of tight closure theory. While it is known that cyclic covers of Gorenstein rings with rational singularities are Cohen-Macaulay, we show this is not true in general in the absence of the Gorenstein hypothesis. Specifically, we show that the canonical cover of a $\mathbb Q$-Gorenstein ring with rational singularities need not be Cohen-Macaulay.
Homological properties of balanced Cohen-Macaulay algebras
Izuru
Mori
1025-1042
Abstract: A balanced Cohen-Macaulay algebra is a connected algebra $A$ having a balanced dualizing complex $\omega_A[d]$ in the sense of Yekutieli (1992) for some integer $d$ and some graded $A$-$A$ bimodule $\omega_A$. We study some homological properties of a balanced Cohen-Macaulay algebra. In particular, we will prove the following theorem: \begin{thm0} Let$A$\space be a Noetherian balanced Cohen-Macaulay algebra, and ... ...s^{r_0}_{j=1} \omega_A(-l_{0j})\to M\to 0. \end{align*}\end{enumerate}\end{thm0} As a corollary, we will have the following characterizations of AS Gorenstein algebras and AS regular algebras: \begin{cor0} Let$A$\space be a Noetherian balanced Cohen-Macaulay algebra. \beg... ...ximal Cohen-Macaulay graded left$A$-module is free. \end{enumerate}\end{cor0}
Noetherian PI Hopf algebras are Gorenstein
Q.-S.
Wu;
J.
J.
Zhang
1043-1066
Abstract: We prove that every noetherian affine PI Hopf algebra has finite injective dimension, which answers a question of Brown (1998).
Expanding maps on infra-nilmanifolds of homogeneous type
Karel
Dekimpe;
Kyung
Bai
Lee
1067-1077
Abstract: In this paper we investigate expanding maps on infra-nilmanifolds. Such manifolds are obtained as a quotient $E\backslash L$, where $L$ is a connected and simply connected nilpotent Lie group and $E$is a torsion-free uniform discrete subgroup of $L {\mathbb o} C$, with $C$ a compact subgroup of $\operatorname{Aut}(L)$. We show that if the Lie algebra of $L$ is homogeneous (i.e., graded and generated by elements of degree 1), then the corresponding infra-nilmanifolds admit an expanding map. This is a generalization of the result of H. Lee and K. B. Lee, who treated the 2-step nilpotent case.
Derivations and invariant forms of Jordan and alternative tori
Erhard
Neher;
Yoji
Yoshii
1079-1108
Abstract: Jordan and alternative tori are the coordinate algebras of extended affine Lie algebras of types ${A}_1$ and ${A}_2$. In this paper we show that the derivation algebra of a Jordan torus is a semidirect product of the ideal of inner derivations and the subalgebra of central derivations. In the course of proving this result, we investigate derivations of the more general class of division graded Jordan and alternative algebras. We also describe invariant forms of these algebras.
Limits of interpolatory processes
W.
R.
Madych
1109-1133
Abstract: Given $N$ distinct real numbers $\nu_1, \ldots, \nu_N$ and a positive approximation of the identity $\phi_{\epsilon}$, which converges weakly to the Dirac delta measure as $\epsilon$goes to zero, we investigate the polynomials $P_{\epsilon}(x)= \sum c_{\epsilon , j} e^{-i \nu_j x}$ which solve the interpolation problem \begin{displaymath}\int P_{\epsilon}(x) e^{i \nu_k x} \phi_{\epsilon}(x)dx=f_{\epsilon,k}, \quad k=1, \ldots, N,\end{displaymath} with prescribed data $f_{\epsilon,1}, \dots, f_{\epsilon,N}$. More specifically, we are interested in the behavior of $P_{\epsilon}(x)$ when the data is of the form $f_{\epsilon, k}=\int f(x) e^{i \nu_k x} \phi_{\epsilon}(x)dx$ for some prescribed function $f$. One of our results asserts that if $f$ is sufficiently nice and $\phi_{\epsilon}$ has sufficiently well-behaved moments, then $P_{\epsilon}$ converges to a limit $P$ which can be completely characterized. As an application we identify the limits of certain fundamental interpolatory splines whose knot set is $\mathbb{Z} \setminus \mathcal{N}$, where $\mathcal{N}$ is an arbitrary finite subset of the integer lattice $\mathbb{Z}$, as their degree goes to infinity.
Maximal functions with polynomial densities in lacunary directions
Kathryn
Hare;
Fulvio
Ricci
1135-1144
Abstract: Given a real polynomial $p(t)$ in one variable such that $p(0)=0$, we consider the maximal operator in $\mathbb{R}^{2}$, \begin{displaymath}M_{p}f(x_{1},x_{2})=\sup _{h>0\,,\,i,j\in \mathbb{Z}}\frac{1... ...t f\big (x_{1}-2^{i}p(t),x_{2}-2^{j}p(t)\big )\big \vert\,dt . \end{displaymath} We prove that $M_{p}$ is bounded on $L^{q}(\mathbb{R}^{2})$ for $q>1$ with bounds that only depend on the degree of $p$.
Singular integrals with rough kernels along real-analytic submanifolds in ${\mathbf{R}}^3$
Dashan
Fan;
Kanghui
Guo;
Yibiao
Pan
1145-1165
Abstract: $L^p$ mapping properties will be established in this paper for singular Radon transforms with rough kernels defined by translates of a real-analytic submanifold in $\mathbf{R}^3$.
Spherical maximal operator on symmetric spaces of constant curvature
Amos
Nevo;
P.
K.
Ratnakumar
1167-1182
Abstract: We prove an endpoint weak-type maximal inequality for the spherical maximal operator applied to radial funcions on symmetric spaces of constant curvature and dimension $n\ge 2$. More explicitly, in the Lorentz space associated with the natural isometry-invariant measure, we show that, for every radial function $f$, \begin{displaymath}\Vert{\mathcal M}f\Vert _{\,n^{\prime},\infty}\leq C_n \Vert f \Vert _{n^{\prime},1},\,\,\,\, n^\prime=\frac{n}{n-1}.\end{displaymath} The proof uses only geometric arguments and volume estimates, and applies uniformly in every dimension.
The Mori cones of moduli spaces of pointed curves of small genus
Gavril
Farkas;
Angela
Gibney
1183-1199
Abstract: We compute the Mori cones of the moduli spaces $\overline M_{g,n}$ of $n$pointed stable curves of genus $g$, when $g$ and $n$ are relatively small. For instance we show that for $g<14$ every curve in $\overline M_g$ is equivalent to an effective combination of the components of the locus of curves with $3g-4$ nodes. We completely describe the cone of nef divisors for the space $\overline M_{0,6}$, thus verifying Fulton's conjecture for this space. Using this description we obtain a classification of all the fibrations of $\overline M_{0,6}$.
On the inversion of the convolution and Laplace transform
Boris
Baeumer
1201-1212
Abstract: We present a new inversion formula for the classical, finite, and asymptotic Laplace transform $\hat f$ of continuous or generalized functions $f$. The inversion is given as a limit of a sequence of finite linear combinations of exponential functions whose construction requires only the values of $\hat f$ evaluated on a Müntz set of real numbers. The inversion sequence converges in the strongest possible sense. The limit is uniform if $f$is continuous, it is in $L^{1}$ if $f\in L^{1}$, and converges in an appropriate norm or Fréchet topology for generalized functions $f$. As a corollary we obtain a new constructive inversion procedure for the convolution transform ${\mathcal K}:f\mapsto k\star f$; i.e., for given $g$ and $k$ we construct a sequence of continuous functions $f_{n}$ such that $k\star f_{n}\to g$.
Infinite partition regular matrices: solutions in central sets
Neil
Hindman;
Imre
Leader;
Dona
Strauss
1213-1235
Abstract: A finite or infinite matrix $A$ is image partition regular provided that whenever ${\mathbb N}$ is finitely colored, there must be some $\vec {x}$with entries from ${\mathbb N}$ such that all entries of $A\vec {x}$ are in the same color class. In contrast to the finite case, infinite image partition regular matrices seem very hard to analyze: they do not enjoy the closure and consistency properties of the finite case, and it is difficult to construct new ones from old. In this paper we introduce the stronger notion of central image partition regularity, meaning that $A$ must have images in every central subset of ${\mathbb N}$. We describe some classes of centrally image partition regular matrices and investigate the extent to which they are better behaved than ordinary image partition regular matrices. It turns out that the centrally image partition regular matrices are closed under some natural operations, and this allows us to give new examples of image partition regular matrices. In particular, we are able to solve a vexing open problem by showing that whenever ${\mathbb N}$ is finitely colored, there must exist injective sequences $\langle x_n\rangle_{n=0}^\infty$ and $\langle z_n\rangle_{n=0}^\infty$ in ${\mathbb N}$ with all sums of the forms $x_n+x_m$ and $z_n+2z_m$ with $n<m$ in the same color class. This is the first example of an image partition regular system whose regularity is not guaranteed by the Milliken-Taylor Theorem, or variants thereof.
Are Hamiltonian flows geodesic flows?
Christopher
McCord;
Kenneth
R.
Meyer;
Daniel
Offin
1237-1250
Abstract: When a Hamiltonian system has a ``Kinetic + Potential'' structure, the resulting flow is locally a geodesic flow. But there may be singularities of the geodesic structure; so the local structure does not always imply that the flow is globally a geodesic flow. In order for a flow to be a geodesic flow, the underlying manifold must have the structure of a unit tangent bundle. We develop homological conditions for a manifold to have such a structure. We apply these criteria to several classical examples: a particle in a potential well, the double spherical pendulum, the Kovalevskaya top, and the $N$-body problem. We show that the flow of the reduced planar $N$-body problem and the reduced spatial 3-body are never geodesic flows except when the angular momentum is zero and the energy is positive.
An extension theorem for separately holomorphic functions with pluripolar singularities
Marek
Jarnicki;
Peter
Pflug
1251-1267
Abstract: Let $D_j\subset\mathbb{C} ^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,\dots,N$. Put \begin{displaymath}X:=\bigcup_{j=1}^N A_1\times\dots\times A_{j-1}\times D_j\tim... ...thbb{C} ^{n_1}\times\dots\times\mathbb{C} ^{n_N}=\mathbb{C} ^n.\end{displaymath} Let $U\subset\mathbb{C} ^n$ be an open neighborhood of $X$ and let $M\subset U$ be a relatively closed subset of $U$. For $j\in\{1,\dots,N\}$ let $\Sigma_j$ be the set of all $M_{(z',\cdot,z'')}:=\{z_j\in\mathbb{C} ^{n_j}: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $\Sigma_1,\dots,\Sigma_N$ are pluripolar. Put \begin{multline*}X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1\times\dots\times A_{j-... ...imes(A_{j+1}\times\dots\times A_N): (z',z'')\notin\Sigma_j\}. \end{multline*} Then there exists a relatively closed pluripolar subset $\widehat{M}\subset\widehat X$ of the ``envelope of holomorphy'' $\widehat{X}\subset\mathbb{C} ^n$ of $X$ such that: $\bullet$ $X\setminus M$ there exists exactly one function $\widehat f$ holomorphic on $\widehat X\setminus\widehat M$ with $\widehat f=f$ on $\widehat M$ is singular with respect to the family of all functions $\widehat f$.
Hyperbolic mean growth of bounded holomorphic functions in the ball
E.
G.
Kwon
1269-1294
Abstract: We consider the hyperbolic Hardy class $\varrho H^{p}(B)$, $0<p<\infty$. It consists of $\phi$ holomorphic in the unit complex ball $B$ for which $\vert \phi \vert < 1$ and \begin{displaymath}\sup _{0<r<1} \, \int _{\partial B} \left \{ \varrho (\phi (r\zeta ), 0)\right \}^{p} \, d\sigma (\zeta ) ~<~ \infty ,\end{displaymath} where $\varrho$denotes the hyperbolic distance of the unit disc. The hyperbolic version of the Littlewood-Paley type $g$-function and the area function are defined in terms of the invariant gradient of $B$, and membership of $\varrho H^{p}(B)$ is expressed by the $L^{p}$ property of the functions. As an application, we can characterize the boundedness and the compactness of the composition operator $\mathcal{C}_{\phi }$, defined by $\mathcal{C}_{\phi }f = f\circ \phi$, from the Bloch space into the Hardy space $H^{p}(B)$.
Erratum to ``Subgroup properties of fully residually free groups''
Ilya
Kapovich
1295-1296
Group actions on graphs related to Krishnan-Sunder subfactors
Bina
Bhattacharyya
433-463
Abstract: We describe the principal graphs of the subfactors studied by Krishnan and Sunder in terms of group actions on Cayley-type graphs. This leads to the construction of a tower of tree algebras, for every positive integer $k$, which are symmetries of the Krishnan-Sunder subfactors of index $k^2$. Using our theory, we prove that the principal graph of the irreducible infinite depth subfactor of index 9 constructed by Krishnan and Sunder is not a tree, contrary to their expectations. We also show that the principal graphs of the Krishnan-Sunder subfactors of index 4 are the affine A and D Coxeter graphs.
The Laplacian MASA in a free group factor
Allan
M.
Sinclair;
Roger
R.
Smith
465-475
Abstract: The Laplacian (or radial) masa in a free group factor is generated by the sum of the generators and their inverses. We show that such a masa $\mathcal{B}$is strongly singular and has Popa invariant $\delta(\mathcal{B}) = 1$. This is achieved by proving that the conditional expectation $\mathbb{E} _{\mathcal{B}}$ onto $\mathcal{B}$ is an asymptotic homomorphism. We also obtain similar results for the free product of discrete groups, each of which contains an element of infinite order.
The co-area formula for Sobolev mappings
Jan
Maly;
David
Swanson;
William
P.
Ziemer
477-492
Abstract: We extend Federer's co-area formula to mappings $f$ belonging to the Sobolev class $W^{1,p}(\mathbb{R}^n;\mathbb{R}^m)$, $1 \le m < n$, $p>m$, and more generally, to mappings with gradient in the Lorentz space $L^{m,1}(\mathbb{R}^n)$. This is accomplished by showing that the graph of $f$ in $\mathbb{R}^{n+m}$is a Hausdorff $n$-rectifiable set.
The radius of metric regularity
A.
L.
Dontchev;
A.
S.
Lewis;
R.
T.
Rockafellar
493-517
Abstract: Metric regularity is a central concept in variational analysis for the study of solution mappings associated with ``generalized equations'', including variational inequalities and parameterized constraint systems. Here it is employed to characterize the distance to irregularity or infeasibility with respect to perturbations of the system structure. Generalizations of the Eckart-Young theorem in numerical analysis are obtained in particular.
The Orevkov invariant of an affine plane curve
Walter
D.
Neumann;
Paul
Norbury
519-538
Abstract: We show that although the fundamental group of the complement of an algebraic affine plane curve is not easy to compute, it possesses a more accessible quotient, which we call the Orevkov invariant.
Linear systems of plane curves with a composite number of base points of equal multiplicity
Anita
Buckley;
Marina
Zompatori
539-549
Abstract: In this article we study linear systems of plane curves of degree $d$ passing through general base points with the same multiplicity at each of them. These systems are known as homogeneous linear systems. We especially investigate for which of these systems, the base points, with their multiplicities, impose independent conditions and which homogeneous systems are empty. Such systems are called non-special. We extend the range of homogeneous linear systems that are known to be non-special. A theorem of Evain states that the systems of curves of degree $d$ with $4^h$ base points with equal multiplicity are non-special. The analogous result for $9^h$ points was conjectured. Both of these will follow, as corollaries, from the main theorem proved in this paper. Also, the case of $4^{h}9^{k}$ points will follow from our result. The proof uses a degeneration technique developed by C. Ciliberto and R. Miranda.
Base loci of linear series are numerically determined
Michael
Nakamaye
551-566
Abstract: We introduce a numerical invariant, called a moving Seshadri constant, which measures the local positivity of a big line bundle at a point. We then show how moving Seshadri constants determine the stable base locus of a big line bundle.
Formulas for tamely ramified supercuspidal characters of $\operatorname{GL}_3$
Tetsuya
Takahashi
567-591
Abstract: Let $F$ denote a $p$-adic local field of residual characteristic $p\ne3$. This article gives formulas, valid on the regular elliptic set, for the irreducible supercuspidal characters of $\operatorname{GL}_3(F)$ which correspond to characters of a ramified Cartan subgroup. In the case in which $F$ does not contain cube roots of unity, i.e., the case in which ramified cubic extensions of degree $3$ over $F$ cannot be Galois, base change results concerning ``simple types" due to Bushnell and Henniart (1996) are used in the proofs.
Resolutions of ideals of quasiuniform fat point subschemes of $\mathbf P^2$
Brian
Harbourne;
Sandeep
Holay;
Stephanie
Fitchett
593-608
Abstract: The notion of a quasiuniform fat point subscheme $Z\subset\mathbf P^2$is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal $I$ defining $Z$ are put forward. In a large range of cases, it is shown that the Hilbert function conjecture implies the resolution conjecture. In addition, the main result gives the first determination of the resolution of the $m$th symbolic power $I(m;n)$ of an ideal defining $n$ general points of $\mathbf P^2$ when both $m$ and $n$ are large (in particular, for infinitely many $m$ for each of infinitely many $n$, and for infinitely many $n$ for every $m>2$). Resolutions in other cases, such as ``fat points with tails'', are also given. Except where an explicit exception is made, all results hold for an arbitrary algebraically closed field $k$. As an incidental result, a bound for the regularity of $I(m;n)$ is given which is often a significant improvement on previously known bounds.
Hyperplane arrangements and linear strands in resolutions
Irena
Peeva
609-618
Abstract: The cohomology ring of the complement of a central complex hyperplane arrangement is the well-studied Orlik-Solomon algebra. The homotopy group of the complement is interesting, complicated, and few results are known about it. We study the ranks for the lower central series of such a homotopy group via the linear strand of the minimal free resolution of the field $\mathbf{C}$ over the Orlik-Solomon algebra.
Test ideals and base change problems in tight closure theory
Ian
M.
Aberbach;
Florian
Enescu
619-636
Abstract: Test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand. Our paper presents results regarding this behavior under flat maps with reasonably nice (but far from smooth) fibers. This involves analyzing, in depth, a special type of ideal of test elements, called the CS test ideal. Besides providing new results, the paper also contains extensions of a theorem by G. Lyubeznik and K. E. Smith on the completely stable test ideal and of theorems by F. Enescu and, independently, M. Hashimoto on the behavior of $F$-rationality under flat base change.
On partitioning the orbitals of a transitive permutation group
Cai
Heng
Li;
Cheryl
E.
Praeger
637-653
Abstract: Let $G$ be a permutation group on a set $\Omega$ with a transitive normal subgroup $M$. Then $G$ acts on the set $\mathrm{Orbl}(M,\Omega)$ of nontrivial $M$-orbitals in the natural way, and here we are interested in the case where $\mathrm{Orbl}(M,\Omega)$ has a partition $\mathcal P$ such that $G$ acts transitively on $\mathcal P$. The problem of characterising such tuples $(M,G,\Omega,\mathcal P)$, called TODs, arises naturally in permutation group theory, and also occurs in number theory and combinatorics. The case where $\vert\mathcal P\vert$ is a prime-power is important in algebraic number theory in the study of arithmetically exceptional rational polynomials. The case where $\vert\mathcal P\vert=2$ exactly corresponds to self-complementary vertex-transitive graphs, while the general case corresponds to a type of isomorphic factorisation of complete graphs, called a homogeneous factorisation. Characterising homogeneous factorisations is an important problem in graph theory with applications to Ramsey theory. This paper develops a framework for the study of TODs, establishes some numerical relations between the parameters involved in TODs, gives some reduction results with respect to the $G$-actions on $\Omega$ and on $\mathcal P$, and gives some construction methods for TODs.
A generalized Minkowski problem with Dirichlet boundary condition
Oliver
C.
Schnurer
655-663
Abstract: We prove the existence of hypersurfaces with prescribed boundary whose Weingarten curvature equals a given function that depends on the normal of the hypersurface.
The $L^p$ Dirichlet problem and nondivergence harmonic measure
Cristian
Rios
665-687
Abstract: We consider the Dirichlet problem \begin{displaymath}\left\{ \begin{array}{rcl} \mathcal{L} u & = & 0\quad\text{ in }D, u & = & g\quad\text{ on }\partial D \end{array}\right.\end{displaymath} for two second-order elliptic operators $\mathcal{L}_k u=\sum_{i,j=1}^na_k^{i,j}(x)\,\partial_{ij} u(x)$, $k=0,1$, in a bounded Lipschitz domain $D\subset\mathbb{R} ^n$. The coefficients $a_k^{i,j}$ belong to the space of bounded mean oscillation ${{BMO}}$ with a suitable small ${{BMO}}$ modulus. We assume that ${\mathcal{L}}_0$ is regular in $L^p(\partial D, d\sigma)$ for some $p$, $1<p<\infty$, that is, $\Vert Nu\Vert _{L^p}\le C\,\Vert g\Vert _{L^p}$ for all continuous boundary data $g$. Here $\sigma$ is the surface measure on $\partial D$ and $Nu$ is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients $\varepsilon^{i,j}(x)=a^{i,j}_1(x)-a^{i,j}_0(x)$ that will assure the perturbed operator $\mathcal{L}_1$ to be regular in $L^q(\partial D,d\sigma)$ for some $q$, $1<q<\infty$.
Estimations $L^p$ des solutions de l'équation des ondes sur certaines variétés coniques
Hong-Quan
Li;
Noël
Lohoue
689-711
Abstract: We prove R. Strichartz's $L^p$ estimates for solutions of the wave equation on some conical manifolds. RÉSUMÉ. On prouve des estimations $L^p$ pour les solutions de l'équation des ondes, analogues aux estimations de R. Strichartz, sur certaines variétés coniques.
From local to global behavior in competitive Lotka-Volterra systems
E.
C.
Zeeman;
M.
L.
Zeeman
713-734
Abstract: In this paper we exploit the linear, quadratic, monotone and geometric structures of competitive Lotka-Volterra systems of arbitrary dimension to give geometric, algebraic and computational hypotheses for ruling out non-trivial recurrence. We thus deduce the global dynamics of a system from its local dynamics. The geometric hypotheses rely on the introduction of a split Liapunov function. We show that if a system has a fixed point $p\in\operatorname{int}{{\mathbf R}^n_+}$ and the carrying simplex of the system lies to one side of its tangent hyperplane at $p$, then there is no nontrivial recurrence, and the global dynamics are known. We translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane. Finally, we derive a computational algorithm for checking the algebraic hypotheses, and we compare this algorithm with the classical Volterra-Liapunov stability theorem for Lotka-Volterra systems.
Axiom A flows with a transverse torus
C.
A.
Morales
735-745
Abstract: Let $X$ be an Axiom A flow with a transverse torus $T$ exhibiting a unique orbit $O$ that does not intersect $T$. Suppose that there is no null-homotopic closed curve in $T$ contained in either the stable or unstable set of $O$. Then we show that $X$ has either an attracting periodic orbit or a repelling periodic orbit or is transitive. In particular, an Anosov flow with a transverse torus is transitive if it has a unique periodic orbit that does not intersect the torus.
Exponential averaging for Hamiltonian evolution equations
Karsten
Matthies;
Arnd
Scheel
747-773
Abstract: We derive estimates on the magnitude of non-adiabatic interaction between a Hamiltonian partial differential equation and a high-frequency nonlinear oscillator. Assuming spatial analyticity of the initial conditions, we show that the dynamics can be transformed to the uncoupled dynamics of an infinite-dimensional Hamiltonian system and an anharmonic oscillator, up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator. The result is derived from an abstract averaging theorem for infinite-dimensional analytic evolution equations in Gevrey spaces. Refining upon a similar result by Neishtadt for analytic ordinary differential equations, the temporal estimate crucially depends on the spatial regularity of the initial condition. The result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data. An application is provided by a system of nonlinear Schrödinger equations, coupled to a rapidly forcing single mode, representing small-scale oscillations. We provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations.
On the nonexistence of closed timelike geodesics in flat Lorentz 2-step nilmanifolds
Mohammed
Guediri
775-786
Abstract: The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold $N/\Gamma ,$ where $N$ is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and $\Gamma$ a discrete subgroup of $N$ acting on $N$ by left translations. For this purpose, we shall first show that if $N$ is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric $g,$ then the restriction of $g$ to the center $Z$of $N$ is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the three-dimensional Heisenberg Lie group $H_{3}$. If $\left( N,g\right)$ is one such group, we prove that no timelike geodesic in $\left( N,g\right)$ can be translated by an element of $N.$ By the way, we rediscover that the Heisenberg Lie group $H_{2k+1}$admits a flat left-invariant Lorentz metric if and only if $k=1.$
Quasiconformal groups, Patterson-Sullivan theory, and local analysis of limit sets
Petra
Bonfert-Taylor;
Edward
C.
Taylor
787-811
Abstract: We extend the part of Patterson-Sullivan theory to discrete quasiconformal groups that relates the exponent of convergence of the Poincaré series to the Hausdorff dimension of the limit set. In doing so we define new bi-Lipschitz invariants that localize both the exponent of convergence and the Hausdorff dimension. We find these invariants help to expose and explain the discrepancy between the conformal and quasiconformal setting of Patterson-Sullivan theory.
Notes on interpolation in the generalized Schur class. II. Nudel$'$man's problem
D.
Alpay;
T.
Constantinescu;
A.
Dijksma;
J.
Rovnyak
813-836
Abstract: An indefinite generalization of Nudel$'$man's problem is used in a systematic approach to interpolation theorems for generalized Schur and Nevanlinna functions with interior and boundary data. Besides known results on existence criteria for Pick-Nevanlinna and Carathéodory-Fejér interpolation, the method yields new results on generalized interpolation in the sense of Sarason and boundary interpolation, including properties of the finite Hilbert transform relative to weights. The main theorem appeals to the Ball and Helton almost-commutant lifting theorem to provide criteria for the existence of a solution to Nudel$'$man's problem.
Analytic models for commuting operator tuples on bounded symmetric domains
Jonathan
Arazy;
Miroslav
Englis
837-864
Abstract: For a domain $\Omega$ in $\mathbb{C} ^{d}$ and a Hilbert space $\mathcal{H}$of analytic functions on $\Omega$ which satisfies certain conditions, we characterize the commuting $d$-tuples $T=(T_{1},\dots ,T_{d})$ of operators on a separable Hilbert space $H$ such that $T^{*}$ is unitarily equivalent to the restriction of $M^{*}$ to an invariant subspace, where $M$ is the operator $d$-tuple $Z\otimes I$ on the Hilbert space tensor product $\mathcal{H} \otimes H$. For $\Omega$ the unit disc and $\mathcal{H}$ the Hardy space $H^{2}$, this reduces to a well-known theorem of Sz.-Nagy and Foias; for $\mathcal{H}$a reproducing kernel Hilbert space on $\Omega \subset \mathbb{C} ^{d}$ such that the reciprocal $1/K(x,\overline{y})$ of its reproducing kernel is a polynomial in $x$ and $\overline y$, this is a recent result of Ambrozie, Müller and the second author. In this paper, we extend the latter result by treating spaces $\mathcal{H}$ for which $1/K$ ceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation) $\mathcal{H} =\mathcal{H} _{\nu }$ on a Cartan domain corresponding to the parameter $\nu$ in the continuous Wallach set, and reproducing kernel Hilbert spaces $\mathcal{H}$ for which $1/K$ is a rational function. Further, we treat also the more general problem when the operator $M$is replaced by $M\oplus W$, $W$ being a certain generalization of a unitary operator tuple. For the case of the spaces $\mathcal{H} _{\nu }$ on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on $\Omega$, which seems to be of an independent interest.
On the minimal free resolution of $n+1$ general forms
J.
Migliore;
R.
M.
Miró-Roig
1-36
Abstract: Let $R = k[x_1,\dots,x_n]$ and let $I$ be the ideal of $n+1$generically chosen forms of degrees $d_1 \leq \dots \leq d_{n+1}$. We give the precise graded Betti numbers of $R/I$ in the following cases: $n=3$; $n=4$ and $\sum_{i=1}^5 d_i$ is even; $n=4$, $\sum_{i=1}^{5} d_i$ is odd and $d_2 + d_3 + d_4 < d_1 + d_5 + 4$; $n$ is even and all generators have the same degree, $a$, which is even; $(\sum_{i=1}^{n+1} d_i) -n$ is even and $d_2 + \dots + d_n < d_1 + d_{n+1} + n$; $(\sum_{i=1}^{n+1} d_i) - n$ is odd, $n \geq 6$ is even, $d_2 + \dots+d_n < d_1 + d_{n+1} + n$ and $d_1 + \dots + d_n - d_{n+1} - n \gg 0$. We give very good bounds on the graded Betti numbers in many other cases. We also extend a result of M. Boij by giving the graded Betti numbers for a generic compressed Gorenstein algebra (i.e., one for which the Hilbert function is maximal, given $n$ and the socle degree) when $n$ is even and the socle degree is large. A recurring theme is to examine when and why the minimal free resolution may be forced to have redundant summands. We conjecture that if the forms all have the same degree, then there are no redundant summands, and we present some evidence for this conjecture.
Iitaka's fibrations via multiplier ideals
Shigeharu
Takayama
37-47
Abstract: We give a new characterization of Iitaka's fibration of algebraic varieties associated to line bundles. Introducing an ``intersection number'' of line bundles and curves by using the notion of multiplier ideal sheaves, Iitaka's fibration can be regarded as a ``numerically trivial fibration'' in terms of this intersection theory.
Nondegenerate multidimensional matrices and instanton bundles
Laura
Costa;
Giorgio
Ottaviani
49-55
Abstract: In this paper we prove that the moduli space of rank $2n$ symplectic instanton bundles on ${\mathbb{P} ^{2n+1}}$, defined from the well-known monad condition, is affine. This result was not known even in the case $n=1$, where by Atiyah, Drinfeld, Hitchin, and Manin in 1978 the real instanton bundles correspond to self-dual Yang Mills $Sp(1)$-connections over the $4$-dimensional sphere. The result is proved as a consequence of the existence of an invariant of the multidimensional matrices representing the instanton bundles.
Inverse problem for upper asymptotic density
Renling
Jin
57-78
Abstract: For a set $A$ of natural numbers, the structural properties are described when the upper asymptotic density of $2A+\{0,1\}$achieves the infimum of the upper asymptotic densities of all sets of the form $2B+\{0,1\}$, where the upper asymptotic density of $B$ is greater than or equal to the upper asymptotic density of $A$. As a corollary, we prove that if the upper asymptotic density of $A$ is less than $1$and the upper asymptotic density of $2A+\{0,1\}$ achieves the infimum, then the lower asymptotic density of $A$ must be $0$.
Abelian groups with layered tiles and the sumset phenomenon
Renling
Jin;
H.
Jerome
Keisler
79-97
Abstract: We prove a generalization of the main theorem in Jin, The sumset phenomenon, about the sumset phenomenon in the setting of an abelian group with layered tiles of cell measures. Then we give some applications of the theorem for multi-dimensional cases of the sumset phenomenon. Several examples are given in order to show that the applications obtained are not vacuous and cannot be improved in various directions. We also give a new proof of Shnirel'man's theorem to illustrate a different approach (which uses the sumset phenomenon) to some combinatorial problems.
On the Jacobi group and the mapping class group of $S^3\times S^3$
Nikolai
A.
Krylov
99-117
Abstract: The paper contains a proof that the mapping class group of the manifold $S^3\times S^3$ is isomorphic to a central extension of the (full) Jacobi group $\Gamma^J$by the group of 7-dimensional homotopy spheres. Using a presentation of the group $\Gamma^J$ and the $\mu$-invariant of the homotopy spheres, we give a presentation of this mapping class group with generators and defining relations. We also compute the cohomology of the group $\Gamma^J$ and determine 2-cocycles that correspond to the mapping class group of $S^3\times S^3$.
A higher Lefschetz formula for flat bundles
Moulay-Tahar
Benameur
119-142
Abstract: In this paper, we prove a fixed point formula for flat bundles. To this end, we use cyclic cocycles which are constructed out of closed invariant currents. We show that such cyclic cocycles are equivariant with respect to isometric longitudinal actions of compact Lie groups. This enables us to prove fixed point formulae in the cyclic homology of the smooth convolution algebra of the foliation.
Levi foliations in pseudoconvex boundaries and vector fields that commute approximately with $\bar\partial$
Emil
J.
Straube;
Marcel
K.
Sucheston
143-154
Abstract: Boas and Straube proved a general sufficient condition for global regularity of the $\bar\partial$-Neumann problem in terms of families of vector fields that commute approximately with $\bar\partial$. In this paper, we study the existence of these vector fields on a compact subset of the boundary whose interior is foliated by complex manifolds. This question turns out to be closely related to properties of interest from the point of view of foliation theory.
On hypersphericity of manifolds with finite asymptotic dimension
A.
N.
Dranishnikov
155-167
Abstract: We prove the following embedding theorems in the coarse geometry: \begin{theorem1}Every metric space$X$\space with bounded geometry whose asympt... ... embedding into the product of$n+1$\space locally finite trees. \end{theorem1} \begin{theorem2}Every metric space$X$\space with bounded geometry whose asympt... ...ding into a non-positively curved manifold of dimension$2n+2$. \end{theorem2} The Corollary is used in the proof of the following. \begin{theorem3}For every uniformly contractible manifold$X$whose asymptotic d... ...\mathbf{R}^{n}$\space is integrally hyperspherical for some$n$. \end{theorem3} Theorem B together with a theorem of Gromov-Lawson implies the result, previously proven by G. Yu (1998), which states that an aspherical manifold whose fundamental group has a finite asymptotic dimension cannot carry a metric of positive scalar curvature. We also prove that if a uniformly contractible manifold $X$ of bounded geometry is large scale uniformly embeddable into a Hilbert space, then $X$ is stably integrally hyperspherical.
Biharmonic lifts by means of pseudo-Riemannian submersions in dimension three
Miguel
A.
Javaloyes Victoria;
Miguel
A.
Meroño Bayo
169-176
Abstract: We study the total lifts of curves by means of a submersion $\pi:M_s^3\rightarrow B_r^2$ that satisfy the condition $\Delta H=\lambda H$analyzing, in particular, the cases in which the submersion has totally geodesic fibres or integrable horizontal distribution. We also consider in detail the case $\lambda=0$ (biharmonic lifts). Moreover, we obtain a biharmonic lift in ${\mathbb R}^3$ by means of a Riemannian submersion that has non-constant mean curvature, getting so a counterexample to the Chen conjecture for ${\mathbb R}^3$ with a non-flat Riemannian metric.
Integration of multivalued operators and cyclic submonotonicity
Aris
Daniilidis;
Pando
Georgiev;
Jean-Paul
Penot
177-195
Abstract: We introduce a notion of cyclic submonotonicity for multivalued operators from a Banach space $X$ to its dual. We show that if the Clarke subdifferential of a locally Lipschitz function is strictly submonotone on an open subset $U$ of $X$, then it is also maximal cyclically submonotone on $U$, and, conversely, that every maximal cyclically submonotone operator on $U$ is the Clarke subdifferential of a locally Lipschitz function, which is unique up to a constant if $U$ is connected. In finite dimensions these functions are exactly the lower C$^{1}$ functions considered by Spingarn and Rockafellar.
Linear parabolic equations with strong singular potentials
Jerome
A.
Goldstein;
Qi
S.
Zhang
197-211
Abstract: Using an extension of a recent method of Cabré and Martel (1999), we extend the blow-up and existence result in the paper of Baras and Goldstein (1984) to parabolic equations with variable leading coefficients under almost optimal conditions on the singular potentials. This problem has been left open in Baras and Goldstein. These potentials lie at a borderline case where standard theories such as the strong maximum principle and boundedness of weak solutions fail. Even in the special case when the leading operator is the Laplacian, we extend a recent result in Cabré and Martel from bounded smooth domains to unbounded nonsmooth domains.
Uniqueness for the determination of sound-soft defects in an inhomogeneous planar medium by acoustic boundary measurements
Luca
Rondi
213-239
Abstract: We consider the inverse problem of determining shape and location of sound-soft defects inside a known planar inhomogeneous and anisotropic medium through acoustic imaging at low frequency. In order to determine the defects, we perform acoustic boundary measurements, with prescribed boundary conditions of different types. We prove that at most two, suitably chosen, measurements allow us to uniquely determine multiple defects under minimal regularity assumptions on the defects and the medium containing them. Finally, we treat applications of these results to the case of inverse scattering.
Another way to say harmonic
Michael
G.
Crandall;
Jianying
Zhang
241-263
Abstract: It is known that solutions of $-\Delta_\infty u=-\sum_{i,j=1}^nu_{x_i} u_{x_j}u_{x_ix_j}=0$, that is, the $\infty$-harmonic functions, are exactly those functions having a comparison property with respect to the family of translates of the radial solutions $G(x)=a\vert x\vert$. We establish a more difficult linear result: a function in ${\mathbb R^n}$ is harmonic if it has the comparison property with respect to sums of $n$ translates of the radial harmonic functions $G(x)=a\vert x\vert^{2-n}$ for $n\not=2$ and $G(x)=b\ln(\vert x\vert)$ for $n=2$. An attempt to generalize these results for $-\Delta_\infty u=0$ ($p=\infty$) and $-\Delta u=0$ ($p=2$) to the general $p$-Laplacian leads to the fascinating discovery that certain sums of translates of radial $p$-superharmonic functions are again $p$-superharmonic. Mystery remains: the class of $p$-superharmonic functions so constructed for $p\not\in\{2,\infty\}$ does not suffice to characterize $p$-subharmonic functions.
On graphic Bernstein type results in higher codimension
Mu-Tao
Wang
265-271
Abstract: Let $\Sigma$ be a minimal submanifold of $\mathbb{R} ^{n+m}$ that can be represented as the graph of a smooth map $f:\mathbb{R} ^n\mapsto\mathbb{R} ^m$. We apply a formula that we derived in the study of mean curvature flow to obtain conditions under which $\Sigma$ must be an affine subspace. Our result covers all known ones in the general case. The conditions are stated in terms of the singular values of $df$.
Matrix-weighted Besov spaces
Svetlana
Roudenko
273-314
Abstract: Nazarov, Treil and Volberg defined matrix $A_p$ weights and extended the theory of weighted norm inequalities on $L^p$ to the case of vector-valued functions. We develop some aspects of Littlewood-Paley function space theory in the matrix weight setting. In particular, we introduce matrix- weighted homogeneous Besov spaces $\dot{B}^{\alpha q}_p(W)$and matrix-weighted sequence Besov spaces $\dot{b}^{\alpha q}_p(W)$, as well as $\dot{b}^{\alpha q}_p(\{A_Q\})$, where the $A_Q$ are reducing operators for $W$. Under any of three different conditions on the weight $W$, we prove the norm equivalences $\Vert \vec{f} \,\Vert_{\dot{B}^{\alpha q}_p(W)} \approx \Vert \{ \vec{s}_Q \}... ... q}_p(W)} \approx \Vert \{ \vec{s}_Q \}_Q \Vert_{\dot{b}^{\alpha q}_p(\{A_Q\})}$, where $\{ \vec{s}_Q \}_Q$ is the vector-valued sequence of $\varphi$-transform coefficients of $\vec{f}$. In the process, we note and use an alternate, more explicit characterization of the matrix $A_p$ class. Furthermore, we introduce a weighted version of almost diagonality and prove that an almost diagonal matrix is bounded on $\dot{b}^{\alpha q}_p(W)$ if $W$ is doubling. We also obtain the boundedness of almost diagonal operators on $\dot{B}^{\alpha q}_p(W)$ under any of the three conditions on $W$. This leads to the boundedness of convolution and non-convolution type Calderón-Zygmund operators (CZOs) on $\dot{B}^{\alpha q}_p(W)$, in particular, the Hilbert transform. We apply these results to wavelets to show that the above norm equivalence holds if the $\varphi$-transform coefficients are replaced by the wavelet coefficients. Finally, we construct inhomogeneous matrix-weighted Besov spaces ${B}^{\alpha q}_p(W)$ and show that results corresponding to those above are true also for the inhomogeneous case.
The space $H^1$ for nondoubling measures in terms of a grand maximal operator
Xavier
Tolsa
315-348
Abstract: Let $\mu$ be a Radon measure on $\mathbb{R} ^d$, which may be nondoubling. The only condition that $\mu$ must satisfy is the size condition $\mu(B(x,r))\leq C\,r^n$, for some fixed $0<n\leq d$. Recently, some spaces of type $BMO(\mu)$ and $H^1(\mu)$ were introduced by the author. These new spaces have properties similar to those of the classical spaces ${BMO}$ and $H^1$defined for doubling measures, and they have proved to be useful for studying the $L^p(\mu)$ boundedness of Calderón-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic Hardy space $H^1(\mu)$ in terms of a maximal operator $M_\Phi$ is given. It is shown that $f$ belongs to $H^1(\mu)$ if and only if $f\in L^1(\mu)$, $\int f\, d\mu=0$ and $M_\Phi f \in L^1(\mu)$, as in the usual doubling situation.
Fullness, Connes' $\chi $-groups, and ultra-products of amalgamated free products over Cartan subalgebras
Yoshimichi
Ueda
349-371
Abstract: Ultra-product algebras associated with amalgamated free products over Cartan subalgebras are investigated. As applications, their Connes' $\chi$-groups are computed in terms of ergodic theory, and also we clarify what condition makes them full factors (i.e., their inner automorphism groups become closed).
Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains
Richard
F.
Bass;
Edwin
A.
Perkins
373-405
Abstract: We consider the operator \begin{displaymath}\sum_{i,j=1}^d \sqrt{x_ix_j}\gamma_{ij}(x) \frac{\partial^2}... ...\partial x_j}+\sum_{i=1}^d b_i(x) \frac{\partial}{\partial x_i}\end{displaymath} acting on functions in $C_b^2(\mathbb{R}^d_+)$. We prove uniqueness of the martingale problem for this degenerate operator under suitable nonnegativity and regularity conditions on $\gamma_{ij}$ and $b_i$. In contrast to previous work, the $b_i$ need only be nonnegative on the boundary rather than strictly positive, at the expense of the $\gamma_{ij}$and $b_i$ being Hölder continuous. Applications to super-Markov chains are given. The proof follows Stroock and Varadhan's perturbation argument, but the underlying function space is now a weighted Hölder space and each component of the constant coefficient process being perturbed is the square of a Bessel process.
Proper actions on cohomology manifolds
Harald
Biller
407-432
Abstract: Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most $2$, then the group is effectively a Lie group.
Inverse spectral theory of finite Jacobi matrices
Peter
C.
Gibson
4703-4749
Abstract: We solve the following physically motivated problem: to determine all finite Jacobi matrices $J$ and corresponding indices $i,j$ such that the Green's function \begin{displaymath}\langle e_j,(zI-J)^{-1}e_i\rangle \end{displaymath} is proportional to an arbitrary prescribed function $f(z)$. Our approach is via probability distributions and orthogonal polynomials. We introduce what we call the auxiliary polynomial of a solution in order to factor the map \begin{displaymath}(J,i,j)\longmapsto [\langle e_j,(zI-J)^{-1}e_i\rangle] \end{displaymath} (where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.
On the number of zeros of nonoscillatory solutions to half-linear ordinary differential equations involving a parameter
Kusano
Takasi;
Manabu
Naito
4751-4767
Abstract: In this paper the following half-linear ordinary differential equation is considered: $\alpha > 0$ is a constant, $\lambda > 0$ is a parameter, and $p(t)$ is a continuous function on $[a, \infty)$, $a > 0$, and $p(t) > 0$ for $t \in [a, \infty)$. The main purpose is to show that precise information about the number of zeros can be drawn for some special type of solutions $x(t; \lambda)$ of (H $_{\lambda})$ such that \begin{displaymath}\lim_{t\to\infty}\frac{x(t; \lambda)}{\sqrt{t}} = 0. \end{displaymath} It is shown that, if $\alpha \geq 1$ and if (H $_{\lambda})$ is strongly nonoscillatory, then there exists a sequence $\{\lambda_{n}\}_{n=1}^{\infty}$ such that $0=\lambda_{0}<\lambda_{1}<\cdots< \lambda_{n}<\cdots$, $\lambda_{n} \to +\infty$ as $n \to \infty$; and $x(t; \lambda)$ with $\lambda = \lambda_n$ has exactly $n-1$ zeros in the interval $(a,\infty)$ and $x(a; \lambda_n) = 0$; and $x(t; \lambda)$ with $\lambda \in (\lambda_{n-1}, \lambda_n)$ has exactly $n-1$ zeros in $(a,\infty)$ and $x(a; \lambda_n) \neq 0$. For the proof of the theorem, we make use of the generalized Prüfer transformation, which consists of the generalized sine and cosine functions.
Sets of uniqueness for spherically convergent multiple trigonometric series
J.
Marshall
Ash;
Gang
Wang
4769-4788
Abstract: A subset $E$ of the $d$-dimensional torus $\mathbb{T} ^{d}$ is called a set of uniqueness, or $U$-set, if every multiple trigonometric series spherically converging to $0$ outside $E$ vanishes identically. We show that all countable sets are $U$-sets and also that $H^{J}$ sets are $U$-sets for every $J$. In particular, $C\times\mathbb{T} ^{d-1}$, where $C$ is the Cantor set, is an $H^{1}$ set and hence a $U$-set. We will say that $E$ is a $U_{A}$-set if every multiple trigonometric series spherically Abel summable to $0$ outside $E$ and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for $U_{A}$ sets. In addition, every $U_{A}$-set has measure $0$, and a countable union of closed $U_{A}$-sets is a $U_{A}$-set.
Ribbon tilings and multidimensional height functions
Scott
Sheffield
4789-4813
Abstract: We fix $n$ and say a square in the two-dimensional grid indexed by $(x,y)$ has color $c$ if $x+y \equiv c \pmod{n}$. A ribbon tile of order $n$ is a connected polyomino containing exactly one square of each color. We show that the set of order-$n$ ribbon tilings of a simply connected region $R$ is in one-to-one correspondence with a set of height functions from the vertices of $R$ to $\mathbb Z^{n}$ satisfying certain difference restrictions. It is also in one-to-one correspondence with the set of acyclic orientations of a certain partially oriented graph. Using these facts, we describe a linear (in the area of $R$) algorithm for determining whether $R$ can be tiled with ribbon tiles of order $n$ and producing such a tiling when one exists. We also resolve a conjecture of Pak by showing that any pair of order-$n$ ribbon tilings of $R$ can be connected by a sequence of local replacement moves. Some of our results are generalizations of known results for order-$2$ ribbon tilings (a.k.a. domino tilings). We also discuss applications of multidimensional height functions to a broader class of polyomino tiling problems.
Lines tangent to $2n-2$ spheres in ${\mathbb R}^n$
Frank
Sottile;
Thorsten
Theobald
4815-4829
Abstract: We show that for $n \ge 3$there are $3 \cdot 2^{n-1}$ complex common tangent lines to $2n-2$ general spheres in $\mathbb{R}^n$ and that there is a choice of spheres with all common tangents real.
Automorphisms of finite order on Gorenstein del Pezzo surfaces
D.-Q.
Zhang
4831-4845
Abstract: In this paper we shall determine all actions of groups of prime order $p$ with $p \ge 5$ on Gorenstein del Pezzo (singular) surfaces $Y$of Picard number 1. We show that every order-$p$ element in $\operatorname{Aut}(Y)$ ( $= \operatorname{Aut}({\widetilde Y})$, ${\widetilde Y}$ being the minimal resolution of $Y$) is lifted from a projective transformation of ${\mathbf{P}}^{2}$. We also determine when $\operatorname{Aut}(Y)$ is finite in terms of $K_{Y}^{2}$, $\operatorname{Sing} Y$ and the number of singular members in $\vert-K_{Y}\vert$. In particular, we show that either $\vert\operatorname{Aut}(Y)\vert = 2^{a}3^{b}$ for some $1 \le a+b \le 7$, or for every prime $p \ge 5$, there is at least one element $g_{p}$ of order $p$ in $\operatorname{Aut}(Y)$ (hence $\vert\operatorname{Aut}(Y)\vert$ is infinite).
Fourier expansion of Eisenstein series on the Hilbert modular group and Hilbert class fields
Claus
Mazanti
Sorensen
4847-4869
Abstract: In this paper we consider the Eisenstein series for the Hilbert modular group of a general number field. We compute the Fourier expansion at each cusp explicitly. The Fourier coefficients are given in terms of completed partial Hecke $L$-series, and from their functional equations, we get the functional equation for the Eisenstein vector. That is, we identify the scattering matrix. When we compute the determinant of the scattering matrix in the principal case, the Dedekind $\xi$-function of the Hilbert class field shows up. A proof in the imaginary quadratic case was given in Efrat and Sarnak, and for totally real fields with class number one a proof was given in Efrat.
Hilbert transforms and maximal functions along variable flat curves
Jonathan
M.
Bennett
4871-4892
Abstract: We study certain Hilbert transforms and maximal functions along variable flat curves in the plane. We obtain their $L^{2}(\mathbb{R} ^{2})$ boundedness by considering the oscillatory singular integrals which arise from an application of a partial Fourier transform.
Nonisotropic strongly singular integral operators
Bassam
Shayya
4893-4907
Abstract: We consider a class of strongly singular integral operators which include those studied by Wainger, and Fefferman and Stein, and extend the results concerning the $L^p$ boundedness of these operators to the nonisotropic setting. We also describe a geometric property of the underlying space which helps us show that our results are sharp.
Spherical nilpotent orbits and the Kostant-Sekiguchi correspondence
Donald
R.
King
4909-4920
Abstract: Let $G$ be a connected, linear semisimple Lie group with Lie algebra $\mathfrak g$, and let ${K_{{}_{\mathbf C}}}~\rightarrow~{\operatorname{Aut} (\mathfrak p_{{}_{\mathbf C}})}$ be the complexified isotropy representation at the identity coset of the corresponding symmetric space. The Kostant-Sekiguchi correspondence is a bijection between the nilpotent $K_{{}_{\mathbf C}}$-orbits in $\mathfrak p_{{}_{\mathbf C}}$ and the nilpotent $G$-orbits in $\mathfrak g$. We show that this correspondence associates each spherical nilpotent $K_{{}_{\mathbf C}}$-orbit to a nilpotent $G$-orbit that is multiplicity free as a Hamiltonian $K$-space. The converse also holds.
A measure-valued analogue of Wiener measure and the measure-valued Feynman-Kac formula
K.
S.
Ryu;
M.
K.
Im
4921-4951
Abstract: In this article, we consider a complex-valued and a measure-valued measure on $C [0,t]$, the space of all real-valued continuous functions on $[0,t]$. Using these concepts, we establish the measure-valued Feynman-Kac formula and we prove that this formula satisfies a Volterra integral equation. The work here is patterned to some extent on earlier works by Kluvanek in 1983 and by Lapidus in 1987, but the present setting requires a number of new concepts and results.
Detection of renewal system factors via the Conley index
Jim
Wiseman
4953-4968
Abstract: Let $N$ be an isolating neighborhood for a map $f$. If we can decompose $N$ into the disjoint union of compact sets $N_1$ and $N_2$, then we can relate the dynamics on the maximal invariant set $\operatorname{Inv} N$ to the shift on two symbols by noting which component of $N$ each iterate of a point $x\in \operatorname{Inv} N$ lies in. We examine a method, based on work by Mischaikow, Szymczak, et al., for using the discrete Conley index to detect explicit subshifts of the shift associated to $N$. In essence, we measure the difference between the Conley index of $\operatorname{Inv}N$and the sum of the indices of $\operatorname{Inv} N_1$ and $\operatorname{Inv} N_2$.
Thick points for intersections of planar sample paths
Amir
Dembo;
Yuval
Peres;
Jay
Rosen;
Ofer
Zeitouni
4969-5003
Abstract: Let $L_n^{X}(x)$ denote the number of visits to $x \in \mathbf{Z} ^2$ of the simple planar random walk $X$, up to step $n$. Let $X'$ be another simple planar random walk independent of $X$. We show that for any $0<b<1/(2 \pi)$, there are $n^{1-2\pi b+o(1)}$ points $x \in \mathbf{Z}^2$ for which $\limsup_{r \rightarrow 0} \mathcal{I} (x,r)/(r^2\vert\log r\vert^4)=a^2$, is almost surely $2-2a$. Here $\mathcal{I}(x,r)$ is the projected intersection local time measure of the disc of radius $r$ centered at $x$ for two independent planar Brownian motions run until time $1$. The proofs rely on a ``multi-scale refinement'' of the second moment method. In addition, we also consider analogous problems where we replace one of the Brownian motions by a transient stable process, or replace the disc of radius $r$centered at $x$ by $x+rK$ for general sets $K$.
Equilibrium existence and topology in some repeated games with incomplete information
Robert
S.
Simon;
Stanislaw
Spiez;
Henryk
Torunczyk
5005-5026
Abstract: This article proves the existence of an equilibrium in any infinitely repeated, un-discounted two-person game of incomplete information on one side where the uninformed player must base his behavior strategy on state-dependent information generated stochastically by the moves of the players and the informed player is capable of sending nonrevealing signals. This extends our earlier result stating that an equilibrium exists if additionally the information is standard. The proof depends on applying new topological properties of set-valued mappings. Given a set-valued mapping $F$ on a compact convex set $P\subset \mathbb R^n$, we give further conditions which imply that every point $p_0\in P$ belongs to the convex hull of a finite subset $P _0$ of the domain of $F$satisfying $\bigcap_{x\in P_0} F(x)\ne \emptyset$.
Location of the Fermat-Torricelli medians of three points
Carlos
Benítez;
Manuel
Fernández;
María
L.
Soriano
5027-5038
Abstract: We prove that a real normed space $X$ with $\dim X\ge 3$ is an inner product space if and only if, for every three points $u,v,w\in X$, the set of points at which the function $x\in X\to \Vert u-x\Vert+\Vert v-x\Vert+\Vert w-x\Vert$attains its minimum (called the set of Fermat-Torricelli medians of the three points) intersects the convex hull of these three points.
Uniform and Lipschitz homotopy classes of maps
Sol
Schwartzman
5039-5047
Abstract: If $X$ is a compact connected polyhedron, we associate with each uniform homotopy class of uniformly continuous mappings from the real line $R$ into $X$ an element of $H_{1} (X, U/U_{0}),$ where $U$ is the space of uniformly continuous functions from $R$ to $R$ and $U_{0}$ is the subspace of bounded uniformly continuous functions. This map from uniform homotopy classes of functions to $H_{1}(X,U/U_{0})$ is surjective. If $X$ is the $n$-dimensional torus, it is bijective, while if $X$ is a compact orientable surface of genus $>1$, it is not injective. In higher dimensions we have to consider smooth Lipschitz homotopy classes of smooth Lipschitz maps from suitable Riemannian manifolds $P$ to compact smooth manifolds $X.$ With each such Lipschitz homotopy class we associate an element of $H_{n} (X, B^+/B_{0}^+),$ where $n$ is the dimension of $P,$ $B$ is the space of bounded continuous functions from the positive real axis to $R,$ and $B_{0}^+$ is the set of all $f\in B^+$ such that $\lim_{t \rightarrow \infty} f(t) = 0.$
Spin structures and codimension two embeddings of $3$-manifolds up to regular homotopy
Osamu
Saeki;
Masamichi
Takase
5049-5061
Abstract: We clarify the structure of the set of regular homotopy classes containing embeddings of a 3-manifold into $5$-space inside the set of all regular homotopy classes of immersions with trivial normal bundles. As a consequence, we show that for a large class of $3$-manifolds $M^3$, the following phenomenon occurs: there exists a codimension two immersion of the $3$-sphere whose double points cannot be eliminated by regular homotopy, but can be eliminated after taking the connected sum with a codimension two embedding of $M^3$. This involves introducing and studying an equivalence relation on the set of spin structures on $M^3$. Their associated $\mu$-invariants also play an important role.
Discrete morse theory and the cohomology ring
Robin
Forman
5063-5085
Abstract: In [5], we presented a discrete Morse Theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which has the same cohomology as the underlying cell complex. In this paper we show how various cohomological operations are induced by maps between Morse cocomplexes. For example, given three discrete Morse functions, we construct a map from the tensor product of the first two Morse cocomplexes to the third Morse cocomplex which induces the cup product on cohomology. All maps are constructed by counting certain configurations of gradient paths. This work is closely related to the corresponding formulas in the smooth category as presented by Betz and Cohen [2] and Fukaya [11], [12].
On the blow-up of heat flow for conformal $3$-harmonic maps
Chao-Nien
Chen;
L.
F.
Cheung;
Y.
S.
Choi;
C.
K.
Law
5087-5110
Abstract: Using a comparison theorem, Chang, Ding, and Ye (1992) proved a finite time derivative blow-up for the heat flow of harmonic maps from $D^2$ (a unit ball in ${\mathbf R}^2$) to $S^2$ (a unit sphere in ${\mathbf R}^3$) under certain initial and boundary conditions. We generalize this result to the case of $3$-harmonic map heat flow from $D^3$ to $S^3$. In contrast to the previous case, our governing parabolic equation is quasilinear and degenerate. Technical issues such as the development of a new comparison theorem have to be resolved.
Diffusions on graphs, Poisson problems and spectral geometry
Patrick
McDonald;
Robert
Meyers
5111-5136
Abstract: We study diffusions, variational principles and associated boundary value problems on directed graphs with natural weightings. We associate to certain subgraphs (domains) a pair of sequences, each of which is invariant under the action of the automorphism group of the underlying graph. We prove that these invariants differ by an explicit combinatorial factor given by Stirling numbers of the first and second kind. We prove that for any domain with a natural weighting, these invariants determine the eigenvalues of the Laplace operator corresponding to eigenvectors with nonzero mean. As a specific example, we investigate the relationship between our invariants and heat content asymptotics, expressing both as special values of an analog of a spectral zeta function.
Associated primes of graded components of local cohomology modules
Markus
P.
Brodmann;
Mordechai
Katzman;
Rodney
Y.
Sharp
4261-4283
Abstract: The $i$-th local cohomology module of a finitely generated graded module $M$ over a standard positively graded commutative Noetherian ring $R$, with respect to the irrelevant ideal $R_+$, is itself graded; all its graded components are finitely generated modules over $R_0$, the component of $R$ of degree $0$. It is known that the $n$-th component $H^i_{R_+}(M)_n$ of this local cohomology module $H^i_{R_+}(M)$ is zero for all $n>> 0$. This paper is concerned with the asymptotic behaviour of $\operatorname{Ass}_{R_0}(H^i_{R_+}(M)_n)$ as $n \rightarrow -\infty$. The smallest $i$ for which such study is interesting is the finiteness dimension $f$ of $M$ relative to $R_+$, defined as the least integer $j$ for which $H^j_{R_+}(M)$ is not finitely generated. Brodmann and Hellus have shown that $\operatorname{Ass}_{R_0}(H^f_{R_+}(M)_n)$ is constant for all $n < < 0$ (that is, in their terminology, $\operatorname{Ass}_{R_0}(H^f_{R_+}(M)_n)$ is asymptotically stable for $n \rightarrow -\infty$). The first main aim of this paper is to identify the ultimate constant value (under the mild assumption that $R$ is a homomorphic image of a regular ring): our answer is precisely the set of contractions to $R_0$ of certain relevant primes of $R$ whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology. Brodmann and Hellus raised various questions about such asymptotic behaviour when $i > f$. They noted that Singh's study of a particular example (in which $f = 2$) shows that $\operatorname{Ass}_{R_0}(H^3_{R_+}(R)_n)$ need not be asymptotically stable for $n \rightarrow -\infty$. The second main aim of this paper is to determine, for Singh's example, $\operatorname{Ass}_{R_0}(H^3_{R_+}(R)_n)$ quite precisely for every integer $n$, and, thereby, answer one of the questions raised by Brodmann and Hellus.
Positivity, sums of squares and the multi-dimensional moment problem
S.
Kuhlmann;
M.
Marshall
4285-4301
Abstract: Let $K$ be the basic closed semi-algebraic set in $\mathbb{R}^n$ defined by some finite set of polynomials $S$ and $T$, the preordering generated by $S$. For $K$ compact, $f$ a polynomial in $n$ variables nonnegative on $K$ and real $\epsilon>0$, we have that $f+\epsilon\in T$ [15]. In particular, the $K$-Moment Problem has a positive solution. In the present paper, we study the problem when $K$ is not compact. For $n=1$, we show that the $K$-Moment Problem has a positive solution if and only if $S$ is the natural description of $K$ (see Section 1). For $n\ge 2$, we show that the $K$-Moment Problem fails if $K$ contains a cone of dimension 2. On the other hand, we show that if $K$is a cylinder with compact base, then the following property holds: \begin{displaymath}(\ddagger)\quad\quad\forall f\in \mathbb{R}[X], f\ge 0 \text{... ...hat }\forall \text{ real } \epsilon>0, f+\epsilon q\in T.\quad \end{displaymath} This property is strictly weaker than the one given in [15], but in turn it implies a positive solution to the $K$-Moment Problem. Using results of [9], we provide many (noncompact) examples in hypersurfaces for which ($\ddagger$) holds. Finally, we provide a list of 8 open problems.
Kähler-Einstein metrics for some quasi-smooth log del Pezzo surfaces
Carolina
Araujo
4303-4312
Abstract: Recently Johnson and Kollár determined the complete list of anticanonically embedded quasi-smooth log del Pezzo surfaces in weighted projective $3$-spaces. They also proved that many of those surfaces admit a Kähler-Einstein metric, and that some of them do not have tigers. The aim of this paper is to settle the question of the existence of Kähler-Einstein metrics and tigers for those surfaces for which the question was left open. In order to do so, we will use techniques developed earlier by Nadel, Demailly and Kollár.
Shifted simplicial complexes are Laplacian integral
Art
M.
Duval;
Victor
Reiner
4313-4344
Abstract: We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs. We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial complex, with equality achieved if and only if the complex is shifted. This generalizes a conjecture of Grone and Merris for graphs.
Tilting theory and the finitistic dimension conjectures
Lidia
Angeleri-Hügel;
Jan
Trlifaj
4345-4358
Abstract: Let $R$ be a right noetherian ring and let $\mathcal{P}^{<\infty}$ be the class of all finitely presented modules of finite projective dimension. We prove that findim $R = n < \infty$ iff there is an (infinitely generated) tilting module $T$ such that pd$T = n$ and $T ^\perp = (\mathcal P^{<\infty})^\perp$. If $R$ is an artin algebra, then $T$ can be taken to be finitely generated iff $\mathcal P^{<\infty}$ is contravariantly finite. We also obtain a sufficient condition for validity of the First Finitistic Dimension Conjecture that extends the well-known result of Huisgen-Zimmermann and Smalø.
On the Representation Theory of Lie Triple Systems
Terrell
L.
Hodge;
Brian
J.
Parshall
4359-4391
Abstract: In this paper, we take a new look at the representation theory of Lie triple systems. We consider both ordinary Lie triple systems and restricted Lie triple systems in the sense of [14]. In a final section, we begin a study of the cohomology of Lie triple systems.
An application of the Littlewood restriction formula to the Kostant-Rallis Theorem
Jeb
F.
Willenbring
4393-4419
Abstract: Consider a symmetric pair $(G,K)$ of linear algebraic groups with $\mathfrak{g} \cong \mathfrak{k} \oplus \mathfrak{p}$, where $\mathfrak{k}$ and $\mathfrak{p}$ are defined as the +1 and -1 eigenspaces of the involution defining $K$. We view the ring of polynomial functions on $\mathfrak{p}$ as a representation of $K$. Moreover, set $\mathcal{P}(\mathfrak{p}) = \bigoplus_{d=0}^\infty \mathcal{P}^d(\mathfrak{p})$, where $\mathcal{P}^d(\mathfrak{p})$ is the space of homogeneous polynomial functions on $\mathfrak{p}$ of degree $d$. This decomposition provides a graded $K$-module structure on $\mathcal{P}(\mathfrak{p})$. A decomposition of $\mathcal{P}^d(\mathfrak{p})$is provided for some classical families $(G,K)$ when $d$ is within a certain stable range. The stable range is defined so that the spaces $\mathcal{P}^d(\mathfrak{p})$are within the hypothesis of the classical Littlewood restriction formula. The Littlewood restriction formula provides a branching rule from the general linear group to the standard embedding of the symplectic or orthogonal subgroup. Inside the stable range the decomposition of $\mathcal{P}^d(\mathfrak{p})$ is interpreted as a $q$-analog of the Kostant-Rallis theorem.
Extensions for finite Chevalley groups II
Christopher
P.
Bendel;
Daniel
K.
Nakano;
Cornelius
Pillen
4421-4454
Abstract: Let $G$ be a semisimple simply connected algebraic group defined and split over the field ${\mathbb{F} }_p$ with $p$ elements, let $G(\mathbb{F} _{q})$ be the finite Chevalley group consisting of the ${\mathbb{F} }_{q}$-rational points of $G$ where $q = p^r$, and let $G_{r}$ be the $r$th Frobenius kernel. The purpose of this paper is to relate extensions between modules in $\text{Mod}(G(\mathbb{F} _{q}))$ and $\text{Mod}(G_{r})$ with extensions between modules in $\text{Mod}(G)$. Among the results obtained are the following: for $r >2$ and $p\geq 3(h-1)$, the $G(\mathbb{F} _{q})$-extensions between two simple $G(\mathbb{F} _{q})$-modules are isomorphic to the $G$-extensions between two simple $p^r$-restricted $G$-modules with suitably ``twisted" highest weights. For $p \geq 3(h-1)$, we provide a complete characterization of $\text{H}^{1}(G(\mathbb{F} _{q}),H^{0}(\lambda))$ where $H^{0}(\lambda)=\text{ind}_{B}^{G} \lambda$ and $\lambda$ is $p^r$-restricted. Furthermore, for $p \geq 3(h-1)$, necessary and sufficient bounds on the size of the highest weight of a $G$-module $V$ are given to insure that the restriction map $\operatorname{H}^{1}(G,V)\rightarrow \operatorname{H}^{1}(G(\mathbb{F} _{q}),V)$ is an isomorphism. Finally, it is shown that the extensions between two simple $p^r$-restricted $G$-modules coincide in all three categories provided the highest weights are ``close" together.
The space of $(\psi,\gamma)$--additive mappings on semigroups
Valerii
A.
Faiziev;
Themistocles
M.
Rassias;
Prasanna
K.
Sahoo
4455-4472
Abstract: In this paper, we introduce the concept of $(\psi,\gamma)$-pseudoadditive mappings from a semigroup into a Banach space, and we provide a generalized solution of Ulam's problem for approximately additive mappings.
Summing inclusion maps between symmetric sequence spaces
Andreas
Defant;
Mieczyslaw
Mastylo;
Carsten
Michels
4473-4492
Abstract: In 1973/74 Bennett and (independently) Carl proved that for $1 \le u \le 2$ the identity map id: $\ell_u \hookrightarrow \ell_2$ is absolutely $(u,1)$-summing, i.e., for every unconditionally summable sequence $(x_n)$in $\ell_u$ the scalar sequence $(\Vert x_n \Vert _{\ell_2})$ is contained in $\ell_u$, which improved upon well-known results of Littlewood and Orlicz. The following substantial extension is our main result: For a $2$-concave symmetric Banach sequence space $E$ the identity map $\text{id}: E \hookrightarrow \ell_2$ is absolutely $(E,1)$-summing, i.e., for every unconditionally summable sequence $(x_n)$ in $E$ the scalar sequence $(\Vert x_n \Vert _{\ell_2})$ is contained in $E$. Various applications are given, e.g., to the theory of eigenvalue distribution of compact operators, where we show that the sequence of eigenvalues of an operator $T$ on $\ell_2$ with values in a $2$-concave symmetric Banach sequence space $E$ is a multiplier from $\ell_2$ into $E$. Furthermore, we prove an asymptotic formula for the $k$-th approximation number of the identity map $\text{id}: \ell_2^n \hookrightarrow E_n$, where $E_n$ denotes the linear span of the first $n$ standard unit vectors in $E$, and apply it to Lorentz and Orlicz sequence spaces.
Classification of compact complex homogeneous spaces with invariant volumes
Daniel
Guan
4493-4504
Abstract: We solve the problem of the classification of compact complex homogeneous spaces with invariant volumes (see Matsushima, 1961).
A Berger-Green type inequality for compact Lorentzian manifolds
Manuel
Gutiérrez;
Francisco
J.
Palomo;
Alfonso
Romero
4505-4523
Abstract: We give a Lorentzian metric on the null congruence associated with a timelike conformal vector field. A Liouville type theorem is proved and a boundedness for the volume of the null congruence, analogous to a well-known Berger-Green theorem in the Riemannian case, will be derived by studying conjugate points along null geodesics. As a consequence, several classification results on certain compact Lorentzian manifolds without conjugate points on its null geodesics are obtained. Finally, several properties of null geodesics of a natural Lorentzian metric on each odd-dimensional sphere have been found.
Two-weight norm inequalities for the Cesàro means of Hermite expansions
Benjamin
Muckenhoupt;
David
W.
Webb
4525-4537
Abstract: An accurate estimate is obtained of the Cesàro kernel for Hermite expansions. This is used to prove two-weight norm inequalities for Cesàro means of Hermite polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and ``unweighted'' cases. An almost everywhere convergence result is obtained as a corollary. It is also shown that the conditions used to prove norm boundedness of the means and most of the conditions used to prove the boundedness of the Cesàro supremum of the means are necessary.
Involutions fixing $\mathbb{RP}^{\text{odd}}\sqcup P(h,i)$, I
Zhi
Lü
4539-4570
Abstract: This paper studies the equivariant cobordism classification of all involutions fixing a disjoint union of an odd-dimensional real projective space $\mathbb{RP}^j$ with its normal bundle nonbounding and a Dold manifold $P(h,i)$ with $h>0$ and $i>0$. For odd $h$, the complete analysis of the equivariant cobordism classes of such involutions is given except that the upper and lower bounds on codimension of $P(h,i)$ may not be best possible; for even $h$, the problem may be reduced to the problem for even projective spaces.
Emergence of the Witt group in the cellular lattice of rational spaces
Kathryn
Hess;
Paul-Eugène
Parent
4571-4583
Abstract: We give an embedding of a quotient of the Witt semigroup into the lattice of rational cellular classes represented by formal $2$-cones between $S^{2n}$ and the two-cell complex $X_n=S^{2n}\cup_{[\iota,\iota]}e^{4n}$ ($n\geq1$).
3-connected planar spaces uniquely embed in the sphere
R.
Bruce
Richter;
Carsten
Thomassen
4585-4595
Abstract: We characterize those locally connected subsets of the sphere that have a unique embedding in the sphere -- i.e., those for which every homeomorphism of the subset to itself extends to a homeomorphism of the sphere. This implies that if $\bar G$ is the closure of an embedding of a 3-connected graph in the sphere such that every 1-way infinite path in $G$ has a unique accumulation point in $\bar G$, then $\bar G$ has a unique embedding in the sphere. In particular, the standard (or Freudenthal) compactification of a 3-connected planar graph embeds uniquely in the sphere.
Growth and ergodicity of context-free languages
Tullio
Ceccherini-Silberstein;
Wolfgang
Woess
4597-4625
Abstract: A language $L$ over a finite alphabet $\boldsymbol\Sigma$ is called growth-sensitive if forbidding any set of subwords $F$ yields a sublanguage $L^{F}$ whose exponential growth rate is smaller than that of $L$. It is shown that every ergodic unambiguous, nonlinear context-free language is growth-sensitive. ``Ergodic'' means for a context-free grammar and language that its dependency di-graph is strongly connected. The same result as above holds for the larger class of essentially ergodic context-free languages, and if growth is considered with respect to the ambiguity degrees, then the assumption of unambiguity may be dropped. The methods combine a construction of grammars for $2$-block languages with a generating function technique regarding systems of algebraic equations.
Brownian-time processes: The PDE connection II and the corresponding Feynman-Kac formula
Hassan
Allouba
4627-4637
Abstract: We delve deeper into our study of the connection of Brownian-time processes (BTPs) to fourth-order parabolic PDEs, which we introduced in a recent joint article with W. Zheng. Probabilistically, BTPs and their cousins BTPs with excursions form a unifying class of interesting stochastic processes that includes the celebrated IBM of Burdzy and other new intriguing processes and is also connected to the Markov snake of Le Gall. BTPs also offer a new connection of probability to PDEs that is fundamentally different from the Markovian one. They solve fourth-order PDEs in which the initial function plays an important role in the PDE itself, not only as initial data. We connect two such types of interesting and new PDEs to BTPs. The first is obtained by running the BTP and then integrating along its path, and the second type of PDEs is related to what we call the Feynman-Kac formula for BTPs. A special case of the second type is a step towards a probabilistic solution to linearized Cahn-Hilliard and Kuramoto-Sivashinsky type PDEs, which we tackle in an upcoming paper.
Gaugeability and conditional gaugeability
Zhen-Qing
Chen
4639-4679
Abstract: New Kato classes are introduced for general transient Borel right processes, for which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Green-tight measures in the classical Brownian motion case. However, the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability. We show that gaugeability, conditional gaugeability and the subcriticality for the associated Schrödinger operators are equivalent for transient Borel right processes with strong duals. Analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric Markov processes. These analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases. Connections with the positivity of the spectral radii of the associated Schrödinger operators are also established.
Scaling coupling of reflecting Brownian motions and the hot spots problem
Mihai
N.
Pascu
4681-4702
Abstract: We introduce a new type of coupling of reflecting Brownian motions in smooth planar domains, called scaling coupling. We apply this to obtain monotonicity properties of antisymmetric second Neumann eigenfunctions of convex planar domains with one line of symmetry. In particular, this gives the proof of the hot spots conjecture for some known types of domains and some new ones.
$ad$-nilpotent $\mathfrak b$-ideals in $sl(n)$ having a fixed class of nilpotence: combinatorics and enumeration
George
E.
Andrews;
Christian
Krattenthaler;
Luigi
Orsina;
Paolo
Papi
3835-3853
Abstract: We study the combinatorics of $ad$-nilpotent ideals of a Borel subalgebra of $sl(n+1,\mathbb C)$. We provide an inductive method for calculating the class of nilpotence of these ideals and formulas for the number of ideals having a given class of nilpotence. We study the relationships between these results and the combinatorics of Dyck paths, based upon a remarkable bijection between $ad$-nilpotent ideals and Dyck paths. Finally, we propose a $(q,t)$-analogue of the Catalan number $C_n$. These $(q,t)$-Catalan numbers count, on the one hand, $ad$-nilpotent ideals with respect to dimension and class of nilpotence and, on the other hand, admit interpretations in terms of natural statistics on Dyck paths.
Inequalities for decomposable forms of degree $n+1$ in $n$ variables
Jeffrey
Lin
Thunder
3855-3868
Abstract: We consider the number of integral solutions to the inequality $\vert F(\mathbf{x}) \vert\le m$, where $F(\mathbf{X} )\in \mathbb{Z} [\mathbf{X} ]$ is a decomposable form of degree $n+1$ in $n$ variables. We show that the number of such solutions is finite for all $m$ only if the discriminant of $F$ is not zero. We get estimates for the number of such solutions that display appropriate behavior in terms of the discriminant. These estimates sharpen recent results of the author for the general case of arbitrary degree.
Hochschild homology criteria for trivial algebra structures
Micheline
Vigué-Poirrier
3869-3882
Abstract: We prove two similar results by quite different methods. The first one deals with augmented artinian algebras over a field: we characterize the trivial algebra structure on the augmentation ideal in terms of the maximality of the dimensions of the Hochschild homology (or cyclic homology) groups. For the second result, let $X$ be a 1-connected finite CW-complex. We characterize the trivial algebra structure on the cohomology algebra of $X$ with coefficients in a fixed field in terms of the maximality of the Betti numbers of the free loop space.
Differential operators on a polarized abelian variety
Indranil
Biswas
3883-3891
Abstract: Let $L$ be an ample line bundle over a complex abelian variety $A$. We show that the space of all global sections over $A$of ${Diff}^{n}_A(L,L)$ and $S^n({Diff}^1_A(L,L))$are both of dimension one. Using this it is shown that the moduli space $M_X$ of rank one holomorphic connections on a compact Riemann surface $X$ does not admit any nonconstant algebraic function. On the other hand, $M_X$ is biholomorphic to the moduli space of characters of $X$, which is an affine variety. So $M_X$ is algebraically distinct from the character variety if $X$ is of genus at least one.
Universal deformation rings and Klein four defect groups
Frauke
M.
Bleher
3893-3906
Abstract: In this paper, the universal deformation rings of certain modular representations of a finite group are determined. The representations under consideration are those which are associated to blocks with Klein four defect groups and whose stable endomorphisms are given by scalars. It turns out that these universal deformation rings are always subquotient rings of the group ring of a Klein four group over the ring of Witt vectors.
Character degrees and nilpotence class of finite $p$-groups: An approach via pro-$p$ groups
A.
Jaikin-Zapirain;
Alexander
Moretó
3907-3925
Abstract: Let $\mathcal{S}$ be a finite set of powers of $p$ containing 1. It is known that for some choices of $\mathcal{S}$, if $P$ is a finite $p$-group whose set of character degrees is $\mathcal{S}$, then the nilpotence class of $P$ is bounded by some integer that depends on $\mathcal{S}$, while for some other choices of $\mathcal{S}$ such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set $\mathcal{S}$ is class bounding if and only if $p\notin\mathcal{S}$. In this article we provide a new approach to this problem. Our main result shows the relevance of certain $p$-adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets $\mathcal{S}$ such that $p\notin\mathcal{S}$.
On the crossing number of positive knots and braids and braid index criteria of Jones and Morton-Williams-Franks
A.
Stoimenow
3927-3954
Abstract: We give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations. In particular, we show that positive braid knots may not have positive minimal (strand number) braid representations, giving a counterpart to results of Franks-Williams and Murasugi. Other examples answer questions of Cromwell on homogeneous and (partially) of Adams on almost alternating knots. We give a counterexample to, and a corrected version of, a theorem of Jones on the Alexander polynomial of 4-braid knots. We also give an example of a knot on which all previously applied braid index criteria fail to estimate sharply (from below) the braid index. A relation between (generalizations of) such examples and a conjecture of Jones that a minimal braid representation has unique writhe is discussed. Finally, we give a counterexample to Morton's conjecture relating the genus and degree of the skein polynomial.
A theory of concordance for non-spherical 3-knots
Vincent
Blanloeil;
Osamu
Saeki
3955-3971
Abstract: Consider a closed connected oriented 3-manifold embedded in the $5$-sphere, which is called a $3$-knot in this paper. For two such knots, we say that their Seifert forms are spin concordant, if they are algebraically concordant with respect to a diffeomorphism between the 3-manifolds which preserves their spin structures. Then we show that two simple fibered 3-knots are geometrically concordant if and only if they have spin concordant Seifert forms, provided that they have torsion free first homology groups. Some related results are also obtained.
Embeddings up to homotopy of two-cones in euclidean space
Pascal
Lambrechts;
Don
Stanley;
Lucile
Vandembroucq
3973-4013
Abstract: We say that a finite CW-complex $X$ embeds up to homotopy in a sphere $S^{n+1}$ if there exists a subpolyhedron $K\subset S^{n+1}$ having the homotopy type of $X$. The main result of this paper is a sufficient condition for the existence of such a homotopy embedding in a given codimension when $X$ is a simply-connected two-cone (a two-cone is the homotopy cofibre of a map between two suspensions). We give different applications of this result: we prove that if $X$is a two-cone then there are no rational obstructions to embeddings up to homotopy in codimension 3. We give also a description of the homotopy type of the boundary of a regular neighborhood of the embedding of a two-cone in a sphere. This enables us to construct a closed manifold $M$ whose Lusternik-Schnirelmann category and cone-length are not affected by removing one point of $M$.
Critical Heegaard surfaces
David
Bachman
4015-4042
Abstract: In this paper we introduce critical surfaces, which are described via a 1-complex whose definition is reminiscent of the curve complex. Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical, then the manifold contains an incompressible surface. Conversely, we also show that if a non-Haken 3-manifold admits at most one Heegaard splitting of each genus, then it does not contain a critical Heegaard surface. In the final section we discuss how this work leads to a natural metric on the space of strongly irreducible Heegaard splittings, as well as many new and interesting open questions.
Spectral asymptotics for Sturm-Liouville equations with indefinite weight
Paul
A.
Binding;
Patrick
J.
Browne;
Bruce
A.
Watson
4043-4065
Abstract: The Sturm-Liouville equation \begin{displaymath}-(py')' + qy =\lambda ry \text{\rm on} [0,l]\end{displaymath} is considered subject to the boundary conditions $O(1/\sqrt{n})$for $\sqrt{\lambda_n}$, or equivalently up to $O(\sqrt{n})$ for $\lambda_n$, the eigenvalues of the above boundary value problem.
The dynamics of expansive invertible onesided cellular automata
Masakazu
Nasu
4067-4084
Abstract: Using textile systems, we prove the conjecture of Boyle and Maass that the dynamical system defined by an expansive invertible onesided cellular automaton is topologically conjugate to a topological Markov shift. We also study expansive leftmost-permutive onesided cellular automata and bipermutive endomorphisms of mixing topological Markov shifts.
Trees and branches in Banach spaces
E.
Odell;
Th.
Schlumprecht
4085-4108
Abstract: An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree $\mathcal{T}$of a certain type on a space $X$ is presumed to have a branch with some property. It is shown that then $X$ can be embedded into a space with an FDD $(E_i)$ so that all normalized sequences in $X$ which are almost a skipped blocking of $(E_i)$ have that property. As an application of our work we prove that if $X$ is a separable reflexive Banach space and for some $1<p<\infty$ and $C<\infty$ every weakly null tree $\mathcal{T}$ on the sphere of $X$ has a branch $C$-equivalent to the unit vector basis of $\ell_p$, then for all $\varepsilon>0$, there exists a subspace of $X$ having finite codimension which $C^2+\varepsilon$ embeds into the $\ell_p$ sum of finite dimensional spaces.
Functional Calculus in Hölder-Zygmund Spaces
G.
Bourdaud;
Massimo
Lanza de Cristoforis
4109-4129
Abstract: In this paper we characterize those functions $f$ of the real line to itself such that the nonlinear superposition operator $T_{f}$ defined by $T_{f}[ g]:= f\circ g$ maps the Hölder-Zygmund space ${\mathcal C}^{s}({\mathbf R}^{n})$ to itself, is continuous, and is $r$ times continuously differentiable. Our characterizations cover all cases in which $s$ is real and $s>0$, and seem to be novel when $s>0$ is an integer.
Weak amenability of module extensions of Banach algebras
Yong
Zhang
4131-4151
Abstract: We start by discussing general necessary and sufficient conditions for a module extension Banach algebra to be $n$-weakly amenable, for $n = 0,1,2,\cdots$. Then we investigate various special cases. All these case studies finally provide us with a way to construct an example of a weakly amenable Banach algebra which is not $3$-weakly amenable. This answers an open question raised by H. G. Dales, F. Ghahramani and N. Grønbæk.
Amenability and exactness for dynamical systems and their $C^{*}$-algebras
Claire
Anantharaman-Delaroche
4153-4178
Abstract: We study the relations between amenability (resp. amenability at infinity) of $C^{*}$-dynamical systems and equality or nuclearity (resp. exactness) of the corresponding crossed products.
Generalized pseudo-Riemannian geometry
Michael
Kunzinger;
Roland
Steinbauer
4179-4199
Abstract: Generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions, we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a ``Fundamental Lemma of (pseudo-) Riemannian geometry'' in this setting and define the notion of geodesics of a generalized metric. Finally, we present applications of the resulting theory to general relativity.
Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains
Xing-Bin
Pan;
Keng-Huat
Kwek
4201-4227
Abstract: We establish an asymptotic estimate of the lowest eigenvalue $\mu (b\mathbf{F})$ of the Schrödinger operator $-\nabla _{b\mathbf{F}}^{2}$ with a magnetic field in a bounded $2$-dimensional domain, where curl $\mathbf{F}$ vanishes non-degenerately, and $b$is a large parameter. Our study is based on an analysis on an eigenvalue variation problem for the Sturm-Liouville problem. Using the estimate, we determine the value of the upper critical field for superconductors subject to non-homogeneous applied magnetic fields, and localize the nucleation of superconductivity.
Harmonic morphisms with one-dimensional fibres on Einstein manifolds
Radu
Pantilie;
John
C.
Wood
4229-4243
Abstract: We prove that, from an Einstein manifold of dimension greater than or equal to five, there are just two types of harmonic morphism with one-dimensional fibres. This generalizes a result of R.L. Bryant who obtained the same conclusion under the assumption that the domain has constant curvature.
Contact reduction
Christopher
Willett
4245-4260
Abstract: In this article I propose a new method for reducing co-oriented contact manifold $M$ equipped with an action of a Lie group $G$ by contact transformations. With a certain regularity and integrality assumption the contact quotient $M_\mu$ at $\mu \in \mathfrak{g}^*$ is a naturally co-oriented contact orbifold which is independent of the contact form used to represent the given contact structure. Removing the regularity and integrality assumptions and replacing them with one concerning the existence of a slice, which is satisfied for compact symmetry groups, results in a contact stratified space; i.e., a stratified space equipped with a line bundle which, when restricted to each stratum, defines a co-oriented contact structure. This extends the previous work of the author and E. Lerman.
Lower central series and free resolutions of hyperplane arrangements
Henry
K.
Schenck;
Alexander
I.
Suciu
3409-3433
Abstract: If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,\Bbbk)$is the cohomology ring of $M$ over a field $\Bbbk$ of characteristic $0$, then the ranks, $\phi_k$, of the lower central series quotients of $\pi_1(M)$ can be computed from the Betti numbers, $b_{ii}=\dim \operatorname{Tor}^A_i(\Bbbk,\Bbbk)_i$, of the linear strand in a minimal free resolution of $\Bbbk$ over $A$. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers,
On the Glauberman and Watanabe correspondences for blocks of finite $p$-solvable groups
M.
E.
Harris;
M.
Linckelmann
3435-3453
Abstract: If $G$ is a finite $p$-solvable group for some prime $p$, $A$ a solvable subgroup of the automorphism group of $G$ of order prime to $\vert G\vert$such that $A$ stabilises a $p$-block $b$ of $G$ and acts trivially on a defect group $P$ of $b$, then there is a Morita equivalence between the block $b$ and its Watanabe correspondent $w(b)$ of $C_{G}(A)$, given by a bimodule $M$ with vertex $\Delta P$ and an endo-permutation module as source, which on the character level induces the Glauberman correspondence (and which is an isotypy by Watanabe's results).
Braid pictures for Artin groups
Daniel
Allcock
3455-3474
Abstract: We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams $A_n$, $B_n=C_n$ and $D_n$ and the affine diagrams $\tilde{A}_n$, $\tilde{B}_n$, $\tilde{C}_n$ and $\tilde{D}_n$ as subgroups of the braid groups of various simple orbifolds. The cases $D_n$, $\tilde{B}_n$, $\tilde{C}_n$ and $\tilde{D}_n$ are new. In each case the Artin group is a normal subgroup with abelian quotient; in all cases except $\tilde{A}_n$ the quotient is finite. We also illustrate the value of our braid calculus by giving a picture-proof of the basic properties of the Garside element of an Artin group of type $D_n$.
Weyl--Titchmarsh $M$-Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators
Steve
Clark;
Fritz
Gesztesy
3475-3534
Abstract: We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general Dirac-type operators on half-lines and on $\mathbb{R}$. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operators and use it to prove a Borg-type theorem.
On the finite-dimensional dynamical systems with limited competition
Xing
Liang;
Jifa
Jiang
3535-3554
Abstract: The asymptotic behavior of dynamical systems with limited competition is investigated. We study index theory for fixed points, permanence, global stability, convergence everywhere and coexistence. It is shown that the system has a globally asymptotically stable fixed point if every fixed point is hyperbolic and locally asymptotically stable relative to the face it belongs to. A nice result is the necessary and sufficient conditions for the system to have a globally asymptotically stable positive fixed point. It can be used to establish the sufficient conditions for the system to persist uniformly and the convergence result for all orbits. Applications are made to time-periodic ordinary differential equations and reaction-diffusion equations.
On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals
László
Hatvani
3555-3571
Abstract: Sufficient conditions are given for the asymptotic stability and uniform asymptotic stability of the zero solution of the nonautonomous FDE's whose right-hand sides can be unbounded functions of the time. The theorems are based upon Lyapunov-Krasovski{\u{\i}}\kern.15em functionals whose derivatives with respect to the equations are negative semidefinite and can vanish at long intervals. The functionals and their derivatives are estimated by either ${x(t)}$, the norm of the instantaneous value of the solutions or $\Vert x_t\Vert _2$, the $L_2$-norm of the phase segment of the solutions. Examples are given to show that the conditions are sharp, and the main theorems with the two different types of estimates are independent and improve earlier results. The theorems are applied to linear and nonlinear retarded FDE's with one delay and with distributed delays.
On the profile of the changing sign mountain pass solutions for an elliptic problem
E.
N.
Dancer;
Shusen
Yan
3573-3600
Abstract: We consider nonlinear elliptic equations with small diffusion and Dirichlet boundary conditions. We construct changing sign solutions with peaks close to the boundary and consider the location of the peak.
Diffusive logistic equation with constant yield harvesting, I: Steady States
Shobha
Oruganti;
Junping
Shi;
Ratnasingham
Shivaji
3601-3619
Abstract: We consider a reaction-diffusion equation which models the constant yield harvesting to a spatially heterogeneous population which satisfies a logistic growth. We prove the existence, uniqueness and stability of the maximal steady state solutions under certain conditions, and we also classify all steady state solutions under more restricted conditions. Exact global bifurcation diagrams are obtained in the latter case. Our method is a combination of comparison arguments and bifurcation theory.
Global existence and nonexistence for nonlinear wave equations with damping and source terms
Mohammad
A.
Rammaha;
Theresa
A.
Strei
3621-3637
Abstract: We consider an initial-boundary value problem for a nonlinear wave equation in one space dimension. The nonlinearity features the damping term $\left\vert u\right\vert^{m-1}u_t$ and a source term of the form $\left\vert u\right\vert^{p-1}u$, with $m,\,p>1$. We show that whenever $m\geq p$, then local weak solutions are global. On the other hand, we prove that whenever $p>m$ and the initial energy is negative, then local weak solutions cannot be global, regardless of the size of the initial data.
Attractors for graph critical rational maps
Alexander
Blokh;
Michal
Misiurewicz
3639-3661
Abstract: We call a rational map $f$ graph critical if any critical point either belongs to an invariant finite graph $G$, or has minimal limit set, or is non-recurrent and has limit set disjoint from $G$. We prove that, for any conformal measure, either for almost every point of the Julia set $J(f)$ its limit set coincides with $J(f)$, or for almost every point of $J(f)$ its limit set coincides with the limit set of a critical point of $f$.
Isomorphisms of function modules, and generalized approximation in modulus
David
Blecher;
Krzysztof
Jarosz
3663-3701
Abstract: For a function algebra $A$ we investigate relations between the following three topics: isomorphisms of singly generated $A$-modules, Morita equivalence bimodules, and ``real harmonic functions'' with respect to $A$. We also consider certain groups which are naturally associated with a uniform algebra $A$. We illustrate the notions considered with several examples.
Compactness properties for families of quasistationary solutions of some evolution equations
Giuseppe
Savaré
3703-3722
Abstract: The following typical problem occurs in passing to the limit in some phase field models: for two sequences of space-time dependent functions $\{\theta_n\}, \{{\raise.3ex\hbox{$\chi$}}_n\}$ (representing, e.g., suitable approximations of the temperature and the phase variable) we know that the sum $\theta_n + {\raise.3ex\hbox{$\chi$}}_n$ converges in some $L^p$-space as $n\uparrow+\infty$ and that the time integrals of a suitable ``space'' functional evaluated on $\theta_n, {\raise.3ex\hbox{$\chi$}}_n$ are uniformly bounded with respect to $n$. Can we deduce that $\theta_n$ and ${\raise.3ex\hbox{$\chi$}}_n$ converge separately? LUCKHAUS (1990) gave a positive answer to this question in the framework of the two-phase Stefan problem with Gibbs-Thompson law for the melting temperature. PLOTNIKOV (1993) proposed an abstract result employing the original idea of Luckhaus and arguments of compactness and reflexivity type. We present a general setting for this and other related problems, providing necessary and sufficient conditions for their solvability: these conditions rely on general topological and coercivity properties of the functionals and the norms involved, and do not require reflexivity.
A note on Meyers' Theorem in $W^{k,1}$
Irene
Fonseca;
Giovanni
Leoni;
Jan
Malý;
Roberto
Paroni
3723-3741
Abstract: Lower semicontinuity properties of multiple integrals \begin{displaymath}u\in W^{k,1}(\Omega;\mathbb{R}^{d})\mapsto\int_{\Omega}f(x,u(x), \cdots,\nabla^{k}u(x))\,dx\end{displaymath} are studied when $f$ may grow linearly with respect to the highest-order derivative, $\nabla^{k}u,$ and admissible $W^{k,1}(\Omega;\mathbb{R}^{d})$ sequences converge strongly in $W^{k-1,1}(\Omega;\mathbb{R}^{d}).$ It is shown that under certain continuity assumptions on $f,$ convexity, $1$-quasiconvexity or $k$-polyconvexity of \begin{displaymath}\xi\mapsto f(x_{0},u(x_{0}),\cdots,\nabla^{k-1}u(x_{0}),\xi)\end{displaymath} ensures lower semicontinuity. The case where $f(x_{0},u(x_{0}),\cdots,\nabla^{k-1}u(x_{0}),\cdot)$ is $k$-quasiconvex remains open except in some very particular cases, such as when $f(x,u(x),\cdots,\nabla^{k}u(x))=h(x)g(\nabla^{k}u(x)).$
Homogeneous weak solenoids
Robbert
Fokkink;
Lex
Oversteegen
3743-3755
Abstract: A (generalized) weak solenoid is an inverse limit space over manifolds with bonding maps that are covering maps. If the covering maps are regular, then we call the inverse limit space a strong solenoid. By a theorem of M.C. McCord, strong solenoids are homogeneous. We show conversely that homogeneous weak solenoids are topologically equivalent to strong solenoids. We also give an example of a weak solenoid that has simply connected path-components, but which is not homogeneous.
Degenerate fibres in the Stone-Cech compactification of the universal bundle of a finite group
David
Feldman;
Alexander
Wilce
3757-3769
Abstract: Applied to a continuous surjection $\pi : E \rightarrow B$ of completely regular Hausdorff spaces $E$ and $B$, the Stone-Cech compactification functor $\beta$ yields a surjection $\beta \pi: \beta E \rightarrow \beta B$. For an $n$-fold covering map $\pi$, we show that the fibres of $\beta \pi$, while never containing more than $n$ points, may degenerate to sets of cardinality properly dividing $n$. In the special case of the universal bundle $\pi:EG \rightarrow BG$ of a $p$-group $G$, we show more precisely that every possible type of $G$-orbit occurs among the fibres of $\beta \pi$. To prove this, we use a weak form of the so-called generalized Sullivan conjecture.
Submersions, fibrations and bundles
Gaël
Meigniez
3771-3787
Abstract: When does a submersion have the homotopy lifting property? When is it a locally trivial fibre bundle? We establish characterizations in terms of consistency in the topology of the neighbouring fibres.
The Chromatic Ext Groups $\Ext_{\Gamma(m+1)}^{0}(BP_{*},M_2^{1})$
Ippei
Ichigi;
Hirofumi
Nakai;
Douglas
C.
Ravenel
3789-3813
Abstract: We compute a certain Ext group related to the chromatic spectral sequence for $T (m)$, the spectrum whose $BP$-homology is $BP_{*}[t_{1},\cdots ,t_{m}]$ for each $m\ge 3$. The answer we get displays a kind of periodicity not seen in the corresponding computation for the sphere spectrum.
Euler characters and submanifolds of constant positive curvature
John
Douglas
Moore
3815-3834
Abstract: This article develops methods for studying the topology of submanifolds of constant positive curvature in Euclidean space. It proves that if $M^n$ is an $n$-dimensional compact connected Riemannian submanifold of constant positive curvature in ${\mathbb E}^{2n-1}$, then $M^n$ must be simply connected. It also gives a conformal version of this theorem.
Sasakian-Einstein structures on $9\#(S^2\times S^3)$
Charles
P.
Boyer;
Krzysztof
Galicki;
Michael
Nakamaye
2983-2996
Abstract: We show that $\scriptstyle{9\char93 (S^2\times S^3)}$ admits an 8-dimensional complex family of inequivalent non-regular Sasakian-Einstein structures. These are the first known Einstein metrics on this 5-manifold. In particular, the bound $\scriptstyle{b_2(M)\leq8}$ which holds for any regular Sasakian-Einstein $\scriptstyle{M}$does not apply to the non-regular case. We also discuss the failure of the Hitchin-Thorpe inequality in the case of 4-orbifolds and describe the orbifold version.
The Bergman metric on a Stein manifold with a bounded plurisubharmonic function
Bo-Yong
Chen;
Jin-Hao
Zhang
2997-3009
Abstract: In this article, we use the pluricomplex Green function to give a sufficient condition for the existence and the completeness of the Bergman metric. As a consequence, we proved that a simply connected complete Kähler manifold possesses a complete Bergman metric provided that the Riemann sectional curvature $\le -A/\rho^2$, which implies a conjecture of Greene and Wu. Moreover, we obtain a sharp estimate for the Bergman distance on such manifolds.
Irreducibility, Brill-Noether loci, and Vojta's inequality
Thomas
J.
Tucker;
with an Appendix by Olivier
Debarre
3011-3029
Abstract: This paper deals with generalizations of Hilbert's irreducibility theorem. The classical Hilbert irreducibility theorem states that for any cover $f$ of the projective line defined over a number field $k$, there exist infinitely many $k$-rational points on the projective line such that the fiber of $f$ over $P$is irreducible over $k$. In this paper, we consider similar statements about algebraic points of higher degree on curves of any genus. We prove that Hilbert's irreducibility theorem admits a natural generalization to rational points on an elliptic curve and thus, via a theorem of Abramovich and Harris, to points of degree 3 or less on any curve. We also present examples that show that this generalization does not hold for points of degree 4 or more. These examples come from an earlier geometric construction of Debarre and Fahlaoui; some additional necessary facts about this construction can be found in the appendix provided by Debarre. We exhibit a connection between these irreducibility questions and the sharpness of Vojta's inequality for algebraic points on curves. In particular, we show that Vojta's inequality is not sharp for the algebraic points arising in our examples.
Birational automorphisms of quartic Hessian surfaces
Igor
Dolgachev;
JongHae
Keum
3031-3057
Abstract: We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16 which embeds naturally in the even unimodular lattice $II_{1,25}$ of rank 26 and signature $(1,25)$. The generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of Kondo. We shall explain how our generators are related to the generators of the group of birational automorphisms of a general quartic Kummer surface which is birationally isomorphic to a special Hessian surface.
Light structures in infinite planar graphs without the strong isoperimetric property
Bojan
Mohar
3059-3074
Abstract: It is shown that the discharging method can be successfully applied on infinite planar graphs of subexponential growth and even on those graphs that do not satisfy the strong edge isoperimetric inequality. The general outline of the method is presented and the following applications are given: Planar graphs with only finitely many vertices of degree $\le 5$ and with subexponential growth contain arbitrarily large finite submaps of the tessellation of the plane or of some tessellation of the cylinder by equilateral triangles. Every planar graph with isoperimetric number zero and with essential minimum degree $\ge3$ has infinitely many edges whose degree sum is at most 15. In particular, this holds for all graphs with minimum degree $\ge3$ and with subexponential growth. The cases without infinitely many edges whose degree sum is $\le14$ (or, similarly, $\le13$ or $\le 12$) are also considered. Several further results are obtained.
Algebraic structure in the loop space homology Bockstein spectral sequence
Jonathan
A.
Scott
3075-3084
Abstract: Let $X$ be a finite, $n$-dimensional, $r$-connected CW complex. We prove the following theorem: If $p \geq n/r$ is an odd prime, then the loop space homology Bockstein spectral sequence modulo $p$ is a spectral sequence of universal enveloping algebras over differential graded Lie algebras.
On certain co--H spaces related to Moore spaces
Manfred
Stelzer
3085-3093
Abstract: We show that certain co-$H$ spaces, constructed by Anick and Gray, carry a homotopy co-associative and co-commutative co-$H$ structure.
Solutions of fully nonlinear elliptic equations with patches of zero gradient: Existence, regularity and convexity of level curves
L.
Caffarelli;
J.
Salazar
3095-3115
Abstract: In this paper, we first construct ``viscosity'' solutions (in the Crandall-Lions sense) of fully nonlinear elliptic equations of the form \begin{displaymath}F(D^{2} u,x) = g(x,u) \text{ on } \{\vert\nabla u\vert \ne 0\}\end{displaymath} In fact, viscosity solutions are surprisingly weak. Since candidates for solutions are just continuous, we only require that the ``test'' polynomials $P$ (those tangent from above or below to the graph of $u$ at a point $x_{0}$) satisfy the correct inequality only if $\vert\nabla P (x_{0})\vert \ne 0$. That is, we simply disregard those test polynomials for which $\vert\nabla P (x_{0})\vert = 0$. Nevertheless, this is enough, by an appropriate use of the Alexandroff-Bakelman technique, to prove existence, regularity and, in two dimensions, for $F = \Delta$, $g = cu$($c>0$) and constant boundary conditions on a convex domain, to prove that there is only one convex patch.
Semilinear Neumann boundary value problems on a rectangle
Junping
Shi
3117-3154
Abstract: We consider a semilinear elliptic equation \begin{displaymath}\Delta u+\lambda f(u)=0, \;\; \mathbf{x}\in \Omega,\;\; \frac{\partial u}{\partial n }=0, \;\; {\mathbf x}\in \partial \Omega, \end{displaymath} where $\Omega$ is a rectangle $(0,a)\times(0,b)$ in $\mathbf{R}^2$. For balanced and unbalanced $f$, we obtain partial descriptions of global bifurcation diagrams in $(\lambda,u)$ space. In particular, we rigorously prove the existence of secondary bifurcation branches from the semi-trivial solutions, which is called dimension-breaking bifurcation. We also study the asymptotic behavior of the monotone solutions when $\lambda\to\infty$. The results can be applied to the Allen-Cahn equation and some equations arising from mathematical biology.
Formation and propagation of singularities for $2\times 2$ quasilinear hyperbolic systems
De-xing
Kong
3155-3179
Abstract: Employing the method of characteristic coordinates and the singularity theory of smooth mappings, in this paper we analyze the long-term behaviour of smooth solutions of general $2\times 2$ quasilinear hyperbolic systems, provide a complete description of the solution close to blow-up points, and investigate the formation and propagation of singularities for $2\times 2$ systems of hyperbolic conservation laws.
Compactness of the solution operator for a linear evolution equation with distributed measures
Ioan
I.
Vrabie
3181-3205
Abstract: The main goal of the present paper is to define the solution operator $(\xi,g)\mapsto u$associated to the evolution equation $du=(Au)dt+dg$, $u(0)=\xi$, where $A$generates a $C_0$-semigroup in a Banach space $X$, $\xi\in X$, $g\in BV([\,a,b\,];X)$, and to study its main properties, such as regularity, compactness, and continuity. Some necessary and/or sufficient conditions for the compactness of the solution operator extending some earlier results due to the author and to BARAS, HASSAN, VERON, as well as some applications to the existence of certain generalized solutions to a semilinear equation involving distributed, or even spatial, measures, are also included. Two concrete examples of elliptic and parabolic partial differential equations subjected to impulsive dynamic conditions on the boundary illustrate the effectiveness of the abstract results.
Ljusternik-Schnirelman theory in partially ordered Hilbert spaces
Shujie
Li;
Zhi-Qiang
Wang
3207-3227
Abstract: We present several variants of Ljusternik-Schnirelman type theorems in partially ordered Hilbert spaces, which assert the locations of the critical points constructed by the minimax method in terms of the order structures. These results are applied to nonlinear Dirichlet boundary value problems to obtain the multiplicity of sign-changing solutions.
Transfer functions of regular linear systems Part II: The system operator and the Lax--Phillips semigroup
Olof
Staffans;
George
Weiss
3229-3262
Abstract: This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as ``Part I''. We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by $\dot x=Ax+Bu$, $y=Cx+Du$ would be the $s$-dependent matrix $S_\Sigma(s)= \left[ {}^{A-sI}_{ \;\,C} { } ^{B}_{D} \right]$. In the general case, $S_\Sigma(s)$ is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks $A-sI$ and $B$, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of $S_\Sigma(s)$ where the right lower block is the feedthrough operator of the system. Using $S_\Sigma(0)$, we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the ``initial time'' is $-\infty$. We also introduce the Lax-Phillips semigroup $\boldsymbol{\mathfrak{T}}$ induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index ${\omega}\in{\mathbb R}$which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of $A$ and also the points where $S_\Sigma(s)$ is not invertible, in terms of the spectrum of the generator of $\boldsymbol{\mathfrak{T}}$(for various values of ${\omega}$). The system $\Sigma$is dissipative if and only if $\boldsymbol{\mathfrak{T}}$(with index zero) is a contraction semigroup.
On the centered Hardy-Littlewood maximal operator
Antonios
D.
Melas
3263-3273
Abstract: We will study the centered Hardy-Littlewood maximal operator acting on positive linear combinations of Dirac deltas. We will use this to obtain improvements in both the lower and upper bounds or the best constant $C$ in the $L^{1}\rightarrow$ weak $L^{1}$ inequality for this operator. In fact we will show that $\frac{11+\sqrt{61}}{12}=1.5675208...\leq C\leq\frac{5} {3}=1.66...$.
On a class of jointly hyponormal Toeplitz operators
Caixing
Gu
3275-3298
Abstract: We characterize when a pair of Toeplitz operators $\mathbf{T}=(T_{\phi },T_{\psi})$ is jointly hyponormal under various assumptions--for example, $\phi$ is analytic or $\phi$ is a trigonometric polynomial or $\phi-\psi$ is analytic. A typical characterization states that $\mathbf{T}=(T_{\phi },T_{\psi})$ is jointly hyponormal if and only if an algebraic relation of $\phi$ and $\psi$ holds and the single Toeplitz operator $T_{\omega}$ is hyponormal, where $\omega$ is a combination of $\phi$ and $\psi$. More general results for an $n$-tuple of Toeplitz operators are also obtained.
Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces
Simon
Gindikin;
Bernhard
Krötz
3299-3327
Abstract: In this paper we define a distinguished boundary for the complex crowns $\Xi \subseteq G_{\mathbb{C} } /K_{\mathbb{C} }$ of non-compact Riemannian symmetric spaces $G/K$. The basic result is that affine symmetric spaces of $G$ can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.
Solvable groups with polynomial Dehn functions
G.
N.
Arzhantseva;
D.
V.
Osin
3329-3348
Abstract: Given a finitely presented group $H$, finitely generated subgroup $B$ of $H$, and a monomorphism $\psi :B\to H$, we obtain an upper bound of the Dehn function of the corresponding HNN-extension $G=\langle H, t\; \vert\; t^{-1}Bt=\psi (B)\rangle$ in terms of the Dehn function of $H$ and the distortion of $B$ in $G$. Using such a bound, we construct first examples of non-polycyclic solvable groups with polynomial Dehn functions. The constructed groups are metabelian and contain the solvable Baumslag-Solitar groups. In particular, this answers a question posed by Birget, Ol'shanskii, Rips, and Sapir.
Hopf modules and the double of a quasi-Hopf algebra
Peter
Schauenburg
3349-3378
Abstract: We give a different proof for a structure theorem of Hausser and Nill on Hopf modules over quasi-Hopf algebras. We extend the structure theorem to a classification of two-sided two-cosided Hopf modules by Yetter-Drinfeld modules, which can be defined in two rather different manners for the quasi-Hopf case. The category equivalence between Hopf modules and Yetter-Drinfeld modules leads to a new construction of the Drinfeld double of a quasi-Hopf algebra, as proposed by Majid and constructed by Hausser and Nill.
Invariant ideals and polynomial forms
D.
S.
Passman
3379-3408
Abstract: Let $K[\mathfrak H]$ denote the group algebra of an infinite locally finite group $\mathfrak H$. In recent years, the lattice of ideals of $K[\mathfrak H]$has been extensively studied under the assumption that $\mathfrak H$ is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra $K[V]$stable under the action of an appropriate automorphism group of $V$. Specifically, in this paper, we let ${\mathfrak{G}}$ be a quasi-simple group of Lie type defined over an infinite locally finite field $F$, and we let $V$ be a finite-dimensional vector space over a field $E$ of the same characteristic $p$. If ${\mathfrak{G}}$ acts nontrivially on $V$ by way of the homomorphism $\phi\colon{\mathfrak{G}}\to\mathrm{GL}(V)$, and if $V$ has no proper ${\mathfrak{G}}$-stable subgroups, then we show that the augmentation ideal $\omega K[V]$ is the unique proper ${\mathfrak{G}}$-stable ideal of $K[V]$ when ${\operatorname{char}} K\neq p$. The proof of this result requires, among other things, that we study characteristic $p$ division rings $D$, certain multiplicative subgroups $G$ of $D^{\bullet}$, and the action of $G$ on the group algebra $K[A]$, where $A$ is the additive group $D^{+}$. In particular, properties of the quasi-simple group ${\mathfrak{G}}$ come into play only in the final section of this paper.
New weighted Rogers-Ramanujan partition theorems and their implications
Krishnaswami
Alladi;
Alexander
Berkovich
2557-2577
Abstract: This paper has a two-fold purpose. First, by considering a reformulation of a deep theorem of Göllnitz, we obtain a new weighted partition identity involving the Rogers-Ramanujan partitions, namely, partitions into parts differing by at least two. Consequences of this include Jacobi's celebrated triple product identity for theta functions, Sylvester's famous refinement of Euler's theorem, as well as certain weighted partition identities. Next, by studying partitions with prescribed bounds on successive ranks and replacing these with weighted Rogers-Ramanujan partitions, we obtain two new sets of theorems - a set of three theorems involving partitions into parts $\not \equiv 0, \pm i$ (mod 6), and a set of three theorems involving partitions into parts $\not \equiv 0, \pm i$ (mod 7), $i=1,2,3$.
Core and residual intersections of ideals
Alberto
Corso;
Claudia
Polini;
Bernd
Ulrich
2579-2594
Abstract: D. Rees and J. Sally defined the core of an $R$-ideal $I$ as the intersection of all (minimal) reductions of $I$. However, it is not easy to give an explicit characterization of it in terms of data attached to the ideal. Until recently, the only case in which a closed formula was known is the one of integrally closed ideals in a two-dimensional regular local ring, due to C. Huneke and I. Swanson. The main result of this paper explicitly describes the core of a broad class of ideals with good residual properties in an arbitrary local Cohen-Macaulay ring. We also find sharp bounds on the number of minimal reductions that one needs to intersect to get the core.
Macaulay style formulas for sparse resultants
Carlos
D'Andrea
2595-2629
Abstract: We present formulas for computing the resultant of sparse polynomials as a quotient of two determinants, the denominator being a minor of the numerator. These formulas extend the original formulation given by Macaulay for homogeneous polynomials.
Surfaces with $p_g=q=3$
Christopher
D.
Hacon;
Rita
Pardini
2631-2638
Abstract: We classify minimal complex surfaces of general type with $p_g=q=3$. More precisely, we show that such a surface is either the symmetric product of a curve of genus $3$ or a free $\mathbb{Z} _2-$quotient of the product of a curve of genus $2$ and a curve of genus $3$. Our main tools are the generic vanishing theorems of Green and Lazarsfeld and the characterization of theta divisors given by Hacon in Corollary 3.4 of Fourier transforms, generic vanishing theorems and polarizations of abelian varieties.
On Fano manifolds with Nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents
Ngaiming
Mok
2639-2658
Abstract: Let $X$ be a Fano manifold of Picard number 1 with numerically effective tangent bundle. According to the principal case of a conjecture of Campana-Peternell's, $X$ should be biholomorphic to a rational homogeneous manifold $G/P$, where $G$ is a simple Lie group, and $P \subset G$ is a maximal parabolic subgroup. In our opinion there is no overriding evidence for the Campana-Peternell Conjecture for the case of Picard number 1 to be valid in its full generality. As part of a general programme that the author has undertaken with Jun-Muk Hwang to study uniruled projective manifolds via their varieties of minimal rational tangents, a new geometric approach is adopted in the current article in a special case, consisting of (a) recovering the generic variety of minimal rational tangents $\mathcal{C}_x$, and (b) recovering the structure of a rational homogeneous manifold from $\mathcal{C}_x$. The author proves that, when $b_4(X) = 1$ and the generic variety of minimal rational tangents is 1-dimensional, $X$ is biholomorphic to the projective plane $\mathbb{P} ^2$, the 3-dimensional hyperquadric $Q^3$, or the 5-dimensional Fano homogeneous contact manifold of type $G_2$, to be denoted by $K(G_2)$. The principal difficulty is part (a) of the scheme. We prove that $\mathcal{C}_x \subset \mathbb{P} T_x(X)$ is a rational curve of degrees $\leq 3$, and show that $d = 1$ resp. 2 resp. 3 corresponds precisely to the cases of $X = \mathbb{P} ^2$ resp. $Q^3$ resp. $K(G_2)$. Let $\mathcal{K}$ be the normalization of a choice of a Chow component of minimal rational curves on $X$. Nefness of the tangent bundle implies that $\mathcal{K}$ is smooth. Furthermore, it implies that at any point $x \in X$, the normalization $\mathcal{K}_x$of the corresponding Chow space of minimal rational curves marked at $x$is smooth. After proving that $\mathcal{K}_x$ is a rational curve, our principal object of study is the universal family $\mathcal{U}$ of $\mathcal{K}$, giving a double fibration $\rho: \mathcal{U} \to \mathcal{K}, \mu: \mathcal{U} \to X$, which gives $\mathbb{P} ^1$-bundles. There is a rank-2 holomorphic vector bundle $V$ on $\mathcal{K}$ whose projectivization is isomorphic to $\rho: \mathcal{U} \to \mathcal{K}$. We prove that $V$ is stable, and deduce the inequality $d \leq 4$ from the inequality $c_1^2(V) \leq 4c_2(V)$ resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of $d = 4$ is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on $V$ in the special case where $c_1^2(V) = 4c_2(V)$.
A Pieri-type formula for isotropic flag manifolds
Nantel
Bergeron;
Frank
Sottile
2659-2705
Abstract: We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from the Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a type $B$ (respectively, type $C$) Schubert polynomial by the Schur $P$-polynomial $p_m$ (respectively, the Schur $Q$-polynomial $q_m$). Geometric constructions and intermediate results allow us to ultimately deduce this formula from formulas for the classical flag manifold. These intermediate results are concerned with the Bruhat order of the infinite Coxeter group ${\mathcal B}_\infty$, identities of the structure constants for the Schubert basis of cohomology, and intersections of Schubert varieties. We show that most of these identities follow from the Pieri-type formula, and our analysis leads to a new partial order on the Coxeter group ${\mathcal B}_\infty$ and formulas for many of these structure constants.
Self-tilting complexes yield unstable modules
Alexander
Zimmermann
2707-2724
Abstract: Let $G$ be a group and $R$ a commutative ring. Let $TrPic_R(RG)$ be the group of isomorphism classes of standard self-equivalences of the derived category of bounded complexes of $RG$-modules. The subgroup $HD_R(G)$ of $TrPic_R(RG)$ consisting of self-equivalences fixing the trivial $RG$-module acts on the cohomology ring $H^*(G,R)$. The action is functorial with respect to $R$. The self-equivalences which are 'splendid' in a sense defined by J. Rickard act naturally with respect to transfer and restriction to centralizers of $p$-subgroups in case $R$ is a field of characteristic $p$. In the present paper we prove that this action of self-equivalences on $H^*(G,R)$ commutes with the action of the Steenrod algebra, and study the behaviour of the action of splendid self-equivalences with respect to Lannes' $T$-functor.
Enright's completions and injectively copresented modules
Steffen
König;
Volodymyr
Mazorchuk
2725-2743
Abstract: Let $\ensuremath{\mathfrak{A}}$be a finite-dimensional simple Lie algebra over the complex numbers. It is shown that a module is complete (or relatively complete) in the sense of Enright if and only if it is injectively copresented by certain injective modules in the BGG-category ${\mathcal O}$. Let $A$ be the finite-dimensional algebra associated to a block of ${\mathcal O}$. Then the corresponding block of the category of complete modules is equivalent to the category of $eAe$-modules for a suitable choice of the idempotent $e$. Using this equivalence, a very easy proof is given for Deodhar's theorem (also proved by Bouaziz) that completion functors satisfy braid relations. The algebra $eAe$ is left properly and standardly stratified. It satisfies a double centralizer property similar to Soergel's ``combinatorial description'' of ${\mathcal O}$. Its simple objects, their characters and their multiplicities in projective or standard objects are determined.
Representations of exceptional simple alternative superalgebras of characteristic 3
M.
C.
López-Díaz;
Ivan
P.
Shestakov
2745-2758
Abstract: We study representations of simple alternative superalgebras $B(1,2)$ and $B(2,4)$. The irreducible bimodules and bimodules with superinvolution over these superalgebras are classified, and some analogues of the Kronecker factorization theorem are proved for alternative superalgebras that contain $B(1,2)$ and $B(4,2)$.
Vertices for characters of $p$-solvable groups
Gabriel
Navarro
2759-2773
Abstract: Suppose that $G$ is a finite $p$-solvable group. We associate to every irreducible complex character $\chi \in \operatorname{Irr}(G)$ of $G$ a canonical pair $(Q,\delta)$, where $Q$ is a $p$-subgroup of $G$ and $\delta \in \operatorname{Irr}(Q)$, uniquely determined by $\chi$ up to $G$-conjugacy. This pair behaves as a Green vertex and partitions $\operatorname{Irr}(G)$ into ``families" of characters. Using the pair $(Q, \delta)$, we give a canonical choice of a certain $p$-radical subgroup $R$ of $G$ and a character $\eta \in \operatorname{Irr}(R)$ associated to $\chi$ which was predicted by some conjecture of G. R. Robinson.
Applications of Langlands' functorial lift of odd orthogonal groups
Henry
H.
Kim
2775-2796
Abstract: Together with Cogdell, Piatetski-Shapiro and Shahidi, we proved earlier the existence of a weak functorial lift of a generic cuspidal representation of $SO_{2n+1}$ to $GL_{2n}$. Recently, Ginzburg, Rallis and Soudry obtained a more precise form of the lift using their integral representation technique, namely, the lift is an isobaric sum of cuspidal representations of $GL_{n_i}$ (more precisely, cuspidal representations of $GL_{2n_i}$ such that the exterior square $L$-functions have a pole at $s=1$). One purpose of this paper is to give a simpler proof of this fact in the case that a cuspidal representation has one supercuspidal component. In a separate paper, we prove it without any condition using a result on spherical unitary dual due to Barbasch and Moy. We give several applications of the functorial lift: First, we parametrize square integrable representations with generic supercuspidal support, which have been classified by Moeglin and Tadic. Second, we give a criterion for cuspidal reducibility of supercuspidal representations of $GL_m\times SO_{2n+1}$. Third, we obtain a functorial lift from generic cuspidal representations of $SO_5$ to automorphic representations of $GL_5$, corresponding to the $L$-group homomorphism $Sp_4(\mathbb{C} )\longrightarrow GL_5(\mathbb{C} )$, given by the second fundamental weight.
Characterization of the Hilbert ball by its automorphism group
Kang-Tae
Kim;
Steven
G.
Krantz
2797-2818
Abstract: Let $\Omega$ be a bounded, convex domain in a separable Hilbert space. The authors prove a version of the theorem of Bun Wong, which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point, then it is biholomorphic to the ball. Key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new ``normal families'' arguments in the construction of the limit biholomorphism.
Regularity properties of solutions of a class of elliptic-parabolic nonlinear Levi type equations
G.
Citti;
A.
Montanari
2819-2848
Abstract: In this paper we prove the smoothness of solutions of a class of elliptic-parabolic nonlinear Levi type equations, represented as a sum of squares plus a vector field. By means of a freezing method the study of the operator is reduced to the analysis of a family $L_{\xi_0}$ of left invariant operators on a free nilpotent Lie group. The fundamental solution $\Gamma_{\xi_0}$ of the operator $L_{\xi_0}$ is used as a parametrix of the fundamental solution of the Levi operator, and provides an explicit representation formula for the solution of the given equation. Differentiating this formula and applying a bootstrap method, we prove that the solution is $C^\infty$.
A Markov partition that reflects the geometry of a hyperbolic toral automorphism
Anthony
Manning
2849-2863
Abstract: We show how to construct a Markov partition that reflects the geometrical action of a hyperbolic automorphism of the $n$-torus. The transition matrix is the transpose of the matrix induced by the automorphism in $u$-dimensional homology, provided this is non-negative. (Here $u$ denotes the expanding dimension.) That condition is satisfied, at least for some power of the original automorphism, under a certain non-degeneracy condition on the Galois group of the characteristic polynomial. The $(^n_u)$ rectangles are constructed by an iterated function system, and they resemble the product of the projection of a $u$-dimensional face of the unit cube onto the unstable subspace and the projection of minus the orthogonal $(n-u)$-dimensional face onto the stable subspace.
A priori degeneracy of one-dimensional rotation sets for periodic point free torus maps
Jaroslaw
Kwapisz
2865-2895
Abstract: Diffeomorphisms of the two torus that are isotopic to the identity have rotation sets that are convex compact subsets of the plane. We show that certain line segments (including all rationally sloped segments with no rational points) cannot be realized as a rotation set.
The inverse problem of the calculus of variations: The use of geometrical calculus in Douglas's analysis
W.
Sarlet;
G.
Thompson;
G.
E.
Prince
2897-2919
Abstract: The main objective of this paper is to work out a full-scale application of the integrability analysis of the inverse problem of the calculus of variations, as developed in recent papers by Sarlet and Crampin. For this purpose, the celebrated work of Douglas on systems with two degrees of freedom is taken as the reference model. It is shown that the coordinate-free, geometrical calculus used in Sarlet and Crampin's general theoretical developments provides effective tools also to do the practical calculations. The result is not only that all subcases distinguished by Douglas can be given a more intrinsic characterization, but also that in most of the cases, the calculations can be carried out in a more efficient way and often lead to sharper conclusions.
On the dimensional structure of hereditarily indecomposable continua
Roman
Pol;
Miroslawa
Renska
2921-2932
Abstract: Any hereditarily indecomposable continuum $X$ of dimension $n\geq 2$ is split into layers $B_r$ consisting of all points in $X$ that belong to some $r$-dimensional continuum but avoid any non-trivial continuum of dimension less than $r$. The subjects of this paper are the dimensional and the descriptive properties of the layers $B_r$.
Transition Probabilities for Symmetric Jump Processes
Richard
F.
Bass;
David
A.
Levin
2933-2953
Abstract: We consider symmetric Markov chains on the integer lattice in $d$ dimensions, where $\alpha \in (0,2)$ and the conductance between $x$ and $y$ is comparable to $\vert x-y\vert^{-(d+\alpha )}$. We establish upper and lower bounds for the transition probabilities that are sharp up to constants.
Commensurability of 1-cusped hyperbolic 3-manifolds
Danny
Calegari;
Nathan
M.
Dunfield
2955-2969
Abstract: We give examples of non-fibered hyperbolic knot complements in homology spheres that are not commensurable to fibered knot complements in homology spheres. In fact, we give many examples of knot complements in homology spheres where every commensurable knot complement in a homology sphere has non-monic Alexander polynomial.
A non-fixed point theorem for Hamiltonian lie group actions
Christopher
Allday;
Volker
Hauschild;
Volker
Puppe
2971-2982
Abstract: We prove that, under certain conditions, if a compact connected Lie group acts effectively on a closed manifold, then there is no fixed point. Because two of the main conditions are satisfied by any Hamiltonian action on a closed symplectic manifold, the theorem applies nicely to such actions. The method of proof, however, is cohomological; and so the result applies more generally.
Subspaces of non-commutative spaces
S.
Paul
Smith
2131-2171
Abstract: This paper concerns the closed points, closed subspaces, open subspaces, weakly closed and weakly open subspaces, and effective divisors, on a non-commutative space.
Cotensor products of modules
L.
Abrams;
C.
Weibel
2173-2185
Abstract: Let $C$ be a coalgebra over a field $k$ and $A$ its dual algebra. The category of $C$-comodules is equivalent to a category of $A$-modules. We use this to interpret the cotensor product $M \square N$ of two comodules in terms of the appropriate Hochschild cohomology of the $A$-bimodule $M \otimes N$, when $A$ is finite-dimensional, profinite, graded or differential-graded. The main applications are to Galois cohomology, comodules over the Steenrod algebra, and the homology of induced fibrations.
Local subgroups and the stable category
Wayne
W.
Wheeler
2187-2205
Abstract: If $G$ is a finite group and $k$ is an algebraically closed field of characteristic $p>0$, then this paper uses the local subgroup structure of $G$to define a category $\mathfrak{L}(G,k)$ that is equivalent to the stable category of all left $kG$-modules modulo projectives. A subcategory of $\mathfrak{L}(G,k)$ equivalent to the stable category of finitely generated $kG$-modules is also identified. The definition of $\mathfrak{L}(G,k)$ depends largely but not exclusively upon local data; one condition on the objects involves compatibility with respect to conjugations by arbitrary group elements rather than just elements of $p$-local subgroups.
Partial regularity for the stochastic Navier-Stokes equations
Franco
Flandoli;
Marco
Romito
2207-2241
Abstract: The effects of random forces on the emergence of singularities in the Navier-Stokes equations are investigated. In spite of the presence of white noise, the paths of a martingale suitable weak solution have a set of singular points of one-dimensional Hausdorff measure zero. Furthermore statistically stationary solutions with finite mean dissipation rate are analysed. For these stationary solutions it is proved that at any time $t$ the set of singular points is empty. The same result holds true for every martingale solution starting from $\mu_0$-a.e. initial condition $u_0$, where $\mu_0$ is the law at time zero of a stationary solution. Finally, the previous result is non-trivial when the noise is sufficiently non-degenerate, since for any stationary solution, the measure $\mu_0$ is supported on the whole space $H$ of initial conditions.
Random points on the boundary of smooth convex bodies
Matthias
Reitzner
2243-2278
Abstract: The convex hull of $n$ independent random points chosen on the boundary of a convex body $K \subset \mathbb{R}^d$ according to a given density function is a random polytope. The expectation of its $i$-th intrinsic volume for $i=1, \dots, d$ is investigated. In the case that the boundary of $K$ is sufficiently smooth, asymptotic expansions for these expected intrinsic volumes as $n \to \infty$ are derived.
Generalized space forms
Neil
N.
Katz;
Kei
Kondo
2279-2284
Abstract: Spaces with radially symmetric curvature at base point $p$ are shown to be diffeomorphic to space forms. Furthermore, they are either isometric to ${\mathbb R^n}$ or $S^n$ under a radially symmetric metric, to ${\mathbb R}{\rm P}^n$ with Riemannian universal covering of $S^n$equipped with a radially symmetric metric, or else have constant curvature outside a metric ball of radius equal to the injectivity radius at $p$.
A momentum construction for circle-invariant Kähler metrics
Andrew
D.
Hwang;
Michael
A.
Singer
2285-2325
Abstract: Examples of Kähler metrics of constant scalar curvature are relatively scarce. Over the past two decades, several workers in geometry and physics have used symmetry reduction to construct complete Kähler metrics of constant scalar curvature by ODE methods. One fruitful idea--the ``Calabi ansatz''--is to begin with an Hermitian line bundle $p:(L,h)\to(M,g_M)$ over a Kähler manifold, and to search for Kähler forms $\omega=p^*\omega_M+dd^c f(t)$ in some disk subbundle, where $t$ is the logarithm of the norm function and $f$ is a function of one variable. Our main technical result (Theorem A) is the calculation of the scalar curvature for an arbitrary Kähler metric $g$ arising from the Calabi ansatz. This suggests geometric hypotheses (which we call ``$\sigma$-constancy'') to impose upon the base metric $g_M$ and Hermitian structure $h$ in order that the scalar curvature of $g$ be specified by solving an ODE. We show that $\sigma$-constancy is ``necessary and sufficient for the Calabi ansatz to work'' in the following sense. Under the assumption of $\sigma$-constancy, the disk bundle admits a one-parameter family of complete Kähler metrics of constant scalar curvature that restrict to $g_M$ on the zero section (Theorems B and D); an analogous result holds for the punctured disk bundle (Theorem C). A simple criterion determines when such a metric is Einstein. Conversely, in the absence of $\sigma$-constancy the Calabi ansatz yields at most one metric of constant scalar curvature, in either the disk bundle or the punctured disk bundle (Theorem E). Many of the metrics constructed here seem to be new, including a complete, negative Einstein-Kähler metric on the disk subbundle of a stable vector bundle over a Riemann surface of genus at least two, and a complete, scalar-flat Kähler metric on $\mathbf{C}^2$.
Isoperimetric regions in cones
Frank
Morgan;
Manuel
Ritoré
2327-2339
Abstract: We consider cones $C = 0\, \times{\kern-10.5pt}\times \,M^n$ and prove that if the Ricci curvature of $C$ is nonnegative, then geodesic balls about the vertex minimize perimeter for given volume. If strict inequality holds, then they are the only stable regions.
Orthogonal polynomials and quadratic extremal problems
J.
M.
McDougall
2341-2357
Abstract: The purpose of this paper is to analyse a class of quadratic extremal problems defined on various Hilbert spaces of analytic functions, thereby generalizing an extremal problem on the Dirichlet space which was solved by S.D. Fisher. Each extremal problem considered here is shown to be connected with a system of orthogonal polynomials. The orthogonal polynomials then determine properties of the extremal function, and provide information about the existence of extremals.
Three-divisible families of skew lines on a smooth projective quintic
Slawomir
Rams
2359-2367
Abstract: We give an example of a family of 15 skew lines on a quintic such that its class is divisible by 3. We study properties of the codes given by arrangements of disjoint lines on quintics.
Twisted face-pairing 3-manifolds
J.
W.
Cannon;
W.
J.
Floyd;
W.
R.
Parry
2369-2397
Abstract: This paper is an enriched version of our introductory paper on twisted face-pairing 3-manifolds. Just as every edge-pairing of a 2-dimensional disk yields a closed 2-manifold, so also every face-pairing $\epsilon$ of a faceted 3-ball $P$ yields a closed 3-dimensional pseudomanifold. In dimension 3, the pseudomanifold may suffer from the defect that it fails to be a true 3-manifold at some of its vertices. The method of twisted face-pairing shows how to correct this defect of the quotient pseudomanifold $P/\epsilon$ systematically. The method describes how to modify $P$ by edge subdivision and how to modify any orientation-reversing face-pairing $\epsilon$ of $P$ by twisting, so as to yield an infinite parametrized family of face-pairings $(Q,\delta)$ whose quotient complexes $Q/\delta$ are all closed orientable 3-manifolds. The method is so efficient that, starting even with almost trivial face-pairings $\epsilon$, it yields a rich family of highly nontrivial, yet relatively simple, 3-manifolds. This paper solves two problems raised by the introductory paper: (1) Replace the computational proof of the introductory paper by a conceptual geometric proof of the fact that the quotient complex $Q/\delta$ of a twisted face-pairing is a closed 3-manifold. We do so by showing that the quotient complex has just one vertex and that its link is the faceted sphere dual to $Q$. (2) The twist construction has an ambiguity which allows one to twist all faces clockwise or to twist all faces counterclockwise. The fundamental groups of the two resulting quotient complexes are not at all obviously isomorphic. Are the two manifolds the same, or are they distinct? We prove the highly nonobvious fact that clockwise twists and counterclockwise twists yield the same manifold. The homeomorphism between them is a duality homeomorphism which reverses orientation and interchanges natural 0-handles with 3-handles, natural 1-handles with 2-handles. This duality result of (2) is central to our further studies of twisted face-pairings. We also relate the fundamental groups and homology groups of the twisted face-pairing 3-manifolds $Q/\delta$ and of the original pseudomanifold $P/\epsilon$ (with vertices removed). We conclude the paper by giving examples of twisted face-pairing 3-manifolds. These examples include manifolds from five of Thurston's eight 3-dimensional geometries.
The one phase free boundary problem for the $p$-Laplacian with non-constant Bernoulli boundary condition
Antoine
Henrot;
Henrik
Shahgholian
2399-2416
Abstract: Our objective, here, is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a Bernoulli-type boundary gradient condition and with the $p$-Laplacian as the governing operator. The main theorems of this paper assert that the exterior and the interior free boundary problem with a Bernoulli law, i.e. with a prescribed pressure $a(x)$ on the ``free'' streamline of the flow, have convex solutions provided the initial domains are convex. The continuous function $a(x)$ is subject to certain convexity properties. In our earlier results we have considered the case of constant $a(x)$. In the lines of the proof of the main results we also prove the semi-continuity (up to the boundary) of the gradient of the $p$-capacitary potentials in convex rings, with $C^1$ boundaries.
Ergodic and Bernoulli properties of analytic maps of complex projective space
Lorelei
Koss
2417-2459
Abstract: We examine the measurable ergodic theory of analytic maps $F$ of complex projective space. We focus on two different classes of maps, Ueda maps of ${\mathbb P}^{n}$, and rational maps of the sphere with parabolic orbifold and Julia set equal to the entire sphere. We construct measures which are invariant, ergodic, weak- or strong-mixing, exact, or automorphically Bernoulli with respect to these maps. We discuss topological pressure and measures of maximal entropy ( $h_{\mu}(F) = h_{top}(F)= \log(\deg F)$). We find analytic maps of ${\mathbb P}^1$ and ${\mathbb P}^2$ which are one-sided Bernoulli of maximal entropy, including examples where the maximal entropy measure lies in the smooth measure class. Further, we prove that for any integer $d>1$, there exists a rational map of the sphere which is one-sided Bernoulli of entropy $\log d$ with respect to a smooth measure.
Hyponormality of trigonometric Toeplitz operators
In
Sung
Hwang;
Woo
Young
Lee
2461-2474
Abstract: In this paper we establish a tractable and explicit criterion for the hyponormality of arbitrary trigonometric Toeplitz operators, i.e., Toeplitz operators $T_{\varphi }$ with trigonometric polynomial symbols $\varphi$. Our criterion involves the zeros of an analytic polynomial $f$ induced by the Fourier coefficients of $\varphi$. Moreover the rank of the selfcommutator of $T_{\varphi }$ is computed from the number of zeros of $f$ in the open unit disk $\mathbb{D}$ and in $\mathbb{C}\setminus \overline{\mathbb{D}}$ counting multiplicity.
Topological dynamics on moduli spaces II
Joseph
P.
Previte;
Eugene
Z.
Xia
2475-2494
Abstract: Let $M$ be an orientable genus $g>0$ surface with boundary $\partial M$. Let $\Gamma$ be the mapping class group of $M$ fixing $\partial M$. The group $\Gamma$ acts on ${\mathcal M}_{\mathcal C} = \operatorname{Hom}_{\mathcal C}(\pi_1(M),\operatorname{SU}(2))/\operatorname{SU}(2),$ the space of $\operatorname{SU}(2)$-gauge equivalence classes of flat $\operatorname{SU}(2)$-connections on $M$ with fixed holonomy on $\partial M$. We study the topological dynamics of the $\Gamma$-action and give conditions for the individual $\Gamma$-orbits to be dense in ${\mathcal M}_{\mathcal C}$.
Algebraic and spectral properties of dual Toeplitz operators
Karel
Stroethoff;
Dechao
Zheng
2495-2520
Abstract: Dual Toeplitz operators on the orthogonal complement of the Bergman space are defined to be multiplication operators followed by projection onto the orthogonal complement. In this paper we study algebraic and spectral properties of dual Toeplitz operators.
Regularized orbital integrals for representations of ${\mathbf{S} \mathbf{L}}(2)$
Jason
Levy
2521-2539
Abstract: Given a finite-dimensional representation of ${\mathbf{S} \mathbf{L}}(2,F)$, on a vector space $V$ defined over a local field $F$ of characteristic zero, we produce a regularization of orbital integrals and determine when the resulting distribution is non-trivial.
Some convolution inequalities and their applications
Daniel
M.
Oberlin
2541-2556
Abstract: We introduce a class of convolution inequalities and study the implications of these inequalities for certain problems in harmonic analysis.
On loop spaces of configuration spaces
F.
R.
Cohen;
S.
Gitler
1705-1748
Abstract: This article gives an analysis of topological and homological properties for loop spaces of configuration spaces. The main topological results are given by certain choices of product decompositions of these spaces, as well as ``twistings" between the factors. The main homological results are given in terms of extensions of the ``infinitesimal braid relations" or ``universal Yang-Baxter Lie relations".
On the structure of $P(n)_\ast P((n))$ for $p=2$
Christian
Nassau
1749-1757
Abstract: We show that $P(n)_\ast(P(n))$ for $p=2$ with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation $\epsilon$ nor the coproduct $\Delta$are multiplicative. As a consequence the algebra structure of $P(n)_\ast(P(n))$ is slightly different from what was supposed to be the case. We give formulas for $\epsilon(xy)$ and $\Delta(xy)$ and show that the inversion of the formal group of $P(n)$is induced by an antimultiplicative involution $\Xi:P(n)\rightarrow P(n)$. Some consequences for multiplicative and antimultiplicative automorphisms of $K(n)$ for $p=2$ are also discussed.
Polar and coisotropic actions on Kähler manifolds
Fabio
Podestà;
Gudlaugur
Thorbergsson
1759-1781
Abstract: The main result of the paper is that a polar action on a compact irreducible homogeneous Kähler manifold is coisotropic. This is then used to give new examples of polar actions and to classify coisotropic and polar actions on quadrics.
Katetov's problem
Paul
Larson;
Stevo
Todorcevic
1783-1791
Abstract: In 1948 Miroslav Katetov showed that if the cube $X^{3}$ of a compact space $X$ satisfies the separation axiom T$_{5}$ then $X$ must be metrizable. He asked whether $X^{3}$ can be replaced by $X^{2}$ in this metrization result. In this note we prove the consistency of this implication.
Asymptotic linear bounds for the Castelnuovo-Mumford regularity
Jürgen
Herzog;
Lê
Tuân
Hoa;
Ngô
Viêt
Trung
1793-1809
Abstract: We prove asymptotic linear bounds for the Castelnuovo-Mumford regularity of certain filtrations of homogeneous ideals whose Rees algebras need not be Noetherian.
Monoidal extensions of a Cohen-Macaulay unique factorization domain
William
J.
Heinzer;
Aihua
Li;
Louis
J.
Ratliff Jr.;
David
E.
Rush
1811-1835
Abstract: Let $A$ be a Noetherian Cohen-Macaulay domain, $b$, $c_1$, $\dots$, $c_g$an $A$-sequence, $J$ = $(b,c_1,\dots,c_g)A$, and $B$ = $A[J/b]$. Then $B$ is Cohen-Macaulay, there is a natural one-to-one correspondence between the sets $\mbox{{Ass}}_B(B/bB)$ and $\mbox{{Ass}}_A(A/J)$, and each $q$ $\in$ $\mbox{{Ass}}_A(A/J)$ has height $g+1$. If $B$ does not have unique factorization, then some height-one prime ideals $P$ of $B$ are not principal. These primes are identified in terms of $J$ and $P \cap A$, and we consider the question of how far from principal they can be. If $A$ is integrally closed, necessary and sufficient conditions are given for $B$ to be integrally closed, and sufficient conditions are given for $B$ to be a UFD or a Krull domain whose class group is torsion, finite, or finite cyclic. It is shown that if $P$is a height-one prime ideal of $B$, then $P \cap A$ also has height one if and only if $b$ $\notin$ $P$ and thus $P \cap A$ has height one for all but finitely many of the height-one primes $P$ of $B$. If $A$ has unique factorization, a description is given of whether or not such a prime $P$ is a principal prime ideal, or has a principal primary ideal, in terms of properties of $P \cap A$. A similar description is also given for the height-one prime ideals $P$ of $B$with $P \cap A$ of height greater than one, if the prime factors of $b$ satisfy a mild condition. If $A$ is a UFD and $b$ is a power of a prime element, then $B$ is a Krull domain with torsion class group if and only if $J$ is primary and integrally closed, and if this holds, then $B$ has finite cyclic class group. Also, if $J$ is not primary, then for each height-one prime ideal $p$ contained in at least one, but not all, prime divisors of $J$, it holds that the height-one prime $pA[1/b] \cap B$ has no principal primary ideals. This applies in particular to the Rees ring ${\mathbf R}$ $=$ $A[1/t, tJ]$. As an application of these results, it is shown how to construct for any finitely generated abelian group $G$, a monoidal transform $B$ = $A[J/b]$ such that $A$ is a UFD, $B$ is Cohen-Macaulay and integrally closed, and $G$ $\cong$ $\mbox{{Cl}}(B)$, the divisor class group of $B$.
Existence of curves with prescribed topological singularities
Thomas
Keilen;
Ilya
Tyomkin
1837-1860
Abstract: Throughout this paper we study the existence of irreducible curves $C$ on smooth projective surfaces $\Sigma$ with singular points of prescribed topological types $\mathcal S_1,\ldots,\mathcal S_r$. There are necessary conditions for the existence of the type $\sum_{i=1}^r \mu(\mathcal S_i)\leq \alpha C^2+\beta C.K+\gamma$ for some fixed divisor $K$on $\Sigma$ and suitable coefficients $\alpha$, $\beta$ and $\gamma$, and the main sufficient condition that we find is of the same type, saying it is asymptotically proper. Ten years ago general results of this quality were not known even for the case $\Sigma=\mathbb P_{\mathbb C}^2$. An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up $\Sigma$ of the form $\mathcal O_{\widetilde{\Sigma}}(\pi^*D-\sum_{i=1}^rm_iE_i)$, deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in $\mathbb P_{\mathbb C}^3$, and K3-surfaces.
On crepant resolution of some hypersurface singularities and a criterion for UFD
Hui-Wen
Lin
1861-1868
Abstract: In this article, we find some diagonal hypersurfaces that admit crepant resolutions. We also give a criterion for unique factorization domains.
Verlinde bundles and generalized theta linear series
Mihnea
Popa
1869-1898
Abstract: In this paper we approach the study of generalized theta linear series on moduli of vector bundles on curves via vector bundle techniques on abelian varieties. We study a naturally defined class of vector bundles on a Jacobian, called Verlinde bundles, in order to obtain information about duality between theta functions and effective global and normal generation on these moduli spaces.
Principal bundles over a projective scheme
Donghoon
Hyeon
1899-1908
Abstract: We prove the existence of a quasi-projective moduli scheme for principal bundles over an arbitrary projective scheme.
Sums of squares in real analytic rings
José
F.
Fernando
1909-1919
Abstract: Let $A$ be an analytic ring. We show: (1) $A$ has finite Pythagoras number if and only if its real dimension is $\leq 2$, and (2) if every positive semidefinite element of $A$ is a sum of squares, then $A$ is real and has real dimension $2$.
Small rational model of subspace complement
Sergey
Yuzvinsky
1921-1945
Abstract: This paper concerns the rational cohomology ring of the complement $M$ of a complex subspace arrangement. We start with the De Concini-Procesi differential graded algebra that is a rational model for $M$. Inside it we find a much smaller subalgebra $D$ quasi-isomorphic to the whole algebra. $D$ is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice $L$whence connecting the De Concini-Procesi model with the Goresky-MacPherson formula for the additive structure of $H^*(M)$. The algebra $D$ has a natural integral version that is a good candidate for an integral model of $M$. If the rational local homology of $L$ can be computed explicitly we obtain an explicit presentation of the ring $H^*(M,{\mathbf Q})$. For example, this is done for the cases where $L$ is a geometric lattice and where $M$ is a $k$-equal manifold.
On the maximal Bochner-Riesz conjecture in the plane for $p<2$
Terence
Tao
1947-1959
Abstract: We give a new estimate on the maximal Bochner-Riesz operator in the plane, for $p<2$; as a corollary we obtain an almost everywhere convergence result for certain Bochner-Riesz means. This work was inspired by discussions with Michael Christ and Chris Sogge.
Nonlinear Cauchy-Riemann operators in $\mathbb{R}^{n}$
Tadeusz
Iwaniec
1961-1995
Abstract: This paper has arisen from an effort to provide a comprehensive and unifying development of the $L^{p}$-theory of quasiconformal mappings in $\mathbb{R}^{n}$. The governing equations for these mappings form nonlinear differential systems of the first order, analogous in many respects to the Cauchy-Riemann equations in the complex plane. This approach demands that one must work out certain variational integrals involving the Jacobian determinant. Guided by such integrals, we introduce two nonlinear differential operators, denoted by $\mathcal{D}^{-}$and $\mathcal{D}^{+}$, which act on weakly differentiable deformations $f:\Omega \to \mathbb{R}^{n}$ of a domain $\Omega \subset \mathbb{R}^{n}$. Solutions to the so-called Cauchy-Riemann equations $\mathcal{D}^{-}f=0$ and $\mathcal{D}^{+}f=0$ are simply conformal deformations preserving and reversing orientation, respectively. These operators, though genuinely nonlinear, possess the important feature of being rank-one convex. Among the many desirable properties, we give the fundamental $L^{p}$-estimate \begin{displaymath}\Vert\mathcal{D}^{+}f\Vert _{p} \le A_{p}(n)\Vert\mathcal{D}^{-}f\Vert _{p}. \end{displaymath} In quest of the best constant $A_{p}(n)$, we are faced with fascinating problems regarding quasiconvexity of some related variational functionals. Applications to quasiconformal mappings are indicated.
Trudinger inequalities without derivatives
Paul
MacManus;
Carlos
Pérez
1997-2012
Abstract: We prove that the Trudinger inequality holds on connected homogeneous spaces for functions satisfying a very weak type of Poincaré inequality. We also illustrate the connection between this result and the John-Nirenberg theorem for BMO.
$A_p$ weights for nondoubling measures in $R^n$ and applications
Joan
Orobitg;
Carlos
Pérez
2013-2033
Abstract: We study an analogue of the classical theory of $A_p(\mu)$weights in $\mathbb{R} ^n$ without assuming that the underlying measure $\mu$is doubling. Then, we obtain weighted norm inequalities for the (centered) Hardy-Littlewood maximal function and corresponding weighted estimates for nonclassical Calderón-Zygmund operators. We also consider commutators of those Calderón- Zygmund operators with bounded mean oscillation functions ($BMO$), extending the main result from R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. Finally, we study self-improving properties of Poincaré-B.M.O. type inequalities within this context; more precisely, we show that if $f$ is a locally integrable function satisfying $\frac{1}{\mu(Q)}\int_{Q} \vert f-f_{Q}\vert d\mu \le a(Q)$ for all cubes $Q$, then it is possible to deduce a higher $L^p$ integrability result for $f$, assuming a certain simple geometric condition on the functional $a$.
A semigroup of operators in convexity theory
Christer
O.
Kiselman
2035-2053
Abstract: We consider three operators which appear naturally in convexity theory and determine completely the structure of the semigroup generated by them. RESUMO. Duongrupo de operatoroj en la teorio pri konvekseco. Ni konsideras tri operatorojn kiuj aperas nature en la teorio pri konvekseco kaj plene determinas la strukturon de la duongrupo generita de ili.
The super order dual of an ordered vector space and the Riesz--Kantorovich formula
Charalambos
D.
Aliprantis;
Rabee
Tourky
2055-2077
Abstract: A classical theorem of F. Riesz and L. V. Kantorovich asserts that if $L$ is a vector lattice and $f$ and $g$are order bounded linear functionals on $L$, then their supremum (least upper bound) $f\lor g$ exists in $L^\sim$ and for each $x\in L_+$ it satisfies the so-called Riesz-Kantorovich formula: \begin{displaymath}\bigl[f\lor g\bigr](x)=\sup\bigl\{f(y)+g(z)\colon y,z\in L_+ \,\hbox{and} \, y+z=x\bigr\}\,. \end{displaymath} Related to the Riesz-Kantorovich formula is the following long-standing problem: If the supremum of two order bounded linear functionals $f$ and $g$ on an ordered vector space exists, does it then satisfy the Riesz-Kantorovich formula? In this paper, we introduce an extension of the order dual of an ordered vector space and provide some answers to this long-standing problem. The ideas regarding the Riesz-Kantorovich formula owe their origins to the study of the fundamental theorems of welfare economics and the existence of competitive equilibrium. The techniques introduced here show that the existence of decentralizing prices for efficient allocations is closely related to the above-mentioned problem and to the properties of the Riesz-Kantorovich formula.
Constructing division rings as module-theoretic direct limits
George
M.
Bergman
2079-2114
Abstract: If $R$ is an associative ring, one of several known equivalent types of data determining the structure of an arbitrary division ring $D$ generated by a homomorphic image of $R$ is a rule putting on all free $R$-modules of finite rank matroid structures (closure operators satisfying the exchange axiom) subject to certain functoriality conditions. This note gives a new description of how $D$ may be constructed from this data. (A classical precursor of this is the construction of $\mathbf Q$ as a field with additive group a direct limit of copies of $\mathbf Z$.) The division rings of fractions of right and left Ore rings, the universal division ring of a free ideal ring, and the concept of a specialization of division rings are then interpreted in terms of this construction.
Certain imprimitive reflection groups and their generic versions
Jian-yi
Shi
2115-2129
Abstract: The present paper is concerned with the connection between the imprimitive reflection groups $G(m,m,n)$, $m\in \mathbb{N}$, and the affine Weyl group $\widetilde {A}_{n-1}$. We show that $\widetilde {A}_{n-1}$ is a generic version of the groups $G(m,m,n)$, $m\in \mathbb{N}$. We introduce some new presentations of these groups which are shown to have some group-theoretic advantages. Then we define the Hecke algebras of these groups and of their braid versions, each in two ways according to two presentations. Finally we give a new description for the affine root system $\overline{\Phi }$ of $\widetilde {A}_{n-1}$ such that the action of $\widetilde {A}_{n-1}$ on $\overline{\Phi }$ is compatible with that of $G(m,m,n)$ on its root system in some sense.
Gaussian bounds for derivatives of central Gaussian semigroups on compact groups
A.
Bendikov;
L.
Saloff-Coste
1279-1298
Abstract: For symmetric central Gaussian semigroups on compact connected groups, assuming the existence of a continuous density, we show that this density admits space derivatives of all orders in certain directions. Under some additional assumptions, we prove that these derivatives satisfy certain Gaussian bounds.
An analogue of minimal surface theory in $\operatorname{SL}(n,\mathbf C)/\operatorname{SU}(n)$
M.
Kokubu;
M.
Takahashi;
M.
Umehara;
K.
Yamada
1299-1325
Abstract: We shall discuss the class of surfaces with holomorphic right Gauss maps in non-compact duals of compact semi-simple Lie groups (e.g. $\operatorname{SL}(n,\mathbf{C})/\operatorname{SU}(n)$), which contains minimal surfaces in $\mathbf{R}^n$ and constant mean curvature $1$ surfaces in $\mathcal{H}^3$. A Weierstrass type representation formula and a Chern-Osserman type inequality for such surfaces are given.
Determinacy and weakly Ramsey sets in Banach spaces
Joan
Bagaria;
Jordi
López-Abad
1327-1349
Abstract: We give a sufficient condition for a set of block subspaces in an infinite-dimensional Banach space to be weakly Ramsey. Using this condition we prove that in the Levy-collapse of a Mahlo cardinal, every projective set is weakly Ramsey. This, together with a construction of W. H. Woodin, is used to show that the Axiom of Projective Determinacy implies that every projective set is weakly Ramsey. In the case of $c_0$ we prove similar results for a stronger Ramsey property. And for hereditarily indecomposable spaces we show that the Axiom of Determinacy plus the Axiom of Dependent Choices imply that every set is weakly Ramsey. These results are the generalizations to the class of projective sets of some theorems from W. T. Gowers, and our paper ``Weakly Ramsey sets in Banach spaces.''
Milnor classes of local complete intersections
J.-P.
Brasselet;
D.
Lehmann;
J.
Seade;
T.
Suwa
1351-1371
Abstract: Let $V$ be a compact local complete intersection defined as the zero set of a section of a holomorphic vector bundle over the ambient space. For each connected component $S$ of the singular set $\operatorname{Sing}(V)$ of $V$, we define the Milnor class $\mu _{*}(V,S)$ in the homology of $S$. The difference between the Schwartz-MacPherson class and the Fulton-Johnson class of $V$ is shown to be equal to the sum of $\mu _{*}(V,S)$ over the connected components $S$ of $\operatorname{Sing}(V)$. This is done by proving Poincaré-Hopf type theorems for these classes with respect to suitable tangent frames. The $0$-degree component $\mu _{0}(V,S)$ coincides with the Milnor numbers already defined by various authors in particular situations. We also give an explicit formula for $\mu _{*}(V,S)$ when $S$ is a non-singular component and $V$ satisfies the Whitney condition along $S$.
Dual decompositions of 4-manifolds
Frank
Quinn
1373-1392
Abstract: This paper concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index $\leq 2$. In dimensions $\geq 5$results of Smale (trivial $\pi _{1}$) and Wall (general $\pi _{1}$) describe analogous decompositions up to diffeomorphism in terms of homotopy type of skeleta or chain complexes. In dimension 4 we show the same data determines decompositions up to 2-deformation of their spines. In higher dimensions spine 2-deformation implies diffeomorphism, but in dimension 4 the fundamental group of the boundary is not determined. Sample results: (1.5) Two 2-complexes are (up to 2-deformation) spines of a dual decomposition of the 4-sphere if and only if they satisfy the conclusions of Alexander-Lefshetz duality ( $H_{1}K\simeq H^{2}L$ and $H_{2}K\simeq H^{1}L$). (3.3) If $(N,\partial N)$ is 1-connected then there is a ``pseudo'' handle decomposition without 1-handles, in the sense that there is a pseudo collar $(M,\partial N)$ (a relative 2-handlebody with spine that 2-deforms to $\partial N$) and $N$ is obtained from this by attaching handles of index $\geq 2$.
Second order Lagrangian Twist systems: simple closed characteristics
J.
B.
Van den Berg;
R.
C.
Vandervorst
1393-1420
Abstract: We consider a special class of Lagrangians that play a fundamental role in the theory of second order Lagrangian systems: Twist systems. This subclass of Lagrangian systems is defined via a convenient monotonicity property that such systems share. This monotonicity property (Twist property) allows a finite dimensional reduction of the variational principle for finding closed characteristics in fixed energy levels. This reduction has some similarities with the method of broken geodesics for the geodesic variational problem on Riemannian manifolds. On the other hand, the monotonicity property can be related to the existence of local Twist maps in the associated Hamiltonian flow. The finite dimensional reduction gives rise to a second order monotone recurrence relation. We study these recurrence relations to find simple closed characteristics for the Lagrangian system. More complicated closed characteristics will be dealt with in future work. Furthermore, we give conditions on the Lagrangian that guarantee the Twist property.
Global existence for a quasi-linear evolution equation with a non-convex energy
Eduard
Feireisl;
Hana
Petzeltová
1421-1434
Abstract: We establish the existence of global in time weak solutions to the initial-boundary value problem related to the dynamics of coherent solid-solid phase transitions in viscoelasticity. The class of the stored energy functionals includes the double well potential, and a general convolution damping term is considered.
Weak amenability of triangular Banach algebras
B.
E.
Forrest;
L.
W.
Marcoux
1435-1452
Abstract: Let ${\mathcal A}$ and ${\mathcal B}$be unital Banach algebras, and let ${\mathcal M}$ be a Banach ${\mathcal A},{\mathcal B}$-module. Then ${\mathcal T} = \left[ \begin{array}{cc} {\mathcal A} & {\mathcal M} 0 & {\mathcal B} \end{array} \right]$ becomes a triangular Banach algebra when equipped with the Banach space norm $\ensuremath {\Vert}\left[ \begin{array}{cc} a & m 0 & b \end{array} \right] \... ...rt} _{{\mathcal M}} + \ensuremath {\Vert} b \ensuremath {\Vert} _{{\mathcal B}}$. A Banach algebra ${\mathcal T}$is said to be $n$-weakly amenable if all derivations from ${\mathcal T}$ into its $n^{\mathrm{th}}$ dual space ${\mathcal T}^{(n)}$ are inner. In this paper we investigate Arens regularity and $n$-weak amenability of a triangular Banach algebra ${\mathcal T}$ in relation to that of the algebras ${\mathcal A}$, ${\mathcal B}$ and their action on the module ${\mathcal M}$.
SRB measures and Pesin's entropy formula for endomorphisms
Min
Qian;
Shu
Zhu
1453-1471
Abstract: We present a formulation of the SRB (Sinai-Ruelle-Bowen) property for invariant measures of $C^2$ endomorphisms (maybe non-invertible and with singularities) of a compact manifold via their inverse limit spaces, and prove that this property is necessary and sufficient for Pesin's entropy formula. This result is a non-invertible endomorphisms version of a result of Ledrappier, Strelcyn and Young.
Wandering orbit portraits
Jan
Kiwi
1473-1485
Abstract: We study a counting problem in holomorphic dynamics related to external rays of complex polynomials. We give upper bounds on the number of external rays that land at a point $z$ in the Julia set of a polynomial, provided that $z$has an infinite forward orbit.
Product systems over right-angled Artin semigroups
Neal
J.
Fowler;
Aidan
Sims
1487-1509
Abstract: We build upon Mac Lane's definition of a tensor category to introduce the concept of a product system that takes values in a tensor groupoid $\mathcal G$. We show that the existing notions of product systems fit into our categorical framework, as do the $k$-graphs of Kumjian and Pask. We then specialize to product systems over right-angled Artin semigroups; these are semigroups that interpolate between free semigroups and free abelian semigroups. For such a semigroup we characterize all product systems which take values in a given tensor groupoid $\mathcal G$. In particular, we obtain necessary and sufficient conditions under which a collection of $k$ $1$-graphs form the coordinate graphs of a $k$-graph.
Tractor calculi for parabolic geometries
Andreas
Cap;
A.
Rod
Gover
1511-1548
Abstract: Parabolic geometries may be considered as curved analogues of the homogeneous spaces $G/P$ where $G$ is a semisimple Lie group and $P\subset G$ a parabolic subgroup. Conformal geometries and CR geometries are examples of such structures. We present a uniform description of a calculus, called tractor calculus, based on natural bundles with canonical linear connections for all parabolic geometries. It is shown that from these bundles and connections one can recover the Cartan bundle and the Cartan connection. In particular we characterize the normal Cartan connection from this induced bundle/connection perspective. We construct explicitly a family of fundamental first order differential operators, which are analogous to a covariant derivative, iterable and defined on all natural vector bundles on parabolic geometries. For an important subclass of parabolic geometries we explicitly and directly construct the tractor bundles, their canonical linear connections and the machinery for explicitly calculating via the tractor calculus.
Block representation type of reduced enveloping algebras
Iain
Gordon;
Alexander
Premet
1549-1581
Abstract: Let $K$ be an algebraically closed field of characteristic $p$, $G$ a connected, reductive $K$-group, $\mathfrak{g}=\text{Lie}(G)$, $\chi\in\mathfrak{g}^*$ and $U_\chi(\mathfrak{g})$ the reduced enveloping algebra of $\mathfrak{g}$ associated with $\chi$. Assume that $G^{(1)}$ is simply-connected, $p$ is good for $G$ and $\mathfrak{g}$ has a non-degenerate $G$-invariant bilinear form. All blocks of $U_\chi(\mathfrak{g})$ having finite and tame representation type are determined.
Nonradial solvability structure of super-diffusive nonlinear parabolic equations
Panagiota
Daskalopoulos;
Manuel
del Pino
1583-1599
Abstract: We study the solvability of the Cauchy problem for the nonlinear parabolic equation \begin{displaymath}\frac {\partial u}{\partial t} = \mbox{div}\, (u^{m-1}\nabla u)\end{displaymath} when $m < 0$ in ${\bf R}^2$, with $u(x,0)= f(x)$ a given nonnegative function. It is known from earlier works of the authors that the asymptotic radial growth $r^{-2/1-m}$, $r=\vert x\vert$ for the spherical averages of $f(x)$ is critical for local solvability, in particular ensuring it if $f$ is radially symmetric. We show that if the initial data $f(x)$ behaves in polar coordinates like $r^{-2/1-m} g(\theta )$, for large $r= \vert x\vert$ with $g$ nonnegative and $2\pi$-periodic, then the following holds: If $g$ vanishes on some interval of length $l^* = \frac {(m-1)\pi}{2m} >0$, then there is no local solution of the initial value problem. On the other hand, if such an interval does not exist, then the initial value problem is locally solvable and the time of existence can be estimated explicitly.
The index of a critical point for densely defined operators of type $(S_+)_L$ in Banach spaces
Athanassios
G.
Kartsatos;
Igor
V.
Skrypnik
1601-1630
Abstract: The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type $(S_{+})_{L}$ acting from a real, reflexive and separable Banach space $X$ into $X^{*}.$ This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in $l^{p},~p>2,$ illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type $(S_{+})).$ Applications to nonlinear Dirichlet problems have appeared elsewhere.
Extremal problems for quasiconformal maps of punctured plane domains
Vladimir
Markovic
1631-1650
Abstract: The main goal of this paper is to give an affirmative answer to the long-standing conjecture which asserts that the affine map is a uniquely extremal quasiconformal map in the Teichmüller space of the complex plane punctured at the integer lattice points. In addition we derive a corollary related to the geometry of the corresponding Teichmüller space. Besides that we consider the classical dual extremal problem which naturally arises in the tangent space of the Teichmüller space. In particular we prove the uniqueness of Hahn-Banach extension of the associated linear functional given on the Bergman space of the integer lattice domain. Several useful estimates related to the local and global properties of integrable meromorphic functions and the delta functional (see the definition below) are also obtained. These estimates are intended to study the behavior of integrable functions near singularities and they are valid in general settings.
Convergence of two-dimensional weighted integrals
Malabika
Pramanik
1651-1665
Abstract: A two-dimensional weighted integral in $\mathbb R^{2}$ is proposed as a tool for analyzing higher-dimensional unweighted integrals, and a necessary and sufficient condition for the finiteness of the weighted integral is obtained.
Monge's transport problem on a Riemannian manifold
Mikhail
Feldman;
Robert
J.
McCann
1667-1697
Abstract: Monge's problem refers to the classical problem of optimally transporting mass: given Borel probability measures $\mu^+ \ne \mu^-$, find the measure-preserving map $s:M \longrightarrow M$ between them which minimizes the average distance transported. Set on a complete, connected, Riemannian manifold $M$ -- and assuming absolute continuity of $\mu^+$ -- an optimal map will be shown to exist. Aspects of its uniqueness are also established.
An estimate for weighted Hilbert transform via square functions
S.
Petermichl;
S.
Pott
1699-1703
Abstract: We show that the norm of the Hilbert transform as an operator on the weighted space $L^2(w)$ is bounded by a constant multiple of the $3/2$ power of the $A_2$ constant of $w$, in other words by $c\, \sup_I (\langle \omega \rangle_I \langle \omega^{-1} \rangle_I)^{3/2}$. We also give a short proof for sharp upper and lower bounds for the dyadic square function.
Stochastic averaging with a flattened Hamiltonian: A Markov process on a stratified space (a whiskered sphere)
Richard
B.
Sowers
853-900
Abstract: We consider a random perturbation of a 2-dimensional Hamiltonian ODE. Under an appropriate change of time, we identify a reduced model, which in some aspects is similar to a stochastically averaged model. The novelty of our problem is that the set of critical points of the Hamiltonian has an interior. Thus we can stochastically average outside this set of critical points, but inside we can make no model reduction. The result is a Markov process on a stratified space which looks like a whiskered sphere (i.e, a 2-dimensional sphere with a line attached). At the junction of the sphere and the line, glueing conditions identify the behavior of the Markov process.
Finely $\mu$-harmonic functions of a Markov process
R.
K.
Getoor
901-924
Abstract: Let $X$ be a Borel right process and $m$ a fixed excessive measure. Given a finely open nearly Borel set $G$ we define an operator $\Lambda_G$ which we regard as an extension of the restriction to $G$ of the generator of $X$. It maps functions on $E$ to (locally) signed measures on $G$ not charging $m$-semipolars. Given a locally smooth signed measure $\mu$ we define $h$ to be (finely) $\mu$-harmonic on $G$ provided $(\Lambda_G + \mu) h = 0$ on $G$ and denote the class of such $h$ by $\mathcal H^\mu_f (G)$. Under mild conditions on $X$ we show that $h \in \mathcal H^\mu_f (G)$ is equivalent to a local ``Poisson'' representation of $h$. We characterize $\mathcal H^\mu_f (G)$ by an analog of the mean value property under secondary assumptions. We obtain global Poisson type representations and study the Dirichlet problem for elements of $\mathcal H^\mu_f (G)$ under suitable finiteness hypotheses. The results take their nicest form when specialized to Hunt processes.
Small profinite structures
Ludomir
Newelski
925-943
Abstract: We propose a model-theoretic framework for investigating profinite structures. We prove that in many cases small profinite structures interpret infinite groups. This corresponds to results of Hrushovski and Peterzil on interpreting groups in locally modular stable and o-minimal structures.
Groups definable in separably closed fields
E.
Bouscaren;
F.
Delon
945-966
Abstract: We consider the groups which are infinitely definable in separably closed fields of finite degree of imperfection. We prove in particular that no new definable groups arise in this way: we show that any group definable in such a field $L$ is definably isomorphic to the group of $L$-rational points of an algebraic group defined over $L$.
Coloring ${\mathbb R}^n$
James
H.
Schmerl
967-974
Abstract: If $1 \leq m \leq n$ and $A \subseteq {\mathbb R}$, then define the graph $G(A,m,n)$ to be the graph whose vertex set is ${\mathbb R}^n$ with two vertices $x,y \in {\mathbb R}^n$ being adjacent iff there are distinct $u,v \in A^m$ such that $\Vert x-y\Vert = \Vert u-v\Vert$. For various $m$ and $n$ and various $A$, typically $A = {\mathbb Q}$ or $A = {\mathbb Z}$, the graph $G(A,m,n)$ can be properly colored with $\omega$ colors. It is shown that in some cases such a coloring $\varphi : {\mathbb R}^n \longrightarrow\omega$ can also have the additional property that if $\alpha : {\mathbb R}^m \longrightarrow{\mathbb R}^n$ is an isometric embedding, then the restriction of $\varphi$ to $\alpha(A^m)$ is a bijection onto $\omega$.
Some combinatorics of binomial coefficients and the Bloch-Gieseker property for some homogeneous bundles
Mei-Chu
Chang
975-992
Abstract: A vector bundle has the Bloch-Gieseker property if all its Chern classes are numerically positive. In this paper we show that the non-ample bundle $\Omega ^{p}_{\mathbb{P}_{n}}(p+1)$ has the Bloch-Gieseker property, except for two cases, in which the top Chern classes are trivial and the other Chern classes are positive. Our method is to reduce the problem to showing, e.g. the positivity of the coefficient of $t^{k}$ in the rational function $\frac{(1+t)^{\binom n p} (1+3t)^{\binom {n}{p-2}} \cdots (1+(p-1)t)^{\binom n2}... ...1+2t)^{\binom {n}{p-1}} (1+4t)^{\binom {n}{p-3}} \cdots (1+pt)^{\binom {n}{1}}}$ (for $p$ even).
A dimension inequality for Cohen-Macaulay rings
Sean
Sather-Wagstaff
993-1005
Abstract: The recent work of Kurano and Roberts on Serre's positivity conjecture suggests the following dimension inequality: for prime ideals $\mathfrak{p}$ and $\mathfrak{q}$ in a local, Cohen-Macaulay ring $(A,\mathfrak{n})$ such that $e(A_{\mathfrak{p}})=e(A)$ we have $\dim(A/\mathfrak{p})+\dim(A/\mathfrak{q})\leq\dim(A)$. We establish this dimension inequality for excellent, local, Cohen-Macaulay rings which contain a field, for certain low-dimensional cases and when $R/\mathfrak{p}$ is regular.
The structure of linear codes of constant weight
Jay
A.
Wood
1007-1026
Abstract: In this paper we determine completely the structure of linear codes over $\mathbb Z/N\mathbb Z$ of constant weight. Namely, we determine exactly which modules underlie linear codes of constant weight, and we describe the coordinate functionals involved. The weight functions considered are: Hamming weight, Lee weight, two forms of Euclidean weight, and pre-homogeneous weights. We prove a general uniqueness theorem for virtual linear codes of constant weight. Existence is settled on a case by case basis.
Symplectic $2$-handles and transverse links
David
T.
Gay
1027-1047
Abstract: A standard convexity condition on the boundary of a symplectic manifold involves an induced positive contact form (and contact structure) on the boundary; the corresponding concavity condition involves an induced negative contact form. We present two methods of symplectically attaching $2$-handles to convex boundaries of symplectic $4$-manifolds along links transverse to the induced contact structures. One method results in concave boundaries and depends on a fibration of the link complement over $S^1$; in this case the handles can be attached with any framing larger than a lower bound determined by the fibration. The other method results in a weaker convexity condition on the new boundary (sufficient to imply tightness of the new contact structure), and in this case the handles can be attached with any framing less than a certain upper bound. These methods supplement methods developed by Weinstein and Eliashberg for attaching symplectic $2$-handles along Legendrian knots.
Splittings of finitely generated groups over two-ended subgroups
Brian
H.
Bowditch
1049-1078
Abstract: We describe a means of constructing splittings of a one-ended finitely generated group over two-ended subgroups, starting with a finite collection of codimension-one two-ended subgroups. In the case where all the two-ended subgroups have two-ended commensurators, we obtain an annulus theorem, and a form of the JSJ splitting of Rips and Sela. The construction uses ideas from the work of Dunwoody, Sageev and Swenson. We use a particular kind of order structure which combines cyclic orders and treelike structures. In the special case of hyperbolic groups, this provides a link between combinarorial constructions, and constructions arising from the topological structure of the boundary. In this context, we recover the annulus theorem of Scott and Swarup. We also show that a one-ended finitely generated group which contains an infinite-order element, and such that every infinite cyclic subgroup is (virtually) codimension-one is a virtual surface group.
Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications
A.
Fursikov;
M.
Gunzburger;
L.
Hou
1079-1116
Abstract: We study trace theorems for three-dimensional, time-dependent solenoidal vector fields. The interior function spaces we consider are natural for solving unsteady boundary value problems for the Navier-Stokes system and other systems of partial differential equations. We describe the space of restrictions of such vector fields to the boundary of the space-time cylinder and construct extension operators from this space of restrictions defined on the boundary into the interior. Only for two exceptional, but useful, values of the spatial smoothness index, the spaces for which we construct extension operators is narrower than the spaces in which we seek restrictions. The trace spaces are characterized by vector fields having different smoothnesses in directions tangential and normal to the boundary; this is a consequence of the solenoidal nature of the fields. These results are fundamental in the study of inhomogeneous boundary value problems for systems involving solenoidal vector fields. In particular, we use the trace theorems in a study of inhomogeneous boundary value problems for the Navier-Stokes system of viscous incompressible flows.
On a stochastic nonlinear equation in one-dimensional viscoelasticity
Jong
Uhn
Kim
1117-1135
Abstract: In this paper we discuss an initial-boundary value problem for a stochastic nonlinear equation arising in one-dimensional viscoelasticity. We propose to use a new direct method to obtain a solution. This method is expected to be applicable to a broad class of nonlinear stochastic partial differential equations.
Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations
Koichiro
Naito
1137-1151
Abstract: In this paper we introduce recurrent dimensions of discrete dynamical systems and we give upper and lower bounds of the recurrent dimensions of the quasi-periodic orbits. We show that these bounds have different values according to the algebraic properties of the frequency and we investigate these dimensions of quasi-periodic trajectories given by solutions of a nonlinear PDE.
Discrete decompositions for bilinear operators and almost diagonal conditions
Loukas
Grafakos;
Rodolfo
H.
Torres
1153-1176
Abstract: Using discrete decomposition techniques, bilinear operators are naturally associated with trilinear tensors. An intrinsic size condition on the entries of such tensors is introduced and is used to prove boundedness for the corresponding bilinear operators on several products of function spaces. This condition should be considered as the direct analogue of an almost diagonal condition for linear operators of Calderón-Zygmund type. Applications include a reduced $T1$ theorem for bilinear pseudodifferential operators and the extension of an $L^p$ multiplier result of Coifman and Meyer to the full range of $H^p$ spaces. The results of this article rely on decomposition techniques developed by Frazier and Jawerth and on the vector valued maximal function estimate of Fefferman and Stein.
Composite Bank-Laine functions and a question of Rubel
J.
K.
Langley
1177-1191
Abstract: A Bank-Laine function is an entire function $E$ satisfying $E = f \circ g$, with $f, g$ entire. Further, we prove that if $h$ is a transcendental entire function of finite order, then there exists a path tending to infinity on which $h$ and all its derivatives tend to infinity, thus establishing for finite order a conjecture of Rubel.
Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds
Emmanuel
Hebey
1193-1213
Abstract: Given $(M,g)$ a smooth compact Riemannian $n$-manifold, $n \ge 3$, we return in this article to the study of the sharp Sobolev-Poincaré type inequality \begin{displaymath}\Vert u\Vert_{2^\star}^2 \le K_n^2\Vert\nabla u\Vert_2^2 + B\Vert u\Vert_1^2\tag*{(0.1)}\end{displaymath} where $2^\star = 2n/(n-2)$ is the critical Sobolev exponent, and $K_n$ is the sharp Euclidean Sobolev constant. Druet, Hebey and Vaugon proved that $(0.1)$ is true if $n = 3$, that $(0.1)$is true if $n \ge 4$ and the sectional curvature of $g$ is a nonpositive constant, or the Cartan-Hadamard conjecture in dimension $n$ is true and the sectional curvature of $g$ is nonpositive, but that $(0.1)$ is false if $n \ge 4$ and the scalar curvature of $g$ is positive somewhere. When $(0.1)$ is true, we define $B(g)$ as the smallest $B$ in $(0.1)$. The saturated form of $(0.1)$ reads as \begin{displaymath}\Vert u\Vert_{2^\star}^2 \le K_n^2\Vert\nabla u\Vert_2^2+B(g)\Vert u\Vert_1^2. \tag*{(0.2)}\end{displaymath} We assume in this article that $n \ge 4$, and complete the study by Druet, Hebey and Vaugon of the sharp Sobolev-Poincaré inequality $(0.1)$. We prove that $(0.1)$ is true, and that $(0.2)$ possesses extremal functions when the scalar curvature of $g$ is negative. A fairly complete answer to the question of the validity of $(0.1)$ under the assumption that the scalar curvature is not necessarily negative, but only nonpositive, is also given.
Scattering poles for asymptotically hyperbolic manifolds
David
Borthwick;
Peter
Perry
1215-1231
Abstract: For a class of manifolds $X$ that includes quotients of real hyperbolic $(n+1)$-dimensional space by a convex co-compact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on $X$ coincide, with multiplicities, with the poles of the meromorphically continued scattering operator for $X$. In order to carry out the proof, we use Shmuel Agmon's perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations.
Spherical unitary highest weight representations
Bernhard
Krötz;
Karl-Hermann
Neeb
1233-1264
Abstract: In this paper we give an almost complete classification of the $H$-spherical unitary highest weight representations of a hermitian Lie group $G$, where $G/H$ is a symmetric space of Cayley type.
Hamburger and Stieltjes moment problems in several variables
F.-H.
Vasilescu
1265-1278
Abstract: In this paper we give solutions to the Hamburger and Stieltjes moment problems in several variables, in algebraic terms, via extended sequences. Some characterizations of the uniqueness of the solutions are also presented.
A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane
Jerry
L.
Bona;
S.
M.
Sun;
Bing-Yu
Zhang
427-490
Abstract: The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem \begin{displaymath}(0.1)\qquad\qquad\quad \left\{ \begin{array}{l} \eta _t+\eta ... ...uad \eta(0,t) =h(t),\end{array}\right. \qquad\qquad\qquad\quad \end{displaymath} studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0.1) by making use of modern methods for the study of nonlinear dispersive wave equations. Speaking technically, local well-posedness is obtained for initial data $\phi$ in the class $H^s(R^+)$ for $s>\frac34$ and boundary data $h$ in $H^{(1+s)/3}_{loc} (R^+)$, whereas global well-posedness is shown to hold for $\phi \in H^s (R^+) , h\in H^{\frac{7+3s}{12}}_{loc} (R^+)$ when $1\leq s\leq 3$, and for $\phi \in H^s(R^+) , h\in H^{(s+1)/3}_{loc} (R^+)$ when $s\geq 3$. In addition, it is shown that the correspondence that associates to initial data $\phi$and boundary data $h$ the unique solution $u$ of (0.1) is analytic. This implies, for example, that solutions may be approximated arbitrarily well by solving a finite number of linear problems.
Quantitative estimates of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries
B.
Canuto;
E.
Rosset;
S.
Vessella
491-535
Abstract: In this paper we obtain quantitative estimates of strong unique continuation for solutions to parabolic equations. We apply these results to prove stability estimates of logarithmic type for an inverse problem consisting in the determination of unknown portions of the boundary of a domain $\Omega$ in $\mathbb{R}^{n}$, from the knowledge of overdetermined boundary data for parabolic boundary value problems.
On the structure of spectra of periodic elliptic operators
Peter
Kuchment;
Sergei
Levendorskiî
537-569
Abstract: The paper discusses the problem of absolute continuity of spectra of periodic elliptic operators. A new result on absolute continuity for a matrix operator of Schrödinger type is obtained. It is shown that all types of operators for which the absolute continuity has previously been established can be reduced to this one. It is also discovered that some natural generalizations stumble upon an obstacle in the form of non-triviality of a certain analytic bundle on the two-dimensional torus.
A classification of hyperpolar and cohomogeneity one actions
Andreas
Kollross
571-612
Abstract: An isometric action of a compact Lie group on a Riemannian manifold is called hyperpolar if there exists a closed, connected submanifold that is flat in the induced metric and meets all orbits orthogonally. In this article, a classification of hyperpolar actions on the irreducible Riemannian symmetric spaces of compact type is given. Since on these symmetric spaces actions of cohomogeneity one are hyperpolar, i.e. normal geodesics are closed, we obtain a classification of the homogeneous hypersurfaces in these spaces by computing the cohomogeneity for all hyperpolar actions. This result implies a classification of the cohomogeneity one actions on compact strongly isotropy irreducible homogeneous spaces.
Sufficient conditions for zero-one laws
Jason
P.
Bell
613-630
Abstract: We generalize a result of Bateman and Erdos concerning partitions, thereby answering a question of Compton. From this result it follows that if $\mathcal{K}$ is a class of finite relational structures that is closed under the formation of disjoint unions and the extraction of components, and if it has the property that the number of indecomposables of size $n$ is bounded above by a polynomial in $n$, then $\mathcal{K}$ has a monadic second order $0$-$1$ law. Moreover, we show that if a class of finite structures with the unique factorization property is closed under the formation of direct products and the extraction of indecomposable factors, and if it has the property that the number of indecomposables of size at most $n$ is bounded above by a polynomial in $\log n$, then this class has a first order $0$-$1$ law. These results cover all known natural examples of classes of structures that have been proved to have a logical $0$-$1$ law by Compton's method of analyzing generating functions.
Polynomials nonnegative on a grid and discrete optimization
Jean
B.
Lasserre
631-649
Abstract: We characterize the real-valued polynomials on $\mathbb R^n$that are nonnegative (not necessarily strictly positive) on a grid $\mathbb K$ of points of $\mathbb R^n$, in terms of a weighted sum of squares whose degree is bounded and known in advance. We also show that the mimimization of an arbitrary polynomial on $\mathbb K$ (a discrete optimization problem) reduces to a convex continuous optimization problem of fixed size. The case of concave polynomials is also investigated. The proof is based on a recent result of Curto and Fialkow on the $\mathbb K$-moment problem.
Skew Schubert functions and the Pieri formula for flag manifolds
Nantel
Bergeron;
Frank
Sottile
651-673
Abstract: We show the equivalence of the Pieri formula for flag manifolds with certain identities among the structure constants for the Schubert basis of the polynomial ring. This gives new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified definition of skew Schur functions, Stanley symmetric functions, and skew Schubert functions (defined here). We also use algebraic geometry to show the coefficient of a monomial in a Schubert polynomial counts certain chains in the Bruhat order, obtainng a combinatorial chain construction of Schubert polynomials.
Tensor product varieties and crystals: $GL$ case
Anton
Malkin
675-704
Abstract: A geometric theory of tensor product for $\mathfrak{gl}_{N}$-crystals is described. In particular, the role of Spaltenstein varieties in the tensor product is explained, and thus a direct (non-combinatorial) proof of the fact that the number of irreducible components of a Spaltenstein variety is equal to a Littlewood-Richardson coefficient (i.e. certain tensor product multiplicity) is obtained.
Tenth order mock theta functions in Ramanujan's lost notebook (IV)
Youn-Seo
Choi
705-733
Abstract: Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved the first six of Ramanujan's tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan's tenth order mock theta function identities which are expressed by mock theta functions and a definite integral. L. J. Mordell's transformation formula for the definite integral plays a key role in the proofs of these identities. Also, the properties of modular forms are used for the proofs of theta function identities.
The A-polynomial from the noncommutative viewpoint
Charles
Frohman;
Razvan
Gelca;
Walter
LoFaro
735-747
Abstract: The paper introduces a noncommutative generalization of the A-polynomial of a knot. This is done using the Kauffman bracket skein module of the knot complement, and is based on the relationship between skein modules and character varieties. The construction is possible because the Kauffman bracket skein algebra of the cylinder over the torus is a subalgebra of the noncommutative torus. The generalized version of the A-polynomial, called the noncommutative A-ideal, consists of a finitely generated ideal of polynomials in the quantum plane. Some properties of the noncommutative A-ideal and its relationships with the A-polynomial and the Jones polynomial are discussed. The paper concludes with the description of the examples of the unknot, and the right- and left-handed trefoil knots.
New bases for Triebel-Lizorkin and Besov spaces
G.
Kyriazis;
P.
Petrushev
749-776
Abstract: We give a new method for construction of unconditional bases for general classes of Triebel-Lizorkin and Besov spaces. These include the $L_p$, $H_p$, potential, and Sobolev spaces. The main feature of our method is that the character of the basis functions can be prescribed in a very general way. In particular, if $\Phi$ is any sufficiently smooth and rapidly decaying function, then our method constructs a basis whose elements are linear combinations of a fixed (small) number of shifts and dilates of the single function $\Phi$. Typical examples of such $\Phi$'s are the rational function $\Phi (\cdot) = (1 + \vert\cdot\vert^2)^{-N}$ and the Gaussian function $\Phi (\cdot) = e^{-\vert\cdot\vert^2}.$ This paper also shows how the new bases can be utilized in nonlinear approximation.
Symmetric approximation of frames and bases in Hilbert spaces
Michael
Frank;
Vern
I.
Paulsen;
Terry
R.
Tiballi
777-793
Abstract: We introduce the symmetric approximation of frames by normalized tight frames extending the concept of the symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces. We prove existence and uniqueness results for the symmetric approximation of frames by normalized tight frames. Even in the case of the symmetric orthogonalization of bases, our techniques and results are new. A crucial role is played by whether or not a certain operator related to the initial frame or basis is Hilbert-Schmidt.
Transverse surfaces and attractors for 3-flows
W.
J.
Colmenarez;
C.
A.
Morales
795-806
Abstract: We prove that a hyperbolic strange attractor of a three-dimensional vector field is a suspension if it exhibits a transverse surface over which the unstable manifold induces a lamination without closed leaves. We also study the topological equivalence of singular attractors exhibiting transverse surfaces for three-dimensional vector fields.
Induced operators on symmetry classes of tensors
Chi-Kwong
Li;
Alexandru
Zaharia
807-836
Abstract: Let $V$ be an $n$-dimensional Hilbert space. Suppose $H$ is a subgroup of the symmetric group of degree $m$, and $\chi: H \rightarrow \mathbb C$ is a character of degree 1 on $H$. Consider the symmetrizer on the tensor space $\bigotimes^m V$ \begin{displaymath}S(v_1\otimes \cdots \otimes v_m) = {1\over \vert H\vert}\sum... ... v_{\sigma^{-1}(1)} \otimes \cdots \otimes v_{\sigma^{-1}(m)} \end{displaymath} defined by $H$ and $\chi$. The vector space \begin{displaymath}V_\chi^m(H) = S(\bigotimes^m V) \end{displaymath} is a subspace of $\bigotimes^m V$, called the symmetry class of tensors over $V$ associated with $H$ and $\chi$. The elements in $V_\chi^m(H)$ of the form $S(v_1\otimes \cdots \otimes v_m)$ are called decomposable tensors and are denoted by $v_1*\cdots * v_m$. For any linear operator $T$ acting on $V$, there is a (unique) induced operator $K(T)$ acting on $V_\chi^m(H)$ satisfying \begin{displaymath}K(T) v_1* \dots *v_m = Tv_1* \cdots * Tv_m. \end{displaymath} In this paper, several basic problems on induced operators are studied.
On inversion of the Bessel and Gelfand transforms
Masaaki
Furusawa;
Joseph
A.
Shalika
837-852
Abstract: We construct the Plancherel measure corresponding to the Bessel model on the split special orthogonal group of $\text{odd degree}\ge 5$ and the Whittaker model on a connected split reductive group in general. As an application we prove the inversion formula which expresses the related integral transform in terms of the Satake transform.
Rational $S^1$-equivariant homotopy theory
Laura
Scull
1-45
Abstract: We give an algebraicization of rational $S^1$-equivariant homotopy theory. There is an algebraic category of `` $\mathbb{T}$-systems'' which is equivalent to the homotopy category of rational $S^1$-simply connected $S^1$-spaces. There is also a theory of ``minimal models'' for $\mathbb{T}$-systems, analogous to Sullivan's minimal algebras. Each $S^1$-space has an associated minimal $\mathbb{T}$-system which encodes all of its rational homotopy information, including its rational equivariant cohomology and Postnikov decomposition.
Cohomology of buildings and finiteness properties of $\widetilde{A}_n$-groups
Jacqui
Ramagge;
Wayne
W.
Wheeler
47-61
Abstract: Borel and Serre calculated the cohomology of the building associated to a reductive group and used the result to deduce that torsion-free $S$-arithmetic groups are duality groups. By replacing their group-theoretic arguments with proofs relying only upon the geometry of buildings, we show that Borel and Serre's approach can be modified to calculate the cohomology of any locally finite affine building. As an application we show that any finitely presented $\widetilde{A}_n$-group is a virtual duality group. A number of other finiteness conditions for $\widetilde{A}_n$-groups are also established.
Construction de certaines opérades et bigèbres associées aux polytopes de Stasheff et hypercubes
Frédéric
Chapoton
63-74
Abstract: Stasheff polytopes, introduced by Stasheff in his study of $H$-spaces, are linked to associativity. The direct sum of their cellular complexes is the underlying complex of the $A_{\infty}$ operad which describes homotopy associative algebras. In particular, there exists a quasi-isomorphism $A_{\infty}\rightarrow \operatorname{As}$. Here, we define on the direct sum of their dual cellular complexes the structure of a differential graded operad. This construction extends the dendriform operad of Loday, which corresponds to the vertices of the polytopes. We also define the structure of a differential graded operad on the direct sum of the dual cellular complexes of the hypercubes. We define a quasi-isomorphism from $\operatorname{As}$ to each of these operads. We also define non-differential variants of the two preceding operads and a morphism from $\operatorname{As}$ to each of these operads. We show that the free algebras have a coproduct which turns them into bialgebras. RÉSUMÉ. Les polytopes de Stasheff, introduits pour l'étude des $H$-espaces, sont liés à l'associativité. La somme directe de leurs complexes cellulaires forme le complexe sous-jacent à l'opérade $A_\infty$ qui décrit les algèbres associatives à homotopie près. En particulier, il existe un quasi-isomorphisme $A_\infty\to \operatorname{As}$. Ici, on munit la somme directe des duaux de leurs complexes cellulaires d'une structure d'opérade différentielle graduée. Cette construction généralise l'opérade des algèbres dendriformes de Loday, qui correspond aux sommets des polytopes. On munit aussi la somme directe des duaux des complexes cellulaires des hypercubes d'une structure d'opérade différentielle graduée. On définit un quasi-isomorphisme de $\operatorname{As}$ dans chacune de ces deux opérades. On construit également des variantes non différentielles des deux opérades précédentes. On définit un morphisme de $\operatorname{As}$ dans chacune de ces opérades et on montre que les algèbres libres sont munies d'un coproduit coassociatif qui en fait des bigèbres.
Localisation homotopique et tour de Taylor pour une catégorie abélienne
Olivier
Renaudin
75-89
Abstract: On indique comment une sous-catégorie colocalisante d'une catégorie abélienne induit une localisation dans la catégorie dérivé. Ceci permet une nouvelle construction de la tour de Taylor d'un foncteur à valeur dans une catégorie de module.
The best constant in the Davis inequality for the expectation of the martingale square function
Donald
L.
Burkholder
91-105
Abstract: A method is introduced for the simultaneous study of the square function and the maximal function of a martingale that can yield sharp norm inequalities between the two. One application is that the expectation of the square function of a martingale is not greater than $\sqrt3$ times the expectation of the maximal function. This gives the best constant for one side of the Davis two-sided inequality. The martingale may take its values in any real or complex Hilbert space. The elementary discrete-time case leads quickly to the analogous results for local martingales $M$ indexed by $[0,\infty)$. Some earlier inequalities are also improved and, closely related, the Lévy martingale is embedded in a large family of submartingales.
APS boundary conditions, eta invariants and adiabatic limits
Xianzhe
Dai
107-122
Abstract: We prove an adiabatic limit formula for the eta invariant of a manifold with boundary. The eta invariant is defined using the Atiyah-Patodi-Singer boundary condition and the underlying manifold is fibered over a manifold with boundary. Our result extends the work of Bismut-Cheeger to manifolds with boundary.
On arithmetic Macaulayfication of Noetherian rings
Takesi
Kawasaki
123-149
Abstract: The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.
Weakly defective varieties
L.
Chiantini;
C.
Ciliberto
151-178
Abstract: A projective variety $X$ is `$k$-weakly defective' when its intersection with a general $(k+1)$-tangent hyperplane has no isolated singularities at the $k+1$ points of tangency. If $X$ is $k$-defective, i.e. if the $k$-secant variety of $X$ has dimension smaller than expected, then $X$ is also $k$-weakly defective. The converse does not hold in general. A classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension. We start this classification here, describing all weakly defective irreducible surfaces. Our method also provides a new proof of the classical Terracini's classification of $k$-defective surfaces.
Polytopal linear retractions
Winfried
Bruns;
Joseph
Gubeladze
179-203
Abstract: We investigate graded retracts of polytopal algebras (essentially the homogeneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are again polytopal algebras and that codimension $1$ retractions factor through retractions preserving the semigroup structure. We expect that these results hold in general. This paper is a part of the project started by the authors in 1999, where we investigate the graded automorphism groups of polytopal algebras. Part of the motivation comes from the observation that there is a reasonable `polytopal' generalization of linear algebra (and, subsequently, that of algebraic $K$-theory).
The multiplier ideals of a sum of ideals
Mircea
Mustata
205-217
Abstract: We prove that if $\underline{\mathbf{a}}$, $\underline{\mathbf{b}}\subseteq\mathcal{O}_X$ are nonzero sheaves of ideals on a complex smooth variety $X$, then for every $\gamma\in{\mathbb Q}_+$we have the following relation between the multiplier ideals of $\underline{\mathbf{a}}$, $\underline{\mathbf{b}}$ and $\underline{\mathbf{a}}+\underline{\mathbf{b}}$: \begin{displaymath}\mathcal{I}\left(X,\gamma\cdot(\underline{\mathbf{a}}+ \under... ...thbf{a}})\cdot\mathcal{I}(X,\beta\cdot \underline{\mathbf{b}}).\end{displaymath} A similar formula holds for the asymptotic multiplier ideals of the sum of two graded systems of ideals. We use this result to approximate at a given point arbitrary multiplier ideals by multiplier ideals associated to zero dimensional ideals. This is applied to compare the multiplier ideals associated to a scheme in different embeddings.
Explicit merit factor formulae for Fekete and Turyn polynomials
Peter
Borwein;
Kwok-Kwong
Stephen
Choi
219-234
Abstract: We give explicit formulas for the $L_{4}$ norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials \begin{displaymath}f_{q}(z) := \sum ^{q-1}_{k=1} \left (\frac{k}{q}\right ) z^{k} \end{displaymath} where $\left (\frac{\cdot }{q}\right )$ is the Legendre symbol. For example for $q$ an odd prime, \begin{displaymath}\Vert f_{q}\Vert _{4}^{4} : = \frac{5q^{2}}{3}-3q+ \frac{4}{3} - 12 (h(-q))^{2} \end{displaymath} where $h(-q)$ is the class number of $\mathbb{Q}(\sqrt {-q})$. Similar explicit formulas are given for various polynomials including an example of Turyn's that is constructed by cyclically permuting the first quarter of the coefficients of $f_{q}$. This is the sequence that has the largest known asymptotic merit factor. Explicitly, \begin{displaymath}R_{q}(z) := \sum ^{q-1}_{k=0} \left (\frac{k+[q/4] }{q}\right ) z^{k} \end{displaymath} where $[\cdot ]$ denotes the nearest integer, satisfies \begin{displaymath}\Vert R_{q}\Vert _{4}^{4} = \frac{7q^{2}}{6}- {q} - \frac{1}{6} - \gamma _{q} \end{displaymath} where \begin{displaymath}\gamma _{q}: = \begin{cases} h(-q) (h(-q)-4) & \text{if} \qu... ...pmod 8, 0 & \text{if} \quad q \equiv 7 \pmod 8. \end{cases}\end{displaymath} Indeed we derive a closed form for the $L_{4}$ norm of all shifted Fekete polynomials \begin{displaymath}f_{q}^{t}(z) := \sum ^{q-1}_{k=0} \left (\frac{k+t}{q}\right ) z^{k}. \end{displaymath} Namely \begin{align*}\Vert f_{q}^{t} \Vert _{4}^{4} &= \frac{1}{3}(5q^{2}+3q+4)+8t^{2}-... ...tyle \sum _{n=1}^{q-1}n\left(\frac{n+t}{q}\right)}\right \vert^{2}, \end{align*} and $\Vert f_{q}^{q-t+1} \Vert _{4}^{4}= \Vert f_{q}^{t} \Vert _{4}^{4}$ if $1 \le t \le (q+1)/2$.
Topological mixing in $CAT\left(-1\right)$-spaces
Charalambos
Charitos;
Georgios
Tsapogas
235-264
Abstract: If $X$ is a proper $CAT\left( -1\right)$-space and $\Gamma$ a non-elementary discrete group of isometries acting properly discontinuously on $X,$ it is shown that the geodesic flow on the quotient space $Y=X/\Gamma$ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary $\partial X$ and the non-wandering set of the flow equals the whole quotient space of geodesics $GY:=GX/\,\Gamma$ (the latter being redundant when $Y$ is compact). Applications include the proof of topological mixing for (A) compact negatively curved polyhedra, (B) compact quotients of proper geodesically complete $CAT\left( -1\right)$-spaces by a one-ended group of isometries and (C) finite $n$-dimensional ideal polyhedra.
On nonlinear oscillations in a suspension bridge system
Zhonghai
Ding
265-274
Abstract: In this paper, we study nonlinear oscillations in a suspension bridge system governed by two coupled nonlinear partial differential equations. By applying the Leray-Schauder degree theory, it is proved that the suspension bridge system has at least two solutions, one is a near-equilibrium oscillation, and the other is a large amplitude oscillation.
Representation type of Hecke algebras of type $A$
Karin
Erdmann;
Daniel
K.
Nakano
275-285
Abstract: In this paper we provide a complete classification of the representation type for the blocks for the Hecke algebra of type $A$, stated in terms of combinatorical data. The computation of the complexity of Young modules is a key component in the proof of this classification result.
The extraspecial case of the $k(GV)\;$ problem
David
Gluck;
Kay
Magaard
287-333
Abstract: Let $E$ be an extraspecial-type group and $V$ a faithful, absolutely irreducible $k[E]$-module, where $k$ is a finite field. Let $G$ be the normalizer in $GL(V)$ of $E$. We show that, with few exceptions, there exists a $v\in V$such that the restriction of $V$ to $C_H(v)$ is self-dual whenever $H\le G$and $(\vert H\vert, \vert V\vert)=1$.
Subgroup properties of fully residually free groups
Ilya
Kapovich
335-362
Abstract: We prove that fully residually free groups have the Howson property, that is the intersection of any two finitely generated subgroups in such a group is again finitely generated. We also establish some commensurability properties for finitely generated fully residually free groups which are similar to those of free groups. Finally we prove that for a finitely generated fully residually free group the membership problem is solvable with respect to any finitely generated subgroup.
Intertwining operator superalgebras and vertex tensor categories for superconformal algebras, II
Yi-Zhi
Huang;
Antun
Milas
363-385
Abstract: We construct the intertwining operator superalgebras and vertex tensor categories for the $N=2$ superconformal unitary minimal models and other related models.
Dynamics of implicit operations and tameness of pseudovarieties of groups
Jorge
Almeida
387-411
Abstract: This work gives a new approach to the construction of implicit operations. By considering ``higher-dimensional'' spaces of implicit operations and implicit operators between them, the projection of idempotents back to one-dimensional spaces produces implicit operations with interesting properties. Besides providing a wealth of examples of implicit operations which can be obtained by these means, it is shown how they can be used to deduce from results of Ribes and Zalesski{\u{\i}}\kern.15em, Margolis, Sapir and Weil, and Steinberg that the pseudovariety of $p$-groups is tame. More generally, for a recursively enumerable extension closed pseudovariety of groups $\mathbf{V}$, if it can be decided whether a finitely generated subgroup of the free group with the pro- $\mathbf{V}$ topology is dense, then $\mathbf{V}$ is tame.
Shellability in reductive monoids
Mohan
S.
Putcha
413-426
Abstract: The purpose of this paper is to extend to monoids the work of Björner, Wachs and Proctor on the shellability of the Bruhat-Chevalley order on Weyl groups. Let $M$ be a reductive monoid with unit group $G$, Borel subgroup $B$ and Weyl group $W$. We study the partially ordered set of $B\times B$-orbits (with respect to Zariski closure inclusion) within a $G\times G$-orbit of $M$. This is the same as studying a $W\times W$-orbit in the Renner monoid $R$. Such an orbit is the retract of a `universal orbit', which is shown to be lexicograhically shellable in the sense of Björner and Wachs.
Orthogonal, symplectic and unitary representations of finite groups
Carl
R.
Riehm
4687-4727
Abstract: Let $K$ be a field, $G$ a finite group, and $\rho: G \to \mathbf{GL}(V)$ a linear representation on the finite dimensional $K$-space $V$. The principal problems considered are: I. Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms $h: V \times V \rightarrow K$ which are $G$-invariant. II. If $h$ is such a form, enumerate the equivalence classes of representations of $G$ into the corresponding group (orthogonal, symplectic or unitary group). III. Determine conditions on $G$ or $K$ under which two orthogonal, symplectic or unitary representations of $G$ are equivalent if and only if they are equivalent as linear representations and their underlying forms are ``isotypically'' equivalent. This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) $KG$-module components of their spaces are equivalent. We assume throughout that the characteristic of $K$ does not divide $2\vert G\vert$. Solutions to I and II are given when $K$ is a finite or local field, or when $K$ is a global field and the representation is ``split''. The results for III are strongest when the degrees of the absolutely irreducible representations of $G$ are odd - for example if $G$ has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index - and, in the case that $K$ is a local or global field, when the representations are split.
Representation type of $q$-Schur algebras
Karin
Erdmann;
Daniel
K.
Nakano
4729-4756
Abstract: In this paper we classify the $q$-Schur algebras having finite, tame or wild representation type and also the ones which are semisimple.
Geometry of chain complexes and outer automorphisms under derived equivalence
Birge
Huisgen-Zimmermann;
Manuel
Saorín
4757-4777
Abstract: The two main theorems proved here are as follows: If $A$ is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of $A$ is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization $\operatorname{Comp}^{A}_{{\mathbf d}}$ of the family of finite $A$-module complexes with fixed sequence ${\mathbf{d}}$ of dimensions and an ``almost projective'' complex $X\in \operatorname{Comp}^{A} _{{\mathbf d}}$, there exists a canonical vector space embedding \begin{displaymath}T_{X}(\operatorname{Comp}^{A}_{{\mathbf{d}}}) / T_{X}(G.X) \... ...atorname{Hom} _{D^{b}(A{\operatorname{\text{-}Mod}})}(X,X[1]), \end{displaymath} where $G$ is the pertinent product of general linear groups acting on $\operatorname{Comp}^{A}_{{\mathbf{d}}}$, tangent spaces at $X$ are denoted by $T_{X}(-)$, and $X$ is identified with its image in the derived category $D^{b} (A{\operatorname{\text{-}Mod}})$.
Random variable dilation equation and multidimensional prescale functions
Julie
Belock;
Vladimir
Dobric
4779-4800
Abstract: A random variable $Z$ satisfying the random variable dilation equation $MZ \overset{d}{=}Z+G$, where $G$ is a discrete random variable independent of $Z$ with values in a lattice $\Gamma \subset$ $\mathbf{R}^{d}$ and weights $\left\{ c_{k}\right\} _{k\in \Gamma }$ and $M$ is an expanding and $\Gamma$-preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density $\varphi$ which will satisfy a dilation equation \begin{displaymath}\varphi \left( x\right) =\left\vert \det M\right\vert \sum_{k\in \Gamma} c_{k}\varphi \left( Mx-k\right) \text{.} \end{displaymath} We have obtained necessary and sufficient conditions for the existence of the density $\varphi$ and a simple sufficient condition for $\varphi$'s existence in terms of the weights $\left\{ c_{k}\right\} _{k\in \Gamma }.$Wavelets in $\mathbf{R}^{d}$ can be generated in several ways. One is through a multiresolution analysis of $L^{2}\left( \mathbf{R}^{d}\right)$ generated by a compactly supported prescale function $\varphi$. The prescale function will satisfy a dilation equation and its lattice translates will form a Riesz basis for the closed linear span of the translates. The sufficient condition for the existence of $\varphi$ allows a tractable method for designing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient condition is necessary in the case when $\varphi$ is a prescale function.
Second class particles as microscopic characteristics in totally asymmetric nearest-neighbor $K$-exclusion processes
Timo
Seppäläinen
4801-4829
Abstract: We prove laws of large numbers for a second class particle in one-dimensional totally asymmetric $K$-exclusion processes, under hydrodynamic Euler scaling. The assumption required is that initially the ambient particle configuration converges to a limiting profile. The macroscopic trajectories of second class particles are characteristics and shocks of the conservation law of the particle density. The proof uses a variational representation of a second class particle, to overcome the problem of lack of information about invariant distributions. But we cannot rule out the possibility that the flux function of the conservation law may be neither differentiable nor strictly concave. To give a complete picture we discuss the construction, uniqueness, and other properties of the weak solution that the particle density obeys.
Centered complexity one Hamiltonian torus actions
Yael
Karshon;
Susan
Tolman
4831-4861
Abstract: We consider symplectic manifolds with Hamiltonian torus actions which are ``almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are ``centered" and the moment map is proper. In particular, this classifies the preimages under the moment map of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmannians $\operatorname{Gr}^+(2,\mathbb R^5)$ and $\operatorname{Gr}^+(2,\mathbb R^6)$ by two equal symplectic balls.
A measurable cardinal with a closed unbounded set of inaccessibles from $o(\kappa)=\kappa$
William
Mitchell
4863-4897
Abstract: We prove that $o(\kappa)=\kappa$ is sufficient to construct a model $V[C]$in which $\kappa$ is measurable and $C$ is a closed and unbounded subset of $\kappa$ containing only inaccessible cardinals of $V$. Gitik proved that $o(\kappa)=\kappa$ is necessary. We also calculate the consistency strength of the existence of such a set $C$ together with the assumption that $\kappa$ is Mahlo, weakly compact, or Ramsey. In addition we consider the possibility of having the set $C$ generate the closed unbounded ultrafilter of $V$ while $\kappa$ remains measurable, and show that Radin forcing, which requires a weak repeat point, cannot be improved on.
Automorphisms of the lattice of $\Pi_1^0$ classes; perfect thin classes and anc degrees
Peter
Cholak;
Richard
Coles;
Rod
Downey;
Eberhard
Herrmann
4899-4924
Abstract: $\Pi_1^0$ classes are important to the logical analysis of many parts of mathematics. The $\Pi_1^0$ classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance, and thin classes. Our main results are an analog of Martin's work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare's work demonstrating that maximal sets form an orbit. In particular, we show that the collection of perfect thin classes (a notion which is definable in the lattice of $\Pi_1^0$ classes) forms an orbit in the lattice of $\Pi_1^0$classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of $\Pi_1^0$classes. We remark that the automorphism result is proven via a $\Delta_3^0$automorphism, and demonstrate that this complexity is necessary.
Conditions imposed by tacnodes and cusps
Joaquim
Roé
4925-4948
Abstract: The study of linear systems of algebraic plane curves with fixed imposed singularities is a classical subject which has recently experienced important progress. The Horace method introduced by A. Hirschowitz has been successfully exploited to prove many $H^1$-vanishing theorems, even in higher dimension. Other specialization techniques, which include degenerations of the plane, are due to Z. Ran and C. Ciliberto and R. Miranda. G. M. Greuel, C. Lossen and E. Shustin use a local specialization procedure together with the Horace method to give the first asymptotically proper general existence criterion for singular curves of low degree. In this paper we develop a specialization method which allows us to compute the dimension of several linear systems as well as to substantially improve the bounds given by Greuel, Lossen and Shustin for curves with tacnodes and cusps.
The curve of ``Prym canonical'' Gauss divisors on a Prym theta divisor
Roy
Smith;
Robert
Varley
4949-4962
Abstract: The Gauss linear system on the theta divisor of the Jacobian of a nonhyperelliptic curve has two striking properties: 1) the branch divisor of the Gauss map on the theta divisor is dual to the canonical model of the curve; 2) those divisors in the Gauss system parametrized by the canonical curve are reducible. In contrast, Beauville and Debarre prove on a general Prym theta divisor of dimension $\ge 3$ all Gauss divisors are irreducible and normal. One is led to ask whether properties 1) and 2) may characterize the Gauss system of the theta divisor of a Jacobian. Since for a Prym theta divisor, the most distinguished curve in the Gauss system is the Prym canonical curve, the natural analog of the canonical curve for a Jacobian, in the present paper we analyze whether the analogs of properties 1) or 2) can ever hold for the Prym canonical curve. We note that both those properties would imply that the general Prym canonical Gauss divisor would be nonnormal. Then we find an explicit geometric model for the Prym canonical Gauss divisors and prove the following results using Beauville's singularities criterion for special subvarieties of Prym varieties: Theorem. For all smooth doubly covered nonhyperelliptic curves of genus $g\ge 5$, the general Prym canonical Gauss divisor is normal and irreducible. Corollary. For all smooth doubly covered nonhyperelliptic curves of genus $g\ge 4$, the Prym canonical curve is not dual to the branch divisor of the Gauss map.
On positivity of line bundles on Enriques surfaces
Tomasz
Szemberg
4963-4972
Abstract: We study linear systems on Enriques surfaces. We prove rationality of Seshadri constants of ample line bundles on Enriques surfaces and provide lower bounds on these numbers. We show the nonexistence of $k$-very ample line bundles on Enriques surfaces of degree $4k+4$ for $k\geq 1$, thus answering an old question of Ballico and Sommese.
Canonical splittings of groups and 3-manifolds
Peter
Scott;
Gadde
A.
Swarup
4973-5001
Abstract: We introduce the notion of a `canonical' splitting over $\mathbb{Z}$ or $\mathbb{Z}\times\mathbb{Z}$ for a finitely generated group $G$. We show that when $G$ happens to be the fundamental group of an orientable Haken manifold $M$ with incompressible boundary, then the decomposition of the group naturally obtained from canonical splittings is closely related to the one given by the standard JSJ-decomposition of $M$. This leads to a new proof of Johannson's Deformation Theorem.
Replacing model categories with simplicial ones
Daniel
Dugger
5003-5027
Abstract: In this paper we show that model categories of a very broad class can be replaced up to Quillen equivalence by simplicial model categories.
The hit problem for the Dickson algebra
Nguyen
H. V.
Hu'ng;
Tran
Ngoc
Nam
5029-5040
Abstract: Let the mod 2 Steenrod algebra, $\mathcal{A}$, and the general linear group, $GL(k,{\mathbb{F} }_2)$, act on $P_{k}:={\mathbb{F} }_2[x_{1},...,x_{k}]$ with $\vert x_{i}\vert=1$ in the usual manner. We prove the conjecture of the first-named author in Spherical classes and the algebraic transfer, (Trans. Amer. Math Soc. 349 (1997), 3893-3910) stating that every element of positive degree in the Dickson algebra $D_{k}:=(P_{k})^{GL(k, {\mathbb{F} }_2)}$ is $\mathcal{A}$-decomposable in $P_{k}$ for arbitrary $k>2$. This conjecture was shown to be equivalent to a weak algebraic version of the classical conjecture on spherical classes, which states that the only spherical classes in $Q_0S^0$ are the elements of Hopf invariant one and those of Kervaire invariant one.
Diophantine approximation, Bessel functions and radially symmetric periodic solutions of semilinear wave equations in a ball
J.
Berkovits;
J.
Mawhin
5041-5055
Abstract: The aim of this paper is to consider the radially-symmetric periodic-Dirichlet problem on $[0,T] \times B^n[a]$ for the equation \begin{displaymath}u_{tt} - \Delta u = f(t,\vert x\vert,u),\end{displaymath} where $\Delta$ is the classical Laplacian operator, and $B^n[a]$ denotes the open ball of center $0$ and radius $a$ in ${\mathbb R}^n.$ When $\alpha = a/T$ is a sufficiently large irrational with bounded partial quotients, we combine some number theory techniques with the asymptotic properties of the Bessel functions to show that $0$ is not an accumulation point of the spectrum of the linear part. This result is used to obtain existence conditions for the nonlinear problem.
Monotonicity of stable solutions in shadow systems
Wei-Ming
Ni;
Peter
Polácik;
Eiji
Yanagida
5057-5069
Abstract: A shadow system appears as a limit of a reaction-diffusion system in which some components have infinite diffusivity. We investigate the spatial structure of its stable solutions. It is known that, unlike scalar reaction-diffusion equations, some shadow systems may have stable nonconstant (monotone) solutions. On the other hand, it is also known that in autonomous shadow systems any nonconstant non-monotone stationary solution is necessarily unstable. In this paper, it is shown in a general setting that any stable bounded (not necessarily stationary) solution is asymptotically homogeneous or eventually monotone in $x$.
La transition vers l'instabilité pour les ondes de choc multi-dimensionnelles
Denis
Serre
5071-5093
Abstract: We consider multi-dimensional shock waves. We study their stability in Hadamard's sense, following Erpenbeck and Majda's strategy. When the unperturbed shock is close to a Lax shock which is already $1$-d unstable, we show, under a generic hypothesis, that it cannot be strongly stable. We also characterize strong instability in terms of a sign of an explicit quadratic form. In most cases, the instability under 1-d perturbations, which occurs for exceptional shock waves, characterizes a transition between weak stability and strong instability in the multi-dimensional setting. RÉSUMÉ. Nous considérons la stabilité des ondes de choc multi-dimensionnelles, en suivant la stratégie d'Erpenbeck et Majda. Lorsque le choc non perturbé est proche d'un choc de Lax longitudinalement instable, nous montrons, moyennant une hypothèse générique, que des ondes de surface sont présentes, empêchant ainsi la stabilité forte. Nous donnons aussi un critère d'instabilité forte en termes de signe d'une certaine forme quadratique. L'instabilité $1$-d d'un choc est en général facile à établir, car elle revêt un caractère exceptionnel. Elle apparaît comme une transition entre la stabilité faible et l'instabilité dans le contexte multi-d.
On the wellposedness of constitutive laws involving dissipation potentials
Wolfgang
Desch;
Ronald
Grimmer
5095-5120
Abstract: We consider a material with memory whose constitutive law is formulated in terms of internal state variables using convex potentials for the free energy and the dissipation. Given the stress at a material point depending on time, existence of a strain and a set of inner variables satisfying the constitutive law is proved. We require strong coercivity assumptions on the potentials, but none of the potentials need be quadratic. As a technical tool we generalize the notion of an Orlicz space to a cone ``normed'' by a convex functional which is not necessarily balanced. Duality and reflexivity in such cones are investigated.
Geometric representation of substitutions of Pisot type
Vincent
Canterini;
Anne
Siegel
5121-5144
Abstract: We prove that a substitutive dynamical system of Pisot type contains a factor which is isomorphic to a minimal rotation on a torus. If the substitution is unimodular and satisfies a certain combinatorial condition, we prove that the dynamical system is measurably conjugate to an exchange of domains in a self-similar compact subset of the Euclidean space.
Invariant measures for parabolic IFS with overlaps and random continued fractions
K.
Simon;
B.
Solomyak;
M.
Urbanski
5145-5164
Abstract: We study parabolic iterated function systems (IFS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no ``overlaps.'' We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.
Maximal semigroups in semi-simple Lie groups
Luiz
A. B.
San Martin
5165-5184
Abstract: The maximal semigroups with nonempty interior in a semi-simple Lie group with finite center are characterized as compression semigroups of subsets in the flag manifolds of the group. For this purpose a convexity theory, called here $\mathcal{B}$-convexity, based on the open Bruhat cells is developed. It turns out that a semigroup with nonempty interior is maximal if and only if it is the compression semigroup of the interior of a $\mathcal{B}$-convex set.
Limit theorems for functionals of mixing processes with applications to $U$-statistics and dimension estimation
Svetlana
Borovkova;
Robert
Burton;
Herold
Dehling
4261-4318
Abstract: In this paper we develop a general approach for investigating the asymptotic distribution of functionals $X_n=f((Z_{n+k})_{k\in\mathbf{Z}})$of absolutely regular stochastic processes $(Z_n)_{n\in \mathbf{Z}}$. Such functionals occur naturally as orbits of chaotic dynamical systems, and thus our results can be used to study probabilistic aspects of dynamical systems. We first prove some moment inequalities that are analogous to those for mixing sequences. With their help, several limit theorems can be proved in a rather straightforward manner. We illustrate this by re-proving a central limit theorem of Ibragimov and Linnik. Then we apply our techniques to $U$-statistics \begin{displaymath}U_n(h) =\frac{1}{{n\choose 2}}\sum_{1\leq i<j\leq n} h(X_i,X_j) \end{displaymath} with symmetric kernel $h:\mathbf{R}\times \mathbf{R}\rightarrow \mathbf{R}$. We prove a law of large numbers, extending results of Aaronson, Burton, Dehling, Gilat, Hill and Weiss for absolutely regular processes. We also prove a central limit theorem under a different set of conditions than the known results of Denker and Keller. As our main application, we establish an invariance principle for $U$-processes $(U_n(h))_{h}$, indexed by some class of functions. We finally apply these results to study the asymptotic distribution of estimators of the fractal dimension of the attractor of a dynamical system.
Uniqueness of volume-minimizing submanifolds calibrated by the first Pontryagin form
Daniel
A.
Grossman;
Weiqing
Gu
4319-4332
Abstract: One way to understand the geometry of the real Grassmann manifold $G_k(\mathbf{R}^{k+n})$ parameterizing oriented $k$-dimensional subspaces of $\mathbf{R}^{k+n}$ is to understand the volume-minimizing subvarieties in each homology class. Some of these subvarieties can be determined by using a calibration. In previous work, one of the authors calculated the set of $4$-planes calibrated by the first Pontryagin form $p_1$ on $G_k(\mathbf{R}^{k+n})$for all $k,n\geq 4$, and identified a family of mutually congruent round $4$-spheres which are consequently homologically volume-minimizing. In the present work, we associate to the family of calibrated planes a Pfaffian system on the symmetry group $SO(k+n,\mathbf R)$, an analysis of which yields a uniqueness result; namely, that any connected submanifold of $G_k(\mathbf{R}^{k+n})$ calibrated by $p_1$ is contained in one of these $4$-spheres. A similar result holds for $p_1$-calibrated submanifolds of the quotient Grassmannian $G_k^\natural(\mathbf{R}^{k+n})$ of non-oriented $k$-planes.
Representation theory and ADHM-construction on quaternion symmetric spaces
Yasuyuki
Nagatomo
4333-4355
Abstract: We determine all irreducible homogeneous bundles with anti-self-dual canonical connections on compact quaternion symmetric spaces. To deform the canonical connections, we give a relation between the representation theory and the theory of monads on the twistor space. The moduli spaces are described via the Bott-Borel-Weil Thereom. The Horrocks bundle is also generalized to higher-dimensional projective spaces.
On the semisimplicity conjecture and Galois representations
Lei
Fu
4357-4369
Abstract: The semisimplicity conjecture says that for any smooth projective scheme $X_0$ over a finite field $\mathbf{F}_q$, the Frobenius correspondence acts semisimply on $H^i(X\otimes_{\mathbf{ F}_q} \mathbf{ F}, \overline{\mathbf{ Q}}_l)$, where $\mathbf{ F}$ is an algebraic closure of $\mathbf{ F}_q$. Based on the works of Deligne and Laumon, we reduce this conjecture to a problem about the Galois representations of function fields. This reduction was also achieved by Laumon a few years ago (unpublished).
$S_{\infty }$ representations and combinatorial identities
Amitai
Regev
4371-4404
Abstract: For various probability measures on the space of the infinite standard Young tableaux we study the probability that in a random tableau, the $(i,j)^{th}$ entry equals a given number $n$. Beside the combinatorics of finite standard tableaux, the main tools here are from the Vershik-Kerov character theory of $S_{\infty}$. The analysis of these probabilities leads to many explicit combinatorial identities, some of which are related to hypergeometric series.
A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves
I.
P.
Goulden;
J.
L.
Harer;
D.
M.
Jackson
4405-4427
Abstract: We determine an expression $\xi^s_g(\gamma)$for the virtual Euler characteristics of the moduli spaces of $s$-pointed real $(\gamma=1/2$) and complex ($\gamma=1$) algebraic curves. In particular, for the space of real curves of genus $g$ with a fixed point free involution, we find that the Euler characteristic is $(-2)^{s-1}(1-2^{g-1})(g+s-2)!B_g/g!$ where $B_g$ is the $g$th Bernoulli number. This complements the result of Harer and Zagier that the Euler characteristic of the moduli space of complex algebraic curves is $(-1)^{s}(g+s-2)!B_{g+1}/(g+1)(g-1)!$ The proof uses Strebel differentials to triangulate the moduli spaces and some recent techniques for map enumeration to count cells. The approach involves a parameter $\gamma$ that permits specialization of the formula to the real and complex cases. This suggests that $\xi^s_g(\gamma)$ itself may describe the Euler characteristics of some related moduli spaces, although we do not yet know what these spaces might be.
Gauss sums and Kloosterman sums over residue rings of algebraic integers
Ronald
Evans
4429-4445
Abstract: Let $\mathcal{O}$ denote the ring of integers of an algebraic number field of degree $m$ which is totally and tamely ramified at the prime $p$. Write $\zeta_q= \exp(2\pi i/q)$, where $q=p^r$. We evaluate the twisted Kloosterman sum \begin{displaymath}\sum\limits_{\alpha\in(\mathcal{O}/q \mathcal{O})^*} \chi(N(\alpha)) \zeta_q^{T(\alpha)+z/N(\alpha)},\end{displaymath} where $T$ and $N$ denote trace and norm, and where $\chi$ is a Dirichlet character (mod $q$). This extends results of Salié for $m=1$ and of Yangbo Ye for prime $m$ dividing $p-1.$ Our method is based upon our evaluation of the Gauss sum \begin{displaymath}\sum\limits_{\alpha\in (\mathcal{O}/q\mathcal{O})^*} \chi(N(\alpha)) \zeta_q^{T(\alpha)},\end{displaymath} which extends results of Mauclaire for $m=1$.
Spherical classes and the Lambda algebra
Nguyen
H. V.
Hu'ng
4447-4460
Abstract: Let $\Gamma^{\wedge}= \bigoplus_k \Gamma_k^{\wedge}$ be Singer's invariant-theoretic model of the dual of the lambda algebra with $H_k(\Gamma^{\wedge})\cong Tor_k^{\mathcal{A}}(\mathbb{F} _2, \mathbb{F} _2)$, where $\mathcal{A}$ denotes the mod 2 Steenrod algebra. We prove that the inclusion of the Dickson algebra, $D_k$, into $\Gamma_k^{\wedge}$ is a chain-level representation of the Lannes-Zarati dual homomorphism \begin{displaymath}\varphi_k^*: \mathbb{F} _2\underset{\mathcal{A}}{\otimes} D_k... ..._k(\mathbb{F} _2, \mathbb{F} _2) \cong H_k(\Gamma^{\wedge})\,. \end{displaymath} The Lannes-Zarati homomorphisms themselves, $\varphi_k$, correspond to an associated graded of the Hurewicz map \begin{displaymath}H:\pi_*^s(S^0)\cong \pi_*(Q_0S^0)\to H_*(Q_0S^0)\,. \end{displaymath} Based on this result, we discuss some algebraic versions of the classical conjecture on spherical classes, which states that Only Hopf invariant one and Kervaire invariant one classes are detected by the Hurewicz homomorphism. One of these algebraic conjectures predicts that every Dickson element, i.e. element in $D_k$, of positive degree represents the homology class $0$ in $Tor^{\mathcal{A}}_k(\mathbb{F} _2,\mathbb{F} _2)$ for $k>2$. We also show that $\varphi_k^*$ factors through $\Fd\underset{\mathcal{A}}{\otimes} Ker\partial_k$, where $\partial_k : \Gamma^{\wedge}_k \to \Gamma^{\wedge}_{k-1}$ denotes the differential of $\Gamma^{\wedge}$. Therefore, the problem of determining $\mathbb{F} _2 \underset{\mathcal{A}}{\otimes} Ker\partial_k$ should be of interest.
Embeddings of $\mathrm{DI}_2$ in $\mathrm{F}_4$
Carles
Broto;
Jesper
M.
Møller
4461-4479
Abstract: We show that there is only one embedding of $\mathrm B\mathrm{DI}_2$ in $\mathrm B\mathrm{F}_4$ at the prime $p=3$, up to self-maps of $\mathrm B\mathrm{DI}_2$. We also describe the effect of the group of self-equivalences of $\mathrm B\mathrm{F}_4$ at the prime $p=3$ on this embedding and then show that the Friedlander exceptional isogeny composed with a suitable Adams map is an involution of $\mathrm B\mathrm{F}_4$ whose homotopy fixed point set coincide with $\mathrm B\mathrm{DI}_2$
Homotopy commutativity of $H$-spaces with finitely generated cohomology
Yusuke
Kawamoto;
James
P.
Lin
4481-4496
Abstract: We show that a simply connected homotopy associative and homotopy commutative mod $3$ $H$-space with finitely generated mod $3$ cohomology is homotopy equivalent to a finite product of $K({\mathbb Z},2)$, $Sp(2)$, the three-connected cover $Sp(2)\langle 3\rangle$and the homotopy fiber $Sp(2)\langle 3;3^i\rangle$of the map $[3^i]:Sp(2)\to K({\mathbb Z},3)$ for $i\ge 1$. Our result also shows that a connected $C_p$-space in the sense of Sugawara with finitely generated mod $p$ cohomology has the homotopy type of a finite product of $K({\mathbb Z},1)$, $K({\mathbb Z},2)$ and $K({\mathbb Z}/p^i,1)$ for $i\ge 1$.
Ground states and spectrum of quantum electrodynamics of nonrelativistic particles
Fumio
Hiroshima
4497-4528
Abstract: A system consisting of finitely many nonrelativistic particles bound on an external potential and minimally coupled to a massless quantized radiation field without the dipole approximation is considered. An ultraviolet cut-off is imposed on the quantized radiation field. The Hamiltonian of the system is defined as a self-adjoint operator in a Hilbert space. The existence of the ground states of the Hamiltonian is established. It is shown that there exist asymptotic annihilation and creation operators. Hence the location of the absolutely continuous spectrum of the Hamiltonian is specified.
Periodic solutions of conservation laws constructed through Glimm scheme
Hermano
Frid
4529-4544
Abstract: We present a periodic version of the Glimm scheme applicable to special classes of $2\times 2$ systems for which a simplication first noticed by Nishida (1968) and further extended by Bakhvalov (1970) and DiPerna (1973) is available. For these special classes of $2\times 2$ systems of conservation laws the simplification of the Glimm scheme gives global existence of solutions of the Cauchy problem with large initial data in $L^\infty\cap BV_{loc}(\mathbb{R} )$, for Bakhvalov's class, and in $L^\infty\cap BV(\mathbb{R} )$, in the case of DiPerna's class. It may also happen that the system is in Bakhvalov's class only at a neighboorhood $\mathcal{V}$ of a constant state, as it was proved for the isentropic gas dynamics by DiPerna (1973), in which case the initial data is taken in $L^\infty\cap BV(\mathbb{R} )$ with $\text{TV}\,(U_0)<\text{const.}$, for some constant which is $O((\gamma-1)^{-1})$ for the isentropic gas dynamics systems. For periodic initial data, our periodic formulation establishes that the periodic solutions so constructed, $u(\cdot ,t)$, are uniformly bounded in $L^\infty\cap BV([0,\ell])$, for all $t>0$, where $\ell$ is the period. We then obtain the asymptotic decay of these solutions by applying a theorem of Chen and Frid in (1999) combined with a compactness theorem of DiPerna in (1983). The question about the decay of Nishida's solution was proposed by Glimm and Lax in (1970) and has remained open since then. The classes considered include the $p$-systems with $p(v)=\gamma v^{-\gamma}$, $-1<\gamma<+\infty$, $\gamma\ne0$, which, for $\gamma\ge 1$, model isentropic gas dynamics in Lagrangian coordinates.
Boundedness and differentiability for nonlinear elliptic systems
Jana
Björn
4545-4565
Abstract: We consider the elliptic system $\operatorname{div} (\mathcal{A}^j (x,u,\nabla u)) = \mathcal{B}^j (x,u,\nabla u)$, $j=1,\ldots,N,$and an obstacle problem for a similar system of variational inequalities. The functions $\mathcal{A}^j$ and $\mathcal{B}^j$ satisfy certain ellipticity and boundedness conditions with a $p$-admissible weight $w$ and exponent $1<p\le2$. The growth of $\mathcal{B}^j$ in $\vert\nabla u\vert$ and $\vert u\vert$ is of order $p-1$. We show that weak solutions of the above systems are locally bounded and differentiable almost everywhere in the classical sense.
Unbounded components of the singular set of the distance function in $\mathbb R^n$
Piermarco
Cannarsa;
Roberto
Peirone
4567-4581
Abstract: Given a closed set $F\subseteq \mathbb{R}^{n}$, the set $\Sigma _{F}$ of all points at which the metric projection onto $F$ is multi-valued is nonempty if and only if $F$ is nonconvex. The authors analyze such a set, characterizing the unbounded connected components of $\Sigma _{F}$. For $F$ compact, the existence of an asymptote for any unbounded component of $\Sigma _{F}$ is obtained.
The Bott--Borel--Weil Theorem for direct limit groups
Loki
Natarajan;
Enriqueta
Rodríguez-Carrington;
Joseph
A.
Wolf
4583-4622
Abstract: We show how highest weight representations of certain infinite dimensional Lie groups can be realized on cohomology spaces of holomorphic vector bundles. This extends the classical Bott-Borel-Weil Theorem for finite-dimensional compact and complex Lie groups. Our approach is geometric in nature, in the spirit of Bott's original generalization of the Borel-Weil Theorem. The groups for which we prove this theorem are strict direct limits of compact Lie groups, or their complexifications. We previously proved that such groups have an analytic structure. Our result applies to most of the familiar examples of direct limits of classical groups. We also introduce new examples involving iterated embeddings of the classical groups and see exactly how our results hold in those cases. One of the technical problems here is that, in general, the limit Lie algebras will have root systems but need not have root spaces, so we need to develop machinery to handle this somewhat delicate situation.
Arithmetic rigidity and units in group rings
F.
E. A.
Johnson
4623-4635
Abstract: For any finite group $G$ the group $U(\mathbf{Z}[G])$ of units in the integral group ring $\mathbf{Z}[G]$ is an arithmetic group in a reductive algebraic group, namely the Zariski closure of $\mathbf{SL}_1(\mathbf{Q}[G])$. In particular, the isomorphism type of the $\mathbf{Q}$-algebra $\mathbf{Q}[G]$ determines the commensurability class of $U(\mathbf{Z}[G])$; we show that, to a large extent, the converse is true. In fact, subject to a certain restriction on the $\mathbf{Q}$-representations of $G$ the converse is exactly true.
Isometries of Hilbert $C^*$-modules
Baruch
Solel
4637-4660
Abstract: Let $X$ and $Y$ be right, full, Hilbert $C^*$-modules over the algebras $A$ and $B$ respectively and let $T:X\to Y$ be a linear surjective isometry. Then $T$ can be extended to an isometry of the linking algebras. $T$ then is a sum of two maps: a (bi-)module map (which is completely isometric and preserves the inner product) and a map that reverses the (bi-)module actions. If $A$(or $B$) is a factor von Neumann algebra, then every isometry $T:X\to Y$ is either a (bi-)module map or reverses the (bi-)module actions.
Metric properties of the group of area preserving diffeomorphisms
Michel
Benaim;
Jean-Marc
Gambaudo
4661-4672
Abstract: Area preserving diffeomorphisms of the 2-disk which are identity near the boundary form a group ${\mathcal D}_2$ which can be equipped, using the $L^2$-norm on its Lie algebra, with a right invariant metric. With this metric the diameter of ${\mathcal D}_2$ is infinite. In this paper we show that ${\mathcal D}_2$ contains quasi-isometric embeddings of any finitely generated free group and any finitely generated abelian free group.
Generic Finiteness for Dziobek Configurations
Richard
Moeckel
4673-4686
Abstract: The goal of this paper is to show that for almost all choices of $n$ masses, $m_i$, there are only finitely many central configurations of the Newtonian $n$-body problem for which the bodies span a space of dimension $n-2$ (such a central configuration is called a Dziobek configuration). The result applies in particular to two-dimensional configurations of four bodies and three-dimensional configurations of five bodies.
Cuntz-Krieger algebras and endomorphisms of finite direct sums of type I$_{\infty }$ factors
Berndt
Brenken
3835-3873
Abstract: A correspondence between algebra endomorphisms of a finite sum of copies of the algebra of all bounded operators on a Hilbert space and representations of certain norm closed $\ast$-subalgebras of bounded operators generated by a finite collection of partial isometries is introduced. Basic properties of this correspondence are investigated after developing some operations on bipartite graphs that usefully describe aspects of this relationship.
Generalized subdifferentials: a Baire categorical approach
Jonathan
M.
Borwein;
Warren
B.
Moors;
Xianfu
Wang
3875-3893
Abstract: We use Baire categorical arguments to construct pathological locally Lipschitz functions. The origins of this approach can be traced back to Banach and Mazurkiewicz (1931) who independently used similar categorical arguments to show that ``almost every continuous real-valued function defined on [0,1] is nowhere differentiable". As with the results of Banach and Mazurkiewicz, it appears that it is easier to show that almost every function possesses a certain property than to construct a single concrete example. Among the most striking results contained in this paper are: Almost every 1-Lipschitz function defined on a Banach space has a Clarke subdifferential mapping that is identically equal to the dual ball; if $\{T_{1}, T_{2},\ldots, T_{n}\}$ is a family of maximal cyclically monotone operators defined on a Banach space $X$ then there exists a real-valued locally Lipschitz function $g$such that $\partial_{0}g(x)=\mbox{co}\{T_{1}(x),T_{2}(x),\ldots, T_{n}(x)\}$for each $x\in X$; in a separable Banach space each non-empty weak$^{*}$compact convex subset in the dual space is identically equal to the approximate subdifferential mapping of some Lipschitz function and for locally Lipschitz functions defined on separable spaces the notions of strong and weak integrability coincide.
Szlenk indices and uniform homeomorphisms
G.
Godefroy;
N.
J.
Kalton;
G.
Lancien
3895-3918
Abstract: We prove some rather precise renorming theorems for Banach spaces with Szlenk index $\omega_0.$ We use these theorems to show the invariance of certain quantitative Szlenk-type indices under uniform homeomorphisms.
Variational principles and mixed multifractal spectra
L.
Barreira;
B.
Saussol
3919-3944
Abstract: We establish a ``conditional'' variational principle, which unifies and extends many results in the multifractal analysis of dynamical systems. Namely, instead of considering several quantities of local nature and studying separately their multifractal spectra we develop a unified approach which allows us to obtain all spectra from a new multifractal spectrum. Using the variational principle we are able to study the regularity of the spectra and the full dimensionality of their irregular sets for several classes of dynamical systems, including the class of maps with upper semi-continuous metric entropy. Another application of the variational principle is the following. The multifractal analysis of dynamical systems studies multifractal spectra such as the dimension spectrum for pointwise dimensions and the entropy spectrum for local entropies. It has been a standing open problem to effect a similar study for the ``mixed'' multifractal spectra, such as the dimension spectrum for local entropies and the entropy spectrum for pointwise dimensions. We show that they are analytic for several classes of hyperbolic maps. We also show that these spectra are not necessarily convex, in strong contrast with the ``non-mixed'' multifractal spectra.
Stochastic processes with sample paths in reproducing kernel Hilbert spaces
Milan
N.
Lukic;
Jay
H.
Beder
3945-3969
Abstract: A theorem of M. F. Driscoll says that, under certain restrictions, the probability that a given Gaussian process has its sample paths almost surely in a given reproducing kernel Hilbert space (RKHS) is either $0$ or $1$. Driscoll also found a necessary and sufficient condition for that probability to be $1$. Doing away with Driscoll's restrictions, R. Fortet generalized his condition and named it nuclear dominance. He stated a theorem claiming nuclear dominance to be necessary and sufficient for the existence of a process (not necessarily Gaussian) having its sample paths in a given RKHS. This theorem - specifically the necessity of the condition - turns out to be incorrect, as we will show via counterexamples. On the other hand, a weaker sufficient condition is available. Using Fortet's tools along with some new ones, we correct Fortet's theorem and then find the generalization of Driscoll's result. The key idea is that of a random element in a RKHS whose values are sample paths of a stochastic process. As in Fortet's work, we make almost no assumptions about the reproducing kernels we use, and we demonstrate the extent to which one may dispense with the Gaussian assumption.
Products of polynomials in uniform norms
Igor
E.
Pritsker
3971-3993
Abstract: We study inequalities connecting a product of uniform norms of polynomials with the norm of their product. This subject includes the well known Gel'fond-Mahler inequalities for the unit disk and Kneser inequality for the segment $[-1,1]$. Using tools of complex analysis and potential theory, we prove a sharp inequality for norms of products of algebraic polynomials over an arbitrary compact set of positive logarithmic capacity in the complex plane. The above classical results are contained in our theorem as special cases. It is shown that the asymptotically extremal sequences of polynomials, for which this inequality becomes an asymptotic equality, are characterized by their asymptotically uniform zero distributions. We also relate asymptotically extremal polynomials to the classical polynomials with asymptotically minimal norms.
A Brunn-Minkowski inequality for the integer lattice
R.
J.
Gardner;
P.
Gronchi
3995-4024
Abstract: A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinalities of the two sets. These bounds in turn imply corresponding new bounds for the lattice point enumerator of the Minkowski sum of two convex lattice polytopes. A Rogers-Shephard type inequality for the lattice point enumerator in the plane is also proved.
Genus $0$ and $1$ Hurwitz numbers: Recursions, formulas, and graph-theoretic interpretations
Ravi
Vakil
4025-4038
Abstract: We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus $0$ and $1$. (Hurwitz numbers essentially count irreducible genus $g$ covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden, Jackson and Vainshtein, and extend results of Hurwitz and many others.
A sharp bound on the size of a connected matroid
Manoel
Lemos;
James
Oxley
4039-4056
Abstract: This paper proves that a connected matroid $M$ in which a largest circuit and a largest cocircuit have $c$ and $c^*$ elements, respectively, has at most $\frac{1}{2}cc^*$ elements. It is also shown that if $e$ is an element of $M$ and $c_e$ and $c^*_e$ are the sizes of a largest circuit containing $e$ and a largest cocircuit containing $e$, then $\vert E(M)\vert \le (c_e -1)(c^*_e - 1) + 1$. Both these bounds are sharp and the first is proved using the second. The second inequality is an interesting companion to Lehman's width-length inequality which asserts that the former inequality can be reversed for regular matroids when $c_e$ and $c^*_e$ are replaced by the sizes of a smallest circuit containing $e$ and a smallest cocircuit containing $e$. Moreover, it follows from the second inequality that if $u$ and $v$ are distinct vertices in a $2$-connected loopless graph $G$, then $\vert E(G)\vert$ cannot exceed the product of the length of a longest $(u,v)$-path and the size of a largest minimal edge-cut separating $u$ from $v$.
Peripheral splittings of groups
B.
H.
Bowditch
4057-4082
Abstract: We define the notion of a ``peripheral splitting'' of a group. This is essentially a representation of the group as the fundamental group of a bipartite graph of groups, where all the vertex groups of one colour are held fixed--the ``peripheral subgroups''. We develop the theory of such splittings and prove an accessibility result. The theory mainly applies to relatively hyperbolic groups with connected boundary, where the peripheral subgroups are precisely the maximal parabolic subgroups. We show that if such a group admits a non-trivial peripheral splitting, then its boundary has a global cut point. Moreover, the non-peripheral vertex groups of such a splitting are themselves relatively hyperbolic. These results, together with results from elsewhere, show that under modest constraints on the peripheral subgroups, the boundary of a relatively hyperbolic group is locally connected if it is connected. In retrospect, one further deduces that the set of global cut points in such a boundary has a simplicial treelike structure.
Limits in the uniform ultrafilters
Joni
Baker;
Kenneth
Kunen
4083-4093
Abstract: Let $u(\kappa)$ be the space of uniform ultrafilters on $\kappa$. If $\kappa$ is regular, then there is an $\mathbf x \in u(\kappa)$which is not an accumulation point of any subset of $u(\kappa)$ of size $\kappa$ or less. $\mathbf x$ is also good, in the sense of Keisler.
Convergence of asymptotic directions
Dinh
The Luc;
Jean-Paul
Penot
4095-4121
Abstract: We study convergence properties of asymptotic directions of unbounded sets in normed spaces. The links between the continuity of a set-valued map and the convergence of asymptotic directions are examined. The results are applied to investigate continuity properties of marginal functions and asymptotic directions of level sets.
Conditional stability estimation for an inverse boundary problem with non-smooth boundary in $\mathcal{R}^3$
J.
Cheng;
Y.
C.
Hon;
M.
Yamamoto
4123-4138
Abstract: In this paper, we investigate an inverse problem of determining a shape of a part of the boundary of a bounded domain in $\mathcal R^3$ by a solution to a Cauchy problem of the Laplace equation. Assuming that the unknown part is a Lipschitz continuous surface, we give a logarithmic conditional stability estimate in determining the part of boundary under reasonably a priori information of an unknown part. The keys are the complex extension and estimates for a harmonic measure.
Hyperbolic conservation laws with stiff reaction terms of monostable type
Haitao
Fan
4139-4154
Abstract: In this paper the zero reaction limit of the hyperbolic conservation law with stiff source term of monostable type \begin{displaymath}\partial _{t} u + \partial _{x} f(u) = \frac {1}{\epsilon } u(1-u)\end{displaymath} is studied. Solutions of Cauchy problems of the above equation with initial value $0\le u_{0}(x)\le 1$ are proved to converge, as $\epsilon \to 0$, to piecewise constant functions. The constants are separated by either shocks determined by the Rankine-Hugoniot jump condition, or a non-shock jump discontinuity that moves with speed $f'(0)$. The analytic tool used is the method of generalized characteristics. Sufficient conditions for the existence and non-existence of traveling waves of the above system with viscosity regularization are given. The reason for the failure to capture the correct shock speed by first order shock capturing schemes when underresolving $\epsilon >0$ is found to originate from the behavior of traveling waves of the above system with viscosity regularization.
On the inverse spectral theory of Schrödinger and Dirac operators
Miklós
Horváth
4155-4171
Abstract: We prove that under some conditions finitely many partially known spectra and partial information on the potential entirely determine the potential. This extends former results of Hochstadt, Lieberman, Gesztesy, Simon and others.
Variational problems on multiply connected thin strips III: Integration of the Ginzburg-Landau equations over graphs
Jacob
Rubinstein;
Michelle
Schatzman
4173-4187
Abstract: We analyze the one-dimensional Ginzburg-Landau functional of superconductivity on a planar graph. In the Euler-Lagrange equations, the equation for the phase can be integrated, provided that the order parameter does not vanish at the vertices; in this case, the minimization of the Ginzburg-Landau functional is equivalent to the minimization of another functional, whose unknowns are a real-valued function on the graph and a finite set of integers.
Invariant distributions supported on the nilpotent cone of a semisimple Lie algebra
Thierry
Levasseur
4189-4202
Abstract: Let $\mathfrak{g}$ be a semisimple complex Lie algebra with adjoint group $G$ and $\mathcal{D}(\mathfrak{g})$ be the algebra of differential operators with polynomial coefficients on $\mathfrak{g}$. If $\mathfrak{g}_0$ is a real form of $\mathfrak{g}$, we give the decomposition of the semisimple $\mathcal{D}(\mathfrak{g})^G$-module of invariant distributions on $\mathfrak{g}_0$ supported on the nilpotent cone.
On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces
Hidénori
Fujiwara;
Gérard
Lion;
Salah
Mehdi
4203-4217
Abstract: Let $G$ be a simply connected connected real nilpotent Lie group with Lie algebra $\mathfrak{g}$, $H$ a connected closed subgroup of $G$ with Lie algebra $\mathfrak{h}$ and $\beta\in\mathfrak{h}^{*}$ satisfying $\beta ([\mathfrak{h},\mathfrak{h} ])=\{0\}$. Let $\chi_{\beta}$ be the unitary character of $H$ with differential $2\sqrt{-1}\pi\beta$ at the origin. Let $\tau\equiv$ $Ind_{H}^{G}\chi_{\beta}$ be the unitary representation of $G$ induced from the character $\chi_{\beta}$ of $H$. We consider the algebra $\mathcal{D}(G,H,\beta)$ of differential operators invariant under the action of $G$ on the bundle with basis $H\backslash G$ associated to these data. We consider the question of the equivalence between the commutativity of $\mathcal{D}(G,H,\beta)$ and the finite multiplicities of $\tau$. Corwin and Greenleaf proved that if $\tau$ is of finite multiplicities, this algebra is commutative. We show that the converse is true in many cases.
On the relation between upper central quotients and lower central series of a group
Graham
Ellis
4219-4234
Abstract: Let $H$ be a group with a normal subgroup $N$ contained in the upper central subgroup $Z_cH$. In this article we study the influence of the quotient group $G=H/N$ on the lower central subgroup $\gamma_{c+1}H$. In particular, for any finite group $G$ we give bounds on the order and exponent of $\gamma_{c+1}H$. For $G$ equal to a dihedral group, or quaternion group, or extra-special group we list all possible groups that can arise as $\gamma_{c+1}H$. Our proofs involve: (i) the Baer invariants of $G$, (ii) the Schur multiplier $\mathcal{M}(L,G)$ of $G$ relative to a normal subgroup $L$, and (iii) the nonabelian tensor product of groups. Some results on the nonabelian tensor product may be of independent interest.
On Herstein's Lie map conjectures, I
K.
I.
Beidar;
M.
Bresar;
M.
A.
Chebotar;
W.
S.
Martindale III
4235-4260
Abstract: We describe surjective Lie homomorphisms from Lie ideals of skew elements of algebras with involution onto noncentral Lie ideals (factored by their centers) of skew elements of prime algebras ${\mathcal{D}}$ with involution, provided that $\operatorname{char}({\mathcal{D}})\not=2$ and ${\mathcal{D}}$ is not PI of low degree. This solves the last remaining open problem of Herstein on Lie isomorphisms module cases of PI rings of low degree. A more general problem on maps preserving any polynomial is also discussed.
Finite state automata: A geometric approach
Benjamin
Steinberg
3409-3464
Abstract: Recently, finite state automata, via the advent of hyperbolic and automatic groups, have become a powerful tool in geometric group theory. This paper develops a geometric approach to automata theory, analogous to various techniques used in combinatorial group theory, to solve various problems on the overlap between group theory and monoid theory. For instance, we give a geometric algorithm for computing the closure of a rational language in the profinite topology of a free group. We introduce some geometric notions for automata and show that certain important classes of monoids can be described in terms of the geometry of their Cayley graphs. A long standing open question, to which the answer was only known in the simplest of cases (and even then was non-trivial), is whether it is true, for a pseudovariety of groups $\mathbf{H}$, that a ${\mathcal J}$-trivial co-extension of a group in $\mathbf{H}$ must divide a semidirect product of a ${\mathcal J}$-trivial monoid and a group in $\mathbf{H}$. We show the answer is affirmative if $\mathbf{H}$ is closed under extension, and may be negative otherwise.
Group cohomology and gauge equivalence of some twisted quantum doubles
Geoffrey
Mason;
Siu-Hung
Ng
3465-3509
Abstract: We study the module category associated to the quantum double of a finite abelian group $G$ twisted by a 3-cocycle, which is known to be a braided monoidal category, and investigate the question of when two such categories are equivalent. We base our discussion on an exact sequence which interweaves the ordinary and Eilenberg-Mac Lane cohomology of $G$. Roughly speaking, this reveals that the data provided by such module categories is equivalent to (among other things) a finite quadratic space equipped with a metabolizer, and also a pair of rational lattices $L\subseteq M$ with $L$ self-dual and integral.
The finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge4$
Cai
Heng
Li
3511-3529
Abstract: A complete classification is given for finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge4$. The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group $\text{M}$, and an infinite family of graphs of valency 5 admitting projective symplectic groups $\text{PSp}(4,p)$ with $p$ prime and $p\equiv\pm1$ (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved.
Jordan curves in the level sets of additive Brownian motion
Robert
C.
Dalang;
T.
Mountford
3531-3545
Abstract: This paper studies the topological and connectivity properties of the level sets of additive Brownian motion. More precisely, for each excursion set of this process from a fixed level, we give an explicit construction of a closed Jordan curve contained in the boundary of this excursion set, and in particular, in the level set of this process.
Markov chains in random environments and random iterated function systems
Örjan
Stenflo
3547-3562
Abstract: We consider random iterated function systems giving rise to Markov chains in random (stationary) environments. Conditions ensuring unique ergodicity and a ``pure type'' characterization of the limiting ``randomly invariant'' probability measure are provided. We also give a dimension formula and an algorithm for simulating exact samples from the limiting probability measure.
Loop-erased walks and total positivity
Sergey
Fomin
3563-3583
Abstract: We consider matrices whose elements enumerate weights of walks in planar directed weighted graphs (not necessarily acyclic). These matrices are totally nonnegative; more precisely, all their minors are formal power series in edge weights with nonnegative coefficients. A combinatorial explanation of this phenomenon involves loop-erased walks. Applications include total positivity of hitting matrices of Brownian motion in planar domains.
Hausdorff convergence and universal covers
Christina
Sormani;
Guofang
Wei
3585-3602
Abstract: We prove that if $Y$ is the Gromov-Hausdorff limit of a sequence of compact manifolds, $M^n_i$, with a uniform lower bound on Ricci curvature and a uniform upper bound on diameter, then $Y$ has a universal cover. We then show that, for $i$ sufficiently large, the fundamental group of $M_i$ has a surjective homeomorphism onto the group of deck transforms of $Y$. Finally, in the non-collapsed case where the $M_i$ have an additional uniform lower bound on volume, we prove that the kernels of these surjective maps are finite with a uniform bound on their cardinality. A number of theorems are also proven concerning the limits of covering spaces and their deck transforms when the $M_i$ are only assumed to be compact length spaces with a uniform upper bound on diameter.
Uniform perfectness of the limit sets of Kleinian groups
Toshiyuki
Sugawa
3603-3615
Abstract: In this note, we show, in a quantitative fashion, that the limit set of a non-elementary Kleinian group is uniformly perfect if the quotient orbifold is of Lehner type, i.e., if the space of integrable holomorphic quadratic differentials on it is continuously contained in the space of (hyperbolically) bounded ones. This result covers the known case when the group is analytically finite. As applications, we present estimates of the Hausdorff dimension of the limit set and the translation lengths in the region of discontinuity for such a Kleinian group. Several examples will also be given.
Maximal degree subsheaves of torsion free sheaves on singular projective curves
E.
Ballico
3617-3627
Abstract: Fix integers $r,k,g$ with $r>k>0$ and $g\ge 2$. Let $X$ be an integral projective curve with $g:=p_a(X)$ and $E$ a rank $r$ torsion free sheaf on $X$which is a flat limit of a family of locally free sheaves on $X$. Here we prove the existence of a rank $k$ subsheaf $A$ of $E$ such that $r(\deg(A))\ge k(\deg (E))-k(r-k)g$. We show that for every $g\ge 9$ there is an integral projective curve $X,X$ not Gorenstein, and a rank 2 torsion free sheaf $E$ on $X$ with no rank 1 subsheaf $A$ with $2(\deg (A))\ge \deg(E)-g$. We show the existence of torsion free sheaves on non-Gorenstein projective curves with other pathological properties.
Orthogonal polynomial eigenfunctions of second-order partial differerential equations
K.
H.
Kwon;
J.
K.
Lee;
L.
L.
Littlejohn
3629-3647
Abstract: In this paper, we show that for several second-order partial differential equations \begin{align*}L[u]&=A(x,y)u_{xx}+2B(x,y)u_{xy}+C(x,y)u_{yy}+D(x,y)u_{x}+E(x,y)u_{y} &=\lambda_{n}u \end{align*} which have orthogonal polynomial eigenfunctions, these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable. This is important since, otherwise, it is very difficult to explicitly find formulas for these polynomial solutions. From this observation and characterization, we are able to produce additional examples of such orthogonal polynomials together with their orthogonality that widens the class found by H. L. Krall and Sheffer in their seminal work in 1967. Moreover, from our approach, we can answer some open questions raised by Krall and Sheffer.
Ill-posedness for the derivative Schrödinger and generalized Benjamin-Ono equations
H.
A.
Biagioni;
F.
Linares
3649-3659
Abstract: Ill-posedness is established for the initial value problem (IVP) associated to the derivative nonlinear Schrödinger equation for data in $H^s(\mathbb R)$, $s<1/2$. This result implies that best result concerning local well-posedness for the IVP is in $H^s(\mathbb R),\, s\ge1/2$. It is also shown that the (IVP) associated to the generalized Benjamin-Ono equation for data below the scaling is in fact ill-posed.
Siegel discs, Herman rings and the Arnold family
Lukas
Geyer
3661-3683
Abstract: We show that the rotation number of an analytically linearizable element of the Arnold family $f_{a,b}(x)=x+a+b\sin(2\pi x)\pmod 1$, $a,b\in{\mathbb R}$, $0<b<1/(2\pi)$, satisfies the Brjuno condition. Conversely, for every Brjuno rotation number there exists an analytically linearizable element of the Arnold family. Along the way we prove the necessity of the Brjuno condition for linearizability of $P_{\lambda,d}(z)=\lambda z(1+z/d)^d$ and $E_\lambda(z)=\lambda z e^z$, $\lambda=e^{2\pi i\alpha}$, at 0. We also investigate the complex Arnold family and classify its possible Fatou components. Finally, we show that the Siegel discs of $P_{\lambda,d}$ and $E_\lambda$ are quasidiscs with a critical point on the boundary if the rotation number is of constant type.
The structure of the Brauer group and crossed products of $C_0(X)$-linear group actions on $C_0(X,\mathcal K)$
Siegfried
Echterhoff;
Ryszard
Nest
3685-3712
Abstract: For a second countable locally compact group $G$ and a second countable locally compact space $X$let $\operatorname{Br}_G(X)$ denote the equivariant Brauer group (for the trivial $G$-space $X$) consisting of all Morita equivalence classes of spectrum fixing actions of $G$ on continuous-trace $C^*$-algebras $A$ with spectrum $\widehat{A}=X$. Extending recent results of several authors, we give a complete description of $\operatorname{Br}_G(X)$ in terms of group cohomology of $G$ and Cech cohomology of $X$. Moreover, if $G$ has a splitting group $H$ in the sense of Calvin Moore, we give a complete description of the $C_0(X)$-bundle structure of the crossed product $A\rtimes_{\alpha}G$ in terms of the topological data associated to the given action $\alpha:G\to \operatorname{Aut} A$and the bundle structure of the group $C^*$-algebra $C^*(H)$ of $H$.
Rotation, entropy, and equilibrium states
Oliver
Jenkinson
3713-3739
Abstract: For a dynamical system $(X,T)$ and function $f:X\to\mathbb{R} ^d$ we consider the corresponding generalised rotation set. This is the convex subset of $\mathbb{R} ^d$ consisting of all integrals of $f$ with respect to $T$-invariant probability measures. We study the entropy $H(\varrho)$of rotation vectors $\varrho$, and relate this to the directional entropy $\mathcal{H}(\varrho)$ of Geller & Misiurewicz. For $(X,T)$ a mixing subshift of finite type, and $f$ of summable variation, we prove that if the rotation set is strictly convex then the functions $\mathcal{H}$ and $H$ are in fact one and the same. For those rotation sets which are not strictly convex we prove that $\mathcal{H}(\varrho)$ and $H(\varrho)$can differ only at non-exposed boundary points $\varrho$.
Canonical symbolic dynamics for one-dimensional generalized solenoids
Inhyeop
Yi
3741-3767
Abstract: We define canonical subshift of finite type covers for Williams' one-dimensional generalized solenoids, and use resulting invariants to distinguish some closely related solenoids.
Berezin transform on real bounded symmetric domains
Genkai
Zhang
3769-3787
Abstract: Let $\mathbb D$ be a bounded symmetric domain in a complex vector space $V_{\mathbb C}$with a real form $V$ and $D=\mathbb D\cap V=G/K$ be the real bounded symmetric domain in the real vector space $V$. We construct the Berezin kernel and consider the Berezin transform on the $L^2$-space on $D$. The corresponding representation of $G$is then unitarily equivalent to the restriction to $G$of a scalar holomorphic discrete series of holomorphic functions on $\mathbb D$ and is also called the canonical representation. We find the spectral symbol of the Berezin transform under the irreducible decomposition of the $L^2$-space.
Crystal bases for $U_{q}(\Gamma(\sigma_{1},\sigma_{2},\sigma_{3}))$
Yi Ming
Zou
3789-3802
Abstract: We construct crystal bases for certain infinite dimensional representations of the $q$-deformation of the Lie superalgebra $\Gamma (\sigma _{1},\sigma _{2}, \sigma _{3})$.
The Serre spectral sequence of a multiplicative fibration
Yves
Félix;
Stephen
Halperin;
Jean-Claude
Thomas
3803-3831
Abstract: In a fibration $\Omega F \overset{\Omega j}{\rightarrow} \Omega X \overset{\Omega \pi}{\rightarrow}\Omega B$ we show that finiteness conditions on $F$ force the homology Serre spectral sequence with $\mathbb{F} _p$-coefficients to collapse at some finite term. This in particular implies that as graded vector spaces, $H_*(\Omega X)$ is ``almost'' isomorphic to $H_*(\Omega B)\otimes H_*(\Omega F)$. One consequence is the conclusion that $X$ is elliptic if and only if $B$ and $F$ are.
Correction to ``Relative Completions of Linear Groups over ${\mathbb Z}[t]$ and ${\mathbb Z}[t,t^{-1}]$''
Kevin
P.
Knudson
3833-3834
Construction and asymptotic stability of structurally stable internal layer solutions
Xiao-Biao
Lin
2983-3043
Abstract: We introduce a geometric/asymptotic method to treat structurally stable internal layer solutions. We consider asymptotic expansions of the internal layer solutions and the critical eigenvalues that determine their stability. Proofs of the existence of exact solutions and eigenvalue-eigenfunctions are outlined. Multi-layered solutions are constructed by a new shooting method through a sequence of pseudo Poincaré mappings that do not require the transversality of the flow to cross sections. The critical eigenvalues are determined by a coupling matrix that generates the SLEP matrix. The transversality of the shooting method is related to the nonzeroness of the critical eigenvalues. An equivalent approach is given to mono-layer solutions. They can be determined by the intersection of a fast jump surface and a slow switching curve, which reduces Fenichel's transversality condition to the slow manifold. The critical eigenvalue is determined by the angle of the intersection. We present three examples. The first treats the critical eigenvalues of the system studied by Angenent, Mallet-Paret & Peletier. The second shows that a key lemma in the SLEP method may not hold. The third is a perturbed activator-inhibitor system that can have any number of mono-layer solutions. Some of the solutions can only be found with the new shooting method.
Compactifying the relative Jacobian over families of reduced curves
Eduardo
Esteves
3045-3095
Abstract: We construct natural relative compactifications for the relative Jacobian over a family $X/S$ of reduced curves. In contrast with all the available compactifications so far, ours admit a Poincaré sheaf after an étale base change. Our method consists of studying the étale sheaf $F$ of simple, torsion-free, rank-1 sheaves on $X/S$, and showing that certain open subsheaves of $F$ have the completeness property. Strictly speaking, the functor $F$ is only representable by an algebraic space, but we show that $F$ is representable by a scheme after an étale base change. Finally, we use theta functions originating from vector bundles to compare our new compactifications with the available ones.
Cohomology of projective space seen by residual complex
I-Chiau
Huang
3097-3114
Abstract: A subcomplex of a residual complex on projective space is constructed for computing the cohomology modules of locally free sheaves. A constructive new proof of the Bott formula is given by explicitly exhibiting bases for the cohomology modules.
Computing the homology of Koszul complexes
Bernhard
Köck
3115-3147
Abstract: Let $R$ be a commutative ring and $I$ an ideal in $R$ which is locally generated by a regular sequence of length $d$. Then, each f. g. projective $R/I$-module $V$ has an $R$-projective resolution $P.$ of length $d$. In this paper, we compute the homology of the $n$-th Koszul complex associated with the homomorphism $P_1 \rightarrow P_0$ for all $n \ge 1$, if $d=1$. This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if $d=2$, we compute the homology of the complex $N\, \operatorname{Sym}^2 \, \Gamma(P.)$ where $\Gamma$ and $N$ denote the functors occurring in the Dold-Kan correspondence.
On the commutation of the test ideal with localization and completion
Gennady
Lyubeznik;
Karen
E.
Smith
3149-3180
Abstract: Let $R$ be a reduced ring that is essentially of finite type over an excellent regular local ring of prime characteristic. Then it is shown that the test ideal of $R$ commutes with localization and, if $R$ is local, with completion, under the additional hypothesis that the tight closure of zero in the injective hull $E$ of the residue field of every local ring of $R$ is equal to the finitistic tight closure of zero in $E$. It is conjectured that this latter condition holds for all local rings of prime characteristic; it is proved here for all Cohen-Macaulay singularities with at most isolated non-Gorenstein singularities, and in general for all isolated singularities. In order to prove the result on the commutation of the test ideal with localization and completion, a ring of Frobenius operators associated to each $R$-module is introduced and studied. This theory gives rise to an ideal of $R$ which defines the non-strongly F-regular locus, and which commutes with localization and completion. This ideal is conjectured to be the test ideal of $R$ in general, and shown to equal the test ideal under the hypothesis that $0_E^*=0_E^{fg*}$in every local ring of $R$.
Classifying subcategories of modules
Mark
Hovey
3181-3191
Abstract: Let $R$ be the quotient of a regular coherent commutative ring by a finitely generated ideal. In this paper, we classify all abelian subcategories of finitely presented $R$-modules that are closed under extensions. We also classify abelian subcategories of arbitrary $R$-modules that are closed under extensions and coproducts, when $R$ is commutative and Noetherian. The method relies on comparison with the derived category of $R$.
Igusa's local zeta functions of semiquasihomogeneous polynomials
W.
A.
Zúñiga-Galindo
3193-3207
Abstract: In this paper, we prove the rationality of Igusa's local zeta functions of semiquasihomogeneous polynomials with coefficients in a non-archimedean local field $K$. The proof of this result is based on Igusa's stationary phase formula and some ideas on Néron $\pi$-desingularization.
Sums of $k$ unit fractions
Christian
Elsholtz
3209-3227
Abstract: Erdos and Straus conjectured that for any positive integer $n\geq 2$ the equation $\frac{4}{n}= \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$has a solution in positive integers $x,\, y$, and $z$. Let $m > k \geq 3$ and \begin{displaymath}E_{m,k}(N)= \, \mid \{ n \leq N \mid \frac{m}{n} = \frac{1}... ...t_k} \text{ has no solution with }t_i \in \mathbb{N}\} \mid . \end{displaymath} We show that parametric solutions can be used to find upper bounds on $E_{m,k}(N)$where the number of parameters increases exponentially with $k$. This enables us to prove \begin{displaymath}E_{m,k}(N) \ll N \exp \left( -c_{m,k} (\log N)^{1-\frac{1}{2^{k-1}-1}} \right) \text{ with } c_{m,k}>0. \end{displaymath} This improves upon earlier work by Viola (1973) and Shen (1986), and is an ``exponential generalization'' of the work of Vaughan (1970), who considered the case $k=3$.
Link complements and the Bianchi modular groups
Mark
D.
Baker
3229-3246
Abstract: We determine the values of $m$ for which the Bianchi modular group $\operatorname{PSL}_2(\mathcal{O}_m)$ contains a link group.
$L^2$-determinant class and approximation of $L^2$-Betti numbers
Thomas
Schick
3247-3265
Abstract: A standing conjecture in $L^2$-cohomology says that every finite $CW$-complex $X$ is of $L^2$-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class $\mathcal G$ of groups containing, e.g., all extensions of residually finite groups with amenable quotients, all residually amenable groups, and free products of these. If, in addition, $X$ is $L^2$-acyclic, we also show that the $L^2$-determinant is a homotopy invariant -- giving a short and easy proof independent of and encompassing all known cases. Under suitable conditions we give new approximation formulas for $L^2$-Betti numbers.
The Siegel modular variety of degree two and level three
J.
William
Hoffman;
Steven
H.
Weintraub
3267-3305
Abstract: Let $\mathcal{A}_{2}(n)$ denote the quotient of the Siegel upper half space of degree two by $\Gamma_{2}(n)$, the principal congruence subgroup of level $n$in $\mathbf{Sp}(4,\mathbf{Z})$. $\mathcal{A}_{2}(n)$ is the moduli space of principally polarized abelian varieties of dimension two with a level $n$structure, and has a compactification $\mathcal{A}_{2}(n)^{\ast}$ first constructed by Igusa. When $n\ge 3$ this is a smooth projective algebraic variety of dimension three. In this work we analyze the topology of $\mathcal{A}_{2}(3)^{\ast}$ and the open subset $\mathcal{A}_{2}(3)$. In this way we obtain the rational cohomology ring of $\Gamma_{2}(3)$. The key is that one has an explicit description of $\mathcal{A}_{2}(3)^{\ast}$: it is the resolution of the 45 nodes on a projective quartic threefold whose equation was first written down about 100 years ago by H. Burkhardt. We are able to compute the zeta function of this variety reduced modulo certain primes.
Bilinear estimates and applications to 2d NLS
J.
E.
Colliander;
J.-M.
Delort;
C.
E.
Kenig;
G.
Staffilani
3307-3325
Abstract: The three bilinearities $u v, \overline{uv},\overline{u}v$ for functions $u, v : \mathbb{R} ^2 \times [0,T] \longmapsto \mathbb{C}$ are sharply estimated in function spaces $X_{s,b}$ associated to the Schrödinger operator $i \partial_t + \Delta$. These bilinear estimates imply local wellposedness results for Schrödinger equations with quadratic nonlinearity. Improved bounds on the growth of spatial Sobolev norms of finite energy global-in-time and blow-up solutions of the cubic nonlinear Schrödinger equation (and certain generalizations) are also obtained.
Theta lifting of holomorphic discrete series: The case of $ U(n, n) \times U(p, q) $
Kyo
Nishiyama;
Chen-bo
Zhu
3327-3345
Abstract: Let $K_{\mathbb{C} }$-module structure of the regular function rings on the closure of the associated nilpotent $K_{\mathbb{C} }$-orbits in $\mathfrak{s}$, where $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{s}$ is a Cartan decomposition. As a consequence, their associated cycles are multiplicity free.
Wandering vector multipliers for unitary groups
Deguang
Han;
D.
Larson
3347-3370
Abstract: A wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system into itself. A special case of unitary system is a discrete unitary group. We prove that for many (and perhaps all) discrete unitary groups, the set of wandering vector multipliers is itself a group. We completely characterize the wandering vector multipliers for abelian and ICC unitary groups. Some characterizations of special wandering vector multipliers are obtained for other cases. In particular, there are simple characterizations for diagonal and permutation wandering vector multipliers. Similar results remain valid for irrational rotation unitary systems. We also obtain some results concerning the wandering vector multipliers for those unitary systems which are the ordered products of two unitary groups. There are applications to wavelet systems.
Projective metrics and mixing properties on towers
Véronique
Maume-Deschamps
3371-3389
Abstract: We study the decay of correlations for towers. Using Birkhoff's projective metrics, we obtain a rate of mixing of the form: \begin{displaymath}c_n (f,g) \leq \text{\rm Ct} \alpha(n) \Vert f \Vert \, \Vert g \Vert_1\end{displaymath} where $\alpha(n)$ goes to zero in a way related to the asymptotic mass of upper floors, $\Vert f\Vert$ is some Lipschitz norm and $\Vert g \Vert_1$ is some $L^1$ norm. The fact that the dependence on $g$ is given by an $L^1$ norm is useful to study asymptotic laws of successive entrance times.
Vector fields with topological stability
Kazumine
Moriyasu;
Kazuhiro
Sakai;
Naoya
Sumi
3391-3408
Abstract: In this paper, we give a characterization of the structurally stable vector fields by making use of the notion of topological stability. More precisely, it is proved that the $C^1$ interior of the set of all topologically stable $C^1$ vector fields coincides with the set of all vector fields satisfying Axiom A and the strong transversality condition.
Discrete Series Characters as Lifts from Two-structure Groups
Rebecca
A.
Herb
2557-2599
Abstract: Let $G$ be a connected reductive Lie group with a relatively compact Cartan subgroup. Then it has relative discrete series representations. The main result of this paper is a formula expressing relative discrete series characters on $G$ as ``lifts'' of relative discrete series characters on smaller groups called two-structure groups for $G$. The two-structure groups are connected reductive Lie groups which are locally isomorphic to the direct product of an abelian group and simple groups which are real forms of $SL(2, \mathbf{C})$ or $SO(5, { {\bf C} })$. They are not necessarily subgroups of $G$, but they ``share'' the relatively compact Cartan subgroup and certain other Cartan subgroups with $G$. The character identity is similar to formulas coming from endoscopic lifting, but the two-structure groups are not necessarily endoscopic groups, and the characters lifted are not stable. Finally, the formulas are valid for non-linear as well as linear groups.
On Bessel distributions for quasi-split groups
Ehud
Moshe
Baruch
2601-2614
Abstract: We show that the Bessel distribution attached by Gelfand and Kazhdan and by Shalika to a generic representation of a quasi-split reductive group over a local field is given by a function when it is restricted to the open Bruhat cell. As in the case of the character distribution, this function is real analytic for archimedean fields and locally constant for non-archimidean fields.
Linear functionals of eigenvalues of random matrices
Persi
Diaconis;
Steven
N.
Evans
2615-2633
Abstract: Let $M_n$ be a random $n \times n$ unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of $M_n$ to converge to a Gaussian limit as $n \rightarrow \infty$. By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenvalues of $M_n$. For example, the integral of this measure against a function with suitably decaying Fourier coefficients converges to a Gaussian limit without any normalisation. Known central limit theorems for the number of eigenvalues in a circular arc and the logarithm of the characteristic polynomial of $M_n$ are also derived from the criterion. Similar results are sketched for Haar distributed orthogonal and symplectic matrices.
Higher type adjunction inequalities for Donaldson invariants
Vicente
Muñoz
2635-2654
Abstract: We prove new adjunction inequalities for embedded surfaces in four-manifolds with non-negative self-intersection number using the Donaldson invariants. These formulas are completely analogous to the ones obtained by Ozsváth and Szabó using the Seiberg-Witten invariants. To prove these relations, we give a fairly explicit description of the structure of the Fukaya-Floer homology of a surface times a circle. As an aside, we also relate the Floer homology of a surface times a circle with the cohomology of some symmetric products of the surface.
Bi-Lipschitz homogeneous curves in $\mathbb{R}^2$ are quasicircles
Christopher
J.
Bishop
2655-2663
Abstract: We show that a bi-Lipschitz homogeneous curve in the plane must satisfy the bounded turning condition, and that this is false in higher dimensions. Combined with results of Herron and Mayer this gives several characterizations of such curves in the plane.
Multiplier ideals of monomial ideals
J.
A.
Howald
2665-2671
Abstract: In this note we discuss a simple algebraic calculation of the multiplier ideal associated to a monomial ideal in affine $n$-space. We indicate how this result allows one to compute not only the multiplier ideal but also the log canonical threshold of an ideal in terms of its Newton polygon.
Invariants and projections of six lines in projective space
Dana
R.
Vazzana
2673-2688
Abstract: Given six lines in $\mathbf{P}^3$, quartics through the six lines define a map from $\mathbf{P}^3$ to $\mathbf{P}^4$, and the image of this map is described in terms of invariants of the six lines. The map can be interpreted as projection of the six lines, and this permits a description of the canonical model of the octic surface which is given by points which project the lines so that they are tangent to a conic. We also define polarity for sets of six lines, and discuss the above map in the case of a self-polar set of lines and in the case of six lines which form a ``double-sixer'' on a cubic surface.
On the shellability of the order complex of the subgroup lattice of a finite group
John
Shareshian
2689-2703
Abstract: We show that the order complex of the subgroup lattice of a finite group $G$ is nonpure shellable if and only if $G$ is solvable. A by-product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups.
Primes in short arithmetic progressions with rapidly increasing differences
P.
D. T. A.
Elliott
2705-2724
Abstract: Primes are, on average, well distributed in short segments of arithmetic progressions, even if the associated moduli grow rapidly.
Centralizers of Iwahori-Hecke algebras
Andrew
Francis
2725-2739
Abstract: To date, integral bases for the centre of the Iwahori-Hecke algebra of a finite Coxeter group have relied on character theoretical results and the isomorphism between the Iwahori-Hecke algebra when semisimple and the group algebra of the finite Coxeter group. In this paper, we generalize the minimal basis approach of an earlier paper, to provide a way of describing and calculating elements of the minimal basis for the centre of an Iwahori-Hecke algebra which is entirely combinatorial in nature, and independent of both the above mentioned theories. This opens the door to further generalization of the minimal basis approach to other cases. In particular, we show that generalizing it to centralizers of parabolic subalgebras requires only certain properties in the Coxeter group. We show here that these properties hold for groups of type $A$ and $B$, giving us the minimal basis theory for centralizers of any parabolic subalgebra in these types of Iwahori-Hecke algebra.
Sur le rang du $2$-groupe de classes de $Q({\sqrt{m}},{\sqrt{d}})$ où $m=2$ ou un premier $p\equiv 1(mod4)$
Abdelmalek
Azizi;
Ali
Mouhib
2741-2752
Abstract: On the rank of the $2$-class group of $Q({\sqrt{m}},{\sqrt{d}})$. Let $d$ be a square-free positive integer and $p$ be a prime such that $p\equiv 1\,(mod\, 4)$. We set $K = Q({\sqrt{m}},{\sqrt{d}})$, where $m=2$ or $m=p$. In this paper, we determine the rank of the $2$-class group of $K$. RÉSUMÉ. Soit $K = Q({\sqrt{m}},{\sqrt{d}})$, un corps biquadratique où $m=2$ ou bien un premier $p\equiv 1\,(mod\,4)$ et $d$ étant un entier positif sans facteurs carrés. Dans ce papier, on détermine le rang du $2$-groupe de classes de $K$.
Degree-one maps between hyperbolic 3-manifolds with the same volume limit
Teruhiko
Soma
2753-2772
Abstract: Suppose that $f_n:M_n\longrightarrow N_n$ $(n\in {\mathbf N})$ are degree-one maps between closed hyperbolic 3-manifolds with \begin{displaymath}\lim_{n\rightarrow \infty} \operatorname{Vol} (M_n)=\lim_{n\rightarrow \infty}{\operatorname{Vol}}(N_n) <\infty. \end{displaymath} Then, our main theorem, Theorem 2, shows that, for all but finitely many $n\in {\mathbf N}$, $f_n$ is homotopic to an isometry. A special case of our argument gives a new proof of Gromov-Thurston's rigidity theorem for hyperbolic 3-manifolds without invoking any ergodic theory. An example in §3 implies that, if the degree of these maps is greater than 1, the assertion corresponding to our theorem does not hold.
$SL_n$-character varieties as spaces of graphs
Adam
S.
Sikora
2773 - 2804
Abstract: An $SL_n$-character of a group $G$ is the trace of an $SL_n$-representation of $G.$ We show that all algebraic relations between $SL_n$-characters of $G$ can be visualized as relations between graphs (resembling Feynman diagrams) in any topological space $X,$ with $\pi_1(X)=G.$ We also show that all such relations are implied by a single local relation between graphs. In this way, we provide a topological approach to the study of $SL_n$-representations of groups. The motivation for this paper was our work with J. Przytycki on invariants of links in 3-manifolds which are based on the Kauffman bracket skein relation. These invariants lead to a notion of a skein module of $M$ which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the $SL_2$-character variety of $\pi_1(M).$This paper provides a generalization of this result to all $SL_n$-character varieties.
A model structure on the category of pro-simplicial sets
Daniel
C.
Isaksen
2805-2841
Abstract: We study the category ${pro-}\mathcal{SS}$ of pro-simplicial sets, which arises in étale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on ${pro-}\mathcal{SS}$ so that it is possible to do homotopy theory in this category. This model structure is closely related to the strict structure of Edwards and Hastings. In order to understand the notion of homotopy groups for pro-spaces we use local systems on pro-spaces. We also give several alternative descriptions of weak equivalences, including a cohomological characterization. We outline dual constructions for ind-spaces.
Spectral lifting in Banach algebras and interpolation in several variables
Gelu
Popescu
2843-2857
Abstract: Let ${\mathcal{A}}$ be a unital Banach algebra and let $J$ be a closed two-sided ideal of ${\mathcal{A}}$. We prove that if any invertible element of ${\mathcal{A}}/J$ has an invertible lifting in ${\mathcal{A}}$, then the quotient homomorphism $\Phi :{\mathcal{A}}\to {\mathcal{A}}/J$ is a spectral interpolant. This result is used to obtain a noncommutative multivariable analogue of the spectral commutant lifting theorem of Bercovici, Foias, and Tannenbaum. This yields spectral versions of Sarason, Nevanlinna-Pick, and Carathéodory type interpolation for $F_{n}^{\infty }\bar \otimes B({\mathcal{K}})$, the WOT-closed algebra generated by the spatial tensor product of the noncommutative analytic Toeplitz algebra $F_{n}^{\infty }$ and $B({\mathcal{K}})$, the algebra of bounded operators on a finite dimensional Hilbert space ${\mathcal{K}}$. A spectral tangential commutant lifting theorem in several variables is considered and used to obtain a spectral tangential version of the Nevanlinna-Pick interpolation for $F_{n}^{\infty }\bar \otimes B({\mathcal{K}})$. In particular, we obtain interpolation theorems for matrix-valued bounded analytic functions on the open unit ball of $\mathbb{C} ^{n}$, in which one bounds the spectral radius of the interpolant and not the norm.
Hyperbolic automorphisms and Anosov diffeomorphisms on nilmanifolds
Karel
Dekimpe
2859-2877
Abstract: We translate the problem of finding Anosov diffeomorphisms on a nilmanifold which is covered by a free nilpotent Lie group into a problem of constructing matrices in $\mathrm{GL}(n,\mathbb{Z})$ whose eigenvalues satisfy certain conditions. Afterwards, we show how this translation can then be solved in some specific situations. The paper starts with a section on polynomial permutations of $\mathbb{Q}^K$, a subject which is of interest on its own.
Livsic theorems for connected Lie groups
M.
Pollicott;
C.
P.
Walkden
2879-2895
Abstract: Let $\phi$ be a hyperbolic diffeomorphism on a basic set $\Lambda$ and let $G$ be a connected Lie group. Let $f : \Lambda \rightarrow G$ be Hölder. Assuming that $f$ satisfies a natural partial hyperbolicity assumption, we show that if $u : \Lambda \rightarrow G$ is a measurable solution to $f=u\phi \cdot u^{-1}$ a.e., then $u$ must in fact be Hölder. Under an additional centre bunching condition on $f$, we show that if $f$ assigns `weight' equal to the identity to each periodic orbit of $\phi$, then $f = u\phi \cdot u^{-1}$ for some Hölder $u$. These results extend well-known theorems due to Livsic when $G$ is compact or abelian.
Derived equivalence in $SL_2(p^2)$
Joseph
Chuang
2897-2913
Abstract: We present a proof that Broué's Abelian Defect Group Conjecture is true for the principal $p$-block of the group $SL_2(p^2)$. Okuyama has independently obtained the same result using a different approach.
Minimal projective resolutions
E.
L.
Green;
Ø.
Solberg;
D.
Zacharia
2915-2939
Abstract: In this paper, we present an algorithmic method for computing a projective resolution of a module over an algebra over a field. If the algebra is finite dimensional, and the module is finitely generated, we have a computational way of obtaining a minimal projective resolution, maps included. This resolution turns out to be a graded resolution if our algebra and module are graded. We apply this resolution to the study of the $\operatorname{Ext}$-algebra of the algebra; namely, we present a new method for computing Yoneda products using the constructions of the resolutions. We also use our resolution to prove a case of the ``no loop'' conjecture.
Galois groups of some vectorial polynomials
Shreeram
S.
Abhyankar;
Nicholas
F. J.
Inglis
2941-2969
Abstract: Previously nice vectorial equations were constructed having various finite classical groups as Galois groups. Here such equations are constructed for the remaining classical groups. The previous equations were genus zero equations. The present equations are strong genus zero.
Invariant ideals of abelian group algebras and representations of groups of Lie type
D.
S.
Passman;
A.
E.
Zalesskii
2971-2982
Abstract: This paper contributes to the general study of ideal lattices in group algebras of infinite groups. In recent years, the second author has extensively studied this problem for $G$ an infinite locally finite simple group. It now appears that the next stage in the general problem is the case of abelian-by-simple groups. Some basic results reduce this problem to that of characterizing the ideals of abelian group algebras stable under certain (simple) automorphism groups. Here we begin the analysis in the case where the abelian group $A$ is the additive group of a finite-dimensional vector space $V$ over a locally finite field $F$ of prime characteristic $p$, and the automorphism group $G$ is a simple infinite absolutely irreducible subgroup of $GL(V)$. Thus $G$ is isomorphic to an infinite simple periodic group of Lie type, and $G$ is realized in $GL(V)$ via a twisted tensor product $\phi$ of infinitesimally irreducible representations. If $S$ is a Sylow $p$-subgroup of $G$ and if $\langle v\rangle$ is the unique line in $V$ stabilized by $S$, then the approach here requires a precise understanding of the linear character associated with the action of a maximal torus $T_G$ on $\langle v\rangle$. At present, we are able to handle the case where $\phi$ is a rational representation with character field equal to $F$.
On supports and associated primes of modules over the enveloping algebras of nilpotent Lie algebras
Boris
Sirola
2131-2170
Abstract: Let $\mathfrak n$ be a nilpotent Lie algebra, over a field of characteristic zero, and $\mathcal U$ its universal enveloping algebra. In this paper we study: (1) the prime ideal structure of $\mathcal U$ related to finitely generated $\mathcal U$-modules $V$, and in particular the set $\operatorname{Ass}V$ of associated primes for such $V$ (note that now $\operatorname{Ass}V$ is equal to the set $\operatorname{Annspec}V$ of annihilator primes for $V$); (2) the problem of nontriviality for the modules $V/\mathcal PV$ when $\mathcal P$ is a (maximal) prime of $\mathcal U$, and in particular when $\mathcal P$ is the augmentation ideal $\mathcal U\mathfrak n$ of $\mathcal U$. We define the support of $V$, as a natural generalization of the same notion from commutative theory, and show that it is the object of primary interest when dealing with (2). We also introduce and study the reduced localization and the reduced support, which enables to better understand the set $\operatorname{Ass}V$. We prove the following generalization of a stability result given by W. Casselman and M. S. Osborne in the case when $\mathfrak N$, $\mathfrak N$ as in the theorem, are abelian. We also present some of its interesting consequences. Theorem. Let $\mathfrak Q$ be a finite-dimensional Lie algebra over a field of characteristic zero, and $\mathfrak N$ an ideal of $\mathfrak Q$; denote by $U(\mathfrak N)$ the universal enveloping algebra of $\mathfrak N$. Let $V$ be a $\mathfrak Q$-module which is finitely generated as an $\mathfrak N$-module. Then every annihilator prime of $V$, when $V$ is regarded as a $U(\mathfrak N)$-module, is $\mathfrak Q$-stable for the adjoint action of $\mathfrak Q$ on $U(\mathfrak N)$.
Groups with two extreme character degrees and their normal subgroups
Gustavo
A.
Fernández-Alcober;
Alexander
Moretó
2171-2192
Abstract: We study the finite groups $G$ for which the set $\operatorname{cd}(G)$ of irreducible complex character degrees consists of the two most extreme possible values, that is, $1$ and $\vert G:Z(G)\vert^{1/2}$. We are easily reduced to finite $p$-groups, for which we derive the following group theoretical characterization: they are the $p$-groups such that $\vert G:Z(G)\vert$ is a square and whose only normal subgroups are those containing $G'$ or contained in $Z(G)$. By analogy, we also deal with $p$-groups such that $\vert G:Z(G)\vert=p^{2n+1}$ is not a square, and we prove that $\operatorname{cd}(G) =\{1,p^n\}$ if and only if a similar property holds: for any $N\trianglelefteq G$, either $G'\le N$ or $\vert NZ(G):Z(G)\vert\le p$. The proof of these results requires a detailed analysis of the structure of the $p$-groups with any of the conditions above on normal subgroups, which is interesting for its own sake. It is especially remarkable that these groups have small nilpotency class and that, if the nilpotency class is greater than $2$, then the index of the centre is small, and in some cases we may even bound the order of $G$.
Simple holonomic modules over rings of differential operators with regular coefficients of Krull dimension 2
V.
Bavula;
F.
van Oystaeyen
2193-2214
Abstract: Let $K$ be an algebraically closed field of characteristic zero. Let $\Lambda$ be the ring of ($K$-linear) differential operators with coefficients from a regular commutative affine domain of Krull dimension $2$ which is the tensor product of two regular commutative affine domains of Krull dimension $1$. Simple holonomic $\Lambda$-modules are described. Let a $K$-algebra $D$ be a regular affine commutative domain of Krull dimension $1$ and ${\cal D} (D)$ be the ring of differential operators with coefficients from $D$. We classify (up to irreducible elements of a certain Euclidean domain) simple ${\cal D}(D)$-modules (the field $K$ is not necessarily algebraically closed).
The monopole equations and $J$-holomorphic curves on weakly convex almost Kähler 4-manifolds
Yutaka
Kanda
2215-2243
Abstract: We prove that a weakly convex almost Kähler 4-manifold contains a compact, non-constant $J$-holomorphic curve if the corresponding monopole invariant is not zero and if the corresponding line bundle is non-trivial.
Non-special, non-canal isothermic tori with spherical lines of curvature
Holly
Bernstein
2245-2274
Abstract: This article examines isothermic surfaces smoothly immersed in Möbius space. It finds explicit examples of non-special, non-canal isothermic tori with spherical lines of curvature in two systems by analyzing Darboux transforms of Dupin tori. In addition, it characterizes the property of spherical lines of curvature in terms of differential equations on the Calapso potential of the isothermic immersion, and investigates the effect of classical transformations on this property.
Nonradial Hörmander algebras of several variables and convolution operators
José
Bonet;
Antonio
Galbis;
Siegfried
Momm
2275-2291
Abstract: A characterization of the closed principal ideals in nonradial Hörmander algebras of holomorphic functions of several variables in terms of the behaviour of the generator is obtained. This result is applied to study the range of convolution operators and ultradifferential operators on spaces of quasianalytic functions of Beurling type. Contrary to what is known to happen in the case of non-quasianalytic functions, an ultradistribution on a space of quasianalytic functions is constructed such that the range of the operator does not contain the real analytic functions.
Gröbner bases, H--bases and interpolation
Thomas
Sauer
2293-2308
Abstract: The paper is concerned with a construction for H-bases of polynomial ideals without relying on term orders. The main ingredient is a homogeneous reduction algorithm which orthogonalizes leading terms instead of completely canceling them. This allows for an extension of Buchberger's algorithm to construct these H-bases algorithmically. In addition, the close connection of this approach to minimal degree interpolation, and in particular to the least interpolation scheme due to de Boor and Ron, is pointed out.
Good ideals in Gorenstein local rings
Shiro
Goto;
Sin-Ichiro
Iai;
Kei-ichi
Watanabe
2309-2346
Abstract: Let $I$ be an $\mathfrak{m}$-primary ideal in a Gorenstein local ring ($A$, $\mathfrak{m}$) with $\dim A = d$, and assume that $I$ contains a parameter ideal $Q$ in $A$ as a reduction. We say that $I$ is a good ideal in $A$ if $G = \sum _{n \geq 0} I^{n}/I^{n+1}$ is a Gorenstein ring with $\mathrm{a} (G) = 1 - d$. The associated graded ring $G$ of $I$ is a Gorenstein ring with $\mathrm{a}(G) = -d$ if and only if $I = Q$. Hence good ideals in our sense are good ones next to the parameter ideals $Q$ in $A$. A basic theory of good ideals is developed in this paper. We have that $I$ is a good ideal in $A$ if and only if $I^{2} = QI$ and $I = Q : I$. First a criterion for finite-dimensional Gorenstein graded algebras $A$ over fields $k$ to have nonempty sets $\mathcal{X}_{A}$ of good ideals will be given. Second in the case where $d = 1$ we will give a correspondence theorem between the set $\mathcal{X}_{A}$ and the set $\mathcal{Y}_{A}$ of certain overrings of $A$. A characterization of good ideals in the case where $d = 2$ will be given in terms of the goodness in their powers. Thanks to Kato's Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show that the structure of the set $\mathcal{X}_{A}$ of good ideals in $A$ heavily depends on $d = \dim A$. The set $\mathcal{X}_{A}$ may be empty if $d \leq 2$, while $\mathcal{X}_{A}$ is necessarily infinite if $d \geq 3$ and $A$contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring $k[X_{1},X_{2},X_{3}]$ in three variables over a field $k$. Examples are given to illustrate the theorems.
Relative Embedding Problems
Elena
V.
Black;
John
R.
Swallow
2347-2370
Abstract: We consider Galois embedding problems $G\twoheadrightarrow H\cong \operatorname{Gal}(X/Z)$ such that a Galois embedding problem $G\twoheadrightarrow \operatorname{Gal}(Y/Z)$ is solvable, where $Y/Z$ is a Galois subextension of $X/Z$. For such embedding problems with abelian kernel, we prove a reduction theorem, first in the general case of commutative $k$-algebras, then in the more specialized field case. We demonstrate with examples of dihedral embedding problems that the reduced embedding problem is frequently of smaller order. We then apply these results to the theory of obstructions to central embedding problems, considering a notion of quotients of central embedding problems, and classify the infinite towers of metacyclic $p$-groups to which the reduction theorem applies.
On the computation of stabilized tensor functors and the relative algebraic $K$-theory of dual numbers
Randy
McCarthy
2371-2390
Abstract: We compute the stabilization of functors from exact categories to abelian groups derived from $n$-fold tensor products. Rationally, this gives a new computation for the relative algebraic $K$-theory of dual numbers.
On the telescopic homotopy theory of spaces
A.
K.
Bousfield
2391-2426
Abstract: In telescopic homotopy theory, a space or spectrum $X$ is approximated by a tower of localizations $L^{f}_{n}X$, $n\ge 0$, taking account of $v_{n}$-periodic homotopy groups for progressively higher $n$. For each $n\ge 1$, we construct a telescopic Kuhn functor $\Phi _{n}$ carrying a space to a spectrum with the same $v_{n}$-periodic homotopy groups, and we construct a new functor $\Theta _{n}$ left adjoint to $\Phi _{n}$. Using these functors, we show that the $n$th stable monocular homotopy category (comprising the $n$th fibers of stable telescopic towers) embeds as a retract of the $n$th unstable monocular homotopy category in two ways: one giving infinite loop spaces and the other giving ``infinite $L^{f}_{n}$-suspension spaces.'' We deduce that Ravenel's stable telescope conjectures are equivalent to unstable telescope conjectures. In particular, we show that the failure of Ravenel's $n$th stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial Johnson-Wilson $E(n)_{*}$-homology but nontrivial $v_{n}$-periodic homotopy groups, showing a fundamental difference between the unstable chromatic and telescopic theories. As a stable chromatic application, we show that each spectrum is $K(n)$-equivalent to a suspension spectrum. As an unstable chromatic application, we determine the $E(n)_{*}$-localizations and $K(n)_{*}$-localizations of infinite loop spaces in terms of $E(n)_{*}$-localizations of spectra under suitable conditions. We also determine the $E(n)_{*}$-localizations and $K(n)_{*}$-localizations of arbitrary Postnikov $H$-spaces.
Equivariant surgery with middle dimensional singular sets. II: Equivariant framed cobordism invariance
Masaharu
Morimoto
2427-2440
Abstract: Let $G$ be a finite group and let $f : X \to Y$ be a degree 1, $G$-framed map such that $X$ and $Y$ are simply connected, closed, oriented, smooth manifolds of dimension $n = 2k \geqq 6$ and such that the dimension of the singular set of the $G$-space $X$ is at most $k$. In the previous article, assuming $f$ is $k$-connected, we defined the $G$-equivariant surgery obstruction $\sigma (f)$ in a certain abelian group. There it was shown that if $\sigma (f) = 0$ then $f$ is $G$-framed cobordant to a homotopy equivalence $f' : X' \to Y$. In the present article, we prove that the obstruction $\sigma (f)$ is a $G$-framed cobordism invariant. Consequently, the $G$-surgery obstruction $\sigma (f)$ is uniquely associated to $f : X \to Y$ above even if it is not $k$-connected.
Model category structures on chain complexes of sheaves
Mark
Hovey
2441-2457
Abstract: The unbounded derived category of a Grothendieck abelian category is the homotopy category of a Quillen model structure on the category of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently by Beke. However, in most cases of interest, such as the category of sheaves on a ringed space or the category of quasi-coherent sheaves on a nice enough scheme, the abelian category in question also has a tensor product. The injective model structure is not well-suited to the tensor product. In this paper, we consider another method for constructing a model structure. We apply it to the category of sheaves on a well-behaved ringed space. The resulting flat model structure is compatible with the tensor product and all homomorphisms of ringed spaces.
Stratified solutions for systems of conservation laws
Andrea
Corli;
Olivier
Gues
2459-2486
Abstract: We study a class of weak solutions to hyperbolic systems of conservation (balance) laws in one space dimension, called stratified solutions. These solutions are bounded and ``regular'' in the direction of a linearly degenerate characteristic field of the system, but not in other directions. In particular, they are not required to have finite total variation. We prove some results of local existence and uniqueness.
Whitney's extension problem for multivariate $C^{1,\omega}$-functions
Yuri
Brudnyi;
Pavel
Shvartsman
2487-2512
Abstract: We prove that the trace of the space $C^{1,\omega}({\mathbb R}^n)$to an arbitrary closed subset $X\subset{\mathbb R}^n$is characterized by the following ``finiteness'' property. A function $f:X\rightarrow{\mathbb R}$belongs to the trace space if and only if the restriction $f\vert _Y$ to an arbitrary subset $Y\subset X$ consisting of at most $3\cdot 2^{n-1}$ can be extended to a function $f_Y\in C^{1,\omega}({\mathbb R}^n)$ such that \begin{displaymath}\sup\{\Vert f_Y\Vert _{C^{1,\omega}}:~Y\subset X, ~\operatorname{card} Y\le 3\cdot 2^{n-1}\}<\infty. \end{displaymath} The constant $3\cdot 2^{n-1}$ is sharp. The proof is based on a Lipschitz selection result which is interesting in its own right.
Topological horseshoes
Judy
Kennedy;
James
A.
Yorke
2513-2530
Abstract: When does a continuous map have chaotic dynamics in a set $Q$? More specifically, when does it factor over a shift on $M$ symbols? This paper is an attempt to clarify some of the issues when there is no hyperbolicity assumed. We find that the key is to define a ``crossing number'' for that set $Q$. If that number is $M$ and $M>1$, then $Q$ contains a compact invariant set which factors over a shift on $M$ symbols.
Complexifications of symmetric spaces and Jordan theory
Wolfgang
Bertram
2531-2556
Abstract: Generalizing Hermitian and pseudo-Hermitian spaces, we define twisted complex symmetric spaces, and we show that they correspond to an algebraic object called Hermitian Jordan triple products. The main topic of this work is to investigate the class of real forms of twisted complex symmetric spaces, called the category of symmetric spaces with twist. We show that this category is equivalent to the category of all real Jordan triple systems, and we can use a work of B.O. Makarevic in order to classify the irreducible spaces. The classification shows that most irreducible symmetric spaces have exactly one twisted complexification. This leads to open problems concerning the relation of Jordan and Lie triple systems.
Vessiot structure for manifolds of $(p,q)$-hyperbolic type: Darboux integrability and symmetry
Peter
J.
Vassiliou
1705-1739
Abstract: It is well known that if a scalar second order hyperbolic partial differential equation in two independent variables is Darboux integrable, then its local Cauchy problem may be solved by ordinary differential equations. In addition, such an equation has infinitely many non-trivial conservation laws. Moreover, Darboux integrable equations have properties in common with infinite dimensional completely integrable systems. In this paper we employ a geometric object intrinsically associated with any hyperbolic partial differential equation, its hyperbolic structure, to study the Darboux integrability of the class $E$ of semilinear second order hyperbolic partial differential equations in one dependent and two independent variables. It is shown that the problem of classifying the Darboux integrable equations in $E$ contains, as a subproblem, that of classifying the manifolds of $(p,q)$-hyperbolic type of rank 4 and dimension $2k+3$, $k\ge2$; $p=2,q\ge 2$. In turn, it is shown that the problem of classifying these manifolds in the two (lowest) cases $(p,q)=(2,2),(2,3)$ contains, as a subproblem, the classification problem for Lie groups. This generalizes classical results of E. Vessiot. The main result is that if an equation in $E$ is (2,2)- or (2,3)-Darboux integrable on the $k$-jets, $k\ge 2$, then its intrinsic hyperbolic structure admits a Lie group of symmetries of dimension $2k-1$ or $2k-2$, respectively. It follows that part of the moduli space for the Darboux integrable equations in $E$ is determined by isomorphism classes of Lie groups. The Lie group in question is the group of automorphisms of the characteristic systems of the given equation which leaves invariant the foliation induced by the characteristic (or, Riemann) invariants of the equation, the tangential characteristic symmetries. The isomorphism class of the tangential characteristic symmetries is a contact invariant of the corresponding Darboux integrable partial differential equation.
A finiteness theorem for harmonic maps into Hilbert Grassmannians
Rodrigo
P.
Gomez
1741-1753
Abstract: In this article we demonstrate that every harmonic map from a closed Riemannian manifold into a Hilbert Grassmannian has image contained within a finite-dimensional Grassmannian.
Pfaffian systems with derived length one. The class of flag systems
María
A.
Cañadas-Pinedo;
Ceferino
Ruiz
1755-1766
Abstract: The incidence relations between a Pfaffian system and the characteristic system of its first derived system lead to obtain a characterization of Pfaffian systems with derived length one. Also, for flag systems, several properties are studied. In particular, an intrinsic proof of a result which determines the class of a system and of all the derived systems is given.
A new affine invariant for polytopes and Schneider's projection problem
Erwin
Lutwak;
Deane
Yang;
Gaoyong
Zhang
1767-1779
Abstract: New affine invariant functionals for convex polytopes are introduced. Some sharp affine isoperimetric inequalities are established for the new functionals. These new inequalities lead to fairly strong volume estimates for projection bodies. Two of the new affine isoperimetric inequalities are extensions of Ball's reverse isoperimetric inequalities.
On the number of ${L}_{\infty\omega_1}$-equivalent non-isomorphic models
Saharon
Shelah;
Pauli
Väisänen
1781-1817
Abstract: We prove that if $\operatorname{ZF}$ is consistent then $\operatorname{ZFC} + \operatorname{GCH}$ is consistent with the following statement: There is for every $k < \omega$ a model of cardinality $\aleph_1$ which is $L_{\infty{\omega_{1}}}$-equivalent to exactly $k$non-isomorphic models of cardinality $\aleph_1$. In order to get this result we introduce ladder systems and colourings different from the ``standard'' counterparts, and prove the following purely combinatorial result: For each prime number $p$ and positive integer $m$ it is consistent with $\operatorname{ZFC} + \operatorname{GHC}$ that there is a ``good'' ladder system having exactly $p^m$ pairwise nonequivalent colourings.
A universal continuum of weight $\aleph$
Alan
Dow;
Klaas
Pieter
Hart
1819-1838
Abstract: We prove that every continuum of weight $\aleph_1$ is a continuous image of the Cech-Stone-remainder $R^*$ of the real line. It follows that under $\mathsf{CH}$ the remainder of the half line $[0,\infty)$ is universal among the continua of weight $\mathfrak{c}$-- universal in the `mapping onto' sense. We complement this result by showing that 1) under $\mathsf{MA}$ every continuum of weight less than $\mathfrak{c}$ is a continuous image of $R^*$, 2) in the Cohen model the long segment of length $\omega_2+1$ is not a continuous image of $R^*$, and 3) $\mathsf{PFA}$ implies that $I_u$ is not a continuous image of $R^*$, whenever $u$ is a $\mathfrak{c}$-saturated ultrafilter. We also show that a universal continuum can be gotten from a $\mathfrak{c}$-saturated ultrafilter on $\omega$, and that it is consistent that there is no universal continuum of weight $\mathfrak{c}$.
Semi-dualizing complexes and their Auslander categories
Lars
Winther
Christensen
1839-1883
Abstract: Let $R$ be a commutative Noetherian ring. We study $R$-modules, and complexes of such, with excellent duality properties. While their common properties are strong enough to admit a rich theory, we count among them such, potentially, diverse objects as dualizing complexes for $R$ on one side, and on the other, the ring itself. In several ways, these two examples constitute the extremes, and their well-understood properties serve as guidelines for our study; however, also the employment, in recent studies of ring homomorphisms, of complexes ``lying between'' these extremes is incentive.
Geometric interpretation of tight closure and test ideals
Nobuo
Hara
1885-1906
Abstract: We study tight closure and test ideals in rings of characteristic $p \gg 0$ using resolution of singularities. The notions of $F$-rational and $F$-regular rings are defined via tight closure, and they are known to correspond with rational and log terminal singularities, respectively. In this paper, we reformulate this correspondence by means of the notion of the test ideal, and generalize it to wider classes of singularities. The test ideal is the annihilator of the tight closure relations and plays a crucial role in the tight closure theory. It is proved that, in a normal $\mathbb Q$-Gorenstein ring of characteristic $p \gg 0$, the test ideal is equal to so-called the multiplier ideal, which is an important ideal in algebraic geometry. This is proved in more general form, and to do this we study the behavior of the test ideal and the tight closure of the zero submodule in certain local cohomology modules under cyclic covering. We reinterpret the results also for graded rings.
A brief proof of a maximal rank theorem for generic double points in projective space
Karen
A.
Chandler
1907-1920
Abstract: We give a simple proof of the following theorem of J. Alexander and A. Hirschowitz: Given a general set of points in projective space, the homogeneous ideal of polynomials that are singular at these points has the expected dimension in each degree of 4 and higher, except in 3 cases.
Arithmetic discriminants and morphisms of curves
Xiangjun
Song;
Thomas
J.
Tucker
1921-1936
Abstract: This paper deals with upper bounds on arithmetic discriminants of algebraic points on curves over number fields. It is shown, via a result of Zhang, that the arithmetic discriminants of algebraic points that are not pull-backs of rational points on the projective line are smaller than the arithmetic discriminants of families of linearly equivalent algebraic points. It is also shown that bounds on the arithmetic discriminant yield information about how the fields of definition $k(P)$ and $k(f(P))$ differ when $P$ is an algebraic point on a curve $C$ and $k(P) \not= k(f(P))$, with at most finitely many exceptions, whenever the degrees of $P$ and $f$ are sufficiently small, relative to the difference between the genera $g(C)$ and $g(C')$. The paper concludes with a detailed analysis of the arithmetic discriminants of quadratic points on bi-elliptic curves of genus 2.
Essential cohomology and extraspecial $p$-groups
Pham
Anh
Minh
1937-1957
Abstract: Let $p$ be an odd prime number and let $G$ be an extraspecial $p$-group. The purpose of the paper is to show that $G$ has no non-zero essential mod-$p$ cohomology (and in fact that $H^{*}(G,\mathbb{F}_{p})$ is Cohen-Macaulay) if and only if $\vert G\vert=27$ and $exp(G)=3$.
Spaces of rational loops on a real projective space
Jacob
Mostovoy
1959-1970
Abstract: We show that the loop spaces on real projective spaces are topologically approximated by the spaces of rational maps $\mathbf{RP}^{1}\rightarrow \mathbf{RP}^{n}$. As a byproduct of our constructions we obtain an interpretation of the Kronecker characteristic (degree) of an ornament via particle spaces.
The limits of refinable functions
Gilbert
Strang;
Ding-Xuan
Zhou
1971-1984
Abstract: A function $\phi$ is refinable ( $\phi \in S$) if it is in the closed span of $\{\phi (2x-k)\}$. This set $S$ is not closed in $L_{2}(\mathbb{R})$, and we characterize its closure. A necessary and sufficient condition for a function to be refinable is presented without any information on the refinement mask. The Fourier transform of every $f\in \overline{S} \setminus S$ vanishes on a set of positive measure. As an example, we show that all functions with Fourier transform supported in $[-{\frac{4}{3}}\pi , {\frac{4}{3}}\pi ]$ are the limits of refinable functions. The relation between a refinable function and its mask is studied, and nonuniqueness is proved. For inhomogeneous refinement equations we determine when a solution is refinable. This result is used to investigate refinable components of multiple refinable functions. Finally, we investigate fully refinable functions for which all translates (by any real number) are refinable.
Ahiezer-Kac type Fredholm determinant asymptotics for convolution operators with rational symbols
Sergio
Albeverio;
Konstantin
A.
Makarov
1985-1993
Abstract: Fredholm determinant asymptotics of convolution operators on large finite intervals with rational symbols having real zeros are studied. The explicit asymptotic formulae obtained can be considered as a direct extension of the Ahiezer-Kac formula to symbols with real zeros.
Vector $A_2$ weights and a Hardy-Littlewood maximal function
Michael
Christ;
Michael
Goldberg
1995-2002
Abstract: An analogue of the Hardy-Littlewood maximal function is introduced, for functions taking values in finite-dimensional Hilbert spaces. It is shown to be $L^2$ bounded with respect to weights in the class $A_2$ of Treil, thereby extending a theorem of Muckenhoupt from the scalar to the vector case.
Vanishing of the third simplicial cohomology group of $l^1(\mathbf{Z}_+)$
Frédéric
Gourdeau;
Michael
C.
White
2003-2017
Abstract: We show that $\mathcal{I}$ is the non-unital Banach algebra $l^1(\mathbf{N})$, and then prove that
Unconditional structures of weakly null sequences
S.
A.
Argyros;
I.
Gasparis
2019-2058
Abstract: The following dichotomy is established for a normalized weakly null sequence in a Banach space: Either every subsequence admits a convex block subsequence equivalent to the unit vector basis of $c_0$, or there exists a subsequence which is boundedly convexly complete.
Irreducible partitions and the construction of quasi-measures
D.
J.
Grubb
2059-2072
Abstract: A quasi-measure is a non-subadditive measure defined on only open or closed subsets of a compact Hausdorf space. We investigate the nature of irreducible partitions as defined by Aarnes and use the results to construct quasi-measures when $g(X)=1$. The cohomology ring is an important tool for this investigation.
A product formula for spherical representations of a group of automorphisms of a homogeneous tree, II
Donald
I.
Cartwright;
Gabriella
Kuhn
2073-2090
Abstract: Let $G=\text{Aut}(T)$ be the group of automorphisms of a homogeneous tree $T$and let $\pi$ be the tensor product of two spherical irreducible unitary representations of $G$. We complete the explicit decomposition of $\pi$commenced in part I of this paper, by describing the discrete series representations of $G$ which appear as subrepresentations of $\pi$.
Varieties of uniserial representations IV. Kinship to geometric quotients
Klaus
Bongartz;
Birge
Huisgen-Zimmermann
2091-2113
Abstract: Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field, and ${\mathbb{S} }$ a finite sequence of simple left $\Lambda$-modules. Quasiprojective subvarieties of Grassmannians, distinguished by accessible affine open covers, were introduced by the authors for use in classifying the uniserial representations of $\Lambda$ having sequence ${\mathbb{S} }$ of consecutive composition factors. Our principal objectives here are threefold: One is to prove these varieties to be `good approximations'--in a sense to be made precise--to geometric quotients of the (very large) classical affine varieties $\operatorname{Mod-Uni} ({\mathbb{S} })$ parametrizing the pertinent uniserial representations, modulo the usual conjugation action of the general linear group. We show that, to some extent, this fills the information gap left open by the frequent non-existence of such quotients. A second goal is that of facilitating the transfer of information among the `host' varieties into which the considered quasi-projective, respectively affine, uniserial varieties are embedded. For that purpose, a general correspondence is established, between Grassmannian varieties of submodules of a projective module $P$ on one hand, and classical varieties of factor modules of $P$ on the other. Our findings are applied towards the third objective, concerning the existence of geometric quotients. The main results are then exploited in a representation-theoretic context: Among other consequences, they yield a geometric characterization of the algebras of finite uniserial type which supplements existing descriptions, but is cleaner and more readily checkable.
Noncrossed products over $k_{\mathfrak{p}}(t)$
Eric
S.
Brussel
2115-2129
Abstract: Noncrossed product division algebras are constructed over rational function fields $k(t)$ over number fields $k$ by lifting from arithmetic completions $k(t)_{\mathfrak{p}}$. The existence of noncrossed products over $\mathfrak{p}$-adic rational function fields $k_{\mathfrak{p}}(t)$ is proved as a corollary.
The topological dynamics of semigroup actions
David
B.
Ellis;
Robert
Ellis;
Mahesh
Nerurkar
1279-1320
Abstract: In these notes we explore the fine structure of recurrence for semigroup actions, using the algebraic structure of compactifications of the acting semigroup.
On the tangent flow of a stochastic differential equation with fast drift
Richard
B.
Sowers
1321-1334
Abstract: We investigate the behavior of the tangent flow of a stochastic differential equation with a fast drift. The state space of the stochastic differential equation is the two-dimensional cylinder. The fast drift has closed orbits, and we assume that the orbit times vary nontrivially with the axial coordinate. Under a nondegeneracy assumption, we find the rate of growth of the tangent flow. The calculations involve a transformation introduced by Pinsky and Wihstutz.
Embedded minimal ends of finite type
Laurent
Hauswirth;
Joaquín
Pérez;
Pascal
Romon
1335-1370
Abstract: We prove that the end of a complete embedded minimal surface in $\mathbb{R} ^3$ with infinite total curvature and finite type has an explicit Weierstrass representation that only depends on a holomorphic function that vanishes at the puncture. Reciprocally, any choice of such an analytic function gives rise to a properly embedded minimal end $E$ provided that it solves the corresponding period problem. Furthermore, if the flux along the boundary vanishes, then the end is $C^0$-asymptotic to a Helicoid. We apply these results to proving that any complete embedded one-ended minimal surface of finite type and infinite total curvature is asymptotic to a Helicoid, and we characterize the Helicoid as the only simply connected complete embedded minimal surface of finite type in $\mathbb{R} ^3$.
Bloch constants in several variables
Huaihui
Chen;
P.
M.
Gauthier
1371-1386
Abstract: We give lower estimates for Bloch's constant for quasiregular holomorphic mappings. A holomorphic mapping of the unit ball $B^n$ into $\mathbf{C}^n$ is $K$-quasiregular if it maps infinitesimal spheres to infinitesimal ellipsoids whose major axes are less than or equal to $K$ times their minor axes. We show that if $f$ is a $K$-quasiregular holomorphic mapping with the normalization $1/12K^{1-1/n}.$ This result is best possible in terms of powers of $K.$ Also, we extend to several variables an analogous result of Landau for bounded holomorphic functions in the unit disk.
Effective estimates on the very ampleness of the canonical line bundle of locally Hermitian symmetric spaces
Sai-Kee
Yeung
1387-1401
Abstract: We study the problem about the very ampleness of the canonical line bundle of compact locally Hermitian symmetric manifolds of non-compact type. In particular, we show that any sufficiently large unramified covering of such manifolds has very ample canonical line bundle, and give estimates on the size of the covering manifold, which is itself a locally Hermitian symmetric manifold, in terms of geometric data such as injectivity radius or degree of coverings.
Representability is not decidable for finite relation algebras
Robin
Hirsch;
Ian
Hodkinson
1403-1425
Abstract: We prove that there is no algorithm that decides whether a finite relation algebra is representable. Representability of a finite relation algebra $\mathcal A$ is determined by playing a certain two player game $G({\mathcal A})$ over `atomic $\mathcal A$-networks'. It can be shown that the second player in this game has a winning strategy if and only if $\mathcal A$ is representable. Let $\tau$ be a finite set of square tiles, where each edge of each tile has a colour. Suppose $\tau$ includes a special tile whose four edges are all the same colour, a colour not used by any other tile. The tiling problem we use is this: is it the case that for each tile $T \in \tau$ there is a tiling of the plane ${\mathbb Z}\times {\mathbb Z}$ using only tiles from $\tau$ in which edge colours of adjacent tiles match and with $T$ placed at $(0,0)$? It is not hard to show that this problem is undecidable. From an instance of this tiling problem $\tau$, we construct a finite relation algebra $RA(\tau)$ and show that the second player has a winning strategy in $G(RA(\tau))$ if and only if $\tau$ is a yes-instance. This reduces the tiling problem to the representation problem and proves the latter's undecidability.
The information encoded in initial ideals
Gunnar
Fløystad;
Mark
L.
Green
1427-1453
Abstract: We consider homogeneous ideals $I$ and the initial ideal $\text{in}(I)$ for the revlex order. First we give a sequence of invariants computed from $I$ giving better and better ``approximations" to the initial ideal and ending in an equivalent description. Then we apply this to different settings in algebraic geometry to understand what information is encoded in the generic initial ideal of the ideal of a projective scheme. We also consider the higher initial ideals as defined in a paper by Fløystad. In particular, we show that giving the generic higher initial ideal of a space curve is equivalent to giving the generic initial ideal of a linked curve.
$K3$ surfaces of genus 8 and varieties of sums of powers of cubic fourfolds
Atanas
Iliev;
Kristian
Ranestad
1455-1468
Abstract: The main result of this paper is that the variety of presentations of a general cubic form $f$ in $6$ variables as a sum of $10$cubes is isomorphic to the Fano variety of lines of a cubic $4$-fold $F'$, in general different from $F=Z(f)$. A general $K3$ surface $S$ of genus $8$determines uniquely a pair of cubic $4$-folds: the apolar cubic $F(S)$ and the dual Pfaffian cubic $F'(S)$ (or for simplicity $F$ and $F'$). As Beauville and Donagi have shown, the Fano variety $\mathcal{F}_{F'}$ and $VSP(F,10)$. Furthermore, we show that $VSP(F,10)$ sets up a $(6,10)$ correspondence between $F'$ and
The automorphism groups of Kummer surfaces associated with the product of two elliptic curves
Jonghae
Keum;
Shigeyuki
Kondo
1469-1487
Abstract: We calculate the automorphism groups of several Kummer surfaces associated with the product of two elliptic curves. We give their generators explicitly.
A characteristic free approach to Brauer algebras
Steffen
König;
Changchang
Xi
1489-1505
Abstract: Brauer algebras arise in representation theory of orthogonal or symplectic groups. These algebras are shown to be iterated inflations of group algebras of symmetric groups. In particular, they are cellular (as had been shown before by Graham and Lehrer). This gives some information about block decomposition of Brauer algebras.
On the representation of unity by binary cubic forms
Michael
A.
Bennett
1507-1534
Abstract: If $F(x,y)$ is a binary cubic form with integer coefficients such that $F(x,1)$has at least two distinct complex roots, then the equation $F(x,y) = 1$possesses at most ten solutions in integers $x$ and $y$, nine if $F$ has a nontrivial automorphism group. If, further, $F(x,y)$ is reducible over $\mathbb{Z}[x,y]$, then this equation has at most $2$ solutions, unless $F(x,y)$ is equivalent under $GL_2(\mathbb{Z})$-action to either $x (x^2-xy-y^2)$ or $x (x^2-2y^2)$. The proofs of these results rely upon the method of Thue-Siegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations $F(x,y)=1$ for $F$ cubic and irreducible of positive discriminant $D_F \leq 10^6$. As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form $F(x,y)=m$ and to Mordell's equation $y^2=x^3+k$, where $m$ and $k$ are nonzero integers.
Shintani functions on $GL(2,{\mathbf{C}})$
Miki
Hirano
1535-1550
Abstract: In this paper, in analogy to the real case, we give a formulation of the Shintani functions on $GL(2,\mathbf{C})$, which have been studied by Murase and Sugano within the theory of automorphic $L$-functions. Also, we obtain the multiplicity one theorem for these functions and an explicit formula in a special case.
Anick's spaces and the double loops of odd primary Moore spaces
Stephen
D.
Theriault
1551-1566
Abstract: Several properties of Anick's spaces are established which give a retraction of Anick's $\Omega T_\infty$ off $\Omega^2P^{2np+1}(p^r)$ if $r\ge2$ and $p\ge5$. The proof is alternate to and more immediate than the two proofs of Neisendorfer's.
On Littlewood's boundedness problem for sublinear Duffing equations
Bin
Liu
1567-1585
Abstract: In this paper, we are concerned with the boundedness of all the solutions and the existence of quasi-periodic solutions for second order differential equations \begin{displaymath}x^{\prime\prime} + g(x) = e(t), \end{displaymath} where the 1-periodic function $e(t)$ is a smooth function and $g(x)$satisfies sublinearity: \begin{displaymath}{sign}(x)\cdot g(x)\to+\infty,\quad g(x)/x\to 0 \quad {as}\,\,\, \vert x\vert\to+\infty. \end{displaymath}
Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth
Avner
Friedman;
Fernando
Reitich
1587-1634
Abstract: In this paper we develop a general technique for establishing analyticity of solutions of partial differential equations which depend on a parameter $\varepsilon$. The technique is worked out primarily for a free boundary problem describing a model of a stationary tumor. We prove the existence of infinitely many branches of symmetry-breaking solutions which bifurcate from any given radially symmetric steady state; these asymmetric solutions are analytic jointly in the spatial variables and in $\varepsilon$.
On the lack of null-controllability of the heat equation on the half-line
Sorin
Micu;
Enrique
Zuazua
1635-1659
Abstract: We consider the linear heat equation on the half-line with a Dirichlet boundary control. We analyze the null-controllability problem. More precisely, we study the class of initial data that may be driven to zero in finite time by means of an appropriate choice of the $L^2$ boundary control. We rewrite the system on the similarity variables that are a common tool when analyzing asymptotic problems. Next, the control problem is reduced to a moment problem which turns out to be critical since it concerns the family of real exponentials $\{e^{jt}\}_{j\geq1}$ in which the usual summability condition on the inverses of the eigenvalues does not hold. Roughly speaking, we prove that controllable data have Fourier coefficients that grow exponentially for large frequencies. This result is in contrast with the existing ones for bounded domains that guarantee that every initial datum belonging to a Sobolev space of negative order may be driven to zero in an arbitrarily small time.
A similarity principle for complex vector fields and applications
S.
Berhanu;
J.
Hounie;
P.
Santiago
1661-1675
Abstract: This paper establishes a similarity principle for a class of non-elliptic, smooth complex vector fields in the plane. This principle is used to prove a uniqueness result for a nonlinear Cauchy problem.
Metrics and embeddings of generalizations of Thompson's group $F$
J.
Burillo;
S.
Cleary;
M.
I.
Stein
1677-1689
Abstract: The distance from the origin in the word metric for generalizations $F(p)$ of Thompson's group $F$ is quasi-isometric to the number of carets in the reduced rooted tree diagrams representing the elements of $F(p)$. This interpretation of the metric is used to prove that every $F(p)$ admits a quasi-isometric embedding into every $F(q)$, and also to study the behavior of the shift maps under these embeddings.
On composite formal power series
Jacques
Chaumat;
Anne-Marie
Chollet
1691-1703
Abstract: Let $F$ be a holomorphic map from ${\mathbb{C}}^{n}$ to ${\mathbb{C}}^{n}$ defined in a neighborhood of $0$ such that $F(0)=0$. If the Jacobian determinant of $F$ is not identically zero, P. M. Eakin et G. A. Harris proved the following result: any formal power series such that ${\mathcal{A}}\circ F$ is analytic is itself analytic. If the Jacobian determinant of $F$ is identically zero, they proved that the previous conclusion is no more true. The authors get similar results in the case of formal power series satifying growth conditions, of Gevrey type for instance. Moreover, the proofs here give, in the analytic case, a control of the radius of convergence of ${\mathcal{A}}$ by the radius of convergence of ${\mathcal{A}}\circ F$. RÉSUMÉ. Soit $F$ une application holomorphe de ${\mathbb{C}}^{n}$dans ${\mathbb{C}}^{n}$ définie dans un voisinage de $0$ et vérifiant $F(0)=0$. Si le jacobien de $F$ n'est pas identiquement nul au voisinage de $0$, P.M. Eakin et G.A. Harris ont établi le résultat suivant: toute série formelle ${\mathcal{A}}$ telle que ${\mathcal{A}}\circ F$ est analytique est elle-même analytique. Si le jacobien de $F$ est identiquement nul, ils montrent que la conclusion précédente est fausse. Les auteurs obtiennent des résultats analogues pour les séries formelles à croissance contrôlée, du type Gevrey par exemple. De plus, les preuves données ici permettent, dans le cas analytique, un contrôle du rayon de convergence de ${\mathcal{A}}$ par celui de ${\mathcal{A}}\circ F$.
On the asymptotic geometry of nonpositively curved graphmanifolds
S.
Buyalo;
V.
Schroeder
853-875
Abstract: In this paper we study the Tits geometry of a 3-dimensional graphmanifold of nonpositive curvature. In particular we give an optimal upper bound for the length of nonstandard components of the Tits metric. In the special case of a $\pi /2$-metric we determine the whole length spectrum of the nonstandard components.
Cyclic coverings and higher order embeddings of algebraic varieties
Thomas
Bauer;
Sandra
Di Rocco;
Tomasz
Szemberg
877-891
Abstract: In the present paper we study higher order embeddings in the context of cyclic coverings. Analyzing the positivity of the line bundle downstairs and its relationship with the branch divisor, we provide criteria for its pull-back to define an embedding of given order. We show that the obtained criteria are sharp. Finally, we apply them to various - sometimes seemingly unrelated-problems in algebraic geometry.
Bicanonical pencil of a determinantal Barlow surface
Yongnam
Lee
893-905
Abstract: In this paper, we study the bicanonical pencil of a Godeaux surface and of a determinantal Barlow surface. This study gives a simple proof for the unobstructedness of deformations of a determinantal Barlow surface. Then we compute the number of hyperelliptic curves in the bicanonical pencil of a determinantal Barlow surface via classical Prym theory.
Special values of multiple polylogarithms
Jonathan
M.
Borwein;
David
M.
Bradley;
David
J.
Broadhurst;
Petr
Lisonek
907-941
Abstract: Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier.
Measuring the tameness of almost convex groups
Susan
Hermiller;
John
Meier
943-962
Abstract: A 1-combing for a finitely presented group consists of a continuous family of paths based at the identity and ending at points $x$ in the 1-skeleton of the Cayley 2-complex associated to the presentation. We define two functions (radial and ball tameness functions) that measure how efficiently a 1-combing moves away from the identity. These functions are geometric in the sense that they are quasi-isometry invariants. We show that a group is almost convex if and only if the radial tameness function is bounded by the identity function; hence almost convex groups, as well as certain generalizations of almost convex groups, are contained in the quasi-isometry class of groups admitting linear radial tameness functions.
A bounding question for almost flat manifolds
Shashidhar
Upadhyay
963-972
Abstract: We study bounding question for almost flat manifolds by looking at the equivalent description of them as infranilmanifolds $\Gamma\backslash L\rtimes G/G$. We show that infranilmanifolds $\Gamma\backslash L \rtimes G/G$ bound if $L$ is a 2-step nilpotent group and $G$ is finite cyclic and acts trivially on the center of the nilpotent Lie group $L$.
A model for the homotopy theory of homotopy theory
Charles
Rezk
973-1007
Abstract: We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory'', or more precisely that the category of such models has a well-behaved internal hom-object.
Properties of Anick's spaces
Stephen
D.
Theriault
1009-1037
Abstract: We prove three useful properties of Anick's space $T^{2n-1}(p^{r})$. First, at odd primes a map from $P^{2n}(p^{r})$ into a homotopy commutative, homotopy associative $H$-space $X$ can be extended to a unique $H$-map from $T^{2n-1}(p^{r})$ into $X$. Second, at primes larger than $3$, $T^{2n-1}(p^{r})$ is itself homotopy commutative and homotopy associative. And third, the first two properties combine to show that the order of the identity map on $T^{2n-1}(p^{r})$ is $p^{r}$.
Transfers of Chern classes in BP-cohomology and Chow rings
Björn
Schuster;
Nobuaki
Yagita
1039-1054
Abstract: The $BP^*$-module structure of $BP^*(BG)$ for extraspecial $2$-groups is studied using transfer and Chern classes. These give rise to $p$-torsion elements in the kernel of the cycle map from the Chow ring to ordinary cohomology first obtained by Totaro.
A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of $\mathbb{S}^n$
Mark
S.
Ashbaugh;
Rafael
D.
Benguria
1055-1087
Abstract: For a domain $\Omega$ contained in a hemisphere of the $n$-dimensional sphere $\mathbb{S}^n$ we prove the optimal result $\lambda_2/\lambda_1(\Omega) \le \lambda_2/\lambda_1(\Omega^{\star})$ for the ratio of its first two Dirichlet eigenvalues where $\Omega^{\star}$, the symmetric rearrangement of $\Omega$ in $\mathbb{S}^n$, is a geodesic ball in $\mathbb{S}^n$ having the same $n$-volume as $\Omega$. We also show that $\lambda_2/\lambda_1$ for geodesic balls of geodesic radius $\theta_1$ less than or equal to $\pi/2$ is an increasing function of $\theta_1$ which runs between the value $(j_{n/2,1}/j_{n/2-1,1})^2$ for $\theta_1=0$ (this is the Euclidean value) and $2(n+1)/n$ for $\theta_1=\pi/2$. Here $j_{\nu,k}$ denotes the $k$th positive zero of the Bessel function $J_{\nu}(t)$. This result generalizes the Payne-Pólya-Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of $\mathbb{S}^n$ and having a fixed value of $\lambda_1$ the one with the maximal value of $\lambda_2$ is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for $\lambda_2/\lambda_1$. Various other results for $\lambda_1$and $\lambda_2$ of geodesic balls in $\mathbb{S}^n$ are proved in the course of our work.
New range theorems for the dual Radon transform
Alexander
Katsevich
1089-1102
Abstract: Three new range theorems are established for the dual Radon transform $R^*$: on $C^\infty$ functions that do not decay fast at infinity (and admit an asymptotic expansion), on $\mathcal{S}(Z_n)$, and on $C_0^\infty(Z_n)$. Here $Z_n:=S^{n-1}\times\mathbb{R}$, and $R^*$ acts on even functions $\mu(\alpha,p)=\mu(-\alpha,-p), (\alpha,p)\in Z_n$.
Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations
Gui-Qiang
Chen;
Hermano
Frid
1103-1117
Abstract: We prove the uniqueness of Riemann solutions in the class of entropy solutions in $L^\infty\cap BV_{loc}$ for the $3\times 3$ system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families. In particular, if the Riemann solutions consist of at most rarefaction waves and contact discontinuities, we show the global $L^2$-stability of the Riemann solutions even in the class of entropy solutions in $L^\infty$with arbitrarily large oscillation for the $3\times 3$ system. We apply our framework established earlier to show that the uniqueness of Riemann solutions implies their inviscid asymptotic stability under $L^1$ perturbation of the Riemann initial data, as long as the corresponding solutions are in $L^\infty$ and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can easily be extended to entropy solutions $U(x,t)$, piecewise Lipschitz in $x$, for any $t>0$.
Two-weight norm inequalities for Cesàro means of Laguerre expansions
Benjamin
Muckenhoupt;
David
W.
Webb
1119-1149
Abstract: Two-weight $L^{p}$ norm inequalities are proved for Cesàro means of Laguerre polynomial series and for the supremum of these means. These extend known norm inequalities, even in the single power weight and ``unweighted'' cases, by including all values of $p\geq1$ for all positive orders of the Cesàro summation and all values of the Laguerre parameter $\alpha>-1$. Almost everywhere convergence results are obtained as a corollary. For the Cesàro means the hypothesized conditions are shown to be necessary for the norm inequalities. Necessity results are also obtained for the norm inequalities with the supremum of the Cesàro means; in particular, for the single power weight case the conditions are necessary and sufficient for summation of order greater than one sixth.
The FBI transform on compact ${\mathcal{C}^\infty}$ manifolds
Jared
Wunsch;
Maciej
Zworski
1151-1167
Abstract: We present a geometric theory of the Fourier-Bros-Iagolnitzer transform on a compact ${\mathcal{C}^\infty}$ manifold $M$. The FBI transform is a generalization of the classical notion of the wave-packet transform. We discuss the mapping properties of the FBI transform and its relationship to the calculus of pseudodifferential operators on $M$. We also describe the microlocal properties of its range in terms of the ``scattering calculus'' of pseudodifferential operators on the noncompact manifold $T^* M$.
Algebraic isomorphisms of limit algebras
A.
P.
Donsig;
T.
D.
Hudson;
E.
G.
Katsoulis
1169-1182
Abstract: We prove that algebraic isomorphisms between limit algebras are automatically continuous, and consider the consequences of this result. In particular, we give partial solutions to a conjecture and an open problem by Power. As a further consequence, we describe epimorphisms between various classes of limit algebras.
Beyond Borcherds Lie algebras and inside
Stephen
Berman;
Elizabeth
Jurisich;
Shaobin
Tan
1183-1219
Abstract: We give a definition for a new class of Lie algebras by generators and relations which simultaneously generalize the Borcherds Lie algebras and the Slodowy G.I.M. Lie algebras. After proving these algebras are always subalgebras of Borcherds Lie algebras, as well as some other basic properties, we give a vertex operator representation for a factor of them. We need to develop a highly non-trivial generalization of the square length two cut off theorem of Goddard and Olive to do this.
Representations as elements in affine composition algebras
Pu
Zhang
1221-1249
Abstract: Let $A$ be the path algebra of a Euclidean quiver over a finite field $k$. The aim of this paper is to classify the modules $M$ with the property $[M]\in \mathcal{C}(A)$, where $\mathcal{C}(A)$ is Ringel's composition algebra. Namely, the main result says that if $\vert k\vert \ne 2, 3$, then $[M]\in \mathcal{C}(A)$ if and only if the regular direct summand of $M$ is a direct sum of modules from non-homogeneous tubes with quasi-dimension vectors non-sincere. The main methods are representation theory of affine quivers, the structure of triangular decompositions of tame composition algebras, and the invariant subspaces of skew derivations. As an application, we see that $\mathcal{C}(A) = \mathcal{H}(A)$ if and only if the quiver of $A$is of Dynkin type.
The combinatorics of Bernstein functions
Thomas
J.
Haines
1251-1278
Abstract: A construction of Bernstein associates to each cocharacter of a split $p$-adic group an element in the center of the Iwahori-Hecke algebra, which we refer to as a Bernstein function. A recent conjecture of Kottwitz predicts that Bernstein functions play an important role in the theory of bad reduction of a certain class of Shimura varieties (parahoric type). It is therefore of interest to calculate the Bernstein functions explicitly in as many cases as possible, with a view towards testing Kottwitz' conjecture. In this paper we prove a characterization of the Bernstein function associated to a minuscule cocharacter (the case of interest for Shimura varieties). This is used to write down the Bernstein functions explicitly for some minuscule cocharacters of $Gl_n$; one example can be used to verify Kottwitz' conjecture for a special class of Shimura varieties (the ``Drinfeld case''). In addition, we prove some general facts concerning the support of Bernstein functions, and concerning an important set called the ``$\mu$-admissible'' set. These facts are compatible with a conjecture of Kottwitz and Rapoport on the shape of the special fiber of a Shimura variety with parahoric type bad reduction.
Some results on increments of the partially observed empirical process
Zacharie
Dindar
427-440
Abstract: The author investigates the almost sure behaviour of the increments of the partially observed, uniform empirical process. Some functional laws of the iterated logarithm are obtained for this process. As an application, new laws of the iterated logarithm are established for kernel density estimators.
Hermitian-Einstein metrics for vector bundles on complete Kähler manifolds
Lei
Ni;
Huaiyu
Ren
441-456
Abstract: In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermitian-Einstein equations on holomorphic vector bundles over strictly pseudoconvex domains.
Analysis and geometry on manifolds with integral Ricci curvature bounds. II
Peter
Petersen;
Guofang
Wei
457-478
Abstract: We extend several geometrical results for manifolds with lower Ricci curvature bounds to situations where one has integral lower bounds. In particular we generalize Colding's volume convergence results and extend the Cheeger-Colding splitting theorem.
Coding into $K$ by reasonable forcing
Ralf-Dieter
Schindler
479-489
Abstract: We present a technique for coding sets ``into $K$,'' where $K$ is the core model below a strong cardinal. Specifically, we show that if there is no inner model with a strong cardinal then any $X\subset\omega_1$ can be made $\boldsymbol{\Delta}^1_3$ (in the codes) in a reasonable and stationary preserving set generic extension.
The completeness of the isomorphism relation for countable Boolean algebras
Riccardo
Camerlo;
Su
Gao
491-518
Abstract: We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen's classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF $C^*$-algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups.
Tame and Wild Coordinates of $K[z][x,y]$
Vesselin
Drensky;
Jie-Tai
Yu
519-537
Abstract: Let $K$ be a field of characteristic zero. We characterize coordinates and tame coordinates in $K[z][x,y]$, i.e. the images of $x$ respectively under all automorphisms and under the tame automorphisms of $K[z][x,y]$. We also construct a new large class of wild automorphisms of $K[z][x,y]$ which maps $x$ to a concrete family of nice looking polynomials. We show that a subclass of this class is stably tame, i.e. becomes tame when we extend its automorphisms to automorphisms of $K[z][x,y,t]$.
An arithmetic property of Fourier coefficients of singular modular forms on the exceptional domain
Shou-Te
Chang;
Minking
Eie
539-556
Abstract: We shall develop the theory of Jacobi forms of degree two over Cayley numbers and use it to construct a singular modular form of weight 4 on the 27-dimensional exceptional domain. Such a singular modular form was obtained by Kim through the analytic continuation of a nonholomorphic Eisenstein series. By applying the results in a joint work with Eie, A. Krieg provided an alternative proof that a function with a Fourier expansion obtained by Kim is indeed a modular form of weight 4. This work provides a systematic and general approach to deal with the whole issue.
Comparing Heegaard and JSJ structures of orientable 3-manifolds
Martin
Scharlemann;
Jennifer
Schultens
557-584
Abstract: The Heegaard genus $g$ of an irreducible closed orientable $3$-manifold puts a limit on the number and complexity of the pieces that arise in the Jaco-Shalen-Johannson decomposition of the manifold by its canonical tori. For example, if $p$ of the complementary components are not Seifert fibered, then $p \leq g-1$. This generalizes work of Kobayashi. The Heegaard genus $g$ also puts explicit bounds on the complexity of the Seifert pieces. For example, if the union of the Seifert pieces has base space $P$ and $f$ exceptional fibers, then $f - \chi(P) \leq 3g - 3 - p$.
Central extensions and generalized plus-constructions
G.
Mislin;
G.
Peschke
585-608
Abstract: We describe the effect of homological plus-constructions on the homotopy groups of Eilenberg-MacLane spaces in terms of universal central extensions.
Asymptotic convergence of the Stefan problem to Hele-Shaw
Fernando
Quirós;
Juan
Luis
Vázquez
609-634
Abstract: We discuss the asymptotic behaviour of weak solutions to the Hele-Shaw and one-phase Stefan problems in exterior domains. We prove that, if the space dimension is greater than one, the asymptotic behaviour is given in both cases by the solution of the Dirichlet exterior problem for the Laplacian in the interior of the positivity set and by a singular, radial and self-similar solution of the Hele-Shaw flow near the free boundary. We also show that the free boundary approaches a sphere as $t\to \infty$, and give the precise asymptotic growth rate for the radius.
Two-parameter spectral averaging and localization for non-monotonic random Schrödinger operators
Dirk
Buschmann;
Günter
Stolz
635-653
Abstract: We prove exponential localization at all energies for two types of one-dimensional random Schrödinger operators: the Poisson model and the random displacement model. As opposed to Anderson-type models, these operators are not monotonic in the random parameters. Therefore the classical one-parameter version of spectral averaging, as used in localization proofs for Anderson models, breaks down. We use the new method of two-parameter spectral averaging and apply it to the Poisson as well as the displacement case. In addition, we apply results from inverse spectral theory, which show that two-parameter spectral averaging works for sufficiently many energies (all but a discrete set) to conclude localization at all energies.
Uniqueness of solution to a free boundary problem from combustion
C.
Lederman;
J.
L.
Vázquez;
N.
Wolanski
655-692
Abstract: We investigate the uniqueness and agreement between different kinds of solutions for a free boundary problem in heat propagation that in classical terms is formulated as follows: to find a continuous function $u(x,t)\geq 0,$ defined in a domain $\mathcal{D} \subset {\mathbb{R}}^{N}\times (0,T)$ and such that \begin{displaymath}\Delta u+\sum a_{i}\,u_{x_{i}}-u_{t}=0\quad \text{in}\quad \mathcal{D}\cap \{u>0\}. \end{displaymath} We also assume that the interior boundary of the positivity set, $\mathcal{D} \cap \partial \{u> \nobreak 0\}$, so-called free boundary, is a regular hypersurface on which the following conditions are satisfied: \begin{displaymath}u=0 ,\quad -\partial u/\partial \nu = C. \end{displaymath} Here $\nu$ denotes outward unit spatial normal to the free boundary. In addition, initial data are specified, as well as either Dirichlet or Neumann data on the parabolic boundary of $\mathcal{D}$. This problem arises in combustion theory as a limit situation in the propagation of premixed flames (high activation energy limit). The problem admits classical solutions only for good data and for small times. Several generalized concepts of solution have been proposed, among them the concepts of limit solution and viscosity solution. We investigate conditions under which the three concepts agree and produce a unique solution.
Tracially AF $C^*$-algebras
Huaxin
Lin
693-722
Abstract: Inspired by a paper of S. Popa and the classification theory of nuclear $C^*$-algebras, we introduce a class of $C^*$-algebras which we call tracially approximately finite dimensional (TAF). A TAF $C^*$-algebra is not an AF-algebra in general, but a ``large'' part of it can be approximated by finite dimensional subalgebras. We show that if a unital simple $C^*$-algebra is TAF then it is quasidiagonal, and has real rank zero, stable rank one and weakly unperforated $K_0$-group. All nuclear simple $C^*$-algebras of real rank zero, stable rank one, with weakly unperforated $K_0$-group classified so far by their $K$-theoretical data are TAF. We provide examples of nonnuclear simple TAF $C^*$-algebras. A sufficient condition for unital nuclear separable quasidiagonal $C^*$-algebras to be TAF is also given. The main results include a characterization of simple rational AF-algebras. We show that a separable nuclear simple TAF $C^*$-algebra $A$ satisfying the Universal Coefficient Theorem and having $K_1(A)=0$ and $K_0(A)=\mathbf{Q}$ is isomorphic to a simple AF-algebra with the same $K$-theory.
Linear maps determining the norm topology
Krzysztof
Jarosz
723-731
Abstract: Let $A$ be a Banach function algebra on a compact space $X$, and let $a\in A$ be such that for any scalar $\lambda$ the element $a+\lambda e$ is not a divisor of zero. We show that any complete norm topology on $A$ that makes the multiplication by $a$ continuous is automatically equivalent to the original norm topology of $A$. Related results for general Banach spaces are also discussed.
Block diagonal polynomials
Verónica
Dimant;
Raquel
Gonzalo
733-747
Abstract: In this paper we introduce and study a certain class of polynomials in spaces with unconditional finite dimensional decomposition. Some applications to the existence of copies of $\ell _\infty$ in spaces of polynomials and to the stabilization of polynomial algebras are given.
Serre's generalization of Nagao's theorem: An elementary approach
A.
W.
Mason
749-767
Abstract: Let $C$ be a smooth projective curve over a field $k$. For each closed point $Q$ of $C$ let $\mathcal{C} = \mathcal{C}(C, Q, k)$be the coordinate ring of the affine curve obtained by removing $Q$from $C$. Serre has proved that $GL_2(\mathcal{C})$ is isomorphic to the fundamental group, $\pi_1(G, T)$, of a graph of groups $(G, T)$, where $T$ is a tree with at most one non-terminal vertex. Moreover the subgroups of $GL_2(\mathcal{C})$attached to the terminal vertices of $T$ are in one-one correspondence with the elements of $\operatorname{Cl}(\mathcal{C})$, the ideal class group of $\mathcal{C}$. This extends an earlier result of Nagao for the simplest case $\mathcal{C} = k[t]$. Serre's proof is based on applying the theory of groups acting on trees to the quotient graph $\overline{X} = GL_2(\mathcal{C}) \backslash X$, where $X$ is the associated Bruhat-Tits building. To determine $\overline{X}$ he makes extensive use of the theory of vector bundles (of rank 2) over $C$. In this paper we determine $\overline{X}$using a more elementary approach which involves substantially less algebraic geometry. The subgroups attached to the edges of $T$ are determined (in part) by a set of positive integers $\mathcal{S}$, say. In this paper we prove that $\mathcal{S}$ is bounded, even when Cl $(\mathcal{C})$ is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of $GL_2(\mathcal{C})$, involving unipotent and elementary matrices.
Isomorphism problems and groups of automorphisms for generalized Weyl algebras
V.
V.
Bavula;
D.
A.
Jordan
769-794
Abstract: We present solutions to isomorphism problems for various generalized Weyl algebras, including deformations of type-A Kleinian singularities and the algebras similar to $U(\mathfrak{sl}_2)$introduced by S. P. Smith. For the former, we generalize results of Dixmier on the first Weyl algebra and the minimal primitive factors of $U(\mathfrak{sl}_2)$ by finding sets of generators for the group of automorphisms.
Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation
Daniel
Tataru
795-807
Abstract: The aim of this article is twofold. First we consider the wave equation in the hyperbolic space $\mathbb H^n$ and obtain the counterparts of the Strichartz type estimates in this context. Next we examine the relationship between semilinear hyperbolic equations in the Minkowski space and in the hyperbolic space. This leads to a simple proof of the recent result of Georgiev, Lindblad and Sogge on global existence for solutions to semilinear hyperbolic problems with small data. Shifting the space-time Strichartz estimates from the hyperbolic space to the Minkowski space yields weighted Strichartz estimates in $\mathbb R^{n} \times \mathbb R$ which extend the ones of Georgiev, Lindblad, and Sogge.
Boundary value problems for higher order parabolic equations
Russell
M.
Brown;
Wei
Hu
809-838
Abstract: We consider a constant coefficient parabolic equation of order $2m$ and establish the existence of solutions to the initial-Dirichlet problem in cylindrical domains. The lateral data is taken from spaces of Whitney arrays which essentially require that the normal derivatives up to order $m-1$ lie in $L^2$ with respect to surface measure. In addition, a regularity result for the solution is obtained if the data has one more derivative. The boundary of the space domain is given by the graph of a Lipschitz function. This provides an extension of the methods of Pipher and Verchota on elliptic equations to parabolic equations.
Correction to ``Optimal factorization of Muckenhoupt weights''
Michael
Brian
Korey
839-851
Abstract: Peter Jones' theorem on the factorization of $A_p$ weights is sharpened for weights with bounds near $1$, allowing the factorization to be performed continuously near the limiting, unweighted case. When $1<p<\infty$ and $w$ is an $A_p$ weight with bound $A_p(w)=1+\varepsilon$, it is shown that there exist $A_1$ weights $u,v$ such that both the formula $w=uv^{1-p}$ and the estimates $A_1(u), A_1(v)=1+\mathcal O(\sqrt\varepsilon)$ hold. The square root in these estimates is also proven to be the correct asymptotic power as $\varepsilon\to 0$.
Standard realizations of crystal lattices via harmonic maps
Motoko
Kotani;
Toshikazu
Sunada
1-20
Abstract: An Eells-Sampson type theorem for harmonic maps from a finite weighted graph is employed to characterize the equilibrium configurations of crystals. It is thus observed that the mimimum principle frames symmetry of crystals.
The Lipschitz continuity of the distance function to the cut locus
Jin-ichi
Itoh;
Minoru
Tanaka
21-40
Abstract: Let $N$ be a closed submanifold of a complete smooth Riemannian manifold $M$ and $U\mbox{{$\nu$}}$ the total space of the unit normal bundle of $N$. For each $v \in U\mbox{{$\nu$}}$, let $\rho(v)$ denote the distance from $N$ to the cut point of $N$ on the geodesic $\gamma_v$ with the velocity vector $\dot\gamma_v(0)=v.$ The continuity of the function $\rho$ on $U\mbox{{$\nu$}}$ is well known. In this paper we prove that $\rho$ is locally Lipschitz on which $\rho$is bounded; in particular, if $M$ and $N$ are compact, then $\rho$ is globally Lipschitz on $U\mbox{{$\nu$}}$. Therefore, the canonical interior metric $\delta$ may be introduced on each connected component of the cut locus of $N,$ and this metric space becomes a locally compact and complete length space.
Projective sets and ordinary differential equations
Alessandro
Andretta;
Alberto
Marcone
41-76
Abstract: We prove that for $n \geq 2$ the set of Cauchy problems of dimension $n$which have a global solution is $\boldsymbol\Sigma_{1}^{1}$-complete and that the set of ordinary differential equations which have a global solution for every initial condition is $\boldsymbol\Pi_{1}^{1}$-complete. The first result still holds if we restrict ourselves to second order equations (in dimension one). We also prove that for $n \geq 2$ the set of Cauchy problems of dimension $n$which have a global solution even if we perturb a bit the initial condition is $\boldsymbol\Pi_{2}^{1}$-complete.
Induced formal deformations and the Cohen-Macaulay property
Phillip
Griffith
77-93
Abstract: The main result states: if $A/B$ is a module finite extension of excellent local normal domains which is unramified in codimension two and if $S/\varkappa S \simeq \hat B$ represents a deformation of the completion of $B$, then there is a corresponding $S$-algebra deformation $T/\varkappa T \simeq \hat A$ such that the ring homomorphism $S \hookrightarrow T$ represents a deformation of $\hat B \hookrightarrow \hat A$. The main application is to the ascent of the arithmetic Cohen-Macaulay property for an étale map $f : X \to Y$ of smooth projective varieties over an algebraically closed field.${}^*$
Degree of strata of singular cubic surfaces
Rafael
Hernández;
María
J.
Vázquez-Gallo
95-115
Abstract: We determine the degree of some strata of singular cubic surfaces in the projective space $\mathbf{P}^3$. These strata are subvarieties of the $\mathbf{P}^{19}$ parametrizing all cubic surfaces in $\mathbf{P}^3$. It is known what their dimension is and that they are irreducible. In 1986, D. F. Coray and I. Vainsencher computed the degree of the 4 strata consisting on cubic surfaces with a double line. To work out the case of isolated singularities we relate the problem with (stationary) multiple-point theory.
Connectivity at infinity for right angled Artin groups
Noel
Brady;
John
Meier
117-132
Abstract: We establish sufficient conditions implying semistability and connectivity at infinity properties for CAT(0) cubical complexes. We use this, along with the geometry of cubical $K(\pi,1)$'s to give a complete description of the higher connectivity at infinity properties of right angled Artin groups. Among other things, this determines which right angled Artin groups are duality groups. Applications to group extensions are also included.
Extension theory of separable metrizable spaces with applications to dimension theory
Alexander
Dranishnikov;
Jerzy
Dydak
133-156
Abstract: The paper deals with generalizing several theorems of the covering dimension theory to the extension theory of separable metrizable spaces. Here are some of the main results: Generalized Eilenberg-Borsuk Theorem. Let $L$ be a countable CW complex. If $X$ is a separable metrizable space and $K\ast L$ is an absolute extensor of $X$ for some CW complex $K$, then for any map $f:A\to K$, $A$ closed in $X$, there is an extension $f':U\to K$ of $f$ over an open set $U$such that $L\in AE(X-U)$. Theorem. Let $K,L$ be countable CW complexes. If $X$ is a separable metrizable space and $K\ast L$ is an absolute extensor of $X$, then there is a subset $Y$ of $X$ such that $K\in AE(Y)$ and $L\in AE(X-Y)$. Theorem. Suppose $G_{i},\ldots ,G_{n}$ are countable, non-trivial, abelian groups and $k>0$. For any separable metrizable space $X$ of finite dimension $\dim X>0$, there is a closed subset $Y$ of $X$ with $\dim _{G_{i}} Y=\max (\dim _{G_{i}} X-k,1)$ for $i=1,\ldots ,n$. Theorem. Suppose $W$ is a separable metrizable space of finite dimension and $Y$ is a compactum of finite dimension. Then, for any $k$, $0<k<\dim W-\dim Y$, there is a closed subset $T$ of $W$such that $\dim T=\dim W-k$ and $\dim (T\times Y)=\dim (W\times Y)-k$. Theorem. Suppose $X$ is a metrizable space of finite dimension and $Y$ is a compactum of finite dimension. If $K\in AE(X)$ and $L\in AE(Y)$ are connected CW complexes, then $K\wedge L\in AE(X\times Y).$
Endofiniteness in stable homotopy theory
Henning
Krause;
Ulrike
Reichenbach
157-173
Abstract: We study endofinite objects in a compactly generated triangulated category in terms of ideals in the category of compact objects. Our results apply in particular to the stable homotopy category. This leads, for example, to a new interpretation of stable splittings for classifying spaces of finite groups.
$(Z_{2})^{k}$-actions whose fixed data has a section
Pedro
L. Q.
Pergher
175-189
Abstract: Given a collection of $2^{k}-1$ real vector bundles $\varepsilon _{a}$ over a closed manifold $F$, suppose that, for some $a_{0}, \varepsilon _{a_{0}}$ is of the form $\varepsilon _{a_{0}}^{\prime }\oplus R$, where $R\to F$ is the trivial one-dimensional bundle. In this paper we prove that if $\bigoplus _{a} \varepsilon _{a} \to F$ is the fixed data of a $(Z_{2})^{k}$-action, then the same is true for the Whitney sum obtained from $\bigoplus _{a} \varepsilon _{a}$ by replacing $\varepsilon _{a_{0}}$ by $\varepsilon _{a_{0}}^{\prime }$. This stability property is well-known for involutions. Together with techniques previously developed, this result is used to describe, up to bordism, all possible $(Z_{2})^{k}$-actions fixing the disjoint union of an even projective space and a point.
Blow up and instability of solitary-wave solutions to a generalized Kadomtsev-Petviashvili equation
Yue
Liu
191-208
Abstract: In this paper we consider a generalized Kadomtsev-Petviashvili equation in the form \begin{equation*}( u_{t} + u_{xxx} + u^{p} u_{x} )_{x} = u_{yy} \quad (x, y) \in R^{2}, t \ge 0. \end{equation*} It is shown that the solutions blow up in finite time for the supercritical power of nonlinearity $p \ge 4/3$ with $p$ the ratio of an even to an odd integer. Moreover, it is shown that the solitary waves are strongly unstable if $2 < p < 4$; that is, the solutions blow up in finite time provided they start near an unstable solitary wave.
On the invariant faces associated with a cone-preserving map
Bit-Shun
Tam;
Hans
Schneider
209-245
Abstract: For an $n\times n$ nonnegative matrix $P$, an isomorphism is obtained between the lattice of initial subsets (of $\{ 1,\cdots,n\}$) for $P$ and the lattice of $P$-invariant faces of the nonnegative orthant $\mathbb{R}^{n}_{+}$. Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If $A$ leaves invariant a polyhedral cone $K$, then for each distinguished eigenvalue $\lambda$ of $A$ for $K$, there is a chain of $m_\lambda$ distinct $A$-invariant join-irreducible faces of $K$, each containing in its relative interior a generalized eigenvector of $A$corresponding to $\lambda$ (referred to as semi-distinguished $A$-invariant faces associated with $\lambda$), where $m_\lambda$ is the maximal order of distinguished generalized eigenvectors of $A$ corresponding to $\lambda$, but there is no such chain with more than $m_\lambda$ members. We introduce the important new concepts of semi-distinguished $A$-invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding $n$ that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.
Spectral theory and hypercyclic subspaces
Fernando
León-Saavedra;
Alfonso
Montes-Rodríguez
247-267
Abstract: A vector $x$ in a Hilbert space $\mathcal{H}$ is called hypercyclic for a bounded operator $T: \mathcal{H} \rightarrow \mathcal{H}$ if the orbit $\{T^{n} x : n \geq 1 \}$ is dense in $\mathcal{H}$. Our main result states that if $T$ satisfies the Hypercyclicity Criterion and the essential spectrum intersects the closed unit disk, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for $T$. The converse is true even if $T$ is a hypercyclic operator which does not satisfy the Hypercyclicity Criterion. As a consequence, other characterizations are obtained for an operator $T$ to have an infinite-dimensional closed subspace of hypercyclic vectors. These results apply to most of the hypercyclic operators that have appeared in the literature. In particular, they apply to bilateral and backward weighted shifts, perturbations of the identity by backward weighted shifts, multiplication operators and composition operators. The main result also applies to the differentiation operator and the translation operator $T:f(z)\rightarrow f(z+1)$ defined on certain Hilbert spaces consisting of entire functions. We also obtain a spectral characterization of the norm-closure of the class of hypercyclic operators which have an infinite-dimensional closed subspace of hypercyclic vectors.
Sharp Sobolev inequalities with lower order remainder terms
Olivier
Druet;
Emmanuel
Hebey;
Michel
Vaugon
269-289
Abstract: Given a smooth compact Riemannian $n$-manifold $(M,g)$, this paper deals with the sharp Sobolev inequality corresponding to the embedding of $H_1^2(M)$ in $L^{2n/(n-2)}(M)$ where the $L^2$ remainder term is replaced by a lower order term.
The number of planar central configurations is finite when $N-1$ mass positions are fixed
Peter
W.
Lindstrom
291-311
Abstract: In this paper, it is proved that for $n>2$ and $n\not=4$, if $n-1$ masses are located at fixed points in a plane, then there are only a finite number of $n$-point central configurations that can be generated by positioning a given additional $n$th mass in the same plane. The result is established by proving an equivalent isolation result for planar central configurations of five or more points. Other general properties of central configurations are established in the process. These relate to the amount of centrality lost when a point mass is perturbed and to derivatives associated with central configurations.
Local derivations on $C^*$-algebras are derivations
B.
E.
Johnson
313-325
Abstract: Kadison has shown that local derivations from a von Neumann algebra into any dual bimodule are derivations. In this paper we extend this result to local derivations from any $C^*$-algebra $\mathfrak{A}$ into any Banach $\mathfrak{A}$-bimodule $\mathfrak{X}$. Most of the work is involved with establishing this result when $\mathfrak{A}$ is a commutative $C^*$-algebra with one self-adjoint generator. A known result of the author about Jordan derivations then completes the argument. We show that these results do not extend to the algebra $C^1[0,1]$ of continuously differentiable functions on $[0,1]$. We also give an automatic continuity result, that is, we show that local derivations on $C^*$-algebras are continuous even if not assumed a priori to be so.
Convergence of the Ruelle operator for a function satisfying Bowen's condition
Peter
Walters
327-347
Abstract: We consider a positively expansive local homeomorphism $T\colon X\to X$ satisfying a weak specification property and study the Ruelle operator $\mathcal{L}_\varphi$ of a real-valued continuous function $\varphi$satisfying a property we call Bowen's condition. We study convergence properties of the iterates $\mathcal{L}_\varphi^n$ and relate them to the theory of equilibrium states.
A product formula for spherical representations of a group of automorphisms of a homogeneous tree, I
Donald
I.
Cartwright;
Gabriella
Kuhn;
Paolo
M.
Soardi
349-364
Abstract: Let $G=\mathrm{Aut}(T)$ be the group of automorphisms of a homogeneous tree $T$, and let $\Gamma$ be a lattice subgroup of $G$. Let $\pi$ be the tensor product of two spherical irreducible unitary representations of $G$. We give an explicit decomposition of the restriction of $\pi$ to $\Gamma$. We also describe the spherical component of $\pi$ explicitly, and this decomposition is interpreted as a multiplication formula for associated orthogonal polynomials.
Dade's invariant conjecture for general linear and unitary groups in non-defining characteristics
Jianbei
An
365-390
Abstract: This paper is part of a program to study the conjecture of E. C. Dade on counting characters in blocks for several finite groups. The invariant conjecture of Dade is proved for general linear and unitary groups when the characteristic of the modular representation is distinct from the defining characteristic of the groups.
On modules of finite upper rank
Dan
Segal
391-410
Abstract: For a group $G$ and a prime $p$, the upper $p$-rank of $G$ is the supremum of the sectional $p$-ranks of all finite quotients of $G$. It is unknown whether, for a finitely generated group $G$, these numbers can be finite but unbounded as $p$ ranges over all primes. The conjecture that this cannot happen if $G$ is soluble is reduced to an analogous `relative' conjecture about the upper $p$-ranks of a `quasi-finitely-generated' module $M$for a soluble minimax group $\Gamma$. The main result establishes a special case of this relative conjecture, namely when the module $M$ is finitely generated and the minimax group $\Gamma$ is abelian-by-polycyclic. The proof depends on generalising results of Roseblade on group rings of polycyclic groups to group rings of soluble minimax groups. (If true in general, the above-stated conjecture would imply the truth of Lubotzky's `Gap Conjecture' for subgroup growth, in the case of soluble groups; the Gap Conjecture is known to be false for non-soluble groups.)
Optimal filtrations on representations of finite dimensional algebras
Lieven
Le Bruyn
411-426
Abstract: We present a representation theoretic description of the non-empty strata in the Hesselink stratification of the nullcone of representations of quivers. We use this stratification to define optimal filtrations on representations of finite dimensional algebras. As an application we investigate the isomorphism problem for uniserial representations.
Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology
Luis
A.
Cordero;
Marisa
Fernández;
Alfred
Gray;
Luis
Ugarte
5405-5433
Abstract: We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if $\Gamma \backslash G$ is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology $H^{\ast ,\ast }_{{\bar{\partial}}}(\Gamma \backslash G)$ is canonically isomorphic to the ${\bar{\partial}}$-cohomology $H^{\ast ,\ast }_{{\bar{\partial}}}({\mathfrak g}^{{\mathbb C}})$of the bigraded complex $(\Lambda ^{\ast ,\ast } ({\mathfrak g}^{{\mathbb C}})^{\ast }, {\bar{\partial}})$of complex valued left invariant differential forms on the nilpotent Lie group $G$.
Weakly o-minimal structures and real closed fields
Dugald
Macpherson;
David
Marker;
Charles
Steinhorn
5435-5483
Abstract: A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.
Curves on normal surfaces
Gunnar
Fløystad
5485-5510
Abstract: We study locally Cohen-Macaulay space curves lying on normal surfaces. We prove some theorems on the behaviour of the cohomology functions and initial ideals of such space curves, which give a basic distinction between those curves and curves lying on non-normal surfaces.
Germs of holomorphic vector fields in $\mathbb{C}^m$ without a separatrix
I.
Luengo;
J.
Olivares
5511-5524
Abstract: We prove the existence of families of germs of holomorphic vector fields in $\mathbb{C}^m$ without a separatrix, in every complex dimension $m$ bigger than or equal to 4.
Ribbon tile invariants
Igor
Pak
5525-5561
Abstract: Let $\mathbf{T}$ be a finite set of tiles, and $\mathcal{B}$ a set of regions $\Gamma$ tileable by $\mathbf{T}$. We introduce a tile counting group $\mathbb{G} (\mathbf{T}, \mathcal{B})$ as a group of all linear relations for the number of times each tile $\tau \in \mathbf{T}$ can occur in a tiling of a region $\Gamma \in \mathcal{B}$. We compute the tile counting group for a large set of ribbon tiles, also known as rim hooks, in a context of representation theory of the symmetric group. The tile counting group is presented by its set of generators, which consists of certain new tile invariants. In a special case these invariants generalize the Conway-Lagarias invariant for tromino tilings and a height invariant which is related to computation of characters of the symmetric group. The heart of the proof is the known bijection between rim hook tableaux and certain standard skew Young tableaux. We also discuss signed tilings by the ribbon tiles and apply our results to the tileability problem.
Asymptotic relations among Fourier coefficients of real-analytic Eisenstein series
Alvaro
Alvarez-Parrilla
5563-5582
Abstract: Following Wolpert, we find a set of asymptotic relations among the Fourier coefficients of real-analytic Eisenstein series. The relations are found by evaluating the integral of the product of an Eisenstein series $\varphi_{ir}$ with an exponential factor along a horocycle. We evaluate the integral in two ways by exploiting the automorphicity of $\varphi_{ir}$; the first of these evaluations immediately gives us one coefficient, while the other evaluation provides us with a sum of Fourier coefficients. The second evaluation of the integral is done using stationary phase asymptotics in the parameter $\lambda (\lambda=\frac{1}{4}+r^2$is the eigenvalue of $\varphi_{ir}$) for a cubic phase. As applications we find sets of asymptotic relations for divisor functions.
Computing the $p$-Selmer group of an elliptic curve
Z.
Djabri;
Edward
F.
Schaefer;
N.
P.
Smart
5583-5597
Abstract: In this paper we explain how to bound the $p$-Selmer group of an elliptic curve over a number field $K$. Our method is an algorithm which is relatively simple to implement, although it requires data such as units and class groups from number fields of degree at most $p^2-1$. Our method is practical for $p=3$, but for larger values of $p$it becomes impractical with current computing power. In the examples we have calculated, our method produces exactly the $p$-Selmer group of the curve, and so one can use the method to find the Mordell-Weil rank of the curve when the usual method of $2$-descent fails.
On the dimension of the product of two compacta and the dimension of their intersection in general position in Euclidean space
A.
N.
Dranishnikov
5599-5618
Abstract: For every two compact metric spaces $X$ and $Y$, both with dimension at most $n-3$, there are dense $G_{\delta}$-subsets of mappings $f:X \to \mathbb{R}^n$ and $g:Y\to \mathbb{R}^n$ with $dimf(X)\cap g(Y)\leq dim(X\times Y)-n$.
Power operations in elliptic cohomology and representations of loop groups
Matthew
Ando
5619-5666
Abstract: Part I of this paper describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve. Part II discusses a relationship between equivariant elliptic cohomology and representations of loop groups. Part III investigates the representation of theoretic considerations which give rise to the power operations discussed in Part I.
On cobordism of manifolds with corners
Gerd
Laures
5667-5688
Abstract: This work sets up a cobordism theory for manifolds with corners and gives an identification with the homotopy of a certain limit of Thom spectra. It thereby creates a geometrical interpretation of Adams-Novikov resolutions and lays the foundation for investigating the chromatic status of the elements so realized. As an application, Lie groups together with their left invariant framings are calculated by regarding them as corners of manifolds with interesting Chern numbers. The work also shows how elliptic cohomology can provide useful invariants for manifolds of codimension 2.
The Farrell-Jones Isomorphism Conjecture for finite covolume hyperbolic actions and the algebraic $K$-theory of Bianchi groups
E.
Berkove;
F.
T.
Farrell;
D.
Juan-Pineda;
K.
Pearson
5689-5702
Abstract: We prove the Farrell-Jones Isomorphism Conjecture for groups acting properly discontinuously via isometries on (real) hyperbolic $n$-space $\mathbb{H} ^n$ with finite volume orbit space. We then apply this result to show that, for any Bianchi group $\Gamma$, $Wh(\Gamma)$, $\tilde K_0(\mathbb{Z}\Gamma)$, and $K_i(\mathbb{Z}\Gamma)$ vanish for $i\leq -1$.
Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents
N.
Ghoussoub;
C.
Yuan
5703-5743
Abstract: We use variational methods to study the existence and multiplicity of solutions for the following quasi-linear partial differential equation: $\left\{ \begin{matrix} {-\triangle_{p} u = \lambda \vert u\vert^{r-2}u + \mu ... ... }, {}} {\hphantom{-} u\vert _{\partial \Omega} = 0, } \end{matrix}\right.$ where $\lambda$ and $\mu$ are two positive parameters and $\Omega$ is a smooth bounded domain in $\mathbf{R}^n$ containing $0$ in its interior. The variational approach requires that $1 < p < n$, $p\leq q\leq p^{*}(s)\equiv \frac{n-s}{n-p}p$ and $p\leq r\leq p^*\equiv p^*(0)=\frac{np}{n-p}$, which we assume throughout. However, the situations differ widely with $q$ and $r$, and the interesting cases occur either at the critical Sobolev exponent ($r=p^*$) or in the Hardy-critical setting ($s=p=q$) or in the more general Hardy-Sobolev setting when $q=\frac{n-s}{n-p}p$. In these cases some compactness can be restored by establishing Palais-Smale type conditions around appropriately chosen dual sets. Many of the results are new even in the case $p=2$, especially those corresponding to singularities (i.e., when $0<s\leq p)$.
The quartile operator and pointwise convergence of Walsh series
Christoph
Thiele
5745-5766
Abstract: The bilinear Hilbert transform is given by \begin{displaymath}H(f,g)(x):= p.v. \int f(x-t)g(x+t)\frac{dt}{t}. \end{displaymath} It satisfies estimates of the type \begin{displaymath}\Vert H(f,g)\Vert _r\le C(s,t)\Vert f\Vert _s \Vert g\Vert _t.\end{displaymath} In this paper we prove such estimates for a discrete model of the bilinear Hilbert transform involving the Walsh Fourier transform. We also reprove the well-known fact that the Walsh Fourier series of a function in $L^p[0,1]$, with $1<p$ converges pointwise almost everywhere. The purpose of this exposition is to clarify the connection between these two results and to present an easy approach to recent methods of time-frequency analysis.
The range of traces on quantum Heisenberg manifolds
Beatriz
Abadie
5767-5780
Abstract: We embed the quantum Heisenberg manifold $D_{\mu\nu}^{c}$ in a crossed product ${C}^*$-algebra. This enables us to show that all tracial states on $D_{\mu\nu}^{c}$ induce the same homomorphism on $K_0(D_{\mu\nu}^{c})$, whose range is the group $\mathbf{Z} +2\mu\mathbf{Z} + 2\nu\mathbf{Z}$.
Contact topology and hydrodynamics III: knotted orbits
John
Etnyre;
Robert
Ghrist
5781-5794
Abstract: We employ the relationship between contact structures and Beltrami fields derived in part I of this series to construct a steady nonsingular solution to the Euler equations on a Riemannian $S^3$ whose flowlines trace out closed curves of all possible knot and link types. Using careful contact-topological controls, we can make such vector fields real-analytic and transverse to the tight contact structure on $S^3$. Sufficient review of concepts is included to make this paper independent of the previous works in this series.
The natural representation of the stabilizer of four subspaces
Jozsef
Horvath;
Roger
Howe
5795-5815
Abstract: Consider the natural action of the general linear group $GL(V)$ on the product of four Grassmann varieties of the vector space $V$. In General linear group action on four Grassmannians we gave an algorithm to construct the generic stabilizer $H$ of this action. In this paper we investigate the structure of $V$ as an $H$-module, and we show the effectiveness of the methods developed there, by applying them to the explicit description of $H$.
Intersection theory on non-commutative surfaces
Peter
Jørgensen
5817-5854
Abstract: Consider a non-commutative algebraic surface, $X$, and an effective divisor $Y$ on $X$, as defined by Van den Bergh. We show that the Riemann-Roch theorem, the genus formula, and the self intersection formula from classical algebraic geometry generalize to this setting. We also apply our theory to some special cases, including the blow up of $X$in a point, and show that the self intersection of the exceptional divisor is $-1$. This is used to give an example of a non-commutative surface with a commutative ${\Bbb P}^1$ which cannot be blown down, because its self intersection is $+1$ rather than $-1$. We also get some results on Hilbert polynomials of modules on $X$.
Quantum $n$-space as a quotient of classical $n$-space
K.
R.
Goodearl;
E.
S.
Letzter
5855-5876
Abstract: The prime and primitive spectra of $\mathcal{O}_{\mathbf q}(k^{n})$, the multiparameter quantized coordinate ring of affine $n$-space over an algebraically closed field $k$, are shown to be topological quotients of the corresponding classical spectra, $\operatorname{spec} \mathcal{O}(k^{n})$ and $\max \mathcal{O}(k^{n})\approx k^{n}$, provided the multiplicative group generated by the entries of $\mathbf{q}$ avoids $-1$.
Correction to ``Geometric groups. I''
Valera
Berestovskii;
Conrad
Plaut;
Cornelius
Stallman
5877
The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature
Frank
Morgan;
Michael
Hutchings;
Hugh
Howards
4889-4909
Abstract: We prove that the least-perimeter way to enclose prescribed area in the plane with smooth, rotationally symmetric, complete metric of nonincreasing Gauss curvature consists of one or two circles, bounding a disc, the complement of a disc, or an annulus. We also provide a new isoperimetric inequality in general surfaces with boundary.
Traces on algebras of parameter dependent pseudodifferential operators and the eta--invariant
Matthias
Lesch;
Markus
J.
Pflaum
4911-4936
Abstract: We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\mathbb{R}$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $\Gamma\subset\mathbb{R} ^p$, we construct a unique ``symbol valued trace'', which extends the $L^2$-trace on operators of small order. This construction is in the spirit of a trace due to Kontsevich and Vishik in the nonparametric case. Our trace allows us to construct various trace functionals in a systematic way. Furthermore, we study the higher-dimensional eta-invariants on algebras with parameter space $\mathbb{R} ^{2k-1}$. Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over $\mathbb{R} ^{2k-1}$. The eta-invariant of this family coincides with the spectral eta-invariant of the operator.
Stability theory, permutations of indiscernibles, and embedded finite models
John
Baldwin;
Michael
Benedikt
4937-4969
Abstract: We show that the expressive power of first-order logic over finite models embedded in a model $M$ is determined by stability-theoretic properties of $M$. In particular, we show that if $M$ is stable, then every class of finite structures that can be defined by embedding the structures in $M$, can be defined in pure first-order logic. We also show that if $M$ does not have the independence property, then any class of finite structures that can be defined by embedding the structures in $M$, can be defined in first-order logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let $I$ be a set of indiscernibles in a model $M$and suppose $(M,I)$ is elementarily equivalent to $(M_1,I_1)$ where $M_1$ is $\vert I_1\vert^+$-saturated. If $M$ is stable and $(M,I)$ is saturated, then every permutation of $I$extends to an automorphism of $M$ and the theory of $(M,I)$ is stable. Let $I$ be a sequence of $<$-indiscernibles in a model $M$, which does not have the independence property, and suppose $(M,I)$ is elementarily equivalent to $(M_1,I_1)$ where $(I_1,<)$ is a complete dense linear order and $M_1$ is $\vert I_1\vert^+$-saturated. Then $(M,I)$-types over $I$are order-definable and if $(M,I)$ is $\aleph_1$-saturated, every order preserving permutation of $I$ can be extended to a back-and-forth system.
Strongly almost disjoint sets and weakly uniform bases
Z.
T.
Balogh;
S.
W.
Davis;
W.
Just;
S.
Shelah;
P.
J.
Szeptycki
4971-4987
Abstract: A combinatorial principle CECA is formulated and its equivalence with GCH + certain weakenings of $\Box_\lambda$ for singular $\lambda$ is proved. CECA is used to show that certain ``almost point-$<\tau$'' families can be refined to point-$< \tau$ families by removing a small set from each member of the family. This theorem in turn is used to show the consistency of ``every first countable $T_1$-space with a weakly uniform base has a point-countable base.''
Effectively dense Boolean algebras and their applications
André
Nies
4989-5012
Abstract: A computably enumerable Boolean algebra ${\mathcal{B}}$ is effectively dense if for each $x \in{\mathcal{B}}$ we can effectively determine an $F(x)\le x$ such that $x \neq 0$ implies $0 < F(x) < x$. We give an interpretation of true arithmetic in the theory of the lattice of computably enumerable ideals of such a Boolean algebra. As an application, we also obtain an interpretation of true arithmetic in all theories of intervals of ${\mathcal{E}}$ (the lattice of computably enumerable sets under inclusion) which are not Boolean algebras. We derive a similar result for theories of certain initial intervals $[{\mathbf{0}},{\mathbf{a}}]$ of subrecursive degree structures, where ${\mathbf{a}}$is the degree of a set of relatively small complexity, for instance a set in exponential time.
A compactification of a family of determinantal Godeaux surfaces
Yongnam
Lee
5013-5023
Abstract: In this paper, we present a geometric description of the compactification of the family of determinantal Godeaux surfaces, via the study of the bicanonical pencil and using classical Prym theory. In particular, we reduce the problem of compactifying the space of bicanonical pencils of determinantal Godeaux surfaces to the compactification of the family of twisted cubic curves in $\mathbb{P}^{3}$ with certain given tangent conditions.
Unipotent groups associated to reduced curves
David
Penniston
5025-5043
Abstract: Let $X$ be a curve defined over an algebraically closed field $k$ with $\operatorname{char}(k)=p>0$. Assume that $X/k$ is reduced. In this paper we study the unipotent part $U$ of the Jacobian $J_{X/k}$. In particular, we prove that if $p$ is large in terms of the dimension of $U$, then $U$ is isomorphic to a product of additive groups $\mathbb{G} _a$.
The density of rational lines on cubic hypersurfaces
Scott
T.
Parsell
5045-5062
Abstract: We provide a lower bound for the density of rational lines on the hypersurface defined by an additive cubic equation in at least 57 variables. In the process, we obtain a result on the paucity of non-trivial solutions to an associated system of Diophantine equations.
Euclidean weights of codes from elliptic curves over rings
José
Felipe
Voloch;
Judy
L.
Walker
5063-5076
Abstract: We construct certain error-correcting codes over finite rings and estimate their parameters. For this purpose, we need to develop some tools, notably an estimate for certain exponential sums and some results on canonical lifts of elliptic curves. These results may be of independent interest.
Infinitely Renormalizable Quadratic Polynomials
Yunping
Jiang
5077-5091
Abstract: We prove that the Julia set of a quadratic polynomial which admits an infinite sequence of unbranched, simple renormalizations with complex bounds is locally connected. The method in this study is three-dimensional puzzles.
Homology manifold bordism
Heather
Johnston;
Andrew
Ranicki
5093-5137
Abstract: The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact $ANR$ homology manifolds of dimension $\geq 6$ is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston. First, we establish homology manifold transversality for submanifolds of dimension $\geq 7$: if $f:M \to P$ is a map from an $m$-dimensional homology manifold $M$ to a space $P$, and $Q \subset P$ is a subspace with a topological $q$-block bundle neighborhood, and $m-q \geq 7$, then $f$ is homology manifold $s$-cobordant to a map which is transverse to $Q$, with $f^{-1}(Q) \subset M$ an $(m-q)$-dimensional homology submanifold. Second, we obtain a codimension $q$ splitting obstruction $s_Q(f) \in LS_{m-q}(\Phi)$ in the Wall $LS$-group for a simple homotopy equivalence $f:M \to P$ from an $m$-dimensional homology manifold $M$ to an $m$-dimensional Poincaré space $P$ with a codimension $q$ Poincaré subspace $Q \subset P$ with a topological normal bundle, such that $s_Q(f)=0$ if (and for $m-q \geq 7$ only if) $f$ splits at $Q$ up to homology manifold $s$-cobordism. Third, we obtain the multiplicative structure of the homology manifold bordism groups $\Omega^H_*\cong\Omega^{TOP}_*[L_0(\mathbb Z)]$.
Spectra of $\text{BP}$-linear relations, $v_n$-series, and $\text{BP}$ cohomology of Eilenberg-Mac Lane spaces
Hirotaka
Tamanoi
5139-5178
Abstract: On Brown-Peterson cohomology groups of a space, we introduce a natural inherent topology, BP topology, which is always complete Hausdorff for any space. We then construct a spectra map which calculates infinite BP-linear sums convergent with respect to the BP topology, and a spectrum which describes infinite sum BP-linear relations in BP cohomology. The mod $p$ cohomology of this spectrum is a cyclic module over the Steenrod algebra with relations generated by products of exactly two Milnor primitives. We show a close relationship between BP-linear relations in BP cohomology and the action of the Milnor primitives on mod $p$ cohomology. We prove main relations in the BP cohomology of Eilenberg-Mac Lane spaces. These are infinite sum BP-linear relations convergent with respect to the BP topology. Using BP fundamental classes, we define $v_{n}$-series which are $v_{n}$-analogues of the $p$-series. Finally, we show that the above main relations come from the $v_{n}$-series.
Center manifolds for smooth invariant manifolds
Shui-Nee
Chow;
Weishi
Liu;
Yingfei
Yi
5179-5211
Abstract: We study dynamics of flows generated by smooth vector fields in ${\mathbb{R} }^n$ in the vicinity of an invariant and closed smooth manifold $Y$. By applying the Hadamard graph transform technique, we show that there exists an invariant manifold (called a center manifold of $Y$) based on the information of the linearization along $Y$, which contains every locally bounded solution and is persistent under small perturbations.
$C^1$ Connecting Lemmas
Lan
Wen;
Zhihong
Xia
5213-5230
Abstract: Like the closing lemma, the connecting lemma is of fundamental importance in dynamical systems. Hayashi recently proved the $C^1$ connecting lemma for stable and unstable manifolds of a hyperbolic invariant set. In this paper, we prove several very general $C^1$ connecting lemmas. We simplify Hayashi's proof and extend the results to more general cases.
Local differentiability of distance functions
R.
A.
Poliquin;
R.
T.
Rockafellar;
L.
Thibault
5231-5249
Abstract: Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets $C$ for which the distance function $d_{C}$ is continuously differentiable everywhere on an open ``tube'' of uniform thickness around $C$. Here a corresponding local theory is developed for the property of $d_{C}$ being continuously differentiable outside of $C$ on some neighborhood of a point $x\in C$. This is shown to be equivalent to the prox-regularity of $C$ at $x$, which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation. Additional characterizations are provided in terms of $d_{C}^{2}$ being locally of class $C^{1+}$ or such that $d_{C}^{2}+\sigma \vert\cdot \vert^{2}$ is convex around $x$ for some $\sigma >0$. Prox-regularity of $C$ at $x$ corresponds further to the normal cone mapping $N_{C}$ having a hypomonotone truncation around $x$, and leads to a formula for $P_{C}$ by way of $N_{C}$. The local theory also yields new insights on the global level of the Clarke-Stern-Wolenski results, and on a property of sets introduced by Shapiro, as well as on the concept of sets with positive reach considered by Federer in the finite dimensional setting.
Optimal factorization of Muckenhoupt weights
Michael
Brian
Korey
5251-5262
Abstract: Peter Jones' theorem on the factorization of $A_p$ weights is sharpened for weights with bounds near $1$, allowing the factorization to be performed continuously near the limiting, unweighted case. When $1<p<\infty$ and $w$ is an $A_p$ weight with bound $A_p(w)=1+\varepsilon$, it is shown that there exist $A_1$ weights $u,v$ such that both the formula $w=uv^{1-p}$ and the estimates $A_1(u), A_1(v)=1+\mathcal O(\sqrt\varepsilon)$ hold. The square root in these estimates is also proven to be the correct asymptotic power as $\varepsilon\to 0$.
A classification of one dimensional almost periodic tilings arising from the projection method
James
A.
Mingo
5263-5277
Abstract: For each irrational number $\alpha$, with continued fraction expansion $[0; a_1,\allowbreak a_2,a_3, \dots ]$, we classify, up to translation, the one dimensional almost periodic tilings which can be constructed by the projection method starting with a line of slope $\alpha$. The invariant is a sequence of integers in the space $X_\alpha = \{(x_i)_{i=1}^\infty \mid x_i \in \{0,1,2, \dots ,a_i\}$ and $x_{i+1} = 0$ whenever $x_i = a_i\}$ modulo the equivalence relation generated by tail equivalence and $(a_1, 0, a_3, 0, \dots ) \sim (0, a_2, 0, a_4, \dots ) \sim (a_1 -1, a_2 - 1, a_3 - 1, \dots )$. Each tile in a tiling $\textsf{T}$, of slope $\alpha$, is coded by an integer $0 \leq x \leq [\alpha]$. Using a composition operation, we produce a sequence of tilings $\textsf{T}_1 = \textsf{T}{}, \textsf{T}_2, \textsf{T}_3, \dots$. Each tile in $\textsf{T}_i$ gets absorbed into a tile in $\textsf{T}_{i+1}$. A choice of a starting tile in $\textsf{T}_1$ will thus produce a sequence in $X_\alpha$. This is the invariant.
Path stability and nonlinear weak ergodic theorems
Yong-Zhuo
Chen
5279-5292
Abstract: Let $\{f_{n} \}$ be a sequence of nonlinear operators. We discuss the asymptotic properties of their inhomogeneous iterates $f_{n} \circ f_{n-1} \circ \cdots \circ f_{1}\,$ in metric spaces, then apply the results to the ordered Banach spaces through projective metrics. Theorems on path stability and nonlinear weak ergodicity are obtained in this paper.
Hypercyclic operators that commute with the Bergman backward shift
Paul
S.
Bourdon;
Joel
H.
Shapiro
5293-5316
Abstract: The backward shift $B$ on the Bergman space of the unit disc is known to be hypercyclic (meaning: it has a dense orbit). Here we ask: ``Which operators that commute with $B$ inherit its hypercyclicity?'' We show that the problem reduces to the study of operators of the form $\varphi(B)$ where $\varphi$ is a holomorphic self-map of the unit disc that multiplies the Dirichlet space into itself, and that the question of hypercyclicity for such an operator depends on how freely $\varphi(z)$ is allowed to approach the unit circle as $\vert z\vert\to 1-$.
Semi-classical limit for random walks
Ursula
Porod;
Steve
Zelditch
5317-5355
Abstract: Let $(G, \mu)$ be a discrete symmetric random walk on a compact Lie group $G$ with step distribution $\mu$ and let $T_{\mu}$ be the associated transition operator on $L^2(G)$. The irreducibles $V_{\rho}$ of the left regular representation of $G$ on $L^2(G)$ are finite dimensional invariant subspaces for $T_{\mu}$ and the spectrum of $T_{\mu}$ is the union of the sub-spectra $\sigma(T_{\mu}\upharpoonleft_{V_{\rho}})$ on the irreducibles, which consist of real eigenvalues $\{ \lambda_{\rho 1},...,\lambda_{\rho \dim V_{\rho}}\}$. Our main result is an asymptotic expansion for the spectral measures \begin{displaymath}m_{\rho}^{\mu}(\lambda) := \frac{1}{\dim V_{\rho}} \sum_{j=1}^{\dim V_{\rho}} \delta(\lambda - \lambda_{\rho j})\end{displaymath} along rays of representations in a positive Weyl chamber $\mathbf{t}^*_+$, i.e. for sequences of representations $k \rho$, $k\in \mathbb{N}$ with $k\rightarrow \infty$. As a corollary we obtain some estimates on the spectral radius of the random walk. We also analyse the fine structure of the spectrum for certain random walks on $U(n)$ (for which $T_{\mu}$ is essentially a direct sum of Harper operators).
Cotorsion theories and splitters
Rüdiger
Göbel;
Saharon
Shelah
5357-5379
Abstract: Let $R$ be a subring of the rationals. We want to investigate self splitting $R$-modules $G$ (that is $\operatorname{Ext}_R(G,G) = 0)$. Following Schultz, we call such modules splitters. Free modules and torsion-free cotorsion modules are classical examples of splitters. Are there others? Answering an open problem posed by Schultz, we will show that there are more splitters, in fact we are able to prescribe their endomorphism $R$-algebras with a free $R$-module structure. As a by-product we are able to solve a problem of Salce, showing that all rational cotorsion theories have enough injectives and enough projectives. This is also basic for answering the flat-cover-conjecture.
The Jantzen sum formula for cyclotomic $q$--Schur algebras
Gordon
James;
Andrew
Mathas
5381-5404
Abstract: The cyclotomic $q$-Schur algebra was introduced by Dipper, James and Mathas, in order to provide a new tool for studying the Ariki-Koike algebra. We here prove an analogue of Jantzen's sum formula for the cyclotomic $q$-Schur algebra. Among the applications is a criterion for certain Specht modules of the Ariki-Koike algebras to be irreducible.
Definably simple groups in o-minimal structures
Y.
Peterzil;
A.
Pillay;
S.
Starchenko
4397-4419
Abstract: Let $\mathbb{G} =\langle G, \cdot\rangle$ be a group definable in an o-minimal structure $\mathcal{M}$. A subset $H$ of $G$ is $\mathbb{G}$-definable if $H$ is definable in the structure $\langle G,\cdot\rangle$(while definable means definable in the structure $\mathcal{M}$). Assume $\mathbb{G}$ has no $\mathbb{G}$-definable proper subgroup of finite index. In this paper we prove that if $\mathbb{G}$has no nontrivial abelian normal subgroup, then $\mathbb{G}$ is the direct product of $\mathbb{G}$-definable subgroups $H_1,\ldots,H_k$ such that each $H_i$ is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin's conjecture.
Simple algebraic and semialgebraic groups over real closed fields
Ya'acov
Peterzil;
Anand
Pillay;
Sergei
Starchenko
4421-4450
Abstract: We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In Definably simple groups in o-minimal structures, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.
Manifolds with minimal radial curvature bounded from below and big volume
Valery
Marenich
4451-4468
Abstract: We prove that a convergence in the Gromov-Hausdorff distance of manifolds with minimal radial curvature bounded from below by 1 to the standard sphere is equivalent to a volume convergence.
Willmore two-spheres in the four-sphere
Sebastián
Montiel
4469-4486
Abstract: Genus zero Willmore surfaces immersed in the three-sphere $\mathbb{S}^3$correspond via the stereographic projection to minimal surfaces in Euclidean three-space with finite total curvature and embedded planar ends. The critical values of the Willmore functional are $4\pi k$, where $k\in\mathbb{N}^*$, with $k\ne 2,3,5,7$. When the ambient space is the four-sphere $\mathbb{S}^4$, the regular homotopy class of immersions of the two-sphere $\mathbb{S}^2$ is determined by the self-intersection number $q\in\mathbb{Z}$; here we shall prove that the possible critical values are $4\pi (\vert q\vert+k+1)$, where $k\in\mathbb{N}$. Moreover, if $k=0$, the corresponding immersion, or its antipodal, is obtained, via the twistor Penrose fibration $\mathbb{P}^3\rightarrow \mathbb{S}^4$, from a rational curve in $\mathbb{P}^3$and, if $k\ne 0$, via stereographic projection, from a minimal surface in $\mathbb{R}^4$ with finite total curvature and embedded planar ends. An immersion lies in both families when the rational curve is contained in some $\mathbb{P}^2\subset\mathbb{P}^3$ or (equivalently) when the minimal surface of $\mathbb{R}^4$ is complex with respect to a suitable complex structure of $\mathbb{R}^4$.
Principal curvatures of isoparametric hypersurfaces in $\mathbb{C}P^{n}$
Liang
Xiao
4487-4499
Abstract: Let $M$ be an isoparametric hypersurface in $\mathbb{C}P^{n}$, and $\overline{M}$ the inverse image of $M$ under the Hopf map. By using the relationship between the eigenvalues of the shape operators of $M$ and $\overline{M}$, we prove that $M$ is homogeneous if and only if either $g$or $l$ is constant, where $g$ is the number of distinct principal curvatures of $M$ and $l$ is the number of non-horizontal eigenspaces of the shape operator on $\overline{M}$.
Symplectic 4-manifolds with Hermitian Weyl tensor
Vestislav
Apostolov;
John
Armstrong
4501-4513
Abstract: It is proved that any compact almost Kähler, Einstein 4-manifold whose fundamental form is a root of the Weyl tensor is necessarily Kähler.
The toric $h$-vectors of partially ordered sets
Margaret
M.
Bayer;
Richard
Ehrenborg
4515-4531
Abstract: An explicit formula for the toric $h$-vector of an Eulerian poset in terms of the $\mathbf{cd}$-index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag $h$-vector. For this, another proof based on Fine's algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, $g_{n/2}(P)=g_{n/2}(P^*)$, is the only equation relating the $h$-vectors of posets and their duals. A result on the $h$-vectors of oriented matroids is given. A simple formula for the $\mathbf{cd}$-index in terms of the flag $h$-vector is derived.
Examples of torsion points on genus two curves
John
Boxall;
David
Grant
4533-4555
Abstract: We describe a method that sometimes determines all the torsion points lying on a curve of genus two defined over a number field and embedded in its Jacobian using a Weierstrass point as base point. We then apply this to the examples $y^{2}=x^{5}+x$, $y^{2}=x^{5}+5\,x^{3}+x$, and $y^{2}-y=x^{5}$.
$\mathbf{C}^{2}$-saddle method and Beukers' integral
Masayoshi
Hata
4557-4583
Abstract: We give good non-quadraticity measures for the values of logarithm at specific rational points by modifying Beukers' double integral. The two-dimensional version of the saddle method, which we call $\mathbf{C}^{2}$-saddle method, is applied.
The Mod-2 cohomology of the Bianchi groups
Ethan
Berkove
4585-4602
Abstract: The Bianchi groups are a family of discrete subgroups of $PSL_2(\mathbb C)$which have group theoretic descriptions as amalgamated products and HNN extensions. Using Bass-Serre theory, we show how the cohomology of these two constructions relates to the cohomology of their pieces. We then apply these results to calculate the mod-2 cohomology ring for various Bianchi groups.
A global approach to fully nonlinear parabolic problems
Athanassios
G.
Kartsatos;
Igor
V.
Skrypnik
4603-4640
Abstract: We consider the general initial-boundary value problem (1) $\displaystyle{\frac{\partial u}{\partial t}-F(x,t,u,\mathcal{D}^{1}u, \mathcal{D}^{2}u)=f(x,t),\quad (x,t)\in Q_{T}\equiv \Omega \times (0,T),}$ (2) $\displaystyle{G(x,t,u,\mathcal{D}^{1}u)=g(x,t),\quad (x,t)\in S_{T}\equiv \partial\Omega \times (0,T),}$ (3) $\displaystyle{u(x,0)=h(x),\quad x\in \Omega,}$ where $\Omega$ is a bounded open set in $\mathcal{R}^{n}$ with sufficiently smooth boundary. The problem (1)-(3) is first reduced to the analogous problem in the space $W^{(4),0}_{p}(Q_{T})$with zero initial condition and \begin{displaymath}f\in W^{(2),0}_{p}(Q_{T}),~g \in W^{(3-\frac{1}{p}),0}_{p}(S_{T}). \end{displaymath} The resulting problem is then reduced to the problem $Au=0,$ where the operator $A:W^{(4),0}_{p}(Q_{T})\to \left [W^{(4),0}_{p}(Q_{T})\right ]^{*}$ satisfies Condition $(S)_{+}.$ This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces. The local and global solvability of the operator equation $Au=0$ are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations.
Invariant foliations near normally hyperbolic invariant manifolds for semiflows
Peter
W.
Bates;
Kening
Lu;
Chongchun
Zeng
4641-4676
Abstract: Let $M$ be a compact $C^1$ manifold which is invariant and normally hyperbolic with respect to a $C^1$ semiflow in a Banach space. Then in an $\epsilon$-neighborhood of $M$ there exist local $C^1$ center-stable and center-unstable manifolds $W^{cs}(\epsilon)$ and $W^{cu}(\epsilon)$, respectively. Here we show that $W^{cs}(\epsilon)$ and $W^{cu}(\epsilon)$ may each be decomposed into the disjoint union of $C^1$ submanifolds (leaves) in such a way that the semiflow takes leaves into leaves of the same collection. Furthermore, each leaf intersects $M$ in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time.
Polynomials that are positive on an interval
Victoria
Powers;
Bruce
Reznick
4677-4692
Abstract: This paper discusses representations of polynomials that are positive on intervals of the real line. An elementary and constructive proof of the following is given: If $h(x), p(x) \in \mathbb{R}[x]$ such that $\{ \alpha \in \mathbb{R} \mid h(\alpha) \geq 0 \} = [-1,1]$ and $p(x) > 0$ on $[-1,1]$, then there exist sums of squares $s(x), t(x) \in \mathbb{R}[x]$ such that $p(x) = s(x) + t(x) h(x)$. Explicit degree bounds for $s$ and $t$ are given, in terms of the degrees of $p$ and $h$ and the location of the roots of $p$. This is a special case of Schmüdgen's Theorem, and extends classical results on representations of polynomials positive on a compact interval. Polynomials positive on the non-compact interval $[0,\infty)$ are also considered.
The optimal differentiation basis and liftings of $L^{{\infty}}$
Jürgen
Bliedtner;
Peter
A.
Loeb
4693-4710
Abstract: There is an optimal way to differentiate measures when given a consistent choice of where zero limits must occur. The appropriate differentiation basis is formed following the pattern of an earlier construction by the authors of an optimal approach system for producing boundary limits in potential theory. Applications include the existence of Lebesgue points, approximate continuity, and liftings for the space of bounded measurable functions - all aspects of the fact that for every point outside a set of measure $0$, a given integrable function has small variation on a set that is ``big'' near the point. This fact is illuminated here by the replacement of each measurable set with the collection of points where the set is ``big'', using a classical base operator. Properties of such operators and of the topologies they generate, e.g., the density and fine topologies, are recalled and extended along the way. Topological considerations are simplified using an extension of base operators from algebras of sets on which they are initially defined to the full power set of the underlying space.
Periodic points of holomorphic maps via Lefschetz numbers
Núria
Fagella;
Jaume
Llibre
4711-4730
Abstract: In this paper we study the set of periods of holomorphic maps on compact manifolds, using the periodic Lefschetz numbers introduced by Dold and Llibre, which can be computed from the homology class of the map. We show that these numbers contain information about the existence of periodic points of a given period; and, if we assume the map to be transversal, then they give us the exact number of such periodic orbits. We apply this result to the complex projective space of dimension $n$ and to some special type of Hopf surfaces, partially characterizing their set of periods. In the first case we also show that any holomorphic map of ${\mathbb CP}(n)$ of degree greater than one has infinitely many distinct periodic orbits, hence generalizing a theorem of Fornaess and Sibony. We then characterize the set of periods of a holomorphic map on the Riemann sphere, hence giving an alternative proof of Baker's theorem.
Endomorphisms of expansive systems on compact metric spaces and the pseudo-orbit tracing property
Masakazu
Nasu
4731-4757
Abstract: We investigate the interrelationships between the dynamical properties of commuting continuous maps of a compact metric space. Let $X$ be a compact metric space. First we show the following. If $\tau: X \rightarrow X$ is an expansive onto continuous map with the pseudo-orbit tracing property (POTP) and if there is a topologically mixing continuous map $\varphi: X \rightarrow X$ with $\tau\varphi = \varphi\tau$, then $\tau$ is topologically mixing. If $\tau: X \rightarrow X$ and $\varphi: X \rightarrow X$ are commuting expansive onto continuous maps with POTP and if $\tau$ is topologically transitive with period $p$, then for some $k$dividing $p$, $X = \bigcup_{i=0}^{l-1} B_i$, where the $B_i$, $0 \leq i \leq l-1$, are the basic sets of $\varphi$ with $l = p/k$ such that all $\varphi\vert B_i : B_i \rightarrow B_i$ have period $k$, and the dynamical systems $(B_i,\varphi\vert B_i)$ are a factor of each other, and in particular they are conjugate if $\tau$ is a homeomorphism. Then we prove an extension of a basic result in symbolic dynamics. Using this and many techniques in symbolic dynamics, we prove the following. If $\tau: X \rightarrow X$ is a topologically transitive, positively expansive onto continuous map having POTP, and $\varphi: X \rightarrow X$ is a positively expansive onto continuous map with $\varphi\tau = \tau\varphi$, then $\varphi$ has POTP. If $\tau:X \rightarrow X$ is a topologically transitive, expansive homeomorphism having POTP, and $\varphi : X \rightarrow X$ is a positively expansive onto continuous map with $\varphi\tau = \tau\varphi$, then $\varphi$ has POTP and is constant-to-one. Further we define `essentially LR endomorphisms' for systems of expansive onto continuous maps of compact metric spaces, and prove that if $\tau: X \rightarrow X$ is an expansive homeomorphism with canonical coordinates and $\varphi$ is an essentially LR automorphism of $(X,\tau)$, then $\varphi$ has canonical coordinates. We add some discussions on basic properties of the essentially LR endomorphisms.
An equivariant Brauer semigroup and the symmetric imprimitivity theorem
Astrid
an Huef;
Iain
Raeburn;
Dana
P.
Williams
4759-4787
Abstract: Suppose that $(X,G)$ is a second countable locally compact transformation group. We let $\operatorname{S}_G(X)$ denote the set of Morita equivalence classes of separable dynamical systems $(A,G,\alpha)$ where $A$ is a $C_{0}(X)$-algebra and $\alpha$ is compatible with the given $G$-action on $X$. We prove that $\operatorname{S}_{G}(X)$ is a commutative semigroup with identity with respect to the binary operation $[A,G,\alpha][B,G,\beta]=[A\otimes_{X}B,G,\alpha\otimes_{X}\beta]$ for an appropriately defined balanced tensor product on $C_{0}(X)$-algebras. If $G$and $H$ act freely and properly on the left and right of a space $X$, then we prove that $\operatorname{S}_{G}(X/H)$ and $\operatorname{S}_{H}(G\backslash X)$ are isomorphic as semigroups. If the isomorphism maps the class of $(A,G,\alpha)$to the class of $(B,H,\beta)$, then $A\rtimes_{\alpha}G$ is Morita equivalent to $B\rtimes_{\beta}H$.
$q$-Krawtchouk polynomials as spherical functions on the Hecke algebra of type $B$
H.
T.
Koelink
4789-4813
Abstract: The Hecke algebra for the hyperoctahedral group contains the Hecke algebra for the symmetric group as a subalgebra. Inducing the index representation of the subalgebra gives a Hecke algebra module, which splits multiplicity free. The corresponding zonal spherical functions are calculated in terms of $q$-Krawtchouk polynomials using the quantised enveloping algebra for ${\mathfrak{sl}}(2,\mathbb{C} )$. The result covers a number of previously established interpretations of ($q$-)Krawtchouk polynomials on the hyperoctahedral group, finite groups of Lie type, hypergroups and the quantum $SU(2)$ group.
Local structure of Schelter-Procesi smooth orders
Lieven
Le Bruyn
4815-4841
Abstract: In this paper we give the étale local classification of Schelter-Procesi smooth orders in central simple algebras. In particular, we prove that if $\Delta$ is a central simple $K$-algebra of dimension $n^2$, where $K$is a field of trancendence degree $d$, then there are only finitely many étale local classes of smooth orders in $\Delta$. This result is a non-commutative generalization of the fact that a smooth variety is analytically a manifold, and so has only one type of étale local behaviour.
A reduced Tits quadratic form and tameness of three-partite subamalgams of tiled orders
Daniel
Simson
4843-4875
Abstract: Let $D$ be a complete discrete valuation domain with the unique maximal ideal ${\mathfrak{p}}$. We suppose that $D$ is an algebra over an algebraically closed field $K$ and $D/{\mathfrak{p}} \cong K$. Subamalgam $D$-suborders $\Lambda ^{\bullet }$ of a tiled $D$-order $\Lambda$ are studied in the paper by means of the integral Tits quadratic form $q_{\Lambda ^{\bullet }}: {\mathbb{Z} }^{n_{1}+2n_{3}+2 } \,\,\longrightarrow {\mathbb{Z} }$. A criterion for a subamalgam $D$-order $\Lambda ^{\bullet }$ to be of tame lattice type is given in terms of the Tits quadratic form $q_{{\Lambda ^{\bullet }}}$ and a forbidden list $\Omega _{1},\ldots ,\Omega _{17}$ of minor $D$-suborders of $\Lambda ^{\bullet }$presented in the tables.
Skein modules and the noncommutative torus
Charles
Frohman;
Razvan
Gelca
4877-4888
Abstract: We prove that the Kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus. The proof is based on Chebyshev polynomials. As an application, we describe the structure of the Kauffman bracket skein module of a solid torus as a module over the algebra of the cylinder over a torus, and recover a result of Hoste and Przytycki about the skein module of a lens space. We establish simple formulas for Jones-Wenzl idempotents in the skein algebra of a cylinder over a torus, and give a straightforward computation of the $n$-th colored Kauffman bracket of a torus knot, evaluated in the plane or in an annulus.
Conformal actions of ${\mathfrak{sl}_n(\mathbb{R})}$ and ${\hbox{SL}_n(\mathbb{R})\ltimes\mathbb{R}^n}$ on Lorentz manifolds
Scot
Adams;
Garrett
Stuck
3913-3936
Abstract: We prove that, for $n\ge3$, a locally faithful action of ${\hbox{SL}_n(\mathbb{R} )\ltimes\mathbb{R} ^n}$ or of $\hbox{SL}_n({\mathbb R})$ by conformal transformations of a connected Lorentz manifold must be a proper action.
Local boundary rigidity of a compact Riemannian manifold with curvature bounded above
Christopher
B.
Croke;
Nurlan
S.
Dairbekov;
Vladimir
A.
Sharafutdinov
3937-3956
Abstract: This paper considers the boundary rigidity problem for a compact convex Riemannian manifold $(M,g)$ with boundary $\partial M$whose curvature satisfies a general upper bound condition. This includes all nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics $g'$ on $M$ there is a $C^{3,\alpha }$-neighborhood of $g$ such that $g$is the unique metric with the given boundary distance-function (i.e. the function that assigns to any pair of boundary points their distance -- as measured in $M$). More precisely, given any metric $g'$ in this neighborhood with the same boundary distance function there is diffeomorphism $\varphi$which is the identity on $\partial M$such that
Some properties of minimal surfaces in singular spaces
Chikako
Mese
3957-3969
Abstract: This paper involves the generalization of minimal surface theory to spaces with singularities. Let $X$ be an NPC space, i.e. a metric space of non-positive curvature. We define a (parametric) minimal surface in $X$ as a conformal energy minimizing map. Using this definition, many properties of classical minimal surfaces can also be observed for minimal surfaces in this general setting. In particular, we will prove the boundary monotonicity property and the isoperimetric inequality for minimal surfaces in $X$.
Uniform densities of regular sequences in the unit disk
Peter
L.
Duren;
Alexander
P.
Schuster;
Kristian
Seip
3971-3980
Abstract: The upper and lower uniform densities of some regular sequences are computed. These densities are used to determine sequences of sampling and interpolation for Bergman spaces of the unit disk.
The nonstationary ideal and the other $\sigma $-ideals on $\omega _{1}$
Jindrich
Zapletal
3981-3993
Abstract: Under Martin's Maximum every $\sigma$-ideal on $\omega _{1}$ is a subset of an ideal Rudin-Keisler reducible to a finite Fubini power of the nonstationary ideal restricted to a positive set.
Analytic types of plane curve singularities defined by weighted homogeneous polynomials
Chunghyuk
Kang
3995-4006
Abstract: We classify analytically isolated plane curve singularities defined by weighted homogeneous polynomials $f(y,z)$, which are not topologically equivalent to homogeneous polynomials, in an elementary way. Moreover, in preparation for the proof of the above analytic classification theorem, assuming that $g(y,z)$ either satisfies the same property as the above $f$ does or is homogeneous, then we prove easily that the weights of the above $g$ determine the topological type of $g$ and conversely. So, this gives another easy proof for the topological classification theorem of quasihomogenous singularities in $\mathbb{C}^{2}$, which was already known. Also, as an application, it can be shown that for a given $h$, where $h(w_{1},\dots ,w_{n})$ is a quasihomogeneous holomorphic function with an isolated singularity at the origin or $h(w_{1})=w^{p}_{1}$ with a positive integer $p$, analytic types of isolated hypersurface singularities defined by $f+h$ are easily classified where $f$ is defined just as above.
Dihedral coverings of algebraic surfaces and their application
Hiro-o
Tokunaga
4007-4017
Abstract: In this article, we study dihedral coverings of algebraic surfaces branched along curves with at most simple singularities. A criterion for a reduced curve to be the branch locus of some dihedral covering is given. As an application we have the following: Let $B$ be a reduced plane curve of even degree $d$ having only $a$ nodes and $b$ cusps. If $2a + 6b > 2d^2 - 6d + 6$, then $\pi_1(\mathbf{P}^2 \setminus B)$ is non-abelian. Note that Nori's result implies that $\pi_1(\mathbf{P}^2 \setminus B)$ is abelian, provided that $2a + 6b < d^2$.
Residues of a Pfaff system relative to an invariant subscheme
F.
Sancho de Salas
4019-4035
Abstract: In this paper we give a purely algebraic construction of the theory of residues of a Pfaff system relative to an invariant subscheme. This construction is valid over an arbitrary base scheme of any characteristic.
Linear systems of plane curves with base points of equal multiplicity
Ciro
Ciliberto;
Rick
Miranda
4037-4050
Abstract: In this article we address the problem of computing the dimension of the space of plane curves of degree $d$with $n$ general points of multiplicity $m$. A conjecture of Harbourne and Hirschowitz implies that when $d \geq 3m$, the dimension is equal to the expected dimension given by the Riemann-Roch Theorem. Also, systems for which the dimension is larger than expected should have a fixed part containing a multiple $(-1)$-curve. We reformulate this conjecture by explicitly listing those systems which have unexpected dimension. Then we use a degeneration technique developed to show that the conjecture holds for all $m \leq 12$.
A condition for the stability of $\mathbb{R}$-covered on foliations of 3-manifolds
Sue
Goodman;
Sandi
Shields
4051-4065
Abstract: We give a sufficient condition for a codimension one, transversely orientable foliation of a closed 3-manifold to have the property that any foliation sufficiently close to it be $\mathbb{R}$-covered. This condition can be readily verified for many examples. Further, if an $\mathbb{R}$-covered foliation has a compact leaf $L$, then any transverse loop meeting $L$ lifts to a copy of the leaf space, and the ambient manifold fibers over $S^1$ with $L$ as fiber.
The Markov spectra for Fuchsian groups
L.
Ya.
Vulakh
4067-4094
Abstract: Applying the Klein model $D^2$ of the hyperbolic plane and identifying the geodesics in $D^2$ with their poles in the projective plane, the author develops a method of determining infinite binary trees in the Markov spectrum for a Fuchsian group. The method is applied to a maximal group commensurable with the modular group and other Fuchsian groups.
A regular space with a countable network and different dimensions
George
Delistathis;
Stephen
Watson
4095-4111
Abstract: In this paper, we construct a regular space with a countable network (even the union of countably many separable metric subspaces) in which $ind$ and $dim$ do not coincide under the assumption of the continuum hypothesis (CH). This gives a consistent negative answer to a question of A.V. Arhangel'skii.
On the homotopy of simplicial algebras over an operad
Benoit
Fresse
4113-4141
Abstract: According to a result of H. Cartan, the homotopy of a simplicial commutative algebra is equipped with divided power operations. In this article, we show how to extend this result to other kinds of algebras. For instance, we prove that the homotopy of a simplicial Lie algebra is equipped with the structure of a restricted Lie algebra.
$p$-central groups and Poincaré duality
Thomas
S.
Weigel
4143-4154
Abstract: In this note we investigate the mod $p$ cohomology ring of finite $p$-central groups with a certain extension property. For $p$ odd it turns out that the structure of the cohomology ring characterizes this class of groups up to extensions by $p'$-groups. For certain examples the cohomology ring can be calculated explicitly. As a by-product one gets an alternative proof of a theorem of M.Lazard which states that the Galois cohomology of a uniformly powerful pro-$p$-group of rank $n$ is isomorphic to $\Lambda [x_{1},..,x_{n}]$.
Representing nonnegative homology classes of $\mathbb{C}P^2\#n\overline{\mathbb{C}P}{}^2$ by minimal genus smooth embeddings
Bang-He
Li
4155-4169
Abstract: For any nonnegative class $\xi$ in $H_2({\mathbb C}P^2\#n{\overline{{\mathbb C}P}}{}^2, {\mathbf Z})$, the minimal genus of smoothly embedded surfaces which represent $\xi$ is given for $n\leq 9$, and in some cases with $n\geq 10$, the minimal genus is also given. For the finiteness of orbits under diffeomorphisms with minimal genus $g$, we prove that it is true for $n\leq 8$ with $g\geq 1$ and for $n\leq 9$ with $g\geq 2$.
The Conley index over a base
Marian
Mrozek;
James
F.
Reineck;
Roman
Srzednicki
4171-4194
Abstract: We construct a generalization of the Conley index for flows. The new index preserves information which in the classical case is lost in the process of collapsing the exit set to a point. The new index has most of the properties of the classical index. As examples, we study a flow with a knotted orbit in ${R}^3$, and the problem of continuing two periodic orbits which are not homotopic as loops.
Resonance problems with respect to the Fucík spectrum
Martin
Schechter
4195-4205
Abstract: We study semilinear boundary value problems which have asymptotic resonance with respect to the linear part. The difficulties for Fucík resonance problems are compounded by the fact that there is no eigenspace with which to work. The present paper uses new linking theorems which can deal with the sets required to obtain critical points.
Robin boundary value problems on arbitrary domains
Daniel
Daners
4207-4236
Abstract: We develop a theory of generalised solutions for elliptic boundary value problems subject to Robin boundary conditions on arbitrary domains, which resembles in many ways that of the Dirichlet problem. In particular, we establish $L_p$-$L_q$-estimates which turn out to be the best possible in that framework. We also discuss consequences to the spectrum of Robin boundary value problems. Finally, we apply the theory to parabolic equations.
A Palais-Smale approach to problems in Esteban-Lions domains with holes
Hwai-Chiuan
F.
Wang
4237-4256
Abstract: Let $\Omega \subset {\mathbb{R} }^{N}$ be the upper half strip with a hole. In this paper, we show there exists a positive higher energy solution of semilinear elliptic equations in $\Omega$ and describe the dynamic systems of solutions of equation $(1)$ in various $\Omega$. We also show there exist at least two positive solutions of perturbed semilinear elliptic equations in $\Omega$.
A sampling theorem on homogeneous manifolds
Isaac
Pesenson
4257-4269
Abstract: We consider a generalization of entire functions of spherical exponential type and Lagrangian splines on manifolds. An analog of the Paley-Wiener theorem is given. We also show that every spectral entire function on a manifold is uniquely determined by its values on some discrete sets of points. The main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using Lagrangian splines.
Estimates for functions of the Laplace operator on homogeneous trees
Michael
Cowling;
Stefano
Meda;
Alberto
G.
Setti
4271-4293
Abstract: In this paper, we study the heat equation on a homogeneous graph, relative to the natural (nearest-neighbour) Laplacian. We find pointwise estimates for the heat and resolvent kernels, and the $L^{p}-L^{q}$ mapping properties of the corresponding operators.
The Noetherian property in some quadratic algebras
Xenia
H.
Kramer
4295-4323
Abstract: We introduce a new class of noncommutative rings called pseudopolynomial rings and give sufficient conditions for such a ring to be Noetherian. Pseudopolynomial rings are standard finitely presented algebras over a field with some additional restrictions on their defining relations--namely that the polynomials in a Gröbner basis for the ideal of relations must be homogeneous of degree 2--and on the Ufnarovskii graph $\Gamma (A)$. The class of pseudopolynomial rings properly includes the generalized skew polynomial rings introduced by M. Artin and W. Schelter. We use the graph $\Gamma (A)$ to define a weaker notion of almost commutative, which we call almost commutative on cycles. We show as our main result that a pseudopolynomial ring which is almost commutative on cycles is Noetherian. A counterexample shows that a Noetherian pseudopolynomial ring need not be almost commutative on cycles.
The $q$-Schur${}^{2}$ algebra
Jie
Du;
Leonard
Scott
4325-4353
Abstract: We study a class of endomomorphism algebras of certain $q$-permutation modules over the Hecke algebra of type $B$, whose summands involve both parabolic and quasi-parabolic subgroups, and prove that these algebras are integrally free and quasi-hereditary, and are stable under base change. Some consequences for decomposition numbers are discussed.
Specializations of Brauer classes over algebraic function fields
Burton
Fein;
Murray
Schacher
4355-4369
Abstract: Let $F$ be either a number field or a field finitely generated of transcendence degree $\ge 1$ over a Hilbertian field of characteristic 0, let $F(t)$ be the rational function field in one variable over $F$, and let ${\alpha }\in \operatorname {Br}(F(t))$. It is known that there exist infinitely many $a\in F$ such that the specialization $t\to a$ induces a specialization ${\alpha }\to \overline {{\alpha }}\in \operatorname {Br}(F)$, where $\overline {{\alpha }}$ has exponent equal to that of ${\alpha }$. Now let $K$ be a finite extension of $F(t)$ and let ${\beta }=\operatorname {res}_{K/F(t)}({\alpha })$. We give sufficient conditions on ${\alpha }$ and $K$ for there to exist infinitely many $a\in F$ such that the specialization $t\to a$has an extension to $K$ inducing a specialization ${\beta }\to \overline {{\beta }}\in \operatorname {Br}(\overline{K})$, $\overline{K}$ the residue field of $K$, where $\overline {{\beta }}$ has exponent equal to that of ${\beta }$. We also give examples to show that, in general, such $a\in F$ need not exist.
Conformally invariant Monge-Ampère equations: Global solutions
Jeff
A.
Viaclovsky
4371-4379
Abstract: In this paper we will examine a class of fully nonlinear partial differential equations which are invariant under the conformal group $SO(n+1,1)$. These equations are elliptic and variational. Using this structure and the conformal invariance, we will prove a global uniqueness theorem for solutions in $\mathbf{R}^n$ with a quadratic growth condition at infinity.
Geometric properties of the sections of solutions to the Monge-Ampère equation
Cristian
E.
Gutiérrez;
Qingbo
Huang
4381-4396
Abstract: In this paper we establish several geometric properties of the cross sections of generalized solutions $\phi$ to the Monge-Ampère equation $\det D^{2}\phi = \mu$, when the measure $\mu$ satisfies a doubling property. A main result is a characterization of the doubling measures $\mu$in terms of a geometric property of the cross sections of $\phi$. This is used to obtain estimates of the shape and invariance properties of the cross sections that are valid under appropriate normalizations.
Jones index theory by Hilbert C{${}^{*}$}-bimodules and K-theory
Tsuyoshi
Kajiwara;
Yasuo
Watatani
3429-3472
Abstract: In this paper we introduce the notion of Hilbert ${\mathrm{C}}^{*}$-bimodules, replacing the associativity condition of two-sided inner products in Rieffel's imprimitivity bimodules by a Pimsner-Popa type inequality. We prove Schur's Lemma and Frobenius reciprocity in this setting. We define minimality of Hilbert ${\mathrm{C}}^{*}$-bimodules and show that tensor products of minimal bimodules are also minimal. For an $A$-$A$ bimodule which is compatible with a trace on a unital ${\mathrm{C}}^{*}$-algebra $A$, its dimension (square root of Jones index) depends only on its $KK$-class. Finally, we show that the dimension map transforms the Kasparov products in $KK(A,A)$ to the product of positive real numbers, and determine the subring of $KK(A,A)$ generated by the Hilbert ${\mathrm{C}}^{*}$-bimodules for a ${\mathrm{C}}^{*}$-algebra generated by Jones projections.
Morse theory for the Yang-Mills functional via equivariant homotopy theory
Ursula
Gritsch
3473-3493
Abstract: In this paper we show the existence of non-minimal critical points of the Yang-Mills functional over a certain family of 4-manifolds $\{ M_{2g} : g=0,1,2, \ldots \}$ with generic $SU(2)$-invariant metrics using Morse and homotopy theoretic methods. These manifolds are acted on fixed point freely by the Lie group $SU(2)$ with quotient a compact Riemann surface of even genus. We use a version of invariant Morse theory for the Yang-Mills functional used by Parker in A Morse theory for equivariant Yang-Mills, Duke Math. J. 66-2 (1992), 337-356 and Råde in Compactness theorems for invariant connections, submitted for publication.
A wall-crossing formula for the signature of symplectic quotients
David
S.
Metzler
3495-3521
Abstract: We use symplectic cobordism, and the localization result of Ginzburg, Guillemin, and Karshon to find a wall-crossing formula for the signature of regular symplectic quotients of Hamiltonian torus actions. The formula is recursive, depending ultimately on fixed point data. In the case of a circle action, we obtain a formula for the signature of singular quotients as well. We also show how formulas for the Poincaré polynomial and the Euler characteristic (equivalent to those of Kirwan can be expressed in the same recursive manner.
On cohomology algebras of complex subspace arrangements
Eva
Maria
Feichtner;
Günter
M.
Ziegler
3523-3555
Abstract: The integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement. We give a combinatorial presentation of the cohomology algebra in the spirit of the Orlik-Solomon result on the cohomology algebras of complex hyperplane arrangements. Our methods are elementary: we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space. We describe simplicial cochains that generate the cohomology. Among them we distinguish a linear basis, study cup product multiplication, and derive an algebra presentation in terms of generators and relations.
Algebraic gamma monomials and double coverings of cyclotomic fields
Pinaki
Das
3557-3594
Abstract: We investigate the properties of algebraic gamma monomials--that is, algebraic numbers which are expressible as monomials in special values of the classical gamma function. Recently Anderson has constructed a double complex ${\mathbb{SK} }$, to compute $H^*(\pm, {\mathbb{U} })$, where ${\mathbb{U} }$ is the universal ordinary distribution. We use the double complex to deduce explicit formulae for algebraic gamma monomials. We provide simple proofs of some previously known results of Deligne on algebraic gamma monomials. Deligne used the theory of Hodge cycles for his results. By contrast, our proofs are constructive and relatively elementary. Given a Galois extension $K/F$, we define a double covering of $K/F$ to be an extension $\tilde{K}/K$ of degree $\leq 2$, such that ${\tilde{K}}/F$ is Galois. We demonstrate that each class ${\mathbf{a}}\in H^2(\pm, {\mathbb{U} })$ gives rise to a double covering of ${\mathbb{Q} }(\zeta_ \infty)/{\mathbb{Q} }$, by ${\mathbb{Q} }(\zeta_ \infty,\sqrt{\sin{\mathbf{a}}})/{\mathbb{Q} }(\zeta_ \infty)$. When ${\mathbf{a}}$ lifts a canonical basis element indexed by two odd primes, we show that this double covering can be non-abelian. However, if ${\mathbf{a}}$ represents any of the canonical basis classes indexed by an odd squarefree positive integer divisible by at least four primes, then the Galois group of ${\mathbb{Q} }(\zeta_ \infty,\sqrt{\sin{\mathbf{a}}})/{\mathbb{Q} }$ is abelian and hence $\sqrt{\sin{\mathbf{a}}} \in {\mathbb{Q} }(\zeta_ \infty)$. The $\sqrt{\sin{\mathbf{a}}}$ may very well be a new supply of abelian units. The relevance of these units to the unit index formula for cyclotomic fields calls for further investigations.
Counting solutions to trinomial Thue equations: a different approach
Emery
Thomas
3595-3622
Abstract: We consider the problem of counting solutions to a trinomial Thue equation -- that is, an equation \begin{equation*}\vert F(x,y)\vert = 1,\tag{$*$} \end{equation*} where $F$ is an irreducible form in $Z[x,y]$ with degree at least three and with three non-zero coefficients. In a 1987 paper J. Mueller and W. Schmidt gave effective bounds for this problem. Their work was based on a series of papers by Bombieri, Bombieri-Mueller and Bombieri-Schmidt, all concerned with the ``Thue-Siegel principle" and its relation to $(*)$. In this paper we give specific numerical bounds for the number of solutions to $(*)$ by a somewhat different approach, the difference lying in the initial step -- solving a certain diophantine approximation problem. We regard this as a real variable extremal problem, which we then solve by elementary calculus.
Geometric flow and rigidity on symmetric spaces of noncompact type
Inkang
Kim
3623-3638
Abstract: In this paper we show that, under a suitable condition, every nonsingular geometric flow on a manifold which is modeled on the Furstenberg boundary of $X$, where $X$ is a symmetric space of non-compact type, induces a torus action, and, in particular, if the manifold is a rational homology sphere, then the flow has a closed orbit.
The geography problem for irreducible spin four-manifolds
B. Doug
Park;
Zoltán
Szabó
3639-3650
Abstract: We study the geography problem for smooth irreducible simply-connected spin four-manifolds. For a large class of homotopy types, we exhibit both symplectic and non-symplectic representatives. We also compute the Seiberg-Witten invariants of all the four-manifolds we construct.
Multiscale decompositions on bounded domains
A.
Cohen;
W.
Dahmen;
R.
DeVore
3651-3685
Abstract: A construction of multiscale decompositions relative to domains $\Omega\subset \mathbb{R} ^d$ is given. Multiscale spaces are constructed on $\Omega$ which retain the important features of univariate multiresolution analysis including local polynomial reproduction and locally supported, stable bases.
Partial differential equations with matricial coefficients and generalized translation operators
N.
H.
Mahmoud
3687-3706
Abstract: Let $\Delta_{\alpha }$ be the Bessel operator with matricial coefficients defined on $(0,\infty )$ by \begin{equation*}\Delta_{\alpha }U(t)=U''(t)+\frac{2\alpha +I}{t}U'(t)\end{equation*} where $\alpha$ is a diagonal matrix and let $q$ be an $n\times n$ matrix-valued function. In this work, we prove that there exists an isomorphism $X$ on the space of even ${\mathcal C}^{\infty}$, $\mathbb{C} ^n$-valued functions which transmutes $\Delta_{\alpha}$and $(\Delta_{\alpha}+q)$. This allows us to define generalized translation operators and to develop harmonic analysis associated with $(\Delta_{\alpha}+q)$. By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.
Weak and Strong Density of Compositions
Luigi
De Pascale;
Eugene
Stepanov
3707-3721
Abstract: The convergence in various topologies of sequences of inner superposition (composition) operators acting between Lebesgue spaces and of their linear combinations is studied. In particular, the sequential density results for the linear span of such operators is proved for the weak, weak continuous and strong operator topologies.
Semiclassical analysis of general second order elliptic operators on bounded domains
E.
N.
Dancer;
J.
López-Gómez
3723-3742
Abstract: In this work we ascertain the semiclassical behavior of the fundamental energy and the ground state of an arbitrary second order elliptic operator, not necessarily selfadjoint, on a bounded domain. Our analysis provides us with substantial improvements of many previous results found in the context of quantum mechanics for $C^\infty$ perturbations of the Laplacian.
The problem of lacunas and analysis on root systems
Yuri
Berest
3743-3776
Abstract: A lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens' principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, Atiyah, Bott and Gårding (1970-73) developed a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. In the present paper we study this problem for one remarkable class of partial differential operators with singular coefficients. These operators stem from the theory of special functions in several variables related to finite root systems (Coxeter groups). The underlying algebraic structure makes it possible to extend many results of the Atiyah-Bott-Gårding theory. We give a generalization of the classical Herglotz-Petrovsky-Leray formulas expressing the fundamental solution in terms of Abelian integrals over properly constructed cycles in complex projective space. Such a representation allows us to employ the Petrovsky topological condition for testing regular (strong) lacunas for the operators under consideration. Some illustrative examples are constructed. A relation between the theory of lacunas and the problem of classification of commutative rings of partial differential operators is discussed.
Mean convergence of orthogonal Fourier series of modified functions
Martin
G.
Grigorian;
Kazaros
S.
Kazarian;
Fernando
Soria
3777-3798
Abstract: We construct orthonormal systems (ONS) which are uniformly bounded, complete, and made up of continuous functions such that some continuous and even some arbitrarily smooth functions cannot be modified so that the Fourier series of the new function converges in the $L^{p}$-metric for any $p > 2.$ We prove also that if $\Phi$ is a uniformly bounded ONS which is complete in all the spaces $L _ {[0,1]} ^{p} , 1 \leq p < \infty$, then there exists a rearrangement $\sigma$ of the natural numbers $\mathbf{N}$such that the system $\Phi _{\sigma }= \{ \phi _{\sigma (n)} \}_{n=1}^{\infty }$ has the strong $L^{p}$-property for all $p>2$; that is, for every $2 \leq p < \infty$ and for every $f \in L _ {[0,1]} ^{p}$ and $\epsilon > 0$there exists a function $f_ \epsilon \in L _ {[0,1]} ^{p}$ which coincides with $f$ except on a set of measure less than $\epsilon$ and whose Fourier series with respect to the system $\Phi _{\sigma }$ converges in $L _ {[0,1]} ^{p} .$
Scattering matrices for the quantum $N$ body problem
Andrew
Hassell
3799-3820
Abstract: Let $H$ be a generalized $N$ body Schrödinger operator with very short range potentials. Using Melrose's scattering calculus, it is shown that the free channel `geometric' scattering matrix, defined via asymptotic expansions of generalized eigenfunctions of $H$, coincides (up to normalization) with the free channel `analytic' scattering matrix defined via wave operators. Along the way, it is shown that the free channel generalized eigenfunctions of Herbst-Skibsted and Jensen-Kitada coincide with the plane waves constructed by Hassell and Vasy and if the potentials are very short range.
Hopf algebras of types $U_q(sl_n)'$ and $O_q(SL_n)'$ which give rise to certain invariants of knots, links and 3-manifolds
Shlomo
Gelaki;
Sara
Westreich
3821-3836
Abstract: In this paper we determine when Lusztig's
Extensions of Hopf Algebras and Lie Bialgebras
Akira
Masuoka
3837-3879
Abstract: Let $\mathfrak{f}$, $\mathfrak{g}$ be finite-dimensional Lie algebras over a field of characteristic zero. Regard $\mathfrak{f}$ and $\mathfrak{g} ^*$, the dual Lie coalgebra of $\mathfrak{g}$, as Lie bialgebras with zero cobracket and zero bracket, respectively. Suppose that a matched pair $(\mathfrak{f} , \mathfrak{g} ^*)$of Lie bialgebras is given, which has structure maps $\rightharpoonup , \rho$. Then it induces a matched pair $(U\mathfrak{f}, U\mathfrak{g}^{\circ},\rightharpoonup ', \rho ')$ of Hopf algebras, where $U\mathfrak{f}$ is the universal envelope of $\mathfrak{f}$ and $U\mathfrak{g}^{\circ}$ is the Hopf dual of $U\mathfrak{g}$. We show that the group $\mathrm{Opext} (U\mathfrak{f},U\mathfrak{g}^{\circ })$of cleft Hopf algebra extensions associated with $(U\mathfrak{f}, U\mathfrak{g} ^{\circ}, \rightharpoonup ', \rho ' )$ is naturally isomorphic to the group $\operatorname{Opext}(\mathfrak{f},\mathfrak{g} ^*)$of Lie bialgebra extensions associated with $(\mathfrak{f}, \mathfrak{g}^*, \rightharpoonup , \rho )$. An exact sequence involving either of these groups is obtained, which is a variation of the exact sequence due to G.I. Kac. If $\mathfrak{g} =[\mathfrak{g} , \mathfrak{g}]$, there follows a bijection between the set $\mathrm{Ext}(U\mathfrak{f} , U\mathfrak{g}^{\circ })$of all cleft Hopf algebra extensions of $U\mathfrak{f}$ by $U\mathfrak{g}^{\circ }$ and the set $\mathrm{Ext}(\mathfrak{f}, \mathfrak{g}^*)$ of all Lie bialgebra extensions of $\mathfrak{f}$ by $\mathfrak{g} ^*$.
Galois embeddings for linear groups
Shreeram
S.
Abhyankar
3881-3912
Abstract: A criterion is given for the solvability of a central Galois embedding problem to go from a projective linear group covering to a vectorial linear group covering.
Heteroclinic cycles for reaction diffusion systems by forced symmetry-breaking
Stanislaus
Maier-Paape;
Reiner
Lauterbach
2937-2991
Abstract: Recently it has been observed, that perturbations of symmetric ODE's can lead to highly nontrivial dynamics. In this paper we want to establish a similar result for certain nonlinear partial differential systems. Our results are applied to equations which are motivated from chemical reactions. In fact we show that the theory applies to the Brusselator on a sphere. To be more precise, we consider solutions of a semi-linear parabolic equation on the 2-sphere. When this equation has an axisymmetric equilibrium $u_\alpha$, the group orbit of $u_\alpha$ (under rotations) gives a whole (invariant) manifold $M$ of equilibria. Under generic conditions we have that, after perturbing our equation by a (small) $L\subset {{{\bf O}(3)}}$-equivariant perturbation, $M$ persists as an invariant manifold $\widetilde M$. However, the flow on $\widetilde M$ is in general no longer trivial. Indeed, we find slow dynamics on $\widetilde M$ and, in the case $L=\mathbb{T}$ (the tetrahedral subgroup of ${{{\bf O}(3)}}$), we observe heteroclinic cycles. In the application to chemical systems we would expect intermittent behaviour. However, for the Brusselator equations this phenomenon is not stable. In order to see it in a physically relevant situation we need to introduce further terms to get a higher codimension bifurcation.
An electromagnetic inverse problem in chiral media
Stephen
R.
McDowall
2993-3013
Abstract: We consider the inverse boundary value problem for Maxwell's equations that takes into account the chirality of a body in ${\mathbb R}^3$. More precisely, we show that knowledge of a boundary map for the electromagnetic fields determines the electromagnetic parameters, namely the conductivity, electric permittivity, magnetic permeability and chirality, in the interior. We rewrite Maxwell's equations as a first order perturbation of the Laplacian and construct exponentially growing solutions, and obtain the result in the spirit of complex geometrical optics.
A criterion for reduction of variables in the Willmore-Chen variational problem and its applications
Manuel
Barros;
Angel
Ferrández;
Pascual
Lucas;
Miguel
A.
Meroño
3015-3027
Abstract: We exhibit a criterion for a reduction of variables for Willmore-Chen submanifolds in conformal classes associated with generalized Kaluza-Klein metrics on flat principal fibre bundles. Our method relates the variational problem of Willmore-Chen with an elasticity functional defined for closed curves in the base space. The main ideas involve the extrinsic conformal invariance of the Willmore-Chen functional, the large symmetry group of generalized Kaluza-Klein metrics and the principle of symmetric criticality. We also obtain interesting families of elasticae in both lens spaces and surfaces of revolution (Riemannian and Lorentzian). We give applications to the construction of explicit examples of isolated Willmore-Chen submanifolds, discrete families of Willmore-Chen submanifolds and foliations whose leaves are Willmore-Chen submanifolds.
Inflection points and topology of surfaces in 4-space
Ronaldo
Alves
Garcia;
Dirce
Kiyomi Hayashida
Mochida;
Maria
del Carmen
Romero Fuster;
Maria
Aparecida
Soares Ruas
3029-3043
Abstract: We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincaré-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.
Projective manifolds with small pluridegrees
Mauro
C.
Beltrametti;
Andrew
J.
Sommese
3045-3064
Abstract: Let $\mathcal{L}$ be a very ample line bundle on a connected complex projective manifold $\mathcal{M}$ of dimension $n\ge 3$. Except for a short list of degenerate pairs $(\mathcal{M},\mathcal{L})$, $\kappa(K_\mathcal{M}+(n-2)\mathcal{L})=n$ and there exists a morphism $\pi: \mathcal{M} \to M$ expressing $\mathcal{M}$ as the blowup of a projective manifold $M$ at a finite set $B$, with $\mathcal{K}_M:=K_M+(n-2)L$ nef and big for the ample line bundle $L:= (\pi _*\mathcal{L})^{**}$. The projective geometry of $(\mathcal{M},\mathcal{L})$ is largely controlled by the pluridegrees $d_j:=L^{n-j}\cdot (K_M+(n-2)L)^j$ for $j=0,\ldots,n$, of $(\mathcal{M},\mathcal{L})$. For example, $d_0+d_1=2g-2$, where $g$ is the genus of a curve section of $(\mathcal{M},\mathcal{L})$, and $d_2$ is equal to the self-intersection of the canonical divisor of the minimal model of a surface section of $(\mathcal{M},\mathcal{L})$. In this article, a detailed analysis is made of the pluridegrees of $(\mathcal{M},\mathcal{L})$. The restrictions found are used to give a new lower bound for the dimension of the space of sections of $\mathcal{K}_M$. The inequalities for the pluridegrees, that are presented in this article, will be used in a sequel to study the sheet number of the morphism associated to $|2(K_\mathcal{M}+ (n-2)\mathcal{L})|$.
A finitely axiomatizable undecidable equational theory with recursively solvable word problems
Dejan
Delic
3065-3101
Abstract: In this paper we construct a finitely based variety, whose equational theory is undecidable, yet whose word problems are recursively solvable, which solves a problem stated by G. McNulty (1992). The construction produces a discriminator variety with the aforementioned properties starting from a class of structures in some multisorted language (which may include relations), axiomatized by a finite set of universal sentences in the given multisorted signature. This result also presents a common generalization of the earlier results obtained by B. Wells (1982) and A. Mekler, E. Nelson, and S. Shelah (1993).
Adams operations, localized Chern characters, and the positivity of Dutta multiplicity in characteristic $0$
Kazuhiko
Kurano;
Paul
C.
Roberts
3103-3116
Abstract: The positivity of the Dutta multiplicity of a perfect complex of $A$-modules of length equal to the dimension of $A$ and with homology of finite length is proven for homomorphic images of regular local rings containing a field of characteristic zero. The proof uses relations between localized Chern characters and Adams operations.
Fundamental groups of moduli and the Grothendieck-Teichmüller group
David
Harbater;
Leila
Schneps
3117-3148
Abstract: Let ${\mathcal{M}}_{0,n}$ denote the moduli space of Riemann spheres with $n$ ordered marked points. In this article we define the group $\operatorname{Out}^{\sharp }_{n}$ of quasi-special symmetric outer automorphisms of the algebraic fundamental group $\widehat \pi _{1}({\mathcal{M}}_{0,n})$ for all $n\ge 4$ to be the group of outer automorphisms respecting the conjugacy classes of the inertia subgroups of $\widehat \pi _{1}({\mathcal{M}}_{0,n})$ and commuting with the group of outer automorphisms of $\widehat \pi _{1}({\mathcal{M}}_{0,n})$ obtained by permuting the marked points. Our main result states that $\operatorname{Out}^{\sharp }_{n}$ is isomorphic to the Grothendieck-Teichmüller group $\widehat {\operatorname{GT}}$for all $n\ge 5$.
Universal Formulae for SU$(n)$ Casson Invariants of Knots
Hans
U.
Boden;
Andrew
Nicas
3149-3187
Abstract: An $\operatorname{SU}(n)$ Casson invariant of a knot is an integer which can be thought of as an algebraic-topological count of the number of characters of $\operatorname{SU}(n)$ representations of the knot group which take a longitude into a given conjugacy class. For fibered knots, these invariants can be characterized as Lefschetz numbers which, for generic conjugacy classes, can be computed using a recursive algorithm of Atiyah and Bott, as adapted by Frohman. Using a new idea to solve the Atiyah-Bott recursion (as simplified by Zagier), we derive universal formulae which explicitly compute the invariants for all $n$. Our technique is based on our discovery that the generating functions associated to the relevant Lefschetz numbers (and polynomials) satisfy certain integral equations.
$K$-theory of projective Stiefel manifolds
Nelza
E.
Barufatti;
Derek
Hacon
3189-3209
Abstract: Using the Hodgkin spectral sequence we calculate $K^{*}(X_{m,k})$, the complex $K$-theory of the projective Stiefel manifold $X_{m,k}$, for $mk$even. For $mk$ odd, we are only able to calculate $K^{0}(X_{m,k})$, but this is sufficient to determine the order of the complexified Hopf bundle over $X_{m,k}$.
A filtration of spectra arising from families of subgroups of symmetric groups
Kathryn
Lesh
3211-3237
Abstract: Let ${\mathcal F}_{n}$ be a family of subgroups of $\Sigma_{n}$ which is closed under taking subgroups and conjugates. Such a family has a classifying space, $B{\mathcal F}_{n}$, and we showed in an earlier paper that a compatible choice of ${\mathcal F}_{n}$ for each $n$ gives a simplicial monoid $\coprod_{n} B{\mathcal F}_{n}$, which group completes to an infinite loop space. In this paper we define a filtration of the associated spectrum whose filtration quotients, given an extra condition on the families, can be identified in terms of the classifying spaces of the families of subgroups that were chosen. This gives a way to go from group theoretic data about the families to homotopy theoretic information about the associated spectrum. We calculate two examples. The first is related to elementary abelian $p$-groups, and the second gives a new expression for the desuspension of $Sp^{m}(S^{0})/Sp^{m-1}(S^{0})$ as a suspension spectrum.
Steiner type formulae and weighted measures of singularities for semi-convex functions
Andrea
Colesanti;
Daniel
Hug
3239-3263
Abstract: For a given convex (semi-convex) function $u$, defined on a nonempty open convex set $\Omega\subset\mathbf{R}^n$, we establish a local Steiner type formula, the coefficients of which are nonnegative (signed) Borel measures. We also determine explicit integral representations for these coefficient measures, which are similar to the integral representations for the curvature measures of convex bodies (and, more generally, of sets with positive reach). We prove that, for $r\in \{0,\ldots,n\}$, the $r$-th coefficient measure of the local Steiner formula for $u$, restricted to the set of $r$-singular points of $u$, is absolutely continuous with respect to the $r$-dimensional Hausdorff measure, and that its density is the $(n-r)$-dimensional Hausdorff measure of the subgradient of $u$. As an application, under the assumptions that $u$ is convex and Lipschitz, and $\Omega$ is bounded, we get sharp estimates for certain weighted Hausdorff measures of the sets of $r$-singular points of $u$. Such estimates depend on the Lipschitz constant of $u$ and on the quermassintegrals of the topological closure of $\Omega$.
Sharp weighted inequalities for the vector-valued maximal function
Carlos
Pérez
3265-3288
Abstract: We prove in this paper some sharp weighted inequalities for the vector-valued maximal function $\overline M_q$ of Fefferman and Stein defined by \begin{displaymath}\overline M_qf(x)=\left(\sum _{i=1}^{\infty}(Mf_i(x))^{q}\right)^{1/q},\end{displaymath} where $M$ is the Hardy-Littlewood maximal function. As a consequence we derive the main result establishing that in the range $1<q<p<\infty$ there exists a constant $C$ such that \begin{displaymath}\int _{\mathbf{R}^{n}}\overline M_qf(x)^p\, w(x)dx\le C\, \int _{\mathbf{R}^n}|f(x)|^{p}_{q}\, M^{[\frac pq]+1}w(x) dx.\end{displaymath} Furthermore the result is sharp since $M^{[\frac pq]+1}$ cannot be replaced by $M^{[\frac pq]}$. We also show the following endpoint estimate \begin{displaymath}w(\{x\in \mathbf{R}^n:\overline M_qf(x)>\lambda\})\,\le \frac C\lambda \int _{\mathbf{R}^n} |f(x)|_q\, Mw(x)dx,\end{displaymath} where $C$ is a constant independent of $\lambda$.
Absolutely continuous S.R.B. measures for random Lasota-Yorke maps
Jérôme
Buzzi
3289-3303
Abstract: A. Lasota and J. A. Yorke proved that a piecewise expanding interval map admits finitely many ergodic absolutely continuous invariant probability measures. We generalize this to the random composition of such maps under conditions which are natural and less restrictive than those previously studied by Morita and Pelikan. For instance our conditions are satisfied in the case of arbitrary random $\beta$-transformations, i.e., $x\mapsto \beta x\mod 1$ on $[0,1]$ where $\beta$ is chosen according to any stationary stochastic process (in particular, not necessarily i.i.d.) with values in $]1,\infty [$. RSESUM´E. A. Lasota et J. A. Yorke ont montré qu'une application de l'intervalle dilatante par morceaux admet un nombre fini de mesures de probabilité invariantes et ergodiques absolument continues. Nous généralisons ce résultat à la composition aléatoire de telles applications sous des conditions naturelles, moins restrictives que celles précédemment envisagées par Morita et Pelikan. Par exemple, nos conditions sont satisfaites par toute $\beta$-transformation aléatoire, i.e., $x\mapsto \beta x\mod 1$ sur $[0,1]$ avec $\beta$ choisi selon un processus stochastique stationnaire quelconque (en particulier, non-nécessairement i.i.d.) à valeurs dans $]1,\infty [$.
Shift equivalence and the Conley index
John
Franks;
David
Richeson
3305-3322
Abstract: In this paper we introduce filtration pairs for an isolated invariant set of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Last, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.
Hyperbolic minimizing geodesics
Daniel
Offin
3323-3338
Abstract: We apply the intersection theory for Lagrangian submanifolds to obtain a Sturm type comparison theorem for linearized Hamiltonian flows. Applications to the theory of geodesics are considered, including a sufficient condition that arclength minimizing closed geodesics, for an $n$-dimensional Riemannian manifold, are hyperbolic under the geodesic flow. This partially answers a conjecture of G. D. Birkhoff.
The characters of the generalized Steinberg representations of finite general linear groups on the regular elliptic set
Allan
J.
Silberger;
Ernst-Wilhelm
Zink
3339-3356
Abstract: Let $k$ be a finite field, $k_{n}\vert k$ the degree $n$ extension of $k$, and $G=\operatorname{GL}_{n}(k)$ the general linear group with entries in $k$. This paper studies the ``generalized Steinberg" (GS) representations of $G$ and proves the equivalence of several different characterizations for this class of representations. As our main result we show that the union of the class of cuspidal and GS representations of $G$ is in natural one-one correspondence with the set of Galois orbits of characters of $k_{n}^{\times }$, the regular orbits of course corresponding to the cuspidal representations. Besides using Green's character formulas to define GS representations, we characterize GS representations by associating to them idempotents in certain commuting algebras corresponding to parabolic inductions and by showing that GS representations are the sole components of these induced representations which are ``generic" (have Whittaker vectors).
Trees and valuation rings
Hans
H.
Brungs;
Joachim
Gräter
3357-3379
Abstract: A subring $B$ of a division algebra $D$ is called a valuation ring of $D$ if $x\in B$ or $x^{-1}\in B$ holds for all nonzero $x$ in $D$. The set $\mathcal{B}$ of all valuation rings of $D$ is a partially ordered set with respect to inclusion, having $D$ as its maximal element. As a graph $\mathcal{B}$ is a rooted tree (called the valuation tree of $D$), and in contrast to the commutative case, $\mathcal{B}$ may have finitely many but more than one vertices. This paper is mainly concerned with the question of whether each finite, rooted tree can be realized as a valuation tree of a division algebra $D$, and one main result here is a positive answer to this question where $D$ can be chosen as a quaternion division algebra over a commutative field.
Hereditary crossed products
Jeremy
Haefner;
Gerald
Janusz
3381-3410
Abstract: We characterize when a crossed product order over a maximal order in a central simple algebra by a finite group is hereditary. We need only concentrate on the cases when the group acts as inner automorphisms and when the group acts as outer automorphisms. When the group acts as inner automorphisms, the classical group algebra result holds for crossed products as well; that is, the crossed product is hereditary if and only if the order of the group is a unit in the ring. When the group is acting as outer automorphisms, every crossed product order is hereditary, regardless of whether the order of the group is a unit in the ring.
A generalized Brauer construction and linear source modules
Robert
Boltje;
Burkhard
Külshammer
3411-3428
Abstract: For a complete discrete valuation ring $\mathcal{O}$ with residue field $F$, a subgroup $H$ of a finite group $G$ and a homomorphism $\varphi: H \to \mathcal{O}^\times$, we define a functor $V \mapsto \overline{\overline{V}} (H,\varphi)$ from the category of $\mathcal{O} G$-modules to the category of $FN_G(H,\varphi)$-modules and investigate its behaviour with respect to linear source modules.
A probabilistic approach to positive harmonic functions in a slab with alternating Dirichlet and Neumann boundary conditions
Ross
G.
Pinsky
2445-2477
Abstract: Let $\Omega = R^{d}\times (-1,1)$, $d\ge 2$, be a $d+1$ dimensional slab. Denote points $z\in R^{d+1}$ by $z=(r,\theta ,y)$, where $(r,\theta )\in [0,\infty )\times S^{d-1}$ and $y\in R$. Denoting the boundary of the slab by $\Gamma =\partial \Omega$, let \begin{displaymath}\Gamma _{D}=\{z=(r,\theta ,y)\in \Gamma : r\in \bigcup _{n=1}^{\infty }(a_{n},b_{n})\},\end{displaymath} where $\{(a_{n},b_{n})\}_{n=1}^{\infty }$is an ordered sequence of intervals on the right half line (that is, $a_{n+1}>b_{n}$). Assume that the lengths of the intervals are bounded and that the spaces between consecutive intervals are bounded and bounded away from zero. Let $\Gamma _{N}=\Gamma -\bar \Gamma _{D}$. Let $C_{B}(\Omega ;\Gamma _{D}, \Gamma _{N})$ and $C_{P}(\Omega ; \Gamma _{D}, \Gamma _{N})$denote respectively the cone of bounded, positive harmonic functions in $\Omega$ and the cone of positive harmonic functions in $\Omega$ which satisfy the Dirichlet boundary condition on $\Gamma _{D}$ and the Neumann boundary condition on $\Gamma _{N}$. Letting $\rho _{n}\equiv b_{n}-a_{n}$, the main result of this paper, under a modest assumption on the sequence $\{\rho _{n}\}$, may be summarized as follows when $d\ge 3$: 1. If $\sum _{n=1}^{\infty }\frac{n}{\vert\log \rho _{n}\vert} <\infty$, then $\mathcal C_B(\Omega,\Gamma_D,\Gamma _N)$ and $\mathcal C_P(\Omega ,\Gamma _D,\Gamma _N)$ are both one-dimensional (as in the case of the Neumann boundary condition on the entire boundary). In particular, this occurs if $\rho _{n}=\exp (-n^{l})$ with $l>2$. 2. If $\sum _{n=1}^{\infty }\frac{n}{\vert\log \rho _{n}\vert} =\infty$and $\sum _{n=1}^{\infty }\frac{\vert\log \rho _{n}\vert^{\frac{1}{2}}}{n^{2}}=\infty$, then $\mathcal C_B(\Omega ,\Gamma _D,\Gamma _N) =\varnothing$ and $\mathcal C_P(\Omega ,\Gamma _D,\Gamma _N)$ is one-dimensional. In particular, this occurs if $\rho _{n}=\exp (-n^{2})$. 3. If $\sum _{n=1}^{\infty }\frac{\vert\log \rho _{n}\vert^{\frac{1}{2}}}{n^{2}}<\infty$, then $\mathcal C_B(\Omega ,\Gamma _D,\Gamma _N)=\varnothing$ and the set of minimal elements generating $\mathcal C_P(\Omega ,\Gamma _D,\Gamma _N)$ is isomorphic to $S^{d-1}$ (as in the case of the Dirichlet boundary condition on the entire boundary). In particular, this occurs if $\rho _{n}=\exp (-n^{l})$with $0\le l<2$. When $d=2$, $\mathcal C_B(\Omega ,\Gamma _D,\Gamma _N)=\varnothing$ as soon as there is at least one interval of Dirichlet boundary condition. The dichotomy for $\mathcal C_P(\Omega ,\Gamma _D,\Gamma _N)$ is as above.
Avoidable algebraic subsets of Euclidean space
James
H.
Schmerl
2479-2489
Abstract: Fix an integer $n\ge 1$ and consider real $n$-dimensional $\mathbb{R}^n$. A partition of $\mathbb{R}^n$ avoids the polynomial $p(x_0,x_1,\dotsc,x_{k-1})\in\mathbb R[x_0,x_1,\dotsc,x_{k-1}]$, where each $x_i$ is an $n$-tuple of variables, if there is no set of the partition which contains distinct $a_0,a_1,\dotsc,a_{k-1}$ such that $p(a_0,a_1,\dotsc,a_{k-1})=0$. The polynomial is avoidable if some countable partition avoids it. The avoidable polynomials are studied here. The polynomial $\|x-y\|^2-\|y-z\|^2$ is an especially interesting example of an avoidable one. We find (1) a countable partition which avoids every avoidable polynomial over $Q$, and (2) a characterization of the avoidable polynomials. An important feature is that both the ``master'' partition in (1) and the characterization in (2) depend on the cardinality of $\mathbb R$.
On Better-Quasi-Ordering Countable Series-Parallel Orders
Stéphan
Thomassé
2491-2505
Abstract: We prove that any infinite sequence of countable series-parallel orders contains an increasing (with respect to embedding) infinite subsequence. This result generalizes Laver's and Corominas' theorems concerning better-quasi-order of the classes of countable chains and trees.
On reflection of stationary sets in $\mathcal{P}_\kappa\lambda$
Thomas
Jech;
Saharon
Shelah
2507-2515
Abstract: Let $\kappa$ be an inaccessible cardinal, and let $E_{0} = \{x \in \mathcal{P}_{\kappa }\kappa ^{+} : \text{cf} \; \lambda _{x} = \text{cf} \; \kappa _{x}\}$ and $E_{1} = \{x \in \mathcal{P}_{\kappa }\kappa ^{+} : \kappa _{x}$ is regular and $\lambda _{x} = \kappa _{x}^{+}\}$. It is consistent that the set $E_{1}$ is stationary and that every stationary subset of $E_{0}$ reflects at almost every $a \in E_{1}$.
On Macaulayfication of Noetherian schemes
Takesi
Kawasaki
2517-2552
Abstract: The Macaulayfication of a Noetherian scheme $X$ is a birational proper morphism from a Cohen-Macaulay scheme to $X$. In 1978 Faltings gave a Macaulayfication of a quasi-projective scheme if its non-Cohen-Macaulay locus is of dimension $0$ or $1$. In the present article, we construct a Macaulayfication of Noetherian schemes without any assumption on the non-Cohen-Macaulay locus. Of course, a desingularization is a Macaulayfication and, in 1964, Hironaka already gave a desingularization of an algebraic variety over a field of characteristic $0$. Our method, however, to construct a Macaulayfication is independent of the characteristic.
Theta line bundles and the determinant of the Hodge bundle
Alexis
Kouvidakis
2553-2568
Abstract: We give an expression of the determinant of the push forward of a symmetric line bundle on a complex abelian fibration, in terms of the pull back of the relative dualizing sheaf via the zero section.
On syzygies of abelian varieties
Elena
Rubei
2569-2579
Abstract: In this paper we prove the following result: Let $X$ be a complex torus and $M$ a normally generated line bundle on $X$; then, for every $p \geq 0$, the line bundle $M^{p+1}$ satisfies Property $N_{p}$ of Green-Lazarsfeld.
Partitions into Primes
Yifan
Yang
2581-2600
Abstract: We investigate the asymptotic behavior of the partition function $p_{\Lambda} (n)$ defined by $\sum ^{\infty }_{n=0}p_{\Lambda} (n)x^{n} =\prod ^{\infty }_{m=1}(1-x^{m})^{-\Lambda (m)}$, where $\Lambda (n)$ denotes the von Mangoldt function. Improving a result of Richmond, we show that $\log p_{\Lambda} (n)=2\sqrt {\zeta (2)n}+O(\sqrt n\exp \{-c(\log n) (\log _{2} n)^{-2/3}(\log _{3} n)^{-1/3}\})$, where $c$ is a positive constant and $\log _{k}$ denotes the $k$ times iterated logarithm. We also show that the error term can be improved to $O(n^{1/4})$ if and only if the Riemann Hypothesis holds.
On Shimura, Shintani and Eichler-Zagier correspondences
M.
Manickam;
B.
Ramakrishnan
2601-2617
Abstract: In this paper, we set up Shimura and Shintani correspondences between Jacobi forms and modular forms of integral weight for arbitrary level and character, and generalize the Eichler-Zagier isomorphism between Jacobi forms and modular forms of half-integral weight to higher levels. Using this together with the known results, we get a strong multiplicity 1 theorem in certain cases for both Jacobi cusp newforms and half-integral weight cusp newforms. As a consequence, we get, among other results, the explicit Waldspurger theorem.
Rigidity of Coxeter groups
Stratos
Prassidis;
Barry
Spieler
2619-2642
Abstract: Let $W$ be a Coxeter group acting properly discontinuously and cocompactly on manifolds $N$ and $M ({\partial}M = {\emptyset})$ such that the fixed point sets of finite subgroups are contractible. Let $f: (N, {\partial}N) \to (M{\times}D^k, M{\times}S^{k-1})$ be a $W$-homotopy equivalence which restricts to a $W$-homeomorphism on the boundary. Under an assumption on the three dimensional fixed point sets, we show that then $f$ is $W$-homotopic to a $W$-homeomorphism.
Homology decompositions for classifying spaces of compact Lie groups
Alexei
Strounine
2643-2657
Abstract: Let $p$ be a prime number and $G$ be a compact Lie group. A homology decomposition for the classifying space $BG$ is a way of building $BG$ up to mod $p$ homology as a homotopy colimit of classifying spaces of subgroups of $G$. In this paper we develop techniques for constructing such homology decompositions. Jackowski, McClure and Oliver (Homotopy classification of self-maps of BG via $G$-actions, Ann. of Math. 135 (1992), 183-270) construct a homology decomposition of $BG$ by classifying spaces of $p$-stubborn subgroups of $G$. Their decomposition is based on the existence of a finite-dimensional mod $p$ acyclic $G$-$CW$-complex with restricted set of orbit types. We apply our techniques to give a parallel proof of the $p$-stubborn decomposition of $BG$which does not use this geometric construction.
Cohomology of uniformly powerful $p$-groups
William
Browder;
Jonathan
Pakianathan
2659-2688
Abstract: In this paper we will study the cohomology of a family of $p$-groups associated to $\mathbb{F}_p$-Lie algebras. More precisely, we study a category $\mathbf{BGrp}$ of $p$-groups which will be equivalent to the category of $\mathbb{F}_p$-bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group $G$ in this category, its $\mathbb{F}_p$-cohomology is that of an elementary abelian $p$-group if and only if it is associated to a Lie algebra. We then proceed to study the exponent of $H^*(G ;\mathbb{Z})$ in the case that $G$ is associated to a Lie algebra $\mathfrak{L}$. To do this, we use the Bockstein spectral sequence and derive a formula that gives $B_2^*$ in terms of the Lie algebra cohomologies of $\mathfrak{L}$. We then expand some of these results to a wider category of $p$-groups. In particular, we calculate the cohomology of the $p$-groups $\Gamma _{n,k}$ which are defined to be the kernel of the mod $p$ reduction $GL_n(\mathbb{Z}/p^{k+1}\mathbb{Z}) \overset{mod}{\longrightarrow} GL_n(\mathbb{F}_p).$
Group actions and group extensions
Ergün
Yalçin
2689-2700
Abstract: In this paper we study finite group extensions represented by special cohomology classes. As an application, we obtain some restrictions on finite groups which can act freely on a product of spheres or on a product of real projective spaces. In particular, we prove that if $(Z/p)^r$ acts freely on $(S^1)^k$, then $r \leq k$.
Problème de Dirichlet pour une équation de Monge-Ampère réelle elliptique dégénérée en dimension $n$
Amel
Atallah
2701-2721
Abstract: RÉSUMÉ. On considère dans un ouvert borné $\Omega$ de $\mathbb{R}^n$, à bord régulier, le problème de Dirichlet \begin{equation*}\left\{ \begin{split} & \det u_{ij}=f(x)\text{ dans }\Omega, & u\vert _{\partial \Omega}=\varphi, \end{split}\right.\tag{1} \end{equation*} où $f\in C^{s_*}(\overline\Omega), \varphi\in C^{s_*+2,\alpha}(\Omega)$, $f$est positive et s'annule sur $\Sigma$ un ensemble fini de points de $\Omega$. On démontre alors sous certaines hypothèses sur $\varphi$ et si $\vert\det \varphi_{ij}-f\vert _{C^{s_*}}$ est assez petit, que le problème (1) possède une solution convexe unique $u\in C^{[s_*-3-n/2]}(\overline\Omega)$. ABSTRACT. We consider in a bounded open set $\Omega$ of $\mathbb{R}^n$, with regular boundary, the Dirichlet problem \begin{equation*}\left\{ \begin{split} & \det u_{ij}=f(x)\text{ in }\Omega, & u\vert _{\partial \Omega}=\varphi, \end{split}\right.\tag{1} \end{equation*} where $f\in C^{s_*}(\overline\Omega), \varphi\in C^{s_*+2,\alpha}(\Omega)$, $f$is positive and vanishes on $\Sigma$, a finite set of points in $\Omega$. We prove, under some hypothesis on $\varphi$ and if $\vert\det \varphi_{ij}-f\vert _{C^{s_*}}$ is sufficiently small, that the problem (1) has a unique convex solution $u\in C^{[s_*-3-n/2]}(\overline\Omega)$.
A new result on the Pompeiu problem
R.
Dalmasso
2723-2736
Abstract: A nonempty bounded open set $\Omega \subset {\mathbb{R}}^{n}$ ($n \geq 2$) is said to have the Pompeiu property if and only if the only continuous function $f$ on ${\mathbb{R}}^{n}$ for which the integral of $f$ over $\sigma (\Omega )$ is zero for all rigid motions $\sigma$ of ${\mathbb{R}}^{n}$ is $f \equiv 0$. We consider a nonempty bounded open set $\Omega \subset {\mathbb{R}}^{n}$ $(n \geq 2)$ with Lipschitz boundary and we assume that the complement of $\overline{\Omega }$ is connected. We show that the failure of the Pompeiu property for $\Omega$ implies some geometric conditions. Using these conditions we prove that a special kind of solid tori in ${\mathbb{R}}^{n}$, $n \geq 3$, has the Pompeiu property. So far the result was proved only for solid tori in ${\mathbb{R}}^{4}$. We also examine the case of planar domains. Finally we extend the example of solid tori to domains in ${\mathbb{R}}^{n}$ bounded by hypersurfaces of revolution.
Quadratic integral games and causal synthesis
Yuncheng
You
2737-2764
Abstract: The game problem for an input-output system governed by a Volterra integral equation with respect to a quadratic performance functional is an untouched open problem. In this paper, it is studied by a new approach called projection causality. The main result is the causal synthesis which provides a causal feedback implementation of the optimal strategies in the saddle point sense. The linear feedback operator is determined by the solution of a Fredholm integral operator equation, which is independent of data functions and control functions. Two application examples are included. The first one is quadratic differential games of a linear system with arbitrary finite delays in the state variable and control variables. The second is the standard linear-quadratic differential games, for which it is proved that the causal synthesis can be reduced to a known result where the feedback operator is determined by the solution of a differential Riccati operator equation.
Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum
Fritz
Gesztesy;
Barry
Simon
2765 - 2787
Abstract: We discuss results where the discrete spectrum (or partial information on the discrete spectrum) and partial information on the potential $q$ of a one-dimensional Schrödinger operator $H=-\frac{d^{2}}{dx^{2}}+q$ determine the potential completely. Included are theorems for finite intervals and for the whole line. In particular, we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of $H$ on a finite interval and knowledge of $q$ over a corresponding fraction of the interval. The methods employed rest on Weyl $m$-function techniques and densities of zeros of a class of entire functions.
Projection orthogonale sur le graphe d'une relation linéaire fermé
Yahya
Mezroui
2789-2800
Abstract: Let ${LR(H)}$ denote the set of all closed linear relations on a Hilbert space $H$ (which contains all closed linear operators on $H$). In this paper, for every $E \in {\mathcal LR(H)}$ we define and study two associated linear operators on $H$, $\cos(E)$ and $\sin(E)$, which play an important role in the study of linear relations. These operators satisfy conditions quite analogous to trigonometric identities (whence their names) and appear, in particular, in the formula that gives the orthogonal projection on the graph of $E$, a formula first established for linear operators by M. H. Stone and extended to linear relations by H. De Snoo. We prove here a slightly modified version of the De Snoo formula. Several other applications of the $\cos(E)$ and $\sin(E)$ operators to operator theory will be given in a forthcoming paper.
Geometry of Banach spaces having shrinking approximations of the identity
Eve
Oja
2801-2823
Abstract: Let $a,c\geq 0$ and let $B$ be a compact set of scalars. We introduce property $M^{\ast }(a,B,c)$ of Banach spaces $X$ by the requirement that \begin{equation*}\limsup _{\nu }\Vert ax_{\nu }^{\ast } +bx^{\ast }+cy^{\ast }\V... ...q \limsup _{\nu }\Vert x_{\nu }^{\ast }\Vert\quad \forall b\in B \end{equation*}whenever $(x_{\nu }^{\ast })$ is a bounded net converging weak$^{\ast }$ to $x^{\ast }$ in $X^{\ast }$ and $\Vert y^{\ast }\Vert\leq \Vert x^{\ast }\Vert$. Using $M^{\ast }(a,B,c)$ with $\max \vert B\vert+c>1$, we characterize the existence of certain shrinking approximations of the identity (in particular, those related to $M$-, $u$-, and $h$-ideals of compact or approximable operators). We also show that the existence of these approximations of the identity is separably determined.
The truncated complex $K$-moment problem
Raúl
Curto;
Lawrence
A.
Fialkow
2825-2855
Abstract: Let $\gamma \equiv \gamma^{\left( 2n\right) }$ denote a sequence of complex numbers $\gamma _{00}, \gamma _{01}, \gamma _{10}, \dots , \gamma _{0,2n}, \dots , \gamma _{2n,0}$ ( $\gamma _{00}>0, \gamma _{ij}=\bar{\gamma}_{ji}$), and let $K$ denote a closed subset of the complex plane $\mathbb{C}$. The Truncated Complex $K$-Moment Problem for $\gamma$ entails determining whether there exists a positive Borel measure $\mu$ on $\mathbb{C}$ such that $\gamma _{ij}=\int \bar{z}^{i}z^{j}\,d\mu$ ( $0\leq i+j\leq 2n$) and $\operatorname{supp}\mu \subseteq K$. For $K\equiv K_{\mathcal{P}}$ a semi-algebraic set determined by a collection of complex polynomials $\mathcal{P} =\left\{ p_{i}\left( z,\bar{z}\right) \right\} _{i=1}^{m}$, we characterize the existence of a finitely atomic representing measure with the fewest possible atoms in terms of positivity and extension properties of the moment matrix $M\left( n\right) \left( \gamma \right)$and the localizing matrices $M_{p_{i}}$. We prove that there exists a $\operatorname{rank}M\left( n\right)$-atomic representing measure for $\gamma ^{\left( 2n\right) }$supported in $K_{\mathcal{P}}$if and only if $M\left( n\right) \geq 0$and there is some rank-preserving extension $M\left( n+1\right)$for which $M_{p_{i}}\left( n+k_{i}\right) \geq 0$, where $\deg p_{i}=2k_{i}$ or $2k_{i}-1$ $(1\leq i\leq m)$.
On the structure of weight modules
Ivan
Dimitrov;
Olivier
Mathieu;
Ivan
Penkov
2857-2869
Abstract: Given any simple Lie superalgebra ${\mathfrak{g}}$, we investigate the structure of an arbitrary simple weight ${\mathfrak{g}}$-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuperalgebra. Then we classify the cuspidal Levi subsuperalgebras of all simple classical Lie superalgebras and of the Lie superalgebra W$(n)$. Most of them are simply Levi subalgebras of ${\mathfrak{g}}_{0}$, in which case the classification of all finite cuspidal representations has recently been carried out by one of us (Mathieu). Our results reduce the classification of the finite simple weight modules over all classical simple Lie superalgebras to classifying the finite cuspidal modules over certain Lie superalgebras which we list explicitly.
Bivariate factorizations connecting Dickson polynomials and Galois theory
Shreeram
S.
Abhyankar;
Stephen
D.
Cohen;
Michael
E.
Zieve
2871-2887
Abstract: In his Ph.D. Thesis of 1897, Dickson introduced certain permutation polynomials whose Galois groups are essentially the dihedral groups. These are now called Dickson polynomials of the first kind, to distinguish them from their variations introduced by Schur in 1923, which are now called Dickson polynomials of the second kind. In the last few decades there have been extensive investigations of both of these types, which are related to the classical Chebyshev polynomials. We give new bivariate factorizations involving both types of Dickson polynomials. These factorizations demonstrate certain isomorphisms between dihedral groups and orthogonal groups, and lead to the construction of explicit equations with orthogonal groups as Galois groups.
The structure of conjugacy closed loops
Kenneth
Kunen
2889-2911
Abstract: We study structure theorems for the conjugacy closed (CC-) loops, a specific variety of G-loops (loops isomorphic to all their loop isotopes). These theorems give a description all such loops of small order. For example, if $p$ and $q$ are primes, $p < q$, and $q-1$ is not divisible by $p$, then the only CC-loop of order $pq$ is the cyclic group of order $pq$. For any prime $q > 2$, there is exactly one non-group CC-loop in order $2q$, and there are exactly three in order $q^2$. We also derive a number of equations valid in all CC-loops. By contrast, every equation valid in all G-loops is valid in all loops.
Infinite convolution products and refinable distributions on Lie groups
Wayne
Lawton
2913-2936
Abstract: Sufficient conditions for the convergence in distribution of an infinite convolution product $\mu_1*\mu_2*\ldots$ of measures on a connected Lie group $\mathcal G$ with respect to left invariant Haar measure are derived. These conditions are used to construct distributions $\phi$ that satisfy $T\phi = \phi$where $T$ is a refinement operator constructed from a measure $\mu$and a dilation automorphism $A$. The existence of $A$ implies $\mathcal G$ is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset $\mathcal K \subset \mathcal G$such that for any open set $\mathcal U$ containing $\mathcal K,$ and for any distribution $f$ on $\mathcal G$ with compact support, there exists an integer $n(\mathcal U,f)$ such that $n \geq n(\mathcal U,f)$implies $\hbox{supp}(T^{n}f) \subset\mathcal U.$If $\mu$ is supported on an $A$-invariant uniform subgroup $\Gamma,$ then $T$ is related, by an intertwining operator, to a transition operator $W$ on $\mathbb C(\Gamma).$ Necessary and sufficient conditions for $T^{n}f$ to converge to $\phi \in L^{2}$, and for the $\Gamma$-translates of $\phi$ to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of $W$ to functions supported on $\Omega := \mathcal K \mathcal K^{-1} \cap \Gamma.$
Lacunarity of self-similar and stochastically self-similar sets
Dimitris
Gatzouras
1953-1983
Abstract: Let $K$ be a self-similar set in $\mathbb R^d$, of Hausdorff dimension $D$, and denote by $\vert K(\epsilon)\vert$the $d$-dimensional Lebesgue measure of its $\epsilon$-neighborhood. We study the limiting behavior of the quantity $\epsilon^{-(d-D)}\vert K(\epsilon)\vert$as $\epsilon\rightarrow 0$. It turns out that this quantity does not have a limit in many interesting cases, including the usual ternary Cantor set and the Sierpinski carpet. We also study the above asymptotics for stochastically self-similar sets. The latter results then apply to zero-sets of stable bridges, which are stochastically self-similar (in the sense of the present paper), and then, more generally, to level-sets of stable processes. Specifically, it follows that, if $K_t$ is the zero-set of a real-valued stable process of index $\alpha\in (1,2]$, run up to time $t$, then $\epsilon^{-1/\alpha}\vert K_t(\epsilon)\vert$converges to a constant multiple of the local time at $0$, simultaneously for all $t\geq 0$, on a set of probability one. The asymptotics for deterministic sets are obtained via the renewal theorem. The renewal theorem also yields asymptotics for the mean $\mathbb E [\vert K(\epsilon)\vert ]$ in the random case, while the almost sure asymptotics in this case are obtained via an analogue of the renewal theorem for branching random walks.
Extending partial automorphisms and the profinite topology on free groups
Bernhard
Herwig;
Daniel
Lascar
1985-2021
Abstract: A class of structures $\mathcal{C}$ is said to have the extension property for partial automorphisms (EPPA) if, whenever $C_1$ and $C_2$ are structures in $\mathcal{C}$, $C_1$ finite, $C_1\subseteq C_2$, and $p_1,p_2,\dotsc,p_n$ are partial automorphisms of $C_1$ extending to automorphisms of $C_2$, then there exist a finite structure $C_3$ in $\mathcal{C}$ and automorphisms $\alpha _1, \alpha _2,\dotsc,\alpha _n$ of $C_3$ extending the $p_i$. We will prove that some classes of structures have the EPPA and show the equivalence of these kinds of results with problems related with the profinite topology on free groups. In particular, we will give a generalisation of the theorem, due to Ribes and Zalesskii stating that a finite product of finitely generated subgroups is closed for this topology.
The distributivity numbers of $\mathcal{P}(\omega)$/fin and its square
Saharon
Shelah;
Otmar
Spinas
2023-2047
Abstract: We show that in a model obtained by forcing with a countable support iteration of Mathias forcing of length $\omega _{2}$, the distributivity number of ${\mathcal{P}}(\omega )$/fin is $\omega _{2}$, whereas the distributivity number of r.o.$({\mathcal{P}}(\omega )$/fin)$^{2}$ is $\omega _{1}$. This answers a problem of Balcar, Pelant and Simon, and others.
Cycling in proofs and feasibility
A.
Carbone
2049-2075
Abstract: There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh's lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles.
A note on elliptic k3 surfaces
JongHae
Keum
2077-2086
Abstract: We study the relationship between an elliptic fibration on an elliptic K3 surface and its Jacobian surface. We give an explicit description of the Picard lattice of the Jacobian surface. Then we use the description to prove that certain K3 surfaces do not admit a non-Jacobian fibration. Moreover, we obtain an inequality involving the determinant of the Picard lattice and the number of components of reducible fibres, which implies, among others, that if an elliptic K3 surface has Picard lattice with relatively small determinant, then every elliptic fibration on it must have a reducible fibre. Some examples of K3 surfaces are discussed.
The geometry of fixed point varieties on affine flag manifolds
Daniel
S.
Sage
2087-2119
Abstract: Let $G$ be a semisimple, simply connected, algebraic group over an algebraically closed field $k$ with Lie algebra $\mathfrak{g}$. We study the spaces of parahoric subalgebras of a given type containing a fixed nil-elliptic element of $\mathfrak{g}\otimes k((\pi))$, i.e. fixed point varieties on affine flag manifolds. We define a natural class of $k^*$-actions on affine flag manifolds, generalizing actions introduced by Lusztig and Smelt. We formulate a condition on a pair $(N,f)$ consisting of $N\in \mathfrak{g}\otimes k((\pi))$ and a $k^*$-action $f$ of the specified type which guarantees that $f$ induces an action on the variety of parahoric subalgebras containing $N$. For the special linear and symplectic groups, we characterize all regular semisimple and nil-elliptic conjugacy classes containing a representative whose fixed point variety admits such an action. We then use these actions to find simple formulas for the Euler characteristics of those varieties for which the $k^*$-fixed points are finite. We also obtain a combinatorial description of the Euler characteristics of the spaces of parabolic subalgebras containing a given element of certain nilpotent conjugacy classes of $\mathfrak{g}$.
Coupled contact systems and rigidity of maximal dimensional variations of Hodge structure
Richárd
Mayer
2121-2144
Abstract: In this article we prove that locally Griffiths' horizontal distribution on the period domain is given by a generalized version of the familiar contact differential system. As a consequence of this description we obtain strong local rigidity properties of maximal dimensional variations of Hodge structure. For example, we prove that if the weight is odd (greater than one) then there is a unique germ of maximal dimensional variation of Hodge structure through every point of the period domain. Similar results hold if the weight is even with the exception of one case.
Some properties of partitions in terms of crank
A.
Bülent
Ek. in
2145 - 2156
Some Theorems on the Rogers--Ramanujan Continued Fraction in Ramanujan's Lost Notebook
Bruce
C.
Berndt;
Sen-Shan
Huang;
Jaebum
Sohn;
Seung
Hwan
Son
2157-2177
Abstract: In his first two letters to G. H. Hardy and in his notebooks, Ramanujan recorded many theorems about the Rogers-Ramanujan continued fraction. In his lost notebook, he offered several further assertions. The purpose of this paper is to provide proofs for many of the claims about the Rogers-Ramanujan and generalized Rogers-Ramanujan continued fractions found in the lost notebook. These theorems involve, among other things, modular equations, transformations, zeros, and class invariants.
Topological Hochschild homology of number rings
Ayelet
Lindenstrauss;
Ib
Madsen
2179-2204
Abstract: We calculate an explicit formula for the topological Hochschild homology of a discrete valuation ring of characteristic zero with finite residue field. From this we deduce the topological Hochschild homology of global number rings.
Relative completions of linear groups over $\mathbb{Z}[t]$ and $\mathbb{Z}[t,t^{-1}]$
Kevin
P.
Knudson
2205-2216
Abstract: We compute the completion of the groups $SL_n({\mathbb Z}[t])$ and $SL_n({\mathbb Z}[t,t^{-1}])$ relative to the obvious homomorphisms to $SL_n({\mathbb Q})$; this is a generalization of the classical Malcev completion. We also make partial computations of the rational second cohomology of these groups.
Normal forms and Hopf bifurcation for partial differential equations with delays
Teresa
Faria
2217-2238
Abstract: The paper addresses the computation of normal forms for some Partial Functional Differential Equations (PFDEs) near equilibria. The analysis is based on the theory previously developed for autonomous retarded Functional Differential Equations and on the existence of center (or other invariant) manifolds. As an illustration of this procedure, two examples of PFDEs where a Hopf singularity occurs on the center manifold are considered.
Schauder estimates for equationswith fractional derivatives
Ph.
Clément;
G.
Gripenberg;
S-O.
Londen
2239-2260
Abstract: The equation \begin{equation*}D^\alpha_t (u-h_1) + D^\beta_x(u-h_2) =f,\quad 0< \alpha,\beta< 1, \quad t,x \geq 0,\tag{$*$} \end{equation*}where $D^\alpha_t$ and $D^\beta_x$ are fractional derivatives of order $\alpha$ and $\beta$ is studied. It is shown that if $f=f(\underline{t}, \underline{x})$, $h_1=h_1(\underline{x})$, and $h_2=h_2(\underline{t})$ are Hölder-continuous and $f(0,0) =0$, then there is a solution such that $D^\alpha_t u$ and $D^\beta_x u$ are Hölder-continuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to ($*$). Finally the solution of ($*$) with $f=1$ is studied.
An algorithmic approach to the construction of homomorphisms induced by maps in homology
Madjid
Allili;
Tomasz
Kaczynski
2261-2281
Abstract: This paper is devoted to giving the theoretical background for an algorithm for computing homomorphisms induced by maps in homology. The principal idea is to insert the graph of a given continuous map $\, f \,$ into a graph of a multi-valued representable map $\, F$. The multi-valued representable maps have well developed continuity properties and admit a finite coding that permits treating them by combinatorial methods. We provide the construction of the homomorphism $\, F_* \,$ induced by $\, F \,$ such that $\, F_* = f_*$. The presented construction does not require subsequent barycentric subdivisions and simplicial approximations of $\, f$. The main motivation for this paper comes from the project of computing the Conley Index for discrete dynamical systems.
From the restricted to the full three--body problem
Kenneth
R.
Meyer;
Dieter
S.
Schmidt
2283-2299
Abstract: The three-body problem with all the classical integrals fixed and all the symmetries removed is called the reduced three-body problem. We use the methods of symplectic scaling and reduction to show that the reduced planar or spatial three-body problem with one small mass is to the first approximation the product of the restricted three-body problem and a harmonic oscillator. This allows us to prove that many of the known results for the restricted problem have generalizations for the reduced three-body problem. For example, all the non-degenerate periodic solutions, generic bifurcations, Hamiltonian-Hopf bifurcations, bridges and natural centers known to exist in the restricted problem can be continued into the reduced three-body problem. The classic normalization calculations of Deprit and Deprit-Bartholomé show that there are two-dimensional KAM invariant tori near the Lagrange point in the restricted problem. With the above result this proves that there are three-dimensional KAM invariant tori near the Lagrange point in the reduced three-body problem.
Locating the first nodal linein the Neumann problem
David
Jerison
2301-2317
Abstract: The location of the nodal line of the first nonconstant Neumann eigenfunction of a convex planar domain is specified to within a distance comparable to the inradius. This is used to prove that the eigenvalue of the partial differential equation is approximated well by the eigenvalue of an ordinary differential equation whose coefficients are expressed solely in terms of the width of the domain. A variant of these estimates is given for domains that are thin strips and satisfy a Lipschitz condition.
A Construction of Homologically Area Minimizing Hypersurfaces with Higher Dimensional Singular Sets
Nathan
Smale
2319-2330
Abstract: We show that a large variety of singular sets can occur for homologically area minimizing codimension one surfaces in a Riemannian manifold. In particular, as a result of Theorem A, if $N$ is smooth, compact $n+1$ dimensional manifold, $n\geq 7$, and if $S$ is an embedded, orientable submanifold of dimension $n$, then we construct metrics on $N$ such that the homologically area minimizing hypersurface $M$, homologous to $S$, has a singular set equal to a prescribed number of spheres and tori of codimension less than $n-7$. Near each component $\Sigma$ of the singular set, $M$ is isometric to a product $C\times \Sigma$, where $C$ is any prescribed, strictly stable, strictly minimizing cone. In Theorem B, other singular examples are constructed.
Conditions for the Existence of SBR Measures for ``Almost Anosov'' Diffeomorphisms
Huyi
Hu
2331-2367
Abstract: A diffeomorphism $f$ of a compact manifold $M$ is called ``almost Anosov'' if it is uniformly hyperbolic away from a finite set of points. We show that under some nondegeneracy condition, every almost Anosov diffeomorphism admits an invariant measure $\mu$ that has absolutely continuous conditional measures on unstable manifolds. The measure $\mu$ is either finite or infinite, and is called SBR measure or infinite SBR measure respectively. Therefore, $\frac{1}{n} \sum _{i=0}^{n-1}\delta _{f^{i}x}$ tends to either an SBR measure or $\delta _{p}$ for almost every $x$ with respect to Lebesgue measure. ($\delta _{x}$ is the Dirac measure at $x$.) For each case, we give sufficient conditions by using coefficients of the third order terms in the Taylor expansion of $f$ at $p$.
Statistical properties for nonhyperbolic maps with finite range structure
Michiko
Yuri
2369-2388
Abstract: We establish the central limit theorem and non-central limit theorems for maps admitting indifferent periodic points (the so-called intermittent maps). We also give a large class of Darling-Kac sets for intermittent maps admitting infinite invariant measures. The essential issue for the central limit theorem is to clarify the speed of $L^1$-convergence of iterated Perron-Frobenius operators for multi-dimensional maps which satisfy Renyi's condition but fail to satisfy the uniformly expanding property. Multi-dimensional intermittent maps typically admit such derived systems. There are examples in section 4 to which previous results on the central limit theorem are not applicable, but our extended central limit theorem does apply.
On the stable module category of a self-injective algebra
Karin
Erdmann;
Otto
Kerner
2389-2405
Abstract: Let $\Lambda$ be a finite-dimensional self-injective algebra. We study the dimensions of spaces of stable homomorphisms between indecomposable $\Lambda$-modules which belong to Auslander-Reiten components of the form $\mathbf{Z}A_\infty$ or $\mathbf{Z}A_\infty/\langle \tau^k\rangle$. The results are applied to representations of finite groups over fields of prime characteristic, especially blocks of wild representation type.
Representations over PID's with three distinguished submodules
Steve
Files;
Rüdiger
Göbel
2407-2427
Abstract: Let $R$ be a principal ideal domain. The $R$-representations with one distinguished submodule are classified by a theorem of Gaußin the case of finite rank, and by the ``Stacked Bases Theorem" of Cohen and Gluck in the case of infinite rank. Results of Hill and Megibben carry this classification even further. The $R$-representations with two distinguished pure submodules have recently been classified by Arnold and Dugas in the finite-rank case, and by the authors for countable rank. Although wild representation type prevails for $R$-representations with three distinguished pure submodules, an extensive category of such objects was recently classified by Arnold and Dugas. We carry their groundbreaking work further, simplifying the proofs of their main results and applying new machinery to study the structure of finite- and infinite-rank representations with two, three, and four distinguished submodules. We also apply these results to the classification of Butler groups, a class of torsion-free abelian groups that has been the focus of many investigations over the last sixteen years.
On the endomorphism monoids of (uniquely) complemented lattices
G.
Grätzer;
J.
Sichler
2429-2444
Abstract: Let $L$ be a lattice with $0$ and $1$. An endomorphism $\varphi$ of $L$ is a $\{0,1\}$-endomorphism, if it satisfies $0\varphi = 0$ and $1\varphi = 1$. The $\{0,1\}$-endomorphisms of $L$ form a monoid. In 1970, the authors proved that every monoid $\mathcal M$ can be represented as the $\{0,1\}$-endomorphism monoid of a suitable lattice $L$ with $0$ and $1$. In this paper, we prove the stronger result that the lattice $L$ with a given $\{0,1\}$-endomorphism monoid $\mathcal M$ can be constructed as a uniquely complemented lattice; moreover, if $\mathcal M$ is finite, then $L$ can be chosen as a finite complemented lattice.
Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology
David
J.
Pengelley;
Frank
Williams
1453-1492
Abstract: The mod 2 Steenrod algebra $\mathcal{A}$ and Dyer-Lashof algebra $\mathcal{R}$ have both striking similarities and differences arising from their common origins in ``lower-indexed'' algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra $\mathcal{K}$, whose module actions are equivalent to, but quite different from, those of $\mathcal{A}$ and $\mathcal{R}$. The exact relationships emerge as ``sheared algebra bijections'', which also illuminate the role of the cohomology of $\mathcal{K}$. As a bialgebra, $\mathcal{K}^{*}$ has a particularly attractive and potentially useful structure, providing a bridge between those of $\mathcal{A^{*}}$ and $\mathcal{R^{*}}$, and suggesting possible applications to the Miller spectral sequence and the $\mathcal{A}$ structure of Dickson algebras.
Rational homotopy theory for non-simply connected spaces
Antonio
Gómez-Tato;
Stephen
Halperin;
Daniel
Tanré
1493-1525
Abstract: We construct an algebraic rational homotopy theory for all connected CW spaces (with arbitrary fundamental group) whose universal cover is rationally of finite type. This construction extends the classical theory in the simply connected case and has two basic properties: (1) it induces a natural equivalence of the corresponding homotopy category to the homotopy category of spaces whose universal cover is rational and of finite type and (2) in the algebraic category, homotopy equivalences are isomorphisms. This algebraisation introduces a new homotopy invariant: a rational vector bundle with a distinguished class of linear connections.
On $BC$ type basic hypergeometric orthogonal polynomials
Jasper
V.
Stokman
1527-1579
Abstract: The five parameter family of Koornwinder's multivariable analogues of the Askey-Wilson polynomials is studied with four parameters generically complex. The Koornwinder polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partly discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain is given for which the partly discrete orthogonality measure is positive. The orthogonality relations and norm evaluations for multivariable $q$-Racah polynomials and multivariable big and little $q$-Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the Koornwinder polynomials. In particular new proofs of several well-known $q$-analogues of the Selberg integral are obtained.
Lorentzian affine hyperspheres with constant affine sectional curvature
Marcus
Kriele;
Luc
Vrancken
1581-1599
Abstract: We study affine hyperspheres $M$ with constant sectional curvature (with respect to the affine metric $h$). A conjecture by M. Magid and P. Ryan states that every such affine hypersphere with nonzero Pick invariant is affinely equivalent to either \begin{displaymath}(x_{1}^{2} \pm x_{2}^{2})(x_{3}^{2}\pm x_{4}^{2})\dots (x_{2m-1}^{2}\pm x_{2m}^{2}) = 1\end{displaymath} or \begin{displaymath}(x_{1}^{2} \pm x_{2}^{2})(x_{3}^{2}\pm x_{4}^{2})\dots (x_{2m-1}^{2}\pm x_{2m}^{2})x_{2m+1} = 1\end{displaymath} where the dimension $n$ satisfies $n=2m-1$ or $n=2m$. Up to now, this conjecture was proved if $M$ is positive definite or if $M$ is a $3$-dimensional Lorentz space. In this paper, we give an affirmative answer to this conjecture for arbitrary dimensional Lorentzian affine hyperspheres.
Sur les intégrales premières dans la classe de Nilsson d'équations différentielles holomorphes
Frédéric
Touzet
1601-1622
Abstract: We study two classes of holomorphic differential equations. The first one is constitued by elements admitting solutions defined in an algebraic way (the so called Liouville class) and the second of elements admitting solutions defined in an analytic way (the Nilsson class). We build up links between these two classes using special properties of the holonomy and its results on the monodromy.
On Chow motives of 3-folds
Pedro
Luis del
Angel;
Stefan
Müller-Stach
1623 - 1633
Abstract: Let $k$ be a field of characteristic zero. For every smooth, projective $k$-variety $Y$ of dimension $n$ which admits a connected, proper morphism $f: Y \to S$ of relative dimension one, we construct idempotent correspondences (projectors) $\pi _{ij}(Y) \in CH^{n}(Y \times Y,\mathbb{Q})$generalizing a construction of Murre. If $n=3$ and the transcendental cohomology group $H^{2}_{\text{tr}}(Y)$ has the property that $H^{2}_{\text{tr}}(Y,\mathbb{C})=f^{*}H^{2}_{\text{tr}}(S,\mathbb{C})+ {\text{Im... ...(S,\mathbb{C}) \otimes H^{1}(Y,\mathbb{C}) \to H^{2}_{\text{tr}}(Y,\mathbb{C}))$, then we can construct a projector $\pi _{2}(Y)$ which lifts the second Künneth component of the diagonal of $Y$. Using this we prove that many smooth projective 3-folds $X$ over $k$ admit a Chow-Künneth decomposition $\Delta =p_{0}+...+p_{6}$ of the diagonal in $CH^{3}(X \times X,{\mathbb{Q}})$.
A theorem on smoothness- Bass-Quillen, Chow groups and intersection multiplicity of Serre
S.
P.
Dutta
1635-1645
Abstract: We describe here an inherent connection of smoothness among the Bass-Quillen conjecture, the Chow-group problem and Serre's Theorem on Intersection Multiplicity. Extension of a theorem of Lindel on smoothness plays a key role in our proof of the Serre-multiplicity theorem in the geometric (resp. unramified) case. We reduce the complete case of the theorem to the above case by using Artin's Approximation. We do not need the concept of ``complete Tor''. Similar proofs are sketched for Quillen's theorem on Chow groups and its extension due to Gillet and Levine.
A bracket power characterization of analytic spread one ideals
L.
J.
Ratliff Jr.;
D.
E.
Rush Jr.
1647-1674
Abstract: The main theorem characterizes, in terms of bracket powers, analytic spread one ideals in local rings. Specifically, let $b_{1},\dots ,b_{g},x$ be regular nonunits in a local (Noetherian) ring $(R,M)$ and assume that $I$ $\subseteq$ $(xR)_{a}$, the integral closure of $xR$, where $I$ $=$ $(b_{1},\dots ,b_{g},x)R$. Then the main result shows that for all but finitely many units $u_{1},\dots ,u_{g}$ in $R$ that are non-congruent modulo $M$ and for all large integers $n$ and $k$ it holds that $I^{jn}$ $=$ $I^{[j]n}$ for $j$ $=$ $1,\dots ,k$ and $j$ not divisible by $char(R/M)$, where $I^{[j]}$ is the $j$-th bracket power $((b_{1}+u_{1}x)^{j}, \dots ,(b_{g}+u_{g}x)^{j},x^{j})R$ of $I$ $=$ $(b_{1}+u_{1}x, \dots ,b_{g}+u_{g}x,x)R$. And, conversely, if there exist positive integers $g$, $n$, and $k$ $\ge$ ${\binom{{n+g} }{{g}}}$ such that $I$ has a basis $\beta _{1},\dots ,\beta _{g} ,x$ such that $I^{kn}$ $=$ $({\beta _{1}}^{k},\dots ,{\beta _{g}}^{k},x^{k})^{n}R$, then $I$ has analytic spread one.
Sharp bounds on Castelnuovo-Mumford regularity
Chikashi
Miyazaki
1675-1686
Abstract: The Castelnuovo-Mumford regularity is one of the most important invariants in studying the minimal free resolution of the defining ideals of the projective varieties. There are some bounds on the Castelnuovo-Mumford regularity of the projective variety in terms of the other basic measures such as dimension, codimension and degree. In this paper we consider an upper bound on the regularity $\operatorname{reg}(X)$ of a nondegenerate projective variety $X$, $\operatorname{reg}(X)\le\lceil(\deg (X) - 1)/\operatorname{codim}(X)\rceil +k \cdot \dim (X)$, provided $X$ is $k$-Buchsbaum for $k \ge 1$, and investigate the projective variety with its Castelnuovo-Mumford regularity having such an upper bound.
Prehomogeneous vector spaces and ergodic theory II
Dave
Witte;
Akihiko
Yukie;
Roger
Zierau
1687-1708
Abstract: We apply M. Ratner's theorem on closures of unipotent orbits to the study of three families of prehomogeneous vector spaces. As a result, we prove analogues of the Oppenheim Conjecture for simultaneous approximation by values of certain alternating bilinear forms in an even number of variables and certain alternating trilinear forms in six and seven variables.
Shintani functions on $GL(2,\mathbf{R})$
Miki
Hirano
1709-1721
Abstract: In this paper, we give a formulation and an explicit formula for Shintani function on $GL(2,{\mathbf{R}})$, which has been studied by Murase and Sugano in the theory of automorphic $L$-functions. In particular, we obtain the multiplicity of this function.
Complexes galoisiens
Yves
Ladegaillerie
1723-1741
Abstract: We construct special $n$-complexes categories which are the object of a Galois theory. Their topological supports are pseudo-manifolds which are branched coverings of spheres. They are a generalization in every dimension of hypercartes. Every category of Galois $n$-complexes is equivalent to a 2-complexes one. Reducing operations are introduced in dimensions two and three. It leads to a notion of irreducible complex which is used in three-dimensions for a simplified generation of 3-manifolds. Résumé. On construit des catégories de $n$-complexes simpliciaux particuliers qui sont l'objet d'une théorie galoisienne. Topologiquement, ce sont des pseudo-variétés revêtements ramifiés de sphères. En particulier, ce sont des généralisations des hypercartes aux dimension supérieures. Tout catégorie de $n$-complexes galoisiens est équivalente à une catégorie de 2-complexes. Des opérations de réduction sont introduites en dimensions deux et trois. Elles mènent à une notion de complexe irréductible qui est utilisée en dimension trois pour obtenir une génération simplifiée des 3-variétés.
A continuous circle of pseudo-arcs filling up the annulus
Janusz
R.
Prajs
1743-1757
Abstract: We prove an early announcement by Knaster on a decomposition of the plane. Then we establish an announcement by Anderson saying that the plane annulus admits a continuous decomposition into pseudo-arcs such that the quotient space is a simple closed curve. This provides a new plane curve, ``a selectible circle of pseudo-arcs", and answers some questions of Lewis.
An approach to symmetrization via polarization
Friedemann
Brock;
Alexander
Yu.
Solynin
1759-1796
Abstract: We prove that the Steiner symmetrization of a function can be approximated in $L^p ({\mathbb R}^n )$ by a sequence of very simple rearrangements which are called polarizations. This result is exploited to develop elementary proofs of many inequalities, including the isoperimetric inequality in Euclidean space. In this way we also obtain new symmetry results for solutions of some variational problems. Furthermore we compare the solutions of two boundary value problems, one of them having a "polarized" geometry and we show some pointwise inequalities between the solutions. This leads to new proofs of well-known functional inequalities which compare the solutions of two elliptic or parabolic problems, one of them having a "Steiner-symmetrized" geometry. The method also allows us to investigate the case of equality in the inequalities. Roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry.
How parabolic free boundaries approximate hyperbolic fronts
Brian
H.
Gilding;
Roberto
Natalini;
Alberto
Tesei
1797-1824
Abstract: A rather complete study of the existence and qualitative behaviour of the boundaries of the support of solutions of the Cauchy problem for nonlinear first-order and second-order scalar conservation laws is presented. Among other properties, it is shown that, under appropriate assumptions, parabolic interfaces converge to hyperbolic ones in the vanishing viscosity limit.
Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems
Julián
López-Gómez
1825 - 1858
A Banach space block finitely universal for monotone bases
E.
Odell;
Th.
Schlumprecht
1859-1888
Abstract: A reflexive Banach space $X$ with a basis $(e_{i})$ is constructed having the property that every monotone basis is block finitely representable in each block basis of $X$.
Local product structure for Equilibrium States
Renaud
Leplaideur
1889-1912
Abstract: The usual way to study the local structure of Equilibrium State of an Axiom-A diffeomorphism or flow is to use the symbolic dynamic and to push results on the manifold. A new geometrical method is given. It consists in proving that Equilibrium States for Hölder-continuous functions are related to other Equilibrium States of some special sub-systems satisfying a sort of expansiveness. Using different kinds of extensions the local product structure of Gibbs-measure is proven.
Periodic Groups Covered by Transitive Subgroups of Finitary Permutations or by Irreducible Subgroups of Finitary Transformations
Felix
Leinen;
Orazio
Puglisi
1913-1934
Abstract: Let $\mathfrak{X}$ be either the class of all transitive groups of finitary permutations, or the class of all periodic irreducible finitary linear groups. We show that almost primitive $\mathfrak{X}$-groups are countably recognizable, while totally imprimitive $\mathfrak{X}$-groups are in general not countably recognizable. In addition we derive a structure theorem for groups all of whose countable subsets are contained in totally imprimitive $\mathfrak{X}$-subgroups. It turns out that totally imprimitive $p$-groups in the class $\mathfrak{X}$ are countably recognizable.
Simple and semisimple Lie algebras and codimension growth
Antonio
Giambruno;
Amitai
Regev;
Michail
V.
Zaicev
1935-1946
Abstract: We study the exponential growth of the codimensions $c_ n^{L}(B)$ of a finite dimensional Lie algebra $B$ over a field of characteristic zero. In the case when $B$ is semisimple we show that $\lim _{n\to \infty } \sqrt[{n}]{c_ n^{L}(B)}$ exists and, when $F$ is algebraically closed, is equal to the dimension of the largest simple summand of $B$. As a result we characterize central-simplicity: $B$ is central simple if and only if $\dim B = \lim _{n\to \infty } \sqrt[{n}]{c_ n^{L} (B)}$.
Correction to ``Harish-Chandra's Plancherel theorem for $\mathfrak{p}-adic groups''
Allan
J.
Silberger
1947-1949
A correction to ``Equivariant cohomology and lower bounds for chromatic numbers''
Igor
Kriz
1951-1952
The calculus of partition sequences, changing cofinalities, and a question of Woodin
Arthur
W.
Apter;
James
M.
Henle;
Stephen
C.
Jackson
969-1003
Abstract: We study in this paper polarized infinite exponent partition relations. We apply our results to constructing a model for the theory ``ZF$+$DC$+\omega _1$ is the only regular, uncountable cardinal $\le\omega _{\omega _1+1}$.'' This gives a partial answer to a question of Woodin.
Decomposition theorems for groups of diffeomorphisms in the sphere
R.
de la Llave;
R.
Obaya
1005-1020
Abstract: We study the algebraic structure of several groups of differentiable diffeomorphisms in $\mathbf{S}^n$. We show that any given sufficiently smooth diffeomorphism can be written as the composition of a finite number of diffeomorphisms which are symmetric under reflection, essentially one-dimensional and about as differentiable as the given one.
Deforming a map into a harmonic map
Deane
Yang
1021-1038
Abstract: This paper presents some existence and uniqueness theorems for harmonic maps between complete noncompact Riemannian manifolds. In particular, we obtain as a corollary a recent result of Hardt-Wolf on the existence of harmonic quasi-isometries of the hyperbolic plane.
Sums of squares of regular functions on real algebraic varieties
Claus
Scheiderer
1039-1069
Abstract: Let $V$ be an affine algebraic variety over $\mathbb{R}$ (or any other real closed field $R$). We ask when it is true that every positive semidefinite (psd) polynomial function on $V$ is a sum of squares (sos). We show that for $\dim V\ge 3$ the answer is always negative if $V$ has a real point. Also, if $V$ is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if $V$ is a smooth surface with only real divisors at infinity. The ``compact'' case is harder. We completely settle the case of smooth curves of genus $\le 1$: If such a curve has a complex point at infinity, then every psd function is sos, provided the field $R$ is archimedean. If $R$ is not archimedean, there are counter-examples of genus $1$.
On the distribution of points in projective space of bounded height
Kwok-Kwong
Choi
1071-1111
Abstract: In this paper we consider the uniform distribution of points in compact metric spaces. We assume that there exists a probability measure on the Borel subsets of the space which is invariant under a suitable group of isometries. In this setting we prove the analogue of Weyl's criterion and the Erdös-Turán inequality by using orthogonal polynomials associated with the space and the measure. In particular, we discuss the special case of projective space over completions of number fields in some detail. An invariant measure in these projective spaces is introduced, and the explicit formulas for the orthogonal polynomials in this case are given. Finally, using the analogous Erdös-Turán inequality, we prove that the set of all projective points over the number field with bounded Arakelov height is uniformly distributed with respect to the invariant measure as the bound increases.
The second bounded cohomology of an amalgamated free product of groups
Koji
Fujiwara
1113-1129
Abstract: We study the second bounded cohomology of an amalgamated free product of groups, and an HNN extension of a group. As an application, we show that a group with infinitely many ends has infinite dimensional second bounded cohomology.
Embeddings in generalized manifolds
J.
L.
Bryant;
W.
Mio
1131-1147
Abstract: We prove that a ($2m-n+1$)-connected map $f\colon M^m\to X^n$ from a compact PL $m$-manifold $M$ to a generalized $n$-manifold $X$ with the disjoint disks property, $3m\le 2n-2$, is homotopic to a tame embedding. There is also a controlled version of this result, as well as a version for noncompact $M$ and proper maps $f$ that are properly ($2m-n+1$)-connected. The techniques developed lead to a general position result for arbitrary maps $f\colon M\to X$, $3m\le 2n-2$, and a Whitney trick for separating $P\hspace*{-1pt}L$submanifolds of $X$ that have intersection number 0, analogous to the well-known results when $X$ is a manifold.
Odd primary $bo$ resolutions and classification of the stable summands of stunted lens spaces
Jesús
González
1149-1169
Abstract: The classification of the stable homotopy types of stunted lens spaces and their stable summands can be obtained by proving the triviality of certain stable classes in the homotopy groups of these spaces. This is achieved in the 2-primary case by Davis and Mahowald using classical Adams spectral sequence techniques. We obtain the odd primary analogue using the corresponding Adams spectral sequence based at the spectrum representing odd primary connective $K$-theory. The methods allow us to answer a stronger problem: the determination of the smallest stunted space where such stable classes remain null homotopic. A technical problem prevents us from giving an answer in all situations; however, in a quantitative way, the number of cases missed is very small.
Characterizations of spectra with $\mathcal{U}$-injective cohomology which satisfy the Brown-Gitler property
David
J.
Hunter;
Nicholas
J.
Kuhn
1171-1190
Abstract: We work in the stable homotopy category of $p$-complete connective spectra having mod $p$ homology of finite type. $H^*(X)$ means cohomology with $\mathbf{Z}/p$ coefficients, and is a left module over the Steenrod algebra $\mathcal{A}$. A spectrum $Z$ is called spacelike if it is a wedge summand of a suspension spectrum, and a spectrum $X$ satisfies the Brown-Gitler property if the natural map $[X,Z] \rightarrow \operatorname{Hom}_{\mathcal{A}}(H^*(Z),H^*(X))$ is onto, for all spacelike $Z$. It is known that there exist spectra $T(n)$ satisfying the Brown-Gitler property, and with $H^*(T(n))$ isomorphic to the injective envelope of $H^*(S^n)$ in the category $\mathcal{U}$ of unstable $\mathcal{A}$-modules. Call a spectrum $X$ standard if it is a wedge of spectra of the form $L \wedge T(n)$, where $L$ is a stable wedge summand of the classifying space of some elementary abelian $p$-group. Such spectra have $\mathcal{U}$-injective cohomology, and all $\mathcal{U}$-injectives appear in this way. Working directly with the two properties of $T(n)$ stated above, we clarify and extend earlier work by many people on Brown-Gitler spectra. Our main theorem is that, if $X$ is a spectrum with $\mathcal{U}$-injective cohomology, the following conditions are equivalent: (A) there exist a spectrum $Y$ whose cohomology is a reduced $\mathcal{U}$-injective and a map $X \rightarrow Y$ that is epic in cohomology, (B) there exist a spacelike spectrum $Z$ and a map $X \rightarrow Z$ that is epic in cohomology, (C) $\epsilon:\Sigma^{\infty}\Omega^{\infty}X \rightarrow X$ is monic in cohomology, (D) $X$ satisfies the Brown-Gitler property, (E) $X$ is spacelike, (F) $X$ is standard. ($M \in \mathcal{U}$ is reduced if it has no nontrivial submodule which is a suspension.) As an application, we prove that the Snaith summands of $\Omega^2S^3$ are Brown-Gitler spectra-a new result for the most interesting summands at odd primes. Another application combines the theorem with the second author's work on the Whitehead conjecture. Of independent interest, enroute to proving that (B) implies (C), we prove that the homology suspension has the following property: if an $n$-connected space $X$ admits a map to an $n$-fold suspension that is monic in mod $p$ homology, then $\epsilon: \Sigma^n\Omega^n X \rightarrow X$ is onto in mod $p$ homology.
The $KO$-theory of toric manifolds
Anthony
Bahri;
Martin
Bendersky
1191-1202
Abstract: Toric manifolds, a topological generalization of smooth projective toric varieties, are determined by an $n$-dimensional simple convex polytope and a function from the set of codimension-one faces into the primitive vectors of an integer lattice. Their cohomology was determined by Davis and Januszkiewicz in 1991 and corresponds with the theorem of Danilov-Jurkiewicz in the toric variety case. Recently it has been shown by Buchstaber and Ray that they generate the complex cobordism ring. We use the Adams spectral sequence to compute the $KO$-theory of all toric manifolds and certain singular toric varieties.
Asymptotics toward the planar rarefaction wave for viscous conservation law in two space dimensions
Masataka
Nishikawa;
Kenji
Nishihara
1203-1215
Abstract: This paper is concerned with the asymptotic behavior of the solution toward the planar rarefaction wave $r(\frac{x}{t})$ connecting $u_{+}$ and $u_{-}$ for the scalar viscous conservation law in two space dimensions. We assume that the initial data $u_{0}(x,y)$ tends to constant states $u_{\pm }$ as $x \rightarrow \pm \infty$, respectively. Then, the convergence rate to $r(\frac{x}{t})$ of the solution $u(t,x,y)$ is investigated without the smallness conditions of $|u_{+}-u_{-}|$ and the initial disturbance. The proof is given by elementary $L^{2}$-energy method.
Symmetry of ground states for a semilinear elliptic system
Henghui
Zou
1217-1245
Abstract: Let $n\ge 3$ and consider the following system \begin{equation*}\Delta \mathbf{u}+\mathbf{f}(\mathbf{u})=0,\quad \mathbf{u}>0,\qquad x\in\mathbf{R}^n.\end{equation*} By using the Alexandrov-Serrin moving plane method, we show that under suitable assumptions every slow decay solution of (I) must be radially symmetric.
Gauge Invariant Eigenvalue Problems in $\mathbb{R}^n$ and in $\mathbb{R}^n_+$
Kening
Lu;
Xing-Bin
Pan
1247-1276
Abstract: This paper is devoted to the study of the eigenvalue problems for the Ginzburg-Landau operator in the entire plane ${\mathbb{R}}^{2}$ and in the half plane ${\mathbb{R}}^{2}_{+}$. The estimates for the eigenvalues are obtained and the existence of the associate eigenfunctions is proved when $curl A$ is a non-zero constant. These results are very useful for estimating the first eigenvalue of the Ginzburg-Landau operator with a gauge-invariant boundary condition in a bounded domain, which is closely related to estimates of the upper critical field in the theory of superconductivity.
Natural extensions for the Rosen fractions
Robert
M.
Burton;
Cornelis
Kraaikamp;
Thomas
A.
Schmidt
1277-1298
Abstract: The Rosen fractions form an infinite family which generalizes the nearest-integer continued fractions. We find planar natural extensions for the associated interval maps. This allows us to easily prove that the interval maps are weak Bernoulli, as well as to unify and generalize results of Diophantine approximation from the literature.
Livsic theorems for hyperbolic flows
C.
P.
Walkden
1299-1313
Abstract: We consider Hölder cocycle equations with values in certain Lie groups over a hyperbolic flow. We extend Livsic's results that measurable solutions to such equations must, in fact, be Hölder continuous.
Random intersections of thick Cantor sets
Roger
L.
Kraft
1315-1328
Abstract: Let $C_{1}$, $C_{2}$ be Cantor sets embedded in the real line, and let $\tau _{1}$, $\tau _{2}$ be their respective thicknesses. If $\tau _{1}\tau _{2}>1$, then it is well known that the difference set $C_{1}-C_{2}$ is a disjoint union of closed intervals. B. Williams showed that for some $t\in \operatorname{int}(C_{1}-C_{2})$, it may be that $C_{1}\cap (C_{2}+t)$ is as small as a single point. However, the author previously showed that generically, the other extreme is true; $C_{1}\cap (C_{2}+t)$ contains a Cantor set for all $t$ in a generic subset of $C_{1}-C_{2}$. This paper shows that small intersections of thick Cantor sets are also rare in the sense of Lebesgue measure; if $\tau _{1}\tau _{2}>1$, then $C_{1}\cap (C_{2}+t)$ contains a Cantor set for almost all $t$ in $C_{1}-C_{2}$.
Isometric Extensions of zero entropy {$\mathbb Z^{\lowercase{d}}$} Loosely Bernoulli Transformations
Aimee
S. A.
Johnson;
Ayse
A.
Sah. in
1329 - 1343
Positive definite spherical functions on Olshanskii domains
Joachim
Hilgert;
Karl-Hermann
Neeb
1345-1380
Abstract: Let $G$ be a simply connected complex Lie group with Lie algebra $\mathfrak{g}$, $\mathfrak{h}$ a real form of $\mathfrak{g}$, and $H$ the analytic subgroup of $G$ corresponding to $\mathfrak{h}$. The symmetric space ${\mathcal{M}}=H\backslash G$ together with a $G$-invariant partial order $\le$ is referred to as an Ol$'$shanskii space. In a previous paper we constructed a family of integral spherical functions $\phi _{\mu }$ on the positive domain ${\mathcal{M}}^{+} := \{Hx\colon Hx\ge H\}$ of ${\mathcal{M}}$. In this paper we determine all of those spherical functions on ${\mathcal{M}}^{+}$ which are positive definite in a certain sense.
The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras
K.
R.
Goodearl;
E.
S.
Letzter
1381-1403
Abstract: We study prime and primitive ideals in a unified setting applicable to quantizations (at nonroots of unity) of $n\times n$ matrices, of Weyl algebras, and of Euclidean and symplectic spaces. The framework for this analysis is based upon certain iterated skew polynomial algebras $A$ over infinite fields $k$ of arbitrary characteristic. Our main result is the verification, for $A$, of a characterization of primitivity established by Dixmier and Moeglin for complex enveloping algebras. Namely, we show that a prime ideal $P$ of $A$ is primitive if and only if the center of the Goldie quotient ring of $A/P$ is algebraic over $k$, if and only if $P$ is a locally closed point - with respect to the Jacobson topology - in the prime spectrum of $A$. These equivalences are established with the aid of a suitable group $\mathcal{H}$ acting as automorphisms of $A$. The prime spectrum of $A$ is then partitioned into finitely many ``$\mathcal{H}$-strata'' (two prime ideals lie in the same $\mathcal{H}$-stratum if the intersections of their $\mathcal{H}$-orbits coincide), and we show that a prime ideal $P$ of $A$ is primitive exactly when $P$ is maximal within its $\mathcal{H}$-stratum. This approach relies on a theorem of Moeglin-Rentschler (recently extended to positive characteristic by Vonessen), which provides conditions under which $\mathcal{H}$ acts transitively on the set of rational ideals within each $\mathcal{H}$-stratum. In addition, we give detailed descriptions of the strata that can occur in the prime spectrum of $A$. For quantum coordinate rings of semisimple Lie groups, results analogous to those obtained in this paper already follow from work of Joseph and Hodges-Levasseur-Toro. For quantum affine spaces, analogous results have been obtained in previous work of the authors.
Double coset density in classical algebraic groups
Jonathan
Brundan
1405-1436
Abstract: We classify all pairs of reductive maximal connected subgroups of a classical algebraic group $G$ that have a dense double coset in $G$. Using this, we show that for an arbitrary pair $(H, K)$ of reductive subgroups of a reductive group $G$ satisfying a certain mild technical condition, there is a dense $H, K$-double coset in $G$ precisely when $G = HK$ is a factorization.
Low-dimensional linear representations of ${Aut} F_n, n \geq 3$
A.
Potapchik;
A.
Rapinchuk
1437-1451
Abstract: We classify all complex representations of $\mathrm{Aut} \: F_n,$ the automorphism group of the free group $F_n$ $(n \geq 3),$ of dimension $\leq 2n - 2.$ Among those representations is a new representation of dimension $n + 1$ which does not vanish on the group of inner automorphisms.
Existence and uniqueness of rectilinear slit maps
Carl
H.
FitzGerald;
Frederick
Weening
485-513
Abstract: We consider a generalization of the parallel slit uniformization in which the angle of inclination of each image slit is assigned independently. Koebe proved that for domains of finite connectivity there is, up to a normalization, a unique rectilinear slit map achieving any given angle assignment. Koebe's theorem is partially extended to domains of infinite connectivity. A uniqueness result is shown for domains of countable connectivity and arbitrary angle assignments, and an existence result is proved for arbitrary domains under the assumption that the angle assignment is continuous and has finite range. In order to prove the existence result a new extremal length tool, called the crossing-module, is introduced. The crossing-module allows greater freedom in the family of admissible arcs than the classical module. Several results known for the module are extended to the crossing-module. A generalization of Jenkins' ${\theta}$ module condition for the parallel slit problem is given for the rectilinear slit problem in terms of the crossing-module and it is shown that rectilinear slit maps satisfying this crossing-module condition exist.
The Calabi invariant and the Euler class
Takashi
Tsuboi
515-524
Abstract: We show the following relationship between the Euler class for the group of the orientation preserving diffeomorphisms of the circle and the Calabi invariant for the group of area preserving diffeomorphisms of the disk which are the identity along the boundary. A diffeomorphism of the circle admits an extension which is an area preserving diffeomorphism of the disk. For a homomorphism $\psi$ from the fundamental group $\langle a_{1}, \cdots , a_{2g} ; [a_{1},a_{2}]\cdots [a_{2g-1},a_{2g}]\rangle$ of a closed surface to the group of the diffeomorphisms of the circle, by taking the extensions $\widetilde {\psi (a}_{i})$ for the generators $a_{i}$, one obtains the product $[\widetilde {\psi (a}_{1}),\widetilde {\psi (a}_{2})]\cdots [\widetilde {\psi (a}_{2g-1}),\widetilde {\psi (a}_{2g})]$ of their commutators, and this is an area preserving diffeomorphism of the disk which is the identity along the boundary. Then the Calabi invariant of this area preserving diffeomorphism is a non-zero multiple of the Euler class of the associated circle bundle evaluated on the fundamental cycle of the surface.
Quantization of presymplectic manifolds and circle actions
Ana
Cannas
da Silva;
Yael
Karshon;
Susan
Tolman
525-552
Abstract: We prove several versions of ``quantization commutes with reduction'' for circle actions on manifolds that are not symplectic. Instead, these manifolds possess a weaker structure, such as a spin$^c$ structure. Our theorems work whenever the quantization data and the reduction data are compatible; this condition always holds if we start from a presymplectic (in particular, symplectic) manifold.
Factorization in generalized power series
Alessandro
Berarducci
553-577
Abstract: The field of generalized power series with real coefficients and exponents in an ordered abelian divisible group $\mathbf{G}$ is a classical tool in the study of real closed fields. We prove the existence of irreducible elements in the ring $\mathbf{R}((\mathbf{G}^{\leq 0}))$ consisting of the generalized power series with non-positive exponents. The following candidate for such an irreducible series was given by Conway (1976): $\sum _n t^{-1/n}+1$. Gonshor (1986) studied the question of the existence of irreducible elements and obtained necessary conditions for a series to be irreducible. We show that Conway's series is indeed irreducible. Our results are based on a new kind of valuation taking ordinal numbers as values. If $\mathbf{G}= (\mathbf{R}, +, 0, \leq)$ we can give the following test for irreducibility based only on the order type of the support of the series: if the order type is either $\omega$ or of the form $\omega^{\omega^\alpha}$ and the series is not divisible by any monomial, then it is irreducible. To handle the general case we use a suggestion of M.-H. Mourgues, based on an idea of Gonshor, which allows us to reduce to the special case $\mathbf{G}=\mathbf{R}$. In the final part of the paper we study the irreducibility of series with finite support.
Resolutions of monomial ideals and cohomology over exterior algebras
Annetta
Aramova;
Luchezar
L.
Avramov;
Jürgen
Herzog
579-594
Abstract: This paper studies the homology of finite modules over the exterior algebra $E$ of a vector space $V$. To such a module $M$ we associate an algebraic set $V_E(M)\subseteq V$, consisting of those $v\in V$ that have a non-minimal annihilator in $M$. A cohomological description of its defining ideal leads, among other things, to complementary expressions for its dimension, linked by a `depth formula'. Explicit results are obtained for $M=E/J$, when $J$ is generated by products of elements of a basis $e_1,\dots,e_n$ of $V$. A (infinite) minimal free resolution of $E/J$ is constructed from a (finite) minimal resolution of $S/I$, where $I$ is the squarefree monomial ideal generated by `the same' products of the variables in the polynomial ring $S=K[x_1,\dots,x_n]$. It is proved that $V_E(E/J)$ is the union of the coordinate subspaces of $V$, spanned by subsets of $\{\,e_1,\dots,e_n\,\}$ determined by the Betti numbers of $S/I$ over $S$.
Automorphism scheme of a finite field extension
Pedro
J. Sancho
de Salas
595-608
Abstract: Let $k\to K$ be a finite field extension and let us consider the automorphism scheme $Aut_kK$. We prove that $Aut_kK$ is a complete $k$-group, i.e., it has trivial centre and any automorphism is inner, except for separable extensions of degree 2 or 6. As a consequence, we obtain for finite field extensions $K_1, K_2$ of $k$, not being separable of degree 2 or 6, the following equivalence: \begin{equation*}K_1\simeq K_2 \Leftrightarrow Aut_kK_1\simeq Aut_kK_2.\end{equation*}
Rarified sums of the Thue-Morse sequence
Michael
Drmota;
Mariusz
Skalba
609-642
Abstract: Let $q$ be an odd number and $S_{q,0}(n)$ the difference between the number of $k<n$, $k\equiv 0\bmod\,q$, with an even binary digit sum and the corresponding number of $k<n$, $k\equiv 0\bmod\,q$, with an odd binary digit sum. A remarkable theorem of Newman says that $S_{3,0}(n)>0$ for all $n$. In this paper it is proved that the same assertion holds if $q$ is divisible by 3 or $q=4^N+1$. On the other hand, it is shown that the number of primes $q\le x$ with this property is $o(x/\log x)$. Finally, analoga for ``higher parities'' are provided.
A hereditarily indecomposable tree-like continuum without the fixed point property
Piotr
Minc
643-654
Abstract: A hereditarily indecomposable tree-like continuum without the fixed point property is constructed. The example answers a question of Knaster and Bellamy.
Closed incompressible surfaces in knot complements
Elizabeth
Finkelstein;
Yoav
Moriah
655-677
Abstract: In this paper we show that given a knot or link $K$ in a $2n$-plat projection with $n\ge 3$ and $m\ge 5$, where $m$ is the length of the plat, if the twist coefficients $a_{i,j}$ all satisfy $|a_{i,j}|>1$ then $S^3-N(K)$ has at least $2n-4$ nonisotopic essential meridional planar surfaces. In particular if $K$ is a knot then $S^3-N(K)$ contains closed incompressible surfaces. In this case the closed surfaces remain incompressible after all surgeries except perhaps along a ray of surgery coefficients in $\mathbb{Z}\oplus\mathbb{Z}$.
Two special cases of Ganea's conjecture
Jeffrey
A.
Strom
679-688
Abstract: Ganea conjectured that for any finite CW complex $X$ and any $k>0$, $\operatorname{cat}(X\times S^k) =\operatorname{cat}(X) + 1$. In this paper we prove two special cases of this conjecture. The main result is the following. Let $X$ be a $(p-1)$-connected $n$-dimensional CW complex (not necessarily finite). We show that if $\operatorname{cat}(X) = \left\lfloor {n \over p} \right\rfloor + 1$ and $n\not\equiv -1 \operatorname{mod} p$(which implies $p>1$), then $\operatorname{cat}(X\times S^k) =\operatorname{cat}(X) +1$. This is proved by showing that $\operatorname{wcat}(X\times S^k) =\operatorname{wcat}(X) + 1$ in a much larger range, and then showing that under the conditions imposed, $\operatorname{cat}(X)=\operatorname{wcat}(X)$. The second special case is an extension of Singhof's earlier result for manifolds.
Products and duality in Waldhausen categories
Michael
S.
Weiss;
Bruce
Williams
689-709
Abstract: The natural transformation $\Xi$ from $\mathbf{L}$-theory to the Tate cohomology of $\mathbb{Z}/2$ acting on $\mathbf{K}$-theory commutes with external products. Corollary: The Tate cohomology of $\mathbb{Z}/2$ acting on the $\mathbf{K}$-theory of any ring with involution is a generalized Eilenberg-Mac Lane spectrum, and it is 4-periodic.
The $\mathcal U$-Lagrangian of a convex function
Claude
Lemaréchal;
François
Oustry;
Claudia
Sagastizábal
711-729
Abstract: At a given point ${\overline{p}}$, a convex function $f$ is differentiable in a certain subspace $\mathcal{U}$ (the subspace along which $\partial f({\overline{p}})$ has 0-breadth). This property opens the way to defining a suitably restricted second derivative of $f$ at ${\overline{p}}$. We do this via an intermediate function, convex on $\mathcal{U}$. We call this function the $\mathcal{U}$-Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization.
Remarks on a Nonlinear Parabolic Equation
Matania
Ben-Artzi;
Jonathan
Goodman;
Arnon
Levy
731-751
Abstract: The equation $u_{t} =\Delta u +\mu |\nabla u |$, $\mu \in \mathbb{R}$, is studied in $\mathbb{R}^{n}$ and in the periodic case. It is shown that the equation is well-posed in $L^{1}$ and possesses regularizing properties. For nonnegative initial data and $\mu <0$ the solution decays in $L^{1}(\mathbb{R}^{n})$ as $t\to \infty$. In the periodic case it tends uniformly to a limit. A consistent difference scheme is presented and proved to be stable and convergent.
Operating points in infinite nonlinear networks approximated by finite networks
Bruce
D.
Calvert;
Armen
H.
Zemanian
753-780
Abstract: Given a nonlinear infinite resistive network, an operating point can be determined by approximating the network by finite networks obtained by shorting together various infinite sets of nodes, and then taking a limit of the nodal potential functions of the finite networks. Initially, by taking a completion of the node set of the infinite network under a metric given by the resistances, limit points are obtained that represent generalized ends, which we call ``terminals,'' of the infinite network. These terminals can be shorted together to obtain a generalized kind of node, a special case of a 1-node. An operating point will involve Kirchhoff's current law holding at 1-nodes, and so the flow of current into these terminals is studied. We give existence and bounds for an operating point that also has a nodal potential function, which is continuous at the 1-nodes. The existence is derived from the said approximations.
$L^p$ estimates for nonvariational hypoelliptic operators with $VMO$ coefficients
Marco
Bramanti;
Luca
Brandolini
781-822
Abstract: Let $X_1,X_2,\ldots,X_q$ be a system of real smooth vector fields, satisfying Hörmander's condition in some bounded domain $\Omega\subset\mathbb{R}^n$ ($n>q$). We consider the differential operator \begin{equation*}\mathcal{L}=\sum _{i=1}^qa_{ij}(x)X_iX_j, \end{equation*} where the coefficients $a_{ij}(x)$ are real valued, bounded measurable functions, satisfying the uniform ellipticity condition: \begin{equation*}\mu|\xi|^2\leq\sum _{i,j=1}^qa_{ij}(x)\xi _i\xi _j\leq\mu^{-1}|\xi|^2 \end{equation*} for a.e. $x\in\Omega$, every $\xi\in\mathbb{R}^q$, some constant $\mu$. Moreover, we assume that the coefficients $a_{ij}$ belong to the space VMO (``Vanishing Mean Oscillation''), defined with respect to the subelliptic metric induced by the vector fields $X_1,X_2,\ldots,X_q$. We prove the following local $\mathcal{L}^p$-estimate: \begin{equation*}\left\|X_iX_jf\right\|_{\mathcal{L}^p(\Omega')}\leq c\left\{\left\|\mathcal{L}f\right\|_{\mathcal{L}^p(\Omega)}+\left\|f\right \|_{\mathcal{L}^p(\Omega)}\right\} \end{equation*} for every $\Omega'\subset\subset\Omega$, $1<p<\infty$. We also prove the local Hölder continuity for solutions to $\mathcal{L}f=g$ for any $g\in\mathcal{L}^p$ with $p$ large enough. Finally, we prove $\mathcal{L}^p$-estimates for higher order derivatives of $f$, whenever $g$ and the coefficients $a_{ij}$ are more regular.
The set of idempotents in the weakly almost periodic compactification of the integers is not closed
B.
Bordbar;
J.
Pym
823-842
Abstract: This paper answers negatively the question of whether the sets of idempotents in the weakly almost periodic compactifications of $(\mathbb{N}, +)$ and $(\mathbb{Z} ,+)$ are closed.
Rates of mixing for potentials of summable variation
Mark
Pollicott
843-853
Abstract: It is well known that for subshifts of finite type and equilibrium measures associated to Hölder potentials we have exponential decay of correlations. In this article we derive explicit rates of mixing for equilibrium states associated to more general potentials.
Banach spaces with the Daugavet property
Vladimir
M.
Kadets;
Roman
V.
Shvidkoy;
Gleb
G.
Sirotkin;
Dirk
Werner
855-873
Abstract: A Banach space $X$ is said to have the Daugavet property if every operator $T:\allowbreak X\to X$ of rank $1$ satisfies $\|\operatorname{Id}+T\| = 1+\|T\|$. We show that then every weakly compact operator satisfies this equation as well and that $X$ contains a copy of $\ell _{1}$. However, $X$ need not contain a copy of $L_{1}$. We also study pairs of spaces $X\subset Y$ and operators $T:\allowbreak X\to Y$ satisfying $\|J+T\|=1+\|T\|$, where $J:\allowbreak X\to Y$ is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with $\|\operatorname{Id}+T\|=1+\|T\|$ is as small as possible and give characterisations in terms of a smoothness condition.
Spin $\mathrm{L}$-functions on $GSp_8$ and $Gsp_{10}$
Daniel
Bump;
David
Ginzburg
875-899
Abstract: The ``spin'' L-function of an automorphic representation of $GSp_{2n}$ is an Euler product of degree $2^{n}$ associated with the spin representation of the L-group $\mathrm{GSpin}(2n+1)$. If $n=4$ or $5$, and the automorphic representation is generic in the sense of having a Whittaker model, the analytic properties of these L-functions are studied by the Rankin-Selberg method.
On the module structure of free Lie algebras
R.
M.
Bryant;
Ralph
Stöhr
901-934
Abstract: We study the free Lie algebra $L$ over a field of non-zero characteristic $p$ as a module for the cyclic group of order $p$ acting on $L$ by cyclically permuting the elements of a free generating set. Our main result is a complete decomposition of $L$ as a direct sum of indecomposable modules.
A quantum octonion algebra
Georgia
Benkart;
José
M.
Pérez-Izquierdo
935-968
Abstract: Using the natural irreducible 8-dimensional representation and the two spin representations of the quantum group $U_{q}$(D$_{4}$) of D$_{4}$, we construct a quantum analogue of the split octonions and study its properties. We prove that the quantum octonion algebra satisfies the q-Principle of Local Triality and has a nondegenerate bilinear form which satisfies a q-version of the composition property. By its construction, the quantum octonion algebra is a nonassociative algebra with a Yang-Baxter operator action coming from the R-matrix of $U_{q}$(D$_{4}$). The product in the quantum octonions is a $U_{q}$(D$_{4}$)-module homomorphism. Using that, we prove identities for the quantum octonions, and as a consequence, obtain at $q = 1$ new ``representation theory'' proofs for very well-known identities satisfied by the octonions. In the process of constructing the quantum octonions we introduce an algebra which is a q-analogue of the 8-dimensional para-Hurwitz algebra.
Functions for parametrization of solutions of an equation in a free monoid
Gennady
S.
Makanin;
Tatiana
A.
Makanina
1-54
Abstract: In this paper we introduce recursive functions \begin{align*}&{}^{\mathbf{Fi}}(x_1,x_2)^{\lambda _1,\dotsc,\lambda _s}\qquad(s\ge 0), &{}^{\mathbf{Th}}(x_1,x_2,x_3)_i^{\lambda _1,\dotsc,\lambda _{2s}} \qquad(i=1,2,3;s\ge 0), &{}^{\mathbf{Ro}}(x_1,x_2,x_3)_i^{\mu _1,\dotsc,\mu _s}\qquad(i=1,2,3;s\ge 0) \end{align*} of the word variables $x_1,x_2,x_3$, natural number variables $\lambda _k$ and variables $\mu _k$ whose values are finite sequences of natural number variables. By means of these functions we give finite expressions for the family of solutions of the equation \begin{displaymath}x_1x_2x_3x_4=\zeta(x_1,x_2,x_3)x_5,\end{displaymath} where $\zeta(x_1,x_2,x_3)$ is an arbitrary word in the alphabet $x_1,x_2,x_3$, in a free monoid.
The metric projection onto the soul
Luis
Guijarro;
Gerard
Walschap
55-69
Abstract: We study geometric properties of the metric projection $\pi :M\to S$ of an open manifold $M$ with nonnegative sectional curvature onto a soul $S$. $\pi$ is shown to be $C^{\infty }$ up to codimension 3. In arbitrary codimensions, small metric balls around a soul turn out to be convex, so that the unit normal bundle of $S$ also admits a metric of nonnegative curvature. Next we examine how the horizontal curvatures at infinity determine the geometry of $M$, and study the structure of Sharafutdinov lines. We conclude with regularity properties of the cut and conjugate loci of $M$.
Even valuations on convex bodies
Daniel
A.
Klain
71-93
Abstract: The notion of even valuation is introduced as a natural generalization of volume on compact convex subsets of Euclidean space. A recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on Grassmannians. This connection can be described in part using generating distributions for symmetric compact convex sets. We also explore some consequences of these characterization results in convex and integral geometry.
Tight closure, plus closure and Frobenius closure in cubical cones
Moira
A.
McDermott
95-114
Abstract: We consider tight closure, plus closure and Frobenius closure in the rings $R = K[[x,y,z]]/(x^{3} + y^{3} +z^{3})$, where $K$ is a field of characteristic $p$ and $p \neq 3$. We use a $\mathbb{Z}_3$-grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring $K[[x,y]]$. We show that Frobenius closure is the same as tight closure in certain classes of ideals when $p \equiv 2 \text{mod} 3$. Since $I^{F} \subseteq IR^{+} \cap R \subseteq I^{*}$, we conclude that $IR^{+} \cap R = I^{*}$ for these ideals. Using injective modules over the ring $R^{\infty }$, the union of all ${p^{e}}$th roots of elements of $R$, we reduce the question of whether $I^{F} = I^{*}$ for $\mathbb{Z}_3$-graded ideals to the case of $\mathbb{Z}_3$-graded irreducible modules. We classify the irreducible $m$-primary $\mathbb{Z}_3$-graded ideals. We then show that $I^{F} = I^{*}$ for most irreducible $m$-primary $\mathbb{Z}_3$-graded ideals in $K[[x,y,z]]/(x^3+y^3+z^3)$, where $K$ is a field of characteristic $p$ and $p \equiv 2 \text{mod} 3$. Hence $I^{*} = IR^{+} \cap R$ for these ideals.
The spectrum of infinite regular line graphs
Tomoyuki
Shirai
115-132
Abstract: Let $G$ be an infinite $d$-regular graph and $L(G)$ its line graph. We consider discrete Laplacians on $G$ and $L(G)$, and show the exact relation between the spectrum of $-\Delta _G$ and that of $-\Delta _{L(G)}$. Our method is also applicable to $(d_1,d_2)$-semiregular graphs, subdivision graphs and para-line graphs.
Cantor sets and numbers with restricted partial quotients
S.
Astels
133-170
Abstract: For $j=1,\dots,k$ let $C_j$ be a Cantor set constructed from the interval $I_j$, and let $\epsilon _j=\pm 1$. We derive conditions under which \begin{equation*}\epsilon _1 C_1+\dots+\epsilon _k C_k = \epsilon _1 I_1+\dots+\epsilon _k I_k \quad\text{and}\quad C_1^{\epsilon _1}\dotsb C_k^{\epsilon _k}= I_1^{\epsilon _1}\dotsb I_k^{\epsilon _k}.\end{equation*} When these conditions do not hold, we derive a lower bound for the Hausdorff dimension of the above sum and product. We use these results to make corresponding statements about the sum and product of sets $F(B_j)$, where $B_j$ is a set of positive integers and $F(B_j)$ is the set of real numbers $x$ such that all partial quotients of $x$, except possibly the first, are members of $B_j$.
Almost normal surfaces in 3-manifolds
Michelle
Stocking
171-207
Abstract: J. H. Rubinstein introduced the theory of almost normal surfaces to solve several homeomorphism problems for 3-manifolds. A. Thompson simplified Rubinstein's algorithm for recognizing the 3-sphere by using almost normal surface theory and thin position. This paper discusses higher genus analogues to A. Thompson's work.
A multiplication in cyclic homology
Kiyoshi
Igusa
209-242
Abstract: We define a multiplication on the cyclic homology of a commutative, cocommutative bialgebra $H$ with ``superproduct.'' In the case when $H$ is a field of characteristic zero the cyclic homology becomes a polynomial algebra in one generator. (The Loday-Quillen multiplication is trivial in that case.)
Absolutely continuous spectrum of perturbed Stark operators
Alexander
Kiselev
243-256
Abstract: We prove new results on the stability of the absolutely continuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stable if the perturbing potential decays at the rate $(1+x) ^{-\frac{1}{3}-\epsilon}$ or if it is continuously differentiable with derivative from the Hölder space $C_{\alpha}(R),$ with any $\alpha>0.$
Orbit equivalence of global attractors of semilinear parabolic differential equations
Bernold
Fiedler;
Carlos
Rocha
257-284
Abstract: We consider global attractors ${\cal A}_f$ of dissipative parabolic equations \begin{equation*}u_t=u_{xx}+f(x,u,u_x) \end{equation*} on the unit interval $0\leq x\leq 1$ with Neumann boundary conditions. A permutation $\pi _f$ is defined by the two orderings of the set of (hyperbolic) equilibrium solutions $u_t\equiv 0$ according to their respective values at the two boundary points $x=0$ and $x=1.$ We prove that two global attractors, ${\cal A}_f$ and ${\cal A}_g$, are globally $C^0$ orbit equivalent, if their equilibrium permutations $\pi _f$ and $\pi _g$ coincide. In other words, some discrete information on the ordinary differential equation boundary value problem $u_t\equiv 0$ characterizes the attractor of the above partial differential equation, globally, up to orbit preserving homeomorphisms.
Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations
José
M.
Arrieta;
Alexandre
N.
Carvalho
285-310
Abstract: We prove a local existence and uniqueness theorem for abstract parabolic problems of the type $\dot x=Ax+f(t,x)$ when the nonlinearity $f$ satisfies certain critical conditions. We apply this abstract result to the Navier-Stokes and heat equations.
Fuchsian Groups, Quasiconformal Groups, and Conical Limit Sets
Peter
W.
Jones;
Lesley
A.
Ward
311-362
Abstract: We construct examples showing that the normalized Lebesgue measure of the conical limit set of a uniformly quasiconformal group acting discontinuously on the disc may take any value between zero and one. This is in contrast to the cases of Fuchsian groups acting on the disc, conformal groups acting discontinuously on the ball in dimension three or higher, uniformly quasiconformal groups acting discontinuously on the ball in dimension three or higher, and discrete groups of biholomorphic mappings acting on the ball in several complex dimensions. In these cases the normalized Lebesgue measure is either zero or one.
Hankel Operators on Bounded Analytic Functions
James
Dudziak;
T.
W.
Gamelin;
Pamela
Gorkin
363-377
Abstract: For $U$ a domain in the complex plane and $g$ a bounded measurable function on $U$, the generalized Hankel operator $S_g$ on $H^\infty(U)$ is the operator of multiplication by $g$ followed by projection into $L^\infty/H^\infty$. Under certain conditions on $U$ we show that either $S_g$ is compact or there is an embedded $\ell^\infty$ on which $S_g$ is bicontinuous. We characterize those $g$'s for which $S_g$ is compact in the case that $U$ is a Behrens roadrunner domain.
Operator ideal norms on $L^p$
L.
Rodríguez-Piazza;
M.
C.
Romero-Moreno
379-395
Abstract: Let $p$ be a real number such that $p \in (1,+\infty )$ and its conjugate exponent $q\not =4,6,8\ldots$. We prove that for an operator $T$ defined on $L^{p}(\lambda )$ with values in a Banach space, the image of the unit ball determines whether $T$ belongs to any operator ideal and its operator ideal norm. We also show that this result fails to be true in the remaining cases of $p$. Finally we prove that when the result holds in finite dimension, the map which associates to the image of the unit ball the operator ideal norm is continuous with respect to the Hausdorff metric.
Vector lattices of weakly compact operators on Banach lattices
Z.
L.
Chen;
A.
W.
Wickstead
397-412
Abstract: A result of Aliprantis and Burkinshaw shows that weakly compact operators from an AL-space into a KB-space have a weakly compact modulus. Groenewegen characterised the largest class of range spaces for which this remains true whenever the domain is an AL-space and Schmidt proved a dual result. Both of these authors used vector-valued integration in their proofs. We give elementary proofs of both results and also characterise the largest class of domains for which the conclusion remains true whenever the range space is a KB-space. We conclude by studying the order structure of spaces of weakly compact operators between Banach lattices to prove results analogous to earlier results of one of the authors for spaces of compact operators.
A local Peter-Weyl theorem
Leonard
Gross
413-427
Abstract: An $Ad\, K$ invariant inner product on the Lie algebra of a compact connected Lie group $K$ extends to a Hermitian inner product on the Lie algebra of the complexified Lie group $K_{c}$. The Laplace-Beltrami operator, $\Delta$, on $K_{c}$ induced by the Hermitian inner product determines, for each number $a>0$, a Green's function $r_{a}$ by means of the identity $(a^{2} -\Delta /4 )^{-1} = r_{a} *$. The Hilbert space of holomorphic functions on $K_{c}$ which are square integrable with respect to $r_{a} (x)dx$ is shown to be finite dimensional. It is spanned by the holomorphic extensions of the matrix elements of those irreducible representations of $K$ whose Casimir operator is appropriately related to $a$.
On the existence of a unipotent support for the irreducible characters of a finite group of Lie type
Meinolf
Geck;
Gunter
Malle
429-456
Abstract: In 1980, Lusztig posed the problem of showing the existence of a unipotent support for the irreducible characters of a finite group of Lie type. This problem was solved by Lusztig in the case where the characteristic of the field over which the group is defined is large enough. The first named author extended this to the case where the characteristic is good. It is the purpose of this paper to remove this condition as well, so that the existence of unipotent supports is established in complete generality.
Finite groups of matrices over group rings
Gerald
Cliff;
Alfred
Weiss
457-475
Abstract: We investigate certain finite subgroups $\Gamma$ of $GL_{n}(\mathbf{Z}\Pi )$, where $\Pi$ is a finite nilpotent group. Such a group $\Gamma$ gives rise to a $\mathbf{Z}[\Gamma \times \Pi]$-module; we study the characters of these modules to limit the structure of $\Gamma$. We also exhibit some exotic subgroups $\Gamma$.
Polynomial Retracts and the Jacobian Conjecture
Vladimir
Shpilrain;
Jie-Tai
Yu
477-484
Abstract: Let $K[x, y]$ be the polynomial algebra in two variables over a field $K$ of characteristic $0$. A subalgebra $R$ of $K[x, y]$ is called a retract if there is an idempotent homomorphism (a retraction, or projection) $\varphi: K[x, y] \to K[x, y]$ such that $\varphi(K[x, y]) = R.$ The presence of other, equivalent, definitions of retracts provides several different methods of studying and applying them, and brings together ideas from combinatorial algebra, homological algebra, and algebraic geometry. In this paper, we characterize all the retracts of $K[x, y]$ up to an automorphism, and give several applications of this characterization, in particular, to the well-known Jacobian conjecture.
Symmetry of properly embedded special Weingarten surfaces in $\mathbf{H}^3$
Ricardo
Sa Earp;
Eric
Toubiana
4693-4711
Abstract: In this paper we prove some existence and uniqueness results about special Weingarten surfaces in hyperbolic space.
Homogeneous spaces with invariant projectively flat affine connections
Hirohiko
Shima
4713-4726
Abstract: We characterize invariant projectively flat affine connections in terms of affine representations of Lie algebras, and show that a homogeneous space admits an invariant projectively flat affine connection if and only if it has an equivariant centro-affine immersion. We give a correspondence between semi-simple symmetric spaces with invariant projectively flat affine connections and central-simple Jordan algebras.
Warped products of metric spaces of curvature bounded from above
Chien-Hsiung
Chen
4727-4740
Abstract: In this work we extend the idea of warped products, which was previously defined on smooth Riemannian manifolds, to geodesic metric spaces and prove the analogue of the theorems on spaces with curvature bounded from above.
A Schwarz lemma for multivalued functions and distortion theorems for Bloch functions with branch points
Ian
Graham;
David
Minda
4741-4752
Abstract: We give a version of the Schwarz lemma for multivalued mappings between hyperbolic plane regions. As in the original work of Nehari on this subject, the derivative must remain bounded near the branch points. Our version of the distance-decreasing principle represents a considerable strengthening of previous results. We apply it to the study of Bloch functions with branch points of specified order. We obtain upper and lower estimates for $|f'|$, an upper estimate for $|f|$, and a lower estimate for the radius of the largest schlicht disk in the image of $f$ centered at $f(0)$. We also obtain some results requiring estimates of second order derivatives of $f$.
Capacity convergence results and applications to a Berstein-Markov inequality
T.
Bloom;
N.
Levenberg
4753-4767
Abstract: Given a sequence $\{E_{j}\}$ of Borel subsets of a given non-pluripolar Borel set $E$ in the unit ball $B$ in $\mathbf{C}^{N}$ with $E \subset \subset B$, we show that the relative capacities $C(E_{j})$ converge to $C(E)$ if and only if the relative (global) extremal functions $u_{E_{j}}^{*}$ ($V_{E_{j}}^{*}$) converge pointwise to $u_{E}^{*}$ ($V_{E}^{*}$). This is used to prove a sufficient mass-density condition on a finite positive Borel measure with compact support $K$ in $\mathbf{C}^{N}$ guaranteeing that the pair $(K,\mu )$ satisfy a Bernstein-Markov inequality. This implies that the $L^{2}-$orthonormal polynomials associated to $\mu$ may be used to recover the global extremal function $V_{K}^{*}$.
On the coefficients of Jacobi sums in prime cyclotomic fields
F.
Thaine
4769-4790
Abstract: Let $p\geq 5$ and $q=pf+1$ be prime numbers, and let $s$ be a primitive root mod $q$. For $1\leq n\leq p-2$, denote by $J_{n}$ the Jacobi sum $-\sum _{k=2}^{q-1}\zeta _p ^{\, \text{ind}_{s}(k)+n\, \text{ind}_{s}(1-k)}$. We study the integers $d_{n,k}$ such that $J_{n}=\sum _{k=0}^{p-1}d_{n,k}\zeta _p ^{k}$ and $\sum _{k=0}^{p-1}d_{n,k}=1$. We give a list of properties that characterize these coefficients. Then we show some of their applications to the study of the arithmetic of $\mathbb {Z} [\zeta _p +\zeta _p ^{-1}]$, in particular to the study of Vandiver's conjecture. For $m\in \mathbb {Z}-q\mathbb {Z}$, let $\rho _{n}(m)$ be the number of distinct roots of $X^{n+1}-X^{n}+m$ in $\mathbb {Z}/q\mathbb {Z}$. We show that $d_{n,k}=f-\sum _{a=0}^{f-1}\rho _{n}(s^{k+pa})$. We give closed formulas for the numbers $d_{1,k}$ and $d_{2,k}$ in terms of quadratic and cubic power residue symbols mod $q$.
Möbius-like groups of homeomorphisms of the circle
Natasa
Kovacevic
4791-4822
Abstract: An orientation preserving homeomorphism of $S^1$ is Möbius-like if it is conjugate in $Homeo(S^1{1})$ to a Möbius transformation. Our main result is: given a (noncyclic) group $G\hookrightarrow Homeo_{+}(S^{1})$ whose every element is Möbius-like, if $G$ has at least one global fixed point, then the whole group $G$ is conjugate in $Homeo(S^1{1})$ to a Möbius group if and only if the limit set of $G$ is all of $S^1$. Moreover, we prove that if the limit set of $G$ is not all of $S^1$, then after identifying some closed subintervals of $S^1$ to points, the induced action of $G$ is conjugate to an action of a Möbius group. Said differently, $G$ is obtained from a group which is conjugate to a Möbius group, by a sort of generalized Denjoy's insertion of intervals. In this case $G$ is isomorphic, as a group, to a Möbius group. This result has another interpretation. Namely, we prove that a group $G$ of orientation preserving homeomorphisms of $\boldsymbol{R}$ whose every element can be conjugated to an affine map (i.e., a map of the form $x \mapsto ax + b$) is just the conjugate of a group of affine maps, up to a certain insertion of intervals. In any case, the group structure of $G$ is the one of an affine group.
Examples of Möbius-like groups which are not Möbius groups
Natasa
Kovacevic
4823-4835
Abstract: In this paper we give two basic constructions of groups with the following properties: (a) $G \hookrightarrow {Homeo_{+}(S^{1})}$, i.e., the group $G$ is acting by orientation preserving homeomorphisms on ${S^{1}}$; (b) every element of $G$ is Möbius-like; (c) ${L(G)}= {S^{1}}$, where ${L(G)}$ denotes the limit set of $G$; (d) $G$ is discrete; (e) $G$ is not a conjugate of a Möbius group. Both constructions have the same basic idea (inspired by Denjoy): we start with a Möbius group $H$ (of a certain type) and then we change the underlying circle upon which $H$ acts by inserting some closed intervals and then extending the group action over the new circle. We denote this new action by $\overline{H}$. Now we form a new group $G$ which is generated by all of $\overline{H}$ and an additional element $g$ whose existence is enabled by the inserted intervals. This group $G$ has all the properties (a) through (e).
Homology of the universal covering of a co-H-space
Norio
Iwase;
Shiroshi
Saito;
Toshio
Sumi
4837-4846
Abstract: The problem 10 posed by Tudor Ganea is known as the Ganea conjecture on a co-H-space, a space with co-H-structure. Many efforts are devoted to show the Ganea conjecture under additional assumptions on the given co-H-structure. In this paper, we show a homological property of co-H-spaces in a slightly general situation. As a corollary, we get the Ganea conjecture for spaces up to dimension 3.
Steepest descent evolution equations: asymptotic behavior of solutions and rate of convergence
R.
Cominetti;
O.
Alemany
4847-4860
Abstract: We study the asymptotic behavior of the solutions of evolution equations of the form $\dot u(t)\in -\partial f(u(t),r(t))$, where $f(\cdot,r)$ is a one-parameter family of approximations of a convex function $f(\cdot)$ we wish to minimize. We investigate sufficient conditions on the parametrization $r(t)$ ensuring that the integral curves $u(t)$ converge when $t\rightarrow\infty$ towards a particular minimizer $u_\infty$ of $f$. The speed of convergence is also investigated, and a result concerning the continuity of the limit point $u_\infty$ with respect to the parametrization $r(\cdot)$ is established. The results are illustrated on different approximation methods. In particular, we present a detailed application to the logarithmic barrier in linear programming.
Boundary value problems on infinite intervals
Jan
Andres;
Grzegorz
Gabor;
Lech
Górniewicz
4861-4903
Abstract: We present two methods, both based on topological ideas, to the solvability of boundary value problems for differential equations and inclusions on infinite intervals. In the first one, related to the rich family of asymptotic problems, we generalize and extend some statements due to the Florence group of mathematicians Anichini, Cecchi, Conti, Furi, Marini, Pera, and Zecca. Thus, their conclusions for differential systems are as well true for inclusions; all under weaker assumptions (for example, the convexity restrictions in the Schauder linearization device can be avoided). In the second, dealing with the existence of bounded solutions on the positive ray, we follow and develop the ideas of Andres, Górniewicz, and Lewicka, who considered periodic problems. A special case of these results was previously announced by Andres. Besides that, the structure of solution sets is investigated. The case of l.s.c. right hand sides of differential inclusions and the implicit differential equations are also considered. The large list of references also includes some where different techniques (like the Conley index approach) have been applied for the same goal, allowing us to envision the full range of recent attacks on the problem stated in the title.
Complex symplectic geometry with applications to ordinary differential operators
W.
N.
Everitt;
L.
Markus
4905-4945
Abstract: Complex symplectic spaces, and their Lagrangian subspaces, are defined in accord with motivations from Lagrangian classical dynamics and from linear ordinary differential operators; and then their basic algebraic properties are established. After these purely algebraic developments, an Appendix presents a related new result on the theory of self-adjoint operators in Hilbert spaces, and this provides an important application of the principal theorems.
Behavior near the boundary of positive solutions of second order parabolic equations. II
E.
B.
Fabes;
M.
V.
Safonov;
Yu
Yuan
4947-4961
Abstract: A boundary backward Harnack inequality is proved for positive solutions of second order parabolic equations in non-divergence form in a bounded cylinder $Q=\Omega \times \left(0,T\right)$ which vanish on $\partial _xQ=\partial\Omega\times \left(0,T\right)$, where $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^n$. This inequality is applied to the proof of the Hölder continuity of the quotient of two positive solutions vanishing on a portion of $\partial _xQ.$
Lower bounds for the absolute value of random polynomials on a neighborhood of the unit circle
S.
V.
Konyagin;
W.
Schlag
4963-4980
Abstract: Let $T(x)=\sum _{j=0}^{n-1}\pm e^{ijx}$ where $\pm$ stands for a random choice of sign with equal probability. The first author recently showed that for any $\epsilon>0$ and most choices of sign, $\min _{x\in[0,2\pi)}|T(x)|<n^{-1/2+\epsilon}$, provided $n$ is large. In this paper we show that the power $n^{-1/2}$ is optimal. More precisely, for sufficiently small $\epsilon>0$ and large $n$ most choices of sign satisfy $\min _{x\in[0,2\pi)}|T(x)|> \epsilon n^{-1/2}$. Furthermore, we study the case of more general random coefficients and applications of our methods to complex zeros of random polynomials.
Optimal individual stability estimates for $C_0$-semigroups in Banach spaces
Volker
Wrobel
4981-4994
Abstract: In a previous paper we proved that the asymptotic behavior of a $C_0$-semigroup is completely determined by growth properties of the resolvent of its generator and geometric properties of the underlying Banach space as described by its Fourier type. The given estimates turned out to be optimal. The method of proof uses complex interpolation theory and reflects the full semigroup structure. In the present paper we show that these uniform estimates have to be replaced by weaker ones, if individual initial value problems and local resolvents are considered because the full semigroup structure is lacking. In a different approach this problem has also been studied by Huang and van Neerven, and a part of our straightforward estimates can be inferred from their results. We mainly stress upon the surprising fact that these estimates turn out to be optimal. Therefore it is not possible to obtain the optimal uniform estimates mentioned above from individual ones. Concerning Hardy-abscissas, individual orbits and their local resolvents behave as badly as general vector valued functions and their Laplace-transforms. This is in strict contrast to the uniform situation of a $C_0$-semigroup itself and the resolvent of its generator where a simple dichotomy holds true.
Conformal iterated function systems with applications to the geometry of continued fractions
R.
Daniel
Mauldin;
Mariusz
Urbanski
4995-5025
Abstract: In this paper we obtain some results about general conformal iterated function systems. We obtain a simple characterization of the packing dimension of the limit set of such systems and introduce some special systems which exhibit some interesting behavior. We then apply these results to the set of values of real continued fractions with restricted entries. We pay special attention to the Hausdorff and packing measures of these sets. We also give direct interpretations of these measure theoretic results in terms of the arithmetic density properties of the set of allowed entries.
When almost multiplicative morphisms are close to homomorphisms
Huaxin
Lin
5027-5049
Abstract: It is shown that approximately multiplicative contractive positive morphisms from $C(X)$ (with dim $X\le 2$) into a simple $C^*$-algebra $A$ of real rank zero and of stable rank one are close to homomorphisms, provided that certain $K$-theoretical obstacles vanish. As a corollary we show that a homomorphism $h: C(X)\to A$ is approximated by homomorphisms with finite dimensional range, if $h$ gives no $K$-theoretical obstacle.
On the Normal Subgroups of $G_2(A)$
Douglas
L.
Costa;
Gordon
E.
Keller
5051-5088
Abstract: We give a characterization theorem for the $E(A)$-normalized subgroups of $G_2(A)$, where $A$ is any commutative ring. This is the last of the simple Chevalley-Demazure group-schemes for which such a theorem is lacking.
*-polynomial identities of matrices with the transpose involution: The low degrees
Alain
D'Amour;
Michel
Racine
5089-5106
Abstract: In this paper, we investigate $*$-polynomial identities of minimal degree for the algebra of $n\times n$ matrices over a field, where $n<5$ and $*$ is the transpose involution. We first present some basic generators, and then proceed to show that all other minimal degree identities can be derived from those.
Dimension formula for graded Lie algebras and its applications
Seok-Jin
Kang;
Myung-Hwan
Kim
4281-4336
Abstract: In this paper, we investigate the structure of infinite dimensional Lie algebras $L=\bigoplus _{\alpha \in \Gamma } L_{\alpha }$ graded by a countable abelian semigroup $\Gamma$ satisfying a certain finiteness condition. The Euler-Poincaré principle yields the denominator identities for the $\Gamma$-graded Lie algebras, from which we derive a dimension formula for the homogeneous subspaces $L_{\alpha }$ $(\alpha \in \Gamma )$. Our dimension formula enables us to study the structure of the $\Gamma$-graded Lie algebras in a unified way. We will discuss some interesting applications of our dimension formula to the various classes of graded Lie algebras such as free Lie algebras, Kac-Moody algebras, and generalized Kac-Moody algebras. We will also discuss the relation of graded Lie algebras and the product identities for formal power series.
Limit theorems for the convex hull of random points in higher dimensions
Irene
Hueter
4337-4363
Abstract: We give a central limit theorem for the number $N_n$ of vertices of the convex hull of $n$ independent and identically distributed random vectors, being sampled from a certain class of spherically symmetric distributions in $\mathbb{R}^d \; (d> 1),$ that includes the normal family. Furthermore, we prove that, among these distributions, the variance of $N_n$ exhibits the same order of magnitude as the expectation as $n \rightarrow \infty.$ The main tools are Poisson approximation of the point process of vertices of the convex hull and (sub/super)-martingales.
The $\overline{\partial}$ problem on domains with piecewise smooth boundaries with applications
Joachim
Michel;
Mei-Chi
Shaw
4365-4380
Abstract: Let $\Omega$ be a bounded domain in $\mathbb C^n$ such that $\Omega$ has piecewise smooth boudnary. We discuss the solvability of the Cauchy-Riemann equation \begin{equation*}\overline{\partial}u=\alpha\quad \text{in}\quad \Omega\tag{0.1} \end{equation*} where $\alpha$ is a smooth $\overline{\partial}$-closed $(p,q)$ form with coefficients $C^\infty$ up to the bundary of $\Omega$, $0\le p\le n$ and $1\le q\le n$. In particular, Equation (0.1) is solvable with $u$ smooth up to the boundary (for appropriate degree $q)$ if $\Omega$ satisfies one of the following conditions: i) $\Omega$ is the transversal intersection of bounded smooth pseudoconvex domains. ii) $\Omega=\Omega _1\setminus\overline\Omega _2$ where $\Omega _2$ is the union of bounded smooth pseudoconvex domains and $\Omega _1$ is a pseudoconvex convex domain with a piecewise smooth boundary. iii) $\Omega=\Omega _1\setminus\overline{\Omega}_2$ where $\Omega _2$ is the intersection of bounded smooth pseudoconvex domains and $\Omega _1$ is a pseudoconvex domain with a piecewise smooth boundary. The solvability of Equation (0.1) with solutions smooth up to the boundary can be used to obtain the local solvability for $\overline{\partial}_b$ on domains with piecewise smooth boundaries in a pseudoconvex manifold.
Arithmetically Buchsbaum divisors on varieties of minimal degree
Uwe
Nagel
4381-4409
Abstract: In this paper we consider integral arithmetically Buchsbaum subschemes of projective space. First we show that arithmetical Buchsbaum varieties of sufficiently large degree have maximal Castelnuovo-Mumford regularity if and only if they are divisors on a variety of minimal degree. Second we determine all varieties of minimal degree and their divisor classes which contain an integral arithmetically Buchsbaum subscheme. Third we investigate these varieties. In particular, we compute their Hilbert function, cohomology modules and (often) their graded Betti numbers and obtain an existence result for smooth arithmetically Buchsbaum varieties.
Some interesting nonspherical tempered representations of graded Hecke algebras
C.
Kriloff
4411-4428
Abstract: Lusztig's presentation of the graded Hecke algebra in terms of generators and relations allows for the definition of algebras associated to noncrystallographic root systems. The representation theory of general graded Hecke algebras is investigated, the expected number of tempered representations for $\mathbb{H}(H_3)$ are accounted for, and it is shown that one of these representations has the unexpected property of being nonspherical despite being the only tempered representation appearing at its infinitesimal character. Additional nonspherical tempered representations of $\mathbb{H}(H_4)$ are also included.
A Littlewood-Richardson rule for factorial Schur functions
Alexander
I.
Molev;
Bruce
E.
Sagan
4429-4443
Abstract: We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Applications to Capelli operators and quantum immanants are also given.
Class numbers of cyclotomic function fields
Li
Guo;
Linghsueh
Shu
4445-4467
Abstract: Let $q$ be a prime power and let ${\mathbb F}_q$ be the finite field with $q$ elements. For each polynomial $Q(T)$ in ${\mathbb F}_q [T]$, one could use the Carlitz module to construct an abelian extension of ${\mathbb F}_q (T)$, called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of ${\mathbb F}_q(T)$, similar to the role played by cyclotomic number fields for abelian extensions of ${\mathbb Q}$. We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in ${\mathbb F}_q [T]$. Two types of properties are obtained for the $l$-parts of the class numbers of the fields in this tower, for a fixed prime number $l$. One gives congruence relations between the $l$-parts of these class numbers. The other gives lower bound for the $l$-parts of these class numbers.
On 2-Generator Subgroups of SO(3)
Charles
Radin;
Lorenzo
Sadun
4469-4480
Abstract: We classify all subgroups of $SO(3)$ that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of $\pi$. In all cases we give a presentation of the subgroup. In most cases the subgroup is the free product, or the amalgamated free product, of cyclic groups or dihedral groups. The relations between the generators are all simple consequences of standard facts about rotations by $\pi$ and $\pi/2$. Embedded in the subgroups are explicit free groups on 2 generators, as used in the Banach-Tarski paradox.
Hecke algebras and cohomotopical Mackey functors
Norihiko
Minami
4481-4513
Abstract: In this paper, we define the concept of the cohomotopical Mackey functor, which is more general than the usual cohomological Mackey functor, and show that Hecke algebra techniques are applicable to cohomotopical Mackey functors. Our theory is valid for any (possibly infinite) discrete group. Some applications to topology are also given.
Densities of idempotent measures and large deviations
Marianne
Akian
4515-4543
Abstract: Considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by Maslov. Then, idempotent measures or integrals with density correspond to supremums of functions for the partial order relation induced by the idempotent structure. In this paper, we give conditions under which an idempotent measure has a density and show by many examples that they are often satisfied. These conditions depend on the lattice structure of the semiring and on the Boolean algebra in which the measure is defined. As an application, we obtain a necessary and sufficient condition for a family of probabilities to satisfy the large deviation principle.
Quadratic differential equations in $\mathbb{Z}_2$-graded algebras
Nora
C.
Hopkins;
Michael
K.
Kinyon
4545-4559
Abstract: Quadratic differential equations whose associated algebra has an automorphism of order two are studied. Under hypotheses that naturally generalize the cases where the even or odd part of the algebra is one dimensional, the following are examined: structure theory of the associated algebra (ideal structure, simplicity, solvability, and nilpotence), derivations and first integrals, trajectories given by derivations, and Floquet decompositions.
The support of the equilibrium measure in the presence of a monomial external field on $[-1,1]$
S.
B.
Damelin;
A.
B. J.
Kuijlaars
4561-4584
Abstract: The support of the equilibrium measure associated with an external field of the form $Q(x) = - cx^{2m+1}$, $x \in [-1,1]$, with $c > 0$ and $m$ a positive integer is investigated. It is shown that the support consists of at most two intervals. This resolves a question of Deift, Kriecherbauer and McLaughlin.
A sharp version of Zhang's theorem on truncating sequences of gradients
Stefan
Müller
4585-4597
Abstract: Let $K \subset \mathbf{R}^{mn}$ be a compact and convex set of $m \times n$ matrices and let $\{u_j\}$ be a sequence in $W_{\operatorname{loc}} ^{1,1}(\mathbf{R}^n;\mathbf{R}^m)$ that converges to $K$ in the mean, i.e. $\int _{\mathbf{R}^n} {\operatorname{dist}} (Du_j, K) \to 0$. I show that there exists a sequence $v_j$ of Lipschitz functions such that $\parallel {\operatorname{dist}} (Dv_j, K)\parallel _\infty \to 0$ and $\mathcal{L}^n (\{u_j \not= v_j\}) \to 0$. This refines a result of Kewei Zhang (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), 313-326), who showed that one may assume $\parallelDv_j \parallel _\infty \le C$. Applications to gradient Young measures and to a question of Kinderlehrer and Pedregal (Arch. Rational Mech. Anal. 115 (1991), 329-365) regarding the approximation of $\mathbf{R} \cup \{+\infty\}$ valued quasiconvex functions by finite ones are indicated. A challenging open problem is whether convexity of $K$ can be replaced by quasiconvexity.
A classification of Baire-1 functions
P.
Kiriakouli
4599-4609
Abstract: In this paper we give some topological characterizations of bounded Baire-1 functions using some ranks. Kechris and Louveau classified the Baire-1 functions to the subclasses $\mathbb{B}^\xi _1(K)$ for every $\xi<\omega _1$ (where $K$ is a compact metric space). The first basic result of this paper is that for $\xi<\omega$, $f\in \mathbb{B}^{\xi+1}_1(K)$ iff there exists a sequence $(f_n)$ of differences of bounded semicontinuous functions on $K$ with $f_n\to f$ pointwise and $\gamma((f_n))\le \omega^\xi$ (where ``$\gamma$'' denotes the convergence rank). This extends the work of Kechris and Louveau who obtained this result for $\xi=1$. We also show that the result fails for $\xi\ge \omega$. The second basic result of the paper involves the introduction of a new ordinal-rank on sequences $(f_n)$, called the $\delta$-rank, which is smaller than the convergence rank $\gamma$. This result yields the following characterization of $\mathbb{B}^\xi _1(K): f\in \mathbb{B}^\xi _1(K)$ iff there exists a sequence $(f_n)$ of continuous functions with $f_n\to f$ pointwise and $\delta((f_n))\le \omega^{\xi-1}$ if $1\le \xi<\omega$, resp. $\delta((f_n))\le \omega^\xi$ if $\xi\ge \omega$.
An $L^p$ a priori estimate for the Tricomi equation in the upper half space
Jong
Uhn
Kim
4611-4628
Abstract: We establish an $L^{p}$ a priori estimate for the Tricomi equation. Our main tool is Mihlin's multiplier theorem combined with well-known estimates of the Newtonian potential.
Admissibility of Weights on Non-normed $*$-Algebras
S.
J.
Bhatt;
A.
Inoue;
H.
Ogi
4629-4656
Abstract: The notion of weights on (topological) $*$-algebras is defined and studied. The primary purpose is to define the notions of admissibility and approximate admissibility of weights, and to investigate when a weight is admissible or approximately admissible. The results obtained are applied to vector weights and tracial weight on unbounded operator algebras, as well as to weights on smooth subalgebras of a C$^*$-algebra.
Quasitriangular $+$ small compact $=$ strongly irreducible
You
Qing
Ji
4657-4673
Abstract: Let $T$ be a bounded linear operator acting on a separable infinite dimensional Hilbert space. Let $\epsilon$ be a positive number. In this article, we prove that the perturbation of $T$ by a compact operator $K$ with $\Vert K\Vert <\epsilon$ can be strongly irreducible if $T$ is a quasitriangular operator with the spectrum $\sigma (T)$ connected. The Main Theorem of this article nearly answers the question below posed by D. A. Herrero. Suppose that $T$ is a bounded linear operator acting on a separable infinite dimensional Hilbert space with $\sigma (T)$ connected. Let $\epsilon >0$ be given. Is there a compact operator $K$ with $\Vert K\Vert <\epsilon$ such that $T+K$ is strongly irreducible?
Decomposition of $B(G)$
Tianxuan
Miao
4675-4692
Abstract: For any locally compact group $G$, let $A(G)$ and $B(G)$ be the Fourier and the Fourier-Stieltjes algebras of $G$, respectively. $B(G)$ is decomposed as a direct sum of $A(G)$ and $B^{s}(G)$, where $B^{s}(G)$ is a subspace of $B(G)$ consisting of all elements $b\in B(G)$ that satisfy the property: for any $\epsilon > 0$ and any compact subset $K\subset G$, there is an $f\in L^{1}(G)$ with $\Vert f\Vert _{C^{*}(G)} \le 1$ and $supp(f) \subset K^{c}$ such that $\vert \langle f, b \rangle \vert > \Vert b\Vert - \epsilon .$ $A(G)$ is characterized by the following: an element $b\in B(G)$ is in $A(G)$ if and only if, for any $\epsilon > 0,$ there is a compact subset $K\subset G$ such that $\vert \langle f, b \rangle \vert < \epsilon$ for all $f\in L^{1}(G)$ with $\Vert f\Vert _{C^{*}(G)} \le 1$ and $supp(f) \subset K^{c}$. Note that we do not assume the amenability of $G$. Consequently, we have $\Vert 1 + a\Vert = 1 + \Vert a\Vert$ for all $a\in A(G)$ if $G$ is noncompact. We will apply this characterization of $B^{s}(G)$ to investigate the general properties of $B^{s}(G)$ and we will see that $B^{s}(G)$ is not a subalgebra of $B(G)$ even for abelian locally compact groups. If $G$ is an amenable locally compact group, then $B^{s}(G)$ is the subspace of $B(G)$ consisting of all elements $b\in B(G)$ with the property that for any compact subset $K\subseteq G$, $\Vert b\Vert = \sup \, \{ \, \Vert a b\Vert : \, a\in A(G), \; supp(a) \subseteq K^{c} \; \text{ and } \; \Vert a\Vert \le 1 \, \}$.
Overgroups of irreducible linear groups, II
Ben
Ford
3869-3913
Abstract: Determining the subgroup structure of algebraic groups (over an algebraically closed field $K$ of arbitrary characteristic) often requires an understanding of those instances when a group $Y$ and a closed subgroup $G$ both act irreducibly on some module $V$, which is rational for $G$ and $Y$. In this paper and in Overgroups of irreducible linear groups, I (J. Algebra 181 (1996), 26-69), we give a classification of all such triples $(G,Y,V)$ when $G$ is a non-connected algebraic group with simple identity component $X$, $V$ is an irreducible $G$-module with restricted $X$-high weight(s), and $Y$ is a simple algebraic group of classical type over $K$ sitting strictly between $X$ and $% \operatorname{SL}(V)$.
Rates of convergence of diffusions with drifted Brownian potentials
Yueyun
Hu;
Zhan
Shi;
Marc
Yor
3915-3934
Abstract: We are interested in the asymptotic behaviour of a diffusion process with drifted Brownian potential. The model is a continuous time analogue to the random walk in random environment studied in the classical paper of Kesten, Kozlov, and Spitzer. We not only recover the convergence of the diffusion process which was previously established by Kawazu and Tanaka, but also obtain all the possible convergence rates. An interesting feature of our approach is that it shows a clear relationship between drifted Brownian potentials and Bessel processes.
The space of complete minimal surfaces with finite total curvature as lagrangian submanifold
Joaquín
Pérez;
Antonio
Ros
3935-3952
Abstract: The space ${\cal M}$ of nondegenerate, properly embedded minimal surfaces in ${\mathbb R}^3$ with finite total curvature and fixed topology is an analytic lagrangian submanifold of ${\mathbb C}^n$, where $n$ is the number of ends of the surface. In this paper we give two expressions, one integral and the other pointwise, for the second fundamental form of this submanifold. We also consider the compact boundary case, and we show that the space of stable nonflat minimal annuli that bound a fixed convex curve in a horizontal plane, having a horizontal end of finite total curvature, is a locally convex curve in the plane ${\mathbb C}$.
Morse homology for generating functions of Lagrangian submanifolds
Darko
Milinkovic
3953-3974
Abstract: The purpose of the paper is to give an alternative construction and the proof of the main properties of symplectic invariants developed by Viterbo. Our approach is based on Morse homology theory. This is a step towards relating the ``finite dimensional'' symplectic invariants constructed via generating functions to the ``infinite dimensional'' ones constructed via Floer theory in Y.-G. Oh, Symplectic topology as the geometry of action functional. I, J. Diff. Geom. 46 (1997), 499-577.
Extendability of Large-Scale Lipschitz Maps
Urs
Lang
3975-3988
Abstract: Let $X,Y$ be metric spaces, $S$ a subset of $X$, and $f \colon S \to Y$ a large-scale lipschitz map. It is shown that $f$ possesses a large-scale lipschitz extension $\bar f \colon X \to Y$ (with possibly larger constants) if $Y$ is a Gromov hyperbolic geodesic space or the cartesian product of finitely many such spaces. No extension exists, in general, if $Y$ is an infinite-dimensional Hilbert space. A necessary and sufficient condition for the extendability of a lipschitz map $f \colon S \to Y$ is given in the case when $X$ is separable and $Y$ is a proper, convex geodesic space.
Chern classes for singular hypersurfaces
Paolo
Aluffi
3989-4026
Abstract: We prove a formula expressing the Chern-Schwartz-MacPherson class of a hypersurface in a nonsingular variety as a variation on another definition of the homology Chern class of singular varieties, introduced by W. Fulton; and we discuss the relation between these classes and others, such as Mather's Chern class and the $\mu$-class we introduced in previous work.
On the depth of the tangent cone and the growth of the Hilbert function
Juan
Elias
4027-4042
Abstract: For a $d-$dimensional Cohen-Macaulay local ring $(R, \mathbf{m})$ we study the depth of the associated graded ring of $R$ with respect to an $ \textbf{ m}$-primary ideal $I$ in terms of the Vallabrega-Valla conditions and the length of $I^{t+1}/JI^{t}$, where $J$ is a $J$ minimal reduction of $I$ and $t\ge 1$. As a corollary we generalize Sally's conjecture on the depth of the associated graded ring with respect to a maximal ideal to $\mathbf{m}$-primary ideals. We also study the growth of the Hilbert function.
Alexander invariants of complex hyperplane arrangements
Daniel
C.
Cohen;
Alexander
I.
Suciu
4043-4067
Abstract: Let $\mathcal{A}$ be an arrangement of $n$ complex hyperplanes. The fundamental group of the complement of $\mathcal{A}$ is determined by a braid monodromy homomorphism, $\alpha:F_{s}\to P_{n}$. Using the Gassner representation of the pure braid group, we find an explicit presentation for the Alexander invariant of $\mathcal{A}$. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of $\mathcal{A}$. We also provide a combinatorial criterion for when these lower bounds are attained.
Inverse eigenvalue problems on directed graphs
Robert
Carlson
4069-4088
Abstract: The differential operators $iD$ and $-D^2 - p$ are constructed on certain finite directed weighted graphs. Two types of inverse spectral problems are considered. First, information about the graph weights and boundary conditions is extracted from the spectrum of $-D^2$. Second, the compactness of isospectral sets for $-D^2 - p$ is established by computation of the residues of the zeta function.
Characterization of Smoothness of Multivariate Refinable Functions in Sobolev Spaces
Rong-Qing
Jia
4089-4112
Abstract: Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the smoothness properties of multivariate refinable functions in Sobolev spaces. We characterize the optimal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant subspace. Several examples are provided to illustrate the general theory.
Remarks about global analytic hypoellipticity
Adalberto
P.
Bergamasco
4113-4126
Abstract: We present a characterization of the operators \begin{displaymath}L=\partial/\partial t+(a(t)+ib(t))\partial/\partial x\end{displaymath} which are globally analytic hypoelliptic on the torus. We give information about the global analytic hypoellipticity of certain overdetermined systems and of sums of squares.
Norm estimates and representations for Calderón-Zygmund operators using averages over starlike sets
David
K.
Watson;
Richard
L.
Wheeden
4127-4171
Abstract: We show that homogeneous singular integrals may be represented in terms of averages over starlike sets. This permits us to use the geometry of starlike sets to derive operator-specific weighted norm inequalities.
The $C^1$ closing lemma for nonsingular endomorphisms equivariant under free actions of finite groups
Xiaofeng
Wang;
Duo
Wang
4173-4182
Abstract: In this paper a closing lemma for $C^1$ nonsingular endomorphisms equivariant under free actions of finite-groups is proved. Hence a recurrent trajectory, as well as all of its symmetric conjugates, of a $C^1$ nonsingular endomorphism equivariant under a free action of a finite group can be closed up simultaneously by an arbitrarily small $C^1$ equivariant perturbation.
Minimal lattice-subspaces
Ioannis
A.
Polyrakis
4183-4203
Abstract: In this paper the existence of minimal lattice-subspaces of a vector lattice $E$ containing a subset $B$ of $E_+$ is studied (a lattice-subspace of $E$ is a subspace of $E$ which is a vector lattice in the induced ordering). It is proved that if there exists a Lebesgue linear topology $\tau$ on $E$ and $E_+$ is $\tau$-closed (especially if $E$ is a Banach lattice with order continuous norm), then minimal lattice-subspaces with $\tau$-closed positive cone exist (Theorem 2.5). In the sequel it is supposed that $B=\{x_1,x_2,\ldots,x_n\}$ is a finite subset of $C_+(\Omega)$, where $\Omega$ is a compact, Hausdorff topological space, the functions $x_i$ are linearly independent and the existence of finite-dimensional minimal lattice-subspaces is studied. To this end we define the function $\beta(t) = \frac{r(t)}{\|r(t)\|_1}$ where $r(t) = \big(x_1(t),x_2(t),\ldots,x_n(t)\big)$. If $R(\beta)$ is the range of $\beta$ and $K$ the convex hull of the closure of $R(\beta)$, it is proved: (i) There exists an $m$-dimensional minimal lattice-subspace containing $B$ if and only if $K$ is a polytope of $\mathbb{R}^n$ with $m$ vertices (Theorem 3.20). (ii) The sublattice generated by $B$ is an $m$-dimensional subspace if and only if the set $R(\beta)$ contains exactly $m$ points (Theorem 3.7). This study defines an algorithm which determines whether a finite-dimensional minimal lattice-subspace (sublattice) exists and also determines these subspaces.
Weighted Laplace transforms and Bessel functions on Hermitian symmetric spaces
Hongming
Ding
4205-4243
Abstract: This paper defines $\pi$-weighted Laplace transforms on the spaces of $\pi$-covariant functions. By the inverse Laplace transform we define operator-valued Bessel functions. We also study the holomorphic discrete series of the automorphism group of a Siegel domain of type II.
Classification of one $\textsf K$-type representations
Dan
Barbasch;
Allen
Moy
4245-4261
Abstract: Suppose $G$ is a simple reductive $p$-adic group with Weyl group $W$. We give a classification of the irreducible representations of $W$ which can be extended to real hermitian representations of the associated graded Hecke algebra $\mathbb{H}$. Such representations correspond to unitary representations of $G$ which have a small spectrum when restricted to an Iwahori subgroup.
A $K$ counterexample machine
Christopher
Hoffman
4263-4280
Abstract: We present a general method for constructing families of measure preserving transformations which are $K$ and loosely Bernoulli with various ergodic theoretical properties. For example, we construct two $K$ transformations which are weakly isomorphic but not isomorphic, and a $K$ transformation with no roots. Ornstein's isomorphism theorem says families of Bernoulli shifts cannot have these properties. The construction uses a combination of properties from maps constructed by Ornstein and Shields, and Rudolph, and reduces the question of isomorphism of two transformations to the conjugacy of two related permutations.
Nonlinear equations and weighted norm inequalities
N.
J.
Kalton;
I.
E.
Verbitsky
3441-3497
Abstract: We study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities. In particular, we obtain explicit criteria for the solvability of the Dirichlet problem \begin{equation*}\begin{split} -& \Delta u = v \, u^{q} + w, \quad u \ge 0 \quad \text{on} \quad \Omega , &u = 0 \quad \text{on} \quad \partial \Omega , \end{split}\end{equation*} on a regular domain $\Omega$ in $\mathbf{R}^{n}$ in the ``superlinear case'' $q > 1$. The coefficients $v, w$ are arbitrary positive measurable functions (or measures) on $\Omega$. We also consider more general nonlinear differential and integral equations, and study the spaces of coefficients and solutions naturally associated with these problems, as well as the corresponding capacities. Our characterizations of the existence of positive solutions take into account the interplay between $v$, $w$, and the corresponding Green's kernel. They are not only sufficient, but also necessary, and are established without any a priori regularity assumptions on $v$ and $w$; we also obtain sharp two-sided estimates of solutions up to the boundary. Some of our results are new even if $v \equiv 1$ and $\Omega$ is a ball or half-space. The corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so-called $3 G$-inequality by an elementary ``integration by parts'' argument. This also gives a new unified proof for some classical inequalities including the Carleson measure theorem for Poisson integrals and trace inequalities for Riesz potentials and Green potentials.
On minimal parabolic functions and time-homogeneous parabolic $h$-transforms
Krzysztof
Burdzy;
Thomas
S.
Salisbury
3499-3531
Abstract: Does a minimal harmonic function $h$ remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin semi-infinite tubes $D\subset \mathbb{R}^{d}$ of variable width and minimal harmonic functions $h$ corresponding to the boundary point of $D$ ``at infinity.'' Suppose $f(u)$ is the width of the tube $u$ units away from its endpoint and $f$ is a Lipschitz function. The answer to the question is affirmative if and only if $\int ^{\infty }f^{3}(u)du = \infty$. If the test fails, there exist parabolic $h$-transforms of space-time Brownian motion in $D$ with infinite lifetime which are not time-homogenous.
Spectral gap estimates on compact manifolds
Kevin
Oden;
Chiung-Jue
Sung;
Jiaping
Wang
3533-3548
Abstract: For a compact Riemannian manifold with boundary, its mass gap is the difference between the first and second smallest Dirichlet eigenvalues. In this paper, taking a variational approach, we obtain an explicit lower bound estimate of the mass gap for any compact manifold in terms of geometric quantities.
Convex functions on Alexandrov surfaces
Yukihiro
Mashiko
3549-3567
Abstract: We investigate the topological structure of Alexandrov surfaces of curvature bounded below which possess convex functions. We do not assume the continuities of these functions. Nevertheless, if the convex functions satisfy a condition of local nonconstancy, then the topological structures of Alexandrov surfaces and the level sets configurations of these functions in question are determined.
A symplectic jeu de taquin bijection between the tableaux of King and of De Concini
Jeffrey
T.
Sheats
3569-3607
Abstract: The definitions, methods, and results are entirely combinatorial. The symplectic jeu de taquin algorithm developed here is an extension of Schützenberger's original jeu de taquin and acts on a skew form of De Concini's symplectic standard tableaux. This algorithm is used to construct a weight preserving bijection between the two most widely known sets of symplectic tableaux. Anticipated applications to Knuth relations and to decomposing symplectic tensor products are indicated.
Weight distributions of geometric Goppa codes
Iwan
M.
Duursma
3609-3639
Abstract: The in general hard problem of computing weight distributions of linear codes is considered for the special class of algebraic-geometric codes, defined by Goppa in the early eighties. Known results restrict to codes from elliptic curves. We obtain results for curves of higher genus by expressing the weight distributions in terms of $L$-series. The results include general properties of weight distributions, a method to describe and compute weight distributions, and worked out examples for curves of genus two and three.
Dense Egyptian fractions
Greg
Martin
3641-3657
Abstract: Every positive rational number has representations as Egyptian fractions (sums of reciprocals of distinct positive integers) with arbitrarily many terms and with arbitrarily large denominators. However, such representations normally use a very sparse subset of the positive integers up to the largest denominator. We show that for every positive rational there exist representations as Egyptian fractions whose largest denominator is at most $N$ and whose denominators form a positive proportion of the integers up to $N$, for sufficiently large $N$; furthermore, the proportion is within a small factor of best possible.
On Vassiliev knot invariants induced from finite type 3-manifold invariants
Matt
Greenwood;
Xiao-Song
Lin
3659-3672
Abstract: We prove that the knot invariant induced by a $\mathbb{Z}$-homology 3-sphere invariant of order $\leq k$ in Ohtsuki's sense, where $k\geq 4$, is of order $\leq k-2$. The method developed in our computation shows that there is no $\mathbb{Z}$-homology 3-sphere invariant of order 5.
Connectedness properties of limit sets
B.
H.
Bowditch
3673-3686
Abstract: We study convergence group actions on continua, and give a criterion which ensures that every global cut point is a parabolic fixed point. We apply this result to the case of boundaries of relatively hyperbolic groups, and consider implications for connectedness properties of such spaces.
Second-order subgradients of convex integral functionals
Mohammed
Moussaoui;
Alberto
Seeger
3687-3711
Abstract: The purpose of this work is twofold: on the one hand, we study the second-order behaviour of a nonsmooth convex function $F$ defined over a reflexive Banach space $X$. We establish several equivalent characterizations of the set $\partial^2F(\overline x,\overline y)$, known as the second-order subdifferential of $F$ at $\overline x$ relative to $\overline y\in \partial F(\overline x)$. On the other hand, we examine the case in which $F=I_f$ is the functional integral associated to a normal convex integrand $f$. We extend a result of Chi Ngoc Do from the space $X=L_{\mathbb R^d}^p$ $(1<p<+\infty)$ to a possible nonreflexive Banach space $X=L_E^p$ $(1\le p<+\infty)$. We also establish a formula for computing the second-order subdifferential $\partial ^2I_f(\overline x,\overline y)$.
A global condition for periodic Duffing-like equations
Piero
Montecchiari;
Margherita
Nolasco;
Susanna
Terracini
3713-3724
Abstract: We study Duffing-like equations of the type $\ddot q= q - \alpha (t)W'(q)$,with $\alpha \in C({\mathbb{R}},{\mathbb{R}})$ periodic. We prove that if the stable and unstable manifolds to the origin do not coincide, then the system exhibits positive topological entropy.
Exact Hausdorff measure and intervals of maximum density for Cantor sets
Elizabeth
Ayer;
Robert
S.
Strichartz
3725-3741
Abstract: Consider a linear Cantor set $K$, which is the attractor of a linear iterated function system (i.f.s.) $S_{j}x = \rho _{j}x+b_{j}$, $j = 1,\ldots ,m$, on the line satisfying the open set condition (where the open set is an interval). It is known that $K$ has Hausdorff dimension $\alpha$ given by the equation $\sum ^{m}_{j=1} \rho ^{\alpha }_{j} = 1$, and that $\mathcal{H}_{\alpha }(K)$ is finite and positive, where $\mathcal{H}_{\alpha }$ denotes Hausdorff measure of dimension $\alpha$. We give an algorithm for computing $\mathcal{H}_{\alpha }(K)$ exactly as the maximum of a finite set of elementary functions of the parameters of the i.f.s. When $\rho _{1} = \rho _{m}$ (or more generally, if $\log \rho _{1}$ and $\log \rho _{m}$ are commensurable), the algorithm also gives an interval $I$ that maximizes the density $d(I) = \mathcal{H}_{\alpha }(K \cap I)/|I|^{\alpha }$. The Hausdorff measure $\mathcal{H}_{\alpha }(K)$ is not a continuous function of the i.f.s. parameters. We also show that given the contraction parameters $\rho _{j}$, it is possible to choose the translation parameters $b_{j}$ in such a way that $\mathcal{H}_{\alpha }(K) = |K|^{\alpha }$, so the maximum density is one. Most of the results presented here were discovered through computer experiments, but we give traditional mathematical proofs.
On the $L^2\rightarrow L^\infty$ norms of spectral multipliers of ``quasi-homogeneous'' operators on homogeneous groups
Adam
Sikora
3743-3755
Abstract: We study the $L^2 \to L^{\infty}$ norms of spectral projectors and spectral multipliers of left-invariant elliptic and subelliptic second-order differential operators on homogeneous Lie groups. We obtain a precise description of the $L^2 \to L^{\infty}$ norms of spectral multipliers for some class of operators which we call quasi-homogeneous. As an application we prove a stronger version of Alexopoulos' spectral multiplier theorem for this class of operators.
Classes of singular integrals along curves and surfaces
Andreas
Seeger;
Stephen
Wainger;
James
Wright;
Sarah
Ziesler
3757-3769
Abstract: This paper is concerned with singular convolution operators in $\mathbb{R}^{d}$, $d\ge 2$, with convolution kernels supported on radial surfaces $y_{d}=\Gamma (|y'|)$. We show that if $\Gamma (s)=\log s$, then $L^{p}$ boundedness holds if and only if $p=2$. This statement can be reduced to a similar statement about the multiplier $m(\tau ,\eta )=|\tau |^{-i\eta }$ in $\mathbb{R}^{2}$. We also construct smooth $\Gamma$ for which the corresponding operators are bounded for $p_{0}<p\le 2$ but unbounded for $p\le p_{0}$, for given $p_{0}\in [1,2)$. Finally we discuss some examples of singular integrals along convex curves in the plane, with odd extensions.
Causal compactification and Hardy spaces
G.
Ólafsson;
B.
Ørsted
3771-3792
Abstract: Let $\mathcal{M}=G/H$ be a irreducible symmetric space of Cayley type. Then $\mathcal{M}$ is diffeomorphic to an open and dense $G$-orbit in the Shilov boundary of $G/K\times G/K$. This compactification of $\mathcal{M}$ is causal and can be used to give answers to questions in harmonic analysis on $\mathcal{M}$. In particular we relate the Hardy space of $\mathcal{M}$ to the classical Hardy space on the bounded symmetric domain $G/K\times G/K$. This gives a new formula for the Cauchy-Szegö kernel for $\mathcal{M}$.
On the degree of groups of polynomial subgroup growth
Aner
Shalev
3793-3822
Abstract: Let $G$ be a finitely generated residually finite group and let $a_n(G)$ denote the number of index $n$ subgroups of $G$. If $a_n(G) \le n^{\alpha}$ for some $\alpha$ and for all $n$, then $G$ is said to have polynomial subgroup growth (PSG, for short). The degree of $G$ is then defined by ${\mathrm{deg}}(G) = \limsup {{\log a_n(G)} \over {\log n}}$. Very little seems to be known about the relation between ${\mathrm{deg}}(G)$ and the algebraic structure of $G$. We derive a formula for computing the degree of certain metabelian groups, which serves as a main tool in this paper. Addressing a problem posed by Lubotzky, we also show that if $H \le G$ is a finite index subgroup, then ${\mathrm{deg}}(G) \le {\mathrm{deg}}(H)+1$. A large part of the paper is devoted to the structure of groups of small degree. We show that $a_n(G)$ is bounded above by a linear function of $n$ if and only if $G$ is virtually cyclic. We then determine all groups of degree less than $3/2$, and reveal some connections with plane crystallographic groups. It follows from our results that the degree of a finitely generated group cannot lie in the open interval $(1, 3/2)$. Our methods are largely number-theoretic, and density theorems à la Chebotarev play essential role in the proofs. Most of the results also rely implicitly on the Classification of Finite Simple Groups.
The trace space and Kauffman's knot invariants
Keqin
Liu
3823-3842
Abstract: The traces in the construction of Kauffman's knot invariants are studied. The trace space is determined for a semisimple finite-dimensional quantum Hopf algebra and the best lower bound of the dimension of the trace space is given for a unimodular finite-dimensional quantum Hopf algebra.
Direct sum decompositions of infinitely generated modules
D.
J.
Benson;
Wayne
W.
Wheeler
3843-3855
Abstract: Almost all of the basic theorems in the representation theory of finite groups have proofs that depend upon the Krull-Schmidt Theorem. Because this theorem holds only for finite-dimensional modules, however, the recent interest in infinitely generated modules raises the question of which results may hold more generally. In this paper we present an example showing that Green's Indecomposability Theorem fails for infinitely generated modules. By developing and applying some general properties of idempotent modules, we are also able to construct explicit examples of modules for which the cancellation property fails.
On the number of terms in the middle of almost split sequences over tame algebras
J.
A.
de la Peña;
M.
Takane
3857-3868
Abstract: Let $A$ be a finite dimensional tame algebra over an algebraically closed field $k$. It has been conjectured that any almost split sequence $0 \to X \to \oplus _{i=1} ^n Y_i \to Z \to 0$ with $Y_i$ indecomposable modules has $n \le 5$ and in case $n=5$, then exactly one of the $Y_i$ is a projective-injective module. In this work we show this conjecture in case all the $Y_i$ are directing modules, that is, there are no cycles of non-zero, non-iso maps $Y_i =M_1 \to M_2 \to \cdots \to M_s=Y_i$ between indecomposable $A$-modules. In case, $Y_1$ and $Y_2$ are isomorphic, we show that $n \le 3$ and give precise information on the structure of $A$.
Model theory of difference fields
Zoé
Chatzidakis;
Ehud
Hrushovski
2997-3071
Abstract: A difference field is a field with a distinguished automorphism $\sigma$. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional formulas when the characteristic is $0$.
Forcing minimal extensions of Boolean algebras
Piotr
Koszmider
3073-3117
Abstract: We employ a forcing approach to extending Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications: 1) We make Fedorchuk's method more flexible, obtaining, for every cardinal $\lambda$ of uncountable cofinality, a consistent example of a Boolean algebra $A_{\lambda }$ whose every infinite homomorphic image is of cardinality $\lambda$ and has a countable dense subalgebra (i.e., its Stone space is a compact S-space whose every infinite closed subspace has weight $\lambda$). In particular this construction shows that it is consistent that the minimal character of a nonprincipal ultrafilter in a homomorphic image of an algebra $A$ can be strictly less than the minimal size of a homomorphic image of $A$, answering a question of J. D. Monk. 2) We prove that for every cardinal of uncountable cofinality it is consistent that $2^{\omega }=\lambda$ and both $A_{\lambda }$ and $A_{\omega _{1}}$ exist. 3) By combining these algebras we obtain many examples that answer questions of J.D. Monk. 4) We prove the consistency of MA + $\neg$CH + there is a countably tight compact space without a point of countable character, complementing results of A. Dow, V. Malykhin, and I. Juhasz. Although the algebra of clopen sets of the above space has no ultrafilter which is countably generated, it is a subalgebra of an algebra all of whose ultrafilters are countably generated. This proves, answering a question of Arhangel$'$skii, that it is consistent that there is a first countable compact space which has a continuous image without a point of countable character. 5) We prove that for any cardinal $\lambda$ of uncountable cofinality it is consistent that there is a countably tight Boolean algebra $A$ with a distinguished ultrafilter $\infty$ such that for every $a\not \ni \infty$ the algebra $A|a$ is countable and $\infty$ has hereditary character $\lambda$.
The maximality of the core model
E.
Schimmerling;
J.
R.
Steel
3119-3141
Abstract: Our main results are: 1) every countably certified extender that coheres with the core model $K$ is on the extender sequence of $K$, 2) $K$ computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of $K$, 4) (joint with W. J. Mitchell) $K\|\kappa$ is universal for mice of height $\le\kappa$ whenever $\kappa\geq\aleph _2$, 5) if there is a $\kappa$ such that $\kappa$ is either a singular countably closed cardinal or a weakly compact cardinal, and $\square _\kappa^{<\omega}$ fails, then there are inner models with Woodin cardinals, and 6) an $\omega$-Erdös cardinal suffices to develop the basic theory of $K$.
Toeplitz operators with PC symbols on general Carleson Jordan curves with arbitrary Muckenhoupt weights
Albrecht
Böttcher;
Yuri
I.
Karlovich
3143-3196
Abstract: We describe the spectra and essential spectra of Toeplitz operators with piecewise continuous symbols on the Hardy space $H^p(\Gamma,\omega)$ in case $1<p<\infty$, $\Gamma$ is a Carleson Jordan curve and $\omega$ is a Muckenhoupt weight in $A_p(\Gamma)$. Classical results tell us that the essential spectrum of the operator is obtained from the essential range of the symbol by filling in line segments or circular arcs between the endpoints of the jumps if both the curve $\Gamma$ and the weight are sufficiently nice. Only recently it was discovered by Spitkovsky that these line segments or circular arcs metamorphose into horns if the curve $\Gamma$ is nice and $\omega$ is an arbitrary Muckenhoupt weight, while the authors observed that certain special so-called logarithmic leaves emerge in the case of arbitrary Carleson curves with nice weights. In this paper we show that for general Carleson curves and general Muckenhoupt weights the sets in question are logarithmic leaves with a halo, and we present final results concerning the shape of the halo.
Bilipschitz homogeneous Jordan curves
Manouchehr
Ghamsari;
David
A.
Herron
3197-3216
Abstract: We characterize bilipschitz homogeneous Jordan curves by utilizing quasihomogeneous parameterizations. We verify that rectifiable bilipschitz homogeneous Jordan curves satisfy a chordarc condition. We exhibit numerous examples including a bilipschitz homogeneous quasicircle which has lower Hausdorff density zero. We examine homeomorphisms between Jordan curves.
A theorem on zeta functions associated with polynomials
Minking
Eie;
Kwang-Wu
Chen
3217-3228
Abstract: Let $\beta =(\beta _{1},\ldots ,\beta _{r})$ be an $r$-tuple of non-negative integers and $P_{j}(X)$ $(j=1,2,\ldots ,n)$ be polynomials in ${\mathbb{R}}[X_{1},\ldots ,X_{r}]$ such that $P_{j}(n)>0$ for all $n\in {\mathbb{N}}^{r}$ and the series \begin{equation*}\sum _{n\in {\mathbb{N}}^{r}} P_{j}(n)^{-s}\end{equation*} is absolutely convergent for Re $s>\sigma _{j}>0$. We consider the zeta functions \begin{equation*}Z(P_{j},\beta ,s)=\sum _{n\in{\mathbb{N}}^{r}}n^{\beta} P_{j}(n)^{-s},\quad \text{Re} s>|\beta |+\sigma _{j}, \quad 1\leq j\leq n.\end{equation*} All these zeta functions $Z(\prod ^{n}_{j=1} P_{j},\beta ,s)$ and $Z(P_{j},\beta ,s)\quad (j=1,2,\ldots ,n)$ are analytic functions of $s$ when Re$\, s$ is sufficiently large and they have meromorphic analytic continuations in the whole complex plane. In this paper we shall prove that \begin{equation*}Z(\prod _{j=1}^{n} P_{j},\beta ,0)=\frac{1}{n} \sum _{j=1}^{n} Z(P_{j},\beta ,0).\end{equation*} As an immediate application, we use it to evaluate the special values of zeta functions associated with products of linear forms as considered by Shintani and the first author.
Deformations of dihedral 2-group extensions of fields
Elena
V.
Black
3229-3241
Abstract: Given a $G$-Galois extension of number fields $L/K$ we ask whether it is a specialization of a regular $G$-Galois cover of $\mathbb{P}^{1}_{K}$. This is the ``inverse" of the usual use of the Hilbert Irreducibility Theorem in the Inverse Galois problem. We show that for many groups such arithmetic liftings exist by observing that the existence of generic extensions implies the arithmetic lifting property. We explicitly construct generic extensions for dihedral $2$-groups under certain assumptions on the base field $k$. We also show that dihedral groups of order $8$ and $16$ have generic extensions over any base field $k$ with characteristic different from $2$.
Knot invariants from symbolic dynamical systems
Daniel
S.
Silver;
Susan
G.
Williams
3243-3265
Abstract: If $G$ is the group of an oriented knot $k$, then the set $\operatorname{Hom} (K, \Sigma )$ of representations of the commutator subgroup $K = [G,G]$ into any finite group $\Sigma$ has the structure of a shift of finite type $\Phi _{\Sigma }$, a special type of dynamical system completely described by a finite directed graph. Invariants of $\Phi _{\Sigma }$, such as its topological entropy or the number of its periodic points of a given period, determine invariants of the knot. When $\Sigma$ is abelian, $\Phi _{\Sigma }$ gives information about the infinite cyclic cover and the various branched cyclic covers of $k$. Similar techniques are applied to oriented links.
Hardy spaces and a Walsh model for bilinear cone operators
John
E.
Gilbert;
Andrea
R.
Nahmod
3267-3300
Abstract: The study of bilinear operators associated to a class of non-smooth symbols can be reduced to ther study of certain special bilinear cone operators to which a time frequency analysis using smooth wave-packets is performed. In this paper we prove that when smooth wave-packets are replaced by Walsh wave-packets the corresponding discrete Walsh model for the cone operators is not only $L^{p}$-bounded, as Thiele has shown in his thesis for the Walsh model corresponding to the bilinear Hilbert transform, but actually improves regularity as it maps into a Hardy space. The same result is expected to hold for the special bilinear cone operators.
An estimate for a first-order Riesz operator on the affine group
Peter
Sjögren
3301-3314
Abstract: On the affine group of the line, which is a solvable Lie group of exponential growth, we consider a right-invariant Laplacian $\Delta$. For a certain right-invariant vector field $X$, we prove that the first-order Riesz operator $X\Delta^{-1/2}$ is of weak type (1, 1) with respect to the left Haar measure of the group. This operator is therefore also bounded on $L^p, \; 1<p\leq 2$. Locally, the operator is a standard singular integral. The main part of the proof therefore concerns the behaviour of the kernel of the operator at infinity and involves cancellation.
Numeration systems and Markov partitions from self similar tilings
Brenda
Praggastis
3315-3349
Abstract: Using self similar tilings we represent the elements of $\mathbb{R}^n$ as digit expansions with digits in $\mathbb{R}^n$ being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the integer lattice. We use the digit expansions inherited from these tilings to give a symbolic representation for the toral automorphisms.
Invariance principles and Gaussian approximation for strictly stationary processes
Dalibor
Volný
3351-3371
Abstract: We show that in any aperiodic and ergodic dynamical system there exists a square integrable process $(f\circ T^{i})$ the partial sums of which can be closely approximated by the partial sums of Gaussian i.i.d. random variables. For $(f\circ T^{i})$ both weak and strong invariance principles hold.
Diffeomorphisms approximated by Anosov on the 2-torus and their SBR measures
Naoya
Sumi
3373-3385
Abstract: We consider the $C^{2}$ set of $C^{2}$ diffeomorphisms of the 2-torus $\mathbb{T}^{2}$, provided the conditions that the tangent bundle splits into the directed sum $T\mathbb{T}^{2}=E^{s}\oplus E^{u}$ of $Df$-invariant subbundles $E^{s}$, $E^{u}$ and there is $0<\lambda <1$ such that $\Vert Df|_{E^{s}}\Vert <\lambda$ and $\Vert Df|_{E^{u}}\Vert \ge 1$. Then we prove that the set is the union of Anosov diffeomorphisms and diffeomorphisms approximated by Anosov, and moreover every diffeomorphism approximated by Anosov in the $C^{2}$ set has no SBR measures. This is related to a result of Hu-Young.
Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan's property (T)
Yehuda
Shalom
3387-3412
Abstract: Let $k$ be any locally compact non-discrete field. We show that finite invariant measures for $k$-algebraic actions are obtained only via actions of compact groups. This extends both Borel's density and fixed point theorems over local fields (for semisimple/solvable groups, resp.). We then prove that for $k$-algebraic actions, finitely additive finite invariant measures are obtained only via actions of amenable groups. This gives a new criterion for Zariski density of subgroups and is shown to have representation theoretic applications. The main one is to Kazhdan's property $(T)$ for algebraic groups, which we investigate and strengthen.
On modules of bounded multiplicities for the symplectic algebras
D.
J.
Britten;
F.
W.
Lemire
3413-3431
Abstract: Simple infinite dimensional highest weight modules having bounded weight multipicities are classified as submodules of a tensor product. Also, it is shown that a simple torsion free module of finite degree tensored with a finite dimensional module is completely reducible.
Gorenstein space with nonzero evaluation map
H.
Gammelin
3433-3440
Abstract: Let $(A,d)$ be a differential graded algebra of finite type, if $H^*(A)$ is a Gorenstein graded algebra, then so is $A$. The purpose of this paper is to prove the converse under some mild hypotheses. We deduce a new characterization of Poincaré duality spaces as well as spaces with a nonzero evaluation map.
Products on $MU$-modules
N.
P.
Strickland
2569-2606
Abstract: Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of $MU$-modules such as $BP$, $K(n)$ and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over $MU[{\textstyle\frac{1}{2}}]_*$ that are concentrated in degrees divisible by $4$; this guarantees that various obstruction groups are trivial. We extend these results to the cases where $2=0$ or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising $2$-local $MU_*$-modules as $MU$-modules.
Brownian sheet images and Bessel-Riesz capacity
Davar
Khoshnevisan
2607-2622
Abstract: We show that the image of a 2-dimensional set under $d$-dimensional, 2-parameter Brownian sheet can have positive Lebesgue measure if and only if the set in question has positive ($d/2$)-dimensional Bessel-Riesz capacity. Our methods solve a problem of J.-P. Kahane.
Vaught's conjecture and the Glimm-Effros property for Polish transformation groups
Greg
Hjorth;
Slawomir
Solecki
2623-2641
Abstract: We extend the original Glimm-Effros theorem for locally compact groups to a class of Polish groups including the nilpotent ones and those with an invariant metric. For this class we thereby obtain the topological Vaught conjecture.
Ultrafilters on $\omega$-their ideals and their cardinal characteristics
Saharon
Shelah;
Jörg
Brendle;
Saharon
Shelah
2643-2674
Abstract: For a free ultrafilter $\mathcal{U}$ on $\omega$ we study several cardinal characteristics which describe part of the combinatorial structure of $\,\mathcal{U}$. We provide various consistency results; e.g. we show how to force simultaneously many characters and many $\pi$-characters. We also investigate two ideals on the Baire space $\omega ^{\omega }$ naturally related to $\mathcal{U}$ and calculate cardinal coefficients of these ideals in terms of cardinal characteristics of the underlying ultrafilter.
CH with no Ostaszewski spaces
Todd
Eisworth;
Judith
Roitman
2675-2693
Abstract: There are models of CH without Ostaszeswki spaces. If $X$ is locally compact and sub-Ostaszewski, there is a forcing $P_X$ which does not add reals and which forces ``$X$ is not sub-Ostaszewski''.
The Quantum Cohomology Ring of Flag Varieties
Ionut
Ciocan-Fontanine
2695-2729
Abstract: We describe the small quantum cohomology ring of complete flag varieties by algebro-geometric methods, as presented in our previous work Quantum cohomology of flag varieties (Internat. Math. Res. Notices, no. 6 (1995), 263-277). We also give a geometric proof of the quantum Monk formula.
A speciality theorem for Cohen-Macaulay space curves
Enrico
Schlesinger
2731-2743
Abstract: We prove a version of the Halphen Speciality Theorem for locally Cohen-Macaulay curves in $\mathbb{P}^3$. To prove the theorem, we strengthen some results of Okonek and Spindler on the spectrum of the ideal sheaf of a curve. As an application, we classify curves $C$ having index of speciality as large as possible once we fix the degree of $C$ and the minimum degree of a surface containing $C$.
Spherical functions and conformal densities on spherically symmetric $CAT(-1)$-spaces
Michel
Coornaert;
Athanase
Papadopoulos
2745-2762
Abstract: Let $X$ be a $CAT(-1)$-space which is spherically symmetric around some point $x_{0}\in X$ and whose boundary has finite positive $s-$dimensional Hausdorff measure. Let $\mu =(\mu _{x})_{x\in X}$ be a conformal density of dimension $d>s/2$ on $\partial X$. We prove that $\mu _{x_{0}}$ is a weak limit of measures supported on spheres centered at $x_{0}$. These measures are expressed in terms of the total mass function of $\mu$ and of the $d-$dimensional spherical function on $X$. In particular, this result proves that $\mu$ is entirely determined by its dimension and its total mass function. The results of this paper apply in particular for symmetric spaces of rank one and semi-homogeneous trees.
An equivariant smash spectral sequence and an unstable box product
Michele
Intermont
2763-2775
Abstract: Let $G$ be a finite group. We construct a first quadrant spectral sequence which converges to the equivariant homotopy groups of the smash product $X \wedge Y$ for suitably connected, based $G$-CW complexes $X$ and $Y$. The $E^2$ term is described in terms of a tensor product functor of equivariant $\Pi$-algebras. A homotopy version of the non-equivariant Künneth theorem and the equivariant suspension theorem of Lewis are both shown to be special cases of the corner of the spectral sequence. We also give a categorical description of this tensor product functor which is analogous to the description in equivariant stable homotopy theory of the box product of Mackey functors. For this reason, the tensor product functor deserves to be called an ``unstable box product''.
Periodic traveling waves and locating oscillating patterns in multidimensional domains
Nicholas
D.
Alikakos;
Peter
W.
Bates;
Xinfu
Chen
2777-2805
Abstract: We establish the existence and robustness of layered, time-periodic solutions to a reaction-diffusion equation in a bounded domain in $\mathbb{R}^n$, when the diffusion coefficient is sufficiently small and the reaction term is periodic in time and bistable in the state variable. Our results suggest that these patterned, oscillatory solutions are stable and locally unique. The location of the internal layers is characterized through a periodic traveling wave problem for a related one-dimensional reaction-diffusion equation. This one-dimensional problem is of independent interest and for this we establish the existence and uniqueness of a heteroclinic solution which, in constant-velocity moving coodinates, is periodic in time. Furthermore, we prove that the manifold of translates of this solution is globally exponentially asymptotically stable.
Bourgin-Yang type theorem and its application to $Z_2$-equivariant Hamiltonian systems
Marek
Izydorek
2807-2831
Abstract: We will be concerned with the existence of multiple periodic solutions of asymptotically linear Hamiltonian systems with the presence of $Z_2$-action. To that purpose we prove a new version of the Bourgin-Yang theorem. Using the notion of the crossing number we also introduce a new definition of the Morse index for indefinite functionals.
``Best possible'' upper and lower bounds for the zeros of the Bessel function $J_\nu(x)$
C.
K.
Qu;
R.
Wong
2833-2859
Abstract: Let $j_{\nu,k}$ denote the $k$-th positive zero of the Bessel function $J_\nu(x)$. In this paper, we prove that for $\nu>0$ and $k=1$, 2, 3, $\ldots$, \begin{displaymath}\nu - \frac{a_k}{2^{1/3}} \nu^{1/3} < j_{\nu,k} < \nu - \frac{a_k}{2^{1/3}} \nu^{1/3} + \frac{3}{20} a_k^2 \frac{2^{1/3}}{\nu^{1/3}} \,. \end{displaymath} These bounds coincide with the first few terms of the well-known asymptotic expansion \begin{displaymath}j_{\nu,k} \sim \nu - \frac{a_k}{2^{1/3}} \nu^{1/3} + \frac{3}{20} a_k^2 \frac{2^{1/3}}{\nu^{1/3}} + \cdots \end{displaymath} as $\nu\to\infty$, $k$ being fixed, where $a_k$ is the $k$-th negative zero of the Airy function $\operatorname{Ai}(x)$, and so are ``best possible''.
Chaotic solutions in differential inclusions: chaos in dry friction problems
Michal
Feckan
2861-2873
Abstract: The existence of a continuum of many chaotic solutions is shown for certain differential inclusions which are small periodic multivalued perturbations of ordinary differential equations possessing homoclinic solutions to hyperbolic fixed points. Applications are given to dry friction problems. Singularly perturbed differential inclusions are investigated as well.
On the enhancement of diffusion by chaos, escape rates and stochastic instability
Pierre
Collet;
Servet
Martínez;
Bernard
Schmitt
2875-2897
Abstract: We consider stochastic perturbations of expanding maps of the interval where the noise can project the trajectory outside the interval. We estimate the escape rate as a function of the amplitude of the noise and compare it with the purely diffusive case. This is done under a technical hypothesis which corresponds to stability of the absolutely continuous invariant measure against small perturbations of the map. We also discuss in detail a case of instability and show how stability can be recovered by considering another invariant measure.
Partial subdifferentials, derivates and Rademacher's Theorem
D.
N.
Bessis;
F.
H.
Clarke
2899-2926
Abstract: In this paper, we present new partial subdifferentiation formulas in nonsmooth analysis, based upon the study of two directional derivatives. Simple applications of these formulas include a new elementary proof of Rademacher's Theorem in ${\mathbb R}^n$, as well as some results on Gâteaux and Fréchet differentiability for locally Lipschitz functions in a separable Hilbert space. RÉSUMÉ. Dans cet article, nous présentons de nouvelles formules de sousdifférentiation partielle en analyse nonlisse, basées sur l'étude de deux dérivées directionnelles. Une simple application de ces formules nous permet d'obtenir une nouvelle preuve élémentaire du théorème de Rademacher dans ${\mathbb R}^{n}$, ainsi que certains résultats sur la différentiabilité Gâteaux ou Fréchet des fonctions localement Lipschitz sur un espace de Hilbert séparable.
Rotation and entropy
William
Geller;
Michal
Misiurewicz
2927-2948
Abstract: For a given map $f: X \to X$ and an observable $\varphi : X \to \mathbb{R} ^{d},$ rotation vectors are the limits of ergodic averages of $\varphi .$ We study which part of the topological entropy of $f$ is associated to a given rotation vector and which part is associated with many rotation vectors. According to this distinction, we introduce directional and lost entropies. We discuss their properties in the general case and analyze them more closely for subshifts of finite type and circle maps.
Compressions of resolvents and maximal radius of regularity
C.
Badea;
M.
Mbekhta
2949-2960
Abstract: Suppose that $\lambda - T$ is left invertible in $L(H)$ for all $\lambda \in \Omega$, where $\Omega$ is an open subset of the complex plane. Then an operator-valued function $L(\lambda)$ is a left resolvent of $T$ in $\Omega$ if and only if $T$ has an extension $\tilde{T}$, the resolvent of which is a dilation of $L(\lambda)$ of a particular form. Generalized resolvents exist on every open set $U$, with $\overline{U}$ included in the regular domain of $T$. This implies a formula for the maximal radius of regularity of $T$ in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by J. Zemánek is obtained.
Equations in the Q-completion of a torsion-free hyperbolic group
O.
Kharlampovich;
E.
Lioutikova;
A.
Myasnikov
2961-2978
Abstract: In this paper we prove the algorithmic solvability of finite systems of equations over the Q-completion of a torsion-free hyperbolic group.
Left-symmetric algebras for $\mathfrak{gl}(n)$
Oliver
Baues
2979-2996
Abstract: We study the classification problem for left-symmetric algebras with commutation Lie algebra ${\mathfrak{gl}}(n)$ in characteristic $0$. The problem is equivalent to the classification of étale affine representations of ${\mathfrak{gl}}(n)$. Algebraic invariant theory is used to characterize those modules for the algebraic group $\operatorname{SL}(n)$ which belong to affine étale representations of ${\mathfrak{gl}}(n)$. From the classification of these modules we obtain the solution of the classification problem for ${\mathfrak{gl}}(n)$. As another application of our approach, we exhibit left-symmetric algebra structures on certain reductive Lie algebras with a one-dimensional center and a non-simple semisimple ideal.
Witten-Helffer-Sjöstrand theory for $S^1$-equivariant cohomology
Hon-kit
Wai
2141-2182
Abstract: Given an $S^1$-invariant Morse function $f$ and an $S^1$-invariant Riemannian metric $g$, a family of finite dimensional subcomplexes $(\widetilde \Omega^*_{inv,sm}(M,t), D(t))$, $t\in [0,\infty)$, of the Witten deformation of the $S^1$-equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian $\widetilde \Delta ^k(t)=D^*_k(t)D_k(t)+D_{k-1}(t)D^*_{k-1}(t)$ as $t\to \infty$. In fact the spectrum of $\widetilde \Delta^k(t)$ can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains $( \widetilde \Omega^*_{inv,sm}(M,t),D(t))$ as the complex of eigenforms corresponding to the small eigenvalues of $\widetilde \Delta(t)$. This permits us to verify the $S^1$-equivariant Morse inequalities. Moreover suppose $f$ is self-indexing and $(f,g)$ satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as $t\to \infty$ to a geometric complex which is induced by $(f,g)$ and calculates the $S^1$-equivariant cohomology of $M$.
Compact Composition Operators on BMOA
P.
S.
Bourdon;
J.
A.
Cima;
A.
L.
Matheson
2183-2196
Abstract: We characterize the compact composition operators on BMOA, the space consisting of those holomorphic functions on the open unit disk $U$ that are Poisson integrals of functions on $\partial U$, that have bounded mean oscillation. We then use our characterization to show that compactness of a composition operator on BMOA implies its compactness on the Hardy spaces (a simple example shows the converse does not hold). We also explore how compactness of the composition operator $C_\phi: \operatorname{BMOA}\rightarrow\operatorname{BMOA}$ relates to the shape of $\phi(U)$ near $\partial U$, introducing the notion of mean order of contact. Finally, we discuss the relationships among compactness conditions for composition operators on BMOA, VMOA, and the big and little Bloch spaces.
Towards a Halphen theory of linear series on curves
L.
Chiantini;
C.
Ciliberto
2197-2212
Abstract: A linear series $g^{N}_{\delta }$ on a curve $C\subset \mathbf{P}^{3}$ is primary when it does not contain the series cut by planes. For such series, we provide a lower bound for the degree $\delta$, in terms of deg($C$), g($C$) and of the number $s=\min \{i:h^{0}\mathcal{I}_{C}(i)\neq 0\}$. Examples show that the bound is sharp. Extensions to the case of general linear series and to the case of curves in higher projective spaces are considered.
The diagonal subring and the Cohen-Macaulay property of a multigraded ring
Eero
Hyry
2213-2232
Abstract: Let $T$ be a multigraded ring defined over a local ring $(A,\mathfrak{m})$. This paper deals with the question how the Cohen-Macaulay property of $T$ is related to that of its diagonal subring $T^\Delta$. In the bigraded case we are able to give necessary and sufficient conditions for the Cohen-Macaulayness of $T$. If $I_1,\dotsc,I_r\subset A$ are ideals of positive height, we can then compare the Cohen-Macaulay property of the multi-Rees algebra $R_A(I_1,\dotsc,I_r)$ with the Cohen-Macaulay property of the usual Rees algebra $R_A(I_1\cdots I_r)$. We also obtain a bound for the joint reduction numbers of two $\mathfrak{m}$-primary ideals in the case the corresponding multi-Rees algebra is Cohen-Macaulay.
Derivatives of Wronskians with applications to families of special Weierstrass points
Letterio
Gatto;
Fabrizio
Ponza
2233-2255
Abstract: Let $\pi :\mathfrak{X}\longrightarrow S$ be a flat proper family of smooth connected projective curves parametrized by some smooth scheme of finite type over $\mathbb{C}$. On every such a family, suitable derivatives along the fibers" (in the sense of Lax) of the relative wronskian, as defined by Laksov and Thorup, are constructed. They are sections of suitable jets extensions of the $g(g+1)/2$-th tensor power of the relative canonical bundle of the family itself. The geometrical meaning of such sections is discussed: the zero schemes of the $(k-1)$-th derivative ($k\geq 1$) of a relative wronskian correspond to families of Weierstrass Points (WP's) having weight at least $k$. The locus in $M_{g}$, the coarse moduli space of smooth projective curves of genus $g$, of curves possessing a WP of weight at least $k$, is denoted by $wt(k)$. The fact that $wt(2)$ has the expected dimension for all $g\geq 2$ was implicitly known in the literature. The main result of this paper hence consists in showing that $wt(3)$ has the expected dimension for all $g\geq 4$. As an application we compute the codimension $2$ Chow ($Q$-)class of $wt(3)$ for all $g\geq 4$, the main ingredient being the definition of the $k$-th derivative of a relative wronskian, which is the crucial tool which the paper is built on. In the concluding remarks we show how this result may be used to get relations among some codimension $2$ Chow ($Q$-)classes in $M_{4}$ ($g\geq 4$), corresponding to varieties of curves having a point $P$ with a suitable prescribed Weierstrass Gap Sequence, relating to previous work of Lax.
A Combinatorial Proof of Bass's Evaluations of the Ihara-Selberg Zeta Function for Graphs
Dominique
Foata;
Doron
Zeilberger
2257-2274
Abstract: We derive combinatorial proofs of the main two evaluations of the Ihara-Selberg zeta function associated with a graph. We give three proofs of the first evaluation all based on the algebra of Lyndon words. In the third proof it is shown that the first evaluation is an immediate consequence of Amitsur's identity on the characteristic polynomial of a sum of matrices. The second evaluation of the Ihara-Selberg zeta function is first derived by means of a sign-changing involution technique. Our second approach makes use of a short matrix-algebra argument.
Dehn surgery on arborescent links
Ying-Qing
Wu
2275-2294
Abstract: This paper studies Dehn surgery on a large class of links, called arborescent links. It will be shown that if an arborescent link $L$ is sufficiently complicated, in the sense that it is composed of at least $4$ rational tangles $T(p_{i}/q_{i})$ with all $q_{i} > 2$, and none of its length 2 tangles are of the form $T(1/2q_{1}, 1/2q_{2})$, then all complete surgeries on $L$ produce Haken manifolds. The proof needs some result on surgery on knots in tangle spaces. Let $T(r/2s, p/2q) = (B, t_{1}\cup t_{2}\cup K)$ be a tangle with $K$ a closed circle, and let $M = B - \operatorname{Int} N(t_{1}\cup t_{2})$. We will show that if $s>1$ and $p \not \equiv \pm 1$ mod $2q$, then $\partial M$ remains incompressible after all nontrivial surgeries on $K$. Two bridge links are a subclass of arborescent links. For such a link $L(p/q)$, most Dehn surgeries on it are non-Haken. However, it will be shown that all complete surgeries yield manifolds containing essential laminations, unless $p/q$ has a partial fraction decomposition of the form $1/(r-1/s)$, in which case it does admit non-laminar surgeries.
Farey polytopes and continued fractions associated with discrete hyperbolic groups
L.
Ya.
Vulakh
2295-2323
Abstract: The known definitions of Farey polytopes and continued fractions are generalized and applied to diophantine approximation in $n$-dimensional euclidean spaces. A generalized Remak-Rogers isolation theorem is proved and applied to show that certain Hurwitz constants for discrete groups acting in a hyperbolic space are isolated. The approximation constant for the imaginary quadratic field of discriminant $-15$ is found.
The iterated transfer analogue of the new doomsday conjecture
Norihiko
Minami
2325-2351
Abstract: A strong general restriction is given on the stable Hurewicz image of the classifying spaces of elementary abelian $p$-groups. In particular, this implies the iterated transfer analogue of the new doomsday conjecture.
Embeddings of open manifolds
Nancy
Cardim
2353-2373
Abstract: Let $TOP(M)$ be the simplicial group of homeomorphisms of $M$. The following theorems are proved. Theorem A. Let $M$ be a topological manifold of dim $\geq$ 5 with a finite number of tame ends $\varepsilon _{i}$, $1\leq i\leq k$. Let $TOP^{ep}(M)$ be the simplicial group of end preserving homeomorphisms of $M$. Let $W_{i}$ be a periodic neighborhood of each end in $M$, and let $p_{i}: W_{i} \to \mathbb{R}$ be manifold approximate fibrations. Then there exists a map $f: TOP^{ep}(M) \to \prod _{i} TOP^{ep}(W_{i})$ such that the homotopy fiber of $f$ is equivalent to $TOP_{cs}(M)$, the simplicial group of homeomorphisms of $M$ which have compact support. Theorem B. Let $M$ be a compact topological manifold of dim $\geq$ 5, with connected boundary $\partial M$, and denote the interior of $M$ by $Int M$. Let $f: TOP(M)\to TOP(Int M)$ be the restriction map and let $\mathcal{G}$ be the homotopy fiber of $f$ over $id_{Int M}$. Then $\pi _{i} \mathcal{G}$ is isomorphic to $\pi _{i} \mathcal{C} (\partial M)$ for $i > 0$, where $\mathcal{C} (\partial M)$ is the concordance space of $\partial M$. Theorem C. Let $q_{0}: W \to \mathbb{R}$ be a manifold approximate fibration with dim $W \geq$ 5. Then there exist maps $\alpha : \pi _{i} TOP^{ep}(W) \to \pi _{i} TOP(\hat W)$ and $\beta : \pi _{i} TOP(\hat W) \to \pi _{i} TOP^{ep}(W)$ for $i >1$, such that $\beta \circ \alpha \simeq id$, where $\hat W$ is a compact and connected manifold and $W$ is the infinite cyclic cover of $\hat W$.
The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor
M.
Grinfeld;
A.
Novick-Cohen
2375-2406
Abstract: In this paper we establish a Morse decomposition of the stationary solutions of the one-dimensional viscous Cahn-Hilliard equation by explicit energy calculations. Strong non-degeneracy of the stationary solutions is proven away from turning points and points of bifurcation from the homogeneous state and the dimension of the unstable manifold is calculated for all stationary states. In the unstable case, the flow on the global attractor is shown to be semi-conjugate to the flow on the global attractor of the Chaffee-Infante equation, and in the metastable case close to the nonlocal reaction-diffusion limit, a partial description of the structure of the global attractor is obtained by connection matrix arguments, employing a partial energy ordering and the existence of a weak lap number principle.
Multivariate matrix refinable functions with arbitrary matrix dilation
Qingtang
Jiang
2407-2438
Abstract: Characterizations of the stability and orthonormality of a multivariate matrix refinable function $\Phi$ with arbitrary matrix dilation $M$ are provided in terms of the eigenvalue and $1$-eigenvector properties of the restricted transition operator. Under mild conditions, it is shown that the approximation order of $\Phi$ is equivalent to the order of the vanishing moment conditions of the matrix refinement mask $\{\mathbf{P}_{\alpha}\}$. The restricted transition operator associated with the matrix refinement mask $\{\mathbf{P}_{\alpha}\}$ is represented by a finite matrix $({\mathcal A} _{Mi-j})_{i, j}$, with ${\mathcal A} _j=|\hbox{det$(M)$}|^{-1}\sum _{\kappa }\mathbf{P}_{\kappa -j}\otimes \mathbf{P}_{\kappa }$ and $\mathbf{P}_{\kappa -j}\otimes \mathbf{P}_{\kappa }$ being the Kronecker product of matrices $\mathbf{P}_{\kappa -j}$ and $\mathbf{P}_{\kappa }$. The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix refinable function $\Phi$ is given in terms of the spectral radius of the restricted transition operator to an invariant subspace. This estimate is analyzed in an example.
Summability of Fourier orthogonal series for Jacobi weight on a ball in $\mathbb{R}^d$
Yuan
Xu
2439-2458
Abstract: Fourier orthogonal series with respect to the weight function $(1-|\mathbf x|^{2})^{\mu - 1/2}$ on the unit ball in $\mathbb{R}^{d}$ are studied. Compact formulae for the sum of the product of orthonormal polynomials in several variables and for the reproducing kernel are derived and used to study the summability of the Fourier orthogonal series. The main result states that the expansion of a continuous function in the Fourier orthogonal series with respect to $(1-|\mathbf x|^{2})^{\mu -1/2}$ is uniformly $(C, \delta )$ summable on the ball if and only if $\delta > \mu + (d-1)/2$.
Hardy inequalities in Orlicz spaces
Andrea
Cianchi
2459-2478
Abstract: We establish a sharp extension, in the framework of Orlicz spaces, of the ($n$-dimensional) Hardy inequality, involving functions defined on a domain $G$, their gradients and the distance function from the boundary of $G$.
Embedded singular continuous spectrum for one-dimensional Schrödinger operators
Christian
Remling
2479-2497
Abstract: We investigate one-dimensional Schrödinger operators with sparse potentials (i.e. the potential consists of a sequence of bumps with rapidly growing barrier separations). These examples illuminate various phenomena related to embedded singular continuous spectrum.
Newton's method on the complex exponential function
Mako
E.
Haruta
2499-2513
Abstract: We show that when Newton's method is applied to the product of a polynomial and the exponential function in the complex plane, the basins of attraction of roots have finite area.
The ideal structure of some analytic crossed products
Miron
Shpigel
2515-2538
Abstract: We study the ideal structure of a class of some analytic crossed products. For an $r$-discrete, principal, minimal groupoid $G$, we consider the analytic crossed product $C^*(G,\sigma)\times _\alpha \mathbb{Z}_+$, where $\alpha$ is given by a cocycle $c$. We show that the maximal ideal space $\mathcal{M}$ of $C^*(G,\sigma)\times _\alpha \mathbb{Z}_+$ depends on the asymptotic range of $c$, $R_\infty(c)$; that is, $\mathcal{M}$ is homeomorphic to $\overline{\mathbb{D}}\mid R_\infty(c)$ for $R_\infty(c)$ finite, and $\cal M$ consists of the unique maximal ideal for $R_\infty(c)=\mathbb{T}$. We also prove that $C^*(G,\sigma)\times _\alpha \mathbb{Z}_+$ is semisimple in both cases, and that $R_\infty(c)$ is invariant under isometric isomorphism.
Representation Theory of Reductive Normal Algebraic Monoids
Stephen
Doty
2539-2551
Abstract: New results in the representation theory of ``semisimple'' algebraic monoids are obtained, based on Renner's monoid version of Chevalley's big cell. (The semisimple algebraic monoids have been classified by Renner.) The rational representations of such a monoid are the same thing as ``polynomial'' representations of the associated reductive group of units in the monoid, and this representation category splits into a direct sum of subcategories by ``homogeneous'' degree. We show that each of these homogeneous subcategories is a highest weight category, in the sense of Cline, Parshall, and Scott, and so equivalent with the module category of a certain finite-dimensional quasihereditary algebra, which we show is a generalized Schur algebra in S. Donkin's sense.
Thermal capacity estimates on the Allen-Cahn equation
Richard
B.
Sowers;
Jang-Mei
Wu
2553-2567
Abstract: We consider the Allen-Cahn equation in a well-known scaling regime which gives motion by mean curvature. A well-known transformation of this PDE, using its standing wave, yields a PDE the solution of which is approximately the distance function to an interface moving by mean curvature. We give bounds on this last fact in terms of thermal capacity. Our techniques hinge upon the analysis of a certain semimartingale associated with a certain PDE (the PDE for the approximate distance function) and an analogue of some results by Bañuelos and Øksendal relating lifetimes of diffusions to exterior capacities.
Isoperimetric Estimates on Sierpinski Gasket Type Fractals
Robert
S.
Strichartz
1705-1752
Abstract: For a compact Hausdorff space $F$ that is pathwise connected, we can define the connectivity dimension $\beta$ to be the infimum of all $b$ such that all points in $F$ can be connected by a path of Hausdorff dimension at most $b$. We show how to compute the connectivity dimension for a class of self-similar sets in $\mathbb{R}^{n}$ that we call point connected, meaning roughly that $F$ is generated by an iterated function system acting on a polytope $P$ such that the images of $P$ intersect at single vertices. This class includes the polygaskets, which are obtained from a regular $n$-gon in the plane by contracting equally to all $n$ vertices, provided $n$ is not divisible by 4. (The Sierpinski gasket corresponds to $n = 3$.) We also provide a separate computation for the octogasket ($n = 8$), which is not point connected. We also show, in these examples, that $\inf \mathcal{H}_{\beta }(\gamma _{x,y})^{1/\beta }$, where the infimum is taken over all paths $\gamma _{x,y}$ connecting $x$ and $y$, and $\mathcal{H}_{\beta }$ denotes Hausdorff measure, is equivalent to the original metric on $F$. Given a compact subset $F$ of the plane of Hausdorff dimension $\alpha$ and connectivity dimension $\beta$, we can define the isoperimetric profile function $h(L)$ to be the supremum of $\mathcal{H}_{\alpha }(F \cap D)$, where $D$ is a region in the plane bounded by a Jordan curve (or union of Jordan curves) $\gamma$ entirely contained in $F$, with $\mathcal{H}_{\beta }(\gamma ) \le L$. The analog of the standard isperimetric estimate is $h(L) \le cL^{\alpha /\beta }$. We are particularly interested in finding the best constant $c$ and identifying the extremal domains where we have equality. We solve this problem for polygaskets with $n = 3,5,6,8$. In addition, for $n = 5,6,8$ we find an entirely different estimate for $h(L)$ as $L \rightarrow \infty$, since the boundary of $F$ has infinite $\mathcal{H}_{\beta }$ measure. We find that the isoperimetric profile function is discontinuous, and that the extremal domains have relatively simple polygonal boundaries. We discuss briefly the properties of minimal paths for the Sierpinski gasket, and the isodiametric problem in the intrinsic metric.
The Hexagonal Honeycomb Conjecture
Frank
Morgan
1753-1763
Abstract: It is conjectured that the planar hexagonal honeycomb provides the least-perimeter way to enclose and separate infinitely many regions of unit area. Various natural formulations of the question are not known to be equivalent. We prove existence for two formulations. Many questions remain open.
The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature
Takashi
Shioya
1765-1801
Abstract: We study the class of closed $2$-dimensional Riemannian manifolds with uniformly bounded diameter and total absolute curvature. Our first theorem states that this class of manifolds is precompact with respect to the Gromov-Hausdorff distance. Our goal in this paper is to completely characterize the topological structure of all the limit spaces of the class of manifolds, which are, in general, not topological manifolds and even may not be locally $2$-connected. We also study the limit of $2$-manifolds with $L^p$-curvature bound for $p \ge 1$.
Convexity properties of holomorphic mappings in $\mathbb{C}^n$
Kevin
A.
Roper;
Ted
J.
Suffridge
1803-1833
Abstract: Not many convex mappings on the unit ball in ${\mathbb C}^n$ for $n>1$ are known. We introduce two families of mappings, which we believe are actually identical, that both contain the convex mappings. These families which we have named the ``Quasi-Convex Mappings, Types A and B'' seem to be natural generalizations of the convex mappings in the plane. It is much easier to check whether a function is in one of these classes than to check for convexity. We show that the upper and lower bounds on the growth rate of such mappings is the same as for the convex mappings.
Reducibility and nonreducibility between $\ell^p$ equivalence relations
Randall
Dougherty;
Greg
Hjorth
1835-1844
Abstract: We show that, for $1 \le p < q < \infty$, the relation of $\ell ^{p}$-equivalence between infinite sequences of real numbers is Borel reducible to the relation of $\ell ^{q}$-equivalence (i.e., the Borel cardinality of the quotient ${\mathbb R}^{{\mathbb N}}/\ell ^{p}$ is no larger than that of ${\mathbb R}^{{\mathbb N}}/\ell ^{q}$), but not vice versa. The Borel reduction is constructed using variants of the triadic Koch snowflake curve; the nonreducibility in the other direction is proved by taking a putative Borel reduction, refining it to a reduction map that is not only continuous but `modular,' and using this nicer map to derive a contradiction.
Criteria of algebraic independence with multiplicities and interpolation determinants
Michel
Laurent;
Damien
Roy
1845-1870
Abstract: We generalize Gel'fond's criterion of algebraic independence by taking into account the values of the derivatives of the polynomials, and show how the new criterion applies to proving results of algebraic independence using interpolation determinants. We also establish a new result of approximation of a transcendental number by algebraic numbers of bounded degree and size. It contains an earlier result of E. Wirsing and also a result announced by A. Durand.
Simple families of Thue inequalities
Günter
Lettl;
Attila
Petho;
Paul
Voutier
1871-1894
Abstract: We use the hypergeometric method to solve three families of Thue inequalities of degree 3, 4 and 6, respectively, each of which is parametrized by an integral parameter. We obtain bounds for the solutions, which are astonishingly small compared to similar results which use estimates of linear forms in logarithms.
Linking forms, reciprocity for Gauss sums and invariants of 3-manifolds
Florian
Deloup
1895-1918
Abstract: We study invariants of $3$-manifolds derived from finite abelian groups equipped with quadratic forms. These invariants arise in Turaev's theory of modular categories and generalize those of H. Murakami, T. Ohtsuki and M. Okada. The crucial algebraic tool is a new reciprocity formula for Gauss sums, generalizing classical formulas of Cauchy, Kronecker, Krazer and Siegel. We use this reciprocity formula to give an explicit formula for the invariants and to generalize them to higher dimensions.
On the number of radially symmetric solutions to Dirichlet problems with jumping nonlinearities of superlinear order
Alfonso
Castro;
Hendrik
J.
Kuiper
1919-1945
Abstract: This paper is concerned with the multiplicity of radially symmetric solutions $u(x)$ to the Dirichlet problem \begin{displaymath}\Delta u+f(u)=h(x)+c\phi(x)\end{displaymath} on the unit ball $\Omega\subset\mathbf R^N$ with boundary condition $u=0$ on $\partial\Omega$. Here $\phi(x)$ is a positive function and $f(u)$ is a function that is superlinear (but of subcritical growth) for large positive $u$, while for large negative $u$ we have that $f'(u)<\mu$, where $\mu$ is the smallest positive eigenvalue for $\Delta\psi+\mu\psi=0$ in $\Omega$ with $\psi=0$ on $\partial\Omega$. It is shown that, given any integer $k\ge 0$, the value $c$ may be chosen so large that there are $2k+1$ solutions with $k$ or less interior nodes. Existence of positive solutions is excluded for large enough values of $c$.
Local tomography with nonsmooth attenuation
A.
I.
Katsevich
1947-1974
Abstract: Local tomography for the Radon transform with nonsmooth attenuation is proposed and justified. The main theoretical tool is analysis of singularities of pseudodifferential operators with nonsmooth symbols. Results of numerical testing of local tomography are presented.
Geometry of Cantor Systems
Yunping
Jiang
1975-1987
Abstract: A Cantor system is defined. The geometry of a certain family of Cantor systems is studied. Such a family arises in dynamical systems as hyperbolicity is created. We prove that the bridge geometry of a Cantor system in such a family is uniformly bounded and that the gap geometry is regulated by the size of the leading gap.
On the contribution of the Coulomb singularity of arbitrary charge to the Dirac Hamiltonian
Jingbo
Xia
1989-2023
Abstract: We study the self-adjoint extensions of the Dirac operator $\alpha \cdot (p - B) + \mu _{0}\beta - W$, where the electrical potential $W$ contains a Coulomb singularity of arbitrary charge and the magnetic potential $B$ is allowed to be unbounded at infinity. We show that if the Coulomb singularity has the form $v(r)/r$ where $v$ has a limit at 0, then, for any self-adjoint extension of the Dirac operator, removing the singularity results in a Hilbert-Schmidt perturbation of its resolvent.
Harnack inequality for the linearized parabolic Monge-Ampère equation
Qingbo
Huang
2025-2054
Abstract: In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampère equation \begin{displaymath}u_{t}-\text{tr}((D^{2}\phi (x))^{-1}D^{2}u)=0\end{displaymath} on parabolic sections associated with $\phi (x)$, under the assumption that the Monge-Ampère measure generated by $\phi$ satisfies the doubling condition on sections and the uniform continuity condition with respect to Lebesgue measure. The theory established is invariant under the group $AT(n)\times AT(1)$, where $AT(n)$ denotes the group of all invertible affine transformations on ${\mathbf{R}}^{n}$.
A Gauss-Kusmin theorem for optimal continued fractions
Karma
Dajani;
Cor
Kraaikamp
2055-2079
Abstract: A Gauss-Kusmin theorem for the Optimal Continued Fraction (OCF) expansion is obtained. In order to do so, first a Gauss-Kusmin theorem is derived for the natural extension of the ergodic system underlying Hurwitz's Singular Continued Fraction (SCF) (and similarly for the continued fraction to the nearer integer (NICF)). Since the NICF, SCF and OCF are all examples of maximal $S$-expansions, it follows from a result of Kraaikamp that the SCF and OCF are metrically isomorphic. This isomorphism is then used to carry over the results for the SCF to any other maximal $S$-expansion, in particular to the OCF. Along the way, a Heilbronn-theorem is obtained for any maximal $S$-expansion.
Conical limit set and Poincaré exponent for iterations of rational functions
Feliks
Przytycki
2081-2099
Abstract: We contribute to the dictionary between action of Kleinian groups and iteration of rational functions on the Riemann sphere. We define the Poincaré exponent $\delta(f,z)=\inf\{\alpha\ge 0:\mathcal{P}(z,\alpha) \le 0\}$, where \begin{equation*}\mathcal{P}(z,\alpha):=\limsup _{n\to\infty}{1\over n}\log\sum _{f^n(x)=z} |(f^n)'(x)|^{-\alpha}. \end{equation*} We prove that $\delta (f,z)$ and $\mathcal{P}(z,\alpha)$ do not depend on $z$, provided $z$ is non-exceptional. $\mathcal{P}$ plays the role of pressure; we prove that it coincides with the Denker-Urbanski pressure if $\alpha\le \delta(f)$. Various notions of ``conical limit set" are considered. They all have Hausdorff dimension equal to $\delta(f)$ which is equal to the hyperbolic dimension of the Julia set and also equal to the exponent of some conformal Patterson-Sullivan measures. In an Appendix we also discuss notions of ``conical limit set" introduced recently by Urbanski and by Lyubich and Minsky.
Symplectic group lattices
Rudolf
Scharlau;
Pham
Huu
Tiep
2101-2139
Abstract: Let $p$ be an odd prime. It is known that the symplectic group $Sp_{2n}(p)$ has two (algebraically conjugate) irreducible representations of degree $(p^{n}+1)/2$ realized over $\mathbb{Q}(\sqrt{{\epsilon}p})$, where ${\epsilon}= (-1)^{(p-1)/2}$. We study the integral lattices related to these representations for the case $p^{n} \equiv 1 \bmod 4$. (The case $p^{n} \equiv 3 \bmod 4$ has been considered in a previous paper.) We show that the class of invariant lattices contains either unimodular or $p$-modular lattices. These lattices are explicitly constructed and classified. Gram matrices of the lattices are given, using a discrete analogue of Maslov index.
These are the differentials of order $n$
Dan
Laksov;
Anders
Thorup
1293-1353
Abstract: We answer P.-A. Meyer's question ``Qu'est ce qu'une différentielle d'ordre $n$?''. In fact, we present a general theory of higher order differentials based upon a construction of universal objects for higher order differentials. Applied to successive tangent spaces on a differentiable manifold, our theory gives the higher order differentials of Meyer as well as several new results on differentials on differentiable manifolds. In addition our approach gives a natural explanation of the quite mysterious multiplicative structure on higher order differentials observed by Meyer. Applied to iterations of the first order Kähler differentials our theory gives an algebra of higher order differentials for any smooth scheme. We also observe that much of the recent work on higher order osculation spaces of varieties fits well into the framework of our theory.
A counterexample concerning the relation between decoupling constants and UMD-constants
Stefan
Geiss
1355-1375
Abstract: For Banach spaces $X$ and $Y$ and a bounded linear operator $T:X \rightarrow Y$ we let $\rho(T):=\inf c$ such that \begin{displaymath}\left( AV_{\theta _l = \pm 1} \left\|\sum\limits _{l=1}^\infty \theta _l \left( \sum\limits _{k=\tau _{l-1}+1}^{\tau _l} h_k T x_k \right)\right\|_{L_2^Y}^2 \right)^{\frac{1}{2}} \le c \left\| \sum\limits _{k=1}^\infty h_k x_k \right\| _{L_2^X} \end{displaymath} for all finitely supported $(x_k)_{k=1}^\infty \subset X$ and all $0 = \tau _0 < \tau _1 < \cdots$, where $(h_k)_{k=1}^\infty \subset L_1[0,1)$ is the sequence of Haar functions. We construct an operator $T:X \rightarrow X$, where $X$ is superreflexive and of type 2, with $\rho(T)<\infty$ such that there is no constant $c>0$ with \begin{displaymath}\sup _{\theta _k = \pm 1} \left\| \sum\limits _{k=1}^\infty \theta _k h_k T x_k \right\| _{L_2^X} \le c \left\| \sum\limits _{k=1}^\infty h_k x_k \right\| _{L_2^X}. \end{displaymath} In particular it turns out that the decoupling constants $\rho(I_X)$, where $I_X$ is the identity of a Banach space $X$, fail to be equivalent up to absolute multiplicative constants to the usual $\operatorname{UMD}$-constants. As a by-product we extend the characterization of the non-superreflexive Banach spaces by the finite tree property using lower 2-estimates of sums of martingale differences.
Transition operators of diffusions reduce zero-crossing
Steven
N.
Evans;
Ruth
J.
Williams
1377-1389
Abstract: If $u(t,x)$ is a solution of a one-dimensional, parabolic, second-order, linear partial differential equation (PDE), then it is known that, under suitable conditions, the number of zero-crossings of the function $u(t,\cdot)$ decreases (that is, does not increase) as time $t$ increases. Such theorems have applications to the study of blow-up of solutions of semilinear PDE, time dependent Sturm Liouville theory, curve shrinking problems and control theory. We generalise the PDE results by showing that the transition operator of a (possibly time-inhomogenous) one-dimensional diffusion reduces the number of zero-crossings of a function or even, suitably interpreted, a signed measure. Our proof is completely probabilistic and depends in a transparent manner on little more than the sample-path continuity of diffusion processes.
Eigenvalue estimate on complete noncompact Riemannian manifolds and applications
Manfredo
P.
do Carmo;
Detang
Zhou
1391-1401
Abstract: We obtain some sharp estimates on the first eigenvalues of complete noncompact Riemannian manifolds under assumptions of volume growth. Using these estimates we study hypersurfaces with constant mean curvature and give some estimates on the mean curvatures.
Geometric groups. I
Valera
Berestovskii;
Conrad
Plaut;
Cornelius
Stallman
1403-1422
Abstract: We define a geometry on a group to be an abelian semigroup of symmetric open sets with certain properties. Examples include well-known structures such as invariant Riemannian metrics on Lie groups, hyperbolic groups, and valuations on fields. In this paper we are mostly concerned with geometries where the semigroup is isomorphic to the positive reals, which for Lie groups come from invariant Finsler metrics. We explore various aspects of these geometric groups, including a theory of covering groups for arcwise connected groups, algebraic expressions for invariant metrics and inner metrics, construction of geometries with curvature bounded below, and finding geometrically significant curves in path homotopy classes.
Universal constraints on the range of eigenmaps and spherical minimal immersions
Gabor
Toth
1423-1443
Abstract: The purpose of this paper is to give lower estimates on the range dimension of spherical minimal immersions in various settings. The estimates are obtained by showing that infinitesimal isometric deformations (with respect to a compact Lie group acting transitively on the domain) of spherical minimal immersions give rise to a contraction on the moduli space of the immersions and a suitable power of the contraction brings all boundary points into the interior of the moduli space.
$C^\ast$-algebras generated by a subnormal operator
Kit
C.
Chan;
Zeljko
Cuckovic
1445-1460
Abstract: Using the functional calculus for a normal operator, we provide a result for generalized Toeplitz operators, analogous to the theorem of Axler and Shields on harmonic extensions of the disc algebra. Besides that result, we prove that if $T$ is an injective subnormal weighted shift, then any two nontrivial subspaces invariant under $T$ cannot be orthogonal to each other. Then we show that the $C^*$-algebra generated by $T$ and the identity operator contains all the compact operators as its commutator ideal, and we give a characterization of that $C^*$-algebra in terms of generalized Toeplitz operators. Motivated by these results, we further obtain their several-variable analogues, which generalize and unify Coburn's theorems for the Hardy space and the Bergman space of the unit ball.
Decomposing Euclidean space with a small number of smooth sets
Juris
Steprans
1461-1480
Abstract: Let the cardinal invariant ${\mathfrak s}_{n}$ denote the least number of continuously smooth $n$-dimensional surfaces into which $(n+1)$-dimensional Euclidean space can be decomposed. It will be shown to be consistent that ${\mathfrak s}_{n}$ is greater than ${\mathfrak s}_{n+1}$. These cardinals will be shown to be closely related to the invariants associated with the problem of decomposing continuous functions into differentiable ones.
Intersections of $\mathbb{Q}$-divisors on Kontsevich's moduli space $\overline{M}_{0,n}(\mathbb{P}^r,d)$ and enumerative geometry
Rahul
Pandharipande
1481-1505
Abstract: The theory of $\mathbb Q$-Cartier divisors on the space of $n$-pointed, genus 0, stable maps to projective space is considered. Generators and Picard numbers are computed. A recursive algorithm computing all top intersection products of $\mathbb Q$-divisors is established. As a corollary, an algorithm computing all characteristic numbers of rational curves in $\mathbb P^r$ is proven (including simple tangency conditions). Computations of these characteristic numbers are carried out in many examples. The degree of the 1-cuspidal rational locus in the linear system of degree $d$ plane curves is explicitly evaluated.
Limit sets of discrete groups of isometries of exotic hyperbolic spaces
Kevin
Corlette;
Alessandra
Iozzi
1507-1530
Abstract: Let $\Gamma$ be a geometrically finite discrete group of isometries of hyperbolic space $\mathcal{H}_{\mathbb{F}}^n$, where $\mathbb{F}= \mathbb{R}, \mathbb{C}, \mathbb{H}$ or $\mathbb{O}$ (in which case $n=2$). We prove that the critical exponent of $\Gamma$ equals the Hausdorff dimension of the limit sets $\Lambda(\Gamma)$ and that the smallest eigenvalue of the Laplacian acting on square integrable functions is a quadratic function of either of them (when they are sufficiently large). A generalization of Hopf ergodicity theorem for the geodesic flow with respect to the Bowen-Margulis measure is also proven.
Supports of derivations, free factorizations, and ranks of fixed subgroups in free groups
George
M.
Bergman
1531-1550
Abstract: For $F$ a free group of finite rank, it is shown that the fixed subgroup of any set $B$ of endomorphisms of $F$ has rank $\leq \operatorname {rank}(F)$, generalizing a recent result of Dicks and Ventura. The proof involves the combinatorics of derivations of groups. Some related questions are examined.
Entropy and periodic points for transitive maps
Ll.
Alsedà;
S.
Kolyada;
J.
Llibre;
L.
Snoha
1551-1573
Abstract: The aim of this paper is to investigate the connection between transitivity, density of the set of periodic points and topological entropy for low dimensional continuous maps. The paper deals with this problem in the case of the $n$-star and the circle among the one-dimensional spaces and in some higher dimensional spaces. Particular attention is paid to triangular maps and to extensions of transitive maps to higher dimensions without increasing topological entropy.
Steady-state bifurcation with Euclidean symmetry
Ian
Melbourne
1575-1603
Abstract: We consider systems of partial differential equations equivariant under the Euclidean group $\mathbf{E}(n)$ and undergoing steady-state bifurcation (with nonzero critical wavenumber) from a fully symmetric equilibrium. A rigorous reduction procedure is presented that leads locally to an optimally small system of equations. In particular, when $n=1$ and $n=2$ and for reaction-diffusion equations with general $n$, reduction leads to a single equation. (Our results are valid generically, with perturbations consisting of relatively bounded partial differential operators.) In analogy with equivariant bifurcation theory for compact groups, we give a classification of the different types of reduced systems in terms of the absolutely irreducible unitary representations of $\mathbf{E}(n)$. The representation theory of $\mathbf{E}(n)$ is driven by the irreducible representations of $\mathbf{O}(n-1)$. For $n=1$, this constitutes a mathematical statement of the `universality' of the Ginzburg-Landau equation on the line. (In recent work, we addressed the validity of this equation using related techniques.) When $n=2$, there are precisely two significantly different types of reduced equation: scalar and pseudoscalar, corresponding to the trivial and nontrivial one-dimensional representations of $\mathbf{O}(1)$. There are infinitely many possibilities for each $n\ge 3$.
Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in $\mathbf R^N$
Der-Chen
Chang;
Galia
Dafni;
Elias
M.
Stein
1605-1661
Abstract: We study two different local $H^p$ spaces, $0 < p \leq 1$, on a smooth domain in $\mathbf{R}^n$, by means of maximal functions and atomic decomposition. We prove the regularity in these spaces, as well as in the corresponding dual BMO spaces, of the Dirichlet and Neumann problems for the Laplacian.
Vertex operators for twisted quantum affine algebras
Naihuan
Jing;
Kailash
C.
Misra
1663-1690
Abstract: We construct explicitly the $q$-vertex operators (intertwining operators) for the level one modules $V(\Lambda _i)$ of the classical quantum affine algebras of twisted types using interacting bosons, where $i=0, 1$ for $A_{2n-1}^{(2)}$ ($n\geq 3$), $i=0$ for $D_4^{(3)}$, $i=0, n$ for $D_{n+1}^{(2)}$ ($n\geq 2$), and $i=n$ for $A_{2n}^{(2)}$ ($n\geq 1$). A perfect crystal graph for $D_4^{(3)}$ is constructed as a by-product.
Haar Measure and the Artin Conductor
Benedict
H.
Gross;
Wee
Teck
Gan
1691-1704
Abstract: Let $G$ be a connected reductive group, defined over a local, non-archimedean field $k$. The group $G(k)$ is locally compact and unimodular. In On the motive of a reductive group, Invent. Math. 130 (1997), by B. H. Gross, a Haar measure $|\omega _G|$ was defined on $G(k)$, using the theory of Bruhat and Tits. In this note, we give another construction of the measure $|\omega _G|$, using the Artin conductor of the motive $M$ of $G$ over $k$. The equivalence of the two constructions is deduced from a result of G. Prasad.
Asymptotic formulae with remainder estimates for eigenvalue branches of the Schrödinger operator $H - \lambda W$ in a gap of $H$
S.
Z.
Levendorskii
857-899
Abstract: The Floquet theory provides a decomposition of a periodic Schrödinger operator into a direct integral, over a torus, of operators on a basic period cell. In this paper, it is proved that the same transform establishes a unitary equivalence between a multiplier by a decaying potential and a pseudo-differential operator on the torus, with an operator-valued symbol. A formula for the symbol is given. As applications, precise remainder estimates and two-term asymptotic formulas for spectral problems for a perturbed periodic Schrödinger operator are obtained.
Erratic solutions of simple delay equations
Bernhard
Lani-Wayda
901-945
Abstract: We give an example of a smooth function $g:\mathbb{R}\longrightarrow \mathbb{R}$ with only one extremum, with $\mathrm{ sign } g(x) = - \mathrm{ sign } g(-x)$ for $x \neq 0$, and the following properties: The delay equation $\dot x (t) = g(x(t-1))$ has an unstable periodic solution and a solution with phase curve transversally homoclinic to the orbit of the periodic solution. The complicated motion arising from this structure, and its robustness under perturbation of $g$, are described in terms of a Poincaré map. The example is minimal in the sense that the condition $g' < 0$ (under which there would be no extremum) excludes complex solution behavior. Based on numerical observations, we discuss the role of the erratic solutions in the set of all solutions.
Discrete threshold growth dynamics are omnivorous for box neighborhoods
Tom
Bohman
947-983
Abstract: In the discrete threshold model for crystal growth in the plane we begin with some set $A_{0} \subset {\mathbf Z}^{2}$ of seed crystals and observe crystal growth over time by generating a sequence of subsets $A_{0} \subset A_{1} \subset A_{2} \subset \dotsb$ of ${\mathbf Z}^{2}$ by a deterministic rule. This rule is as follows: a site crystallizes when a threshold number of crystallized points appear in the site's prescribed neighborhood. The growth dynamics generated by this model are said to be omnivorous if $A_{0}$ finite and $A_{i+1} \neq A_{i} \; \forall i$ imply $\bigcup _{i=0}^{\infty} A_{i} = {\mathbf Z}^{2}$. In this paper we prove that the dynamics are omnivorous when the neighborhood is a box (i.e. when, for some fixed $\rho$, the neighborhood of $z$ is $\{ x \in {\mathbf Z}^{2} : \|x-z\|_{\infty} \le \rho\})$. This result has important implications in the study of the first passage time when $A_{0}$ is chosen randomly with a sparse Bernoulli density and in the study of the limiting shape to which $n^{-1}A_{n}$ converges.
Dual kinematic formulas
Gaoyong
Zhang
985-995
Abstract: We establish kinematic formulas for dual quermassintegrals of star bodies and for chord power integrals of convex bodies by using dual mixed volumes. These formulas are extensions of the fundamental kinematic formula involving quermassintegrals to the cases of dual quermassintegrals and chord power integrals. Applications to geometric probability are considered.
Compatible complex structures on almost quaternionic manifolds
D.
V.
Alekseevsky;
S.
Marchiafava;
M.
Pontecorvo
997-1014
Abstract: On an almost quaternionic manifold $(M^{4n},Q)\;$we study the integrability of almost complex structures which are compatible with the almost quaternionic structure $Q$. If $n\geq 2$, we prove that the existence of two compatible complex structures $I_{1}, I_{2}\neq \pm I_{1}$ forces $(M^{4n},Q)\;$to be quaternionic. If $n=1$, that is $(M^{4},Q)=(M^{4},[g],or)$ is an oriented conformal 4-manifold, we prove a maximum principle for the angle function $\langle I_{1},I_{2}\rangle$ of two compatible complex structures and deduce an application to anti-self-dual manifolds. By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure $\mathbb J$ on the twistor space $Z$ of an almost quaternionic manifold $(M^{4n},Q)\;$and show that $\mathbb J$ is a complex structure if and only if $Q$ is quaternionic. This is a natural generalization of the Penrose twistor constructions.
A remarkable formula for counting nonintersecting lattice paths in a ladder with respect to turns
C.
Krattenthaler;
M.
Prohaska
1015 - 1042
On graphs with a metric end space
Kerstin
Waas
1043-1062
Abstract: R. Diestel conjectured that an infinite graph contains a topologically end-faithful forest if and only if its end space is metrizable. We prove this conjecture for uniform end spaces.
On the diophantine equation $(x^3-1)/(x-1)=(y^n-1)/(y-1)$
Maohua
Le
1063-1074
Abstract: In this paper we prove that the equation $(x^3-1)/(x-1)=$ $(y^n-1)/(y-1)$, $x,y,n\in\mathbb{N}$, $x>1$, $y>1$, $n>3$, has only the solutions $(x,y,n)=(5,2,5)$ and $(90,2,13)$ with $y$ is a prime power. The proof depends on some new results concerning the upper bounds for the number of solutions of the generalized Ramanujan-Nagell equations.
On the non-vanishing of cubic twists of automorphic $L$-series
Xiaotie
She
1075-1094
Abstract: Let $f$ be a normalised new form of weight $2$ for $\Gamma _{0} (N)$ over ${\mathbb{Q}}$ and $F$, its base change lift to $\mathbb{Q}(\sqrt {-3})$. A sufficient condition is given for the nonvanishing at the center of the critical strip of infinitely many cubic twists of the $L$-function of $F$. There is an algorithm to check the condition for any given form. The new form of level $11$ is used to illustrate our method.
Group extensions and tame pairs
Michael
L.
Mihalik
1095-1107
Abstract: Tame pairs of groups were introduced to study the missing boundary problem for covers of compact 3-manifolds. In this paper we prove that if $1\to A\to G\to B\to 1$ is an exact sequence of infinite finitely presented groups or if $G$ is an ascending HNN-extension with base $A$ and $H$ is a certain type of finitely presented subgroup of $A$, then the pair $(G,H)$ is tame. Also we develop a technique for showing certain groups cannot be the fundamental group of a compact 3-manifold. In particular, we give an elementary proof of the result of R. Bieri, W. Neumann and R. Strebel: A strictly ascending HNN-extension cannot be the fundamental group of a compact 3-manifold.
A Weakly Chainable Tree-Like Continuum without the Fixed Point Property
Piotr
Minc
1109-1121
Abstract: An example of a fixed points free map is constructed on a tree-like, weakly chainable continuum.
A generalization of Snaith-type filtration
Greg
Arone
1123-1150
Abstract: In this paper we describe the Goodwillie tower of the stable homotopy of a space of maps from a finite-dimensional complex to a highly enough connected space. One way to view it is as a partial generalization of some well-known results on stable splittings of mapping spaces in terms of configuration spaces.
Riesz transforms for $1\le p\le 2$
Thierry
Coulhon;
Xuan
Thinh
Duong
1151-1169
Abstract: It has been asked (see R. Strichartz, Analysis of the Laplacian$\dotsc$, J. Funct. Anal. 52 (1983), 48-79) whether one could extend to a reasonable class of non-compact Riemannian manifolds the $L^p$ boundedness of the Riesz transforms that holds in ${\mathbb R}^n$. Several partial answers have been given since. In the present paper, we give positive results for $1\leq p\leq 2$ under very weak assumptions, namely the doubling volume property and an optimal on-diagonal heat kernel estimate. In particular, we do not make any hypothesis on the space derivatives of the heat kernel. We also prove that the result cannot hold for $p>2$ under the same assumptions. Finally, we prove a similar result for the Riesz transforms on arbitrary domains of ${\mathbb R}^n$.
Building blocks for quadratic Julia sets
Joachim
Grispolakis;
John
C.
Mayer;
Lex
G.
Oversteegen
1171-1201
Abstract: We obtain results on the structure of the Julia set of a quadratic polynomial $P$ with an irrationally indifferent fixed point $z_0$ in the iterative dynamics of $P$. In the Cremer point case, under the assumption that the Julia set is a decomposable continuum, we obtain a building block structure theorem for the corresponding Julia set $J=J(P)$: there exists a nowhere dense subcontinuum $B\subset J$ such that $P(B)=B$, $B$ is the union of the impressions of a minimally invariant Cantor set $A$ of external rays, $B$ contains the critical point, and $B$ contains both the Cremer point $z_0$ and its preimage. In the Siegel disk case, under the assumption that no impression of an external ray contains the boundary of the Siegel disk, we obtain a similar result. In this case $B$ contains the boundary of the Siegel disk, properly if the critical point is not in the boundary, and $B$ contains no periodic points. In both cases, the Julia set $J$ is the closure of a skeleton $S$ which is the increasing union of countably many copies of the building block $B$ joined along preimages of copies of a critical continuum $C$ containing the critical point. In addition, we prove that if $P$ is any polynomial of degree $d\ge 2$ with a Siegel disk which contains no critical point on its boundary, then the Julia set $J(P)$ is not locally connected. We also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected Julia set have homeomorphic Julia sets.
Invariant Measures for Set-Valued Dynamical Systems
Walter
Miller;
Ethan
Akin
1203-1225
Abstract: A continuous map on a compact metric space, regarded as a dynamical system by iteration, admits invariant measures. For a closed relation on such a space, or, equivalently, an upper semicontinuous set-valued map, there are several concepts which extend this idea of invariance for a measure. We prove that four such are equivalent. In particular, such relation invariant measures arise as projections from shift invariant measures on the space of sample paths. There is a similarly close relationship between the ideas of chain recurrence for the set-valued system and for the shift on the sample path space.
Germs of Kloosterman Integrals for $GL(3)$
Hervé
Jacquet;
Yangbo
Ye
1227-1255
Abstract: In an earlier paper we introduced the concept of Shalika germs for certain Kloosterman integrals. We compute explicitly the germs in the case of the group $GL(3)$.
On locally linearly dependent operators and derivations
Matej
Bresar;
Peter
Semrl
1257-1275
Abstract: The first section of the paper deals with linear operators $T_i:U\longrightarrow V$, $i = 1,\ldots,n$, where $U$ and $V$ are vector spaces over an infinite field, such that for every $u \in U$, the vectors $T_1 u,\ldots,T_n u$ are linearly dependent modulo a fixed finite dimensional subspace of $V$. In the second section, outer derivations of dense algebras of linear operators are discussed. The results of the first two sections of the paper are applied in the last section, where commuting pairs of continuous derivations $d,g$ of a Banach algebra ${\cal A}$ such that $(dg)(x)$ is quasi-nilpotent for every $x \in {\cal A}$ are characterized.
Composition algebras over rings of genus zero
S.
Pumplün
1277-1292
Abstract: The theory of composition algebras over locally ringed spaces and some basic results from algebraic geometry are used to characterize composition algebras over open dense subschemes of curves of genus zero.
Quadratic homology
Hans-Joachim
Baues
429-475
Abstract: We describe axioms for a `quadratic homology theory' which generalize the classical axioms of homology. As examples we consider quadratic homology theories induced by 2-excisive homotopy functors in the sense of Goodwillie and the homology of a space with coefficients in a square group which generalizes the homology of a space with coefficients in an abelian group.
Global character formulae for compact Lie groups
A.
H.
Dooley;
N.
J.
Wildberger
477-495
Abstract: We introduce the concept of a modulator, which leads to a family of character formulae, each generalizing the Kirillov formula. For a suitable choice of modulator, this enables one to understand the Plancherel measure of a compact Lie group as arising from a partition of the identity on the dual of its Lie algebra.
On some Banach space properties sufficient for weak normal structure and their permanence properties
Brailey
Sims;
Michael
A.
Smyth
497-513
Abstract: We consider Banach space properties that lie between conditions introduced by Bynum and Landes. These properties depend on the metric behavior of weakly convergent sequences. We also investigate the permanence properties of these conditions.
The stability of the equilibrium of reversible systems
Bin
Liu
515-531
Abstract: In this paper, we consider the system \begin{displaymath}\dot x=a(t)y^{2m+1}+f_1(x,y,t),\quad\dot y=-b(t)x^{2n+1}+f_2(x,y,t),\end{displaymath} where $m,n\in \mathbf Z_+$, $m+n\ge 1$, $a(t)$ and $b(t)$ are continuous, even and 1-periodic in the time variable $t$; $f_1$ and $f_2$ are real analytic in a neighbourhood of the origin $(0,0)$ of $(x,y)$-plane and continuous 1-periodic in $t$. We also assume that the above system is reversible with respect to the involution $G\colon(x,y)\mapsto(-x,y)$. A sufficient and necessary condition for the stability in the Liapunov sense of the equilibrium $(x,y)=(0,0)$ is given.
Homogeneous projective varieties with degenerate secants
Hajime
Kaji
533-545
Abstract: The secant variety of a projective variety $X$ in $\mathbb{P}$, denoted by $\operatorname{Sec}X$, is defined to be the closure of the union of lines in $\mathbb{P}$ passing through at least two points of $X$, and the secant deficiency of $X$ is defined by $\delta := 2 \dim X + 1 - \dim \operatorname{Sec}X$. We list the homogeneous projective varieties $X$ with $\delta > 0$ under the assumption that $X$ arise from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety $X$ with $\operatorname{Sec}X \not = \mathbb{P}$ and $\delta > 8$, and the $E_{6}$-variety is the only homogeneous projective variety with largest secant deficiency $\delta = 8$. This gives a negative answer to a problem posed by R. Lazarsfeld and A. Van de Ven if we restrict ourselves to homogeneous projective varieties.
The nilpotence height of $P_t^s$ for odd primes
Ismet
Karaca
547-558
Abstract: K. G. Monks has recently shown that the element $P^{s}_{t}$ has nilpotence height $2[\frac{s}{t}] + 2$ in the mod $2$ Steenrod algebra. Here the method and result are generalized to show that for an odd prime $p$ the element $P^{s}_{t}$ has nilpotence height $p[\frac{s}{t}] + p$ in the mod $p$ Steenrod algebra.
Regularity of solutions to the measurable Livsic equation
M.
Pollicott;
M.
Yuri
559-568
Abstract: In this note we give generalisations of Livsic's result that a priori measurable solutions to cocycle equations must in fact be more regular. We go beyond the original continuous hyperbolic examples of Livsic to consider examples of this phenomenon in the context of: (a) $\beta$-transformations; (b) rational maps; and (c) planar maps with indifferent periodic points. Such examples are not immediately covered by Livsic's original approach either due to a lack of continuity or hyperbolicity.
Windows of given area with minimal heat diffusion
Jochen
Denzler
569-580
Abstract: For a bounded Lipschitz domain $\Omega$, we show the existence of a measurable set $D\subset \partial\Omega$ of given area such that the first eigenvalue of the Laplacian with Dirichlet conditions on $D$ and Neumann conditions on $\partial \Omega \setminus D$ becomes minimal. If $\Omega$ is a ball, $D$ will be a spherical cap.
Golubev series for solutions of elliptic equations
Ch.
Dorschfeldt;
N.
N.
Tarkhanov
581-594
Abstract: Let $P$ be an elliptic system with real analytic coefficients on an open set $X\subset {\Bbb R}^{n},$ and let $\Phi$ be a fundamental solution of $P.$ Given a locally connected closed set $\sigma \subset X,$ we fix some massive measure $m$ on $\sigma$. Here, a non-negative measure $m$ is called massive, if the conditions $s \subset \sigma$ and $m(s)=0$ imply that $\overline{\sigma \setminus s} = \sigma .$ We prove that, if $f$ is a solution of the equation $Pf =0$ in $X \setminus \sigma ,$ then for each relatively compact open subset $U$ of $X$ and every $1<p<\infty$ there exist a solution $f_{e}$ of the equation in $U$ and a sequence $f_{\alpha }$ ($\alpha \in {\Bbb N}^{n}_{0}$) in $L^{p} (\sigma \cap U, m)$ satisfying $\| \alpha ! f_{\alpha } \|^{1/|\alpha|}_{L^{p} (\sigma \cap U,m)} \rightarrow 0$ such that $f(x) = f_{e} (x) +\sum _{\alpha}\int _{\sigma \cap U} D^{\alpha }_{y} \Phi (x,y) f_{\alpha } (y) dm(y)$ for $x \in U \setminus \sigma .$ This complements an earlier result of the second author on representation of solutions outside a compact subset of $X.$
Maximal ideals in modular group algebras of the finitary symmetric and alternating groups
Alexander
Baranov;
Alexander
Kleshchev
595-617
Abstract: The main result of the paper is a description of the maximal ideals in the modular group algebras of the finitary symmetric and alternating groups (provided the characteristic $p$ of the ground field is greater than 2). For the symmetric group there are exactly $p-1$ such ideals and for the alternating group there are $(p-1)/2$ of them. The description is obtained in terms of the annihilators of certain systems of the `completely splittable' irreducible modular representations of the finite symmetric and alternating groups. The main tools used in the proofs are the modular branching rules (obtained earlier by the second author) and the `Mullineux conjecture' proved recently by Ford-Kleshchev and Bessenrodt-Olsson. The results obtained are relevant to the theory of PI-algebras. They are used in a later paper by the authors and A. E. Zalesskii on almost simple group algebras and asymptotic properties of modular representations of symmetric groups.
Gap estimates of the spectrum of Hill's equation and action variables for KdV
T.
Kappeler;
B.
Mityagin
619-646
Abstract: Consider the Schrödinger equation $-y'' + Vy = \lambda y$ for a potential $V$ of period 1 in the weighted Sobolev space $(N \in \mathbb{Z}_{\ge 0}, \omega \in \mathbb{R}_{\ge 0})$ \begin{displaymath}H^{N, \omega}(S^1; \mathbb{C}) := \{ f(x) = \sum^{\infty}_{k= - \infty} \Hat {\Hat f}(k) e^{i 2 \pi kx} \bigg | \parallel f \parallel _{N, \omega} < \infty \}\end{displaymath} where $\Hat {\Hat f}(k) (k \in \mathbb{Z})$ denote the Fourier coefficients of $f$ when considered as a function of period 1, \begin{displaymath}\parallel f \parallel _{N, \omega} := \bigg ( \sum _k (1+| k|)^{2N} e^{2 \omega | k |} | \ \Hat{\Hat{f}} (k) |^2 \bigg )^{^{1}/2} < \infty ,\end{displaymath} and where $S^1$ is the circle of length 1. Denote by $\lambda _k \equiv \lambda _k (V) (k \ge 0)$ the periodic eigenvalues of $- \frac{d^2}{dx^2} + V$ when considered on the interval $[0,2],$ with multiplicities and ordered so that $Re \lambda _j \le Re \lambda _{j+1} (j \ge 0).$ We prove the following result. Theorem. For any bounded set ${\mathcal B} \subseteq H^{N, \omega} (S^1; \mathbb{C}),$ there exist $n_0 \ge 1$ and $M \ge 1$ so that for $k \ge n_0$ and $V \in {\mathcal B}$, the eigenvalues $\lambda _{2k}, \lambda _{2k -1}$ are isolated pairs, satisfying (with $\{ \lambda _{2k}, \lambda _{2k-1} \} = \{ \lambda^+_k , \lambda^-_k \})$ (i) $\sum _{k \ge n_0} (1+k)^{2N} e^{2 \omega k} | \lambda _k^+ - \lambda^-_k |^2 \le M$, (ii) $\sum _{k \ge n_0} (1 + k)^{2 N+1} e^{2 \omega k} \bigg | (\lambda^+_k - \lambda^-_k) -2 \sqrt{\Hat{\Hat{V}} (k) \Hat{\Hat{V}}(-k)} \bigg |^2 \le M$.
Exact categories and vector space categories
Peter
Dräxler;
Idun
Reiten;
Sverre
O.
Smal\o;
Øyvind
Solberg;
with an appendix by
B. Keller
647-682
Abstract: In a series of papers additive subbifunctors $F$ of the bifunctor $\operatorname{Ext}_{\Lambda} ( , )$ are studied in order to establish a relative homology theory for an artin algebra $\Lambda$. On the other hand, one may consider the elements of $F(X,Y)$ as short exact sequences. We observe that these exact sequences make $\operatorname{mod}\Lambda$ into an exact category if and only if $F$ is closed in the sense of Butler and Horrocks. Concerning the axioms for an exact category we refer to Gabriel and Roiter's book. In fact, for our general results we work with subbifunctors of the extension functor for arbitrary exact categories. In order to study projective and injective objects for exact categories it turns out to be convenient to consider categories with almost split exact pairs, because many earlier results can easily be adapted to this situation. Exact categories arise in representation theory for example if one studies categories of representations of bimodules. Representations of bimodules gained their importance in studying questions about representation types. They appear as domains of certain reduction functors defined on categories of modules. These reduction functors are often closely related to the functor $\operatorname{Ext}_{\Lambda}( , )$ and in general do not preserve at all the usual exact structure of $\operatorname{mod}\Lambda$. By showing the closedness of suitable subbifunctors of $\operatorname{Ext}_{\Lambda}( , )$ we can equip $\operatorname{mod}\Lambda$ with an exact structure such that some reduction functors actually become `exact'. This allows us to derive information about the projective and injective objects in the respective categories of representations of bimodules appearing as domains, and even show that almost split sequences for them exist. Examples of such domains appearing in practice are the subspace categories of a vector space category with bonds. We provide an example showing that existence of almost split sequences for them is not a general fact but may even fail if the vector space category is finite.
The heat kernel weighted Hodge Laplacian on noncompact manifolds
Edward
L.
Bueler
683-713
Abstract: On a compact orientable Riemannian manifold, the Hodge Laplacian $\triangle$ has compact resolvent, therefore a spectral gap, and the dimension of the space $\mathcal{H}^{p} = \ker \triangle ^{p}$ of harmonic $p$-forms is a topological invariant. By contrast, on complete noncompact Riemannian manifolds, $\triangle$ is known to have various pathologies, among them the absence of a spectral gap and either ``too large'' or ``too small'' a space $\mathcal{H}^{p}$. In this article we use a heat kernel measure $d\mu$ to determine the space of square-integrable forms and to construct the appropriate Laplacian $\triangle _{\mu }$. We recover in the noncompact case certain results of Hodge's theory of $\triangle$ in the compact case. If the Ricci curvature of a noncompact connected Riemannian manifold $M$ is bounded below, then this ``heat kernel weighted Laplacian'' $\triangle _{\mu }$ acts on functions on $M$ in precisely the manner we would wish, that is, it has a spectral gap and a one-dimensional kernel. We prove that the kernel of $\triangle _{\mu }$ on $n$-forms is zero-dimensional on $M$, as we expect from topology, if the Ricci curvature is nonnegative. On Euclidean space, there is a complete Hodge theory for $\triangle _{\mu }$. Weighted Laplacians also have a duality analogous to Poincaré duality on noncompact manifolds. Finally, we show that heat kernel-like measures give desirable spectral properties (compact resolvent) in certain general cases. In particular, we use measures with Gaussian decay to justify the statement that every topologically tame manifold has a strong Hodge decomposition.
Galois coverings of selfinjective algebras by repetitive algebras
Andrzej
Skowronski;
Kunio
Yamagata
715-734
Abstract: In the representation theory of selfinjective artin algebras an important role is played by selfinjective algebras of the form $\widehat {B}/G$ where $\widehat {B}$ is the repetitive algebra of an artin algebra $B$ and $G$ is an admissible group of automorphisms of $\widehat {B}$. If $B$ is of finite global dimension, then the stable module category $\underline{\operatorname{mod}} \widehat {B}$ of finitely generated $\widehat {B}$-modules is equivalent to the derived category $D^{b} (\operatorname{mod} B)$ of bounded complexes of finitely generated $B$-modules. For a selfinjective artin algebra $A$, an ideal $I$ and $B=A/I$, we establish a criterion for $A$ to admit a Galois covering $F: \widehat {B}\to \widehat {B}/G=A$ with an infinite cyclic Galois group $G$. As an application we prove that all selfinjective artin algebras $A$ whose Auslander-Reiten quiver $\Gamma _{A}$ has a non-periodic generalized standard translation subquiver closed under successors in $\Gamma _{A}$ are socle equivalent to the algebras $\widehat {B}/G$, where $B$ is a representation-infinite tilted algebra and $G$ is an infinite cyclic group of automorphisms of $\widehat{B}$.
On Siegel modular forms of half-integral weights and Jacobi forms
Koichi
Takase
735-780
Abstract: We will establish a bijective correspondence between the space of the cuspidal Jacobi forms and the space of the half-integral weight Siegel cusp forms which is compatible with the action of the Hecke operators. This correspondence is based on a bijective correspondence between the irreducible unitary representations of a two-fold covering group of a symplectic group and a Jacobi group (that is, a semidirect product of a symplectic group and a Heisenberg group). The classical results due to Eichler-Zagier and Ibukiyama will be reconsidered from our representation theoretic point of view.
The nonarchimedean theta correspondence for $\mathrm{GSp}(2)$ and $\mathrm{GO}(4)$
Brooks
Roberts
781-811
Abstract: In this paper we consider the theta correspondence between the sets $\operatorname{Irr} (\operatorname{GSp} (2,k))$ and $\operatorname{Irr} (\operatorname{GO} (X))$ when $k$ is a nonarchimedean local field and $\dim _{k} X =4$. Our main theorem determines all the elements of $\operatorname{Irr} (\operatorname{GO} (X))$ that occur in the correspondence. The answer involves distinguished representations. As a corollary, we characterize all the elements of $\operatorname{Irr} (\operatorname{O} (X))$ that occur in the theta correspondence between $\operatorname{Irr} (\operatorname{Sp} (2,k))$ and $\operatorname{Irr} (\operatorname{O} (X))$. We also apply our main result to prove a case of a new conjecture of S.S. Kudla concerning the first occurrence of a representation in the theta correspondence.
Projective structures with discrete holonomy representations
Hiroshige
Shiga;
Harumi
Tanigawa
813-823
Abstract: Let $K(X)$ denote the set of projective structures on a compact Riemann surface $X$ whose holonomy representations are discrete. We will show that each component of the interior of $K(X)$ is holomorphically equivalent to a complex submanifold of the product of Teichmüller spaces and the holonomy representation of every projective structure in the interior of $K(X)$ is a quasifuchsian group.
Conjugacy classes of $SU(h,\mathcal O_S)$ in $SL(2,\mathcal O_S)$
Donald
G.
James
825-835
Abstract: Let $K$ be a quadratic extension of a global field $F$, of characteristic not two, and $\mathcal{O}_S$ the integral closure in $K$ of a Dedekind ring of $S$-integers $\mathfrak{O}_S$ in $F$. Then $PSL(2, \mathcal{O}_S)$ is isomorphic to the spinorial kernel $O'(L)$ for an indefinite quadratic $\mathfrak{O}_S$-lattice $L$ of rank 4. The isomorphism is used to study the conjugacy classes of unitary groups $PSU(h,\mathcal{O}_S)$ of primitive odd binary hermitian matrices $h$ under the action of $PSL(2, \mathcal{O}_S)$.
Distribution semigroups and abstract Cauchy problems
Peer
Christian
Kunstmann
837-856
Abstract: We present a new definition of distribution semigroups, covering in particular non-densely defined generators. We show that for a closed operator $A$ in a Banach space $E$ the following assertions are equivalent: (a) $A$ generates a distribution semigroup; (b) the convolution operator $\delta'\otimes I-\delta\otimes A$ has a fundamental solution in ${\mathcal D}'(L(E,D))$ where $D$ denotes the domain of $A$ supplied with the graph norm and $I$ denotes the inclusion $D\to E$; (c) $A$ generates a local integrated semigroup. We also show that every generator of a distribution semigroup generates a regularized semigroup.
Simplicity and the stable rank of some free product C*-algebras
Kenneth
J.
Dykema
1-40
Abstract: A necessary and sufficient condition for the simplicity of the C$^{*}$-algebra reduced free product of finite dimensional abelian algebras is found, and it is proved that the stable rank of every such free product is 1. Related results about other reduced free products of C$^{*}$-algebras are proved.
An algorithm for calculating the Nielsen number on surfaces with boundary
Joyce
Wagner
41-62
Abstract: Let $f:M\to M$ be a self-map of a hyperbolic surface with boundary. The Nielsen number, $N(f)$, depends only on the induced map $f_{\#}$ of the fundamental group, which can be viewed as a free group on $n$ generators, $a_1,\dotsc,a_n$. We determine conditions for fixed points to be in the same fixed point class and if these conditions are enough to determine the fixed point classes, we say that $f_{\#}$ is $W$-characteristic. We define an algebraic condition on the $f_{\#}(a_i)$ and show that ``most'' maps satisfy this condition and that all maps which satisfy this condition are $W$-characteristic. If $f_{\#}$ is $W$-characteristic, we present an algorithm for calculating $N(f)$ and prove that the inequality $|L(f)-\chi(M)|\le N(f)-\chi(M)$ holds, where $L(f)$ denotes the Lefschetz number of $f$ and $\chi(M)$ the Euler characteristic of $M$, thus answering in part a question of Jiang and Guo.
Rotating an interval and a circle
Alexander
Blokh;
Michal
Misiurewicz
63-78
Abstract: We compare periodic orbits of circle rotations with their counterparts for interval maps. We prove that they are conjugate via a map of modality larger by at most 2 than the modality of the interval map. The proof is based on observation of trips of inhabitants of the Green Islands in the Black Sea.
Maximal function estimates of solutions to general dispersive partial differential equations
Hans
P.
Heinig;
Sichun
Wang
79-108
Abstract: Let $u(x,t)=(S_\Omega f)(x,t)$ be the solution of the general dispersive initial value problem: \begin{displaymath}\partial _tu-i\Omega(D)u=0, \quad u(x,0)=f(x), \qquad (x,t)\in \mathbb{R}^n \times \mathbb{R}\end{displaymath} and $S^{**}_\Omega f$ the global maximal operator of $S_\Omega f$. Sharp weighted $L^p$-esimates for $S^{**}_\Omega f$ with $f\in H_s(\mathbb{R}^n)$ are given for general phase functions $\Omega$.
Large time asymptotics of solutions to the generalized Benjamin-Ono equation
Nakao
Hayashi;
Pavel
I.
Naumkin
109-130
Abstract: We study the asymptotic behavior for large time of solutions to the Cauchy problem for the generalized Benjamin-Ono (BO) equation: $u_{t} + (|u|^{\rho -1}u)_{x} + \mathcal{H} u_{xx} = 0$, where $\mathcal{H}$ is the Hilbert transform, $x, t \in {\mathbf{R}}$, when the initial data are small enough. If the power $\rho$ of the nonlinearity is greater than $3$, then the solution of the Cauchy problem has a quasilinear asymptotic behavior for large time. In the case $\rho=3$ critical for the asymptotic behavior i.e. for the modified Benjamin-Ono equation, we prove that the solutions have the same $L^{\infty }$ time decay as in the corresponding linear BO equation. Also we find the asymptotics for large time of the solutions of the Cauchy problem for the BO equation in the critical and noncritical cases. For the critical case, we prove the existence of modified scattering states if the initial function is sufficiently small in certain weighted Sobolev spaces. These modified scattering states differ from the free scattering states by a rapidly oscillating factor.
Classifying 2-groups by coclass
M.
F.
Newman;
E.
A.
O'Brien
131-169
Abstract: Now that the conjectures of Leedham-Green and Newman have been proved, we probe deeper into the classification of $p$-groups using coclass. We determine the pro-$2$-groups of coclass at most 3 and use these to classify the 2-groups of coclass at most 3 into families. Using extensive computational evidence, we make some detailed conjectures about the structure of these families. We also conjecture that the 2-groups of arbitrary fixed coclass exhibit similar behaviour.
Monomial Bases for $H^*(\mathbf{C}P^\infty\times\mathbf{C}P^\infty)$ over $\mathcal A(p)$
M.
D.
Crossley
171-192
Abstract: We consider the polynomial algebra $H^*(\mathbf{C} P^\infty \times \mathbf{C} P^\infty; \mathbf{F}_p)$ as a module over the mod $p$ Steenrod algebra, $\mathcal{A}(p)$, $p$ being an odd prime. We give a minimal set of generators consisting of monomials and characterise all such `monomial bases'.
Reducibility of some induced representations of $p$-adic unitary groups
Fiona
Murnaghan;
Joe
Repka
193-210
Abstract: In this paper we study reducibility of those representations of quasi-split unitary $p$-adic groups which are parabolically induced from supercuspidal representations of general linear groups. For a supercuspidal representation associated via Howe's construction to an admissible character, we show that in many cases a criterion of Goldberg for reducibility of the induced representation reduces to a simple condition on the admissible character.
Turnpike property for extremals of variational problems with vector-valued functions
A.
J.
Zaslavski
211-231
Abstract: In this paper we study the structure of extremals $\nu\colon[0,T]\to R^n$ of variational problems with large enough $T$, fixed end points and an integrand $f$ from a complete metric space of functions. We will establish the turnpike property for a generic integrand $f$. Namely, we will show that for a generic integrand $f$, any small $\varepsilon>0$ and an extremal $\nu\colon[0,T]\to R^n$ of the variational problem with large enough $T$, fixed end points and the integrand $f$, for each $\tau\in[L_1, T-L_1]$ the set $\{\nu(t)\colon t\in[\tau,\tau+L_2]\}$ is equal to a set $H(f)$ up to $\varepsilon$ in the Hausdorff metric. Here $H(f)\subset R^n$ is a compact set depending only on the integrand $f$ and $L_1>L_2>0$ are constants which depend only on $\varepsilon$ and $|\nu(0)|$, $|\nu(T)|$.
Properties of some families of hypergeometric orthogonal polynomials in several variables
J.
F.
van Diejen
233-270
Abstract: Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable Wilson, continuous Hahn and Jacobi type polynomials, respectively. For each class of polynomials we provide systems of difference (or differential) equations, recurrence relations, and expressions for the (squared) norms of the polynomials in question.
Density doubling, double-circulants, and new sphere packings
Alexander
Vardy
271-283
Abstract: New nonlattice sphere packings in dimensions 20, 22, and 44-47 that are denser than the best previously known sphere packings were recently discovered. We extend these results, showing that the density of many sphere packings in dimensions just below a power of 2 can be doubled using orthogonal binary codes. This produces new dense sphere packings in $\mathbb R^n$ for $n = 25,26,\dots,31$ and $55,56,\dots,63$. For $n = 27,28,29,30$ the resulting packings are denser than any packing previously known.
Existence and uniqueness for a degenerate parabolic equation with $L^1$-data
F.
Andreu;
J.
M.
Mazón;
S.
Segura
de León;
J.
Toledo
285-306
Abstract: In this paper we study existence and uniqueness of solutions for the boundary-value problem, with initial datum in $L^{1}(\Omega )$, \begin{equation*}u_{t} = \mathrm{div} {\hbox{$\mathbf a$}}(x,Du) \quad \text{in } (0, \infty ) \times \Omega, \end{equation*} \begin{equation*}-{\frac{{\partial u} }{{\partial \eta _{a}}}} \in \beta (u) \quad \text{on } (0, \infty ) \times \partial \Omega,\end{equation*} \begin{equation*}u(x, 0) = u_{0}(x) \quad \text{in }\Omega ,\end{equation*} where a is a Carathéodory function satisfying the classical Leray-Lions hypothesis, $\partial / {\partial \eta _{a}}$ is the Neumann boundary operator associated to ${\hbox{$\mathbf a$}}$, $Du$ the gradient of $u$ and $\beta$ is a maximal monotone graph in ${\mathbb{R}}\times {\mathbb{R}}$ with $0 \in \beta (0)$.
The Krein-Milman theorem in operator convexity
Corran
Webster;
Soren
Winkler
307-322
Abstract: We generalize the Krein-Milman theorem to the setting of matrix convex sets of Effros-Winkler, extending the work of Farenick-Morenz on compact C$^*$-convex sets of complex matrices and the matrix state spaces of C$^*$-algebras. An essential ingredient is to prove the non-commutative analogue of the fact that a compact convex set $K$ may be thought of as the state space of the space of continuous affine functions on $K$.
Operator valued weights without structure theory
Tony
Falcone;
Masamichi
Takesaki
323-341
Abstract: A result of Haagerup, generalizing a theorem of Takesaki, states the following: If ${\mathcal{N}}\subset {\mathcal{M}}$ are von Neumann algebras, then there exists a faithful, normal and semi-finite (fns) operator valued weight $T \colon {\mathcal{M}}_{+} \rightarrow \widehat {{\mathcal{N}}_{+}}$ if and only if there exist fns weights $\tilde \varphi$ on ${\mathcal{M}}$ and $\varphi$ on ${\mathcal{N}}$ satisfying $\sigma ^{\varphi }_{t}(x) = \sigma ^{\tilde \varphi }_{t}(x) \, \forall x \in {\mathcal{N}} , t \in \mathbb{R}$. In fact, $T$ can be chosen such that $\tilde \varphi = \varphi \circ T$; $T$ is then uniquely determined by this condition. We present a proof of the above which does not use any structure theory.
Multi-separation, centrifugality and centripetality imply chaos
Jiehua
Mai
343-351
Abstract: Let $I$ be an interval. $I$ need not be compact or bounded. Let $f:I\rightarrow I$ be a continuous map, and $(x_0, x_1, \cdots, x_n)$ be a trajectory of $f$ with $x_n\leq x_0<x_1$ or $x_1<x_0\leq x_n$. Then there is a point $v\in I$ such that $\min\{x_0, \cdots, x_n\}<v=f(v)<\max\{x_0, \cdots, x_n\}$. A point $y\in I$ is called a centripetal point of $f$ relative to $v$ if $v<f(y)<y$, and $y$ is centrifugal if $f(y)<y<v$ or $v<y<f(y)$. In this paper we prove that if there exist $k$ centripetal points of $f$ in $\{x_0, \cdots, x_{n-1}\}, k\geq 1$, then $f$ has periodic points of some odd ($\not= 1$) period $p\leq (n-2)/k+2$. In addition, we also prove that if $(x_0, x_1, \cdots, x_{n-1}$) is multi-separated by Fix($f$), or there exists a centrifugal point of $f$ in $\{x_0, \cdots, x_{n-1}\}$, then $f$ is turbulent and hence has periodic points of all periods.
Scrambled sets of continuous maps of 1-dimensional polyhedra
Jiehua
Mai
353-362
Abstract: Let $K$ be a 1-dimensional simplicial complex in $R^3$ without isolated vertexes, $X = |K|$ be the polyhedron of $K$ with the metric $d_K$ induced by $K$, and $f:X\rightarrow X$ be a continuous map. In this paper we prove that if $K$ is finite, then the interior of every scrambled set of $f$ in $X$ is empty. We also show that if $K$ is an infinite complex, then there exist continuous maps from $X$ to itself having scrambled sets with nonempty interiors, and if $X = R$ or $R_+$, then there exist $C^\infty$ maps of $X$ with the whole space $X$ being a scrambled set.
On sectional genus of quasi-polarized 3-folds
Yoshiaki
Fukuma
363-377
Abstract: Let $X$ be a smooth projective variety over $\mathbb{C}$ and $L$ a nef-big (resp. ample) divisor on $X$. Then $(X,L)$ is called a quasi-polarized (resp. polarized) manifold. Then we conjecture that $g(L)\geq q(X)$, where $g(L)$ is the sectional genus of $L$ and $q(X)=\operatorname{dim}H^{1}(\mathcal{O}_{X})$ is the irregularity of $X$. In general it is unknown whether this conjecture is true or not, even in the case of $\operatorname{dim}X=2$. For example, this conjecture is true if $\operatorname{dim}X=2$ and $\operatorname{dim}H^{0}(L)>0$. But it is unknown if $\operatorname{dim}X\geq 3$ and $\operatorname{dim}H^{0}(L)>0$. In this paper, we prove $g(L)\geq q(X)$ if $\operatorname{dim}X=3$ and $\operatorname{dim}H^{0}(L)\geq 2$. Furthermore we classify polarized manifolds $(X,L)$ with $\operatorname{dim}X=3$, $\operatorname{dim}H^{0}(L)\geq 3$, and $g(L)=q(X)$.
Combinatorial families that are exponentially far from being listable in Gray code sequence
Ted
Chinburg;
Carla
D.
Savage;
Herbert
S.
Wilf
379-402
Abstract: Let $S(n)$ be a collection of subsets of $\{1,...,n\}$. In this paper we study numerical obstructions to the existence of orderings of $S(n)$ for which the cardinalities of successive subsets satisfy congruence conditions. Gray code orders provide an example of such orderings. We say that an ordering of $S(n)$ is a Gray code order if successive subsets differ by the adjunction or deletion of a single element of $\{1,\ldots,n\}$. The cardinalities of successive subsets in a Gray code order must alternate in parity. It follows that if $d(S(n))$ is the difference between the number of elements of $S(n)$ having even (resp. odd) cardinality, then $|d(S(n))| - 1$ is a lower bound for the cardinality of the complement of any subset of $S(n)$ which can be listed in Gray code order. For $g \ge 2$, the collection $B(n,g)$ of $g$-blockfree subsets of $\{1,\ldots,n\}$ is defined to be the set of all subsets $S$ of $\{1,\ldots,n\}$ such that $|a-b| \ge g$ if $a,b \in S$ and $a \ne b$. We will construct a Gray code order for $B(n,2)$. In contrast, for $g > 2$ we find the precise (positive) exponential growth rate of $d(B(n,g))$ with $n$ as $n \to \infty$. This implies $B(n,g)$ is far from being listable in Gray code order if $n$ is large. Analogous results for other kinds of orderings of subsets of $B(n,g)$ are proved using generalizations of $d(B(n,g))$ . However, we will show that for all $g$, one can order $B(n,g)$ so that successive elements differ by the adjunction and/or deletion of an integer from $\{1,\ldots,n\}$. We show that, over an $A$-letter alphabet, the words of length $n$ which contain no block of $k$ consecutive letters cannot, in general, be listed so that successive words differ by a single letter. However, if $k>2$ and $A>2$ or if $k=2$ and $A>3$, such a listing is always possible.
On the unitary dual of $Spin(2n,\mathbb C)$
Alfredo
O.
Brega
403-415
Abstract: In this paper we begin a systematic study of the unitarity question for genuine representations of the group $Spin (2 n, \mathbb{C})$. The main result is that we find a class of unitary representations (mostly isolated) analogous to the special unipotent representations defined by D. Barbasch and D. Vogan. In particular the full unitary dual of $Spin (2 n, \mathbb{C})$ should be obtainable from this set by complementary series.
Ergodic sequences in the Fourier-Stieltjes algebra and measure algebra of a locally compact group
Anthony
To-Ming
Lau;
Viktor
Losert
417-428
Abstract: Let $G$ be a locally compact group. Blum and Eisenberg proved that if $G$ is abelian, then a sequence of probability measures on $G$ is strongly ergodic if and only if the sequence converges weakly to the Haar measure on the Bohr compactification of $G.$ In this paper, we shall prove an extension of Blum and Eisenberg's Theorem for ergodic sequences in the Fourier-Stieltjes algebra of $G.$ We shall also give an improvement to Milnes and Paterson's more recent generalization of Blum and Eisenberg's result to general locally compact groups, and we answer a question of theirs on the existence of strongly (or weakly) ergodic sequences of measures on $G.$
On the existence of multiple steady-state solutions in the theory of electrodiffusion. Part I: the nonelectroneutral case. Part II: a constructive method for the electroneutral case
Fatiha
Alabau
4709-4756
Abstract: We give a constructive method for giving examples of doping functions and geometry of the device for which the nonelectroneutral voltage driven equations have multiple solutions. We show in particular, by performing a singular perturbation analysis of the current driven equations that if the electroneutral voltage driven equations have multiple solutions then the nonelectroneutral voltage driven equations have multiple solutions for sufficiently small normed Debye length. We then give a constructive method for giving examples of data for which the electroneutral voltage driven equations have multiple solutions.
Convergence of Polynomial Level Sets
J.
Ferrera
4757-4773
Abstract: In this paper we give a characterization of pointwise and uniform convergence of sequences of homogeneous polynomials on a Banach space by means of the convergence of their level sets. Results are obtained both in the real and the complex cases, as well as some generalizations to the nonhomogeneous case and to holomorphic functions in the complex case. Kuratowski convergence of closed sets is used in order to characterize pointwise convergence. We require uniform convergence of the distance function to get uniform convergence of the sequence of polynomials.
The Stable Homotopy Types of Stunted Lens Spaces mod 4
Huajian
Yang
4775-4798
Abstract: Let $L^{n+k}_n$ be the mod $4$ stunted lens space $L^{n+k}/L^{n-1}$. Let $\nu(m)$ denote the exponent of $2$ in $m$, and $\phi (k)$ the number of integers $j$ satisfying $j\equiv 0,1, 2, 4 (\operatorname{mod}8)$, and $0< j\leq k$. In this paper we complete the classification of the stable homotopy types of mod $4$ stunted lens spaces. The main result (Theorem 1.3 (i)) is that, under some appropriate conditions, $L^{n+k}_n$ and $L^{m+k}_m$ are stably equivalent iff $\nu(n-m)\geq \phi(k)+\delta$, where $\delta=-1, 0$ or $1$.
Symmetric functional differential equations and neural networks with memory
Jianhong
Wu
4799-4838
Abstract: We establish an analytic local Hopf bifurcation theorem and a topological global Hopf bifurcation theorem to detect the existence and to describe the spatial-temporal pattern, the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry. We apply these general results to obtain the coexistence of multiple large-amplitude wave solutions for the delayed Hopfield-Cohen-Grossberg model of neural networks with a symmetric circulant connection matrix.
Baire and $\sigma$-Borel characterizations of weakly compact sets in $M(T)$
T.
V.
Panchapagesan
4839-4847
Abstract: Let $T$ be a locally compact Hausdorff space and let $M(T)$ be the Banach space of all bounded complex Radon measures on $T$. Let $\mathcal{B}_o(T)$ and $\mathcal{B}_c(T)$ be the $\sigma$-rings generated by the compact $G_\delta$ subsets and by the compact subsets of $T$, respectively. The members of $\mathcal{B}_o(T)$ are called Baire sets of $T$ and those of $\mathcal{B}_c(T)$ are called $\sigma$-Borel sets of $T$ (since they are precisely the $\sigma$-bounded Borel sets of $T$). Identifying $M(T)$ with the Banach space of all Borel regular complex measures on $T$, in this note we characterize weakly compact subsets $A$ of $M(T)$ in terms of the Baire and $\sigma$-Borel restrictions of the members of $A$. These characterizations permit us to give a generalization of a theorem of Dieudonné which is stronger and more natural than that given by Grothendieck.
Characterizations of weakly compact operators on $C_o(T)$
T.
V.
Panchapagesan
4849-4867
Abstract: Let $T$ be a locally compact Hausdorff space and let $C_o(T)= \{f\,: T \rightarrow \mathbb{C}$, $f$ is continuous and vanishes at infinity} be provided with the supremum norm. Let $\mathcal{B}_c(T)$ and $\mathcal{B}_o(T)$ be the $\sigma$-rings generated by the compact subsets and by the compact $G_\delta$ subsets of $T$, respectively. The members of $\mathcal{B}_c(T)$ are called $\sigma$-Borel sets of $T$ since they are precisely the $\sigma$-bounded Borel sets of $T$. The members of $\mathcal{B}_o(T)$ are called the Baire sets of $T$. $M(T)$ denotes the dual of $C_o(T)$. Let $X$ be a quasicomplete locally convex Hausdorff space. Suppose $u: C_o(T) \rightarrow X$ is a continuous linear operator. Using the Baire and $\sigma$-Borel characterizations of weakly compact sets in $M(T)$ as given in a previous paper of the author's and combining the integration technique of Bartle, Dunford and Schwartz, we obtain 35 characterizations for the operator $u$ to be weakly compact, several of which are new. The independent results on the regularity and on the regular Borel extendability of $\sigma$-additive $X$-valued Baire measures are deduced as an immediate consequence of these characterizations. Some other applications are also included.
Homology of the configuration spaces of quasi-equilateral polygon linkages
Yasuhiko
Kamiyama;
Michishige
Tezuka;
Tsuguyoshi
Toma
4869-4896
Abstract: We consider the configuration space $M_{n,r}$ of quasi-equilateral polygon linkages with $n$ vertices each edge having length $1$ except for one fixed edge having length $r \; (r \geq 0)$ in the Euclidean plane $\mathbf{R}^{2}.$ In this paper, we determine $H_{\ast }(M_{n,r}; \mathrm{\bf Z})$.
Two dimensional elliptic equation with critical nonlinear growth
Takayoshi
Ogawa;
Takashi
Suzuki
4897-4918
Abstract: We study the asymptotic behavior of solutions to a semilinear elliptic equation associated with the critical nonlinear growth in two dimensions. \begin{equation*}\left\{ \begin{array}{cc} -\Delta u= \lambda ue^{u^2}, u>0 & \text{in} \ \Omega , u=0 & \text{on} \ \partial \Omega , \end{array} \right. \tag{1.1} \end{equation*} where $\Omega$ is a unit disk in $\mathbb{R}^2$ and $\lambda$ denotes a positive parameter. We show that for a radially symmetric solution of (1.1) satisfies \begin{equation*}\int _{D}\left\vert\nabla u\right\vert^{2}dx\rightarrow 4\pi,\quad\lambda \searrow 0. \end{equation*} Moreover, by using the Pohozaev identity to the rescaled equation, we show that for any finite energy radially symmetric solutions to (1.1), there is a rescaled asymptotics such as \begin{equation*}u_m^2(\gamma _m x)-u_m^2 (\gamma _m)\to 2\log\frac{2}{1+|x|^2} \quad\text{as }\lambda _m\searrow 0 \end{equation*} locally uniformly in $x\in\mathbb R^2$. We also show some extensions of the above results for general two dimensional domains.
Kähler Differentials, the $T$-functor, and a Theorem of Steinberg
W.
G.
Dwyer;
C.
W.
Wilkerson
4919-4930
Abstract: Let $T$ be the functor on the category of unstable algebras over the Steenrod algebra constructed by Lannes. We use an argument involving Kähler differentials to show that $T$ preserves polynomial algebras. This leads to new and relatively simple proofs of some topological and algebraic theorems.
The singular limit of a vector-valued reaction-diffusion process
Lia
Bronsard;
Barbara
Stoth
4931-4953
Abstract: We study the asymptotic behaviour of the solution to the vector-valued reaction-diffusion equation \begin{equation*}\varepsilon {\partial _{t}}\varphi -\varepsilon \triangle \varphi + {\frac{1}{\varepsilon }} \tilde W_{,\varphi } (\varphi ) = 0 \quad \text{ in } \Omega _{T}, \end{equation*} where $\varphi _{\varepsilon }=\varphi :\Omega _{T}:=(0,T)\times \Omega \longrightarrow \mathbf{R}^{2}$. We assume that the the potential $\tilde W$ depends only on the modulus of $\varphi$ and vanishes along two concentric circles. We present a priori estimates for the solution $\varphi$, and, in the spatially radially symmetric case, we show rigorously that in the singular limit as $\varepsilon \to 0$, two phases are created. The interface separating the bulk phases evolves by its mean curvature, while $\varphi$ evolves according to a harmonic map flow on the respective circles, coupled across the interfaces by a jump condition in the gradient.
The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature
Bo
Guan
4955-4971
Abstract: In this paper we extend the well known results on the existence and regularity of solutions of the Dirichlet problem for Monge-Ampère equations in a strictly convex domain to an arbitrary smooth bounded domain in $\mathbb R^n$ as well as in a general Riemannian manifold. We prove for the nondegenerate case that a sufficient (and necessary) condition for the classical solvability is the existence of a subsolution. For the totally degenerate case we show that the solution is in $C^{1,1} (\overline {\Omega})$ if the given boundary data extends to a locally strictly convex $C^2$ function on $\overline {\Omega}$. As an application we prove some existence results for spacelike hypersurfaces of constant Gauss-Kronecker curvature in Minkowski space spanning a prescribed boundary.
An index for periodic orbits of local semidynamical systems
Christian
C.
Fenske
4973-4991
Abstract: We define an index of Fuller type counting the number of periodic orbits of a semiflow on an ANR by a suitable approximation process.
Global analytic regularity for sums of squares of vector fields
Paulo
D.
Cordaro;
A.
Alexandrou
Himonas
4993-5001
Abstract: We consider a class of operators in the form of a sum of squares of vector fields with real analytic coefficients on the torus and we show that the zero order term may influence their global analytic hypoellipticity. Also we extend a result of Cordaro-Himonas.
The lifting of an exponential sum to a cyclic algebraic number field of prime degree
Yangbo
Ye
5003-5015
Abstract: Let $E$ be a cyclic algebraic number field of prime degree. We prove an identity which lifts an exponential sum similar to the Kloosterman sum to an exponential sum taken over certain algebraic integers in $E$.
Degenerate principal series and local theta correspondence
Soo
Teck
Lee;
Chen-bo
Zhu
5017-5046
Abstract: In this paper we determine the structure of the natural $\widetilde{U}(n,n)$ module ${\Omega^{p,q}(l)}$ which is the Howe quotient corresponding to the determinant character $\det^l$ of $U(p,q)$. We first give a description of the tempered distributions on $M_{p+q,n}(\mathbb C)$ which transform according to the character $\det^{-l}$ under the linear action of $U(p,q)$. We then show that after tensoring with a character, ${\Omega^{p,q}(l)}$ can be embedded into one of the degenerate series representations of $U(n,n)$. This allows us to determine the module structure of ${\Omega^{p,q}(l)}$. Moreover we show that certain irreducible constituents in the degenerate series can be identified with some of these representations ${\Omega^{p,q}(l)}$ or their irreducible quotients. We also compute the Gelfand-Kirillov dimensions of the irreducible constituents of the degenerate series.
Character sums associated to finite Coxeter groups
Jan
Denef;
François
Loeser
5047-5066
Abstract: The main result of this paper is a character sum identity for Coxeter arrangements over finite fields which is an analogue of Macdonald's conjecture proved by Opdam.
On Shintani zeta functions for $\mathrm{GL}(2)$
Akihiko
Yukie
5067-5094
Abstract: In this paper we consider an analogue of the zeta function for not necessarily prehomogeneous representations of $\text{GL}(2)$ and compute some of the poles.
A generalized Dedekind-Mertens lemma and its converse
Alberto
Corso;
William
Heinzer;
Craig
Huneke
5095-5109
Abstract: We study content ideals of polynomials and their behavior under multiplication. We give a generalization of the Lemma of Dedekind-Mertens and prove the converse under suitable dimensionality restrictions.
On transversality with deficiency and a conjecture of Sard
Carlos
Biasi;
Osamu
Saeki
5111-5122
Abstract: Let $f : M \to N$ be a $C^{r}$ map between $C^{r}$ manifolds $(r \geq 1)$ and $K$ a $C^{r}$ manifold. In this paper, by using the Sard theorem, we study the topological properties of the space of $C^{r}$ maps $g : K \to N$ which satisfy a certain transversality condition with respect to $f$ in a weak sense. As an application, by considering the case where $K$ is a point, we obtain some new results about the topological properties of $f(R_{q}(f))$, where $R_{q}(f)$ is the set of points of $M$ where the rank of the differential of $f$ is less than or equal to $q$. In particular, we show a result about the topological dimension of $f(R_{q}(f))$, which is closely related to a conjecture of Sard concerning the Hausdorff measure of $f(R_{q}(f))$.
Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition. Part I: Controlled hysteresis
Goong
Chen;
Sze-Bi
Hsu;
Jianxin
Zhou;
Guanrong
Chen;
Giovanni
Crosta
4265-4311
Abstract: The study of nonlinear vibrations/oscillations in mechanical and electronic systems has always been an important research area. While important progress in the development of mathematical chaos theory has been made for finite dimensional second order nonlinear ODEs arising from nonlinear springs and electronic circuits, the state of understanding of chaotic vibrations for analogous infinite dimensional systems is still very incomplete. The 1-dimensional vibrating string satisfying $w_{tt}- w_{xx}=0$ on the unit interval $x \in (0,1)$ is an infinite dimensional harmonic oscillator. Consider the boundary conditions: at the left end $x=0$, the string is fixed, while at the right end $x=1$, a nonlinear boundary condition $w_{x}= \alpha w_t - \beta w_{t}^{3}, \alpha, \beta>0$, takes effect. This nonlinear boundary condition behaves like a van der Pol oscillator, causing the total energy to rise and fall within certain bounds regularly or irregularly. We formulate the problem into an equivalent first order hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary condition. Since the solution of the first order hyperbolic system depends completely on this nonlinear relation and its iterates, the problem is reduced to a discrete iteration problem of the type $u_{n+1}=F(u_n)$, where $F$ is the nonlinear reflection relation. We say that the PDE system is chaotic if the mapping $F$ is chaotic as an interval map. Algebraic, asymptotic and numerical techniques are developed to tackle the cubic nonlinearities. We then define a rotation number, following J.P. Keener , and obtain denseness of orbits and periodic points by either directly constructing a shift sequence or by applying results of M.I. Malkin to determine the chaotic regime of $\alpha$ for the nonlinear reflection relation $F$, thereby rigorously proving chaos. Nonchaotic cases for other values of $\alpha$ are also classified. Such cases correspond to limit cycles in nonlinear second order ODEs. Numerical simulations of chaotic and nonchaotic vibrations are illustrated by computer graphics.
Projectivity, transitivity and AF-telescopes
Terry
A.
Loring;
Gert
K.
Pedersen
4313-4339
Abstract: Continuing our study of projective $C^{*}$-algebras, we establish a projective transitivity theorem generalizing the classical Glimm-Kadison result. This leads to a short proof of Glimm's theorem that every $C^{*}$-algebra not of type I contains a $C^{*}$-subalgebra which has the Fermion algebra as a quotient. Moreover, we are able to identify this subalgebra as a generalized mapping telescope over the Fermion algebra. We next prove what we call the multiplier realization theorem. This is a technical result, relating projective subalgebras of a multiplier algebra $M(A)$ to subalgebras of $M(E)$, whenever $A$ is a $C^{*}$-subalgebra of the corona algebra $C(E)=M(E)/E$. We developed this to obtain a closure theorem for projective $C^{*}$-algebras, but it has other consequences, one of which is that if $A$ is an extension of an MF (matricial field) algebra (in the sense of Blackadar and Kirchberg) by a projective $C^{*}$-algebra, then $A$ is MF. The last part of the paper contains a proof of the projectivity of the mapping telescope over any AF (inductive limit of finite-dimensional) $C^{*}$-algebra. Translated to generators, this says that in some cases it is possible to lift an infinite sequence of elements, satisfying infinitely many relations, from a quotient of any $C^{*}$-algebra.
Second-order conditions in extremal problems. The abnormal points
A.
V.
Arutyunov
4341-4365
Abstract: In this paper we study a minimization problem with constraints and obtain first- and second-order necessary conditions for a minimum. Those conditions - as opposed to the known ones - are also informative in the abnormal case. We have introduced the class of 2-normal constraints and shown that for them the ``gap" between the sufficient and the necessary conditions is as minimal as possible. It is proved that a 2-normal mapping is generic.
Eigenfunctions of the Laplacian on rotationally symmetric manifolds
Michel
Marias
4367-4375
Abstract: Eigenfunctions of the Laplacian on a negatively curved, rotationally symmetric manifold $M=(\mathbf{R}^n,ds^2),$ $ds^2=dr^2+f(r)^2d\theta ^2,$ are constructed explicitly under the assumption that an integral of $f(r)$ converges. This integral is the same one which gives the existence of nonconstant harmonic functions on $M.$
The real field with convergent generalized power series
Lou
van den Dries;
Patrick
Speissegger
4377-4421
Abstract: We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on $[0,1]$ by a series $\sum c_{n} x^{\alpha _{n}}$ with $0 \leq \alpha _{n} \rightarrow \infty$ and $\sum |c_{n}| r^{\alpha _{n}} < \infty$ for some $r>1$ is definable. This expansion is polynomially bounded.
Bordism of spin 4-manifolds with local action of tori
Piotr
Mikrut
4423-4444
Abstract: We prove that bordism group of spin $4$-manifolds with singular $T$-structure, the notion introduced by Cheeger and Gromov, is an infinite cyclic group and is detected by singnature. In particular we obtain that the theorem of Atiyah and Hirzebruch of vanishing of Â-genus in case of $S^{1}$ action on spin $4n$-manifolds is not valid in case of $T$-structures on spin $4$-manifolds.
A $q$-deformation of a trivial symmetric group action
Phil
Hanlon;
Richard
P.
Stanley
4445-4459
Abstract: Let $\mathcal{A}$ be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system $A_{n-1}$. Let $B=B(q)$ be the Varchenko matrix for this arrangement with all hyperplane parameters equal to $q$. We show that $B$ is the matrix with rows and columns indexed by permutations with $\sigma, \tau$ entry equal to $q^{i(\sigma \tau^{-1})}$ where $i(\pi)$ is the number of inversions of $\pi$. Equivalently $B$ is the matrix for left multiplication on $\mathbb{C}\mathfrak{S}_n$ by \begin{displaymath}\Gamma _n(q)=\sum _{\pi \in \mathfrak{S}_n} q^{i(\pi)} \pi . \end{displaymath} Clearly $B$ commutes with the right-regular action of $\mathfrak{S}_n$ on $\mathbb{C}\mathfrak{S}_n$. A general theorem of Varchenko applied in this special case shows that $B$ is singular exactly when $q$ is a $j(j-1)^{st}$ root of $1$ for some $j$ between $2$ and $n$. In this paper we prove two results which partially solve the problem (originally posed by Varchenko) of describing the $\mathfrak{S}_n$-module structure of the nullspace of $B$ in the case that $B$ is singular. Our first result is that \begin{displaymath}\ker(B) = \mathrm{ind}^{\mathfrak{S}_n}_{\mathfrak{S}_{n-1}} (\mathrm{Lie}_{n-1}) /\mathrm{Lie}_n\end{displaymath} in the case that $q = e^{2\pi i/n(n-1)}$ where Lie$_n$ denotes the multilinear part of the free Lie algebra with $n$ generators. Our second result gives an elegant formula for the determinant of $B$ restricted to the virtual $\mathfrak{S}_n$-module $P^\eta$ with characteristic the power sum symmetric function $p_\eta(x)$.
On the zeros of a polynomial and its derivatives
Piotr
Pawlowski
4461-4472
Abstract: If $p(z)$ is univariate polynomial with complex coefficients having all its zeros inside the closed unit disk, then the Gauss-Lucas theorem states that all zeros of $p'(z)$ lie in the same disk. We study the following question: what is the maximum distance from the arithmetic mean of all zeros of $p(z)$ to a nearest zero of $\displaystyle{p'(z)}$? We obtain bounds for this distance depending on degree. We also show that this distance is equal to $\frac{1}{3}$ for polynomials of degree 3 and polynomials with real zeros.
Trigonometric moment problems for arbitrary finite subsets of $\mathbb Z^n$
Jean-Pierre
Gabardo
4473-4498
Abstract: We consider finite subsets $\Lambda \subset \mathbf{Z}^{n}$ satisfying the extension property, i.e. the property that every collection $\{c_{\mathbf{k}}\}_{\mathbf{k} \in \Lambda - \Lambda }$ of complex numbers which is positive-definite on $\Lambda$ is the restriction to $\Lambda - \Lambda$ of the Fourier coefficients of some positive measure on $\mathbf{T}^{n}$. A simple algebraic condition on the set of trigonometric polynomials with non-zero coefficients restricted to $\Lambda$ is shown to imply the failure of the extension property for $\Lambda$. This condition is used to characterize the one-dimensional sets satisfying the extension property and to provide many examples of sets failing to satisfy it in higher dimensions. Another condition, in terms of unitary matrices, is investigated and is shown to be equivalent to the extension property. New two-dimensional examples of sets satisfying the extension property are given as well as explicit examples of collections for which the extension property fails.
Trace on the boundary for solutions of nonlinear differential equations
E.
B.
Dynkin;
S.
E.
Kuznetsov
4499-4519
Abstract: Let $L$ be a second order elliptic differential operator in $\mathbb{R}^{d}$ with no zero order terms and let $E$ be a bounded domain in $\mathbb{R}^{d}$ with smooth boundary $\partial E$. We say that a function $h$ is $L$-harmonic if $Lh=0$ in $E$. Every positive $L$-harmonic function has a unique representation \begin{equation*}h(x)=\int _{\partial E} k(x,y) \nu (dy), \end{equation*} where $k$ is the Poisson kernel for $L$ and $\nu$ is a finite measure on $\partial E$. We call $\nu$ the trace of $h$ on $\partial E$. Our objective is to investigate positive solutions of a nonlinear equation \begin{equation*}L u=u^{\alpha }\quad \text{in } E \end{equation*} for $1<\alpha \le 2$ [the restriction $\alpha \le 2$ is imposed because our main tool is the $\alpha$-superdiffusion which is not defined for $\alpha >2$]. We associate with every solution $u$ a pair $(\Gamma ,\nu )$, where $\Gamma$ is a closed subset of $\partial E$ and $\nu$ is a Radon measure on $O=\partial E\setminus \Gamma$. We call $(\Gamma ,\nu )$ the trace of $u$ on $\partial E$. $\Gamma$ is empty if and only if $u$ is dominated by an $L$-harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. We establish necessary and sufficient conditions for a pair $(\Gamma ,\nu )$ to be a trace, and we give a probabilistic formula for the maximal solution with a given trace.
Solutions of nonlinear differential equations on a Riemannian manifold and their trace on the Martin boundary
E.
B.
Dynkin;
S.
E.
Kuznetsov
4521-4552
Abstract: Let $L$ be a second order elliptic differential operator on a Riemannian manifold $E$ with no zero order terms. We say that a function $h$ is $L$-harmonic if $Lh=0$. Every positive $L$-harmonic function has a unique representation \begin{equation*}h(x)=\int _{E'} k(x,y) \nu (dy), \end{equation*} where $k$ is the Martin kernel, $E'$ is the Martin boundary and $\nu$ is a finite measure on $E'$ concentrated on the minimal part $E^{*}$ of $E'$. We call $\nu$ the trace of $h$ on $E'$. Our objective is to investigate positive solutions of a nonlinear equation \begin{equation*}L u=u^{\alpha }\quad \text{on } E \tag{*} \end{equation*} for $1<\alpha \le 2$ [the restriction $\alpha \le 2$ is imposed because our main tool is the $(L,\alpha )$-superdiffusion, which is not defined for $\alpha >2$]. We associate with every solution $u$ of (*) a pair $(\Gamma ,\nu )$, where $\Gamma$ is a closed subset of $E'$ and $\nu$ is a Radon measure on $O=E'\setminus \Gamma$. We call $(\Gamma ,\nu )$ the trace of $u$ on $E'$. $\Gamma$ is empty if and only if $u$ is dominated by an $L$-harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. In an earlier paper, we investigated the case when $L$ is a second order elliptic differential operator in $\mathbb{R}^{d}$ and $E$ is a bounded smooth domain in $\mathbb{R}^{d}$. We obtained necessary and sufficient conditions for a pair $(\Gamma ,\nu )$ to be a trace, and we gave a probabilistic formula for the maximal solution with a given trace. The general theory developed in the present paper is applicable, in particular, to elliptic operators $L$ with bounded coefficients in an arbitrary bounded domain of $\mathbb{R}^{d}$, assuming only that the Martin boundary and the geometric boundary coincide.
Consequences of contractible geodesics on surfaces
J.
Denvir;
R.
S.
Mackay
4553-4568
Abstract: The geodesic flow of any Riemannian metric on a geodesically convex surface of negative Euler characteristic is shown to be semi-equivalent to that of any hyperbolic metric on a homeomorphic surface for which the boundary (if any) is geodesic. This has interesting corollaries. For example, it implies chaotic dynamics for geodesic flows on a torus with a simple contractible closed geodesic, and for geodesic flows on a sphere with three simple closed geodesics bounding disjoint discs.
The Santaló-regions of a convex body
Mathieu
Meyer;
Elisabeth
Werner
4569-4591
Abstract: Motivated by the Blaschke-Santaló inequality, we define for a convex body $K$ in $\mathbf{R}^n$ and for $t \in \mathbf{R}$ the Santaló-regions $S(K,t)$ of $K$. We investigate the properties of these sets and relate them to a concept of affine differential geometry, the affine surface area of $K$.
The Dynkin-Lamperti arc-sine laws for measure preserving transformations
Maximilian
Thaler
4593-4607
Abstract: Arc-sine laws in the sense of renewal theory are proved for return time processes generated by transformations with infinite invariant measure on sets satisfying a type of Darling-Kac condition, and an application to real transformations with indifferent fixed points is discussed.
Lower bounds for dimensions of representation varieties
Andy
R.
Magid
4609-4621
Abstract: The set of $n$-dimensional complex representations of a finitely generated group $\Gamma$ form a complex affine variety $R_{n}(\Gamma )$. Suppose that $\rho $ is such a representation and consider the associated representation $Ad \circ \rho$ on $n \times n$ complex matrices obtained by following $\rho$ with conjugation of matrices. Then it is shown that the dimension of $R_{n}(\Gamma )$ at $\rho$ is at least the difference of the complex dimensions of $Z^{1}(\Gamma , Ad \circ \rho )$ and $H^{2}(\Gamma , Ad \circ \rho )$. It is further shown that in the latter cohomology $\Gamma$ may be replaced by various proalgebraic groups associated to $\Gamma$ and $\rho$.
Atomic maps and the Chogoshvili-Pontrjagin claim
M.
Levin;
Y.
Sternfeld
4623-4632
Abstract: It is proved that all spaces of dimension three or more disobey the Chogoshvili-Pontrjagin claim. This is of particular interest in view of the recent proof (in Certain 2-stable embeddings, by Dobrowolski, Levin, and Rubin, Topology Appl. 80 (1997), 81-90) that two-dimensional ANRs obey the claim. The construction utilizes the properties of atomic maps which are maps whose fibers ($=$point inverses) are atoms ($=$hereditarily indecomposable continua). A construction of M. Brown is applied to prove that every finite dimensional compact space admits an atomic map with a one-dimensional range.
Subelliptic harmonic maps
Jürgen
Jost;
Chao-Jiang
Xu
4633-4649
Abstract: We study a nonlinear harmonic map type system of subelliptic PDE. In particular, we solve the Dirichlet problem with image contained in a convex ball.
Singular limit of solutions of the porous medium equation with absorption
Kin
Ming
Hui
4651-4667
Abstract: We prove that as $m\to \infty$ the solutions $u^{(m)}$ of $u_{t}=\Delta u^{m}-u^{p}$, $(x,t)\in R^{n}\times (0,T)$, $T>0$, $m>1$, $p>1$, $u\ge 0$, $u(x,0)=f(x)\in L^{1}(R^{n})\cap L^{\infty }(R^{n})$, converges in $L^{1}_{loc}(R^{n}\times (0,T))$ to the solution of the ODE $v_{t}=-v^{p}$, $v(x,0)=g(x)$, where $g\in L^{1}(R^{n})$, $0\le g\le 1$, satisfies $g-\Delta \widetilde {g}=f$ in $\mathcal{D}'(R^{n})$ for some function $\widetilde {g}\in L^{\infty }_{loc}(R^{n})$, $\widetilde {g}\ge 0$, satisfying $\widetilde {g}(x)=0$ whenever $g(x)<1$ for a.e. $x\in R^{n}$, $\int _{E}\widetilde {g}dx\le C|E|^{2/n}$ for $n\ge 3$ and $\int _{E}|\nabla \widetilde {g}|dx\le C|E|^{1/2}$ for $n=2$, where $C>0$ is a constant and $E$ is any measurable subset of $R^{n}$.
A class of parabolic $k$-subgroups associated with symmetric $k$-varieties
A.
G.
Helminck;
G.
F.
Helminck
4669-4691
Abstract: Let $G$ be a connected reductive algebraic group defined over a field $k$ of characteristic not 2, $\sigma$ an involution of $G$ defined over $k$, $H$ a $k$-open subgroup of the fixed point group of $\sigma$, $G_k$ (resp. $H_k$) the set of $k$-rational points of $G$ (resp. $H$) and $G_k/H_k$ the corresponding symmetric $k$-variety. A representation induced from a parabolic $k$-subgroup of $G$ generically contributes to the Plancherel decomposition of $L^2(G_k/H_k)$ if and only if the parabolic $k$-subgroup is $\sigma$-split. So for a study of these induced representations a detailed description of the $H_k$-conjucagy classes of these $\sigma$-split parabolic $k$-subgroups is needed. In this paper we give a description of these conjugacy classes for general symmetric $k$-varieties. This description can be refined to give a more detailed description in a number of cases. These results are of importance for studying representations for real and $\mathfrak p$-adic symmetric $k$-varieties.
Model aspherical manifolds with no periodic maps
Wim
Malfait
4693-4708
Abstract: A. Borel proved that, if the fundamental group $E$ of an aspherical manifold $M$ is centerless and the outer automorphism group of $E$ is torsion-free, then $M$ admits no periodic maps, or equivalently, there are no non-trivial finite groups of homeomorphisms acting effectively on $M$. In the literature, taking off from this result, several examples of (rather complex) aspherical manifolds exhibiting this total lack of periodic maps have been presented. In this paper, we investigate to what extent the converse of Borel's result holds for aspherical manifolds $M$ arising from Seifert fiber space constructions. In particular, for e.g. flat Riemannian manifolds, infra-nilmanifolds and infra-solvmanifolds of type (R), it turns out that having a centerless fundamental group with torsion-free outer automorphism group is also necessary to conclude that all finite groups of affine diffeomorphisms acting effectively on the manifold are trivial. Finally, we discuss the problem of finding (less complex) examples of such aspherical manifolds with no periodic maps.
A nonlinear Fokker-Planck equation modelling the approach to thermal equilibrium in a homogeneous plasma
M.
Escobedo;
M.
A.
Herrero;
J.
J. L.
Velazquez
3837-3901
Abstract: This work deals with the problem consisting in the equation \begin{equation*}{\frac{\partial f}{\partial t}} ={\frac{1}{x^{2}}}{\frac{\partial }{\partial x}} [x^{4}({\frac{\partial f}{\partial x}}+f+f^{2})], \quad \hbox {when}\quad x\in (0,\infty ), t>0, \tag*{(1)}\end{equation*} together with no-flux conditions at $x=0$ and $x=+\infty$, i.e. \begin{equation*}x^{4}({\frac{\partial f}{\partial x}}+f+f^{2})=0\quad \hbox {as}\hskip 0.2cm x\mathop{\longrightarrow } 0 \hskip 0.2cm\hbox {or}\hskip 0.2cmx\mathop{\longrightarrow } +\infty . \tag*{(2)}\end{equation*} Such a problem arises as a kinetic approximation to describe the evolution of the radiation distribution $f(x, t)$ in a homogeneous plasma when radiation interacts with matter via Compton scattering. We shall prove that there exist solutions of $(1)$, $(2)$ which develop singularities near $x=0$ in a finite time, regardless of how small the initial number of photons $N(0)=\int _{0}^{+\infty }x^{2}f(x, 0)dx$ is. The nature of such singularities is then analyzed in detail. In particular, we show that the flux condition $(2)$ is lost at $x=0$ when the singularity unfolds. The corresponding blow-up pattern is shown to be asymptotically of a shock wave type. In rescaled variables, it consists in an imploding travelling wave solution of the Burgers equation near $x=0$, that matches a suitable diffusive profile away from the shock. Finally, we also show that, on replacing $(2)$ near $x=0$ as determined by the manner of blow-up, such solutions can be continued for all times after the onset of the singularity.
Stability results on interpolation scales of quasi-Banach spaces and applications
Nigel
Kalton;
Marius
Mitrea
3903-3922
Abstract: We investigate the stability of Fredholm properties on interpolation scales of quasi-Banach spaces. This analysis is motivated by problems arising in PDE's and several applications are presented.
Local and global properties of limit sets of foliations of quasigeodesic Anosov flows
Sérgio
R.
Fenley
3923-3941
Abstract: A nonsingular flow is quasigeodesic when all flow lines are efficient in measuring distances in relative homotopy classes. We analyze the quasigeodesic property for Anosov flows in $3$-manifolds which have negatively curved fundamental group. We show that this property implies that limit sets of stable and unstable leaves (in the universal cover) vary continuously in the sphere at infinity. It also follows that the union of the limit sets of all stable (or unstable) leaves is not the whole sphere at infinity. Finally, under the quasigeodesic hypothesis we completely determine when limit sets of leaves in the universal cover can intersect. This is done by determining exactly when flow lines in the universal cover share an ideal point.
Sum of squares manifolds: The expressibility of the Laplace-Beltrami operator on pseudo-Riemannian manifolds as a sum of squares of vector fields
Wilfried
H.
Paus
3943-3966
Abstract: In this paper, we investigate under what circumstances the Laplace-Beltrami operator on a pseudo-Riemannian manifold can be written as a sum of squares of vector fields, as is naturally the case in Euclidean space. We show that such an expression exists globally on one-dimensional manifolds and can be found at least locally on any analytic pseudo-Riemannian manifold of dimension greater than two. For two-dimensional manifolds this is possible if and only if the manifold is flat. These results are achieved by formulating the problem as an exterior differential system and applying the Cartan-Kähler theorem to it.
On the connection between the existence of zeros and the asymptotic behavior of resolvents of maximal monotone operators in reflexive Banach spaces
Athanassios
G.
Kartsatos
3967-3987
Abstract: A more systematic approach is introduced in the theory of zeros of maximal monotone operators $T:X\supset D(T)\to 2^{X^{*}}$, where $X$ is a real Banach space. A basic pair of necessary and sufficient boundary conditions is given for the existence of a zero of such an operator $T$. These conditions are then shown to be equivalent to a certain asymptotic behavior of the resolvents or the Yosida resolvents of $T$. Furthermore, several interesting corollaries are given, and the extendability of the necessary and sufficient conditions to the existence of zeros of locally defined, demicontinuous, monotone mappings is demonstrated. A result of Guan, about a pathwise connected set lying in the range of a monotone operator, is improved by including non-convex domains. A partial answer to Nirenberg's problem is also given. Namely, it is shown that a continuous, expansive mapping $T$ on a real Hilbert space $H$ is surjective if there exists a constant $\alpha \in (0,1)$ such that $\langle Tx-Ty,x-y\rangle \ge -\alpha \|x-y\|^{2},~x,~y\in H.$ The methods for these results do not involve explicit use of any degree theory.
Rigidity and topological conjugates of topologically tame Kleinian groups
Ken'ichi
Ohshika
3989-4022
Abstract: Minsky proved that two Kleinian groups $G_1$ and $G_2$ are quasi-conformally conjugate if they are freely indecomposable, the injectivity radii at all points of $\bold{H}^3/G_1$, $\bold{H}^3/G_2$ are bounded below by a positive constant, and there is a homeomorphism $h$ from a topological core of $\bold{H}^3/G_1$ to that of $\bold{H}^3/G_2$ such that $h$ and $h^{-1}$ map ending laminations to ending laminations. We generalize this theorem to the case when $G_1$ and $G_2$ are topologically tame but may be freely decomposable under the same assumption on the injectivity radii. As an application, we prove that if a Kleinian group is topologically conjugate to another Kleinian group which is topologically tame and not a free group, and both Kleinian groups satisfy the assumption on the injectivity radii as above, then they are quasi-conformally conjugate.
Infinite type homeomorphisms of the circle and convergence of Fourier series
Antônio
Zumpano
4023-4040
Abstract: We consider the problem of convergence of Fourier series when we make a change of variable. Under a certain reasonable hypothesis, we give a necessary and sufficient condition for a homeomorphism of the circle to transform absolutely convergent Fourier series into uniformly convergent Fourier series.
Test ideals in quotients of $F$-finite regular local rings
Janet
Cowden
Vassilev
4041-4051
Abstract: Let $S$ be an $F$-finite regular local ring and $I$ an ideal contained in $S$. Let $R=S/I$. Fedder proved that $R$ is $F$-pure if and only if $(I^{[p]}:I) \nsubseteq \mathfrak{m}^{[p]}$. We have noted a new proof for his criterion, along with showing that $(I^{[q]}:I) \subseteq (\tau ^{[q]}:\tau )$, where $\tau$ is the pullback of the test ideal for $R$. Combining the the $F$-purity criterion and the above result we see that if $R=S/I$ is $F$-pure then $R/\tau$ is also $F$-pure. In fact, we can form a filtration of $R$, $I \subseteq \tau = \tau _{0} \subseteq \tau _{1} \subseteq \ldots \subseteq \tau _{i} \subseteq \ldots$ that stabilizes such that each $R/\tau _{i}$ is $F$-pure and its test ideal is $\tau _{i+1}$. To find examples of these filtrations we have made explicit calculations of test ideals in the following setting: Let $R=T/I$, where $T$ is either a polynomial or a power series ring and $I= P_{1} \cap \ldots \cap P_{n}$ is generated by monomials and the $R/P_{i}$ are regular. Set $J = \Sigma (P_{1} \cap \ldots \cap \hat {P_{i}} \cap \ldots \cap P_{n})$. Then $J=\tau =\tau _{par}$.
Regularity of solutions to a contact problem
Russell
M.
Brown;
Zhongwei
Shen;
Peter
Shi
4053-4063
Abstract: We consider a variational inequality for the Lamé system which models an elastic body in contact with a rigid foundation. We give conditions on the domain and the contact set which allow us to prove regularity of solutions to the variational inequality. In particular, we show that the gradient of the solution is a square integrable function on the boundary.
Self-similar measures and intersections of Cantor sets
Yuval
Peres;
Boris
Solomyak
4065-4087
Abstract: It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g. when both sets are the middle-$\alpha$ Cantor set and $\alpha \in ({1 \over 3}, \frac 12)$. We show that for any compact set $K$ and for a.e. $\alpha \in (0,1)$, the arithmetic sum of $K$ and the middle-$\alpha$ Cantor set does indeed have positive Lebesgue measure when the sum of their Hausdorff dimensions exceeds 1. In this case we also determine the essential supremum, as the translation parameter $t$ varies, of the dimension of the intersection of $K+t$ with the middle-$\alpha$ Cantor set. We also establish a new property of the infinite Bernoulli convolutions $\nu _\lambda^p$ (the distributions of random series $\sum _{n=0}^\infty \pm \lambda^n ,$ where the signs are chosen independently with probabilities $(p,1-p)$). Let $1 \leq q_1<q_2 \leq 2$. For $p \neq \frac 12$ near $\frac 12$ and for a.e. $\lambda$ in some nonempty interval, $\nu _\lambda^p$ is absolutely continuous and its density is in $L^{q_1}$ but not in $L^{q_2}$. We also answer a question of Kahane concerning the Fourier transform of $\nu _\lambda ^{\scriptscriptstyle 1/2}$.
An inverse problem for scattering by a doubly periodic structure
Gang
Bao;
Zhengfang
Zhou
4089-4103
Abstract: Consider scattering of electromagnetic waves by a doubly periodic structure $S=\{x_3=f(x_1, x_2)\}$ with $f(x_1+n_1\Lambda _1, x_2+n_2\Lambda _2)=f(x_1, x_2)$ for integers $n_1$, $n_2$. Above the structure, the medium is assumed to be homogeneous with a constant dielectric coefficient. The medium is a perfect conductor below the structure. An inverse problem arises and may be described as follows. For a given incident plane wave, the tangential electric field is measured away from the structure, say at $x_3=b$ for some large $b$. To what extent can one determine the location of the periodic structure that separates the dielectric medium from the conductor? In this paper, results on uniqueness and stability are established for the inverse problem. A crucial step in our proof is to obtain a lower bound for the first eigenvalue of the following problem in a convex domain $\Omega$: \begin{displaymath}\left\{ \begin{array}{l} - \triangle u = \lambda u \quad \text{in} \quad \Omega, \nabla \cdot u = 0 \quad \text{in} \quad \Omega, n \times u = 0 \quad \text{on} \quad \partial \Omega. \end{array} \right. \end{displaymath}
On asymptotic approximations of the residual currents
Alekos
Vidras;
Alain
Yger
4105-4125
Abstract: We use a ${\cal D}$-module approach to discuss positive examples for the existence of the unrestricted limit of the integrals involved in the approximation to the Coleff-Herrera residual currents in the complete intersection case. Our results also provide asymptotic developments for these integrals.
The Ext class of an approximately inner automorphism
Akitaka
Kishimoto;
Alex
Kumjian
4127-4148
Abstract: Let $A$ be a simple unital $A\mathbf{T}$ algebra of real rank zero. It is shown below that the range of the natural map from the approximately inner automorphism group to $KK(A, A)$ coincides with the kernel of the map $KK(A, A) \rightarrow \bigoplus _{i=0}^{1} \operatorname{Hom}(K_i(A), K_i(A))$.
On free actions, minimal flows, and a problem by Ellis
Vladimir
G.
Pestov
4149-4165
Abstract: We exhibit natural classes of Polish topological groups $G$ such that every continuous action of $G$ on a compact space has a fixed point, and observe that every group with this property provides a solution (in the negative) to a 1969 problem by Robert Ellis, as the Ellis semigroup $E(U)$ of the universal minimal $G$-flow $U$, being trivial, is not isomorphic with the greatest $G$-ambit. Further refining our construction, we obtain a Polish topological group $G$ acting freely on the universal minimal flow $U$ yet such that ${\mathcal S}(G)$ and $E(U)$ are not isomorphic. We also display Polish topological groups acting effectively but not freely on their universal minimal flows. In fact, we can produce examples of groups of all three types having any prescribed infinite weight. Our examples lead to dynamical conclusions for some groups of importance in analysis. For instance, both the full group of permutations $S(X)$ of an infinite set, equipped with the pointwise topology, and the unitary group $U(\mathcal{H})$ of an infinite-dimensional Hilbert space with the strong operator topology admit no free action on a compact space, and the circle $\mathbb{S}^{1}$ forms the universal minimal flow for the topological group ${\operatorname {Homeo}\,}_{+}(\mathbb{S}^{1})$ of orientation-preserving homeomorphisms. It also follows that a closed subgroup of an amenable topological group need not be amenable.
The structure of indecomposable injectives in generic representation theory
Geoffrey
M. L.
Powell
4167-4193
Abstract: This paper considers the structure of the injective objects $I_{V_{n}}$ in the category $\mathcal F$ of functors between ${\mathbb F}_2$-vector spaces. A co-Weyl object $J_\lambda$ is defined, for each simple functor $F_\lambda$ in $\mathcal F$. A functor is defined to be $J$-good if it admits a finite filtration of which the quotients are co-Weyl objects. Properties of $J$-good functors are considered and it is shown that the indecomposable injectives in $\mathcal F$ are $J$-good. A finiteness result for proper sub-functors of co-Weyl objects is proven, using the polynomial filtration of the shift functor $\tilde\Delta: \mathcal F \rightarrow \mathcal F$. This research is motivated by the Artinian conjecture due to Kuhn, Lannes and Schwartz.
On the rigidity theorem for elliptic genera
Anand
Dessai;
Rainer
Jung
4195-4220
Abstract: We give a detailed proof of the rigidity theorem for elliptic genera. Using the Lefschetz fixed point formula we carefully analyze the relation between the characteristic power series defining the elliptic genera and the equivariant elliptic genera. We show that equivariant elliptic genera converge to Jacobi functions which are holomorphic. This implies the rigidity of elliptic genera. Our approach can be easily modified to give a proof of the rigidity theorem for the elliptic genera of level $N$.
Computations in generic representation theory: maps from symmetric powers to composite functors
Nicholas
J.
Kuhn
4221-4233
Abstract: If ${\bold F}_q$ is the finite field of order $q$ and characteristic $p$, let ${\cal F}(q)$ be the category whose objects are functors from finite dimensional ${\bold F}_q$-vector spaces to ${\bold F}_q$-vector spaces, and with morphisms the natural transformations between such functors. Important families of objects in ${\cal F}(q)$ include the families $S_n, S^n, \Lambda^n, \Bar{S}^n$, and $cT^n$, with $c \in {\bold F}_q[\Sigma _n]$, defined by $S_n(V) = (V^{\otimes n})^{\Sigma _n}$,$S^n(V) = V^{\otimes n}/\Sigma _n$, $\Lambda^n(V) = n^{th} \text{ exterior power of } V$, $\Bar{S}^*(V) = S^*(V)/(p^{th} \text{ powers})$, and $cT^n(V) = c(V^{\otimes n})$. Fixing $F$, we discuss the problem of computing $\operatorname{Hom}_{{\cal F}(q)}(S_m, F \circ G)$, for all $m$, given knowledge of $\operatorname{Hom}_{{\cal F}(q)}(S_m, G)$ for all $m$. When $q = p$, we get a complete answer for any functor $F$ chosen from the families listed above. Our techniques involve Steenrod algebra technology, and, indeed, our most striking example, when $F=S^n$, arose in recent work on the homology of iterated loopspaces.
Universal maps on trees
Carl
Eberhart;
J.
B.
Fugate
4235-4251
Abstract: A map $f:R \to S$ of continua $R$ and $S$ is called a universal map from $R$ to $S$ if for any map $g:R \to S$, $f(x) = g(x)$ for some point $x \in R$. When $R$ and $S$ are trees, we characterize universal maps by reducing to the case of light minimal universal maps. The characterization uses the notions of combinatorial map and folded subedge of $R$.
Chung's law for integrated Brownian motion
Davar
Khoshnevisan;
Zhan
Shi
4253-4264
Abstract: The small ball problem for the integrated process of a real-valued Brownian motion is solved. In sharp contrast to more standard methods, our approach relies on the sample path properties of Brownian motion together with facts about local times and Lévy processes.
On the subgroup structure of exceptional groups of Lie type
Martin
W.
Liebeck;
Gary
M.
Seitz
3409-3482
Abstract: We study finite subgroups of exceptional groups of Lie type, in particular maximal subgroups. Reduction theorems allow us to concentrate on almost simple subgroups, the main case being those with socle $X(q)$ of Lie type in the natural characteristic. Our approach is to show that for sufficiently large $q$ (usually $q>9$ suffices), $X(q)$ is contained in a subgroup of positive dimension in the corresponding exceptional algebraic group, stabilizing the same subspaces of the Lie algebra. Applications are given to the study of maximal subgroups of finite exceptional groups. For example, we show that all maximal subgroups of sufficiently large order arise as fixed point groups of maximal closed subgroups of positive dimension.
Connected finite loop spaces with maximal tori
J.
M.
Møller;
D.
Notbohm
3483-3504
Abstract: Finite loop spaces are a generalization of compact Lie groups. However, they do not enjoy all of the nice properties of compact Lie groups. For example, having a maximal torus is a quite distinguished property. Actually, an old conjecture, due to Wilkerson, says that every connected finite loop space with a maximal torus is equivalent to a compact connected Lie group. We give some more evidence for this conjecture by showing that the associated action of the Weyl group on the maximal torus always represents the Weyl group as a crystallographic group. We also develop the notion of normalizers of maximal tori for connected finite loop spaces, and prove for a large class of connected finite loop spaces that a connected finite loop space with maximal torus is equivalent to a compact connected Lie group if it has the right normalizer of the maximal torus. Actually, in the cases under consideration the information about the Weyl group is sufficient to give the answer. All this is done by first studying the analogous local problems.
A growth dichotomy for o-minimal expansions of ordered groups
Chris
Miller;
Sergei
Starchenko
3505-3521
Abstract: Let $\mathfrak{R}$ be an o-minimal expansion of a divisible ordered abelian group $(R,<,+,0,1)$ with a distinguished positive element $1$. Then the following dichotomy holds: Either there is a $0$-definable binary operation $\cdot$ such that $(R,<,+,\cdot ,0,1)$ is an ordered real closed field; or, for every definable function $f:R\to R$ there exists a $0$-definable $\lambda \in \{0\}\cup \operatorname{Aut}(R,+)$ with $\lim _{x\to +\infty }[f(x)-\lambda (x)]\in R$. This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure $\mathfrak{M}:=(M,<,\dots )$ there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) $\mathfrak{M}$-definable groups with underlying set $M$.
Periodic billiard orbits are dense in rational polygons
M.
Boshernitzan;
G.
Galperin;
T.
Krüger;
S.
Troubetzkoy
3523-3535
Abstract: We show that periodic orbits are dense in the phase space for billiards in polygons for which the angle between each pair of sides is a rational multiple of $\pi.$
Poincaré embedding of the diagonal
Yanghyun
Byun
3537-3553
Abstract: There is a Poincaré embedding structure on the diagonal $X\rightarrow X\times X$ under the conditions: i) $X$ is formed by gluing two compact smooth manifolds along their boundaries using a homotopy equivalence and ii) a square-root closed condition is satisfied by the fundamental groupoid of the boundary.
Copies of $c_0$ and $\ell_\infin$ in topological Riesz spaces
Lech
Drewnowski;
Iwo
Labuda
3555-3570
Abstract: The paper is concerned with order-topological characterizations of topological Riesz spaces, in particular spaces of measurable functions, not containing Riesz isomorphic or linearly homeomorphic copies of $c_{0}$ or $\ell _{\infty }$.
Induction theorems on the stable rationality of the center of the ring of generic matrices
Esther
Beneish
3571-3585
Abstract: Following Procesi and Formanek, the center of the division ring of $n\times n$ generic matrices over the complex numbers $\mathbf C$ is stably equivalent to the fixed field under the action of $S_n$, of the function field of the group algebra of a $ZS_n$-lattice, denoted by $G_n$. We study the question of the stable rationality of the center $C_n$ over the complex numbers when $n$ is a prime, in this module theoretic setting. Let $N$ be the normalizer of an $n$-sylow subgroup of $S_n$. Let $M$ be a $ZS_n$-lattice. We show that under certain conditions on $M$, induction-restriction from $N$ to $S_n$ does not affect the stable type of the corresponding field. In particular, $\mathbf C (G_n)$ and $\mathbf C(ZG\otimes _{ZN}G_n)$ are stably isomorphic and the isomorphism preserves the $S_n$-action. We further reduce the problem to the study of the localization of $G_n$ at the prime $n$; all other primes behave well. We also present new simple proofs for the stable rationality of $C_n$ over $\mathbf C$, in the cases $n=5$ and $n=7$.
Subvarieties of $\mathcal{SU}_C(2)$ and $2\theta$-divisors in the Jacobian
W.
M.
Oxbury;
C.
Pauly;
E.
Previato
3587-3614
Abstract: We explore some of the interplay between Brill-Noether subvarieties of the moduli space ${\mathcal{SU}}_C(2,K)$ of rank 2 bundles with canonical determinant on a smooth projective curve and $2\theta$-divisors, via the inclusion of the moduli space into $|2\Theta|$, singular along the Kummer variety. In particular we show that the moduli space contains all the trisecants of the Kummer and deduce that there are quadrisecant lines only if the curve is hyperelliptic; we show that for generic curves of genus $<6$, though no higher, bundles with $>2$ sections are cut out by $\Gamma _{00}$; and that for genus 4 this locus is precisely the Donagi-Izadi nodal cubic threefold associated to the curve.
Quantum cohomology of projective bundles over $\mathbb P^n$
Zhenbo
Qin;
Yongbin
Ruan
3615-3638
Abstract: In this paper we study the quantum cohomology ring of certain projective bundles over the complex projective space $\mathbb{P}^{n}$. Using excessive intersection theory, we compute the leading coefficients in the relations among the generators of the quantum cohomology ring structure. In particular, Batyrev's conjectural formula for quantum cohomology of projective bundles associated to direct sum of line bundles over $\mathbb{P}^{n}$ is partially verified. Moreover, relations between the quantum cohomology ring structure and Mori's theory of extremal rays are observed. The results could shed some light on the quantum cohomology for general projective bundles.
On zeta functions and Iwasawa modules
Jangheon
Oh
3639-3655
Abstract: We study the relation between zeta-functions and Iwasawa modules. We prove that the Iwasawa modules $X^{-}_{k({\zeta }_{p})}$ for almost all $p$ determine the zeta function ${\zeta }_{k}$ when $k$ is a totally real field. Conversely, we prove that the $\lambda$-part of the Iwasawa module $X_{k}$ is determined by its zeta-function ${\zeta }_{k}$ up to pseudo-isomorphism for any number field $k.$ Moreover, we prove that arithmetically equivalent CM fields have also the same ${\mu }^{-}$-invariant.
On the conjectures of J. Thompson and O. Ore
Erich
W.
Ellers;
Nikolai
Gordeev
3657-3671
Abstract: If $G$ is a finite simple group of Lie type over a field containing more than $8$ elements (for twisted groups $^{l} X_{n} (q^{l})$ we require $q > 8$, except for $^{2} B_{2} (q^{2})$, $^{2} G_{2} (q^{2})$, and $^{2} F_{4} (q^{2})$, where we assume $q^{2} > 8$), then $G$ is the square of some conjugacy class and consequently every element in $G$ is a commutator.
Self-duality operators on odd dimensional manifolds
Houhong
Fan
3673-3706
Abstract: In this paper we construct a new elliptic operator associated to any nowhere zero vector field on an odd-dimensional manifold and study its index theory. It turns out this operator has several geometric applications to conformal vector fields, self-dual vector fields, locally free $S^{1}$-actions and transversal hypersurfaces of these vector fields in an odd-dimensional manifold. In particular, we reveal a non-stable phenomena about the existence of conformal vector fields and self-dual vector fields in odd dimensions above 3. This is in sharp contrast to the stable phenomena about the existence of nowhere zero vector fields in odd dimensions. Besides these applications, the index formula of this new operator also gives the formulas for the dimensions of self-duality cohomology groups and for the virtual dimensions of the moduli spaces of anti-self-dual connections on 5-cobordisms, which are introduced in author's previous papers.
Hypercyclicity in the scattering theory for linear transport equation
H.
Emamirad
3707-3716
Abstract: We show how the hypercyclicity of the transport semigroup can intervene in the scattering theory to characterize the density property of the Lax and Phillips representation theorem and conversely, how the existence of the wave operators of the scattering theory can be used for recovering the hypercyclicity of the absorbing transport group in some weighted $L^{1}$ spaces.
Convergence of random walks on the circle generated by an irrational rotation
Francis
Edward
Su
3717-3741
Abstract: Fix $\alpha \in [0,1)$. Consider the random walk on the circle $S^1$ which proceeds by repeatedly rotating points forward or backward, with probability $\frac 12$, by an angle $2\pi\alpha$. This paper analyzes the rate of convergence of this walk to the uniform distribution under ``discrepancy'' distance. The rate depends on the continued fraction properties of the number $\xi=2\alpha$. We obtain bounds for rates when $\xi$ is any irrational, and a sharp rate when $\xi$ is a quadratic irrational. In that case the discrepancy falls as $k^{-\frac 12}$ (up to constant factors), where $k$ is the number of steps in the walk. This is the first example of a sharp rate for a discrete walk on a continuous state space. It is obtained by establishing an interesting recurrence relation for the distribution of multiples of $\xi$ which allows for tighter bounds on terms which appear in the Erdös-Turán inequality.
Curve-straightening and the Palais-Smale condition
Anders
Linnér
3743-3765
Abstract: This paper considers the negative gradient trajectories associated with the modified total squared curvature functional $\int k^{2} +\nu ds$. The focus is on the limiting behavior as $\nu$ tends to zero from the positive side. It is shown that when $\nu =0$ spaces of curves exist in which some trajectories converge and others diverge. In one instance the collection of critical points splits into two subsets. As $\nu$ tends to zero the critical curves in the first subset tend to the critical points present when $\nu =0$. Meanwhile, all the critical points in the second subset have lengths that tend to infinity. It is shown that this is the only way the Palais-Smale condition fails in the present context. The behavior of the second class of critical points supports the view that some of the trajectories are `dragged' all the way to `infinity'. When the curves are rescaled to have constant length the Euler figure eight emerges as a `critical point at infinity'. It is discovered that a reflectional symmetry need not be preserved along the trajectories. There are examples where the length of the curves along the same trajectory is not a monotone function of the flow-time. It is shown how to determine the elliptic modulus of the critical curves in all the standard cases. The modulus $p$ must satisfy $2E(p)/K(p)=1\pm |g|/\widetilde L$ when the space is limited to curves of fixed length $\widetilde L$ and the endpoints are separated by the vector $g$.
Tessellations of solvmanifolds
Dave
Witte
3767-3796
Abstract: Let $A$ be a closed subgroup of a connected, solvable Lie group $G$, such that the homogeneous space $A\backslash G$ is simply connected. As a special case of a theorem of C. T. C. Wall, it is known that every tessellation $A\backslash G/\Gamma$ of $A\backslash G$ is finitely covered by a compact homogeneous space $G'/\Gamma'$. We prove that the covering map can be taken to be very well behaved - a ``crossed" affine map. This establishes a connection between the geometry of the tessellation and the geometry of the homogeneous space. In particular, we see that every geometrically-defined flow on $A\backslash G/\Gamma$ that has a dense orbit is covered by a natural flow on $G'/\Gamma'$.
Homoclinic Solutions and Chaos in Ordinary Differential Equations with Singular Perturbations
Joseph
Gruendler
3797-3814
Abstract: Ordinary differential equations are considered which contain a singular perturbation. It is assumed that when the perturbation parameter is zero, the equation has a hyperbolic equilibrium and homoclinic solution. No restriction is placed on the dimension of the phase space or on the dimension of intersection of the stable and unstable manifolds. A bifurcation function is established which determines nonzero values of the perturbation parameter for which the homoclinic solution persists. It is further shown that when the vector field is periodic and a transversality condition is satisfied, the homoclinic solution to the perturbed equation produces a transverse homoclinic orbit in the period map. The techniques used are those of exponential dichotomies, Lyapunov-Schmidt reduction and scales of Banach spaces. A much simplified version of this latter theory is developed suitable for the present case. This work generalizes some recent results of Battelli and Palmer.
Operations and Spectral Sequences. I
James
M.
Turner
3815-3835
Abstract: Using methods developed by W. Singer and J. P. May, we describe a systematic approach to showing that many spectral sequences, determined by a filtration on a complex whose homology has an action of operations, possess a compatible action of the same operations. As a consequence, we obtain W. Singer's result for Steenrod operations on Serre spectral sequence and extend A. Bahri's action of Dyer-Lashof operations on the second quadrant Eilenberg-Moore spectral sequence.
On the existence of convex classical solutions for multilayer free boundary problems with general nonlinear joining conditions
Andrew
Acker
2981-3020
Abstract: We prove the existence of convex classical solutions for a general multidimensional, multilayer free-boundary problem. The geometric context of this problem is a nested family of closed, convex surfaces. Except for the innermost and outermost surfaces, which are given, these surfaces are interpreted as unknown layer-interfaces, where the layers are the bounded annular domains between them. Each unknown interface is characterized by a quite general nonlinear equation, called a joining condition, which relates the first derivatives (along the interface) of the capacitary potentials in the two adjoining layers, as well as the spatial variables. A well-known special case of this problem involves several stationary, immiscible, two-dimensional flows of ideal fluid, related along their interfaces by Bernoulli's law.
Small subalgebras of Steenrod and Morava stabilizer algebras
N.
Yagita
3021-3041
Abstract: Let $P(j)$ (resp. $S(n)_{(j)})$ be the subalgebra of the Steenrod algebra $\mathcal{A}_p$ (resp. $n$th Morava stabilizer algebra) generated by reduced powers $\mathcal{P}^{p^i}$, $0\le i\le j$ (resp. $t_i$, $1\le i\le j)$. In this paper we identify the dual $P(j-1)^*$ of $P(j-1)$ (resp. $S(n)_{(j)}$, for $j\le n)$ with some Frobenius kernel (resp. $F_{p^n}$-points) of a unipotent subgroup $G(j+1)$ of the general linear algebraic group $GL_{j+1}$. Using these facts, we get the additive structure of $H^*(P(1))=\operatorname{Ext}_{P(1)}(Z/p,Z/p)$ for odd primes.
Morita equivalence for crossed products by Hilbert $C^\ast$-bimodules
Beatriz
Abadie;
Søren
Eilers;
Ruy
Exel
3043-3054
Abstract: We introduce the notion of the crossed product $A \rtimes _X{\Bbb{Z}}$ of a $C^*$-algebra $A$ by a Hilbert $C^*$-bimodule $X$. It is shown that given a $C^*$-algebra $B$ which carries a semi-saturated action of the circle group (in the sense that $B$ is generated by the spectral subspaces $B_0$ and $B_1$), then $B$ is isomorphic to the crossed product $B_0 \rtimes _{B_1}{\Bbb{Z}}$. We then present our main result, in which we show that the crossed products $A \rtimes _X{\Bbb{Z}}$ and $B \rtimes _Y{\Bbb{Z}}$ are strongly Morita equivalent to each other, provided that $A$ and $B$ are strongly Morita equivalent under an imprimitivity bimodule $M$ satisfying $X\otimes _A M \simeq M\otimes _B Y$ as $A-B$ Hilbert $C^*$-bimodules. We also present a six-term exact sequence for $K$-groups of crossed products by Hilbert $C^*$-bimodules.
Homogeneity in powers of subspaces of the real line
L.
Brian
Lawrence
3055-3064
Abstract: Working in ZFC, we prove that for every zero-dimensional subspace $S$ of the real line, the Tychonoff power ${}^\omega S$ is homogeneous ($\omega$ denotes the nonnegative integers). It then follows as a corollary that ${}^\omega S$ is homogeneous whenever $S$ is a separable zero-dimensional metrizable space. The question of homogeneity in powers of this type was first raised by Ben Fitzpatrick, and was subsequently popularized by Gary Gruenhage and Hao-xuan Zhou.
Liouvillian integration and Bernoulli foliations
D.
Cerveau;
P.
Sad
3065-3081
Abstract: Analytic foliations in the 2-dimensional complex projective space with algebraic invariant curves are studied when the holonomy groups of these curves are solvable. It is shown that such a condition leads to the existence of a Liouville type first integral, and, under ``generic'' extra conditions, it is proven that these foliations can be defined by Bernoulli equations.
Equations for the Jacobian of a hyperelliptic curve
Paul
van Wamelen
3083-3106
Abstract: We give an explicit embedding of the Jacobian of a hyperelliptic curve, $y^2 = f(x)$, into projective space such that the image is isomorphic to the Jacobian over the splitting field of $f$. The embedding is a modification of the usual embedding by theta functions with half integer characteristics.
Cusp forms for congruence subgroups of $Sp_n(\mathbb{Z})$ and theta functions
Yaacov
Kopeliovich
3107-3118
Abstract: In this paper we use theta functions with rational characteristic to construct cusp forms for congruence subgroups $\Gamma _g(p)$ of $Sp(g,\mathbb Z)$.The action of the quotient group $Sp(g,\mathbb Z_p)$ on these forms is conjugate to the linear action of $Sp(g,\mathbb Z_p)$ on $(\mathbb Z_p)^{2g}$. We show that these forms are higher-dimensional analogues of the Fricke functions.
Complicated dynamics of parabolic equations with simple gradient dependence
Martino
Prizzi;
Krzysztof
P.
Rybakowski
3119-3130
Abstract: Let $\Omega \subset \mathbb R^{2}$ be a smooth bounded domain. Given positive integers $n$, $k$ and $q_{l}~\le ~l$, $l=1$, ..., $k$, consider the semilinear parabolic equation \begin{alignat*}{2} u_{t}&=u_{xx}+u_{yy}+a(x,y)u+ \smash{\sum _{l=1}^{k}}a_{l}(x,y) u^{l-q_{l}}(u_{y})^{q_{l}},&\quad &t>0, (x,y)\in \Omega,\tag{E} u&=0,&\quad& t>0, (x,y)\in \partial \Omega . \end{alignat*} where $a(x,y)$ and $a_{l}(x,y)$ are smooth functions. By refining and extending previous results of Polácik we show that arbitrary $k$-jets of vector fields in $\mathbb R^{n}$ can be realized in equations of the form (E). In particular, taking $q_{l}\equiv 1$ we see that very complicated (chaotic) behavior is possible for reaction-diffusion-convection equations with linear dependence on $\nabla u$.
Convergence of Madelung-like lattice sums
David
Borwein;
Jonathan
M.
Borwein;
Christopher
Pinner
3131-3167
Abstract: We make a general study of the convergence properties of lattice sums, involving potentials, of the form occurring in mathematical chemistry and physics. Many specific examples are studied in detail. The prototype is Madelung's constant for NaCl: \begin{equation*}\sum _{-\infty}^{\infty} \frac{(-1)^{n+m+p}} {\sqrt{n^2+m^2+p^2}} = -1.74756459 \cdots, \end{equation*} presuming that one appropriately interprets the summation proccess.
Similarity to a contraction, for power-bounded operators with finite peripheral spectrum
Ralph
deLaubenfels
3169-3191
Abstract: Suppose $T$ is a power-bounded linear opertor on a Hilbert space with finite peripheral spectrum (spectrum on the unit circle). Several sufficient conditions are given for $T$ to be similar to a contraction. A natural growth condition on the resolvent in half-planes tangent to the unit circle at the peripheral spectrum is shown to be equivalent to $T$ having an $H^\infty(\mathcal P)\cap C(\overline{\mathcal P})$ functional calculus, for some open polygon $\mathcal P$ contained in the unit disc, which, in turn, is equivalent to $T$ being similar to a contraction with numerical range contained in a closed polygon in the closed unit disc. Having certain orbits of $T$ be square summable also implies that $T$ is similar to a contraction.
On hyper Kähler manifolds associated to Lagrangian Kähler submanifolds of $T^\ast\mathbb{C}^n$
Vicente
Cortés
3193-3205
Abstract: For any Lagrangian Kähler submanifold $M \subset T^*\mathbb{C}^n$, there exists a canonical hyper Kähler metric on $T^*M$. A Kähler potential for this metric is given by the generalized Calabi Ansatz of the theoretical physicists Cecotti, Ferrara and Girardello. This correspondence provides a method for the construction of (pseudo) hyper Kähler manifolds with large automorphism group. Using it, an interesting class of pseudo hyper Kähler manifolds of complex signature $(2,2n)$ is constructed. For any manifold $N$ in this class a group of automorphisms with a codimension one orbit on $N$ is specified. Finally, it is shown that the bundle of intermediate Jacobians over the moduli space of gauged Calabi Yau 3-folds admits a natural pseudo hyper Kähler metric of complex signature $(2,2n)$.
Based algebras and standard bases for quasi-hereditary algebras
Jie
Du;
Hebing
Rui
3207-3235
Abstract: Quasi-hereditary algebras can be viewed as a Lie theory approach to the theory of finite dimensional algebras. Motivated by the existence of certain nice bases for representations of semisimple Lie algebras and algebraic groups, we will construct in this paper nice bases for (split) quasi-hereditary algebras and characterize them using these bases. We first introduce the notion of a standardly based algebra, which is a generalized version of a cellular algebra introduced by Graham and Lehrer, and discuss their representation theory. The main result is that an algebra over a commutative local noetherian ring with finite rank is split quasi-hereditary if and only if it is standardly full-based. As an application, we will give an elementary proof of the fact that split symmetric algebras are not quasi-hereditary unless they are semisimple. Finally, some relations between standardly based algebras and cellular algebras are also discussed.
On the best constant for Hardy's inequality in $\mathbb{R}^n$
Moshe
Marcus;
Victor
J.
Mizel;
Yehuda
Pinchover
3237-3255
Abstract: Let $\Omega$ be a domain in $\mathbb R^n$ and $p\in (1,\infty)$. We consider the (generalized) Hardy inequality $\int _\Omega |\nabla u|^p\geq K\int _\Omega |u/\delta |^p$, where $\delta (x)=\operatorname{dist}\,(x,\partial \Omega )$. The inequality is valid for a large family of domains, including all bounded domains with Lipschitz boundary. We here explore the connection between the value of the Hardy constant $\mu _p(\Omega )=\inf _{\stackrel{\circ}{W}_{1,p}(\Omega )}\left (\int _\Omega |\nabla u|^p\,/\,\int _\Omega |u/\delta |^p \right )$ and the existence of a minimizer for this Rayleigh quotient. It is shown that for all smooth $n$-dimensional domains, $\mu _p(\Omega )\leq c_p$, where $c_p=(1-{1\over p})^p$ is the one-dimensional Hardy constant. Moreover it is shown that $\mu _p(\Omega )=c_p$ for all those domains not possessing a minimizer for the above Rayleigh quotient. Finally, for $p=2$, it is proved that $\mu _2(\Omega )<c_2=1/4$ if and only if the Rayleigh quotient possesses a minimizer. Examples show that strict inequality may occur even for bounded smooth domains, but $\mu _p=c_p$ for convex domains.
Criteria for $\bar{d}$-continuity
Zaqueu
Coelho;
Anthony
N.
Quas
3257-3268
Abstract: Bernoullicity is the strongest mixing property that a measure-theoretic dynamical system can have. This is known to be intimately connected to the so-called $\bar d$ metric on processes, introduced by Ornstein. In this paper, we consider families of measures arising in a number of contexts and give conditions under which the measures depend $\bar d$-continuously on the parameters. At points where there is $\bar d$-continuity, it is often straightforward to establish that the measures have the Bernoulli property.
A family of quantum projective spaces and related $q$-hypergeometric orthogonal polynomials
Mathijs
S.
Dijkhuizen;
Masatoshi
Noumi
3269-3296
Abstract: A one-parameter family of two-sided coideals in $\mathcal{U}_{q} (\mathfrak{g}\mathfrak{l}(n))$ is defined and the corresponding algebras of infinitesimally right invariant functions on the quantum unitary group $U_{q}(n)$ are studied. The Plancherel decomposition of these algebras with respect to the natural transitive $U_{q}(n)$-action is shown to be the same as in the case of a complex projective space. By computing the radial part of a suitable Casimir operator, we identify the zonal spherical functions (i.e. infinitesimally bi-invariant matrix coefficients of finite-dimensional irreducible representations) as Askey-Wilson polynomials containing two continuous and one discrete parameter. In certain limit cases, the zonal spherical functions are expressed as big and little $q$-Jacobi polynomials depending on one discrete parameter.
Quantized enveloping algebras for Borcherds superalgebras
Georgia
Benkart;
Seok-Jin
Kang;
Duncan
Melville
3297-3319
Abstract: We construct quantum deformations of enveloping algebras of Borcherds superalgebras, their Verma modules, and their irreducible highest weight modules.
Lie ideals in triangular operator algebras
T.
D.
Hudson;
L.
W.
Marcoux;
A.
R.
Sourour
3321-3339
Abstract: We study Lie ideals in two classes of triangular operator algebras: nest algebras and triangular UHF algebras. Our main results show that if ${\mathfrak L}$ is a closed Lie ideal of the triangular operator algebra ${\mathcal A}$, then there exist a closed associative ideal ${\mathcal K}$ and a closed subalgebra ${\mathfrak D}_{\mathcal K}$ of the diagonal ${\mathcal A}\cap {\mathcal A}^*$ so that ${\mathcal K}\subseteq {\mathfrak L}\subseteq {\mathcal K}+ {\mathfrak D}_{\mathcal K}$.
Discrete series characters and two-structures
Rebecca
A.
Herb
3341-3369
Abstract: Let $G$ be a connected semisimple real Lie group with compact Cartan subgroup. Harish-Chandra gave formulas for discrete series characters which are completely explicit except for certain interger constants appearing in the numerators. The main result of this paper is a new formula for these constants using two-structures. The new formula avoids endoscopy and stable discrete series entirely, expressing (unaveraged) discrete series constants directly in terms of (unaveraged) discrete series constants corresponding to two-structures of noncompact type.
Weighted norm inequalities for integral operators
Igor
E.
Verbitsky;
Richard
L.
Wheeden
3371-3391
Abstract: We consider a large class of positive integral operators acting on functions which are defined on a space of homogeneous type with a group structure. We show that any such operator has a discrete (dyadic) version which is always essentially equivalent in norm to the original operator. As an application, we study conditions of ``testing type,'' like those initially introduced by E. Sawyer in relation to the Hardy-Littlewood maximal function, which determine when a positive integral operator satisfies two-weight weak-type or strong-type $(L^{p}, L^{q})$ estimates. We show that in such a space it is possible to characterize these estimates by testing them only over ``cubes''. We also study some pointwise conditions which are sufficient for strong-type estimates and have applications to solvability of certain nonlinear equations.
Filling-invariants at infinity for manifolds of nonpositive curvature
Noel
Brady;
Benson
Farb
3393-3405
Abstract: In this paper we construct and study isoperimetric functions at infinity for Hadamard manifolds. These quasi-isometry invariants give a measure of the spread of geodesics in such a manifold.
Correction to ``Fox calculus, symplectic forms, and moduli spaces''
Valentino
Zocca
3407
The classification of the simple modular Lie algebras: VI. Solving the final case
H.
Strade
2553-2628
Abstract: We investigate the structure of simple Lie algebras $L$ over an algebraically closed field of characteristic $p>7$. Let $T$ denote a torus in the $p$-envelope of $L$ in $\operatorname{Der}L$ of maximal dimension. We classify all $L$ for which every 1-section with respect to every such torus $T$ is solvable. This settles the remaining case of the classification of these algebras.
The Regular Complex in the $BP\rangle 1 \langle$-Adams Spectral Sequence
Jesús
González
2629-2664
Abstract: We give a complete description of the quotient complex ${\cal C}$ obtained by dividing out the ${\mathbb F}_{p}$ Eilenberg-Mac Lane wedge summands in the first term of the $BP\langle 1\rangle$-Adams spectral sequence for the sphere spectrum $S^0$. We also give a detailed computation of the cohomology groups $H^{s,t}({\cal C})$ and obtain as a consequence a vanishing line of slope $(p^{2}-p-1)^{-1}$ in their usual $(t-s, s)$ representation. These calculations are interpreted as giving general simple conditions to lift homotopy classes through a $BP\langle 1 \rangle$ resolution of $S^0$.
Generators and relations of direct products of semigroups
E.
F.
Robertson;
N.
Ruskuc;
J.
Wiegold
2665-2685
Abstract: The purpose of this paper is to give necessary and sufficient conditions for the direct product of two semigroups to be finitely generated, and also for the direct product to be finitely presented. As a consequence we construct a semigroup $S$ of order 11 such that $S\times T$ is finitely generated but not finitely presented for every finitely generated infinite semigroup $T$. By way of contrast we show that, if $S$ and $T$ belong to a wide class of semigroups, then $S\times T$ is finitely presented if and only if both $S$ and $T$ are finitely presented, exactly as in the case of groups and monoids.
A weak-type inequality for differentially subordinate harmonic functions
Changsun
Choi
2687-2696
Abstract: Assuming an extra condition, we decrease the constant in the sharp inequality of Burkholder $\mu(|v|\ge 1)\le 2\|u\|_1$ for two harmonic functions $u$ and $v$. That is, we prove the sharp weak-type inequality $\mu(|v|\ge 1)\le K\|u\|_1$ under the assumptions that $|v(\xi)|\le |u(\xi)|$, $|\nabla v|\le|\nabla u|$ and the extra assumption that $\nabla u\cdot\nabla v=0$. Here $\mu$ is the harmonic measure with respect to $\xi$ and the constant $K$ is the one found by Davis to be the best constant in Kolmogorov's weak-type inequality for conjugate functions.
Periodic orbits in magnetic fields and Ricci curvature of Lagrangian systems
Abbas
Bahri;
Iskander
A.
Taimanov
2697-2717
Abstract: A Lagrangian system describing a motion of a charged particle on a Riemannian manifold is studied. For this flow an analog of a Ricci curvature is introduced, and for Ricci positively curved flows the existence of periodic orbits is proved.
The $\Pi_3$-theory of the computably enumerable Turing degrees is undecidable
Steffen
Lempp;
André
Nies;
Theodore
A.
Slaman
2719-2736
Abstract: We show the undecidability of the $\Pi _{3}$-theory of the partial order of computably enumerable Turing degrees.
Picard groups and infinite matrix rings
Gene
Abrams;
Jeremy
Haefner
2737-2752
Abstract: We describe a connection between the Picard group of a ring with local units $T$ and the Picard group of the unital overring $End(_TT)$. Using this connection, we show that the three groups $Pic(R)$, $Pic(FM(R))$, and $Pic(RFM(R))$ are isomorphic for any unital ring $R$. Furthermore, each element of $Pic(RFM(R))$ arises from an automorphism of $RFM(R)$, which yields an isomorphsm between $Pic(RFM(R))$ and $Out(RFM(R))$. As one application we extend a classical result of Rosenberg and Zelinsky by showing that the group $Out_R(RFM(R))$ is abelian for any commutative unital ring $R$.
Scattering theory for twisted automorphic functions
Ralph
Phillips
2753-2778
Abstract: The purpose of this paper is to develop a scattering theory for twisted automorphic functions on the hyperbolic plane, defined by a cofinite (but not cocompact) discrete group $\Gamma$ with an irreducible unitary representation $\rho$ and satisfying $u(\gamma z)=\rho(\gamma)u(z)$. The Lax-Phillips approach is used with the wave equation playing a central role. Incoming and outgoing subspaces are employed to obtain corresponding unitary translation representations, $R_-$ and $R_+$, for the solution operator. The scattering operator, which maps $R_-f$ into $R_+f$, is unitary and commutes with translation. The spectral representation of the scattering operator is a multiplicative operator, which can be expressed in terms of the constant term of the Eisenstein series. When the dimension of $\rho$ is one, the elements of the scattering operator cannot vanish. However when $\dim(\rho)>1$ this is no longer the case.
On the measure theoretic structure of compact groups
S.
Grekas;
S.
Mercourakis
2779-2796
Abstract: If $G$ is a compact group with $w(G)=a\geq \omega$, we show the following results: (i) There exist direct products $\displaystyle{\prod _{\xi<a}G_{\xi}, \prod _{\xi<a}H_{\xi}}$ of compact metric groups and continuous open surjections $\displaystyle{\prod _{\xi<a}G_{\xi} \stackrel{p}{\rightarrow }G \stackrel{q}{\rightarrow }\prod _{\xi<a}H_{\xi}}$ with respect to Haar measure; and (ii) the Haar measure on $G$ is Baire and at the same time Jordan isomorphic to the Haar measure on a direct product of compact Lie groups. Applications of the above results in measure theory are given.
A Hilbert-Nagata theorem in noncommutative invariant theory
Mátyás
Domokos;
Vesselin
Drensky
2797-2811
Abstract: Nagata gave a fundamental sufficient condition on group actions on finitely generated commutative algebras for finite generation of the subalgebra of invariants. In this paper we consider groups acting on noncommutative algebras over a field of characteristic zero. We characterize all the T-ideals of the free associative algebra such that the algebra of invariants in the corresponding relatively free algebra is finitely generated for any group action from the class of Nagata. In particular, in the case of unitary algebras this condition is equivalent to the nilpotency of the algebra in Lie sense. As a consequence we extend the Hilbert-Nagata theorem on finite generation of the algebra of invariants to any finitely generated associative algebra which is Lie nilpotent. We also prove that the Hilbert series of the algebra of invariants of a group acting on a relatively free algebra with a non-matrix polynomial identity is rational, if the action satisfies the condition of Nagata.
The Castelnuovo regularity of the Rees algebra and the associated graded ring
Ngô
Viêt
Trung
2813-2832
Abstract: It is shown that there is a close relationship between the invariants characterizing the homogeneous vanishing of the local cohomology and the Koszul homology of the Rees algebra and the associated graded ring of an ideal. From this it follows that these graded rings share the same Castelnuovo regularity and the same relation type. The main result of this paper is however a simple characterization of the Castenuovo regularity of these graded rings in terms of any reduction of the ideal. This characterization brings new insights into the theory of $d$-sequences.
On the averages of Darboux functions
Aleksander
Maliszewski
2833-2846
Abstract: Let $\mathbf{A}$ be the family of functions which can be written as the average of two comparable Darboux functions. In 1974 A. M. Bruckner, J. G. Ceder, and T. L. Pearson characterized the family $\mathbf{A} $ and showed that if $\alpha \ge 2$, then $\mathbf{A} \cap {\mathbf B}_\alpha$ is the family of the averages of comparable Darboux functions in Baire class $\alpha$. They also asked whether the latter result holds true also for $\alpha =1$. The main goal of this paper is to answer this question in the negative and to characterize the family of the averages of comparable Darboux Baire one functions.
Kruzkov's estimates for scalar conservation laws revisited
F.
Bouchut;
B.
Perthame
2847-2870
Abstract: We give a synthetic statement of Kruzkov-type estimates for multi-dimensional scalar conservation laws. We apply it to obtain various estimates for different approximation problems. In particular we recover for a model equation the rate of convergence in $h^{1/4}$ known for finite volume methods on unstructured grids. Les estimations de Kruzkov pour les lois de conservation scalaires revisitées Résumé Nous donnons un énoncé synthétique des estimations de type de Kruzkov pour les lois de conservation scalaires multidimensionnelles. Nous l'appliquons pour obtenir d'estimations nombreuses pour problèmes différents d'approximation. En particulier, nous retrouvons pour une équation modèle la vitesse de convergence en $h^{1/4}$ connue pour les méthodes de volumes finis sur des maillages non structurés.
A Note on the Monomial Conjecture
S.
P.
Dutta
2871-2878
Abstract: Several cases of the monomial conjecture are proved. An equivalent form of the direct summand conjecture is discussed.
Bounds for multiplicities
Jürgen
Herzog;
Hema
Srinivasan
2879-2902
Abstract: Let $R=K[x_1,x_2,\ldots, x_n]$ and $S=R/I$ be a homogeneous $K$-algebra. We establish bounds for the multiplicity of certain homogeneous $K$-algebras $S$ in terms of the shifts in a free resolution of $S$ over $R$. Huneke and we conjectured these bounds as they generalize the formula of Huneke and Miller for the algebras with pure resolution, the simplest case. We prove these conjectured bounds for various algebras including algebras with quasi-pure resolutions. Our proof for this case gives a new and simple proof of the Huneke-Miller formula. We also settle these conjectures for stable and square free strongly stable monomial ideals $I$. As a consequence, we get a bound for the regularity of $S$. Further, when $S$ is not Cohen-Macaulay, we show that the conjectured lower bound fails and prove the upper bound for almost Cohen-Macaulay algebras as well as algebras with a $p$-linear resolution.
Projective threefolds on which $\mathbf{SL}(2)$ acts with 2-dimensional general orbits
T.
Nakano
2903-2924
Abstract: The birational geometry of projective threefolds on which $\mathbf{SL}(2)$ acts with 2-dimensional general orbits is studied from the viewpoint of the minimal model theory of projective threefolds. These threefolds are closely related to the minimal rational threefolds classified by Enriques, Fano and Umemura. The main results are (i) the $\mathbf{SL}(2)$-birational classification of such threefolds and (ii) the classification of relatively minimal models in the fixed point free cases.
A Singular Quasilinear Anisotropic Elliptic Boundary Value Problem. II
Y.
S.
Choi;
P.
J.
McKenna
2925-2937
Abstract: Let $\Omega \subset \mathbf{R}^N$ with $N \geq 2$. We consider the equations \begin{displaymath}\begin{array}{rcl} \displaystyle \sum _{i=1}^{N} u^{a_i} \frac{\partial^2 u}{\partial x_i^2} +p(\mathbf{x})& = & 0, u|_{\partial\Omega} & = & 0, \end{array} \end{displaymath} with $a_1 \geq a_2 \geq .... \geq a_N \geq 0$ and $a_1>a_N$. We show that if $\Omega$ is a convex bounded region in $\mathbf{R}^N$, there exists at least one classical solution to this boundary value problem. If the region is not convex, we show the existence of a weak solution. Partial results for the existence of classical solutions for non-convex domains in $\mathbf{R}^2$ are also given.
A probabilistic approach to some of Euler's number theoretic identities
Don
Rawlings
2939-2951
Abstract: Probabilistic proofs and interpretations are given for the $q$-binomial theorem, $q$-binomial series, two of Euler's fundamental partition identities, and for $q$-analogs of product expansions for the Riemann zeta and Euler phi functions. The underlying processes involve Bernoulli trials with variable probabilities. Also presented are several variations on the classical derangement problem inherent in the distributions considered.
Sum theorems for monotone operators and convex functions
S.
Simons
2953-2972
Abstract: In this paper, we derive sufficient conditions for the sum of two or more maximal monotone operators on a reflexive Banach space to be maximal monotone, and we achieve this without any renorming theorems or fixed-point-related concepts. In the course of this, we will develop a generalization of the uniform boundedness theorem for (possibly nonreflexive) Banach spaces. We will apply this to obtain the Fenchel Duality Theorem for the sum of two or more proper, convex lower semicontinuous functions under the appropriate constraint qualifications, and also to obtain additional results on the relation between the effective domains of such functions and the domains of their subdifferentials. The other main tool that we use is a standard minimax theorem.
Pairs of monotone operators
S.
Simons
2973-2980
Abstract: This note is an addendum to Sum theorems for monotone operators and convex functions. In it, we prove some new results on convex functions and monotone operators, and use them to show that several of the constraint qualifications considered in the preceding paper are, in fact, equivalent.
Equivariant Novikov conjecture for groups acting on Euclidean buildings
Donggeng
Gong
2141-2183
Abstract: We prove the equivariant Novikov conjecture for groups acting on Euclidean buildings by using an equivariant Hilsum-Skandalis method. We also obtain an equivariant version of the Connes-Gromov-Moscovici theorem for almost flat $C^{*}$-algebra bundles.
Carathéodory-Toeplitz and Nehari problems for matrix valued almost periodic functions
Leiba
Rodman;
Ilya
M.
Spitkovsky;
Hugo
J.
Woerdeman
2185-2227
Abstract: In this paper the positive and strictly contractive extension problems for almost periodic matrix functions are treated. We present necessary and sufficient conditions for the existence of extensions in terms of Toeplitz and Hankel operators on Besicovitch spaces and Lebesgue spaces. Furthermore, when a solution exists a special extension (the band extension) is constructed which enjoys a maximum entropy property. A linear fractional parameterization of the set of all extensions is also provided. The techniques used in the proofs include factorizations of matrix valued almost periodic functions and a general algebraic scheme called the band method.
Topological conditions for the existence of absorbing Cantor sets
Henk
Bruin
2229-2263
Abstract: This paper deals with strange attractors of S-unimodal maps $f$. It generalizes earlier results in the sense that very general topological conditions are given that either i) guarantee the existence of an absorbing Cantor set provided the critical point of $f$ is sufficiently degenerate, or ii) prohibit the existence of an absorbing Cantor set altogether. As a by-product we obtain very weak topological conditions that imply the existence of an absolutely continuous invariant probability measure for $f$.
Periodic orbits of the restricted three-body problem
Salem
Mathlouthi
2265-2276
Abstract: We prove, using a variational formulation, the existence of an infinity of periodic solutions of the restricted three-body problem. When the problem has some additional symmetry (in particular, in the autonomous case), we prove the existence of at least two periodic solutions of minimal period $T$, for every $T>0$. We also study the bifurcation problem in a neighborhood of each closed orbit of the autonomous restricted three-body problem.
Weak*-closedness of subspaces of Fourier-Stieltjes algebras and weak*-continuity of the restriction map
M.
B.
Bekka;
E.
Kaniuth;
A.
T.
Lau;
G.
Schlichting
2277-2296
Abstract: Let $G$ be a locally compact group and $B(G)$ the Fourier-Stieltjes algebra of $G$. We study the problem of how weak*-closedness of some translation invariant subspaces of $B(G)$ is related to the structure of $G$. Moreover, we prove that for a closed subgroup $H$ of $G$, the restriction map from $B(G)$ to $B(H)$ is weak*-continuous only when $H$ is open in $G$.
Singularity of self-similar measures with respect to Hausdorff measures
Manuel
Morán;
José-Manuel
Rey
2297-2310
Abstract: Besicovitch (1934) and Eggleston (1949) analyzed subsets of points of the unit interval with given frequencies in the figures of their base-$p$ expansions. We extend this analysis to self-similar sets, by replacing the frequencies of figures with the frequencies of the generating similitudes. We focus on the interplay among such sets, self-similar measures, and Hausdorff measures. We give a fine-tuned classification of the Hausdorff measures according to the singularity of the self-similar measures with respect to those measures. We show that the self-similar measures are concentrated on sets whose frequencies of similitudes obey the Law of the Iterated Logarithm.
The irrationality of $\log(1+1/q) \log(1-1/q)$
Masayoshi
Hata
2311-2327
Abstract: We shall show that the numbers $1, \log (1+ 1/q), \log (1-1/q)$ and $\log (1+1/q)\log (1-1/q)$ are linearly independent over $\mathbf{Q}$ for any natural number $q \ge 54$. The key is to construct explicit Padé-type approximations using Legendre-type polynomials.
Approximation of the equilibrium distribution by distributions of equal point charges with minimal energy
J.
Korevaar;
M.
A.
Monterie
2329-2348
Abstract: Let $\omega$ denote the classical equilibrium distribution (of total charge $1$) on a convex or $C^{1,\alpha }$-smooth conductor $K$ in $\mathbb{R}^{q}$ with nonempty interior. Also, let $\omega _{N}$ be any $N$th order ``Fekete equilibrium distribution'' on $K$, defined by $N$ point charges $1/N$ at $N$th order ``Fekete points''. (By definition such a distribution minimizes the energy for $N$-tuples of point charges $1/N$ on $K$.) We measure the approximation to $\omega$ by $\omega _{N}$ for $N \to \infty$ by estimating the differences in potentials and fields, \begin{equation*}U^{\omega }-U^{\omega _{N}}\quad \text{\rm and}\quad {\mathcal{E}}^{\omega }-{\mathcal{E}}^{\omega _{N}},\end{equation*} both inside and outside the conductor $K$. For dimension $q \geq 3$ we obtain uniform estimates ${\mathcal{O}}(1/N^{1/(q-1)})$ at distance $\geq \varepsilon >0$ from the outer boundary $\Sigma$ of $K$. Observe that ${\mathcal{E}}^{\omega }=0$ throughout the interior $\Omega$ of $\Sigma$ (Faraday cage phenomenon of electrostatics), hence ${\mathcal{E}}^{\omega _{N}}={\mathcal{O}}(1/N^{1/(q-1)})$ on the compact subsets of $\Omega$. For the exterior $\Omega ^{\infty }$ of $\Sigma$ the precise results are obtained by comparison of potentials and energies. Admissible sets $K$ have to be regular relative to capacity and their boundaries must allow good Harnack inequalities. For the passage to interior estimates we develop additional machinery, including integral representations for potentials of measures on Lipschitz boundaries $\Sigma$ and bounds on normal derivatives of interior and exterior Green functions. Earlier, one of us had considered approximations to the equilibrium distribution by arbitrary distributions $\mu _{N}$ of equal point charges on $\Sigma$. In that context there is an important open problem for the sphere which is discussed at the end of the paper.
Strongly indefinite systems with critical Sobolev exponents
Josephus
Hulshof;
Enzo
Mitidieri;
Robertus
vanderVorst
2349-2365
Abstract: We consider an elliptic system of Hamiltonian type on a bounded domain. In the superlinear case with critical growth rates we obtain existence and positivity results for solutions under suitable conditions on the linear terms. Our proof is based on an adaptation of the dual variational method as applied before to the scalar case.
Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators
Roger
D.
Nussbaum;
Bertram
Walsh
2367-2391
Abstract: For $\Sigma$ a compact subset of $\mathbf{C}$ symmetric with respect to conjugation and $f: \Sigma \to \mathbf{C}$ a continuous function, we obtain sharp conditions on $f$ and $\Sigma$ that insure that $f$ can be approximated uniformly on $\Sigma$ by polynomials with nonnegative coefficients. For $X$ a real Banach space, $K \subseteq X$ a closed but not necessarily normal cone with $\overline{K - K} = X$, and $A: X \to X$ a bounded linear operator with $A[K] \subseteq K$, we use these approximation theorems to investigate when the spectral radius $\text{\rm r}(A)$ of $A$ belongs to its spectrum $\sigma (A)$. A special case of our results is that if $X$ is a Hilbert space, $A$ is normal and the 1-dimensional Lebesgue measure of $\sigma (i(A - A^{*}))$ is zero, then $\text{\rm r}(A) \in \sigma (A)$. However, we also give an example of a normal operator $A = - U -\alpha I$ (where $U$ is unitary and $\alpha > 0$) for which $A[K] \subseteq K$ and $\text{\rm r}(A) \notin \sigma (A)$.
Eigenfunctions of the Weil representation of unitary groups of one variable
Tonghai
Yang
2393-2407
Abstract: In this paper, we construct explicit eigenfunctions of the local Weil representation on unitary groups of one variable in the $p$-adic case when $p$ is odd. The idea is to use the lattice model, and the results will be used to compute special values of certain Hecke $L$-functions in separate papers. We also recover Moen's results on when a local theta lifting from $U(1)$ to itself does not vanish.
Necessary conditions for constrained optimization problems with semicontinuous and continuous data
Jonathan
M.
Borwein;
Jay
S.
Treiman;
Qiji
J.
Zhu
2409-2429
Abstract: We consider nonsmooth constrained optimization problems with semicontinuous and continuous data in Banach space and derive necessary conditions without constraint qualification in terms of smooth subderivatives and normal cones. These results, in different versions, are set in reflexive and smooth Banach spaces.
On homological properties of singular braids
Vladimir
V.
Vershinin
2431-2455
Abstract: Homology of objects which can be considered as singular braids, or braids with crossings, is studied. Such braids were introduced in connection with Vassiliev's theory of invariants of knots and links. The corresponding algebraic objects are the braid-permutation group $BP_{n}$ of R. Fenn, R. Rimányi and C. Rourke and the Baez-Birman monoid $SB_{n}$ which embeds into the singular braid group $SG_{n}$. The following splittings are proved for the plus-constructions of the classifying spaces of the infinite braid-permutation group and the singular braid group \begin{equation*}\mathbb{Z}\times BBP_{\infty }^{+}\simeq \Omega ^{\infty }S^{\infty }\times S^{1} \times Y, \end{equation*} \begin{equation*}\mathbb{Z}\times BSG_{\infty }^{+}\simeq S^{1}\times \Omega ^{2} S^{2}\times W, \end{equation*} where $Y$ is an infinite loop space and $W$ is a double loop space.
Fixed point sets of deformations of polyhedra with local cut points
Peter
Wolfenden
2457-2471
Abstract: A locally finite simplicial complex $X$ is said to be 2-dimensionally connected if $X - \{\text{local cut points of } X \}$ is connected. Such spaces exhibit ``classical'' behavior in that they all admit deformations with one fixed point, and they admit fixed point free deformations if and only if the Euler characteristic is zero. A result of G.-H. Shi implies that, for non 2-dimensionally connected spaces, the fixed point sets of deformations are equivalent to the fixed point sets of certain combinatorial maps which he calls good displacements. U. K. Scholz combined Shi's results with a theorem of P. Hall to obtain a characterization of all finite simplicial complexes which admit fixed point free deformations. In this paper we begin by explicitly capturing the combinatorial structure of a non 2-dimensionally connected polyhedron in a bipartite graph. We then apply an extended version of Hall's theorem to this graph to get a realization theorem which gives necessary and sufficient conditions for the existence of a deformation with a prescribed finite fixed point set. Scholz's result, and a characterization of all finite simplicial complexes without fixed point free deformations that admit deformations with a single fixed point follow immediately from this realization theorem.
Abelian subgroups of pro-$p$ Galois groups
Antonio
José
Engler;
Jochen
Koenigsmann
2473-2485
Abstract: It is proved that non-trivial normal abelian subgroups of the Galois group of the maximal Galois $p$-extension of a field $F$ (where $p$ is an odd prime) arise from $p$-henselian valuations with non-$p$-divisible value group, provided $\# (\dot {F}/\dot {F}^{p})\geq p^{2}$ and $F$ contains a primitive $p$-th root of unity. Also, a generalization to arbitrary prime-closed Galois-extensions is given.
On the distribution of mass in collinear central configurations
Peter
W.
Lindstrom
2487-2523
Abstract: Moulton's Theorem says that given an ordering of masses, $m_1,m_2, \dotsc,m_n$, there exists a unique collinear central configuration with center of mass at the origin and moment of inertia equal to 1. This theorem allows us to ask the questions: What is the distribution of mass in this standardized collinear central configuration? What is the behavior of the distribution as $n\to\infty$? In this paper, we define continuous configurations, prove a continuous version of Moulton's Theorem, and then, in the spirit of limit theorems in probability theory, prove that as $n\to\infty$, under rather general conditions, the discrete mass distributions of the standardized collinear central configurations have distribution functions which converge uniformly to the distribution function of the unique continuous standardized collinear central configuration which we determine.
Geometric aspects of multiparameter spectral theory
Luzius
Grunenfelder;
Tomaz
Kosir
2525-2546
Abstract: The paper contains a geometric description of the dimension of the total root subspace of a regular multiparameter system in terms of the intersection multiplicities of its determinantal hypersurfaces. The new definition of regularity used here is proved to restrict to the familiar definition in the linear case. A decomposability problem is also considered. It is shown that the joint root subspace of a multiparameter system may be decomposable even when the root subspace of each member is indecomposable.
An intersection number for the punctual Hilbert scheme of a surface
Geir
Ellingsrud;
Stein
Arild
Strømme
2547-2552
Abstract: We compute the intersection number between two cycles $A$ and $B$ of complementary dimensions in the Hilbert scheme $H$ parameterizing subschemes of given finite length $n$ of a smooth projective surface $S$. The $(n+1)$-cycle $A$ corresponds to the set of finite closed subschemes the support of which has cardinality 1. The $(n-1)$-cycle $B$ consists of the closed subschemes the support of which is one given point of the surface. Since $B$ is contained in $A$, indirect methods are needed. The intersection number is $A.B=(-1)^{n-1}n$, answering a question by H. Nakajima.
Boundary slopes of punctured tori in 3-manifolds
C.
McA.
Gordon
1713-1790
Abstract: Let $M$ be an irreducible 3-manifold with a torus boundary component $T$, and suppose that $r,s$ are the boundary slopes on $T$ of essential punctured tori in $M$, with their boundaries on $T$. We show that the intersection number $\Delta(r,s)$ of $r$ and $s$ is at most $8$. Moreover, apart from exactly four explicit manifolds $M$, which contain pairs of essential punctured tori realizing $\Delta(r,s)=8,8,7$ and 6 respectively, we have $\Delta(r,s)\le 5$. It follows immediately that if $M$ is atoroidal, while the manifolds $M(r), M(s)$ obtained by $r$- and $s$-Dehn filling on $M$ are toroidal, then $\Delta(r,s)\le 8$, and $\Delta(r,s)\le 5$ unless $M$ is one of the four examples mentioned above. Let $\mathcal{H}_0$ be the class of 3-manifolds $M$ such that $M$ is irreducible, atoroidal, and not a Seifert fibre space. By considering spheres, disks and annuli in addition to tori, we prove the following. Suppose that $M\in \mathcal{H}_0$, where $\partial M$ has a torus component $T$, and $\partial M-T\ne \varnothing$. Let $r,s$ be slopes on $T$ such that $M(r), M(s)\notin \mathcal{H}_0$. Then $\Delta(r,s)\le 5$. The exterior of the Whitehead sister link shows that this bound is best possible.
Integration of singular braid invariants and graph cohomology
Michael
Hutchings
1791-1809
Abstract: We prove necessary and sufficient conditions for an arbitrary invariant of braids with $m$ double points to be the ``$m^{th}$ derivative'' of a braid invariant. We show that the ``primary obstruction to integration'' is the only obstruction. This gives a slight generalization of the existence theorem for Vassiliev invariants of braids. We give a direct proof by induction on $m$ which works for invariants with values in any abelian group. We find that to prove our theorem, we must show that every relation among four-term relations satisfies a certain geometric condition. To find the relations among relations we show that $H_1$ of a variant of Kontsevich's graph complex vanishes. We discuss related open questions for invariants of links and other things.
Integral representation of continuous comonotonically additive functionals
Lin
Zhou
1811-1822
Abstract: In this paper, I first prove an integral representation theorem: Every quasi-integral on a Stone lattice can be represented by a unique upper-continuous capacity. I then apply this representation theorem to study the topological structure of the space of all upper-continuous capacities on a compact space, and to prove the existence of an upper-continuous capacity on the product space of infinitely many compact Hausdorff spaces with a collection of consistent finite marginals.
Kronecker conjugacy of polynomials
Peter
Müller
1823-1850
Abstract: Let $f,g\in \mathbb{Z}[X]$ be non-constant polynomials with integral coefficients. In 1968 H. Davenport raised the question as to when the value sets $f(\mathbb{Z})$ and $g(\mathbb{Z})$ are the same modulo all but finitely many primes. The main progress until now is M. Fried's result that $f$ and $g$ then differ by a linear substitution, provided that $f$ is functionally indecomposable. We extend this result to polynomials $f$ of composition length $2$. Also, we study the analog when $\mathbb{Z}$ is replaced by the integers of a number field. The above number theoretic property translates to an equivalent property of subgroups of a finite group, known as Kronecker conjugacy, a weakening of conjugacy which has been studied by various authors under different assumptions and in other contexts. We also give a simplified and strengthened version of the Galois theoretic translation to finite groups.
On the hyperbolicity of small cancellation groups and one-relator groups
S.
V.
Ivanov;
P.
E.
Schupp
1851-1894
Abstract: In the article, a result relating to maps (= finite planar connected and simply connected 2-complexes) that satisfy a $C(p)\&T(q)$ condition (where $(p,q)$ is one of $(3,6)$, $(4,4)$, $(6,3)$ which correspond to regular tessellations of the plane by triangles, squares, hexagons, respectively) is proven. On the base of this result a criterion for the Gromov hyperbolicity of finitely presented small cancellation groups satisfying non-metric $C(p)\&T(q)$-conditions is obtained and a complete (and explicit) description of hyperbolic groups in some classes of one-relator groups is given: All one-relator hyperbolic groups with $> 0$ and $\le 3$ occurrences of a letter are indicated; it is shown that a finitely generated one-relator group $G$ whose reduced relator $R$ is of the form $R \equiv a T_{0} a T_{1} \dots a T_{n-1}$, where the words $T_{i}$ are distinct and have no occurrences of the letter $a^{\pm 1}$, is not hyperbolic if and only if one has in the free group that (1) $n=2$ and $T_{0} T_{1}^{-1}$ is a proper power; (2) $n = 3$ and for some $i$ it is true (with subscripts $\operatorname{mod} 3$) that $T_{i} T_{i+1}^{-1} T_{i} T_{i+2}^{-1} = 1$; (3) $n = 4$ and for some $i$ it is true (with subscripts $\operatorname{mod} 4$) that $T_{i} T_{i+1}^{-1} T_{i+2} T_{i+3}^{-1} = 1$.
A convexity theorem in the scattering theory for the Dirac operator
K.
L.
Vaninsky
1895-1911
Abstract: The Dirac operator enters into zero curvature representation for the cubic nonlinear Schrödinger equation. We introduce and study a conformal map from the upper half-plane of the spectral parameter of the Dirac operator into itself. The action variables turn out to be limiting boundary values of the imaginary part of this map. We describe the image of the momentum map (convexity theorem) in the simplest case of a potential from the Schwartz class. We apply this description to the invariant manifolds for the nonlinear Schrödinger equation.
Linkage and sums of ideals
Mark
R.
Johnson
1913-1930
Abstract: It is shown (under mild conditions) that the sum of transversal ideals in a regular local ring cannot lie in the linkage class of a complete intersection. For a sum of geometrically linked Cohen-Macaulay ideals, we compute the depths of the conormal module and the first Koszul homology. As applications, we construct general examples of ideals which are strongly Cohen-Macaulay, strongly nonobstructed but not in the linkage class of a complete intersection, and Gorenstein ideals which are strongly nonobstructed but not syzygetic.
The $L_2$-localization of $W(n)$
Robert
D.
Thompson
1931-1944
Abstract: In this paper we analyze the localization of $W(n)$, the fiber of the double suspension map $S^{2n-1}\to \Omega^{2}S^{2n+1}$, with respect to $E(2)$. If four cells at the bottom of $D_pM^{2np-1}$, the $p$th extended power spectrum of the Moore spectrum, are collapsed to a point, then one obtains a spectrum $C$. Let $QM^{2np-1}\to QC$ be the James-Hopf map followed by the collapse map. Then we show that the secondary suspension map $BW(n)\to QM^{2np-1}$ has a lifting to the fiber of $QM^{2np-1}\to QC$ and this lifting is shown to be a $v_2$-periodic equivalence, hence an $E(2)$-equivalence.
Extension and approximation of CR functions on tube manifolds
André
Boivin;
Roman
Dwilewicz
1945-1956
Abstract: A complete generalization of the classical Bochner theorem for infinite tubes is given.
Generalized Hestenes' Lemma and extension of functions
Massimo
Ferrarotti;
Leslie
C.
Wilson
1957-1975
Abstract: Suppose we have an $m$-jet field on $V\subset \mathbf{R}^{n}$ which is a Whitney field on the nonsingular part $M$ of $V$. We show that, under certain hypotheses about the relationship between geodesic and euclidean distance on $V$, if the field is flat enough at the singular part $S$, then it is a Whitney field on $V$ (the order of flatness required depends on the coefficients in the hypotheses). These hypotheses are satisfied when $V$ is subanalytic. In Section II, we show that a $C^{2}$ function $f$ on $M$ can be extended to one on $V$ if the differential $df$ goes to $0$ faster than the order of divergence of the principal curvatures of $M$ and if the first covariant derivative of $df$ is sufficiently flat. For the general case of $C^{m}$ functions with $m >2$, we give a similar result for $\operatorname{codim} M=1$ in Section III.
Hyperbolic complete minimal surfaces with arbitrary topology
F.
J.
López
1977-1990
Abstract: We show a method to construct orientable minimal surfaces in $\Bbb R^3$ with arbitrary topology. This procedure gives complete examples of two different kinds: surfaces whose Gauss map omits four points of the sphere and surfaces with a bounded coordinate function. We also apply these ideas to construct stable minimal surfaces with high topology which are incomplete or complete with boundary.
Prime and semiprime semigroup algebras of cancellative semigroups
M.
V.
Clase
1991-2007
Abstract: Necessary and sufficient conditions are given for a semigroup algebra of a cancellative semigroup to be prime and semiprime. These conditions were proved necessary by Okninski; our contribution is to show that they are also sufficient. The techniques used in the proof are a new variation on the $\Delta$-methods which were developed originally for group algebras.
Composition factors of indecomposable modules
Maria
Izabel
Ramalho Martins
2009-2031
Abstract: Let $\Lambda$ be a connected, basic finite dimensional algebra over an algebraically closed field. Our main aim is to prove that if $\Lambda$ is biserial, its ordinary quiver has no loop and every indecomposable $\Lambda$-module is uniquely determined by its composition factors, then each indecomposable $\Lambda$-module is multiplicity-free.
Equivalence of norms on operator space tensor products of $C^\ast$-algebras
Ajay
Kumar;
Allan
M.
Sinclair
2033-2048
Abstract: The Haagerup norm $\Vert \cdot \Vert _{h}$ on the tensor product $A\otimes B$ of two $C^*$-algebras $A$ and $B$ is shown to be Banach space equivalent to either the Banach space projective norm $\Vert \cdot \Vert _{\gamma }$ or the operator space projective norm $\Vert \cdot \Vert _{\wedge }$ if and only if either $A$ or $B$ is finite dimensional or $A$ and $B$ are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are equivalent on $A\otimes B$ if and only if $A$ or $B$ is subhomogeneous.
Carleson conditions for asymptotic weights
Michael
Brian
Korey
2049-2069
Abstract: The doubling and $A_\infty$ conditions are characterized in terms of convolution with rapidly decreasing kernels. The Carleson-measure criterion for $A_\infty$ of Fefferman, Kenig, and Pipher is extended to the case when all bounds become optimally small in the asymptotic limit.
Local spectra and individual stability of uniformly bounded $C_0$-semigroups
Charles
J. K.
Batty;
Jan
van Neerven;
Frank
Räbiger
2071-2085
Abstract: We study the asymptotic behaviour of individual orbits $T(\cdot )x$ of a uniformly bounded $C_{0}$-semigroup $\{T(t)\}_{t\ge 0}$ with generator $A$ in terms of the singularities of the local resolvent $(\lambda -A)^{-1}x$ on the imaginary axis. Among other things we prove individual versions of the Arendt-Batty-Lyubich-Vu theorem and the Katznelson-Tzafriri theorem.
Tauberian theorems and stability of solutions of the Cauchy problem
Charles
J. K.
Batty;
Jan
van Neerven;
Frank
Räbiger
2087-2103
Abstract: Let $f : \mathbb{R}_{+} \to X$ be a bounded, strongly measurable function with values in a Banach space $X$, and let $iE$ be the singular set of the Laplace transform $\widetilde f$ in $i\mathbb{R}$. Suppose that $E$ is countable and $\alpha \left \| \int _{0}^{\infty }e^{-(\alpha + i\eta ) u} f(s+u) \, du \right \| \to 0$ uniformly for $s\ge 0$, as $\alpha \searrow 0$, for each $\eta$ in $E$. It is shown that \begin{displaymath}\left \| \int _{0}^{t} e^{-i\mu u} f(u) \, du - \widetilde f(i\mu ) \right \| \to 0\end{displaymath} as $t\to \infty$, for each $\mu$ in $\mathbb{R} \setminus E$; in particular, $\|f(t)\| \to 0$ if $f$ is uniformly continuous. This result is similar to a Tauberian theorem of Arendt and Batty. It is obtained by applying a result of the authors concerning local stability of bounded semigroups to the translation semigroup on $BUC(\mathbb{R}_{+}, X)$, and it implies several results concerning stability of solutions of Cauchy problems.
Minimizing the Laplacian of a function squared with prescribed values on interior boundaries- Theory of polysplines
Ognyan
Iv.
Kounchev
2105-2128
Abstract: In this paper we consider the minimization of the integral of the Laplacian of a real-valued function squared (and more general functionals) with prescribed values on some interior boundaries $\Gamma$, with the integral taken over the domain D. We prove that the solution is a biharmonic function in $D$ except on the interior boundaries $\Gamma$, and satisfies some matching conditions on $\Gamma$. There is a close analogy with the one-dimensional cubic splines, which is the reason for calling the solution a polyspline of order 2, or biharmonic polyspline. Similarly, when the quadratic functional is the integral of $(\Delta ^{q}f)^{2}, \, q$ a positive integer, then the solution is a polyharmonic function of order $2q, \, \Delta ^{2q}f(x) = 0,$ for $x \in D\setminus \Gamma$, satisfying matching conditions on $\Gamma$, and is called a polyspline of order $2q$. Uniqueness and existence for polysplines of order $2q$, provided that the interior boundaries $\Gamma$ are sufficiently smooth surfaces and $\partial D \subseteq \Gamma$, is proved. Three examples of data sets $\Gamma$ possessing symmetry are considered, in which the computation of polysplines is reduced to computation of one-dimensional $L-$splines.
The average edge order of triangulations of 3-manifolds with boundary
Makoto
Tamura
2129-2140
Abstract: Feng Luo and Richard Stong introduced the average edge order $\mu _0(K)$ of a triangulation $K$ and showed in particular that for closed 3-manifolds $\mu _0(K)$ being less than 4.5 implies that $K$ is on $S^3$. In this paper, we establish similar results for 3-manifolds with non-empty boundary; in particular it is shown that $\mu _0(K)$ being less than 4 implies that $K$ is on the 3-ball.
The residual spectrum of $U(2,2)$
Takyua
Kon-No
1285 - 1358
Covers of algebraic varieties III. The discriminant of a cover of degree 4 and the trigonal construction
G.
Casnati
1359-1378
Abstract: For each Gorenstein cover $\varrho \colon X\to Y$ of degree $4$ we define a scheme $\Delta (X)$ and a generically finite map $\Delta (\varrho )\colon \Delta (X)\to Y$ of degree $3$ called the discriminant of $\varrho$. Using this construction we deal with smooth degree $4$ covers $\varrho \colon X\to {{\mathbb P}^{n}_{\mathbb{C}}}$ with $n\ge 5$. Moreover we also generalize the trigonal construction of S. Recillas.
Geometric families of constant reductions and the Skolem property
Barry
Green
1379-1393
Abstract: Let $F|K$ be a function field in one variable and $\mathcal V$ be a family of independent valuations of the constant field $K.$ Given $v\in \mathcal V ,$ a valuation prolongation $\mathrm v$ to $F$ is called a constant reduction if the residue fields $F\mathrm v |Kv$ again form a function field of one variable. Suppose $t\in F$ is a non-constant function, and for each $v\in \mathcal V$ let $V_{t}$ be the set of all prolongations of the Gauß valuation $v_{t}$ on $K(t)$ to $F.$ The union of the sets $V_{t}$ over all $v\in \mathcal V$ is denoted by ${{\mathchoice {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}} {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptscriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}}}_{t}.$ The aim of this paper is to study families of constant reductions ${{\mathchoice {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}} {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptscriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}}}$ of $F$ prolonging the valuations of $\mathcal V$ and the criterion for them to be principal, that is to be sets of the type ${{\mathchoice {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}} {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptscriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}}}_{t}.$ The main result we prove is that if either $\mathcal V$ is finite and each $v\in \mathcal V$ has rational rank one and residue field algebraic over a finite field, or if $\mathcal V$ is any set of non-archimedean valuations of a global field $K$ satisfying the strong approximation property, then each geometric family of constant reductions ${{\mathchoice {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}} {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptscriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}} }$ prolonging $\mathcal V$ is principal. We also relate this result to the Skolem property for the existence of $\mathcal V$-integral points on varieties over $K,$ and Rumely's existence theorem. As an application we give a birational characterization of arithmetic surfaces $\mathcal X /S$ in terms of the generic points of the closed fibre. The characterization we give implies the existence of finite morphisms to $\mathbb P ^{1}_{S}.$
Extreme points of the distance function on convex surfaces
Tudor
Zamfirescu
1395-1406
Abstract: We first see that, in the sense of Baire categories, many convex surfaces have quite large cut loci and infinitely many relative maxima of the distance function from a point. Then we find that, on any convex surface, all these extreme points lie on a single subtree of the cut locus, with at most three endpoints. Finally, we confirm (both in the sense of measure and in the sense of Baire categories) Steinhaus' conjecture that ``almost all" points admit a single farthest point on the surface.
The behavior of the heat operator on weighted Sobolev spaces
G.
N.
Hile;
C.
P.
Mawata
1407-1428
Abstract: Denoting by ${\mathcal{H}}$ the heat operator in $R^{n+1}$, we investigate its properties as a bounded operator from one weighted Sobolev space to another. Our main result gives conditions on the weights under which ${\mathcal{H}}$ is an injection, a surjection, or an isomorphism. We also describe the range and kernel of ${\mathcal{H}}$ in all the cases. Our results are analogous to those obtained by R. C. McOwen for the Laplace operator in $R^{n}$.
Fox calculus, symplectic forms, and moduli spaces
Valentino
Zocca
1429-1466
Abstract: An ``open pre-symplectic form'' on surfaces with boundary and glueing formulae are provided to symplectically integrate the symplectic form on the deformation space of representations of the fundamental group of a Riemann surface into a reductive Lie group $G$.
Berezin's quantization on flag manifolds and spherical modules
Alexander
V.
Karabegov
1467-1479
Abstract: We show that the theory of spherical Harish-Chandra modules naturally gives rise to Berezin's symbol quantization on generalized flag manifolds. It provides constructions of symbol algebras and of their representations for covariant and contravariant symbols, and also for symbols which so far have no explicit definition. For all these symbol algebras we give a general proof of the correspondence principle.
Limit theorems for random transformations and processes in random environments
Yuri
Kifer
1481-1518
Abstract: I derive general relativized central limit theorems and laws of iterated logarithm for random transformations both via certain mixing assumptions and via the martingale differences approach. The results are applied to Markov chains in random environments, random subshifts of finite type, and random expanding in average transformations where I show that the conditions of the general theorems are satisfied and so the corresponding (fiberwise) central limit theorems and laws of iterated logarithm hold true in these cases. I consider also a continuous time version of such limit theorems for random suspensions which are continuous time random dynamical systems.
The trace of jet space $J^k\Lambda^\omega$ to an arbitrary closed subset of $\mathbb{R}^n$
Yuri
Brudnyi;
Pavel
Shvartsman
1519-1553
Abstract: The classical Whitney extension theorem describes the trace $J^k|_X$ of the space of $k$-jets generated by functions from $C^k(\mathbb R^n)$ to an arbitrary closed subset $X\subset\mathbb R^n$. It establishes existence of a bounded linear extension operator as well. In this paper we investigate a similar problem for the space $C^k\Lambda^\omega(\mathbb R^n)$ of functions whose higher derivatives satisfy the Zygmund condition with majorant $\omega$. The main result states that the vector function $\vec f=(f_\alpha \colon X\to\mathbb R)_{|\alpha |\le k}$ belongs to the corresponding trace space if the trace $\vec f|_Y$ to every subset $Y\subset X$ of cardinality $3\cdot 2^\ell$, where $\ell=(\begin{smallmatrix}n+k-1 k+1\end{smallmatrix})$, can be extended to a function $f_Y\in C^k\Lambda^\omega(\mathbb R^n)$ and $\sup _Y|f_Y|_{C^k\Lambda^\omega}<\infty$. The number $3\cdot 2^l$ generally speaking cannot be reduced. The Whitney theorem can be reformulated in this way as well, but with a two-pointed subset $Y\subset X$. The approach is based on the theory of local polynomial approximations and a result on Lipschitz selections of multivalued mappings.
Linearization, Dold-Puppe stabilization, and Mac Lane's $Q$-construction
Brenda
Johnson;
Randy
McCarthy
1555-1593
Abstract: In this paper we study linear functors, i.e., functors of chain complexes of modules which preserve direct sums up to quasi-isomorphism, in order to lay the foundation for a further study of the Goodwillie calculus in this setting. We compare the methods of Dold and Puppe, Mac Lane, and Goodwillie for producing linear approximations to functors, and establish conditions under which these methods are equivalent. In addition, we classify linear functors in terms of modules over an explicit differential graded algebra. Several classical results involving Dold-Puppe stabilization and Mac Lane's $Q$-construction are extended or given new proofs.
Commuting Toeplitz operators with pluriharmonic symbols
Dechao
Zheng
1595-1618
Abstract: By making use of $\mathcal M$-harmonic function theory, we characterize commuting Toeplitz operators with bounded pluriharmonic symbols on the Bergman space of the unit ball or on the Hardy space of the unit sphere in $n$-dimensional complex space.
$n$-unisolvent sets and flat incidence structures
Burkard
Polster
1619-1641
Abstract: For the past forty years or so topological incidence geometers and mathematicians interested in interpolation have been studying very similar objects. Nevertheless no communication between these two groups of mathematicians seems to have taken place during that time. The main goal of this paper is to draw attention to this fact and to demonstrate that by combining results from both areas it is possible to gain many new insights about the fundamentals of both areas. In particular, we establish the existence of nested orthogonal arrays of strength $n$, for short nested $n$-OAs, that are natural generalizations of flat affine planes and flat Laguerre planes. These incidence structures have point sets that are ``flat'' topological spaces like the Möbius strip, the cylinder, and strips of the form $I \times \mathbb{R}$, where $I$ is an interval of $\mathbb{R}$. Their circles (or lines) are subsets of the point sets homeomorphic to the circle in the first two cases and homeomorphic to $I$ in the last case. Our orthogonal arrays of strength $n$ arise from $n$-unisolvent sets of half-periodic functions, $n$-unisolvent sets of periodic functions, and $n$-unisolvent sets of functions $I\to \mathbb{R}$, respectively. Associated with every point $p$ of a nested $n$-OA, $n>1$, is a nested $(n-1)$-OA-the derived $(n-1)$-OA at the point $p$. We discover that, in our examples that arise from $n$-unisolvent sets of $n-1$ times differentiable functions that solve the Hermite interpolation problem, deriving in our geometrical sense coincides with deriving in the analytical sense.
Geometric properties of the double-point divisor
Bo
Ilic
1643-1661
Abstract: The locus of double points obtained by projecting a variety $X^{n} \subset \mathbf P^N$ to a hypersurface in $\mathbf{P}^{n+1}$ moves in a linear system which is shown to be ample if and only if $X$ is not an isomorphic projection of a Roth variety. Such Roth varieties are shown to exist, and some of their geometric properties are determined.
Comultiplications on free groups and wedges of circles
Martin
Arkowitz;
Mauricio
Gutierrez
1663-1680
Abstract: By means of the fundamental group functor, a co-H-space structure or a co-H-group structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group $F$. We consider individual comultiplications on $F$ and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of $F$. For a comultiplication $m$ of $F$ we define a subset $\Delta _{m} \subseteq F$ of quasi-diagonal elements which is basic to our investigation of associativity. The subset $\Delta _{m}$ can be determined algorithmically and contains the set of diagonal elements $D_{m}$. We show that $D_{m}$ is a basis for the largest subgroup $A_{m}$ of $F$ on which $m$ is associative and that $A_{m}$ is a free factor of $F$. We also give necessary and sufficient conditions for a comultiplication $m$ on $F$ to be a coloop in terms of the Fox derivatives of $m$ with respect to a basis of $F$. In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group $\operatorname{Aut} F$ on the set of comultiplications of $F$. We give many examples to illustrate these notions. We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles.
Algebraic invariant curves for the Liénard equation
Henryk
Zoladek
1681-1701
Abstract: Odani has shown that if $\deg g\leq \deg f$ then after deleting some trivial cases the polynomial system $\dot {x}=y,\,\,\dot {y}=-f(x)y-g(x)$ does not have any algebraic invariant curve. Here we almost completely solve the problem of algebraic invariant curves and algebraic limit cycles of this system for all values of $\deg f$ and $\deg g$. We give also a simple presentation of Yablonsky's example of a quartic limit cycle in a quadratic system.
A new degree bound for vector invariants of symmetric groups
P.
Fleischmann
1703-1712
Abstract: Let $R$ be a commutative ring, $V$ a finitely generated free $R$-module and $G\le GL_R(V)$ a finite group acting naturally on the graded symmetric algebra $A=S(V)$. Let $\beta(V,G)$ denote the minimal number $m$, such that the ring $A^G$ of invariants can be generated by finitely many elements of degree at most $m$. For $G=\Sigma _n$ and $V(n,k)$, the $k$-fold direct sum of the natural permutation module, one knows that $\beta(V(n,k),\Sigma _n) \le n$, provided that $n!$ is invertible in $R$. This was used by E. Noether to prove $\beta(V,G) \le |G|$ if $|G|! \in R^*$. In this paper we prove $\beta(V(n,k),\Sigma _n) \le max\{n,k(n-1)\}$ for arbitrary commutative rings $R$ and show equality for $n=p^s$ a prime power and $R = \mathbb Z$ or any ring with $n\cdot 1_R=0$. Our results imply \begin{equation*}\beta(V,G)\le max\{|G|, \operatorname{rank}(V)(|G|-1)\}\end{equation*} for any ring with $|G| \in R^*$.
Poisson transforms on vector bundles
An
Yang
857-887
Abstract: Let $G$ be a connected real semisimple Lie group with finite center, and $K$ a maximal compact subgroup of $G$. Let $(\tau,V)$ be an irreducible unitary representation of $K$, and $G\times _K\,V$ the associated vector bundle. In the algebra of invariant differential operators on $G\times _K\,V$ the center of the universal enveloping algebra of $\operatorname{Lie}(G)$ induces a certain commutative subalgebra $Z_\tau$. We are able to determine the characters of $Z_\tau$. Given such a character we define a Poisson transform from certain principal series representations to the corresponding space of joint eigensections. We prove that for most of the characters this map is a bijection, generalizing a famous conjecture by Helgason which corresponds to $\tau$ the trivial representation.
Algebraic transition matrices in the Conley index theory
Robert
Franzosa;
Konstantin
Mischaikow
889-912
Abstract: We introduce the concept of an algebraic transition matrix. These are degree zero isomorphisms which are upper triangular with respect to a partial order. It is shown that all connection matrices of a Morse decomposition for which the partial order is a series-parallel admissible order are related via a conjugation with one of these transition matrices. This result is then restated in the form of an existence theorem for global bifurcations. Simple examples of how these results can be applied are also presented.
Metric completions of ordered groups and $K_0$ of exchange rings
E.
Pardo
913-933
Abstract: We give a description of the closure of the natural affine continuous function representation of $K_0(R)$ for any exchange ring $R$. This goal is achieved by extending the results of Goodearl and Handelman, about metric completions of dimension groups, to a more general class of pre-ordered groups, which includes $K_0$ of exchange rings. As a consequence, the results about $K_0^+$ of regular rings, which the author gave in an earlier paper, can be extended to a wider class of rings, which includes $C^*$-algebras of real rank zero, among others. Also, the framework of pre-ordered groups developed here allows other potential applications.
Sobolev estimates for operators given by averages over cones
Scipio
Cuccagna
935-946
Abstract: We prove a result related to work by A. Greenleaf and G. Uhlmann concerning Sobolev estimates for operators given by averages over cones. This is done using the almost orthogonality lemma of Cotlar and Stein, and the van der Corput lemma on oscillatory integrals.
Equations in a free $\mathbf Q$-group
O.
Kharlampovich;
A.
Myasnikov
947-974
Abstract: An algorithm is constructed that decides if a given finite system of equations over a free $\mathbf{Q}$-group has a solution, and if it does, finds a solution.
Euler products associated to metaplectic automorphic forms on the 3-fold cover of ${GSp}(4)$
Thomas
Goetze
975-1011
Abstract: If $\phi$ is a generic cubic metaplectic form on GSp(4), that is also an eigenfunction for all the Hecke operators, then corresponding to $\phi$ is an Euler product of degree 4 that has a functional equation and meromorphic continuation to the whole complex plane. This correspondence is obtained by convolving $\phi$ with the cubic $\theta$-function on GL(3) in a Shimura type Rankin-Selberg integral.
Tight contact structures on solid tori
Sergei
Makar-Limanov
1013-1044
Abstract: In this paper we study properties of tight contact structures on solid tori. In particular we discuss ways of distinguishing two solid tori with tight contact structures. We also give examples of unusual tight contact structures on solid tori. We prove the existence of a $\mathbb{Z}$-valued and a $\mathbb{R}/2\pi\mathbb{Z}$-valued invariant of a closed solid torus. We call them the self-linking number and the rotation number respectively. We then extend these definitions to the case of an open solid torus. We show that these invariants exhibit certain monotonicity properties with respect to inclusion. We also prove a number of results which give sufficient conditions for two solid tori to be contactomorphic. At the same time we discuss various ways of constructing a tight contact structure on a solid torus. We then produce examples of solid tori with tight contact structures and calculate self-linking and rotation numbers for these tori. These examples show that the invariants we defined do not give a complete classification of tight contact structure on open solid tori. At the end, we construct a family of tight contact structure on a solid torus such that the induced contact structure on a finite-sheeted cover of that solid torus is no longer tight. This answers negatively a question asked by Eliashberg in 1990. We also give an example of tight contact structure on an open solid torus which cannot be contactly embedded into a sphere with the standard contact structure, another example of unexpected behavior.
On the elliptic equation $\Delta u+ku-Ku^p=0$ on complete Riemannian manifolds and their geometric applications
Peter
Li;
Luen-fai
Tam;
DaGang
Yang
1045-1078
Abstract: We study the elliptic equation $\Delta u + ku - Ku^{p} = 0$ on complete noncompact Riemannian manifolds with $K$ nonnegative. Three fundamental theorems for this equation are proved in this paper. Complete analyses of this equation on the Euclidean space ${\mathbf{R}}^{n}$ and the hyperbolic space ${\mathbf{H}}^{n}$ are carried out when $k$ is a constant. Its application to the problem of conformal deformation of nonpositive scalar curvature will be done in the second part of this paper.
Galois rigidity of pro-$l$ pure braid groups of algebraic curves
Hiroaki
Nakamura;
Naotake
Takao
1079-1102
Abstract: In this paper, Grothendieck's anabelian conjecture on the pro-$l$ fundamental groups of configuration spaces of hyperbolic curves is reduced to the conjecture on those of single hyperbolic curves. This is done by estimating effectively the Galois equivariant automorphism group of the pro-$l$ braid group on the curve. The process of the proof involves the complete determination of the groups of graded automorphisms of the graded Lie algebras associated to the weight filtration of the braid groups on Riemann surfaces.
Green's function, harmonic transplantation, and best Sobolev constant in spaces of constant curvature
C.
Bandle;
A.
Brillard;
M.
Flucher
1103-1128
Abstract: We extend the method of harmonic transplantation from Euclidean domains to spaces of constant positive or negative curvature. To this end the structure of the Green's function of the corresponding Laplace-Beltrami operator is investigated. By means of isoperimetric inequalities we derive complementary estimates for its distribution function. We apply the method of harmonic transplantation to the question of whether the best Sobolev constant for the critical exponent is attained, i.e. whether there is an extremal function for the best Sobolev constant in spaces of constant curvature. A fairly complete answer is given, based on a concentration-compactness argument and a Pohozaev identity. The result depends on the curvature.
On the equivariant Morse complex of the free loop space of a surface
Nancy
Hingston
1129-1141
Abstract: We prove two theorems about the equivariant topology of the free loop space of a surface. The first deals with the nondegenerate case and says that the ``ordinary'' Morse complex can be given an $O(2)$-action in such a way that it carries the $O(2)$-homotopy type of the free loop space. The second says that, in terms of topology, the iterates of an isolated degenerate closed geodesic ``look like'' the continuous limit of the iterates of a finite, fixed number of nondegenerate closed geodesics.
Dyadic equivalence to completely positive entropy
Adam
Fieldsteel;
J.
Roberto
Hasfura-Buenaga
1143-1166
Abstract: We show that every free ergodic action of $\bigoplus _1^\infty {\mathbb Z}_2$ of positive entropy is dyadically equivalent to an action with completely positive entropy.
The homological degree of a module
Wolmer
V.
Vasconcelos
1167-1179
Abstract: A homological degree of a graded module $M$ is an extension of the usual notion of multiplicity tailored to provide a numerical signature for the module even when $M$ is not Cohen-Macaulay. We construct a degree, $\operatorname{hdeg}(M)$, that behaves well under hyperplane sections and the modding out of elements of finite support. When carried out in a local algebra this degree gives a simulacrum of complexity à la Castelnuovo-Mumford's regularity. Several applications for estimating reduction numbers of ideals and predictions on the outcome of Noether normalizations are given.
Necessary conditions for optimal control problems with state constraints
R.
B.
Vinter;
H.
Zheng
1181-1204
Abstract: Necessary conditions of optimality are derived for optimal control problems with pathwise state constraints, in which the dynamic constraint is modelled as a differential inclusion. The novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid. An Euler Lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded, nonconvex sets and satisfies a mild `one-sided' Lipschitz continuity hypothesis. We recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by Ioffe. For problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one-sided Lipschitz hypothesis is replaced by a milder, local hypothesis. A recent `dualization' theorem permits us to infer a strengthened form of the Hamiltonian inclusion from the Euler Lagrange condition. The necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by Loewen and Rockafellar to achieve related necessary conditions for state constrained problems, and improve on available results in certain respects even when specialized to the state constraint free case. Proofs make use of recent `decoupling' ideas of the authors, which reduce the optimization problem to one to which Pontryagin's maximum principle is applicable, and a refined penalization technique to deal with the dynamic constraint.
Values of Gaussian hypergeometric series
Ken
Ono
1205-1223
Abstract: Let $p$ be prime and let $GF(p)$ be the finite field with $p$ elements. In this note we investigate the arithmetic properties of the Gaussian hypergeometric functions \begin{equation*}_{2}F_{1}(x)=_{2} F_{1} \left ( \begin{matrix}\phi , & \phi & \epsilon \end{matrix} | x \right ) \ {\text{\rm and}} \ _{3}F_{2}(x)= _{3}F_{2} \left ( \begin{matrix}\phi , & \phi , & \phi & \epsilon , & \epsilon \end{matrix} | x \right ), \end{equation*} where $\phi$ and $\epsilon$ respectively are the quadratic and trivial characters of $GF(p).$ For all but finitely many rational numbers $x=\lambda ,$ there exist two elliptic curves $_{2}E_{1}(\lambda )$ and $_{3}E_{2}(\lambda )$ for which these values are expressed in terms of the trace of the Frobenius endomorphism. We obtain bounds and congruence properties for these values. We also show, using a theorem of Elkies, that there are infinitely many primes $p$ for which $_{2}F_{1}(\lambda )$ is zero; however if $\lambda \neq -1,0, \frac{1}{2}$ or $2$, then the set of such primes has density zero. In contrast, if $\lambda \neq 0$ or $1$, then there are only finitely many primes $p$ for which $_{3}F_{2}(\lambda ) =0.$ Greene and Stanton proved a conjecture of Evans on the value of a certain character sum which from this point of view follows from the fact that $_{3}E_{2}(8)$ is an elliptic curve with complex multiplication. We completely classify all such CM curves and give their corresponding character sums in the sense of Evans using special Jacobsthal sums. As a consequence of this classification, we obtain new proofs of congruences for generalized Apéry numbers, as well as a few new ones, and we answer a question of Koike by evaluating $_{3}F_{2}(4)$ over every $GF(p).$
The possible orders of solutions of linear differential equations with polynomial coefficients
Gary
G.
Gundersen;
Enid
M.
Steinbart;
Shupei
Wang
1225-1247
Abstract: We find specific information about the possible orders of transcendental solutions of equations of the form $f^{(n)}+p_{n-1}(z)f^{(n-1)}+\cdots +p_{0}(z)f=0$, where $p_0(z), p_1(z),\dots, p_{n-1}(z)$ are polynomials with $p_0(z) \not\equiv 0$. Several examples are given.
Bilinear operators on Herz-type Hardy spaces
Loukas
Grafakos;
Xinwei
Li;
Dachun
Yang
1249-1275
Abstract: The authors prove that bilinear operators given by finite sums of products of Calderón-Zygmund operators on $\mathbb{R}^{n}$ are bounded from $H\dot K_{q_{1}}^{\alpha _{1},p_{1}}\times H\dot K_{q_{2}}^{\alpha _{2},p_{2}}$ into $H\dot K_{q}^{\alpha ,p}$ if and only if they have vanishing moments up to a certain order dictated by the target space. Here $H\dot K_{q}^{\alpha ,p}$ are homogeneous Herz-type Hardy spaces with $1/p=1/p_{1}+1/p_{2},$ $0<p_{i}\le \infty ,$ $1/q=1/q_{1}+1/q_{2},$ $1<q_{1},q_{2}<\infty ,$ $1\le q<\infty ,$ $\alpha =\alpha _{1}+\alpha _{2}$ and $-n/q_{i}<\alpha _{i}<\infty$. As an application they obtain that the commutator of a Calderón-Zygmund operator with a BMO function maps a Herz space into itself.
A classification theorem for Albert algebras
R.
Parimala;
R.
Sridharan;
Maneesh
L.
Thakur
1277-1284
Abstract: Let $k$ be a field whose characteristic is different from 2 and 3 and let $L/k$ be a quadratic extension. In this paper we prove that for a fixed, degree 3 central simple algebra $B$ over $L$ with an involution $\sigma$ of the second kind over $k$, the Jordan algebra $J(B,\sigma,u,\mu)$, obtained through Tits' second construction is determined up to isomorphism by the class of $(u,\mu)$ in $H^1(k,SU(B,\sigma))$, thus settling a question raised by Petersson and Racine. As a consequence, we derive a ``Skolem Noether'' type theorem for Albert algebras. We also show that the cohomological invariants determine the isomorphism class of $J(B,\sigma,u,\mu)$, if $(B,\sigma)$ is fixed.
Stability of multiple-pulse solutions
Björn
Sandstede
429-472
Abstract: In this article, stability of multiple-pulse solutions in semilinear parabolic equations on the real line is studied. A system of equations is derived which determines stability of $N$-pulses bifurcating from a stable primary pulse. The system depends only on the particular bifurcation leading to the existence of the $N$-pulses. As an example, existence and stability of multiple pulses are investigated if the primary pulse converges to a saddle-focus. It turns out that under suitable assumptions infinitely many $N$-pulses bifurcate for any fixed $N>1$. Among them are infinitely many stable ones. In fact, any number of eigenvalues between 0 and $N-1$ in the right half plane can be prescribed.
Comparison theorems and orbit counting in hyperbolic geometry
Mark
Pollicott;
Richard
Sharp
473-499
Abstract: In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a mixture of ideas from both ``thermodynamic'' ergodic theory and the automaton associated to strongly Markov groups.
Lévy processes in semisimple Lie groups and stability of stochastic flows
Ming
Liao
501-522
Abstract: We study the asymptotic stability of stochastic flows on compact spaces induced by Levy processes in semisimple Lie groups. It is shown that the Lyapunov exponents can be determined naturally in terms of root structure of the Lie group and there is an open subset whose complement has a positive codimension such that, after a random rotation, each of its connected components is shrunk to a single moving point exponentially under the flow.
Asymptotics for minimal discrete energy on the sphere
A.
B. J.
Kuijlaars;
E.
B.
Saff
523-538
Abstract: We investigate the energy of arrangements of $N$ points on the surface of the unit sphere $S^d$ in $\mathbf{R}^{d+1}$ that interact through a power law potential $V = 1/r^s ,$ where $s > 0$ and $r$ is Euclidean distance. With $\mathcal{E}_d(s,N)$ denoting the minimal energy for such $N$-point arrangements we obtain bounds (valid for all $N$) for $\mathcal{E}_d(s,N)$ in the cases when $0 < s < d$ and $2 \leq d < s$. For $s = d$, we determine the precise asymptotic behavior of $\mathcal{E}_d(d,N)$ as $N \rightarrow \infty$. As a corollary, lower bounds are given for the separation of any pair of points in an $N$-point minimal energy configuration, when $s \geq d \geq 2$. For the unit sphere in $\mathbf{R}^3$ $(d = 2)$, we present two conjectures concerning the asymptotic expansion of $\mathcal{E}_2(s,N)$ that relate to the zeta function $\zeta _L(s)$ for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the first of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of $\zeta _L(s)$ when $0 < s < 2$ (the divergent case).
Extremal vectors and invariant subspaces
Shamim
Ansari;
Per
Enflo
539-558
Abstract: For a bounded linear operator on Hilbert space we define a sequence of so-called minimal vectors in connection with invariant subspaces and show that this presents a new approach to invariant subspaces. In particular, we show that for any compact operator $K$ some weak limit of the sequence of minimal vectors is noncyclic for all operators commuting with $K$ and that for any normal operator $N$, the norm limit of the sequence of minimal vectors is noncyclic for all operators commuting with $N$. Thus, we give a new and more constructive proof of existence of invariant subspaces. The sequence of minimal vectors does not seem to converge in norm for an arbitrary bounded linear operator. We will prove that if $T$ belongs to a certain class ${\mathcal{C}}$ of operators, then the sequence of such vectors converges in norm, and that if $T$ belongs to a subclass of ${\mathcal{C}}$, then the norm limit is cyclic.
Relativity of the spectrum and discrete groups on hyperbolic spaces
N.
Mandouvalos
559-569
Abstract: We give a simple proof of the analytic continuation of the resolvent kernel for a convex cocompact Kleinian group.
Hyperbolic groups and free constructions
O.
Kharlampovich;
A.
Myasnikov
571-613
Abstract: It is proved that the property of a group to be hyperbolic is preserved under HHN-extensions and amalgamated free products provided the associated (amalgamated) subgroups satisfy certain conditions. Some more general results about the preservation of hyperbolicity under graph products are also obtained. Using these results we describe the $\mathbf{Q}$-completion $(\mathbf{Q}$ is the field of rationals) $G^{\mathbf{Q}}$ of a torsion-free hyperbolic group $G$ as a union of an effective chain of hyperbolic subgroups, and solve the conjugacy problem in $G^{\mathbf{Q}}$.
The approximate functional formula for the theta function and Diophantine Gauss sums
E.
A.
Coutsias;
N.
D.
Kazarinoff
615-641
Abstract: We consider the polygonal lines in the complex plane $\Bbb{C}$ whose $N$-th vertex is defined by $S_N = \sum _{n=0}^{N\,'} \exp(i\omega \pi n^2)$ (with $\omega \in \Bbb{R}$), where the prime means that the first and last terms in the sum are halved. By introducing the discrete curvature of the polygonal line, and by exploiting the similarity of segments of the line, for small $\omega$, to Cornu spirals (C-spirals), we prove the precise renormalization formula \begin{equation}\begin{split} &\left| \sum _{k=0}^{N}\,' \exp(i\omega \pi k^2) -\frac{\exp(sgn(\omega )i\pi /4)}{\sqrt{|\omega |}} \sum _{k=0}^n \,' \exp(-i\frac{\pi}{\omega} k^2)\right| &\qquad\leq C \left|\frac{\omega N - n}{\omega}\right|, \ 0<|\omega | <1, \end{split} \end{equation} where $N=[[n/\omega]]$, the nearest integer to $n/\omega$ and $1<C<3.14$ . This formula, which sharpens Hardy and Littlewood's approximate functional formula for the theta function, generalizes to irrationals, as a Diophantine inequality, the well-known sum formula of Gauss. The geometrical meaning of the relation between the two limits is that the first sum is taken to a point of inflection of the corresponding C-spirals. The second sum replaces whole C-spirals of the first by unit vectors times scale and phase factors. The block renormalization procedure implied by this replacement is governed by the circle map \begin{equation}\omega \rightarrow -\frac{1}{\omega} \pmod 2 , \omega \in ]-1,+1[ \setminus \{0\}, \end{equation} whose orbits are analyzed by expressing $\omega$ as an even continued fraction.
Derivations, isomorphisms, and second cohomology of generalized Witt algebras
Dragomir
Z. DJ
Okovic;
Kaming
Zhao
643-664
Abstract: Generalized Witt algebras, over a field $F$ of characteristic $0$, were defined by Kawamoto about 12 years ago. Using different notations from Kawamoto's, we give an essentially equivalent definition of generalized Witt algebras $W=W(A,T,\varphi)$ over $F$, where the ingredients are an abelian group $A$, a vector space $T$ over $F$, and a map $\varphi:T\times A\to K$ which is linear in the first variable and additive in the second one. In this paper, the derivations of any generalized Witt algebra $W=$$W(A,T,\varphi)$, with the right kernel of $\varphi$ being $0$, are explicitly described; the isomorphisms between any two simple generalized Witt algebras are completely determined; and the second cohomology group $H^2(W,F)$ for any simple generalized Witt algebra is computed. The derivations, the automorphisms and the second cohomology groups of some special generalized Witt algebras have been studied by several other authors as indicated in the references.
On Non-hyperbolic Quasi-convex Spaces
Rafael
Oswaldo
Ruggiero
665-687
Abstract: We show that if the universal covering of a compact Riemannian manifold with no conjugate points is a quasi-convex metric space then the following assertion holds: Either the universal covering of the manifold is a hyperbolic geodesic space or it contains a quasi-isometric immersion of $Z\times Z$.
Comparing Heegaard splittings -the bounded case
Hyam
Rubinstein;
Martin
Scharlemann
689-715
Abstract: In a recent paper we used Cerf theory to compare strongly irreducible Heegaard splittings of the same closed irreducible orientable 3-manifold. This captures all irreducible splittings of non-Haken 3-manifolds. One application is a solution to the stabilization problem for such splittings: If $p \leq q$ are the genera of two splittings, then there is a common stabilization of genus $5p + 8q - 9$. Here we show how to obtain similar results even when the 3-manifold has boundary.
Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials
Feliks
Przytycki
717-742
Abstract: We prove that for every rational map on the Riemann sphere $f:\overline{\mathbb{C}} \to \overline{\mathbb{C}}$, if for every $f$-critical point $c\in J$ whose forward trajectory does not contain any other critical point, the growth of $|(f^{n})'(f(c))|$ is at least of order $\exp Q \sqrt n$ for an appropriate constant $Q$ as $n\to \infty$, then $\operatorname{HD}_{\operatorname {ess}} (J)=\alpha _{0}=\operatorname{HD} (J)$. Here $\operatorname{HD}_{\operatorname {ess}} (J)$ is the so-called essential, dynamical or hyperbolic dimension, $\operatorname{HD} (J)$ is Hausdorff dimension of $J$ and $\alpha _{0}$ is the minimal exponent for conformal measures on $J$. If it is assumed additionally that there are no periodic parabolic points then the Minkowski dimension (other names: box dimension, limit capacity) of $J$ also coincides with $\operatorname{HD}(J)$. We prove ergodicity of every $\alpha$-conformal measure on $J$ assuming $f$ has one critical point $c\in J$, no parabolic, and $\sum _{n=0}^{\infty }|(f^{n})'(f(c))|^{-1} <\infty$. Finally for every $\alpha$-conformal measure $\mu$ on $J$ (satisfying an additional assumption), assuming an exponential growth of $|(f^{n})'(f(c))|$, we prove the existence of a probability absolutely continuous with respect to $\mu$, $f$-invariant measure. In the Appendix we prove $\operatorname{HD}_{\operatorname {ess}} (J)=\operatorname {HD} (J)$ also for every non-renormalizable quadratic polynomial $z\mapsto z^{2}+c$ with $c$ not in the main cardioid in the Mandelbrot set.
$L^p$ and operator norm estimates for the complex time heat operator on homogeneous trees
Alberto
G.
Setti
743-768
Abstract: Let $\mathfrak{X}$ be a homogeneous tree of degree greater than or equal to three. In this paper we study the complex time heat operator ${\mathcal{H}}_{\zeta }$ induced by the natural Laplace operator on $\mathfrak{X}$. We prove comparable upper and lower bounds for the $L^{p}$ norms of its convolution kernel $h_{\zeta }$ and derive precise estimates for the $L^{p}\text{--}L^{r}$ operator norms of ${\mathcal{H}}_{\zeta }$ for $\zeta$ belonging to the half plane $\text{Re}\,\zeta \geq 0.$ In particular, when $\zeta$ is purely imaginary, our results yield a description of the mapping properties of the Schrödinger semigroup on $\mathfrak{X}$.
Contiguous relations, continued fractions and orthogonality
Dharma
P.
Gupta;
David
R.
Masson
769-808
Abstract: We examine a special linear combination of balanced very-well-poised ${_{10} \phi _{9}}$ basic hypergeometric series that is known to satisfy a transformation. We call this $\Phi$ and show that it satisfies certain three-term contiguous relations. From two of these contiguous relations for $\Phi$ we obtain fifty-six pairwise linearly independent solutions to a three-term recurrence that generalizes the recurrence for Askey-Wilson polynomials. The associated continued fraction is evaluated using Pincherle's theorem. From this continued fraction we are able to derive a discrete system of biorthogonal rational functions. This ties together Wilson's results for rational biorthogonality, Watson's $q$-analogue of Ramanujan's Entry 40 continued fraction, and a conjecture of Askey concerning the latter. Some new $q$-series identities are also obtained. One is an important three-term transformation for $\Phi$'s which generalizes all the known two- and three-term ${_{8} \phi _{7}}$ transformations. Others are new and unexpected quadratic identities for these very-well-poised ${_{8} \phi _{7}}$'s.
Recognizing constant curvature discrete groups in dimension 3
J.
W.
Cannon;
E.
L.
Swenson
809-849
Abstract: We characterize those discrete groups $G$ which can act properly discontinuously, isometrically, and cocompactly on hyperbolic $3$-space ${\mathbb H}^3$ in terms of the combinatorics of the action of $G$ on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the Sullivan-Tukia theorem on groups which act uniformly quasiconformally on the $2$-sphere.
Erratum to ``Orthogonal calculus''
Michael
S.
Weiss
851-855
Minimal sets and varieties
Keith
A.
Kearnes;
Emil
W.
Kiss;
Matthew
A.
Valeriote
1-41
Abstract: The aim of this paper is twofold. First some machinery is established to reveal the structure of abelian congruences. Then we describe all minimal, locally finite, locally solvable varieties. For locally solvable varieties, this solves problems 9 and 10 of Hobby and McKenzie. We generalize part of this result by proving that all locally finite varieties generated by nilpotent algebras that have a trivial locally strongly solvable subvariety are congruence permutable.
Decomposition theorems and approximation by a ``floating" system of exponentials
E.
S.
Belinskii
43-53
Abstract: The main problem considered in this paper is the approximation of a trigonometric polynomial by a trigonometric polynomial with a prescribed number of harmonics. The method proposed here gives an opportunity to consider approximation in different spaces, among them the space of continuous functions, the space of functions with uniformly convergent Fourier series, and the space of continuous analytic functions. Applications are given to approximation of the Sobolev classes by trigonometric polynomials with prescribed number of harmonics, and to the widths of the Sobolev classes. This work supplements investigations by Maiorov, Makovoz and the author where similar results were given in the integral metric.
Rumely's local global principle for algebraic ${\mathrm P}{\mathcal S}{\mathrm C}$ fields over rings
Moshe
Jarden;
Aharon
Razon
55-85
Abstract: Let $\mathcal{S}$ be a finite set of rational primes. We denote the maximal Galois extension of $\mathbb{Q}$ in which all $p\in \mathcal{S}$ totally decompose by $N$. We also denote the fixed field in $N$ of $e$ elements $\sigma _{1},\ldots , \sigma _{e}$ in the absolute Galois group $G( \mathbb{Q})$ of $\mathbb{Q}$ by $N( {\boldsymbol \sigma })$. We denote the ring of integers of a given algebraic extension $M$ of $\mathbb{Q}$ by $\mathbb{Z}_{M}$. We also denote the set of all valuations of $M$ (resp., which lie over $S$) by $\mathcal{V}_{M}$ (resp., $\mathcal{S}_{M}$). If $v\in \mathcal{V}_{M}$, then $O_{M,v}$ denotes the ring of integers of a Henselization of $M$ with respect to $v$. We prove that for almost all ${\boldsymbol \sigma }\in G( \mathbb{Q})^{e}$, the field $M=N( {\boldsymbol \sigma })$ satisfies the following local global principle: Let $V$ be an affine absolutely irreducible variety defined over $M$. Suppose that $V(O_{M,v})\not =\varnothing$ for each $v\in \mathcal{V}_{M}\backslash \mathcal{S}_{M}$ and $V_{\mathrm{sim}}(O_{M,v})\not =\varnothing$ for each $v\in \mathcal{S}_{M}$. Then $V(O_{M})\not =\varnothing$. We also prove two approximation theorems for $M$.
Realizing homology boundary links with arbitrary patterns
Paul
Bellis
87-100
Abstract: Homology boundary links have become an increasingly important class of links, largely due to their significance in the ongoing concordance classification of links. Tim Cochran and Jerome Levine defined an algebraic object called a pattern associated to a homology boundary link which can be used to study the deviance of a homology boundary link from being a boundary link. Since a pattern is a set of $m$ elements which normally generates the free group of rank $m$, any invariants which detect non-trivial patterns can be applied to the purely algebraic question of when such a set is a set of conjugates of a generating set for the free group. We will give a constructive geometric proof that all patterns are realized by some homology boundary link $L^n$ in $S^{n+2}$. We shall also prove an analogous existence theorem for calibrations of $\mathbb{E}$-links, a more general and less understood class of links tha homology boundary links.
Weighted ergodic theorems for mean ergodic $L_1$-contractions
Dogan
Çömez;
Michael
Lin;
James
Olsen
101-117
Abstract: It is shown that any bounded weight sequence which is good for all probability preserving transformations (a universally good weight) is also a good weight for any $L_{1}$-contraction with mean ergodic (ME) modulus, and for any positive contraction of $L_{p}$ with $1 < p <\infty$. We extend the return times theorem by proving that if $S$ is a Dunford-Schwartz operator (not necessarily positive) on a Lebesgue space, then for any $g$ bounded measurable $\{S^{n} g(\omega )\}$ is a universally good weight for a.e. $\omega .$ We prove that if a bounded sequence has "Fourier coefficents", then its weighted averages for any $L_{1}$-contraction with mean ergodic modulus converge in $L_{1}$-norm. In order to produce weights, good for weighted ergodic theorems for $L_{1}$-contractions with quasi-ME modulus (i.e., so that the modulus has a positive fixed point supported on its conservative part), we show that the modulus of the tensor product of $L_{1}$-contractions is the product of their moduli, and that the tensor product of positive quasi-ME $L_{1}$-contractions is quasi-ME.
A condition on the value function both necessary and sufficient for full regularity of minimizers of one-dimensional variational problems
M.
A.
Sychev;
V.
J.
Mizel
119-133
Abstract: We study two-point Lagrange problems for integrands $L= L(t,u,v)$: \begin{equation}\begin{split} F[u]=\int _a^b L(t,u(t),\dot u(t))&\,dt \to \inf, & u\in\mathcal A=\{v\in W^{1,1} ([a,b];\mathbb R^n)|v(a)=A,v(b)=B\}. \end{split}\tag{P}\label{tagp} \end{equation} Under very weak regularity hypotheses [$L$ is Hölder continuous and locally elliptic on each compact subset of $\mathbb R\times\mathbb R^n\times\mathbb R^n$] we obtain, when $L$ is of superlinear growth in $v$, a characterization of problems in which the minimizers of (P) are $C^1$-regular for all boundary data. This characterization involves the behavior of the value function $S$: $\mathbb R\times\mathbb R^n\times\mathbb R\times\mathbb R^n\to\mathbb R$ defined by $S(a,A,b,B)=\inf _{\mathcal A} F$. Namely, all minimizers for (P) are $C^1$-regular in neighborhoods of $a$ and $b$ if and only if $S$ is Lipschitz continuous at $(a,A,b,B)$. Consequently problems (P) possessing no singular minimizers are characterized in cases where not even a weak form of the Euler-Lagrange equations is available for guidance. Full regularity results for problems where $L$ is nearly autonomous, nearly independent of $u$, or jointly convex in $(u,v)$ are presented.
Divisor spaces on punctured Riemann surfaces
Sadok
Kallel
135-164
Abstract: In this paper, we study the topology of spaces of $n$-tuples of positive divisors on (punctured) Riemann surfaces which have no points in common (the divisor spaces). These spaces arise in connection with spaces of based holomorphic maps from Riemann surfaces to complex projective spaces. We find that there are Eilenberg-Moore type spectral sequences converging to their homology. These spectral sequences collapse at the $E^2$ term, and we essentially obtain complete homology calculations. We recover for instance results of F. Cohen, R. Cohen, B. Mann and J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), 163-221. We also study the homotopy type of certain mapping spaces obtained as a suitable direct limit of the divisor spaces. These mapping spaces, first considered by G. Segal, were studied in a special case by F. Cohen, R. Cohen, B. Mann and J. Milgram, who conjectured that they split. In this paper, we show that the splitting does occur provided we invert the prime two.
Factorisation in nest algebras. II
M.
Anoussis;
E.
G.
Katsoulis
165-183
Abstract: The main result of this paper is Theorem 5, which provides a necessary and sufficient condition on a positive operator $A$ for the existence of an operator $B$ in the nest algebra $AlgN$ of a nest $N$ satisfying $A=BB^{*}$ (resp. $A=B^{*}B)$. In Section 3 we give a new proof of a result of Power concerning outer factorisation of operators. We also show that a positive operator $A$ has the property that there exists for every nest $N$ an operator $B_N$ in $AlgN$ satisfying $A=B_NB_N^{*}$ (resp. $A=B_N^{*}B_N$) if and only if $A$ is a Fredholm operator. In Section 4 we show that for a given operator $A$ in $B(H)$ there exists an operator $B$ in $AlgN$ satisfying $AA^{*}=BB^{*}$ if and only if the range $r(A)$ of $A$ is equal to the range of some operator in $AlgN$. We also determine the algebraic structure of the set of ranges of operators in $AlgN$. Let $F_r(N)$ be the set of positive operators $A$ for which there exists an operator $B$ in $AlgN$ satisfying $A=BB^{*}$. In Section 5 we obtain information about this set. In particular we discuss the following question: Assume $A$ and $B$ are positive operators such that $A\leq B$ and $A$ belongs to $F_r(N)$. Which further conditions permit us to conclude that $B$ belongs to $F_r(N)$?
New subfactors from braid group representations
Juliana
Erlijman
185-211
Abstract: This paper is about the construction of new examples of pairs of subfactors of the hyperfinite II$_{1}$ factor, and the computation of their indices and relative commutants. The construction is done in general by considering unitary braid representations with certain properties that are satisfied in natural examples. We compute the indices explicitly for the particular cases in which the braid representations are obtained in connection with representation theory of Lie algebras of types A,B,C,D.
Filling by holomorphic curves in symplectic 4-manifolds
Rugang
Ye
213-250
Abstract: We develop a general framework for embedded (immersed) $J$-holomorphic curves and a systematic treatment of the theory of filling by holomorphic curves in 4-dimensional symplectic manifolds. In particular, a deformation theory and an intersection theory for $J$-holomorphic curves with boundary are developed. Bishop's local filling theorem is extended to almost complex manifolds. Existence and uniqueness of global fillings are given complete proofs. Then they are extended to the situation with nontrivial $J$-holomorphic spheres, culminating in the construction of singular fillings.
Geometry of families of nodal curves on the blown-up projective plane
Gert-Martin
Greuel;
Christoph
Lossen;
Eugenii
Shustin
251-274
Abstract: Let $\mathbb P^2_r\,$ be the projective plane blown up at $r$ generic points. Denote by $E_0,E_1,\ldots ,E_r$ the strict transform of a generic straight line on $\mathbb P^2$ and the exceptional divisors of the blown-up points on $\mathbb P^2_r$ respectively. We consider the variety $V_{irr}(d;\,d_1,\ldots,d_r;\,k)$ of all irreducible curves $C$ in $|dE_0-\sum _{i=1}^{r} d_iE_i|$ with $k$ nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility to families of reducible curves. For $r\leq 9$ we give the complete answer concerning the existence of nodal curves in $V_{irr}(d;\,d_1,\ldots,d_r;\,k)$.
On the classification of irregular surfaces of general type with nonbirational bicanonical map
Fabrizio
Catanese;
Ciro
Ciliberto;
Margarida
Mendes
Lopes
275-308
Abstract: The present paper is devoted to the classification of irregular surfaces of general type with $p_{g}\geq 3$ and nonbirational bicanonical map. Our main result is that, if $S$ is such a surface and if $S$ is minimal with no pencil of curves of genus $2$, then $S$ is the symmetric product of a curve of genus $3$, and therefore $p_{g}=q=3$ and $K^{2}=6$. Furthermore we obtain some results towards the classification of minimal surfaces with $p_{g}=q=3$. Such surfaces have $6\leq K^{2}\leq 9$, and we show that $K^{2}=6$ if and only if $S$ is the symmetric product of a curve of genus $3$. We also classify the minimal surfaces with $p_{g}=q=3$ with a pencil of curves of genus $2$, proving in particular that for those one has $K^{2}=8$.
Wandering vectors for irrational rotation unitary systems
Deguang
Han
309-320
Abstract: An abstract characterization for those irrational rotation unitary systems with complete wandering subspaces is given. We prove that an irrational rotation unitary system has a complete wandering vector if and only if the von Neumann algebra generated by the unitary system is finite and shares a cyclic vector with its commutant. We solve a factorization problem of Dai and Larson negatively for wandering vector multipliers, and strengthen this by showing that for an irrational rotation unitary system $\mathcal{U}$, every unitary operator in $w^{*}(\mathcal{U})$ is a wandering vector multiplier. Moreover, we show that there is a class of wandering vector multipliers, induced in a natural way by pairs of characters of the integer group $\mathbb{Z}$, which fail to factor even as the product of a unitary in $\mathcal{U}'$ and a unitary in $w^{*}(\mathcal{U})$. Incomplete maximal wandering subspaces are also considered, and some questions are raised.
Widths of Subgroups
Rita
Gitik;
Mahan
Mitra;
Eliyahu
Rips;
Michah
Sageev
321-329
Abstract: We say that the width of an infinite subgroup $H$ in $G$ is $n$ if there exists a collection of $n$ essentially distinct conjugates of $H$ such that the intersection of any two elements of the collection is infinite and $n$ is maximal possible. We define the width of a finite subgroup to be $0$. We prove that a quasiconvex subgroup of a negatively curved group has finite width. It follows that geometrically finite surfaces in closed hyperbolic $3$-manifolds satisfy the $k$-plane property for some $k$.
Sur la multiplicité de la première valeur propre de l'opérateur de Schrödinger avec champ magnétique sur la sphère
Gérard
Besson;
Bruno
Colbois;
Gilles
Courtois
331-345
Abstract: L'objet de cet article est d'étudier la multiplicité de la première valeur propre de l'opérateur de Schrödinger avec champ magnétique sur la sphère $S^{2}$, et, répondant en cela à une question posée par Y. Colin de Verdière, de montrer d'une part que cette multiplicité peut être arbitrairement grande, mais que, d'autre part, elle est toujours bornée en fonction de la courbure de la connexion associée.ABSTRACT. The purpose of this text is to study the first eigenvalue of the Schrödinger operator with magnetic field on the 2-sphere and to show that its multiplicity can be arbitrarily high. We also show that this multiplicity is bounded in terms of the curvature of the corresponding connection. This answers a question asked by Y. Colin de Verdière.
On differential equations for Sobolev-type Laguerre polynomials
J.
Koekoek;
R.
Koekoek;
H.
Bavinck
347-393
Abstract: The Sobolev-type Laguerre polynomials $\{L_n^{\alpha,M,N}(x)\}_{n=0}^{\infty}$ are orthogonal with respect to the inner product \begin{displaymath}\langle f,g\rangle\;=\frac{1}{\Gamma(\alpha+1)}\int _0^{\infty}x^{\alpha}e^{-x}f(x)g(x)dx+Mf(0)g(0)+ Nf'(0)g'(0),\end{displaymath} where $\alpha>-1$, $M\ge 0$ and $N\ge 0$. In 1990 the first and second author showed that in the case $M>0$ and $N=0$ the polynomials are eigenfunctions of a unique differential operator of the form \begin{displaymath}M\sum _{i=1}^{\infty}a_i(x)D^i+xD^2+(\alpha+1-x)D,\end{displaymath} where $\left\{a_i(x)\right\}_{i=1}^{\infty}$ are independent of $n$. This differential operator is of order $2\alpha+4$ if $\alpha$ is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form \begin{align}&M\sum _{i=0}^{\infty}a_i(x)y^{(i)}(x)+ N\sum _{i=0}^{\infty}b_i(x)y^{(i)}(x)\nonumber &\hspace{1cm}{}+MN\sum _{i=0}^{\infty}c_i(x)y^{(i)}(x)+ xy''(x)+(\alpha +1-x)y'(x)+ny(x)=0,\nonumber \end{align} where the coefficients $\left\{a_i(x)\right\}_{i=1}^{\infty}$, $\left\{b_i(x)\right\}_{i=1}^{\infty}$ and $\left\{c_i(x)\right\}_{i=1}^{\infty}$ are independent of $n$ and the coefficients $a_0(x)$, $b_0(x)$ and $c_0(x)$ are independent of $x$, satisfied by the Sobolev-type Laguerre polynomials $\{L_n^{\alpha,M,N}(x)\}_{n=0}^{\infty}$. Further, we show that in the case $M=0$ and $N>0$ the polynomials are eigenfunctions of a linear differential operator, which is of order $2\alpha+8$ if $\alpha$ is a nonnegative integer and of infinite order otherwise. Finally, we show that in the case $M>0$ and $N>0$ the polynomials are eigenfunctions of a linear differential operator, which is of order $4\alpha+10$ if $\alpha$ is a nonnegative integer and of infinite order otherwise.
Double Walsh series with coefficients of bounded variation of higher order
Chang-Pao
Chen;
Ching-Tang
Wu
395-417
Abstract: Let $D_{j}^{k}(x)$ denote the Cesàro sums of order $k$ of the Walsh functions. The estimates of $D_{j}^{k}(x)$ given by Fine back in 1949 are extended to the case $k>2$. As a corollary, the following properties are established for the rectangular partial sums of those double Walsh series whose coefficients satisfy conditions of bounded variation of order $(p,0), (0,p)$, and $(p,p)$ for some $p\ge 1$: (a) regular convergence; (b) uniform convergence; (c) $L^{r}$-integrability and $L^{r}$-metric convergence for $0<r<1/p$; and (d) Parseval's formula. Extensions to those with coefficients of generalized bounded variation are also derived.
Local Boundary Regularity of the Szego Projection and Biholomorphic Mappings of Non-Pseudoconvex Domains
Peiming
Ma
419-428
Abstract: It is shown that the Szeg\H{o} projection $S$ of a smoothly bounded domain $\Omega$, not necessarily pseudoconvex, satisfies local regularity estimates at certain boundary points, provided that condition $R$ holds for $\Omega$. It is also shown that any biholomorphic mapping $f:\Omega \rightarrow D$ between smoothly bounded domains extends smoothly near such points, provided that a weak regularity assumption holds for $D$.
The Jacobson radical of group rings of locally finite groups
D.
S.
Passman
4693-4751
Abstract: This paper is the final installment in a series of articles, started in 1974, which study the semiprimitivity problem for group algebras $K[G]$ of locally finite groups. Here we achieve our goal of describing the Jacobson radical ${\mathcal{J}}K[{G}]$ in terms of the radicals ${\mathcal{J}}K[{A}]$ of the group algebras of the locally subnormal subgroups $A$ of $G$. More precisely, we show that if $\operatorname{char} K=p>0$ and if $\mathbb{O}_{p}(G)=1$, then the controller of ${\mathcal{J}}K[{G}]$ is the characteristic subgroup $\mathbb{S}^{p}(G)$ generated by the locally subnormal subgroups $A$ of $G$ with $A=\mathbb{O}^{p'}(A)$. In particular, we verify a conjecture proposed some twenty years ago and, in so doing, we essentially solve one half of the group ring semiprimitivity problem for arbitrary groups. The remaining half is the more difficult case of finitely generated groups. This article is effectively divided into two parts. The first part, namely the material in Sections 2-6, covers the group theoretic aspects of the proof and may be of independent interest. The second part, namely the work in Sections 7-12, contains the group ring and ring theoretic arguments and proves the main result. As usual, it is necessary for us to work in the more general context of twisted group algebras and crossed products. Furthermore, the proof ultimately depends upon results which use the Classification of the Finite Simple Groups.
Nonexistence and uniqueness of positive solutions of Yamabe type equations on nonpositively curved manifolds
Bruno
Bianchini;
Marco
Rigoli
4753-4774
Abstract: We prove nonexistence and uniqueness of positive $C^{2}$-solutions of the elliptic equation $\Delta u +a(x)u - K(x)u^{\sigma }=0$, $ \sigma >1$, on a nonpositively curved, complete manifold $(M,g)$ .
Restriction of stable bundles in characteristic $p$
Tohru
Nakashima
4775-4786
Abstract: Let $k$ be an algebraically closed field of characteristic $p>0$. Let $X$ be a nonsingular projective variety defined over $k$ and $H$ an ample line bundle on $X$. We shall prove that there exists an explicit number $m_{0}$ such that if $E$ is a $\mu$-stable vector bundle of rank at most three, then the restriction $E_{\vert D}$ is $\mu$-stable for all $m\geq m_{0}$ and all smooth irreducible divisors $D\in \vert mH\vert$. This result has implications to the geometry of the moduli space of $\mu$-stable bundles on a surface or a projective space.
A theorem of the Dore-Venni type for noncommuting operators
Sylvie
Monniaux;
Jan
Prüss
4787-4814
Abstract: A theorem of the Dore-Venni type for the sum of two closed linear operators is proved, where the operators are noncommuting but instead satisfy a certain commutator condition. This result is then applied to obtain optimal regularity results for parabolic evolution equations $\dot{u}(t)+L(t)u(t)=f(t)$ and evolutionary integral equations $u(t)+\int _0^ta(t-s)L(s)u(s)ds = g(t)$ which are nonautonomous. The domains of the involved operators $L(t)$ may depend on $t$, but $L(t)^{-1}$ is required to satisfy a certain smoothness property. The results are then applied to parabolic partial differential and integro-differential equations.
On measures ergodic with respect to an analytic equivalence relation
Alain
Louveau;
Gabriel
Mokobodzki
4815-4823
Abstract: In this paper, we prove that the set of probability measures which are ergodic with respect to an analytic equivalence relation is an analytic set. This is obtained by approximating analytic equivalence relations by measures, and is used to give an elementary proof of an ergodic decomposition theorem of Kechris.
Lexicographic TAF Algebras
Justin
R.
Peters;
Yiu
Tung
Poon
4825-4855
Abstract: Lexicographic TAF algebras constitute a class of triangular AFalgebras which are determined by a countable ordered set $\Omega$, a dimension function, and a third parameter. While some of the important examples of TAF algebras belong to the class, most algebras in this class have not been studied. The semigroupoid of the algebra, the lattice of invariant projections, the Jacobson radical, and for some cases the automorphism group are computed. Necessary and sufficient conditions for analyticity are given. The results often involve the order properties of the set $\Omega$.
Frobenius extensions of subalgebras of Hopf algebras
D.
Fischman;
S.
Montgomery;
H.-J.
Schneider
4857-4895
Abstract: We consider when extensions $S\subset R$ of subalgebras of a Hopf algebra are $\beta$-Frobenius, that is Frobenius of the second kind. Given a Hopf algebra $H$, we show that when $S\subset R$ are Hopf algebras in the Yetter-Drinfeld category for $H$, the extension is $\beta$-Frobenius provided $R$ is finite over $S$ and the extension of biproducts $S\star H\subset R\star H$ is cleft. More generally we give conditions for an extension to be $\beta$-Frobenius; in particular we study extensions of integral type, and consider when the Frobenius property is inherited by the subalgebras of coinvariants. We apply our results to extensions of enveloping algebras of Lie coloralgebras, thus extending a result of Bell and Farnsteiner for Lie superalgebras.
Dehn surgery on knots in solid tori creating essential annuli
Chuichiro
Hayashi;
Kimihiko
Motegi
4897-4930
Abstract: Let $M$ be a $3$-manifold obtained by performing a Dehn surgery on a knot in a solid torus. In the present paper we study when $M$ contains a separating essential annulus. It is shown that $M$ does not contain such an annulus in the majority of cases. As a corollary, we prove that symmetric knots in the $3$-sphere which are not periodic knots of period $2$ satisfy the cabling conjecture. This is an improvement of a result of Luft and Zhang. We have one more application to a problem on Dehn surgeries on knots producing a Seifert fibred manifold over the $2$-sphere with exactly three exceptional fibres.
On the rational homotopy type of function spaces
Edgar
H.
Brown Jr.;
Robert
H.
Szczarba Jr.
4931-4951
Abstract: The main result of this paper is the construction of a minimal model for the function space $\mathcal {F}(X,Y)$ of continuous functions from a finite type, finite dimensional space $X$ to a finite type, nilpotent space $Y$ in terms of minimal models for $X$ and $Y$. For the component containing the constant map, $\pi _{*}(\mathcal {F}(X,Y))\otimes Q =\pi _{*}(Y)\otimes H^{-*}(X;Q)$ in positive dimensions. When $X$ is formal, there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for $Y$ and the coproduct of $H_{*}(X;Q)$. We also give a version of the main result for the space of cross sections of a fibration.
Lower bounds for derivatives of polynomials and Remez type inequalities
Tamás
Erdélyi;
Paul
Nevai
4953-4972
Abstract: P. Turán [Über die Ableitung von Polynomen, Comositio Math. 7 (1939), 89-95] proved that if all the zeros of a polynomial $p$ lie in the unit interval $I% \overset {\text {def}}{=} [-1,1]$, then $\|p'\|_{L^{\infty }(I)}\ge {\sqrt {\deg (p)}}/{6}\; \|p\|_{L^{\infty }(I)}\;$. Our goal is to study the feasibility of $\lim _{{n\to \infty }% }{\|p_{n}'\|_{X}} / {\|p_{n}\|_{Y}} =\infty$ for sequences of polynomials $\{p_{n}\}_{n\in \mathbb N% }$ whose zeros satisfy certain conditions, and to obtain lower bounds for derivatives of (generalized) polynomials and Remez type inequalities for generalized polynomials in various spaces.
Gorenstein algebras, symmetric matrices, self-linked ideals, and symbolic powers
Steven
Kleiman;
Bernd
Ulrich
4973-5000
Abstract: Inspired by recent work in the theory of central projections onto hypersurfaces, we characterize self-linked perfect ideals of grade $2$ as those with a Hilbert-Burch matrix that has a maximal symmetric subblock. We also prove that every Gorenstein perfect algebra of grade $1$ can be presented, as a module, by a symmetric matrix. Both results are derived from the same elementary lemma about symmetrizing a matrix that has, modulo a nonzerodivisor, a symmetric syzygy matrix. In addition, we establish a correspondence, roughly speaking, between Gorenstein perfect algebras of grade $1$ that are birational onto their image, on the one hand, and self-linked perfect ideals of grade $2$ that have one of the self-linking elements contained in the second symbolic power, on the other hand. Finally, we provide another characterization of these ideals in terms of their symbolic Rees algebras, and we prove a criterion for these algebras to be normal.
Partition identities involving gaps and weights
Krishnaswami
Alladi
5001-5019
Abstract: We obtain interesting new identities connecting the famous partition functions of Euler, Gauss, Lebesgue, Rogers-Ramanujan and others by attaching weights to the gaps between parts. The weights are in general multiplicative. Some identities involve powers of 2 as weights and yield combinatorial information about some remarkable partition congruences modulo powers of 2.
On the number of geodesic segments connecting two points on manifolds of non-positive curvature
Paul
Horja
5021-5030
Abstract: We prove that on a complete Riemannian manifold $M$ of dimension $n$ with sectional curvature $K_M < 0$, two points which realize a local maximum for the distance function (considered as a function of two arguments) are connected by at least $2n+1$ geodesic segments. A simpler version of the argument shows that if one of the points is fixed and $K_M \leq 0$ then the two points are connected by at least $n+1$ geodesic segments. The proof uses mainly the convexity properties of the distance function for metrics of negative curvature.
Covering Sato-Levine invariants
Gui-Song
Li
5031-5042
Abstract: Two covering versions of the Sato-Levine invariant are constructed which provide obstructions to certain two-component oriented links in the 3-sphere being link concordant to boundary links. These covering invariants are rational functions one of which detects both nonamphicheirality and noninvertibility of oriented links.
Non-Archimedean Nevanlinna theory in several variables and the non-Archimedean Nevanlinna inverse problem
William
Cherry;
Zhuan
Ye
5043-5071
Abstract: Cartan's method is used to prove a several variable, non-Archimedean, Nevanlinna Second Main Theorem for hyperplanes in projective space. The corresponding defect relation is derived, but unlike in the complex case, we show that there can only be finitely many non-zero non-Archimedean defects. We then address the non-Archimedean Nevanlinna inverse problem, by showing that given a set of defects satisfying our conditions and a corresponding set of hyperplanes in projective space, there exists a non-Archimedean analytic function with the given defects at the specified hyperplanes, and with no other defects.
Hausdorff dimension, pro-$p$ groups, and Kac-Moody algebras
Yiftach
Barnea;
Aner
Shalev
5073-5091
Abstract: Every finitely generated profinite group can be given the structure of a metric space, and as such it has a well defined Hausdorff dimension function. In this paper we study Hausdorff dimension of closed subgroups of finitely generated pro-$p$ groups $G$. We prove that if $G$ is $p$-adic analytic and $H \le _c G$ is a closed subgroup, then the Hausdorff dimension of $H$ is $\dim H/\dim G$ (where the dimensions are of $H$ and $G$ as Lie groups). Letting the spectrum $% \operatorname {Spec}(G)$ of $G$ denote the set of Hausdorff dimensions of closed subgroups of $G$, it follows that the spectrum of $p$-adic analytic groups is finite, and consists of rational numbers. We then consider some non-$p$-adic analytic groups $G$, and study their spectrum. In particular we investigate the maximal Hausdorff dimension of non-open subgroups of $G$, and show that it is equal to $1 - {1 \over {d+1}}$ in the case of $G = SL_d(F_p[[t]])$ (where $p > 2$), and to $1/2$ if $G$ is the so called Nottingham group (where $p >5$). We also determine the spectrum of $SL_2(F_p[[t]])$ ($p>2$) completely, showing that it is equal to $[0,2/3] \cup \{ 1 \}$. Some of the proofs rely on the description of maximal graded subalgebras of Kac-Moody algebras, recently obtained by the authors in joint work with E. I. Zelmanov.
On a General Form of the Second Main Theorem
Min
Ru
5093-5105
Abstract: We give a proof of a general form of the Second Main Theorem for holomorphic curves with a good error term. Two applications of this general form are also provided.
Semidirect products of regular semigroups
Peter
R.
Jones;
Peter
G.
Trotter
4265-4310
Abstract: Within the usual semidirect product $S*T$ of regular semigroups $S$ and $T$ lies the set $\text {Reg}\,% (S*T)$ of its regular elements. Whenever $S$ or $T$ is completely simple, $\text {Reg}\,% (S*T)$ is a (regular) subsemigroup. It is this `product' that is the theme of the paper. It is best studied within the framework of existence (or e-) varieties of regular semigroups. Given two such classes, ${\mathbf U}%$ and ${\mathbf V}%$, the e-variety ${\mathbf U}*{\mathbf V}%$ generated by $\{\text {Reg}\,% (S*T) : S \in {\mathbf U}% , T \in {\mathbf V}% \}$ is well defined if and only if either ${\mathbf U} %$ or ${\mathbf V}%$ is contained within the e-variety ${\mathbf {CS}}%$ of completely simple semigroups. General properties of this product, together with decompositions of many important e-varieties, are obtained. For instance, as special cases of general results the e-variety $L{\mathbf I}%$ of locally inverse semigroups is decomposed as ${\mathbf I}% * {\mathbf {RZ}}%$, where ${\mathbf I}%$ is the variety of inverse semigroups and ${\mathbf {RZ}}%$ is that of right zero semigroups; and the e-variety ${\mathbf {ES}}%$ of $E$-solid semigroups is decomposed as ${\mathbf {CR}}*{\mathbf G}%$, where ${\mathbf {CR}}%$ is the variety of completely regular semigroups and ${\mathbf G}%$ is the variety of groups. In the second half of the paper, a general construction is given for the e-free semigroups (the analogues of free semigroups in this context) in a wide class of semidirect products ${\mathbf U}% * {\mathbf V}%$ of the above type, as a semidirect product of e-free semigroups from ${\mathbf U}%$ and ${\mathbf V}%$, ``cut down to regular generators''. Included as special cases are the e-free semigroups in almost all the known important e-varieties, together with a host of new instances. For example, the e-free locally inverse semigroups, $E$-solid semigroups, orthodox semigroups and inverse semigroups are included, as are the e-free semigroups in such sub-e-varieties as strict regular semigroups, $E$-solid semigroups for which the subgroups of its self-conjugate core lie in some given group variety, and certain important varieties of completely regular semigroups. Graphical techniques play an important role, both in obtaining decompositions and in refining the descriptions of the e-free semigroups in some e-varieties. Similar techniques are also applied to describe the e-free semigroups in a different `semidirect' product of e-varieties, recently introduced by Auinger and Polák. The two products are then compared.
Doi-Hopf modules, Yetter-Drinfel'd modules and Frobenius type properties
S.
Caenepeel;
G.
Militaru;
Shenglin
Zhu
4311-4342
Abstract: We study the following question: when is the right adjoint of the forgetful functor from the category of $(H,A,C)$-Doi-Hopf modules to the category of $A$-modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that $C\otimes A$ and the smash product $A\# C^*$ are isomorphic as $(A, A\# C^*)$-bimodules. The isomorphism can be described using a generalized type of integral. Our results may be applied to some specific cases. In particular, we study the case $A=H$, and this leads to the notion of $k$-Frobenius $H$-module coalgebra. In the special case of Yetter-Drinfel'd modules over a field, the right adjoint is also a left adjoint of the forgetful functor if and only if $H$ is finite dimensional and unimodular.
Essential embedding of cyclic modules in projectives
José
L. Gómez
Pardo;
Pedro
A. Guil
Asensio
4343-4353
Abstract: Let $R$ be a ring and $E = E(R_R)$ its injective envelope. We show that if every simple right $R$-module embeds in $R_R$ and every cyclic submodule of $E_R$ is essentially embeddable in a projective module, then $R_R$ has finite essential socle. As a consequence, we prove that if each finitely generated right $R$-module is essentially embeddable in a projective module, then $R$ is a quasi-Frobenius ring. We also obtain several other applications and, among them: a) we answer affirmatively a question of Al-Huzali, Jain, and López-Permouth, by showing that a right CEP ring (i.e., a ring $R$ such that every cyclic right module is essentially embeddable in a projective module) is always right artinian; b) we prove that if $R$ is right FGF (i.e., any finitely generated right $R$-module embeds in a free module) and right CS, then $R$ is quasi-Frobenius.
$\beta$-expansions with deleted digits for Pisot numbers $\beta$
Steven
P.
Lalley
4355-4365
Abstract: An algorithm is given for computing the Hausdorff dimension of the set(s) $\Lambda =\Lambda (\beta ,D)$ of real numbers with representations $x=\sum _{n=1}^\infty d_n \beta ^{-n}$, where each $d_n \in D$, a finite set of ``digits'', and $\beta >0$ is a Pisot number. The Hausdorff dimension is shown to be $\log \lambda /\log \beta$, where $\lambda$ is the top eigenvalue of a finite 0-1 matrix $A$, and a simple algorithm for generating $A$ from the data $\beta ,D$ is given.
Some ramifications of a theorem of Boas and Pollard concerning the completion of a set of functions in $L^2$
K.
S.
Kazarian;
Robert
E.
Zink
4367-4383
Abstract: About fifty years ago, R. P. Boas and Harry Pollard proved that an orthonormal system that is completable by the adjunction of a finite number of functions also can be completed by multiplying the elements of the given system by a fixed, bounded, nonnegative measurable function. In subsequent years, several variations and extensions of this theorem have been given by a number of other investigators, and this program is continued here. A mildly surprising corollary of one of the results is that the trigonometric and Walsh systems can be multiplicatively transformed into quasibases for $L^{1}[0,1]$.
The transfer and symplectic cobordism
Malkhaz
Bakuradze
4385-4399
Abstract: The main result of this paper is the nilpotency fomula $\phi _{i}^{4} =0$, $\forall i\geq 1$ for N. Ray classes $\phi _{i}$ in the torsion of the symplectic bordism ring $MSp_{*}$
Order evaluation of products of subsets in finite groups and its applications. II
Z.
Arad;
M.
Muzychuk
4401-4414
Abstract: In this paper we give a new estimate of the cardinality of the product of subsets $AB$ in a finite non-abelian simple group, where $A$ is normal and $B$ is arbitrary. This estimate improves the one given in J. Algebra 182 (1996), 577-603.
Approximation by harmonic functions
Evgeny
A.
Poletsky
4415-4427
Abstract: For a compact set $X\subset \mathbb R^n$ we construct a restoring covering for the space $h(X)$ of real-valued functions on $X$ which can be uniformly approximated by harmonic functions. Functions from $h(X)$ restricted to an element $Y$ of this covering possess some analytic properties. In particular, every nonnegative function $f\in h(Y)$, equal to 0 on an open non-void set, is equal to 0 on $Y$. Moreover, when $n=2$, the algebra $H(Y)$ of complex-valued functions on $Y$ which can be uniformly approximated by holomorphic functions is analytic. These theorems allow us to prove that if a compact set $X\subset \mathbb C$ has a nontrivial Jensen measure, then $X$ contains a nontrivial compact set $Y$ with analytic algebra $H(Y)$.
On composite twisted unknots
Chaim
Goodman-Strauss
4429-4463
Abstract: Following Mathieu, Motegi and others, we consider the class of possible composite twisted unknots as well as pairs of composite knots related by twisting. At most one composite knot can arise from a particular $V$-twisting of an unknot; moreover a twisting of the unknot cannot be composite if we have applied more than a single full twist. A pair of composite knots can be related through at most one full twist for a particular $V$-twisting, or one summand was unaffected by the twist, or the knots were the right and left handed granny knots. Finally a conjectured characterization of all composite twisted unknots that do arise is given.
Only single twists on unknots can produce composite knots
Chuichiro
Hayashi;
Kimihiko
Motegi
4465-4479
Abstract: Let $K$ be a knot in the $3$-sphere $S^{3}$, and $D$ a disc in $S^{3}$ meeting $K$ transversely more than once in the interior. For non-triviality we assume that $\vert K \cap D \vert \ge 2$ over all isotopy of $K$. Let $K_{n}$($\subset S^{3}$) be a knot obtained from $K$ by cutting and $n$-twisting along the disc $D$ (or equivalently, performing $1/n$-Dehn surgery on $\partial D$). Then we prove the following: (1) If $K$ is a trivial knot and $K_{n}$ is a composite knot, then $\vert n \vert \le 1$; (2) if $K$ is a composite knot without locally knotted arc in $S^{3} - \partial D$ and $K_{n}$ is also a composite knot, then $\vert n \vert \le 2$. We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.
Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string
Marianna
A.
Shubov
4481-4499
Abstract: We consider a one-dimensional wave equation, which governs the vibrations of a damped string with spatially nonhomogeneous density and damping coefficients. We introduce a family of boundary conditions depending on a complex parameter $h$. Corresponding to different values of $h$, the problem describes either vibrations of a finite string or propagation of elastic waves on an infinite string. Our main object of interest is the family of non-selfadjoint operators $A_h$ in the energy space of two-component initial data. These operators are the generators of the dynamical semigroups corresponding to the above boundary-value problems. We show that the operators $A_h$ are dissipative, simple, maximal operators, which differ from each other by rank-one perturbations. We also prove that the operator $A_1 (h=1)$ coincides with the generator of the Lax-Phillips semigroup, which plays an important role in the aforementioned scattering problem. The results of this work are applied in our two forthcoming papers both to the proof of the Riesz basis property of the eigenvectors and associated vectors of the operators $A_h$ and to establishing the exact and approximate controllability of the system governed by the damped wave equation.
Nonsymmetric systems on nonsmooth planar domains
G.
C.
Verchota;
A.
L.
Vogel
4501-4535
Abstract: We study boundary value problems, in the sense of Dahlberg, for second order constant coefficient strongly elliptic systems. In this class are systems without a variational formulation, viz. the nonsymmetric systems. Various similarities and differences between this subclass and the symmetrizable systems are examined in nonsmooth domains.
Compact groups and fixed point sets
Alex
Chigogidze;
Karl
H.
Hofmann;
John
R.
Martin
4537-4554
Abstract: Some structure theorems for compact abelian groups are derived and used to show that every closed subset of an infinite compact metrizable group is the fixed point set of an autohomeomorphism. It is also shown that any metrizable product containing a positive-dimensional compact group as a factor has the property that every closed subset is the fixed point set of an autohomeomorphism.
Coloring graphs with fixed genus and girth
John
Gimbel;
Carsten
Thomassen
4555-4564
Abstract: It is well known that the maximum chromatic number of a graph on the orientable surface $S_g$ is $\theta (g^{1/2})$. We prove that there are positive constants $c_1,c_2$ such that every triangle-free graph on $S_g$ has chromatic number less than $c_2(g/\log (g))^{1/3}$ and that some triangle-free graph on $S_g$ has chromatic number at least $c_1\frac {g^{1/3}}{\log (g)}$. We obtain similar results for graphs with restricted clique number or girth on $S_g$ or $N_k$. As an application, we prove that an $S_g$-polytope has chromatic number at most $O(g^{3/7})$. For specific surfaces we prove that every graph on the double torus and of girth at least six is 3-colorable and we characterize completely those triangle-free projective graphs that are not 3-colorable.
Virtually free groups with finitely many outer automorphisms
Martin
R.
Pettet
4565-4587
Abstract: Let $G$ be a finitely generated virtually free group. From a presentation of $G$ as the fundamental group of a finite graph of finite-by-cyclic groups, necessary and sufficient conditions are derived for the outer automorphism group of $G$ to be finite. Two versions of the characterization are given, both effectively verifiable from the graph of groups. The more purely group theoretical criterion is expressed in terms of the structure of the normalizers of the edge groups (Theorem 5.10); the other version involves certain finiteness conditions on the associated $G$-tree (Theorem 5.16). Coupled with an earlier result, this completes a description of the finitely generated groups whose full automorphism groups are virtually free.
Curves of maximum genus in range A and stick-figures
Edoardo
Ballico;
Giorgio
Bolondi;
Philippe
Ellia;
Rosa
Maria
Mirò-Roig
4589-4608
Abstract: In this paper we show the existence of smooth connected space curves not contained in a surface of degree less than $m$, with genus maximal with respect to the degree, in half of the so-called range A. The main tool is a technique of deformation of stick-figures due to G. Fløystad.
Existence of conservation laws and characterization of recursion operators for completely integrable systems
Joseph
Grifone;
Mohamad
Mehdi
4609-4633
Abstract: Using the Spencer-Goldschmidt version of the Cartan-Kähler theorem, we give conditions for (local) existence of conservation laws for analytical quasi-linear systems of two independent variables. This result is applied to characterize the recursion operator (in the sense of Magri) of completely integrable systems.
On the Kolyvagin cup product
Amnon
Besser
4635-4657
Abstract: We define a new cohomological operation, which we call the Kolyvagin cup product, that is a generalization of the derivative operator introduced by Kolyvagin in his work on Euler systems. We show some of the basic properties of this operation. We also define a higher dimensional derivative in certain cases and a dual operation which we call the Kolyvagin cap product and which generalizes a computation of Rubin.
Cohen-Macaulay Section Rings
Zhou
Caijun
4659-4667
Abstract: In this paper, we study the section rings of sheaves of Cohen-Macaulay algebras (over a field $F$) on a ranked poset. A necessary and sufficient condition for these rings to be Cohen-Macaulay will be given. This is a further generalization of a result of Yuzvinsky, which generalizes Reisner's theorem concerning Stanley-Reisner rings.
Contractions on a manifold polarized by an ample vector bundle
Marco
Andreatta;
Massimiliano
Mella
4669-4683
Abstract: A complex manifold $X$ of dimension $n$ together with an ample vector bundle $E$ on it will be called a generalized polarized variety. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the positivity (the nefness or ampleness) of the adjoint bundle in the case $r := rank (E) = (n-2)$. If $r\geq (n-1)$ this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski. If $K_X+detE$ is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map $\pi :X \longrightarrow W$ from $X$ onto a normal projective variety $W$ with connected fiber and such that $K_X + det(E) = \pi^*H$, for some ample line bundle $H$ on $W$. We describe those contractions for which $dimF \leq (r-1)$. We extend this result to the case in which $X$ has log terminal singularities. In particular this gives Mukai's conjecture 1 for singular varieties. We consider also the case in which $dimF = r$ for every fiber and $\pi$ is birational.
Correction and extension of ``Measurable quotients of unipotent translations on homogeneous spaces''
Dave
Witte
4685-4688
Abstract: The statements of Main Theorem 1.1 and Theorem 2.1 of the author's paper [Trans. Amer. Math. Soc. 345 (1994), 577-594] should assume that $\Gamma$ is discrete and $G$ is connected. (Corollaries 1.3, 5.6, and 5.8 are affected similarly.) These restrictions can be removed if the conclusions of the results are weakened to allow for the possible existence of transitive, proper subgroups of $G$. In this form, the results can be extended to the setting where $G$ is a product of real and $p$-adic Lie groups.
Correction to ``Bifurcation of minimal surfaces in Riemannian manifolds''
Jürgen
Jost;
Xianqing
Li-Jost;
Xiao-Wei
Peng
4689-4690
Erratum to ``On a quadratic-trigonometric functional equation and some applications"
J.
K.
Chung;
B.
R.
Ebanks;
C.
T.
Ng;
P.
K.
Sahoo
4691 - 4691
Asymptotic prime-power divisibility of binomial, generalized binomial, and multinomial coefficients
John
M.
Holte
3837-3873
Abstract: This paper presents asymptotic formulas for the abundance of binomial, generalized binomial, multinomial, and generalized multinomial coefficients having any given degree of prime-power divisibility. In the case of binomial coefficients, for a fixed prime $p$, we consider the number of $(x, y)$ with $0 \leq x, y < p^n$ for which $\binom {x+y}{x}$ is divisible by $p^{zn}$ (but not $p^{zn+1}$) when $zn$ is an integer and $\alpha < z < \beta$, say. By means of a classical theorem of Kummer and the probabilistic theory of large deviations, we show that this number is approximately $p^{n D((\alpha , \beta ))}$, where $D((\alpha , \beta )) := \sup \{ D(z) : \alpha < z < \beta \}$ and $D$ is given by an explicit formula. We also develop a ``$p$-adic multifractal'' theory and show how $D$ may be interpreted as a multifractal spectrum of divisibility dimensions. We then prove that essentially the same results hold for a large class of the generalized binomial coefficients of Knuth and Wilf, including the $q$-binomial coefficients of Gauss and the Fibonomial coefficients of Lucas, and finally we extend our results to multinomial coefficients and generalized multinomial coefficients.
Incompressible reacting flows
Joel
D.
Avrin
3875-3892
Abstract: We establish steady-state convergence results for a system of reaction-convection-diffusion equations that model in particular combustion phenomena in the presence of nontrivial incompressible fluid motion. Despite the presence of the convection terms, we find that the asymptotic behavior of the system is identical to the case we have previously considered in which the velocity field was set equal to zero. In particular we are again able to establish the convergence of solutions to steady-states and to explicitly calculate the steady-states from the initial and boundary data. Key to our analysis is the establishment of high-order uniform bounds on the temperature and mass fraction components, a process significantly complicated by the presence of the convection terms.
Spherical classes and the algebraic transfer
Nguyen
H. V.
Hu'ng
3893 - 3910
Degenerate parabolic equations with initial data measures
Daniele
Andreucci
3911-3923
Abstract: We address the problem of existence of solutions to degenerate (and nondegenerate) parabolic equations under optimal assumptions on the initial data, which are assumed to be measures. The requirements imposed on the initial data are connected both with the degeneracy of the principal part of the equation, and with the form of the nonlinear forcing term. The latter depends on the space gradient of a power of the solution. Applications to related problems are also outlined.
Two Decompositions in Topological Combinatorics with Applications to Matroid Complexes
Manoj
K.
Chari
3925-3943
Abstract: This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of M-shellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the rank-numbers of M-shellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex ear-decomposition, and, using results of Kalai and Stanley on $h$-vectors of simplicial polytopes, we show that $h$-vectors of pure rank-$d$ simplicial complexes that have this property satisfy $h_{0} \leq h_{1} \leq \cdots \leq h_{[d/2]}$ and $h_{i} \leq h_{d-i}$ for $0 \leq i \leq [d/2]$. We then show that the abstract simplicial complex formed by the collection of independent sets of a matroid (or matroid complex) admits a special type of convex ear-decomposition called a PS ear-decomposition. This enables us to construct an associated M-shellable poset, whose set of rank-numbers is the $h$-vector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi that the $h$-vector of a matroid complex satisfies the above two sets of inequalities.
Shellable nonpure complexes and posets. II
Anders
Björner;
Michelle
L.
Wachs
3945-3975
Abstract: This is a direct continuation of Shellable Nonpure Complexes and Posets. I, which appeared in Transactions of the American Mathematical Society 348 (1996), 1299-1327.
The Wills conjecture
Noah
Samuel
Brannen
3977-3987
Abstract: Two strengthenings of the Wills conjecture, an extension of Bonnesen's inradius inequality to $n$-dimensional space, are obtained. One is the sharpest of the known strengthenings of the conjecture in three dimensions; the other employs techniques which are fundamentally different from those utilized in the other proofs.
Monge-Ampère equations relative to a Riemannian metric
A.
Atallah;
C.
Zuily
3989-4006
Abstract: We prove that in a bounded strictly convex open set $\Omega$ in $\mathbb {R}^n$, the problem \begin{displaymath}\begin {cases} \det \nabla ^2u=f(x), u|_{\partial \Omega }=\varphi , \end {cases}\end{displaymath} where $f>0,f\in C^\infty (\overline \Omega ), \varphi \in C^\infty (\partial \Omega )$, has a unique strictly convex solution $u\in C^\infty (\overline \Omega )$. This result extends to an arbitrary metric a theorem which has been proved by Caffarelli-Nirenberg-Spruck in the case of the Euclidean metric.
Four-Manifolds With Surface Fundamental Groups
Alberto
Cavicchioli;
Friedrich
Hegenbarth;
Dusan
Repovs
4007-4019
Abstract: We study the homotopy type of closed connected topological $4$-manifolds whose fundamental group is that of an aspherical surface $F$. Then we use surgery theory to show that these manifolds are $s$-cobordant to connected sums of simply-connected manifolds with an $\mathbb {S}^{2}$-bundle over $F$.
Graded Lie Algebras of Maximal Class
A.
Caranti;
S.
Mattarei;
M.
F.
Newman
4021-4051
Abstract: We study graded Lie algebras of maximal class over a field $\mathbf {F}$ of positive characteristic $p$. A. Shalev has constructed infinitely many pairwise non-isomorphic insoluble algebras of this kind, thus showing that these algebras are more complicated than might be suggested by considering only associated Lie algebras of p-groups of maximal class. Here we construct $| \mathbf {F}|^{\aleph _{0}}$ pairwise non-isomorphic such algebras, and $\max \{| \mathbf {F}|, \aleph _{0} \}$ soluble ones. Both numbers are shown to be best possible. We also exhibit classes of examples with a non-periodic structure. As in the case of groups, two-step centralizers play an important role.
A hypergeometric function approach to the persistence problem of single sine-Gordon breathers
Jochen
Denzler
4053-4083
Abstract: It is shown that for an interesting class of perturbation functions, at most one of the continuum of sine-Gordon breathers can persist for the perturbed equation. This question is much more subtle than the question of persistence of large portions of the family, because analytic continuation arguments in the amplitude parameter are no longer available. Instead, an asymptotic analysis of the obstructions to persistence for large Fourier orders is made, and it is connected to the asymptotic behaviour of the Taylor coefficients of the perturbation function by means of an inverse Laplace transform and an integral transform whose kernel involves hypergeometric functions in a way that is degenerate in that asymptotic analysis involves a splitting monkey saddle. Only first order perturbation theory enters into the argument. The reasoning can in principle be carried over to other perturbation functions than the ones considered here.
The Szego curve, zero distribution and weighted approximation
Igor
E.
Pritsker;
Richard
S.
Varga
4085-4105
Abstract: In 1924, Szeg\H{o} showed that the zeros of the normalized partial sums, $s_{n}(nz)$, of $e^{z}$ tended to what is now called the Szeg\H{o} curve $S$, where \begin{displaymath}S:= \left \{ z \in {\mathbb {C}}:|ze^{1-z}|=1 \text { and } |z| \leq 1 \right \}. \end{displaymath} Using modern methods of weighted potential theory, these zero distribution results of Szeg\H{o} can be essentially recovered, along with an asymptotic formula for the weighted partial sums $\{e^{-nz}s_{n} (nz)\}^{\infty }_{n=0}$. We show that $G:= {\operatorname {Int}} S$ is the largest universal domain such that the weighted polynomials $e^{-nz} P_{n}(z)$ are dense in the set of functions analytic in $G$. As an example of such results, it is shown that if $f(z)$ is analytic in $G$ and continuous on $\overline {G}$ with $f(1)=0$, then there is a sequence of polynomials $\left \{P_{n}(z)\right \}^{\infty }_{n=0}$, with $\deg P_{n} \leq n$, such that \begin{displaymath}\displaystyle \lim_{n \rightarrow \infty } \|e^{-nz} P_{n}(z)-f(z)\|_{\overline {G}} =0, \end{displaymath} where $\| \cdot \|_{\overline {G}}$ denotes the supremum norm on $\overline {G}$. Similar results are also derived for disks.
Asymptotic analysis for linear difference equations
Katsunori
Iwasaki
4107-4142
Abstract: We are concerned with asymptotic analysis for linear difference equations in a locally convex space. First we introduce the profile operator, which plays a central role in analyzing the asymptotic behaviors of the solutions. Then factorial asymptotic expansions for the solutions are given quite explicitly. Finally we obtain Gevrey estimates for the solutions. In a forthcoming paper we will develop the theory of cohomology groups for recurrence relations. The main results in this paper lay analytic foundations of such an algebraic theory, while they are of intrinsic interest in the theory of finite differences.
On the convergence of $\sum c_nf(nx)$ and the Lip 1/2 class
István
Berkes
4143-4158
Abstract: We investigate the almost everywhere convergence of $\sum c_{n} f(nx)$, where $f$ is a measurable function satisfying \begin{equation*}f(x+1) = f(x), \qquad \int _{0}^{1} f(x) \, dx =0.\end{equation*} By a known criterion, if $f$ satisfies the above conditions and belongs to the Lip $\alpha$ class for some $\alpha > 1/2$, then $\sum c_{n} f(nx)$ is a.e. convergent provided $\sum c_{n}^{2} < +\infty$. Using probabilistic methods, we prove that the above result is best possible; in fact there exist Lip 1/2 functions $f$ and almost exponentially growing sequences $(n_{k})$ such that $\sum c_{k} f(n_{k} x)$ is a.e. divergent for some $(c_{k})$ with $\sum c_{k}^{2} < +\infty$. For functions $f$ with Fourier series having a special structure, we also give necessary and sufficient convergence criteria. Finally we prove analogous results for the law of the iterated logarithm.
Equilibria of set-valued maps on nonconvex domains
H.
Ben-El-Mechaiekh;
W.
Kryszewski
4159-4179
Abstract: We present new theorems on the existence of equilibria (or zeros) of convex as well as nonconvex set-valued maps defined on compact neighborhood retracts of normed spaces. The maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets. Among other things, we show that if $K$ is a compact neighborhood retract with nontrivial Euler characteristic in a Banach space $E$ , and $\Phi :K\longrightarrow 2^E$ is an upper hemicontinuous set-valued map with nonempty closed convex values satisfying the tangency condition \begin{equation*}\Phi (x)\cap T_K^r(x)\neq \emptyset \text { for all }x\in K, \end{equation*} then there exists $x_0\in K$ such that $0\in \Phi (x_0).$ Here, $T_K^r(x)$ denotes a new concept of retraction tangent cone to $K$ at $x$ suited for compact neighborhood retracts. When $K$ is locally convex at $x,T_K^r(x)$ coincides with the usual tangent cone of convex analysis. Special attention is given to neighborhood retracts having ``lipschitzian behavior'', called $L-$retracts below. This class of sets is very broad; it contains compact homeomorphically convex subsets of Banach spaces, epi-Lipschitz subsets of Banach spaces, as well as proximate retracts. Our results thus generalize classical theorems for convex domains, as well as recent results for nonconvex sets.
On the conjecture of Birch and Swinnerton-Dyer
Cristian
D.
Gonzalez-Avilés
4181-4200
Abstract: In this paper we complete Rubin's partial verification of the conjecture for a large class of elliptic curves with complex multiplication by ${\mathbb {Q}}(\sqrt {-7})$.
Hamiltonian torus actions on symplectic orbifolds and toric varieties
Eugene
Lerman;
Susan
Tolman
4201-4230
Abstract: In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly, we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each open facet and that all such orbifolds are algebraic toric varieties.
Existence of positive solutions for some problems with nonlinear diffusion
A.
Cañada;
P.
Drábek;
J.
L.
Gámez
4231-4249
Abstract: In this paper we study the existence of positive solutions for problems of the type \begin{equation*}\begin {array}{cl} -\Delta _pu(x)=u(x)^{q-1}h(x,u(x)), & x\in \Omega , u(x)=0, & x\in \partial \Omega , \end {array} \end{equation*} where $\Delta _p$ is the $p$-Laplace operator and $p,q>1$. If $p=2$, such problems arise in population dynamics. Making use of different methods (sub- and super-solutions and a variational approach), we treat the cases $p=q$, $p<q$ and $p>q$, respectively. Also, some systems of equations are considered.
Tame Combings of Groups
Michael
L.
Mihalik;
Steven
T.
Tschantz
4251-4264
Abstract: In this paper, we introduce the idea of tame combings for finitely presented groups. If $M$ is a closed irreducible 3-manifold and $\pi _{1}(M)$ is tame combable, then the universal cover of $M$ is homeomorphic to ${\mathbb {R}}^{3}$. We show that all asynchronously automatic and all semihyperbolic groups are tame combable.
Elliptic three-folds II: Multiple fibres
Mark
Gross
3409-3468
Abstract: Let $f:X\rightarrow S$ be an elliptic fibration with a section, where $S$ is a projective surface and $X$ is a projective threefold. We determine when it is possible to perform a logarithmic transformation along a closed subset $Z\subseteq S$ to obtain a new elliptic fibration $f':X'\rightarrow S$ which now has multiple fibres along $Z$. This is done in the setting of Ogg-Shafarevich theory. We find a number of obstructions to performing such a logarithmic transformation, the very last of which takes values in the torsion part of the codimension 2 Chow group of $X$.
On perfect isometries and isotypies in alternating groups
Paul
Fong;
Morton
E.
Harris
3469-3516
Abstract: Perfect isometries and isotypies are constructed for alternating groups between blocks with abelian defect groups and the Brauer correspondents of these blocks. These perfect isometries and isotypies satisfy additional compatibility conditions which imply that an extended Broué conjecture holds for the principal block of an almost simple group with an abelian Sylow $p$-subgroup and a generalized Fitting subgroup isomorphic to an alternating group.
Monoid Hecke algebras
Mohan
S.
Putcha
3517-3534
Abstract: This paper concerns the monoid Hecke algebras $\mathcal {H}$ introduced by Louis Solomon. We determine explicitly the unities of the orbit algebras associated with the two-sided action of the Weyl group $W$. We use this to: find a description of the irreducible representations of $\mathcal {H}$, find an explicit isomorphism between $\mathcal {H}$ and the monoid algebra of the Renner monoid $R$, extend the Kazhdan-Lusztig involution and basis to $\mathcal {H}$, and prove, for a $W\times W$ orbit of $R$, the existence (conjectured by Renner) of generalized Kazhdan-Lusztig polynomials.
Endomorphism algebras of peak $I$-spaces over posets of infinite prinjective type
Rüdiger
Göbel;
Warren
May
3535-3567
Abstract: We will derive a general result for $R$-categories which allows us to derive the existence of large objects with prescribed endomorphism algebras from the existence of small families. This theorem is based on an earlier result of S. Shelah in which he established the existence of indecomposable abelian groups of any cardinality. We will apply this `Shelah-elevator' for abelian groups and - which is our main concern - for prescribing endomorphism algebras of peak $I$-spaces which are classified by a recent result of Simson.
Spherical functions on symmetric cones
P.
Sawyer
3569-3584
Abstract: In this note, we obtain a recursive formula for the spherical functions associated with the symmetric cone of a formally real Jordan algebra. We use this formula as an inspiration for a similar recursive formula involving the Jack polynomials.
Absolute Borel sets and function spaces
Witold
Marciszewski;
Jan
Pelant
3585-3596
Abstract: An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Frolík for characterizing Cech-complete spaces. We also show that the absolute Borel class of $X$ is determined by the uniform structure of the space of continuous functions $C_{p}(X)$; however the case of absolute $G_{\delta }$ metric spaces is still open. More precisely, we prove that, for metrizable spaces $X$ and $Y$, if $\Phi : C_{p}(X) \rightarrow C_{p}(Y)$ is a uniformly continuous surjection and $X$ is an absolute Borel set of multiplicative (resp., additive) class $\alpha$, $\alpha >1$, then $Y$ is also an absolute Borel set of the same class. This result is new even if $\Phi$ is a linear homeomorphism, and extends a result of Baars, de Groot, and Pelant which shows that the \v{C}ech-completeness of a metric space $X$ is determined by the linear structure of $C_{p}(X)$.
Polynomial structures on polycyclic groups
Karel
Dekimpe;
Paul
Igodt
3597-3610
Abstract: We know, by recent work of Benoist and of Burde & Grunewald, that there exist polycyclic-by-finite groups $G$, of rank $h$ (the examples given were in fact nilpotent), admitting no properly discontinuous affine action on $\mathbb {R}^h$. On the other hand, for such $G$, it is always possible to construct a properly discontinuous smooth action of $G$ on $\mathbb {R}^h$. Our main result is that any polycyclic-by-finite group $G$ of rank $h$ contains a subgroup of finite index acting properly discontinuously and by polynomial diffeomorphisms of bounded degree on $\mathbb {R}^h$. Moreover, these polynomial representations always appear to contain pure translations and are extendable to a smooth action of the whole group $G$.
Arithmeticity, discreteness and volume
F.
W.
Gehring;
C.
Maclachlan;
G.
J.
Martin;
A.
W.
Reid
3611-3643
Abstract: We give an arithmetic criterion which is sufficient to imply the discreteness of various two-generator subgroups of ${PSL}(2,\mathbf {c})$. We then examine certain two-generator groups which arise as extremals in various geometric problems in the theory of Kleinian groups, in particular those encountered in efforts to determine the smallest co-volume, the Margulis constant and the minimal distance between elliptic axes. We establish the discreteness and arithmeticity of a number of these extremal groups, the associated minimal volume arithmetic group in the commensurability class and we study whether or not the axis of a generator is simple. We then list all ``small'' discrete groups generated by elliptics of order $2$ and $n$, $n=3,4,5,6,7$.
The nonexistence of expansive homeomorphisms of a class of continua which contains all decomposable circle-like continua
Hisao
Kato
3645-3655
Abstract: A homeomorphism $f:X \to X$ of a compactum $X$ with metric $d$ is expansive if there is $c > 0$ such that if $x, y \in X$ and $x \not = y$, then there is an integer $n \in % \mathbf {Z}$ such that $d(f^{n}(x),f^{n}(y)) > c$. It is well-known that $p$-adic solenoids $S_p$ ($p\geq 2$) admit expansive homeomorphisms, each $S_p$ is an indecomposable continuum, and $S_p$ cannot be embedded into the plane. In case of plane continua, the following interesting problem remains open: For each $1 \leq n \leq 3$, does there exist a plane continuum $X$ so that $X$ admits an expansive homeomorphism and $X$ separates the plane into $n$ components? For the case $n=2$, the typical plane continua are circle-like continua, and every decomposable circle-like continuum can be embedded into the plane. Naturally, one may ask the following question: Does there exist a decomposable circle-like continuum admitting expansive homeomorphisms? In this paper, we prove that a class of continua, which contains all chainable continua, some continuous curves of pseudo-arcs constructed by W. Lewis and all decomposable circle-like continua, admits no expansive homeomorphisms. In particular, any decomposable circle-like continuum admits no expansive homeomorphism. Also, we show that if $f:X\to X$ is an expansive homeomorphism of a circle-like continuum $X$, then $f$ is itself weakly chaotic in the sense of Devaney.
The class number one problem for some non-abelian normal CM-fields
Stéphane
Louboutin;
Ryotaro
Okazaki;
Michel
Olivier
3657-3678
Abstract: Let ${\bf N}$ be a non-abelian normal CM-field of degree $4p,$ $p$ any odd prime. Note that the Galois group of ${\bf N}$ is either the dicyclic group of order $4p,$ or the dihedral group of order $4p.$ We prove that the (relative) class number of a dicyclic CM-field of degree $4p$ is always greater then one. Then, we determine all the dihedral CM-fields of degree $12$ with class number one: there are exactly nine such CM-fields.
Quadratic optimal control of stable well-posed linear systems
Olof
J.
Staffans
3679-3715
Abstract: We consider the infinite horizon quadratic cost minimization problem for a stable time-invariant well-posed linear system in the sense of Salamon and Weiss, and show that it can be reduced to a spectral factorization problem in the control space. More precisely, we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution. If both the system and the spectral factor are regular, then the feedback operator can be expressed in terms of the Riccati operator, and the Riccati operator is a positive self-adjoint solution of an algebraic Riccati equation. This Riccati equation is similar to the usual algebraic Riccati equation, but one of its coefficients varies depending on the subspace in which the equation is posed. Similar results are true for unstable systems, as we have proved elsewhere.
Asymptotic behaviour of reproducing kernels of weighted Bergman spaces
Miroslav
Englis
3717-3735
Abstract: Let $\Omega$ be a domain in $\mathbb {C}^{n}$, $F$ a nonnegative and $G$ a positive function on $\Omega$ such that $1/G$ is locally bounded, $A^{2}_{\alpha }$ the space of all holomorphic functions on $\Omega$ square-integrable with respect to the measure $F^{\alpha }G\,d\lambda$, where $d\lambda$ is the $2n$-dimensional Lebesgue measure, and $K_{\alpha }(x,y)$ the reproducing kernel for $A^{2}_{\alpha }$. It has been known for a long time that in some special situations (such as on bounded symmetric domains $\Omega$ with $G=\text {\bf 1}$ and $F=\,$the Bergman kernel function) the formula \begin{equation*}\lim _{\alpha \to +\infty }K_{\alpha }(x,x)^{1/\alpha }=1/F(x) \tag {$*$} \end{equation*} holds true. [This fact even plays a crucial role in Berezin's theory of quantization on curved phase spaces.] In this paper we discuss the validity of this formula in the general case. The answer turns out to depend on, loosely speaking, how well the function $-\log F$ can be approximated by certain pluriharmonic functions lying below it. For instance, ($*$) holds if $-\log F$ is convex (and, hence, can be approximated from below by linear functions), for any function $G$. Counterexamples are also given to show that in general ($*$) may fail drastically, or even be true for some $x$ and fail for the remaining ones. Finally, we also consider the question of convergence of $K_{\alpha }(x,y)^{1/\alpha }$ for $x\neq y$, which leads to an unexpected result showing that the zeroes of the reproducing kernels are affected by the smoothness of $F$: for instance, if $F$ is not real-analytic at some point, then $K_{\alpha }(x,y)$ must have zeroes for all $\alpha$ sufficiently large.
The Brauer group of Yetter-Drinfel'd module algebras
S.
Caenepeel;
F.
Van
Oystaeyen;
Y.
H.
Zhang
3737-3771
Abstract: Let $H$ be a Hopf algebra with bijective antipode. In a previous paper, we introduced $H$-Azumaya Yetter-Drinfel'd module algebras, and the Brauer group ${\mathrm {BQ}}(k,H)$ classifying them. We continue our study of ${\mathrm {BQ}}(k,H)$, and we generalize some properties that were previously known for the Brauer-Long group. We also investigate separability properties for $H$-Azumaya algebras, and this leads to the notion of strongly separable $H$-Azumaya algebra, and to a new subgroup of the Brauer group ${\mathrm {BQ}}(k,H)$.
Boundary limits and non-integrability of $\mathcal M$-subharmonic functions in the unit ball of $\mathbb C^n (n\ge 1)$
Manfred
Stoll
3773-3785
Abstract: In this paper we consider weighted non-tangential and tangential boundary limits of non-negative functions on the unit ball $B$ in ${{\mathbb {C}}^{\vphantom {P}}}^{n}$ that are subharmonic with respect to the Laplace-Beltrami operator $\widetilde {\varDelta }$ on $B$. Since the operator $\widetilde {\varDelta }$ is invariant under the group $\mathcal {M}$ of holomorphic automorphisms of $B$, functions that are subharmonic with respect to $\widetilde {\varDelta }$ are usually referred to as $\mathcal {M}$-subharmonic functions. Our main result is as follows: Let $f$ be a non-negative $\mathcal {M}$-subharmonic function on $B$ satisfying \begin{equation*}\int _{B} (1-|z|^{2})^{\gamma }f^{p}(z)\,d\lambda (z)< \infty \end{equation*} for some $p> 0$ and some $\gamma >\min \{n,pn\}$, where $\lambda$ is the $\mathcal {M}$-invariant measure on $B$. Suppose $\tau \ge 1$. Then for a.e. $\zeta \in S$, \begin{equation*}f^{p}(z)= o\left ((1-|z|^{2})^{n/\tau -\gamma }\right ) \end{equation*} uniformly as $z\to \zeta$ in each $\mathcal {T}_{\tau ,\alpha }(\zeta )$, where for $\alpha >0$ ($\alpha >\frac {1}{2}$ when $\tau =1$) \begin{equation*}\mathcal {T}_{\tau ,\alpha }(\zeta ) = \{z\in B: |1-\langle z,\zeta \rangle |^{\tau } <\alpha (1-|z|^{2}) \}. \end{equation*} We also prove that for $\gamma \le \min \{n,pn\}$ the only non-negative $\mathcal {M}$-subharmonic function satisfying the above integrability criteria is the zero function.
Line bundle Laplacians over isospectral nilmanifolds
Dorothee
Schueth
3787-3802
Abstract: We show that nontrivial isospectral deformations of a big class of compact Riemannian two-step nilmanifolds can be distinguished from trivial deformations by the behaviour of bundle Laplacians on certain non-flat hermitian line bundles over these manifolds.
A construction of codimension three arithmetically Gorenstein subschemes of projective space
Juan
C.
Migliore;
Chris
Peterson
3803-3821
Abstract: This paper presents a construction method for a class of codimension three arithmetically Gorenstein subschemes of projective space. These schemes are obtained from degeneracy loci of sections of certain specially constructed rank three reflexive sheaves. In contrast to the structure theorem of Buchsbaum and Eisenbud, we cannot obtain every arithmetically Gorenstein codimension three subscheme by our method. However, certain geometric applications are facilitated by the geometric aspect of this construction, and we discuss several examples of this in the final section.
On Frobenius algebras and the quantum Yang-Baxter equation
K.
I.
Beidar;
Y.
Fong;
A.
Stolin
3823-3836
Abstract: It is shown that every Frobenius algebra over a commutative ring determines a class of solutions of the quantum Yang-Baxter equation, which forms a subbimodule of its tensor square. Moreover, this subbimodule is free of rank one as a left (right) submodule. An explicit form of a generator is given in terms of the Frobenius homomorphism. It turns out that the generator is invertible in the tensor square if and only if the algebra is Azumaya.
Time-dependent coating flows in a strip, Part I: The linearized problem
Avner
Friedman;
Juan
J. L.
Velázquez
2981-3074
Abstract: This work is concerned with time-dependent coating flow in a strip $0 < y < 1$. The Navier-Stokes equations are satisfied in the fluid region, the bottom substrate $y = 0$ is moving with fixed velocity $(U,0)$, and fluid is entering the strip through the upper boundary $y = 1$. The free boundary has the form $y = f(x,t)$ for $-\infty < x < R(t)$, where $R(t)$ is the moving contact point. Our objective is to prove that if the initial data are close to those of a stationary solution (the existence of such a solution was established by the authors in an earlier paper) then the time-dependent problem has a unique solution with smooth free boundary, at least for a small time interval. In this Part I we study the linearized problem, about the stationary solution, and obtain sharp estimates for the solution and its derivatives. These estimates will be used in Part II to establish existence and uniqueness for the full nonlinear problem.
Tetragonal curves, scrolls and $K3$ surfaces
James
N.
Brawner
3075-3091
Abstract: In this paper we establish a theorem which determines the invariants of a general hyperplane section of a rational normal scroll of arbitrary dimension. We then construct a complete intersection surface on a four-dimensional scroll and prove it is regular with a trivial dualizing sheaf. We determine the invariants for which the surface is nonsingular, and hence a $K3$ surface. A general hyperplane section of this surface is a tetragonal curve; we use the first theorem to determine for which tetragonal invariants such a construction is possible. In particular we show that for every genus $g\geq 7$ there is a tetragonal curve of genus $g$ that is a hyperplane section of a $K3$ surface. Conversely, if the tetragonal invariants are not sufficiently balanced, then the complete intersection must be singular. Finally we determine for which additional sets of invariants this construction gives a tetragonal curve as a hyperplane section of a singular canonically trivial surface, and discuss the connection with other recent results on canonically trivial surfaces.
The second variation of nonorientable minimal submanifolds
Marty
Ross
3093-3104
Abstract: Suppose $M$ is a complete nonorientable minimal submanifold of a Riemannian manifold $N$. We derive a second variation formula for the area of $M$ with respect to certain perturbations, giving a sufficient condition for the instability of $M$. Some simple applications are given: we show that the totally geodesic $\mathbb {R} \mathbb {P}^{2}$ is the only stable surface in $\mathbb {R} \mathbb {P}^{3}$, and we show the non-existence of stable nonorientable cones in $\mathbb {R}^{4}$. We reproduce and marginally extend some known results in the truly non-compact setting.
Matrix polynomials and the index problem for elliptic systems
B.
Rowley
3105-3148
Abstract: The main new results of this paper concern the formulation of algebraic conditions for the Fredholm property of elliptic systems of P.D.E.'s with boundary values, which are equivalent to the Lopatinskii condition. The Lopatinskii condition is reformulated in a new algebraic form (based on matrix polynomials) which is then used to study the existence of homotopies of elliptic boundary value problems. The paper also contains an exposition of the relevant parts of the theory of matrix polynomials and the theory of elliptic systems of P.D.E.'s.
An index formula for elliptic systems in the plane
B.
Rowley
3149-3179
Abstract: An index formula is proved for elliptic systems of P.D.E.'s with boundary values in a simply connected region $\Omega$ in the plane. Let $\mathcal {A}$ denote the elliptic operator and $\mathcal {B}$ the boundary operator. In an earlier paper by the author, the algebraic condition for the Fredholm property, i.e. the Lopatinskii condition, was reformulated as follows. On the boundary, a square matrix function $\Delta ^{+}_{{\mathcal {B}}}$ defined on the unit cotangent bundle of $\partial \Omega$ was constructed from the principal symbols of the coefficients of the boundary operator and a spectral pair for the family of matrix polynomials associated with the principal symbol of the elliptic operator. The Lopatinskii condition is equivalent to the condition that the function $\Delta ^{+}_{{\mathcal {B}}}$ have invertible values. In the present paper, the index of $({\mathcal {A}},{\mathcal {B}})$ is expressed in terms of the winding number of the determinant of $\Delta ^{+}_{{\mathcal {B}}}$.
An infinite dimensional Morse theory with applications
Wojciech
Kryszewski;
Andrzej
Szulkin
3181-3234
Abstract: In this paper we construct an infinite dimensional (extraordinary) cohomology theory and a Morse theory corresponding to it. These theories have some special properties which make them useful in the study of critical points of strongly indefinite functionals (by strongly indefinite we mean a functional unbounded from below and from above on any subspace of finite codimension). Several applications are given to Hamiltonian systems, the one-dimensional wave equation (of vibrating string type) and systems of elliptic partial differential equations.
The quantum analog of a symmetric pair: a construction in type $(C_n,A_1\times C_{n-1})$
Welleda
Baldoni;
Pierluigi
Möseneder
Frajria
3235-3276
Abstract: Let $\mathcal {I}$ be the ideal in the enveloping algebra of $\mathfrak {sp}(n,\mathbb C)$ generated by the maximal compact subalgebra of $\mathfrak {sp}(n-1,1)$. In this paper we construct an analog of $\mathcal I$ in the quantized enveloping algebra $\mbox {$\mathfrak {U}$}$ corresponding to a type $C_{n}$ diagram at generic $q$. We find generators for $\mathcal {I}$ and explicit bases for $\mbox {$\mathfrak {U}$}/\mathcal {I}$.
On the second adjunction mapping. The case of a $1$-dimensional image
Mauro
C.
Beltrametti;
Andrew
J.
Sommese
3277-3302
Abstract: Let $\widehat {L}$ be a very ample line bundle on an $n$-dimensional projective manifold $\widehat {X}$, i.e., assume that $\widehat {L}\approx i^*{\mathcal O}_{{\mathbb P}^{N}}(1)$ for some embedding $i:\widehat {X}\hookrightarrow {\mathbb P}^{N}$. In this article, a study is made of the meromorphic map, $\widehat {\varphi } : \widehat {X}\to \Sigma$, associated to $|K_{\widehat {X}}+(n-2)\widehat {L}|$ in the case when the Kodaira dimension of $K_{\widehat {X}}+(n-2)\widehat {L}$ is $\ge 3$, and $\widehat {\varphi }$ has a $1$-dimensional image. Assume for simplicity that $n=3$. The first main result of the paper shows that $\widehat \varphi$ is a morphism if either $h^0(K_{\widehat X}+\widehat L)\geq 7$ or $\kappa (\widehat {X})\geq 0$. The second main result of this paper shows that if $\kappa (\widehat X)\ge 0$, then the genus, $g(f)$, of a fiber, $f$, of the map induced by $\widehat \varphi$ on hyperplane sections is $\leq 6$. Moreover, if $h^0(K_{\widehat X}+\widehat L)\ge 21$ then $g(f)\leq 5$, a connected component $F$ of a general fiber of $\widehat \varphi$ is either a $K3$ surface or the blowing up at one point of a $K3$ surface, and $h^1({\mathcal O}_{\widehat X})\le 1$. Finally the structure of the finite to one part of the Remmert-Stein factorization of $\widehat \varphi$ is worked out.
Kernel of locally nilpotent $R$-derivations of $R[X,Y]$
S.
M.
Bhatwadekar;
Amartya
K.
Dutta
3303-3319
Abstract: In this paper we study the kernel of a non-zero locally nilpotent $R$-derivation of the polynomial ring $R[X,Y]$ over a noetherian integral domain $R$ containing a field of characteristic zero. We show that if $R$ is normal then the kernel has a graded $R$-algebra structure isomorphic to the symbolic Rees algebra of an unmixed ideal of height one in $R$, and, conversely, the symbolic Rees algebra of any unmixed height one ideal in $R$ can be embedded in $R[X,Y]$ as the kernel of a locally nilpotent $R$-derivation of $R[X,Y]$. We also give a necessary and sufficient criterion for the kernel to be a polynomial ring in general.
Strict definiteness of integrals via complete monotonicity of derivatives
L.
Mattner
3321-3342
Abstract: Let $k$ be a nonnegative integer and let $\varphi : (0,\infty ) \rightarrow \Bbb R$ be a $C^\infty$ function with $(-)^k\cdot \varphi ^{(k)}$ completely monotone and not constant. If $\sigma \neq 0$ is a signed measure on any euclidean space $\Bbb R^d$, with vanishing moments up to order $k-1$, then the integral $\int _{\Bbb R^d} \int _{\Bbb R^d} \varphi ( \|x-y\|^2 ) \, d\sigma (x) d\sigma (y)$ is strictly positive whenever it exists. For general $d$ no larger class of continuous functions $\varphi$ seems to admit the same conclusion. Examples and applications are indicated. A section on ''bilinear integrability'' might be of independent interest.
Extensions of modules over Weyl algebras
S.
C.
Coutinho
3343-3352
Abstract: In this paper we calculate some $\mathrm {Ext}$ groups of singular modules over the complex Weyl algebra $A_{n}$. In particular we determine conditions under which $\mathrm {Ext}$ is an infinite dimensional vector space when $n =2$ or $3$.
Hecke algebras, $U_qsl_n$, and the Donald-Flanigan conjecture for $S_n$
Murray
Gerstenhaber;
Mary
E.
Schaps
3353-3371
Abstract: The Donald-Flanigan conjecture asserts that the integral group ring $\mathbb {Z}G$ of a finite group $G$ can be deformed to an algebra $A$ over the power series ring $\mathbb {Z}[[t]]$ with underlying module $\mathbb {Z}G[[t]]$ such that if $p$ is any prime dividing $\#G$ then $A\otimes _{\mathbb {Z}[[t]]}\overline {\mathbb {F}_{p}((t))}$ is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of $\mathbb {C}G.$ We prove this for $G = S_{n}$ using the natural representation of its Hecke algebra $\mathcal {H}$ by quantum Yang-Baxter matrices to show that over $\mathbb {Z}[q]$ localized at the multiplicatively closed set generated by $q$ and all $i_{q^{2}} = 1+q^{2} + q^{4} + \dots + q^{2(i-1)}, i = 1,2,\dots , n$, the Hecke algebra becomes a direct sum of total matric algebras. The corresponding ``canonical" primitive idempotents are distinct from Wenzl's and in the classical case ($q=1$), from those of Young.
Kaehler structures on $K_{\mathbf C}/(P,P)$
Meng-Kiat
Chuah
3373-3390
Abstract: Let $K$ be a compact connected semi-simple Lie group, let $G = K_{\mathbf C}$, and let $G = KAN$ be an Iwasawa decomposition. To a given $K$-invariant Kaehler structure $\omega$ on $G/N$, there corresponds a pre-quantum line bundle ${\mathbf L}$ on $G/N$. Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections ${\mathcal O}({\mathbf L})$ as a $K$-representation space. We defined a $K$-invariant $L^2$-structure on ${\mathcal O}({\mathbf L})$, and let $H_\omega \subset {\mathcal O}({\mathbf L})$ denote the space of square-integrable holomorphic sections. Then $H_\omega$ is a unitary $K$-representation space, but not all unitary irreducible $K$-representations occur as subrepresentations of $H_\omega$. This paper serves as a continuation of that work, by generalizing the space considered. Let $B$ be a Borel subgroup containing $N$, with commutator subgroup $(B,B)=N$. Instead of working with $G/N = G/(B,B)$, we consider $G/(P,P)$, for all parabolic subgroups $P$ containing $B$. We carry out a similar construction, and recover in $H_\omega$ the unitary irreducible $K$-representations previously missing. As a result, we use these holomorphic sections to construct a model for $K$: a unitary $K$-representation in which every irreducible $K$-representation occurs with multiplicity one.
Extreme points in triangular UHF algebras
Timothy
D.
Hudson;
Elias
G.
Katsoulis;
David
R.
Larson
3391-3400
Abstract: We examine the strongly extreme point structure of the unit balls of triangular UHF algebras. The semisimple triangular UHF algebras are characterized as those for which this structure is minimal in the sense that every strongly extreme point belongs to the diagonal. In contrast to this, for the class of full nest algebras we prove a Krein-Milman type theorem which asserts that every operator in the open unit ball of the algebra is a convex combination of strongly extreme points.
Matrix extensions and eigenvalue completions, the generic case
William
Helton;
Joachim
Rosenthal;
Xiaochang
Wang
3401-3408
Abstract: In this paper we provide new necessary and sufficient conditions for the so-called eigenvalue completion problem.
Liouville type theorems for fourth order elliptic equations in a half plane
Avner
Friedman;
Juan
J. L.
Velázquez
2537-2603
Abstract: Consider an elliptic equation $\omega \Delta\varphi -\Delta ^2\varphi =0$ in the half plane $\{(x,\,y),\,-\infty <x<\infty ,\,y>0\}$ with boundary conditions $\varphi =\varphi _y=0$ if $y=0,\,x>0$ and $B_j\varphi =0$ if $y=0,\,x<0$ where $B_j$ $(j=2,3)$ are second and third order differential operators. It is proved that if $Re\,\omega \geq0,\,\omega \neq0$ and, for some $\varepsilon >0$, $|\varphi |\leq Cr^\alpha$ if $r=\sqrt {x^2+y^2}\to \infty ,\quad |\varphi |\leq Cr^\beta$ if $r\to 0$ where $\alpha =n+\frac {1}{2}-\varepsilon \,,\quad \beta=n+\frac {1}{2}+\varepsilon$ for some nonnegative integer $n$, then $\varphi \equiv0$. Results of this type are also established in case $\omega =0$ under different conditions on $\alpha$ and $\beta$; furthermore, in one case $B_3\varphi$ has a lower order term which depends nonlocally on $\varphi$. Such Liouville type theorems arise in the study of coating flow; in fact, they play a crucial role in the analysis of the linearized version of this problem. The methods developed in this paper are entirely different for the two cases (i) $Re\,\omega \geq0,\,\omega \neq0$ and (ii) $\omega =0$; both methods can be extended to other linear elliptic boundary value problems in a half plane.
Analytic subgroups of $t$-modules
Robert
Tubbs
2605-2617
Abstract: In this paper we study the structure of analytic subgroups and of $t$-submodules of $t$-modules.
The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems
Di
Dong;
Yiming
Long
2619-2661
Abstract: In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.
Verma type modules of level zero for affine Lie algebras
Viatcheslav
Futorny
2663-2685
Abstract: We study the structure of Verma type modules of level zero induced from non-standard Borel subalgebras of an affine Kac-Moody algebra. For such modules in ``general position'' we describe the unique irreducible quotients, construct a BGG type resolution and prove the BGG duality in certain categories. All results are extended to generalized Verma type modules of zero level.
Generalized Weil's reciprocity law and multiplicativity theorems
András
Némethi
2687-2697
Abstract: Fix a one-dimensional group variety $G$ with Euler-characteristic $\chi (G)=0$, and a quasi-projective variety $Y$, both defined over $\bold {C}$. For any $f\in Hom(Y,G)$ and constructible sheaf ${\cal F}$ on $Y$, we construct an invariant $c_{{\cal F}}(f)\in G$, which provides substantial information about the topology of the fiber-structure of $f$ and the structure of ${\cal F}$ along the fibers of $f$. Moreover, $c_{{\cal F}}:Hom(Y,G)\to G$ is a group homomorphism.
Isomorphism of lattices of recursively enumerable sets
Todd
Hammond
2699-2719
Abstract: Let $\omega = \{\,0,1,2,\ldots \,\}$, and for $A\subseteq \omega$, let $\mathcal E^A$ be the lattice of subsets of $\omega$ which are recursively enumerable relative to the ``oracle'' $A$. Let $(\mathcal E^A)^*$ be $\mathcal E^A/\mathcal I$, where $\mathcal I$ is the ideal of finite subsets of $\omega$. It is established that for any $A,B\subseteq \omega$, $(\mathcal E^A)^*$ is effectively isomorphic to $(\mathcal E^B)^*$ if and only if $A'\equiv _T B'$, where $A'$ is the Turing jump of $A$. A consequence is that if $A'\equiv _T B'$, then $\mathcal E^A\cong \mathcal E^B$. A second consequence is that $(\mathcal E^A)^*$ can be effectively embedded into $(\mathcal E^B)^*$ preserving least and greatest elements if and only if $A'\leq _T B'$.
A combinatorial correspondence related to Göllnitz' (big) partition theorem and applications
Krishnaswami
Alladi
2721-2735
Abstract: In recent work, Alladi, Andrews and Gordon discovered a key identity which captures several fundamental theorems in partition theory. In this paper we construct a combinatorial bijection which explains this key identity. This immediately leads to a better understanding of a deep theorem of Göllnitz, as well as Jacobi's triple product identity and Schur's partition theorem.
Specification on the interval
Jérôme
Buzzi
2737-2754
Abstract: We study the consequences of discontinuities on the specification property for interval maps. After giving a necessary and sufficient condition for a piecewise monotonic, piecewise continuous map to have this property, we show that for a large and natural class of families of such maps (including the $\beta$-transformations), the set of parameters for which the specification property holds, though dense, has zero Lebesgue measure. Thus, regarding the specification property, the general case is at the opposite of the continuous case solved by A.M. Blokh (Russian Math. Surveys 38 (1983), 133-134) (for which we give a proof).
Tarski's finite basis problem via $\mathbf A(\mathcal T)$
Ross
Willard
2755-2774
Abstract: R. McKenzie has recently associated to each Turing machine ${\mathcal T}$ a finite algebra $\mathbf {A} ({\mathcal T})$ having some remarkable properties. We add to the list of properties, by proving that the equational theory of $\mathbf {A}({\mathcal T})$ is finitely axiomatizable if ${\mathcal T}$ halts on the empty input. This completes an alternate (and simpler) proof of McKenzie's negative answer to A. Tarski's finite basis problem. It also removes the possibility, raised by McKenzie, of using $\mathbf {A}({\mathcal T})$ to answer an old question of B. Jónsson.
Symmetric Gibbs measures
Karl
Petersen;
Klaus
Schmidt
2775-2811
Abstract: We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures-a version of de Finetti's theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, `canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.
Invariant cocycles, random tilings and the super-$K$ and strong Markov properties
Klaus
Schmidt
2813-2825
Abstract: We consider $1$-cocycles with values in locally compact, second countable abelian groups on discrete, nonsingular, ergodic equivalence relations. If such a cocycle is invariant under certain automorphisms of these relations, we show that the skew product extension defined by the cocycle is ergodic. As an application we obtain an extension of many recent results of the author and K. Petersen to higher-dimensional shifts of finite type, and prove a transitivity result concerning rearrangements of certain random tilings.
A homotopy classification of certain 7-manifolds
Bernd
Kruggel
2827-2843
Abstract: This paper gives a homotopy classification of Wallach spaces and a partial homotopy classification of closely related spaces obtained by free $S^1$-actions on $SU(3)$ and on $S^3\times S^5$.
On the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank
Haseo
Ki
2845-2870
Abstract: We show that the Denjoy rank and the Zalcwasser rank are incomparable. We construct for any countable ordinal $\alpha$ differentiable functions $f$ and $g$ such that the Zalcwasser rank and the Kechris-Woodin rank of $f$ are $\alpha +1$ but the Denjoy rank of $f$ is 2 and the Denjoy rank and the Kechris-Woodin rank of $g$ are $\alpha +1$ but the Zalcwasser rank of $g$ is 1. We then derive a theorem that shows the surprising behavior of the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank.
Euler-Lagrange and Hamiltonian formalisms in dynamic optimization
Alexander
Ioffe
2871-2900
Abstract: We consider dynamic optimization problems for systems governed by differential inclusions. The main focus is on the structure of and interrelations between necessary optimality conditions stated in terms of Euler-Lagrange and Hamiltonian formalisms. The principal new results are: an extension of the recently discovered form of the Euler-Weierstrass condition to nonconvex valued differential inclusions, and a new Hamiltonian condition for convex valued inclusions. In both cases additional attention was given to weakening Lipschitz type requirements on the set-valued mapping. The central role of the Euler type condition is emphasized by showing that both the new Hamiltonian condition and the most general form of the Pontriagin maximum principle for equality constrained control systems are consequences of the Euler-Weierstrass condition. An example is given demonstrating that the new Hamiltonian condition is strictly stronger than the previously known one.
On the Faber coefficients of functions univalent in an ellipse
E.
Haliloglu
2901-2916
Abstract: Let $E$ be the elliptical domain \begin{displaymath}E=\{x+iy: \frac {x^{2}}{(5/4)^{2}}+ \frac {y^{2}}{(3/4)^{2}}<1 \}.\end{displaymath} Let $S(E)$ denote the class of functions $F(z)$ analytic and univalent in $E$ and satisfying the conditions $F(0)=0$ and $F'(0)=1$. In this paper, we obtain global sharp bounds for the Faber coefficients of the functions $F(z)$ in certain related classes and subclasses of $S(E).$
The local dimensions of the Bernoulli convolution associated with the golden number
Tian-You
Hu
2917-2940
Abstract: Let $X_1,X_2,\dotsc$ be a sequence of i.i.d. random variables each taking values of 1 and $-1$ with equal probability. For $1/2<\rho <1$ satisfying the equation $1-\rho -\dotsb -\rho ^s=0$, let $\mu$ be the probability measure induced by $S=\sum _{i=1}^\infty \rho ^iX_i$. For any $x$ in the range of $S$, let \begin{displaymath}d(\mu ,x)=\lim _{r\to 0^+}\log \mu([x-r,x+r])/\log r\end{displaymath} be the local dimension of $\mu$ at $x$ whenever the limit exists. We prove that \begin{displaymath}\alpha ^*=-\frac {\log 2}{\log \rho}\quad \text{and}\quad \alpha _*=-\frac {\log \delta }{s\log \rho}-\frac {\log 2}{\log \rho},\end{displaymath} where $\delta =(\sqrt {5}-1)/2$, are respectively the maximum and minimum values of the local dimensions. If $s=2$, then $\rho$ is the golden number, and the approximate numerical values are $\alpha ^*\approx 1.4404$ and $\alpha _*\approx 0.9404$.
Bloch constants of bounded symmetric domains
Genkai
Zhang
2941-2949
Abstract: Let $D_{1}$ and $D_{2}$ be two irreducible bounded symmetric domains in the complex spaces $V_{1}$ and $V_{2}$ respectively. Let $E$ be the Euclidean metric on $V_{2}$ and $h$ the Bergman metric on $V_{1}$. The Bloch constant $b(D_{1}, D_{2})$ is defined to be the supremum of $E(f^{\prime }(z)x, f^{\prime }(z)x)^{\frac {1}{2}}/h_{z}(x, x)^{1/2}$, taken over all the holomorphic functions $f: D_{1}\to D_{2}$ and $z\in D_{1}$, and nonzero vectors $x\in V_{1}$. We find the constants for all the irreducible bounded symmetric domains $D_{1}$ and $D_{2}$. As a special case we answer an open question of Cohen and Colonna.
An isometry theorem for quadratic differentials on Riemann surfaces of finite genus
Nikola
Lakic
2951-2967
Abstract: Assume both $X$ and $Y$ are Riemann surfaces which are subsets of compact Riemann surfaces $X_1$ and $Y_1,$ respectively, and that the set $ X_1 - X$ has infinitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic differentials on $X$ and $Y$ are the ones induced by conformal homeomorphisms and complex constants of modulus 1. It follows that every biholomorphic map from the Teichmüller space of $X$ onto the Teichmüller space of $Y$ is induced by some quasiconformal map of $X$ onto $Y$. Consequently we can find an uncountable set of Riemann surfaces whose Teichmüller spaces are not biholomorphically equivalent.
Subgroups of finite soluble groups inducing the same permutation character
Norberto
Gavioli
2969-2980
Abstract: In this paper there are found necessary and sufficient conditions that a pair of solvable finite groups, say $G$ and $K$, must satisfy for the existence of a solvable finite group $L$ containing two isomorphic copies of $G$ and $H$ inducing the same permutation character. Also a construction of $L$ is given as an iterated wreath product, with respect to their actions on their natural modules, of finite one-dimensional affine groups.
Ramanujan's class invariants, Kronecker's limit formula, and modular equations
Bruce
C.
Berndt;
Heng Huat
Chan;
Liang-Cheng
Zhang
2125-2173
Abstract: In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated by Heinrich Weber, but approximately half of them had not been heretofore determined. G. N. Watson wrote several papers devoted to the calculation of class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan's class invariants remained to be verified. Five were verified by the authors in a recent paper. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are two classes per genus. The authors devised three methods to calculate these thirteen class invariants. The first depends upon Kronecker's limit formula, the second employs modular equations, and the third uses class field theory to make Watson's ``empirical method''rigorous.
A fixed point index for generalized inward mappings of condensing type
Kunquan
Lan;
Jeffrey
Webb
2175-2186
Abstract: A fixed point index is defined for mappings defined on a cone $K$ which do not necessarily take their values in $K$ but satisfy a weak type of boundary condition called generalized inward. This class strictly includes the well-known weakly inward class. New results for existence of multiple fixed points are established.
Resultants and the algebraicity of the join pairing on Chow varieties
Judith
Plümer
2187-2209
Abstract: The Chow/Van der Waerden approach to algebraic cycles via resultants is used to give a purely algebraic proof for the algebraicity of the complex suspension. The algebraicity of the join pairing on Chow varieties then follows. The approach implies a more algebraic proof of Lawson's complex suspension theorem in characteristic 0. The continuity of the action of the linear isometries operad on the group completion of the stable Chow variety is a consequence.
Proximity inequalities and bounds for the degree of invariant curves by foliations of \mathbb{P}_{\mathbb{C}}^2$
Antonio
Campillo;
Manuel
M.
Carnicer
2211-2228
Abstract: In this paper we prove that if $C$ is a reduced curve which is invariant by a foliation $\mathcal F$ in the complex projective plane then one has $\partial ^{\underline {\circ }} C\leq \partial^{\underline {\circ }} \mathcal F+2+a$ where $a$ is an integer obtained from a concrete problem of imposing singularities to projective plane curves. If $\mathcal F$ is nondicritical or if $C$ has only nodes as singularities, then one gets $a=0$ and we recover known bounds. We also prove proximity formulae for foliations and we use these formulae to give relations between local invariants of the curve and the foliation.
$L^2$-homology over traced *-algebras
William
L.
Paschke
2229-2251
Abstract: Given a unital complex *-algebra $A$, a tracial positive linear functional $\tau$ on $A$ that factors through a *-representation of $A$ on Hilbert space, and an $A$-module $M$ possessing a resolution by finitely generated projective $A$-modules, we construct homology spaces $H_k(A,\tau ,M)$ for $k = 0, 1, \ldots$. Each is a Hilbert space equipped with a *-representation of $A$, independent (up to unitary equivalence) of the given resolution of $M$. A short exact sequence of $A$-modules gives rise to a long weakly exact sequence of homology spaces. There is a Künneth formula for tensor products. The von Neumann dimension which is defined for $A$-invariant subspaces of $L^2(A,\tau )^n$ gives well-behaved Betti numbers and an Euler characteristic for $M$ with respect to $A$ and $\tau$.
Second variation of superminimal surfaces into self-dual Einstein four-manifolds
Sebastián
Montiel;
Francisco
Urbano
2253-2269
Abstract: The index of a compact orientable superminimal surface of a self-dual Einstein four-manifold $M$ with positive scalar curvature is computed in terms of its genus and area. Also a lower bound of its nullity is obtained. Applications to the cases $M=\mathbb {S}^4$ and $M=\mathbb {C}\mathbb {P}^2$ are given, characterizing the standard Veronese immersions and their twistor deformations as those with lowest index.
Discrete tomography: Determination of finite sets by X-rays
R.
J.
Gardner;
Peter
Gritzmann
2271-2295
Abstract: We study the determination of finite subsets of the integer lattice ${\Bbb Z}^n$, $n\ge 2$, by X-rays. In this context, an X-ray of a set in a direction $u$ gives the number of points in the set on each line parallel to $u$. For practical reasons, only X-rays in lattice directions, that is, directions parallel to a nonzero vector in the lattice, are permitted. By combining methods from algebraic number theory and convexity, we prove that there are four prescribed lattice directions such that convex subsets of ${\Bbb Z}^n$ (i.e., finite subsets $F$ with $F={\Bbb Z}^n\cap {\mathrm {conv}}\,F$) are determined, among all such sets, by their X-rays in these directions. We also show that three X-rays do not suffice for this purpose. This answers a question of Larry Shepp, and yields a stability result related to Hammer's X-ray problem. We further show that any set of seven prescribed mutually nonparallel lattice directions in ${\Bbb Z}^2$ have the property that convex subsets of ${\Bbb Z}^2$ are determined, among all such sets, by their X-rays in these directions. We also consider the use of orthogonal projections in the interactive technique of successive determination, in which the information from previous projections can be used in deciding the direction for the next projection. We obtain results for finite subsets of the integer lattice and also for arbitrary finite subsets of Euclidean space which are the best possible with respect to the numbers of projections used.
Doodle groups
Mikhail
Khovanov
2297-2315
Abstract: A doodle is a finite number of closed curves without triple intersections on an oriented surface. There is a ``fundamental'' group, naturally associated with a doodle. In this paper we study these groups, in particular, we show that fundamental groups of some doodles are automatic and give examples of doodles whose fundamental groups have non-trivial center.
Algebras associated to elliptic curves
Darin
R.
Stephenson
2317-2340
Abstract: This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one. In particular, we study the elliptic algebras $A(+)$, $A(-)$, and $A({\mathbf {a}})$, where ${\mathbf {a}}\in \mathbb {P}^{2}$, which were first defined in an earlier paper. We omit a set $S\subset \mathbb {P}^2$ consisting of 11 specified points where the algebras $A({\mathbf {a}})$ become too degenerate to be regular. Theorem. Let $A$ represent $A(+)$, $A(-)$ or $A({\mathbf {a}})$, where ${\mathbf {a}} \in \mathbb {P}^2\setminus S$. Then $A$ is an Artin-Schelter regular algebra of global dimension three. Moreover, $A$ is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. This, combined with our earlier results, completes the classification.
Unramified cohomology and Witt groups of anisotropic Pfister quadrics
R.
Sujatha
2341-2358
Abstract: The unramified Witt group of an anisotropic conic over a field $k$, with $char~k \neq 2$, defined by the form $\langle 1,-a,-b\rangle$ is known to be a quotient of the Witt group $W(k)$ of $k$ and isomorphic to $W( {k})/\langle 1,-a,-b,ab \rangle W( {k})$. We compute the unramified cohomology group $H^{3}_{nr}{k({C})}$, where $C$ is the three dimensional anisotropic quadric defined by the quadratic form $\langle 1,-a,-b,ab,-c\rangle$ over $k$. We use these computations to study the unramified Witt group of $C$.
A finiteness theorem for low-codimensional nonsingular subvarieties of quadrics
Mark
Andrea A.
de Cataldo
2359-2370
Abstract: We prove that there are only finitely many families of codimension two nonsingular subvarieties of quadrics $\mathcal {Q}^{n}$ which are not of general type, for $n=5$ and $n\geq 7$. We prove a similar statement also for the case of higher codimension.
On singly-periodic minimal surfaces with planar ends
Joaquín
Pérez
2371-2389
Abstract: The spaces of nondegenerate properly embedded minimal surfaces in quotients of ${\mathbf R}^3$ by nontrivial translations or by screw motions with nontrivial rotational part, fixed finite topology and planar type ends, are endowed with natural structures of finite dimensional real analytic manifolds. This nondegeneracy is defined in terms of Jacobi functions. Riemann's minimal examples are characterized as the only nondegenerate surfaces with genus one in their corresponding spaces. We also give natural immersions of these spaces into certain complex Euclidean spaces which turn out to be Lagrangian immersions with respect to the standard symplectic structures.
The stretch of a foliation and geometric superrigidity
Raul
Quiroga-Barranco
2391-2426
Abstract: We consider compact smooth foliated manifolds with leaves isometrically covered by a fixed symmetric space of noncompact type. Such objects can be considered as compact models for the geometry of the symmetric space. Based on this we formulate and solve a geometric superrigidity problem for foliations that seeks the existence of suitable isometric totally geodesic immersions. To achieve this we consider the heat flow equation along the leaves of a foliation, a Bochner formula on foliations and a geometric invariant for foliations with leafwise Riemannian metrics called the stretch. We obtain as applications a metric rigidity theorem for foliations and a rigidity type result for Riemannian manifolds whose geometry is only partially symmetric.
On roots of random polynomials
Ildar
Ibragimov;
Ofer
Zeitouni
2427-2441
Abstract: We study the distribution of the complex roots of random polynomials of degree $n$ with i.i.d. coefficients. Using techniques related to Rice's treatment of the real roots question, we derive, under appropriate moment and regularity conditions, an exact formula for the average density of this distribution, which yields appropriate limit average densities. Further, using a different technique, we prove limit distribution results for coefficients in the domain of attraction of the stable law.
Some uniqueness and exact multiplicity results for a predator-prey model
Yihong
Du;
Yuan
Lou
2443-2475
Abstract: In this paper, we consider positive solutions of a predator-prey model with diffusion and under homogeneous Dirichlet boundary conditions. It turns out that a certain parameter $m$ in this model plays a very important role. A good understanding of the existence, stability and number of positive solutions is gained when $m$ is large. In particular, we obtain various results on the exact number of positive solutions. Our results for large $m$ reveal interesting contrast with that for the well-studied case $m=0$, i.e., the classical Lotka-Volterra predator-prey model.
Degenerations of K3 surfaces in projective space
Francisco
Javier
Gallego;
B.
P.
Purnaprajna
2477-2492
Abstract: The purpose of this article is to study a certain kind of numerical K3 surfaces, the so-called K3 carpets. These are double structures on rational normal scrolls with trivial dualizing sheaf and irregularity $0$. As is deduced from our study, K3 carpets can be obtained as degenerations of smooth K3 surfaces. We also study the Hilbert scheme near the locus parametrizing K3 carpets, characterizing those K3 carpets whose corresponding Hilbert point is smooth. Contrary to the case of canonical ribbons, not all K3 carpets are smooth points of the Hilbert scheme.
The Floer homotopy type of height functions on complex Grassmann manifolds
David
E.
Hurtubise
2493-2505
Abstract: A family of Floer functions on the infinite dimensional complex Grassmann manifold is defined by taking direct limits of height functions on adjoint orbits of unitary groups. The Floer cohomology of a generic function in the family is computed using the Schubert calculus. The Floer homotopy type of this function is computed and the Floer cohomology which was computed algebraically is recovered from the Floer homotopy type. Certain non-generic elements of this family of Floer functions were shown to be related to the symplectic action functional on the universal cover of the loop space of a finite dimensional complex Grassmann manifold in the author's preprint The Floer homotopy type of complex Grassmann manifolds.
Differential operators on Stanley-Reisner rings
J.
R.
Tripp
2507-2523
Abstract: Let $k$ be an algebraically closed field of characteristic zero, and let $R=k[x_{1},\dots ,x_{n}]$ be a polynomial ring. Suppose that $I$ is an ideal in $R$ that may be generated by monomials. We investigate the ring of differential operators $\mathcal {D}(R/I)$ on the ring $R/I$, and $\mathcal {I}_{R}(I)$, the idealiser of $I$ in $R$. We show that $\mathcal {D}(R/I)$ and $\mathcal {I}_{R}(I)$ are always right Noetherian rings. If $I$ is a square-free monomial ideal then we also identify all the two-sided ideals of $\mathcal {I}_{R}(I)$. To each simplicial complex $\Delta$ on $V=\{v_{1},\dots ,v_{n}\}$ there is a corresponding square-free monomial ideal $I_{\Delta }$, and the Stanley-Reisner ring associated to $\Delta$ is defined to be $k[\Delta ]=R/I_{\Delta }$. We find necessary and sufficient conditions on $\Delta$ for $\mathcal {D}(k[\Delta ])$ to be left Noetherian.
Herz-Schur multipliers and weakly almost periodic functions on locally compact groups
Guangwu
Xu
2525-2536
Abstract: For a locally compact group $G$ and $1<p<\infty$, let $A_{p}(G)$ be the Herz-Figà-Talamanca algebra and $B_{p}(G)$ the Herz-Schur multipliers of $G$, and $MA_{p}(G)$ the multipliers of $A_{p}(G)$. Let $W(G)$ be the algebra of continuous weakly almost periodic functions on $G$. In this paper, we show that (1), if $G$ is a noncompact nilpotent group or a noncompact [IN]-group, then $W(G)/B_{p}(G)^{-}$ contains a linear isometric copy of $l^{\infty }({\mathbb {N}})$; (2), for a noncommutative free group $F, B_{p}(F)$ is a proper subset of ${MA_{p}(F)\cap {W(F)}}$.
Structural properties of universal minimal dynamical systems for discrete semigroups
Bohuslav
Balcar;
Frantisek
Franek
1697-1724
Abstract: We show that for a discrete semigroup $S$ there exists a uniquely determined complete Boolean algebra $B(S)$ - the algebra of clopen subsets of $M(S)$. $M(S)$ is the phase space of the universal minimal dynamical system for $S$ and it is an extremally disconnected compact Hausdorff space. We deal with this connection of semigroups and complete Boolean algebras focusing on structural properties of these algebras. We show that $B(S)$ is either atomic or atomless; that $B(S)$ is weakly homogenous provided $S$ has a minimal left ideal; and that for countable semigroups $B(S)$ is semi-Cohen. We also present a class of what we call group-like semigroups that includes commutative semigroups, inverse semigroups, and right groups. The group reflection $G(S)$ of a group-like semigroup $S$ can be constructed via universal minimal dynamical system for $S$ and, moreover, $B(S)$ and $B(G(S))$ are the same.
$C^*$-extreme points in the generalised state spaces of a $C^*$-algebra
Douglas
R.
Farenick;
Phillip
B.
Morenz
1725-1748
Abstract: In this paper we study the space $S_{H}(A)$ of unital completely positive linear maps from a $C^{*}$-algebra $A$ to the algebra $B(H)$ of continuous linear operators on a complex Hilbert space $H$. The state space of $A$, in this notation, is $S_{\mathbb {C}}(A)$. The main focus of our study concerns noncommutative convexity. Specifically, we examine the $C^{*}$-extreme points of the $C^{*}$-convex space $S_{H}(A)$. General properties of $C^{*}$-extreme points are discussed and a complete description of the set of $C^{*}$-extreme points is given in each of the following cases: (i) the cases $S_{{\mathbb {C}}^{2}}(A)$, where $A$ is arbitrary ; (ii) the cases $S_{{\mathbb {C}}^{r}}(A)$, where $A$ is commutative; (iii) the cases $S_{{\mathbb {C}}^{r}}(M_{n})$, where $M_{n}$ is the $C^{*}$-algebra of $n\times n$ complex matrices. An analogue of the Krein-Milman theorem will also be established.
A Characterization of Minimal Locally Finite Varieties
Keith
A.
Kearnes;
Ágnes
Szendrei
1749-1768
Abstract: In this paper we describe a one-variable Mal'cev-like condition satisfied by any locally finite minimal variety. We prove that a locally finite variety is minimal if and only if it satisfies this Mal'cev-like condition and it is generated by a strictly simple algebra which is nonabelian or has a trivial subalgebra. Our arguments show that the strictly simple generator of a minimal locally finite variety is unique, it is projective and it embeds into every member of the variety. We give a new proof of the structure theorem for strictly simple abelian algebras that generate minimal varieties.
Reversion of power series and the extended Raney coefficients
Charles
Ching-An
Cheng;
James
H.
McKay;
Jacob
Towber;
Stuart
Sui-Sheng
Wang;
David
L.
Wright
1769-1782
Abstract: In direct as well as diagonal reversion of a system of power series, the reversion coefficients may be expressed as polynomials in the coefficients of the original power series. These polynomials have coefficients which are natural numbers (Raney coefficients). We provide a combinatorial interpretation for Raney coefficients. Specifically, each such coefficient counts a certain collection of ordered colored trees. We also provide a simple determinantal formula for Raney coefficients which involves multinomial coefficients.
Correlation dimension for iterated function systems
Wai
Chin;
Brian
Hunt;
James
A.
Yorke
1783-1796
Abstract: The correlation dimension of an attractor is a fundamental dynamical invariant that can be computed from a time series. We show that the correlation dimension of the attractor of a class of iterated function systems in $\mathbf {R}^N$ is typically uniquely determined by the contraction rates of the maps which make up the system. When the contraction rates are uniform in each direction, our results imply that for a corresponding class of deterministic systems the information dimension of the attractor is typically equal to its Lyapunov dimension, as conjected by Kaplan and Yorke.
On trace identities and universal eigenvalue estimates for some partial differential operators
Evans
M.
Harrell II;
Joachim
Stubbe II
1797-1809
Abstract: In this article, we prove and exploit a trace identity for the spectra of Schrödinger operators and similar operators. This identity leads to universal bounds on the spectra, which apply to low-lying eigenvalues, eigenvalue asymptotics, and to partition functions (traces of heat operators). In many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated. Special cases corresponding to known inequalities include those of Hile and Protter and of Yang.
Bodies with similar projections
G.
D.
Chakerian;
E.
Lutwak
1811-1820
Abstract: Aleksandrov's projection theorem characterizes centrally symmetric convex bodies by the measures of their orthogonal projections on lower dimensional subspaces. A general result proved here concerning the mixed volumes of projections of a collection of convex bodies has the following corollary. If $K$ is a convex body in ${\mathbb {R}}^{n}$ whose projections on $r$-dimensional subspaces have the same $r$-dimensional volume as the projections of a centrally symmetric convex body $M$, then the Quermassintegrals satisfy $W_{j}(M)\ge W_{j}(K)$, for $0\le j < n-r$, with equality, for any $j$, if and only if $K$ is a translate of $M$. The case where $K$ is centrally symmetric gives Aleksandrov's projection theorem.
The homotopy groups of the $L_2$-localized Toda-Smith spectrum $V(1)$ at the prime 3
Katsumi
Shimomura
1821-1850
Abstract: In this paper, we try to compute the homotopy groups of the $L_2$-localized Toda-Smith spectrum $V(1)$ at the prime 3 by using the Adams-Novikov spectral sequence, and have almost done so. This computation involves non-trivial differentials $d_5$ and $d_9$ of the Adams-Novikov spectral sequence, different from the case $p>3$. We also determine the homotopy groups of some $L_2$-localized finite spectra relating to $V(1)$. We further show some of the non-trivial differentials on elements relating so-called $\beta$-elements in the Adams-Novikov spectral sequence for $\pi _*(S^0)$.
Small cancellation groups and translation numbers
Ilya
Kapovich
1851-1875
Abstract: In this paper we prove that C(4)-T(4)-P, C(3)-T(6)-P and C(6)-P small cancellation groups are translation discrete in the strongest possible sense and that in these groups for any $g$ and any $n$ there is an algorithm deciding whether or not the equation $x^{n}=g$ has a solution. There is also an algorithm for calculating for each $g$ the maximum $n$ such that $g$ is an $n$-th power of some element. We also note that these groups cannot contain isomorphic copies of the group of $p$-adic fractions and so in particular of the group of rational numbers. Besides we show that for $C^{\prime \prime }(4)-T(4)$ and $C''(3)-T(6)$ groups all translation numbers are rational and have bounded denominators.
Boundary value maps, Szegö maps and intertwining operators
L.
Barchini
1877-1900
Abstract: We consider one series of unitarizable representations, the cohomological induced modules $A_{\mathfrak {q}}(\lambda )$ with dominant regular infinitesimal character. The minimal $K$-type $(\tau , V)$ of $A_{\mathfrak {q}}(\lambda )$ determines a homogeneous vector bundle $V_{\tau } \longrightarrow G/K$. The derived functor modules can be realized on the solution space of a first order differential operator $\mathcal {D}_{\mathfrak {l}}^{\lambda }$ on $V_{\tau }$. Barchini, Knapp and Zierau gave an explicit integral map $\mathcal {S}$ from the derived functor module, realized in the Langlands classification, into the space of smooth sections of the vector bundle $V_{\tau } \longrightarrow G/K$. In this paper we study the asymptotic behavior of elements in the image of $\mathcal {S}$. We obtain a factorization of the standard intertwining opeartors into the composition of the Szegö map $\mathcal {S}$ and a passage to boundary values.
On the extremal structure of the unit balls of Banach spaces of weakly continuous functions and their duals
Zhibao
Hu;
Mark
A.
Smith
1901-1918
Abstract: A sufficient and then a necessary condition are given for a function to be an extreme point of the unit ball of the Banach space $C(K,(X,w))$ of continuous functions, under the supremum norm, from a compact Hausdorff topological space $K$ into a Banach space $X$ equipped with its weak topology $w$. Strongly extreme points of the unit ball of $C(K,(X,w))$ are characterized as the norm-one functions that are uniformly strongly extreme point valued on a dense subset of $K$. It is shown that a variety of stronger types of extreme points (e.g. denting points) never exist in the unit ball of $C(K,(X,w))$. Lastly, some naturally arising and previously known extreme points of the unit ball of $C(K,(X,w))^{*}$ are shown to actually be strongly exposed points.
Higher-dimensional virtual diagonals and ideal cohomology for triangular algebras
Alan
L. T.
Paterson;
Roger
R.
Smith
1919-1943
Abstract: We investigate the cohomology of non-self-adjoint algebras using virtual diagonals and their higher-dimensional generalizations. We show that infinite dimensional nest algebras always have non-zero second cohomology by showing that they cannot possess 2-virtual diagonals. In the case of the upper triangular atomic nest algebra we exhibit concrete modules for non-vanishing cohomology.
Principal Eigenvalues with Indefinite Weight Functions
Zhiren
Jin
1945-1959
Abstract: Both existence and non-existence results for principal eigenvalues of an elliptic operator with indefinite weight function have been proved. The existence of a continuous family of principal eigenvalues is demonstrated.
Singular set of some Kähler orbifolds
Thalia
D.
Jeffres
1961-1971
Abstract: We consider some examples of orbifolds with positive first Chern class. Applying a result of Ding and Tian, we show that the singularities must be very mild if the orbifold admits a Kähler-Einstein metric.
Partial regularity of solutions to a class of degenerate systems
Xiangsheng
Xu
1973-1992
Abstract: We consider the system $\displaystyle \frac{\partial u }{\partial t}-\Delta u=\sigma \left ( u\right ) \left | \nabla \varphi \right | ^2$, $\mathrm {div}\left ( \sigma \left ( u\right ) \nabla \varphi \right ) =0$ in $Q_T\equiv \Omega \times \left ( 0,T\right ]$ coupled with suitable initial-boundary conditions, where $\Omega$ is a bounded domain in $\mathbf {R}^N$ with smooth boundary and $\sigma \left ( u\right )$ is a continuous and positive function of $u$. Our main result is that under some conditions on $\sigma$ there exists a relatively open subset $Q_0$ of $Q_T$ such that $u$ is locally Hölder continuous on $Q_0$, the interior of $Q_T\backslash Q_0$ is empty, and $u$ is essentially bounded on $Q_T\backslash Q_0$.
Classification problems for shifts on modules over a principal ideal domain
Fabio
Fagnani;
Sandro
Zampieri
1993-2006
Abstract: In this paper we study symbolic dynamics over alphabets which are modules over a principal ideal domain, considering topological shifts which are also submodules. Our main result is the classification, up to algebraic and topological conjugacy, of the torsion-free, transitive, finite memory shifts.
Menas' Result is Best Possible
Arthur
W.
Apter;
Saharon
Shelah
2007-2034
Abstract: Generalizing some earlier techniques due to the second author, we show that Menas' theorem which states that the least cardinal $\kappa$ which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not $2^{\kappa }$ supercompact is best possible. Using these same techniques, we also extend and give a new proof of a theorem of Woodin and extend and give a new proof of an unpublished theorem due to the first author.
Periodic solutions to nonlinear one dimensional wave equation with $X$-dependent coefficients
V.
Barbu;
N.
H.
Pavel
2035-2048
Abstract: This paper deals with $t$-periodicity and regularity of solutions to the one dimensional nonlinear wave equation with $x$-dependent coefficients
Wavelet transform and orthogonal decomposition of space $L^2$ on the Cartan domain $BDI(q=2)$
Qingtang
Jiang
2049-2068
Abstract: Let $G=\left ({\mathbb {R}}^{*}_{+}\times SO_{0}(1, n)\right ) \ltimes {\mathbb {R}}^{n+1}$ be the Weyl-Poincaré group and $KAN$ be the Iwasawa decomposition of $SO_{0}(1, n)$ with $K=SO(n)$. Then the ``affine Weyl-Poincaré group'' $G_{a}=\left ({\mathbb {R}}^{*}_{+}\times AN\right ) \ltimes {\mathbb {R}}^{n+1}$ can be realized as the complex tube domain $\Pi ={\mathbb {R}}^{n+1}+iC$ or the classical Cartan domain $BDI(q=2)$. The square-integrable representations of $G$ and $G_{a}$ give the admissible wavelets and wavelet transforms. An orthogonal basis $\{ \psi _{k}\}$ of the set of admissible wavelets associated to $G_{a}$ is constructed, and it gives an orthogonal decomposition of $L^{2}$ space on $\Pi$ (or the Cartan domain $BDI(q=2)$) with every component $A_{k}$ being the range of wavelet transforms of functions in $H^{2}$ with $\psi _{k}$.
Twin trees and $\lambda_{\Lambda}$-gons
Curtis
D.
Bennett
2069-2084
Abstract: We define a natural generalization of generalized $n$-gons to the case of $\Lambda$-graphs (where $\Lambda$ is a totally ordered abelian group and $0<\lambda \in \Lambda$). We term these objects $\lambda _{\Lambda }$-gons. We then show that twin trees as defined by Ronan and Tits can be viewed as $(1,0)_{\Lambda }$-gons, where $\Lambda = Z \times Z$ is ordered lexicographically. This allows us to then generalize twin trees to the case of $\Lambda$-trees. Finally, we give a free construction of $\lambda _{\Lambda }$-gons in the cases where $\Lambda$ is discrete and has a subgroup of index $2$ that does not contain the minimal element of $\Lambda$.
Duality of restriction and induction for $C^*$-coactions
S.
Kaliszewski;
John
Quigg;
Iain
Raeburn
2085-2113
Abstract: Consider a coaction $\delta$ of a locally compact group $G$ on a $C^*$- algebra $A$, and a closed normal subgroup $N$ of $G$. We prove, following results of Echterhoff for abelian $G$, that Mansfield's imprimitivity between $A\times _{\delta |}G/N$ and $A\times _\delta G\times _{\hat {\delta } ,r}N$ implements equivalences between Mansfield induction of representations from $A\times _{\delta |}G/N$ to $A\times _\delta G$ and restriction of representations from $A\times _\delta G\times _{\hat {\delta } ,r}N$ to $A\times _\delta G$, and between restriction of representations from $A\times _\delta G$ to $A\times _{\delta |}G/N$ and Green induction of representations from $A\times _\delta G$ to $A\times _\delta G\times _{\hat {\delta } ,r}N$. This allows us to deduce properties of Mansfield induction from the known theory of ordinary crossed products.
Another note on Weyl's theorem
Robin
Harte;
Woo
Young
Lee
2115-2124
Abstract: ``Weyl's theorem holds" for an operator $T$ on a Banach space $X$ when the complement in the spectrum of the ``Weyl spectrum" coincides with the isolated points of spectrum which are eigenvalues of finite multiplicity. This is close to, but not quite the same as, equality between the Weyl spectrum and the ``Browder spectrum", which in turn ought to, but does not, guarantee the spectral mapping theorem for the Weyl spectrum of polynomials in $T$. In this note we try to explore these distinctions.
Some combinatorial aspects of reduced words in finite Coxeter groups
John
R.
Stembridge
1285-1332
Abstract: We analyze the structure of reduced expressions in the Coxeter groups $A_n$, $B_n$ and $D_n$. Several special classes of elements are singled out for their connections with symmetric functions or the theory of $P$-partitions. Membership in these special classes is characterized in a variety of ways, including forbidden patterns, forbidden subwords, and by the form of canonically chosen reduced words.
Disjoint paths, planarizing cycles, and spanning walks
Xingxing
Yu
1333-1358
Abstract: We study the existence of certain disjoint paths in planar graphs and generalize a theorem of Thomassen on planarizing cycles in surfaces. Results are used to prove that every 5-connected triangulation of a surface with sufficiently large representativity is hamiltonian, thus verifying a conjecture of Thomassen. We also obtain results about spanning walks in graphs embedded in a surface with large representativity.
Randomness and semigenericity
John
T.
Baldwin;
Saharon
Shelah
1359-1376
Abstract: Let $L$ contain only the equality symbol and let $L^+$ be an arbitrary finite symmetric relational language containing $L$. Suppose probabilities are defined on finite $L^+$ structures with `edge probability' $n^{-\alpha }$. By $T^{\alpha }$, the almost sure theory of random $L^+$-structures we mean the collection of $L^+$-sentences which have limit probability 1. $T_{\alpha }$ denotes the theory of the generic structures for ${\mathbf {K}} _{\alpha }$ (the collection of finite graphs $G$ with $\delta _{\alpha }(G) =|G| - \alpha \cdot |\text { edges of$G$}|$ hereditarily nonnegative). Theorem.. $T^{\alpha }$, the almost sure theory of random $L^+$-structures, is the same as the theory $T_{\alpha }$ of the ${\mathbf {K}} _{\alpha }$-generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.
There are no piecewise linear maps of type $2^{\infty}$
Víctor
Jiménez
López;
L'ubomír
Snoha
1377-1387
Abstract: The aim of this paper is to show that there are no piecewise linear maps of type $2^{\infty }$. For this purpose we use the fact that any piecewise monotone map of type $2^{\infty }$ has an infinite $\omega$-limit set which is a subset of a doubling period solenoid. Then we prove that piecewise linear maps cannot have any doubling period solenoids.
Group actions on arrangements of linear subspaces and applications to configuration spaces
Sheila
Sundaram;
Volkmar
Welker
1389-1420
Abstract: For an arrangement of linear subspaces in ${\mathbb R} ^n$ that is invariant under a finite subgroup of the general linear group $Gl_n({\mathbb R} )$ we develop a formula for the $G$-module structure of the cohomology of the complement ${\mathcal M} _{\mathcal A}$. Our formula specializes to the well known Goresky-MacPherson theorem in case $G = 1$, but for $G \neq 1$ the formula shows that the $G$-module structure of the complement is not a combinatorial invariant. As an application we are able to describe the free part of the cohomology of the quotient space ${\mathcal M} _{\mathcal A} /G$. Our motivating examples are arrangements in ${\mathbb C} ^n$ that are invariant under the action of $S_n$ by permuting coordinates. A particular case is the ``$k$-equal'' arrangement, first studied by Björner, Lovász, and Yao motivated by questions in complexity theory. In these cases ${\mathcal M} _{\mathcal A}$ and ${\mathcal M} _{\mathcal A} /S_n$ are spaces of ordered and unordered point configurations in ${\mathbb C} ^n$ many of whose properties are reduced by our formulas to combinatorial questions in partition lattices. More generally, we treat point configurations in ${\mathbb R} ^d$ and provide explicit results for the ``$k$-equal'' and the ``$k$-divisible'' cases.
Convex integral functionals
Nikolaos
S.
Papageorgiou
1421-1436
Abstract: We study nonlinear integral functionals determined by normal convex integrands. First we obtain expressions for their convex conjugate, their $\varepsilon$-subdifferential $(\varepsilon \ge 0)$ and their $\varepsilon$-directional derivative. Then we derive a necessary and sufficient condition for the existence of an approximate solution for the continuous infimal convolution. We also obtain general conditions which guarantee the interchangeability of the conditional expectation and subdifferential operators. Finally we examine the conditional expectation of random sets.
$p$-adic Power Series which Commute under Composition
Hua-Chieh
Li
1437-1446
Abstract: When two noninvertible series commute to each other, they have same set of roots of iterates. Most of the results of this paper will be concerned with the problem of which series commute with a given noninvertible series. Our main theorem is a generalization of Lubin's result about isogenies of formal groups.
Integer translation of meromorphic functions
Jeong
H.
Kim;
Lee
A.
Rubel
1447-1462
Abstract: Let $G$ be a given open set in the complex plane. We prove that there is an entire function such that its integer translations forms a normal family in a neighborhood of $z$ exactly for $z$ in $G$ if and only if $G$ is periodic with period 1, i.e., $z\pm 1\in G$ for all $z\in G$.
Essential laminations in $I$-bundles
Mark
Brittenham
1463-1485
Abstract: We show that, with a few familiar exceptions, every essential lamination in an interval-bundle over a closed surface can be isotoped to lie everywhere transverse to the $I$-fibers of the bundle.
On The Homotopy Type of $BG$ for Certain Finite 2-Groups $G$
Carlos
Broto;
Ran
Levi
1487-1502
Abstract: We consider the homotopy type of classifying spaces $BG$, where $G$ is a finite $p$-group, and we study the question whether or not the mod $p$ cohomology of $BG$, as an algebra over the Steenrod algebra together with the associated Bockstein spectral sequence, determine the homotopy type of $BG$. This article is devoted to producing some families of finite 2-groups where cohomological information determines the homotopy type of $BG$.
The Group of Galois Extensions Over Orders in $KC_{p^2}$
Robert
Underwood
1503-1514
Abstract: In this paper we characterize all Galois extensions over $H$ where $H$ is an arbitrary $R$-Hopf order in $KC_{p^{2}}$. We conclude that the abelian group of $H$-Galois extensions is isomorphic to a certain quotient of units groups in $R\times R$. This result generalizes the classification of $H$-Galois extensions, where $H\subset KC_{p}$, due to Roberts, and also to Hurley and Greither.
A four-dimensional deformation of a numerical Godeaux surface
Caryn
Werner
1515-1525
Abstract: A numerical Godeaux surface is a surface of general type with invariants $p_g =q =0$ and $K^2 =1$. In this paper the moduli space of a numerical Godeaux surface with order two torsion is computed to be eight-dimensional; whether or not the moduli space of such a surface is irreducible is still unknown. The surface in this paper is constructed as one member of a four parameter family of double planes. There is a natural involution on the surface, inherited from the double plane construction, which acts on the moduli space. We show that the invariant subspace is four-dimensional and coincides with the family of double planes.
Intersection Lawson homology
Pawel
Gajer
1527-1550
Abstract: The aim of this paper is to construct and describe basic properties of a theory that unifies Lawson homology and intersection homology. It is shown that this theory has a localization sequence, is functorial, satisfies a property analogous to the Lawson Suspension Theorem, and is equipped with an operation analogous to the Friedlander-Mazur $\mathbf s$-operation.
Parabolic Higgs bundles and Teichmüller spaces for punctured surfaces
Indranil
Biswas;
Pablo
Arés-Gastesi;
Suresh
Govindarajan
1551-1560
Abstract: In this paper we study the relation between parabolic Higgs vector bundles and irreducible representations of the fundamental group of punctured Riemann surfaces established by Simpson. We generalize a result of Hitchin, identifying those parabolic Higgs bundles that correspond to Fuchsian representations. We also study the Higgs bundles that give representations whose image is contained, after conjugation, in SL($k,\mathbb R$). We compute the real dimension of one of the components of this space of representations, which in the absence of punctures is the generalized Teichmüller space introduced by Hitchin, and which in the case of $k=2$ is the usual Teichmüller space of the punctured surface.
On Poincaré Type Inequalities
Roger
Chen;
Peter
Li
1561-1585
Abstract: Using estimates of the heat kernel we prove a Poincaré inequality for star-shape domains on a complete manifold. The method also gives a lower bound for the gap of the first two Neumann eigenvalues of a Schrödinger operator.
On the cohomology of split extensions of finite groups
Stephen
F.
Siegel
1587-1609
Abstract: Let $G=H\rtimes Q$ be a split extension of finite groups. A theorem of Charlap and Vasquez gives an explicit description of the differentials $d_2$ in the Lyndon-Hochschild-Serre spectral sequence of the extension with coefficients in a field $k$. We generalize this to give an explicit description of all the $d_r$ ($r\geq 2$) in this case. The generalization is obtained by associating to the group extension a new twisting cochain, which takes values in the $kG$-endomorphism algebra of the minimal $kH$-projective resolution induced from $H$ to $G$. This twisting cochain not only determines the differentials, but also allows one to construct an explicit $kG$-projective resolution of $k$.
Cohomological construction of quantized universal enveloping algebras
Joseph
Donin;
Steven
Shnider
1611-1632
Abstract: Given an associative algebra $A$ and the category $\mathcal C$ of its finite dimensional modules, additional structures on the algebra $A$ induce corresponding ones on the category $\mathcal C$. Thus, the structure of a rigid quasi-tensor (braided monoidal) category on $Rep_{A}$ is induced by an algebra homomorphism $A\to A\otimes A$ (comultiplication), coassociative up to conjugation by $\Phi \in A^{\otimes 3}$ (associativity constraint) and cocommutative up to conjugation by $\mathcal R\in A^{\otimes 2}$ (commutativity constraint), together with an antiautomorphism (antipode) $S$ of $A$ satisfying the compatibility conditions. A morphism of quasi-tensor structures is given by an element $F\in A^{\otimes 2}$ with suitable induced actions on $\Phi$, $\mathcal R$ and $S$. Drinfeld defined such a structure on $A=U(\mathcal G)[[h]]$ for any semisimple Lie algebra $\mathcal {G}$ with the usual comultiplication and antipode but nontrivial $\mathcal R$ and $\Phi$, and proved that the corresponding quasi-tensor category is isomomorphic to the category of representations of the Drinfeld-Jimbo (DJ) quantum universal enveloping algebra (QUE), $U_{h}(\mathcal G)$. In the paper we give a direct cohomological construction of the $F$ which reduces $\Phi$ to the trivial associativity constraint, without any assumption on the prior existence of a strictly coassociative QUE. Thus we get a new approach to the DJ quantization. We prove that $F$ can be chosen to satisfy some additional invariance conditions under (anti)automorphisms of $U(\mathcal G )[[h]]$, in particular, $F$ gives an isomorphism of rigid quasi-tensor categories. Moreover, we prove that for pure imaginary values of the deformation parameter, the elements $F$, $R$ and $\Phi$ can be chosen to be formal unitary operators on the second and third tensor powers of the regular representation of the Lie group associated to $\mathcal G$ with $\Phi$ depending only on even powers of the deformation parameter. In addition, we consider some extra properties of these elements and give their interpretation in terms of additional structures on the relevant categories.
Topological conjugacy of linear endomorphisms of the 2-torus
Roy
Adler;
Charles
Tresser;
Patrick
A.
Worfolk
1633-1652
Abstract: We describe two complete sets of numerical invariants of topological conjugacy for linear endomorphisms of the two-dimensional torus, i.e., continuous maps from the torus to itself which are covered by linear maps of the plane. The trace and determinant are part of both complete sets, and two candidates are proposed for a third (and last) invariant which, in both cases, can be understood from the topological point of view. One of our invariants is in fact the ideal class of the Latimer-MacDuffee-Taussky theory, reformulated in more elementary terms and interpreted as describing some topology. Merely, one has to look at how closed curves on the torus intersect their image under the endomorphism. Part of the intersection information (the intersection number counted with multiplicity) can be captured by a binary quadratic form associated to the map, so that we can use the classical theories initiated by Lagrange and Gauss. To go beyond the intersection number, and shortcut the classification theory for quadratic forms, we use the rotation number of Poincaré.
De Rham cohomology of logarithmic forms on arrangements of hyperplanes
Jonathan
Wiens;
Sergey
Yuzvinsky
1653-1662
Abstract: The paper is devoted to computation of the cohomology of the complex of logarithmic differential forms with coefficients in rational functions whose poles are located on the union of several hyperplanes of a linear space over a field of characteristic zero. The main result asserts that for a vast class of hyperplane arrangements, including all free and generic arrangements, the cohomology algebra coincides with the Orlik-Solomon algebra. Over the field of complex numbers, this means that the cohomologies coincide with the cohomologies of the complement of the union of the hyperplanes. We also prove that the cohomologies do not change if poles of arbitrary multiplicity are allowed on some of the hyperplanes. In particular, this gives an analogue of the algebraic de Rham theorem for an arbitrary arrangement over an arbitrary field of zero characteristic.
One and two dimensional Cantor-Lebesgue type theorems
J.
Marshall
Ash;
Gang
Wang
1663-1674
Abstract: Let $\varphi (n)$ be any function which grows more slowly than exponentially in $n,$ i.e., $\mathop {limsup}\limits _{n\rightarrow \infty }\varphi (n)^{1/n}\leq 1.$ There is a double trigonometric series whose coefficients grow like $\varphi (n),$ and which is everywhere convergent in the square, restricted rectangular, and one-way iterative senses. Given any preassigned rate, there is a one dimensional trigonometric series whose coefficients grow at that rate, but which has an everywhere convergent partial sum subsequence. There is a one dimensional trigonometric series whose coefficients grow like $\varphi (n),$ and which has the everywhere convergent partial sum subsequence $S_{2^j}.$ For any $p>1,$ there is a one dimensional trigonometric series whose coefficients grow like $\varphi (n^{\frac {p-1}p}),$ and which has the everywhere convergent partial sum subsequence $S_{[j^p]}.$ All these examples exhibit, in a sense, the worst possible behavior. If $m_j$ is increasing and has arbitrarily large gaps, there is a one dimensional trigonometric series with unbounded coefficients which has the everywhere convergent partial sum subsequence $S_{m_j}.$
Primitive higher order embeddings of abelian surfaces
Th.
Bauer;
T.
Szemberg
1675-1683
Abstract: In recent years several concepts of higher order embeddings have been studied: $k$-spannedness, $k$-very ampleness and $k$-jet ampleness. In the present note we consider primitive line bundles on abelian surfaces and give numerical criteria which allow to check whether a given ample line bundle satisfies these properties.
Nonexistence of global solutions of a nonlinear hyperbolic system
Keng
Deng
1685-1696
Abstract: Consider the initial value problem \begin{equation*}\begin {array}{llll} u_{tt} = \Delta u+\vert v\vert ^{p}, & v_{tt} = \Delta v +\vert u\vert ^{q}, &x\in \mathbb {R}^{n},&t>0, [2\jot ] u(x,0)=f(x),&v(x,0)=h(x),&{}&{} [2\jot ] u_{t}(x,0) = g(x), &v_{t}(x,0) = k(x), &{}&{} \end {array} \end{equation*} with $1\le n\le 3$ and $p,q>0$. We show that there exists a bound $B(n) (\le \infty )$ such that if $1<pq<B(n)$ all nontrivial solutions with compact support blow up in finite time.
The conjugacy problem for groups of alternating prime tame links is polynomial-time
Karin
Johnsgard
857-901
Abstract: An alternating projection of a prime link can to used to produce a group presentation (of the link group under free product with the infinite cyclic group) with some useful peculiarities, including small cancellation conditions. In this presentation, conjugacy diagrams are shown to have the form of a tiling of squares in the Euclidean plane in one of a limited number of shapes. An argument based on the shape of the link projection is used to show that the tiling requires no more square tiles than a linear function of word length (with constant multiple based on the number of link crossings). It follows that the computation time for testing conjugacy of two group elements (previously known to be no worse than exponential) is bounded by a cubic polynomial. This bounds complexity in the original link group.
A Characterization of Finitely Decidable Congruence Modular Varieties
Pawel
M.
Idziak
903-934
Abstract: For every finitely generated, congruence modular variety $\mathcal {V}$ of finite type we find a finite family $\cal R$ of finite rings such that the variety $\mathcal {V}$ is finitely decidable if and only if $\mathcal {V}$ is congruence permutable and residually small, all solvable congruences in finite algebras from $\mathcal {V}$ are Abelian, each congruence above the centralizer of the monolith of a subdirectly irreducible algebra $\mathbf {A}$ from $\mathcal {V}$ is comparable with all congruences of $\mathbf {A}$, each homomorphic image of a subdirectly irreducible algebra with a non-Abelian monolith has a non-Abelian monolith, and, for each ring $R$ from $\cal R$, the variety of $R$-modules is finitely decidable.
The homology representations of the $k$-equal partition lattice
Sheila
Sundaram;
Michelle
Wachs
935-954
Abstract: We determine the character of the action of the symmetric group on the homology of the induced subposet of the lattice of partitions of the set $\{1,2,\ldots ,n\}$ obtained by restricting block sizes to the set $\{1,k,k+1,\ldots \}$. A plethystic formula for the generating function of the Frobenius characteristic of the representation is given. We combine techniques from the theory of nonpure shellability, recently developed by Björner and Wachs, with symmetric function techniques, developed by Sundaram, for determining representations on the homology of subposets of the partition lattice.
Existence and nonexistence of global positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary
Mingxin
Wang;
Yonghui
Wu
955-971
Abstract: This paper deals with the existence and nonexistence of global positive solutions to $u_t=\Delta \ln(1+u)$ in $\Omega \times (0, +\infty )$, \begin{displaymath}\frac {\partial \ln(1+u)}{\partial n}=\sqrt {1+u}(\ln (1+u))^{\alpha } \quad \text{on} \partial \Omega \times (0, +\infty ),\end{displaymath} and $u(x, 0)=u_0(x)$ in $\Omega$. Here $\alpha \geq 0$ is a parameter, $\Omega \subset\mathbb {R}^N$ is a bounded smooth domain. After pointing out the mistakes in Global behavior of positive solutions to nonlinear diffusion problems with nonlinear absorption through the boundary, SIAM J. Math. Anal. 24 (1993), 317-326, by N. Wolanski, which claims that, for $\Omega =B_R$ the ball of $\mathbb {R}^N$, the positive solution exists globally if and only if $\alpha \leq 1$, we reconsider the same problem in general bounded domain $\Omega$ and obtain that every positive solution exists globally if and only if $\alpha \leq {1/2}$.
Examples of asymptotic $\ell_1$ Banach spaces
S.
A.
Argyros;
I.
Deliyanni
973-995
Abstract: Two examples of asymptotic $\ell _{1}$ Banach spaces are given. The first, $X_{u}$, has an unconditional basis and is arbitrarily distortable. The second, $X$, does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson's.
Extremal properties of Rademacher functions with applications to the Khintchine and Rosenthal inequalities
T.
Figiel;
P.
Hitczenko;
W.
B.
Johnson;
G.
Schechtman;
J.
Zinn
997-1027
Abstract: The best constant and the extreme cases in an inequality of H.P. Rosenthal, relating the $p$ moment of a sum of independent symmetric random variables to that of the $p$ and $2$ moments of the individual variables, are computed in the range $2<p\le 4$. This complements the work of Utev who has done the same for $p>4$. The qualitative nature of the extreme cases turns out to be different for $p<4$ than for $p>4$. The method developed yields results in some more general and other related moment inequalities.
The Mizohata complex
Abdelhamid
Meziani
1029-1062
Abstract: This paper deals with the local solvability of systems of first order linear partial differential equations defined by a germ $\omega$ at $0\in \mathbb {R}^{n+1}$ of a $\mathbb {C}$-valued, formally integrable ($\omega \wedge d\omega =0$), 1-form with nondegenerate Levi form. More precisely, the size of the obstruction to the solvability, for $(q-1)$-forms $u$, of the equation \begin{equation*}du\wedge \omega =\eta \wedge \omega ,\end{equation*} where $\eta$ is a given $q$-form satisfying $d\eta \wedge \omega =0$ is estimated in terms of the De Rham cohomology relative to $\omega$
Structure of Lorentzian tori with a killing vector field
Miguel
Sánchez
1063-1080
Abstract: All Lorentzian tori with a non-discrete group of isometries are characterized and explicitly obtained. They can lie into three cases: (a) flat, (b) conformally flat but non-flat, and (c) geodesically incomplete. A detailed study of many of their properties (including results on the logical dependence of the three kinds of causal completeness, on geodesic connectedness and on prescribed curvature) is carried out. The incomplete case is specially analyzed, and several known examples and results in the literature are generalized from a unified point of view.
Signed Quasi-Measures
D.
J.
Grubb
1081-1089
Abstract: Let $X$ be a compact Hausdorff space and let $\mathcal A$ denote the subsets of $X$ which are either open or closed. A quasi-linear functional is a map $\rho :C(X)\rightarrow \mathbf R$ which is linear on singly generated subalgebras and such that $|\rho (f)|\leq M\|f\|$ for some $M<\infty$. There is a one-to-one correspondence between the quasi-linear functional on $C(X)$ and the set functions $\mu :\mathcal A \rightarrow \mathbf R$ such that i) $\mu (\emptyset )=0$, ii) If $A,B,A\cup B\in \mathcal A$ with $A$ and $B$ disjoint, then $\mu (A\cup B)=\mu (A)+\mu (B)$, iii) There is an $M<\infty$ such that whenever $\{U_\alpha \}$ are disjoint open sets, $\displaystyle \sum |\mu (U_\alpha )|\leq M$, and iv) if $U$ is open and $\varepsilon >0$, there is a compact $K\subseteq U$ such that whenever $V\subseteq U\setminus K$ is open, then $|\mu (V)|<\varepsilon$. The space of quasi-linear functionals is investigated and quasi-linear maps between two $C(X)$ spaces are studied.
Hilbert-Kunz functions and Frobenius functors
Shou-Te
Chang
1091-1119
Abstract: We study the asymptotic behavior as a function of $e$ of the lengths of the cohomology of certain complexes. These complexes are obtained by applying the $e$-th iterated Frobenius functor to a fixed finite free complex with only finite length cohomology and then tensoring with a fixed finitely generated module. The rings involved here all have positive prime characteristic. For the zeroth homology, these functions also contain the class of Hilbert-Kunz functions that a number of other authors have studied. This asymptotic behavior is connected with certain intrinsic dimensions introduced in this paper: these are defined in terms of the Krull dimensions of the Matlis duals of the local cohomology of the module. There is a more detailed study of this behavior when the given complex is a Koszul complex.
A bound on the geometric genus of projective varieties verifying certain flag conditions
Vincenzo
Di Gennaro
1121-1151
Abstract: Fix integers $n,r,s_{1},...,s_{l}$ and let $\mathcal {S}(n,r;s_{1},...,s_{l})$ be the set of all integral, projective and nondegenerate varieties $V$ of degree $s_{1}$ and dimension $n$ in the projective space $\mathbf {P}^{r}$, such that, for all $i=2,...,l$, $V$ does not lie on any variety of dimension $n+i-1$ and degree $<s_{i}$. We say that a variety $V$ satisfies a flag condition of type $(n,r;s_{1},...,s_{l})$ if $V$ belongs to $\mathcal {S}(n,r;s_{1},...,s_{l})$. In this paper, under the hypotheses $s_{1}>>...>>s_{l}$, we determine an upper bound $G^{h}(n,r;s_{1},...,s_{l})$, depending only on $n,r,s_{1},...,s_{l}$, for the number $G(n,r;s_{1},...,s_{l}):= {max} {\{} p_{g}(V) : V\in \mathcal {S}(n,r;s_{1},...,s_{l}){\}}$, where $p_{g}(V)$ denotes the geometric genus of $V$. In case $n=1$ and $l=2$, the study of an upper bound for the geometric genus has a quite long history and, for $n\geq 1$, $l=2$ and $s_{2}=r-n$, it has been introduced by Harris. We exhibit sharp results for particular ranges of our numerical data $n,r,s_{1},...,s_{l}$. For instance, we extend Halphen's theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimension $n$ in $\mathbf {P}^{n+3}$ as the smooth varieties of maximal geometric genus with respect to appropriate flag condition. This result applies to smooth surfaces in $\mathbf {P}^{5}$. Next we discuss how far $G^{h}(n,r;s_{1},...,s_{l})$ is from $G(n,r;s_{1},...,s_{l})$ and show a sort of lifting theorem which states that, at least in certain cases, the varieties $V\in \mathcal {S}(n,r;s_{1},...,s_{l})$ of maximal geometric genus $G(n,r;s_{1},...,s_{l})$ must in fact lie on a flag such as $V=V_{s_{1}}^{n}\subset V_{s_{2}}^{n+1}\subset ...\subset V_{s_{l}}^{n+l-1}\subset {\mathbf {P}^{r}}$, where $V^{j}_{s}$ denotes a subvariety of $\mathbf {P}^{r}$ of degree $s$ and dimension $j$. We also discuss further generalizations of flag conditions, and finally we deduce some bounds for Castelnuovo's regularity of varieties verifying flag conditions.
Strassen theorems for a class of iterated processes
Endre
Csáki;
Antónia
Földes;
Pál
Révész
1153-1167
Abstract: A general direct Strassen theorem is proved for a class of stochastic processes and applied for iterated processes such as $W(L_t)$, where $W(\cdot )$ is a standard Wiener process and $L_.$ is a local time of a Lévy process independent from $W(\cdot )$.
Mean-boundedness and Littlewood-Paley for separation-preserving operators
Earl
Berkson;
T.
A.
Gillespie
1169-1189
Abstract: Suppose that $(\Omega ,\mathcal {M},\mu )$ is a $\sigma$-finite measure space, $1<p<\infty$, and $T: L^{p}(\mu )\to L^{p}(\mu )$ is a bounded, invertible, separation-preserving linear operator such that the linear modulus of $T$ is mean-bounded. We show that $T$ has a spectral representation formally resembling that for a unitary operator, but involving a family of projections in $L^{p}(\mu )$ which has weaker properties than those associated with a countably additive Borel spectral measure. This spectral decomposition for $T$ is shown to produce a strongly countably spectral measure on the ``dyadic sigma-algebra'' of $\mathbb {T}$, and to furnish $L^{p}(\mu )$ with abstract analogues of the classical Littlewood-Paley and Vector-Valued M. Riesz Theorems for $\ell ^{p}(\mathbb {Z})$.
Anticanonical Rational Surfaces
Brian
Harbourne
1191-1208
Abstract: A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven over an algebraically closed field of arbitrary characteristic. Applications, to be treated in separate papers, include questions involving: points in good position, birational models of rational surfaces in projective space, and resolutions for 0-dimensional subschemes of $\mathbf {P}^{2}$ defined by complete ideals.
The image of the $BP$ Thom map for Eilenberg-MacLane spaces
Hirotaka
Tamanoi
1209-1237
Abstract: Fundamental classes in $BP$ cohomology of Eilenberg-MacLane spaces are defined. The image of the Thom map from $BP$ cohomology to mod-$p$ cohomology is determined for arbitrary Eilenberg-MacLane spaces. This image is a polynomial subalgebra generated by infinitely many elements obtained by applying a maximum number of Milnor primitives to the fundamental class in mod-$p$ cohomology. This subalgebra in mod $p$ cohomology is invariant under the action of the Steenrod algebra, and it is annihilated by all Milnor primitives. We also show that $BP$ cohomology determines Morava $K$ cohomology for Eilenberg-MacLane spaces.
Estimation of spectral gap for elliptic operators
Mu-Fa
Chen;
Feng-Yu
Wang
1239-1267
Abstract: A variational formula for the lower bound of the spectral gap of an elliptic operator is presented in the paper for the first time. The main known results are either recovered or improved. A large number of new examples with sharp estimate are illustrated. Moreover, as an application of the march coupling, the Poincaré inequality with respect to the absolute distribution of the process is also studied.
Rings with FZP
P.
R.
Fuchs;
C.
J.
Maxson;
G.
F.
Pilz
1271-1284
Abstract: In this paper we investigate the problem of characterizing those rings \begin{math}R\end{math} such that every nonzero polynomial with coefficients from \begin{math}R\end{math} has a finite number of zeros in \begin{math}R\end{math}. Particular attention is directed to the class of skew polynomial domains.
Cylindric Partitions
Ira
M.
Gessel;
C.
Krattenthaler
429-479
Abstract: A new object is introduced into the theory of partitions that generalizes plane partitions: cylindric partitions. We obtain the generating function for cylindric partitions of a given shape that satisfy certain row bounds as a sum of determinants of $q$-binomial coefficients. In some special cases these determinants can be evaluated. Extending an idea of Burge (J. Combin. Theory Ser. A 63 (1993), 210-222), we count cylindric partitions in two different ways to obtain several known and new summation and transformation formulas for basic hypergeometric series for the affine root system $\widetilde A_{r}$. In particular, we provide new and elementary proofs for two $\widetilde A_{r}$ basic hypergeometric summation formulas of Milne (Discrete Math. 99 (1992), 199-246).
Coherent functors, with application to torsion in the Picard group
David
B.
Jaffe
481-527
Abstract: Let $A$ be a commutative noetherian ring. We investigate a class of functors from $\langle \langle$commutative $A$-algebras$\rangle \rangle$ to $\langle \langle$sets$\rangle \rangle$, which we call coherent. When such a functor $F$ in fact takes its values in $\langle \langle$abelian groups$\rangle \rangle$, we show that there are only finitely many prime numbers $p$ such that ${}_pF(A)$ is infinite, and that none of these primes are invertible in $A$. This (and related statements) yield information about torsion in $\operatorname {Pic} (A)$. For example, if $A$ is of finite type over $\mathbb {Z}$, we prove that the torsion in $\operatorname {Pic} (A)$ is supported at a finite set of primes, and if ${}_p\operatorname {Pic} (A)$ is infinite, then the prime $p$ is not invertible in $A$. These results use the (already known) fact that if such an $A$ is normal, then $\operatorname {Pic} (A)$ is finitely generated. We obtain a parallel result for a reduced scheme $X$ of finite type over $\mathbb {Z}$. We classify the groups which can occur as the Picard group of a scheme of finite type over a finite field.
The generalized Berg theorem and BDF-theorem
Huaxin
Lin
529-545
Abstract: Let $A$ be a separable simple $AF$-algebra with finitely many extreme traces. We give a necessary and sufficient condition for an essentially normal element $x\in M(A)$, i.e., $\pi (x)$ is normal ($\pi : M(A)\to M(A)/A$ is the quotient map), having the form $y+a$ for some normal element $y\in M(A)$ and $a\in A.$ We also show that a normal element $x\in M(A)$ can be quasi-diagonalized if and only if the Fredholm index $ind(\lambda -x)=0$ for all $\lambda \not\in sp(\pi (x)).$ In the case that $A$ is a simple $C^*$-algebra of real rank zero, with stable rank one and with continuous scale, $K_1(A)=0,$ and $K_0(A)$ has countable rank, we show that a normal element $x\in M(A)$ with zero Fredholm index can be written as \begin{equation*}x=\sum _{n=1}^{\infty }\lambda _n(e_n-e_{n-1})+a, \end{equation*} where $\{e_n\}$ is an (increasing) approximate identity for $A$ consisting of projections, $\{\lambda _n\}$ is a bounded sequence of numbers and $a\in A$ with $\|a\|<\epsilon$ for any given $\epsilon >0.$
Jet Cohomology of Isolated Hypersurface Singularities and Spectral Sequences
Xiao
Er
Jian
547-577
Abstract: We study jet cohomology of isolated hypersurface singularities defined by partial differential forms and prove formulas to compute jet cohomology groups by linear algebra.
On Matroids Representable over $GF(3)$ and Other Fields
Geoff
Whittle
579-603
Abstract: The matroids that are representable over $GF(3)$ and some other fields depend on the choice of field. This paper gives matrix characterisations of the classes that arise. These characterisations are analogues of the characterisation of regular matroids as the ones that can be represented over the rationals by a totally-unimodular matrix. Some consequences of the theory are as follows. A matroid is representable over $GF(3)$ and $GF(5)$ if and only if it is representable over $GF(3)$ and the rationals, and this holds if and only if it is representable over $GF(p)$ for all odd primes $p$. A matroid is representable over $GF(3)$ and the complex numbers if and only if it is representable over $GF(3)$ and $GF(7)$. A matroid is representable over $GF(3)$, $GF(4)$ and $GF(5)$ if and only if it is representable over every field except possibly $GF(2)$. If a matroid is representable over $GF(p)$ for all odd primes $p$, then it is representable over the rationals.
Congruences, Trees, and $p$-adic Integers
Wolfgang
M.
Schmidt;
C.
L.
Stewart
605-639
Abstract: Let $f$ be a polynomial in one variable with integer coefficients, and $p$ a prime. A solution of the congruence $f(x) \equiv 0 (\text {mod} \,p)$ may branch out into several solutions modulo $p^{2}$, or it may be extended to just one solution, or it may not extend to any solution. Again, a solution modulo $p^{2}$ may or may not be extendable to solutions modulo $p^{3}$, etc. In this way one obtains the ``solution tree'' $T = T(f)$ of congruences modulo $p^{\lambda }$ for $\lambda = 1,2,\ldots$. We will deal with the following questions: What is the structure of such solution trees? How many ``isomorphism classes'' are there of trees $T(f)$ when $f$ ranges through polynomials of bounded degree and height? We will also give bounds for the number of solutions of congruences $f(x) \equiv 0 (\text {mod} \,p^{\lambda })$ in terms of $p, \lambda$ and the degree of $f$.
Locally conformal Kähler structures in quaternionic geometry
Liviu
Ornea;
Paolo
Piccinni
641-655
Abstract: We consider compact locally conformal quaternion Kähler manifolds $M$. This structure defines on $M$ a canonical foliation, which we assume to have compact leaves. We prove that the local quaternion Kähler metrics are Ricci-flat and allow us to project $M$ over a quaternion Kähler orbifold $N$ with fibers conformally flat 4-dimensional real Hopf manifolds. This fibration was known for the subclass of locally conformal hyperkähler manifolds; in this case we make some observations on the fibers' structure and obtain restrictions on the Betti numbers. In the homogeneous case $N$ is shown to be a manifold and this allows a classification. Examples of locally conformal quaternion Kähler manifolds (some with a global complex structure, some locally conformal hyperkähler) are the Hopf manifolds quotients of $\mathbb H^n-\{0\}$ by the diagonal action of appropriately chosen discrete subgroups of $CO^+(4)$.
Tangential flatness and global rigidity of higher rank lattice actions
Nantian
Qian
657-673
Abstract: We establish the continuous tangential flatness for orientable weakly Cartan actions of higher rank lattices. As a corollary, we obtain the global rigidity of Anosov Cartan actions.
Confluence of cycles for hypergeometric functions on $Z_{2,n+1}$
Yoshishige
Haraoka
675-712
Abstract: The hypergeometric function of general type, which is a generalization of the classical confluent hypergeometric functions, admits an integral representation derived from a character of a linear abelian group. For the hypergeometric function on the space of $2\times (n+1)$ matrices, a basis of cycles for the integral is constructed by a limit process, which is called a process of confluence. The determinant of the period matrix is explicitly evaluated to show the independence of the cycles.
A Lie theoretic Galois theory for the spectral curves of an integrable system. II
Andrew
McDaniel;
Lawrence
Smolinsky
713-746
Abstract: In the study of integrable systems of ODE's arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. Many of these systems are algebraically completely integrable in that they linearize on the Jacobian of a spectral curve. In an earlier paper the authors gave a classification of the spectral curves in terms of the Weyl group and arranged the spectral curves in a hierarchy. This paper examines the Jacobians of the spectral curves, again exploiting the Weyl group action. A hierarchy of Jacobians will give a basis of comparison for flows from various representations. A construction of V. Kanev is generalized and the Jacobians of the spectral curves are analyzed for abelian subvarieties. Prym-Tjurin varieties are studied using the group ring of the Weyl group $W$ and the Hecke algebra of double cosets of a parabolic subgroup of $W.$ For each algebra a subtorus is identified that agrees with Kanev's Prym-Tjurin variety when his is defined. The example of the periodic Toda lattice is pursued.
Symmetric powers of complete modules over a two-dimensional regular local ring
Daniel
Katz;
Vijay
Kodiyalam
747-762
Abstract: Let $(R,m)$ be a two-dimensional regular local ring with infinite residue field. For a finitely generated, torsion-free $R$-module $A$, write $A_{n}$ for the $n$th symmetric power of $A$, mod torsion. We study the modules $A_{n}$, $n \geq 1$, when $A$ is complete (i.e., integrally closed). In particular, we show that $B\cdot A = A_{2}$, for any minimal reduction $B \subseteq A$ and that the ring $\oplus _{n \geq 1} A_{n}$ is Cohen-Macaulay.
Enriched $P$-Partitions
John
R.
Stembridge
763-788
Abstract: An (ordinary) $P$-partition is an order-preserving map from a partially ordered set to a chain, with special rules specifying where equal values may occur. Examples include number-theoretic partitions (ordered and unordered, strict or unrestricted), plane partitions, and the semistandardtableaux associated with Schur's $S$-functions. In this paper, we introduce and develop a theory of enriched $P$-partitions; like ordinary $P$-partitions, these are order-preserving maps from posets to chains, but with different rules governing the occurrence of equal values. The principal examples of enriched $P$-partitions given here are the tableaux associated with Schur's $Q$-functions. In a sequel to this paper, further applications related to commutation monoids and reduced words in Coxeter groups will be presented.
Limiting subhessians, limiting subjets and their calculus
Alexander
D.
Ioffe;
Jean-Paul
Penot
789-807
Abstract: We study calculus rules for limiting subjets of order two. These subjets are obtained as limits of sequences of subjets, a subjet of a function $f$ at some point $x$ being the Taylor expansion of a twice differentiable function which minorizes $f$ and coincides with $f$ at $x$. These calculus rules are deduced from approximate (or fuzzy) calculus rules for subjets of order two. In turn, these rules are consequences of delicate results of Crandall-Ishii-Lions. We point out the similarities and the differences with the case of first order limiting subdifferentials.
Stiefel-Whitney classes and the conormal cycle of a singular variety
Joseph
H. G.
Fu;
Clint
McCrory
809-835
Abstract: A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety $X$ is given by means of the conormal cycle of an embedding of $X$ in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. We also show that, for a complex analytic variety, these classes are the mod 2 reductions of the Chern-MacPherson classes.
Congruences between Modular Forms, Cyclic Isogenies of Modular Elliptic Curves, and Integrality of $p$-adic $L$-Functions
Shu-Leung
Tang
837-856
Abstract: Let $\Gamma$ be a congruence subgroup of type $(N_1,N_2)$ and of level $N$. We study congruences between weight 2 normalized newforms $f$ and Eisenstein series $E$ on $\Gamma$ modulo a prime $\gp$ above a rational prime $p$. Assume that $p\nmid 6N$, $E$ is a common eigenfunction for all Hecke operators and $f$ is ordinary at $\gp$. We show that the abelian variety associated to $f$ and the cuspidal subgroup associated to $E$ intersect non-trivially in their $p$-torsion points. Let $A$ be a modular elliptic curve over $\Q$ with good ordinary reduction at $p$. We apply the above result to show that an isogeny of degree divisible by $p$ from the optimal curve $A_1$ in the $\Q$-isogeny class of elliptic curves containing $A$ to $A$ extends to an étale morphism of Néron models over $\Z _p$ if $p>7$. We use this to show that $p$-adic distributions associated to the $p$-adic $L$-functions of $A$ are $\Z _p$-valued.
Homotopy coherent category theory
Jean-Marc
Cordier;
Timothy
Porter
1-54
Abstract: This article is an introduction to the categorical theory of homotopy coherence. It is based on the construction of the homotopy coherent analogues of end and coend, extending ideas of Meyer and others. The paper aims to develop homotopy coherent analogues of many of the results of elementary category theory, in particular it handles a homotopy coherent form of the Yoneda lemma and of Kan extensions. This latter area is linked with the theory of generalised derived functors.
Expansive Subdynamics
Mike
Boyle;
Douglas
Lind
55-102
Abstract: This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let $\alpha$ be a continuous action of ${\mathbb Z}^d$ on an infinite compact metric space. For each subspace $V$ of ${\mathbb R}^d$ we introduce a notion of expansiveness for $\alpha$ along $V$, and show that there are nonexpansive subspaces in every dimension $\le d-1$. For each $k\le d$ the set ${\mathbb E} _k(\alpha )$ of expansive $k$-dimensional subspaces is open in the Grassmann manifold of all $k$-dimensional subspaces of ${\mathbb R}^d$. Various dynamical properties of $\alpha$ are constant, or vary nicely, within a connected component of ${\mathbb E} _k(\alpha )$, but change abruptly when passing from one expansive component to another. We give several examples of this sort of ``phase transition,'' including the topological and measure-theoretic directional entropies studied by Milnor, zeta functions, and dimension groups. For $d=2$ we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an ${\mathbb E} _1(\alpha )$. The unresolved case is that of the complement of a single irrational direction. Algebraic examples using commuting automorphisms of compact abelian groups are an important source of phenomena, and we study several instances in detail. We conclude with a set of problems and research directions suggested by our analysis.
On the strong equality between supercompactness and strong compactness
Arthur
W.
Apter;
Saharon
Shelah
103-128
Abstract: We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if $V \models$ ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension $V[{G}] \models$ ZFC + GCH in which, (a) (preservation) for $\kappa \le \lambda$ regular, if $V \models ``\kappa$ is $\lambda$ supercompact'', then $V[G] \models ``\kappa$ is $\lambda$ supercompact'' and so that, (b) (equivalence) for $\kappa \le \lambda$ regular, $V[{G}] \models ``\kappa$ is $\lambda$ strongly compact'' iff $V[{G}] \models ``\kappa$ is $\lambda$ supercompact'', except possibly if $\kappa$ is a measurable limit of cardinals which are $\lambda$ supercompact.
Integration of Correspondences on Loeb Spaces
Yeneng
Sun
129-153
Abstract: We study the Bochner and Gel$^{\prime }$fand integration of Banach space valued correspondences on a general Loeb space. Though it is well known that the Lyapunov type result on the compactness and convexity of the integral of a correspondence and the Fatou type result on the preservation of upper semicontinuity by integration are in general not valid in the setting of an infinite dimensional space, we show that exact versions of these two results hold in the case we study. We also note that our results on a hyperfinite Loeb space capture the nature of the corresponding asymptotic results for the large finite case; but the unit Lebesgue interval fails to provide such a framework.
On the Ornstein-Uhlenbeck operator in $L^2$ spaces with respect to invariant measures
Alessandra
Lunardi
155-169
Abstract: We consider a class of elliptic and parabolic differential operators with unbounded coefficients in $\mathbb R^n$, and we study the properties of the realization of such operators in suitable weighted $L^2$ spaces.
Bifurcation problems for the $p$-Laplacian in $R^n$
Pavel
Drábek;
Yin
Xi
Huang
171-188
Abstract: In this paper we consider the bifurcation problem \begin{equation*}-\text {div } (|{\nabla } u|^{p-2}{\nabla } u)={\lambda } g(x)|u|^{p-2}u+f({\lambda } , x, u), \end{equation*} in ${R^N}$ with $p>1$. We show that a continuum of positive solutions bifurcates out from the principal eigenvalue ${\lambda } _{1}$ of the problem \begin{equation*}-\text {div } (|{\nabla } u|^{p-2}{\nabla } u)={\lambda } g(x)|u|^{p-2}u. \end{equation*} Here both functions $f$ and $g$ may change sign.
$\beta\mathbf{nbc}$-bases for cohomology of local systems on hyperplane complements
Michael
Falk;
Hiroaki
Terao
189-202
Abstract: We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes ${\mathcal A}$. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only non-vanishing cohomology group, under some nonresonance conditions on the local system, for any arrangement ${\mathcal A}$. The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain ``no-broken-circuits" bases of ${\mathcal A}$. The basic forms depend on the local system, but any two bases constructed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system. In certain special cases we show the existence of bases of monomial logarithmic forms.
Shadowing orbits of ordinary differential equations on invariant submanifolds
Brian
A.
Coomes
203-216
Abstract: A finite time shadowing theorem for autonomous ordinary differential equations is presented. Under consideration is the case were there exists a twice continuously differentiable function $g$ mapping phase space into $\mathbb {R}^{m}$ with the property that for a particular regular value $\boldsymbol c$ of $g$ the submanifold $g^{-1}(\boldsymbol c)$ is invariant under the flow. The main theorem gives a condition which implies that an approximate solution lying close to $g^{-1}(\boldsymbol c)$ is uniformly close to a true solution lying in $g^{-1}(\boldsymbol c)$. Applications of this theorem to computer generated approximate orbits are discussed.
Essentially Normal Operator + Compact Operator = Strongly Irreducible Operator
Chunlan
Jiang;
Shunhua
Sun;
Zongyao
Wang
217-233
Abstract: It is shown that given an essentially normal operator $T$ with connected spectrum, there exists a compact operator $K$ such that $T+K$ is strongly irreducible.
Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces
Eleonor
Harboure;
Oscar
Salinas;
Beatriz
Viviani
235-255
Abstract: Necessary and sufficient conditions are given for the fractional integral operator $I_\alpha$ to be bounded from weighted strong and weak $% L^p$ spaces within the range $p\geq n/\alpha$ into suitable weighted $BMO$ and Lipschitz spaces. We also characterize the weights for which $% I_\alpha$ can be extended to a bounded operator from weighted $BMO$ into a weighted Lipschitz space of order $\alpha$. Finally, under an additional assumption on the weight, we obtain necessary and sufficient conditions for the boundedness of $I_\alpha$ between weighted Lipschitz spaces.
Multidimensional stability of planar travelling waves
Todd
Kapitula
257-269
Abstract: The multidimensional stability of planar travelling waves for systems of reaction-diffusion equations is considered in the case that the diffusion matrix is the identity. It is shown that if the wave is exponentially orbitally stable in one space dimension, then it is stable for $x\in % \mathbf {R}^n,\,n\ge 2$. Furthermore, it is shown that the perturbation of the wave decays like $t^{-(n-1)/4}$ as $t\to \infty$. The result is proved via an application of linear semigroup theory.
Every semigroup is isomorphic to a transitive semigroup of binary relations
Ralph
McKenzie;
Boris
M.
Schein
271-285
Abstract: Every (finite) semigroup is isomorphic to a transitive semigroup of binary relations (on a finite set).
Sums of Three or More Primes
J.
B.
Friedlander;
D.
A.
Goldston
287-310
Abstract: It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error $\sum _{p \le x} \log p - x$ in the Prime Number Theorem, such bounds being within a factor of $(\log x)^{2}$ of each other and this fact being equivalent to the Riemann Hypothesis. In this paper we show that, provided ``Riemann Hypothesis'' is replaced by ``Generalized Riemann Hypothesis'', results of similar (often greater) precision hold in the case of the corresponding formula for the representation of an integer as the sum of $k$ primes for $k \ge 4$, and, in a mean square sense, for $k \ge 3$. We also sharpen, in most cases to best possible form, the original estimates of Hardy and Littlewood which were based on the assumption of a ``Quasi-Riemann Hypothesis''. We incidentally give a slight sharpening to a well-known exponential sum estimate of Vinogradov-Vaughan.
Automorphism Groups and Invariant Subspace Lattices
Paul
S.
Muhly;
Baruch
Solel
311-330
Abstract: Let $(B,\mathbf {R},\alpha )$ be a $C^{*}$- dynamical system and let $% A=B^\alpha ([0,\infty ))$ be the analytic subalgebra of $B$. We extend the work of Loebl and the first author that relates the invariant subspace structure of $\pi (A),$ for a $C^{*}$-representation $\pi$ on a Hilbert space $\mathcal {H}_\pi$, to the possibility of implementing $\alpha$ on $% \mathcal {H}_\pi .$ We show that if $\pi$ is irreducible and if lat $\pi (A)$ is trivial, then $\pi (A)$ is ultraweakly dense in $\mathcal {L(H}_\pi ).$ We show, too, that if $A$ satisfies what we call the strong Dirichlet condition, then the ultraweak closure of $\pi (A)$ is a nest algebra for each irreducible representation $\pi .$ Our methods give a new proof of a ``density'' theorem of Kaftal, Larson, and Weiss and they sharpen earlier results of ours on the representation theory of certain subalgebras of groupoid $C^{*}$-algebras.
On the Complete Integrability of some Lax Equations on a Periodic Lattice
Luen-Chau
Li
331-372
Abstract: We consider some Lax equations on a periodic lattice with $N=2$ sites under which the monodromy matrix evolves according to the Toda flows. To establish their integrability (in the sense of Liouville) on generic symplectic leaves of the underlying Poisson structure, we construct the action-angle variables explicitly. The action variables are invariants of certain group actions. In particular, one collection of these invariants is associated with a spectral curve and the linearization of the associated Hamilton equations involves interesting new feature. We also prove the injectivity of the linearization map into real variables and solve the Hamilton equations generated by the invariants via factorization problems.
Decomposition of Birational Toric Maps in Blow-Ups and Blow-Downs
Jaroslaw
Wlodarczyk
373-411
Abstract: We prove that a toric birational map between two complete smooth toric varieties of the same dimension can be decomposed in a sequence of equivariant blow-ups and blow-downs along smooth centers.
Linear isometries between subspaces of continuous functions
Jesús
Araujo;
Juan
J.
Font
413-428
Abstract: We say that a linear subspace $A$ of $C_0 (X)$ is strongly separating if given any pair of distinct points $x_1, x_2$ of the locally compact space $X$, then there exists $f \in A$ such that $\left | f(x_1 ) \right | \neq \left | f(x_2 ) \right |$. In this paper we prove that a linear isometry $T$ of $A$ onto such a subspace $B$ of $C_0(Y)$ induces a homeomorphism $h$ between two certain singular subspaces of the Shilov boundaries of $B$ and $A$, sending the Choquet boundary of $B$ onto the Choquet boundary of $A$. We also provide an example which shows that the above result is no longer true if we do not assume $A$ to be strongly separating. Furthermore we obtain the following multiplicative representation of $T$: $(Tf)(y)=a(y)f(h(y))$ for all $y \in \partial B$ and all $f \in A$, where $a$ is a unimodular scalar-valued continuous function on $\partial B$. These results contain and extend some others by Amir and Arbel, Holszty\'{n}ski, Myers and Novinger. Some applications to isometries involving commutative Banach algebras without unit are announced.
On biunitary permutation matrices and some subfactors of index 9
Uma
Krishnan;
V.
S.
Sunder
4691-4736
Abstract: This paper is devoted to a study of the subfactors arising from vertex models constructed out of `biunitary' matrices which happen to be permutation matrices. After a discussion on the computation of the higher relative commutants of the associated subfactor (in the members of the tower of Jones' basic construction), we discuss the principal graphs of these subfactors for small sizes $N=k \leq 3$ of the vertex model. Of the 18 possibly inequivalent such biunitary matrices when $N = 3$, we compute the principal graphs completely in 15 cases, all of which turn out to be finite. In the last section, we prove that two of the three remaining cases lead to subfactors of infinite depth and discuss their principal graphs.
On the group of homotopy equivalences of a manifold
Hans
Joachim
Baues
4737-4773
Abstract: We consider the group of homotopy equivalences $\mathcal E(M)$ of a simply connected manifold $M$ which is part of the fundamental extension of groups due to Barcus-Barratt. We show that the kernel of this extension is always a finite group and we compute this kernel for various examples. This leads to computations of the group $\mathcal E(M)$ for special manifolds $M$, for example if $M$ is a connected sum of products $S^n\times S^m$ of spheres. In particular the group $\mathcal E(S^n\times S^n)$ is determined completely. Also the connection of $\mathcal E(M)$ with the group of isotopy classes of diffeomorphisms of $M$ is studied.
The geometry of uniserial representations of finite dimensional algebras. III: Finite uniserial type
Birge
Huisgen-Zimmermann
4775-4812
Abstract: A description is given of those sequences $\mathbf {S}= (S(0),S(1),\dots ,S(l))$ of simple modules over a finite dimensional algebra for which there are only finitely many uniserial modules with consecutive composition factors $S(0),\dots , S(l)$. Necessary and sufficient conditions for an algebra to permit only a finite number of isomorphism types of uniserial modules are derived. The main tools in this investigation are the affine algebraic varieties parametrizing the uniserial modules with composition series $\mathbf {S}$.
Multiplication of natural number parameters and equations in a free semigroup
Gennady
S.
Makanin
4813-4824
Abstract: This paper deals with the problem of describing the set $M$ of all solutions of an equation over a free semigroup $S$. The standard way to do this involves the introduction of auxiliary equations containing polynomials in natural number parameters of arbitrarily high degree. Since $S$ has a solvable word problem, $M$ must be computable. However, $M$ cannot necessarily be computed from the standard description of $M$. The present paper shows that the only polynomials needed to describe $M$ are just products of one parameter by a linear combination of some other parameters. The resulting simplification of the standard description of $M$ clearly can be used to compute $M$.
Multiplicative $\eta$-quotients
Yves
Martin
4825-4856
Abstract: Let $\eta (z)$ be the Dedekind $\eta $-function. In this work we exhibit all modular forms of integral weight $f(z) = \eta (t_1z)^{r_1}\eta (t_2z)^{r_2}\dots \eta(t_sz)^{r_s}$, for positive integers $s$ and $t_j$ and arbitrary integers $r_j$, such that both $f(z)$ and its image under the Fricke involution are eigenforms of all Hecke operators. We also relate most of these modular forms with the Conway group $2 % \mathrm {Co}_1$ via a generalized McKay-Thompson series.
On extension of cocycles to normalizer elements, outer conjugacy, and related problems
Alexandre
I.
Danilenko;
Valentin
Ya.
Golodets
4857-4882
Abstract: Let $T$ be an ergodic automorphism of a Lebesgue space and $\alpha$ a cocycle of $T$ with values in an Abelian locally compact group $G$. An automorphism $\theta$ from the normalizer $N[T]$ of the full group $[T]$ is said to be compatible with $\alpha$ if there is a measurable function $\varphi : X \to G$ such that $\alpha (\theta x, \theta T\theta ^{-1}) = - \varphi (x) + \alpha (x, T) + \varphi (Tx)$ at a.e. $x$. The topology on the set $D(T, \alpha )$ of all automorphisms compatible with $\alpha$ is introduced in such a way that $D(T , \alpha )$ becomes a Polish group. A complete system of invariants for the $\alpha$-outer conjugacy (i.e. the conjugacy in the quotient group $D(T, \alpha )/[T])$ is found. Structure of the cocycles compatible with every element of $N[T]$ is described.
Spectral averaging, perturbation of singular spectra, and localization
J.
M.
Combes;
P.
D.
Hislop;
E.
Mourre
4883-4894
Abstract: A spectral averaging theorem is proved for one-parameter families of self-adjoint operators using the method of differential inequalities. This theorem is used to establish the absolute continuity of the averaged spectral measure with respect to Lebesgue measure. This is an important step in controlling the singular continuous spectrum of the family for almost all values of the parameter. The main application is to the problem of localization for certain families of random Schrödinger operators. Localization for a family of random Schrödinger operators is established employing these results and a multi-scale analysis.
Characterizations of Kadec-Klee properties in symmetric spaces of measurable functions
V.
I.
Chilin;
P.
G.
Dodds;
A.
A.
Sedaev;
F.
A.
Sukochev
4895-4918
Abstract: We present several characterizations of Kadec-Klee properties in symmetric function spaces on the half-line, based on the $K$-functional of J. Peetre. In addition to the usual Kadec-Klee property, we study those symmetric spaces for which sequential convergence in measure (respectively, local convergence in measure) on the unit sphere coincides with norm convergence.
Ergodic properties of real cocycles and pseudo-homogeneous Banach spaces
M.
Lemanczyk;
F.
Parreau;
D.
Volný
4919-4938
Abstract: Given an irrational rotation, in the space of real bounded variation functions it is proved that there are ergodic cocycles whose small perturbations remain ergodic; in fact, the set of ergodic cocycles has nonempty dense interior. Given a pseudo-homogeneous Banach space and an irrational rotation, we study the set of elements satisfying the mean ergodic theorem. Once such a space is not homogeneous, we prove it is not reflexive and not separable. In ``natural" cases, up to $L^1$-cohomology, the only elements satisfying the mean ergodic theorem are those from the closure of trigonometric polynomials. For pseudo-homogeneous spaces admitting a Koksma's inequality ergodicity of the corresponding cylinder flows can be deduced from spectral properties of some circle extensions. In particular this is the case of Lebesgue spectrum (in the orthocomplement of the space of eigenfunctions) for the circle extension.
A deformation of flat conformal structures
Hiroyasu
Izeki
4939-4964
Abstract: We consider deformations of flat conformal structures from a viewpoint of connected sum decomposition of conformally flat manifolds.
The ergodic theory of discrete isometry groups on manifolds of variable negative curvature
Chengbo
Yue
4965-5005
Abstract: This paper studies the ergodic theory at infinity of an arbitrary discrete isometry group $\Gamma$ acting on any Hadamard manifold $H$ of pinched variable negative curvature. Most of the results obtained by Sullivan in the constant curvature case are generalized to the case of variable curvature. We describe connections between measures supported on the limit set of $\Gamma$, dynamics of the geodesic flow and the geometry of $M=H/ \Gamma$. We explore the relationship between the growth exponent of the group, the Hausdorff dimension of the limit set and the topological entropy of the geodesic flow. The equivalence of various descriptions of an analogue of the Hopf dichotomy is proved. As applications, we settle a question of J. Feldman and M. Ratner about the horocycle flow on a finite volume surface of negative curvature and obtain an asymptotic formula for the counting function of lattice points. At the end of this paper, we apply our results to the study of some rigidity problems. More applications to Mostow rigidity for discrete subgroups of rank 1 noncompact semisimple Lie groups with infinite covolume will be published in subsequent papers by the author.
The inverse problem of the calculus of variations for scalar fourth-order ordinary differential equations
M.
E.
Fels
5007-5029
Abstract: A simple invariant characterization of the scalar fourth-order ordinary differential equations which admit a variational multiplier is given. The necessary and sufficient conditions for the existence of a multiplier are expressed in terms of the vanishing of two relative invariants which can be associated with any fourth-order equation through the application of Cartan's equivalence method. The solution to the inverse problem for fourth-order scalar equations provides the solution to an equivalence problem for second-order Lagrangians, as well as the precise relationship between the symmetry algebra of a variational equation and the divergence symmetry algebra of the associated Lagrangian.
Spatial chaotic structure of attractors of reaction-diffusion systems
V.
Afraimovich;
A.
Babin;
S.-N.
Chow
5031-5063
Abstract: The dynamics described by a system of reaction-diffusion equations with a nonlinear potential exhibits complicated spatial patterns. These patterns emerge from preservation of homotopy classes of solutions with bounded energies. Chaotically arranged stable patterns exist because of realizability of all elements of a fundamental homotopy group of a fixed degree. This group corresponds to level sets of the potential. The estimates of homotopy complexity of attractors are obtained in terms of geometric characteristics of the potential and other data of the problem.
Integral type linear functionals on ordered cones
Walter
Roth
5065-5085
Abstract: We introduce linear functionals on an ordered cone that are minimal with respect to a given subcone. Using concepts developed for Choquet theory we observe that the properties of these functionals resemble those of positive Radon measures on locally compact spaces. Other applications include monotone functionals on cones of convex sets, H-integrals on H-cones in abstract potential theory, and classical Choquet theory itself.
Lévy group action and invariant measures on $\beta\mathbb{N}$
Martin
Blümlinger
5087-5111
Abstract: For $f\in \ell ^{\infty }( \mathbb {N})$ let $Tf$ be defined by $Tf(n)=\frac {1}{n}\sum _{i=1}^{n}f(i)$. We investigate permutations $g$ of $ \mathbb {N}$, which satisfy $Tf(n)-Tf_{g}(n)\to 0$ as $n\to \infty$ with $f_{g}(n)=f(gn)$ for $f\in \ell ^{\infty }( \mathbb {N})$ (i.e. $g$ is in the Lévy group $\mathcal {G})$, or for $f$ in the subspace of Cesàro-summable sequences. Our main interest are $ \mathcal {G}$-invariant means on $\ell ^{\infty }( \mathbb {N})$ or equivalently $\mathcal {G}$-invariant probability measures on $\beta \mathbb {N}$. We show that the adjoint $T^{*}$ of $T$ maps measures supported in $\beta \mathbb {N} \setminus \mathbb {N}$ onto a weak*-dense subset of the space of $ \mathcal {G}$-invariant measures. We investigate the dynamical system $( \mathcal {G}, \beta \mathbb {N})$ and show that the support set of invariant measures on $\beta \mathbb {N}$ is the closure of the set of almost periodic points and the set of non-topologically transitive points in $\beta \mathbb {N}\setminus \mathbb {N}$. Finally we consider measures which are invariant under $T^{*}$.
Covering the integers by arithmetic sequences. II
Zhi-Wei
Sun
4279-4320
Abstract: Let $A= \{a_{s}+n_{s}\mathbb {Z}\}^{k}_{s=1}$ ($n_{1} \leqslant \cdots \leqslant n_{k})$ be a system of arithmetic sequences where $a_{1}, \cdots ,a_{k}\in \mathbb {Z}$ and $n_{1},\cdots ,n_{k}\in \mathbb {Z}^{+}$. For $m\in \mathbb {Z}^{+}$ system $A$ will be called an (exact) $m$-cover of $\mathbb {Z}$ if every integer is covered by $A$ at least (exactly) $m$ times. In this paper we reveal further connections between the common differences in an (exact) $m$-cover of $\mathbb {Z}$ and Egyptian fractions. Here are some typical results for those $m$-covers $A$ of $\mathbb {Z}$: (a) For any $m_{1},\cdots ,m_{k}\in \mathbb {Z}^{+}$ there are at least $m$ positive integers in the form $\Sigma _{s\in I} m_{s}/n_{s}$ where $I \subseteq \{1,\cdots ,k\}$. (b) When $n_{k-l}<n_{k-l+1}= \cdots =n_{k}$ ($0<l<k)$, either $l \geqslant n_{k}/n_{k-l}$ or $\Sigma ^{k-l}_{s=1}1/n_{s} \geqslant m$, and for each positive integer $\lambda <n_{k}/n_{k-l}$ the binomial coefficient $\binom l{ \lambda }$ can be written as the sum of some denominators $>1$ of the rationals $\Sigma _{s\in I}1/n_{s}- \lambda /n_{k}, I \subseteq \{1,\cdots ,k\}$ if $A$ forms an exact $m$-cover of $\mathbb {Z}$. (c) If $\{a_{s}+n_{s}\mathbb {Z}\}^{k}_{\substack {s=1 s\not =t}}$ is not an $m$-cover of $\mathbb {Z}$, then $\Sigma _{s\in I}1/n_{s}, I \subseteq \{1,\cdots ,k\}\setminus \{t\}$ have at least $n_{t}$ distinct fractional parts and for each $r=0,1,\cdots ,n_{t}-1$ there exist $I_{1},I_{2} \subseteq \{1,\cdots ,k\}\setminus \{t\}$ such that $r/n_{t} \equiv \Sigma _{s\in I_{1}}1/n_{s}-\Sigma _{s\in I_{2}}1/n_{s}$ (mod 1). If $A$ forms an exact $m$-cover of $\mathbb {Z}$ with $m=1$ or $n_{1}< \cdots <n_{k-l}<n_{k-l+1}= \cdots =n_{k}$ ($l>0$) then for every $t=1, \cdots ,k$ and $r=0,1,\cdots ,n_{t}-1$ there is an $I \subseteq \{1,\cdots ,k\}$ such that $\Sigma _{s\in I}1/n_{s} \equiv r/n_{t}$ (mod 1).
A variational problem for surfaces in Laguerre geometry
Emilio
Musso;
Lorenzo
Nicolodi
4321-4337
Abstract: We consider the variational problem defined by the functional $\int {\frac {{H^{2}-K}}{{K}}}dA$ on immersed surfaces in Euclidean space. Using the invariance of the functional under the group of Laguerre transformations, we study the extremal surfaces by the method of moving frames.
Functorial structure of units in a tensor product
David
B.
Jaffe
4339-4353
Abstract: The behavior of units in a tensor product of rings is studied, as one factor varies. For example, let $k$ be an algebraically closed field. Let $A$ and $B$ be reduced rings containing $k$, having connected spectra. Let $u\in A\otimes _k\,B$ be a unit. Then $u=a\otimes b$ for some units $a\in A$ and $b\in B$. Here is a deeper consequence, stated for simplicity in the affine case only. Let $k$ be a field, and let $\varphi :R\to S$ be a homomorphism of finitely generated $k$-algebras such that $\operatorname {Spec}(\varphi )$ is dominant. Assume that every irreducible component of $\operatorname {Spec}(R_{\operatorname {red}})$ or $\operatorname {Spec}(S_{\operatorname {red}})$ is geometrically integral and has a rational point. Let $B\to C$ be a faithfully flat homomorphism of reduced $k$-algebras. For $A$ a $k$-algebra, define $Q(A)$ to be $(S\otimes _k\,A)^*/(R\otimes _k\,A)^*$. Then $Q$ satisfies the following sheaf property: the sequence \begin{displaymath}0\to Q(B)\to Q(C)\to Q(C\otimes _B\,C)\end{displaymath} is exact. This and another result are used to prove (5.2) of [7].
The Morse spectrum of linear flows on vector bundles
Fritz
Colonius;
Wolfgang
Kliemann
4355-4388
Abstract: For a linear flow $\Phi$ on a vector bundle $\pi : E \rightarrow S$ a spectrum can be defined in the following way: For a chain recurrent component $\mathcal {M}$ on the projective bundle $\mathbb {P} E$ consider the exponential growth rates associated with (finite time) $(\varepsilon ,T)$-chains in $\mathcal {M}$, and define the Morse spectrum $\Sigma _{Mo}(\mathcal {M},\Phi )$ over $\mathcal {M}$ as the limits of these growth rates as $\varepsilon \rightarrow 0$ and $T \rightarrow \infty$. The Morse spectrum $\Sigma _{Mo}(\Phi )$ of $\Phi$ is then the union over all components $\mathcal {M}\subset \mathbb {P}E$. This spectrum is a synthesis of the topological approach of Selgrade and Salamon/Zehnder with the spectral concepts based on exponential growth rates, such as the Oseledec spectrum or the dichotomy spectrum of Sacker/Sell. It turns out that $\Sigma _{Mo}(\Phi )$ contains all Lyapunov exponents of $\Phi$ for arbitrary initial values, and the $\Sigma _{Mo}(\mathcal {M},\Phi )$ are closed intervals, whose boundary points are actually Lyapunov exponents. Using the fact that $\Phi$ is cohomologous to a subflow of a smooth linear flow on a trivial bundle, one can prove integral representations of all Morse and all Lyapunov exponents via smooth ergodic theory. A comparison with other spectral concepts shows that, in general, the Morse spectrum is contained in the topological spectrum and the dichotomy spectrum, but the spectral sets agree if the induced flow on the base space is chain recurrent. However, even in this case, the associated subbundle decompositions of $E$ may be finer for the Morse spectrum than for the dynamical spectrum. If one can show that the (closure of the) Floquet spectrum (i.e. the Lyapunov spectrum based on periodic trajectories in $\mathbb {P} E$) agrees with the Morse spectrum, then one obtains equality for the Floquet, the entire Oseledec, the Lyapunov, and the Morse spectrum. We present an example (flows induced by $C^{\infty }$ vector fields with hyperbolic chain recurrent components on the projective bundle) where this fact can be shown using a version of Bowen's Shadowing Lemma.
The Lyapunov spectrum of families of time-varying matrices
Fritz
Colonius;
Wolfgang
Kliemann
4389-4408
Abstract: For $L^{\infty }$-families of time varying matrices centered at an unperturbed matrix, the Lyapunov spectrum contains the Floquet spectrum obtained by considering periodically varying piecewise constant matrices. On the other hand, it is contained in the Morse spectrum of an associated flow on a vector bundle. A closer analysis of the Floquet spectrum based on geometric control theory in projective space and, in particular, on control sets, is performed. Introducing a real parameter $\rho \ge 0$, which indicates the size of the $L^{\infty }$-perturbation, we study when the Floquet spectrum, the Morse spectrum, and hence the Lyapunov spectrum all coincide. This holds, if an inner pair condition is satisfied, for all up to at most countably many $\rho$-values.
Projectively bounded Fréchet measures
Ron
C.
Blei
4409-4432
Abstract: A scalar valued set function on a Cartesian product of $\sigma$-algebras is a Fréchet measure if it is a scalar measure independently in each coordinate. A basic question is considered: is it possible to construct products of Fréchet measures that are analogous to product measures in the classical theory? A Fréchet measure is said to be projectively bounded if it satisfies a Grothendieck type inequality. It is shown that feasibility of products of Fréchet measures is linked to the projective boundedness property. All Fréchet measures in a two dimensional framework are projectively bounded, while there exist Fréchet measures in dimensions greater than two that are projectively unbounded. A basic problem is considered: when is a Fréchet measure projectively bounded? Some characterizations are stated. Applications to harmonic and stochastic analysis are given.
Packing dimension and Cartesian products
Christopher
J.
Bishop;
Yuval
Peres
4433-4445
Abstract: We show that for any analytic set $A$ in $\mathbf {R}^d$, its packing dimension $\dim _{\mathrm {P}}(A)$ can be represented as $\; \sup _B \{ \dim _{\mathrm {H}} (A \times B) -\dim _{\mathrm {H}}(B) \} \, , \,$, where the supremum is over all compact sets $B$ in $\mathbf {R}^d$, and $\dim _{\mathrm {H}}$ denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if $\dim _{\mathrm {P}} (A) < d$. In contrast, we show that the dual quantity $\; \inf _B \{ \dim _{\mathrm {P}}(A \times B) -\dim _{\mathrm {P}}(B) \} \, , \,$, is at least the ``lower packing dimension'' of $A$, but can be strictly greater. (The lower packing dimension is greater than or equal to the Hausdorff dimension.)
The dynamical properties of Penrose tilings
E.
Arthur
Robinson Jr.
4447-4464
Abstract: The set of Penrose tilings, when provided with a natural compact metric topology, becomes a strictly ergodic dynamical system under the action of $% \mathbf {R}^2$ by translation. We show that this action is an almost 1:1 extension of a minimal $% \mathbf {R}^2$ action by rotations on $% \mathbf {T}^4$, i.e., it is an $% \mathbf {R}^2$ generalization of a Sturmian dynamical system. We also show that the inflation mapping is an almost 1:1 extension of a hyperbolic automorphism on $% \mathbf {T}^4$. The local topological structure of the set of Penrose tilings is described, and some generalizations are discussed.
Eigenvalue asymptotics and exponential decay of eigenfunctions for Schrödinger operators with magnetic fields
Zhongwei
Shen
4465-4488
Abstract: We consider the Schrödinger operator with magnetic field, \begin{equation*}H=(\frac {1}{i}\nabla -{\overset {\rightharpoonup }{a}}(x))^{2}+V(x) \ \text { in } \mathbb {R}^{n}. \end{equation*} Assuming that $V\ge 0$ and $|\text {curl}\, \overset {\rightharpoonup }{a}|+V+1$ is locally in certain reverse Hölder class, we study the eigenvalue asymptotics and exponential decay of eigenfunctions.
Abstract functions with continuous differences and Namioka spaces
Bolis
Basit;
Hans
Günzler
4489-4500
Abstract: Let $G$ be a semigroup and a topological space. Let $X$ be an Abelian topological group. The right differences $\triangle _{h} \varphi$ of a function $\varphi : G \to X$ are defined by $\triangle _{h}\varphi (t) = \varphi (th) - \varphi (t)$ for $h,t \in G$. Let $\triangle _{h} \varphi$ be continuous at the identity $e$ of $G$ for all $h$ in a neighbourhood $U$ of $e$. We give conditions on $X$ or range $\varphi$ under which $\varphi$ is continuous for any topological space $G$. We also seek conditions on $G$ under which we conclude that $\varphi$ is continuous at $e$ for arbitrary $X$. This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.
Existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature
Xu-Jia
Wang
4501-4524
Abstract: Let $f(x)$ be a given positive function in $R^{n+1}$. In this paper we consider the existence of convex, closed hypersurfaces $X$ so that its Gauss-Kronecker curvature at $x\in X$ is equal to $f(x)$. This problem has variational structure and the existence of stable solutions has been discussed by Tso (J. Diff. Geom. 34 (1991), 389--410). Using the Mountain Pass Lemma and the Gauss curvature flow we prove the existence of unstable solutions to the problem.
The fixed-point property for simply connected plane continua
Charles
L.
Hagopian
4525-4548
Abstract: We answer a question of R. Ma\'{n}ka by proving that every simply-connected plane continuum has the fixed-point property. It follows that an arcwise-connected plane continuum has the fixed-point property if and only if its fundamental group is trivial. Let $M$ be a plane continuum with the property that every simple closed curve in $M$ bounds a disk in $M$. Then every map of $M$ that sends each arc component into itself has a fixed point. Hence every deformation of $M$ has a fixed point. These results are corollaries to the following general theorem. If $M$ is a plane continuum, $\mathcal {D}$ is a decomposition of $M$, and each element of $\mathcal {D}$ is simply connected, then every map of $M$ that sends each element of $\mathcal {D}$ into itself has a fixed point.
The boundary of iterates in Euclidean growth models
Janko
Gravner
4549-4559
Abstract: This paper defines a general Euclidean growth model via a translation invariant, monotone and local transformation on Borel subsets of $\mathbf {R}^2$. The main result gives a geometric condition for the boundary curvature of the iterates to go to 0. Consequences include estimates for the speed of convergence to the asymptotic shape, and a result about survival of Euclidean deterministic forest fires.
Krull-Schmidt fails for serial modules
Alberto
Facchini
4561-4575
Abstract: We answer a question posed by Warfield in 1975: the Krull-Schmidt Theorem does not hold for serial modules, as we show via an example. Nevertheless we prove a weak form of the Krull-Schmidt Theorem for serial modules (Theorem 1.9). And we show that the Grothendieck group of the class of serial modules of finite Goldie dimension over a fixed ring $R$ is a free abelian group.
Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète
Qing
Liu
4577-4610
Abstract: Let $C$ be a hyperelliptic curve of genus $g\ge 1$ over a discrete valuation field $K$. In this article we study the models of $C$ over the ring of integers $\mathcal {O}_{K}$ of $K$. To each Weierstrass model (that is a projective model arising from a hyperelliptic equation of $C$ with integral coefficients), one can associate a (valuation of) discriminant. Then we give a criterion for a Weierstrass model to have minimal discriminant. We show also that in the most cases, the minimal regular model of $C$ over $\mathcal {O}_{K}$ dominates every minimal Weierstrass model. Some classical facts concerning Weierstrass models over $\mathcal {O}_{K}$ of elliptic curves are generalized to hyperelliptic curves, and some others are proved in this new setting.
Infinite products of finite simple groups
Jan
Saxl;
Saharon
Shelah;
Simon
Thomas
4611-4641
Abstract: We classify the sequences $\langle S_{n} \mid n \in \mathbb {N} \rangle$ of finite simple nonabelian groups such that $\prod$$_{n}$ $S_{n}$ has uncountable cofinality.
Curvature invariants, differential operators and local homogeneity
Friedbert
Prüfer;
Franco
Tricerri;
Lieven
Vanhecke
4643-4652
Abstract: We first prove that a Riemannian manifold $(M,g)$ with globally constant additive Weyl invariants is locally homogeneous. Then we use this result to show that a manifold $(M,g)$ whose Laplacian commutes with all invariant differential operators is a locally homogeneous space.
The stability of foliations of orientable 3-manifolds covered by a product
Sandra
L.
Shields
4653-4671
Abstract: We examine the relationship between codimension one foliations that are covered by a trivial product of hyperplanes and the branched surfaces that can be constructed from them. We present a sufficient condition on a branched surface constructed from a foliation to guarantee that all $C^1$ perturbations of the foliation are covered by a trivial product of hyperplanes. We also show that a branched surface admits a strictly positive weight system if and only if it can be constructed from a fibration over $S^1$.
Harish-Chandra's Plancherel theorem for $\frak p$-adic groups
Allan
J.
Silberger
4673-4686
Abstract: Let $G$ be a reductive $\mathfrak {p}$-adic group. In his paper, ``The Plancherel Formula for Reductive $\mathfrak {p}$-adic Groups", Harish-Chandra summarized the theory underlying the Plancherel formula for $G$ and sketched a proof of the Plancherel theorem for $G$. One step in the proof, stated as Theorem 11 in Harish-Chandra's paper, has seemed an elusively difficult step for the reader to supply. In this paper we prove the Plancherel theorem, essentially, by proving a special case of Theorem 11. We close by deriving a version of Theorem 11 from the Plancherel theorem.
Erratum à ``Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d'un groupe réductif p-adique''
Anne-Marie
Aubert
4687-4690
Multi-bump orbits homoclinic to resonance bands
Tasso
J.
Kaper;
Gregor
Kovacic
3835-3887
Abstract: We establish the existence of several classes of multi-bump orbits homoclinic to resonance bands for completely-integrable Hamiltonian systems subject to small-amplitude Hamiltonian or dissipative perturbations. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The homoclinic orbits, which include multi-bump \v{S}ilnikov orbits, connect equilibria and periodic orbits in the resonance band. The main tools we use in the existence proofs are the exchange lemma with exponentially small error and the existence theory of orbits homoclinic to resonance bands which make only one fast excursion away from the resonance bands.
A new uncountably categorical group
Andreas
Baudisch
3889-3940
Abstract: We construct an uncountably categorical group with a geometry that is not locally modular. It is not possible to interpret a field in this group. We show the group is CM-trivial.
On the homology spectral sequence for topological Hochschild homology
Thomas
J.
Hunter
3941-3953
Abstract: Marcel Bökstedt has computed the homotopy type of the topological Hochschild homology of $\Bbb Z/p$ using his definition of topological Hochschild homology for a functor with smash product. Here we show that easy conceptual proofs of his main technical result of are possible in the context of the homotopy theory of \begin{math}S\end{math}-algebras as introduced by Elmendorf, Kriz, Mandell and May. We give algebraic arguments based on naturality properties of the topological Hochschild homology spectral sequence. In the process we demonstrate the utility of the unstable ``lower'' notation for the Dyer-Lashof algebra.
Sharp upper bound for the first non-zero Neumann eigenvalue for bounded domains in rank-1 symmetric spaces
A.
R.
Aithal;
G.
Santhanam
3955-3965
Abstract: In this paper, we prove that for a bounded domain $\Omega$ in a rank-$1$ symmetric space, the first non-zero Neumann eigenvalue $\mu _{1}(\Omega )\leq \mu _{1}(B(r_{1}))$ where $B(r_{1})$ denotes the geodesic ball of radius $r_{1}$ such that \begin{equation*}vol(\Omega )=vol(B(r_{1}))\end{equation*} and equality holds iff $\Omega =B(r_{1})$. This result generalises the works of Szego, Weinberger and Ashbaugh-Benguria for bounded domains in the spaces of constant curvature.
Murnaghan-Nakayama rules for characters of Iwahori-Hecke algebras of classical type
Tom
Halverson;
Arun
Ram
3967-3995
Abstract: In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of the Iwahori-Hecke algebras of types $A_{n-1}, B_n$, and $D_n$. Our method is a generalization of a derivation of the Murnaghan-Nakayama formula for the irreducible characters of the symmetric group given by Curtis Greene. Greene's approach is to sum up the diagonal entries of the matrices of certain cycle permutations in Young's seminormal representations. The analogues of the Young seminormal representations for the Iwahori-Hecke algebras of types $A_{n-1},B_n$, and $D_n$ were given by Hoefsmit.
Invariants of piecewise-linear 3-manifolds
John
W.
Barrett;
Bruce
W.
Westbury
3997-4022
Abstract: This paper presents an algebraic framework for constructing invariants of closed oriented 3-manifolds by taking a state sum model on a triangulation. This algebraic framework consists of a tensor category with a condition on the duals which we have called a spherical category. A significant feature is that the tensor category is not required to be braided. The main examples are constructed from the categories of representations of involutive Hopf algebras and of quantised enveloping algebras at a root of unity.
On Baire-1/4 functions
Vassiliki
Farmaki
4023-4041
Abstract: We give descriptions of the spaces $D(K)$ (i.e. the space of differences of bounded semicontinuous functions on $K$) and especially of $B_{1/4}(K)$ (defined by Haydon, Odell and Rosenthal) as well as for the norms which are defined on them. For example, it is proved that a bounded function on a metric space $K$ belongs to $B_{1/4}(K)$ if and only if the $\omega ^{ % \mathrm {th}}$-oscillation, $\mathrm {osc}_{\omega }f$, of $f$ is bounded and in this case $\| f\|_{1/4}=\|\, |f|+ \widetilde {\mathrm {osc}}_{\omega } f\|_{\infty }$. Also, we classify $B_{1/4}(K)$ into a decreasing family $(S_{\xi }(K))_{1\leq \xi <\omega _1}$ of Banach spaces whose intersection is equal to $D(K)$ and $S_1 (K)=B_{1/4}(K)$. These spaces are characterized by spreading models of order $\xi$ equivalent to the summing basis of $c_0$, and for every function $f$ in $S_{\xi }(K)$ it is valid that $\mathrm {osc}_{\omega ^{\xi }}f$ is bounded. Finally, using the notion of null-coefficient of order $\xi$ sequence, we characterize the Baire-1 functions not belonging to $S_{\xi }(K)$.
Transfinite multifractal dimension spectrums
Stanley
C.
Williams
4043-4081
Abstract: The first order theory of the decomposition of measures with respect to dimension which has been developed by Kahane, Katznelson, Cutler, and others is extended through transfinite recursion to a $\omega _1$-order theory. Necessary and sufficient conditions for a finite regular Borel measure on $% [0,d]^{\omega _1}$ to be a $\omega _1$-order multispectrum for a finite Borel measure on $\mathbb {R}^d$ is given.
Competitive exclusion and coexistence for competitive systems on ordered Banach spaces
S.
B.
Hsu;
H.
L.
Smith;
Paul
Waltman
4083-4094
Abstract: The dynamics of competitive maps and semiflows defined on the product of two cones in respective Banach spaces is studied. It is shown that exactly one of three outcomes is possible for two viable competitors. Either one or the other population becomes extinct while the surviving population approaches a steady state, or there exists a positive steady state representing the coexistence of both populations.
Extremal problems and symmetrization for plane ring domains
A.
Yu.
Solynin;
M.
Vuorinen
4095-4112
Abstract: We show that Teichmüller's classical lower bound for the capacity of a ring domain, obtained by circular symmetrization, can be replaced by an explicit one which is almost always better. The proof is based on a duplication formula for the solution of an associated extremal problem. Some inequalities are obtained for conformal invariants.
Single generator problem
Jun-ichi
Tanaka
4113-4129
Abstract: Using the Stone-\v{C}ech compactification $\beta \mathbf Z$ of integers, we introduce a free extension of an almost periodic flow. Together with some properties of outer functions, we see that, in a certain class of ergodic Hardy spaces $H^p(\mu )$, $1\le p\le \infty$, the corresponding subspaces $H_0^p(\mu )$ are all singly generated. This shows the existence of maximal weak-$^*$ Dirichlet algebras, different from $H^\infty$ of the disc, for which the single generator problem is settled.
Wiener's test for space-time random walks and its applications
Yasunari
Fukai;
Kôhei
Uchiyama
4131-4152
Abstract: This paper establishes a criterion for whether a $d$-dimensional random walk on the integer lattice $\mathbf {Z}^{d}$ visits a space-time subset infinitely often or not. It is a precise analogue of Wiener's test for regularity of a boundary point with respect to the classical Dirichlet problem. The test obtained is applied to strengthen the harder half of Kolmogorov's test for the random walk.
Hyponormality and spectra of Toeplitz operators
Douglas
R.
Farenick;
Woo
Young
Lee
4153-4174
Abstract: This paper concerns algebraic and spectral properties of Toeplitz operators $T_{\varphi }$, on the Hardy space $H^{2}({\mathbb {T}})$, under certain assumptions concerning the symbols $\varphi \in L^{\infty }({\mathbb {T}})$. Among our algebraic results is a characterisation of normal Toeplitz opertors with polynomial symbols, and a characterisation of hyponormal Toeplitz operators with polynomial symbols of a prescribed form. The results on the spectrum are as follows. It is shown that by restricting the spectrum, a set-valued function, to the set of all Toeplitz operators, the spectrum is continuous at $T_{\varphi }$, for each quasicontinuous $\varphi$. Secondly, we examine under what conditions a classic theorem of H. Weyl, which has extensions to hyponormal and Toeplitz operators, holds for all analytic functions of a single Toeplitz operator with continuous symbol.
Algebraic surfaces with log canonical singularities and the fundamental groups of their smooth parts
D.-Q.
Zhang
4175-4184
Abstract: Let $(S, \Delta )$ be a log surface with at worst log canonical singularities and reduced boundary $\Delta$ such that $-(K_{S}+\Delta )$ is nef and big. We shall prove that $S^{o} = S - Sing S - \Delta$ either has finite fundamental group or is affine-ruled. Moreover, $\pi _{1}(S^{o})$ and the structure of $S$ are determined in some sense when $\Delta = 0$.
On polarized surfaces $(X,L)$ with $h^0(L)>0$, $\kappa (X)=2$, and $g(L)=q(X)$
Yoshiaki
Fukuma
4185-4197
Abstract: Let $X$ be a smooth projective surface over $\mathbb {C}$ and $L$ an ample Cartier divisor on $X$. If the Kodaira dimension $\kappa (X)\leq 1$ or $\operatorname {dim}H^{0}(L)>0$, the author proved $g(L)\geq q(X)$, where $q(X)=\operatorname {dim}H^{1}(\mathcal {O}_{X})$. If $\kappa (X)\leq 1$, then the author studied $(X,L)$ with $g(L)=q(X)$. In this paper, we study the polarized surface $(X,L)$ with $\kappa (X)=2$, $g(L)=q(X)$, and $\operatorname {dim}H^{0}(L)>0$.
K-theoretic classification for certain inductive limit $Z_2$ actions on real rank zero $C^*$-algebras
Hongbing
Su
4199-4230
Abstract: In this paper a K-theoretic classification is given of the C$^*$-algebra dynamical systems $(A, \alpha , Z_2)= \lim \limits_\to (A_n, {\alpha }_n, Z_2)$ where $A$ is of real rank zero, each $A_n$ is a finite direct sum of matrix algebras over finite connected graphs, and each $\alpha _n$ is induced by an action on each component of the spectrum of $A_n$. Corresponding to the trivial actions is the K-theoretic classification for real rank zero C$^*$-algebras that can be expressed as finite direct sums of matrix algebras over finite graphs obtained in Mem. Amer. Math. Soc. no. 547, vol. 114.
Separable Banach space theory needs strong set existence axioms
A.
James
Humphreys;
Stephen
G.
Simpson
4231-4255
Abstract: We investigate the strength of set existence axioms needed for separable Banach space theory. We show that a very strong axiom, $\Pi ^1_1$ comprehension, is needed to prove such basic facts as the existence of the weak-$*$ closure of any norm-closed subspace of $\ell _1=c_0^*$. This is in contrast to earlier work in which theorems of separable Banach space theory were proved in very weak subsystems of second order arithmetic, subsystems which are conservative over Primitive Recursive Arithmetic for $\Pi ^0_2$ sentences. En route to our main results, we prove the Krein-\v{S}mulian theorem in $\mathsf {ACA}_0$, and we give a new, elementary proof of a result of McGehee on weak-$*$ sequential closure ordinals.
Gross spaces
Saharon
Shelah;
Otmar
Spinas
4257-4277
Abstract: A Gross space is a vector space $E$ of infinite dimension over some field $F$, which is endowed with a symmetric bilinear form $\Phi :E^{2} \rightarrow F$ and has the property that every infinite dimensional subspace $U\subseteq E$ satisfies dim$U^{\perp }<$ dim$E$. Gross spaces over uncountable fields exist (in certain dimensions). The existence of a Gross space over countable or finite fields (in a fixed dimension not above the continuum) is independent of the axioms of ZFC. Here we continue the investigation of Gross spaces. Among other things, we show that if the cardinal invariant b equals $\omega _{1}$, a Gross space in dimension $\omega _{1}$ exists over every infinite field, and that it is consistent that Gross spaces exist over every infinite field but not over any finite field. We also generalize the notion of a Gross space and construct generalized Gross spaces in ZFC.
Krull-Schmidt theorems in dimension 1
Lawrence
S.
Levy;
Charles
J.
Odenthal
3391-3455
Abstract: Let $\Lambda$ be a semiprime, module-finite algebra over a commutative noetherian ring $R$ of Krull dimension 1. We find necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated $\Lambda$-modules, and necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated torsionfree $\Lambda$-modules (called ``$\Lambda$-lattices'' in integral representation theory, and ``maximal Cohen-Macaulay modules'' in the dimension-one situation in commutative algebra).
Package deal theorems and splitting orders in dimension 1
Lawrence
S.
Levy;
Charles
J.
Odenthal
3457-3503
Abstract: Let $\Lambda$ be a module-finite algebra over a commutative noetherian ring $R$ of Krull dimension 1. We determine when a collection of finitely generated modules over the localizations $\Lambda _{\mathbf {m}}$, at maximal ideals of $R$, is the family of all localizations $M_{\mathbf {m}}$ of a finitely generated $\Lambda$-module $M$. When $R$ is semilocal we also determine which finitely generated modules over the $J(R)$-adic completion of $\Lambda$ are completions of finitely generated $\Lambda$-modules. If $\Lambda$ is an $R$-order in a semisimple artinian ring, but not contained in a maximal such order, several of the basic tools of integral representation theory behave differently than in the classical situation. The theme of this paper is to develop ways of dealing with this, as in the case of localizations and completions mentioned above. In addition, we introduce a type of order called a ``splitting order'' of $\Lambda$ that can replace maximal orders in many situations in which maximal orders do not exist.
Drinfel´d algebra deformations, homotopy comodules and the associahedra
Martin
Markl;
Steve
Shnider
3505-3547
Abstract: The aim of this work is to construct a cohomology theory controlling the deformations of a general Drinfel'd algebra $A$ and thus finish the program which began in [13], [14]. The task is accomplished in three steps. The first step, which was taken in the aforementioned articles, is the construction of a modified cobar complex adapted to a non-coassociative comultiplication. The following two steps each involves a new, highly non-trivial, construction. The first construction, essentially combinatorial, defines a differential graded Lie algebra structure on the simplicial chain complex of the associahedra. The second construction, of a more algebraic nature, is the definition of a map of differential graded Lie algebras from the complex defined above to the algebra of derivations on the bar resolution. Using the existence of this map and the acyclicity of the associahedra we can define a so-called homotopy comodule structure (Definition 3.3 below) on the bar resolution of a general Drinfel'd algebra. This in turn allows us to define the desired cohomology theory in terms of a complex which consists, roughly speaking, of the bimodule and bicomodule maps from the bar resolution to the modified cobar resolution. The complex is bigraded but not a bicomplex as in the Gerstenhaber-Schack theory for bialgebra deformations. The new components of the coboundary operator are defined via the constructions mentioned above. The results of the paper were announced in [12].
Homology and some homotopy decompositions for the James filtration on spheres
Paul
Selick
3549-3572
Abstract: The filtrations on the James construction on spheres, $J_{k}\left (S^{2n}\right )$, have played a major role in the study of the double suspension $S^{2n-1}\to \Omega ^2 S^{2n+1}$ and have been used to get information about the homotopy groups of spheres and Moore spaces and to construct product decompositions of related spaces. In this paper we calculate $H_*\left ( \Omega J_{k}\left (S^{2n}\right ); {\mathbb {Z}}/p{\mathbb {Z}}\right )$ for odd primes $p$. When $k$ has the form $p^t-1$, the result is well known, but these are exceptional cases in which the homology has polynomial growth. We find that in general the homology has exponential growth and in some cases also has higher $p$-torsion. The calculations are applied to construct a $p$-local product decomposition of $\Omega J_{k}\left (S^{2n}\right )$ for $k<p^2-p$ which demonstrates a mod $p$ homotopy exponent in these cases.
Constructing product fibrations by means of a generalization of a theorem of Ganea
Paul
Selick
3573-3589
Abstract: A theorem of Ganea shows that for the principal homotopy fibration $\Omega B\to F\to E$ induced from a fibration $F\to E\to B$, there is a product decomposition $\Omega (E/F)\approx \Omega B\times \Omega (F*\Omega B)$. We will determine the conditions for a fibration $X\to Y\to Z$ to yield a product decomposition $\Omega (Z/Y)\approx X\times \Omega (X*Y)$ and generalize it to pushouts. Using this approach we recover some decompositions originally proved by very computational methods. The results are then applied to produce, after localization at an odd prime $p$, homotopy decompositions for $\Omega {J_{k}\left (S^{2n}\right )}$ for some $k$ which include the cases $k=p^{t}$. The factors of $\Omega {J_{p^{t}}\left (S^{2n}\right )}$ consist of the homotopy fibre of the attaching map $S^{2np^{t}-1}\to {J_{p^{t}-1}\left (S^{2n}\right )}$ for ${J_{p^{t}}\left (S^{2n}\right )}$ and combinations of spaces occurring in the Snaith stable decomposition of $\Omega ^{2} S^{2n+1}$.
Combinatorial $B_n$-analogues of Schubert polynomials
Sergey
Fomin;
Anatol
N.
Kirillov
3591-3620
Abstract: Combinatorial $B_{n}$-analogues of Schubert polynomials and corresponding symmetric functions are constructed and studied. The development is based on an exponential solution of the type $B$ Yang-Baxter equation that involves the nilCoxeter algebra of the hyperoctahedral group.
Continuous-trace groupoid $C^*$-algebras. III
Paul
S.
Muhly;
Jean
N.
Renault;
Dana
P.
Williams
3621-3641
Abstract: Suppose that ${\mathcal {G}}$ is a second countable locally compact groupoid with a Haar system and with abelian isotropy. We show that the groupoid $C^{\displaystyle *}$-algebra $C^{\displaystyle *} ({\mathcal {G}},\lambda )$ has continuous trace if and only if there is a Haar system for the isotropy groupoid ${\mathcal {A}}$ and the action of the quotient groupoid ${\mathcal {G}}/{\mathcal {A}}$ is proper on the unit space of ${\mathcal {G}}$.
Exact controllability and stabilizability of the Korteweg-de Vries equation
David
L.
Russell;
Bing-Yu
Zhang
3643-3672
Abstract: In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation \begin{equation*}% \partial _t % u + % u % \partial _x u + % \partial _x^3 u = f \tag {i}\label {star} \end{equation*} on the interval $0\leq x\leq 2\pi , \, t\geq 0$, with periodic boundary conditions \begin{equation*}\partial ^k_x u(2\pi , t ) = \partial ^k_x u(0,t) , \quad k=0,1,2, \tag {ii}\label {2star} \end{equation*} where the distributed control $f\equiv f(x,t)$ is restricted so that the ``volume'' $\int ^{2\pi }_0 u(x,t) dx$ of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control $f$ is allowed to act on the whole spatial domain $(0,2\pi )$, it is shown that the system is globally exactly controllable, i.e., for given $T> 0$ and functions $\phi (x)$, $\psi (x)$ with the same ``volume'', one can alway find a control $f$ so that the system (i)--(ii) has a solution $u(x,t)$ satisfying \begin{displaymath}u(x,0) = \phi (x) , \qquad \quad u(x,T) = \psi (x) .\end{displaymath} If the control $f$ is allowed to act on only a small subset of the domain $(0,2\pi )$, then the same result still holds if the initial and terminal states, $\psi$ and $\phi$, have small ``amplitude'' in a certain sense. In the case of closed loop control, the distributed control $f$ is assumed to be generated by a linear feedback law conserving the ``volume'' while monotonically reducing $\int ^{2\pi }_0 u(x,t)^2 dx$. The solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state. As in the open loop control case, a small amplitude assumption is needed if the control is allowed to act on only a small subdomain. The smoothing property of the periodic (linear) KdV equation discovered recently by Bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper.
Optimal natural dualities. II: General theory
B.
A.
Davey;
H.
A.
Priestley
3673-3711
Abstract: A general theory of optimal natural dualities is presented, built on the test algebra technique introduced in an earlier paper. Given that a set $R$ of finitary algebraic relations yields a duality on a class of algebras $\mathcal {A} = \operatorname {\mathbb {I}\mathbb {S}\mathbb {P}}( \underline {M})$, those subsets $R'$ of $R$ which yield optimal dualities are characterised. Further, the manner in which the relations in $R$ are constructed from those in $R'$ is revealed in the important special case that $\underline {M}$ generates a congruence-distributive variety and is such that each of its subalgebras is subdirectly irreducible. These results are obtained by studying a certain algebraic closure operator, called entailment, definable on any set of algebraic relations on $\underline {M}$. Applied, by way of illustration, to the variety of Kleene algebras and to the proper subvarieties $\mathbf {B}_{n}$ of pseudocomplemented distributive lattices, the theory improves upon and illuminates previous results.
Minimal isometric immersions of inhomogeneous spherical space forms into spheres--- a necessary condition for existence
Christine
M.
Escher
3713-3732
Abstract: Although much is known about minimal isometric immersions into spheres of homogeneous spherical space forms, there are no results in the literature about such immersions in the dominant case of inhomogeneous space forms. For a large class of these, we give a necessary condition for the existence of such an immersion of a given degree. This condition depends only upon the degree and the fundamental group of the space form and is given in terms of an explicitly computable function. Evaluating this function shows that neither $L(5,2)$ nor $L(8,3)$ admit a minimal isometric immersion into any sphere if the degree of the immersion is less than $28$, or less than $20$, respectively.
On representations of affine Kac-Moody groups and related loop groups
Yu
Chen
3733-3743
Abstract: We demonstrate a one to one correspondence between the irreducible projective representations of an affine Kac-Moody group and those of the related loop group, which leads to the results that every non-trivial representation of an affine Kac-Moody group must have its degree greater than or equal to the rank of the group and that the equivalence appears if and only if the group is of type $A_{n}^{(1)}$ for some $n\ge 1$. Moreover the characteristics of the base fields for the non-trivial representations are found being always zero.
Maximal subgroups in finite and profinite groups
Alexandre
V.
Borovik;
Laszlo
Pyber;
Aner
Shalev
3745-3761
Abstract: We prove that if a finitely generated profinite group $G$ is not generated with positive probability by finitely many random elements, then every finite group $F$ is obtained as a quotient of an open subgroup of $G$. The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann. of Math. 137 (1993), 203--220, we thenprove that a finite group $G$ has at most $|G|^c$ maximal soluble subgroups, and show that this result is rather useful in various enumeration problems.
Seifert manifolds with fiber spherical space forms
Jong Bum
Lee;
Kyung
Bai
Lee;
Frank
Raymond
3763-3798
Abstract: We study the Seifert fiber spaces modeled on the product space $S^3 \times \mathbb {R}^2$. Such spaces are ``fiber bundles'' with singularities. The regular fibers are spherical space-forms of $S^3$, while singular fibers are finite quotients of regular fibers. For each of possible space-form groups $\Delta$ of $S^3$, we obtain a criterion for a group extension $\varPi$ of $\Delta$ to act on $S^3 \times \mathbb {R}^2$ as weakly $S^3$-equivariant maps, which gives rise to a Seifert fiber space modeled on $S^3 \times \mathbb {R}^2$ with weakly $S^3$-equivariant maps $\mathrm {TOP}_{S^3}(S^3 \times \mathbb {R}^2)$ as the universal group. In the course of proving our main results, we also obtain an explicit formula for $H^2(Q; \mathbb {Z})$ for a cocompact crystallographic or Fuchsian group $Q$. Most of our methods for $S^3$ apply to compact Lie groups with discrete center, and we state some of our results in this general context.
Rook theory, compositions, and zeta functions
James
Haglund
3799-3825
Abstract: A new family of Dirichlet series having interesting combinatorial properties is introduced. Although they have no functional equation or Euler product, under the Riemann Hypothesis it is shown that these functions have no zeros in $\mathrm {Re}(s)>1/2$. Some identities in the ring of formal power series involving rook theory and continued fractions are developed.
On the ordering of $n$-modal cycles
Chris
Bernhardt
3827-3834
Abstract: The forcing relation on $n$-modal cycles is studied. If $\alpha$ is an $n$-modal cycle then the $n$-modal cycles with block structure that force $\alpha$ form a $2^n$-horseshoe above $\alpha$. If $n$-modal $\beta$ forces $\alpha$, and $\beta$ does not have a block structure over $\alpha$, then $\beta$ forces a $2$-horseshoe of simple extensions of $\alpha$.
Half De Rham complexes and line fields on odd-dimensional manifolds
Houhong
Fan
2947-2982
Abstract: In this paper we introduce a new elliptic complex on an odd-dimensional manifold with a self-dual line field. The notion of a self-dual line field is a generalization of the notion of a conformal line field. Ellipticity, Fredholm properties and Hodge decompositions of these new complexes are proved both in the case of a closed manifold and in the case of a manifold with boundary. The cohomology groups of these elliptic complexes are computed in some cases. In addition, in this paper, we generalize the notion of an anti-self-dual connection on a smooth 4-manifold to a 3-manifold with a line field and a smooth 5-manifold with a line field. The above new elliptic complexes can be twisted by anti-self-dual connections in dimensions 3 and 5, but only by flat connections in dimensions above 5. This reveals a special feature of dimensions 3 and 5.
An algebraic approach to multiparameter spectral theory
Luzius
Grunenfelder;
Tomaz
Kosir
2983-2998
Abstract: Root subspaces for multiprameter eigenvalue problems are described using coalgebraic techniques. An algorithm is given to construct bases for the root subspaces.
Topological entropy of standard type monotone twist maps
Oliver
Knill
2999-3013
Abstract: We study invariant measures of families of monotone twist maps $S_{\gamma }(q,p)$ $=$ $(2q-p+ \gamma \cdot V'(q),q)$ with periodic Morse potential $V$. We prove that there exist a constant $C=C(V)$ such that the topological entropy satisfies $h_{top}(S_{\gamma }) \geq \log (C \cdot \gamma )/3$. In particular, $h_{top}(S_{\gamma }) \to \infty$ for $|\gamma | \to \infty$. We show also that there exist arbitrary large $\gamma$ such that $S_{\gamma }$ has nonuniformly hyperbolic invariant measures $\mu _{\gamma }$ with positive metric entropy. For large $\gamma$, the measures $\mu _{\gamma }$ are hyperbolic and, for a class of potentials which includes $V(q)=\sin (q)$, the Lyapunov exponent of the map $S$ with invariant measure $\mu _{\gamma }$ grows monotonically with $\gamma$.
The Schwarzian derivative for maps between manifolds with complex projective connections
Robert
Molzon;
Karen
Pinney
Mortensen
3015-3036
Abstract: In this paper we define, in two equivalent ways, the Schwarzian derivative of a map between complex manifolds equipped with complex projective connections. Also, a new, coordinate-free definition of complex projective connections is given. We show how the Schwarzian derivative is related to the projective structure of the manifolds, to projective linear transformations, and to complex geodesics.
Cohomology of the complement of a free divisor
Francisco
J.
Castro-Jiménez;
Luis
Narváez-Macarro;
David
Mond
3037-3049
Abstract: We prove that if $D$ is a ``strongly quasihomogeneous" free divisor in the Stein manifold $X$, and $U$ is its complement, then the de Rham cohomology of $U$ can be computed as the cohomology of the complex of meromorphic differential forms on $X$ with logarithmic poles along $D$, with exterior derivative. The class of strongly quasihomogeneous free divisors, introduced here, includes free hyperplane arrangements and the discriminants of stable mappings in Mather's nice dimensions (and in particular the discriminants of Coxeter groups).
Compact self-dual Hermitian surfaces
Vestislav
Apostolov;
Johann
Davidov;
Oleg
Muskarov
3051-3063
Abstract: In this paper, we obtain a classification (up to conformal equivalence) of the compact self-dual Hermitian surfaces. As an application, we prove that every compact Hermitian surface of pointwise constant holomorphic sectional curvature with respect to either the Riemannian or the Hermitian connection is Kähler.
Extensions of codimension one immersions
Christian
Pappas
3065-3083
Abstract: We give a method for constructing all of the extensions of an immersion, and determine the CW structure and diffeomorphism type of each.
On vanishing of characteristic numbers in Poincaré complexes
Yanghyun
Byun
3085-3095
Abstract: Let $G_r(X)\subset \pi _r(X)$ be the evaluation subgroup as defined by Gottlieb. Assume the Hurewicz map $G_r(X)\rightarrow H_r(X; R)$ is non-trivial and $R$ is a field. We will prove: if $X$ is a Poincaré complex oriented in $R$-coefficient, all the characteristic numbers of $X$ in $R$-coefficient vanish. Similarly, if $R=Z_p$ and $X$ is a $Z_p$-Poincaré complex, then all the mod $p$ Wu numbers vanish. We will also show that the existence of a non-trivial derivation on $H^*(X; Z_p)$ with some suitable conditions implies vanishing of mod $p$ Wu numbers.
Presentation and central extensions of mapping class groups
Sylvain
Gervais
3097-3132
Abstract: We give a presentation of the mapping class group $\mathcal {M}$ of a (possibly bounded) surface, considering either all twists or just non-separating twists as generators. We also study certain central extensions of $\mathcal {M}$. One of them plays a key role in studying TQFT functors, namely the mapping class group of a $p_1$-structure surface. We give a presentation of this extension.
Conjugate points and shocks in nonlinear optimal control
N.
Caroff;
H.
Frankowska
3133-3153
Abstract: We investigate characteristics of the Hamilton-Jacobi-Bellman equation arising in nonlinear optimal control and their relationship with weak and strong local minima. This leads to an extension of the Jacobi conjugate points theory to the Bolza control problem. Necessary and sufficient optimality conditions for weak and strong local minima are stated in terms of the existence of a solution to a corresponding matrix Riccati differential equation.
A group of paths in $\mathbb{R}^2$
Richard
Kenyon
3155-3172
Abstract: We define a group structure on the set of compact ``minimal'' paths in $\mathbb {R} ^2$. We classify all finitely generated subgroups of this group $G$: they are free products of free abelian groups and surface groups. Moreover, each such group occurs in $G$. The subgroups of $G$ isomorphic to surface groups arise from certain topological $1$-forms on the corresponding surfaces. We construct examples of such $1$-forms for cohomology classes corresponding to certain eigenvectors for the action on cohomology of a pseudo-Anosov diffeomorphism. Using $G$ we construct a non-polygonal tiling problem in $\mathbb {R} ^2$, that is, a finite set of tiles whose corresponding tilings are not equivalent to those of any set of polygonal tiles. The group $G$ has applications to combinatorial tiling problems of the type: given a set of tiles $P$ and a region $R$, can $R$ be tiled by translated copies of tiles in $P$?
The automorphism group of a coded system
Doris
Fiebig;
Ulf-Rainer
Fiebig
3173-3191
Abstract: We give a general construction of coded systems with an automorphism group isomorphic to $\mathbf Z\oplus G$ where $G$ is any preassigned group which has a ``continuous block presentation'' (the isomorphism will map the shift to $(1,e_G))$. Several applications are given. In particular, we obtain automorphism groups of coded systems which are abelian, which are finitely generated and one which contains $\mathbf Z[1/2]$. We show that any group which occurs as a subgroup of the automorphism group of some subshift with periodic points dense already occurs for some synchronized system.
Packing measure of the sample paths of fractional Brownian motion
Yimin
Xiao
3193-3213
Abstract: Let $X(t) (t \in % \mathbf {R})$ be a fractional Brownian motion of index $% \alpha$ in $% \mathbf {R}^d.$ If $1 < % \alpha d$, then there exists a positive finite constant $K$ such that with probability 1, \begin{displaymath}\hbox {$\phi$-$p(X([0,t]))$} = Kt \ \hbox {for any } t > 0 ,\end{displaymath} where $% \phi (s) = s^{\frac 1 {% \alpha }}/ (\log \log \frac 1 s)^{\frac 1 {2 % \alpha }}$ and $\phi$-$p (X([0,t]))$ is the $\phi$-packing measure of $X([0,t])$.
Which families of $l$-modal maps are full?
R.
Galeeva;
S.
van Strien
3215-3221
Abstract: In this paper we shall show that certain conditions which are sufficient for a family of one-dimensional maps to be full cannot be dispensed with.
Foxby duality and Gorenstein injective and projective modules
Edgar
E.
Enochs;
Overtoun
M. G.
Jenda;
Jinzhong
Xu
3223-3234
Abstract: In 1966, Auslander introduced the notion of the $G$-dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of $G$-dimensions. It seemed appropriate to call the modules with $G$-dimension 0 Gorenstein projective, so the basic problem was to define Gorenstein injective modules. These were defined in Math. Z. 220 (1995), 611--633 and were shown to have properties predicted by Auslander's results. The way we define Gorenstein injective modules can be dualized, and so we can define Gorenstein projective modules (i.e. modules of $G$-dimension 0) whether the modules are finitely generated or not. The investigation of these modules and also Gorenstein flat modules was continued by Enochs, Jenda, Xu and Torrecillas. However, to get good results it was necessary to take the base ring Gorenstein. H.-B. Foxby introduced a duality between two full subcategories in the category of modules over a local Cohen-Macaulay ring admitting a dualizing module. He proved that the finitely generated modules in one category are precisely those of finite $G$-dimension. We extend this result to modules which are not necessarily finitely generated and also prove the dual result, i.e. we characterize the modules in the other class defined by Foxby. The basic result of this paper is that the two classes involved in Foxby's duality coincide with the classes of those modules having finite Gorenstein projective and those having finite Gorenstein injective dimensions. We note that this duality then allows us to extend many of our results to the original Auslander setting.
An $\omega_2$-minimal Boolean algebra
Mariusz
Rabus
3235-3244
Abstract: For every linear order $L$ we define a notion of $L$-minimal Boolean algebra and then give a consistent example of an $\omega _{2}$-minimal algebra. The Stone space $X$ of our algebra contains a point $\{*\}$ such that $X-\{*\}$ is an example of a countably tight, initially $\aleph _{1}$-compact, non-compact space. This answers a question of Dow and van Douwen.
Computation of Nielsen numbers for maps of closed surfaces
O.
Davey;
E.
Hart;
K.
Trapp
3245-3266
Abstract: Let $X$ be a closed surface, and let $f: X \rightarrow X$ be a map. We would like to determine $\text {Min}(f):= \mathrm {min} \{ | \mathrm {Fix}g| : g \simeq f\}.$ Nielsen fixed point theory provides a lower bound $N(f)$ for $\text {Min}(f)$, called the Nielsen number, which is easy to define geometrically and is difficult to compute. We improve upon an algebraic method of calculating $N(f)$ developed by Fadell and Husseini, so that the method becomes algorithmic for orientable closed surfaces up to the distinguishing of Reidemeister orbits. Our improvement makes tractable calculations of Nielsen numbers for many maps on surfaces of negative Euler characteristic. We apply the improved method to self-maps on the connected sum of two tori including classes of examples for which no other method is known. We also include the application of this algebraic method to maps on the Klein bottle $K$. Nielsen numbers for maps on $K$ were first calculated (geometrically) by Halpern. We include a sketch of Halpern's never published proof that $N(f)= \text {Min}(f)$ for all maps $f$ on $K$.
On quadratic forms of height two and a theorem of Wadsworth
Detlev
W.
Hoffmann
3267-3281
Abstract: Let $\varphi$ and $\psi$ be anisotropic quadratic forms over a field $F$ of characteristic not $2$. Their function fields $F(% \varphi )$ and $F(\psi )$ are said to be equivalent (over $F$) if $% \varphi \otimes F(\psi )$ and $\psi \otimes F(% \varphi )$ are isotropic. We consider the case where $\dim % \varphi =2^n$ and $% \varphi$ is divisible by an $(n-2)$-fold Pfister form. We determine those forms $\psi$ for which $% \varphi$ becomes isotropic over $F(\psi )$ if $n\leq 3$, and provide partial results for $n\geq 4$. These results imply that if $F(% \varphi )$ and $F(\psi )$ are equivalent and $\dim % \varphi =\dim \psi$, then $% \varphi$ is similar to $\psi$ over $F$. This together with already known results yields that if $% \varphi$ is of height $2$ and degree $1$ or $2$, and if $\dim % \varphi =\dim \psi$, then $F(% \varphi )$ and $F(\psi )$ are equivalent iff $F(% \varphi )$ and $F(\psi )$ are isomorphic over $F$.
On multiplicities in polynomial system solving
M.
G.
Marinari;
H.
M.
Möller;
T.
Mora
3283-3321
Abstract: This paper deals with the description of the solutions of zero dimensional systems of polynomial equations. Based on different models for describing solutions, we consider suitable representations of a multiple root, or more precisely suitable descriptions of the primary component of the system at a root. We analyse the complexity of finding the representations and of algorithms which perform transformations between the different representations.
Quadratic forms for the 1-D semilinear Schrödinger equation
Carlos
E.
Kenig;
Gustavo
Ponce;
Luis
Vega
3323-3353
Abstract: This paper is concerned with 1-D quadratic semilinearSchrödinger equations. We study local well posedness in classical Sobolev space $H^s$ of the associated initial value problem and periodic boundary value problem. Our main interest is to obtain the lowest value of $s$ which guarantees the desired local well posedness result. We prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered.
Weierstrass points on cyclic covers of the projective line
Christopher
Towse
3355-3378
Abstract: We are interested in cyclic covers of the projective line which are totally ramified at all of their branch points. We begin with curves given by an equation of the form $y^{n}=f(x)$, where $f$ is a polynomial of degree $d$. Under a mild hypothesis, it is easy to see that all of the branch points must be Weierstrass points. Our main problem is to find the total Weierstrass weight of these points, $BW$. We obtain a lower bound for $BW$, which we show is exact if $n$ and $d$ are relatively prime. As a fraction of the total Weierstrass weight of all points on the curve, we get the following particularly nice asymptotic formula (as well as an interesting exact formula): \begin{equation*}\lim _{d\to \infty }\frac {BW}{g^{3}-g}=\frac {n+1}{3(n-1)^{2}}, \end{equation*} where $g$ is the genus of the curve. In the case that $n=3$ (cyclic trigonal curves), we are able to show in most cases that for sufficiently large primes $p$, the branch points and the non-branch Weierstrass points remain distinct modulo $p$.
$S$-integral points of $\mathbb{P}^n-\{2n+1$ hyperplanes in general position over number fields and function fields\}
Julie
T.-Y.
Wang
3379-3389
Abstract: For the number field case we will give an upper bound on the number of the $S$-integral points in $\mathbb {P}^n(K)-\{ 2n+1\text { hyperplanes in general}$$\text {position}\}$. The main tool here is the explicit upper bound of the number of solutions of $S$-unit equations (Invent. Math. 102 (1990), 95--107). For the function field case we will give a bound on the height of the $S$-integral points in $\mathbb {P}^n(K)-\{ 2n+1\text { hyperplanes in general position}\}$. We will also give a bound for the number of ``generators" of those $S$-integral points. The main tool here is the $S$-unit Theorem by Brownawell and Masser (Proc. Cambridge Philos. Soc. 100 (1986), 427--434).
On functions in the little Bloch space and inner functions
S.
Rohde
2519-2531
Abstract: We prove that analytic functions in the little Bloch space assume every value as a radial limit on a set of Hausdorff dimension one, unless they have radial limits on a set of positive measure. The analogue for inner functions in the little Bloch space is also proven, and characterizations of various classes of Bloch functions in terms of their level sets are given.
Propagation of Gevrey Regularity for a Class of Hypoelliptic Equations
Antonio
Bove;
David
S.
Tartakoff
2533-2575
Abstract: We prove results on the propagation of Gevrey and analytic wave front sets for a class of $C^\infty$ hypoelliptic equations with double characteristics.
Global (and local) analyticity for second order operators constructed from rigid vector fields on products of tori
David
S.
Tartakoff
2577-2583
Abstract: We prove global analytic hypoellipticity on a product of tori for partial differential operators which are constructed as rigid (variable coefficient) quadratic polynomials in real vector fields satisfying the Hörmander condition and where $P$ satisfies a ``maximal'' estimate. We also prove an analyticity result that is local in some variables and global in others for operators whose prototype is \begin{displaymath}P=\left ( \frac \partial {\partial x_1}\right ) ^2+\left ( \frac \partial { \partial x_2}\right ) ^2+\left ( a(x_1,x_2)\frac \partial {\partial t}\right )^2 \end{displaymath} (with analytic $a(x),a(0)=0$, naturally, but not identically zero). The results, because of the flexibility of the methods, generalize recent work of Cordaro and Himonas in [4] and Himonas in [8] which showed that certain operators known not to be locally analytic hypoelliptic (those of Baouendi and Goulaouic [1], Hanges and Himonas [6], and Christ [3]) were globally analytic hypoelliptic on products of tori.
Immersed $n$-manifolds in $\mathbf{R}^{2n}$ and the double points of their generic projections into $\mathbf{R}^{2n-1}$
Osamu
Saeki;
Kazuhiro
Sakuma
2585-2606
Abstract: We give two congruence formulas concerning the number of non-trivial double point circles and arcs of a smooth map with generic singularities --- the Whitney umbrellas --- of an $n$-manifold into $\text {\bf R}^{2n-1}$, which generalize the formulas by Szücs for an immersion with normal crossings. Then they are applied to give a new geometric proof of the congruence formula due to Mahowald and Lannes concerning the normal Euler number of an immersed $n$-manifold in $\text {\bf R}^{2n}$. We also study generic projections of an embedded $n$-manifold in $\text {\bf R}^{2n}$ into $\text {\bf R}^{2n-1}$ and prove an elimination theorem of Whitney umbrella points of opposite signs, which is a direct generalization of a recent result of Carter and Saito concerning embedded surfaces in $\text {\bf R}^{4}$. The problem of lifting a map into $\text {\bf R}^{2n-1}$ to an embedding into $\text {\bf R}^{2n}$ is also studied.
Simplifying stable mappings into the plane from a global viewpoint
Mahito
Kobayashi;
Osamu
Saeki
2607-2636
Abstract: Let $f : M \to \text {\bf R}^{2}$ be a $C^{\infty }$ stable map of an $n$-dimensional manifold into the plane. The main purpose of this paper is to define a global surgery operation on $f$ which simplifies the configuration of the critical value set and which does not change the diffeomorphism type of the source manifold $M$. For this purpose, we also study the quotient space $W_{f}$ of $f$, which is the space of the connected components of the fibers of $f$, and we completely determine its local structure for arbitrary dimension $n$ of the source manifold $M$. This is a completion of the result of Kushner, Levine and Porto for dimension 3 and that of Furuya for orientable manifolds of dimension 4. We also pay special attention to dimension 4 and obtain a simplification theorem for stable maps whose regular fiber is a torus or a 2-sphere, which is a refinement of a result of Kobayashi.
Rotation Vectors and Fixed Points of Area Preserving Surface Diffeomorphisms
John
Franks
2637-2662
Abstract: We consider the (homological) rotation vectors for area preserving diffeomorphisms of compact surfaces which are homotopic to the identity. There are two main results. The first is that if $0$ is in the interior of the convex hull of the rotation vectors for such a diffeomorphism then $f$ has a fixed point of positive index. The second result asserts that if $f$ has a vanishing mean rotation vector then $f$ has a fixed point of positive index. There are several applications including a new proof of the Arnold conjecture for area preserving diffeomorphisms of compact surfaces.
Extremal functions for Moser's inequality
Kai-Ching
Lin
2663-2671
Abstract: Let $\Omega$ be a bounded smooth domain in $R^{n}$, and $u(x)$ a $C^{1}$ function with compact support in $\Omega$. Moser's inequality states that there is a constant $c_{o}$, depending only on the dimension $n$, such that \begin{equation*}% \frac {1}{|\Omega |} \int _{\Omega } e^{n \omega _{n-1}^{\frac {1}{n-1}} u^{\frac {n}{n-1}}}\, dx \leq c_{o}% , \end{equation*} where $|\Omega |$ is the Lebesgue measure of $\Omega$, and $\omega _{n-1}$ the surface area of the unit ball in $R^{n}$. We prove in this paper that there are extremal functions for this inequality. In other words, we show that the \begin{equation*}% \sup \{\frac {1}{|\Omega |} \int _{\Omega } e^{n \omega _{n-1}^{\frac {1}{n-1}} u^{\frac {n}{n-1}}}\, dx: u \in W_{o}^{1,n}, \|\nabla u\|_{n} \leq 1 \} % \end{equation*} is attained. Earlier results include Carleson-Chang (1986, $\Omega$ is a ball in any dimension) and Flucher (1992, $\Omega$ is any domain in 2-dimensions).
Spectral Convergence for Degenerating Sequences of Three Dimensional Hyperbolic Manifolds
Lizhen
Ji
2673-2688
Abstract: For degenerating sequences of three dimensional hyperbolic manifolds of finite volume, we prove convergence of their eigenfunctions, heat kernel and spectral measure.
Reciprocity Laws in the Verlinde Formulae for the Classical Groups
W.
M.
Oxbury;
S.
M. J.
Wilson
2689-2710
Abstract: The Verlinde formula is computed for each of the simply-connected classical Lie groups, and it is shown that the resulting formula obeys certain reciprocity laws with respect to the exchange of the rank and the level. Some corresponding dualities between spaces of sections of theta line bundles over moduli spaces of $G$-bundles on curves are conjectured but not proved.
Low codimensional submanifolds of Euclidean space with nonnegative isotropic curvature
Francesco
Mercuri;
Maria
Helena
Noronha
2711-2724
Abstract: In this paper we study compact submanifolds of Euclidean space with nonnegative isotropic curvature and low codimension. We determine their homology completely in the case of hypersurfaces and for some low codimensional conformally flat immersions.
Divisors on Generic Complete Intersections in Projective Space
Geng
Xu
2725-2736
Abstract: Let $V$ be a generic complete intersection of hypersurfaces of degree $d_{1}, d_{2}, \cdots , d_{m}$ in $n$-dimensional projective space. We study the question when a divisor on $V$ is nonrational or of general type, and give an alternative proof of a result of Ein. We also give some improvement of Ein's result in the case $d_{1}+d_{2}+\cdots + d_{m}=n+2$.
On complete nonorientable minimal surfaces with low total curvature
Francisco
J.
Lopez
2737-2758
Abstract: We classify complete nonorientable minimal surfaces in $\mathbb R^3$ with total curvature $-8\pi$.
On Jacobian Ideals Invariant by a Reducible $s\ell(2,\mathbf{C})$ Action
Yung
Yu
2759-2791
Abstract: This paper deals with a reducible $s\ell (2, \mathbf {C})$ action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau Conjecture: suppose that $s\ell (2, \mathbf {C})$ acts on the formal power series ring via $(0.1)$. Then $I(f)=(\ell _{i_{1}})\oplus (\ell _{i_{2}})\oplus \cdots \oplus (\ell _{i_{s}})$ modulo some one dimensional $s\ell (2, \mathbf {C})$ representations where $(\ell _{i})$ is an irreducible $s\ell (2, \mathbf {C})$ representation of dimension $\ell _{i}$ or empty set and $\{\ell _{i_{1}},\ell _{i_{2}},\ldots ,\ell _{i_{s}}\}\subseteq \{\ell _{1},\ell _{2},\ldots ,\ell _{r}\}$. Unlike classical invariant theory which deals only with irreducible action and 1--dimensional representations, we treat the reducible action and higher dimensional representations succesively.
Finite-dimensional lattice-subspaces of $C(\Omega)$ and curves of $\mathbb{R}^n$
Ioannis
A.
Polyrakis
2793-2810
Abstract: Let $x_1,\dotsc ,x_n$ be linearly independent positive functions in $C(\Omega )$, let $X$ be the vector subspace generated by the $x_i$ and let $\beta$ denote the curve of $\mathbb R^n$ determined by the function $\beta (t)=\frac {1}{z(t)} (x_1(t),x_2(t),\dotsc ,x_n(t))$, where $z(t)=x_1(t)+x_2(t)+\dotsb +x_n(t)$. We establish that $X$ is a vector lattice under the induced ordering from $C(\Omega )$ if and only if there exists a convex polygon of $\mathbb R^n$ with $n$ vertices containing the curve $\beta$ and having its vertices in the closure of the range of $\beta$. We also present an algorithm which determines whether or not $X$ is a vector lattice and in case $X$ is a vector lattice it constructs a positive basis of $X$. The results are also shown to be valid for general normed vector lattices.
On a parabolic equation with a singular lower order term
Qi
Zhang
2811-2844
Abstract: We obtain the existence of the weak Green's functions of parabolic equations with lower order coefficients in the so called parabolic Kato class which is being proposed as a natural generalization of the Kato class in the study of elliptic equations. As a consequence we are able to prove the existence of solutions of some initial boundary value problems. Moreover, based on a lower and an upper bound of the Green's function, we prove a Harnack inequality for the non-negative weak solutions.
Strong laws for $L$- and $u$-statistics
J.
Aaronson;
R.
Burton;
H.
Dehling;
D.
Gilat;
T.
Hill;
B.
Weiss
2845-2866
Abstract: Strong laws of large numbers are given for $L$-statistics (linear combinations of order statistics) and for $U$-statistics (averages of kernels of random samples) for ergodic stationary processes, extending classical theorems of Hoeffding and of Helmers for iid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.
On Gelfand-Kirillov Transcendence Degree
James
J.
Zhang
2867-2899
Abstract: We study some basic properties of the Gelfand-Kirillov transcendence degree and compute the transcendence degree of various infinite-dimensional division algebras including quotient division algebras of quantized algebras related to quantum groups, 3-dimensional Artin-Schelter regular algebras and the 4-dimensional Sklyanin algebra.
Properties of extremal functions for some nonlinear functionals on Dirichlet spaces
Alec
Matheson;
Alexander
R.
Pruss
2901-2930
Abstract: Let $\mathfrak {B}$ be the set of holomorphic functions $f$ on the unit disc $D$ with $f(0)=0$ and Dirichlet integral $(1/\pi ) \iint _{D} |f'|^{2}$ not exceeding one, and let $\mathfrak {b}$ be the set of complex-valued harmonic functions $f$ on the unit disc with $f(0)=0$ and Dirichlet integral $(1/2)(1/\pi ) \iint _{D} |\nabla f|^{2}$ not exceeding one. For a (semi)continuous function $\Phi :[0,\infty ) \to \mathbb {R}$, define the nonlinear functional $\Lambda _{\Phi }$ on $\mathfrak {B}$ or $\mathfrak {b}$ by $\Lambda _{\Phi }(f)={\frac {1}{2\pi }} \int _{0}^{2\pi }\Phi (|f(e^{i\theta })|)\,d\theta$. We study the existence and regularity of extremal functions for these functionals, as well as the weak semicontinuity properties of the functionals. We also state a number of open problems.
Representations of monoids arising from finite groups of Lie type
A.
Salwa
2931-2945
Abstract: A class of finite monoids $M$ constructed from a group $G$ of Lie type is considered. We describe the irreducible complex representations and prove the complete reducibility of the representations of $M$. The sandwich matrix of $M$ is decomposed into a product of matrices corresponding to maximal parabolic subgroups of $G$.
Hyperbolic groups and their quotients of bounded exponents
S.
V.
Ivanov;
A.
Yu.
Ol'shanskii
2091-2138
Abstract: In 1987, Gromov conjectured that for every non-elementary hyperbolic group $G$ there is an $n =n(G)$ such that the quotient group $G/G^{n}$ is infinite. The article confirms this conjecture. In addition, a description of finite subgroups of $G/G^{n}$ is given, it is proven that the word and conjugacy problem are solvable in $G/G^{n}$ and that $\bigcap _{k=1}^{\infty }G^{k} = \{ 1\}$. The proofs heavily depend upon prior authors' results on the Gromov conjecture for torsion free hyperbolic groups and on the Burnside problem for periodic groups of even exponents.
An Application of Convex Integration to Contact Geometry
Hansjörg
Geiges;
Jesús
Gonzalo
2139-2149
Abstract: We prove that every closed, orientable $3$-manifold $M$ admits a parallelization by the Reeb vector fields of a triple of contact forms with equal volume form. Our proof is based on Gromov's convex integration technique and the $h$-principle. Similar methods can be used to show that $M$ admits a parallelization by contact forms with everywhere linearly independent Reeb vector fields. We also prove a generalization of this latter result to higher dimensions. If $M$ is a closed $(2n+1)$-manifold with contact form $\omega$ whose contact distribution $\ker \omega$ admits $k$ everywhere linearly independent sections, then $M$ admits $k+1$ linearly independent contact forms with linearly independent Reeb vector fields.
Bott's vanishing theorem for regular Lie algebroids
Jan
Kubarski
2151-2167
Abstract: Differential geometry has discovered many objects which determine Lie algebroids playing a role analogous to that of Lie algebras for Lie groups. For example: --- differential groupoids, --- principal bundles, --- vector bundles, --- actions of Lie groups on manifolds, --- transversally complete foliations, --- nonclosed Lie subgroups, --- Poisson manifolds, --- some complete closed pseudogroups. We carry over the idea of Bott's Vanishing Theorem to regular Lie algebroids (using the Chern-Weil homomorphism of transitive Lie algebroids investigated by the author) and, next, apply it to new situations which are not described by the classical version, for example, to the theory of transversally complete foliations and nonclosed Lie subgroups in order to obtain some topological obstructions for the existence of involutive distributions and Lie subalgebras of some types (respectively).
Cesàro Summability of Two-dimensional Walsh-Fourier Series
Ferenc
Weisz
2169-2181
Abstract: We introduce p-quasi-local operators and the two-dimensionaldyadic Hardy spaces $H_{p}$ defined by the dyadic squares. It is proved that, if a sublinear operator $T$ is p-quasi-local and bounded from $L_{\infty }$ to $L_{\infty }$, then it is also bounded from $H_{p}$ to $L_{p}$ $(0<p \leq 1)$. As an application it is shown that the maximal operator of the Cesàro means of a martingale is bounded from $H_{p}$ to $L_{p}$ $(1/2<p \leq \infty )$ and is of weak type (1,1) provided that the supremum in the maximal operator is taken over a positive cone. So we obtain the dyadic analogue of a summability result with respect to two-dimensional trigonometric Fourier series due to Marcinkievicz and Zygmund; more exactly, the Cesàro means of a function $f \in L_{1}$ converge a.e. to the function in question, provided again that the limit is taken over a positive cone. Finally, it is verified that if we take the supremum in a cone, but for two-powers, only, then the maximal operator of the Cesàro means is bounded from $H_{p}$ to $L_{p}$ for every $0<p \leq \infty$.
$\Omega$-inverse limit stability theorem
Hiroshi
Ikeda
2183-2200
Abstract: We prove that if an endomorphism $f$ satisfies weak Axiom A and the no-cycles condition then $f$ is $\Omega$-inverse limit stable. This result is a generalization of Smale's $\Omega$-stability theorem from diffeomorphisms to endomorphisms.
Real connective K-theory and the quaternion group
Dilip
Bayen;
Robert
R.
Bruner
2201-2216
Abstract: Let $ko$ be the real connective K-theory spectrum. We compute $ko_*BG$ and $ko^*BG$ for groups $G$ whose Sylow 2-subgroup is quaternion of order 8. Using this we compute the coefficients $t(ko)^G_*$ of the $G$ fixed points of the Tate spectrum $t(ko)$ for $G = Sl_2(3)$ and $G = Q_8$. The results provide a counterexample to the optimistic conjecture of Greenlees and May [9, Conj. 13.4], by showing, in particular, that $t(ko)^G$ is not a wedge of Eilenberg-Mac Lane spectra, as occurs for groups of prime order.
Relatively free invariant algebras of finite reflection groups
Mátyás
Domokos
2217-2234
Abstract: Let $G$ be a finite subgroup of $Gl_{n}(K)$ $(K$ is a field of characteristic $0$ and $n\geq 2)$ acting by linear substitution on a relatively free algebra $K\langle x_{1},\hdots ,x_{n}\rangle /I$ of a variety of unitary associative algebras. The algebra of invariants is relatively free if and only if $G$ is a pseudo-reflection group and $I$ contains the polynomial $[[x_{2},x_{1}],x_{1}].$
The Structure and Enumeration of Link Projections
Martin
Bridgeman
2235-2248
Abstract: We define a decomposition of link projections whose pieces we call atoroidal graphs. We describe a surgery operation on these graphs and show that all atoroidal graphs can be generated by performing surgery repeatedly on a family of well-known link projections. This gives a method of enumerating atoroidal graphs and hence link projections by recomposing the pieces of the decomposition.
Special values of symmetric hypergeometric functions
Francesco
Baldassarri
2249-2289
Abstract: We discuss the $p$-adic formula (0.3) of P. Th. Young, in the framework of Dwork's theory of the hypergeometric equation. We show that it gives the value at 0 of the Frobenius automorphism of the unit root subcrystal of the hypergeometric crystal. The unit disk at 0 is in fact singular for the differential equation under consideration, so that it's not a priori clear that the Frobenius structure should extend to that disk. But the singularity is logarithmic, and it extends to a divisor with normal crossings relative to $\mathbf {Z}_{p}$ in $\mathbf {P}^{1}_{\mathbf {Z}_{p}}$. We show that whenever the unit root subcrystal of the hypergeometric system has generically rank 1, it actually extends as a logarithmic $F$-subcrystal to the unit disk at 0. So, in these optics, ``singular classes are not supersingular''. If, in particular, the holomorphic solution at 0 is bounded, the extended logarithmic $F$-crystal has no singularity in the residue class of 0, so that it is an $F$-crystal in the usual sense and the Frobenius operation is holomorphic. We examine in detail its analytic form.
$C^*$-Algebras with the Approximate Positive Factorization Property
G.
J.
Murphy;
N.
C.
Phillips
2291-2306
Abstract: We say that a unital $\mathrm {C}^{*}$-algebra $A$ has the approximate positive factorization property (APFP) if every element of $A$ is a norm limit of products of positive elements of $A$. (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of $\mathrm {C}^{*}$-algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial $K_{1}$, and stable rank 1. In the unital case, the $K_{0}$ group separates the tracial states. The APFP passes to matrix algebras, and if $I$ is an ideal in $A$ such that $I$ and $A/I$ have the APFP, then so does $A$. We also give some new examples of $\mathrm {C}^{*}$-algebras with the APFP, including type $\mathrm {II}_{1}$ factors and infinite-dimensional simple unital direct limits of homogeneous $\mathrm {C}^{*}$-algebras with slow dimension growth, real rank zero, and trivial $K_{1}$ group. Simple direct limits of homogeneous $\mathrm {C}^{*}$-algebras with slow dimension growth which have the APFP must have real rank zero, but we also give examples of (nonsimple) unital algebras with the APFP which do not have real rank zero. Our analysis leads to the introduction of a new concept of rank for a $\mathrm {C}^{*}$-algebra that may be of interest in the future.
Boundary and Lens Rigidity of Lorentzian Surfaces
Lars
Andersson;
Mattias
Dahl;
Ralph
Howard
2307-2329
Abstract: Let $g$ be a Lorentzian metric on the plane $\r ^2$ that agrees with the standard metric $g_0=-dx^2+dy^2$ outside a compact set and so that there are no conjugate points along any time-like geodesic of $(\r ^2,g)$. Then $(\r ^2,g)$ and $(\r ^2,g_0)$ are isometric. Further, if $(M,g)$ and $(M^*,g^*)$ are two dimensional compact time oriented Lorentzian manifolds with space--like boundaries and so that all time-like geodesics of $(M,g)$ maximize the distances between their points and $(M,g)$ and $(M^*,g^*)$ are ``boundary isometric'', then there is a conformal diffeomorphism between $(M,g)$ and $(M^*,g^*)$ and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.
Composition operators between Bergman and Hardy spaces
Wayne
Smith
2331-2348
Abstract: We study composition operators between weighted Bergman spaces. Certain growth conditions for generalized Nevanlinna counting functions of the inducing map are shown to be necessary and sufficient for such operators to be bounded or compact. Particular choices for the weights yield results on composition operators between the classical unweighted Bergman and Hardy spaces.
Multiplicity results for periodic solutions of second order ODEs with asymmetric nonlinearities
C.
Rebelo;
F.
Zanolin
2349-2389
Abstract: We prove various results on the existence and multiplicity of harmonic and subharmonic solutions to the second order nonautonomous equation $x'' + g(x) = s + w(t,x)$, as $s\to +\infty$ or $s\to - \infty ,$ where $g$ is a smooth function defined on a open interval $]a,b[\subset {\mathbb {R}}.$ The hypotheses we assume on the nonlinearity $g(x)$ allow us to cover the case $b=+\infty$ (or $a = -\infty$) and $g$ having superlinear growth at infinity, as well as the case $b < +\infty$ (or $a > -\infty$) and $g$ having a singularity in $b$ (respectively in $a$). Applications are given also to situations like $g'(-\infty ) \not = g'(+\infty )$ (including the so-called ``jumping nonlinearities''). Our results are connected to the periodic Ambrosetti - Prodi problem and related problems arising from the Lazer - McKenna suspension bridges model.
Smooth classification of geometrically finite one-dimensional maps
Yunping
Jiang
2391-2412
Abstract: The scaling function of a one-dimensional Markov map is defined and studied. We prove that the scaling function of a non-critical geometrically finite one-dimensional map is Hölder continuous, while the scaling function of a critical geometrically finite one-dimensional map is discontinuous. We prove that scaling functions determine Lipschitz conjugacy classes, and moreover, that the scaling function and the exponents and asymmetries of a geometrically finite one-dimensional map are complete $C^{1}$-invariants within a mixing topological conjugacy class.
Total absolute curvature and tightness of noncompact manifolds
Martin
van Gemmeren
2413-2426
Abstract: In the first part we prove an extension of the Chern-Lashof inequality for noncompact immersed manifolds with finitely many ends. For this we give a lower bound of the total absolute curvature in terms of topological invariants of the manifold. In the second part we discuss tightness properties for such immersions. Finally, we give an upper bound for the substantial codimension.
Tensor products over abelian $W^*$-algebras
Bojan
Magajna
2427-2440
Abstract: Tensor products of C$^*-$algebras over an abelian W$^*-$algebra $Z$ are studied. The minimal C$^*-$norm on $A\odot _ZB$ is shown to be just the quotient of the minimal C$^*-$norm on $A\odot B$ if $A$ or $B$ is exact.
Fine structure of the space of spherical minimal immersions
Hillel
Gauchman;
Gabor
Toth
2441-2463
Abstract: The space of congruence classes of full spherical minimal immersions $f:S^m\to S^n$ of a given source dimension $m$ and algebraic degree $p$ is a compact convex body $\mathcal {M}_m^p$ in a representation space $\mathcal {F}_m^p$ of the special orthogonal group $SO(m+1)$. In Ann. of Math. 93 (1971), 43--62 DoCarmo and Wallach gave a lower bound for $\mathcal {F}_m^p$ and conjectured that the estimate was sharp. Toth resolved this ``exact dimension conjecture'' positively so that all irreducible components of $\mathcal {F}_m^p$ became known. The purpose of the present paper is to characterize each irreducible component $V$ of $\mathcal {F}_m^p$ in terms of the spherical minimal immersions represented by the slice $V\cap \mathcal {M}_m^p$. Using this geometric insight, the recent examples of DeTurck and Ziller are located within $\mathcal {M}_m^p$.
Prime spectra of quantum semisimple groups
K.
A.
Brown;
K.
R.
Goodearl
2465-2502
Abstract: We study the prime ideal spaces of the quantized function algebras $R_{q}[G]$, for $G$ a semisimple Lie group and $q$ an indeterminate. Our method is to examine the structure of algebras satisfying a set of seven hypotheses, and then to demonstrate, using work of Joseph, Hodges and Levasseur, that the algebras $R_{q}[G]$ satisfy this list of assumptions. Rings satisfying the assumptions are shown to satisfy normal separation, and therefore Jategaonkar's strong second layer condition. For such rings much representation-theoretic information is carried by the graph of links of the prime spectrum, and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions. Homogeneity is a key feature -- it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms. Bounds on the ranks of these groups are obtained in the case of $R_{q}[G]$. In the final section the results are specialized to the case $G= SL_{n}(\mathbb {C})$, where detailed calculations can be used to illustrate the general results. As a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine $n$-space satisfy our axiom scheme when the group generated by the parameters is torsionfree.
Hardy spaces and twisted sectors for geometric models
Pietro
Poggi-Corradini
2503-2518
Abstract: We study the one-to-one analytic maps $\sigma$ that send the unit disc into a region $G$ with the property that $\lambda G\subset G$ for some complex number $\lambda$, $0<|\lambda |<1$. These functions arise in iteration theory, giving a model for the self-maps of the unit disk into itself, and in the study of composition operators as their eigenfunctions. We show that for such functions there are geometrical conditions on the image region $G$ that characterize their rate of growth, i.e. we prove that $\sigma \in\bigcap _{p<\infty }H^p$ if and only if $G$ does not contain a twisted sector. Then, we examine the connection with composition operators, and further investigate the no twisted sector condition. Finally, in the Appendix, we give a different proof of a result of J. Shapiro about the essential norm of a composition operator.
New results in the perturbation theory of maximal monotone and $M$-accretive operators in Banach spaces
Athanassios
G.
Kartsatos
1663-1707
Abstract: Let $X$ be a real Banach space and $G$ a bounded, open and convex subset of $X.$ The solvability of the fixed point problem $(*)~Tx+Cx \owns x$ in $D(T)\cap \overline{G}$ is considered, where $T:X\supset D(T)\to 2^{X}$ is a possibly discontinuous $m$-dissipative operator and $C: \overline{G}\to X$ is completely continuous. It is assumed that $X$ is uniformly convex, $D(T)\cap G \not = \emptyset$ and $(T+C)(D(T)\cap \partial G)\subset \overline{G}.$ A result of Browder, concerning single-valued operators $T$ that are either uniformly continuous or continuous with $X^{*}$ uniformly convex, is extended to the present case. Browder's method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let $\Gamma = \{\beta :\mathcal{R}_{+}\to \mathcal{R}_{+}~;~\beta (r)\to 0\text{ as }r\to \infty \}.$ The effect of a weak boundary condition of the type $\langle u+Cx,x\rangle \ge -\beta (\|x\|)\|x\|^{2}$ on the range of operators $T+C$ is studied for $m$-accretive and maximal monotone operators $T.$ Here, $\beta \in \Gamma ,~x\in D(T)$ with sufficiently large norm and $u\in Tx.$ Various new eigenvalue results are given involving the solvability of $Tx+ \lambda Cx\owns 0$ with respect to $(\lambda ,x)\in (0,\infty )\times D(T).$ Several results do not require the continuity of the operator $C.$ Four open problems are also given, the solution of which would improve upon certain results of the paper.
A class of counterexamples to the Gel'fand-Kirillov conjecture
Jacques
Alev;
Alfons
Ooms;
Michel
Van den Bergh
1709-1716
Abstract: Let $G$ be a connected non-special semisimple algebraic group and let $W$ be a finite dimensional $G$-representation such that $W$ has trivial generic stabilizer. Let $\mathfrak{g}=\text{Lie}(G)$. Then the semi-direct product $\mathfrak{g}\oplus W$ is a counter-example to the Gel´fand-Kirillov conjecture.
Simultaneous rational approximation to binomial functions
Michael
A.
Bennett
1717-1738
Abstract: We apply Padé approximation techniques to deduce lower bounds for simultaneous rational approximation to one or more algebraic numbers. In particular, we strengthen work of Osgood, Fel´dman and Rickert, proving, for example, that \begin{displaymath}\max \left\{ \left| \sqrt{2} - p_{1}/q \right| , \left| \sqrt{3} - p_{2}/q \right| \right\} > q^{-1.79155} \end{displaymath} for $q > q_{0}$ (where the latter is an effective constant). Some of the Diophantine consequences of such bounds will be discussed, specifically in the direction of solving simultaneous Pell's equations and norm form equations.
Orientation-reversing involutions on handlebodies
John
Kalliongis;
Darryl
McCullough
1739-1755
Abstract: The observation that the quotient orbifold of an orientation- reversing involution on a 3-dimensional handlebody has the structure of a compression body leads to a strong classification theorem, and general structure theorems. The structure theorems decompose the action along invariant discs into actions on handlebodies which preserve the $I$-fibers of some $I$-bundle structure. As applications, various results of R. Nelson are proved without restrictive hypotheses.
Groups Quasi-isometric to Complex Hyperbolic Space
Richard
Chow
1757-1769
Abstract: We show that any finitely generated group quasi-isometric to complex hyperbolic space is a finite extension of a properly discontinuous, cocompact subgroup of the isometry group.
The Bergman kernel function of some Reinhardt domains
Sheng
Gong;
Xuean
Zheng
1771-1803
Abstract: The boundary behavior of the Bergman Kernel function of some Reinhardt domains is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points $(z,\bar z)$. Let $D$ be the Reinhardt domain \begin{displaymath}D=\left\{ z\in\mathbf C^n\,|\,\|z\|_\alpha=\sum_{j=1}^n|z_j|^{2/\al_j}<1\right\} \end{displaymath} where $\al_j>0$, $j=1,2,\dots, n$; and let $K(z,\bar w)$ be the Bergman kernel function of $D$. Then there exist two positive constants $m$ and $M$ and a function $F$ such that \begin{displaymath}mF(z,\bar z) \le K(z,\bar z)\le MF(z,\bar z) \end{displaymath} holds for every $z\in D$. Here \begin{displaymath}F(z,\bar z)=(-r(z))^{-n-1} \prod_{j=1}^n (-r(z)+|z_j|^{2/\al_j})^{1-\al_j} \end{displaymath} and $r(z)=\|z\|_\alpha-1$ is the defining function for $D$. The constants $m$ and $M$ depend only on $\alpha=(\al_1,\dots, \al_n)$ and $n$, not on $z$.
Prox-regular functions in variational analysis
R.
A.
Poliquin;
R.
T.
Rockafellar
1805-1838
Abstract: The class of prox-regular functions covers all l.s.c., proper, convex functions, lower-$\mathcal{C}^{2}$ functions and strongly amenable functions, hence a large core of functions of interest in variational analysis and optimization. The subgradient mappings associated with prox-regular functions have unusually rich properties, which are brought to light here through the study of the associated Moreau envelope functions and proximal mappings. Connections are made between second-order epi-derivatives of the functions and proto-derivatives of their subdifferentials. Conditions are identified under which the Moreau envelope functions are convex or strongly convex, even if the given functions are not.
Decompositions in Quantum Logic
John
Harding
1839-1862
Abstract: We present a method of constructing an orthomodular poset from a relation algebra. This technique is used to show that the decompositions of any algebraic, topological, or relational structure naturally form an orthomodular poset, thereby explaining the source of orthomodularity in the ortholattice of closed subspaces of a Hilbert space. Several known methods of producing orthomodular posets are shown to be special cases of this result. These include the construction of an orthomodular poset from a modular lattice and the construction of an orthomodular poset from the idempotents of a ring. Particular attention is paid to decompositions of groups and modules. We develop the notion of a norm on a group with operators and of a projection on such a normed group. We show that the projections of a normed group with operators form an orthomodular poset with a full set of states. If the group is abelian and complete under the metric induced by the norm, the projections form a $\sigma$-complete orthomodular poset with a full set of countably additive states. We also describe some properties special to those orthomodular posets constructed from relation algebras. These properties are used to give an example of an orthomodular poset which cannot be embedded into such a relational orthomodular poset, or into an orthomodular lattice. It had previously been an open question whether every orthomodular poset could be embedded into an orthomodular lattice.
Diagrams up to cohomology
W.
G.
Dwyer;
C.
W.
Wilkerson
1863-1883
Abstract: We compute (under suitable assumptions) how many ways there are to take a diagram in the homotopy category of spaces and perturb it to get another diagram which looks the same up to cohomology. Sometimes there are no perturbations. This can shed light on the question of whether the $p$-completion of the classifying space of a particular connected compact Lie group is determined up to homotopy by cohomological data.
Noncomplete linear systems on abelian varieties
Christina
Birkenhake
1885-1908
Abstract: Let $X$ be a smooth projective variety. Every embedding $X\hookrightarrow \mathbb{P}_N$ is the linear projection of an embedding defined by a complete linear system. In this paper the geometry of such not necessarily complete embeddings is investigated in the special case of abelian varieites. To be more precise, the properties $N_p$ of complete embeddings are extended to arbitrary embeddings, and criteria for these properties to be satisfied are elaborated. These results are applied to abelian varieties. The main result is: Let $(X,L)$ be a general polarized abelian variety of type $(d_1,\dots,d_g)$ and $p\ge1$, $n\ge 2p+2$ such that $nd_g\ge 6$ is even, and $c\le n^{g-1}$. The general subvector space $V\subseteq H^0(L^n)$ of codimension $c$ satisfies the property $N_p$.
On homomorphisms from a fixed representation to a general representation of a quiver
William
Crawley-Boevey
1909-1919
Abstract: We study the dimension of the space of homomorphisms from a given representation $X$ of a quiver to a general representation of dimension vector $\beta$. We prove a theorem about this number, and derive two corollaries concerning its asymptotic behaviour as $\beta$ increases. These results are related to work of A. Schofield on homological epimorphisms from the path algebra to a simple artinian ring.
Hyperfinite transversal theory. II
Bosko
Zivaljevic
1921-1938
Abstract: We continue the investigation of validity of Hall's theorem in the case of the Loeb space $L({\mathcal{H}})$ of an internal, uniformly distributed, hyperfinite measure space ${\mathcal{H}}=(\Omega ,{\mathcal{A}},\mu )$ initiated in1992 by the author. Some new classes of graphs are introduced for which the measure theoretic version of Hall's theorem still holds.
On transversely flat conformal foliations with good measures
Taro
Asuke
1939-1958
Abstract: Transversely flat conformal foliations with good transverse invariant measures are Riemannian in the $C^{1+{\operatorname{Lip}}}$ sense. In particular, transversely similar foliations with good measures are transversely Riemannian as transversely $C^{\omega }$-foliations.
Linear additive functionals of superdiffusions and related nonlinear P.D.E.
E.
B.
Dynkin;
S.
E.
Kuznetsov
1959-1987
Abstract: Let $L$ be a second order elliptic differential operator in a bounded smooth domain $D$ in $\mathbb{R}^{d}$ and let $1<\alpha \le 2$. We get necessary and sufficient conditions on measures $\eta , \nu$ under which there exists a positive solution of the boundary value problem \begin{equation*}\begin{gathered} -Lv+v^{\alpha }=\eta \quad \text{ in } D, v=\nu \quad \text{ on } \partial D. \end{gathered}\tag{*} \end{equation*} The conditions are stated both analytically (in terms of capacities related to the Green's and Poisson kernels) and probabilistically (in terms of branching measure-valued processes called $(L,\alpha )$-superdiffusions). We also investigate a closely related subject --- linear additive functionals of superdiffusions. For a superdiffusion in an arbitrary domain $E$ in $\mathbb{R}^{d}$, we establish a 1-1 correspondence between a class of such functionals and a class of $L$-excessive functions $h$ (which we describe in terms of their Martin integral representation). The Laplace transform of $A$ satisfies an integral equation which can be considered as a substitute for (*).
Fuchsian Subgroups of Bianchi Groups
D.
G.
James;
C.
Maclachlan
1989-2002
Abstract: A maximal non-elementary Fuchsian subgroup of a Bianchi group $PSL(2,O_d)$ has an invariant circle or straight line under its linear fractional action on the complex plane, to which is associated a positive integer $D$, the discriminant, which, in turn, is an invariant of the wide commensurability class of the Fuchsian subgroup. In this paper, for all Bianchi groups, we classify the conjugacy classes of these maximal Fuchsian subgroups by determining the number with given discriminant.
Fractal Dimensions and Random Transformations
Yuri
Kifer
2003-2038
Abstract: I start with random base expansions of numbers from the interval $[0,1]$ and, more generally, vectors from $[0,1]^{d}$, which leads to random expanding transformations on the $d$-dimensional torus $\mathbb{T}^{d}$. As in the classical deterministic case of Besicovitch and Eggleston I find the Hausdorff dimension of random sets of numbers with given averages of occurrences of digits in these expansions, as well as of general closed sets ``invariant'' with respect to these random transformations, generalizing the corresponding deterministic result of Furstenberg. In place of the usual entropy which emerges (as explained in Billingsley's book) in the Besicovitch-Eggleston and Furstenberg cases, the relativised entropy of random expanding transformations comes into play in my setup. I also extend to the case of random transformations the Bowen-Ruelle formula for the Hausdorff dimension of repellers.
Higher Lefschetz Traces and Spherical Euler Characteristics
Ross
Geoghegan;
Andrew
Nicas;
John
Oprea
2039-2062
Abstract: Higher analogs of the Euler characteristic and Lefschetz number are introduced. It is shown that they possess a variety of properties generalizing known features of those classical invariants. Applications are then given. In particular, it is shown that the higher Euler characteristics are obstructions to homotopy properties such as the TNCZ condition, and to a manifold being homologically Kähler.
Regularity of Lorentzian Busemann Functions
Gregory
J.
Galloway;
Arnaldo
Horta
2063-2084
Abstract: A general theory of regularity for Lorentzian Busemann functions in future timelike geodesically complete spacetimes is presented. This treatment simplifies and extends the local regularity developed by Eschenburg, Galloway and Newman to prove the Lorentzian splitting theorem. Criteria for global regularity are obtained and used to improve results in the literature pertaining to a conjecture of Bartnik.
$k$-spaces and Borel filters on the set of integers
Jean
Calbrix
2085-2090
Abstract: We say that a countable, Hausdorff, topological space with one and only one accumulation point is a point-space. For such a space, we give several properties which are equivalent to the property of being a k-space. We study some free filters on the set of integers and we determine if the associated point-spaces are k-spaces or not. We show that the filters of Lutzer-van Mill-Pol are k-filters. We deduce that, for each countable ordinal ${\alpha \geq 2}$, there exists a free filter of true additive class ${\alpha }$ (Baire's classification) and a free filter of true multiplicative class ${\alpha }$ for which the associated point-spaces are k-spaces but not ${\aleph _{0}}$, the existence being true in the additive case for ${\alpha =1}$. In particular, we answer negatively a question raised in J. Calbrix, C. R. Acad. Sci. Paris 305 (1987), 109--111.
Nonsmooth sequential analysis in Asplund spaces
Boris
S.
Mordukhovich;
Yongheng
Shao
1235-1280
Abstract: We develop a generalized differentiation theory for nonsmooth functions and sets with nonsmooth boundaries defined in Asplund spaces. This broad subclass of Banach spaces provides a convenient framework for many important applications to optimization, sensitivity, variational inequalities, etc. Our basic normal and subdifferential constructions are related to sequential weak-star limits of Fréchet normals and subdifferentials. Using a variational approach, we establish a rich calculus for these nonconvex limiting objects which turn out to be minimal among other set-valued differential constructions with natural properties. The results obtained provide new developments in infinite dimensional nonsmooth analysis and have useful applications to optimization and the geometry of Banach spaces.
Duality and Polynomial Testing of Tree Homomorphisms
P.
Hell;
J.
Nesetril;
X.
Zhu
1281-1297
Abstract: Let $H$ be a fixed digraph. We consider the $H$-colouring problem, i.e., the problem of deciding which digraphs $G$ admit a homomorphism to $H$. We are interested in a characterization in terms of the absence in $G$ of certain tree-like obstructions. Specifically, we say that $H$ has tree duality if, for all digraphs $G$, $G$ is not homomorphic to $H$ if and only if there is an oriented tree which is homomorphic to $G$ but not to $H$. We prove that if $H$ has tree duality then the $H$-colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the $\underline X$-property studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when $H$ itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads $H$ for which the $H$-colouring problem is $NP$-complete. We contrast these with several families of oriented triads $H$ which have tree duality, or bounded treewidth duality, and hence polynomial $H$-colouring problems. If $P \neq NP$, then no oriented triad $H$ with an $NP$-complete $H$-colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad $H$. We prove that none of the oriented triads $H$ with $NP$-complete $H$-colouring problems given in the companion paper has tree duality.
Shellable Nonpure Complexes and Posets. I
Anders
Björner;
Michelle
L.
Wachs
1299-1327
Abstract: The concept of shellability of complexes is generalized by deleting the requirement of purity (i.e., that all maximal faces have the same dimension). The usefulness of this level of generality was suggested by certain examples coming from the theory of subspace arrangements. We develop several of the basic properties of the concept of nonpure shellability. Doubly indexed $f$-vectors and $h$-vectors are introduced, and the latter are shown to be nonnegative in the shellable case. Shellable complexes have the homotopy type of a wedge of spheres of various dimensions, and their Stanley-Reisner rings admit a combinatorially induced direct sum decomposition. The technique of lexicographic shellability for posets is similarly extended from pure posets (all maximal chains of the same length) to the general case. Several examples of nonpure lexicographically shellable posets are given, such as the $k$-equal partition lattice (the intersection lattice of the $k$-equal subspace arrangement) and the Tamari lattices of binary trees. This leads to simplified computation of Betti numbers for the $k$-equal arrangement. It also determines the homotopy type of intervals in a Tamari lattice and in the lattice of number partitions ordered by dominance, thus strengthening some known Möbius function formulas. The extension to regular CW complexes is briefly discussed and shown to be related to the concept of lexicographic shellability.
On universal relations in gauge theory
Selman
Akbulut
1329-1355
Abstract: In this paper we study the algebraic topology of gauge group, and as a corollary we deduce some universal relations among Donaldson polynomials of smooth 4-manifolds.
The space of $\omega$-limit sets of a continuous map of the interval
Alexander
Blokh;
A.
M.
Bruckner;
P.
D.
Humke;
J.
Smítal
1357-1372
Abstract: We first give a geometric characterization of $\omega$-limit sets. We then use this characterization to prove that the family of $\omega$-limit sets of a continuous interval map is closed with respect to the Hausdorff metric. Finally, we apply this latter result to other dynamical systems.
Twisted Groups and Locally Toroidal Regular Polytopes
Peter
McMullen;
Egon
Schulte
1373-1410
Abstract: In recent years, much work has been done on the classification of abstract regular polytopes by their local and global topological type. Abstract regular polytopes are combinatorial structures which generalize the well-known classical geometric regular polytopes and tessellations. In this context, the classical theory is concerned with those which are of globally or locally spherical type. In a sequence of papers, the authors have studied the corresponding classification of abstract regular polytopes which are globally or locally toroidal. Here, this investigation of locally toroidal regular polytopes is continued, with a particular emphasis on polytopes of ranks $5$ and $6$. For large classes of such polytopes, their groups are explicitly identified using twisting operations on quotients of Coxeter groups. In particular, this leads to new classification results which complement those obtained elsewhere. The method is also applied to describe certain regular polytopes with small facets and vertex-figures.
Some remarks on a probability limit theorem for continued fractions
Jorge
D.
Samur
1411-1428
Abstract: It is shown that if a certain condition on the variances of the partial sums is satisfied then a theorem of Philipp and Stout, which implies the asymptotic fluctuation results known for independent random variables, can be applied to some quantities related to continued fractions. Previous results on the behavior of the approximation by the continued fraction convergents to a random real number are improved.
Analysis of the Wu metric. I: The case of convex Thullen domains
C.
K.
Cheung;
Kang-Tae
Kim
1429-1457
Abstract: We present an explicit description of the Wu metric on the convex Thullen domains which turns out to be the first natural example of a purely Hermitian, non-Kählerian invariant metric. Also, we show that the Wu metric on these Thullen domains is in fact real analytic everywhere except along a lower dimensional subvariety, and is $C^{1}$ smooth overall. Finally, we show that the holomorphic curvature of the Wu metric on these Thullen domains is strictly negative where the Wu metric is real analytic, and is strictly negative everywhere in the sense of current.
Comparative asymptotics for perturbed orthogonal polynomials
Franz
Peherstorfer;
Robert
Steinbauer
1459-1486
Abstract: Let $\{\Phi_n\}_{n\in\mathbb N_0}$ and $\{\widetilde\Phi_n\}_{n\in\mathbb N_0}$ be such systems of orthonormal polynomials on the unit circle that the recurrence coefficients of the perturbed polynomials $\widetilde\Phi_n$ behave asymptotically like those of $\Phi_n$. We give, under weak assumptions on the system $\{\Phi_n\}_{n\in\mathbb N_0}$ and the perturbations, comparative asymptotics as for $\widetilde\Phi_n^*(z)/ \Phi_n^*(z)$ etc., $\Phi_n^*(z):= z^n\bar\Phi_n(\frac 1z)$, on the open unit disk and on the circumference mainly off the support of the measure $\sigma$ with respect to which the $\Phi_n$'s are orthonormal. In particular these results apply if the comparative system $\{\Phi_n\} _{n\in\mathbb N_0}$ has a support which consists of several arcs of the unit circumference, as in the case when the recurrence coefficients are (asymptotically) periodic.
A Periodic Point Free Homeomorphism of a Tree-Like Continuum
Piotr
Minc
1487-1519
Abstract: An example of a homeomorphism without periodic points is constructed on a tree-like continuum.
Whitehead test modules
Jan
Trlifaj
1521-1554
Abstract: A (right $R$-) module $N$ is said to be a Whitehead test module for projectivity (shortly: a p-test module) provided for each module $M$, $Ext_R(M,N)=0$ implies $M$ is projective. Dually, i-test modules are defined. For example, $\mathbb{Z}$ is a p-test abelian group iff each Whitehead group is free. Our first main result says that if $R$ is a right hereditary non-right perfect ring, then the existence of p-test modules is independent of ZFC + GCH. On the other hand, for any ring $R$ , there is a proper class of i-test modules. Dually, there is a proper class of p-test modules over any right perfect ring. A non-semisimple ring $R$ is said to be fully saturated ($\kappa$-saturated) provided that all non-projective ($\le\kappa$-generated non-projective) modules are i-test. We show that classification of saturated rings can be reduced to the indecomposable ones. Indecomposable 1-saturated rings fall into two classes: type I, where all simple modules are isomorphic, and type II, the others. Our second main result gives a complete characterization of rings of type II as certain generalized upper triangular matrix rings, $GT(1,n,p,S,T)$. The four parameters involved here are skew-fields $S$ and $T$, and natural numbers $n,p$. For rings of type I, we have several partial results: e.g. using a generalization of Bongartz Lemma, we show that it is consistent that each fully saturated ring of type I is a full matrix ring over a local quasi-Frobenius ring. In several recent papers, our results have been applied to Tilting Theory and to the Theory of $\ast$-modules.
Further Nice Equations for Nice Groups
Shreeram
S.
Abhyankar
1555-1577
Abstract: Nice sextinomial equations are given for unramified coverings of the affine line in nonzero characteristic $p$ with P$\Omega ^{-}(2m,q)$ and $\Omega ^{-}(2m,q)$ as Galois groups where $m>3$ is any integer and $q>1$ is any power of $p>2$.
A Groenewold-Van Hove Theorem for $S^2$
Mark
J.
Gotay;
Hendrik
Grundling;
C.
A.
Hurst
1579-1597
Abstract: We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold $S^2$ which is irreducible on the su(2) subalgebra generated by the components $\{S_1,S_2,S_3\}$ of the spin vector. In fact there does not exist such a quantization of the Poisson subalgebra $\cal P$ consisting of polynomials in $\{S_1,S_2,S_3\}$. Furthermore, we show that the maximal Poisson subalgebra of $\cal P$ containing $\{1,S_1,S_2,S_3\}$ that can be so quantized is just that generated by $\{1,S_1,S_2,S_3\}$.
Transfer operators acting on Zygmund functions
Viviane
Baladi;
Yunping
Jiang;
Oscar
E.
Lanford III
1599-1615
Abstract: We obtain a formula for the essential spectral radius $\rho _{\text{ess}}$ of transfer-type operators associated with families of $C^{1+\delta }$ diffeomorphisms of the line and Zygmund, or Hölder, weights acting on Banach spaces of Zygmund (respectively Hölder) functions. In the uniformly contracting case the essential spectral radius is strictly smaller than the spectral radius when the weights are positive.
Epigraphical and Uniform Convergence of Convex Functions
Jonathan
M.
Borwein;
Jon
D.
Vanderwerff
1617-1631
Abstract: We examine when a sequence of lsc convex functions on a Banach space converges uniformly on bounded sets (resp. compact sets) provided it converges Attouch-Wets (resp. Painlevé-Kuratowski). We also obtain related results for pointwise convergence and uniform convergence on weakly compact sets. Some known results concerning the convergence of sequences of linear functionals are shown to also hold for lsc convex functions. For example, a sequence of lsc convex functions converges uniformly on bounded sets to a continuous affine function provided that the convergence is uniform on weakly compact sets and the space does not contain an isomorphic copy of $\ell _{1}$.
Totally real submanifolds in $S^6(1)$ satisfying Chen's equality
Franki
Dillen;
Luc
Vrancken
1633-1646
Abstract: In this paper, we study 3-dimensional totally real submanifolds of $S^{6}(1)$. If this submanifold is contained in some 5-dimensional totally geodesic $S^{5}(1)$, then we classify such submanifolds in terms of complex curves in $\mathbb{C}P^{2}(4)$ lifted via the Hopf fibration $S^{5}(1)\to \mathbb{C}P^{2}(4)$. We also show that such submanifolds always satisfy Chen's equality, i.e. $\delta _{M} = 2$, where $\delta _{M}(p)=\tau (p)-\inf K(p)$ for every $p\in M$. Then we consider 3-dimensional totally real submanifolds which are linearly full in $S^{6}(1)$ and which satisfy Chen's equality. We classify such submanifolds as tubes of radius $\pi /2$ in the direction of the second normal space over an almost complex curve in $S^{6}(1)$.
Cohomological dimension and metrizable spaces. II
Jerzy
Dydak
1647-1661
Abstract: The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : $\dim (A\cup B)\le \dim A+\dim B+1$. Theorem. Suppose $A,B$ are subsets of a metrizable space and $K$ and $L$ are CW complexes. If $K$ is an absolute extensor for $A$ and $L$ is an absolute extensor for $B$, then the join $K*L$ is an absolute extensor for $A\cup B$. As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension: Theorem. Suppose $A,B$ are subsets of a metrizable space. Then \begin{equation*}\dim _{{\mathbf R} }(A\cup B)\le \dim _{{\mathbf R} }A+\dim _{{\mathbf R} }B+1 \end{equation*} for any ring ${\mathbf R}$ with unity and \begin{equation*}\dim _{G}(A\cup B)\le \dim _{G}A+\dim _{G}B+2\end{equation*} for any abelian group $G$. The second part of the paper is devoted to the question of existence of universal spaces: Theorem. Suppose $\{K_{i}\}_{i\ge 1}$ is a sequence of CW complexes homotopy dominated by finite CW complexes. Then a. Given a separable, metrizable space $Y$ such that $K_{i}\in AE(Y)$, $i\ge 1$, there exists a metrizable compactification $c(Y)$ of $Y$ such that $K_{i}\in AE(c(Y))$, $i\ge 1$. b. There is a universal space of the class of all compact metrizable spaces $Y$ such that $K_{i}\in AE(Y)$ for all $i\ge 1$. c. There is a completely metrizable and separable space $Z$ such that $K_{i}\in AE(Z)$ for all $i\ge 1$ with the property that any completely metrizable and separable space $Z'$ with $K_{i}\in AE(Z')$ for all $i\ge 1$ embeds in $Z$ as a closed subset.
Structural properties of the one-dimensional drift-diffusion models for semiconductors
Fatiha
Alabau
823-871
Abstract: This paper is devoted to the analysis of the one-dimensional current and voltage drift-diffusion models for arbitrary types of semiconductor devices and under the assumption of vanishing generation recombination. We show in the course of this paper, that these models satisfy structural properties, which are due to the particular form of the coupling of the involved systems. These structural properties allow us to prove an existence and uniqueness result for the solutions of the current driven model together with monotonicity properties with respect to the total current $I$, of the electron and hole current densities and of the electric field at the contacts. We also prove analytic dependence of the solutions on $I$. These results allow us to establish several qualitative properties of the current voltage characteristic. In particular, we give the nature of the (possible) bifurcation points of this curve, we show that the voltage function is an analytic function of the total current and we characterize the asymptotic behavior of the currents for large voltages. As a consequence, we show that the currents never saturate as the voltage goes to $\pm \infty$, contrary to what was predicted by numerical simulations by M. S. Mock (Compel. 1 (1982), pp. 165--174). We also analyze the drift-diffusion models under the assumption of quasi-neutral approximation. We show, in particular, that the reduced current driven model has at most one solution, but that it does not always have a solution. Then, we compare the full and the reduced voltage driven models and we show that, in general, the quasi-neutral approximation is not accurate for large voltages, even if no saturation phenomenon occurs. Finally, we prove a local existence and uniqueness result for the current driven model in the case of small generation recombination terms.
Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions
I.
P.
Goulden;
D.
M.
Jackson
873-892
Abstract: A power series is introduced that is an extension to three sets of variables of the Cauchy sum for Jack symmetric functions in the Jack parameter $\alpha.$ We conjecture that the coefficients of this series with respect to the power sum basis are nonnegative integer polynomials in $b$, the Jack parameter shifted by $1$. More strongly, we make the Matchings-Jack Conjecture, that the coefficients are counting series in $b$ for matchings with respect to a parameter of nonbipartiteness. Evidence is presented for these conjectures and they are proved for two infinite families. The coefficients of a second series, essentially the logarithm of the first, specialize at values $1$ and $2$ of the Jack parameter to the numbers of hypermaps in orientable and locally orientable surfaces, respectively. We conjecture that these coefficients are also nonnegative integer polynomials in $b$, and we make the Hypermap-Jack Conjecture, that the coefficients are counting series in $b$ for hypermaps in locally orientable surfaces with respect to a parameter of nonorientability.
Iterated Spectra of Numbers---Elementary, Dynamical, and Algebraic Approaches
Vitaly
Bergelson;
Neil
Hindman;
Bryna
Kra
893-912
Abstract: $IP^*$ sets and central sets are subsets of $\mathbb N$ which arise out of applications of topological dynamics to number theory and are known to have rich combinatorial structure. Spectra of numbers are often studied sets of the form $\{[n\alpha+\gamma]\colon n\in\mathbb N\}$. Iterated spectra are similarly defined with $n$ coming from another spectrum. Using elementary, dynamical, and algebraic approaches we show that iterated spectra have significantly richer combinatorial structure than was previously known. For example we show that if $\alpha>0$ and $0<\gamma<1$, then $\{[n\alpha+\gamma]\colon n\in\mathbb N\}$ is an $IP^*$ set and consequently contains an infinite sequence together with all finite sums and products of terms from that sequence without repetition.
Distinguished representations and quadratic base change for $GL(3)$
Herve
Jacquet;
Yangbo
Ye
913-939
Abstract: Let $E/F$ be a quadratic extension of number fields. Suppose that every real place of $F$ splits in $E$ and let $H$ be the unitary group in 3 variables. Suppose that $\Pi$ is an automorphic cuspidal representation of $GL(3,E_{\mathbb{A}})$. We prove that there is a form $\phi$ in the space of $\Pi$ such that the integral of $\phi$ over $H(F)\setminus H(F_{\mathbb{A}})$ is non zero. Our proof is based on earlier results and the notion, discussed in this paper, of Shalika germs for certain Kloosterman integrals.
Geometrizing Infinite Dimensional Locally Compact Groups
Conrad
Plaut
941-962
Abstract: We study groups having invariant metrics of curvature bounded below in the sense of Alexandrov. Such groups are a generalization of Lie groups with invariant Riemannian metrics, but form a much larger class. We prove that every locally compact, arcwise connected, first countable group has such a metric. These groups may not be (even infinite dimensional) manifolds. We show a number of relationships between the algebraic and geometric structures of groups equipped with such metrics. Many results do not require local compactness.
R-torsion and zeta functions for analytic Anosov flows on 3-manifolds
Héctor
Sánchez-Morgado
963-973
Abstract: We improve previous results relating R-torsion, for an acyclic representation of the fundamental group, with a special value of the torsion zeta function of an analytic Anosov flow on a 3-manifold. By using the new techniques of Rugh and Fried we get rid of the unpleasent assumptions about the regularity of the invariant foliations.
An Extension of Lomonosov's Techniques to Non-compact Operators
Aleksander
Simonic
975-995
Abstract: The aim of this work is to generalize Lomonosov's techniques in order to apply them to a wider class of not necessarily compact operators. We start by establishing a connection between the existence of invariant subspaces and density of what we define as the associated Lomonosov space in a certain function space. On a Hilbert space, approximation with Lomonosov functions results in an extended version of Burnside's Theorem. An application of this theorem to the algebra generated by an essentially self-adjoint operator $A$ yields the existence of vector states on the space of all polynomials restricted to the essential spectrum of $A$. Finally, the invariant subspace problem for compact perturbations of self-adjoint operators acting on a real Hilbert space is translated into an extreme problem and the solution is obtained upon differentiating certain real-valued functions at their extreme.
Partially hyperbolic fixed points with constraints
Patrick
Bonckaert
997-1011
Abstract: We investigate the local conjugacy, at a partially hyperbolic fixed point, of a diffeomorphism (vector field) to its normally linear part in the presence of constraints, where the change of variables also must satisfy the constraints. The main result is applied to vector fields respecting a singular foliation, encountered, by F. Dumortier and R. Roussarie, in the desingularization of families of vector fields.
Quasiadditivity and measure property of capacity and the tangential boundary behavior of harmonic functions
H.
Aikawa;
A.
A.
Borichev
1013-1030
Abstract: We show that if a set $E$ is dispersely decomposed into subsets, then the capacity of $E$ is comparable to the summation of the capacities of the subsets. From this fact it is derived that the Lebesgue measure of a certain expanded set is estimated by the capacity of $E$. These properties hold for classical capacities, $L^{p}$-capacities and energy capacities of general kernels. The estimation is applied to the boundary behavior of harmonic functions. We introduce a boundary thin set and show a fine limit type boundary behavior of harmonic functions. We show that a thin set does not meet essentially Nagel-Stein and Nagel-Rudin-Shapiro type approaching regions at almost all bounary points.
The behavior of Fourier transforms for nilpotent Lie groups
Ronald
L.
Lipsman;
Jonathan
Rosenberg
1031-1050
Abstract: We study weak analogues of the Paley-Wiener Theorem for both the scalar-valued and the operator-valued Fourier transforms on a nilpotent Lie group $G$. Such theorems should assert that the appropriate Fourier transform of a function or distribution of compact support on $G$ extends to be ``holomorphic'' on an appropriate complexification of (a part of) $\hat G$. We prove the weak scalar-valued Paley-Wiener Theorem for some nilpotent Lie groups but show that it is false in general. We also prove a weak operator-valued Paley-Wiener Theorem for arbitrary nilpotent Lie groups, which in turn establishes the truth of a conjecture of Moss. Finally, we prove a conjecture about Dixmier-Douady invariants of continuous-trace subquotients of $C^{*}(G)$ when $G$ is two-step nilpotent.
Operator Semigroup Compactifications
H.
D.
Junghenn
1051-1073
Abstract: A weakly continuous, equicontinuous representation of a semitopological semigroup $S$ on a locally convex topological vector space $X$ gives rise to a family of operator semigroup compactifications of $S$, one for each invariant subspace of $X$. We consider those invariant subspaces which are maximal with respect to the associated compactification possessing a given property of semigroup compactifications and show that under suitable hypotheses this maximality is preserved under the formation of projective limits, strict inductive limits and tensor products.
Real analysis related to the Monge-Ampère equation
Luis
A.
Caffarelli;
Cristian
E.
Gutiérrez
1075-1092
Abstract: In this paper we consider a family of convex sets in $\mathbf{R}^{n}$, $\mathcal{F}= \{S(x,t)\}$, $x\in \mathbf{R}^{n}$, $t>0$, satisfying certain axioms of affine invariance, and a Borel measure $\mu$ satisfying a doubling condition with respect to the family $\mathcal{F}.$ The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of $\mathcal{F}.$ This is achieved by showing first a Besicovitch-type covering lemma for the family $\mathcal{F}$ and then using the doubling property of the measure $\mu .$ The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to $\mathcal{F}.$
Affine Dupin Surfaces
Ross
Niebergall;
Patrick
J.
Ryan
1093-1115
Abstract: In this paper we study nondegenerate affine surfaces in ${\mathbb R} ^{3}$ whose affine principal curvatures are constant along their lines of curvature. We give a complete local classification of these surfaces assuming that the lines of curvature are planar, and there are no umbilics.
A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials
Minking
Eie
1117-1136
Abstract: Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields. In this paper, we shall improve Shintani's formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [4] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise.
Even Linkage Classes
Scott
Nollet
1137-1162
Abstract: In this paper we generalize the $\mathcal{E}$ and $\mathcal{N}$-type resolutions used by Martin-Deschamps and Perrin for curves in $\mathbb{P}^{3}$ to subschemes of pure codimension in projective space, and shows that these resolutions are interchanged by the mapping cone procedure under a simple linkage. Via these resolutions, Rao's correspondence is extended to give a bijection between even linkage classes of subschemes of pure codimension two and stable equivalence classes of reflexive sheaves $\mathcal{E}$ satisfying $H^{1}_{*}( \mathcal{E})=0$ and $\mathop{\mathcal{E}xt}^{1}( \mathcal{E}^{\vee }, \mathcal{O})=0$. Further, these resolutions are used to extend the work of Martin-Deschamps and Perrin for Cohen-Macaulay curves in $\mathbb{P}^{3}$ to subschemes of pure codimension two in $\mathbb{P}^{n}$. In particular, even linkage classes of such subschemes satisfy the Lazarsfeld-Rao property and any minimal subscheme for an even linkage class links directly to a minimal subscheme for the dual class.
Geometric Isomorphisms between Infinite Dimensional Teichmüller Spaces
Clifford
J.
Earle;
Frederick
P.
Gardiner
1163-1190
Abstract: Let $X$ and $Y$ be the interiors of bordered Riemann surfaces with finitely generated fundamental groups and nonempty borders. We prove that every holomorphic isomorphism of the Teichmüller space of $X$ onto the Teichmüller space of $Y$ is induced by a quasiconformal homeomorphism of $X$ onto $Y$. These Teichmüller spaces are not finite dimensional and their groups of holomorphic automorphisms do not act properly discontinuously, so the proof presents difficulties not present in the classical case. To overcome them we study weak continuity properties of isometries of the tangent spaces to Teichmüller space and special properties of Teichmüller disks.
Topological centers of certain dual algebras
Anthony
To-Ming
Lau;
Ali
Ülger
1191-1212
Abstract: Let $A$ be a Banach algebra with a bounded approximate identity. Let $Z_1$ and $\widetilde Z_2$ be, respectively, the topological centers of the algebras $A^{**}$ and $(A^*A)^*$. In this paper, for weakly sequentially complete Banach algebras, in particular for the group and Fourier algebras $L^1(G)$ and $A(G)$, we study the sets $Z_1$, $\widetilde Z_2$, the relations between them and with several other subspaces of $A^{**}$ or $A^*$.
On the Poles of Rankin-Selberg Convolutions of Modular Forms
Xian-jin
Li
1213-1234
Abstract: The Rankin-Selberg convolution is usually normalized by the multiplication of a zeta factor. One naturally expects that the non-normalized convolution will have poles where the zeta factor has zeros, and that these poles will have the same order as the zeros of the zeta factor. However, this will only happen if the normalized convolution does not vanish at the zeros of the zeta factor. In this paper, we prove that given any point inside the critical strip, which is not equal to $\frac{1}{2}$ and is not a zero of the Riemann zeta function, there exist infinitely many cusp forms whose normalized convolutions do not vanish at that point.
Berezin Quantization and Reproducing Kernels on Complex Domains
Miroslav
Englis
411-479
Abstract: Let $\Omega$ be a non-compact complex manifold of dimension $n$, $\omega =\partial \overline{\partial }\Psi$ a Kähler form on $\Omega$, and $K_\alpha ( x,\overline{y})$ the reproducing kernel for the Bergman space $A^2_\alpha$ of all analytic functions on $\Omega$ square-integrable against the measure $e^{-\alpha \Psi } |\omega ^n|$. Under the condition \begin{equation*}K_\alpha ( x,\overline{x})= \lambda _\alpha e^{\alpha \Psi (x)} \end{equation*} F. A. Berezin [Math. USSR Izvestiya 8 (1974), 1109--1163] was able to establish a quantization procedure on $(\Omega ,\omega )$ which has recently attracted some interest. The only known instances when the above condition is satisfied, however, are just $\Omega = \mathbf{C} ^n$ and $\Omega$ a bounded symmetric domain (with the euclidean and the Bergman metric, respectively). In this paper, we extend the quantization procedure to the case when the above condition is satisfied only asymptotically, in an appropriate sense, as $\alpha \to +\infty$. This makes the procedure applicable to a wide class of complex Kähler manifolds, including all planar domains with the Poincaré metric (if the domain is of hyperbolic type) or the euclidean metric (in the remaining cases) and some pseudoconvex domains in $\mathbf{C}^n$. Along the way, we also fix two gaps in Berezin's original paper, and discuss, for $\Omega$ a domain in $\mathbf{C}^n$, a variant of the quantization which uses weighted Bergman spaces with respect to the Lebesgue measure instead of the Kähler-Liouville measure $|\omega ^n|$.
A Construction of the Level 3 Modules for the Affine Lie Algebra $A_2^{(2)}$ and a New Combinatorial Identity of the Rogers-Ramanujan Type Amer. Math. Soc. 348 (1996), pp. 481-501.
Stefano
Capparelli
481-501
Abstract: We obtain a vertex operator construction of level 3 standard representations for the affine Lie algebra $A_2^{(2)}$. As a corollary, we also get new conbinatorial identities.
A Multivariate Faa di Bruno Formula with Applications
G.
M.
Constantine;
T.
H.
Savits
503-520
Abstract: A multivariate Faa di Bruno formula for computing arbitrary partial derivatives of a function composition is presented. It is shown, by way of a general identity, how such derivatives can also be expressed in the form of an infinite series. Applications to stochastic processes and multivariate cumulants are then delineated.
Isomorphisms of adjoint Chevalley groups over integral domains
Yu
Chen
521-541
Abstract: It is shown that every automorphism of an adjoint Chevalley group over an integral domain containing the rational number field is uniquely expressible as the product of a ring automorphism, a graph automorphism and an inner automorphism while every isomorphism between simple adjoint Chevalley groups can be expressed uniquely as the product of a ring isomorphism, a graph isomorphism and an inner automorphism. The isomorphisms between the elementary subgroups are also found having analogous expressions.
Regularity and Algebras of Analytic Functions in Infinite Dimensions
R.
M.
Aron;
P.
Galindo;
D.
García;
M.
Maestre
543-559
Abstract: A Banach space $E$ is known to be Arens regular if every continuous linear mapping from $E$ to $E^{\prime}$ is weakly compact. Let $U$ be an open subset of $E$, and let $H_b(U)$ denote the algebra of analytic functions on $U$ which are bounded on bounded subsets of $U$ lying at a positive distance from the boundary of $U.$ We endow $H_b(U)$ with the usual Fréchet topology. $M_b(U)$ denotes the set of continuous homomorphisms $\phi:H_b(U) \to {\mathbb{C}}$. We study the relation between the Arens regularity of the space $E$ and the structure of $M_b(U)$.
An Index Theory For Quantum Dynamical Semigroups
B.
V. Rajarama
Bhat
561-583
Abstract: W. Arveson showed a way of associating continuous tensor product systems of Hilbert spaces with endomorphism semigroups of type I factors. We do the same for general quantum dynamical semigroups through a dilation procedure. The product system so obtained is the index and its dimension is a numerical invariant for the original semigroup.
A cascade decomposition theory with applications to Markov and exchangeable cascades
Edward
C.
Waymire;
Stanley
C.
Williams
585-632
Abstract: A multiplicative random cascade refers to a positive $T$-martingale in the sense of Kahane on the ultrametric space $T = { \{ 0,1,\dots ,b-1 \} }^{\mathbf{N}}.$ A new approach to the study of multiplicative cascades is introduced. The methods apply broadly to the problems of: (i) non-degeneracy criterion, (ii) dimension spectra of carrying sets, and (iii) divergence of moments criterion. Specific applications are given to cascades generated by Markov and exchangeable processes, as well as to homogeneous independent cascades.
Harmonic Bergman Functions on Half-Spaces
Wade
C.
Ramey;
HeungSu
Yi
633-660
Abstract: We study harmonic Bergman functions on the upper half-space of $\bold{R}^n$. Among our main results are: The Bergman projection is bounded for the range $1< p < \infty$; certain nonorthogonal projections are bounded for the range $1\leq p < \infty$; the dual space of the Bergman $L^1$-space is the harmonic Bloch space modulo constants; harmonic conjugation is bounded on the Bergman spaces for the range $1\leq p < \infty$; the Bergman norm is equivalent to a ``normal derivative norm'' as well as to a ``tangential derivative norm''.
A Tranversality Theorem for Holomorphic Mappings and Stability of Eisenman-Kobayashi Measures
Sh.
Kaliman;
M.
Zaidenberg
661-672
Abstract: We show that Thom's Transversality Theorem is valid for holomorphic mappings from Stein manifolds. More precisely, given such a mapping $f:S\rightarrow M$ from a Stein manifold $S$ to a complex manifold $M$ and given an analytic subset $A$ of the jet space $J^{k} (S, M), \; f$ can be approximated in neighborhoods of compacts by holomorphic mappings whose $k$-jet extensions are transversal to $A$. As an application the stability of Eisenman-Kobayshi intrinsic $k$-measures with respect to deleting analytic subsets of codimension $>k$ is proven. This is a generalization of the Campbell-Howard-Ochiai-Ogawa theorem on stability of Kobayashi pseudodistances.
An existence result for linear partial differential equations with $C^\infty$ coefficients in an algebra of generalized functions
Todor
Todorov
673-689
Abstract: We prove the existence of solutions for essentially all linear partial differential equations with $C^\infty$-coefficients in an algebra of generalized functions, defined in the paper. In particular, we show that H. Lewy's equation has solutions whenever its right-hand side is a classical $C^\infty$-function.
Characterizations of generalized Hermite and sieved ultraspherical polynomials
Holger
Dette
691-711
Abstract: A new characterization of the generalized Hermite polyno- mials and of the orthogonal polynomials with respect to the measure $|x|^\gamma (1-x^2)^{1/2}dx$ is derived which is based on a ``reversing property" of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of Laguerre and Jacobi polynomials. A similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind. These results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order.
Asymptotic Expansion for Layer Solutions of a Singularly Perturbed Reaction-Diffusion System
Xiao-Biao
Lin
713-753
Abstract: For a singularly perturbed $n$-dimensional system of reaction-- diffusion equations, assuming that the 0th order solutions possess boundary and internal layers and are stable in each regular and singular region, we construct matched asymptotic expansions for formal solutions in all the regular, boundary, internal and initial layers to any desired order in $\epsilon$. The formal solution shows that there is an invariant manifold of wave-front-like solutions that attracts other nearby solutions. We also give conditions for the wave-front-like solutions to converge slowly to stationary solutions on that manifold.
Regularity theory and traces of $\mathcal{A}$-harmonic functions
Pekka
Koskela;
Juan
J.
Manfredi;
Enrique
Villamor
755-766
Abstract: In this paper we discuss two different topics concerning $\mathcal{A}$-harmonic functions. These are weak solutions of the partial differential equation \begin{equation*}\text{div}(\mathcal{A}(x,\nabla u))=0,\end{equation*} where $\alpha (x)|\xi |^{p-1}\le \langle \mathcal{A}(x,\xi ),\xi \rangle \le \beta (x) |\xi |^{p-1}$ for some fixed $p\in (1,\infty )$, the function $\beta$ is bounded and $\alpha (x)>0$ for a.e. $x$. First, we present a new approach to the regularity of $\mathcal{A}$-harmonic functions for $p>n-1$. Secondly, we establish results on the existence of nontangential limits for $\mathcal{A}$-harmonic functions in the Sobolev space $W^{1,q}(\mathbb{B})$, for some $q>1$, where $\mathbb{B}$ is the unit ball in $\mathbb{R}^n$. Here $q$ is allowed to be different from $p$.
On $CR$-mappings between algebraic Cauchy-Riemann manifolds and separate algebraicity for holomorphic functions
Ruslan
Sharipov;
Alexander
Sukhov
767-780
Abstract: We prove the algebraicity of smooth $CR$-mappings between algebraic Cauchy-Riemann manifolds. A generalization of separate algebraicity principle is established.
Radial Solutions to a Dirichlet Problem Involving Critical Exponents when $N=6$
Alfonso
Castro;
Alexandra
Kurepa
781-798
Abstract: In this paper we show that, for each $\lambda > 0$, the set of radially symmetric solutions to the boundary value problem \begin{equation*}\begin{split} -\Delta u(x) & = % \lambda u(x) + u(x)\vert u(x)\vert,\quad x\in B := \{x\in R^6\colon\Vert x\Vert < 1\}, u(x) & = % 0, \quad x\in\partial B, \end{split} \end{equation*} is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.
Factorizations of simple algebraic groups
Martin
W.
Liebeck;
Jan
Saxl;
Gary
M.
Seitz
799-822
Abstract: We determine all factorizations of simple algebraic groups as the product of two maximal closed connected subgroups. Additional results are established which drop the maximality assumption, and applications are given to the study of subgroups of classical groups transitive on subspaces of a given dimension.
Weakly repelling fixpoints and the connectivity of wandering domains
Walter
Bergweiler;
Norbert
Terglane
1-12
Abstract: It is proved that if a transcendental meromorphic function $f$ has a multiply-connected wandering domain, then $f$ has a fixpoint $z_0$ such that $|f'(z_0)|>1$ or $f'(z_0)=1$. Entire functions with a multiply-connected wandering domain have infinitely many such fixpoints. These results are used to show that solutions of certain differential equations do not have wandering domains at all.
Invertibility preserving linear maps on $\mathcal{L}(X)$
A.
R.
Sourour
13-30
Abstract: For Banach spaces $X$ and $Y$, we show that every unital bijective invertibility preserving linear map between $\mathcal L(X)$ and $\mathcal L(Y)$ is a Jordan isomorphism. The same conclusion holds for maps between $\mathbb CI+ \mathcal K(X)$ and $\mathbb CI+\mathcal K(Y)$.
The space of invariant functions on a finite Lie algebra
G.
I.
Lehrer
31-50
Abstract: We show that the operations of Fourier transform and duality on the space of adjoint-invariant functions on a finite Lie algebra commute with each other. This result is applied to give formulae for the Fourier transform of a ``Brauer function''---i.e. one whose value at $X$ depends only on the semisimple part $X_s$ of $X$ and for the dual of the convolution of any function with the Steinberg function. Geometric applications include the evaluation of the characters of the Springer representations of Weyl groups and the study of the equivariant cohomology of local systems on $G/T$, where $T$ is a maximal torus of the underlying reductive group $G$.
Measure-Valued Solutions to Initial-Boundary Value Problems for Certain Systems of Conservation Laws: Existence and Dynamics
Hermano
Frid
51-76
Abstract: A framework for studying initial-boundary value problems for systems of conservation laws, in what concerns to the existence of measure-valued solutions and their asymptotic behavior, is developed here with the helpful introduction of a class of flux maps which allow a rather complete treatment of these questions including systems of practical importance as those arising in multiphase flow in porous media. The systems of this class may, in general, admit umbilic points, submanifolds where genuine nonlinearity fails, as well as elliptic regions. We prove the existence of measure-valued solutions by using the vanishing viscosity method and, also, finite difference schemes. The main result about the dynamics of the measure-valued solutions is that for certain special boundary values, given by constant states, the time-averages of these m-v solutions converge weakly to the Dirac measure concentrated at those states, for a.e. space variable. The rate of convergence of the time-averages of the expected values can be estimated by properties of the flux maps only.
Polynomial structures for nilpotent groups
Karel
Dekimpe;
Paul
Igodt;
Kyung
Bai
Lee
77-97
Abstract: If a polycyclic-by-finite rank-$K$ group $\Gamma$ admits a faithful affine representation making it acting on $\Bbb R^{K}$ properly discontinuously and with compact quotient, we say that $\Gamma$ admits an affine structure. In 1977, John Milnor questioned the existence of affine structures for such groups $\Gamma$. Very recently examples have been obtained showing that, even for torsion-free, finitely generated nilpotent groups $N$, affine structures do not always exist. It looks natural to consider affine structures as examples of polynomial structures of degree one. We introduce the concept of a canonical type polynomial structure for polycyclic-by-finite groups. Using the algebraic framework of the Seifert Fiber Space construction and a nice cohomology vanishing theorem, we prove the existence and uniqueness (up to conjugation) of canonical type polynomial structures for virtually finitely generated nilpotent groups. Applying this uniqueness to a result going back to Mal$'$cev, it follows that, for torsion-free, finitely generated nilpotent groups, each canonical polynomial structure is expressed in polynomials of limited degree. The minimal degree needed for obtaining a polynomial structure will determine the ``affine defect number''. We prove that the known counterexamples to Milnor's question have the smallest possible affine defect, i.e. affine defect number equal to one.
Nonnegative Radix Representations for the Orthant $R^n_+$
Jeffrey
C.
Lagarias;
Yang
Wang
99-117
Abstract: Let $A$ be a nonnegative real matrix which is expanding, i.e. with all eigenvalues $|\lambda| > 1$, and suppose that $|\det(A)|$ is an integer. Let ${\mathcal D}$ consist of exactly $|\det(A)|$ nonnegative vectors in $\R^n$. We classify all pairs $(A, {\mathcal D})$ such that every $x$ in the orthant $\R^n_+$ has at least one radix expansion in base $A$ using digits in ${\mathcal D}$. The matrix $A$ must be a diagonal matrix times a permutation matrix. In addition $A$ must be similar to an integer matrix, but need not be an integer matrix. In all cases the digit set $\mathcal D$ can be diagonally scaled to lie in $\Z^n$. The proofs generalize a method of Odlyzko, previously used to classify the one--dimensional case.
Gelfand-Kirillov dimension in Jordan Algebras
C.
Martinez
119-126
Abstract: In this paper we study Gelfand-Kirillov dimension in Jordan algebras. In particular we will relate Gelfand-Kirillov (GK for short) dimensions of a special Jordan algebra and its associative enveloping algebra and also the GK dimension of a Jordan algebra and the GK dimension of its universal multiplicative enveloping algebra.
Complex convexity in Lebesgue-Bochner Function Spaces
Patrick
N.
Dowling;
Zhibao
Hu;
Douglas
Mupasiri
127-139
Abstract: Complex geometric properties of continuously quasi-normed spaces are introduced and their relationship to their analogues in real Banach spaces is discussed. It is shown that these properties lift from a continuously quasi-normed space $X$ to $L^p(\mu , X)$, for $0 < p < \infty$. Local versions of these properties and results are also considered.
Subgroups of $GL(n^2, \bold C)$ containing $PSU(n)$
V.
P.
Platonov;
D.
Z.
Ðokovic
141-152
Abstract: Let ${\operatorname{PSU}}(n)$ be the image of the unitary group ${\operatorname{U}}(n)$ under the representation $x\to axa^{-1}$ on the space $M_n({\mathbf C} )$ of $n$ by $n$ complex matrices. We classify all connected Lie subgroups of ${\operatorname{GL}}(n^2,{\mathbf C} )$ containing ${\operatorname{PSU}}(n)$. We use this result to obtain a description of all abstract overgroups of ${\operatorname{PSU}}(n)$ in ${\operatorname{GL}}(n^2,{\mathbf C} )$. We apply this classification to solve the problem of describing all invertible linear transformations of $M_n({\mathbf C} )$ which preserve the set of normal matrices. Our results can be applied to solve many other problems of similar nature.
Some New Homogeneous Einstein Metrics on Symmetric Spaces
Megan
M.
Kerr
153-171
Abstract: We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, we consider symmetric spaces with rank$(M)> 1$, not isometric to a compact Lie group. Whenever there exists a closed proper subgroup $G$ of Isom$(M)$ acting transitively on $M$ we find all $G$-homogeneous (non-symmetric) Einstein metrics on $M$.
On the variances of occupation times of conditioned Brownian motion
Biao
Zhang
173-185
Abstract: We extend some bounds on the variance of the lifetime of two--dimensional Brownian motion, conditioned to exit a planar domain at a given point, to certain domains in higher dimensions. We also give a short ``analytic'' proof of some existing results.
Failure of normality in the box product of uncountably many real lines
L.
Brian
Lawrence
187-203
Abstract: We prove in ZFC that the box product of $\omega_1$ many copies of $\omega+1$ is neither normal nor collectionwise Hausdorff. As an addendum to the proof, we show that if the cardinality of the continuum is $2^{\omega_1}$, then these properties also fail in the closed subspace consisting of all functions which assume the value $\omega$ on all but countably many indices.
A Concordance Extension Theorem
Joel
L.
Jones
205-218
Abstract: Let $p\:E\to B$ be a manifold approximate fibration between closed manifolds, where $\dim(E)\ge 4$, and let $M(p)$ be the mapping cylinder of $p$. In this paper it is shown that if $g\colon B\times I\to B\times I$ is any concordance on $B$, then there exists a concordance $G\:M(p)\times I \to M(p)\times I$ such that $G|B\times I=g$ and $G|E\times\{0\}\times I= id_{E\times I}$. As an application, if $N^n$ and $M^{n+j}$ are closed manifolds where $N$ is a locally flat submanifold of $M$ and $n\ge 5$ and $j\ge 1$, then a concordance $g\colon N\times I\to N\times I$ extends to a concordance $G\:M\times I\to M\times I$ on $M$ such that $G|N\times I=g$. This uses the fact that under these hypotheses there exists a manifold approximate fibration $p\colon E\to N$, where $E$ is a closed $(n+j-1)$-manifold, such that the mapping cylinder $M(p)$ is homeomorphic to a closed neighborhood of $N$ in $M$ by a homeomorphism which is the identity on $N$.
A problem of Foldes and Puri on the Wiener process
Z.
Shi
219-228
Abstract: Let $W$ be a real-valued Wiener process starting from 0, and $\tau (t)$ be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of $X(t)=\int _0^{\tau (t)} {\text{\bf 1}\hskip -1.25pt\mathrm{l}}_{\{ | W(u)| \le \alpha t\} }du$ as $t$ tends to infinity, i.e. they ask: how long does $W$ stay in a tube before ``crossing very much" a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for $X$.
Second Root Vectors for Multiparameter Eigenvalue Problems of Fredholm Type
Paul
Binding;
Tomaz
Kosir
229-249
Abstract: A class of multiparameter eigenvalue problems involving (generally) non self-adjoint and unbounded operators is studied. A basis for the second root subspace, at eigenvalues of Fredholm type, is computed in terms of the underlying multiparameter system. A self-adjoint version of this result is given under a weak definiteness condition, and Sturm-Liouville and finite-dimensional examples are considered.
Uniform harmonic approximation of bounded functions
Stephen
J.
Gardiner
251-265
Abstract: Let $\Omega$ be an open set in $\mathbb R^n$ and $E$ be a relatively closed subset of $\Omega$. We characterize those pairs $(\Omega,E)$ which have the following property: every function which is bounded and continuous on $E$ and harmonic on $E^0$ can be uniformly approximated by functions harmonic on $\Omega$. Several related results concerning both harmonic and superharmonic approximation are also established.
Global smooth solutions for a class of parabolic integrodifferential equations
Hans
Engler
267-290
Abstract: The existence and uniqueness of smooth global large data solutions of a class of quasilinear partial integrodifferential equations in one space and one time dimension are proved, if the integral kernel behaves like $t^{-\alpha }$ near $t=0$ with $\alpha > 2/3$. An existence and regularity theorem for linear equations with variable coefficients that are related to this type is also proved in arbitrary space dimensions and with no restrictions for $\alpha$.
Powers in Finitely Generated Groups
E.
Hrushovski;
P.
H.
Kropholler;
A.
Lubotzky;
A.
Shalev
291-304
Abstract: In this paper we study the set $\G^n$ of $n^{th}$-powers in certain finitely generated groups $\G$. We show that, if $\G$ is soluble or linear, and $\G^n$ contains a finite index subgroup, then $\G$ is nilpotent-by-finite. We also show that, if $\G$ is linear and $\G^n$ has finite index (i.e. $\G$ may be covered by finitely many translations of $\G^n$), then $\G$ is soluble-by-finite. The proof applies invariant measures on amenable groups, number-theoretic results concerning the $S$-unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.
On the Well-Posedness of the Kirchhoff String
Alberto
Arosio;
Stefano
Panizzi
305-330
Abstract: Let us consider the Cauchy problem for the quasilinear hyperbolic integro-differential equation \begin{displaymath}% {% \begin{array}{ll} u_{tt}-m \left(\g{ \int_{_{\p{\Omega}}}} |\bigtriangledown_{x}u|^{2} \, dx \right) \bigtriangleup_{x}u= f(x,t) \: & \,(x\in \, \Omega, \, t \G 0),\qquad\qquad\qquad u(\cdot ,t)_{|\partial\Omega} =0 &\,(t\, \geq \,0), \end{array} } \end{displaymath} where $\; \Omega \;$ is an open subset of $\; \Reali^{n} \;$ and $\, m \,$ is a positive function of one real variable which is continuously differentiable. We prove the well-posedness in the Hadamard sense (existence, uniqueness and continuous dependence of the local solution upon the initial data) in Sobolev spaces of low order.
Defect zero blocks for finite simple groups
Andrew
Granville;
Ken
Ono
331-347
Abstract: We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a $p$-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero $p-$blocks remained unclassified were the alternating groups $A_{n}$. Here we show that these all have a $p$-block with defect 0 for every prime $p\geq 5$. This follows from proving the same result for every symmetric group $S_{n}$, which in turn follows as a consequence of the $t$-core partition conjecture, that every non-negative integer possesses at least one $t$-core partition, for any $t\geq 4$. For $t\geq 17$, we reduce this problem to Lagrange's Theorem that every non-negative integer can be written as the sum of four squares. The only case with $t<17$, that was not covered in previous work, was the case $t=13$. This we prove with a very different argument, by interpreting the generating function for $t$-core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne's Theorem (née the Weil Conjectures). We also consider congruences for the number of $p$-blocks of $S_{n}$, proving a conjecture of Garvan, that establishes certain multiplicative congruences when $5\leq p \leq 23$. By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime $p$ and positive integer $m$, the number of $p-$blocks with defect 0 in $S_n$ is a multiple of $m$ for almost all $n$. We also establish that any given prime $p$ divides the number of $p-$modularly irreducible representations of $S_{n}$, for almost all $n$.
Uniqueness theorems in inverse spectral theory for one-dimensional Schrödinger operators
F.
Gesztesy;
B.
Simon
349-373
Abstract: New unique characterization results for the potential $V(x)$ in connection with Schrödinger operators on $\mathbb{R}$ and on the half-line $[0,\infty )$ are proven in terms of appropriate Krein spectral shift functions. Particular results obtained include a generalization of a well-known uniqueness theorem of Borg and Marchenko for Schrödinger operators on the half-line with purely discrete spectra to arbitrary spectral types and a new uniqueness result for Schrödinger operators with confining potentials on the entire real line.
Negative Flows of the potential KP-hierarchy
Guido
Haak
375-390
Abstract: We construct a Grassmannian-like formulation for the potential KP-hierarchy including additional ``negative'' flows. Our approach will generalize the notion of a $\tau$-function to include negative flows. We compare the resulting hierarchy with results by Hirota, Satsuma and Bogoyavlenskii.
The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds
Manuel
Ritoré;
Antonio
Ros
391-410
Abstract: It is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in (non fixed) flat three manifolds are compact in a strong sense: given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the $C^k$ topology. These limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds. Some applications to the isoperimetric problem are given.
A geometric approach to regular perturbation theory with an application to hydrodynamics
Carmen
Chicone
4559-4598
Abstract: The Lyapunov-Schmidt reduction technique is used to prove a persistence theorem for fixed points of a parameterized family of maps. This theorem is specialized to give a method for detecting the existence of persistent periodic solutions of perturbed systems of differential equations. In turn, this specialization is applied to prove the existence of many hyperbolic periodic solutions of a steady state solution of Euler's hydrodynamic partial differential equations. Incidentally, using recent results of S. Friedlander and M. M. Vishik, the existence of hyperbolic periodic orbits implies the steady state solutions of the Eulerian partial differential equation are hydrodynamically unstable. In addition, a class of the steady state solutions of Euler's equations are shown to exhibit chaos.
A genealogy for finite kneading sequences of bimodal maps on the interval
John
Ringland;
Charles
Tresser
4599-4624
Abstract: We generate all the finite kneading sequences of one of the two kinds of bimodal map on the interval, building each sequence uniquely from a pair of shorter ones. There is a single pair at generation 0, with members of length $ 1$. Concomitant with this genealogy of kneading sequences is a unified genealogy of all the periodic orbits. (See Figure 0.) Figure 0. Loci of some finite kneading sequences for a two-parameter cubic family
A categorical approach to matrix Toda brackets
K. A.
Hardie;
K. H.
Kamps;
H. J.
Marcum
4625-4649
Abstract: In this paper we give a categorical treatment of matrix Toda brackets, both in the pre- and post-compositional versions. Explicitly the setting in which we work is, à la Gabriel-Zisman, a $2$-category with zeros. The development parallels that in the topological setting but with homotopy groups replaced by nullity groups of invertible $ 2$-morphisms. A central notion is that of conjugation of $2$-morphisms. Our treatment of matrix Toda brackets is carried forward to the point of establishing appropriate indeterminacies.
Topological entropy for finite invariant subsets of $Y$
Shi Hai
Li;
Xiang Dong
Ye
4651-4661
Abstract: Let $Y$ be the space $\{ z \in {\mathbf{C}}:{z^3} \in [0,1]\}$ with a metric defined by the arc length. Suppose that $f$ is a continuous map from $Y$ to itself and $P$ is a finite $f$-invariant subset. In this paper we construct a continuous map ${C_P}$ from $Y$ to itself satisfying ${C_P}{\vert _P} = f{\vert _P}$ which achieves the infimum topological entropies of continuous maps from $ Y$ to itself which agree with $f$ on $P$.
Periodic orbits of $n$-body type problems: the fixed period case
Hasna
Riahi
4663-4685
Abstract: This paper gives a proof of the existence and multiplicity of periodic solutions to Hamiltonian systems of the form $\displaystyle ({\text{A}})\quad {\text{ }}\left\{ {\begin{array}{*{20}{c}} {{m_... ...q) = 0} {q(t + T) = q(t),\quad \forall t \in \Re .} \end{array} } \right.$ where $ {q_i} \in {\Re ^\ell },\ell \geqslant 3,1 \leqslant i \leqslant n,q = ({q_1}, \ldots ,{q_n})$ and with $ {V_{ij}}(t,\xi )$ $ T$-periodic in $ t$ and singular in $ \xi$ at $\xi = 0$ Under additional hypotheses on $V$, when (A) is posed as a variational problem, the corresponding functional, $I$, is shown to have an unbounded sequence of critical values if the singularity of $V$ at 0 is strong enough. The critical points of $I$ are classical $T$-periodic solutions of (A). Then, assuming that $ I$ has only non-degenerate critical points, up to translations, Morse type inequalities are proved and used to show that the number of critical points with a fixed Morse index $ k$ grows exponentially with $k$, at least when $k \equiv 0,1( \mod \ell - 2)$. The proof is based on the study of the critical points at infinity done by the author in a previous paper and generalizes the topological arguments used by A. Bahri and P. Rabinowitz in their study of the $3$-body problem. It uses a recent result of E. Fadell and S. Husseini on the homology of free loop spaces on configuration spaces. The detailed proof is given for the $4$-body problem then generalized to the $ n$-body problem.
$\scr A$-generators for the Dickson algebra
\textviet{Nguyễn H. V.}
Hu’ng;
Franklin P.
Peterson
4687-4728
Abstract: Let ${D_k}$ denote the Dickson algebra in $ k$ variables over the field of two elements. We study the problem of determining a minimal set of generators for ${D_k}$ as a module over the Steenrod algebra $\mathcal{A}$. This is easy for $k = 1$ and $2$. In this paper we answer this question for $k = 3$ and $4$ and give techniques which may help solve the problem for general $k$.
Courbures scalaires des vari\'et\'es d'invariant conforme n\'egatif
Antoine
Rauzy
4729-4745
Abstract: In this paper, we are interested in the problem of prescribing the scalar curvature on a compact riemannian manifold of negative conformal invariant. We give a necessary and sufficient condition when the prescribed function $ f$ is nonpositive. When $\sup(f) > 0$, we merely find a sufficient condition. This is the subject of the first theorem. In the second one, we prove the multiplicity of the solutions of subcritical (for the Sobolev imbeddings) elliptic equations. In another article [8], we will prove the multiplicity of the solutions of the prescribing curvature problem, i.e. for a critical elliptic equation.
Shadow forms of Brasselet-Goresky-MacPherson
Belkacem
Bendiffalah
4747-4763
Abstract: Brasselet, Goresky and MacPherson constructed an explicit morphism, providing a De Rham isomorphism between the intersection homology of a singular variety $X$ and the cohomology of some complex of differential forms, called "shadow forms" and generalizing Whitney forms, on the smooth part of $X$. The coefficients of shadow forms are integrals of Dirichlet type. We find an explicit formula for them; from that follows an alternative proof of Brasselet, Goresky and MacPherson's theorem. Next, we give a duality formula and a product formula for shadow forms and construct the correct algebra structure, for which shadow forms yield a morphism.
Equivalence relations induced by actions of Polish groups
Sławomir
Solecki
4765-4777
Abstract: We give an algebraic characterization of those sequences $({H_n})$ of countable abelian groups for which the equivalence relations induced by Borel (or, equivalently, continuous) actions of ${H_0} \times {H_1} \times {H_2} \times \cdots$ are Borel. In particular, the equivalence relations induced by Borel actions of $ {H^\omega }$, $ H$ countable abelian, are Borel iff $H \simeq { \oplus _p}({F_p} \times \mathbb{Z}{({p^\infty })^{{n_p}}})$, where $ {F_p}$ is a finite $ p$-group, $\mathbb{Z}({p^\infty })$ is the quasicyclic $ p$-group, ${n_p} \in \omega$, and $p$ varies over the set of all primes. This answers a question of R. L. Sami by showing that there are Borel actions of Polish abelian groups inducing non-Borel equivalence relations. The theorem also shows that there exist non-locally compact abelian Polish groups all of whose Borel actions induce only Borel equivalence relations. In the process of proving the theorem we generalize a result of Makkai on the existence of group trees of arbitrary height.
Periods for transversal maps via Lefschetz numbers for periodic points
A.
Guillamon;
X.
Jarque;
J.
Llibre;
J.
Ortega;
J.
Torregrosa
4779-4806
Abstract: Let $f:M \to M$ be a ${C^1}$ map on a ${C^1}$ differentiable manifold. The map $ f$ is called transversal if for all $m \in \mathbb{N}$ the graph of ${f^m}$ intersects transversally the diagonal of $M \times M$ at each point $(x,x)$ such that $x$ is a fixed point of ${f^m}$. We study the set of periods of $f$ by using the Lefschetz numbers for periodic points. We focus our study on transversal maps defined on compact manifolds such that their rational homology is $ {H_0} \approx \mathbb{Q}$, ${H_1} \approx \mathbb{Q} \oplus \mathbb{Q}$ and ${H_k} \approx \{ 0\}$ for $ k \ne 0,1$.
Radially symmetric internal layers in a semilinear elliptic system
Manuel A.
del Pino
4807-4837
Abstract: Let $B$ denote the unit ball in ${R^N},\quad N \geqslant 1$. We consider the problem of finding nonconstant solutions to a class of elliptic systems including the Gierer and Meinhardt model of biological pattern formation, $\displaystyle (1.1)\qquad {\varepsilon ^2}\Delta u - u + \frac{{{u^2}}} {{1 + k{u^2}}} + p = 0\quad {\text{in}}B,$ $\displaystyle (1.2)\quad D\Delta v - v + {u^2} = 0\quad {\text{in}}B,$ $\displaystyle (1.3)\quad \frac{{\partial u}} {{\partial n}} = 0 = \frac{{\partial v}} {{\partial n}}\quad {\text{on}}\partial B,$ where $ \varepsilon$, $ D$, $k$ and $\rho$ denote positive constants and $n$ the unit outer normal to $\partial B$. Assuming that the parameters $ \rho$, $k$ are small and $D$ large, we construct a family of radially symmetric solutions to (1.1)-(1.3) indexed by the parameter $ \varepsilon$, which exhibits an internal layer in $B$, as $ \varepsilon \to 0$.
Multiple solutions for a semilinear elliptic equation
Manuel A.
del Pino;
Patricio L.
Felmer
4839-4853
Abstract: Let $\Omega$ be a bounded, smooth domain in ${\mathbb{R}^N}$, $N \geqslant 1$. We consider the problem of finding nontrivial solutions to the elliptic boundary value problem \begin{displaymath}\begin{array}{*{20}{c}} {\Delta u + \lambda u = h(x)\vert u{\... ...a } {u = 0\quad {\text{on}}\partial \Omega } \end{array} \end{displaymath} where $h \geqslant 0$, $h\not\equiv0$ is Hölder continuous on $\overline \Omega $ and $p > 1$, $\lambda$ are constants. Let ${\Omega _0}$ denote the interior of the set where $ h$ vanishes in $ \Omega$. We assume $ h > 0$ a.e. on $\Omega \backslash {\Omega _0}$ and consider the eigenvalues ${\lambda _i}(\Omega )$ and ${\lambda _i}({\Omega _0})$ of the Dirichlet problem in $\Omega$ and $ {\Omega _0}$ respectively. We prove that no nontrivial solution of the equation exists if $\lambda$ satisfies, for some $k \geqslant 1$, $\displaystyle {\lambda _k}({\Omega _0}) \leqslant \lambda \leqslant {\lambda _{k + 1}}(\Omega )$ On the other hand, if, for some nonnegative integers $ s$, $k$ with $ s \geqslant k + 1$, $ \lambda$ satisfies $\displaystyle {\lambda _s}(\Omega ) < \lambda < {\lambda _{k + 1}}({\Omega _0})$ then the equation above possesses at least $ s - k$ pairs of nontrivial solutions. For the proof of these results we use a variational approach. In particular, the existence result takes advantage of the even character of the associated functional.
On the ideal class group of real biquadratic fields
Patrick J.
Sime
4855-4876
Abstract: We discuss the structure of the ideal class group of real biquadratic fields $K$, concentrating on the case that the $4$-rank of the ideal class groups of the quadratic subfields of $K$ is 0. In this case, we give estimates for the $ 4$-rank of the ideal class group of $K$. As an example, let $K = \mathbb{Q}(\sqrt p ,\sqrt {627} )$, where $ p$ is a prime satisfying certain congruence conditions. The $ 2$-primary part of the ideal class group of $K$ is then isomorphic to $ {(\mathbb{Z}/4\mathbb{Z})^2},\mathbb{Z}/4\mathbb{Z} \times {(\mathbb{Z}/2\mathbb{Z})^2}$, or $ {(\mathbb{Z}/2\mathbb{Z})^4}$. Further, each of the above occurs infinitely often.
Linear Chevalley estimates
Ti
Wang
4877-4898
Abstract: A Chevalley estimate for a germ of an analytic mapping $ f$ is a function $ l:\mathbb{N} \to \mathbb{N}$ such that if the composite with $ f$ of a germ of an analytic function on the target vanishes to order at least $ l(k)$, then it vanishes on the image to order at least $k$. Work of Izumi revealed the equivalence between regularity of a mapping (in the sense of Gabrielov, see $\S1$) and the existence of a linear Chevalley estimate $l(k)$. Bierstone and Milman showed that uniformity of the Chevalley estimate is fundamental to several analytic and geometric problems on the images of mappings. The central topic of this article is uniformity of linear Chevalley estimates for regular mappings. We first establish the equivalence between uniformity of a linear Chevalley estimate and uniformity of a "linear product estimate" on the image: A linear product estimate on a local analytic ring (or, equivalently, on a germ of an analytic space) means a bound on the order of vanishing of a product of elements which is linear with respect to the sum of the orders of its factors. We study the linear product estimate in the central case of a hypersurface (i.e., the zero set of an analytic function). Our results show that a linear product estimate is equivalent to an explicit estimate concerning resultants. In the special case of hypersurfaces of multiplicity $2$, this allows us to prove uniformity of linear product estimates.
On the set of periods for $\sigma$ maps
M. Carme
Leseduarte;
Jaume
Llibre
4899-4942
Abstract: Let $\sigma$ be the topological graph shaped like the letter $\sigma$. We denote by 0 the unique branching point of $\sigma$, and by $ {\mathbf{O}}$ and ${\mathbf{I}}$ the closures of the components of $ \sigma \backslash \{ 0\}$ homeomorphics to the circle and the interval, respectively. A continuous map from $\sigma$ into itself satisfying that $ f$ has a fixed point in ${\mathbf{O}}$, or $f$ has a fixed point and $f(0) \in {\mathbf{I}}$ is called a $\sigma$ map. These are the continuous self-maps of $\sigma$ whose sets of periods can be studied without the notion of rotation interval. We characterize the sets of periods of all $\sigma$ maps.
Discontinuous robust mappings are approximatable
Shu Zhong
Shi;
Quan
Zheng;
Deming
Zhuang
4943-4957
Abstract: The concepts of robustness of sets and and functions were introduced to form the foundation of the theory of integral global optimization. A set $A$ of a topological space $X$ is said to be robust iff ${\text{cl}}A = {\text{cl}}$ int $ A$. A mapping $ f:X \to Y$ is said to be robust iff for each open set ${U_Y}$ of $Y$, $ {f^{ - 1}}({U_Y})$ is robust. We prove that if $X$ is a Baire space and $Y$ satisfies the second axiom of countability, then a mapping $f:X \to Y$ is robust iff it is approximatable in the sense that the set of points of continuity of $ f$ is dense in $ X$ and that for any other point $x \in X$, $(x,f(x))$ is the limit of $\{ ({x_\alpha },f({x_\alpha }))\}$, where for all $\alpha$, $ {x_\alpha }$ is a continuous point of $f$. This result justifies the notion of robustness.
Binary forms, hypergeometric functions and the Schwarz-Christoffel mapping formula
Michael A.
Bean
4959-4983
Abstract: In a previous paper, it was shown that if $F$ is a binary form with complex coefficients having degree $n \geqslant 3$ and discriminant ${D_F} \ne 0$, and if ${A_F}$ is the area of the region $ \left\vert {F(x,y)} \right\vert \leqslant 1$ in the real affine plane, then $ {\left\vert {{D_F}} \right\vert^{1/n(n - 1)}}{A_F} \leqslant 3B(\frac{1} {3},\frac{1} {3})$, where $B(\frac{1} {3},\frac{1} {3})$ denotes the Beta function with arguments of 1/3. This inequality was derived by demonstrating that the sequence $\{ {M_n}\}$ defined by ${M_n} = \max \vert{D_F}{\vert^{1/n(n - 1)}}{A_F}$, where the maximum is taken over all forms of degree $n$ with ${D_F} \ne 0$, is decreasing, and then by showing that ${M_3} = 3B(\frac{1} {3},\frac{1} {3})$. The resulting estimate, ${A_F} \leqslant 3B(\frac{1} {3},\frac{1} {3})$ for such forms with integer coefficients, has had significant consequences for the enumeration of solutions of Thue inequalities. This paper examines the related problem of determining precise values for the sequence $ \{ {M_n}\}$. By appealing to the theory of hypergeometric functions, it is shown that ${M_4} = {2^{7/6}}B(\frac{1} {4},\frac{1} {2})$ and that ${M_4}$ is attained for the form $XY({X^2} - {Y^2})$. It is also shown that there is a correspondence, arising from the Schwarz-Christoifel mapping formula, between a particular collection of binary forms and the set of equiangular polygons, with the property that ${A_F}$ is the perimeter of the polygon corresponding to $F$. Based on this correspondence and a representation theorem for $\vert{D_F}{\vert^{1/n(n - 1)}}{A_F}$, it is conjectured that ${M_n} = D_{F_n^ * }^{1/n(n - 1)}{A_{F_n^*}}$, where $F_n^*(X,Y) = \prod_{k = 1}^n \left(X \sin\left(\frac{k\pi}{n}\right) - Y \cos\left(\frac{k\pi}{n}\right)\right)$, and that the limiting value of the sequence $\{ {M_n}\}$ is $2\pi$.
The variations of Hodge structure of maximal dimension with associated Hodge numbers $h\sp {2,0}>2$ and $h\sp {1,1}=2q+1$ do not arise from geometry
Azniv
Kasparian
4985-5007
Abstract: The specified variations are proved to be covered by a bounded contractible domain $\Omega$. After classifying the analytic boundary components of $\Omega$ with respect to a fixed realization, the group of the biholomorphic automorphisms ${\text{Aut}}\Omega$ and the ${\text{Aut}}\Omega $-orbit structure of $ \Omega$ are found explicitly. Then $\Omega$ is shown to admit no quasiprojective arithmetic quotients, whereas the lack of geometrically arising variations, covered by $\Omega$.
$e$-invariants and finite covers. II
Larry
Smith
5009-5021
Abstract: Let $\widetilde{M} \downarrow M$ be a finite covering of closed framed manifolds. By the Pontrijagin-Thom construction both $ \widetilde{M}$ and $ M$ define elements in the stable homotopy ring of spheres $\pi _*^s$. Associated to $\widetilde{M}$ and $M$ are their $e$invariants $ {e_L}(\widetilde{M})$, $ {e_L}(M) \in \mathbb{Q}/\mathbb{Z}$. If $\widetilde{N} \downarrow N$ is a finite covering of closed oriented manifolds, then there is a related invariant ${I_\Delta }(\widetilde{N} \downarrow N) \in \mathbb{Q}$ of the diffeomorphism class of the covering. In a previous paper we examined the relation between these invariants. We reduced the determination of $ {e_L}(\widetilde{M}) - p{e_L}(M)$, as well as ${I_\Delta }(\widetilde{N} \downarrow N)$, for a $ p$-fold cover, to the evaluation of certain sums of roots of unity. In this sequel we show how the invariant theory of the cyclic group $\mathbb{Z}/p$ may be used to evaluate these rums. For example we obtain $\displaystyle \sum\limits_{\mathop {{\zeta ^p} = 1}\limits_{\zeta \ne 1} } {\fr... ... - 1}})}} {{(1 - \zeta )(1 - {\zeta ^{ - 1}})}} = \frac{{(p - 1)(p - 2)}} {3}}$ which may be used to determine the value of $ {I_\Delta }$ in degrees congruent to $3$ $\mod 2(p - 1)$ for odd primes $ p$.
Ramanujan's theories of elliptic functions to alternative bases
Bruce C.
Berndt;
S.
Bhargava;
Frank G.
Garvan
4163-4244
Abstract: In his famous paper on modular equations and approximations to $ \pi$, Ramanujan offers several series representations for $1/\pi$, which he claims are derived from "corresponding theories" in which the classical base $q$ is replaced by one of three other bases. The formulas for $1/\pi$ were only recently proved by J. M. and P. B. Borwein in 1987, but these "corresponding theories" have never been heretofore developed. However, on six pages of his notebooks, Ramanujan gives approximately 50 results without proofs in these theories. The purpose of this paper is to prove all of these claims, and several further results are established as well.
On the classification of $(n-k+1)$-connected embeddings of $n$-manifolds into $(n+k)$-manifolds in the metastable range
Rong
Liu
4245-4258
Abstract: For an $(n - k + 1)$-connected map $f$ from a connected smooth $ n$-manifold $M$ to a connected smooth $ (n + k)$-manifold $ V$, where $M$ is closed, we work out the isotopy group ${[M \subset V]_f}$ in the metastable range $n \leqslant 2k - 4$. To prove our results, we develop the Hurewicz-type theorems which provide us with the efficient methods of computing the homology groups with local coefficients from the homotopy groups.
The order bidual of almost $f$-algebras and $d$-algebras
S. J.
Bernau;
C. B.
Huijsmans
4259-4275
Abstract: It is shown in this paper that the second order dual $A''$ of an Archimedean (almost) $ f$-algebra $A$, equipped with the Arens multiplication, is again an (almost) $f$-algebra. Also, the order continuous bidual $ (A')_n'$ of an Archimedean $ d$-algebra $A$ is a $d$-algebra. Moreover, if the $d$-algebra $A$ is commutative or has positive squares, then $ A''$ is again a $ d$-algebra.
Formes diff\'erentielles non commutatives et cohomologie \`a coefficients arbitraires
Max
Karoubi
4277-4299
Abstract: The purpose of the paper is to promote a new definition of cohomology, using the theory of non commutative differential forms, introduced already by Alain Connes and the author in order to study the relation between $ K$-theory and cyclic homology. The advantages of this theory in classical Algebraic Topology are the following: A much simpler multiplicative structure, where the symmetric group plays an important role. This is important for cohomology operations and the investigation of a model for integral homotopy types (Formes différentielles non commutatives et opérations de Steenrod, Topology, to appear). These considerations are of course related to the theory of operads. A better relation between de Rham cohomology (defined through usual differential forms on a manifold) and integral cohomology, thanks to a "non commutative integration". A new definition of Deligne cohomology which can be generalized to manifolds provided with a suitable filtration of their de Rham complex. In this paper, the theory is presented in the framework of simplicial sets. With minor modifications, the same results can be obtained in the topological category, thanks essentially to the Dold-Thom theorem (Formes topologiques non commutatives, Ann. Sci. Ecole Norm. Sup., to appear). A summary of this paper has been presented to the French Academy: CR Acad. Sci. Paris 316 (1993), 833-836.
Characterizations of Bergman spaces and Bloch space in the unit ball of ${\bf C}\sp n$
Cai Heng
Ouyang;
Wei Sheng
Yang;
Ru Han
Zhao
4301-4313
Abstract: In this paper we prove that, in the unit ball $B$ of $ {{\mathbf{C}}^n}$, a holomorphic function $f$ is in the Bergman space $L_a^p(B),\;0 < p < \infty$, if and only if $\displaystyle \int_B {\vert\tilde \nabla } f(z){\vert^2}\vert f(z){\vert^{p - 2}}{(1 - \vert z{\vert^2})^{n + 1}}d\lambda (z) < \infty ,$ where $\tilde \nabla$ and $\lambda$ denote the invariant gradient and invariant measure on $B$, respectively. Further, we give some characterizations of Bloch functions in the unit ball $ B$, including an exponential decay characterization of Bloch functions. We also give the analogous results for $\operatorname{BMOA} (\partial B)$ functions in the unit ball.
The alternative torus and the structure of elliptic quasi-simple Lie algebras of type $A\sb 2$
Stephen
Berman;
Yun
Gao;
Yaroslav
Krylyuk;
Erhard
Neher
4315-4363
Abstract: We present the complete classification of the tame irreducible elliptic quasi-simple Lie algebras of type ${A_2}$, and in particular, specialize on the case where the coordinates are not associative. Here the coordinates are Cayley-Dickson algebras over Laurent polynomial rings in $\nu \geqslant 3$ variables, which we call alternative tori. In giving our classification we need to present much information on these alternative tori and the Lie algebras coordinatized by them.
The complex zeros of random polynomials
Larry A.
Shepp;
Robert J.
Vanderbei
4365-4384
Abstract: Mark Kac gave an explicit formula for the expectation of the number, ${\nu _n}(\Omega )$, of zeros of a random polynomial, $\displaystyle {P_n}(z) = \sum\limits_{j = 0}^{n - 1} {{\eta _j}{z^j}} ,$ in any measurable subset $\Omega$ of the reals. Here, $ {\eta _0}, \ldots ,{\eta _{n - 1}}$ are independent standard normal random variables. In fact, for each $n > 1$, he obtained an explicit intensity function $ {g_n}$ for which $\displaystyle {\mathbf{E}}{\nu _n}(\Omega ) = \int_\Omega {{g_n}(x)\,dx.}$ Here, we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset $\Omega$ of the complex plane $\mathbb{C}$. Namely, we show that $\displaystyle {\mathbf{E}}{\nu _n}(\Omega ) = \int_\Omega {{h_n}(x,y)\,dxdy + \int_{\Omega \cap \mathbb{R}} {{g_n}(x)\,dx,} }$ where ${h_n}$ is an explicit intensity function. We also study the asymptotics of ${h_n}$ showing that for large $n$ its mass lies close to, and is uniformly distributed around, the unit circle.
On invariants for $\omega\sb 1$-separable groups
Paul C.
Eklof;
Matthew
Foreman;
Saharon
Shelah
4385-4402
Abstract: We study the classification of $ {\omega _1}$-separable groups by using Ehrenfeucht-Fraïssé games and prove a strong classification result assuming PFA, and a strong non-structure theorem assuming $\diamondsuit $.
Wiman-Valiron theory in two variables
P. C.
Fenton
4403-4412
Abstract: Inequalities are obtained for the coefficients of the Taylor series of an entire function of two complex variables and used to obtain an inequality for the maximum modulus of the function in terms of the maximum term of the series.
On minimal sets of scalar parabolic equations with skew-product structures
Wen Xian
Shen;
Yingfei
Yi
4413-4431
Abstract: Skew-product semi-flow ${\Pi _t}:X \times Y \to X \times Y$ which is generated by $\displaystyle \left\{ \begin{gathered}{u_t} = {u_{xx}} + f(y \cdot \,t,x,u,{u_x... ...D\;{\text{or }}N\;{\text{boundary conditions}} \end{gathered} \right.$ is considered, where $X$ is an appropriate subspace of $ {H^2}(0,1),\;(Y,\,\mathbb{R})$ is a minimal flow with compact phase space. It is shown that a minimal set $E \subset X \times Y$ of ${\Pi _t}$ is an almost $1{\text{ - }}1$ extension of $Y$, that is, set $ {Y_0} = \{ y \in Y\vert\operatorname{card} (E \subset {P^{ - 1}}(y)) = 1\}$ is a residual subset of $Y$, where $ P:X \times Y \to Y$ is the natural projection. Consequently, if $(Y,\mathbb{R})$ is almost periodic minimal, then any minimal set $ E \subset X \times Y$ of $ {\Pi _t}$ is an almost automorphic minimal set. It is also proved that dynamics of $ {\Pi _t}$ is closed in the category of almost automorphy, that is, a minimal set $ E \subset X \times Y$ of $ {\Pi _t}$ is almost automorphic minimal if and only if $(Y,\mathbb{R})$ is almost automorphic minimal. Asymptotically almost periodic parabolic equations and certain coupled parabolic systems are discussed. Examples of nonalmost periodic almost automorphic minimal sets are provided.
Uniformisations partielles et crit\`eres \`a la Hurewicz dans le plan
Dominique
Lecomte
4433-4460
Abstract: Résumé: On donne des caractérisations des boréliens potentiellement d'une classe de Wadge donnée, parmi les boréliens à coupes verticales dénombrables d'un produit de deux espaces polonais. Pour ce faire, on utilise des résultats d'uniformisation partielle.
On some subalgebras of $B(c\sb 0)$ and $B(l\sb 1)$
F. P.
Cass;
J. X.
Gao
4461-4470
Abstract: For a non-reflexive Banach space $X$ and $w \in {X^{{\ast}{\ast}}}$, two families of subalgebras of $ B(X),\;{\Gamma _w} = \{ T \in B(X)\vert{T^{{\ast}{\ast}}}w = kw\;{\text{for some}}\;k \in \mathbb{C}{\text{\} }}$, and $ {\Omega _w} = \{ T \in B(X)\vert{T^{{\ast}{\ast}}}w \in w \oplus \hat X\}$ for $w \in {X^{{\ast}{\ast}}}\backslash \hat X$ with $ {\Omega _w} = B(X)$ for $w \in \hat X$, were defined originally by Wilansky. We consider $X = {c_0}$ and $X = {l_1}$ and investigate relationships between the subalgebras for different $w \in {X^{{\ast}{\ast}}}$. We prove in the case of ${c_0}$ that, for $w \in {X^{{\ast}{\ast}}}\backslash \hat X$, all $ {\Gamma _w}$'s are isomorphic and all $ {\Omega _w}$ 's are isomorphic. For $X = {l_1}$, where it is known that not all ${\Gamma _w}$'s are isomorphic and not all ${\Omega _w}$ 's are isomorphic, we show, surprisingly, that subalgebras associated with a Dirac measure on $\beta \mathbb{N}\backslash \mathbb{N}$, regarded as a functional on $ l_1^{\ast}$, are isomorphic to those associated with some Banach limit (i.e., a translation invariant extended limit). We also obtain a representation for the operators in the subalgebras $\{ \cap {\Gamma _f}\vert f\;{\text{is a Banach limit}}\}$ and $\{ \cap {\Omega _f}\vert f\;{\text{is a Banach limit}}\}$ of $B({l_1})$.
The Kechris-Woodin rank is finer than the Zalcwasser rank
Haseo
Ki
4471-4484
Abstract: For each differentiable function $f$ on the unit circle, the Kechris-Woodin rank measures the failure of continuity of the derivative function $f'$ while the Zalcwasser rank measures how close the Fourier series of $f$ is to being a uniformly convergent series. We show that the Kechris-Woodin rank is finer than the Zalcwasser rank. Roughly speaking, small ranks mean the function is well behaved and big ranks imply bad behavior. For each countable ordinal, we explicitly construct a continuous function with everywhere convergent Fourier series such that the Zalcwasser rank of the function is bigger than the ordinal.
Minimization problems for noncoercive functionals subject to constraints
Khoi Le
Vy;
Klaus
Schmitt
4485-4513
Abstract: We consider noncoercive functionals on a reflexive Banach space and establish minimization theorems for such functionals on smooth constraint manifolds. These results in turn yield critical point theorems for certain classes of homogeneous functionals. Several applications to the study of boundary value problems for quasilinear elliptic equations are included.
${\bf R}$-trees, small cancellation, and convergence
Andrew
Chermak
4515-4531
Abstract: The "metric small cancellation hypotheses" of combinatorial group theory imply, when satisfied, that a given presentation has a solvable Word Problem via Dehn's Algorithm. The present work both unifies and generalizes the small cancellation hypotheses, and analyzes them by means of group actions on trees, leading to the strengthening of some classical results.
Triangles of groups
Andrew
Chermak
4533-4558
Abstract: Given a certain commutative diagram of groups and monomorphisms, does there necessarily exist a group in which the given diagram is essentially a diagram of subgroups and inclusions? In general, the answer is negative, but J. Corson, and Gersten and Stallings have shown that in the case of a "non-spherical triangle" of groups the answer is positive. This paper improves on these results by weakening the non-sphericality requirement.
Sharp inequalities, the functional determinant, and the complementary series
Thomas P.
Branson
3671-3742
Abstract: Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functional determinants of natural differential operators. The setting is that of compact Riemannian manifolds. We concentrate especially on the conformally flat case, and obtain formulas in dimensions $2$, $4$, and $6$ for the functional determinants of operators which are well behaved under conformal change of metric. The two-dimensional formulas are due to Polyakov, and the four-dimensional formulas to Branson and Ørsted; the method is sufficiently streamlined here that we are able to present the sixdimensional case for the first time. In particular, we solve the extremal problems for the functional determinants of the conformal Laplacian and of the square of the Dirac operator on ${S^2}$, and in the standard conformal classes on ${S^4}$ and ${S^6}$. The ${S^2}$ results are due to Onofri, and the $ {S^4}$ results to Branson, Chang, and Yang; the ${S^6}$ results are presented for the first time here. Recent results of Graham, Jenne, Mason, and Sparling on conformally covariant differential operators, and of Beckner on sharp Sobolev and Moser-Trudinger type inequalities, are used in an essential way, as are a computation of the spectra of intertwining operators for the complementary series of $ {\text{S}}{{\text{O}}_0}(m + 1,1)$, and the precise dependence of all computations on the dimension. In the process of solving the extremal problem on ${S^6}$, we are forced to derive a new and delicate conformally covariant sharp inequality, essentially a covariant form of the Sobolev embedding $ L_1^2({S^6})\hookrightarrow {L^3}({S^6})$ for section spaces of trace free symmetric two-tensors.
Orthogonal calculus
Michael
Weiss
3743-3796
Abstract: Orthogonal calculus is a calculus of functors, similar to Goodwillie's calculus. The functors in question take finite dimensional real vector spaces (with an inner product) to pointed spaces. Prime example: $F(V) = BO(V)$, where $O(V)$ is the orthogonal group of $V$. In this example, and in general, first derivatives in the orthogonal calculus reproduce and generalize much of the theory of Stiefel-Whitney classes. Similarly, second derivatives in the orthogonal calculus reproduce and generalize much of the theory of Pontryagin classes.
Coexistence states and global attractivity for some convective diffusive competing species models
Julián
López-Gómez;
José C.
Sabina de Lis
3797-3833
Abstract: In this paper we analyze the dynamics of a general competing species model with diffusion and convection. Regarding the interaction coefficients between the species as continuation parameters, we obtain an almost complete description of the structure and stability of the continuum of coexistence states. We show that any asymptotically stable coexistence state lies in a global curve of stable coexistence states and that Hopf bifurcations or secondary bifurcations only may occur from unstable coexistence states. We also characterize whether a semitrivial coexistence state or a coexistence state is a global attractor. The techniques developed in this work can be applied to obtain generic properties of general monotone dynamical systems.
Banach spaces with the $2$-summing property
A.
Arias;
T.
Figiel;
W. B.
Johnson;
G.
Schechtman
3835-3857
Abstract: A Banach space $ X$ has the $2$-summing property if the norm of every linear operator from $X$ to a Hilbert space is equal to the $2$-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent of the scalar field: the property is self-dual and any space with the property is a finite dimensional space of maximal distance to the Hilbert space of the same dimension. In the case of real scalars only the real line and real $\ell _\infty ^2$ have the $2$-summing property. In the complex case there are more examples; e.g., all subspaces of complex $\ell _\infty ^3$ and their duals.
Automorphism group schemes of basic matrix invariants
William C.
Waterhouse
3859-3872
Abstract: For $3 \leqslant k < n,\quad {\text{let}}\quad {E_k}(X)$ be the polynomial in ${n^2}$ variables defined by ${\text{det}}(X + \lambda I) = \sum {{E_k}(X){\lambda ^{n - k}}}$. Let $R$ be a ring containing a field of characteristic $p \geqslant 0$. If $p$ does not divide $n - k + 1$, the invertible linear transformations on matrices preserving ${E_k}(X)$ up to scalars are (in essence) just the obvious ones arising from scaling, similarities, and transposition. If the power ${p^s}$ dividing $n - k + 1$ is greater than $k$, then we have these elements times maps of the form $ X \mapsto X + f(X)I$. When smaller powers ${p^s}$ divide $n - k + 1$, the group scheme is like the first with an infinitesimal part of the second. One corollary is that every division algebra of finite dimension $ {n^2} > 4$ over its center carries a canonical cubic form that determines it up to antiisomorphism.
On spectral geometry of minimal surfaces in $\bold C{\rm P}\sp n$
Yi Bing
Shen
3873-3889
Abstract: By employing the standard isometric imbedding of $C{P^n}$ into the Euclidean space, a classification theorem for full, minimal, $2$-type surfaces in $C{P^n}$ that are not $\pm$ holomorphic is given. All such compact minimal surfaces are either totally real minimal surfaces in $C{P^2}$ or totally real superminimal surfaces in $ C{P^3}$ and $C{P^4}$. In the latter case, they are locally unique. Moreover, some eigenvalue inequalities for compact minimal surfaces of $C{P^n}$ with constant Kaehler angle are shown.
Kazhdan-Lusztig conjecture for generalized Kac-Moody algebras. II. Proof of the conjecture
Satoshi
Naito
3891-3919
Abstract: Generalized Kac-Moody algebras were introduced by Borcherds in the study of Conway and Norton's moonshine conjectures for the Monster sporadic simple group. In this paper, we prove the Kazhdan-Lusztig conjecture for generalized Kac-Moody algebras under a certain mild condition, by using a generalization (to the case of generalized Kac-Moody algebras) of Jantzen's character sum formula. Our (main) formula generalizes the celebrated result for the case of Kac-Moody algebras, and describes the characters of irreducible highest weight modules over generalized Kac-Moody algebras in terms of the "extended" Kazhdan-Lusztig polynomials.
Focusing at a point and absorption of nonlinear oscillations
J.-L.
Joly;
G.
Métivier;
J.
Rauch
3921-3969
Abstract: Several recent papers give rigorous justifications of weakly nonlinear geometric optics. All of them consider oscillating wave trains on domains where focusing phenomena do not exist, either because the space dimension is equal to one, or thanks to a coherence assumption on the phases. This paper is devoted to a study of some nonlinear effects of focusing. In a previous paper, the authors have given a variety of examples which show how focusing in nonlinear equations can spoil even local existence in the sense that the domain of existence shrinks to zero as the wavelength decreases to zero. On the other hand, there are many problems for which global existence is known and in those cases it is natural to ask what happens to oscillations as they pass through a focus. The main goal of this paper is to present such a study for some strongly dissipative semilinear wave equations and spherical wavefronts which focus at the origin. We show that the strongly nonlinear phenomenon which is produced is that oscillations are killed by the simultaneous action of focusing and dissipation. Our study relies on the analysis of Young measures and two-scale Young measures associated to sequences of solutions. The main step is to prove that these measures satisfy appropriate transport equations. Then, their variances are shown to satisfy differential inequalities which imply a propagation result for their support.
Cyclic Sullivan-de Rham forms
Christopher
Allday
3971-3982
Abstract: For a simplicial set $X$ the Sullivan-de Rham forms are defined to be the simplicial morphisms from $X$ to a simplicial rational commutative graded differential algebra (cgda)$\nabla$. However $\nabla$ is a cyclic cgda in a standard way. And so, when $X$ is a cyclic set, one has a cgda of cyclic morphisms from $X$ to $\nabla$. It is shown here that the homology of this cgda is naturally isomorphic to the rational cohomology of the orbit space of the geometric realization $\left\vert X \right\vert$ with its standard circle action. In addition, a cyclic cgda $ \nabla C$ is introduced; and it is shown that the homology of the cgda of cyclic morphisms from $X$ to $\nabla C$ is naturally isomorphic to the rational equivariant (Borel construction) cohomology of $\left\vert X \right\vert$.
$L\sb 2(q)$ and the rank two Lie groups: their construction in light of Kostant's conjecture
Mark R.
Sepanski
3983-4021
Abstract: This paper deals with certain aspects of a conjecture made by B. Kostant in 1983 relating the Coxeter number to the occurrence of the simple finite groups $L(2,q)$ in simple complex Lie groups. A unified approach to Kostant's conjecture that yields very general results for the rank two case is presented.
$L\sp p$ spectra of pseudodifferential operators generating integrated semigroups
Matthias
Hieber
4023-4035
Abstract: Consider the $ {L^p}$-realization $ {\text{O}}{{\text{p}}_p}(a)$ of a pseudodifferential operator with symbol $a \in S_{\rho ,0}^m$ having constant coefficients. We show that for a certain class of symbols the spectrum of $ {\text{O}}{{\text{p}}_p}(a)$ is independent of $p$. This implies that $ {\text{O}}{{\text{p}}_p}(a)$ generates an $N$-times integrated semigroup on ${L^p}({\mathbb{R}^n})$ for a certain $N$ if and only if $ \rho ({\text{O}}{{\text{p}}_p}(a)) \ne \emptyset$ and the numerical range of $ a$ is contained in a left half-plane. Our method allows us also to construct examples of operators generating integrated semigroups on $ {L^p}({\mathbb{R}^n})$ if and only if $p$ is sufficiently close to $2$.
Classifying spaces and Dirac operators coupled to instantons
Marc
Sanders
4037-4072
Abstract: Let $M(k,SU(l))$ denote the moduli space of based gauge equivalence classes of $SU(l)$ instantons on principal bundles over $ {S^4}$ with second Chern class equal to $k$. In this paper we use Dirac operators coupled to such connections to study the topology of these moduli spaces as $l$ increases relative to $k$. This "coupling" procedure produces maps ${\partial _u}:M(k,SU(l)) \to BU(k)$, and we prove that in the limit over $l$ such maps recover Kirwan's $[$K$]$ homotopy equivalence $M(k,SU) \simeq BU(k)$. We also compute, for any $ k$ and $l$, the image of the homology map ${({\partial _u})_ * }:{H_ * }(M(k,SU(l));Z) \to {H_ * }(BU(k);Z)$. Finally, we prove all the analogous results for $Sp(l)$ instantons.
Branched circle packings and discrete Blaschke products
Tomasz
Dubejko
4073-4103
Abstract: In this paper we introduce the notion of discrete Blaschke products via circle packing. We first establish necessary and sufficient conditions for the existence of finite branched circle packings. Next, discrete Blaschke products are defined as circle packing maps from univalent circle packings that properly fill $D = \{ z:\left\vert z \right\vert < 1\}$ to the corresponding branched circle packings that properly cover $D$. It is verified that such maps have all geometric properties of their classical counterparts. Finally, we show that any classical finite Blaschke product can be approximated uniformly on compacta of $ D$ by discrete ones.
On permutations of lacunary intervals
Kathryn E.
Hare;
Ivo
Klemes
4105-4127
Abstract: Let $\{ {I_j}\}$ be an interval partition of the integers and consider the Littlewood-Paley type square function $S(f) = {(\sum {\left\vert {{f_j}} \right\vert^2})^{1/2}}$ where $ {\hat f_j} = \hat f\chi {I_j}$. We prove that if the lengths $\ell ({I_j})$ of the intervals ${I_j}$ satisfy $\ell ({I_{j + 1}})/\ell ({I_j}) \to \infty$, then $ {\left\Vert {S(f)} \right\Vert _p} \approx {\left\Vert f \right\Vert _p}$ for $1 < p < \infty$. As these intervals need not be adjacent, such partitions can be thought of as permutations of lacunary intervals. This work generalizes the specific partition considered in a previous paper [H2]. We conjecture that it suffices to assume $ \ell ({I_{j + 1}})/\ell ({I_j}) \geqslant \lambda > 1$, and we also conjecture a necessary and sufficient condition.
Transition time analysis in singularly perturbed boundary value problems
Freddy
Dumortier;
Bert
Smits
4129-4145
Abstract: The paper deals with the boundary value problem $\varepsilon \ddot x + x\dot x - {x^2} = 0$, with $ x(0) = A,x(T) = B$ for $ A,B,T > 0$ and $\varepsilon > 0$ close to zero. It is shown that for $T$ sufficiently big, the problem has exactly three solutions, two of which reach negative values. Solutions reaching negative values occur for $T \geqslant T(\varepsilon ) > 0$ and we show that asymptotically for $\varepsilon \to 0,\quad T(\varepsilon ) \sim - {\text{ln}}(\varepsilon )$, $ {\text{i}}{\text{.e}}{\text{.}}\quad {\text{li}}{{\text{m}}_{\varepsilon \to 0}} - \frac{{T(\varepsilon )}} {{{\text{ln(}}\varepsilon {\text{)}}}} = 1$. The main tools are transit time analysis in the Liénard plane and normal form techniques. As such the methods are rather qualitative and useful in other similar problems.
Mean value property and subdifferential criteria for lower semicontinuous functions
Didier
Aussel;
Jean-Noël
Corvellec;
Marc
Lassonde
4147-4161
Abstract: We define an abstract notion of subdifferential operator and an associated notion of smoothness of a norm covering all the standard situations. In particular, a norm is smooth for the Gâteaux (Fréchet, Hadamard, Lipschitz-smooth) subdifferential if it is Gâteaux (Fréchet, Hadamard, Lipschitz) smooth in the classical sense, while on the other hand any norm is smooth for the Clarke-Rockafellar subdifferential. We then show that lower semicontinuous functions on a Banach space satisfy an Approximate Mean Value Inequality with respect to any subdifferential for which the norm is smooth, thus providing a new insight on the connection between the smoothness of norms and the subdifferentiability properties of functions. The proof relies on an adaptation of the "smooth" variational principle of Borwein-Preiss. Along the same vein, we derive subdifferential criteria for coercivity, Lipschitz behavior, conemonotonicity, quasiconvexity, and convexity of lower semicontinuous functions which clarify, unify and extend many existing results for specific subdifferentials.
Curves in Grassmannians
David
Perkinson
3179-3246
Abstract: Curves in Grassmannians are analyzed using the special structure of the tangent bundle of a Grassmannian, resulting in a theory of inflections or Weierstrass behavior. A duality theorem is established, generalizing the classical duality theorem for projective plane curves. The appendices summarize basic information about principal parts bundles and their application to studying the inflections of curves in projective space.
Hybrid spaces with interesting cohomology
Kathryn
Lesh
3247-3262
Abstract: Let $p$ be an odd prime, and let $ R$ be a polynomial algebra over the Steenrod algebra with generators in dimensions prime to $p$. To such an algebra is associated a $p$-adic pseudoreflection group $ W$, and we assume that $ W$ is of order prime to $ p$ and irreducible. Adjoin to $R$ a one-dimensional element $z$, and give $R[z]$ an action of the Steenrod algebra by $\beta z = 0$ and $\beta x = (\left\vert x \right\vert/2)zx$ for an even dimensional element $x$. We show that the subalgebra of elements of $ R[z]$ consisting of elements of degree greater than one is realized uniquely, up to homotopy, as the cohomology of a $ p$-complete space. This space can be thought of as a cross between spaces studied by Aguade, Broto, and Notbohm, and the Clark-Ewing examples, further studied by Dwyer, Miller, and Wilkerson.
Continuous functions on extremally disconnected spaces
J.
Vermeer
3263-3285
Abstract: Using results and techniques due to Abramovich, Arenson and Kitover it is shown that each fixed-point set of a selfmap of a compact extremally disconnected space is a retract of that space, and that the retraction can be constructed from the particular selfmap itself. Also, the closure of the set of periodic points turns out to be a retract of the space. Several decomposition theorems for arbitrary selfmaps on extremally disconnected spaces are obtained similar to the theorem of Frolík on embeddings. Conditions are obtained under which the set of fixed points is clopen.
On Euler characteristics associated to exceptional divisors
Willem
Veys
3287-3300
Abstract: Let $k$ be an algebraically closed field and $f \in k[{x_1}, \ldots ,{x_{n + 1}}]$. Fix an embedded resolution $h:X \to {\mathbb{A}^{n + 1}}\quad {\text{of}}\quad {f^{ - 1}}\{ 0\}$ and denote by ${E_i}$, $i \in S$, the irreducible components of ${h^{ - 1}}({f^{ - 1}}\{ 0\} )$ with multiplicity ${N_i}$ in the divisor of $ f{\text{o}}h$. Put also $ {\mathop E\limits^{\text{o}} _i}: = {E_i}\backslash { \cup _{j \ne i}}{E_j}$, and denote by $\chi ({E_i})$ its Euler characteristic. Several conjectures concerning Igusa's local zeta function and the topological zeta function of $ f$ motivate the study of Euler characteristics associated to subsets ${ \cup _{i \in T}}{E_i}$ of ${ \cup _{i \in S}}{E_i}$, which are maximal connected with respect to the property that $ d\vert{N_i}$ for all $ i \in T$. Here $d \in \mathbb{N},d > 1$. We prove that if $ h$ maps ${ \cup _{i \in T}}{E_i}$ to a point, then $\displaystyle {( - 1)^n}\sum\limits_{i \in T} {\chi ({{\mathop E\limits^{\text{o}} }_i}) \geqslant 0}$ This generalizes a well-known result for curves. We also prove some vanishing results concerning the $\chi ({\mathop E\limits^{\text{o}} _i})$ for such a maximal connected subset ${ \cup _{i \in T}}{E_i}$ and give an application on the above-mentioned zeta functions, yielding some confirmation of the holomorphy conjecture for those zeta functions.
Stability of optimal-order approximation by bivariate splines over arbitrary triangulations
C. K.
Chui;
D.
Hong;
R. Q.
Jia
3301-3318
Abstract: Let $\Delta$ be a triangulation of some polygonal domain in $ {\mathbb{R}^2}$ and $S_k^r(\Delta )$, the space of all bivariate $ {C^r}$ piecewise polynomials of total degree $ \leqslant k$ on $ \Delta$. In this paper, we construct a local basis of some subspace of the space $S_k^r(\Delta )$, where $k \geqslant 3r + 2$, that can be used to provide the highest order of approximation, with the property that the approximation constant of this order is independent of the geometry of $\Delta$ with the exception of the smallest angle in the partition. This result is obtained by means of a careful choice of locally supported basis functions which, however, require a very technical proof to justify their stability in optimal-order approximation. A new formulation of smoothness conditions for piecewise polynomials in terms of their ${\text{B}}$-net representations is derived for this purpose.
The approximation theorem and the $K$-theory of generalized free products
Roland
Schwänzl;
Ross E.
Staffeldt
3319-3345
Abstract: We use methods of abstract algebraic $K$-theory as developed by Friedhelm Waldhausen to give a new derivation of the decomposition theorem for the algebraic $K$-theory of a generalized free product ring. The result takes the form of a fibration sequence which relates the algebraic $K$-theory of such a ring with the algebraic $ K$-theory of its factors, plus a Nil-term.
Intertwining operators associated to the group $S\sb 3$
Charles F.
Dunkl
3347-3374
Abstract: For any finite reflection group $G$ on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections in $G$. There exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients, for all but a singular set of parameter values (containing only certain negative rational numbers). This paper constructs an integral transform implementing the intertwining operator for the group ${S_3}$, the symmetric group on three objects, for parameter value $ \geqslant \frac{1} {2}$. The transform is realized as an absolutely continuous measure on a compact subset of ${M_2}({\mathbf{R}})$, which contains the group as a subset of its boundary. The construction of the integral formula involves integration over the unitary group $U(3)$.
Global uniqueness for a two-dimensional semilinear elliptic inverse problem
Victor
Isakov;
Adrian I.
Nachman
3375-3390
Abstract: For a general class of nonlinear Schrödinger equations $- \Delta u + a(x,u) = 0$ in a bounded planar domain $\Omega$ we show that the function $a(x,u)$ can be recovered from knowledge of the corresponding Dirichlet-to-Neumann map on the boundary $ \partial \Omega$.
Automorphisms of spaces with finite fundamental group
Georgia
Triantafillou
3391-3403
Abstract: Let $X$ be a finite CW-complex with finite fundamental group. We show that the group ${\text{aut}}(X)$ of homotopy classes of self-homotopy equivalences of $X$ is commensurable to an arithmetic group. If in addition $X$ is an oriented manifold then the subgroup $ {\text{au}}{{\text{t}}_t}(X)$ of homotopy classes of tangential homotopy equivalences is commensurable to an arithmetic group. Moreover if $X$ is a smooth manifold of dimension $\geqslant 5$ then the subgroup ${\text{diff}}(X)$ of ${\text{aut}}(X)$ the elements of which are represented by diffeomorphisms is also commensurable to an arithmetic group.
Global surjectivity of submersions via contractibility of the fibers
Patrick J.
Rabier
3405-3422
Abstract: We give a sufficient condition for a ${C^1}$ submersion $F:X \to Y$, $X$ and $Y$ real Banach spaces, to be surjective with contractible fibers ${F^{ - 1}}(y)$. Roughly speaking, this condition "interpolates" two well-known but unrelated hypotheses corresponding to the two extreme cases: Hadamard's criterion when $Y \simeq X$ and $F$ is a local diffeomorphism, and the Palais-Smale condition when $Y = \mathbb{R}$. These results may be viewed as a global variant of the implicit function theorem, which unlike the local one does not require split kernels. They are derived from a deformation theorem tailored to fit functionals with a norm-like nondifferentiability.
The structure of MFD shock waves for rectilinear motion in some models of plasma
Mahmoud
Hesaaraki
3423-3452
Abstract: The mathematical question of the existence of structure for "fast", "slow" and "intermediate" MFD shock waves in the case of rectilinear motion in some model of plasma is stated in terms of a six-dimensional system of ordinary differential equations, which depends on five viscosity parameters. In this article we shall show that this system is gradient-like. Then by using the Conley theory we prove that the fast and the slow shocks always possess structure. Moreover, the intermediate shocks do not admit structure. Some limiting cases for singular viscosities are investigated. In particular, we show how the general results in the classical one fluid MHD theory are obtained when "the plasma viscosities" $\beta$ and $\chi$ tend to zero.
Test ideals in local rings
Karen E.
Smith
3453-3472
Abstract: It is shown that certain aspects of the theory of tight closure are well behaved under localization. Let $J$ be the parameter test ideal for $ R$, a complete local Cohen-Macaulay ring of positive prime characteristic. For any multiplicative system $U \subset R$, it is shown that $J{U^{ - 1}}R$ is the parameter test ideal for ${U^{ - 1}}R$. This is proved by proving more general localization results for the here-introduced classes of " $ {\text{F}}$-ideals" of $ R$ and " ${\text{F}}$-submodules of the canonical module" of $ R$, which are annihilators of $R$ modules with an action of Frobenius. It also follows that the parameter test ideal cannot be contained in any parameter ideal of $R$.
The $7$-connected cobordism ring at $p=3$
Mark A.
Hovey;
Douglas C.
Ravenel
3473-3502
Abstract: In this paper, we study the cobordism spectrum $MO\left\langle 8 \right\rangle$ at the prime $3$. This spectrum is important because it is conjectured to play the role for elliptic cohomology that Spin cobordism plays for real $K$-theory. We show that the torsion is all killed by $ 3$, and that the Adams-Novikov spectral sequence collapses after only $ 2$ differentials. Many of our methods apply more generally.
Rates of eigenvalues on a dumbbell domain. Simple eigenvalue case
José M.
Arrieta
3503-3531
Abstract: We obtain the first term in the asymptotic expansion of the eigenvalues of the Laplace operator in a typical dumbbell domain in $ {\mathbb{R}^2}$. This domain consists of two disjoint domains ${\Omega ^L}$, $ {\Omega ^R}$ joined by a channel $ {R_\varepsilon }$ of height of the order of the parameter $\varepsilon$. When an eigenvalue approaches an eigenvalue of the Laplacian in ${\Omega ^L} \cup {\Omega ^R}$, the order of convergence is $ \varepsilon$, while if the eigenvalue approaches an eigenvalue which comes from the channel, the order is weaker: $\varepsilon \left\vert {{\text{ln}}\varepsilon } \right\vert$. We also obtain estimates on the behavior of the eigenfunctions.
Bounded geodesics of Riemann surfaces and hyperbolic manifolds
J. L.
Fernández;
M. V.
Melián
3533-3549
Abstract: We study the set of bounded geodesics of hyperbolic manifolds. For general Riemann surfaces and for hyperbolic manifolds with some finiteness assumption on their geometry we determine its Hausdorff dimension. Some applications to diophantine approximation are included.
Integrally closed modules over two-dimensional regular local rings
Vijay
Kodiyalam
3551-3573
Abstract: This paper is based on work of Rees on integral closures of modules and initiates the study of integrally closed modules over two-dimensional regular local rings in analogy with the classical theory of complete ideals of Zariski. The main results can be regarded as generalizations of Zariski's product theorem. They assert that the tensor product mod torsion of integrally closed modules is integrally closed, that the symmetric algebra mod torsion of an integrally closed module is a normal domain and that the first Fitting ideal of an integrally closed module is an integrally closed ideal. A construction of indecomposable integrally closed modules is also given. The primary technical tool is a study of the Buchsbaum-Rim multiplicity.
Elliptic equations of order $2m$ in annular domains
Robert
Dalmasso
3575-3585
Abstract: In this paper we study the existence of positive radial solutions for some semilinear elliptic problems of order $ 2m$ in an annulus with Dirichlet boundary conditions. We consider a nonlinearity which is either sublinear or the sum of a sublinear and a superlinear term.
Circle bundles and the Kreck-Stolz invariant
Xianzhe
Dai;
Wei Ping
Zhang
3587-3593
Abstract: We present a direct analytic calculation of the $s$-invariant of Kreck-Stolz for circle bundles, by evaluating the adiabatic limits of $ \eta$ invariants. We believe that this method should have wider applications.
A note on singularities in semilinear problems
Mohammed
Guedda;
Mokhtar
Kirane
3595-3603
Abstract: We consider the equation $ \Delta u - \frac{1} {2}x.\Delta u - \frac{u} {{q - 1}} + {u^q} = 0,{\text{for}}q > 1$. We study the isolated singularities and present a nonlinear technique, and give a complete classification.
The zero-sets of the radial-limit functions of inner functions
Charles L.
Belna;
Robert D.
Berman;
Peter
Colwell;
George
Piranian
3605-3612
Abstract: A set $ E$ on the unit circle is the zero-set of the radial-limit function of some inner function if and only if $E$ is a countable intersection of ${F_\sigma }$-sets of measure 0.
Finite generalized triangle groups
J.
Howie;
V.
Metaftsis;
R. M.
Thomas
3613-3623
Abstract: We give an almost complete classification of those generalized triangle groups that are finite, building on previous results of Baumslag, Morgan and Shalen [1], Conder [4], Rosenberger [12] and Levin and Rosenberger [11]. There are precisely two groups for which we cannot decide whether or not they are finite.
All finite generalized triangle groups
L.
Lévai;
G.
Rosenberger;
B.
Souvignier
3625-3627
Abstract: We complete the classification of those generalized triangle groups that are finite.
Sobolev orthogonal polynomials and spectral differential equations
I. H.
Jung;
K. H.
Kwon;
D. W.
Lee;
L. L.
Littlejohn
3629-3643
Abstract: We find necessary and sufficient conditions for a spectral differential equation $\displaystyle {L_N}[y](x) = \sum\limits_{i = 1}^N {{\ell _i}(x){y^{(i)}}(x) = {\lambda _n}y(x)}$ to have Sobolev orthogonal polynomials of solutions, which are orthogonal relative to the Sobolev (pseudo-) inner product $\displaystyle \phi (p,q) = \int_\mathbb{R}^{} {pqd\mu + \int_\mathbb{R}^{} {p'q'dv,} }$ where $d\mu$ and $dv$ are signed Borel measures having finite moments. This result generalizes a result by H. L. Krall, which handles the case when $dv = 0$.
Harmonic diffeomorphisms between Hadamard manifolds
Peter
Li;
Luen-Fai
Tam;
Jiaping
Wang
3645-3658
Abstract: In this paper, we study the Dirichlet problem at infinity for harmonic maps between complete hyperbolic Hadamard surfaces. We will address the existence and uniqueness questions relating to the problem. In particular, we generalize results in the work of Li-Tam and Wan.
On the cohomology of $\Gamma\sb p$
Yining
Xia
3659-3670
Abstract: Let ${\Gamma _g}$ denote the mapping class group of genus $ g$. In this paper, we calculate $p$-torsion of Farrell cohomology $ {\widehat{H}^*}({\Gamma_p})$ for any odd prime $p$.
Errata to: ``Hypersurfaces with constant mean curvature in the complex hyperbolic space'' [Trans. Amer. Math. Soc. {\bf 339} (1993), no. 2, 685--702; MR1123452 (93m:53065)]
S.
Fornari;
K.
Frensel;
J.
Ripoll
The large deviation principle for a general class of queueing systems. I
Paul
Dupuis;
Richard S.
Ellis
2689-2751
Abstract: We prove the existence of a rate function and the validity of the large deviation principle for a general class of jump Markov processes that model queueing systems. A key step in the proof is a local large deviation principle for tubes centered at a class of piecewise linear, continuous paths mapping [0,1] into $[0,1]$. In order to prove certain large deviation limits, we represent the large deviation probabilities as the minimal cost functions of associated stochastic optimal control problems and use a subadditivity--type argument. We give a characterization of the rate function that can be used either to evaluate it explicitly in the cases where this is possible or to compute it numerically in the cases where an explicit evaluation is not possible.
On Lam\'e operators which are pull-backs of hypergeometric ones
Bruno
Chiarellotto
2753-2780
Abstract: We give a method that would allow one to calculate the number of Lamé operators, $ {\mathcal{L}_n}$, $n \in {\mathbf{N}}$, with prescribed finite monodromy and do the calculation for the case $n = 1$. We find a bound for the degree over $ {\mathbf{Q}}$ of the field of definition of the coefficients of a Lamé operator with prescribed finite monodromy and give examples of Lamé operators with finite monodromy. Finally we study Lamé operators with infinite monodromy and generic second order differential operators which are pull-backs of hypergeometric ones under algebraic maps.
On the $K$-theory of crystallographic groups
Georgios
Tsapogas
2781-2794
Abstract: For any crystallographic group $\Gamma$ we show that the groups ${K_i}(\Gamma )$ are isomorphic, via the forget control map, to the controlled $K$-groups ${K_i}{(\Gamma )_c}$, for all $i \leqslant 1$ and for an appropriate choice of the control map. By using this result and under a mild hypothesis on the crystallographic group $ \Gamma$, it is proved that ${K_i}(\Gamma ) = 0$ for all $i \leqslant - 2$ and ${N^j}{K_i}(\Gamma ) = 0$ for all $i \leqslant - 1$ and $j > 0$.
The index of determinacy for measures and the $l\sp 2$-norm of orthonormal polynomials
Christian
Berg;
Antonio J.
Duran
2795-2811
Abstract: For determinate measures $\mu$ having moments of every order we define and study an index of determinacy which checks the stability of determinacy under multiplication by even powers of $\vert t - z\vert$ for $z$ a complex number. Using this index of determinacy, we solve the problem of determining for which $z \in \mathbb{C}$ the sequence $ {(p_n^{(m)}(z))_n}(m \in \mathbb{N})$ belongs to ${\ell ^2}$, where $ {({p_n})_n}$ is the sequence of orthonormal polynomials associated with the measure $\mu$.
Immersions and embeddings in domains of holomorphy
Avner
Dor
2813-2849
Abstract: Let ${D_1}$ be a bounded smooth strongly pseudoconvex domain in $ {\mathbb{C}^N}$ and let $ {D_2}$ be a domain of holomorphy in $ {\mathbb{C}^M}(2 \leqslant N,5 \leqslant M,2N \leqslant M)$. There exists then a proper holomorphic immersion from ${D_1}$ to ${D_2}$. Furthermore if $ {\mathbf{PI}}({D_1},{D_2})$ is the set of proper holomorphic immersions from $ {D_1}$ to ${D_2}$ and $A({D_1},{D_2})$ is the set of holomorphic maps from $ {D_1}$ to ${D_2}$ that are continuous on the boundary, then the closure of $ {\mathbf{PI}}({D_1},{D_2})$ in the topology of uniform convergence on compacta contains $ A({D_1},{D_2})$. The approximating proper maps can be made tangent to any finite order of contact at a given point. The same result was obtained for proper holomorphic maps, in one codimension, when the target domain has a plurisubharmonic exhaustion function with no saddle critical points. This includes the case where the target domain is convex. Density in a weaker sense was derived in one codimension when the critical points are contained in a compact subset of the target domain. This occurs (for example) when the target domain is bounded weakly pseudoconvex with ${C^2}$-smooth boundary. If the target domain is strongly pseudoconvex then the approximating proper holomorphic maps can also be made continuous on the boundary. A lesser degree of pseudoconvexity is required from the target domain when the codimension is larger than the minimal. A domain in ${\mathbb{C}^L}$ is called "$ M$dimensional-pseudoconvex" (where $ L \geqslant M$) if it has a smooth exhaustion function $r$ such that every point $w$ in this domain has some $ M$-dimensional complex affine subspace going through this point for which $ r$, restricted to this subspace, is strictly plurisubharmonic in $ w$. In the result mentioned above the assumption that the target domain is pseudoconvex in ${\mathbb{C}^M}(M \geqslant 2N,5)$ can be substituted for the assumption that the domain is "$ M$-dimensional-pseudoconvex". Similarly, the assumption that the target domain ${D_2}$ is "$(N + 1)$-dimensional-pseudoconvex" and all the critical points of some appropriate exhaustion function are "$(N + 1)$-dimensional-convex" (defined in a similar manner) yields that the closure of the set of proper holomorphic maps from ${D_1}$ to ${D_2}$ contains $ A({D_1},{D_2})$. All the results are obtained with embeddings when the Euclidean dimensions are such that ${\dim _\mathbb{C}}({D_2}) \geqslant 2{\dim _\mathbb{C}}({D_1}) + 1$. Thus, in this case, when one of the assumptions mentioned above is fulfilled, then the closure of the set of embeddings from ${D_1}$ to ${D_2}$ contains $ A({D_1},{D_2})$.
A principle of linearized stability for nonlinear evolution equations
Nobuyuki
Kato
2851-2868
Abstract: We present a principle of linearized stability of stationary solutions to nonlinear evolution equation in Banach spaces. The well-known semilinear case is shown to fit into our framework. Applications to nonlinear population dynamics and to functional differential equations are also considered.
When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide?
Szymon
Dolecki;
Gabriele H.
Greco;
Alojzy
Lechicki
2869-2884
Abstract: A topology is called consonant if the corresponding upper Kuratowski topology on closed sets coincides with the co-compact topology, equivalently if each Scott open set is compactly generated. It is proved that Čechcomplete topologies are consonant and that consonance is not preserved by passage to $ {G_\delta }$-sets, quotient maps and finite products. However, in the class of the regular spaces, the product of a consonant topology and of a locally compact topology is consonant. The latter fact enables us to characterize the topologies generated by some $\Gamma$-convergences.
Induced $C\sp *$-algebras and Landstad duality for twisted coactions
John C.
Quigg;
Iain
Raeburn
2885-2915
Abstract: Suppose $ N$ is a closed normal subgroup of a locally compact group $G$. A coaction $:A \to M(A \otimes {C^ * }(N))$ of $N$ on a ${C^ * }$-algebra $A$ can be inflated to a coaction $\delta$ of $G$ on $A$, and the crossed product $A{ \times_\delta }G$ is then isomorphic to the induced ${C^ * }$-algebra Ind$_N^G A{\times_\epsilon}N$. We prove this and a natural generalization in which $A{ \times_\epsilon}N$ is replaced by a twisted crossed product $A{ \times _{G/N}}G$; in case $G$ is abelian, we recover a theorem of Olesen and Pedersen. We then use this to extend the Landstad duality of the first author to twisted crossed products, and give several applications. In particular, we prove that if $\displaystyle 1 \to N \to G \to G/N \to 1$ is topologically trivial, but not necessarily split as a group extension, then every twisted crossed product $A{ \times _{G/N}}G$ is isomorphic to a crossed product of the form $A \times N$.
Comparison of certain $H\sp \infty$-domains of holomorphy
Ulf
Backlund
2917-2926
Abstract: We study open sets defined by certain global plurisubharmonic functions in $ {\mathbb{C}^N}$. We examine how the fact that the connected components of the sets are $ {H^\infty }$-domains of holomorphy is related to the structure of the set of discontinuity points of the global defining functions and to polynomial convexity.
Orbits of rank one and parallel mean curvature
Carlos
Olmos
2927-2939
Abstract: Let ${M^n}(n \geqslant 2)$ be a (extrinsic) homogeneous irreducible full submanifold of Euclidean space with $rank(M) = k \geqslant 1$ (i.e., it admits $k \geqslant 1$ locally defined, linearly independent parallel normal vector fields). We prove that $M$ must be contained in a sphere. This result toghether with previous work of the author about homogeneous submanifolds of higher rank imply, in particular, the following theorem: A homogeneous irreducible submanifold of Euclidean space with parallel mean curvature vector is either minimal, or minimal in a sphere, or an orbit of the isotropy representation of a simple symmetric space.
The structure of the reverse H\"older classes
David
Cruz-Uribe;
C. J.
Neugebauer
2941-2960
Abstract: In this paper we study the structure of the class of functions $ (R{H_s})$ which satisfy the reverse Hölder inequality with exponent $ s > 1$. To do so we introduce a new operator, the minimal operator, which is analogous to the Hardy-Littlewood maximal operator, and a new class of functions, $(R{H_\infty })$, which plays the same role for $ (R{H_s})$ that the class $ ({A_1})$ does for the $ ({A_p})$ classes.
An analogue of the Jacobson-Morozov theorem for Lie algebras of reductive groups of good characteristics
Alexander
Premet
2961-2988
Abstract: Let $\mathfrak{g}$ be the Lie algebra of a connected reductive group $G$ over an algebraically closed field of characteristic $p > 0$. Suppose that ${G^{(1)}}$ is simply connected and $p$ is good for the root system of $ G$. Given a one-dimensional torus $ \lambda \subset G$ let $ \mathfrak{g}(\lambda ,1)$ denote the weight component of ${\text{Ad(}}\lambda {\text{)}}$ corresponding to weight $i \in X(\lambda ) \cong \mathbb{Z}$. It is proved in the paper that, for any nonzero nilpotent element $e \in \mathfrak{g}$, there is a one-dimentional torus ${\lambda _e} \subset G$ such that $e \in \mathfrak{g}({\lambda _e},2)$ and ${\text{Ker}}{\text{ad}}e \subseteq { \oplus _{i \geqslant 0}}\mathfrak{g}({\lambda _e},i)$.
Test problems for operator algebras
Edward A.
Azoff
2989-3001
Abstract: Kaplansky's test problems, originally formulated for abelian groups, concern the relationship between isomorphism and direct sums. They provide a "reality check" for purported structure theories. The present paper answers Kaplansky's problems in operator algebraic contexts including unitary equivalence of von Neumann algebras and equivalence of representations of (non self-adjoint) matrix algebras. In particular, it is shown that matrix algebras admitting similar ampliations are themselves similar.
An explicit theory of heights
E. V.
Flynn
3003-3015
Abstract: We consider the problem of explicitly determining the naive height constants for Jacobians of hyperelliptic curves. For genus $> 1$, it is impractical to apply Hilbert's Nullstellensatz directly to the defining equations of the duplication law; we indicate how this technical difficulty can be overcome by use of isogenies. The height constants are computed in detail for the Jacobian of an arbitrary curve of genus $2$, and we apply the technique to compute generators of $\mathcal{J}(\mathbb{Q})$, the Mordell-Weil group for a selection of rank $1$ examples.
Periodic orbits for the planar Newtonian three-body problem coming from the elliptic restricted three-body problems
Jaume
Llibre;
Donald G.
Saari
3017-3030
Abstract: Through the introduction of a new coordinate system followed by a Poincaré compactification, a new relationship is developed to connect the planar three-body problem with the various planar restricted three-body systems. This framework is further used to develop new conditions for the continuation of symmetric periodic orbits from elliptic restricted systems to the full planar three-body problem.
Local uniqueness in the inverse conductivity problem with one measurement
G.
Alessandrini;
V.
Isakov;
J.
Powell
3031-3041
Abstract: We prove local uniqueness of a domain $D$ entering the conductivity equation $ {\text{div}}((1 + \chi (D))\nabla u) = 0$ in a bounded planar domain $\Omega$ given the Cauchy data for $ u$ on a part of $\partial \Omega$. The main assumption is that $ \nabla u$ has zero index on $\partial \Omega $ which is easy to guarantee by choosing special boundary data for $ u$. To achieve our goals we study index of critical points of $u$ on $ \partial \Omega$.
Hadamard convexity and multiplicity and location of zeros
Faruk F.
Abi-Khuzam
3043-3051
Abstract: We consider certain questions arising from a paper of Hayman concerning quantitative versions of the Hadamard three-circle theorem for entire functions. If $b(r)$ denotes the second derivative of $\log M(r)$ with respect to $ \log r$, the principal contributions of this work are (i) a characterization of those entire $f$ with nonnegative Maclaurin coefficients for which $ \lim \sup b(r) = \frac{1} {4}$ and (ii) some exploration of the relationship between multiple zeros of $f$ and the growth of $b(r)$.
On the relational basis of Cayley's theorem and of similar representations for algebras
Hassan
Sedaghat
3053-3060
Abstract: Considering a binary operation as a ternary relation permits certain sections of the latter (which are functions) to be used in representing an abstract semigroup as a family of the self-maps of its underlying set under function composition. The idea is thus seen to be entirely similar to the way that the sections of a partial ordering under set inclusion represent the (abstract) partially ordered set. An extension of this procedure yields a uniform set of representation theorems for a large class of associative algebras.
Multiple viscous solutions for systems of conservation laws
A. V.
Azevedo;
D.
Marchesin
3061-3077
Abstract: We exhibit an example of mechanism responsible for multiple solutions in the Riemann problem for a mixed elliptic-hyperbolic type system of two quadratic polynomial conservation laws. In this example, multiple solutions result from folds in the set of Riemann solutions. The multiple solutions occur despite the fact that they all satisfy the viscous profile entropy criterion. The failure of this criterion to provide uniqueness is evidence in support of a need for conceptual change in the theory of shock waves for a system of conservation laws.
On the number of solutions of a third-order boundary value problem
Eva
Rovderová
3079-3092
Abstract: This paper deals with the number of solutions of the third-order boundary value problem $ y''' = f(t,y,y',y'')$, $y(0) = {A_0}$, $y''(T) = B$. This number of solutions is investigated in connection with the number of zeros of a solution for the corresponding variational problem.
Irreducible semigroups of functionally positive nilpotent operators
Yong
Zhong
3093-3100
Abstract: For each irrational number $ \theta \in (0,1)$, we construct a semigroup $ {\mathcal{S}_\theta }$ of nilpotent operators on ${\mathcal{S}^2}([0,1])$ that are also partial isometries and positive in the sense that the operator maps nonnegative functions to nonnegative functions. We prove that each semigroup ${\mathcal{S}_\theta }$ is discrete in the norm topology and hence norm-closed and that the weak closure of $ {\mathcal{S}_\theta }$ is independent of $ {\mathcal{S}_\theta }$. We show that each semigroup ${\mathcal{S}_\theta }$ has no nontrivial invariant subspaces.
Power regular operators
Aharon
Atzmon
3101-3109
Abstract: We show that for a wide class of operators $T$ on a Banach space, including the class of decomposable operators, the sequence $\left\{ {{{\left\Vert {{T^n}x} \right\Vert}^{1/n}}} \right\}_{n = 1}^\infty $ converges for every $ x$ in the space to the spectral radius of the restriction of $ T$ to the subspace $ \vee _{n = 0}^\infty \{ {T^n}x\}$.
A new functional equation of Pexider type related to the complex exponential function
Hiroshi
Haruki;
Themistocles M.
Rassias
3111-3119
Abstract: The purpose of this paper is to solve a new functional equation, characteristic for the complex exponential function, which contains four unknown entire functions and to solve, as an application, three further functional equations.
Sub-self-similar sets
K. J.
Falconer
3121-3129
Abstract: A compact set $E \subseteq {{\mathbf{R}}^n}$ is called sub-self-similar if $E \subseteq \bigcup\nolimits_{i = 1}^m {{S_i}(E)}$, where the ${S_i}$ are similarity transfunctions. We consider various examples and constructions of such sets and obtain formulae for their Hausdorff and box dimensions, generalising those for self-similar sets.
A right countably sigma-CS ring with ACC or DCC on projective principal right ideals is left Artinian and QF-$3$
Dinh Van Huynh
3131-3139
Abstract: A module $ M$ is called a CS module if every submodule of $M$ is essential in a direct summand of $ M$. A ring $R$ is said to be right (countably) $ \Sigma$-CS if any direct sum of (countably many) copies of the right $ R$-module $R$ is CS. It is shown that for a right countably $\Sigma$-CS ring $R$ the following are equivalent: (i) $ R$ is right $\Sigma$-CS, (ii) $R$ has ACC or DCC on projective principal right ideals, (iii) $R$ has finite right uniform dimension and ACC or DCC holds on projective uniform principal right ideals of $R$, (iv) $R$ is semiperfect. From results of Oshiro [12], [13], under these conditions, $R$ is left artinian and QF-$3$. As a consequence, a ring $R$ is quasi-Frobenius if it is right countably $\Sigma$-CS, semiperfect and no nonzero projective right ideals are contained in its Jacobson radical.
Stable range one for rings with many idempotents
Victor P.
Camillo;
Hua-Ping
Yu
3141-3147
Abstract: An associative ring $ R$ is said to have stable range $1$ if for any $a$, $b \in R$ satisfying $ aR + bR = R$, there exists $y \in R$ such that $ a + by$ by is a unit. The purpose of this note is to prove the following facts. Theorem $3$: An exchange ring $R$ has stable range $1$ if and only if every regular element of $ R$ is unit-regular. Theorem $5$: If $R$ is a strongly $\pi$-regular ring with the property that all powers of every regular element are regular, then $ R$ has stable range $ 1$. The latter generalizes a recent result of Goodearl and Menal [$5$].
Convergence of diagonal Pad\'e approximants for functions analytic near $0$
D. S.
Lubinsky
3149-3157
Abstract: For functions analytic in a neighbourhood of 0, we show that at least for a subsequence of the diagonal Padé approximants, the point 0 attracts a zero proportion of the poles. The same is true for every "sufficiently dense" diagonal subsequence. Consequently these subsequences have a convergence in capacity type property, which is possibly the correct analogue of the Nuttall-Pommerenke theorem in this setting.
Left annihilators characterized by GPIs
Tsiu Kwen
Lee
3159-3165
Abstract: Let $R$ be a semiprime ring with extended centroid $C$, $U$ the right Utumi quotient ring of $ R$, $S$ a subring of $U$ containing $R$ and ${\rho _1}$, ${\rho _2}$ two right ideals of $R$. In the paper we show that $ {l_S}({\rho _1}) = {l_S}({\rho _2})$ if and only if ${\rho _1}$ and ${\rho _2}$ satisfy the same generalized polynomial identities (GPIs) with coefficients in $ SC$, where ${l_S}({\rho _i})$ denotes the left annihilator of ${\rho _i}$ in $S$. As a consequence of the result, if $ \rho$ is a right ideal of $ R$ such that ${l_R}(\rho ) = 0$, then $\rho$ and $U$ satisfy the same GPIs with coefficients in the two-sided Utumi quotient ring of $R$.
Construction of homomorphisms of $M$-continuous lattices
Xiao Quan
Xu
3167-3175
Abstract: We present a direct approach to constructing homomorphisms of $ M$-continuous lattices, a generalization of continuous lattices, into the unit interval, and show that an $M$-continuous lattice has sufficiently many homomorphisms into the unit interval to separate the points.
Equilibria and quasiequilibria for infinite collections of interacting Fleming-Viot processes
Donald A.
Dawson;
Andreas
Greven;
Jean
Vaillancourt
2277-2360
Abstract: In this paper of infinite systems of interacting measure-valued diffusions each with state space $\mathcal{P}\left( {[0,1]} \right)$, the set of probability measures on [0, 1], is constructed and analysed (Fleming-Viot systems). These systems arise as diffusion limits of population genetics models with infinitely many possible types of individuals (labelled by [0, 1]), spatially distributed over a countable collection of sites and evolving as follows. Individuals can migrate between sites and after an exponential waiting time a colony replaces its population by a new generation where the types are assigned by resampling from the empirical distribution of types at this site. It is proved that, depending on recurrence versus transience properties of the migration mechanism, the system either clusters as $t \to \infty$, that is, converges in distribution to a law concentrated on the states in which all components are equal to some ${\delta _u}, u \in [0,1]$, or the system approaches a nontrivial equilibrium state. The properties of the equilibrium states, respectively the cluster formation, are studied by letting a parameter in the migration mechanism tend to infinity and explicitly identifying the limiting dynamics in a sequence of different space-time scales. These limiting dynamics have stationary states which are quasi-equilibria of the original system, that is, change only in longer time scales. Properties of these quasi-equilibria are derived and related to the global equilibrium process for large $N$. Finally we establish that the Fleming-Viot systems are the unique dynamics which remain invariant under the associated space-time renormalization procedure.
Principally polarized ordinary abelian varieties over finite fields
Everett W.
Howe
2361-2401
Abstract: Deligne has shown that there is an equivalence from the category of ordinary abelian varieties over a finite field $k$ to a category of ${\mathbf{Z}}$-modules with additional structure. We translate several geometric notions, including that of a polarization, into Deligne's category of ${\mathbf{Z}}$-modules. We use Deligne's equivalence to characterize the finite group schemes over $ k$ that occur as kernels of polarizations of ordinary abelian varieties in a given isogeny class over $k$. Our result shows that every isogeny class of simple odd-dimensional ordinary abelian varieties over a finite field contains a principally polarized variety. We use our result to completely characterize the Weil numbers of the isogeny classes of two-dimensional ordinary abelian varieties over a finite field that do not contain principally polarized varieties. We end by exhibiting the Weil numbers of several isogeny classes of absolutely simple four-dimensional ordinary abelian varieties over a finite field that do not contain principally polarized varieties.
Ranges of perturbed maximal monotone and $m$-accretive operators in Banach spaces
Zhengyuan
Guan;
Athanassios G.
Kartsatos
2403-2435
Abstract: A more comprehensive and unified theory is developed for the solvability of the inclusions $S \subset \overline {R(A + B)}$, int $S \subset R(A + B)$, where $A:X \supset D(A) \to {2^Y}$, $B:X \supset D(B) \to Y$ and $S \subset X$. Here, $X$ is a real Banach space and $Y = X$ or $Y = {X^*}$. Mainly, $A$ is either maximal monotone or maccretive, and $ B$ is either pseudo-monotone or compact. Cases are also considered where $ A$ has compact resolvents and $B$ is continuous and bounded. These results are then used to obtain more concrete sets in the ranges of sums of such operators $A$ and $B$. Various results of Browder, Calvert and Gupta, Gupta, Gupta and Hess, and Kartsatos are improved and/or extended. The methods involve the application of a basic result of Browder, concerning pseudo-monotone perturbation of maximal monotone operators, and the Leray-Schauder degree theory.
Conformal hypersurfaces with the same Gauss map
Marcos
Dajczer;
E.
Vergasta
2437-2456
Abstract: In this paper we provide a complete classification of all hypersurfaces of Euclidean space which admit conformal deformations, other than the ones obtained through conformal diffeomorphisms of the ambient space, preserving the Gauss map.
L\'evy type characterization of stable laws for free random variables
Vittorino
Pata
2457-2472
Abstract: We give a description of stable probability measures relative to free additive convolution. The definition of domain of attraction is given, and a proof is provided of the noncommutative analogue of Lévy Theorem.
Sur les structures de contact r\'eguli\`eres en dimension trois
Amine
Hadjar
2473-2480
Abstract: Let $\mathcal{M}$ be a compact and oriented $3$-manifold with boundary, endowed with a free $ {\mathbb{S}^1}$ action. We give a characterization of germs of invariant contact structures along $ \partial \mathcal{M}$ which are extendable to $ \mathcal{M}$ as regular contact structures.
Quasilinear elliptic equations with VMO coefficients
Dian K.
Palagachev
2481-2493
Abstract: Strong solvability and uniqueness in Sobolev space ${W^{2,n}}(\Omega )$ are proved for the Dirichlet problem $\displaystyle \left\{ {_{u = \varphi \quad {\text{on}}\partial \Omega .}^{{a^{i... ...{D_{ij}}u + b(x,u,Du) = 0\quad {\text{a}}{\text{.e}}{\text{.}}\Omega }} \right.$ It is assumed that the coefficients of the quasilinear elliptic operator satisfy Carathéodory's condition, the ${a^{ij}}$ are $V\, M\, O$ functions with respect to $x$, and structure conditions on $ b$ are required. The main results are derived by means of the Aleksandrov-Pucci maximum principle and Leray-Schauder's fixed point theorem via a priori estimate for the $ {L^{2n}}$-norm of the gradient.
On the conditional expectation and convergence properties of random sets
Nikolaos S.
Papageorgiou
2495-2515
Abstract: In this paper we study random sets, with values in a separable Banach space. First we establish several useful properties of the set-valued conditional expectation and then prove some convergence theorems for set-valued amarts and uniform amarts, using the weak, Kuratowski-Mosco and Hausdorff modes of set convergence.
On the ideal class groups of imaginary abelian fields with small conductor
Kuniaki
Horie;
Hiroko
Ogura
2517-2532
Abstract: Let $k$ be any imaginary abelian field with conductor not exceeding 100, where an abelian field means a finite abelian extension over the rational field $ {\mathbf{Q}}$ contained in the complex field. Let $C(k)$ denote the ideal class group of $ k$, ${C^ - }(k)$ the kernel of the norm map from $ C(k)$ to the ideal class group of the maximal real subfield of $k$, and $f(k)$ the conductor of $k;f(k) \leqslant 100$. Proving a preliminary result on $2$-ranks of ideal class groups of certain imaginary abelian fields, this paper determines the structure of the abelian group $ {C^ - }(k)$ and, under the condition that either $[k:{\mathbf{Q}}] \leqslant 23$ or $f(k)$ is not a prime $\geqslant 71$, determines the structure of $ C(k)$.
Heegaard splittings of Seifert fibered spaces with boundary
Jennifer
Schultens
2533-2552
Abstract: We give the classification theorem for Heegaard splittings of fiberwise orientable Seifert fibered spaces with nonempty boundary. A thin position argument yields a reducibility result which, by induction, shows that all Heegaard splittings of such manifolds are vertical in the sense of Lustig-Moriah. Algebraic arguments allow a classification of the vertical Heegaard splittings.
On a semilinear elliptic Neumann problem with asymmetric nonlinearities
J.-P.
Gossez;
P.
Omari
2553-2562
Abstract: We consider the Neumann problem (1.1) below. We extend the range of applicability of the sharp nonresonance condition derived in [Go-Om] so as, in particular, allow asymmetric nonlinearities.
Characterization of summability points of N\"orlund methods
Karl-Goswin
Grosse-Erdmann;
Karin
Stadtmüller
2563-2574
Abstract: By a theorem of F. Leja any regular Nörlund method $ (N,p)$ sums a given power series $f$ at most at countably many points outside its disc of convergence. This result was recently extended to a class of non-regular Nörlund methods by K. Stadtmüller. In this paper we obtain a more detailed picture showing how possible points of summability and the value of summation depend on $ p$ and $f$.
Viewing parallel projection methods as sequential ones in convex feasibility problems
G.
Crombez
2575-2583
Abstract: We show that the parallel projection method with variable weights and one variable relaxation coefficient for obtaining a point in the intersection of a finite number of closed convex sets in a given Hilbert space may be interpreted as a semi-alternating sequential projection method in a suitably newly constructed Hilbert space. As such, convergence results for the parallel projection method may be derived from those which may be constructed in the semi-alternating sequential case.
The Toeplitz theorem and its applications to approximation theory and linear PDEs
Rong Qing
Jia
2585-2594
Abstract: We take an algebraic approach to the problem of approximation by dilated shifts of basis functions. Given a finite collection $ \Phi$ of compactly supported functions in $ {L_p}({\mathbb{R}^s})\quad (1 \leqslant p \leqslant \infty )$, we consider the shift-invariant space $S$ generated by $\Phi$ and the family $({S^h}:h > 0)$, where ${S^h}$ is the $h$-dilate of $S$. We prove that $ ({S^h}:h > 0)$ provides $ {L_p}$-approximation order $ r$ only if $S$ contains all the polynomials of total degree less than $r$. In particular, in the case where $ \Phi$ consists of a single function $\varphi$ with its moment $\int {\varphi \ne 0}$, we characterize the approximation order of $ ({S^h}:h > 0)$ by showing that the above condition on polynomial containment is also sufficient. The above results on approximation order are obtained through a careful analysis of the structure of shift-invariant spaces. It is demonstrated that a shiftinvariant space can be described by a certain system of linear partial difference equations with constant coefficients. Such a system then can be reduced to an infinite system of linear equations, whose solvability is characterized by an old theorem of Toeplitz. Thus, the Toeplitz theorem sheds light into approximation theory. It is also used to give a very simple proof for the well-known Ehrenpreis principle about the solvability of a system of linear partial differential equations with constant coefficients.
On $C\sp *$-algebras associated to the conjugation representation of a locally compact group
Eberhard
Kaniuth;
Annette
Markfort
2595-2606
Abstract: For a locally compact group $G$, let $ {\gamma _G}$ denote the conjugation representation of $G$ in ${L^2}(G)$. In this paper we are concerned with nuclearity of ${C^*}$-algebras associated to ${\gamma _G}$ and the question of when these are of bounded representation type.
Differential identities
Bernard
Beauzamy;
Jérôme
Dégot
2607-2619
Abstract: We deal here with homogeneous polynomials in many variables and their hypercube representation, introduced in [5]. Associated with this representation there is a norm (Bombieri's norm) and a scalar product. We investigate differential identities connected with this scalar product. As a corollary, we obtain Bombieri's inequality (originally proved in [4]), with significant improvements. The hypercube representation of a polynomial was elaborated in order to meet the requests of massively parallel computation on the "Connection Machine" at Etablissement Technique Central de l'Armement; we see here once again (after [3] and [5]) the theoretical power of the model.
$\Pi\sp 1\sb 1$ functions are almost internal
Boško
Živaljević
2621-2632
Abstract: In Analytic mappings on hyperfinite sets [Proc. Amer. Math. Soc. 2 (1993), 587-596] Henson and Ross asked for what hyperfinite sets $S$ and $T$ does there exists a bijection $f$ from $S$ onto $T$ whose graph is a projective subset of $S \times T$? In particular, when is there a $ \Pi _1^1$ bijection from $ S$ onto $T$? In this paper we prove that given an internal, bounded measure $\mu$, any $\Pi _1^1$ function is $L(\mu )$ a.e. equal to an internal function, where $ L(\mu )$ is the Loeb measure associated with $\mu$. It follows that if two $\Pi _1^1$ subsets $S$ and $T$ of a hyperfinite set $X$ are $\Pi _1^1$ bijective, then $S$ and $T$ have the same measure for every uniformly distributed counting measure $\mu$. When $S$ and $T$ are internal it turns out that any $\Pi _1^1$ bijection between them must already be Borel. We also prove that if a $ \Pi _1^1$ graph in the product of two hyperfinite sets $X$ and $Y$ is universal for all internal subsets of $ Y$, then $\vert X\vert \geqslant {2^{\vert Y\vert}}$, which is a partial answer to Henson and Ross's Problem 1.5. At the end we prove some standard results about the projections and a structure of co-proper $K$-analytic subsets of the product of two completely regular Hausdorff topological spaces with open vertical sections. We were able to prove the above results by revealing the structure of $ \Pi _1^1$ subsets of the products $X \times Y$ of two internal sets $ X$ and $Y$, all of whose $Y$-sections are $\Sigma _1^0(\kappa )$ sets.
On a singular quasilinear anisotropic elliptic boundary value problem
Y. S.
Choi;
A. C.
Lazer;
P. J.
McKenna
2633-2641
Abstract: We consider the problem $\displaystyle {u^a}{u_{xx}} + {u^b}{u_{yy}} + p({\mathbf{x}}) = 0$ with $a \geqslant 0$, $b \geqslant 0$, on a smooth convex bounded region in ${{\mathbf{R}}^2}$ with Dirichlet boundary conditions. We show that if the positive function $ p$ is uniformly bounded away from zero, then the problem has a classical solution.
Affine transformations and analytic capacities
Thomas
Dowling;
Anthony G.
O’Farrell
2643-2655
Abstract: Analytic capacities are set functions defined on the plane which may be used in the study of removable singularities, boundary smoothness and approximation of analytic functions belonging to some function space. The symmetric concrete Banach spaces form a class of function spaces that includes most spaces usually studied. The Beurling transform is a certain singular integral operator that has proved useful in analytic function theory. It is shown that the analytic capacity associated to each Beurling-invariant symmetric concrete Banach space behaves reasonably under affine transformation of the plane. It is not known how general analytic capacities behave under affine maps.
Involutory Hopf algebras
D. S.
Passman;
Declan
Quinn
2657-2668
Abstract: In 1975, Kaplansky conjectured that a finite-dimensional semisimple Hopf algebra is necessarily involutory. Twelve years later, Larson and Radford proved the conjecture in characterisitic 0 and obtained significant partial results in positive characteristics. The goal of this paper is to offer an efficient proof of these results using rather minimal prerequisites, no "harpoons", and gratifyingly few "hits".
Variational formulas on Lipschitz domains
Alan R.
Elcrat;
Kenneth G.
Miller
2669-2678
Abstract: A rigorous treatment is given of variational formulas for solutions of certain Dirichlet problems for the Laplace operator on Lipschitz domains under interior variations. In particular we extend well-known variational formulas for the torsional rigidity and for capacity from the class of $ {C^1}$ domains to the class of Lipschitz domains. A motivation for this work comes from the use of variational methods in the study of Prandtl-Batchelor flows in fluid mechanics.
Compact composition operators on the Bloch space
Kevin
Madigan;
Alec
Matheson
2679-2687
Abstract: Necessary and sufficient conditions are given for a composition operator $ {C_\phi }f = f{\text{o}}\phi$ to be compact on the Bloch space $\mathcal{B}$ and on the little Bloch space ${\mathcal{B}_0}$. Weakly compact composition operators on $ {\mathcal{B}_0}$ are shown to be compact. If $\phi \in {\mathcal{B}_0}$ is a conformal mapping of the unit disk $ \mathbb{D}$ into itself whose image $ \phi (\mathbb{D})$ approaches the unit circle $ \mathbb{T}$ only in a finite number of nontangential cusps, then ${C_\phi }$ is compact on ${\mathcal{B}_0}$. On the other hand if there is a point of $ \mathbb{T} \cap \phi (\mathbb{D})$ at which $ \phi (\mathbb{D})$ does not have a cusp, then ${C_\phi }$ is not compact.
The structure of Mackey functors
Jacques
Thévenaz;
Peter
Webb
1865-1961
Abstract: Mackey functors are a framework having the common properties of many natural constructions for finite groups, such as group cohomology, representation rings, the Burnside ring, the topological $ {\text{K}}$-theory of classifying spaces, the algebraic ${\text{K}}$-theory of group rings, the Witt rings of Galois extensions, etc. In this work we first show that the Mackey functors for a group may be identified with the modules for a certain algebra, called the Mackey algebra. The study of Mackey functors is thus the same thing as the study of the representation theory of this algebra. We develop the properties of Mackey functors in the spirit of representation theory, and it emerges that there are great similarities with the representation theory of finite groups. In previous work we had classified the simple Mackey functors and demonstrated semisimplicity in characteristic zero. Here we consider the projective Mackey functors (in arbitrary characteristic), describing many of their features. We show, for example, that the Cartan matrix of the Mackey algebra may be computed from a decomposition matrix in the same way as for group representations. We determine the vertices, sources and Green correspondents of the projective and simple Mackey functors, as well as providing a way to compute the Ext groups for the simple Mackey functors. We parametrize the blocks of Mackey functors and determine the groups for which the Mackey algebra has finite representation type. It turns out that these Mackey algebras are direct sums of simple algebras and Brauer tree algebras. Throughout this theory there is a close connection between the properties of the Mackey functors, and the representations of the group on which they are defined, and of its subgroups. The relationships between these representations are exactly the information encoded by Mackey functors. This observation suggests the use of Mackey functors in a new way, as tools in group representation theory.
Complete quotient Boolean algebras
Akihiro
Kanamori;
Saharon
Shelah
1963-1979
Abstract: For $I$ a proper, countably complete ideal on the power set $ \mathcal{P}(x)$ for some set $X$, can the quotient Boolean algebra $\mathcal{P}(X)/I$ be complete? We first show that, if the cardinality of $X$ is at least $ {\omega _3}$, then having completeness implies the existence of an inner model with a measurable cardinal. A well-known situation that entails completeness is when the ideal $I$ is a (nontrivial) ideal over a cardinal $\kappa$ which is $ {\kappa ^ + }$-saturated. The second author had established the sharp result that it is consistent by forcing to have such an ideal over $ \kappa = {\omega _1}$ relative to the existence of a Woodin cardinal. Augmenting his proof by interlacing forcings that adjoin Boolean suprema, we establish, relative to the same large cardinal hypothesis, the consistency of: ${2^{{\omega _1}}} = {\omega _3}$ and there is an ideal ideal $I$ over $ {\omega _1}$ such that $ \mathcal{P}({\omega _1})/I$ is complete. (The cardinality assertion implies that there is no ideal over ${\omega _1}$ which is $ {\omega _2}$-saturated, and so completeness of the Boolean algebra and saturation of the ideal has been separated.)
The spectrum of the Hodge Laplacian for a degenerating family of hyperbolic three manifolds
Jozef
Dodziuk;
Jeffrey
McGowan
1981-1995
Abstract: We consider a sequence $ ({M_n})_{n = 1}^\infty$ of compact hyperbolic manifolds converging to a complete hyperbolic manifold ${M_0}$ with cusps. The Laplace operator acting on the space of ${L^2}$ differential forms on ${M_0}$ has continuous spectrum filling the half-line $[0,\infty )$. One expects therefore that the spectra of this operator on ${M_n}$ accumulate to produce the continuous spectrum of the limiting manifold. We prove that this is the case and obtain a sharp estimate of the rate of accumulation.
The direct decompositions of a group $G$ with $G/G'$ finitely generated
Francis
Oger
1997-2010
Abstract: We consider the class $\mathcal{C}$ which consists of the groups $ M$ with $M/M'$ finitely generated which satisfy the maximal condition on direct factors. It is well known that any $ \mathcal{C}$-group has a decomposition in finite direct product of indecomposable groups, and that two such decompositions are not necessarily equivalent up to isomorphism, even for a finitely generated nilpotent group. Here, we show that any $ \mathcal{C}$-group has only finitely many nonequivalent decompositions. In order to prove this result, we introduce, for $\mathcal{C}$-groups, a slightly different notion of decomposition, that we call $J$-decomposition; we show that this decomposition is necessarily unique. We also obtain, as consequences of the properties of $J$-decompositions, several generalizations of results of R. Hirshon. For instance, we have $ \mathbb{Z} \times G \cong \mathbb{Z} \times H$ for any groups $G$, $H$ which satisfy $M \times G \cong M \times H$ for a $\mathcal{C}$-group $M$.
Interpretation of Lavrentiev phenomenon by relaxation: the higher order case
Marino
Belloni
2011-2023
Abstract: We consider integral functionals of the calculus of variations of the form $u \in {W^{n,\infty }}(0,1)$, and we show that the relaxed functional $F$ with respect to the weak $ W_{{\text{loc}}}^{n,1}(0,1)$ convergence can be written as
Smooth sets for a Borel equivalence relation
Carlos E.
Uzcátegui A.
2025-2039
Abstract: We study some properties of smooth Borel sets with respect to a Borel equivalence relation, showing some analogies with the collection of countable sets from a descriptive set theoretic point of view. We found what can be seen as an analog of the hyperarithmetic points in the context of smooth sets. We generalize a theorem of Weiss from ${\mathbf{Z}}$-actions to actions by arbitrary countable groups. We show that the $\sigma$-ideal of closed smooth sets is $ \Pi _1^1$ non-Borel.
Singular Jacobi forms
Jae-Hyun
Yang
2041-2049
Abstract: We introduce the differential operator ${M_{g,h,\mathcal{M}}}$ characterizing singular Jacobi forms. We also characterize singular Jacobi forms by the weight of the associated rational representation of the general linear group. And we provide eigenfunctions of the differential operator $ {M_{g,h,\mathcal{M}}}$.
Generalized $(t,s)$-sequences, Kronecker-type sequences, and Diophantine approximations of formal Laurent series
Gerhard
Larcher;
Harald
Niederreiter
2051-2073
Abstract: The theory of $ (t,s)$-sequences leads to powerful constructions of low-discrepancy sequences in an $s$-dimensional unit cube. We generalize this theory in order to cover arbitrary sequences constructed by the digital method and, in particular, the Kronecker-type sequences introduced by the second author. We define diophantine approximation constants for formal Laurent series over finite fields and show their connection with the distribution properties of Kronecker-type sequences. The main results include probabilistic theorems on the distribution of sequences constructed by the digital method and on the diophantine approximation character of $s$-tuples of formal Laurent series over finite fields.
$L\sp p$ theory of differential forms on manifolds
Chad
Scott
2075-2096
Abstract: In this paper, we establish a Hodge-type decomposition for the $ {L^p}$ space of differential forms on closed (i.e., compact, oriented, smooth) Riemannian manifolds. Critical to the proof of this result is establishing an ${L^p}$ estimate which contains, as a special case, the ${L^2}$ result referred to by Morrey as Gaffney's inequality. This inequality helps us show the equivalence of the usual definition of Sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds. We also prove the $ {L^p}$ boundedness of Green's operator which we use in developing the $ {L^p}$ theory of the Hodge decomposition. For the calculus of variations, we rigorously verify that the spaces of exact and coexact forms are closed in the ${L^p}$ norm. For nonlinear analysis, we demonstrate the existence and uniqueness of a solution to the $A$-harmonic equation.
The connection matrix in Morse-Smale flows. II
James F.
Reineck
2097-2110
Abstract: Given a connection matrix for a Morse-Smale flow on a compact manifold, if there are no periodic orbits of equal or adjacent indices related in the partial order, we show that the periodic orbits can be replaced by doubly connected rest points in such a way that the given connection matrix induces the unique connection matrix for the resulting flow. It follows that for this class of flows, all nonuniqueness in the connection matrix is a consequence of the continuation theorem for connection matrices.
A constructive proof of the Poincar\'e-Birkhoff theorem
Yong
Li;
Zheng Hua
Lin
2111-2126
Abstract: In this paper, with the use of the homotopy method, a constructive proof of the Poincaré-Birkhoff theorem is given. This approach provides a global method for finding fixed points of area-preserving maps and periodic solutions of Duffing equations.
Geometry of strictly convex domains and an application to the uniform estimate of the $\overline\partial$-problem
Ten Ging
Chen
2127-2137
Abstract: In this paper, we construct a nice defining function $\rho$ for a bounded smooth strictly convex domain $\Omega$ in ${R^n}$ with explicit gradient and Hessian estimates near the boundary $ \partial \Omega$ of $ \Omega$. From the approach, we deduce that any two normals through $\partial \Omega$ do not intersect in any tubular neighborhood of $ \partial \Omega$ with radius which is less than $ \frac{1} {K}$, where $ K$ is the maximum principal curvature of $ \partial \Omega$. Finally, we apply such $\rho$ to obtain an explicit upper bound of the constant $ {C_\Omega }$ in the Henkin's estimate $ {\left\Vert {{H_\Omega }f} \right\Vert _{{L^\infty }(\Omega )}} \leqslant {C_\Omega }{\left\Vert f \right\Vert _{{L^\infty }(\Omega )}}$ of the $\partial$-problem on strictly convex domains $ \Omega$ in ${{\mathbf{C}}^n}$.
The regionally proximal relation
Joseph
Auslander;
David B.
Ellis;
Robert
Ellis
2139-2146
Abstract: Sufficient conditions for the regionally proximal relation $ Q(X)$ of a minimal flow to be an equivalence relation are obtained in terms of the group $ \mathcal{G}(X)$ of the flow and various groups which depend only on the acting group $T$.
Algebraic field theory on curved manifolds
Martin
Olesen
2147-2160
Abstract: In this paper we set up an algebraic framework for the study of quantum field theory in a class of manifolds, which includes Minkowski space and the Kruskal spacetime. The formalism provides a unifying framework for studying problems of Bisognano-Wichmann type, e.g., Hawking radiation in black hole geometries. Analogously to flat spacetime, we establish a correspondence between isometries of certain wedge domains of spacetime and the modular structure of the local algebras. Under an ergodic hypothesis, the wedge algebras are shown to be type III factors as expected, and we derive a result concerning factorization of the equilibrium state. This result generalizes a similar one obtained by Sewell in [Ann. Phys. 141 (1982), 201-224]. Finally an example of a quantum field theory satisfying the basic axioms is constructed. The local algebras are field algebras of bosonic free field solutions to the Klein-Gordon equation twisted through a PCT-like conjugation, and we show that this model realizes the abstract properties developed on the axiomatic basis.
Some inequalities of algebraic polynomials with nonnegative coefficients
Weiyu
Chen
2161-2167
Abstract: Let ${S_n}$ be the collection of all algebraic polynomials of degree $ \leqslant n$ with nonnegative coefficients. In this paper we discuss the extremal problem $\omega (x)$ is a positive and integrable function. This problem is solved completely in the cases $\displaystyle ({\text{i}})[a,b] = [ - 1,1],\omega (x) = {(1 - {x^2})^\alpha },\alpha > - 1;$ $\displaystyle ({\text{ii}})[a,b) = [0,\infty ),\omega (x) = {x^\alpha }{e^{ - x}},\alpha > - 1;$ $\displaystyle ({\text{iii}})(a,b) = ( - \infty ,\infty ),\omega (x) = {e^{ - \alpha {x^2}}},\alpha > 0.$ The second case was solved by Varma for some values of $\alpha$ and by Milovanović completely. We provide a new proof here in this case.
Bounded point evaluation in ${\bf C}\sp n$
R. M.
Range;
M. I.
Stessin
2169-2177
Abstract: A positive Borel measure $\mu$ on a domain $\Omega \in {{\mathbf{C}}^n}$ is said to be in $ \mathcal{R}(\Omega )$, if point evaluations at every $p \in \Omega$ are locally uniformly bounded in $ {L^2}(\mu )$-norm. It is proved that the multiplication of a measure in $ \mathcal{R}(\Omega )$ by a function decreasing no faster than a power of a holomorphic function produces a measure in $\mathcal{R}(\Omega )$. Some applications to classical Hardy and Bergman spaces are given.
Dualit\'e dans le groupe de Grothendieck de la cat\'egorie des repr\'esentations lisses de longueur finie d'un groupe r\'eductif $p$-adique
Anne-Marie
Aubert
2179-2189
Abstract: We define an involution on the Grothendieck ring of the category of finite length smooth representations of a $ p$-adic algebraic group, which is a direct analogue Curtis-Alvis duality for finite groups of Lie type. This involution commutes with taking the contragredient, with parabolic induction and, up a few twists, with truncation. It also preserves the irreducible representations up to sign.
Simultaneous triangularizability, near commutativity and Rota's theorem
A. A.
Jafarian;
H.
Radjavi;
P.
Rosenthal;
A. R.
Sourour
2191-2199
Abstract: In this paper we consider simultaneously triangularizable collections of compact operators and show that similarities of any finite subcollection can be made arbitrarily close to commuting normal operators. As a consequence, we obtain a variant of a theorem of G.-C. Rota. Also, we give some sufficient conditions for simultaneous triangularization of collections of compact operators. Finally, several counterexamples are given.
On the Littlewood-Paley-Stein $g$-function
Stefano
Meda
2201-2212
Abstract: We consider semigroups $({T_t})$, which are contractive on $ {L^p}(M)$ for all $q \in [1,2)$. We give an example (on symmetric spaces of the noncompact type) which shows that the Littlewood-Paley-Stein $ g$-function associated to the infinitesimal generator of $({T_t})$ may be unbounded on ${L^q}(M)$ and on
An access theorem for analytic functions
Marvin
Ortel
2213-2223
Abstract: Suppose that $\mathcal{M}$ is an analytic manifold, ${m_0} \in \mathcal{M},f:\mathcal{M} \to \mathbb{R}$, and $f$ is analytic. Then at least one of the following three statements is true: (1) ${m_0}$ is a local maximum of $f$. (2) There is a continuous path $ \sigma :[0,1] \to \mathcal{M}$ such that $ \sigma (0) = {m_0}$, $f \circ \sigma$ is strictly increasing on $ [0,1]$, and $\sigma (1)$ is a local maximum of $f$. (3) There is a continuous path $ \sigma :[0,1) \to \mathcal{M}$ with these properties: $\sigma (0) = {m_0};f \circ \sigma$ is strictly increasing on $[0,1)$; whenever $K$ is a compact subset of $\mathcal{M}$, there is a corresponding number $d(K) \in [0,1)$ such that $\sigma (t) \notin K$ for all $t \in [d(K),1)$.
A covering cocycle which does not grow linearly
Kathleen M.
Madden
2225-2234
Abstract: A cocycle $h:X \times {Z^m} \to {R^n}$ of a $ {Z^m}$ action on a compact metric space, provides an ${R^n}$ suspension flow (analogous to a flow under a function) on a space ${X_h}$ which may not be Hausdorff or even $ {T_1}$. Linear growth of $ h$ guarantees that $ {X_h}$ is a Hausdorff space; when $m = n$, linear growth is a consequence of $ {X_h}$ being Hausdorff and a covering condition. This paper contains the construction of a cocycle $h:X \times Z \to {R^2}$ which does not grow linearly yet produces a locally compact Hausdorff space with the covering condition. The $Z$ action used in the construction is a substitution minimal set.
Distinct degree factorizations for polynomials over a finite field
Arnold
Knopfmacher;
Richard
Warlimont
2235-2243
Abstract: Let $ {\widetilde{\mathbb{F}}_q}[X]$ denote the multiplicative semigroup of monic polynomials in one indeterminate $X$, over a finite field ${\mathbb{F}_q}$. We determine for each fixed $ q$ and fixed $ n$ the probability that a polynomial of degree $n$ in $ {\mathbb{F}_q}[X]$ has irreducible factors of distinct degrees only. These results are of relevance to various polynomial factorization algorithms.
Conditions on the logarithmic derivative of a function implying boundedness
T. H.
MacGregor;
F.
Rønning
2245-2254
Abstract: In this paper we investigate functions analytic and nonvanishing in the unit disk, with the property that the logarithmic derivative is contained in some domain $\Omega$. We obtain conditions on $ \Omega$ which imply that the functions are bounded and that their first derivatives belong to ${H^p}$ for some $p \geqslant 1$. For certain domains $\Omega$ the sufficient conditions that we give are also, in some sense, necessary. Examples of domains to which the results apply are given.
Existence of extremal metrics on compact almost homogeneous K\"ahler manifolds with two ends
Daniel
Guan
2255-2262
Abstract: In this note we prove the existence and the uniqueness of extremal metrics in every Kähler class of any compact almost homogeneous Kähler manifold with two ends by considering the scalar curvature equations, those manifolds might not be projective. We also prove that there are extremal metrics in some Kähler classes of a completion of the multicanonical line bundle of a Kähler-Einstein manifold of positive Ricci curvature.
The Connes spectrum of group actions and group gradings for certain quotient rings
James
Osterburg;
Xue
Yao
2263-2275
Abstract: Let $H$ be a finite-dimensional, semisimple Hopf algebra over an algebraically closed field $ K$ where $H$ is either commutative or cocommutative. We let $A$ be an $H$-module algebra which is semiprime right Goldie. We show that the Connes spectrum of $H$ acting on $A$ is the Connes spectrum of $H$ acting on the classical quotient ring of $ A$. In our last section, we define a symmetric quotient ring and show that the Connes spectrum of the ring and its quotient ring are the same. Finally, we apply our results to finite group actions and group gradings.
Local subgroups of the Monster and odd code loops
Thomas M.
Richardson
1453-1531
Abstract: The main result of this work is an explicit construction of $ p$-local subgroups of the Monster, the largest sporadic simple group. The groups constructed are the normalizers in the Monster of certain subgroups of order $3^{2}$, $5^{2}$, and $7^{2}$ and have shapes $\displaystyle {3^{2 + 5 + 10}}\cdot ({M_{11}} \times GL(2,3)),\quad {5^{2 + 2 + 4}}\cdot {S_3} \times GL(2,5)),\quad {\text{and}}{7^{2 + 1 + 2}}\cdot GL(2,7)$ . These groups result from a general construction which proceeds in three steps. We start with a self-orthogonal code $C$ of length $n$ over the field $ {\mathbb{F}_p}$, where $ p$ is an odd prime. The first step is to define a code loop $L$ whose structure is based on $ C$. The second step is to define a group $N$ of permutations of functions from $\mathbb{F}_p^2$ to $L$. The final step is to show that $N$ has a normal subgroup $ K$ of order ${p^2}$. The result of this construction is the quotient group $N/K$ of shape ${p^{2 + m + 2m}}(S \times GL(2,p))$, where $m + 1 = \dim (C)$ and $S$ is the group of permutations of Aut$(C)$. To show that the groups we construct are contained in the Monster, we make use of certain lattices $\Lambda (C)$, defined in terms of the code $ C$. One step in demonstrating this is to show that the centralizer of an element of order $p$ in $N/K$ is contained in the centralizer of an element of order $p$ in the Monster. The lattices are useful in this regard since a quotient of the automorphism group of the lattice is a composition factor of the appropriate centralizer in the Monster. This work was inspired by a similar construction using code loops based on binary codes that John Conway used to construct a subgroup of the Monster of shape $ {2^{2 + 11 + 22}}\cdot ({M_{24}} \times GL(2,2))$.
Geometrical evolution of developed interfaces
Piero
de Mottoni;
Michelle
Schatzman
1533-1589
Abstract: Consider the reaction-diffusion equation in ${\mathbb{R}^N} \times {\mathbb{R}^ + }:{u_t} - {h^2}\Delta u + \varphi (u) = 0;\varphi$ is the derivative of a bistable potential with wells of equal depth and $h$ is a small parameter. If the initial data has an interface, we give an asymptotic expansion of arbitrarily high order and error estimates valid up to time $O({h^{ - 2}})$. At lowest order, the interface evolves normally, with a velocity proportional to the mean curvature. Soit l'équation de réaction-diffusion dans ${\mathbb{R}^N} \times {\mathbb{R}^ + },\quad {u_t} - {h^2}\Delta u + \varphi (u) = 0$, avec $ \varphi$ la dérivée d'un potentiel bistable à puits également profonds et $h$ un petit paramètre. Pour une condition initiale possédant une interface, on donne un développement asymptotique d'ordre arbitrairement élevé, ainsi que des estimations d'erreur valides jusqu'à un temps en $O({h^{ - 2}})$. A l'ordre le plus bas, l'interface évolue normalement, à une vitesse proportionnelle à la courbure moyenne.
Schur's partition theorem, companions, refinements and generalizations
Krishnaswami
Alladi;
Basil
Gordon
1591-1608
Abstract: Schur's partition theorem asserts the equality $S(n) = {S_1}(n)$, where $S(n)$ is the number of partitions of $n$ into distinct parts $\equiv 1,2(\mod 3)$ and ${S_1}(n)$ is the number of partitions of $ n$ into parts with minimal difference $3$ and no consecutive multiples of $3$. Using a computer search Andrews found a companion result $ S(n) = {S_2}(n)$, where $ {S_2}(n)$ is the number of partitions of $n$ whose parts ${e_i}$ satisfy ${e_i} - {e_{i + 1}} \geqslant 3,2or5$ according as ${e_i} \equiv 1,2$ or $(\bmod 3)$. By means of a new technique called the method of weighted words, a combinatorial as well as a generating function proof of both these theorems are given simultaneously. It is shown that ${S_1}(n)$ and ${S_2}(n)$ are only two of six companion partition functions ${S_j}(n),j = 1,2, \ldots 6$, all equal to $ S(n)$. A three parameter refinement and generalization of these results is obtained.
On the decomposition of Langlands subrepresentations for a group in the Harish-Chandra class
Eugenio
Garnica-Vigil
1609-1648
Abstract: When a group $ G$ is in the Harish-Chandra class, the goal of classifying its tempered representations and the goal of decomposing the Langlands subrepresentation for any of its standard representations are equivalent. The main result of this work is given in Theorem (5.3.5) that consists of a formula for decomposing any Langlands subrepresentation for the group $G$. The classification of tempered representations is a consequence of this theorem (Corollary (5.3.6)).
On the period-two-property of the majority operator in infinite graphs
Gadi
Moran
1649-1667
Abstract: A self-mapping $ M:X \to X$ of a nonempty set $X$ has the Period-Two-Property (p2p) if ${M^2}x = x$ holds for every $M$-periodic point $x \in X$. Let $X$ be the set of all $\{ 0,1\}$-labelings $x:V \to \{ 0,1\}$ of the set of vertices $ V$ of a locally finite connected graph $G$. For $x \in X$ let $Mx \in X$ label $v \in V$ by the majority bit that $x$ applies to its neighbors, retaining $ \upsilon$'s $ x$-label in case of a tie. We show that $M$ has the p2p if there is a finite bound on the degrees in $G$ and $\frac{1} {n}\log {b_n} \to 0$, where ${b_n}$ is the number of $\upsilon \in V$ at a distance at most $ n$ from a fixed vertex ${\upsilon _0} \in V$.
Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions
Konstantin
Mischaikow;
Hal
Smith;
Horst R.
Thieme
1669-1685
Abstract: From the work of C. Conley, it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set. Here, we improve a result of L. Markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow. In the special case that there is a Lyapunov function for the limiting semiflow, sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the Lyapunov function.
Singular limit of solutions of $u\sb t=\Delta u\sp m-A\cdot \nabla (u\sp q/q)$ as $q\to\infty$
Kin Ming
Hui
1687-1712
Abstract: We will show that the solutions of ${u_t} = \Delta {u^m} - A\nabla ({u^q}/q)$ in $ {R^n} \times (0,T),T > 0,m > 1,u(x,0) = f(x) \in {L^1}({R^n}) \cap {L^\infty }({R^n})$ converge weakly in ${({L^\infty }(G))^ * }$ for any compact subset $ G$ of ${R^n} \times (0,T)$ as $ q \to \infty$ to the solution of the porous medium equation ${\upsilon _t} = \Delta {\upsilon ^m}$ in ${R^n} \times (0,T)$ with $\upsilon (x,0) = g(x)$ where $ g \in {L^1}({R^n}),0 \leqslant g \leqslant 1$, satisfies $\tilde{g}(x) \in {L^1}({R^n}),\quad \tilde{g}(x) \geqslant 0$ such that $g(x) = f(x),\quad \tilde{g}(x) = 0$ whenever $ g(x) < 1$ a.e. $x \in {R^n}$. The convergence is uniform on compact subsets of $ {R^n} \times (0,T)\quad {\text{if}}\quad f \in {C_0}({R^n})$.
Inequalities for zero-balanced hypergeometric functions
G. D.
Anderson;
R. W.
Barnard;
K. C.
Richards;
M. K.
Vamanamurthy;
M.
Vuorinen
1713-1723
Abstract: The authors study certain monotoneity and convexity properties of the Gaussian hypergeometric function and those of the Euler gamma function.
Some recurrence formulas for spherical polynomials on tube domains
Gen Kai
Zhang
1725-1734
Abstract: For a tube domain $ G/K$ we study the tensor products of two spherical representations of the maximal compact group $K$ and the product of the corresponding spherical polynomials. When one of these is a fundamental representation, we prove that the spherical representations appear with multiplicity at most one and we then find all the coefficients in the recurrence formula for the product of the spherical polynomials. This generalizes the previous result of L. Vretare and proves for certain cases a conjecture of R. Stanley on Jack symmetric polynomials.
Spectral and Fredholm properties of operators in elementary nest algebras
Bruce A.
Barnes;
Jon M.
Clauss
1735-1741
Abstract: Some spectral and Fredholm properties are proved for linear operators which leave invariant certain nests of closed subspaces.
On closed minimal submanifolds in pinched Riemannian manifolds
Hong Wei
Xu
1743-1751
Abstract: In this paper, we first prove a generalized Simons integral inequality for closed minimal submanifolds in a Riemannian manifold. Second, we prove a pinching theorem for closed minimal submanifolds in a complete simply connected pinched Riemannian manifold, which generalizes the results obtained by S. S. Chern, M. do Carmo, and S. Kobayashi and A. M. Li and J. M. Li respectively. Finally, we obtain a distribution theorem for the square norm of the second fundamental form of $M$ under the assumption that $M$ is a minimal submanifold with parallel second fundamental form in a Riemannian manifold.
On the $L\sp 2$ inequalities involving trigonometric polynomials and their derivatives
Weiyu
Chen
1753-1761
Abstract: In this note we study the upper bound of the integral $\displaystyle \int_0^\pi {{{({t^{(k)}}(x))}^2}w(x)} dx$ where $t(x)$ is a trigonometric polynomial with real coefficients such that $\left\Vert t \right\Vert\infty \leqslant 1$ and $w(x)$ is a nonnegative function defined on $ [0,\pi ]$. When $w(x) = \sin ^jx$, where $j$ is a positive integer, we obtain the exact upper bound for the above integral.
Hausdorff measure and level sets of typical continuous mappings in Euclidean spaces
Bernd
Kirchheim
1763-1777
Abstract: We determine the Hausdorff dimension of level sets and of sets of points of multiplicity for mappings in a residual subset of the space of all continuous mappings from ${\mathbb{R}^n}$ to $ {\mathbb{R}^m}$.
Groupoids associated with endomorphisms
Valentin
Deaconu
1779-1786
Abstract: To a compact Hausdorff space which covers itself, we associate an $ r$-discrete locally compact Hausdorff groupoid. Its ${{\mathbf{C}}^ * }$-algebra carries an action of the circle allowing it to be regarded as a crossed product by an endomorphism and as a generalization of the Cuntz algebra ${O_p}$. We consider examples related to coverings of the circle and of a Heisenberg $ 3$-manifold.
Holomorphic martingales and interpolation between Hardy spaces: the complex method
P. F. X.
Müller
1787-1792
Abstract: A probabilistic proof is given to identify the complex interpolation space of $ {H^1}(\mathbb{T})$ and $ {H^\infty }(\mathbb{T})$ as $ {H^p}(\mathbb{T})$.
Hypersurfaces in space forms satisfying the condition $\Delta x=Ax+B$
Luis J.
Alías;
Angel
Ferrández;
Pascual
Lucas
1793-1801
Abstract: In this work we study and classify pseudo-Riemannian hypersurfaces in pseudo-Riemannian space forms which satisfy the condition $ \Delta x = Ax + B$, where $ A$ is an endomorphism, $ B$ is a constant vector, and $x$ stands for the isometric immersion. We prove that the family of such hypersurfaces consists of open pieces of minimal hypersurfaces, totally umbilical hypersurfaces, products of two nonflat totally umbilical submanifolds, and a special class of quadratic hypersurfaces.
Examples of $B(D,\lambda)$-refinable and weak $\overline\theta$-refinable spaces
Stephen H.
Fast;
J. C.
Smith
1803-1809
Abstract: In 1980, J. C. Smith asked for examples which would demonstrate the relationships between the properties $B(D,\lambda )$-refinability, $B(D,{\omega _0})$-refinability, and weak $\bar \theta $-refinability. This paper gives such examples in the class of ${T_4}$ spaces.
A measure theoretical subsequence characterization of statistical convergence
Harry I.
Miller
1811-1819
Abstract: The concept of statistical convergence of a sequence was first introduced by H. Fast. Statistical convergence was generalized by R. C. Buck, and studied by other authors, using a regular nonnegative summability matrix $ A$ in place of $ {C_1}$. The main result in this paper is a theorem that gives meaning to the statement: $S = \{ {s_n}\} $ converges to $ L$ statistically $ (T)$ if and only if "most" of the subsequences of $S$ converge, in the ordinary sense, to $ L$. Here $T$ is a regular, nonnegative and triangular matrix. Corresponding results for lacunary statistical convergence, recently defined and studied by J. A. Fridy and C. Orhan, are also presented.
Standard Lyndon bases of Lie algebras and enveloping algebras
Pierre
Lalonde;
Arun
Ram
1821-1830
Abstract: It is well known that the standard bracketings of Lyndon words in an alphabet $A$ form a basis for the free Lie algebra ${\text{Lie}}(A)$ generated by $A$. Suppose that $\mathfrak{g} \cong {\text{Lie}}(A)/J$ is a Lie algebra given by a generating set $ A$ and a Lie ideal $ J$ of relations. Using a Gröbner basis type approach we define a set of "standard" Lyndon words, a subset of the set Lyndon words, such that the standard bracketings of these words form a basis of the Lie algebra $\mathfrak{g}$. We show that a similar approach to the universal enveloping algebra $\mathfrak{g}$ naturally leads to a Poincaré-Birkhoff-Witt type basis of the enveloping algebra of $\mathfrak{g}$. We prove that the standard words satisfy the property that any factor of a standard word is again standard. Given root tables, this property is nearly sufficient to determine the standard Lyndon words for the complex finite-dimensional simple Lie algebras. We give an inductive procedure for computing the standard Lyndon words and give a complete list of the standard Lyndon words for the complex finite-dimensional simple Lie algebras. These results were announced in [LR].
On the oscillation of differential equations with an oscillatory coefficient
B. J.
Harris;
Q.
Kong
1831-1839
Abstract: We derive lower bounds for the distance between consecutive zeros of solutions of
Tate cohomology of periodic $K$-theory with reality is trivial
Lisbeth
Fajstrup
1841-1846
Abstract: We calculate the $ RO(\mathbb{Z}/2)$-graded spectrum for Atiyah's periodic $K$-theory with reality and the Tate cohomology associated to it. The latter is shown to be trivial.
Uniformly ergodic multioperators
M.
Mbekhta;
F.-H.
Vasilescu
1847-1854
Abstract: A version of the uniform ergodic theorem valid for commuting multioperators is given.
Statistical inference based on the possibility and belief measures
Yuan Yan
Chen
1855-1863
Abstract: In statistical inference, we infer the population parameter based on the realization of sample statistics. This can be considered in the framework of inductive inference. We showed, in Chen (1993), that if we measure a parameter by the possibility (or belief) measure, we can have an inductive inference similar to the Bayesian inference in belief update. In this article we apply this inference to statistical estimation and hypotheses evaluation (testing) for some parametric models, and compare them to the classical statistical inferences for both one-sample and two-sample problems.
Stochastic control problems and spherical functions on symmetric spaces
T. E.
Duncan;
H.
Upmeier
1083-1130
Abstract: A family of explicitly solvable stochastic control problems is formulated and solved in noncompact symmetric spaces. The symmetric spaces include all of the classical spaces and four of the exceptional spaces. The stochastic control problems are the control of Brownian motion in these symmetric spaces by a drift vector field. For each symmetric space a family of stochastic control problems is formulated by using spherical functions in the cost functionals. These spherical functions are explicitly described and are polynomials in suitable coordinates. A generalization to abstract root systems is given.
On a quadratic-trigonometric functional equation and some applications
J. K.
Chung;
B. R.
Ebanks;
C. T.
Ng;
P. K.
Sahoo
1131-1161
Abstract: Our main goal is to determine the general solution of the functional equation \begin{displaymath}\begin{array}{*{20}{c}} {{f_1}(xy) + {f_2}(x{y^{ - 1}}) = {f_... ...,} {{f_i}(txy) = {f_i}(tyx)\qquad (i = 1,2)} \end{array} \end{displaymath} where ${f_i}$ are complex-valued functions defined on a group. This equation contains, among others, an equation of H. Swiatak whose general solution was not known until now and an equation studied by K.S. Lau in connection with a characterization of Rao's quadratic entropies. Special cases of this equation also include the Pexider, quadratic, d'Alembert and Wilson equations.
Norm estimates for radially symmetric solutions of semilinear elliptic equations
Ryuji
Kajikiya
1163-1199
Abstract: The semilinear elliptic equation $ \Delta u + f(u) = 0$ in $ {R^n}$ with the condition $ {\lim _{\vert x\vert \to \infty }}u(x) = 0$ is studied, where $n \geqslant 2$ and $f(u)$ has a superlinear and subcritical growth at $u = \pm \infty$. For example, the functions $f(u) = \vert u{\vert^{p - 1}}u - u\;(1 < p < \infty \;{\text{if}}\;n = 2,\;1 < p < (n + 2)/(n - 2)\;{\text{if}}\;n \geqslant 3)$ and $f(u) = u\log \vert u\vert$ are treated. The $ {L^2}$ and ${H^1}$ norm estimates $ {C_1}{(k + 1)^{n/2}} \leqslant \vert\vert u\vert{\vert _{{L^2}}} \leqslant \vert\vert u\vert{\vert _{{H^1}}} \leqslant {C_2}{(k + 1)^{n/2}}$ are established for any radially symmetric solution $u$ which has exactly $k \geqslant 0$ zeros in the interval $0 \leqslant \vert x\vert < \infty$. Here ${C_1},\;{C_2} > 0$ are independent of $ u$ and $k$.
Hopf constructions and higher projective planes for iterated loop spaces
Nicholas J.
Kuhn;
Michael
Slack;
Frank
Williams
1201-1238
Abstract: We define a category, $ \mathcal{H}_p^n$ (for each $ n$ and $p$), of spaces with strong homotopy commutativity properties. These spaces have just enough structure to define the $\bmod p$ Dyer-Lashof operations for $ n$-fold loop spaces. The category $ \mathcal{H}_p^n$ is very convenient for applications since its objects and morphisms are defined in a homotopy invariant way. We then define a functor, $P_p^n$, from $ \mathcal{H}_p^n$ to the homotopy category of spaces and show $P_p^n$ to be left adjoint to the $ n$-fold loop space functor. We then show how one can exploit this adjointness in cohomological calculations to yield new results about iterated loop spaces.
Attractors in inhomogeneous conservation laws and parabolic regularizations
Hai Tao
Fan;
Jack K.
Hale
1239-1254
Abstract: The asymptotic behavior of inhomogeneous conservation laws is considered. The attractor of the equation is characterized. The relationship between attractors of the equation and that of its parabolic regularization is studied.
Verifiable conditions for openness and regularity of multivalued mappings in Banach spaces
A.
Jourani;
L.
Thibault
1255-1268
Abstract: This paper establishes verifiable conditions in terms of approximate subdifferentials implying openness and metric regularity of multivalued mappings in Banach spaces. The results are then applied to derive Lagrange multipliers for general nonsmooth vector optimization problems.
Partial extensions of Attouch's theorem with applications to proto-derivatives of subgradient mappings
A. B.
Levy;
R.
Poliquin;
L.
Thibault
1269-1294
Abstract: Attouch's Theorem, which gives on a reflexive Banach space the equivalence between the Mosco epi-convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients, has many important applications in convex optimization. In particular, generalized derivatives have been defined in terms of the epi-convergence or graph convergence of certain difference quotient mappings, and Attouch's Theorem has been used to relate these various generalized derivatives. These relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems. We prove in a Hilbert space several "partial extensions" of Attouch's Theorem to functions more general than convex; these functions are called primal-lower-nice. Furthermore, we use our extensions to derive a relationship between the second-order epi-derivatives of primal-lower-nice functions and the proto-derivative of their associated subgradient mappings.
The derivatives of homotopy theory
Brenda
Johnson
1295-1321
Abstract: We construct a functor of spaces, ${M_n}$, and show that its multilinearization is equivalent to the $n$th layer of the Taylor tower of the identity functor of spaces. This allows us to identify the derivatives of the identity functor and determine their homotopy type.
When does unique local support ensure convexity?
Donald Francis
Young
1323-1329
Abstract: A basic theorem of convex analysis states that a real-valued function on an open interval of the real line is convex and differentiable if at each point of its domain there exists a unique supporting line. In this paper we show that the same conclusion can be drawn under the weaker hypothesis that there exists a unique locally supporting line at each point. We also show by counterexample that convexity cannot be concluded under analogous circumstances for $ f:S \to \mathbb{R}$, where $S \subset {\mathbb{R}^n}$ is open and convex, if $n > 1$.
Cauchy-Green type formulae in Clifford analysis
John
Ryan
1331-1341
Abstract: A Cauchy integral formula is constructed for solutions to the polynomial Dirac equation $({D^k} + \sum\nolimits_{m = 0}^{k - 1} {{b_m}{D^m})f = 0}$, where each ${b_m}$ is a complex number, $ D$ is the Dirac operator in ${R^n}$, and $f$ is defined on a domain in $^{{R^n}}$ and takes values in a complex Clifford algebra. Some basic properties for the solutions to this equation, arising from the integral formula, are described, including an approximation theorem. We also introduce a Bergman kernel for square integrable solutions to $ (D + \lambda )f = 0$ over bounded domains with piecewise ${C^1}$, or Lipschitz, boundary.
A convergence theorem for Riemannian submanifolds
Zhong Min
Shen
1343-1350
Abstract: In this paper we study the convergence of Riemannian submanifolds. In particular, we prove that any sequence of closed submanifolds with bounded normal curvature and volume in a closed Riemannian manifold subconverge to a closed submanifold in the $ {C^{1,\alpha }}$ topology. We also obtain some applications to irreducible homogeneous manifolds and pseudo-holomorphic curves in symplectic manifolds.
A discrete transform and Triebel-Lizorkin spaces on the bidisc
Wei
Wang
1351-1364
Abstract: We use a discrete transform to study the Triebel-Lizorkin spaces on bidisc $\dot F_p^{\alpha q},\,\dot f_p^{\alpha q}$ and establishes the boundedness of transform $ {S_\phi }:\dot F_p^{\alpha q} \to \dot f_p^{\alpha q}$ and $ {T_\psi }:\dot f_p^{\alpha q} \to \dot F_p^{\alpha q}$. We also define the almost diagonal operator and prove its boundedness. With the use of discrete transform and Journé lemma, we get the atomic decomposition of $\dot f_p^{\alpha q}$ for $0 < p \leqslant 1,\,p \leqslant q < \infty$. The atom supports on an open set, not a rectangle. Duality $ {(\dot f_1^{\alpha q})^{\ast}} = \dot f_\infty ^{ - \alpha q'},\,\tfrac{1} {q} + \tfrac{1} {{q'}} = 1,\,q > 1,\,\alpha \in R$, is established, too. The case for $\dot F_p^{\alpha q}$ is similar.
A norm convergence result on random products of relaxed projections in Hilbert space
H. H.
Bauschke
1365-1373
Abstract: Suppose $ X$ is a Hilbert space and ${C_1}, \ldots ,{C_N}$ are closed convex intersecting subsets with projections ${P_1}, \ldots ,{P_N}$. Suppose further $ r$ is a mapping from $\mathbb{N}$ onto $ \{ 1, \ldots ,N\}$ that assumes every value infinitely often. We prove (a more general version of) the following result: If the $ N$-tuple $({C_1}, \ldots ,{C_N})$ is "innately boundedly regular", then the sequence $({x_n})$, defined by $\displaystyle {x_0} \in X\;{\text{arbitrary,}}\quad {x_{n + 1}}: = {P_{r(n)}}{x_n},\quad {\text{for all}}\;n \geqslant 0,$ converges in norm to some point in $\cap _{i = 1}^N{C_i}$. Examples without the usual assumptions on compactness are given. Methods of this type have been used in areas like computerized tomography and signal processing.
Asymptotic stability in functional-differential equations by Liapunov functionals
Bo
Zhang
1375-1382
Abstract: We consider the asymptotic stability in a system of functional differential equations $ {I_n} = [{S_n},{t_n}]$. We also show that it is not necessary to require a uniform upper bound on $V$ for nonuniform asymptotic stability.
The de Branges-Rovnyak model with finite-dimensional coefficients
James
Guyker
1383-1389
Abstract: A characterization in terms of the canonical model spaces of L. de Branges and J. Rovnyak is obtained for Hilbert spaces of formal power series with vector coefficients which satisfy a difference-quotient inequality, thereby extending the closed ideal theorems of A. Beurling and P. D. Lax.
Remarks on some integrals and series involving the Stirling numbers and $\zeta(n)$
Li-Chien
Shen
1391-1399
Abstract: From the perspective of the well-known identity $\displaystyle {}_2{F_1}(a,b;c;1) = \frac{{\Gamma (c)\Gamma (c - a - b)}} {{\Gamma (c - a)\Gamma (c - b)}},$ we clarify the connections between the Stirling numbers $s_k^n$ and the Riemann zeta function $\zeta (n)$. As a consequence, certain series and integrals can be evaluated in terms of $ \zeta (n)$ and $ s_k^n$.
The exposed points of the set of invariant means
Tianxuan
Miao
1401-1408
Abstract: Let $G$ be a $\sigma$-compact infinite locally compact group, and let $LIM$ be the set of left invariant means on ${L^\infty }(G)$. We prove in this paper that if $ G$ is amenable as a discrete group, then $LIM$ has no exposed points. We also give another proof of the Granirer theorem that the set $ LIM(X,G)$ of $ G$-invariant means on ${L^\infty }(X,\beta ,p)$ has no exposed points, where $G$ is an amenable countable group acting ergodically as measure-preserving transformations on a nonatomic probability space $ (X,\beta ,p)$.
Inverse theorems for subset sums
Melvyn B.
Nathanson
1409-1418
Abstract: Let $A$ be a finite set of integers. For $h \geqslant 1$, let ${S_h}(A)$ denote the set of all sums of $ h$ distinct elements of $ A$. Let $S(A)$ denote the set of all nonempty sums of distinct elements of $A$. The direct problem for subset sums is to find lower bounds for $ \vert{S_h}(A)\vert$ and $ \vert S(A)\vert$ in terms of $\vert A\vert$. The inverse problem for subset sums is to determine the structure of the extremal sets $A$ of integers for which $ \vert{S_h}(A)\vert$ and $ \vert S(A)\vert$ are minimal. In this paper both the direct and the inverse problem for subset sums are solved.
Groups with no free subsemigroups
P.
Longobardi;
M.
Maj;
A. H.
Rhemtulla
1419-1427
Abstract: We look at groups which have no (nonabelian) free subsemigroups. It is known that a finitely generated solvable group with no free subsemigroup is nilpotent-by-finite. Conversely nilpotent-by-finite groups have no free subsemigroups. Torsion-free residually finite- $ p$ groups with no free subsemigroups can have very complicated structure, but with some extra condition on the subsemigroups of such a group one obtains satisfactory results. These results are applied to ordered groups, right-ordered groups, and locally indicable groups.
Matrix variate $\theta$-generalized normal distribution
A. K.
Gupta;
T.
Varga
1429-1437
Abstract: In this paper, the matrix variate $\theta$-generalized normal distribution is introduced. Then its properties are studied. In particular, it is proved that this distribution has maximal entropy in a certain class of distributions.
Differential operators, $n$-branch curve singularities and the $n$-subspace problem
R. C.
Cannings;
M. P.
Holland
1439-1451
Abstract: Let $R$ be the coordinate ring of a smooth affine curve over an algebraically closed field of characteristic zero $k$. For $S$ a subalgebra of $R$ with integral closure $R$ denote by $ \mathcal{D}(S)$ the ring of differential operators on $S$ and by $H(S)$ the finite-dimensional factor of $ \mathcal{D}(S)$ by its unique minimal ideal. The theory of diagonal $ n$-subspace systems is introduced. This is used to show that if $A$ is a finite-dimensional $ k$-algebra and $t \geqslant 1$ is any integer there exists such an $ S$ with $\displaystyle H(S) \cong \left( {\begin{array}{*{20}{c}} A & {\ast} 0 & {{M_t}(k)} \end{array} } \right).$ Further, the Morita classes of $H(S)$ are classified for curves with few branches, and it is shown how to lift Morita equivalences from $ H(S)$ to $\mathcal{D}(S)$.
Errata to: ``Analysis and applications of holomorphic functions in higher dimensions''
R. Z.
Yeh
Ribbons and their canonical embeddings
Dave
Bayer;
David
Eisenbud
719-756
Abstract: We study double structures on the projective line and on certain other varieties, with a view to having a nice family of degenerations of curves and K3 surfaces of given genus and Clifford index. Our main interest is in the canonical embeddings of these objects, with a view toward Green's Conjecture on the free resolutions of canonical curves. We give the canonical embeddings explicitly, and exhibit an approach to determining a minimal free resolution.
Clifford indices of ribbons
David
Eisenbud;
Mark
Green
757-765
Abstract: We present a theory of "limit linear series" for rational ribbons-- that is, for schemes that are double structures on $ {P^1}$. This allows us to define a "linear series Clifford index" for ribbons. Our main theorem shows that this is the same as the Clifford index of ribbons studied by Eisenbud-Bayer in this same volume. This allows us to prove that the Clifford index is semicontinuous in degenerations from a smooth curve to a ribbon. A result of Fong [1993] then shows that ribbons may be deformed to smooth curves of the same Clifford index. Thus the Canonical Curve Conjecture of Green [1984] would follow, at least for a general smooth curve of each Clifford index, from the corresponding statement for ribbons.
Geometry of dots and ropes
Karen A.
Chandler
767-784
Abstract: An $\alpha $-dot is the first infinitesimal neighbourhood of a point with respect to an $(\alpha - 1)$-dimensional affine space. We define a notion of uniform position for a collection of dots in projective space, which in particular holds for a collection of dots arising as a general plane section of a higher-dimensional scheme. We estimate the Hilbert function of such a collection of dots, with the result that Theorem 1. Let $\Gamma$ be a collection of $ d$ $\alpha $-dots in uniform position in ${\mathbb{P}^n},\alpha \geqslant 2$. Then the Hilbert function $ {h_\Gamma }$ of $ \Gamma$ satisfies $\displaystyle {h_\Gamma }(r) \geqslant \min (rn + 1,2d) + (\alpha - 2)\min ((r - 1)n - 1,\,d)$ for $r \geqslant 3$. Equality occurs for some $r$ with $ rn + 2 \leqslant 2d$ if and only if ${\Gamma _{{\text{red}}}}$ is contained in a rational normal curve $C$, and the tangent directions to this curve at these points are all contained in $ \Gamma$. Equality occurs for some $r$ with $ (r - 1)n \leqslant d$ if and only if $\Gamma$ is contained in the first infinitesimal neighbourhood of $C$ with respect to a subbundle, of rank $\alpha - 1$ and of maximal degree, of the normal bundle of $C$ in $ {\mathbb{P}^n}$. This implies an upper bound on the degree of a subbundle of rank $\alpha - 1$ of the normal bundle of an irreducible nondegenerate smooth curve of degree $ d$ in ${\mathbb{P}^n}$, by a Castelnuovo argument.
On the Ramsey property of families of graphs
N.
Sauer
785-833
Abstract: For graphs $ A$ and $B$ the relation $A \to (B)_r^1$ means that for every $r$-coloring of the vertices of $ A$ there is a monochromatic copy of $B$ in $A$. $\operatorname{Forb} ({G_1},{G_2}, \ldots ,{G_n})$ is the family of graphs which do not embed any one of the graphs ${G_1},{G_2}, \ldots ,{G_n}$, a family $\mathcal{F}$ of graphs has the Ramsey property if for every graph $ B \in \mathcal{F}$ there is a graph $ A \in \mathcal{F}$ such that $A \to (B)_r^1$. Nešetřil and Rödl (1976) have proven that if either both graphs $G$ and $K$ are two-connected or the complements of both graphs $G$ and $K$ are two-connected then $\operatorname{Forb} (G,K)$ has the Ramsey property. We prove that if $ \overline G$ is disconnected and $K$ is disconnected then $\operatorname{Forb} (G,K)$ does not have the Ramsey property, except for four pairs of graphs $ (G,K)$. A family $\mathcal{F}$ of finite graphs is an age if there is a countable graph $G$ whose set of finite induced subgraphs is $\mathcal{F}$. We characterize those pairs of graphs $(G,H)$ for which $\operatorname{Forb} (G,H)$ is not an age but has the Ramsey property.
On the dimension and the index of the solution set of nonlinear equations
P. S.
Milojević
835-856
Abstract: We study the covering dimension and the index of the solution set to multiparameter nonlinear and semilinear operator equations involving Fredholm maps of positive index. The classes of maps under consideration are (pseudo) $A$-proper and either approximation-essential or equivariant approximation-essential. Applications are given to semilinear elliptic BVP's.
On the general notion of fully nonlinear second-order elliptic equations
N. V.
Krylov
857-895
Abstract: The general notion of fully nonlinear second-order elliptic equation is given. Its relation to so-called Bellman equations is investigated. A general existence theorem for the equations like ${P_m}({u_{{x^i}{x^j}}}) = \sum\nolimits_{k = 0}^{m - 1} {{c_k}(x){P_k}({u_{{x^i}{x^j}}})}$ is obtained as an example of an application of the general notion of fully nonlinear elliptic equations.
Some new observations on the G\"ollnitz-Gordon and Rogers-Ramanujan identities
Krishnaswami
Alladi
897-914
Abstract: Two new, short and elementary proofs of the Göllnitz-Gordon identities are presented by considering the odd-even split of the Euler Pentagonal Series and the Triangular Series of Gauss. Using this approach the equality of certain shifted partition functions are established. Next, the odd and even parts of the famous Rogers-Ramanujan series are shown to have interesting product representations ($\bmod 80$). From this, new shifted partition identities are derived.
Infinitesimal bending and twisting in one-dimensional dynamics
Frederick P.
Gardiner
915-937
Abstract: An infinitesimal theory for bending and earthquaking in one-dimensional dynamics is developed. It is shown that any tangent vector to Teichmüller space is the initial data for a bending and for an earthquaking ordinary differential equation. The discussion involves an analysis of infinitesimal bendings and earthquakes, the Hilbert transform, natural bounded linear operators from a Banach space of measures on the Möbius strip to tangent vectors to Teichmüller space, and the construction of a nonlinear right inverse for these operators. The inverse is constructed by establishing an infinitesimal version of Thurston's earthquake theorem.
Invariants of locally conformally flat manifolds
Thomas
Branson;
Peter
Gilkey;
Juha
Pohjanpelto
939-953
Abstract: Let $M$ be a locally conformally flat manifold with metric $g$. Choose a local coordinate system on $ M$ so $g = {e^{2h}}x\,dx \circ dx$ where $dx \circ dx$ is the Euclidean standard metric. A polynomial $P$ in the derivatives of $h$ with coefficients depending smoothly on $ h$ is a local invariant for locally conformally flat structures if the expression $ P({h_X})$ is independent of the choice of $X$. Form valued local invariants are defined similarly. In this paper, we study the properties of the associated de Rham complex. We show that any invariant form can be obtained from the previously studied local invariants of Riemannian structures by restriction. We show the cohomology of the de Rham complex of local invariants is trivial. We also obtain the following characterization of the Euler class. Suppose that for an invariant polynomial $P$, the integral $\int_{{T^m}} {P\vert d{v_g}\vert}$ vanishes for any locally conformally flat metric $g$ on the torus ${T^m}$. Then up to the divergence of an invariantly defined one form, the polynomial $ P$ is a constant multiple of the Euler integrand.
On the morphology of $\gamma$-expansions with deleted digits
Mike
Keane;
Meir
Smorodinsky;
Boris
Solomyak
955-966
Abstract: We investigate the size of the set of reals which can be represented in base $\gamma$ using only the digits $0,1,3$. It is shown that this set has Lebesgue measure zero for $\gamma \leqslant 1/3$ and equals an interval for $\gamma \geqslant 2/5$. Our main goal is to prove that it has Lebesgue measure zero for a certain countable subset of $(1/3,2/5)$.
The Hausdorff dimension of $\lambda$-expansions with deleted digits
Mark
Pollicott;
Károly
Simon
967-983
Abstract: In this article we examine the continuity of the Hausdorff dimension of the one parameter family of Cantor sets $ \Lambda (\lambda ) = \{ \sum\nolimits_{k = 1}^\infty {{i_k}{\lambda ^k}:{i_k} \in S\} }$, where $S \subset \{ 0,1, \ldots ,(n - 1)\}$. In particular, we show that for almost all (Lebesgue) $ \lambda \in [\tfrac{1} {n},\tfrac{1} {l}]$ we have that ${\dim _H}(\Lambda (\lambda )) = \frac{{\log l}} {{ - \log \lambda }}$ where $l = \operatorname{Card} (S)$. In contrast, we show that under appropriate conditions on $S$ we have that for a dense set of $\lambda \in [\tfrac{1} {n},\tfrac{1} {l}]$ we have $ {\dim _H}(\Lambda (\lambda )) < \frac{{\log l}} {{ - \log \lambda }}$.
Crossed products of ${\rm II}\sb 1$-subfactors by strongly outer actions
Carl
Winsløw
985-991
Abstract: We study the crossed product $A \rtimes G \supseteq B \rtimes G$ of an inclusion $A \supseteq B$ of type $ {\text{I}}{{\text{I}}_1}$-factors by a discrete strongly outer action $ G$. In particular, we find conditions under which the strong amenability of $A \supseteq B$ implies that of $A \rtimes G \supseteq B \rtimes G$, and vice versa.
Hypercyclic weighted shifts
Héctor N.
Salas
993-1004
Abstract: An operator $ T$ acting on a Hilbert space is hypercyclic if, for some vector $ x$ in the space, the orbit $ \{ {T^n}x:n \geqslant 0\}$ is dense. In this paper we characterize hypercyclic weighted shifts in terms of their weight sequences and identify the direct sums of hypercyclic weighted shifts which are also hypercyclic. As a consequence, we show within the class of weighted shifts that multi-hypercyclic shifts and direct sums of fixed hypercyclic shifts are both hypercyclic. For general hypercyclic operators the corresponding questions were posed by D. A. Herrero, and they still remain open. Using a different technique we prove that $I + T$ is hypercyclic whenever $ T$ is a unilateral backward weighted shift, thus answering in more generality a question recently posed by K. C. Chan and J. H. Shapiro.
On Cappell-Shaneson's homology $L$-classes of singular algebraic varieties
Shoji
Yokura
1005-1012
Abstract: S. Cappell and J. Shaneson (Stratifiable maps and topological invariants, J. Amer. Math. Soc. 4 (1991), 521-551) have recently developed a theory of homology $ L$-classes, extending Goresky-MacPherson's homology $L$-classes. In this paper we show that Cappell-Shaneson's homology $L$-classes for compact complex, possibly singular, algebraic varieties can be interpreted as a unique natural transformation from a covariant cobordism function $ \Omega$ to the ${\mathbf{Q}}$-homology functor $ {H_{\ast}}(;{\mathbf{Q}})$ satisfying a certain normalization condition, just like MacPherson's Chern classes and Baum-Fulton-MacPherson's Todd classes.
Distinguished K\"ahler metrics on Hirzebruch surfaces
Andrew D.
Hwang;
Santiago R.
Simanca
1013-1021
Abstract: Let ${\mathcal{F}_n}$ be a Hirzebruch surface, $n \geqslant 1$. Using the family of extremal metrics on these surfaces constructed by Calabi [1], we study a closely related scale-invariant variational problem, and show that only ${\mathcal{F}_1}$ admits an extremal Kähler metric which is critical for this new functional. Applying a result of Derdzinski [3], we prove that this metric cannot be conformally equivalent to an Einstein metric on $ {\mathcal{F}_1}$. When $n = 2$, we show there is a critical orbifold metric on the space obtained from ${\mathcal{F}_2}$ by blowing down the negative section.
Formation of diffusion waves in a scalar conservation law with convection
Kevin R.
Zumbrun
1023-1032
Abstract: We study the scalar conservation law, $c( - \infty )$ and $c( + \infty )$, behavior that has been observed numerically in solutions of the full equations. The interesting aspect of the analysis is that the asymptotic state of the solution is not known a priori, in contrast to cases treated previously.
A note on norm inequalities for integral operators on cones
Ke Cheng
Zhou
1033-1041
Abstract: Norm inequalities for the Riemann-Liouville operator ${R_r}f(x) = \int_{\langle 0,x\rangle } {\Delta _V^{r - 1}(x - t)f(t)dt}$ and Weyl operator ${W_r}f(x) = \int_{\langle x,\infty \rangle } {\Delta _V^{r - 1}(t - x)f(t)dt}$ on cones in ${R^d}$ have been obtained in the case $r \geqslant 1$ [7]. In this note, these inequalities are further extended to the case $r < 1$. The question of whether the Hardy operator $Hf(x) = \int_{\langle 0,x\rangle } {f(t)dt}$ on cones is bounded from ${L^p}(\Delta _V^\alpha (X))$ to ${L^q}(\Delta _V^\beta (x))\;(q < p)$ is also solved.
Tensor product of difference posets
Anatolij
Dvurečenskij
1043-1057
Abstract: A tensor product of difference posets, which generalize orthoalgebras and orthomodular posets, is defined, and an equivalent condition is presented. In particular, we show that a tensor product for difference posets with a sufficient system of probability measures exists, as well as a tensor product of any difference poset and any Boolean algebra, which is isomorphic to a bounded Boolean power.
A note on the problem of prescribing Gaussian curvature on surfaces
Wei Yue
Ding;
Jia Quan
Liu
1059-1066
Abstract: The problem of existence of conformal metrics with Gaussian curvature equal to a given function $K$ on a compact Riemannian $2$-manifold $M$ of negative Euler characteristic is studied. Let ${K_0}$ be any nonconstant function on $M$ with $ \max {K_0} = 0$, and let $ {K_\lambda } = {K_0} + \lambda$. It is proved that there exists a ${\lambda ^{\ast}} > 0$ such that the problem has a solution for $ K = {K_\lambda }$ iff $\lambda \in ( - \infty ,{\lambda ^{\ast}}]$. Moreover, if $\lambda \in (0,{\lambda ^{\ast}})$, then the problem has at least $2$ solutions.
Functions with bounded spectrum
Ha Huy
Bang
1067-1080
Abstract: Let $0 < p \leqslant \infty ,\,f(x) \in {L_p}({\mathbb{R}^n})$, and $ \operatorname{supp} Ff$ be bounded, where $F$ is the Fourier transform. We will prove in this paper that the sequence $ \vert\vert{D^\alpha }f\vert\vert _p^{1/\vert\alpha \vert},\,\alpha \geqslant 0$, has the same behavior as the sequence $\mathop {\lim }\limits_{\xi \in \operatorname{supp} Ff} \vert{\xi ^\alpha }{\vert^{1/\vert\alpha \vert}}$, $ \alpha \geqslant 0$. In other words, if we know all "far points" of $\operatorname{supp} Ff$, we can wholly describe this behavior without any concrete calculation of $\vert\vert{D^\alpha }f\vert{\vert _p},\,\alpha \geqslant 0$. A Paley-Wiener-Schwartz theorem for a nonconvex case, which is a consequence of the result, is given.
Applications of simplicial $M$-sets to proper and strong shape theories
L. J.
Hernández Paricio
363-409
Abstract: In this paper we have tried to reduce the classical classification problems for spaces and maps of the proper category and of the strong shape category to similar problems in the homotopy category of simplicial sets or in the homotopy category of simplicial $M$-sets, which $M$ is the monoid of proper selfmaps of the discrete space $ \mathbb{N}$ of nonnegative integers. Given a prospace (prosimplicial set) $ Y$, we have constructed a simplicial set ${\overline {\mathcal{P}} ^R}Y$ such that the Hurewicz homotopy groups of ${\overline {\mathcal{P}} ^R}Y$ are the Grossman homotopy groups of $Y$. For the case of the end prospace $Y = \varepsilon X$ of a space $X$, we obtain Brown's proper homotopy groups; and for the Vietoris prospace $Y = VX$ (introduced by Porter) of a compact metrisable space $X$, we have Quigley's inward groups. The simplicial subset ${\overline {\mathcal{P}} ^R}Y$ of a tower $ Y$ contains, as a simplicial subset, the homotopy limit ${\lim ^R}Y$. The inclusion $ {\lim ^R}Y \to {\overline {\mathcal{P}} ^R}Y$ induces many relations between the homotopy and (co)homology invariants of the prospace $ Y$. Using the functor $ {\overline {\mathcal{P}} ^R}$ we prove Whitehead theorems for proper homotopy, prohomotopy, and strong shape theories as a particular case of the standard Whitehead theorem. The algebraic condition is given in terms of Brown's proper groups, Grossman's homotopy groups and Quigley's inward groups, respectively. In all these cases an equivalent cohomological condition can be given by taking twisted coefficients. The "singular" homology groups of ${\overline {\mathcal{P}} ^R}Y$ provide homology theories for the Brown, Grossman and Quigley homotopy groups that satisfy Hurewicz theorems in the corresponding settings. However, there are other homology theories for the homotopy groups above satisfying other Hurewicz theorems. We also analyse the notion of $\overline {\mathcal{P}} $-movable prospace. For a $ \overline {\mathcal{P}}$-movable tower we prove easily (without $ {\lim ^1}$ functors) that the strong homotopy groups agree with the Čech homotopy groups and the Grossman homotopy groups are determined by the Čech (or strong) groups by the formula $^G{\pi_q} = \overline{\mathcal{P}} \check{\pi}_q$. This implies that the algebraic condition of the Whitehead theorem can be given in terms of strong (Čech) groups when the condition of $\overline {\mathcal{P}} $-movability is included. We also study homology theories for the strong (Steenrod) homotopy groups which satisfy Hurewicz theorems but in general do not agree with the corresponding Steenrod-Sitnikov homology theories.
A free-boundary problem for the heat equation arising in flame propagation
Luis A.
Caffarelli;
Juan L.
Vázquez
411-441
Abstract: We introduce a new free-boundary problem for the heat equation, of interest in combustion theory. It is obtained in the description of laminar flames as an asymptotic limit for high activation energy. The problem asks for the determination of a domain in space-time, $\Omega \subset {{\mathbf{R}}^n} \times (0,T)$, and a function $u(x,t) \geqslant 0$ defined in $\Omega$, such that ${u_t} = \Delta u$ in $ \Omega ,\;u$ takes certain initial conditions, $u(x,0) = {u_0}(x)$ for $x \in {\Omega _0} = \partial \Omega \cap \{ t = 0\}$, and two conditions are satisfied on the free boundary $\Gamma = \partial \Omega \cap \{ t > 0\} :u = 0$ and ${u_\nu } = - 1$, where ${u_\nu }$ denotes the derivative of $ u$ along the spatial exterior normal to $\Gamma$. We approximate this problem by means of a certain regularization on the boundary and prove the existence of a weak solution under suitable assumptions on the initial data.
Epi-derivatives of integral functionals with applications
Philip D.
Loewen;
Harry H.
Zheng
443-459
Abstract: We study first- and second-order epi-differentiability for integral functionals defined on ${L^2}[0,T]$, and apply the results to obtain first- and second-order necessary conditions for optimality in free endpoint control problems.
The relative Burnside module and the stable maps between classifying spaces of compact Lie groups
Norihiko
Minami
461-498
Abstract: Tom Dieck's Burnside ring of compact Lie groups is generalized to the relative case: For any $G \triangleright N$, a compact Lie group and its normal subgroup $A(G \triangleright N)$ is defined to be an appropriate set of the equivalence classes of compact $ G$-ENR's with free $ N$-action, in such a way that $\psi :A(G \triangleright N) \simeq \pi _{G/N}^0({S^0};B{(N,G)_ + })$, where $B(N,G)$ is the classifying space of principal $(N,G)$-bundle. Under the "product" situation, i.e. $G = F \times K,\;N = K,\;A(F \times K \triangleright K)$ is also denoted by $A(F,K)$, as it turns out to be the usual $ A(F,K)$ when both $ F$ and $K$ are finite. Then a couple of applications are given to the study of stable maps between classifying spaces of compact Lie groups: a conceptual proof of Feshbach's double coset formula, and a density theorem on the map $\alpha _p^ \wedge :A(L,H)_p^ \wedge \to \{ B{L_{ + ,}}B{H_ + }\} _p^ \wedge$ for any compact Lie groups $L,\;K$ when $p$ is odd. (Some restriction is applied to $ L$ when $p = 2$.) This latter result may be regarded as the pushout of Feshbach's density theorem and the theorem of May-Snaith-Zelewski, over the celebrated Carlsson solution of Segal's Burnside ring conjecture.
Functional rotation numbers for one-dimensional maps
A. M.
Blokh
499-513
Abstract: We introduce functional rotation numbers and sets for one-dimensional maps (we call them $f$-rotation numbers and sets) and deduce some of their properties (density of ${\text{f}}$-rotation numbers of periodic points in the $ {\text{f}}$-rotation set, conditions for the connectedness of the ${\text{f}}$-rotation set) from the spectral decomposition theorem for one-dimensional maps.
An optimal condition for the LIL for trigonometric series
I.
Berkes
515-530
Abstract: By a classical theorem (Salem-Zygmund [6], Erdős-Gàl [3]), if $ ({n_k})$ is a sequence of positive integers satisfying $ {n_{k + 1}}/{n_k} \geqslant q > 1\;(k = 1,2, \ldots )$ then $(\cos {n_k}x)$ obeys the law of the iterated logarithm, i.e., (1) $\displaystyle \mathop {\lim \sup }\limits_{N \to \infty } {(N\log \log N)^{ - 1... ...\limits_{k \leqslant N} {\cos {n_k}x = 1\quad {\text{a}}{\text{.e}}{\text{.}}}$ It is also known (Takahashi [7, 8]) that the Hadamard gap condition ${n_{k + 1}}/{n_k} \geqslant q > 1$ can be essentially weakened here but the problem of finding the precise gap condition for the LIL (1) has remained open. In this paper we find, using combinatorial methods, an optimal gap condition for the upper half of the LIL, i.e., the inequality $\leqslant 1$ in (1).
The vanishing viscosity method in one-dimensional thermoelasticity
Gui Qiang
Chen;
Constantine M.
Dafermos
531-541
Abstract: The vanishing viscosity method is applied to the system of conservation laws of mass, momentum, and energy for a special class of one-dimensional thermoelastic media that do not conduct heat. Two types of vanishing "viscosity" are considered: Newtonian and artificial, in both cases accompanied by vanishing heat conductivity. It is shown that in either case one can pass to the zero viscosity limit by the method of compensated compactness, provided that velocity and pressure are uniformly bounded. Oscillations in the entropy field may propagate along the linearly degenerate characteristic field but do not affect the compactness of the velocity field or the pressure field. A priori bounds on velocity and pressure are established, albeit only for the case of artificial viscosity.
Entire functions, in the classification of differentiable germs tangent to the identity, in one or two variables
Patrick
Ahern;
Jean-Pierre
Rosay
543-572
Abstract: This paper presents a survey and some (hopefully) new facts on germs of maps tangent to the identity (in $\mathbb{R},\mathbb{C},$ or ${\mathbb{R}^2}$), (maps $f$ defined near 0, such that $f(0) = 0$, and $f'(0)$ is the identity). Proofs are mostly original. The paper is mostly oriented towards precise examples and the questions of descriptions of members in the conjugacy class, flows, $k$th root. It happened that entire functions provide clear and easy examples. However they should be considered just as a tool, not as the main topic. For example in Proposition $2$ the function $z \mapsto z + {z^2}$ should be better thought of as the map $(x,y) \to (x + {x^2} - {y^2},y + 2xy)$.
Diophantine approximation in ${\bf R}\sp n$
L. Ya.
Vulakh
573-585
Abstract: A modification of the Ford geometric approach to the problem of approximation of irrational real numbers by rational fractions is developed. It is applied to find an upper bound for the Hurwitz constant for a discrete group acting in a hyperbolic space. A generalized Khinchine's approximation theorem is also given.
Weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains
Ji Ye
Yu
587-614
Abstract: The purpose of this paper is to study the existence of weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains in ${\mathbb{C}^n}$ and to explore the connections between the limits and the Levi invariants. The main result extends Graham's result on strongly pseudoconvex domains to a large class of weakly pseudoconvex domains.
Contiguity relations for generalized hypergeometric functions
Alan
Adolphson;
Bernard
Dwork
615-625
Abstract: It is well known that the hypergeometric functions $\displaystyle _2{F_1}(\alpha \pm 1,\beta ,\gamma ;t),{\quad _2}{F_1}(\alpha ,\beta \pm 1,\gamma ;t),{\quad _2}{F_1}(\alpha ,\beta ,\gamma \pm 1;t),$ which are contiguous to $_2{F_1}(\alpha ,\beta ,\gamma ;t)$, can be expressed in terms of $\displaystyle _2{F_1}(\alpha ,\beta ,\gamma ;t)\quad {\text{and}}{\quad _2}F_1^\prime (\alpha ,\beta ,\gamma ;t).$ We explain how to derive analogous formulas for generalized hypergeometric functions. Our main point is that such relations can be deduced from the geometry of the cone associated in a recent paper by B. Dwork and F. Loeser to a generalized hypergeometric series.
Assessing prediction error in autoregressive models
Ping
Zhang;
Paul
Shaman
627-637
Abstract: Assessing prediction error is a problem which arises in time series analysis. The distinction between the conditional prediction error $e$ and the unconditional prediction error $ E(e)$ has not received much attention in the literature. Although one can argue that the conditional version is more practical, we show in this article that assessing $e$ is nearly impossible. In particular, we use the correlation coefficient $ \operatorname{corr} (\hat e,e)$, where $\hat e$ is an estimate of $e$, as a measure of performance and show that ${\lim _{T \to \infty }}\sqrt T \operatorname{corr} (\hat e,e) = C$ where $T$ is the sample size and $C > 0$ is some constant. Furthermore, the value of $C$ is large only when the process is extremely non-Gaussian or nearly nonstationary.
Prime ideals in polynomial rings over one-dimensional domains
William
Heinzer;
Sylvia
Wiegand
639-650
Abstract: Let $R$ be a one-dimensional integral domain with only finitely many maximal ideals and let $ x$ be an indeterminate over $R$. We study the prime spectrum of the polynomial ring $R[x]$ as a partially ordered set. In the case where $R$ is countable we classify $\operatorname{Spec} (R[x])$ in terms of splitting properties of the maximal ideals ${\mathbf{m}}$ of $R$ and the valuative dimension of ${R_{\mathbf{m}}}_{}$.
On the existence of global Tchebychev nets
Sandra L.
Samelson;
W. P.
Dayawansa
651-660
Abstract: Let $S$ be a complete, open simply connected surface. Suppose that the integral of the Gauss curvature over arbitrary measurable sets is less than $ \pi /2$ in magnitude. We show that the surface admits a global Tchebychev net.
Analytic Fourier-Feynman transforms and convolution
Timothy
Huffman;
Chull
Park;
David
Skoug
661-673
Abstract: In this paper we develop an ${L_p}$ Fourier-Feynman theory for a class of functionals on Wiener space of the form $F(x) = f(\int_0^T {{\alpha _1}dx, \ldots ,\int_0^T {{\alpha _n}dx)} }$. We then define a convolution product for functionals on Wiener space and show that the Fourier-Feynman transform of the convolution product is a product of Fourier-Feynman transforms.
On the tangential interpolation problem for $H\sb 2$ functions
Daniel
Alpay;
Vladimir
Bolotnikov;
Yossi
Peretz
675-686
Abstract: The aim of this paper is to solve a matrix-valued version of the Nevanlinna-Pick interpolation problem for $ {H_2}$ functions. We reduce this problem to a Nevanlinna-Pick interpolation problem for Schur functions and obtain a linear fractional transformation which describes the set of all solutions.
Convexity of the ideal boundary for complete open surfaces
Jin-Whan
Yim
687-700
Abstract: For complete open surfaces admitting total curvature, we define several kinds of convexity for the ideal boundary, and provide examples of each of them. We also prove that a surface with most strongly convex ideal boundary is in fact a generalization of a Hadamard manifold in the sense that the ideal boundary consists entirely of Busemann functions.
On the embedded primary components of ideals. IV
William
Heinzer;
L. J.
Ratliff;
Kishor
Shah
701-708
Abstract: The results in this paper expand the fundamental decomposition theory of ideals pioneered by Emmy Noether. Specifically, let $I$ be an ideal in a local ring $(R,M)$ that has $M$ as an embedded prime divisor, and for a prime divisor $P$ of $I$ let $I{C_P}(I)$ be the set of irreducible components $ q$ of $I$ that are $P$-primary (so there exists a decomposition of $ I$ as an irredundant finite intersection of irreducible ideals that has $ q$ as a factor). Then the main results show: (a) $I{C_M}(I) = \cup \{ I{C_M}(Q);Q\;{\text{is a }}\operatorname{MEC} {\text{ of }}I\}$ ($Q$ is a MEC of $I$ in case $Q$ is maximal in the set of $M$-primary components of $I$); (b) if $I = \cap \{ {q_i};i = 1, \ldots ,n\}$ is an irredundant irreducible decomposition of $ I$ such that $ {q_i}$ is $M$-primary if and only if $i = 1, \ldots ,k < n$, then $\cap \{ {q_i};i = 1, \ldots ,k\}$ is an irredundant irreducible decomposition of a MEC of $ I$, and, conversely, if $ Q$ is a MEC of $ I$ and if $\cap \{ {Q_j};j = 1, \ldots ,m\}$ (resp., $ \cap \{ {q_i};i = 1, \ldots ,n\}$) is an irredundant irreducible decomposition of $ Q$ (resp., $I$) such that ${q_1}, \ldots ,{q_k}$ are the $M$-primary ideals in $\{ {q_1}, \ldots ,{q_n}\} $, then $m = k$ and $( \cap \{ {q_i};i = k + 1, \ldots ,n\} ) \cap ( \cap \{ {Q_j};j = 1, \ldots ,m\} )$ is an irredundant irreducible decomposition of $I$; (c) $I{C_M}(I) = \{ q,q\;{\text{is maximal in the set of ideals that contain }}I\;{\text{and do not contain }}I:M\}$; (d) if $Q$ is a MEC of $I$, then $I{C_M}(Q) = \{ q;Q \subseteq q \in I{C_M}(I)\}$; (e) if $J$ is an ideal that lies between $I$ and an ideal $Q \in I{C_M}(I)$, then $J = \cap \{ q;J \subseteq q \in I{C_M}(I)\}$; and, (f) there are no containment relations among the ideals in $ \cup \{ I{C_P}(I)$; $ P$ is a prime divisor of $ I$}.
Random quadratic forms
John
Gregory;
H. R.
Hughes
709-717
Abstract: The results of Boyce for random Sturm-Liouville problems are generalized to random quadratic forms. Order relationships are proved between the means of eigenvalues of a random quadratic form and the eigenvalues of an associated mean quadratic form. Finite-dimensional and infinite-dimensional examples that show these are the best possible results are given. Also included are some results for a general approximation theory for random quadratic forms.
Conormal and piecewise smooth solutions to quasilinear wave equations
Seong Joo
Kang
1-35
Abstract: In this paper, we show first that if a solution $u$ of the equation ${P_2}(t,x,u,Du,D)u = f(t,x,u,Du)$, where ${P_2}(t,x,u,Du,D)$ is a second order strictly hyperbolic quasilinear operator, is conormal with respect to a single characteristic hypersurface $\Sigma$ of ${P_2}$ in the past and $\Sigma$ is smooth in the past, then $\Sigma$ is smooth and $u$ is conormal with respect to $ \Sigma$ for all time. Second, let ${\Sigma _0}$ and ${\Sigma _1}$ be characteristic hypersurfaces of ${P_2}$ which intersect transversally and let $ \Gamma = {\Sigma _0} \cap {\Sigma _1}$. If $ {\Sigma _0}$ and ${\Sigma _1}$ are smooth in the past and $ u$ is conormal with repect to $ \{ {\Sigma _0},{\Sigma _1}\}$ in the past, then $\Gamma$ is smooth, and $u$ is conormal with respect to $\{ {\Sigma _0},{\Sigma _1}\}$ locally in time outside of $\Gamma$, even though $ {\Sigma _0}$ and ${\Sigma _1}$ are no longer necessarily smooth across $\Gamma$. Finally, we show that if $u(0,x)$ and ${\partial _t}u(0,x)$ are in an appropriate Sobolev space and are piecewise smooth outside of $\Gamma$, then $u$ is piecewise smooth locally in time outside of ${\Sigma _0} \cup {\Sigma _1}$.
Further results on fixpoints and zeros of entire functions
Jian Hua
Zheng;
Chung-Chun
Yang
37-50
Abstract: In this paper, a quantitative estimation on the number of zeros of the function $f \circ g(z) - \alpha (z)$ is derived, where $f$ and $g$ are transcendental entire functions and $\alpha (z)$ a nonconstant polynomial. As an application of this and a further step towards an affirmative answer to a conjecture of Baker, a quantitative estimation on the number of period points of exact order $ n$ of ${f_n}$ ($n$th iterate of $f$) is obtained.
Bifurcation of minimal surfaces in Riemannian manifolds
Jürgen
Jost;
Xianqing
Li-Jost;
Xiao Wei
Peng
51-62
Abstract: We study the bifurcation of closed minimal surfaces in Riemannian manifolds through higher order variations of the area functional and relate it to elementary catastrophes.
Torsion classes and a universal constraint on Donaldson invariants for odd manifolds
Selman
Akbulut;
Tom
Mrowka;
Yongbin
Ruan
63-76
Abstract: This paper studies the topology of the gauge group and gives $ \bmod \,2$ universal relations along Donaldson polynomials of smooth $ 4$-manifolds, generalizing Y. Ruan's previous related result.
On the coefficient groups of equivariant $K$-theory
Yimin
Yang
77-98
Abstract: We calculated the coefficient groups of equivariant $K$-theory for any cyclic group, and we proved that, for any compact Lie group, the coefficient groups can only have $2$-torsion. If there is any $2$-torsion, it is detected by $ 2$-primary finite subgroups of $G$. The rationalization of the coefficient groups then can be easily calculated.
Homology operations on a new infinite loop space
Burt
Totaro
99-110
Abstract: Boyer et al. [1] defined a new infinite loop space structure on the space ${M_0} = {\prod _{n \geqslant 1}}K({\mathbf{Z}},2n)$ such that the total Chern class map $BU \to {M_0}$ is an infinite loop map. This is a sort of Riemann-Roch theorem without denominators: for example, it implies Fulton-MacPherson's theorem that the Chern classes of the direct image of a vector bundle $E$ under a given finite covering map are determined by the rank and Chern classes of $E$. We compute the Dyer-Lashof operations on the homology of ${M_0}$. They provide a new explanation for Kochman's calculation of the operations on the homology of $BU$, and they suggest a possible characterization of the infinite loop structure on ${M_0}$.
A new measure of growth for countable-dimensional algebras. I
John
Hannah;
K. C.
O’Meara
111-136
Abstract: A new dimension function on countable-dimensional algebras (over a field) is described. Its dimension values lie in the unit interval [0, 1]. Since the free algebra on two generators turns out to have dimension 0 (although conceivably some Noetherian algebras might have positive dimension!), this dimension function promises to distinguish among algebras of infinite $ GK$dimension.
Circle actions on rational homology manifolds and deformations of rational homotopy types
Martin
Raussen
137-153
Abstract: The aim of this paper is to follow up the program set in [LR85, Rau92], i.e., to show the existence of nontrivial group actions ("symmetries") on certain classes of manifolds. More specifically, given a manifold $X$ with submanifold $F$, I would like to construct nontrivial actions of cyclic groups on $X$ with $F$ as fixed point set. Of course, this is not always possible, and a list of necessary conditions for the existence of an action of the circle group $T = {S^1}$ on $X$ with fixed point set $F$ was established in [Rau92]. In this paper, I assume that the rational homotopy types of $ F$ and $X$ are related by a deformation in the sense of [A1178] between their (Sullivan) models as graded differential algebras (cf. [Sul77, Hal83]). Under certain additional assumptions, it is then possible to construct a rational homotopy description of a $ T$-action on the complement $X\backslash F$ that fits together with a given $T$-bundle action on the normal bundle of $ F$ in $X$. In a subsequent paper [Rau94], I plan to show how to realize this $T$-action on an actual manifold $ Y$ rationally homotopy equivalent to $X$ with fixed point set $F$ and how to "propagate" all but finitely many of the restricted cyclic group actions to $ X$ itself.
Localization and genus in group theory
G.
Peschke
155-174
Abstract: We provide a unifying category theoretical framework to discuss various kinds of local global phenomena. Specializing to localization of groups at sets of primes $ P$, we identify a large class of groups for which localization supports a passage from local information to global information. Local global principles for groups in this class are established and used to calculate certain homomorphism sets as well as splittings of epimorphisms and monomorphisms from local data.
Free ideals of one-relator graded Lie algebras
John P.
Labute
175-188
Abstract: In this paper we show that a one-relator graded Lie algebra $\mathfrak{g} = L/(r)$, over a principal ideal domain $K$, has a homogeneous ideal $\mathfrak{h}$ with $ \mathfrak{g}/\mathfrak{h}$ a free $K$-module of finite rank if the relator $ r$ is not a proper multiple of another element in the free Lie algebra $ L$. As an application, we deduce that the center of a one-relator Lie algebra over $ K$ is trivial if the rank of $L$ is greater than two. As another application, we find a new class of one-relator pro-$ p$-groups which are of cohomological dimension $2$.
Residue classes of Lagrangian subbundles and Maslov classes
Haruo
Suzuki
189-202
Abstract: For Lagrangian subbundles with singularities in symplectic vector bundles, explicit formulas of relation between their residue classes and Maslov classes outside singularities are obtained. Then a Lagrangian subbundle with singularity is constructed where all possible Maslov classes are nonzero but residue classes vanish for dimension $> 2$. Moreover, a Lagrangian immersion with singularity is constructed, where the similar property for the associated Maslov classes and residue classes is shown.
On parametric evolution inclusions of the subdifferential type with applications to optimal control problems
Nikolaos S.
Papageorgiou
203-231
Abstract: In this paper we study parametric evolution inclusions of the subdifferential type and their applications to the sensitivity analysis of nonlinear, infinite dimensional optimal control problems. The parameter appears in all the data of the problem, including the subdifferential operator. First we establish several continuity results for the solution multifunction of the subdifferential inclusion. Then we study how these results can be used to examine the sensitivity properties (variational stability) of certain broad classes of nonlinear infinite dimensional optimal control problems. Some examples are worked out in detail, illustrating the applicability of our work. These include obstacle problems (with time varying obstacles), optimal control of distributed parameter systems, and differential variational inequalities.
Generalizations of Browder's degree theory
Shou Chuan
Hu;
Nikolaos S.
Papageorgiou
233-259
Abstract: The starting point of this paper is the recent important work of F. E. Browder, who extended degree theory to operators of monotone type. The degree function of Browder is generalized to maps of the form $T + f + G$, where $T$ is maximal monotone, $f$ is of class ${(S)_ + }$ bounded, and $ G( \cdot )$ is an u.s.c. compact multifunction. It is also generalized to maps of the form $f + {N_G}$, with $f$ of class ${(S)_ + }$ and ${N_G}$ the Nemitsky operator of a multifunction $ G(x,r)$ satisfying various types of sign conditions. Some examples are also included to illustrate the abstract results.
Cohomologically symplectic spaces: toral actions and the Gottlieb group
Gregory
Lupton;
John
Oprea
261-288
Abstract: Aspects of symplectic geometry are explored from a homotopical viewpoint. In particular, the question of whether or not a given toral action is Hamiltonian is shown to be independent of geometry. Rather, a new homotopical obstruction is described which detects when an action is Hamiltonian. This new entity, the ${\lambda _{\hat \alpha }}$-invariant, allows many results of symplectic geometry to be generalized to manifolds which are only cohomologically symplectic in the sense that there is a degree $ 2$ cohomology class which cups to a top class. Furthermore, new results in symplectic geometry also arise from this homotopical approach.
Singularities produced in conormal wave interactions
Linda M.
Holt
289-315
Abstract: Three problems on the interactions of conormal waves are considered. Two are examples which demonstrate that nonlinear spreading of singularities can occur when the waves are conormal. In one case, two of the waves are tangential, and the other wave is transversal to the first two. The third result is a noninteraction theorem. It is shown that under certain conditions, no nonlinear spreading of the singularities will occur.
Polar $\sigma$-ideals of compact sets
Gabriel
Debs
317-338
Abstract: Let $E$ be a metric compact space. We consider the space $ \mathcal{K}(E)$ of all compact subsets of $E$ endowed with the topology of the Hausdorff metric and the space $ \mathcal{M}(E)$ of all positive measures on $E$ endowed with its natural ${w^{\ast}}$-topology. We study $\sigma $-ideals of $\mathcal{K}(E)$ of the form $I = {I_P} = \{ K \in \mathcal{K}(E):\mu (K) = 0,\;\forall \mu \in P\}$ where $P$ is a given family of positive measures on $E$. If $M$ is the maximal family such that $I = {I_M}$, then $M$ is a band. We prove that several descriptive properties of $I$: being Borel, and having a Borel basis, having a Borel polarity-basis, can be expressed by properties of the band $M$ or of the orthogonal band $M'$.
Curvature conditions on Riemannian manifolds with Brownian harmonicity properties
H. R.
Hughes
339-361
Abstract: The time and place that Brownian motion on a Riemannian manifold first exits a normal ball of radius $ \varepsilon$ is considered and a general procedure is given for computing asymptotic expansions, as $ \varepsilon$ decreases to zero, for joint moments of the first exit time and place random variables. It is proven that asymptotic versions of exit time and place distribution properties that hold on harmonic spaces are equivalent to certain curvature conditions for harmonic spaces. In particular, it is proven that an asymptotic mean value condition involving first exit place is equivalent to certain levels of curvature conditions for harmonic spaces depending on the order of the asymptotics. Also, it is proven that an asymptotic uncorrelated condition for first exit time and place is equivalent to weaker curvature conditions at corresponding orders of asymptotics.
On the local regularity of solutions in linear viscoelasticity of several space dimensions
Jong Uhn
Kim
359-398
Abstract: In this paper we discuss the local regularity of solutions of a nonlocal system of equations which describe the motion of a viscoelastic medium in several space dimensions. Our main tool is the microlocal analysis combined with MacCamy's trick and the argument of the classical energy method.
On the theory of internal waves of permanent form in fluids of great depth
C. J.
Amick
399-419
First- and second-order necessary conditions for control problems with constraints
Zsolt
Páles;
Vera
Zeidan
421-453
Abstract: Second-order necessary conditions are developed for an abstract nonsmooth control problem with mixed state-control equality and inequality constraints as well as a constraint of the form $G(x,u) \in \Gamma$, where $ \Gamma$ is a closed convex set of a Banach space with nonempty interior. The inequality constraints $ g(s,x,u) \leqslant 0$ depend on a parameter $s$ belonging to a compact metric space $S$. The equality constraints are split into two sets of equations $K(x,u) = 0$ and $ H(x,u) = 0$, where the first equation is an abstract control equation, and $ H$ is assumed to have a full rank property in $u$. The objective function is ${\max _{t \in T}}f(t,x,u)$ where $ T$ is a compact metric space, $f$ is upper semicontinuous in $ t$ and Lipschitz in $ (x,u)$. The results are in terms of a function $\sigma$ that disappears when the parameter spaces $ T$ and $S$ are discrete. We apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints. Thus we obtain a generalization and extension of the existing results on this problem.
Porous sets and null sets for elliptic harmonic measures
Jang-Mei
Wu
455-473
Abstract: We give a genuinely $ n$-dimensional construction of uniformly elliptic operators $L$ in $ \mathbb{R}_ + ^n$ (of divergence form, and of nondivergence form), which have positive $L$-harmonic measures on a class of porous sets on $\partial \mathbb{R}_ + ^n$ with zero surface measure. The porosity condition given is sharp. The earlier methods were all two dimensional.
An elliptic regularity coefficient estimate for a problem arising from a frequency domain treatment of waves
Xiaobing
Feng;
Dongwoo
Sheen
475-487
Abstract: We consider a sequence of noncoercive elliptic problems, which are the wave equation in the frequency domain, in a rectangular or cubic domain with an absorbing boundary condition. The elliptic regularity coefficient depends on the frequency, and it has a singularity for both zero and infinite frequency. In this paper we derive an elliptic regularity estimate as the frequency tends to zero and infinity.
$L\sp p$-boundedness of pseudo-differential operators of class $S\sb {0,0}$
I. L.
Hwang;
R. B.
Lee
489-510
Abstract: We study the $ {L^p}$-boundedness of pseudo-differential operators with the support of their symbols being contained in $E \times {{\mathbf{R}}^n}$, where $ E$ is a compact subset of $ {{\mathbf{R}}^n}$, and their symbols have derivatives with respect to $ x$ only up to order $ k$, in the Hölder continuous sense, where $k > n/2$ (the case $1 < p \leqslant 2$) and $k > n/p$ (the case $2 < p < \infty$). We also give a new proof of the ${L^p}$-boundedness, $ 1 < p < \infty$, of pseudo-differential operators of class $ S_{0,0}^m$, where $m = m(p) = - n\vert 1/p - 1/2\vert$, and $ a \in S_{0,0}^m$ satisfies $\vert\partial _x^\alpha \partial _\xi ^\beta a(x,\xi )\vert \leqslant {C_{\alpha ,\beta }}{\langle \xi \rangle ^m}$ for $(x,\xi ) \in {{\mathbf{R}}^n} \times {{\mathbf{R}}^n},\vert\alpha \vert \leqslant k$ and $1 < p \leqslant 2$) and $2 < p < \infty $).
Generalization of the Whitney-Mahowald theorem
Bang He
Li
511-521
Abstract: The Whitney-Mahowald theorem gave normal Euler number $(\bmod \,4)$ for embeddings of a closed $ 2n$-manifold in Euclidean $ 4n$-space. We generalize this theorem to embeddings of closed $2n$-manifolds in an oriented $ 4n$-manifold with an approach in the framework of unoriented bordism groups of maps.
Global oscillatory waves for second order quasilinear wave equations
Paul
Godin
523-547
Abstract: In this paper we prove the global existence and describe the asymptotic behaviour of a family of oscillatory solutions of Cauchy problems for a class of scalar second order quasilinear wave equations, when the space dimension is odd and at least equal to $3$. If time is bounded, corresponding results for quasilinear first order systems were obtained by Guès; to prove our results we reduce our problems to bounded time problems with the help of a conformal inversion. To obtain global results, suitable geometric assumptions must be made on the set where the oscillations are concentrated at initial time.
Algebras associated to the Young-Fibonacci lattice
Soichi
Okada
549-568
Abstract: The algebra ${\mathcal{F}_n}$ generated by ${E_1},\; \ldots \;,\;{E_{n - 1}}$ subject to the defining relations $ E_i^2 = {x_i}{E_i}\;(i = 1,\; \ldots \;,\;n - 1),\;{E_{i + 1}}{E_i}{E_{i + 1}}... ...\; \ldots \;,\;n - 2),\;{E_i}{E_j} = {E_j}{E_i}\;(\vert i - j\vert \geqslant 2)$ is shown to be a semisimple algebra of dimension $n!$ if the parameters ${x_1},\; \ldots \;,\;{x_{n - 1}},\;{y_1},\; \ldots \;,\;{y_{n - 2}}$ are generic. We also prove that the Bratteli diagram of the tower ${({\mathcal{F}_n})_{n \geqslant 0}}$ of these algebras is the Hasse diagram of the Young-Fibonacci lattice, which is an interesting example, as well as Young's lattice, of a differential poset introduced by $\operatorname{R}$. Stanley. A Young-Fibonacci analogue of the ring of symmetric functions is given and studied.
Modular Schur functions
Grant
Walker
569-604
Abstract: A new family of symmetric functions is considered. These functions are analogous to the classical Schur functions, but depend on an integer modulus $p \geqslant 2$, as well as on a partition $ \lambda$. In the case where $p$ is prime, certain of these functions are shown to be irreducible characters of the general linear group $ GL(n,K)$ in the natural characteristic $p$ of the field $K$. This dualises a wellknown criterion of G. D. James for such characters to be given by classical Schur functions.
Upper bound for distortion of capacity under conformal mapping
Robert E.
Thurman
605-616
Abstract: For a finitely-connected domain $\Omega$ containing $\infty$, with boundary $\Gamma$, the logarithmic capacity $d(\Gamma )$ is invariant under normalized conformal maps of $\Omega$. But the capacity of a subset $A \subset \Gamma$ will likely be distorted by such a map. Duren and Schiffer showed that the sharp lower bound for the distortion of the capacity of such a set is the so-called "Robin capacity" of the set $ A$. We present here the sharp upper bound for the distortion, in terms of conformal invariants of $\Omega$: the harmonic measures of the boundary components of $\Omega$ and the periods of their harmonic conjugates (the Riemann matrix), and the capacity of $ \Gamma$. In particular, the upper bound depends only on knowing which components of $\Gamma$ contain parts of $A$, not on the specific distribution of $ A$. An extremal configuration is described explicitly for a special case.
The nilpotency class of finite groups of exponent $p$
Michael
Vaughan-Lee
617-640
Abstract: We investigate the properties of Lie algebras of characteristic $ p$ which satisfy the Engel identity $x{y^n} = 0$ for some $n < p$. We establish a criterion which (when satisfied) implies that if $a$ and $b$ are elements of an Engel-$n$ Lie algebra $L$ then $ a{b^{n - 2}}$ generates a nilpotent ideal of $L$. We show that this criterion is satisfied for $n = 6,\,p = 7$, and we deduce that if $ G$ is a finite $ m$-generator group of exponent $7$ then $G$ is nilpotent of class at most $51{m^8}$.
Geometric invariants for Seifert fibred $3$-manifolds
Ming Qing
Ouyang
641-659
Abstract: In this paper, we obtain a formula for the $\eta$-invariant of the signature operator for some circle bundles over Riemannian $2$-orbifolds. We then apply it to Seifert fibred $3$-manifolds endowed with one of the six Seifert geometries. By using a relation between the Chern-Simons invariant and the $\eta$-invariant, we also derive some elementary formulae for the Chern-Simons invariant of these manifolds. As applications, we show that some families of these manifolds cannot be conformally immersed into the Euclidean space ${{\mathbf{E}}^4}$.
BMO on strongly pseudoconvex domains: Hankel operators, duality and $\overline\partial$-estimates
Huiping
Li;
Daniel H.
Luecking
661-691
Abstract: We study the condition that characterizes the symbols of bounded Hankel operators on the Bergman space of a strongly pseudoconvex domain and show that it is equivalent to $BMO$ plus analytic. (Here we mean the Bergman metric $BMO$ of Berger, Coburn and Zhu.) In the course of the proof we obtain new $\overline \partial $-estimates that may be of independent interest. Some applications include a decomposition of $BMO$ similar to the classical ${L^\infty } + \widetilde{{L^\infty }}$, and two characterizations of the dual of $VMO$ (which is also a predual of $ BMO$). In addition, we obtain some partial results on the boundedness of Hankel operators in ${L^1}$ norm.
The Mackey obstruction and the coadjoint orbits
Zongyi
Li
693-705
Abstract: This paper studies the Mackey obstruction representation theory at the coadjoint orbit level. It shows how to get rid of such obstructions and to get orbits of the "little groups". Such little group data is essential for inductive construction of coadjoint orbits of general Lie groups.
On square-preserving isometries of convolution algebras
Sadahiro
Saeki
707-718
Abstract: Let $S$ and $S'$ be two semigroups, each contained in a locally compact group. Under certain conditions on $S$ and $S'$, we shall characterize those isometric additive surjections
Nonproduct type analytic TUHF algebras
Belisario A.
Ventura
719-738
Abstract: We construct examples of nonproduct type real valued cocycles on a UHF groupoid, and show that the analytic triangular algebras associated to those cocycles, can only correspond to nonproduct type cocycles.
$F$-regularity, test elements, and smooth base change
Melvin
Hochster;
Craig
Huneke
1-62
Abstract: This paper deals with tight closure theory in positive characteristic. After a good deal of preliminary work in the first five sections, including a treatment of $ F$-rationality and a treatment of $F$-regularity for Gorenstein rings, a very widely applicable theory of test elements for tight closure is developed in $\S6$ and is then applied in $\S7$ to prove that both tight closure and $ F$-regularity commute with smooth base change under many circumstances (where "smooth" is used to mean flat with geometrically regular fibers). For example, it is shown in $\S6$ that for a reduced ring $ R$ essentially of finite type over an excellent local ring of characteristic $ p$, if $c$ is not in any minimal prime of $ R$ and ${R_c}$ is regular, then $c$ has a power that is a test element. It is shown in $\S7$ that if $S$ is a flat $R$-algebra with regular fibers and $R$ is $F$-regular then $S$ is $F$-regular. The general problem of showing that tight closure commutes with smooth base change remains open, but is reduced here to showing that tight closure commutes with localization.
$q$-Hermite polynomials, biorthogonal rational functions, and $q$-beta integrals
M. E. H.
Ismail;
D. R.
Masson
63-116
Abstract: We characterize the solutions of the indeterminate moment problem associated with the continuous $q$-Hermite polynomials when $ q > 1$ in terms of their Stieltjes transforms. The extremal measures are found explicitly. An analog of the Askey-Wilson integral is evaluated. It involves integrating a kernel, similar to the Askey-Wilson kernel, against any solution of the $q$-Hermite moment problem, provided that certain integrability conditions hold. This led to direct evaluation of several $q$-beta integrals and their discrete analogs as well as a generalization of Bailey's ${}_6{\psi _6}$, sum containing infinitely many parameters. A system of biorthogonal rational functions is also introduced.
The profile near blowup time for solution of the heat equation with a nonlinear boundary condition
Bei
Hu;
Hong-Ming
Yin
117-135
Abstract: This paper studies the blowup profile near the blowup time for the heat equation ${u_t} = \Delta u$ with the nonlinear boundary condition $ {u_n} = {u^p}$ on $\partial \Omega \times [0,T)$. Under certain assumptions, the exact rate of the blowup is established. It is also proved that the blowup will not occur in the interior of the domain. The asymptotic behavior near the blowup point is also studied.
Univalent functions and the Pompeiu problem
Nicola
Garofalo;
Fausto
Segàla
137-146
Abstract: In this paper we prove a result on the Pompeiu problem. If the Schwarz function $\Phi$ of the boundary of a simply-connected domain $ \Omega \subset {\mathbb{R}^2}$ extends meromorphically into a certain portion $ E$ of $\Omega$ with a pole at some point ${z_0} \in E$, then $\Omega$ has the Pompeiu property unless $ \Phi$ is a Möbius transformation, in which case $\Omega$ is a disk.
On the fundamental periods of Hilbert modular forms
Ze-Li
Dou
147-158
Abstract: The main purpose of this paper is to establish the existence of fundamental periods of primitive cusp forms of Hilbert modular type of several variables, as well as the relationship between those fundamental periods and the special values of the associated $L$-functions. These results, together with some recent results of Shimura, give us the means of translating with ease results concerning periods of automorphic forms derived from various points of view. We also verify several conjectures of Shimura on the properties of such fundamental periods.
Optimal drift on $[0,1]$
Susan
Lee
159-175
Abstract: Consider one-dimensional diffusions on the interval $[0,1]$ of the form $d{X_t} = d{B_t} + b({X_t})dt$, with 0 a reflecting boundary, $ b(x) \geqslant 0$, and $\int_0^1 {b(x)dx = 1}$. In this paper, we show that there is a unique drift which minimizes the expected time for ${X_t}$ to hit $1$, starting from ${X_0} = 0$. In the deterministic case $d{X_t} = b({X_t})dt$, the optimal drift is the function which is identically equal to $1$. By contrast, if $d{X_t} = d{B_t} + b({X_t})dt$, then the optimal drift is the step function which is $ 2$ on the interval $ [1/4,3/4]$ and is 0 otherwise. We also solve this problem for arbitrary starting point ${X_0} = {x_0}$ and find that the unique optimal drift depends on the starting point, $ {x_0}$, in a curious manner.
Commutator theory without join-distributivity
Paolo
Lipparini
177-202
Abstract: We develop Commutator Theory for congruences of general algebraic systems (henceforth called algebras) assuming only the existence of a ternary term $d$ such that $d(a,b,b)[\alpha ,\alpha ]a[\alpha ,\alpha ]d(b,b,a)$, whenever $\alpha$ is a congruence and $a\alpha b$. Our results apply in particular to congruence modular and $n$-permutable varieties, to most locally finite varieties, and to inverse semigroups. We obtain results concerning permutability of congruences, abelian and solvable congruences, connections between congruence identities and commutator identities. We show that many lattices cannot be embedded in the congruence lattice of algebras satisfying our hypothesis. For other lattices, some intervals are forced to be abelian, and others are forced to be nonabelian. We give simplified proofs of some results about the commutator in modular varieties, and generalize some of them to single algebras having a modular congruence lattice.
Geometric consequences of extremal behavior in a theorem of Macaulay
Anna
Bigatti;
Anthony V.
Geramita;
Juan C.
Migliore
203-235
Abstract: F. S. Macaulay gave necessary and sufficient conditions on the growth of a nonnegative integer-valued function which determine when such a function can be the Hilbert function of a standard graded $k$-algebra. We investigate some algebraic and geometric consequences which arise from the extremal cases of Macaulay's theorem. Our work also builds on the fundamental work of G. Gotzmann. Our principal applications are to the study of Hilbert functions of zero-schemes with uniformity conditions. As a consequence, we have new strong limitations on the possible Hilbert functions of the points which arise as a general hyperplane section of an irreducible curve.
Singular polynomials for finite reflection groups
C. F.
Dunkl;
M. F. E.
de Jeu;
E. M.
Opdam
237-256
Abstract: The Dunkl operators involve a multiplicity function as parameter. For generic values of this function the simultaneous kernel of these operators, acting on polynomials, is equal to the constants. For special values, however, this kernel is larger. We determine these singular values completely and give partial results on the representations of $G$ that occur in this kernel.
On the oblique derivative problem for diffusion processes and diffusion equations with H\"older continuous coefficients
Masaaki
Tsuchiya
257-281
Abstract: On a ${C^2}$-domain in a Euclidean space, we consider the oblique derivative problem for a diffusion equation and assume the coefficients of the diffusion and boundary operators are Hölder continuous. We then prove the uniqueness of diffusion processes and fundamental solutions corresponding to the problem. For the purpose, obtaining a stochastic representation of some solutions to the problem plays a key role; in our situation, a difficulty arises from the absence of a fundamental solution with ${C^2}$-smoothness up to the boundary. It is overcome by showing some stability of a fundamental solution and a diffusion process, respectively, under approximation of the domain. In particular, the stability of the fundamental solution is verified through construction: it is done by applying the parametrix method twice to a parametrix with explicit expression.
Distribution of partial sums of the Taylor development of rational functions
V.
Nestoridis
283-295
Abstract: Let $f$ be a rational function regular at 0, which is not a polynomial; let $ {S_N}(z),\;N = 0,1,2, \ldots ,z \in \mathbb{C}$, denote the partial sums of the Taylor development of $f$. We investigate the angular distribution of the sequence ${S_N}(z),\;N = 0,1,2, \ldots$, around $ f(z)$. We show that for all $z$ in the plane, except a denumerable union of straight lines passing through 0, this angular distribution exists and is uniform.
Arithmetic calculus of Fourier transforms by Igusa local zeta functions
Tatsuo
Kimura
297-306
Abstract: We show the possibility of explicit calculation of the Fourier transforms of complex powers of relative invariants of some prehomogeneous vector spaces over $ \mathbb{R}$ by using the explicit form of $p$-adic Igusa local zeta functions.
Asymptotics for orthogonal rational functions
A.
Bultheel;
P.
González-Vera;
E.
Hendriksen;
O.
Njåstad
307-329
Abstract: Let $\{ {\alpha _n}\}$ be a sequence of (not necessarily distinct) points in the open unit disk, and let $\displaystyle {B_0} = 1,\quad {B_n}(z) = \prod\limits_{m = 1}^n {\frac{{\overli... ...alpha _m} - z)}} {{(1 - \overline {{\alpha _m}} z}}),\qquad n = 1,2, \ldots ,}$ ( $\frac{{\overline {{\alpha _n}} }} {{\vert{\alpha _n}\vert}} = - 1$ when $ {\alpha _n} = 0$). Let $ \mu$ be a finite (positive) Borel measure on the unit circle, and let $\{ {\varphi _n}(z)\}$ be orthonormal functions obtained by orthogonalization of $\{ {B_n}:n = 0,1,2, \ldots \}$ with respect to $ \mu$. Boundedness and convergence properties of the reciprocal orthogonal functions $\varphi _n^*(z) = {B_n}(z)\overline {{\varphi _n}(1/\overline z )}$ and the reproducing kernels $ {k_n}(z,w) = \sum\nolimits_{m = 0}^n {{\varphi _m}(z)\overline {{\varphi _m}(w)} }$ are discussed in the situation $\vert{\alpha _n}\vert \leqslant R < 1$ for all $n$, in particular their relationship to the Szegö condition ${L_2}(\mu )$ of the system $ \{ {\varphi _n}(z):n = 0,1,2, \ldots \}$. Limit functions of $\varphi _n^{\ast}(z)$ and ${k_n}(z,w)$ are obtained. In particular, if a subsequence $ \{ {\alpha _{n(s)}}\}$ converge to $\alpha$, then the subsequence $\{ \varphi _{n(s)}^{\ast}(z)\}$ converges to $\displaystyle {e^{i\lambda }}\frac{{\sqrt {1 - \vert\alpha {\vert^2}} }} {{1 - ... ...ine \alpha z}}\frac{1} {{{\sigma _{\mu (z)}}}},\qquad \lambda \in {\mathbf{R}},$ where $\{ {k_n}(z,w)\}$ converge to $1/(1 - z\overline w ){\sigma _\mu }(z)\overline {{\sigma _\mu }(w)}$. The results generalize corresponding results from the classical Szegö theory (concerned with the polynomial situation ${\alpha _n} = 0$ for all $n$).
On orthogonal polynomials with respect to varying measures on the unit circle
K.
Pan
331-340
Abstract: Let $\{ {\phi _n}(d\mu )\}$ be a system of orthonormal polynomials on the unit circle with respect to $ d\mu$ and $\{ {\psi _{n,m}}(d\mu )\} $ be a system of orthonormal polynomials on the unit circle with respect to the varying measures $d\mu /\vert{w_n}(z){\vert^2},\;z = {e^{i\theta }}$, where $\{ {w_n}(z)\}$ is a sequence of polynomials, $\deg {w_n} = n$, whose zeros ${w_{n,1}}, \ldots ,{w_{n,n}}$ lie in $ \vert z\vert < 1$ The asymptotic behavior of the ratio of the two systems on and outside the unit circle is obtained.
Conical limit points and groups of divergence type
Sungbok
Hong
341-357
Abstract: We use the Patterson-Sullivan measure to generalize Agard's theorem to all groups of divergence type. As a consequence, we prove that for a nonelementary group $ \Gamma$ of divergence type, the conical limit set has positive Patterson-Sullivan measure.
A reimbedding algorithm for Casson handles
Žarko
Bižaca
435-510
Abstract: An algorithmic proof of Freedman's Reimbedding Theorem [F2] is given. This reimbedding algorithm produces an explicit description of an imbedded Casson tower with seven levels inside an arbitrary Casson tower with six levels. Our approach is similar to Freedman's original idea, but we also make essential use of the grope technology from [FQ]. The reimbedding algorithm is applied to obtain an explicitly described Casson handle inside an arbitrary six-level tower (Theorem A), a description of a family of exotic Casson handles (Theorem B) and an explicitly constructed exotic $ {\mathbb{R}^4}$.
Extremal properties of Green functions and A. Weitsman's conjecture
Alexander
Fryntov
511-525
Abstract: A new version of the symmetrization theorem is proved. Using a modification of the $\ast$-function of Baernstein we construct an operator which maps a family of $\delta$-subharmonic functions defined on an annulus into a family of subharmonic functions on an annular sector. Applying this operator to the Green function of special domains we prove A. Weitsman's conjecture linked with exact estimates of the Green functions of these domains.
Well-posedness and stabilizability of a viscoelastic equation in energy space
Olof J.
Staffans
527-575
Abstract: We consider the well-posedness and exponential stabilizability of the abstract Volterra integrodifferential system \begin{displaymath}\begin{array}{*{20}{c}} {v\prime (t) = - {D^\ast}\sigma (t) +... ... }^t {a(t - s)Dv(s)ds,\quad t \geq 0,} } \end{array} \end{displaymath} in ilbeubert space. In a typical viscoelastic interpretation of this equation one lets v represent velocit, $v\prime$ acceleratio $\sigma$, stres, $- {D^ \ast }\sigma$ the divergence of the stres, $v \geq 0$ pure viscosity (usually equal to zero) Dv the time derivative of the strain, and a the linear stress relaxation modulus of the material. The problems that we treat are one-dimensional in the sense that we require a to be scalar. First we prove well-posedness in a new semigroup setting, where the history component of the state space describes the absorbed energy of the system rather than the history of the function v. To get the well-posedness we need extremely weak assumptions on the kernel; it suffices if the system is "passive", i.e., a is of positive type; it may even be a distribution. The system is exponentially stabilizable with a finite dimensional continuous feedback if and only if the essential growth rate of the original system is negative. Under additional assumptions on the kernel we prove that this is indeed the case. The final part of the treatment is based on a new class of kernels. These kernels are of positive type, but they need not be completely monotone. Still, they have many properties similar to those of completely monotone kernels, and a number of results that have been proved earlier for completely monotone kernels can be extended to the new class.
Measurable quotients of unipotent translations on homogeneous spaces
Dave
Witte
577-594
Abstract: Let U be a nilpotent, unipotent subgroup of a Lie group G, and let $\Gamma$ be a closed subgroup of G. Marina Ratner showed that every ergodic U-invariant probability measure on the homogeneous space $\Gamma \backslash G$ is of a simple algebraic form. We use this fundamental new result to show that every measurable quotient of the U-action on $\Gamma \backslash G$ is of a simple algebraic form. Roughly speaking, any quotient is a double-coset space $ \Lambda \backslash G/K$.
Hilbert 90 theorems over division rings
T. Y.
Lam;
A.
Leroy
595-622
Abstract: Hilbert's Satz 90 is well-known for cyclic extensions of fields, but attempts at generalizations to the case of division rings have only been partly successful. Jacobson's criterion for logarithmic derivatives for fields equipped with derivations is formally an analogue of Satz 90, but the exact relationship between the two was apparently not known. In this paper, we study triples (K, S, D) where S is an endomorphism of the division ring K, and D is an S-derivation. Using the technique of Ore extensions $K[t,S,D]$, we characterize the notion of (S, D)-algebraicity for elements $a \in K$, and give an effective criterion for two elements $a,b \in K$ to be (S, D)-conjugate, in the case when the (S, D)-conjugacy class of a is algebraic. This criterion amounts to a general Hilbert 90 Theorem for division rings in the (K, S, D)-setting, subsuming and extending all known forms of Hilbert 90 in the literature, including the aforementioned Jacobson Criterion. Two of the working tools used in the paper, the Conjugation Theorem (2.2) and the Composite Function Theorem (2.3), are of independent interest in the theory of Ore extensions.
Notes on ruled symplectic $4$-manifolds
Dusa
McDuff
623-639
Abstract: A symplectic 4-manifold $(V,\omega )$ is said to be ruled if it is the total space of a fibration whose fibers are 2-spheres on which the symplectic form does not vanish. This paper develops geometric methods for analysing the symplectic structure of these manifolds, and shows how this structure is related to that of a generic complex structure on V. It is shown that each V admits a unique ruled symplectic form up to pseudo-isotopy (or deformation). Moreover, if the base is a sphere or if V is the trivial bundle over the torus, all ruled cohomologous forms are isotopic. For base manfolds of higher genus this remains true provided that a cohomological conditon on the form is satisfied: one needs the fiber to be "small" relative to the base. These results correct the statement of Theorem 1.3 in The structure of rational and ruled symplectic manifolds, J. Amer. Math. Soc. 3 (1990), 679-712, and give more details of some of the proofs.
Foundations of BQO theory
Alberto
Marcone
641-660
Abstract: In this paper we study the notion of better-quasi-ordering (bqo) originally defined by Nash-Williams [14]. In particular we consider the approximation to this concept given by the notion of $\alpha$-wqo, for $\alpha$ a countable indecomposable ordinal [15]. We prove that if a quasi-ordering Q is $ \alpha$-wqo then ${Q^{ < \alpha }}$ is wqo, thereby obtaining a new proof of Nash-Williams' theorem that Q bqo implies $\tilde Q$ (the set of all countable transfinite sequences of elements of Q) bqo. We show that for $ \alpha < \alpha \prime ,\alpha \prime$-wqo is properly stronger than $ \alpha$-wqo. We also show that the definition of $\alpha$-wqo (and therefore also of bqo) can be modified by considering only barriers with a nice additional property. In the last part of the paper we establish a conjecture of Clote [3] by proving that the set of indices for recursive bqos is complete $ \Pi _2^1$.
Sensitivity analysis of solutions to generalized equations
A. B.
Levy;
R. T.
Rockafellar
661-671
Abstract: Generalized equations are common in the study of optimization through nonsmooth analysis. For instance, variational inequalities can be written as generalized equations involving normal cone mappings, and have been used to represent first-order optimality conditions associated with optimization problems. Therefore, the stability of the solutions to first-order optimality conditions can be determined from the differential properties of the solutions of parameterized generalized equations. In finite-dimensions, solutions to parameterized variational inequalities are known to exhibit a type of generalized differentiability appropriate for multifunctions. Here it is shown, in a Banach space setting, that solutions to a much broader class of parameterized generalized equations are "differentiable" in a similar sense.
Factoring $L$-functions as products of $L$-functions
Douglas
Grenier
673-692
Abstract: We will demonstrate two factorizations of L-functions associated with automorphic forms on $GL(n,\mathbb{R})$, where one factor is a Riemann zetafunction and the other is an L-function associated to an automorphic form for $GL(n - 1,\mathbb{R})$. These will be obtained by establishing the commutation of the Hecke operators and the $\Phi$-operator, a homomorphism from automorphic forms on $ GL(n,\mathbb{R})$ to automorphic forms on $ GL(n - 1,\mathbb{R})$.
Dynamics near the essential singularity of a class of entire vector fields
Kevin
Hockett;
Sita
Ramamurti
693-703
Abstract: We investigate the dynamics near the essential singularity at infinity for a class of zero-free entire vector fields of finite order, i.e., those of the form $f(z) = {e^{P(z)}}$ where $ P(z) = {z^d}$ or $P(z) = a{z^2} + bz + c$. We show that the flow generated by such a vector field has a "bouquet of flowers" attached to the point at infinity.
Empirical distribution functions and strong approximation theorems for dependent random variables. A problem of Baker in probabilistic number theory
Walter
Philipp
705-727
Abstract: Let $ \mathcal{T} = \{ {q_1}, \ldots ,{q_\tau }\}$ be a finite set of coprime integers and let $ \{ {n_1},{n_2}, \ldots \}$ denote the mutiplicative semigroup generated by $\mathcal{T}$, and arranged in increasing order. Let $ {D_N}(\omega )$ denote the discrepancy of the sequence $\{ {n_k}\omega \} _{k = 1}^N\bmod 1$, $\omega \in [0,1)$. In this paper we solve a problem posed by R.C. Baker [3], by proving that for all $ \omega$ except on a set of Lebesgue measure 0 $\displaystyle \frac{1}{4} \leq \mathop {\lim \sup }\limits_{N \to \infty } \frac{{N{D_N}(\omega )}}{{\sqrt {N\log \log N} }} \leq C.$ Here the constant C only depends on the total number of primes involved in the prime factorization of ${q_1}, \ldots ,{q_\tau }$. The lower bound is obtained from a strong approximation theorem for the partial sums of the sequence $ \{ \cos 2\pi {n_k}\omega \} _{k = 1}^\infty$ by sums of independent standard normal random variables.
On the dependence of analytic solutions of partial differential equations on the right-hand side
Siegfried
Momm
729-752
Abstract: Given a nonzero polynomial $P(z) = \sum\nolimits_{\vert\alpha \vert \leq m} {{a_\alpha }{z^\alpha }}$ on ${\mathbb{C}^N}$, Martineau proved in the 1960s that for each convex domain G of ${\mathbb{C}^N}$ the partial differential operator $P(D)f = \sum\nolimits_{\vert\alpha \vert \leq m} {{a_\alpha }{f^{(\alpha )}}}$ acting on the Fréchet space $A(G)$ of all analytic functions on G is surjective. In the present paper it is investigated whether solutions f of the equation $P(D)f = g$ can be chosen as $ f = R(g)$ with a continuous linear operator $ R:A(G) \to A(G)$. For bounded G we give a necessary and sufficient condition for the existence of such an R.
Rigidity of ergodic volume-preserving actions of semisimple groups of higher rank on compact manifolds
Guillaume
Seydoux
753-776
Abstract: Let M be a compact manifold, H a semisimple Lie group of higher rank (e.g., $H = SL(n,{\mathbf{R}})$ with $n \geq 3$) and $a \in \mathcal{A}(H,M)$ an ergodic H-action on M which preserves a volume v. Such an H-action is conjectured to be "locally rigid": if $a \prime$ is a sufficiently $ {C^1}$-small perturbation of a, then there should exist a diffeomorphism $\Phi$ of the manifold M which conjugates $a \prime$ to a. This conjecture would imply that if $\omega$ is an a-invariant geometrical structure on M, then there should exist an. $a \prime$-invariant geometrical structure $ \omega \prime$ on M of the same type. Using Kazhdan property, superrigidity for cocycles, and Sobolev spaces techniques we prove, under suitable conditions, two such results with $\omega = v$ and with $\omega$ a Riemannian metric along the leaves of a foliation of M.
Centered bodies and dual mixed volumes
Gao Yong
Zhang
777-801
Abstract: We establish a number of characterizations and inequalities for intersection bodies, polar projection bodies and curvature images of projection bodies in $ {{\mathbf{R}}^n}$ by using dual mixed volumes. One of the inequalities is between the dual Quermassintegrals of centered bodies and the dual Quermassintegrals of their central $(n - 1)$-slices. It implies Lutwak's affirmative answer to the Busemann-Petty problem when the body with the smaller sections is an intersection body. We introduce and study the intersection body of order i of a star body, which is dual to the projection body of order i of a convex body. We show that every sufficiently smooth centered body is a generalized intersection body. We discuss a type of selfadjoint elliptic differential operator associated with a convex body. These operators give the openness property of the class of curvature functions of convex bodies. They also give an existence theorem related to a well-known uniqueness theorem about deformations of convex hypersurfaces in global differential geometry.
Boundary behavior of the Bergman kernel function on some pseudoconvex domains in ${\bf C}\sp n$
Sanghyun
Cho
803-817
Abstract: Let $\Omega$ be a bounded pseudoconvex domain in $ {\mathbb{C}^n}$ with smooth defining function r and let ${z_0} \in b\Omega$ be a point of finite type. We also assume that the Levi form $\partial \bar \partial r(z)$ of $b\Omega$ has $(n - 2)$-positive eigenvalues at $ {z_0}$. Then we get a quantity which bounds from above and below the Bergman kernel function in a small constant and large constant sense.
Classifications of Baire-$1$ functions and $c\sb 0$-spreading models
V.
Farmaki
819-831
Abstract: We prove that if for a bounded function f defined on a compact space K there exists a bounded sequence $ ({f_n})$ of continuous functions, with spreading model of order $\xi$, $1 \leq \xi < {\omega _1}$, equivalent to the summing basis of ${c_0}$, converging pointwise to f, then ${r_{{\text{ND}}}}(f) > {\omega ^\xi }$ (the index $ {r_{{\text{ND}}}}$ as defined by A. Kechris and A. Louveau). As a corollary of this result we have that the Banach spaces ${V_\xi }(K)$, $1 \leq \xi < {\omega _1}$, which previously defined by the author, consist of functions with rank greater than ${\omega ^\xi }$. For the case $ \xi = 1$ we have the equality of these classes. For every countable ordinal number $\xi$ we construct a function S with $ {r_{{\text{ND}}}}(S) > {\omega ^\xi }$. Defining the notion of null-coefficient sequences of order $\xi$, $ 1 \leq \xi < {\omega _1}$, we prove that every bounded sequence $ ({f_n})$ of continuous functions converging pointwise to a function f with ${r_{{\text{ND}}}}(f) \leq {\omega ^\xi }$ is a null-coefficient sequence of order $\xi$. As a corollary to this we have the following ${c_0}$-spreading model theorem: Every nontrivial, weak-Cauchy sequence in a Banach space either has a convex block subsequence generating a spreading model equivalent to the summing basis of ${c_0}$ or is a null-coefficient sequence of order 1.
Stable vector bundles on algebraic surfaces
Wei-Ping
Li;
Zhenbo
Qin
833-852
Abstract: We prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of a certain moduli space of stable bundles on a rational ruled surface.
The representation of binary quadratic forms by positive definite quaternary quadratic forms
A. G.
Earnest
853-863
Abstract: A quadratic $\mathbb{Z}$-lattice L of rank n is denned to be k-regular for a positive integer $k \leq n$ if L globally represents all quadratic $ \mathbb{Z}$-lattices of rank k which are represented everywhere locally by L. It is shown that there exist only finitely many isometry classes of primitive positive definite quadratic $ \mathbb{Z}$-lattices of rank 4 which are 2-regular.
On power subgroups of profinite groups
Consuelo
Martínez
865-869
Abstract: In this paper we prove that if G is a finitely generated pro-(finite nilpotent) group, then every subgroup $ {G^n}$, generated by nth powers of elements of G, is closed in G. It is also obtained, as a consequence of the above proof, that if G is a nilpotent group generated by m elements $ {x_1}, \ldots ,{x_m}$, then there is a function $f(m,n)$ such that if every word in $x_i^{ \pm 1}$ of length $ \leq f(m,n)$ has order n, then G is a group of exponent n. This question had been formulated by Ol'shansky in the general case and, in this paper, is proved in the solvable case and the problem is reduced to the existence of such function for finite simple groups.
Normal tree orders for infinite graphs
J.-M.
Brochet;
R.
Diestel
871-895
Abstract: A well-founded tree T denned on the vertex set of a graph G is called normal if the endvertices of any edge of G are comparable in T. We study how normal trees can be used to describe the structure of infinite graphs. In particular, we extend Jung's classical existence theorem for trees of height $ \omega$ to trees of arbitrary height. Applications include a structure theorem for graphs without large complete topological minors. A number of open problems are suggested.
A class of exceptional polynomials
Stephen D.
Cohen;
Rex W.
Matthews
897-909
Abstract: We present a class of indecomposable polynomials of non prime-power degree over the finite field of two elements which are permutation polynomials on infinitely many finite extensions of the field. The associated geometric monodromy groups are the simple groups $PS{L_2}({2^k})$, where $k \geq 3$ and odd. (The first member of this class was previously found by P. Müller [17]. This realises one of only two possibilities for such a class which remain following deep work of Fried, Guralnick and Saxl [7]. The other is associated with $PS{L_2}({3^k})$, $k \geq 3$ , and odd in fields of characteristic 3.
The Selberg trace formula for ${\rm SL}(3,{\bf Z})\backslash{\rm SL}(3,{\bf R})/{\rm SO}(3,{\bf R})$
D. I.
Wallace
1-36
Abstract: In this paper we compute the trace formula for $SL(3,\mathbb{Z})$ in detail and refine it to a greater extent than has previously been done. We show that massive cancellation occurs in the parabolic terms, leading to a far simpler formula than had been thought possible.
Finite Group Actions on Siegel Modular Spaces
K. F.
Lai;
Ronnie
Lee
37-45
Abstract: The theory of nonabelian cohomology is used to show that the set of fixed points of a finite group acting on a Siegel modular space is a union of Shimura varieties
Differential equations for symmetric generalized ultraspherical polynomials
Roelof
Koekoek
47-72
Abstract: We look for differential equations satisfied by the generalized Jacobi polynomials $\{ P_n^{\alpha ,\beta ,M,N}(x)\} _{n = 0}^\infty$ which are orthogonal on the interval $[- 1,1]$ with respect to the weight function $\displaystyle \frac{{\Gamma (\alpha + \beta + 2)}}{{{2^{\alpha + \beta + 1}}\Ga... ...ta + 1)}}{(1 - x)^\alpha }{(1 + x)^\beta } + M\delta (x + 1) + N\delta (x - 1),$ where $ \alpha > - 1$, $\beta > - 1$, $M \geq 0$, and $N \geq 0$. In the special case that $\beta = \alpha$ and $N = M$ we find all differential equations of the form $\displaystyle \sum\limits_{i = 0}^\infty {{c_i}(x){y^{(i)}}(x) = 0,\quad y(x) = P_n^{\alpha ,\alpha ,M,M}(x),}$ where the coefficients $\{ {c_i}(x)\} _{i = 1}^\infty$ are independent of the degree n. We show that if $ M > 0$ only for nonnegative integer values of $\alpha$ there exists exactly one differential equation which is of finite order $2\alpha + 4$. By using quadratic transformations we also obtain differential equations for the polynomials $\{ P_n^{\alpha, \pm 1/2,0,N}(x)\} _{n = 0}^\infty$ for all $ \alpha > - 1$ and $ N \geq 0$.
Nonlinear quantum fields in $\geq 4$ dimensions and cohomology of the infinite Heisenberg group
J.
Pedersen;
I. E.
Segal;
Z.
Zhou
73-95
Abstract: Aspects of the cohomology of the infinite-dimensional Heisenberg group as represented on the free boson field over a given Hilbert space are treated. The 1-cohomology is shown to be trivial in certain spaces of generalized vectors. From this derives a canonical quantization mapping from classical (unquantized) forms to generalized operators on the boson field. An example, applied here to scalar relativistic fields, is the quantization of a given classical interaction Lagrangian or Hamiltonian, i.e., the establishment and characterization of corresponding boson field operators. For example, if $\phi$ denotes the free massless scalar field in d-dimensional Minkowski space ($d \geq 4$, even) and if q is an even integer greater than or equal to 4, then ${\smallint _{{{\mathbf{M}}_0}}}:\phi {(X)^q}:dX$ exists as a nonvanishing, Poincaré invariant, hermitian, selfadjointly extendable operator, where : $ \phi {(X)^q}$ : denotes the Wick power. Applications are also made to the rigorous establishment of basic symbolic operators in heuristic quantum field theory, including certain massive field theories; to a class of pseudo-interacting fields obtained by substituting the free field into desingularized expressions for the total Hamiltonian in the conformally invariant case $d = q = 4$ and to corresponding scattering theory.
Eigenvalues and eigenspaces for the twisted Dirac operator over ${\rm SU}(N,1)$ and ${\rm Spin}(2N,1)$
Esther
Galina;
Jorge
Vargas
97-113
Abstract: Let X be a symmetric space of noncompact type whose isometry group is either $SU(n,1)$ or $ Spin(2n,1)$. Then the Dirac operator D is defined on $ {L^2}$-sections of certain homogeneous vector bundles over X. Using representation theory we obtain explicitly the eigenvalues of D and describe the eigenspaces in terms of the discrete series.
Braid groups and left distributive operations
Patrick
Dehornoy
115-150
Abstract: The decidability of the word problem for the free left distributive law is proved by introducing a structure group which describes the underlying identities. This group is closely connected with Artin's braid group ${B_\infty }$. Braid colourings associated with free left distributive structures are used to show the existence of a unique ordering on the braids which is compatible with left translation and such that every generator $ {\sigma _i}$ is preponderant over all $ {\sigma _k}$ with $k > i$. This ordering is a linear ordering.
Analysis and applications of holomorphic functions in higher dimensions
R. Z.
Yeh
151-177
Abstract: Holomorphic functions in ${R^n}$ are defined to generalize those in $ {R^2}$. A Taylor formula and a Cauchy integral formula are presented. An application of the Taylor formula to the kernel of the Cauchy integral formula results in Taylor series expansions of holomorphic functions. Real harmonic functions are expanded in series of homogeneous harmonic polynomials.
Th\'eor\`eme de Ney-Spitzer sur le dual de ${\rm SU}(2)$
Philippe
Biane
179-194
Abstract: Let $\phi$ be a central, noneven, positive type function on $ {\text{SU}}(2)$ with $\phi (e) < 1$. For any polynomial function p on $ {\text{SU}}(2)$, let $ V(p)$ be the left convolution operator by $ p/(1 - \phi )$ on $ {L^2}({\text{SU}}(2))$, we prove that $V(p)/V(1)$ is a pseudodifferential operator of order 0 and give an explicit formula for its principal symbol. This is interpreted in terms of Martin compactification of a quantum random walk.
Lattice-ordered algebras that are subdirect products of valuation domains
Melvin
Henriksen;
Suzanne
Larson;
Jorge
Martinez;
R. G.
Woods
195-221
Abstract: An f-ring (i.e., a lattice-ordered ring that is a subdirect product of totally ordered rings) A is called an SV-ring if $A/P$ is a valuation domain for every prime ideal P of A . If M is a maximal $ \ell$-ideal of A, then the rank of A at M is the number of minimal prime ideals of A contained in M , rank of A is the sup of the ranks of A at each of its maximal $\ell$-ideals. If the latter is a positive integer, then A is said to have finite rank, and if $A = C(X)$ is the ring of all real-valued continuous functions on a Tychonoff space, the rank of X is defined to be the rank of the f-ring $ C(X)$, and X is called an SV-space if $C(X)$ is an ST-ring. X has finite rank k iff k is the maximal number of pairwise disjoint cozero sets with a point common to all of their closures. In general f-rings these two concepts are unrelated, but if A is uniformly complete (in particular, if $ A = C(X)$) then if A is an SV-ring then it has finite rank. Showing that this latter holds makes use of the theory of finite-valued lattice-ordered (abelian) groups. These two kinds of rings are investigated with an emphasis on the uniformly complete case. Fairly powerful machinery seems to have to be used, and even then, we do not know if there is a compact space X of finite rank that fails to be an SV-space.
Semigroups and stability of nonautonomous differential equations in Banach spaces
Nguyen Van Minh
223-241
Abstract: This paper is concerned with nonautonomous differential equations in Banach spaces. Using the theory of semigroups of linear and nonlinear operators one investigates the semigroups of weighted translation operators associated with the underlying equations. Necessary and sufficient conditions for different types of stability are given in terms of spectral properties of the translation operators and the differential operators associated with the equations.
Kinematic formulas for mean curvature powers of hypersurfaces and Hadwiger's theorem in ${\bf R}\sp {2n}$
Jia Zu
Zhou
243-262
Abstract: We first discuss the theory of hypersurfaces and submanifolds in the m-dimensional Euclidean space leading up to high dimensional analogues of the classical Euler's and Meusnier's theorems. Then we deduce the kinematic formulas for powers of mean curvature of the $ (m - 2)$-dimensional intersection submanifold $ {S_0} \cap g{S_1}$ of two $ {C^2}$-smooth hypersurfaces ${S_0}$, ${S_1}$, i.e., ${\smallint _G}({\smallint _{{S_0} \cap g{S_1}}}{H^{2k}}d\sigma )dg$. Many well-known results, for example, the C-S. Chen kinematic formula and Crofton type formulas are easy consequences of our kinematic formulas. As direct applications of our formulas, we obtain analogues of Hadwiger's theorem in ${\mathbb{R}^{2n}}$, i.e., sufficient conditions for one domain $ {K_\beta }$ to contain, or to be contained in, another domain ${K_\alpha }$.
Asymptotic measures for skew products of Bernoulli shifts with generalized north pole--south pole diffeomorphisms
D. K.
Molinek
263-291
Abstract: We study asymptotic measures for a certain class of dynamical systems. In particular, for $T:{\Sigma _2} \times M \to {\Sigma _2} \times M$, a skew product of the Bernoulli shift with a generalized north pole-south pole diffeomorphism, we describe the limits of the following two sequences of measures: (1) iterates under T of the product of Bernoulli measure with Lebesgue measure, $T_\ast ^n(\mu \times m)$, and (2) the averages of iterates of point mass measures, $\frac{1}{n}\Sigma _{k = 0}^{n - 1}{\delta _{{T^k}(w,x)}}$. We give conditions for the limit of each sequence to exist. We also determine the subsequential limits in case the sequence does not converge. We exploit several properties of null recurrent Markov Chains and apply them to the symmetric random walk on the integers. We also make use of Strassen's Theorem as an aid in determining subsequential limits.
Connections with exotic holonomy
Lorenz J.
Schwachhöfer
293-321
Abstract: Berger [Ber] partially classified the possible irreducible holonomy representations of torsion free connections on the tangent bundle of a manifold. However, it was shown by Bryant [Bry] that Berger's list is incomplete. Connections whose holonomy is not contained on Berger's list are called exotic. We investigate a certain 4-dimensional exotic holonomy representation of $Sl(2,\mathbb{R})$. We show that connections with this holonomy are never complete and do not exist on compact manifolds. We give explicit descriptions of these connections on an open dense set and compute their groups of symmetry.
On the additive formulae of the theta functions and a collection of Lambert series pertaining to the modular equations of degree $5$
Li-Chien
Shen
323-345
Abstract: We examine the connection between the additive formulae of the theta functions, the Fourier series expansion of the 12 elliptic functions, and the logarithmic derivatives of the theta functions. As an application, we study the Lambert series related to the modular equations of degree 5 and many interesting identities of Ramanujan are found in this process.
A homotopy invariance theorem in coarse cohomology and $K$-theory
Nigel
Higson;
John
Roe
347-365
Abstract: We introduce a notion of homotopy which is appropriate to the coarse geometry and topology studied by the second author in [7]. We prove the homotopy invariance of coarse cohomology, and of the K-theory of the $ {C^\ast}$-algebra associated to a coarse structure on a space. We apply our homotopy invariance results to show that if M is a Hadamard manifold then the inverse of the exponential map at any point 0 induces an isomorphism between the K-theory groups of the ${C^\ast}$-algebras associated to M and its tangent space at 0 (see Theorem 7.9). This result is consistent with a coarse version of the Baum-Connes conjecture.
Writing integers as sums of products
Charles E.
Chace
367-379
Abstract: In this paper we obtain an asymptotic expression for the number of ways of writing an integer N as a sum of k products of l factors, valid for $k \geq 3$ and $l \geq 2$. The proof is an application of the Hardy-Littlewood method, and uses recent results from the divisor problem for arithmetic progressions.
The spectra of random pseudo-differential operators
Jingbo
Xia
381-411
Abstract: We study the spectra of random pseudo-differential operators generated by the same symbol function on different $ {L^2}$-spaces. Our results generalize the spectral coincidence theorem of S. Kozlov and M. Shubin (Math. USSRSb. 51 (1985), 455-471) for elliptic operators of positive order associated with ergodic systems. Because of our new approach, we are able to treat operators of arbitrary order and associated with arbitrary dynamical systems. Furthermore, we characterize the spectra of these operators in terms of certain naturally obtained Borel measures on R.
A family of real $2\sp n$-tic fields
Yuan Yuan
Shen;
Lawrence C.
Washington
413-434
Abstract: We study the family of polynomials $\displaystyle {P_n}(X;a) = \Re ({(X + i)^{{2^n}}}) - \frac{a}{{{2^n}}}\Im ({(X + i)^{{2^n}}})$ and determine when ${P_n}(X;a)$, $a \in \mathbb{Z}$, is irreducible. The roots are all real and are permuted cyclically by a linear fractional transformation defined over the real subfield of the ${2^n}$th cyclotomic field. The families of fields we obtain are natural extensions of those studied by M.-N. Gras and Y.-Y. Shen, but in general the present fields are non-Galois for $n \geq 4$. From the roots we obtain a set of independent units for the Galois closure that generate an "almost fundamental piece" of the full group of units. Finally, we discuss the two examples where our fields are Galois, namely $a = \pm {2^n}$ and $a = \pm {2^4} \bullet 239$.
The functional determinant of a four-dimensional boundary value problem
Thomas P.
Branson;
Peter B.
Gilkey
479-531
Abstract: Working on four-dimensional manifolds with boundary, we consider, elliptic boundary value problems (A, B), A being the interior and B the boundary operator. These problems (A, B) should be valued in a tensorspinor bundle; should depend in a universal way on a Riemannian metric g and be formally selfadjoint; should behave in an appropriate way under conformal change $g \to {\Omega ^2}g$, $\Omega$ a smooth positive function; and the leading symbol of A should be positive definite. We view the functional determinant det $ {A_B}$ of such a problem as a functional on a conformal class $\{ {\Omega ^2}g\}$, and develop a formula for the quotient of the determinant at $ {\Omega ^2}g$ by that at g. (Analogous formulas are known to be intimately related to physical string theories in dimension two, and to sharp inequalities of borderline Sobolev embedding and Moser-Trudinger types for the boundariless case in even dimensions.) When the determinant in a background metric ${g_0}$ is explicitly computable, the result is a formula for the determinant at each metric ${\Omega ^2}{g_0}$ (not Just a quotient of determinants). For example, we compute the functional determinants of the Dirichlet and Robin (conformally covariant Neumann) problems for the Laplacian in the ball $ {B^4}$, using our general quotient formulas in the case of the conformal Laplacian, together with an explicit computation on the hemisphere ${H^4}$.
Densely defined selections of multivalued mappings
M. M.
Čoban;
P. S.
Kenderov;
J. P.
Revalski
533-552
Abstract: Rather general suficient conditions are found for a given multivalued mapping $F:X \to Y$ to possess an upper semicontinuous and compact-valued selection G which is defined on a dense $ {G_\delta }$-subset of the domain of F. The case when the selection G is single-valued (and continuous) is also investigated. The results are applied to prove some known as well as new results concerning generic differentiability of convex functions, Lavrentieff type theorem, generic well-posedness of optimization problems and generic non-multivaluedness of metric projections and antiprojections.
On $U$-rank $2$ types
Ludomir
Newelski
553-581
Abstract: Let T be a superstable theory with $< {2^{{\aleph _0}}}$ countable models. We study some special types $p \in S(\emptyset )$ of U-rank 2 called skeletal (cf. [Bu4]). We reduce an eventual version of the problem of counting isomorphism types of sets $p(M)$ for countable M to a problem from linear algebra.
Subanalytic functions
Adam
Parusiński
583-595
Abstract: We prove a strong version of rectilinearization theorem for subanalytic functions. Then we use this theorem to study the properties of arc-analytic functions.
Partition identities and labels for some modular characters
G. E.
Andrews;
C.
Bessenrodt;
J. B.
Olsson
597-615
Abstract: In this paper we prove two conjectures on partitions with certain conditions. A motivation for this is given by a problem in the modular representation theory of the covering groups ${\hat S_n}$ of the finite symmetric groups $ {S_n}$ in characteristic 5. One of the conjectures (Conjecture B below) has been open since 1974, when it was stated by the first author in his memoir [A3]. Recently the second and third author (jointly with A. O. Morris) arrived at essentially the same conjecture from a completely different direction. Their paper [BMO] was concerned with decomposition matrices of $ {\hat S_n}$ in characteristic 3. A basic difficulty for obtaining similar results in characteristic 5 (or larger) was the lack of a class of partitions which would be "natural" character labels for the modular characters of these groups. In this connection two conjectures were stated (Conjectures A and ${B^\ast}$ below), whose solutions would be helpful in the characteristic 5 case. One of them, Conjecture $ {{\text{B}}^\ast}$, is equivalent to the old Conjecture B mentioned above. Conjecture A is concerned with a possible inductive definition of the set of partitions which should serve as the required labels.
Best comonotone approximation
Frank
Deutsch;
Jun
Zhong
617-627
Abstract: A general theory of best comonotone approximation in $ C[a,b]$ by elements of an n-dimensional extended Chebyshev subspace is described. In particular, theorems on the existence, (in general) nonuniqueness, and characterization of best comonotone approximations are established.
The algorithmic theory of finitely generated metabelian groups
Gilbert
Baumslag;
Frank B.
Cannonito;
Derek J. S.
Robinson
629-648
Abstract: Algorithms are constructed which, when an explicit presentation of a finitely generated metabelian group G in the variety $ {\mathcal{A}^2}$ is given, produce finitary presentations for the derived subgroup $G\prime$, the centre $Z(G)$, the Fitting subgroup $\operatorname{Fit}(G)$, and the Frattini subgroup $\varphi (G)$. Additional algorithms of independent interest are developed for commutative algebra which construct the associated set of primes $\operatorname{Ass}(M)$ of a finitely generated module M over a finitely generated commutative ring R, and the intersection ${\varphi _R}(M)$ of the maximal submodules of M.
Nonorientable $4$-manifolds with fundamental group of order $2$
Ian
Hambleton;
Matthias
Kreck;
Peter
Teichner
649-665
Abstract: In this paper we classify nonorientable topological closed 4-manifolds with fundamental group $ \mathbb{Z}/2$ up to homeomorphism. Our results give a complete list of such manifolds, and show how they can be distinguished by explicit invariants including characteristic numbers and the $\eta$-invariant associated to a normal $ Pin^c$-structure by the spectral asymmetry of a certain Dirac operator. In contrast to the oriented case, there exist homotopy equivalent nonorientable topological 4-manifolds which are stably homeomorphic (after connected sum with ${S^2} \times {S^2}$) but not homeomorphic.
Approximation properties for group $C\sp *$-algebras and group von Neumann algebras
Uffe
Haagerup;
Jon
Kraus
667-699
Abstract: Let G be a locally compact group, let $ C_r^\ast(G)$ (resp. ${\text{VN}}(G)$) be the ${C^\ast}$-algebra (resp. the von Neumann algebra) associated with the left regular representation l of G, let $A(G)$ be the Fourier algebra of G, and let $ {M_0}A(G)$ be the set of completely bounded multipliers of $ A(G)$. With the completely bounded norm, ${M_0}A(G)$ is a dual space, and we say that G has the approximation property (AP) if there is a net $\{ {u_\alpha }\}$ of functions in $ A(G)$ (with compact support) such that $ {u_\alpha } \to 1$ in the associated weak $^\ast$-topology. In particular, G has the AP if G is weakly amenable ( $\Leftrightarrow A(G)$ has an approximate identity that is bounded in the completely bounded norm). For a discrete group $\Gamma$, we show that $\Gamma$ has the ${\text{AP}} \Leftrightarrow C_r^\ast(\Gamma )$ has the slice map property for subspaces of any ${C^\ast}$-algebra $\Leftrightarrow {\text{VN}}(\Gamma )$ has the slice map property for $\sigma$-weakly closed subspaces of any von Neumann algebra (Property $ {S_\sigma }$). The semidirect product of weakly amenable groups need not be weakly amenable. We show that the larger class of groups with the AP is stable with respect to semidirect products, and more generally, this class is stable with respect to group extensions. We also obtain some results concerning crossed products. For example, we show that the crossed product $ M{ \otimes _\alpha }G$ of a von Neumann algebra M with Property ${S_\sigma }$ by a group G with the AP also has Property $ {S_\sigma }$.
Homotopy groups in Lie foliations
Enrique
Macias-Virgós
701-711
Abstract: According to the results of Fédida and Molino [9], the structure of a G-Lie foliation F on a compact manifold M can be described by means of four locally trivial fibre bundles. In this paper we study the relations that those fibrations imply among the (rational) homotopy groups of: the manifold M, the generic leaf L, its closure $N = \bar L$, the basic manifold W, the Lie group G, and the structural Lie group H. Also, we prove that those relations are a particular case of an algebraic result concerning generalized homology theories.
Dirichlet problem at infinity for harmonic maps: rank one symmetric spaces
Harold
Donnelly
713-735
Abstract: Given a symmetric space M, of rank one and noncompact type, one compactifies M by adding a sphere at infinity, to obtain a manifold $M\prime$ with boundary. If $\bar M$ is another rank one symmetric space, suppose that $f:\partial M\prime \to \partial \bar M\prime$ is a continuous map. The Dirichlet problem at infinity is to construct a proper harmonic map $u:M \to \bar M$ with boundary values f. This paper concerns existence, uniqueness, and boundary regularity for this Dirichlet problem.
Measures of chaos and a spectral decomposition of dynamical systems on the interval
B.
Schweizer;
J.
Smítal
737-754
Abstract: Let $f:[0,1] \to [0,1]$ be continuous. For $x,y \in [0,1]$, the upper and lower (distance) distribution functions, $ F_{xy}^\ast$ and $ {F_{xy}}$, are defined for any $t \geq 0$ as the lim sup and lim inf as $n \to \infty$ of the average number of times that the distance $ \vert{f^i}(x) - {f^i}(y)\vert$ between the trajectories of x and y is less than t during the first n iterations. The spectrum of f is the system $\Sigma (f)$ of lower distribution functions which is characterized by the following properties: (1) The elements of $ \Sigma (f)$ are mutually incomparable; (2) for any $F \in \Sigma (f)$, there is a perfect set ${P_F} \ne \emptyset$ such that ${F_{uv}} = F$ and $F_{uv}^\ast \equiv 1$ for any distinct u, $v \in {P_F}$; (3) if S is a scrambled set for f, then there are F, G in $\Sigma (f)$ and a decomposition $S = {S_F} \cup {S_G}$ (${S_G}$ may be empty) such that ${F_{uv}} \geq F$ if u, $v \in {S_F}$ and $ {F_{uv}} \geq G$ if u, $v \in {S_G}$. Our principal results are: (1) If f has positive topological entropy, then $\Sigma (f)$ is nonempty and finite, and any $F \in \Sigma (f)$ is zero on an interval $[0,\varepsilon]$, where $\varepsilon > 0$ (and hence any ${P_F}$ is a scrambled set in the sense of Li and Yorke). (2) If f has zero topological entropy, then $ \Sigma (f) = \{ F\}$ where $F \equiv 1$. It follows that the spectrum of f provides a measure of the degree of chaos of f. In addition, a useful numerical measure is the largest of the numbers $\int_0^1 {(1 - F(t))dt} $, where $F \in \Sigma (f)$.
On an integral representation for the genus series for $2$-cell embeddings
D. M.
Jackson
755-772
Abstract: An integral representation for the genus series for maps on oriented surfaces is derived from the combinatorial axiomatisation of 2-cell embeddings in orientable surfaces. It is used to derive an explicit expression for the genus series for dipoles. The approach can be extended to vertex-regular maps in general and, in this way, may shed light on the genus series for quadrangulations. The integral representation is used in conjunction with an approach through the group algebra, $ \mathbb{C}{\mathfrak{G}_n}$, of the symmetric group [11] to obtain a factorisation of certain Gaussian integrals.
The dynamics of continuous maps of finite graphs through inverse limits
Marcy
Barge;
Beverly
Diamond
773-790
Abstract: Suppose that $ f:G \to G$ is a continuous piecewise monotone function on a finite graph G. Then the following are equivalent: (i) f has positive topological entropy; (ii) there are disjoint intervals ${I_1}$, and ${I_2}$ and a positive integer n with $\displaystyle {I_1} \cup {I_2} \subseteq {f^n}({I_1}) \cap {f^n}({I_2});$ (iii) the inverse limit space constructed by using f on G as a single bonding map contains an indecomposable subcontinuum. This result generalizes known results for the interval and circle.
Composition operators with closed range
Nina
Zorboska
791-801
Abstract: We characterize the closed-range composition operators on weighted Bergman spaces in terms of the ranges of the inducing maps on the unit disc. The method uses Nevanlinna's counting function and Luecking's results on inequalities on Bergman spaces.
Amenable actions of groups
Scot
Adams;
George A.
Elliott;
Thierry
Giordano
803-822
Abstract: The equivalence between different characterizations of amenable actions of a locally compact group is proved. In particular, this answers a question raised by R. J. Zimmer in 1977.
Infinitesimally stable endomorphisms
Hiroshi
Ikeda
823-833
Abstract: It is well known that infinitesimal stability of diffeomorphisms is an open property. However, infinitesimal stability of endomorphisms is not an open property. So we consider the interior of the set of all infinitesimally stable endomorphisms. We prove that if f belongs to the interior of the set of all infinitesimally stable endomorphisms, then f is $\Omega$-stable. This means a generalization of Smale's $\Omega$-stability theorem for diffeomorphisms. Moreover, it is proved that for Anosov endomorphisms structural stability is equivalent to lying in the interior of the set of infinitesimally stable endomorphisms.
There is just one rational cone-length
Octavian
Cornea
835-848
Abstract: We show that the homotopic nilpotency of the algebra of piecewise polynomial forms on a simply-connected, finite type, CW-complex coincides with the strong L.S. category of the rationalization of that space. This is used to prove that, in the rational, simply-connected context all reasonable notions of cone-length agree. Both these two results are obtained as parts of a more general and functorial picture.
Completely continuous composition operators
Joseph A.
Cima;
Alec
Matheson
849-856
Abstract: A composition operator ${T_b}f = f \circ b$ is completely continuous on ${H^1}$ if and only if $\vert b\vert < 1$ a.e. If the adjoint operator $ T_b^\ast$ is completely continuous on VMOA, then ${T_b}$ is completely continuous on $ {H^1}$. Examples are given to show that the converse fails in general. Two results are given concerning the relationship between the complete continuity of an operator and of its adjoint in the presence of certain separability conditions on the underlying Banach space.
The Picard group, closed geodesics and zeta functions
Mark
Pollicott
857-872
Abstract: In this article we consider the Picard group $ {\text{SL}}(2,\mathbb{Z}[i])$, viewed as a discrete subgroup of the isometries of hyperbolic space. We fix a canonical choice of generators and then construct a Markov partition for the action of the group on the sphere at infinity. Our main application is to the study of the zeta function associated to the associated three-dimensional hyperbolic manifold.
Notes sur la propri\'et\'e de Namioka
Ahmed
Bouziad
873-883
Abstract: We show that the class of co-Namioka compacts is stable under the arbitrary product if and only if it is stable under the finished product. We also prove that if X is a Valdivia compact space, then for every co-Namioka compact Y the product $X \times Y$ is co-Namioka. Several examples of co-Namioka compacts are given.
Hochschild homology in a braided tensor category
John C.
Baez
885-906
Abstract: An r-algebra is an algebra A over k equipped with a Yang-Baxter operator $R:A \otimes A \to A \otimes A$ such that $R(1 \otimes a) = a \otimes 1$, $R(a \otimes 1) = 1 \otimes a$, and the quasitriangularity conditions $ R(m \otimes I) = (I \otimes m)(R \otimes I)(I \otimes R)$ and $R(I \otimes m) = (m \otimes I)(I \otimes R)(R \otimes I)$ hold, where $ m:A \otimes A \to A$ is the multiplication map and $I:A \to A$ is the identity. R-algebras arise naturally as algebra objects in a braided tensor category of k-modules (e.g., the category of representations of a quantum group). If $m = m{R^2}$, then A is both a left and right module over the braided tensor product $ {A^e} = A\hat \otimes {A^{{\text{op}}}}$, where $ {A^{{\text{op}}}}$ is simply A equipped with the "opposite" multiplication map ${m^{{\text{op}}}} = mR$. Moreover, there is an explicit chain complex computing the braided Hochschild homology ${H^R}(A) = \operatorname{Tor}^{{A^e}}(A,A)$. When $m = mR$ and ${R^2} = {\text{id}}_{A \otimes A}$, this chain complex admits a generalized shuffle product, and there is a homomorphism from the r-commutative differential forms $ {\Omega _R}(A)$ to $ {H^R}(A)$.
A distortion theorem for biholomorphic mappings in ${\bf C}\sp 2$
Roger W.
Barnard;
Carl H.
FitzGerald;
Sheng
Gong
907-924
Abstract: Let ${J_f}$ be the Jacobian of a normalized biholomorphic mapping f from the unit ball $ {B^2}$ into ${\mathbb{C}^2}$. An expression for the $\log \det {J_f}$ is determined by considering the series expansion for the renormalized mappings F obtained from f under the group of holomorphic automorphisms of ${B^2}$. This expression is used to determine a bound for $\vert\det {J_f}\vert$ and $\vert\arg \det {J_f}\vert$ for f in a compact family X of normalized biholomorphic mappings from ${B^2}$ into $ {\mathbb{C}^2}$ in terms of a bound $C(X)$ of a certain combination of second-order coefficients. Estimates are found for $C(X)$ for the specific family X of normalized convex mappings from ${B^2}$ into ${\mathbb{C}^2}$.
The Jacobson radical of a CSL algebra
Kenneth R.
Davidson;
John Lindsay
Orr
925-947
Abstract: Extrapolating from Ringrose's characterization of the Jacobson radical of a nest algebra, Hopenwasser conjectured that the radical of a CSL algebra coincides with the Ringrose ideal (the closure of the union of zero diagonal elements with respect to finite sublattices). A general interpolation theorem is proved that reduces this conjecture for completely distributive lattices to a strictly combinatorial problem. This problem is solved for all width two lattices (with no restriction of complete distributivity), verifying the conjecture in this case.
The Cauchy problem in ${\bf C}\sp N$ for linear second order partial differential equations with data on a quadric surface
Gunnar
Johnsson
1-48
Abstract: By means of a method developed essentially by Leray some global existence results are obtained for the problem referred to in the title. The partial differential equations are required to have constant principal part and the initial surface to be irreducible and not everywhere characteristic. The Cauchy data are assumed to be given by entire functions. Under these conditions the location of all possible singularities of solutions are determined. The sets of singularities can be divided into two types, K- and L-singularities. K, the set of K-singularities, is the global version of the characteristic tangent defined by Leray. The L-sets are here quadric surfaces which, in contrast to the Ksets, allow unbounded singularities. The L-sets are in turn divided into three types: initial, asymptotic, and latent singularities. The initial singularities appear when the characteristic points of the initial surface are exceptional according to Leray's local theory. These sets of singularity intersect the initial surface at characteristic points. The asymptotic case, where the set of singularities does not cut the initial surface, can be viewed as projectively equivalent to the initial case, the intersection taking place at infinite characteristic points. Finally the latent singularities are sets which intersect the initial surface, but where the solutions do not develop singularities initially. In the case of the Laplace equation with data on a real quadric surface it is shown that the K-singularities and the asymptotic singularities occur on the classical focal sets defined by Poncelet, Plücker, Darboux et al. There are also latent singularities appearing in coordinate subspaces of $ {\mathbb{R}^N}$. As a corollary a new proof is given of the fact that ellipsoids have the Pompeiu property.
The Inverse Stable Range Functor
Robert S. Y.
Young
49-56
Abstract: We give an inverse construction of the stable range for general flows which may or may not admit an invariant measure. The inverse map is then shown to be a right inverse functor of the stable range functor.
Finitely generated Kleinian groups in $3$-space and $3$-manifolds of infinite homotopy type
L.
Potyagaĭlo
57-77
Abstract: We prove the existence of a finitely generated Kleinian group $N \subset S{O_ + }(1,4)$ acting freely on an invariant component $ \Omega \subset {S^3}$ without parabolic elements such that the fundamental group ${\pi _1}(\Omega /N)$ is not finitely generated. Moreover, N is a finite index subgroup of a Kleinian group ${N_0}$ which has infinitely many conjugacy classes of elliptic elements.
Integer points on curves of genus two and their Jacobians
David
Grant
79-100
Abstract: Let C be a curve of genus 2 defined over a number field, and $ \Theta$ the image of C embedded into its Jacobian J. We show that the heights of points of J which are integral with respect to $ {[2]_\ast}\Theta$ can be effectively bounded. As a result, if P is a point on C, and $\bar P$ its image under the hyperelliptic involution, then the heights of points on C which are integral with respect to P and $\bar P$ can be effectively bounded, in such a way that we can isolate the dependence on P, and show that if the height of P is bigger than some bound, then there are no points which are S-integral with respect to P and $\bar P$. We relate points on C which are integral with respect to P to points on J which are integral with respect to $ \Theta$, and discuss approaches toward bounding the heights of the latter.
Homoclinic loop and multiple limit cycle bifurcation surfaces
L. M.
Perko
101-130
Abstract: This paper establishes the existence and analyticity of homoclinic loop bifurcation surfaces $ \mathcal{H}$ and multiplicity-two, limit cycle bifurcation surfaces $\mathcal{C}$ for planar systems depending on two or more parameters; it determines the side of $\mathcal{H}$ or $ \mathcal{C}$ on which limit cycles occur; and it shows that if $\mathcal{H}$ and $ \mathcal{C}$ intersect, then typically they do so at a flat contact.
Graphs with the circuit cover property
Brian
Alspach;
Luis
Goddyn;
Cun Quan
Zhang
131-154
Abstract: A circuit cover of an edge-weighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly $p(e)$ circuits in the multiset. A nonnegative integer valued weight vector p is admissible if the total weight of any edge-cut is even, and no edge has more than half the total weight of any edge-cut containing it. A graph G has the circuit cover property if (G, p) has a circuit cover for every admissible weight vector p. We prove that a graph has the circuit cover property if and only if it contains no subgraph homeomorphic to Petersen's graph. In particular, every 2-edge-connected graph with no subgraph homeomorphic to Petersen's graph has a cycle double cover.
Spectral analysis for the generalized Hermite polynomials
Allan M.
Krall
155-172
Abstract: The operator theory associated with the Hermite polynomials does not extend to the generalized Hermite polynomials because the even and odd polynomials satisfy different differential equations. We show that this leads to two problems, each of interest on its own. We then weld them together to form a united spectral expansion. In addition, the exponent $\mu$ in the weight $\vert x{\vert^{2\mu }}{e^{ - {x^2}}}$ has traditionally always been greater than $- \frac{1}{2}$. We show what happens if $\mu \leq - \frac{1}{2}$. Finally, we examine the differential equations in left-definite spaces.
Representable $K$-theory of smooth crossed products by ${\bf R}$ and ${\bf Z}$
N. Christopher
Phillips;
Larry B.
Schweitzer
173-201
Abstract: We show that the Thorn isomorphism and the Pimsner-Voiculescu exact sequence both hold for smooth crossed products of Fréchet algebras by $ \mathbb{R}$ and $\mathbb{Z}$ respectively. We also obtain the same results for ${L^1}$-crossed products of Banach algebras by $\mathbb{R}$ and $ \mathbb{Z}$.
Minimal torsion in isogeny classes of elliptic curves
Raymond
Ross
203-215
Abstract: Let K be a field finitely generated over its prime field, and let $ w(K)$ denote the number of roots of unity in K. If K is of characteristic 0, then there is an integer D, divisible only by those primes dividing $w(K)$, such that for any elliptic curve $ E/K$ without complex multiplication over K, there is an elliptic curve $ E\prime/K$ isogenous to E such that $ E\prime{(K)_{{\text{tors}}}}$ is of order dividing D. In case K admits a real embedding, we show $D = 2$, and a nonuniform result is proved in positive characteristic.
Generalised Castelnuovo inequalities
Liam A.
Donohoe
217-260
Abstract: Given a Riemann surface of genus p, denoted by ${X_p}$, admitting j linear series of dimension r and degree n Accola derived a polynomial function $f(j,n,r)$ so that $p \leq f(j,n,r)$ and exhibited plane models of Riemann surfaces attaining equality in the inequality. In this paper we provide a classification of all such ${X_p}$ when $r \geq 6$. In addition we classify curves, $ {X_p}$, of maximal genus when ${X_p}$ admits two linear series which have a common dimension but different degrees.
The Brown-Peterson homology of Mahowald's $X\sb k$ spectra
Dung Yung
Yan
261-289
Abstract: We compute the Brown-Peterson homology of Mahowald's ${X_k}$ spectrum which is the Thom spectrum induced from $ \Omega {J_{{2^k} - 1}}{S^2} \to {\Omega ^2}{S^3} - {\text{BO}}$, and the edge homomorphism of the Adams-Novikov spectral sequence for $ {\pi _\ast}({X_k})$. We then compute the nonnilpotent elements of ${\pi _\ast}({X_k})$.
$3$-primary $v\sb 1$-periodic homotopy groups of $F\sb 4$ and $E\sb 6$
Martin
Bendersky;
Donald M.
Davis
291-306
Abstract: We compute the 3-primary ${v_1}$-periodic homotopy groups of the exceptional Lie groups ${F_4}$ and ${E_6}$. The unstable Novikov spectral sequence is used for the most delicate part of the analysis.
Mean value inequalities in Hilbert space
F. H.
Clarke;
Yu. S.
Ledyaev
307-324
Abstract: We establish a new mean value theorem applicable to lower semi-continuous functions on Hilbert space. A novel feature of the result is its "multidirectionality": it compares the value of a function at a point to its values on a set. We then discuss some refinements and consequences of the theorem, including applications to calculus, flow invariance, and generalized solutions to partial differential equations. Résumé. On établit un nouveau théorème de la valeur moyenne qui s'applique aux fonctions semicontinues inférieurement sur un espace de Hilbert. On déduit plusieurs conséquences du résultat portant, par exemple, sur les fonctions monotones et sur les solutions généralisées des équations aux dérivées partielles.
A general view of reflexivity
Don
Hadwin
325-360
Abstract: Various concepts of reflexivity for an algebra or linear space of operators have been studied by operator theorists and algebraists. This paper contains a very general version of reflexivity based on dual pairs of vector spaces over a Hausdorff field. The special cases include topological, algebraic and approximate reflexivity. In addition general versions of hyperreflexivity and direct integrals are introduced. We prove general versions of many known (and some new) theorems, often with simpler proofs.
Groups and fields interpretable in separably closed fields
Margit
Messmer
361-377
Abstract: We prove that any infinite group interpretable in a separably closed field F of finite Eršov-invariant is definably isomorphic to an F-algebraic group. Using this result we show that any infinite field K interpretable in a separably closed field F is itself separably closed; in particular, in the finite invariant case K is definably isomorphic to a finite extension of F.
Structural instability of exponential functions
Zhuan
Ye
379-389
Abstract: We first prove some equivalent statements on J-stability of families of critically finite entire functions. Then, with these in hand, a conjecture concerning stability of the family of exponential functions is affirmatively answered in some cases.
A $4$-dimensional Kleinian group
B. H.
Bowditch;
G.
Mess
391-405
Abstract: We give an example of a 4-dimensional Kleinian group which is finitely generated but not finitely presented, and is a subgroup of a cocompact Kleinian group.
A Banach space not containing $c\sb 0, l\sb 1$ or a reflexive subspace
W. T.
Gowers
407-420
Abstract: An infinite-dimensional Banach space is constructed which does not contain ${c_0}$, ${l_1}$ or an infinite-dimensional reflexive subspace. In fact, it does not even contain $ {l_1}$ or an infinite-dimensional subspace with a separable dual.
A differential operator for symmetric functions and the combinatorics of multiplying transpositions
I. P.
Goulden
421-440
Abstract: By means of irreducible characters for the symmetric group, formulas have previously been given for the number of ways of writing permutations in a given conjugacy class as products of transpositions. These formulas are alternating sums of binomial coefficients and powers of integers. Combinatorial proofs are obtained in this paper by analyzing the action of a partial differential operator for symmetric functions.
On nonlinear delay differential equations
A.
Iserles
441-477
Abstract: We examine qualitative behaviour of delay differential equations of the form $\displaystyle y\prime (t) = h(y(t),\;y(qt)),\quad y(0) = {y_0},$ where h is a given function and $q > 0$. We commence by investigating existence of periodic solutions in the case of $h(u,v) = f(u) + p(v)$, where f is an analytic function and p a polynomial. In that case we prove that, unless q is a rational number of a fairly simple form, no nonconstant periodic solutions exist. In particular, in the special case when f is a linear function, we rule out periodicity except for the case when $q = 1/\deg p$. If, in addition, p is a quadratic or a quartic, we show that this result is the best possible and that a nonconstant periodic solution exists for $q = \frac{1}{2}$ or $ \frac{1}{4}$, respectively. Provided that g is a bivariate polynomial, we investigate solutions of the delay differential equation by expanding them into Dirichlet series. Coefficients and arguments of these series are derived by means of a recurrence relation and their index set is isomorphic to a subset of planar graphs. Convergence results for these Dirichlet series rely heavily upon the derivation of generating functions of such graphs, counted with respect to certain recursively-defined functionals. We prove existence and convergence of Dirichlet series under different general conditions, thereby deducing much useful information about global behaviour of the solution.
On the core of a cone-preserving map
Bit Shun
Tam;
Hans
Schneider
479-524
Abstract: This is the third of a sequence of papers in an attempt to study the Perron-Frobenius theory of a nonnegative matrix and its generalizations from the cone-theoretic viewpoint. Our main object of interest here is the core of a cone-preserving map. If A is an $n \times n$ real matrix which leaves invariant a proper cone K in $ {\mathbb{R}^n}$, then by the core of A relative to K, denoted by $ {\text{core}}_K(A)$, we mean the convex cone $\bigcap\nolimits_{i = 1}^\infty {{A^i}K}$. It is shown that when $ {\text{core}}_K(A)$ is polyhedral, which is the case whenever K is, then $ {\text{core}}_K(A)$ is generated by the distinguished eigenvectors of positive powers of A. The important concept of a distinguished A-invariant face is introduced, which corresponds to the concept of a distinguished class in the nonnegative matrix case. We prove a significant theorem which describes a one-to-one correspondence between the distinguished A-invariant faces of K and the cycles of the permutation induced by A on the extreme rays of ${\text{core}}_K(A)$, provided that the latter cone is nonzero, simplicial. By an interplay between cone-theoretic and graph-theoretic ideas, the extreme rays of the core of a nonnegative matrix are fully described. Characterizations of K-irreducibility or A-primitivity of A are also found in terms of $ {\text{core}}_K(A)$. Several equivalent conditions are also given on a matrix with an invariant proper cone so that its spectral radius is an eigenvalue of index one. An equivalent condition in terms of the peripheral spectrum is also found on a real matrix A with the Perron-Schaefer condition for which there exists a proper invariant cone K suchthat $ {\text{core}}_K(A)$ is polyhedral, simplicial, or a single ray. A method of producing a large class of invariant proper cones for a matrix with the Perron-Schaefer condition is also offered.
Minimal models and degenerations of surfaces with Kodaira number zero
Bruce
Crauder;
David R.
Morrison
525-558
Abstract: For Ueno's degenerations of abelian surfaces and for analytic flowerpot degenerations of surfaces with Kodaira number zero, we find analytically $ \mathbb{Q}$-factorial terminal minimal models. All contractions, singularities and flips are given by explicit geometric constructions.
A maximally pathological Brouwer homeomorphism
Edward Warwick
Daw
559-573
Abstract: This paper constructs a Brouwer homeomorphism which does not act properly discontinuously on any nonempty invariant closed connected set in $ {\mathbb{R}^2}$ .
Minimal displacement of points under holomorphic mappings and fixed point properties for unions of convex sets
Tadeusz
Kuczumow;
Simeon
Reich;
Adam
Stachura
575-586
Abstract: Let D be an open convex bounded subset of a complex Banach space $ (X,\left\Vert \cdot \right\Vert)$, and let C be the union of a finite number of closed convex sets lying strictly inside D. Using the Kuratowski measure of noncompactness with respect to the Kobayashi distance in D, we first show that if $f:D \to D$ is a holomorphic mapping which leaves C invariant, and if the Lefschetz number $\lambda ({f_{\vert C}}) \ne 0$, then $\inf \{ \left\Vert {x - f(x)} \right\Vert:x \in C\} = 0$. We then deduce several new fixed point theorems for holomorphic and nonexpansive mappings.
The orders of solutions of the Kummer system of congruences
Ladislav
Skula
587-607
Abstract: A new method concerning solutions of the Kummer system of congruences (K) (modulo an odd prime l) is developed. This method is based on the notion of the Stickelberger ideal. By means of this method a new proof of Pollaczek's and Morishima's assertion on solutions of (K) of orders 3, 6 and 4 $\bmod\; l$ is given. It is also shown that in case there is a solution of $(K)\, \not\equiv \, 0, \pm 1\;\pmod l$, then for the index of irregularity $i(l)$ of the prime l we have $i(l) \geq [\sqrt[3]{{l/2}}]$.
Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis
Boris
Mordukhovich
609-657
Abstract: In this paper we develop a stability theory for broad classes of parametric generalized equations and variational inequalities in finite dimensions. These objects have a wide range of applications in optimization, nonlinear analysis, mathematical economics, etc. Our main concern is Lipschitzian stability of multivalued solution maps depending on parameters. We employ a new approach of nonsmooth analysis based on the generalized differentiation of multivalued and nonsmooth operators. This approach allows us to obtain effective sufficient conditions as well as necessary and sufficient conditions for a natural Lipschitzian behavior of solution maps. In particular, we prove new criteria for the existence of Lipschitzian multivalued and single-valued implicit functions.
Fractal properties of invariant subsets for piecewise monotonic maps on the interval
Franz
Hofbauer;
Mariusz
Urbański
659-673
Abstract: Let T be a piecewise monotonic transformation on [0, 1] and let A be a T-invariant subset, which has positive topological entropy and satisfies the Darboux property. A general existence theorem for conformal measures on A is proved. This is then used to show equality of the dynamical dimension of A and the minimal zero of a certain pressure function.
Weak solutions of hyperbolic-parabolic Volterra equations
Gustaf
Gripenberg
675-694
Abstract: The existence of a global weak solution, satisfying certain a priori ${L^\infty }$-bounds, of the equation ${u_t}(t,x) = \int _0^tk(t - s){(\sigma ({u_x}))_x}(s,x)ds + f(t,x)$ is established. The kernel k is locally integrable and log-convex, and $\sigma \prime$ has only one local minimum which is positive.
On realization of Bj\"orner's ``continuous partition lattice'' by measurable partitions
Mark D.
Haiman
695-711
Abstract: Björner [1] showed how a construction by von Neumann of examples of continuous geometries can be adapted to construct a continuous analogue of finite partition lattices. Björner's construction realizes the continuous partition lattice abstractly, as a completion of a direct limit of finite lattices. Here we give an alternative construction realizing a continuous partition lattice concretely as a lattice of measurable partitions. This new lattice contains the Björner lattice and shares its key properties. Furthermore its automorphism group is the full automorphism group $ \pmod 0$ of the unit interval with Lebesgue measure, whereas, as we show, the Björner lattice possesses only a proper subgroup of these automorphisms.
Le th\'eor\`eme de Fermat-Goss
Laurent
Denis
713-726
Abstract: The analogue of the Fermat equation and of the Fermat conjecture is studied by Goss [G], on the rational function fields in characteristic $p > 0$. We prove here that this equation has no nontrivial solutions. When $q = 2$, the method uses the canonical height on the t-module constructed in [D]. This method also gives finiteness theorems for some generalization of the Fermat equation in higher dimension.
Powers of ideals generated by quadratic sequences
K.
Raghavan
727-747
Abstract: Huneke's conjecture that weak d-sequences generate ideals of quadratic type is proved. The proof suggests the definition of quadratic sequences, which are more general than weak d-sequences yet simpler to define and handle, in addition to being just as useful. We extend the theory of d-sequences and weak d-sequences to quadratic sequences. Results of Costa on sequences of linear type are generalized. An example of a two-dimensional local domain in which every system of parameters is a d-sequence in some order but which nevertheless fails to be Buchsbaum is given. A criterion is established for when equality holds in Burch's inequality for an ideal generated by a quadratic sequence.
On a two-dimensional elliptic problem with large exponent in nonlinearity
Xiaofeng
Ren;
Juncheng
Wei
749-763
Abstract: A semilinear elliptic equation on a bounded domain in ${R^2}$ with large exponent in the nonlinear term is studied in this paper. We investigate positive solutions obtained by the variational method. It turns put that the constrained minimizing problem possesses nice asymptotic behavior as the nonlinear exponent, serving as a parameter, gets large. We shall prove that $ {c_p}$, the minimum of energy functional with the nonlinear exponent equal to p, is like ${(8\pi e)^{1/2}}{p^{ - 1/2}}$ as p tends to infinity. Using this result, we shall prove that the variational solutions remain bounded uniformly in p. As p tends to infinity, the solutions develop one or two peaks. Precisely the solutions approach zero except at one or two points where they stay away from zero and bounded from above. Then we consider the problem on a special class of domains. It turns out that the solutions then develop only one peak. For these domains, the solutions enlarged by a suitable quantity behave like a Green's function of $- \Delta$. In this case we shall also prove that the peaks must appear at a critical point of the Robin function of the domain.
Hodge decompositions and Dolbeault complexes on normal surfaces
Jeffrey
Fox;
Peter
Haskell
765-778
Abstract: Give the smooth subset of a normal singular complex projective surface the metric induced from the ambient projective space. The ${L^2}$ cohomology of this incomplete manifold is isomorphic to the surface's intersection cohomology, which has a natural Hodge decomposition. This paper identifies Dolbeault complexes whose $\bar \partial$-closed and $\bar \partial$-coclosed forms represent the classes of pure type in the corresponding Hodge decomposition of ${L^2}$ cohomology.
Flows in fibers
Jun-ichi
Tanaka
779-804
Abstract: Let ${H^\infty }(\Delta )$ be the algebra of all bounded analytic functions on the open unit disc $ \Delta$, and let $ \mathfrak{M}({H^\infty }(\Delta ))$ be the maximal ideal space of ${H^\infty }(\Delta )$. Using a flow, we represent a reasonable portion of a fiber in $ \mathfrak{M}({H^\infty }(\Delta ))$. This indicates a relation between the corona theorem and the individual ergodic theorem. As an application, we answer a question of Forelli [3] by showing that there exists a minimal flow on which the induced uniform algebra is not a Dirichlet algebra. The proof rests on the fact that the closure of a nonhomeomorphic part in $\mathfrak{M}({H^\infty }(\Delta ))$ may contain a homeomorphic copy of $\mathfrak{M}({H^\infty }(\Delta ))$. Taking suitable factors, we may construct a lot of minimal flows which are not strictly ergodic.
Geometry of $2$-step nilpotent groups with a left invariant metric. II
Patrick
Eberlein
805-828
Abstract: We obtain a partial description of the totally geodesic submanifolds of a 2-step, simply connected nilpotent Lie group with a left invariant metric. We consider only the case that N is nonsingular; that is, $ {\text{ad}}\xi :\mathcal{N} \to \mathcal{Z}$ is surjective for all elements $\xi \in \mathcal{N} - \mathcal{Z}$, where $\mathcal{N}$ denotes the Lie algebra of N and $\mathcal{Z}$ denotes the center of $\mathcal{N}$. Among other results we show that if H is a totally geodesic submanifold of N with $ \dim H \geq 1 + \dim \mathcal{Z}$, then H is an open subset of $ g{N^\ast}$, where g is an element of H and ${N^\ast}$ is a totally geodesic subgroup of N. We find simple and useful criteria that are necessary and sufficient for a subalgebra ${\mathcal{N}^\ast}$ of $ \mathcal{N}$ to be the Lie algebra of a totally geodesic subgroup $ {N^\ast}$. We define and study the properties of a Gauss map of a totally geodesic submanifold H of N. We conclude with a characterization of 2-step nilpotent Lie groups N of Heisenberg type in terms of the abundance of totally geodesic submanifolds of N.
Entropy solutions for diffusion-convection equations with partial diffusivity
M.
Escobedo;
J. L.
Vázquez;
Enrike
Zuazua
829-842
Abstract: We consider the Cauchy problem for the following scalar conservation law with partial viscosity $\displaystyle {u_t} = {\Delta _x}u + {\partial _y}(f(u)),\quad (x,y) \in {{\mathbf{R}}^N},t > 0.$ The existence of solutions is proved by the vanishing viscosity method. By introducing a suitable entropy condition we prove uniqueness of solutions. This entropy condition is inspired by the entropy criterion introduced by Kruzhkov for hyperbolic conservation laws but it takes into account the effect of diffusion.
Escape rate for $2$-dimensional Brownian motion conditioned to be transient with application to Zygmund functions
Elizabeth Ann
Housworth
843-852
Abstract: The escape rate of a 2-dimensional Brownian motion conditioned to be transient is determined to be $P\{ X(t) < f(t)$ i.o. as $t \uparrow \infty \} = 0$ or 1 according as $\sum\nolimits_{n = 1}^\infty {{e^{ - n}}\log f({e^{{e^n}}}) < }$ or $= \infty$. The result is used to construct a complex-valued Zygmund function--as a lacunary series--whose graph does not have $\sigma$-finite linear Hausdorff measure. This contrasts the result of Mauldin and Williams that the graphs of all real-valued Zygmund functions have $ \sigma$-finite linear Hausdorff measure.
Quasiconformal mappings and Ahlfors-David curves
Paul
MacManus
853-881
Abstract: We show that if $ \rho$ is a quasiconformal mapping of the plane whose dilatation satisfies a certain quadratic Carleson measure condition relative to an Ahlfors-David curve $\Gamma$ then $\rho$ is differentiable almost everywhere on $ \Gamma$ and $ \log \vert\rho \prime \vert \in {\text{BMO}}$. When $\Gamma$ is chord-arc we show that its image is a Bishop-Jones curve. If the Carleson norm is small then we show that $\rho$ is absolutely continuous on $\Gamma$, the image of $\Gamma$ is an Ahlfors-David curve, and $\rho \prime = {e^a}$, where $a \in {\text{BMO}}$ with a small norm.
Une dichotomie de Hopf pour les flots g\'eod\'esiques associ\'es aux groupes discrets d'isom\'etries des arbres
M.
Coornaert;
A.
Papadopoulos
883-898
Abstract: Let X be a complete locally compact metric tree and $ \Gamma$ a group of isometries of X acting properly on this space. The space of bi-infinite geodesics in X constitutes a space GX on which $\Gamma$ acts properly. Let $\Omega$ be the quotient of GX by this action. The geodesic flow associated to $ \Gamma$ is the flow on $ \Omega$ which is the quotient of the geodesic flow on GX, defined by the time-shift on geodesics. To any $\Gamma $-conformal measure on the boundary $ \partial X$ there is an associated measure m on $\Omega$ which is invariant by the geodesic flow. We prove the following results: The geodesic flow on $(\Omega ,m)$ is either conservative or dissipative. If it is conservative, then it is ergodic, If it is dissipative, then it is not ergodic unless it is measurably conjugate to the action of $\mathbb{R}$ on itself by conjugation. We prove also a dichotomy in terms of the conical limit set $ {\Lambda _c} \subset \partial X$ of $\Gamma$: the flow on $ (\Omega ,m)$ is conservative if and only if $\mu ({\Lambda _c}) = \mu (\partial X)$, and it is dissipative if and only if $\mu ({\Lambda _c}) = 0$. The results are analogous to results of E. Hopf and D. Sullivan in the case of Riemannian manifolds of constant negative curvature.
On the force between rotating co-axial black holes
Gilbert
Weinstein
899-906
Abstract: We study the force between rotating coaxial black holes, as it was defined in [9 and 10]. We show that under a certain limit, the force is attractive, and in fact tends to infinity. This lends support to the conjecture that the force is always positive.
Radially symmetric solutions to a Dirichlet problem involving critical exponents
Alfonso
Castro;
Alexandra
Kurepa
907-926
Abstract: In this paper we answer, for $N = 3,4$, the question raised in [1] on the number of radially symmetric solutions to the boundary value problem $- \Delta u(x) = \lambda u(x) + u(x)\vert u(x){\vert^{4/(N - 2)}}$, $x \in B: = \{ x \in {R^N}:\left\Vert x \right\Vert < 1\}$, $u(x) = 0$, $x \in \partial B$, where $ \Delta$ is the Laplacean operator and $ \lambda > 0$. Indeed, we prove that if $N = 3,4$, then for any $ \lambda > 0$ this problem has only finitely many radial solutions. For $ N = 3,4,5$ we show that, for each $\lambda > 0$, the set of radially symmetric solutions is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.
Holomorphic motions and Teichm\"uller spaces
C. J.
Earle;
I.
Kra;
S. L.
KrushkalЬ
927-948
Abstract: We prove an equivariant form of Slodkowski's theorem that every holomorphic motion of a subset of the extended complex plane $\hat{\mathbb{C}}$ extends to a holomorphic motion of $ \widehat{\mathbb{C}}$. As a consequence we prove that every holomorphic map of the unit disc into Teichmüller space lifts to a holomorphic map into the space of Beltrami forms. We use this lifting theorem to study the Teichmüller metric.
Interpolation of weighted and vector-valued Hardy spaces
Serguei V.
Kisliakov;
Quan Hua
Xu
1-34
Abstract: Real and complex interpolation methods, when applied to the couple $ ({H^{{p_0}}}({E_0};{w_0}),{H^{{p_1}}}({E_1};{w_1}))$, give what is expected if $ {E_0}$ and ${E_1}$ are quasi-Banach lattices of measurable functions satisfying certain mild conditions and if $\log (w_0^{1/{p_0}}w_1^{ - 1/{p_1}}) \in {\text{BMO}}\,({w_0},{w_1}$ being weights on the unit circle). The last condition is in fact necessary. (It is expected, of course, that the resulting spaces coincide with the subspaces of analytic functions in the corresponding interpolation spaces for the couple $ ({L^{{p_0}}}({E_0};{w_0}),{L^{{p_1}}}({E_1};{w_1})).)$
Some cubic modular identities of Ramanujan
J. M.
Borwein;
P. B.
Borwein;
F. G.
Garvan
35-47
Abstract: There is a beautiful cubic analogue of Jacobi's fundamental theta function identity: $\theta _3^4 = \theta _4^4 + \theta _2^4$. It is $\displaystyle {\left({\sum\limits_{n,m = - \infty }^\infty {{q^{{n^2} + nm + {m... ...+ (n + \frac{1}{3})(m + \frac{1}{3}) + {{(m + \frac{1}{3})}^2}}}} } \right)^3}.$ Here $\omega = \exp (2\pi i/3)$. In this note we provide an elementary proof of this identity and of a related identity due to Ramanujan. We also indicate how to discover and prove such identities symbolically.
Generalized Casson invariants for ${\rm SO}(3),\;{\rm U}(2),\;{\rm Spin}(4),$ and ${\rm SO}(4)$
Cynthia L.
Curtis
49-86
Abstract: We investigate Casson-type invariants corresponding to the low-rank groups $ {\text{SO}}(3)$, ${\text{SU}}(2) \times {S^1}$, ${\text{U}}(2)$, $ {\text{Spin}}(4)$ and ${\text{SO}}(4)$. The invariants are defined following an approach similar to those of K. Walker and S. Cappell, R. Lee, and E. Miller. We obtain a description for each of the invariants in terms of the $ {\text{SU}}(2)$-invariant. Thus, all of them may be calculated using formulae for the $ {\text{SU}}(2)$-invariant. In defining these invariants, we offer methods which should prove useful for studying the invariants for other non-simply-connected groups once the invariants for the simply-connected covering groups are known.
Intersections of analytically and geometrically finite subgroups of Kleinian groups
James W.
Anderson
87-98
Abstract: We consider the intersection of pairs of subgroups of a Kleinian group of the second kind K whose limit sets intersect, where one subgroup G is analytically finite and the other J is geometrically finite, possibly infinite cyclic. In the case that J is infinite cyclic generated by M, we show that either some power of M lies in G or there is a doubly cusped parabolic element Q of G with the same fixed point as M. In the case that J is nonelementary, we show that the intersection of the limit sets of G and J is equal to the limit set of the intersection $G \cap J$ union with a subset of the rank 2 parabolic fixed points of K. Hence, in both cases, the limit set of the intersection is essentially equal to the intersection of the limit sets. The main facts used in the proof are results of Beardon and Pommerenke [4] and Canary [6], which yield that the Poincaré metric on the ordinary set of an analytically finite Kleinian group G is comparable to the Euclidean distance to the limit set of G.
On superquadratic elliptic systems
Djairo G.
de Figueiredo;
Patricio L.
Felmer
99-116
Abstract: In this article we study the existence of solutions for the elliptic system \begin{displaymath}\begin{array}{*{20}{c}} { - \Delta u = \frac{{\partial H}}{{\... ...quad v = 0\quad {\text{on}}\;\partial \Omega .} \end{array} \end{displaymath} where $\Omega$ is a bounded open subset of ${\mathbb{R}^N}$ with smooth boundary $\partial \Omega$, and the function $ H:{\mathbb{R}^2} \times \bar \Omega \to \mathbb{R}$, is of class ${C^1}$. We assume the function H has a superquadratic behavior that includes a Hamiltonian of the form $\displaystyle H(u,v) = \vert u{\vert^\alpha } + \vert v{\vert^\beta }\quad {\te... ... \frac{1}{\alpha } + \frac{1}{\beta } < 1\;{\text{with}}\;\alpha > 1,\beta > 1.$ We obtain existence of nontrivial solutions using a variational approach through a version of the Generalized Mountain Pass Theorem. Existence of positive solutions is also discussed.
On transformation group $C\sp *$-algebras with continuous trace
Siegfried
Echterhoff
117-133
Abstract: In this paper we answer some questions posed by Dana Williams in [19] concerning the problem under which conditions the transformation group ${C^ \ast }$-algebra ${C^\ast}(G,\Omega )$ of a locally compact transformation group (G, $\Omega$) has continuous trace. One consequence will be, for compact G, that ${C^\ast}(G,\Omega )$ has continuous trace if and only if the stabilizer map is continuous. We also give a complete solution to the problem if G is discrete.
Bounded holomorphic functions on bounded symmetric domains
Joel M.
Cohen;
Flavia
Colonna
135-156
Abstract: Let D be a bounded homogeneous domain in ${\mathbb{C}^n}$, and let $\Delta$ denote the open unit disk. If $z \in D$ and $f:D \to \Delta$ is holomorphic, then ${\beta _f}(z)$ is defined as the maximum ratio $\vert{\nabla _z}(f)x\vert/{H_z}{(x,\bar x)^{1/2}}$, where x is a nonzero vector in ${\mathbb{C}^n}$ and ${H_z}$ is the Bergman metric on D. The number ${\beta _f}(z)$ represents the maximum dilation of f at z. The set consisting of all ${\beta _f}(z)$ for $z \in D$ and $f:D \to \Delta$ holomorphic, is known to be bounded. We let ${c_D}$, be its least upper bound. In this work we calculate ${c_D}$ for all bounded symmetric domains having no exceptional factors and give indication on how to handle the general case. In addition we describe the extremal functions (that is, the holomorphic functions f for which $ {\beta _f} = {c_D}$) when D contains $\Delta$ as a factor, and show that the class of extremal functions is very large when $\Delta$ is not a factor of D.
The hexagonal packing lemma and Rodin Sullivan conjecture
Dov
Aharonov
157-167
Abstract: The Hexagonal Packing Lemma of Rodin and Sullivan [6] states that ${s_n} \to 0$ as $n \to \infty$. Rodin and Sullivan conjectured that $ {s_n} = O(1/n)$. This has been proved by Z-Xu He [2]. Earlier, the present author proved the conjecture under some additional restrictions [1]. In the following we are able to remove these restrictions, and thus give an alternative proof of the RS conjecture. The proof is based on our previous article [1]. It is completely different from the proof of He, and it is mainly based on discrete potential theory, as developed by Rodin for the hexagonal case [4].
Amenable relations for endomorphisms
J. M.
Hawkins
169-191
Abstract: We give necessary and sufficient conditions for an endomorphism to admit an equivalent invariant $\sigma$-finite measure in terms of a generalized Perron-Frobenius operator. The assumptions are that the endomorphism is nonsingular (preserves sets of measure zero), conservative, and finite-to-1. We study two orbit equivalence relations associated to an endomorphism, and their connections to nonsingularity, ergodicity, and exactness. We also discuss Radon-Nikodym derivative cocycles for the relations and the endomorphism, and relate these to the Jacobian of the endomorphism.
Multivariate orthogonal polynomials and operator theory
Yuan
Xu
193-202
Abstract: The multivariate orthogonal polynomials are related to a family of commuting selfadjoint operators. The spectral theorem for these operators is used to prove that a polynomial sequence satisfying a vector-matrix form of the three-term relation is orthonormal with a determinate measure.
Noetherian properties of skew polynomial rings with binomial relations
Tatiana
Gateva-Ivanova
203-219
Abstract: In this work we study standard finitely presented associative algebras over a fixed field K. A restricted class of skew polynomial rings with quadratic relations considered in an earlier work of M. Artin and W. Schelter will be studied. We call them binomial skew polynomial algebras. We establish necessary and sufficient conditions for such an algebra to be a Noetherian domain.
Product recurrence and distal points
J.
Auslander;
H.
Furstenberg
221-232
Abstract: Recurrence is studied in the context of actions of compact semigroups on compact spaces. (An important case is the action of the Stone-Čech compactification of an acting group.) If the semigroup E acts on the space X and F is a closed subsemigroup of E, then x in X is said to be F-recurrent if $px = x$ for some $p \in F$, and product F-recurrent if whenever y is an F-recurrent point (in some space Y on which E acts) the point (x, y) in the product system is F-recurrent. The main result is that, under certain conditions, a point is product F-recurrent if and only if it is a distal point.
Amenability and the structure of the algebras $A\sb p(G)$
Brian
Forrest
233-243
Abstract: A number of characterizations are given of the class of amenable locally compact groups in terms of the ideal structure of the algebras ${A_p}(G)$. An almost connected group is amenable if and only if for some $ 1 < p < \infty$ and some closed ideal I of ${A_p}(G)$, I has a bounded approximate identity. Furthermore, G is amenable if and only if every derivation of ${A_p}(G)$ into a Banach ${A_p}(G)$-bimodule is continuous.
Microlocal analysis of some isospectral deformations
F.
Marhuenda
245-275
Abstract: We study the microlocal structure of the examples of isospectral deformations of Riemannian manifolds given by D. DeTurck and C. Gordon in [DeT-Gl]. The Schwartz kernel of the intertwining operators considered by them may be written as an oscillatory integral with a singular phase function and product type amplitude. In certain instances, we identify them as belonging to the space of Fourier integral operators associated with various pairwise intersecting Lagrangians. After formulating a class of operators incorporating the most relevant features of the operators above, we establish a composition calculus for this class and show that is not necessary to introduce new Lagrangians in the composition.
Witt equivalence of global fields. II. Relative quadratic extensions
Kazimierz
Szymiczek
277-303
Abstract: This paper explores the consequences of the Hasse Principle for Witt equivalence of global fields in the case of relative quadratic extensions. We are primarily interested in generating the Witt equivalence classes of quadratic extensions of a given number field, and we study the structure of the class, the number of classes, and the structure of the set of classes. Along the way, we reprove several results obtained earlier in the absolute case of the rational ground field, giving unified and short proofs based on the Hasse Principle.
Semirigid spaces
Věra
Trnková
305-325
Abstract: Semirigid spaces are introduced and used as a means to construct two metrizable spaces with isomorphic monoids of continuous self-maps and nonisomorphic clones; this resolves Problem 1 in [13]. The clone of any free variety of a given type with sufficiently many constants is shown to be isomorphic to the clone of a metrizable (semirigid) space.
Periodic orbits for Hamiltonian systems in cotangent bundles
Christophe
Golé
327-347
Abstract: We prove the existence of at least $ \operatorname{cl}(M)$ periodic orbits for certain time-dependent Hamiltonian systems on the cotangent bundle of an arbitrary compact manifold M. These Hamiltonians are not necessarily convex but they satisfy a certain boundary condition given by a Riemannian metric on M. We discretize the variational problem by decomposing the time-1 map into a product of "symplectic twist maps". A second theorem deals with homotopically non-trivial orbits of negative curvature.
Pseudocircles in dynamical systems
Judy A.
Kennedy;
James A.
Yorke
349-366
Abstract: We construct an example of a $ {C^\infty }$ map on a 3-manifold which has an invariant set with an uncountable number of components, each of which is a pseudocircle. Furthermore, any map which is sufficiently close (in the ${C^1}$-metric) to the constructed map has a similar set.
Calculating discriminants by higher direct images
Jerzy
Weyman
367-389
Abstract: The author uses the homological algebra to construct for any line bundle $\mathcal{L}$ on a nonsingular projective variety X the complex $\mathbb{F}(\mathcal{L})$ whose determinant is equal to the equation of the dual variety ${X^{\text{V}}}$. This generalizes the Cayley-Koszul complexes defined by Gelfand, Kapranov and Zelevinski. The formulas for the codimension and degree of ${X^{\text{V}}}$ in terms of complexes $\mathbb{F}(\mathcal{L})$ are given. In the second part of the article the general technique is applied to classical discriminants and hyperdeterminants.
Fibrations of classifying spaces
Kenshi
Ishiguro;
Dietrich
Notbohm
391-415
Abstract: We investigate fibrations of the form $Z \to Y \to X$, where two of the three spaces are classifying spaces of compact connected Lie groups. We obtain certain finiteness conditions on the third space which make it also a classifying space. Our results allow to express some of the basic notions in group theory in terms of homotopy theory, i.e., in terms of classifying spaces. As an application we prove that every retract of the classifying space of a compact connected Lie group is again a classifying space.
On the number of singularities in meromorphic foliations on compact complex surfaces
Edoardo
Ballico
417-432
Abstract: Here we study meromorphic foliations with singularities on a smooth compact complex surface. Aim: study the locations of the singularities, using vector bundle techniques and techniques introduced in the cohomological study of projective geometry.
${\germ F}$-categories and ${\germ F}$-functors in the representation theory of Lie algebras
Ben
Cox
433-453
Abstract: The fields of algebra and representation theory contain abundant examples of functors on categories of modules over a ring. These include of course Horn, Ext, and Tor as well as the more specialized examples of completion and localization used in the setting of representation theory of a semisimple Lie algebra. In this article we let $\mathfrak{a}$ be a Lie subalgebra of a Lie algebra $\mathfrak{g}$ and $\Gamma$ be a functor on some category of $\mathfrak{a}$-modules. We then consider the following general question: For a $ \mathfrak{g}$-module E what hypotheses on $\Gamma$ and E are sufficient to insure that $\Gamma (E)$ admits a canonical structure as a $\mathfrak{g}$-module? The article offers an answer through the introduction of the notion of $\mathfrak{F}$-categories and $\mathfrak{F}$-functors. The last section of the article treats various examples of this theory.
Solutions to the quantum Yang-Baxter equation arising from pointed bialgebras
David E.
Radford
455-477
Abstract: Let $R:M \otimes M \to M \otimes M$ be a solution to the quantum Yang-Baxter equation, where M is a finite-dimensional vector space over a field k. We introduce a quotient $ {A^{{\text{red}}}}(R)$ of the bialgebra $A(R)$ constructed by Fadeev, Reshetihkin and Takhtajan, whose characteristics seem to more faithfully reflect properties R possesses as a linear operator. We characterize all R such that ${A^{{\text{red}}}}(R)$ is a pointed bialgebra, and we determine all solutions R to the quantum Yang-Baxter equation when ${A^{{\text{red}}}}(R)$ is pointed and $\dim M = 2$ (with a few technical exceptions when k has characteristic 2). Extensions of such solutions to the quantum plane are studied.
Projective resolutions and Poincar\'e duality complexes
D. J.
Benson;
Jon F.
Carlson
447-488
Abstract: Let k be a field lof characteristic $p > 0$ and let G be a finite group. We investigate the structure of the cohomology ring ${H^\ast}(G,k)$ in relation to certain spectral sequences determined by systems of homogeneous parameters for the cohomology ring. Each system of homogeneous parameters is associated to a complex of projective kG-modules which is homotopically equivalent to a Poincaré duality complex. The initial differentials in the hypercohomology spectral sequence of the complex are multiplications by the parameters, while the higher differentials are matric Massey products. If the cohomology ring is Cohen-Macaulay, then the duality of the complex assures that the Poincaré series for the cohomology satisfies a certain functional equation. The structure of the complex also implies the existence of cohomology classes which are in relatively large degrees but are not in the ideal generated by the parameters. We consider several other questions concerned with the minimal projective resolutions and the convergence of the spectral sequence.
A variational principle in Kre\u\i n space
Paul
Binding;
Branko
Najman
489-499
Abstract: A variational characterization, involving a max-inf of the Rayleigh quotient, is established for certain eigenvalues of a wide class of definitizable selfadjoint operators Q in a Krein space. The operator Q may have continuous spectrum and nonreal and nonsemisimple eigenvalues; in particular it may have embedded eigenvalues. Various applications are given to selfadjoint linear and quadratic eigenvalue problems with weak definiteness assumptions.
Approximate solutions to first and second order quasilinear evolution equations via nonlinear viscosity
Juan R.
Esteban;
Pierangelo
Marcati
501-521
Abstract: We shall consider a model problem for the fully nonlinear parabolic equation $\displaystyle {u_t} + F(x,t,u,Du,\varepsilon {D^2}u) = 0$ and we study both the approximating degenerate second order problem and the related first order equation, obtained by the limit as $\varepsilon \to 0$. The strong convergence of the gradients is provided by semiconcavity unilateral bounds and by the supremum bounds of the gradients. In this way we find solutions in the class of viscosity solutions of Crandall and Lions.
M\"untz systems and orthogonal M\"untz-Legendre polynomials
Peter
Borwein;
Tamás
Erdélyi;
John
Zhang
523-542
Abstract: The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system $\{ {x^{{\lambda _0}}},{x^{{\lambda _1}}}, \ldots \}$ with respect to Lebesgue measure on [0, 1]. In this paper, differential and integral recurrence formulae for the Müntz-Legendre polynomials are obtained. Interlacing and lexicographical properties of their zeros are studied, and the smallest and largest zeros are universally estimated via the zeros of Laguerre polynomials. The uniform convergence of the Christoffel functions is proved equivalent to the nondenseness of the Müntz space on [0, 1], which implies that in this case the orthogonal Müntz-Legendre polynomials tend to 0 uniformly on closed subintervals of [0, 1). Some inequalities for Müntz polynomials are also investigated, most notably, a sharp ${L^2}$ Markov inequality is proved.
On the solvability of systems of inclusions involving noncompact operators
P.
Nistri;
V. V.
Obukhovskiĭ;
P.
Zecca
543-562
Abstract: We consider the solvability of a system $\displaystyle \left\{ {\begin{array}{*{20}{c}} {y \in \bar F(x,y),} {x \in \bar G(x,y)} \end{array} } \right.$ of set-valued maps in two different cases. In the first one, the map $(x,y) - \circ \bar F(x,y)$ is supposed to be closed graph with convex values and condensing in the second variable and $(x,y) - \circ \bar G(x,y)$ is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, single-valued map), satisfying a condensivity condition in the first variable. In the second case $\bar F$ is as before with compact, not necessarily convex, values and $\bar G$ is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set $x - \circ S(x)$ of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period.
Noether's theorem for Hopf orders in group algebras
David M.
Weinraub
563-574
Abstract: Let K be a local field with valuation ring R of residue characteristic p containing a primitive pth root of unity $ {\zeta _p}$. We state an analog to Noether's Theorem for modules over R-Hopf algebras and use induction techniques to deduce a criterion for this analog to hold. We then construct a family of noncommutative Hopf algebras which satisfy the criterion.
Transference for radial multipliers and dimension free estimates
P.
Auscher;
M. J.
Carro
575-593
Abstract: For a large class of radial multipliers on $ {L^p}({{\mathbf{R}}^{\mathbf{n}}})$, we obtain bounds that do not depend on the dimension n. These estimates apply to well-known multiplier operators and also give another proof of the boundedness of the Hardy-Littlewood maximal function over Euclidean balls on $ {L^p}({{\mathbf{R}}^{\mathbf{n}}})$, $p \geq 2$, with constant independent of the dimension. The proof is based on the corresponding result for the Riesz transforms and the method of rotations.
Scattering theory for semilinear wave equations with small data in two space dimensions
Kimitoshi
Tsutaya
595-618
Abstract: We study scattering theory for the semilinear wave equation ${u_{tt}} - \Delta u = \vert u{\vert^{p - 1}}u$ in two space dimensions. We show that if $p > {p_0} = (3 + \sqrt {17} )/2$, the scattering operator exists for smooth and small data. The lower bound ${p_0}$ of p is considered to be optimal (see Glassey [6, 7], Schaeffer [18]). Our result is an extension of the results by Strauss [19], Klainerman [10], and Mochizuki and Motai [14, 15]. The construction of the scattering operator for small data does not follow directly from the proofs in [7, 13, 20 and 22] concerning the global existence of solutions for the Cauchy problem of the above equation with small initial data given at $t = 0$ in two space dimensions, because we have to consider the integral equation with unbounded integral region associated to the above equation: $\displaystyle u(x,t) = u_0^ - (x,t) + \frac{1}{{2\pi }}\int_{ - \infty }^t {\in... ...t^{p - 1}}u)(y,s)}}{{\sqrt {{{(t - s)}^2} - \vert x - y{\vert^2}} }}dy\;ds,} }$ for $t \in R$, where $ u_0^ - (x,t)$ is a solution of $ {u_{tt}} - \Delta u = 0$ which $u(x,t)$ approaches asymptotically as $t \to - \infty$. The proof of the basic estimate for the above integral equation is more difficult and complicated than that for the Cauchy problem of $ {u_{tt}} - \Delta u = \vert u{\vert^{p - 1}}u$ in two space dimensions.
The structure of the set of singular points of a codimension $1$ differential system on a $5$-manifold
P.
Mormul;
M. Ya.
Zhitomirskiĭ
619-629
Abstract: Generic modules V of vector fields tangent to a 5-dimensional smooth manifold M, generated locally by four not necessarily linearly independent fields $ {X_1}$, ${X_2}$, ${X_3}$, ${X_4}$, are considered. Denoting by $\omega$ the 1-form $ {X_4}\lrcorner{X_3}\lrcorner{X_2}\lrcorner{X_1}\lrcorner\mathop \Omega \limits^5$ conjugated to V ( $\mathop \Omega \limits^5 $ is a fixed local volume form on M), the loci of singular behavior of $V:{M_{\deg }}(V) = \{ p \in M\vert\omega (p) = 0\}$ and ${M_{{\text{sing}}}}(V) = \{ p \in M\vert\omega \wedge {(d\omega )^2}(p) = 0\}$ are handled. The local classification of this pair of sets is carried out (outside a curve and a discrete set in ${M_{\deg }}(V)$) up to a smooth diffeomorphism. In the most complicated case, around points of a codimension 3 submanifold of M, ${M_{{\text{sing}}}}(V)$ turns out to be diffeomorphic to the Cartesian product of ${\mathbb{R}^2}$ and the Whitney's umbrella in ${\mathbb{R}^3}$.
On the Gorenstein property of Rees and form rings of powers of ideals
M.
Herrmann;
J.
Ribbe;
S.
Zarzuela
631-643
Abstract: In this paper we determine the exponents n for which the Rees ring $R({I^n})$ and the form ring ${\text{gr}}_{A}({I^n})$ are Gorenstein rings, where I is a strongly Cohen-Macaulay ideal of linear type (including complete and almost complete intersections) or an $ \mathfrak{m}$-primary ideal in a local ring A with maximal ideal $\mathfrak{m}$.
Nonlinear stability of rarefaction waves for a viscoelastic material with memory
Harumi
Hattori
645-669
Abstract: In this paper we will discuss the stability of rarefaction waves for a viscoelastic material with memory. The rarefaction waves for which the stability is tested are not themselves solutions to the integrodifferential equations (1.1) governing the viscoelastic material. They are solutions to a related equilibrium system of conservation laws given by (1.11). We shall show that if the forcing term and the past history are small and if the initial data are close to the rarefaction waves, the solutions to (1.1) will approach the rarefaction waves in sup norm as the time goes to infinity.
Gromov's compactness theorem for pseudo holomorphic curves
Rugang
Ye
671-694
Abstract: We give a complete proof for Gromov's compactness theorem for pseudo holomorphic curves both in the case of closed curves and curves with boundary.
An almost strongly minimal non-Desarguesian projective plane
John T.
Baldwin
695-711
Abstract: There is an almost strongly minimal projective plane which is not Desarguesian.
Ulam-Zahorski problem on free interpolation by smooth functions
A.
Olevskiĭ
713-727
Abstract: Let f be a function belonging to $ {C^n}[0,1]$. Is it possible to find a smoother function $g \in {C^{n + 1}}$ (or at least ${C^{n + \varepsilon }}$) which has infinitely many points of contact of maximal order n with f (or at least arbitrarily many such points with fixed norm ${\left\Vert g \right\Vert _{{C^{n + \varepsilon }}}}$)? It turns out that for n = 0 and 1 the answer is positive, but if $n \geq 2$, it is negative. This gives a complete solution to the Ulam-Zahorski question on free interpolation on perfect sets.
On the discriminant of a hyperelliptic curve
P.
Lockhart
729-752
Abstract: The minimal discriminant of a hyperelliptic curve is defined and used to generalize much of the arithmetic theory of elliptic curves. Over number fields this leads to a higher genus version of Szpiro's Conjecture. Analytically, the discriminant is shown to be related to Siegel modular forms of higher degree.
A proof of $C\sp 1$ stability conjecture for three-dimensional flows
Sen
Hu
753-772
Abstract: We give a proof of the ${C^1}$ stability conjecture for three-dimensional flows, i.e., prove that there exists a hyperbolic structure over the $\Omega$ set for the structurally stable three-dimensional flows. Mañé's proof for the discrete case motivates our proof and we find his perturbation techniques crucial. In proving this conjecture we have overcome several new difficulties, e.g., the change of period after perturbation, the ergodic closing lemma for flows, the existence of dominated splitting over $\Omega \backslash \mathcal{P}$ where $\mathcal{P}$ is the set of singularities for the flow, the discontinuity of the contracting rate function on singularities, etc. Based on these we finally succeed in separating the singularities from the other periodic orbits for the structurally stable systems, i.e., we create unstable saddle connections if there are accumulations of periodic orbits on the singularities.
Positive harmonic functions vanishing on the boundary for the Laplacian in unbounded horn-shaped domains
Dimitry
Ioffe;
Ross
Pinsky
773-791
Abstract: Denote points $\bar x \in {R^{d + 1}}$, $d \geq 2$, by $\bar x = (\rho ,\theta ,z)$, where $\rho > 0$, $\theta \in {S^{d - 1}}$, and $z \in R$. Let $a:[0,\infty ) \to (0,\infty )$ be a nondecreasing $ {C^2}$-function and define the "horn-shaped" domain $ \Omega = \{ \bar x = (\rho ,\theta ,z):\vert z\vert < a(\rho )\}$ and its unit "cylinder" $D = \{ \bar x = (\rho ,\theta ,z) \in \Omega :\rho < 1\}$. Under appropriate regularity conditions on a, we prove the following theorem: (i) If ${\smallint ^\infty }a(\rho )/{\rho ^2}d\rho = \infty$, then the Martin boundary at infinity for $\frac{1}{2}\Delta$ in $\Omega$ is a single point, (ii) If ${\smallint ^\infty }a(\rho )/{\rho ^2}d\rho < \infty$, then the Martin boundary at infinity for $ \frac{1}{2}\Delta$ in $ \Omega$ is homeomorphic to ${S^{d - 1}}$. More specifically, a sequence $\{ ({\rho _n},{\theta _n},{z_n})\} _{n = 1}^\infty \subset \Omega $ satisfying $ {\lim _{n \to \infty }}{\rho _n} = \infty$ is a Martin sequence if and only if $ {\lim _{n \to \infty }}{\theta _n}$ exists on $ {S^{d - 1}}$. From (i), it follows that the cone of positive harmonic functions in $\Omega$ vanishing continuously on $\partial \Omega$ is one-dimensional. From (ii), it follows easily that the cone of positive harmonic functions on $\Omega$ vanishing continuously on $\partial \Omega$ is generated by a collection of minimal elements which is homeomorphic to ${S^{d - 1}}$. In particular, the above result solves a problem stated by Kesten, who asked what the Martin boundary is for $ \frac{1}{2}\Delta$ in $ \Omega$ in the case $ a(\rho ) = 1 + {\rho ^\gamma }$, $0 < \gamma < 1$. Our method of proof involves an analysis as $\rho \to \infty$ of the exit distribution on $\partial D$ for Brownian motion starting from $(\rho ,\theta ,z) \in \Omega$ and conditioned to hit D before exiting $ \Omega$.
The theory of Jacobi forms over the Cayley numbers
M.
Eie;
A.
Krieg
793-805
Abstract: As a generalization of the classical theory of Jacobi forms we discuss Jacobi forms on $\mathcal{H} \times {\mathbb{C}^8}$, which are related with integral Cayley numbers. Using the Selberg trace formula we give a simple explicit formula for the dimension of the space of Jacobi forms. The orthogonal complement of the space of cusp forms is shown to be spanned by certain types of Eisenstein series.
A controlled plus construction for crumpled laminations
R. J.
Daverman;
F. C.
Tinsley
807-826
Abstract: Given a closed n-manifold M $(n > 4)$ and a finitely generated perfect subgroup P of $ {\pi _1}(M)$, we previously developed a controlled version of Quillen's plus construction, namely a cobordism (W, M, N) with the inclusion $ j:N \mapsto W$ a homotopy equivalence and kernel of ${i_\char93 }:{\pi _1}(M) \mapsto {\pi _1}(W)$ equalling the smallest normal subgroup of ${\pi _1}(M)$ containing P together with a closed map $ p:W \mapsto [0,1]$ such that ${p^{ - 1}}(t)$ is a closed n-manifold for every $t \in [0,1]$ and, in particular, $M = {p^{ - 1}}(0)$ and $N = {p^{ - 1}}(1)$. We accomplished this by constructing an acyclic map of manifolds $f:M \mapsto N$ having the right fundamental groups, and W arose as the mapping cylinder of f with a collar attached along N. The main result here presents a condition under which the desired controlled plus construction can still be accomplished in many cases even when ${\pi _1}(M)$ contains no finitely generated perfect subgroups. By-products of these results include a new method for constructing wild embeddings of codimension one manifolds and a better understanding of perfect subgroups of finitely presented groups.
Transfer functions of regular linear systems. I. Characterizations of regularity
George
Weiss
827-854
Abstract: We recall the main facts about the representation of regular linear systems, essentially that they can be described by equations of the form $\dot x(t) = Ax(t) + Bu(t)$, $y(t) = Cx(t) + Du(t)$, like finite dimensional systems, but now A, B and C are in general unbounded operators. Regular linear systems are a subclass of abstract linear systems. We define transfer functions of abstract linear systems via a generalization of a theorem of Fourés and Segal. We prove a formula for the transfer function of a regular linear system, which is similar to the formula in finite dimensions. The main result is a (simple to state but hard to prove) necessary and sufficient condition for an abstract linear system to be regular, in terms of its transfer function. Other conditions equivalent to regularity are also obtained. The main result is a consequence of a new Tauberian theorem, which is of independent interest.
Block Jacobi matrices and zeros of multivariate orthogonal polynomials
Yuan
Xu
855-866
Abstract: A commuting family of symmetric matrices are called the block Jacobi matrices, if they are block tridiagonal. They are related to multivariate orthogonal polynomials. We study their eigenvalues and joint eigenvectors. The joint eigenvalues of the truncated block Jacobi matrices correspond to the common zeros of the multivariate orthogonal polynomials.
Classification of rank-2 ample and spanned vector bundles on surfaces whose zero loci consist of general points
Atsushi
Noma
867-894
Abstract: Let X be an n-dimensional smooth projective variety over an algebraically closed field k of characteristic zero, and E an ample and spanned vector bundle of rank n on X. To study the geometry of (X, E) in view of the zero loci of global sections of E, Ballico introduces a numerical invariant $s(E)$. The purposes of this paper are to give a cohomological interpretation of $s(E)$, and to classify ample and spanned rank-2 bundles E on smooth complex surfaces X with $ s(E) = 2{c_2}(E)$, or $2{c_2}(E) - 1$; namely ample and spanned 2-bundles whose zero loci of global sections consist of general $ {c_2}(E)$ points or general ${c_2}(E) - 1$ points plus one. As an application of these classification, we classify rank-2 ample and spanned vector bundles E on smooth complex projective surfaces with ${c_2}(E) = 2$.
Rational homotopy of the space of self-maps of complexes with finitely many homotopy groups
Samuel B.
Smith
895-915
Abstract: For simply connected CW complexes X with finitely many, finitely generated homotopy groups,$^{1}$ the path components of the function space $ M(X,X)$ of free self-maps of X are all of the same rational homotopy type if and only if all the k-invariants of X are of finite order. In case X is rationally a two-stage Postnikov system the space ${M_0}(X,X)$ of inessential self-maps of X has the structure of rational H-space if and only if the k-invariants of X are of finite order.
Multiplier Hopf algebras
A.
Van Daele
917-932
Abstract: In this paper we generalize the notion of Hopf algebra. We consider an algebra A, with or without identity, and a homomorphism $\Delta$ from A to the multiplier algebra $M(A \otimes A)$ of $A \otimes A$. We impose certain conditions on $\Delta$ (such as coassociativity). Then we call the pair $ (A,\Delta )$ a multiplier Hopf algebra. The motivating example is the case where A is the algebra of complex, finitely supported functions on a group G and where $(\Delta f)(s,t) = f(st)$ with $ s,t \in G$ and $ f \in A$. We prove the existence of a counit and an antipode. If A has an identity, we have a usual Hopf algebra. We also consider the case where A is a $ \ast$-algebra. Then we show that (a large enough) subspace of the dual space can also be made into a $\ast$-algebra.
Weights for classical groups
Jian Bei
An
1-42
Abstract: This paper proves the Alperin's weight conjecture for the finite unitary groups when the characteristic r of modular representation is odd. Moreover, this paper proves the conjecture for finite odd dimensional special orthogonal groups and gives a combinatorial way to count the number of weights, block by block, for finite symplectic and even dimensional special orthogonal groups when r and the defining characteristic of the groups are odd.
Second order differentiability of convex functions in Banach spaces
Jonathan M.
Borwein;
Dominikus
Noll
43-81
Abstract: We present a second order differentiability theory for convex functions on Banach spaces.
Separation and coding
Stephen
Watson
83-106
Abstract: We construct a normal collectionwise Hausdorff space which is not collectionwise normal with respect to copies of [0,1]. We do this by developing a general theory of coding properties into topological spaces. We construct a para-Lindelöf regular space in which para-Lindelöf is coded directly rather than $\sigma $-para-Lindelöf and normal. We construct a normal collectionwise Hausdorff space which is not collectionwise normal in which collectionwise Hausdorff is coded directly rather than obtained as a side-effect to countable approximation. We also show that the Martin's axiom example of a normal space which is not collectionwise Hausdorff is really just a kind of "dual" of Bing's space.
Indecomposable generalized Cohen-Macaulay modules
Mihai
Cipu;
Jürgen
Herzog;
Dorin
Popescu
107-136
Abstract: The aim of this paper is to study the indecomposable modules which are Cohen-Macaulay on the punctured spectrum of a local ring, and to describe some of their invariants such as their local cohomology groups and ranks. One of our main concerns is to find indecomposable quasi-Buchsbaum modules of high rank with prescribed cohomology over a regular local ring.
Growth functions for some nonautomatic Baumslag-Solitar groups
Marcus
Brazil
137-154
Abstract: The growth function of a group is a generating function whose coefficients $ {a_n}$ are the number of elements in the group whose minimum length as a word in the generators is n. In this paper we use finite state automata to investigate the growth function for the Baumslag-Solitar group of the form $ \langle a,b\vert{a^{ - 1}}ba = {a^2}\rangle$ based on an analysis of its combinatorial and geometric structure. In particular, we obtain a set of length-minimal normal forms for the group which, although it does not form the language of a finite state automata, is nevertheless built up in a sufficiently coherent way that the growth function can be shown to be rational. The rationality of the growth function of this group is particularly interesting as it is known not to be synchronously automatic. The results in this paper generalize to the groups $\langle a,b\vert{a^{ - 1}}ba = {a^m}\rangle$ for all positive integers m.
On the generalized Benjamin-Ono equation
Carlos E.
Kenig;
Gustavo
Ponce;
Luis
Vega
155-172
Abstract: We study well-posedness of the initial value problem for the generalized Benjamin-Ono equation $ {\partial _t}u + {u^k}{\partial _x}u - {\partial _x}{D_x}u = 0$, $k \in {\mathbb{Z}^ + }$, in Sobolev spaces ${H^s}(\mathbb{R})$. For small data and higher nonlinearities $(k \geq 2)$ new local and global (including scattering) results are established. Our method of proof is quite general. It combines several estimates concerning the associated linear problem with the contraction principle. Hence it applies to other dispersive models. In particular, it allows us to extend the results for the generalized Benjamin-Ono to nonlinear Schrödinger equations (or systems) of the form $ {\partial _t}u - i\partial _x^2u + P(u,{\partial _x}u,\bar u,{\partial _x}\bar u) = 0$.
Vaught's conjecture for varieties
Bradd
Hart;
Sergei
Starchenko;
Matthew
Valeriote
173-196
Abstract: We prove that if $\mathcal{V}$ is a superstable variety or one with few countable models then $\mathcal{V}$ is the varietal product of an affine variety and a combinatorial variety. Vaught's conjecture for varieties is an immediate consequence.
Operations on resolutions and the reverse Adams spectral sequence
David A.
Blanc
197-213
Abstract: We describe certain operations on resolutions in abelian categories, and apply them to calculate part of a reverse Adams spectral sequence, going "from homotopy to homology", for the space $ {\mathbf{K}}(\mathbb{Z}/2,n)$. This calculation is then used to deduce that there is no space whose homotopy groups are the reduction $\bmod \; 2$ of ${\pi _\ast}{{\mathbf{S}}^r}$. As another application of the operations we give a short proof of T. Y. Lin's theorem on the infinite projective dimension of all nonfree $\pi $-modules.
Noncharacteristic embeddings of the $n$-dimensional torus in the $(n+2)$-dimensional torus
David
Miller
215-240
Abstract: We construct certain exotic embeddings of the n-torus $ {T^n}$ in ${T^{n + 2}}$ in the standard homotopy class. We turn an embedding $f:{T^n} \to {T^{n + 2}}$ characteristic if there exists some map $\alpha :{T^{n + 2}} \to {T^{n + 2}}$ in the standard homotopy class with the property that $ \alpha \; \circ \;f:{T^n} \to {T^{n + 2}}$ is the standard coordinate inclusion and $\alpha ({T^{n + 2}} - f({T^n})) \subset {T^{n + 2}} - {T^n}$. We find examples of noncharacteristic embeddings, f, in dimensions $n = 4k + 1$, $n \geq 5$, and show that these examples are not even cobordant to characteristic embeddings. We let G denote the fundamental group of the complement of the standard coordinate inclusion, ${T^{n + 2}} - {T^n}$. Then we can associate to f a real-valued signature function on the set of j-dimensional unitary representations of $ \bar G$, where $ \bar G$ denotes the fundamental group of the localization of ${T^{n + 2}} - {T^n}$ with respect to homology with local coefficients in $\mathbb{Z}[{\mathbb{Z}^{n + 2}}]$. This function is a cobordism invariant which has certain periodicity properties for characteristic embeddings. We verify that this periodicity does not hold for our examples, f, implying that they are not characteristic. Additional results include a proof that the examples, f, become cobordant to characteristic embeddings upon taking the cartesian product with the identity map on a circle.
The $H\sp 2$ corona problem and $\overline\partial\sb b$ in weakly pseudoconvex domains
Mats
Andersson
241-255
Abstract: We derive a Bochner-Kodaira-Nakano-Morrey-Kohn-Hörmander type equality in holomorphic vector bundles and obtain ${L^2}$-estimates for ${\bar \partial _b}$ in a pseudoconvex domain that admits a plurisubharmonic ${C^2}$ defining function. We combine these with the trick in Wolff's proof of the corona theorem and obtain a ${H^2}$-corona theorem in such a domain.
Homology and cohomology of $\Pi$-algebras
W. G.
Dwyer;
D. M.
Kan
257-273
Abstract: We study a type of homological algebra associated to the collection of all homotopy groups of a space (just as the theory of group homology is associated to the fundamental group).
Varieties of commutative semigroups
Andrzej
Kisielewicz
275-306
Abstract: In this paper, we describe all equational theories of commutative semigroups in terms of certain well-quasi-orderings on the set of finite sequences of nonnegative integers. This description yields many old and new results on varieties of commutative semigroups. In particular, we obtain also a description of the lattice of varieties of commutative semigroups, and we give an explicit uniform solution to the word problems for free objects in all varieties of commutative semigroups.
Wavelets of multiplicity $r$
T. N. T.
Goodman;
S. L.
Lee
307-324
Abstract: A multiresolution approximation ${({V_m})_{m \in {\mathbf{Z}}}}$ of ${L^2}({\mathbf{R}})$ is of multiplicity $r > 0$ if there are r functions ${\phi _1}, \ldots ,{\phi _r}$ whose translates form a Riesz basis for ${V_0}$. In the general theory we derive necessary and sufficient conditions for the translates of ${\phi _1}, \ldots ,{\phi _r},\;{\psi _1}, \ldots ,{\psi _r}$ to form a Riesz basis for ${V_1}$. The resulting reconstruction and decomposition sequences lead to the construction of dual bases for ${V_0}$ and its orthogonal complement ${W_0}$ in ${V_1}$. The general theory is applied in the construction of spline wavelets with multiple knots. Algorithms for the construction of these wavelets for some special cases are given.
Harmonic diffeomorphisms of the hyperbolic plane
Kazuo
Akutagawa
325-342
Abstract: In this paper, we consider the Dirichlet problem at infinity for harmonic maps between the Poincaré model D of the hyperbolic plane ${\mathbb{H}^2}$, and solve this when given boundary data are ${C^4}$ immersions of $ D(\infty )$, the boundary at infinity of D, to $ D(\infty )$. Also, we present a construction of nonconformal harmonic diffeomorphisms of D, and give a complete description of the boundary behavior, including their first derivatives.
Smooth extensions for finite CW complexes
Guihua
Gong
343-358
Abstract: In this paper, we have completely classified the ${C_n}$-smooth elements of $\operatorname{Ext} (X)$ modulo torsion for X being an arbitrary finite CW complex.
Best uniform approximation by solutions of elliptic differential equations
P. M.
Gauthier;
D.
Zwick
359-374
Abstract: We investigate best uniform approximations to continuous functions on compact subsets of $ {\mathbb{R}^n}$ by solutions of elliptic differential equations and, in particular, by harmonic functions. An axiomatic setting general enough to encompass problems of this kind is given, and in this context we extend necessary and sufficient conditions for best harmonic approximation on precompact Jordan domains to arbitrary compact sets and to more general classes of solutions of linear elliptic differential equations.
A Cameron-Martin type quasi-invariance theorem for pinned Brownian motion on a compact Riemannian manifold
Bruce K.
Driver
375-395
Abstract: The results in Driver [13] for quasi-invariance of Wiener measure on the path space of a compact Riemannian manifold (M) are extended to the case of pinned Wiener measure. To be more explicit, let $h:[0,1] \to {T_0}M$ be a ${C^1}$ function where M is a compact Riemannian manifold, $o \in M$ is a base point, and ${T_o}M$ is the tangent space to M at $o \in M$. Let $W(M)$ be the space of continuous paths from [0,1] into M, $\nu$ be Wiener measure on $W(M)$ concentrated on paths starting at $ o \in M$, and ${H_s}(\omega )$ denote the stochastic-parallel translation operator along a path $\omega \in W(M)$ up to "time" s. (Note: ${H_s}(\omega )$ is only well defined up to $ \nu$-equivalence.) For $\omega \in W(M)$ let ${X^h}(\omega )$ denote the vector field along $ \omega$ given by $ X_s^h(\omega ) \equiv {H_s}(\omega )h(s)$ for each $ s \in [0,1]$. One should interpret ${X^h}$ as a vector field on $W(M)$. The vector field ${X^h}$ induces a flow ${S^h}(t, \bullet ):W(M) \to W(M)$ which leaves Wiener measure $(\nu )$ quasi-invariant, see Driver [13]. It is shown in this paper that the same result is valid if $ h(1) = 0$ and the Wiener measure $(\nu )$ is replaced by a pinned Wiener measure $({\nu _e})$. (The measure ${\nu _e}$ is proportional to the measure $ \nu$ conditioned on the set of paths which start at $o \in M$ and end at a fixed end point $ e \in M$.) Also as in [13], one gets an integration by parts formula for the vector-fields ${X^h}$ defined above.
Infinite families of isomorphic nonconjugate finitely generated subgroups
F. E. A.
Johnson
397-406
Abstract: Let $ \langle \;,\;\rangle :L \times L \to \mathbb{Z}$ be a nondegenerate symmetric bilinear form on a finitely generated free abelian group L which splits as an orthogonal direct sum $(L,\;\langle \;,\;\rangle ) \cong ({L_1},\;\langle \;,\;\rangle ) \bot ({L_2},\;\langle \;,\;\rangle ) \bot ({L_3},\;\langle \;,\;\rangle )$ in which $ ({L_1},\;\langle \;,\;\rangle )$ has signature (2, 1), $({L_2},\;\langle \;,\;\rangle )$ has signature (n, 1) with $n \geq 2$, and $({L_3},\;\langle \;,\;\rangle )$ is either zero or indefinite with $ {\text{rk}}_\mathbb{Z}({L_3}) \geq 3$. We show that the integral automorphism group $ {\operatorname{Aut} _\mathbb{Z}}(L,\;\langle \;,\;\rangle )$ contains an infinite family of mutually isomorphic finitely generated subgroups ${({\Gamma _\sigma })_{\sigma \in \Sigma }}$, no two of which are conjugate. In the simplest case, when ${L_3} = 0$, the groups ${\Gamma _\sigma }$ are all normal subdirect products in a product of free groups or surface groups. The result can be seen as a failure of the rigidity property for subgroups of infinite covolume within the corresponding Lie group $ {\operatorname{Aut} _\mathbb{Z}}(L{ \otimes _\mathbb{Z}}\mathbb{R},\;\langle \;,\;\rangle \otimes 1)$.
Tur\'an inequalities and zeros of Dirichlet series associated with certain cusp forms
J. B.
Conrey;
A.
Ghosh
407-419
Abstract: The "Turan inequalities" are a countably infinite set of conditions about the power series coefficients of certain entire functions which are necessary in order for the function to have only real zeros. We give a one-parameter family of generalized Dirichlet series, each with functional equation, for which the Turan inequalities hold for the associated $\xi$-function (normalized so that the critical line is the real axis). For a discrete set of values of the parameter the Dirichlet series has an Euler product and is the L-series associated to a modular form. For these we expect the analogue of the Riemann Hypothesis to hold. For the rest of the values of the parameter we do not expect an analogue of the Riemann Hypothesis. We show for one particular value of the parameter that the Dirichlet series in fact has zeros within the region of absolute convergence.
Abel's theorem for twisted Jacobians
Donu
Arapura;
Kyungho
Oh
421-433
Abstract: A twisted version of the Abel-Jacobi map, associated to a local system with finite monodromy on a smooth projectve complex curve, is introduced. An analogue of Abel's theorem characterizing the kernel of this map is proved. The proof, which is new even in the classical case, involves reinterpreting the Abel-Jacobi map in the language of mixed Hodge structures and their extensions.
Intersection bodies and the Busemann-Petty problem
R. J.
Gardner
435-445
Abstract: It is proved that the answer to the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies in d-dimensional Euclidean space $ {\mathbb{E}^d}$ is negative for a given d if and only if certain centrally symmetric convex bodies exist in ${\mathbb{E}^d}$ which are not intersection bodies. It is also shown that a cylinder in ${\mathbb{E}^d}$ is an intersection body if and only if $d \leq 4$, and that suitably smooth axis-convex bodies of revolution are intersection bodies when $ d \leq 4$. These results show that the Busemann-Petty problem has a negative answer for $d \geq 5$ and a positive answer for $d = 3$ and $d = 4$ when the body with smaller sections is a body of revolution.
$p$-adic hyperbolic planes and modular forms
John A.
Rhodes
469-504
Abstract: For $K$ a number field and ${\mathbf{p}}$ a finite prime of $K$, we define a ${\mathbf{p}}$-adic hyperbolic plane and study its geometry under the action of $ G{L_2}({K_{\mathbf{p}}})$. Seeking an operator with properties analogous to those of the non-Euclidean Laplacian of the classical hyperbolic plane, we investigate the fundamental invariant integral operator, the Hecke operator ${T_{\mathbf{p}}}$. Letting $S$ be a finite set of primes of a totally real $K$ (including all the infinite ones), a modular group $\Gamma (S)$ is defined. This group acts discontinuously on a product of classical and ${\mathbf{p}}$-adic hyperbolic planes. $ S$-modular forms and their associated Dirichlet series are studied.
Division algebra coproducts of index $n$
Michel
Van den Bergh;
Aidan
Schofield
505-517
Abstract: Given a family of separable finite dimensional extensions $\{ {L_i}\}$ of a field $k$, we construct a division algebra $ {n^2}$ over its center which is freely generated over $k$ by the fields $ \{ {L_i}\}$.
Decompositions for relatively normal lattices
James B.
Hart;
Constantine
Tsinakis
519-548
Abstract: Continuing the work begun in Snodgrass and Tsinakis [26, 27], we develop a family of decomposition theorems for classes of relatively normal lattices. These results subsume and are inspired by known decomposition theorems in order-algebra due to P. Conrad and D. B. McAlister. As direct corollaries of the main results, we obtain corresponding decomposition theorems for classes of partially ordered sets.
Basic constructions in the $K$-theory of homotopy ring spaces
R.
Schwänzl;
R. M.
Vogt
549-584
Abstract: Using the language of category theory and universal algebra we formalize the passage from the permutative category of finitely generated free $R$-modules to the algebraic $K$-theory $KR$ of $R$ and thus make it applicable to homotopy ring spaces. As applications we construct a Waldhausen type of algebraic $K$-theory for arbitrary homotopy ring spaces, show its equivalence with constructions of May and Steiner, prove its Morita invariance and show that the algebraic $K$-theory $KX$ of an $ {E_\infty }$ ring $ X$ is itself an ${E_\infty }$ ring. Finally we investigate the monomial map $ Q(BX_ + ^\ast) \to KX$.
Finite and countable additivity of topological properties in nice spaces
V. V.
Tkachuk
585-601
Abstract: Let $Q \in$ character $\leq \tau$, pseudocharacter $ \leq \tau$, tightness $ \leq \tau$, weight $ \leq \tau$ , $ {P_\tau }$-property, discreteness, Fréchet-Urysohn property, sequentiality, radiality, pseudoradiality, local compactness, $k$-property. If ${X^n} = \cup \{ {X_i}:i \in n\}$, ${X_i} \vdash Q$ for all $i \in n$ then $ X \vdash Q$ (i.e. the property $Q$ is $n$-additive in ${X^n}$ for any $ X \in {T_3}$). Metrizability is $n$-additive in ${X^n}$ provided $X$ is compact or $ c(X) = \omega$. ${\text{ANR}}$-property is closely $n$-additive in ${X^n}$ if $X$ is compact ("closely" means additivity in case $ {X_i}$ is closed in $ {X^n}$). If $Q \in$ metrizability, character $ \leq \tau$, pseudocharacter $\leq \tau$, diagonal number $\leq \tau$ , $i$-weight $\leq \tau$, pseudoweight $\leq \tau$, local compactness then $ Q$ is finitely additive in any topological group.
Characteristic cycles of discrete series for ${\bf R}$-rank one groups
Jen-Tseh
Chang
603-622
Abstract: We determine the characteristic cycles of the discrete series representations for connected $ \mathbb{R}$-rank one linear groups. The computation is made through the moment maps; we determine their fibers and the cohomology in question case by case. The multiplicity of the discrete series, in terms of their Harish-Chandra parameters, is given by recursive formulae; for groups of type $ A$ and $B$ closed formulae are obtained.
On the hyperbolic Kac-Moody Lie algebra $HA\sb 1\sp {(1)}$
Seok-Jin
Kang
623-638
Abstract: In this paper, using a homological theory of graded Lie algebras and the representation theory of $A_1^{(1)}$, we compute the root multiplicities of the hyperbolic Kac-Moody Lie algebra $HA_1^{(1)}$ up to level $4$ and deduce some interesting combinatorial identities.
Iterated floor function, algebraic numbers, discrete chaos, Beatty subsequences, semigroups
Aviezri S.
Fraenkel
639-664
Abstract: For a real number $ \alpha$, the floor function $ \left\lfloor \alpha \right\rfloor$ is the integer part of $\alpha$. The sequence $\{ \left\lfloor {m\alpha } \right\rfloor :m = 1,2,3, \ldots \}$ is the Beatty sequence of $\alpha$. Identities are proved which express the sum of the iterated floor functional $ {A^i}$ for $1 \leq i \leq n$, operating on a nonzero algebraic number $\alpha$ of degree $\leq n$, in terms of only $ {A^1} = \left\lfloor {m\alpha } \right\rfloor ,m$ and a bounded term. Applications include discrete chaos (discrete dynamical systems), explicit construction of infinite nonchaotic subsequences of chaotic sequences, discrete order (identities), explicit construction of nontrivial Beatty subsequences, and certain arithmetical semigroups. (Beatty sequences have a large literature in combinatorics. They have also been used in nonperiodic tilings (quasicrystallography), periodic scheduling, computer vision (digital lines), and formal language theory.)
Perverse sheaves and finite-dimensional algebras
K.
Vilonen
665-676
Abstract: We study finite dimensional algebras which arise from categories of perverse sheaves on complex algebraic varieties.
Stability of special instanton bundles on ${\bf P}\sp {2n+1}$
Vincenzo
Ancona;
Giorgio
Ottaviani
677-693
Abstract: We prove that the special instanton bundles of rank $2n$ on ${\mathbb{P}^{2n + 1}}(\mathbb{C})$ with a symplectic structure studied by Spindler and Trautmann are stable in the sense of Mumford-Takemoto. This implies that the generic special instanton bundle is stable. Moreover all instanton bundles on ${\mathbb{P}^5}$ are stable. We get also the stability of other related vector bundles.
Operator semigroups for functional-differential equations with delay
W. M.
Ruess;
W. H.
Summers
695-719
Abstract: We show that a strongly continuous operator semigroup can be associated with the functional differential delay equation (FDE) $\displaystyle \left\{ {\begin{array}{*{20}{c}} {x\prime(t) + ax(t) + Bx(t) \bac... ...{\mathbb{R}^ - }}} = \varphi \in E} & {} \end{array} } \right.$ under local conditions which give wide latitude to those subsets of the state space $ X$ and initial data space $ E$, respectively, where (a) the (generally multivalued) operator $B \subseteq X \times X$ is defined and accretive, and (b) the historyresponsive function $F:D(F) \subseteq E \to X$ is defined and Lipschitz continuous. The associated semigroup is then used to investigate existence and uniqueness of solutions to (FDE). By allowing the domain of the solution semigroup to be restricted according to specific local properties of $B$ and $F$, moreover, our methods automatically lead to assertions on flow invariance. We illustrate our results through applications to the Goodwin oscillator and a single species population model.
The structure of a random graph at the point of the phase transition
Tomasz
Łuczak;
Boris
Pittel;
John C.
Wierman
721-748
Abstract: Consider the random graph models $G(n,\char93 \;{\text{edges}} = M)$ and $ G(n,\operatorname{Prob}({\text{edge}}) = p)$ with $M = M(n) = (1 + \lambda {n^{ - 1/3}})n/2$ and $p = p(n) = (1 + \lambda {n^{ - 1/3}})/n$. For $l \geq - 1$ define an $l$-component of a random graph as a component which has exactly $l$ more edges than vertices. Call an $ l$-component with $ l \geq 1$ a complex component. For both models, we show that when $ \lambda$ is constant, the expected number of complex components is bounded, almost surely (a.s.) each of these components (if any exist) has size of order ${n^{2/3}}$, and the maximum value of $ l$ is bounded in probability. We prove that a.s. the largest suspended tree in each complex component has size of order ${n^{2/3}}$, and deletion of all suspended trees results in a "smoothed" graph of size of order ${n^{1/3}}$, with the maximum vertex degree $ 3$. The total number of branching vertices, i.e., of degree $3$, is bounded in probability. Thus, each complex component is almost surely topologically equivalent to a $3$-regular multigraph of a uniformly bounded size. Lengths of the shortest cycle and of the shortest path between two branching vertices of a smoothed graph are each of order ${n^{1/3}}$. We find a relatively simple integral formula for the limit distribution of the numbers of complex components, which implies, in particular, that all values of the "complexity spectrum" have positive limiting probabilities. We also answer questions raised by Erdös and Rényi back in 1960. It is proven that there exists $p(\lambda )$, the limiting planarity probability, with $ 0 < p(\lambda ) < 1$, $p( - \infty ) = 1$, $p(\infty ) = 0$. In particular, $G(n,M)\quad (G(n,p),{\text{resp}}.)$ is almost surely nonplanar iff $(M - n/2){n^{ - 2/3}} \to \infty \;((np - 1){n^{ - 1/3}}) \to \infty ,{\text{resp}}.)$.
Orthogonal polynomials and hypergroups. II. The symmetric case
R.
Lasser
749-770
Abstract: The close relationship between orthogonal polynomial sequences and polynomial hypergroups is further studied in the case of even weight function, cf. [18]. Sufficient criteria for the recurrence relation of orthogonal polynomials are given such that a polynomial hypergroup structure is determined on $ {\mathbb{N}_0}$. If the recurrence coefficients are convergent the dual spaces are determined explicitly. The polynomial hypergroup structure is revealed and investigated for associated ultraspherical polynomials, Pollaczek polynomials, associated Pollaczek polynomials, orthogonal polynomials with constant monic recursion formula and random walk polynomials.
A singular representation of $E\sb 6$
B.
Binegar;
R.
Zierau
771-785
Abstract: Algebraic properties of a singular representation of ${{\mathbf{E}}_6}$ are studied. This representation has the Joseph ideal as its annihilator and it remains irreducible when restricted to ${{\mathbf{F}}_4}$.
Approximation from shift-invariant subspaces of $L\sb 2(\bold R\sp d)$
Carl
de Boor;
Ronald A.
DeVore;
Amos
Ron
787-806
Abstract: A complete characterization is given of closed shift-invariant subspaces of $ {L_2}({\mathbb{R}^d})$ which provide a specified approximation order. When such a space is principal (i.e., generated by a single function), then this characterization is in terms of the Fourier transform of the generator. As a special case, we obtain the classical Strang-Fix conditions, but without requiring the generating function to decay at infinity. The approximation order of a general closed shift-invariant space is shown to be already realized by a specifiable principal subspace.
On convergence and closedness of multivalued martingales
Zhen Peng
Wang;
Xing Hong
Xue
807-827
Abstract: In this paper, various convergence theorems and criteria of closedness of multivalued martingales, submartingales, and supermartingales are proved.
Uniqueness theorems for parametrized algebraic curves
Peter
Hall
829-840
Abstract: Let ${L_1}, \ldots ,{L_n}$ be lines in ${\mathbb{P}^2}$ and let $f,g:{\mathbb{P}^1} \to {\mathbb{P}^2}$ be nonconstant algebraic maps. For certain configurations of lines ${L_1}, \ldots ,{L_n}$, the hypothesis that, for $i = 1, \ldots ,n$, the inverse images $ {f^{ - 1}}({L_i})$ and ${g^{ - 1}}({L_i})$ are equal, not necessarily with the same multiplicities, implies that $ f$ is identically equal to $ g$.
Carleson measures on spaces of homogeneous type
Steven C.
Gadbois;
William T.
Sledd
841-862
Abstract: Let $X$ be a space of homogeneous type in the sense of Coifman and Weiss $[{\text{CW}}2]$ and let ${X^ + } = X \times {{\mathbf{R}}^ + }$. A positive function $F$ on ${X^ + }$ is said to have horizontal bounded ratio $({\text{HBR}})$ on ${X^ + }$ if there is a constant ${A_F}$ so that $F(x,t) \leq {A_F}F(y,t)$ whenever $\rho (x,y) < t$. (By Harnack's inequality, a well-known example is any positive harmonic function in the upper half plane.) $ {\text{HBR}}$ is a rich class that is closed under a wide variety of operations and it provides useful majorants for many classes of functions that are encountered in harmonic analysis. We are able to prove theorems in spaces of homogeneous type for functions in $ {\text{HBR}}$ which are analogous to the classical Carleson measure theorems and to extend these results to the functions which they majorize. These results may be applied to obtain generalizations of the original Carleson measure theorem, and of results of Flett's which contain the Hardy-Littlewood theorems on intermediate spaces of analytic functions. Hörmander's generalization of Carleson's theorem is included and it is possible to extend those results to the atomic ${H^p}$ spaces of Coifman and Weiss.
Invariant subspaces of the Dirichlet shift and pseudocontinuations
Stefan
Richter;
Carl
Sundberg
863-879
Abstract: In this paper we study extremal functions for invariant subspaces $\mathcal{M}$ of the Dirichlet shift, i.e., solutions $\varphi$ of the extremal problem $\displaystyle \sup \{ \vert{f^{(n)}}(0)\vert/{\left\Vert f \right\Vert _D}:f \in \mathcal{M},f \ne 0\}$ . Here $n$ is the smallest nonnegative integer such that the sup is positive. It is known that such a function $ \varphi$ generates $\mathcal{M}$. We show that the derivative $(z\varphi )\prime$ has a pseudocontinuation to the exterior disc. This pseudocontinuation is an analytic continuation exactly near those points of the unit circle where $\varphi$ is bounded away from zero. We also show that the radial limit of the absolute value of an extremal function exists at every point of the unit circle. Some of our results are valid for all functions that are orthogonal to a nonzero invariant subspace.
Local asymptotic distribution of zeros of orthogonal polynomials
Vilmos
Totik;
Joseph L.
Ullman
881-894
Abstract: Converse results, which state a relation (inequality) for measures from that on their logarithmic potentials, are applied to local density of zeros of orthogonal polynomials when the measure of orthogonality is a general one with compact support. It will be shown that if the measure is sufficiently thick on a part of its support, then on that part the density of the zeros will be at least as large as the equilibrium measure of the support. A corresponding upper estimate on the distribution of the zeros will also be proved. All of our estimates are sharp, and they localize several well-known results.
Regularity of algebras related to the Sklyanin algebra
J. T.
Stafford
895-916
Abstract: This paper continues the research of [SS] by finding further examples of (Artin-Schelter) regular rings of dimension four. Unlike the threedimensional case studied in [ATV1, 2], these examples show that the fourdimensional regular rings are not uniquely determined by their associated geometric data. Indeed, we find a one-parameter family of regular algebras associated to this data.
On invertible bimodules and automorphisms of noncommutative rings
Robert M.
Guralnick;
Susan
Montgomery
917-937
Abstract: In this article, we attempt to generalize the result that for a commutative ring $R$ the outer automorphism group of $ R$-automorphisms of $ {M_n}(R)$ is abelian of exponent $n$. It is shown that a slightly weaker stable version of the result is still valid for affine semiprime noetherian pi rings. We also show that the automorphism group of an affine commutative domain of positive dimension acts faithfully on the spectrum of the domain. We investigate other questions involving bimodules and automorphisms and extend a result of Smith on the first Weyl algebra as a fixed ring.
Nonsimple, ribbon fibered knots
Katura
Miyazaki
1-44
Abstract: The connected sum of an arbitrary knot and its mirror image is a ribbon knot, however the converse is not necessarily true for all ribbon knots. We prove that the converse holds for any ribbon fibered knot which is a connected sum of iterated torus knots, knots with irreducible Alexander polynomials, or cables of such knots. This gives a practical method to detect nonribbon fibered knots. The proof uses a characterization of homotopically ribbon, fibered knots by their monodromies due to Casson and Gordon. We also study when cable fibered knots are ribbon and results which support the following conjecture. Conjecture. If a $(p,q)$ cable of a fibered knot $k$ is ribbon where $p(> 1)$ is the winding number of a cable in ${S^1} \times {D^2}$, then $ q = \pm 1$ and $ k$ is ribbon.
Topological applications of generic huge embeddings
Franklin D.
Tall
45-68
Abstract: In the Foreman-Laver model obtained by huge cardinal collapse, for many $ \Phi ,\Phi ({\aleph _1})$ implies $ \Phi ({\aleph _2})$. There are a variety of set-theoretic and topological applications, in particular to paracompactness. The key tools are generic huge embeddings and preservation via $\kappa$-centred forcing. We also formulate "potent axioms" à la Foreman which enable us to transfer from $ {\aleph _1}$ to all cardinals. One such axiom implies that all ${\aleph _1}$-collectionwise normal Moore spaces are metrizable. It also implies (as does Martin's Maximum) that a first countable generalized ordered space is hereditarily paracompact iff every subspace of size $ {\aleph _1}$ is paracompact.
Some $q$-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras
Robert A.
Gustafson
69-119
Abstract: Multidimensional generalizations of beta type integrals of Barnes, Ramanujan, Askey-Wilson, and others are evaluated. These integrals are analogues of the summation theorems for multilateral hypergeometric series associated to the simple Lie algebras of classical type and type $ {G_2}$. Many of these integrals can also be written as group integrals over a compact Lie group or conjugation invariant integrals over the corresponding Lie algebra.
Nonfibering spherical $3$-orbifolds
William D.
Dunbar
121-142
Abstract: Among the finite subgroups of $SO(4)$, members of exactly $21$ conjugacy classes act on $ {S^3}$ preserving no fibration of ${S^3}$ by circles. We identify the corresponding spherical $3$-orbifolds, i.e., for each such ${\mathbf{G}} < SO(4)$, we describe the embedded trivalent graph $\{ x \in {S^3}:\exists {\mathbf{I}} \ne {\mathbf{g}} \in {\mathbf{G}}$ s.t. ${\mathbf{g}}(x) = x\} /{\mathbf{G}}$ in the topological space $ {S^3}/{\mathbf{G}}$ (which turns out to be homeomorphic to $ {S^3}$ in all cases). Explicit fundamental domains (of Dirichlet type) are described for $9$ of the groups, together with the identifications to be made on the boundary. The remaining $ 12$ spherical orbifolds are obtained as mirror images or (branched) covers of these.
Co-Hopficity of Seifert-bundle groups
F.
González-Acuña;
R.
Litherland;
W.
Whitten
143-155
Abstract: A group $ G$ is cohopfian, if every monomorphism $G \to G$ is an automorphism. In this paper, we answer the cohopficity question for the fundamental groups of compact Seifert fiber spaces (or Seifert bundles, in the current vernacular). If $ M$ is a closed Seifert bundle, then the following are equivalent: (a) $ {\pi _1}M$ is cohopfian; (b) $M$ does not cover itself nontrivially; (c) $ M$ admits a geometric structure modeled on ${S^3}$ or on $ {\tilde{\text{SL}_2\mathbf{R}}}$. If $M$ is a compact Seifert bundle with nonempty boundary, then ${\pi _1}M$ is not cohopfian.
An explicit Plancherel formula for ${\rm U}(2,1)$
David
Jabon;
C. David
Keys;
Allen
Moy
157-171
Abstract: The admissible duals of quasi-split unitary groups over nonarchimedean fields are determined. The set of irreducible unitarizable representations, and the Plancherel measure on the unitary dual, is given explicitly.
The planar closing lemma for chain recurrence
Maria Lúcia Alvarenga
Peixoto;
Charles Chapman
Pugh
173-192
Abstract: If $z$ is a chain recurrent point of a $ {C^r}$ planar flow, all of whose fixed points are hyperbolic, then it is proved that the orbit through $z$ becomes periodic under a perturbation that is $ {C^r}$ small in the Whitney topology.
The structure of hyperfinite Borel equivalence relations
R.
Dougherty;
S.
Jackson;
A. S.
Kechris
193-225
Abstract: We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed.
Contributions to the classification of simple modular Lie algebras
Georgia
Benkart;
J. Marshall
Osborn;
Helmut
Strade
227-252
Abstract: We develop results directed towards the problem of classifying the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic $ p > 7$. A $ 1$-section of such a Lie algebra relative to a torus $T$ of maximal absolute toral rank possesses a unique subalgebra maximal with respect to having a composition series with factors which are abelian or classical simple. In this paper we show that the sum $ Q$ of those compositionally classical subalgebras is a subalgebra. This extends to the general case a crucial step in the classification by Block and Wilson of the restricted simple Lie algebras. We derive properties of the filtration which can be constructed using $Q$ and obtain structural information about the $ 1$-sections and $ 2$-sections of $ Q$ relative to $ T$. We further classify all those algebras in which $Q$ is solvable.
The Brownian motion and the canonical stochastic flow on a symmetric space
Ming
Liao
253-274
Abstract: We study the limiting behavior of Brownian motion ${x_t}$ on a symmetric space $V = G/K$ of noncompact type and the asymptotic stability of the canonical stochastic flow ${F_t}$ on $O(V)$. We show that almost surely, $ {x_t}$ has a limiting direction as it goes to infinity. The study of the asymptotic stability of ${F_t}$ is reduced to the study of the limiting behavior of the adjoint action on the Lie algebra $\mathcal{G}$ of $G$ by the horizontal diffusion in $G$. We determine the Lyapunov exponents and the associated filtration of ${F_t}$ in terms of root space structure of $ \mathcal{G}$.
A functional from geometry with applications to discrepancy estimates and the Radon transform
Allen D.
Rogers
275-313
Abstract: Estimates of discrepancy, or irregularities of distribution, are obtained for measures without atoms. Two estimators are used, the half-space, or separation, discrepancy ${D_S}$ and a geometric functional ${I^\alpha }$. A representation formula for the generalized energy integral ${I^\alpha }$ is developed. Norm inequalities for the Radon transform are obtained as an application of the continuous discrepancy results. Integral geometric notions play a prominent role.
Representations of the symmetric group in deformations of the free Lie algebra
A. R.
Calderbank;
P.
Hanlon;
S.
Sundaram
315-333
Abstract: We consider, for a given complex parameter $\alpha$, the nonassociative product defined on the tensor algebra of $n$-dimensional complex vector space by $ [x,y] = x \otimes y - \alpha y \otimes x$. For $k$ symbols $ {x_1}, \ldots ,{x_k}$, the left-normed bracketing is defined recursively to be the bracketing sequence ${b_k}$, where $ {b_1} = {x_1}$, ${b_2} = [{x_1},{x_2}]$, and ${b_k} = [{b_{k - 1}},{x_k}]$. The linear subspace spanned by all multilinear left-normed bracketings of homogeneous degree $n$, in the basis vectors ${v_1}, \ldots ,{v_n}$ of ${\mathbb{C}^n}$, is then an ${S_n}$-module $ {V_n}(\alpha )$. Note that $ {V_n}(1)$ is the Lie representation $ \operatorname{Lie}_n$ of $ {S_n}$ afforded by the $ n$th-degree multilinear component of the free Lie algebra. Also, ${V_n}(- 1)$ is the subspace of simple Jordan products in the free associative algebra as studied by Robbins [Ro]. Among our preliminary results is the observation that when $\alpha$ is not a root of unity, the module ${V_n}(\alpha )$ is simply the regular representation. Thrall [T] showed that the regular representation of the symmetric group ${S_n}$ can be written as a direct sum of tensor products of symmetrised Lie modules ${V_\lambda }$. In this paper we determine the structure of the representations ${V_n}(\alpha )$ as a sum of a subset of these ${V_\lambda }$. The $ {V_\lambda }$, indexed by the partitions $\lambda$ of $n$, are defined as follows: let ${m_i}$ be the multiplicity of the part $ i$ in $\lambda$, let $\operatorname{Lie}_i$ be the Lie representation of $ {S_i}$, and let ${\iota _k}$ denote the trivial character of the symmetric group ${S_k}$. Let ${\iota _{{m_i}}}[\operatorname{Lie}_i]$ denote the character of the wreath product ${S_{{m_i}}}[{S_i}]$ of $ {S_{{m_i}}}$ acting on $ {m_i}$ copies of $ {S_i}$. Then ${V_\lambda }$ is isomorphic to the $ {S_n}$-module $\displaystyle ({\iota _{{m_1}}}[\operatorname{Lie}_1] \otimes \cdots \otimes {\... ...S_{m_1}}[{S_1}] \times \cdots \times {S_{{m_i}}}[{S_i}] \times \cdots }^{S_n}}.$ Our theorem now states that when $\alpha$ is a primitive $p$th root of unity, the ${S_n}$-module $ {V_n}(\alpha )$ is isomorphic to the direct sum of those ${V_\lambda }$, where $\lambda$ runs over all partitions $\lambda$ of $n$ such that no part of $\lambda$ is a multiple of $p$.
Polish group actions and the Vaught conjecture
Ramez L.
Sami
335-353
Abstract: We consider the topological Vaught conjecture: If a Polish group $ G$ acts continuously on a Polish space $S$, then $S$ has either countably many or perfectly many orbits. We show 1. The conjecture is true for Abelian groups. 2. The conjecture is true whenever $ G$, $S$ are recursively presented, the action of $G$ is recursive and, for $x \in S$ the orbit of $x$ is of Borel multiplicative rank $\leq \omega _1^x$. Assertion $1$ holds also for analytic $S$. Specializing $G$ to a closed subgroup of $ \omega !$, we prove that nonempty invariant Borel sets, not having perfectly many orbits, have orbits of about the same Borel rank. An upper bound is derived for the Borel rank of orbits when the set of orbits is finite.
Besov spaces on closed subsets of ${\bf R}\sp n$
Alf
Jonsson
355-370
Abstract: Motivated by the need in boundary value problems for partial differential equations, classical trace theorems characterize the trace to a subset $F$ of $ {\mathbb{R}^n}$ of Sobolev spaces and Besov spaces consisting of functions defined on $ {\mathbb{R}^n}$, if $ F$ is a linear subvariety $ {\mathbb{R}^d}$ of ${\mathbb{R}^n}$ or a $d$-dimensional smooth submanifold of ${\mathbb{R}^n}$. This was generalized in [2] to the case when $F$ is a $d$-dimensional fractal set of a certain type. In this paper, traces are described when $ F$ is an arbitrary closed set. The result may also be looked upon as a Whitney extension theorem in ${L^p}$.
The $H\sp p$-corona theorem for the polydisc
Kai-Ching
Lin
371-375
Abstract: Let ${H^p} = {H^p}({D^n})$ denote the usual Hardy spaces on the polydisc ${D^n}$. We prove in this paper the following theorem: Suppose ${f_1},{f_2}, \ldots ,{f_n} \in {H^\infty },{\left\Vert {{f_j}} \right\Vert _{{H^\infty }}} \leq 1$, and $ \sum\nolimits_{j = 1}^m {\vert{f_j}(z)\vert} \geq \delta > 0$. Then for every $g$ in ${H^p}$, $1 < p < \infty$, there are $ {H^p}$ functions $g,g, \ldots ,{g_m}$ such that $\sum\nolimits_{j = 1}^m {{f_j}(z){g_j}(z) = g(z)}$. Moreover, we have ${\left\Vert {{g_j}} \right\Vert _{{H^p}}} \leq c(m,n,\delta ,p){\left\Vert g \right\Vert _{{H^p}}}$. (When $p = 2,n = 1$, this theorem is known to be equivalent to Carleson's corona theorem.)
A weak characteristic pair for end-irreducible $3$-manifolds
Bobby Neal
Winters
377-403
Abstract: This extends a weakened version of the Characteristic Pair Theorem of Jaco, Shalen, and Johannson to a large subclass of the class of end-irreducible $ 3$-manifolds. The Main Theorem of this paper states that if $(W,w)$ is a noncompact $3$-manifold pair (where $W$ is a noncompact $3$-manifold that has an exhausting sequence with certain nice properties and where $ w$ is incompressible in $ W$), then there is a Seifert pair $(\Sigma ,\Phi )$ contained in $ (W,w)$ such that any $ 2$-manifold that is strongly essential in $(W,w)$ and each of whose components is a torus, an annulus, an open annulus, or a half-open annulus is isotopic in $(W,w)$ into $ (\Sigma ,\Phi )$.
Circular units of function fields
Frederick F.
Harrop
405-421
Abstract: A unit index-class number formula is proved for subfields of cyclotomic function fields in analogy with similar results for subfields of cyclotomic number fields.
Subgroup rigidity in finite-dimensional group algebras over $p$-groups
Gary
Thompson
423-447
Abstract: In 1986, Roggenkamp and Scott proved in [RS1] Theorem 1.1. Let $G$ be a finite $p$-group for some prime $p$, and $S$ a local or semilocal Dedekind domain of characteristic 0 with a unique maximal ideal containing $p$ (for example, $S = {\mathbb{Z}_p}$ where ${\mathbb{Z}_p}$ is the $p$-adic integers). If $ H$ is a subgroup of the normalized units of $SG$ with $\vert H\vert = \vert G\vert$, then $ H$ is conjugate to $ G$ by an inner automorphism of $SG$. In the Appendix of a later paper [S], Scott outlined a possible proof of a related result: Theorem 1.3. Let $S$ be a complete, discrete valuation domain of characteristic 0 having maximal ideal $ \wp$ and residue field $F \cong S/\wp $ of characteristic $ p$. Let $ G$ be a finite $ p$-group, and let $ U$ be a finite group of normalized units in $SG$. Then there is a unit $w$ in $SG$ such that $wU{w^{ - 1}} \leq G$. The author later filled in that outline to give a complete proof of Theorem 1.3 and, at the urging of Scott, has been able to extend that result to Theorem 1.2. Let $S$ be a complete, discrete valuation ring of characteristic 0 having maximal ideal $ \wp$ containing $ p$. Let $ A$ be a local $ S$-algebra that is finitely generated as an $S$-module, and let $G$ be a finite $p$-group. Then any finite, normalized subgroup of the $S$-algebra $\mathcal{A} = A{ \otimes _S}SG$ is conjugate to a subgroup of $G$.
On the rank and the crank modulo $4$ and $8$
Richard
Lewis;
Nicolas
Santa-Gadea
449-465
Abstract: In this paper we prove some identities, conjectured by Lewis, for the rank and crank of partitions concerning the modulo $4$ and $8$. These identities are similar to Dyson's identities for the rank modulo $5$ and $7$ which give a combinatorial interpretation to Ramanujan's partition congruences. For this, we use multisection of series and some of the results that Watson established for the third order mock theta functions.
Corrections to: ``First steps in descriptive theory of locales'' [Trans. Amer. Math. Soc. {\bf 327} (1991), no. 1, 353--371; MR1091230 (92b:54078)]
John
Isbell
467-468
Symmetries of planar growth functions. II
William J.
Floyd
447-502
Abstract: Let $G$ be a finitely generated group, and let $\Sigma$ be a finite generating set of $ G$. The growth function of $(G,\Sigma )$ is the generating function $ f(z) = \sum\nolimits_{n = 0}^\infty {{a_n}{z^n}}$, where ${a_n}$ is the number of elements of $ G$ with word length $ n$ in $\Sigma$. Suppose that $G$ is a cocompact group of isometries of Euclidean space $ {\mathbb{E}^2}$ or hyperbolic space $ {\mathbb{H}^2}$, and that $ D$ is a fundamental polygon for the action of $G$. The full geometric generating set for $ (G,D)$ is $\{ g \in G:g \ne 1$ and $gD \cap D \ne \emptyset \} $. In this paper the recursive structure for the growth function of $(G,\Sigma )$ is computed, and it is proved that the growth function $f$ is reciprocal $ (f(z) = f(1/z))$ except for some exceptional cases when $D$ has three, four, or five sides.
Homogeneous chaos, $p$-forms, scaling and the Feynman integral
G. W.
Johnson;
G.
Kallianpur
503-548
Abstract: In a largely heuristic but fascinating recent paper, Hu and Meyer have given a "formula" for the Feynman integral of a random variable $f$ on Wiener space in terms of the expansion of $ f$ in Wiener chaos. The surprising properties of scaling in Wiener space make the problem of rigorously connecting this formula with the usual definition of the analytic Feynman integral a subtle one. One of the main tools in carrying this out is our definition of the 'natural extension' of $ p$th homogeneous chaos in terms of the 'scale-invariant lifting' of $ p$-forms on the white noise space ${L^2}({\mathbb{R}_ + })$ connected with Wiener space. The key result in our development says that if $ {f_p}$ is a symmetric function in ${L^2}(\mathbb{R}_ + ^p)$ and ${\psi _p}({f_p})$ is the associated $p$-form on ${L^2}({\mathbb{R}_ + })$, then ${\psi _p}({f_p})$ has a scaled $ {L^2}$-lifting if and only if the '$k$th limiting trace' of ${f_p}$ exists for $k = 0,1, \ldots ,[p/2]$. This necessary and sufficient condition for the lifting of a $p$-form on white noise space to a random variable on Wiener space is a worthwhile contribution to white noise theory apart from any connection with the Feynman integral since $p$-forms play a role in white noise calculus analogous to the role played by $p$th homogeneous chaos in Wiener calculus. Various $k$-traces arise naturally in this subject; we study some of their properties and relationships. The limiting $k$-trace plays the most essential role for us.
On a theorem of Muckenhoupt and Wheeden and a weighted inequality related to Schr\"odinger operators
C.
Pérez
549-562
Abstract: We extend in several directions a theorem of B. Muckenhoupt and R. Wheeden relating the ${L^p}$-norms of Riesz potentials and fractional maximal operators. We apply these results to give a simple proof and sharpen a weighted inequality for Schrödinger operators of Chang, Wilson and Wolff.
Normality in $X\sp 2$ for compact $X$
G.
Gruenhage;
P. J.
Nyikos
563-586
Abstract: In 1977, the second author announced the following consistent negative answer to a question of Katětov: Assuming $ {\text{MA}} + \neg {\text{CH}}$, there is a compact nonmetric space $ X$ such that $ {X^2}$ is hereditarily normal. We give the details of this example, and construct another example assuming $ {\text{CH}}$. We show that both examples can be constructed so that ${X^2}\backslash \Delta$ is perfectly normal. We also construct in $ {\text{ZFC}}$ a compact nonperfectly normal $X$ such that $ {X^2}\backslash \Delta$ is normal.
A qualitative uncertainty principle for unimodular groups of type ${\rm I}$
Jeffrey A.
Hogan
587-594
Abstract: It has long been known that if $f \in {L^2}({{\mathbf{R}}^n})$ and the supports of and its Fourier transform $ \hat f$ are bounded then $f = 0$ almost everywhere. More recently it has been shown that the same conclusion can be reached under the weaker condition that the supports of $f$ and $\hat f$ have finite measure. These results may be thought of as qualitative uncertainty principles since they limit the "concentration" of the Fourier transform pair $ (f,\hat f)$. Little is known, however, of analogous results for functions on locally compact groups. A qualitative uncertainty principle is proved here for unimodular groups of type I.
$EHP$ spectra and periodicity. I. Geometric constructions
Brayton
Gray
595-616
Abstract: The techniques used in $EHP$ calculation are studied, and lead to the notion of an $EHP$ spectrum. A simple inductive procedure suggests the existence of higher order $EHP$ spectra in which the first differential corresponds to ${v_n}$ multiplication. The next case $(n = 1)$ is constructed using the work of Cohen, Moore, and Neisendorfer. Some of the expected universal properties are proven.
$EHP$ spectra and periodicity. II. $\Lambda$-algebra models
Brayton
Gray
617-640
Abstract: The results of part I suggest that for small $m$, the Smith-Toda spectrum $V(m)$ can be approximated by spaces having universal mapping properties and interlocking fibrations. For each $m$, a $\Lambda$-algebra model representing the Bousfield-Kan $E\prime$ term for these spaces is constructed, and all of the ideal results are proven on the chain level.
Solutions to the nonautonomous bistable equation with specified Morse index. I. Existence
Nicholas D.
Alikakos;
Peter W.
Bates;
Giorgio
Fusco
641-654
Abstract: We investigate the existence of unstable solutions of specified Morse index for the equation ${\varepsilon ^2}{u_{xx}} - f(x,u) = 0$ on a finite interval and Neumann boundary conditions.
A Frobenius characterization of rational singularity in $2$-dimensional graded rings
Richard
Fedder
655-668
Abstract: A ring $ R$ is said to be $ F$-rational if, for every prime $P$ in $R$, the local ring ${R_P}$ has the property that every system of parameters ideal is tightly closed (as defined by Hochster-Huneke). It is proved that if $R$ is a $2$-dimensional graded ring with an isolated singularity at the irrelevant maximal ideal $m$, then the following are equivalent: (1) $R$ has a rational singularity at $ m$. (2) $R$ is $F$-rational. (3) $a(R) < 0$. Here $a(R)$ (as defined by Goto-Watanabe) denotes the least nonvanishing graded piece of the local cohomology module ${H_m}(R)$. The proof of this result relies heavily on the properties of derivations of $ R$, and suggests further questions in that direction; paradigmatically, if one knows that $D(a)$ satisfies a certain property for every derivation $D$, what can one conclude about the original ring element $a$?
Cohomology classes associated to anomalies
Gregory Lambros
Peterson
669-704
Abstract: One of the proposed settings for the description of anomalies in the setting of gauge field theories is a local bicomplex associated to a principal fiber bundle $G \to P \to M$. It is a bigraded algebra with two differentials which is invariantly defined, i.e. independent of local coordinates of $ M$. I denote it by $S_M^{ \bullet , \bullet }$. Briefly, $ S_M^{p,q}$ consists of local $p$-multilinear operators from the gauge algebra into $ q$-forms on $M$ which depend on a connection $ A$ in a local manner; local means that the operators decrease supports. The gauge algebra is the Lie algebra of the gauge group, which consists of diffeomorphisms of $ P$ that respect the action of $G$ and cover the identity diffeomorphism of $ M$. In this setting, the anomalies are described as integrals over $ M$ whose integrands can be shown to be representatives of total cohomology classes in $ {H^1}(S_M^{ \bullet , \bullet })$. The main reason for restricting to a local bicomplex is due to Peetre's theorem. It states that local operators are differential operators over open sets $U \subset M$. This property is both mathematically natural and required by physical considerations. This paper explores the computation of the total cohomology of the local bicomplex by beginning with the coordinate description of the differential operators and then determining which of these differential operators can be used to construct invariantly defined objects. What is accomplished is the description of the differential operators which are invariant under the action of the local diffeomorphisms of ${\mathbb{R}^n}$ and the computation of their total cohomology over open sets $ U \subset M$. The main result is that $\displaystyle H_d^ \bullet ({(S_U^{ \bullet , \bullet })^{{\operatorname{Diff}_... ...}({\mathbb{R}^n})}}) \simeq H_d^ \bullet (W{(\mathfrak{g})_{[\tfrac{n} {2}]}}),$ where ${(S_U^{ \bullet , \bullet })^{{{\operatorname{Diff}}_{{\text{loc}}}}({\mathbb{R}^n})}}$ denotes the invariant differential operators over the open set $U$ and $ W{(\mathfrak{g})_{[\tfrac{n} {2}]}}$ is the Weil algebra of $\mathfrak{g}$, the Lie algebra of $G$ truncated at $[\tfrac{n} {2}]$, the greatest integer less than or equal to half the dimension of $M$. This shows that the cohomology groups over open sets are nonzero only in the range $n \leq q \leq n + r$ where $r$ is the dimension of the Lie algebra $\mathfrak{g}$ , and in this range they are all finite dimensional. This result is globalized in the special case that the associated fiber bundle $ \operatorname{ad}^\ast\;P$ is trivializable.
Vojta's refinement of the subspace theorem
Wolfgang M.
Schmidt
705-731
Abstract: Vojta's refinement of the Subspace Theorem says that given linearly independent linear forms $ {L_1}, \ldots , {L_n}$ in $ n$ variables with algebraic coefficients, there is a finite union $U$ of proper subspaces of ${\mathbb{Q}^n}$, such that for any $\varepsilon > 0$ the points $ \underline{\underline x} \in {\mathbb{Z}^n}\backslash \{ \underline{\underline 0} \}$ with (1) $ \vert{L_1}(\underline{\underline x} ) \cdots {L_n}(\underline{\underline x} )\vert\; < \;\vert\underline{\underline x} {\vert^{ - \varepsilon }}$ lie in $U$, with finitely many exceptions which will depend on $ \varepsilon$ . Put differently, if $ X(\varepsilon )$ is the set of solutions of (1), if $\bar X(\varepsilon )$ is its closure in the subspace topology (whose closed sets are finite unions of subspaces) and if $ \bar X\prime(\varepsilon )$ consists of components of dimension $> 1$ , then $\bar X\prime(\varepsilon ) \subset U$ . In the present paper it is shown that $ \bar X\prime(\varepsilon )$ is in fact constant when $ \varepsilon$ lies outside a simply described finite set of rational numbers. More generally, let $k$ be an algebraic number field and $ S$ finite set of absolute values of $k$ containing the archimedean ones. For $\upsilon \in S$ let $L_1^\upsilon, \ldots ,L_m^\upsilon$ be linear forms with coefficients in $k$, and for $\underline{\underline x} \in {K^n}\backslash \{ \underline{\underline 0} \}$ with height $ {H_k}(\underline{\underline x} ) > 1$ define ${a_{\upsilon i}}(\underline{\underline x} )$ by $\vert L_i^\upsilon(\underline{\underline x} )\vert _\upsilon/\vert\underline{\... ...{\underline x} )^{ - {a_{\upsilon i}}(\underline{\underline x} )/{d_\upsilon}}}$ where the ${d_\upsilon}$ are the local degrees. The approximation set $A$ consists of tuples $\underline{\underline a} = \{ {a_{\upsilon i}}\} \;(\upsilon \in S,1 \leqq i \leqq m)$ such that for every neighborhood $O$ of $\underline{\underline a} $ the points $\underline{\underline x} $ with $\{ {a_{{v_i}}}\{ \underline{\underline x} )\} \in O$ are dense in the subspace topology. Then $ A$ is a polyhedron whose vertices are rational points.
Harmonic analysis and ultracontractivity
Michael
Cowling;
Stefano
Meda
733-752
Abstract: Let ${({T_t})_{t > 0}}$ be a symmetric contraction semigroup on the spaces ${L^p}(M)\;(1 \leq p \leq \infty )$, and let the functions $\phi$ and $\psi$ be "regularly related". We show that ${({T_t})_{t > 0}}$ is $\phi$-ultracontractive, i.e., that ${({T_t})_{t > 0}}$ satisfies the condition $ {\left\Vert {{T_t}f} \right\Vert _\infty } \leq C\phi {(t)^{ - 1}}{\left\Vert f \right\Vert _1}$ for all $f$ in ${L^1}(M)$ and all $t$ in $ {{\mathbf{R}}^ + }$, if and only if the infinitesimal generator $\mathcal{G}$ has Sobolev embedding properties, namely, ${\left\Vert {\psi {{(\mathcal{G})}^{ - \alpha }}f} \right\Vert _q} \leq C{\left\Vert f \right\Vert _p}$ for all $f$ in ${L^p}(M)$, whenever $1 < p < q < \infty$ and $\alpha = 1/p - 1/q$ . We establish some new spectral multiplier theorems and maximal function estimates. In particular, we give sufficient conditions on $ m$ for $m(\mathcal{G})$ to map ${L^p}(M)$ to ${L^q}(M)$, and for the example where there exists $ \mu$ in ${{\mathbf{R}}^ + }$ such that $\phi (t) = {t^\mu }$ for all $t$ in ${{\mathbf{R}}^ + }$ , we give conditions which ensure that the maximal function ${\sup _{t > 0}}\vert{t^\alpha }{T_t}f( \bullet )\vert$ is bounded.
On the discrete series of generalized Stiefel manifolds
Jian-Shu
Li
753-766
Abstract: A study of the discrete series of generalized Stiefel manifolds is made using the oscillator representation. New infinite families of such discrete series are constructed.
Lyapunov graphs and flows on surfaces
K. A.
de Rezende;
R. D.
Franzosa
767-784
Abstract: In this paper, a characterization of Lyapunov graphs associated to smooth flows on surfaces is presented. We first obtain necessary and sufficient conditions for a Lyapunov graph to be associated to Morse-Smale flows and then generalize them to smooth flows. The methods employed in the proofs are of interest in their own right for they introduce the use of the Conley index in this context. Moreover, an algorithmic geometric construction of flows on surfaces is described.
Prescribing scalar curvatures on the conformal classes of complete metrics with negative curvature
Zhi Ren
Jin
785-810
Abstract: Let $({M^n},g)$ be a complete noncompact Riemannian manifold with the curvature bounded between two negative constants. Given a function $K$ on ${M^n}$, in terms of the behaviors of $K$ at infinite, we give a fairly complete answer to when the $K$ can be the scalar curvature function of a complete metric ${g_1}$ which is conformal to $g$.
Weighted norm inequalities for general operators on monotone functions
Shanzhong
Lai
811-836
Abstract: In this paper we characterize the weights $w,v$ for which $ {\left\Vert {{S_\phi }f} \right\Vert _{p,w}} \leq C{\left\Vert f \right\Vert _{q,v}}$, for $f$ nonincreasing, where $ {S_\phi }f = \smallint _0^\infty {\phi (x,y)f(y)dy} $.
Threshold growth dynamics
Janko
Gravner;
David
Griffeath
837-870
Abstract: We study the asymptotic shape of the occupied region for monotone deterministic dynamics in $d$-dimensional Euclidean space parametrized by a threshold $ \theta > 0$, and a Borel set $ \mathcal{N} \subset {\mathbb{R}^d}$ with positive and finite Lebesgue measure. If $ {A_n}$ denotes the oocupied set of the dynamics at integer time $n$, then $ {A_{n + 1}}$ is obtained by adjoining any point $x$ for which the volume of overlap between $x + \mathcal{N}$ and ${A_n}$ exceeds $\theta$. Except in some degenerate cases, we prove that ${n^{ - 1}}{A_n}$ converges to a unique limiting "shape" $L$ starting from any bounded initial region $ {A_0}$ that is suitably large. Moreover, $L$ is computed as the polar transform for $ 1/w$, where $w$ is an explicit width function that depends on $ \mathcal{N}$ and $ \theta$. It is further shown that $L$ describes the limiting shape of wave fronts for certain cellular automaton growth rules related to lattice models of excitable media, as the threshold and range of interaction increase suitably. In the case of box $({l^\infty })$ neighborhoods on ${\mathbb{Z}^2}$, these limiting shapes are calculated and the dependence of their anisotropy on $ \theta$ is examined. Other specific two- and three-dimensional examples are also discussed in some detail.
Function spaces of completely metrizable spaces
Jan
Baars;
Joost
de Groot;
Jan
Pelant
871-883
Abstract: Let $X$ and $Y$ be metric spaces and let $\phi :{C_p}(X) \to {C_p}(Y)$ (resp. $ \phi :C_p^\ast(X) \to C_p^\ast(Y)$) be a continuous linear surjection. We prove that $Y$ is completely metrizable whenever $ X$ is. As a corollary we obtain that complete metrizability is preserved by ${l_p}$ (resp. $l_p^\ast$-equivalence) in the class of all metric spaces. This solves Problem 35 in [2] (raised by Arhangel'skiĭ).
Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions
Boris
Mordukhovich
1-35
Abstract: We consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior. These properties have many important applications to various problems in nonlinear analysis, optimization, control theory, etc., especially for studying sensitivity and stability questions with respect to perturbations of initial data and parameters. We establish interrelations between these properties and prove effective criteria for their fulfillment stated in terms of robust generalized derivatives for multifunctions and nonsmooth mappings. The results obtained provide complete characterizations of the properties under consideration in a general setting of closed-graph multifunctions in finite dimensions. They ensure new information even in the classical cases of smooth single-valued mappings as well as multifunctions with convex graphs.
Elementary duality of modules
Ivo
Herzog
37-69
Abstract: Let $R$ be a ring. A formula $\varphi ({\mathbf{x}})$ in the language of left $ R$-modules is called a positive primitive formula (ppf) if it is of the form $ \exists {\mathbf{y}}\left({AB} \right)\left(\begin{array}{*{20}{c}}x y \end{array} \right) = 0$ where $ A$ and $B$ are matrices of appropriate size with entries in $R$. We apply Prest's notion of $D\varphi ({\mathbf{x}})$, the ppf in the language of right $R$-modules dual to $\varphi$, to show that the model theory of left $ R$-modules as developed by Ziegler [Z] is in some sense dual to the model theory of right $R$-modules. We prove that the topologies on the left and right Ziegler spectra are "isomorphic" (Proposition 4.4). When the lattice of ppfs is well behaved, there is a homeomorphism $D$ between the left and right Ziegler spectra which assigns to a given pure-injective indecomposable left $R$-module $U$ the dual pure-injective indecomposable right $R$-module $DU$. Theorem 6.6 asserts that given a complete theory $ T$ of left $R$-modules, there is a dual complete theory $DT$ of right $R$-modules with corresponding Baur-Garavaglia-Monk invariants. In the end, we give some conditions on a pure-injective indecomposable $ _RU$ which ensure that its dual $DU$ may be represented as a hom set of the form $ {\operatorname{Hom}_S}{(_R}{U_S},{E_S})$ where $S$ is some ring making $_R{U_S}$ into a bimodule and ${E_S}$ is injective.
Sieved orthogonal polynomials. VII. Generalized polynomial mappings
Jairo A.
Charris;
Mourad E. H.
Ismail
71-93
Abstract: Systems of symmetric orthogonal polynomials whose recurrence relations are given by compatible blocks of second-order difference equations are studied in detail. Applications are given to the theory of the recently discovered sieved orthogonal polynomials. The connection with polynomial mappings is examined. An example of a family of orthogonal polynomials having discrete masses in the interior of the spectrum is included.
Homogeneous foliations of spheres
Duojia
Lu
95-102
Abstract: In this paper we discuss Riemannian foliations of the round sphere. We prove that there are no homogeneous Riemannian foliations of the round sphere with dimensions of the leaves bigger than three.
Solutions containing a large parameter of a quasi-linear hyperbolic system of equations and their nonlinear geometric optics approximation
Atsushi
Yoshikawa
103-126
Abstract: It is well known that a quasi-linear first order strictly hyperbolic system of partial differential equations admits a formal approximate solution with the initial data ${\lambda ^{ - 1}}{a_0}(\lambda x \bullet \eta ,x){r_1}(\eta ),\lambda > 0,x,\eta \in {{\mathbf{R}}^n}, \eta \ne 0$. Here $ {r_1}(\eta )$ is a characteristic vector, and $ {a_0}(\sigma ,x)$ is a smooth scalar function of compact support. Under the additional requirements that $n = 2$ or $3$ and that $ {a_0}(\sigma ,x)$ have the vanishing mean with respect to $\sigma$, it is shown that a genuine solution exists in a time interval independent of $ \lambda$, and that the formal solution is asymptotic to the genuine solution as $\lambda \to \infty$.
Completions and fibrations for topological monoids
Paulo
Lima-Filho
127-147
Abstract: We show that, for a certain class of topological monoids, there is a homotopy equivalence between the homotopy theoretic group completion ${M^ + }$ of a monoid $M$ in that class and the topologized Grothendieck group $\tilde M$ associated to $M$. The class under study is broad enough to include the Chow monoids effective cycles associated to a projective algebraic variety and also the infinite symmetric products of finite $ {\text{CW}}$-complexes. We associate principal fibrations to the completions of pairs of monoids, showing the existence of long exact sequences for the naïve approach to Lawson homology [Fri91, LF91a]. Another proof of the Eilenberg-Steenrod axioms for the functors $X \mapsto {\tilde{SP}}(X)$ in the category of finite $ {\text{CW}}$-complexes (Dold-Thom theorem [DT56]) is obtained.
Universal cover of Salvetti's complex and topology of simplicial arrangements of hyperplanes
Luis
Paris
149-178
Abstract: Let $V$ be a real vector space. An arrangement of hyperplanes in $V$ is a finite set $ \mathcal{A}$ of hyperplanes through the origin. A chamber of $\mathcal{A}$ is a connected component of $V - ({ \cup _{H \in \mathcal{A}}}H)$. The arrangement $ \mathcal{A}$ is called simplicial if ${ \cap _{H \in \mathcal{A}}}H = \{ 0\}$ and every chamber of $ \mathcal{A}$ is a simplicial cone. For an arrangement $ \mathcal{A}$ of hyperplanes in $V$, we set $\displaystyle M(\mathcal{A}) = {V_\mathbb{C}} - \left({\bigcup\limits_{H \in \mathcal{A}} {{H_\mathbb{C}}} } \right),$ where ${V_\mathbb{C}} = \mathbb{C} \otimes V$ is the complexification of $V$, and, for $ H \in \mathcal{A}$ , ${H_\mathbb{C}}$ is the complex hyperplane of ${V_\mathbb{C}}$ spanned by $H$. Let $ \mathcal{A}$ be an arrangement of hyperplanes of $V$. Salvetti constructed a simplicial complex $ \operatorname{Sal}(\mathcal{A})$ and proved that $ \operatorname{Sal}(\mathcal{A})$ has the same homotopy type as $M(\mathcal{A})$. In this paper we give a new short proof of this fact. Afterwards, we define a new simplicial complex $ \hat{\operatorname{Sal}}(\mathcal{A})$ and prove that there is a natural map $p:\hat {\operatorname{Sal}}(\mathcal{A}) \to \operatorname{Sal}(\mathcal{A})$ which is the universal cover of $ \operatorname{Sal}(\mathcal{A})$. At the end, we use $ \hat{\operatorname{Sal}}(\mathcal{A})$ to give a new proof of Deligne's result: "if $\mathcal{A}$ is a simplicial arrangement of hyperplanes, then $ M(\mathcal{A})$ is a $ K(\pi ,1)$ space." Namely, we prove that $ \hat{\operatorname{Sal}}(\mathcal{A})$ is contractible if $\mathcal{A}$ is a simplicial arrangement.
Multivariate discrete splines and linear Diophantine equations
Rong Qing
Jia
179-198
Abstract: In this paper we investigate the algebraic properties of multivariate discrete splines. It turns out that multivariate discrete splines are closely related to linear diophantine equations. In particular, we use a solvability condition for a system of linear diophantine equations to obtain a necessary and sufficient condition for the integer translates of a discrete box spline to be linearly independent. In order to understand the local structure of discrete splines we develop a general theory for certain systems of linear partial difference equations. Using this theory we prove that the integer translates of a discrete box spline are locally linearly independent if and only if they are linearly independent.
Set convergences. An attempt of classification
Yves
Sonntag;
Constantin
Zălinescu
199-226
Abstract: We endow families of nonempty closed subsets of a metric space with uniformities defined by semimetrics. Such structure is completely determined by a class (which is a family of closed sets) and a type (which is a semimetric). Two types are sufficient to define (and classify) almost all convergences known till now. These two types offer the possibility of defining other set convergences.
The nef value and defect of homogeneous line bundles
Dennis M.
Snow
227-241
Abstract: Formulas for the nef value of a homogeneous line bundle are derived and applied to the classification of homogeneous spaces with positive defect and to the classification of complete homogeneous real hypersurfaces of projective space.
Stability and dimension---a counterexample to a conjecture of Chogoshvili
Yaki
Sternfeld
243-251
Abstract: For every $n \geq 2$ we construct an $ n$-dimensional compact subset $X$ of some Euclidean space $E$ so that none of the canonical projections of $E$ on its two-dimensional coordinate subspaces has a stable value when restricted to $ X$. This refutes a longstanding claim due to Chogoshvili. To obtain this we study the lattice of upper semicontinuous decompositions of $X$ and in particular its sublattice that consists of monotone decompositions when $X$ is hereditarily indecomposable.
Estimates for operator norms on weighted spaces and reverse Jensen inequalities
Stephen M.
Buckley
253-272
Abstract: We examine the dependence on the ${A_p}$ norm of $w$ of the operator norms of singular integrals, maximal functions, and other operators in ${L^p}(w)$. We also examine connections between some fairly general reverse Jensen inequalities and the ${A_p}$ and $R{H_p}$ weight conditions.
Weighted norm inequalities for Vilenkin-Fourier series
Wo-Sang
Young
273-291
Abstract: Let ${S_n}f$ be the $n$th partial sum of the Vilenkin-Fourier series of $f \in {L^1}$. For $1 < p < \infty$, we characterize all weight functions $w$ such that if $f \in {L^p}(w)$, ${S_n}f$ converges to $f$ in ${L^p}(w)$. We also determine all weight functions $w$ such that $ \{ {S_n}\}$ is uniformly of weak type $(1,1)$ with respect to $w$.
Singular integral operators on $C\sp 1$ manifolds
Jeff E.
Lewis;
Renata
Selvaggi;
Irene
Sisto
293-308
Abstract: We show that the kernel of a singular integral operator is real analytic in ${{\mathbf{R}}^n}\backslash \{ 0\}$ iff the symbol [Fourier transform] is real analytic in $ {{\mathbf{R}}^n}\backslash \{ 0\}$. The singular integral operators with continuous coefficients and real analytic kernels (symbols) form an operator algebra with the usual symbolic calculus. The symbol is invariantly defined under ${C^1}$ changes of coordinates.
Unipotent representations and reductive dual pairs over finite fields
Jeffrey
Adams;
Allen
Moy
309-321
Abstract: Consider the representation correspondence for a reductive dual pair $({G_1},{G_2})$ over a finite field. We consider the question of how the correspondence behaves for unipotent representations. In the special case of cuspidal unipotent representations, and a certain fundamental situation, that of "first occurrence", the representation correspondence takes a cuspidal unipotent representation of ${G_1}$ to one of ${G_2}$. This should serve as a fundamental case in studying the correspondence in general over both finite and local fields.
The Gorensteinness of the symbolic blow-ups for certain space monomial curves
Shiro
Goto;
Koji
Nishida;
Yasuhiro
Shimoda
323-335
Abstract: Let ${\mathbf{p}} = {\mathbf{p}}({n_1},{n_2},{n_3})$ denote the prime ideal in the formal power series ring $A = k[[X,Y,Z]]$ over a field $ k$ defining the space monomial curve $X = {T^{{n_1}}}$, $Y = {T^{{n_2}}}$ , and $Z = {T^{{n_3}}}$ with $ {\text{GCD}}({n_1},{n_2},{n_3}) = 1$. Then the symbolic Rees algebras ${R_s}({\mathbf{p}}) = { \oplus _{n \geq 0}}{{\mathbf{p}}^{(n)}}$ are Gorenstein rings for the prime ideals ${\mathbf{p}} = {\mathbf{p}}({n_1},{n_2},{n_3})$ with $\min \{ {n_1},{n_2},{n_3}\} = 4$ and ${\mathbf{p}} = {\mathbf{p}}(m,m + 1,m + 4)$ with $m \ne 9,13$ . The rings ${R_s}({\mathbf{p}})$ for ${\mathbf{p}} = {\mathbf{p}}(9,10,13)$ and $ {\mathbf{p}} = {\mathbf{p}}(13,14,17)$ are Noetherian but non-Cohen-Macaulay, if $\operatorname{ch}\,k = 3$ .
On a conjecture regarding nonstandard uniserial modules
Paul C.
Eklof;
Saharon
Shelah
337-351
Abstract: We consider the question of which valuation domains (of cardinality ${\aleph _1}$) have nonstandard uniserial modules. We show that a criterion conjectured by Osofsky is independent of ${\text{ZFC}} + {\text{GCH}}$.
On automorphisms of matrix invariants
Zinovy
Reichstein
353-371
Abstract: Let ${Q_{m,n}}$ be the space of $m$-tuples of $n \times n$-matrices modulo the simultaneous conjugation action of $PG{L_n}$. Let ${Q_{m,n}}(\tau)$ be the set of points of $ {Q_{m,n}}$ of representation type $\tau$. We show that for $ m \geq n + 1$ the group $ \operatorname{Aut}({Q_{m,n}})$ of representation type preserving algebraic automorphisms of ${Q_{m,n}}$ acts transitively on each ${Q_{m,n}}(\tau)$. Moreover, the action of $ \operatorname{Aut}({Q_{m,n}})$ on the Zariski open subset ${Q_{m,n}}(1,n)$ of ${Q_{m,n}}$ is $s$-transitive for every positive integer $ s$. We also prove slightly weaker analogues of these results for all $ m \geq 3$.
Harmonic measures on covers of compact surfaces of nonpositive curvature
M.
Brin;
Y.
Kifer
373-393
Abstract: Let $M$ be the universal cover of a compact nonflat surface $N$ of nonpositive curvature. We show that on the average the Brownian motion on $M$ behaves similarly to the Brownian motion on negatively curved manifolds. We use this to prove that harmonic measures on the sphere at infinity have positive Hausdorff dimension and if the geodesic flow on $N$ is ergodic then the harmonic and geodesic measure classes at infinity are singular unless the curvature is constant.
Supercuspidal representations and Poincar\'e series over function fields
Daniel
Bump;
Shuzo
Takahashi
395-413
Abstract: In this paper, we will give a new construction of certain cusp forms on $ GL(2)$ over a rational function field. The forms which we construct are analogs of holomorphic modular forms, in that the local representations at the infinite place are in the discrete series. The novelty of our approach is that we are able to give a very explicit construction of these forms as certain 'Poincaré series.' We will also study the exponential sums which arise in the Fourier expansions of these Poincaré series.
Sandwich matrices, Solomon algebras, and Kazhdan-Lusztig polynomials
Mohan S.
Putcha
415-428
Abstract: Sandwich matrices have proved to be of importance in semigroup theory for the last 50 years. The work of the author on algebraic monoids leads to sandwich matrices in group theory. In this paper, we find some connections between sandwich matrices and the Hecke algebras (for monoids) introduced recently by Louis Solomon. At the local level we then obtain an explicit isomorphism between Solomon's Hecke algebra and the complex monoid algebra of the Renner monoid. In the simplest case of monoids associated with a Borel subgroup, we find that the entries of the inverse of the sandwich matrix, as well as those of the related structure matrix of Solomon's Hecke algebra are 'almost' the polynomials $ {R_{x,y}}$ associated with the Kazhdan-Lusztig polynomials.
Removing index $0$ fixed points for area preserving maps of two-manifolds
Edward E.
Slaminka
429-445
Abstract: Using the method of free modifications developed by M. Brown and extended to area preserving homeomorphisms, we prove the following fixed point removal theorem. Theorem. Let $h:M \to M$ be an orientation preserving, area preserving homeomorphism of an orientable two-manifold $M$ having an isolated fixed point $ p$ of index 0. Given any open neighborhood $ N$ of $ p$ such that $N \cap \operatorname{Fix}(h) = p$, there exists an area preserving homeomorphism $ \hat h$ such that (i) $\displaystyle \hat h = h\;on\;\overline {M - N} $ and (ii) $\hat h$ is fixed point free on $ N$. Two applications of this theorem are the second fixed point for the topological version of the Conley-Zehnder theorem on the two-torus, and a new proof of the second fixed point for the Poincaré-Birkhoff Fixed Point Theorem.
Kac-Moody Lie algebras, spectral sequences, and the Witt formula
Seok-Jin
Kang
463-493
Abstract: In this work, we develop a homological theory for the graded Lie algebras, which gives new information on the structure of the Lorentzian Kac-Moody Lie algebras. The technique of the Hochschild-Serre spectral sequences offers a uniform method of studying the higher level root multiplicities and the principally specialized affine characters of Lorentzian Kac-Moody Lie algebras.
Uniqueness of positive radial solutions of $\Delta u+f(u)=0$ in ${\bf R}\sp n$. II
Kevin
McLeod
495-505
Abstract: We prove a uniqueness result for the positive solution of $\Delta u + f(u) = 0$ in ${\mathbb{R}^n}$ which goes to 0 at $\infty$. The result applies to a wide class of nonlinear functions $f$, including the important model case $f(u) = - u + {u^p}$ , $1 < p < (n + 2)/(n - 2)$. The result is proved by reducing to an initial-boundary problem for the ${\text{ODE}}\;u'' + (n - 1)/r + f(u) = 0$ and using a shooting method.
Continuous dependence of nonmonotonic discontinuous differential equations
Daniel C.
Biles
507-524
Abstract: Continuous dependence of solutions for a class of nonmonotonic, discontinuous differential equations is studied. First, a local existence theorem due to $Z$. Wu is extended to a larger class. Then, a result concerning continuous dependence for this larger class is proven. This employs a type of convergence similar to Gihman's Convergence Criterion, which is defined to be $\displaystyle {\text{For all}}\;a,b\;{\text{and}}\;y\quad \mathop {\lim }\limits_{n \to \infty } \int_a^b {{f_n}(s,y)ds = } \int_a^b {{f_\infty }(s,y)\,ds}.$ The significance of Gihman's Convergence Criterion is that for certain classes of differential equations it has been found to be necessary and sufficient for continuous dependence. Finally, examples are presented to motivate and clarify this continuous dependence result.
Determinants of Laplacians on the space of conical metrics on the sphere
Hala Khuri
King
525-536
Abstract: On a compact surface with smooth boundary, the determinant of the Laplacian associated to a smooth metric on the surface (with Dirichlet boundary conditions if the boundary is nonempty) is a well-defined isospectral invariant. As a function on the moduli space of such surfaces, it is a smooth function whose boundary behavior in certain cases is well understood; see [OPS and K]. In this paper, we restrict ourselves to a certain class of singular metrics on closed surfaces called conical metrics. We show that the determinant of the associated Laplacian is still well defined and that it is a real analytic function on a suitably restricted subset of the space of conical metrics on the sphere.
An isoperimetric inequality for Artin groups of finite type
Kay
Tatsuoka
537-551
Abstract: We show that Artin groups of finite type satisfy a quadratic isoperimetric inequality. Moreover we describe an explicit algorithm to solve the word problem in quadratic time.
The cohomology algebra of a commutative group scheme
Robert
Fossum;
William
Haboush
553-565
Abstract: Let $k$ be a commutative ring with unit of characteristic $p > 0$ and let $G = \operatorname{Spec}(A)$ be an affine commutative group scheme over $k$. Let ${{\text{H}}^ \bullet }(G)$ be the graded Hochschild algebraic group cohomology algebra and, for $ M$ a rational $ G$-module, let ${{\text{H}}^ \bullet }(G,M)$ denote the graded Hochschild cohomology ${{\text{H}}^ \bullet }(G)$-module. We show that $ {{\text{H}}^ \bullet }(G)$ is, in general, a graded Hopf algebra. When $G = {{\mathbf{G}}_{a,k}}$, let ${\alpha _{{p^\nu }}}$ denote the subgroup of ${p^\nu }$-nilpotents and let ${{\text{F}}_\nu }$ denote the $\nu$th power of the Frobenius. We show that for any finite $M$ that there is a $\nu$ such that $\displaystyle {{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}},M) \cong {{\text{H}}^... ...otimes _k}{\text{F}}_{\nu }^\ast ({{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}}))$ where ${\text{F}}_\nu ^\ast$ is the endomorphism of $ {{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}})$ induced by ${F_v}$. As a consequence, we can show that ${{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}},M)$ is a finitely generated module over $ {{\text{H}}^ \bullet }({{\mathbf{G}}_{a,k}})$ when $M$ is a finite dimensional vector space over $k$.
The Ehrenfeucht-Fra\"\i ss\'e-game of length $\omega\sb 1$
Alan
Mekler;
Saharon
Shelah;
Jouko
Väänänen
567-580
Abstract: Let $\mathfrak{A}$ and $ \mathfrak{B}$ be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fraïssé-game of length $ {\omega _1}$ of $\mathfrak{A}$ and $ \mathfrak{B}$ which we denote by $ {\mathcal{G}_{{\omega _1}}}(\mathfrak{A},\mathfrak{B})$. This game is like the ordinary Ehrenfeucht-Fraïssé-game of ${L_{\omega \omega }}$ except that there are ${\omega _1}$ moves. It is clear that ${\mathcal{G}_{{\omega _1}}}(\mathfrak{A},\mathfrak{B})$ is determined if $ \mathfrak{A}$ and $\mathfrak{B}$ are of cardinality $\leq {\aleph _1}$. We prove the following results: Theorem 1. If $V = L$, then there are models $\mathfrak{A}$ and $\mathfrak{B}$ of cardinality ${\aleph _2}$ such that the game ${\mathcal{G}_{{\omega _1}}}(\mathfrak{A},\mathfrak{B})$ is nondetermined. Theorem 2. If it is consistent that there is a measurable cardinal, then it is consistent that ${\mathcal{G}_{{\omega _1}}}(\mathfrak{A},\mathfrak{B})$ is determined for all $\mathfrak{A}$ and $ \mathfrak{B}$ of cardinality $ \leq {\aleph _2}$. Theorem 3. For any $\kappa \geq {\aleph _3}$ there are $\mathfrak{A}$ and $ \mathfrak{B}$ of cardinality $\kappa$ such that the game ${\mathcal{G}_{{\omega _1}}}(\mathfrak{A},\mathfrak{B})$ is nondetermined.
Certain hypergeometric series related to the root system $BC$
R. J.
Beerends;
E. M.
Opdam
581-609
Abstract: We show that the generalized hypergeometric function $_2{F_1}$ of matrix argument is the series expansion at the origin of a special case of the hypergeometric function associated with the root system of type $ BC$. In addition we prove that the Jacobi polynomials of matrix argument correspond to the Jacobi polynomials associated with the root system of type $BC$. We also give a precise relation between Jack polynomials and the Jacobi polynomials associated with the root system of type $A$. As a side result one obtains generalized hook-length formulas which are related to Harish-Chandra's ${\mathbf{c}}$-function and one can prove a conjecture due to Macdonald relating two inner products on a space of symmetric functions.
On the evaluation map
Aniceto
Murillo
611-622
Abstract: The evaluation map of a differential graded algebra or of a space is described under two different approaches. This concept turns out to have geometric implications: (i) A $ 1$-connected topological space, with finite-dimensional rational homotopy, has finite-dimensional rational cohomology if and only if it has nontrivial evaluation map. (ii) Let $ E\xrightarrow{\rho }B$ be a fibration of simplyconnected spaces. If the rational cohomology of the fibre is finite dimensional and the evaluation map of the base is different from zero, then the evaluation map of the total space is nonzero. Also, if $\rho$ is surjective in rational homotopy and the evaluation map of $E$ is nontrivial, then the evaluation map of the fibre is different from zero.
Finitely decidable congruence modular varieties
Joohee
Jeong
623-642
Abstract: A class $\mathcal{V}$ of algebras of the same type is said to be finitely decidable iff the first order theory of the class of finite members of $\mathcal{V}$ is decidable. Let $\mathcal{V}$ be a congruence modular variety. In this paper we prove that if $\mathcal{V}$ is finitely decidable, then the following hold. (1) Each finitely generated subvariety of $\mathcal{V}$ has a finite bound on the cardinality of its subdirectly irreducible members. (2) Solvable congruences in any locally finite member of $\mathcal{V}$ are abelian. In addition we obtain various necessary conditions on the congruence lattices of finite subdirectly irreducible algebras in $\mathcal{V}$.
On property I for knots in $S\sp 3$
Xingru
Zhang
643-657
Abstract: This paper deals with the question of which knot surgeries on $ {S^3}$ can yield $ 3$-manifolds homeomorphic to, or with the same fundamental group as, the Poincaré homology $3$-sphere.
Homogeneity for open partitions of pairs of reals
Qi
Feng
659-684
Abstract: We prove a partition theorem for analytic sets of reals, namely, if $A \subseteq \mathbb{R}$ is analytic and $ {[A]^2} = {K_0} \cup {K_1}$ with ${K_0}$ relatively open, then either there is a perfect 0-homogeneous subset or $A$ is a countable union of $1$-homogeneous subsets. We also show that such a partition property for coanalytic sets is the same as that each uncountable coanalytic set contains a perfect subset. A two person game for this partition property is also studied. There are some applications of such partition properties.
Hypersurfaces with constant mean curvature in the complex hyperbolic space
Susana
Fornari;
Katia
Frensel;
Jaime
Ripoll
685-702
Abstract: A classical theorem of A. D. Alexandrov characterized round spheres is extended to the complex hyperbolic space $ {\mathbf{C}}{{\mathbf{H}}^2}$ of constant holomorphic sectional curvature. A detailed description of the horospheres and equidistant hypersurfaces in $ {\mathbf{C}}{{\mathbf{H}}^2}$ determining in particular their stability, is also given.
Geometric curvature bounds in Riemannian manifolds with boundary
Stephanie B.
Alexander;
I. David
Berg;
Richard L.
Bishop
703-716
Abstract: An Alexandrov upper bound on curvature for a Riemannian manifold with boundary is proved to be the same as an upper bound on sectional curvature of interior sections and of sections of the boundary which bend away from the interior. As corollaries those same sectional curvatures are related to estimates for convexity and conjugate radii; the Hadamard-Cartan theorem and Yau's isoperimetric inequality for spaces with negative curvature are generalized.
Noncommutative K\"othe duality
Peter G.
Dodds;
Theresa K.-Y.
Dodds;
Ben
de Pagter
717-750
Abstract: Using techniques drawn from the classical theory of rearrangement invariant Banach function spaces we develop a duality theory in the sense of Köthe for symmetric Banach spaces of measurable operators affiliated with a semifinite von Neumann algebra equipped with a distinguished trace. A principal result of the paper is the identification of the Köthe dual of a given Banach space of measurable operators in terms of normality.
Gauss map of minimal surfaces with ramification
Min
Ru
751-764
Abstract: We prove that for any complete minimal surface $M$ immersed in ${R^n}$, if in $ C{P^{n - 1}}$ there are $q > n(n + 1)/2$ hyperplanes ${H_j}$ in general position such that the Gauss map of $M$ is ramified over ${H_j}$ with multiplicity at least ${e_j}$ for each $j$ and $\displaystyle \sum\limits_{j = 1}^q {\left({1 - \frac{{(n - 1)}} {{{e_j}}}} \right) > n(n + 1)/2}$ , then $ M$ must be flat.
Bass numbers of local cohomology modules
Craig L.
Huneke;
Rodney Y.
Sharp
765-779
Abstract: Let $A$ be a regular local ring of positive characteristic. This paper is concerned with the local cohomology modules of $A$ itself, but with respect to an arbitrary ideal of $A$. The results include that all the Bass numbers of all such local cohomology modules are finite, that each such local cohomology module has finite set of associated prime ideals, and that, whenever such a local cohomology module is Artinian, then it must be injective. (This last result had been proved earlier by Hartshorne and Speiser under the additional assumptions that $A$ is complete and contains its residue field which is perfect.) The paper ends with some low-dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true.
Brown-Peterson and ordinary cohomology theories of classifying spaces for compact Lie groups
Akira
Kono;
Nobuaki
Yagita
781-798
Abstract: The Steenrod algebra structures of $ {H^\ast}(BG;Z/p)$ for compact Lie groups are studied. Using these, Brown-Peterson cohomology and Morava $K$-theory are computed for many concrete cases. All these cases have properties similar as torsion free Lie groups or finite groups, e.g., $B{P^{odd}}(BG) = 0$.
Zeros of the successive derivatives of Hadamard gap series
Robert M.
Gethner
799-807
Abstract: A complex number $ z$ is in the final set of an analytic function $f$, as defined by Pólya, if every neighborhood of $z$ contains zeros of infinitely many $ {f^{(n)}}$. If $ f$ is a Hadamard gap series, then the part of the final set in the open disk of convergence is the origin along with a union of concentric circles.
The number of irreducible factors of a polynomial. I
Christopher G.
Pinner;
Jeffrey D.
Vaaler
809-834
Abstract: Let $F(x)$ be a polynomial with coefficients in an algebraic number field $k$. We estimate the number of irreducible cyclotomic factors of $F$ in $k[x]$, the number of irreducible noncyclotomic factors of $F$, the number of $n$th roots of unity among the roots of $F$, and the number of primitive $ n$th roots of unity among the roots of $F$. All of these quantities are counted with multiplicity and estimated by expressions which depend explicitly on $k$, on the degree of $F$ and height of $F$, and (when appropriate) on $ n$. We show by constructing examples that some of our results are essentially sharp.
Uniform algebras generated by holomorphic and pluriharmonic functions
Alexander J.
Izzo
835-847
Abstract: It is shown that if ${f_1}, \ldots ,{f_n}$ are pluriharmonic on ${B_n}$ (the open unit ball in ${\mathbb{C}^n})$ and ${C^1}$ on $ {\bar B_n}$, and the $n \times n$ matrix $(\partial {f_j}/\partial {\bar z_k})$ is invertible at every point of ${B_n}$, then the norm-closed algebra generated by the ball algebra $ A({\bar B_n})$ and ${f_1}, \ldots ,{f_n}$ is equal to $C({\bar B_n})$. Extensions of this result to more general strictly pseudoconvex domains are also presented.
On the Toda and Kac-van Moerbeke systems
F.
Gesztesy;
H.
Holden;
B.
Simon;
Z.
Zhao
849-868
Abstract: Given a solution of the Toda lattice we explicitly construct a solution of the Kac-van Moerbeke system related to each other by a Miura-type transformation. As an illustration of our method we derive the $N$-soliton solutions of the Kac-van Moerbeke lattice.
When Cantor sets intersect thickly
Brian R.
Hunt;
Ittai
Kan;
James A.
Yorke
869-888
Abstract: The thickness of a Cantor set on the real line is a measurement of its "size". Thickness conditions have been used to guarantee that the intersection of two Cantor sets is nonempty. We present sharp conditions on the thicknesses of two Cantor sets which imply that their intersection contains a Cantor set of positive thickness.
The trace of the heat kernel in Lipschitz domains
Russell M.
Brown
889-900
Abstract: We establish the existence of an asymptotic expansion as $t \to {0^ + }$ for the trace of the heat kernel for the Neumann Laplacian in a bounded Lipschitz domain. The proof of an asymptotic expansion for the heat kernel for the Dirichlet Laplacian is also sketched. The treatment of the Dirichlet Laplacian extends work of Brossard and Carmona who obtained the same result in ${C^1}$-domains.
Inequalities for mixed projection bodies
Erwin
Lutwak
901-916
Abstract: Mixed projection bodies are related to ordinary projection bodies (zonoids) in the same way that mixed volumes are related to ordinary volume. Analogs of the classical inequalities from the Brunn-Minkowski Theory (such as the Minkowski, Brunn-Minkowski, and Aleksandrov-Fenchel inequalities) are developed for projection and mixed projection bodies.
Holomorphic extension and decomposition from a totally real manifold
Zai Fei
Ye
1-33
Abstract: This paper is to develop an elementary cohomological approach for decomposing a function into boundary values of holomorphic functions and for discussing the corresponding microlocal analysis and hyperfunction theory.
Irredundant sets in Boolean algebras
Stevo
Todorčević
35-44
Abstract: It is shown that every uncountable Boolean algebra $A$ contains an uncountable subset $ I$ such that no $ a$ of $I$ is in the subalgebra generated by $I\backslash \{ a\}$ using an additional axiom of set theory. It is also shown that a use of some such axiom is necessary.
$\Delta$-sets
R. W.
Knight
45-60
Abstract: A model of ${\text{ZFC}}$ is constructed in which there exists a subset of the Moore plane that is countably paracompact but not normal. The method used in the construction is forcing using uncountable sets of finite partial functions, $ {\omega _1}$ and ${\omega _2}$ are shown to be preserved using a fusion lemma.
Classification of the Tor-algebras of codimension four almost complete intersections
Andrew R.
Kustin
61-85
Abstract: Let $(R,m,k)$ be a local ring in which $ 2$ is a unit. Assume that every element of $k$ has a square root in $k$. We classify the algebras $ \operatorname{Tor}_ \bullet ^R(R/J,k)$ as $J$ varies over all grade four almost complete intersection ideals in $R$ . The analogous classification has already been found when $J$ varies over all grade four Gorenstein ideals [21], and when $J$ varies over all ideals of grade at most three [5, 30]. The present paper makes use of the classification, in [21], of the Tor-algebras of codimension four Gorenstein rings, as well as the (usually nonminimal) $ {\text{DG}}$-algebra resolution of a codimension four almost complete intersection which is produced in [25 and 26].
Complex multiplication cycles and a conjecture of Be\u\i linson and Bloch
Chad
Schoen
87-115
Abstract: A generalization of the conjecture of Birch and Swinnerton-Dyer is investigated using complex multiplication cycles on a particular Kuga fiber variety. A weak finiteness result consistent with the conjecture is proved. The image of complex multiplication cycles under the Abel-Jacobi map is computed explicitly. The results provide numerical evidence supporting the conjecture. They also give evidence for a relationship between complex multiplication cycles and a modular form of weight $5/2$ and raise questions for further investigation.
Localizing with respect to self-maps of the circle
Carles
Casacuberta;
Georg
Peschke
117-140
Abstract: We describe a general procedure to construct idempotent functors on the pointed homotopy category of connected ${\text{CW}}$-complexes, some of which extend $ P$-localization of nilpotent spaces, at a set of primes $P$. We focus our attention on one such functor, whose local objects are ${\text{CW}}$-complexes $X$ for which the $p$th power map on the loop space $\Omega X$ is a self-homotopy equivalence if $p \notin P$. We study its algebraic properties, its behaviour on certain spaces, and its relation with other functors such as Bousfield's homology localization, Bousfield-Kan completion, and Quillen's plus-construction.
Periodic seeded arrays and automorphisms of the shift
Ezra
Brown
141-161
Abstract: The automorphism group $\operatorname{Aut}({\Sigma _2})$ of the full $ 2$-shift is conjectured to be generated by the shift and involutions. We approach this problem by studying a certain family of automorphisms whose order was unknown, but which we show to be finite and for which we find factorizations as products of involutions. The result of this investigation is the explicit construction of a subgroup $\mathcal{H}$ of $\operatorname{Aut}({\Sigma _2})$ ; $\mathcal{H}$ is generated by certain involutions $ {g_n}$, and turns out to have a number of curious properties. For example, $ {g_n}$ and ${g_k}$ commute unless $n$ and $k$ are consecutive integers, the order of $ {g_{n + k}} \circ \cdots \circ {g_k}$ is independent of $k$, and $ \mathcal{H}$ contains elements of all orders. The investigation is aided by the development of results about certain new types of arrays of 0's and $1$'s called periodic seeded arrays, as well as the use of Boyle and Krieger's work on return numbers and periodic points.
Cayley-Bacharach schemes and their canonical modules
Anthony V.
Geramita;
Martin
Kreuzer;
Lorenzo
Robbiano
163-189
Abstract: A set of $ s$ points in ${\mathbb{P}^d}$ is called a Cayley-Bacharach scheme ( ${\text{CB}}$-scheme), if every subset of $s - 1$ points has the same Hilbert function. We investigate the consequences of this "weak uniformity." The main result characterizes $ {\text{CB}}$-schemes in terms of the structure of the canonical module of their projective coordinate ring. From this we get that the Hilbert function of a $ {\text{CB}}$-scheme $ X$ has to satisfy growth conditions which are only slightly weaker than the ones given by Harris and Eisenbud for points with the uniform position property. We also characterize ${\text{CB}}$-schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points. Applications include efficient algorithms for checking whether a given set of points is a $ {\text{CB}}$-scheme, results about generic hyperplane sections of arithmetically Cohen-Macaulay curves and inequalities for the Hilbert functions of Cohen-Macaulay domains.
$3$-manifolds which admit finite group actions
Shi Cheng
Wang
191-203
Abstract: We prove several results which support the following conjectures: $ (1)$ Any smooth action of a finite group on a geometric $3$-manifold can be conjugated to preserve the geometric structure. $(2)$ Every irreducible closed $3$-manifold $M$ with infinite $ {\pi _1}(M)$ is finitely covered by a Haken $3$-manifold.
A metric deformation and the first eigenvalue of Laplacian on $1$-forms
Takashi
Otofuji
205-220
Abstract: We search for a higher-dimensional analogue of Calabi's example of a metric deformation, quoted by Cheeger, which inspired him to prove an inequality between the first eigenvalue of the Laplacian on functions and an isoperimetric constant. We construct an example of a metric deformation on ${S^n}$, ${n} \geq 5$, where the first eigenvalue of the Laplacian on functions remains bounded above from zero, and the first eigenvalue of the Laplacian on $1$-forms tends to zero. This metric deformation makes the sphere in the limit into a manifold with a cone singularity, which is an intermediate point on a path of deformation from an (${S^n}$, some metric) to an ( ${S^{n - 1}} \times {S^1}$, some metric).
Analyse quasi-sure et l'estimation du noyau de la chaleur pour temps petit
Shizan
Fang
221-241
Abstract: The Ito functional can be redefined out of a slim set by the natural way. Quasi-sure analysis is used to deal with the heat kernel asymptotic problems.
$\omega$-chaos and topological entropy
Shi Hai
Li
243-249
Abstract: We present a new concept of chaos, $\omega$-chaos, and prove some properties of $ \omega$-chaos. Then we prove that $\omega$-chaos is equivalent to positive entropy on the interval. We also prove that $\omega$-chaos is equivalent to the definition of chaos given by Devaney on the interval.
$2$-weights for unitary groups
Jian Bei
An
251-278
Abstract: This paper gives a description of the local structures of $2$-radical subgroups in a finite unitary group and proves Alperin's weight conjecture for finite unitary groups when the characteristic of modular representation is even.
Approximation of approximate fibrations by bundle maps
Y. H.
Im
279-295
Abstract: In this paper, we give some conditions under which approximate fibrations can be approximated by locally trivial bundle maps.
Reflecting Brownian motion in a cusp
R. Dante
DeBlassie;
Ellen H.
Toby
297-321
Abstract: Let $C$ be the cusp $\{ (x,y):x \geq 0$, $- {x^\beta } \leq y \leq {x^\beta }\}$ where $ \beta > 1$. Set $\partial {C_1} = \{ (x,y):x \geq 0, y = - {x^\beta }\}$ and $\partial {C_2} = \{ (x,y):x \geq 0$, $y = {x^\beta }\}$. We study the existence and uniqueness in law of reflecting Brownian motion in $ C$. The angle of reflection at $ \partial {C_j}\backslash \{ 0\}$ (relative to the inward unit normal) is a constant ${\theta _j} \in \left( { - \frac{\pi } {2},\frac{\pi } {2}} \right)$, and is positive iff the direction of reflection has a negative first component in all sufficiently small neighborhoods of 0. When $ {\theta _1} + {\theta _2} \leq 0$, existence and uniqueness in law hold. When $ {\theta _1} + {\theta _2} > 0$, existence fails. We also obtain results for a large class of asymmetric cusps. We make essential use of results of Warschawski on the differentiability at the boundary of conformal maps.
Some complete $\Sigma\sp 1\sb 2$ sets in harmonic analysis
Howard
Becker;
Sylvain
Kahane;
Alain
Louveau
323-336
Abstract: We prove that several specific pointsets are complete $\Sigma _2^1$ (complete PCA). For example, the class of ${N_0}$-sets, which is a hereditary class of thin sets that occurs in harmonic analysis, is a pointset in the space of compact subsets of the unit circle; we prove that this pointset is complete $\Sigma _2^1$. We also consider some other aspects of descriptive set theory, such as the nonexistence of Borel (and consistently with $ {\text{ZFC}}$, the nonexistence of universally measurable) uniformizing functions for several specific relations. For example, there is no Borel way (and consistently, no measurable way) to choose for each ${N_0}$-set, a trigonometric series witnessing that it is an ${N_0}$-set.
The Koebe semigroup and a class of averaging operators on $H\sp p({\bf D})$
Aristomenis G.
Siskakis
337-350
Abstract: We study on the Hardy space ${H^p}$ the operators ${T_F}$ given by $\displaystyle {T_F}(f)(z) = \frac{1} {z}\int_0^z {f(\zeta )\frac{1} {{F(\zeta )}}\;d\zeta }$ where $ F(z)$ is analytic on the unit disc $ \mathbb{D}$ and has $ \operatorname{Re} F(z) \geq 0$. Each such operator is closely related to a strongly continuous semigroup of weighted composition operators. By studying first an extremal such semigroup (the Koebe semigroup) we are able to obtain the upper bound ${\left\Vert {{T_F}} \right\Vert _p} \leq 2p\operatorname{Re} (1/F(0)) + \vert\operatorname{Im} (1/F(0))\vert$ for the norm. We also show that $ {T_F}$ is compact on $ {H^p}$ if and only if the measure $\mu$ in the Herglotz representation of $ 1/F$ is continuous.
Probing L-S category with maps
Barry
Jessup
351-360
Abstract: For any map $X\xrightarrow{f}Y$, we introduce two new homotopy invariants, $ {\text{dcat}}\;f$ and ${\text{rcat}}\;f$. The classical category ${\text{cat}}\;f$ is a lower bound for both, while $ {\text{dcat}}\;f \leq {\text{cat}}\;X$ and ${\text{rcat}}\;f \leq {\text{cat}}\;Y$. When $ Y$ is an Eilenberg-Mac Lane space, $f$ represents a cohomology class and ${\text{dcat}}\;f$ often gives a good estimate for $ {\text{cat}}\;X$. We prove that if $\Omega \in {H^n}(M;\mathbb{Z})$ is the fundamental class of a compact, simply connected $ n$-manifold, then $ {\text{dcat}}\;\Omega = {\text{cat}}\;M$. Similarly, when $X$ is sphere, then $f$ is a homotopy class and while ${\text{cat}}\;f = 1$, ${\text{rcat}}\;f$ can be a good approximation to ${\text{cat}}\;Y$. We show that if $ \alpha \in {\pi _2}(\mathbb{C}{P^n})$ is nonzero, then ${\text{rcat}}\;\alpha = n$. Rational analogues are introduced and we prove that for $u \in {H^\ast}(X;\mathbb{Q})$, ${\text{dcat}_0}\;u = 1 \Leftrightarrow {u^2} = 0$ and $u$ is spherical.
The limiting behavior of the Kobayashi-Royden pseudometric
Shulim
Kaliman
361-371
Abstract: We study the limit of the sequence of Kobayashi metrics of Riemann surfaces (when these Riemann surfaces form an analytic fibration in such a way that the total space of fibration becomes a complex surface), as the fibers approach the center fiber which is not in general smooth. We prove that if the total space is a Stein surface and the smooth part of the center fiber contains a component biholomorphic to a quotient of the disk by a Fuchsian group of first kind, then the Kobayashi metrics of the near-by fibers converge to the Kobayashi metric of this component as fibers tend to the center fiber.
Rees algebras of ideals having small analytic deviation
Sam
Huckaba;
Craig
Huneke
373-402
Abstract: In this article we identify two large families of ideals of a Cohen-Macaulay (sometimes Gorenstein) local ring whose Rees algebras are Cohen-Macaulay. Our main results imply, for example, that if $(R,M)$ is a regular local ring and $P$ is a prime ideal of $R$ such that ${P^n}$ is unmixed for all $n \geq 1$, then the Rees algebra $ R[Pt]$ is Cohen-Macaulay if either $ \dim (R/P) = 2$, or $\dim (R/P) = 3,R/P$ is Cohen-Macaulay, and $ R/P$ is integrally closed.
The constrained least gradient problem in ${\bf R}\sp n$
Peter
Sternberg;
Graham
Williams;
William P.
Ziemer
403-432
Abstract: We consider the constrained least gradient problem $\displaystyle \inf \left\{ {\int_\Omega {\vert\nabla u\vert dx:u \in {C^{0,1}}(... ...a u\vert \leq 1\;{\text{a.e.}},u = g\;{\text{on}}\;\partial \Omega } } \right\}$ which arises as the relaxation of a nonconvex problem in optimal design. We establish the existence of a solution by an explicit construction in which each level set is required to solve an obstacle problem. We also establish the uniqueness of solutions and discuss their structure.
Calcul du spectre d'une nilvari\'et\'e de rang deux et applications
Hubert
Pesce
433-461
Abstract: Résumé. On calcule, en utilisant la théorie des orbites de Kirillov, le spectre d'une nilvariété compacte de rang deux. Puis on utilise ce calcul pour étudier et caractériser les déformations isospectrales de ces variétés.
Asymptotic homotopy cycles for flows and $\Pi \sb 1$ de Rham theory
Diego
Benardete;
John
Mitchell
495-535
Abstract: We define the asymptotic homotopy of trajectories of flows on closed manifolds. These homotopy cycles take values in the $2$-step nilpotent Lie group which is associated to the fundamental group by means of Malcev completion. The cycles are an asymptotic limit along the orbit of the product integral of a Lie algebra valued $ 1$-form. Propositions 5.1-5.7 show how the formal properties of our theory parallel the properties of the asymptotic homology cycles of Sol Schwartzman. In particular, asymptotic homotopy is an invariant of topological conjugacy, and, in certain cases, of topological equivalence. We compute the asymptotic homotopy of those measure-preserving flows on Heisenberg manifolds which lift from the torus ${T^2}$ (Theorem 8.1), and then show how this invariant distinguishes up to topological equivalence certain of these flows which are indistinguishable homologically (Theorem 9.1). We also compute the asymptotic homotopy of those geodesic flows for Heisenberg manifolds which come from left invariant metrics on the Heisenberg group (Example 8.1), and then show how this invariant distinguishes up to topological conjugacy certain of these flows which are indistinguishable homologically.
An atriodic simple-$4$-od-like continuum which is not simple-triod-like
Piotr
Minc
537-552
Abstract: The paper contains an example of a continuum $K$ such that $K$ is the inverse limit of simple $4$-ods, $K$ cannot be represented as the inverse limit of simple triods and each proper subcontinuum of $ K$ is an arc.
Critical LIL behavior of the trigonometric system
I.
Berkes
553-585
Abstract: It is a classical fact that for rapidly increasing $({n_k})$ the sequence $(\cos {n_k}x)$ behaves like a sequence of i.i.d. random variables. Actually, this almost i.i.d. behavior holds if $({n_k})$ grows faster than ${e^{c\sqrt k }}$; below this speed we have strong dependence. While there is a large literature dealing with the almost i.i.d. case, practically nothing is known on what happens at the critical speed ${n_k} \sim {e^{c\sqrt k }}$ (critical behavior) and what is the probabilistic nature of $(\cos {n_k}x)$ in the strongly dependent domain. In our paper we study the critical LIL behavior of $(\cos {n_k}x)$ i.e., we investigate how classical fluctuational theorems like the law of the iterated logarithm and the Kolmogorov-Feller test turn to nonclassical laws in the immediate neighborhood of $ {n_k} \sim {e^{c\sqrt k }}$.
Fragments of bounded arithmetic and bounded query classes
Jan
Krajíček
587-598
Abstract: We characterize functions and predicates $ \Sigma _{i + 1}^b$-definable in $S_2^i$. In particular, predicates $\Sigma _{i + 1}^b$-definable in $ S_2^i$ are precisely those in bounded query class ${P^{\Sigma _i^p}}[O(\log n)]$ (which equals to $ \operatorname{Log}\;{\text{Space}}^{\Sigma _i^p}$ by [B-H, W]). This implies that $S_2^i \ne T_2^i$ unless $ {P^{\Sigma _i^p}}[O(\log n)] = \Delta _{i + 1}^p$. Further we construct oracle $ A$ such that for all $ i \geq 1$: ${P^{\Sigma _i^p(A)}}[O(\log n)] \ne \Delta _{i + 1}^p(A)$. It follows that $S_2^i(\alpha ) \ne T_2^i(\alpha )$ for all $i \geq 1$. Techniques used come from proof theory and boolean complexity.
L'espace des plongements d'un arc dans une surface
Robert
Cauty
599-614
Abstract: We prove that the space of embeddings of an arc into a surface without boundary $M$ is homeomorphic to the product $U(M) \times {l^2}$, where $U(M)$ is the unit tangent bundle of $M$.
Parabolic systems: the ${\rm GF}(3)$-case
Thomas
Meixner
615-637
Abstract: Parabolic systems defined over $GF(q)$ have been classified by Timmesfeld for $ q \geq 4$ and by Stroth for $q = 2$ (see references). We deal with the case $ q = 3$.
Wavelets in wandering subspaces
T. N. T.
Goodman;
S. L.
Lee;
W. S.
Tang
639-654
Abstract: Mallat's construction, via a multiresolution approximation, of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions. The connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in Hilbert space is exploited in the general setting. An example of multiresolution approximation generated by cardinal Hermite $B$-splines is constructed.
Optimal natural dualities
B. A.
Davey;
H. A.
Priestley
655-677
Abstract: The authors showed previously that for each of the varieties $ {{\mathbf{B}}_n}(3 \leq n < \omega )$ of pseudocomplemented distributive lattices there exists a natural duality given by a set of $p(n) + 3$ binary algebraic relations, where $ p(n)$ denotes the number of partitions of $n$. This paper improves this result by establishing that an optimal set of $n + 3$ of these relations suffices. This is achieved by the use of "test algebras": it is shown that redundancy among the relations of a duality for a prevariety generated by a finite algebra may be decided by testing the duality on the relations, qua algebras.
Martin and end compactifications for non-locally finite graphs
Donald I.
Cartwright;
Paolo M.
Soardi;
Wolfgang
Woess
679-693
Abstract: We consider a connected graph, having countably infinite vertex set $ X$, which is permitted to have vertices of infinite degree. For a transient irreducible transition matrix $P$ corresponding to a nearest neighbor random walk on $X$, we study the associated harmonic functions on $X$ and, in particular, the Martin compactification. We also study the end compactification of the graph. When the graph is a tree, we show that these compactifications coincide; they are a disjoint union of $ X$, the set of ends, and the set of improper vertices--new points associated with vertices of infinite degree. Other results proved include a solution of the Dirichlet problem in the context of the end compactification of a general graph. Applications are given to, e.g., the Cayley graph of a free group on infinitely many generators.
Product formulas and convolutions for angular and radial spheroidal wave functions
William C.
Connett;
Clemens
Markett;
Alan L.
Schwartz
695-710
Abstract: Product formulas for angular spheroidal wave functions on $[0,\pi ]$ and for radial spheroidal wave functions on $ [0,\infty)$ are presented, which generalize results for the ultraspherical polynomials and functions as well as for the Mathieu functions. Although these functions cannot be given in closed form, the kernels of the product formulas are represented in an explicit, and surprisingly simple way in terms of Bessel functions so that the exact range of positivity can easily be read off. The formulas are used to introduce two families of convolution structures on $ [0,\pi ]$ and $[0,\infty)$, many of which provide new hypergroups. We proceed from the fact that the spheroidal wave functions are eigenfunctions of Sturm-Liouville equations of confluent Heun type and employ a partial differential equation technique based on Riemann's integration method.
Loop space homology of spaces of small category
Yves
Félix;
Jean-Claude
Thomas
711-721
Abstract: Only little is known concerning ${H_\ast}(\Omega X;{\mathbf{k}})$, the loop space homology of a finite $ {\text{CW}}$ complex $ X$ with coefficients in a field $ {\mathbf{k}}$. A space $ X$ is called an $ r$-cone if there exists a filtration $\ast = {X_0} \subset {X_1} \subset \cdots \subset {X_r} = X$, such that ${X_i}$ has the homotopy type of the cofibre of a map from a wedge of sphere into ${X_{i - 1}}$. Denote by ${A_X}$ the sub-Hopf algebra image of ${H_\ast}(\Omega {X_1})$. We prove then that for a graded $r$-cone, $r \leq 3$, there exists an isomorphism ${A_X} \otimes T(U)\xrightarrow{ \cong }{H_\ast}(\Omega X)$.
On Dubrovin valuation rings in crossed product algebras
Darrell
Haile;
Patrick
Morandi
723-751
Abstract: Let $F$ be a field and let $V$ be a valuation ring in $ F$. If $A$ is a central simple $ F$-algebra then $ V$ can be extended to a Dubrovin valuation ring in $A$. In this paper we consider the structure of Dubrovin valuation rings with center $V$ in crossed product algebras $ (K/F,G,f)$ where $ K/F$ is a finite Galois extension with Galois group $G$ unramified over $V$ and $f$ is a normalized two-cocycle. In the case where $V$ is indecomposed in $K$ we introduce a family of orders naturally associated to $f$, examine their basic properties, and determine which of these orders is Dubrovin. In the case where $ V$ is decomposed we determine the structure in the case of certain special discrete, finite rank valuations.
Hyponormal Toeplitz operators and extremal problems of Hardy spaces
Takahiko
Nakazi;
Katsutoshi
Takahashi
753-767
Abstract: The symbols of hyponormal Toeplitz operators are completely described and those are also studied, being related with the extremal problems of Hardy spaces. Moreover, we discuss Halmos's question about a subnormal Toeplitz operator when the self-commutator is finite rank.
Composition operators between algebras of differentiable functions
Joaquín M.
Gutiérrez;
José G.
Llavona
769-782
Abstract: Let $E$, $F$ be real Banach spaces, $U \subseteq E$ and $V \subseteq F$ nonvoid open subsets and ${C^k}(U)$ the algebra of real-valued $k$-times continuously Fréchet differentiable functions on $U$, endowed with the compact open topology of order $k$. It is proved that, for $m \geq p$, the nonzero continuous algebra homomorphisms $A:{C^m}(U) \to {C^p}(V)$ are exactly those induced by the mappings $g:V \to U$ satisfying $\phi \circ g \in {C^p}(V)$ for each $\phi \in {E^\ast}$, in the sense that $A(f) = f \circ g$ for every $f \in {C^m}(U)$. Other homomorphisms are described too. It is proved that a mapping $g:V \to {E^{\ast \ast}}$ belongs to $ {C^k}(V,({E^{\ast \ast}},{w^\ast}))$ if and only if $\phi \circ g \in {C^k}(V)$ for each $\phi \in {E^\ast}$. It is also shown that if a mapping $g:V \to E$ verifies $\phi \circ g \in {C^k}(V)$ for each $\phi \in {E^\ast}$, then $g \in {C^{k - 1}}(V,E)$.
Induced connections on $S\sp 1$-bundles over Riemannian manifolds
G.
D’Ambra
783-798
Abstract: Let $(V,g)$ and $(W,h)$ be Riemannian manifolds and consider two $ {S^1}$-bundles $X \to V$ and $Y \to W$ with connections $\Gamma$ on $X$ and $\nabla$ on $Y$ respectively. We study maps $X \to Y$ which induce both connections and metrics. Our study relies on Nash's implicit function theorem for infinitesimally invertible differential operators. We show, for the case when $ Y \to W = {\mathbf{C}}{P^q}$ is the Hopf bundle, that if $2q \geq n(n + 1)/2 + 3n$ then there exists a nonempty open subset in the space of $ {C^\infty }$-pairs $(g,\Gamma)$ on $V$ which can be induced from $(h,\nabla)$ on $ {\mathbf{C}}{P^Q}$.
Algebraic particular integrals, integrability and the problem of the center
Dana
Schlomiuk
799-841
Abstract: In this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields. We first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center: these curves control the changes in the systems as parameters vary. The bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered, namely the real four-dimensional projective space. Next, we consider the known four algebraic conditions for the center for quadratic vector fields. One of them says that the system is Hamiltonian, a condition which has a clear geometric meaning. We determine the geometric meaning of the remaining other three algebraic conditions (I), (II), (III). We show that a quadratic system with a weak focus $F$, possessing algebraic particular integrals not passing through $F$ of the following types, satisfies in some coordinate axes the condition (I), (II) or (III) respectively and hence has a center at $F$: either a parabola and an irreducible cubic particular integral having only one point at infinity, coinciding with the one of the parabola; or a straight line and an irreducible conic curve; or distinct straight lines (possibly with complex coefficients). We show that each one of these geometric properties is generic for systems satisfying the corresponding algebraic condition for the center. Another version of this result in terms of real algebraic curves is given. These results make clear the many facets of the problem of the center in the quadratic case, in particular the question of integrability and form a basis for analogous investigations for the general problem of the center for cubic systems.
On manifolds with nonnegative curvature on totally isotropic 2-planes
Walter
Seaman
843-855
Abstract: We prove that a compact orientable $2n$-dimensional Riemannian manifold, with second Betti number nonzero, nonnegative curvature on totally isotropic $2$-planes, and satisfying a positivity-type condition at one point, is necessarily Kähler, with second Betti number $1$. Using the methods of Siu and Yau, we prove that if the positivity condition is satisfied at every point, then the manifold is biholomorphic to complex projective space.
A tom Dieck theorem for strong shape theory
Bernd
Günther
857-870
Abstract: We consider an appropriate class of locally finite closed coverings of spaces, for which the strong shape of the elements of the covering and of their intersections determine the strong shape of the whole space. Conclusions concerning shape dimension and spaces having the strong shape of ${\text{CW}}$-complexes are drawn, and a Leray spectral sequence for strong homology is given.
Ricci flow, Einstein metrics and space forms
Rugang
Ye
871-896
Abstract: The main results in this paper are: (1) Ricci pinched stable Riemannian metrics can be deformed to Einstein metrics through the Ricci flow of R. Hamilton; (2) (suitably) negatively pinched Riemannian manifolds can be deformed to hyperbolic space forms through Ricci flow; and (3) ${L^2}$-pinched Riemannian manifolds can be deformed to space forms through Ricci flow.
The geometry of Julia sets
Jan M.
Aarts;
Lex G.
Oversteegen
897-918
Abstract: The long term analysis of dynamical systems inspired the study of the dynamics of families of mappings. Many of these investigations led to the study of the dynamics of mappings on Cantor sets and on intervals. Julia sets play a critical role in the understanding of the dynamics of families of mappings. In this paper we introduce another class of objects (called hairy objects) which share many properties with the Cantor set and the interval: they are topologically unique and admit only one embedding in the plane. These uniqueness properties explain the regular occurrence of hairy objects in pictures of Julia sets--hairy objects are ubiquitous. Hairy arcs will be used to give a complete topological description of the Julia sets of many members of the exponential family.
Parametrizing smooth compactly supported wavelets
Raymond O.
Wells
919-931
Abstract: In this paper a concrete parameter space for the compactly supported wavelet systems of Daubechies is constructed. For wavelet systems with $N$ (generic) nonvanishing coefficients the parameter space is a closed convex set in ${{\mathbf{R}}^{(N - 2)/2}}$, which can be explicitly described in the Fourier transform domain. The moment-free wavelet systems are subsets obtained by the intersection of the parameter space and an affine subspace of ${{\mathbf{R}}^{(N - 2)/2}}$.
Fibered products of homogeneous continua
Karen
Villarreal
933-939
Abstract: In this paper, we construct homogeneous continua by using a fibered product of a homogeneous continuum $X$ with itself. The space $ X$ must have a continuous decomposition into continua, and it must possess a certain type of homogeneity property with respect to this decomposition. It is known that the points of any one-dimensional, homogeneous continuum can be "blown up" into pseudo-arcs to form a new continuum with a continuous decomposition into pseudo-arcs. We will show that these continua can be used in the above construction. Finally, we will show that the continuum constructed by using the pseudo-arcs, the circle of pseudo-arcs, or the solenoid of pseudo-arcs is not homeomorphic to any known homogeneous continuum.
Extending the $t$-design concept
A. R.
Calderbank;
P.
Delsarte
941-952
Abstract: Let $\mathfrak{B}$ be a family of $k$-subsets of a $v$-set $V$, with $ 1 \leq k \leq v/2$. Given only the inner distribution of $\mathfrak{B}$, i.e., the number of pairs of blocks that meet in $j$ points (with $j = 0,1, \ldots ,k$), we are able to completely describe the regularity with which $\mathfrak{B}$ meets an arbitrary $t$-subset of $V$, for each order $t$ (with $ 1 \leq t \leq v/2$). This description makes use of a linear transform based on a system of dual Hahn polynomials with parameters $ v$, $k$, $t$. The main regularity parameter is the dimension of a well-defined subspace of ${\mathbb{R}^{t + 1}}$, called the $t$-form space of $\mathfrak{B}$. (This subspace coincides with ${\mathbb{R}^{t + 1}}$ if and only if $\mathfrak{B}$ is a $t$-design.) We show that the $t$-form space has the structure of an ideal, and we explain how to compute its canonical generator.
An inverse boundary value problem for Schr\"odinger operators with vector potentials
Zi Qi
Sun
953-969
Abstract: We consider the Schrödinger operator for a magnetic potential $ \vec A$ and an electric potential $q$, which are supported in a bounded domain in ${\mathbb{R}^n}$ with $n \geq 3$. We prove that knowledge of the Dirichlet to Neumann map associated to the Schrödinger operator determines the magnetic field $\operatorname{rot}(\vec A)$ and the electric potential $q$ simultaneously, provided $\operatorname{rot}(\vec A)$ is small in the ${L^\infty }$ topology.
Characterization of eigenfunctions of the Laplacian by boundedness conditions
Robert S.
Strichartz
971-979
Abstract: If $ {\{ {f_k}(x)\} _{k \in \mathbb{Z}}}$ is a doubly infinite sequence of functions on $ {\mathbb{R}^n}$ which are uniformly bounded and such that $\Delta {f_k} = {f_{k + 1}}$, then $\Delta {f_0} = - {f_0}$. This generalizes a theorem of Roe $(n = 1)$. The analogous statement is true on the Heisenberg group, but false in hyperbolic space.
The structure of a nonlinear elliptic operator
P. T.
Church;
E. N.
Dancer;
J. G.
Timourian
1-42
Abstract: Consider the nonlinear Dirichlet problem $(1) - \Delta u - \lambda u + {u^3} = g$, for $u:\Omega \to \mathbb{R}$, $u\vert\partial \Omega = 0$, and $\Omega \subset {\mathbb{R}^n}$ connected and bounded, and let $ {\lambda _i}$ be the $ i$th eigenvalue of $- \Delta u$ on $\Omega$ with $ u\vert\partial \Omega = 0$, $ (i = 1,2, \ldots )$. Define a map ${A_\lambda }:H \to H\prime $ by $ {A_\lambda }(u) = - \Delta u - \lambda u + {u^3}$, for either the Sobolev space $ W_0^{1,2}(\Omega ) = H = H\prime$ (if $n \leq 4)$ or the Hölder spaces $ C_0^{2,\alpha }(\bar \Omega ) = H$ and ${C^{0,\alpha }}(\bar \Omega ) = H\prime$ (if $\partial \Omega $ is ${C^{2,\alpha }}$ ), and define $ A:H \times \mathbb{R} \to H\prime \times \mathbb{R}$ by $A(u,\lambda ) = ({A_\lambda }(u),\lambda )$. Let $ G:{\mathbb{R}^2} \times E \to {\mathbb{R}^2} \times E$ be the global cusp map given by $ G(s,t,v) = ({s^3} - ts,t,v)$, and let $F:\mathbb{R} \times E \to \mathbb{R} \times E$ be the global fold map given by $F(t,v) = ({t^2},v)$, where $E$ is any Fréchet space. Theorem 1. If $H = H\prime = W_0^{1,2}(\Omega )$, assume in addition that $ n \leqslant 3$. There exit $\varepsilon > 0$ and homeomorphisms $ \alpha$ and $ \beta$ such that the following diagram commutes:
Equiconvergence theorems for Fourier-Bessel expansions with applications to the harmonic analysis of radial functions in Euclidean and non-Euclidean spaces
Leonardo
Colzani;
Antonio
Crespi;
Giancarlo
Travaglini;
Marco
Vignati
43-55
Abstract: We shall prove an equiconvergence theorem between Fourier-Bessel expansions of functions in certain weighted Lebesgue spaces and the classical cosine Fourier expansions of suitable related functions. These weighted Lebesgue spaces arise naturally in the harmonic analysis of radial functions on euclidean spaces and we shall use the equiconvergence result to deduce sharp results for the pointwise almost everywhere convergence of Fourier integrals of radial functions in the Lorentz spaces $ {L^{p,q}}({{\mathbf{R}}^n})$. Also we shall briefly apply the above approach to the study of the harmonic analysis of radial functions on noneuclidean hyperbolic spaces.
Generalizations of the wave equation
J. Marshall
Ash;
Jonathan
Cohen;
C.
Freiling;
Dan
Rinne
57-75
Abstract: The main result of this paper is a generalization of the property that, for smooth $u$, ${u_{xy}} = 0$ implies $(\ast)$ $\displaystyle u(x,y) = a(x) + b(y).$ Any function having generalized unsymmetric mixed partial derivative identically zero is of the form $(\ast)$. There is a function with generalized symmetric mixed partial derivative identically zero not of the form $(\ast)$, but $(\ast)$ does follow here with the additional assumption of continuity. These results connect to the theory of uniqueness for multiple trigonometric series. For example, a double trigonometric series is the $ {L^2}$ generalized symmetric mixed partial derivative of its formal $ (x,y)$-integral.
Andr\'e permutations, lexicographic shellability and the $cd$-index of a convex polytope
Mark
Purtill
77-104
Abstract: The $cd$-index of a polytope was introduced by Fine; it is an integer valued noncommutative polynomial obtained from the flag-vector. A result of Bayer and Fine states that for any integer "flag-vector," the existence of the $cd$-index is equivalent to the holding of the generalized Dehn-Sommerville equations of Bayer and Billera for the flag-vector. The coefficients of the $ cd$-index are conjectured to be nonnegative. We show a connection between the $cd$-index of a polytope $ \mathcal{P}$ and any $ CL$-shelling of the lattice of faces of $ \mathcal{P}$ ; this enables us to prove that each André polynomial of Foata and Schützenberger is the $cd$-index of a simplex. The combinatorial interpretation of this $cd$-index can be extended to cubes, simplicial polytopes, and some other classes (which implies that the $cd$-index has nonnegative coefficients for these polytopes). In particular, we show that any polytope of dimension five or less has a positive $ cd$-index.
Super efficiency in vector optimization
J. M.
Borwein;
D.
Zhuang
105-122
Abstract: We introduce a new concept of efficiency in vector optimization. This concept, super efficiency, is shown to have many desirable properties. In particular, we show that in reasonable settings the super efficient points of a set are norm-dense in the efficient frontier. We also provide a Chebyshev characterization of super efficient points for nonconvex sets and a scalarization theory when the underlying set is convex.
On the characterization of a Riemann surface by its semigroup of endomorphisms
A.
Erëmenko
123-131
Abstract: Suppose $ {D_1}$ and ${D_2}$ be Riemann surfaces which have bounded nonconstant holomorphic functions. Denote by $E({D_i})$, $i = 1,2$, the semigroups of all holomorphic endomorphisms. If $ \phi :E({D_1}) \to E({D_2})$ is an isomorphism of semigroups then there exists a conformal or anticonformal isomorphism $\psi :{D_1} \to {D_2}$ such that $ \phi$ is the conjugation by $\psi$. Also the semigroup of injective endomorphisms as well as some parabolic surfaces are considered.
Asymptotic behavior for a coalescence problem
Oscar
Bruno;
Avner
Friedman;
Fernando
Reitich
133-158
Abstract: Consider spherical particles of volume $x$ having paint on a fraction $y$ of their surface area. The particles are assumed to be homogeneously distributed at each time $t$, so that one can introduce the density number $ n(x,y,t)$. When collision between two particles occurs, the particles will coalesce if and only if they happen to touch each other, at impact, at points which do not belong to the painted portions of their surfaces. Introducing a dynamics for this model, we study the evolution of $n(x,y,t)$ and, in particular, the asymptotic behavior of the mass $ xn(x,y,t)dx$ as $t \to \infty$.
Rigidity of invariant complex structures
Isabel
Dotti Miatello
159-172
Abstract: A Kähler solvmanifold is a connected Kähler manifold $ (M,j,\left\langle , \right\rangle )$ admitting a transitive solvable group of automorphisms. In this paper we study the isomorphism classes of Kähler structures $(j,\left\langle , \right\rangle )$ turning $ M$ into a Kähler solvmanifold. In the case when $(M,j,\left\langle , \right\rangle )$ is irreducible and simply connected we show that any Kähler structure on $M$, having the same group of automorphisms, is isomorphic to $(j,\left\langle , \right\rangle )$.
The weighted Hardy's inequality for nonincreasing functions
Vladimir D.
Stepanov
173-186
Abstract: The purpose of this paper is to give an alternative proof of recent results of M. Arino and B. Muckenhoupt [1] and E. Sawyer [8], concerning Hardy's inequality for nonincreasing functions and related applications to the boundedness of some classical operators on general Lorentz spaces. Our approach will extend the results of [1,8] to the values of the parameters which are inaccessible by the methods of these papers.
The Noetherian property in rings of integer-valued polynomials
Robert
Gilmer;
William
Heinzer;
David
Lantz
187-199
Abstract: Let $D$ be a Noetherian domain, $ D\prime$ its integral closure, and $ \operatorname{Int}(D)$ its ring of integer-valued polynomials in a single variable. It is shown that, if $D\prime$ has a maximal ideal $M\prime$ of height one for which $D\prime /M\prime$ is a finite field, then $\operatorname{Int}(D)$ is not Noetherian; indeed, if $ M\prime$ is the only maximal ideal of $D\prime$ lying over $M\prime \cap D$, then not even $ \operatorname{Spec}(\operatorname{Int}(D))$ is Noetherian. On the other hand, if every height-one maximal ideal of $ D\prime$ has infinite residue field, then a sufficient condition for $ \operatorname{Int}(D)$ to be Noetherian is that the global transform of $ D$ is a finitely generated $ D$-module.
Complex geodesics and iterates of holomorphic maps on convex domains in ${\bf C}\sp n$
Peter R.
Mercer
201-211
Abstract: We study complex geodesics $ f:\Delta \to \Omega$, where $\Delta$ is the unit disk in ${\mathbf{C}}$ and $\Omega$ belongs to a class of bounded convex domains in $ {{\mathbf{C}}^n}$ with no boundary regularity assumption. Along with continuity up to the boundary, existence of such complex geodesics with two prescribed values $z$, $ w \in \bar \Omega$ is established. As a consequence we obtain some new results from iteration theory of holomorphic self maps of bounded convex domains in $ {{\mathbf{C}}^n}$.
Zeta regularized products
J. R.
Quine;
S. H.
Heydari;
R. Y.
Song
213-231
Abstract: If ${\lambda _k}$ is a sequence of nonzero complex numbers, then we define the zeta regularized product of these numbers, $\prod\nolimits_k {{\lambda _k}}$, to be $\exp ( - Z\prime (0))$ where $ Z(s) = \sum\nolimits_{k = 0}^\infty {\lambda _k^{ - s}} $. We assume that $ Z(s)$ has analytic continuation to a neighborhood of the origin. If ${\lambda _k}$ is the sequence of positive eigenvalues of the Laplacian on a manifold, then the zeta regularized product is known as $\det \prime \Delta$, the determinant of the Laplacian, and $\prod\nolimits_k {({\lambda _k} - \lambda )}$ is known as the functional determinant. The purpose of this paper is to discuss properties of the determinant and functional determinant for general sequences of complex numbers. We discuss asymptotic expansions of the functional determinant as $\lambda \to - \infty$ and its relationship to the Weierstrass product. We give some applications to the theory of Barnes' multiple gamma functions and elliptic functions. A new proof is given for Kronecker's limit formula and the product expansion for Barnes' double Stirling modular constant.
Sets of determination for harmonic functions
Stephen J.
Gardiner
233-243
Abstract: Let $h$ denote a positive harmonic function on the open unit ball $B$ of Euclidean space ${{\mathbf{R}}^n}\;(n \geq 2)$. This paper characterizes those subsets $E$ of $B$ for which ${\sup _E}H/h = {\sup _B}H/h$ or ${\inf _E}H/h = {\inf _B}H/h$ for all harmonic functions $H$ belonging to a specified class. In this regard we consider the classes of positive harmonic functions, differences of positive harmonic functions, and harmonic functions with a one-sided quasi-boundedness condition. We also consider the closely related question of representing functions on the sphere $\partial B$ as sums of Poisson kernels corresponding to points in $E$.
Chains, null-chains, and CR geometry
Lisa K.
Koch
245-261
Abstract: A system of distinguished curves distinct from chains is defined on indefinite nondegenerate $ {\text{CR}}$ hypersurfaces; the new curves are called null-chains. The properties of these curves are explored, and it is shown that two sufficiently nearby points of any nondegenerate ${\text{CR}}$ hypersurface can be connected by either a chain or a null-chain.
Intersection cohomology of $S\sp 1$-actions
Gilbert
Hector;
Martin
Saralegi
263-288
Abstract: Given a free action of the circle $ {{\mathbf{S}}^1}$ on a differentiable manifold $M$, there exists a long exact sequence that relates the cohomology of $M$ with the cohomology of the manifold $M/{{\mathbf{S}}^1}$. This is the Gysin sequence. This result is still valid if we allow the action to have stationary points. In this paper we are concerned with actions where fixed points are allowed. Here the quotient space $ M/{{\mathbf{S}}^1}$ is no longer a manifold but a stratified pseudomanifold (in terms of Goresky and MacPherson). We get a similar Gysin sequence where the cohomology of $M/{{\mathbf{S}}^1}$ is replaced by its intersection cohomology. As in the free case, the connecting homomorphism is given by the product with the Euler class $ [e]$. Also, the vanishing of this class is related to the triviality of the action. In this Gysin sequence we observe the phenomenon of perversity shifting. This is due to the allowability degree of the Euler form.
On complete manifolds of nonnegative $k$th-Ricci curvature
Zhong Min
Shen
289-310
Abstract: In this paper we establish some vanishing and finiteness theorems for the topological type of complete open riemannian manifolds under certain positivity conditions for curvature. Key tools are comparison techniques and Morse Theory of Busemann and distance functions.
Global phase structure of the restricted isosceles three-body problem with positive energy
Kenneth
Meyer;
Qiu Dong
Wang
311-336
Abstract: We study a restricted three-body problem with special symmetries: the restricted isosceles three-body problem. For positive energy the energy manifold is partially compactified by adding boundary manifolds corresponding to infinity and triple collision. We use a new set of coordinates which are a variation on the McGehee coordinates of celestial mechanics. These boundary manifolds are used to study the global phase structure of this gradational system. The orbits are classified by intersection number, that is the number of times the infinitesimal body cross the line of syzygy before escaping to infinity.
Local integrability of Mizohata structures
Jorge
Hounie;
Pedro
Malagutti
337-362
Abstract: In this work we study the local integrability of strongly pseudoconvex Mizohata structures of rank $n > 2$ (and co-rank $1$). These structures are locally generated in an appropriate coordinate system $({t_1}, \ldots ,{t_n},x)$ by flat perturbations of Mizohata vector fields ${M_j} = \frac{\partial } {{\partial {t_j}}} - i{t_j}\frac{\partial } {{\partial x}}$, $j = 1, \ldots ,n$. For this, we first prove the global integrability of small perturbations of the structure generated by $\frac{\partial } {{\partial \bar z}} + {\sigma _1}\frac{\partial } {{\partial z}}$, $\frac{\partial } {{\partial {\theta _{n - 1}}}} + {\sigma _j}\frac{\partial } {{\partial z}}$, $ j = 2, \ldots ,n$, defined over a manifold $ {\mathbf{C}} \times S$, where $S$ is simply connected.
Inverse scattering for singular potentials in two dimensions
Zi Qi
Sun;
Gunther
Uhlmann
363-374
Abstract: We consider the Schrödinger equation for a compactly supported potential having jump type singularities at a subdomain of $ {\mathbb{R}^2}$. We prove that knowledge of the scattering amplitude at a fixed energy, determines the location of the singularity as well as the jump across the curve of discontinuity. This result follows from a similar result for the Dirichlet to Neumann map associated to the Schrödinger equation for a compactly supported potential with the same type of singularities.
Twists of Hilbert modular forms
Thomas R.
Shemanske;
Lynne H.
Walling
375-403
Abstract: The theory of newforms for Hilbert modular forms is summarized including a statement of a strong multiplicity-one theorem and a characterization of newforms as eigenfunctions for a certain involution whose Dirichlet series has a prescribed Euler product. The general question of twisting Hilbert modular newforms by arbitrary Hecke characters is considered and the exact level of a character twist of a Hilbert modular form is determined. Conditions under which the twist of a newform is a newform are given. Applications include a strengthening in the elliptic modular case of a theorem of Atkin and Li's regarding the characterization of imprimitive newforms as well as its generalization to the Hilbert modular case, and a decomposition theorem for certain spaces of newforms as the direct sum of twists of spaces of newforms of lower level.
Regularity properties of solutions to transmission problems
Luis
Escauriaza;
Jin Keun
Seo
405-430
Abstract: We show that the gradients of solutions to certain elliptic and parabolic transmission problems with internal Lipschitz boundary and constant coefficients at each side of the internal boundary are square integrable along the internal boundary.
Characterization of automorphisms on the Barrett and the Diederich-Forn\ae ss worm domains
So-Chin
Chen
431-440
Abstract: In this paper we show that every automorphism on either the Barrett or the Diederich-Fornaess worm domains is given by a rotation in $w$-variable. In particular, any automorphism on either one of these two domains can be extended smoothly up to the boundary.
Classification of singularities for blowing up solutions in higher dimensions
J. J. L.
Velázquez
441-464
Abstract: Consider the Cauchy problem (P) $\displaystyle \left\{ {\begin{array}{*{20}{c}} {{u_t} - \Delta u = {u^p}} ... ...text{when}}\;x \in {\mathbb{R}^N},} & {} \end{array} } \right.$ where $p > 1$, and ${u_0}(x)$ is a continuous, nonnegative and bounded function. It is known that, under fairly general assumptions on ${u_0}(x)$, the unique solution of $({\text{P}})$, $u(x,t)$, blows up in a finite time, by which we mean that $\displaystyle \mathop {\lim \sup }\limits_{t \uparrow T} \left( {\mathop {\sup }\limits_{x \in {\mathbb{R}^N}} \;u(x,t)} \right) = + \infty .$ In this paper we shall assume that $u(x,t)$ blows up at $x = 0$, $ t = T < + \infty$ , and derive the possible asymptotic behaviours of $ u(x,t)$ as $(x,t) \to (0,T)$, under general assumptions on the blow-up rate.
Imprimitive Gaussian sums and theta functions over a number field
Jacob
Nemchenok
465-478
Abstract: We obtain a reduction formula for an imprimitive Gaussian sum with a numerical character in an algebraic number field, i.e. a formula that expresses that sum as a product of several elementary factors times a primitive, proper, normed Gaussian sum (formulae (16) and (19)). We also introduce Gaussian sums with Hecke characters and derive a similar reduction formula for them. The derivation is based on an inversion formula for a multivariable theta function associated with the number field, twisted with the numerical character.
Mixing properties of a class of Bernoulli-processes
Doris
Fiebig
479-493
Abstract: We prove that stationary very weak Bernoulli processes with rate $ O(1/n)\;({\text{VWB}}\,O(1/n))$ are strictly very weak Bernoulli with rate $ O(1/n)$. Furthermore we discuss the relation between ${\text{VWB}}\;O(1/n)$ and the classical mixing properties for countable state processes. In particular, we show that $ {\text{VWB}}\,O(1/n)$ implies $\phi$-mixing.
Ergodic theory for Markov fibred systems and parabolic rational maps
Jon
Aaronson;
Manfred
Denker;
Mariusz
Urbański
495-548
Abstract: A parabolic rational map of the Riemann sphere admits a non atomic $ h$-conformal measure on its Julia set where $h =$ the Hausdorff dimension of the Julia set and satisfies $1/2 < h < 2$. With respect to this measure the rational map is conservative, exact and there is an equivalent $\sigma$-finite invariant measure. Finiteness of the measure is characterised. Central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one. A theory of Markov fibred systems is developed, and parabolic rational maps are considered within this framework.
On the Cauchy problem for reaction-diffusion equations
Xuefeng
Wang
549-590
Abstract: The simplest model of the Cauchy problem considered in this paper is the following $(\ast)$ \begin{displaymath}\begin{array}{*{20}{c}} {{u_t} = \Delta u + {u^p},} & ... ... & {\phi \geq 0,\phi \,\not\equiv\,0.} \end{array} \;\end{displaymath} It is well known that when $1 < p \leq (n + 2)/n$, the local solution of $ (\ast)$ blows up in finite time as long as the initial value $\phi$ is nontrivial; and when $p > (n + 2)/n$, if $\phi$ is "small", $(\ast)$ has a global classical solution decaying to zero as $t \to + \infty $, while if $\phi$ is "large", the local solution blows up in finite time. The main aim of this paper is to obtain optimal conditions on $\phi$ for global existence and to study the asymptotic behavior of those global solutions. In particular, we prove that if $n \geq 3$, $ p > n/(n - 2)$, $\displaystyle 0 \leq \phi (x) \leq \lambda {u_s}(x) = \lambda {\left( {\frac{{2... ... \frac{n} {{n - 2}}} \right)} \right)^{1/(p - 1)}}\vert x{\vert^{ - 2/(p - 1)}}$ (${u_s}$ is a singular equilibrium of $ (\ast)$) where $0 < \lambda < 1$, then $(\ast)$ has a (unique) global classical solution $u$ with $0 \leq u \leq \lambda {u_s}$ and $\displaystyle u(x,t) \leq {(({\lambda ^{1 - p}} - 1)(p - 1)t)^{ - 1/(p - 1)}}.$ (This result implies that ${u_0} \equiv 0$ is stable w.r.t. to a weighted ${L^\infty }$ topology when $n \geq 3$ and $p > n/(n - 2)$.) We also obtain some sufficient conditions on $\phi$ for global nonexistence and those conditions, when combined with our global existence result, indicate that for $\phi$ around ${u_s}$, we are in a delicate situation, and when $ p$ is fixed, ${u_0} \equiv 0$ is "increasingly stable" as the dimension $ n \uparrow + \infty$. A slightly more general version of $(\ast)$ is also considered and similar results are obtained.
The Alexander and Markov theorems via diagrams for links in $3$-manifolds
Paul A.
Sundheim
591-607
Abstract: Let $M$ be a $3$-manifold with an open book decomposition. We obtain a new proof that a link in $M$ has a braided form and that two braided forms are related by a sequence of two Markov moves for $ M$ by generalizing Morton's approach for links in ${S^3}$.
Baire class $1$ selectors for upper semicontinuous set-valued maps
V. V.
Srivatsa
609-624
Abstract: Let $T$ be a metric space and $X$ a Banach space. Let $F:T \to X$ be a set-valued map assuming arbitrary values and satisfying the upper semicontinuity condition: $\{ t \in T:F(t) \cap C \ne \emptyset \}$ is closed for each weakly closed set $C$ in $X$. Then there is a sequence of norm-continuous functions converging pointwise (in the norm) to a selection for $F$. We prove a statement of similar precision and generality when $X$ is a metric space.
Applying coordinate products to the topological identification of normed spaces
Robert
Cauty;
Tadeusz
Dobrowolski
625-649
Abstract: Using the $ {l^2}$-products we find pre-Hilbert spaces that are absorbing sets for all Borelian classes of order $ \alpha \geq 1$. We also show that the following spaces are homeomorphic to $\Sigma^\infty$, the countable product of the space $\Sigma = \{(x_n) \in R^\infty: (x_n)$ is bounded}: (1) every coordinate product $\prod_C H_n$ of normed spaces $H_n$ in the sense of a Banach space $ C$, where each $ H_n$ is an absolute $F_{\sigma\delta}$-set and infinitely many of the $ H_n$'s are ${Z_\sigma }$-spaces, (2) every function space $\tilde{L}^p = \cap_{p\prime <p}L^{p\prime}$ with the ${L^q}$-topology, $0<q<p \leq \infty$, (3) every sequence space ${\tilde l^p} = { \cap _{p < p\prime}}{l^{p\prime}}$ with the $l^q$-topology, $ 0 \leq p < q < \infty$. We also note that each additive and multiplicative Borelian class of order $\alpha \geq 2$, each projective class, and the class of nonprojective spaces contain uncountably many topologically different pre-Hilbert spaces which are $ Z_\sigma$-spaces.
Chebyshev type estimates for Beurling generalized prime numbers. II
Wen-Bin
Zhang
651-675
Abstract: Let $N(x)$ be the distribution function of the integers in a Beurling generalized prime system. The Chebyshev type estimates for Beurling generalized prime numbers in the general case $\displaystyle N(x) = x\sum\limits_{\nu = 1}^n {{A_\nu }} {\log ^{{\rho _\nu } - 1}}x + O(x{\log ^{ - \gamma }}x)$ is a long standing question. In this paper we shall give an affirmative answer to the question by proving that the Chebyshev type estimates $\displaystyle 0 < \mathop {\lim \inf }\limits_{x \to \infty } \frac{{\psi (x)}}... ...quad \mathop {\lim \sup }\limits_{x \to \infty } \frac{{\psi (x)}}{x} < \infty$ hold even under weaker condition $\displaystyle \int_1^\infty {{x^{ - 1}}} \left\{ {\mathop {\sup }\limits_{x < \... ...n {{A_\nu }} {{\log }^{{\rho _\nu } - 1}}y} \right\vert} \right\}\,dx < \infty$ with $ \rho_n=\tau \geq 1$, $ 0<\rho_1<\rho_2 <\cdots < \rho_n$, and $A_n > 0$. This generalizes a result of Diamond and a result of the present author.
Computing the equations of a variety
Michela
Brundu;
Mike
Stillman
677-690
Abstract: Let $X \subset {\mathbb{P}^n}$ be a projective variety or subscheme, and let $ \mathcal{F}$ be an invertible sheaf on $X$. A set of global sections of $\mathcal{F}$ determines a map from a Zariski open subset of $X$ to $ {\mathbb{P}^r}$. The purpose of this paper is to find, given $X$ and $ \mathcal{F}$, the homogeneous ideal defining the image in ${\mathbb{P}^r}$ of this rational map. We present algorithms to compute the ideal of the image. These algorithms can be implemented using only the computation of Gröbner bases and syzygies, and they have been implemented in our computer algebra system Macaulay. Our methods generalize to include the case when $X$ is an arbitrary projective scheme and $\mathcal{F}$ is generically invertible.
Varieties of topological geometries
Hansjoachim
Groh
691-702
Abstract: A variety of topological geometries is either A. a projective variety $ \mathcal{L}(F)$ over some topological field $F$, or B. a matchstick variety $\mathcal{M}(X)$ over some topological space $ X$. As a main tool for showing this, we prove a structure theorem for arbitrary topological geometries.
The structure of random partitions of large integers
Bert
Fristedt
703-735
Abstract: Random partitions of integers are treated in the case where all partitions of an integer are assumed to have the same probability. The focus is on limit theorems as the number being partitioned approaches $\infty$. The limiting probability distribution of the appropriately normalized number of parts of some small size is exponential. The large parts are described by a particular Markov chain. A central limit theorem and a law of large numbers holds for the numbers of intermediate parts of certain sizes. The major tool is a simple construction of random partitions that treats the number being partitioned as a random variable. The same technique is useful when some restriction is placed on partitions, such as the requirement that all parts must be distinct.
A generalization of the Airy integral for $f''-z\sp nf=0$
Gary G.
Gundersen;
Enid M.
Steinbart
737-755
Abstract: It is well known that the Airy integral is a solution of the Airy differential equation $ f'' - zf = 0$ and that the Airy integral is a contour integral function with special properties. We show that there exist analogous special contour integral solutions of the more general equation $f'' - {z^n}f = 0$ where $n$ is any positive integer. Related results are given.
Kloosterman sums for Chevalley groups
Romuald
Dąbrowski
757-769
Abstract: A generalization of Kloosterman sums to a simply connected Chevalley group $G$ is discussed. These sums are parameterized by pairs $(w,t)$ where $w$ is an element of the Weyl group of $G$ and $t$ is an element of a $ {\mathbf{Q}}$-split torus in $G$. The $ SL(2,{\mathbf{Q}})$-Kloosterman sums coincide with the classical Kloosterman sums and $ SL(r,{\mathbf{Q}})$-Kloosterman sums, $r \geq 3$, coincide with the sums introduced in [B-F-G,F,S]. Algebraic properties of the sums are proved by root system methods. In particular an explicit decomposition of a general Kloosterman sum over ${\mathbf{Q}}$ into the product of local $ p$-adic factors is obtained. Using this factorization one can show that the Kloosterman sums corresponding to a toral element, which acts trivially on the highest weight space of a fundamental irreducible representation, splits into a product of Kloosterman sums for Chevalley groups of lower rank.
The finite part of singular integrals in several complex variables
Xiaoqin
Wang
771-793
Abstract: A divergent integral can sometimes be handled by assigning to it as its value the finite part in the sense of Hadamard. This is done by expanding the integral over the complement of a symmetric neighborhood of a singularity in powers of the radius, and throwing away the negative powers. In this paper the finite part of a singular integral of Cauchy type is defined, and this is then used to describe the boundary behavior of derivatives of a Cauchy-type integral. The finite part of a singular integral of Bochner-Martinelli type is studied, and an extension of the Plemelj jump formulas is shown to hold.
Totally monotone functions with applications to the Bergman space
B.
Korenblum;
R.
O’Neil;
K.
Richards;
K.
Zhu
795-806
Abstract: Using a theorem of S. Bernstein [1] we prove a special case of the following maximum principle for the Bergman space conjectured by B. Korenblum [3]: There exists a number $\delta \in (0,1)$ such that if $f$ and $g$ are analytic functions on the open unit disk ${\mathbf{D}}$ with $\vert f(z)\vert \leq \vert g(z)\vert$ on $\delta \leq \vert z\vert < 1$ then $ {\left\Vert f \right\Vert _2} \leq {\left\Vert g \right\Vert _2}$, where $ {\left\Vert {} \right\Vert _2}$ is the ${L^2}$ norm with respect to area measure on ${\mathbf{D}}$. We prove the above conjecture when either $f$ or $g$ is a monomial; in this case we show that the optimal constant $\delta$ is greater than or equal to $1/\sqrt 3 $.
Computing the Mordell-Weil rank of Jacobians of curves of genus two
Daniel M.
Gordon;
David
Grant
807-824
Abstract: We derive the equations necessary to perform a two-descent on the Jacobians of curves of genus two with rational Weierstrass points. We compute the Mordell-Weil rank of the Jacobian of some genus two curves defined over the rationals, and discuss the practicality of using this method.
A subcategory of TOP
Alan
Dow;
Stephen
Watson
825-837
Abstract: We consider the smallest class of topological spaces which contains the converging sequence and which is closed under the operations of taking arbitrary sums, quotients and finite products. We show that if there is a model of set-theory in which there is a measurable cardinal then there is a model in which this class does not contain all topological spaces. In addition, we prove that it is consistent that this class does contain all topological spaces--in fact much more, a large cardinal is needed to produce a model of set theory in which this class is proper.
${\rm T}1$ theorems for Besov and Triebel-Lizorkin spaces
Y.-S.
Han;
Steve
Hofmann
839-853
Abstract: We give simple proofs of the $T1$ theorem in the general context of Besov spaces and (weighted) Triebel-Lizorkin spaces. Our approach yields some new results for kernels satisfying weakened regularity conditions, while also recovering previously known results.
Quantitative rectifiability and Lipschitz mappings
Guy
David;
Stephen
Semmes
855-889
Abstract: The classical notion of rectifiability of sets in ${{\mathbf{R}}^n}$ is qualitative in nature, and in this paper we are concerned with quantitative versions of it. This issue arises in connection with $ {L^p}$ estimates for singular integral operators on sets in ${{\mathbf{R}}^n}$. We give a criterion for one reasonably natural quantitative rectifiability condition to hold, and we use it to give a new proof of a theorem in [D3]. We also give some results on the geometric properties of a certain class of sets in ${{\mathbf{R}}^n}$ which can be viewed as generalized hypersurfaces. Along the way we shall encounter some questions concerning the behavior of Lipschitz functions, with regard to approximation by affine functions in particular. We shall also discuss an amusing variation of the classical Lipschitz and bilipschitz conditions, which allow some singularities forbidden by the classical conditions while still forcing good behavior on substantial sets.
Combinatorics of triangulations of $3$-manifolds
Feng
Luo;
Richard
Stong
891-906
Abstract: In this paper, we study the average edge order of triangulations of closed $ 3$-manifolds and show in particular that the average edge order being less than $ 4.5$ implies that triangulation is on the $3$-sphere.
A structural criterion for the existence of infinite central $\Lambda(p)$ sets
Kathryn E.
Hare;
David C.
Wilson
907-925
Abstract: We classify the compact, connected groups which have infinite central $\Lambda (p)$ sets, arithmetically characterize central $\Lambda (p)$ sets on certain product groups, and give examples of $ \Lambda (p)$ sets which are non-Sidon and have unbounded degree. These sets are intimately connected with Figà-Talamanca and Rider's examples of Sidon sets, and stem from the existence of families of tensor product representations of almost simple Lie groups whose decompositions into irreducibles are rank-independent.
On the existence and uniqueness of solutions of M\"obius equations
Xingwang
Xu
927-945
Abstract: A generalization of the Schwarzian derivative to conformal mappings of Riemannian manifolds has naturally introduced the corresponding overdetermined differential equation which we call the Möbius equation. We are interested in study of the existence and uniqueness of the solution of the Möbius equation. Among other things, we show that, for a compact manifold, if Ricci curvature is nonpositive, for a complete noncompact manifold, if the scalar curvature is a positive constant, then the differential equation has only constant solutions. We also study the nonhomogeneous equation in an $n$-dimensional Euclidean space.
Higher-dimensional analogues of Fuchsian subgroups of ${\rm PSL}(2,\germ o)$
L. Ya.
Vulakh
947-963
Abstract: The problem of classification of $2 \times 2$ indefinite Hermitian matrices over orders in Clifford algebras is considered. The unit groups of these matrices are analogous to maximal arithmetic Fuchsian subgroups of $ {\text{PSL}}(2,\mathfrak{o})$ where $ \mathfrak{o}$ is an order in a quadratic number field.
Rational orbits on three-symmetric products of abelian varieties
A.
Alzati;
G. P.
Pirola
965-980
Abstract: Let $A$ be an $n$-dimensional Abelian variety, $n \geq 2$; let ${\text{CH}_0}(A)$ be the group of zero-cycles of $ A$, modulo rational equivalence; by regarding an effective, degree $ k$, zero-cycle, as a point on ${S^k}(A)$ (the $k$-symmetric product of $A$), and by considering the associated rational equivalence class, we get a map $\gamma :{S^k}(A) \to {\text{CH}_0}(A)$, whose fibres are called $\gamma$-orbits. For any $n \geq 2$, in this paper we determine the maximal dimension of the $\gamma$-orbits when $k = 2$ or $3$ (it is, respectively, $1$ and $2$), and the maximal dimension of families of $ \gamma$-orbits; moreover, for generic $A$, we get some refinements and in particular we show that if $ \dim (A) \geq 4$, $ {S^3}(A)$ does not contain any $\gamma$-orbit; note that it implies that a generic Abelian four-fold does not contain any trigonal curve. We also show that our bounds are sharp by some examples. The used technique is the following: we have considered some special families of Abelian varieties: $ {A_t} = {E_t} \times B$ ($ {E_t}$ is an elliptic curve with varying moduli) and we have constructed suitable projections between $ {S^k}({A_t})$ and $ {S^k}(B)$ which preserve the dimensions of the families of $\gamma $-orbits; then we have done induction on $n$. For $n = 2$ the proof is based upon the papers of Mumford and Roitman on this topic.
New invariant Einstein metrics on generalized flag manifolds
Andreas
Arvanitoyeorgos
981-995
Abstract: A generalized flag manifold (or a Kählerian $ C$-space) is a homogeneous space $G/K$ whose isotropy subgroup $K$ is the centralizer of a torus in $ G$. These spaces admit a finite number of Kähler-Einstein metrics. We present new non-Kahler Einstein metrics for certain quotients of $U(n)$, $SO(2n)$ and ${G_2}$. We also examine the isometry question for these metrics.
A construction of the supercuspidal representations of ${\rm GL}\sb n(F),\;F\;p$-adic
Lawrence
Corwin
1-58
Abstract: Let $F$ be a nondiscrete, locally compact, non-Archimedean field. In this paper, we construct all irreducible supercuspidal representations of $G = {\text{GL}_n}(F)$ For each such representation $ \pi$ (which we may as well assume is unitary), we give a subgroup $ J$ of $G$ that is compact mod the center $ Z$ of $G$ and a (finite-dimensional) representation $\sigma$ of $J$ such that inducing $\sigma$ to $G$ gives $\pi$. The proof that all supercuspidals have been constructed appeals to a theorem (the Matching Theorem) that has been proved by global methods.
Propagation of singularities, Hamilton-Jacobi equations and numerical applications
Eduard
Harabetian
59-71
Abstract: We consider applications of Hamilton-Jacobi equations for which the initial data is only assumed to be in ${L^\infty }$. Such problems arise for example when one attempts to describe several characteristic singularities of the compressible Euler equations such as contact and acoustic surfaces, propagating from the same discontinuous initial front. These surfaces represent the level sets of solutions to a Hamilton-Jacobi equation which belongs to a special class. For such Hamilton-Jacobi equations we prove the existence and regularity of solutions for any positive time and convergence to initial data along rays of geometrical optics at any point where the gradient of the initial data exists. Finally, we present numerical algorithms for efficiently capturing singular fronts with complicated topologies such as corners and cusps. The approach of using Hamilton-Jacobi equations for capturing fronts has been used in [14] for fronts propagating with curvature-dependent speed.
Quantization of K\"ahler manifolds. II
Michel
Cahen;
Simone
Gutt;
John
Rawnsley
73-98
Abstract: We use Berezin's dequantization procedure to define a formal $ \ast$-product on a dense subalgebra of the algebra of smooth functions on a compact homogeneous Kähler manifold $M$. We prove that this formal $ \ast$-product is convergent when $M$ is a hermitian symmetric space.
An example of a two-term asymptotics for the ``counting function'' of a fractal drum
Jacqueline
Fleckinger-Pellé;
Dmitri G.
Vassiliev
99-116
Abstract: In this paper we study the spectrum of the Dirichlet Laplacian in a bounded domain $\Omega \subset {\mathbb{R}^n}$ with fractal boundary $ \partial \Omega$. We construct an open set $ \mathcal{Q}$ for which we can effectively compute the second term of the asymptotics of the "counting function" $N(\lambda ,\mathcal{Q})$, the number of eigenvalues less than $\lambda$. In this example, contrary to the M. V. Berry conjecture, the second asymptotic term is proportional to a periodic function of In $ \lambda$, not to a constant. We also establish some properties of the $ \zeta$-function of this problem. We obtain asymptotic inequalities for more general domains and in particular for a connected open set $\mathcal{O}$ derived from $ \mathcal{Q}$. Analogous periodic functions still appear in our inequalities. These results have been announced in $[{\text{FV}}]$.
Divisors on symmetric products of curves
Alexis
Kouvidakis
117-128
Abstract: For a curve with general moduli, the Neron-Severi group of its symmetric products is generated by the classes of two divisors $x$ and $\theta$. In this paper we give bounds for the cones of effective and ample divisors in the $ x\theta$-plane.
Minimal hypersurfaces of ${\bf R}\sp {2m}$ invariant by ${\rm SO}(m)\times {\rm SO}(m)$
Hilário
Alencar
129-141
Abstract: Let $G = {\text{SO}}(m) \times {\text{SO}}(m)$ act in the standard way on $ {{\mathbf{R}}^m} \times {{\mathbf{R}}^m}$. We describe all complete minimal hypersurfaces of ${{\mathbf{R}}^m}\backslash \{ 0\}$ which are invariant under $G$ for $m = 2$, $3$ . We also show that the unique minimal hypersurface of $ {{\mathbf{R}}^{2m}}$ which is invariant under $G$ and passes through the origin of ${{\mathbf{R}}^{2m}}$ is the minimal quadratic cone.
Formal moduli of modules over local $k$-algebras
Allan
Adler;
Pradeep
Shukla
143-158
Abstract: We determine explicitly the formal moduli space of certain complete topological modules over a topologically finitely generated local $k$-algebra $R$, not necessarily commutative, where $ k$ is a field. The class of topological modules we consider include all those of finite rank over $k$ and some of infinite rank as well, namely those with a Schauder basis in the sense of $\S1$. This generalizes the results of [Sh], where the result was obtained in a different way in case the ring $R$ is the completion of the local ring of a plane curve singularity and the module is ${k^n}$. Along the way, we determine the ring of infinite matrices which correspond to the endomorphisms of the modules with Schauder bases. We also introduce functions called "growth functions" to handle explicit epsilonics involving the convergence of formal power series in noncommuting variables evaluated at endomorphisms of our modules. The description of the moduli space involves the study of a ring of infinite series involving possibly infinitely many variables and which is different from the ring of power series in these variables in either the wide or the narrow sense. Our approach is beyond the methods of [Sch] which were used in [Sh] and is more conceptual.
Residual finiteness of color Lie superalgebras
Yu. A.
Bahturin;
M. V.
Zaicev
159-180
Abstract: A (color) Lie superalgebra $L$ over a field $K$ of characteristic $\ne 2, 3$ is called residually finite if any of its nonzero elements remains nonzero in a finite-dimensional homomorphic image of $L$. In what follows we are looking for necessary and sufficient conditions under which all finitely generated Lie superalgebras satisfying a fixed system of identical relations are residually finite. In the case $\operatorname{char}\;K = 0$ we show that a variety $ V$ satisfies this property if and only if $V$ does not contain all center-by-metabelian algebras and every finitely generated algebra of $ V$ has nilpotent commutator subalgebra.
Triangulations in M\"obius geometry
Feng
Luo
181-193
Abstract: We prove that a conformally flat closed manifold of dimension at least three adimts a hyperbolic, spherical or similarity structure in the conformally flat class if and only if the manifold has a smooth triangulation so that all codimension one Simplexes are in some codimension one spheres.
Symmetrization with respect to a measure
Friedmar
Schulz;
Virginia
Vera de Serio
195-210
Abstract: In this paper we study the spherical symmetric rearrangement ${u^\ast}$ of a nonnegative measurable function $u$ on $ {\mathbb{R}^n}$ with respect to a measure given by a nonhomogeneous density distribution $p$. Conditions on $u$ are given which guarantee that $ {u^\ast}$ is continuous, of bounded variation, or absolutely continuous on lines, i.e., Sobolev regular. The energy inequality is proven in $n = 2$ dimensions by employing a Carleman type isoperimetric inequality if $\log p$ is subharmonic. The energy equality is settled via a reduction to the case of a homogeneous mass density.
Classification of all parabolic subgroup-schemes of a reductive linear algebraic group over an algebraically closed field
Christian
Wenzel
211-218
Abstract: Let $G$ be a reductive linear algebraic group over an algebraically closed field $K$. The classification of all parabolic subgroups of $G$ has been known for many years. In that context subgroups of $G$ have been understood as varieties, i.e. as reduced schemes. Also several nontrivial nonreduced subgroup schemes of $G$ are known, but until now nobody knew how many there are and what there structure is. Here I give a classification of all parabolic subgroup schemes of $G$ in $\operatorname{char}(K) > 3$ .
Cohomological dimension and metrizable spaces
Jerzy
Dydak
219-234
Abstract: The purpose of this paper is to address several problems posed by V. I. Kuzminov [Ku] regarding cohomological dimension of noncompact spaces. In particular, we prove the following results: Theorem A. Suppose $X$ is metrizable and $G$ is the direct limit of the direct system $\{ {G_s},{h_{s\prime ,s}},S\}$ of abelian groups. Then, $\displaystyle {\dim _G}X \leq \max \{ {\dim _{{G_s}}}X\vert s \in S\}$ . Theorem B. Let $X$ be a metrizable space and let $ G$ be an abelian group. Let $l = \{ p\vert p \cdot (G/\operatorname{Tor}G) \ne G/\operatorname{Tor}G\}$. (a) If $G = \operatorname{Tor}G$, then ${\dim _G}X = \max \{ {\dim _H}X\vert H \in \sigma (G)\}$, (b) ${\dim _G}X = \max \{ {\dim _{\operatorname{Tor}G}}X,{\dim _{G/\operatorname{Tor}G}}X\}$, (c) ${\dim _G}X \geq {\dim _\mathbb{Q}}X$ if $ G \ne \operatorname{Tor}G$, (d) ${\dim _G}X \geq {\dim _{{{\hat{\mathbb{Z}}}_l}}}X$, where $ {\hat{\mathbb{Z}}_l}$ is the group of $l$-adic integers, (e) $\max ({\dim _G}X,{\dim _\mathbb{Q}}X + 1) \geq \max \{ {\dim _H}X\vert H \in \sigma (G)\}$, (f) ${\dim _G}X \leq \,{\dim _{{\mathbb{Z}_l}}}X \leq \,{\dim _G}X + 1$ if $G \ne 0$ is torsion-free. Theorem B generalizes a well-known result of M. F. Bockstein [B].
Algebraic convergence of Schottky groups
Richard D.
Canary
235-258
Abstract: A discrete faithful representation of the free group on $g$ generators ${F_g}$ into $\operatorname{Isom}_ + ({{\mathbf{H}}^3})$ is said to be a Schottky group if $ ({{\mathbf{H}}^3} \cup {D_\Gamma })/\Gamma$ is homeomorphic to a handlebody $ {H_g}$ (where ${D_\Gamma }$ is the domain of discontinuity for $ \Gamma$'s action on the sphere at infinity for $ {{\mathbf{H}}^3}$). Schottky space $ {\mathcal{S}_g}$, the space of all Schottky groups, is parameterized by the quotient of the Teichmüller space $\mathcal{T}({S_g})$ of the closed surface of genus $ g$ by ${\operatorname{Mod} _0}({H_g})$ where $ {\operatorname{Mod} _0}({H_g})$ is the group of (isotopy classes of) homeomorphisms of ${S_g}$ which extend to homeomorphisms of $ {H_g}$ which are homotopic to the identity. Masur exhibited a domain $\mathcal{O}({H_g})$ of discontinuity for $ {\operatorname{Mod} _0}({H_g})$'s action on $PL({S_g})$ (the space of projective measured laminations on ${S_g}$), so $\mathcal{B}({H_g}) = \mathcal{O}({H_g})/{\operatorname{Mod} _0}({H_g})$ may be appended to ${\mathcal{S}_g}$ as a boundary. Thurston conjectured that if a sequence $ \{ {\rho _i}:{F_g} \to \operatorname{Isom}_ + ({{\mathbf{H}}^3})\}$ of Schottky groups converged into $\mathcal{B}({H_g})$, then it converged as a sequence of representations, up to subsequence and conjugation. In this paper, we prove Thurston's conjecture in the case where ${H_g}$ is homeomorphic to $S \times I$ and the length ${l_{{N_i}}}({(\partial S)^\ast})$ in ${N_i} = {{\mathbf{H}}^3}/{\rho _i}({F_g})$ of the closed geodesic(s) in the homotopy class of the boundary of $S$ is bounded above by some constant $K$.
Nonwandering structures at the period-doubling limit in dimensions $2$ and $3$
Marcy M.
Barge;
Russell B.
Walker
259-277
Abstract: A Cantor set supporting an adding machine is the simplest nonwandering structure that can occur at the conclusion of a sequence of perioddoubling bifurcations of plane homeomorphisms. In some families this structure is persistent. In this manuscript it is shown that no plane homeomorphism has nonwandering Knaster continua on which the homeomorphism is semiconjugate to the adding machine. Using a theorem of M. Brown, a three-space homeomorphism is constructed which has an invariant set, $ \Lambda$, the product of a Knaster continuum and a Cantor set. $\Lambda$ is chainable, supports positive entropy but contains only power-of-two periodic orbits. And the homeomorphism restricted to $\Lambda$ is semiconjugate to the adding machine. Lastly, a zero topological entropy ${C^\infty }$ disk diffeomorphism is constructed which has large nonwandering structures over a generalized adding machine on a Cantor set.
The rectifiable metric on the set of closed subspaces of Hilbert space
Lawrence G.
Brown
279-289
Abstract: Consider the set of selfadjoint projections on a fixed Hilbert space. It is well known that the connected components, under the norm topology, are the sets $\{ p:{\text{rank}}\;p = \alpha ,{\text{rank}}(1 - p) = \beta \}$, where $\alpha$ and $\beta$ are appropriate cardinal numbers. On a given component, instead of using the metric induced by the norm, we can use the rectifiable metric $ {d_r}$ which is defined in terms of the lengths of rectifiable paths or, equivalently in this case, the lengths of $\varepsilon$-chains. If $\left\Vert {p - q} \right\Vert < 1$, then ${d_r}(p,q) = {\sin ^{ - 1}}(\left\Vert {p - q} \right\Vert)$, but if $\left\Vert {p - q} \right\Vert = 1$, ${d_r}(p,q)$ can have any value in $\left[ {\frac{\pi } {2},\pi } \right]$ (assuming $\alpha$ and $\beta$ are infinite). If ${d_r}(p,q) \ne \frac{\pi } {2}$, a minimizing path joining $p$ and $q$ exists; but if ${d_r}(p,q) = \frac{\pi } {2}$, a minimizing path exists if and only if $ {\text{rank}}(p \wedge (1 - q)) = {\text{rank}}(q \wedge (1 - p))$.
Symmetries of homotopy complex projective three spaces
Mark
Hughes
291-304
Abstract: We study symmetry properties of six-dimensional, smooth, closed manifolds which are homotopy equivalent to ${\mathbf{C}}{P^3}$. There are infinitely differentiably distinct such manifolds. It is known that if $m$ is an odd prime, infinitely many homotopy $ {\mathbf{C}}{P^3}$'s admit $ {{\mathbf{Z}}_m}$-actions whereas only the standard ${\mathbf{C}}{P^3}$ admits an action of the group $ {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$. We study the intermediate case of ${{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$-actions and show that infinitely many homotopy ${\mathbf{C}}{P^3}$'s do admit $ {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$-actions for a fixed prime $m$. The major tool involved is equivariant surgery theory. Using a transversality argument, we construct normal maps for which the relevant surgery obstructions vanish allowing the construction of $ {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$-actions on homotopy ${\mathbf{C}}{P^3}$'s which are $ {{\mathbf{Z}}_m} \times {{\mathbf{Z}}_m}$-homotopy equivalent to a specially chosen linear action on $ {\mathbf{C}}{P^3}$. A key idea is to exploit an extra bit of symmetry which is built into our set-up in a way that forces the signature obstruction to vanish. By varying the parameters of our construction and calculating Pontryagin classes, we may construct actions on infinitely many differentiably distinct homotopy $ {\mathbf{C}}{P^3}$'s as claimed.
The canonical compactification of a finite group of Lie type
Mohan S.
Putcha;
Lex E.
Renner
305-319
Abstract: Let $G$ be a finite group of Lie type. We construct a finite monoid $ \mathcal{M}$ having $ G$ as the group of units. $\mathcal{M}$ has properties analogous to the canonical compactification of a reductive group. The complex representation theory of $\mathcal{M}$ yields Harish-Chandra's philosophy of cuspidal representations of $G$. The main purpose of this paper is to determine the irreducible modular representations of $\mathcal{M}$. We then show that all the irreducible modular representations of $G$ come (via the 1942 work of Clifford) from the one-dimensional representations of the maximal subgroups of $ \mathcal{M}$. This yields a semigroup approach to the modular representation theory of $G$, via the full rank factorizations of the 'sandwich matrices' of $ \mathcal{M}$. We then determine the irreducible modular representations of any finite monoid of Lie type.
Some geometric properties on the Fourier and Fourier-Stieltjes algebras of locally compact groups, Arens regularity and related problems
Anthony To Ming
Lau;
Ali
Ülger
321-359
Abstract: Let $G$ be a locally compact topological group and $ A(G)\;[B(G)]$ be, respectively, the Fourier and Fourier-Stieltjes algebras of $G$. It is one of the purposes of this paper to investigate the $ {\text{RNP}}$ (= Radon-Nikodym property) and some other geometric properties such as weak $RNP$, the Dunford-Pettis property and the Schur property on the algebras $A(G)$ and $B(G)$, and to relate these properties to the properties of the multiplication operator on the group $ {C^\ast}$-algebra ${C^\ast}(G)$. We also investigate the problem of Arens regularity of the projective tensor products $ {C^\ast}(G)\hat \otimes A$, when $ B(G) = {C^\ast}{(G)^\ast}$ has the $ {\text{RNP}}$ and $ A$ is any $ {C^\ast}$-algebra. Some related problems on the measure algebra, the group algebra and the algebras ${A_p}(G)$, $P{F_p}(G)$, $P{M_p}(G)\;(1 < p < \infty )$ are also discussed.
The Martin boundary in non-Lipschitz domains
Richard F.
Bass;
Krzysztof
Burdzy
361-378
Abstract: The Martin boundary with respect to the Laplacian and with respect to uniformly elliptic operators in divergence form can be identified with the Euclidean boundary in ${C^\gamma }$ domains, where $\displaystyle \gamma (x) = bx\log \log (1/x)/\log \log \log (1/x),$ $b$ small. A counterexample shows that this result is very nearly sharp.
Characterization of completions of unique factorization domains
Raymond C.
Heitmann
379-387
Abstract: It is shown that a complete local ring is the completion of a unique factorization domain if and only if it is a field, a discrete valuation ring, or it has depth at least two and no element of its prime ring is a zerodivisor. It is also shown that the Normal Chain Conjecture is false and that there exist local noncatenary UFDs.
On the resolution of certain graded algebras
M. P.
Cavaliere;
M. E.
Rossi;
G.
Valla
389-409
Abstract: Let $A = R/I$ be a graded algebra over the polynomial ring $R = k[{X_0}, \ldots ,{X_n}]$. Some properties of the numerical invariants in a minimal free resolution of $A$ are discussed in the case $A$ is a "Short Graded Algebra". When $ A$ is the homogeneous coordinate ring of a set of points in generic position in the projective space, several result are obtained on the line traced by some conjectures proposed by Green and Lazarsfeld in [GL] and Lorenzini in [L1]
Conjugate loci of totally geodesic submanifolds of symmetric spaces
J. M.
Burns
411-425
Abstract: The conjugate and cut loci of fixed point sets of involutions which fix the origin of a compact symmetric space are studied. The first conjugate locus is described in terms of roots and weights of certain representations. When the first conjugate locus and the cut locus agree, we study Morse functions which give a simple decomposition of the symmetric space. We describe for some examples the topological implications of our results.
Topological properties of $q$-convex sets
Guido
Lupacciolu
427-435
Abstract: We discuss the topological properties of a certain class of compact sets in a $q$-complete complex manifold $M$. These sets--which we call $ q$-convex in $ M$--include, for $ q = 0$, the $\mathcal{O}(M)$-convex compact sets in a Stein manifold. Then we show applications of the topological results to the subjects of removable singularities for ${\bar \partial _b}$.
Taylor series with limit-points on a finite number of circles
Emmanuel S.
Katsoprinakis
437-450
Abstract: Let $ S(z):\sum\nolimits_{n = 0}^\infty {{a_n}{z_n}}$ be a power series with complex coefficients. For each $z$ in the unit circle $T = \{ z \in \mathbb{C}:\vert z\vert = 1\}$ we denote by $L(z)$ the set of limit-points of the sequence $\{ {s_n}(z)\} $ of the partial sums of $ S(z)$. In this paper we examine Taylor series for which the set $L(z)$, for $z$ in an infinite subset of $T$, is the union of a finite number, uniformly bounded in $z$, of concentric circles. We show that, if in addition $ \lim \inf \vert{a_n}\vert\; > 0$, a complete characterization of these series in terms of their coefficients is possible (see Theorem 1).
Polynomial identities in graded group rings, restricted Lie algebras and $p$-adic analytic groups
Aner
Shalev
451-462
Abstract: Let $G$ be any finitely generated group, and let $K$ be a field of characteristic $ p > 0$. It is shown that the graded group ring $\operatorname{gr}(KG)$ satisfies a nontrivial polynomial identity if and only if the pro-$p$ completion of $G$ is $p$-adic analytic, i.e. can be given the structure of a Lie group over the $p$-adic field $ {\mathbb{Q}_p}$. The proof applies theorems of Lazard, Quillen and Passman, as well as results on Engel Lie algebras and on dimension subgroups in positive characteristic.
Algebraic cycles and approximation theorems in real algebraic geometry
J.
Bochnak;
W.
Kucharz
463-472
Abstract: Let $M$ be a compact ${C^\infty }$ manifold. A theorem of Nash-Tognoli asserts that $M$ has an algebraic model, that is, $ M$ is diffeomorphic to a nonsingular real algebraic set $X$. Let $ H_{{\text{alg}}}^k(X,\mathbb{Z}/2)$ denote the subgroup of ${H^k}(X,\mathbb{Z}/2)$ of the cohomology classes determined by algebraic cycles of codimension $ k$ on $X$. Assuming that $M$ is connected, orientable and $\dim\,M \geq 5$, we prove in this paper that a subgroup $G$ of $ {H^2}(M,\mathbb{Z}/2)$ is isomorphic to $ H_{{\text{alg}}}^2(X,\mathbb{Z}/2)$ for some algebraic model $X$ of $M$ if and only if ${w_2}(TM)$ is in $G$ and each element of $G$ is of the form $ {w_2}(\xi )$ for some real vector bundle $\xi$ over $M$, where ${w_2}$ stands for the second Stiefel-Whitney class. A result of this type was previously known for subgroups $G$ of $ {H^1}(M,\mathbb{Z}/2)$.
Composition algebras over algebraic curves of genus zero
Holger P.
Petersson
473-493
Abstract: We rephrase the classical theory of composition algebras over fields, particularly the Cayley-Dickson Doubling Process and Zorn's Vector Matrices, in the setting of locally ringed spaces. Fixing an arbitrary base field, we use these constructions to classify composition algebras over (complete smooth) curves of genus zero. Applications are given to composition algebras over function fields of genus zero and polynomial rings.
A selection theorem for topological convex structures
M.
van de Vel
463-496
Abstract: A continuous selection theorem has been obtained for multivalued functions, the values of which are convex sets of certain synthetic convex structures. Applications are given related with superextensions, (semi)lattices, spaces of order arcs, trees, Whitney levels in hyperspaces, and geometric topology. Applications to traditional convexity in vector spaces involve Beer's approximation theorem and a fixed point theorem of Dugundji-Granas. Some other applications (a.o. an invariant arc theorem) appear elsewhere.
Attracting and repelling point pairs for vector fields on manifolds. I
Gabriele
Meyer
497-507
Abstract: Consider a compact, connected, $n$-dimensional, triangulable manifold $ M$ without boundary, embedded in $ {{\mathbf{R}}^{n + 1}}$ and a continuous vector field on $M$, given as a map $f$ from $M$ to ${S^n}$ of degree not equal to 0 or ${( - 1)^{n + 1}}$. In this paper it is shown that there exists at least one pair of points $ x$, $y \in M$ satisfying both $f(x) = - f(y)$ and $f(x) = \frac{{x - y}} {{\vert\vert x - y\vert\vert}}$. Geometrically, this means, that the points and the vectors lie on one straight line and the vector field is "repelling". Similarly, if the degree of $ f$ is not equal to 0 or $ 1$, then there exists at least one "attracting" pair of points $x$, $y \in M$ satisfying both $f(x) = - f(y)$ and $f(x) = \frac{{y - x}} {{\vert\vert y - x\vert\vert}}$. The total multiplicities are $ \frac{{k \bullet (k + {{( - 1)}^n})}} {2}$ for repelling pairs and $\frac{{k \bullet (k - 1)}} {2}$ for attracting pairs. In the proof, we work with close simplicial approximations of the map $f$, using Simplicial, Singular and Čech Homology Theory, Künneth's Theorem, Hopf's Classification Theorem and the algebraic intersection number between two $n$-dimensional homology cycles in a $ 2n$-dimensional space. In the case of repelling pairs, we intersect the graph of $ f$ in $M \times {S^n}$ with the set of points $(x,\frac{{x - y}} {{\vert\vert x - y\vert\vert}}) \in M \times {S^n}$, where $x$ and $y$ satisfy that $ f(x) = - f(y)$. In order to show that this set carries the homology $ (k,k) \in {H_n}(M \times {S^n},{\mathbf{Z}})$, we study the set ${A_f} \equiv \{ (x,y) \in M \times M\vert f(x) = - f(y)\}$ in a simplicial setting. Let ${f_j}$ be a close simplicial approximation of $f$. It can be shown, that ${A_{{f_j}}}$ is a homology cycle of dimension $ n$ with a natural triangulation and a natural orientation and that $ {A_f}$ and ${A_f}_j$ carry the same homology.
The product of two normal initially $\kappa$-compact spaces
Eric K.
van Douwen
509-521
Abstract: We prove that it is independent from $ {\text{ZFC}}$ that for every cardinal $\kappa$ the following statements are equivalent: (a) $\kappa$ is singular; (b) initial $ \kappa$-compactness (defined above the introduction) is productive; (c) initial $\kappa$-compactness is finitely productive; and (d) the product of two initially $\kappa $-compact normal spaces is initially $\kappa$-compact. In particular, MA$$ implies that there are two countably compact normal spaces whose product is not countably compact.
Semilinear evolution equations in Banach spaces with application to parabolic partial differential equations
Samuel M.
Rankin
523-535
Abstract: A theory for a class of semilinear evolution equations in Banach spaces is developed which when applied to certain parabolic partial differential equations with nonlinear terms in divergence form gives strong solutions even for nondifferentiable data.
Higher order commutators for vector-valued Calder\'on-Zygmund operators
Carlos
Segovia;
José L.
Torrea
537-556
Abstract: Weighted norm estimates for higher order commutators are obtained. The proof, that remain valid in the vector-valued case, are obtained as an application of some extrapolation results. The vector-valued version of the commutator theorem is applied to the Carleson operator, U.M.D. Banach spaces, approximate identities and maximal operators.
Cohen-Macaulayness of semi-invariants for tori
Michel
Van den Bergh
557-580
Abstract: In this paper we give a new method, in terms of one-parameter subgroups, to study semi-invariants for algebraic tori. In some cases we obtain extensions to results in [7]. In other cases we obtain different proofs.
Failure of cancellation for direct sums of line bundles
Richard G.
Swan
581-605
Abstract: In answer to a question of Murthy and Wiegand, examples are given of finitely generated projective modules $L$ of rank $1$ over a commutative ring $R$ such that $L \oplus {L^{ - 1}}$ is stably free but not free. Examples are also given of projective modules for which the determinant map det: $\operatorname{Aut}(P) \to {R^{\ast} }$ is not onto. Some related topological questions are also discussed.
The value semigroups of prime divisors of the second kind in $2$-dimensional regular local rings
Sunsook
Noh
607-619
Abstract: In this paper, it is shown that the value semigroup of a prime divisor of the second kind on a $2$-dimensional regular local ring is symmetric. Further, a necessary and sufficient condition for two prime divisors of the second kind on a $ 2$-dimensional regular local ring to have the same value semigroup is obtained.
Mountain impasse theorem and spectrum of semilinear elliptic problems
Kyril
Tintarev
621-629
Abstract: This paper studies a minimax problem for functionals in Hilbert space in the form of $G(u) = \frac{1} {2}\rho \vert\vert u\vert{\vert^2} - g(u)$, where $g(u)$ is Fréchet differentiable with weakly continuous derivative. If $G$ has a "mountain pass geometry" it does not necessarily have a critical point. Such a case is called, in this paper, a "mountain impasse". This paper states that in a case of mountain impasse, there exists a sequence ${u_j} \in H$ such that $\displaystyle g\prime ({u_j}) = {\rho _j}{u_j},\quad {\rho _j} \to \rho ,\vert\vert{u_j}\vert\vert \to \infty ,$ and $G({u_j})$ approximates the minimax value from above. If $\displaystyle \gamma (t) = \mathop {\sup }\limits_{\vert\vert u\vert{\vert^2} = t} \;g(u)$ and $\displaystyle {J_0} = \left( {2\mathop {\inf }\limits_{{t_2} > {t_1} > 0} \frac... ...{t_1} > 0} \frac{{\gamma ({t_2}) - \gamma ({t_1})}} {{{t_2} - {t_1}}}} \right),$ then $ g\prime (u) = \rho u$ has a nonzero solution $u$ for a dense subset of $\rho \in {J_0}$.
On a conjecture of Lin-Ni for a semilinear Neumann problem
Adimurthi;
S. L.
Yadava
631-637
Abstract: Let $\Omega$ be a bounded domain in ${\mathbb{R}^n}$ $ (n \geq 3)$ and $\lambda > 0$. We consider \begin{displaymath}\begin{array}{*{20}{c}} { - \Delta u + \lambda u = {u^{(n + 2... ... }} = 0} & {{\text{on}}} \; {\partial \Omega ,} \end{array} \end{displaymath} and show that for $\lambda$ sufficiently small, the minimal energy solutions are only constants.
A unique continuation property on the boundary for solutions of elliptic equations
Zhi Ren
Jin
639-653
Abstract: We prove the following conclusion: if $u$ is a harmonic function on a smooth domain $ \Omega$ in ${R^n}$ , $n \geq 3$ , or a solution of a general second-order linear elliptic equation on a domain $\Omega$ in ${R^2}$, and if there are ${x_0} \in \partial \Omega $ and constants $ a$, $b > 0$ such that $\vert u(x)\vert \leq a\exp \{ - b/\vert x - {x_0}\vert\}$ for $ x \in \Omega$, $\vert x - {x_0}\vert$ small, then $u = 0$ in $\Omega$ . The decay rate in our results is best possible by the example that $u =$ real part of $\exp \{ - 1/{z^\alpha }\} $ , $0 < \alpha < 1$ , is harmonic but not identically zero in the right complex half-plane.
An indirect method in the calculus of variations
F. H.
Clarke
655-673
Abstract: This article presents a new approach to the issue of the existence of solutions to the basic problem in the calculus of variations. The method is indirect, and applies to certain classes of of problems with slow or no growth, in addition to those satisfying the traditional coercivity condition. The proof hinges upon showing with the help of nonsmooth analysis that a certain value function is constant. Examples are given to illustrate the applicability of the results and the necessity of the hypotheses.
Toeplitz operators and weighted Wiener-Hopf operators, pseudoconvex Reinhardt and tube domains
Norberto
Salinas
675-699
Abstract: The notion of weighted Wiener-Hopf operators is introduced. Their relationship with Toeplitz operators acting on the space of holomorphic functions which are square integrable with respect to a given "symmetric" measure is discussed. The groupoid approach is used in order to present a general program for studying the $ {C^{\ast} }$-algebra generated by weighted Wiener-Hopf operators associated with a solid cone of a second countable locally compact Hausdorff group. This is applied to the case when the group is the dual of a connected locally compact abelian Lie group and the measure is "well behaved" in order to produce a geometric groupoid which is independent of the representation. The notion of a Reinhardt-tube domain $\Omega$ appears thus naturally, and a decomposition series of the corresponding ${C^{\ast} }$-algebra is presented in terms of groupoid ${C^{\ast} }$-algebras associated with various parts of the boundary of the domain $ \Omega$.
Weak solutions of the porous medium equation in a cylinder
Björn E. J.
Dahlberg;
Carlos E.
Kenig
701-709
Abstract: We show that if $D \subset {{\mathbf{R}}^n}$ is a bounded domain with smooth boundary, and $u \in {L^m}(D \times (\varepsilon ,T))$, $u \geq 0$, solves $\frac{{\partial u}} {{\partial t}} = \Delta {u^m}$, $m > 1$, in the sense of distributions on $D \times (0,T)$, and vanishes on $\partial D \times (0,T)$ in a suitable weak sense, then $u$ is Hölder continuous in $\overline D \times (0,T)$.
Weak solutions of the porous medium equation
Björn E. J.
Dahlberg;
Carlos E.
Kenig
711-725
Abstract: We show that if $u \geq 0$, $u \in L_{{\text{loc}}}^m(\Omega )$, $\Omega \subset {{\mathbf{R}}^{n + 1}}$ solves $\partial u/\partial t = \Delta {u^m}$, $m > 1$ , in the sense of distributions, then $u$ is locally Hölder continuous in $\Omega$.
Uniqueness in bounded moment problems
Hans G.
Kellerer
727-757
Abstract: Let $(X,\mathfrak{A},\mu )$ be a $\sigma$-finite measure space and $\mathcal{K}$ be a linear subspace of ${\mathcal{L}_1}(\mu )$ with $\mathcal{K} = X$. The following inverse problem is treated: Which sets $A \in \mathfrak{A}$ are " $ \mathcal{K}$-determined" within the class of all functions $g \in {\mathcal{L}_\infty }(\mu )$ satisfying $ 0 \leq g \leq 1$ , i.e. when is $g = {1_A}$ the unique solution of $ \smallint fg\;d\mu = \smallint f{1_A}\;d\mu$, $f \in \mathcal{K}?$ Recent results of Fishburn et al. and Kemperman show that the condition $A = \{ f \geq 0\}$ for some $f \in \mathcal{K}$ is sufficient but not necessary for uniqueness. To obtain a complete characterization of all $ \mathcal{K}$-determined sets, $\mathcal{K}$ has to be enlarged to some hull $ {\mathcal{K}^{\ast} }$ by extending the usual weak convergence to limits not in $ {\mathcal{L}_1}(\mu )$. Then one of the main results states that $A$ is $ \mathcal{K}$-determined if and only if there is a representation $A = \{ {f^{\ast} } > 0\}$ and $X\backslash A = \{ {f^{\ast} } < 0\}$ for some ${f^{\ast}} \in {\mathcal{K}^{\ast} }$ .
Branched surfaces and attractors. I. Dynamic branched surfaces
Joe
Christy
759-784
Abstract: We show how, using ideas of R. F. Williams about branched surfaces, hyperbolic attractors of flows on three manifolds may be classified up to topological equivalence on an isolating neighborhood by a finite combinatorial object, a swaddled graph.
On injectivity in locally presentable categories
Jiří
Adámek;
Jiří
Rosický
785-804
Abstract: Classes of objects injective w.r.t. specified morphisms are known to be closed under products and retracts. We prove the converse: a class of objects in a locally presentable category is an injectivity class iff it is closed under products and retracts. This result requires a certain large-cardinal principle. We characterize classes of objects injective w.r.t. a small collection of morphisms: they are precisely the accessible subcategories closed under products and $\kappa$-filtered colimits. Assuming the (large-cardinal) Vopênka's principle, the accessibility can be left out. As a corollary, we solve a problem of ${\text{L}}$. Fuchs concerning injectivity classes of abelian groups. Finally, we introduce a weak concept of reflectivity, called cone reflectivity, and we prove that under Vopênka's principle all subcategories of locally presentable categories are cone reflective. Several open questions are formulated, e.g., does each topological space have a largest (non-${T_2}$) compactification?
Supercuspidal representations and the theta correspondence. II. ${\rm SL}(2)$ and the anisotropic ${\rm O}(3)$
David
Manderscheid
805-816
Abstract: A parametrization is given of the local theta correspondence attached to the reductive dual pair $ ({\text{SL}}_2(F),\;{\text{O}}(F))$ where $F$ is a nonarchimedean local field of odd residual characteristic and $ {\text{O}}$ is the orthogonal group of a ternary quadratic form which is anisotropic over $F$. The parametrization is in terms of inducing data. Various lattice models of the oscillator representation are used.
On Lagrange interpolation at disturbed roots of unity
Charles K.
Chui;
Xie Chang
Shen;
Le Fan
Zhong
817-830
Abstract: Let ${z_{nk}} = {e^{i{t_{nk}}}}$, $0 \leq {t_{n0}} < \cdots < {t_{nn}} < 2\pi$, $f$ a function in the disc algebra $A$, and $ {\mu _n} = \max \{ \vert{t_{nk}} - 2k\pi /(n + 1)\vert:0 \leq k \leq n\}$. Denote by ${L_n}(f;\; \cdot )$ the polynomial of degree $n$ that agrees with $f$ at $\{ {z_{nk}}:k = 0, \ldots ,n\}$ . In this paper, we prove that for every $p$, $0 < p < \infty$, there exists a ${\delta _p} > 0$, such that $\vert\vert{L_n}(f;\cdot) - f\vert{\vert _p} = O(\omega (f;\frac{1} {n}))$ whenever ${\mu _n} \leq {\delta _p}/(n + 1)$. It must be emphasized that $ {\delta _p}$ necessarily depends on $p$, in the sense that there exists a family $ \{ {z_{nk}}:k = 0, \ldots ,n\}$ with ${\mu _n} = {\delta _2}/(n + 1)$ and such that $\vert\vert{L_n}(f;\cdot) - f\vert{\vert _2} = O(\omega (f;\frac{1} {n}))$ for all $f \in A$, but $\sup \{ \vert\vert{L_n}(f;\cdot)\vert{\vert _p}:f \in A,\vert\vert f\vert{\vert _\infty } = 1\}$ diverges for sufficiently large values of $p$. In establishing our estimates, we also derive a Marcinkiewicz-Zygmund type inequality for $\{ {z_{nk}}\}$.
Remarks on prescribing Gauss curvature
Xingwang
Xu;
Paul C.
Yang
831-840
Abstract: We study the nonlinear partial differential equation for the problem of prescribing Gauss curvature $K$ on ${S^2}$ . We give an example of a rotationally symmetric $K$ for which the Kazdan-Warner obstruction is satisfied but the equation has no rotationally symmetric solution. On the other hand, we give a simple sufficient condition for solvability of the equation when $K$ is rotationally symmetric. Finally we give a sufficient condition for solvability when $ K$ is not necessarily rotationally symmetric.
$W\sp {2,p}$-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients
Filippo
Chiarenza;
Michele
Frasca;
Placido
Longo
841-853
Abstract: We prove a well-posedness result in the class ${W^{2,p}} \cap W_0^{1,p}$ for the Dirichlet problem $\displaystyle \left\{ {\begin{array}{*{20}{c}} {Lu = f} & {{\text{a.e.}}\;{\tex... ...Omega }, {u = 0} & {{\text{on}}\;\partial \Omega }. \end{array} } \right.$ We assume the coefficients of the elliptic nondivergence form equation that we study are in ${\text{VMO}} \cap {L^\infty }$ .
Unlinking via simultaneous crossing changes
Martin
Scharlemann
855-868
Abstract: Given two distinct crossings of a knot or link projection, we consider the question: Under what conditions can we obtain the unlink by changing both crossings simultaneously? More generally, for which simultaneous twistings at the crossings is the genus reduced? Though several examples show that the answer must be complicated, they also suggest the correct necessary conditions on the twisting numbers.
Weighted norm inequalities for homogeneous singular integrals
Javier
Duoandikoetxea
869-880
Abstract: We prove weighted norm inequalities for homogeneous singular integrals when only a size condition is assumed on the restriction of the kernel to the unit sphere. The same results hold for the operator obtained by modifying the centered Hardy-Littlewood maximal operator over balls with a degree zero homogeneous function and also for the maximal singular integral.
Functional equations satisfied by intertwining operators of reductive groups
Chen-bo
Zhu
881-899
Abstract: This paper generalizes a recent work of Vogan and Wallach [VW] in which they derived a difference equation satisfied by intertwining operators of reductive groups. We show that, associated with each irreducible finite-dimensional representation, there is a functional equation relating intertwining operators. In this way, we obtain natural relations between intertwining operators for different series of induced representations.
Groups of dualities
Georgi D.
Dimov;
Walter
Tholen
901-913
Abstract: For arbitrary categories $ \mathcal{A}$ and $\mathcal{B}$ , the "set" of isomorphism-classes of dualities between $ \mathcal{A}$ and $\mathcal{B}$ carries a natural group structure. In case $\mathcal{A}$ and $\mathcal{B}$ admit faithful representable functors to Set, this structure can often be described quite concretely in terms of "schizophrenic objects" (in the sense of Johnstone's book on "Stone Spaces"). The general theory provided here allows for a concrete computation of that group in case $ \mathcal{A} = \mathcal{B} = \mathcal{C}$ is the category of all compact and all discrete abelian groups: it is the uncountable group of algebraic automorphisms of the circle $\mathbb{R}/\mathbb{Z}$ , modulo its subgroup ${\mathbb{Z}_2}$ of continuous automorphisms.
Small solutions to inhomogeneous linear equations over number fields
Robbin
O’Leary;
Jeffrey D.
Vaaler
915-931
Abstract: We consider a system of $M$ independent, inhomogeneous linear equations in $N > M$ variables having coefficients in an algebraic number field $k$ . We give a best possible lower bound on the inhomogeneous height of a solution vector in $ {k^N}$ and determine when a solution exists in ${({\mathcal{O}_S})^N}$, where ${\mathcal{O}_S}$ is the ring of $S$-integers in $k$ . If such a system has a solution vector in $ {({\mathcal{O}_S})^N}$, we show that it has a solution $ \vec \zeta$ in ${({\mathcal{O}_S})^N}$ such that the inhomogeneous height of $\vec \zeta$ is relatively small. We give an explicit upper bound for this height in terms of the heights of the matrices defining the linear system. Our method uses geometry of numbers over adele spaces and local to global arguments.
Algebraic cycles and the Hodge structure of a Kuga fiber variety
B. Brent
Gordon
933-947
Abstract: Let $\tilde A$ denote a smooth compactification of the $k$-fold fiber product of the universal family ${A^1} \to M$ of elliptic curves with level $ N$ structure. The purpose of this paper is to completely describe the algebraic cycles in and the Hodge structure of the Betti cohomology ${H^{\ast} }(\tilde A,\mathbb{Q})$ of $ \tilde A$ , for by doing so we are able (a) to verify both the usual and generalized Hodge conjectures for $\tilde A$ ; (b) to describe both the kernel and the image of the Abel-Jacobi map from algebraic cycles algebraically equivalent to zero (modulo rational equivalence) into the Griffiths intermediate Jacobian; and (c) to verify Tate's conjecture concerning the algebraic cycles in the étale cohomology $H_{{\text{et}}}^{\ast} (\tilde A \otimes \bar{\mathbb{Q}},{\mathbb{Q}_l})$. The methods used lead also to a complete description of the Hodge structure of the Betti cohomology ${H^{\ast} }({E^k},\mathbb{Q})$ of the $k$-fold product of an elliptic curve $ E$ without complex multiplication, and a verification of the generalized Hodge conjecture for ${E^k}$ .
Detecting algebraic (in)dependence of explicitly presented functions (some applications of Nevanlinna theory to mathematical logic)
R. H.
Gurevič
1-67
Abstract: We consider algebraic relations between explicitly presented analytic functions with particular emphasis on Tarski's high school algebra problem. The part not related directly to Tarski's high school algebra problem. Let $ U$ be a connected complex-analytic manifold. Denote by $\mathcal{F}(U)$ the minimal field containing all functions meromorphic on $U$ and closed under exponentiation $f \mapsto {e^f}$. Let ${f_j} \in \mathcal{F}(U)$, ${p_j} \in \mathcal{M}(U) - \{ 0\}$ for $1 \leq j \leq m$, and ${g_k} \in \mathcal{F}(U)$, ${q_k} \in \mathcal{M}(U) - \{ 0\}$ for $1 \leq k \leq n$ (where $\mathcal{M}(U)$ is the field of functions meromorphic on $U$). Let ${f_i} - {f_j} \notin \mathcal{H}(U)$ for $ i \ne j$ and $ {g_k} - {g_l} \notin \mathcal{H}(U)$ for $k \ne 1$ (where $ \mathcal{H}(U)$ is the ring of functions holomorphic on $U$). If all zeros and singularities of $\displaystyle h = \frac{{\sum\nolimits_{j = 1}^m {{p_j}{e^{{f_j}}}} }} {{\sum\nolimits_{k = 1}^n {{q_k}{e^{{g_k}}}} }}$ are contained in an analytic subset of $ U$ then $m = n$ and there exists a permutation $ \sigma$ of $\{ 1, \ldots ,m\}$ such that $h = ({p_j}/{q_{\sigma (j)}}) \cdot {e^{{f_j} - {g_{\sigma (j)}}}}$ for $ 1 \leq j \leq m$. When $ h \in \mathcal{M}(U)$, additionally ${f_j} - {g_{\sigma (j)}} \in \mathcal{H}(U)$ for all $j$ . On Tarski's high school algebra problem. Consider $L = \{$-terms in variables and $1$, $+$, $\cdot$, $ \uparrow \}$ , where $\uparrow :a$, $ b \mapsto {a^b}$ for positive $a$, $b$. Each term $t \in L$ naturally determines a function $ \bar t$ : $ {({{\mathbf{R}}_ + })^n} \to {{\mathbf{R}}_ - }$ , where $n$ is the number of variables involved. For $S \subset L$ put $\bar S = \{ \bar t\vert t \in S\}$ . (i) We describe the algebraic structure of $\bar \Lambda$ and $ \bar{\mathcal{L}}$ , where $\Lambda = \{ t \in L\vert$ if $u \uparrow v$ occurs as a subterm of $t$ then either $u$ is a variable or $u$ contains no variables at all, and $\mathcal{L} = \{ t \in L\vert$ if $u \uparrow v$ occurs as a subterm of $t$ then $ u \in \Lambda \}$. Of these, $\bar \Lambda $ is a free semiring with respect to addition and multiplication but $\bar{\mathcal{L}}$ is free only as a semigroup with respect to addition. A function $\bar t \in \bar S$ is called $+$-prime in $\bar S$ if $\bar t \ne \bar u + \bar v$ for all $u$, $v \in S$ and is called multiplicatively prime in $ \bar S$ if $ \bar t = \bar u \cdot \bar v \Rightarrow \bar u = 1$ or $\bar v = 1$ for $u$, $v \in S$. A function is called $( + , \cdot )$-prime in $\bar S$ if it is both $+$-prime and multiplicatively prime in $ \bar S$. A function in $\bar \Lambda$ is said to have content $ 1$ if it is not divisible by constants in $ {\mathbf{N}} - \{ 1\}$ or by $\ne 1 ( + , \cdot )$-primes of $\bar \Lambda$ . The product of functions of content $1$ has content $1$ . Let $P$ be the multiplicative subsemigroup of $\bar \Lambda$ of functions of content $ 1$ . Then $\bar{\mathcal{L}}$ as a semiring is isomorphic to the semigroup semiring $\bar \Lambda ({ \oplus _f}{P_f})$, where each $ {P_f}$ is a copy of $ P$ and $f$ ranges over the $\ne 1\; + $-primes of $\bar{\mathcal{L}}$. (ii) We prove that if $ t$, $u \in \mathcal{L}$ and ${{\mathbf{R}}_ + } \vdash t = u$ (i.e., if $\bar t = \bar u$) then Tarski's "high school algebra" identities $ \vdash t = u$. This result covers a conjecture of C. W. Henson and L. A. Rubel. (Note: this result does not generalize to arbitrary $ t$, $u \in L$ . Moreover, the equational theory of $({{\mathbf{R}}_ + };\;1, + , \cdot , \uparrow )$ is not finitely axiomatizable.
Volumes of vector fields on spheres
Sharon L.
Pedersen
69-78
Abstract: In this paper we study the problem: What is the unit vector field of smallest volume on an odd-dimensional sphere? We exhibit on each sphere a unit vector field with singularity which has exceptionally small volume on spheres of dimension greater than four. We conjecture that this volume is the infimum for volumes of bona fide unit vector fields, and is only achieved by the singular vector field. We generalize the construction of the singular vector field to give a family of cycles in Stiefel manifolds, each of which is a smooth manifold except for one singular point. Except for some low-dimensional cases, the tangent cones at these singular points are volume-minimizing; and half of the cones are nonorientable. Thus, we obtain a new family of nonorientable volume-minimizing cones.
Reye constructions for nodal Enriques surfaces
A.
Conte;
A.
Verra
79-100
Abstract: A classical Reye congruence $X$ is an Enriques surface of rational equivalence class $(3,7)$ in the grassmannian $G(1,3)$ of lines of ${{\mathbf{P}}^3}$. $X$ is the locus of lines of ${{\mathbf{P}}^3}$ which are included in two quadrics of $W=$ web of quadrics. A generalization to $ G(1,t)$ is given (1) for each $t > 2$ there exist Enriques surfaces $ X$ of class $(t,3t - 2)$ in $G(1,t)$, (2) the determinant of the dual of the universal bundle on $X$ is ${\mathcal{O}_X}(2E + R + {K_X})$, with $ E=$ isolated elliptic curve, ${R^2} = - 2$, $E \cdot R = t$, (3) $X$ parameterizes lines of ${{\mathbf{P}}^t}$ which are included in a codimension $2$ subsystem of $W$, $W=$ linear system of quadrics of dimension $\left( \begin{array}{*{20}{c}} t 2 \end{array} \right)$. The paper includes a description of the variety of trisecant lines to a smooth Enriques surface of degree $10$ in $ {{\mathbf{P}}^5}$ .
Nonstandard topology on function spaces with applications to hyperspaces
Hermann
Render
101-119
Abstract: In this paper the techniques of Nonstandard Analysis are used to study topologies on the set of all continuous functions. We obtain nonstandard characterizations for conjoining and splitting topologies and we give a complete description of the monads of the compact-open topology which leads to very elegant and simple proofs of some important results. For example we prove a generalized Ascoli Theorem where the image space is only Hausdorff or regular. Then we apply our results to the hyperspace and solve questions of Arens and Dugundji, Wattenberg and Topsøe. Finally we discuss real compact spaces and the continuity of the diagonal function.
Algebraic shift equivalence and primitive matrices
Mike
Boyle;
David
Handelman
121-149
Abstract: Motivated by symbolic dynamics, we study the problem, given a unital subring $S$ of the reals, when is a matrix $A$ algebraically shift equivalent over $S$ to a primitive matrix? We conjecture that simple necessary conditions on the nonzero spectrum of $ A$ are sufficient, and establish the conjecture in many cases. If $ S$ is the integers, we give some lower bounds on sizes of realizing primitive matrices. For Dedekind domains, we prove that algebraic shift equivalence implies algebraic strong shift equivalence.
Zero-equivalence in function fields defined by algebraic differential equations
John
Shackell
151-171
Abstract: We consider function fields obtained as towers over the field of rational functions, each extension being by a solution of an algebraic differential equation. On the assumption that an oracle exists for the constants, we present two algorithms for determining whether a given expression is functionally equivalent to zero in such a field. The first, which uses Gröbner bases, has the advantage of theoretical simplicity, but is liable to involve unnecessary computations. The second method is designed with a view to eliminating these.
Geometry of weight diagrams for ${\rm U}(n)$
Eng-Chye
Tan
173-192
Abstract: We study the geometry of the weight diagrams for irreducible representations of $U(n)$. Multiplicity-one weights are shown to have nice geometric characterizations. We then apply our results to study multiplicity-one $ K$-types of principal representations of $U(n,n)$.
Countable closed ${\rm LFC}$-groups with $p$-torsion
Felix
Leinen
193-217
Abstract: Let $LFC$ be the class of all locally $ FC$-groups. We study the existentially closed groups in the class $LF{C_p}$ of all $LFC$-groups $H$ whose torsion subgroup $T(H)$ is a $p$-group. Differently from the situation in $ LFC$, every existentially closed $LF{C_p}$-group is already closed in $ LF{C_p}$, and there exist ${2^{{\aleph _0}}}$ countable closed $ LF{C_P}$-groups $ G$. However, in the countable case, $T(G)$ is up to isomorphism always a unique locally finite $p$-group with similar properties as the unique countable existentially closed locally finite $ p$-group ${E_p}$.
Conformal metrics with prescribed Gaussian curvature on $S\sp 2$
Kuo-Shung
Cheng;
Joel A.
Smoller
219-251
Abstract: We consider on $ {S^2}$ the problem of which functions $K$ can be the scalar curvature of a metric conformal to the standard metric on ${S^2}$. We assume that $K$ is a function of one variable, and we obtain a necessary and sufficient condition for the problem to be solvable. We also obtain several new sufficient conditions on $k$ (which are easy to check), in order that the problem be solvable.
${\rm GL}(4,{\bf R})$-Whittaker functions and ${}\sb 4F\sb 3(1)$ hypergeometric series
Eric
Stade
253-264
Abstract: In this paper we consider spaces of $ {\text{GL}}(4,\mathbb{R})$-Whittaker functions, which are special functions that arise in the study of $ {\text{GL}}(4,\mathbb{R})$ automorphic forms. Our main result is to determine explicitly the series expansion for a $ {\text{GL}}(4,\mathbb{R})$-Whittaker function that is "fundamental," in that it may be used to generate a basis for the space of all $ {\text{GL}}(4,\mathbb{R})$-Whittaker functions of fixed eigenvalues. The series that we find in the case of $ {\text{GL}}(4,\mathbb{R})$ is particularly interesting in that its coefficients are not merely ratios of Gamma functions, as they are in the lower-rank cases. Rather, these coefficients are themselves certain series-- namely, they are finite hypergeometric series of unit argument. We suspect that this is a fair indication of what will happen in the general case of $ {\text{GL}}(n,\mathbb{R})$.
On Klein's combination theorem. IV
Bernard
Maskit
265-294
Abstract: This paper contains an expansion of the combination theorems to cover the following problems. New rank $1$ parabolic subgroups are produced, while, as in previous versions, all elliptic and parabolic elements are tracked. A proof is given that the combined group is analytically finite if and only if the original groups are; in the analytically finite case, we also give a formula for the hyperbolic area of the combined group (i.e., the hyperbolic area of the set of discontinuity on the $2$-sphere modulo $G$) in terms of the hyperbolic areas of the original groups. There is also a new variation on the first combination theorem in which the common subgroup has finite index in one of the two groups.
Artinian subrings of a commutative ring
Robert
Gilmer;
William
Heinzer
295-310
Abstract: Given a commutative ring $R$, we investigate the structure of the set of Artinian subrings of $R$. We also consider the family of zero-dimensional subrings of $R$. Necessary and sufficient conditions are given in order that every zero-dimensional subring of a ring be Artinian. We also consider closure properties of the set of Artinian subrings of a ring with respect to intersection or finite intersection, and the condition that the set of Artinian subrings of a ring forms a directed family.
Decidable discriminator varieties from unary classes
Ross
Willard
311-333
Abstract: Let $\mathcal{K}$ be a class of (universal) algebras of fixed type. $ {\mathcal{K}^t}$ denotes the class obtained by augmenting each member of $\mathcal{K}$ by the ternary discriminator function $(t(x,y,z) = x$ if $x \ne y,t(x,x,z) = z)$, while $\vee ({\mathcal{K}^t})$ is the closure of ${\mathcal{K}^t}$ under the formation of subalgebras, homomorphic images, and arbitrary Cartesian products. For example, the class of Boolean algebras is definitionally equivalent to $\vee ({\mathcal{K}^t})$ where $\mathcal{K}$ consists of a two-element algebra whose only operations are the two constants. Any equationally defined class (that is, variety) of algebras which is equivalent to some $\vee ({\mathcal{K}^t})$ is known as a discriminator variety. Building on recent work of S. Burris, R. McKenzie, and M. Valeriote, we characterize those locally finite universal classes $ \mathcal{K}$ of unary algebras of finite type for which the first-order theory of $ \vee ({\mathcal{K}^t})$ is decidable.
Self-similar measures and their Fourier transforms. II
Robert S.
Strichartz
335-361
Abstract: A self-similar measure on $ {{\mathbf{R}}^n}$ was defined by Hutchinson to be a probability measure satisfying $({\ast})$ $\displaystyle \mu = \sum\limits_{j = 1}^m {{a_j}\mu \circ S_j^{ - 1}}$ , where $ {S_j}x = {\rho _j}{R_j}x + {b_j}$ is a contractive similarity $(0 < {\rho _j} < 1,{R_j}$ orthogonal) and the weights ${a_j}$ satisfy $0 < {a_j} < 1,\sum\nolimits_{j = 1}^m {{a_j} = 1}$. By analogy, we define a self-similar distribution by the same identity $( {\ast} )$ but allowing the weights ${a_j}$ to be arbitrary complex numbers. We give necessary and sufficient conditions for the existence of a solution to $ ( {\ast} )$ among distributions of compact support, and show that the space of such solutions is always finite dimensional. If $ F$ denotes the Fourier transformation of a self-similar distribution of compact support, let $\displaystyle H(R) = \frac{1}{{{R^{n - \beta }}}}\int_{\vert x\vert \leq R} {\vert F(x){\vert^2}dx,}$ where $\beta$ is defined by the equation $\sum\nolimits_{j = 1}^m {\rho _j^{ - \beta }\vert{a_j}{\vert^2} = 1}$. If $\rho _j^{{\nu _j}} = \rho $ for some fixed $ \rho$ and ${\nu _j}$ positive integers we say the $\{ {\rho _j}\} $ are exponentially commensurable. In this case we prove (under some additional hypotheses) that $H(R)$ is asymptotic (in a suitable sense) to a bounded function $ \tilde H(R)$ that is bounded away from zero and periodic in the sense that $ \tilde H(\rho R) = \tilde H(R)$ for all $R > 0$. If the $ \{ {\rho _j}\}$ are exponentially incommensurable then ${\lim _{R \to \infty }}H(R)$ exists and is nonzero.
Rational approximations to the dilogarithm
Masayoshi
Hata
363-387
Abstract: The irrationality proof of the values of the dilogarithmic function $ {L_2}(z)$ at rational points $z = 1/k$ for every integer $k \in ( - \infty , - 5] \cup [7,\infty )$ is given. To show this we develop the method of Padé-type approximations using Legendre-type polynomials, which also derives good irrationality measures of $ {L_2}(1/k)$. Moreover, the linear independence over $ {\mathbf{Q}}$ of the numbers $1$, $\log (1 - 1/k)$, and ${L_2}(1/k)$ is also obtained for each integer $ k \in ( - \infty , - 5] \cup [7,\infty )$ .
An extension theorem for closing maps of shifts of finite type
Jonathan
Ashley
389-420
Abstract: If there exists some right-closing factor map $\pi :{\Sigma _A} \to {\Sigma _B}$ between aperiodic shifts of finite type, then any right-closing map $ \varphi :X \to {\Sigma _B}$ from any shift of finite type $X$ contained in ${\Sigma _A}$ can be extended to a right-closing factor map from all of $ {\Sigma _A}$ onto ${\Sigma _B}$. We prove this and give some consequences.
Hyperbolic structures for surfaces of infinite type
Ara
Basmajian
421-444
Abstract: Our main objective is to understand the geometry of hyperbolic structures on surfaces of infinite type. In particular, we investigate the properties of surfaces called flute spaces which are constructed from infinite sequences of "pairs of pants," each glued to the next along a common boundary geodesic. Necessary and sufficient conditions are supplied for a flute space to be constructed using only "tight pants," along with sufficient conditions on when the hyperbolic structure is complete. An infinite version of the Klein-Maskit combination theorem is derived. Finally, using the above constructions a number of applications to the deformation theory of infinite type hyperbolic surfaces are examined.
Witt's extension theorem for mod four valued quadratic forms
Jay A.
Wood
445-461
Abstract: The $\bmod \,4$ valued quadratic forms defined by E. H. Brown, Jr. are studied. A classification theorem is proven which states that these forms are determined by two things: whether or not their associated bilinear form is alternating, and the $\sigma $-invariant of Brown (which generalizes the Arf invariant of an ordinary quadratic form). Particular attention is paid to a generalization of Witt's extension theorem for quadratic forms. Some applications to selforthogonal codes are sketched, and an exposition of some unpublished work of E. Prange on Witt's theorem is provided in an appendix.
A composite coincidence degree with applications to boundary value problems of neutral equations
L. H.
Erbe;
W.
Krawcewicz;
J. H.
Wu
459-478
Abstract: We present a topological degree theory for the nonlinear problem $L(I - B)(x) = G(x)$ with applications to a class of boundary value problems of neutral equations, where $ L$ is an unbounded Fredholm operator of index zero, $B$ is condensing and $G$ is $L$-compact.
$7$-dimensional nilpotent Lie algebras
Craig
Seeley
479-496
Abstract: All $ 7$-dimensional nilpotent Lie algebras over $ \mathbb{C}$ are determined by elementary methods. A multiplication table is given for each isomorphism class. Distinguishing features are given, proving that the algebras are pairwise nonisomorphic. Moduli are given for the infinite families which are indexed by the value of a complex parameter.
Bass series of local ring homomorphisms of finite flat dimension
Luchezar L.
Avramov;
Hans-Bjørn
Foxby;
Jack
Lescot
497-523
Abstract: Nontrivial relations between Bass numbers of local commutative rings are established in case there exists a local homomorphism $\phi :R \to S$ which makes $S$ into an $R$-module of finite flat dimension. In particular, it is shown that an inequality $ \mu _R^{i + {\text{depth}}\;R} \leq \mu _s^{i + {\text{depth}}\;S}$ holds for all $i \in \mathbb{Z}$. This is a consequence of an equality involving the Bass series $I_R^M(t) = \sum\nolimits_{i \in \mathbb{Z}} {\mu _R^i(M){t^i}}$ of a complex $M$ of $R$-modules which has bounded above and finite type homology and the Bass series of the complex of $ S$-modules $ M{\underline{\underline \otimes } _R}S$, where $ \underline{\underline{\otimes}}$ denotes the derived tensor product. It is proved that there is an equality of formal Laurent series $ I_s^{M{{\underline{\underline \otimes } }_R}S}(t) = I_R^M(t){I_{F(\phi )}}(t)$, where $F(\phi)$ is the fiber of $\phi$ considered as a homomorphism of commutative differential graded rings. Coefficientwise inequalities are deduced for $ I_S^{M{{\underline{\underline \otimes } }_R}S}(t)$, and Golod homomorphisms are characterized by one of them becoming an equality.
Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings
Matej
Brešar
525-546
Abstract: Biadditive mappings $B: R \times R \to R$ where $ R$ is a prime ring with certain additional properties, satisfying $B(x,x)x = xB(x,x)$ for all $x \in R$, are characterized. As an application we determine the structures of commutativity-preserving mappings, Lie isomorphisms, and Lie derivations of certain prime rings.
The Martin kernel and infima of positive harmonic functions
Zoran
Vondraček
547-557
Abstract: Let $D$ be a bounded Lipschitz domain in $ {{\mathbf{R}}^n}$ and let $ K(x,z)$, $x \in D$, $z \in \partial D$, be the Martin kernel based at $ {x_0} \in D$. For $ x,y \in D$, let $ k(x,y) = \inf \{ h(x):h\;$positive$\;$harmonic$\;$in$\; D, h(y) = 1\}$. We show that the function $ k$ completely determines the family of positive harmonic functions on $ D$. Precisely, for every $z \in \partial D$, ${\lim _{y \to z}}k(x,y)/k({x_0},y) = K(x,z)$. The same result is true for second-order uniformly elliptic operators and Schrödinger operators.
On the Brauer group of toric varieties
Frank R.
DeMeyer;
Timothy J.
Ford
559-577
Abstract: We compute the cohomological Brauer group of a normal toric variety whose singular locus has codimension less than or equal to $2$ everywhere.
Rosenlicht fields
John
Shackell
579-595
Abstract: Let $\phi$ satisfy an algebraic differential equation over $ {\mathbf{R}}$. We show that if $\phi$ also belongs to a Hardy field, it possesses an asymptotic form which must be one of a restricted number of types. The types depend only on the order of the differential equation. For a particular equation the types are still more restricted. In some cases one can conclude that no solution of the given equation lies in a Hardy field, and in others that a particular asymptotic form is the only possibility for such solutions. This therefore gives a new method for obtaining asymptotic solutions of nonlinear differential equations. The techniques used are in part derived from the work of Rosenlicht in Hardy fields.
Excluding infinite trees
P. D.
Seymour;
Robin
Thomas
597-630
Abstract: For each infinite cardinal $\kappa$ we give several necessary and sufficient conditions for a graph not to contain a minor isomorphic to the infinite $\kappa$-branching tree in terms of a certain kind of a "tree-decomposition," in terms of a "path-decomposition," and also in terms of a "cops-and-robber game." We also give necessary and sufficient conditions for a graph not to contain a subgraph isomorphic to a subdivision of the same tree.
Spectral symmetry of the Dirac operator in the presence of a group action
H. D.
Fegan;
B.
Steer
631-647
Abstract: Let $G$ be a compact Lie group of rank two or greater which acts on a spin manifold $M$ of dimension $4k + 3$ through isometries with finite isotropy subgroups at each point. Define the Dirac operator, $P$, on $M$ relative to the split connection. Then we show that $P$ has spectral $G$-symmetry. This is first established in a number of special cases which are both of interest in their own right and necessary to establish the more general case. Finally we consider changing the connection and show that for the Levi-Civita connection the equivariant eta function evaluated at zero is constant on $G$.
Fractal dimensions and singularities of the Weierstrass type functions
Tian You
Hu;
Ka-Sing
Lau
649-665
Abstract: A new type of fractal measures $ {\mathcal{K}^s}$, $1 \leq s \leq 2$, defined on the subsets of the graph of a continuous function is introduced. The $\mathcal{K}$-dimension defined by this measure is 'closer' to the Hausdorff dimension than the other fractal dimensions in recent literatures. For the Weierstrass type functions defined by $W(x) = \sum\nolimits_0^\infty {{\lambda ^{ - \alpha i}}g({\lambda ^i}x)}$, where $\lambda > 1$, $0 < \alpha < 1$, and $g$ is an almost periodic Lipschitz function of order greater than $\alpha$, it is shown that the $\mathcal{K}$-dimension of the graph of $ W$ equals to $2 - \alpha$, this conclusion is also equivalent to certain rate of the local oscillation of the function. Some problems on the 'knot' points and the nondifferentiability of $W$ are also discussed.
Estimates for some Kakeya-type maximal operators
Jose
Barrionuevo
667-682
Abstract: We use an abstract version of a theorem of Kolmogorov-Seliverstov-Paley to obtain sharp ${L^2}$ estimates for maximal operators of the form: $\displaystyle {\mathcal{M}_\mathcal{B}}f(x) = \mathop {\sup }\limits_{x \in S \in \mathcal{B}} \frac{1}{{\vert S\vert}}\int_S {\vert f(x - y)\vert dy}$ . We consider the cases where $ \mathcal{B}$ is the class of all rectangles in $ {{\mathbf{R}}^n}$ congruent to some dilate of ${[0,1]^{n - 1}} \times [0,{N^{ - 1}}]$; the class congruent to dilates of ${[0,{N^{ - 1}}]^{n - 1}} \times [0,1]$ ; and, in $ {{\mathbf{R}}^2}$ , the class of all rectangles with longest side parallel to a particular countable set of directions that include the lacunary and the uniformly distributed cases.
The Mandelbrot set and $\sigma$-automorphisms of quotients of the shift
Pau
Atela
683-703
Abstract: In this paper we study how certain loops in the parameter space of quadratic complex polynomials give rise to shift-automorphisms of quotients of the set $ {\Sigma _2}$ of sequences on two symbols. The Mandelbrot set ${\mathbf{M}}$ is the set of parameter values for which the Julia set of the corresponding polynomial is connected. Blanchard, Devaney, and Keen have shown that closed loops in the complement of the Mandelbrot set give rise to shift-automorphisms of ${\Sigma _2}$ , i.e., homeomorphisms of ${\Sigma _2}$ that commute with the shift map. We study what happens when the loops are not entirely in the complement of the Mandelbrot set. We consider closed loops that cross the Mandelbrot set at a single main bifurcation point, surrounding a component of ${\mathbf{M}}$ attached to the main cardioid. If $ n$ is the period of this component, we identify a period- $n$ orbit of $ {\Sigma _2}$ to a single point. The loop determines a shift-automorphism of this quotient space of $ {\Sigma _2}$ . We give these maps explicitly.
Piecewise ${\rm SL}\sb 2{\bf Z}$ geometry
Peter
Greenberg
705-720
Abstract: Piecewise $ {\text{SL}}_2{\mathbf{Z}}$ geometry studies properties of the plane invariant under $ {\text{pl}}$-homeomorphisms which, locally, have the form $x \mapsto Ax + b$ , with $A \in {\text{SL}}_2{\mathbf{Z}}$, $b \in {{\mathbf{Q}}^2}$ , and whose singular lines are rational. In this paper, invariants of polygons are obtained, relations with Pick's theorem are described, and a conjecture is posed.
Harmonic calculus on p.c.f. self-similar sets
Jun
Kigami
721-755
Abstract: The main object of this paper is the Laplace operator on a class of fractals. First, we establish the concept of the renormalization of difference operators on post critically finite (p.c.f. for short) self-similar sets, which are large enough to include finitely ramified self-similar sets, and extend the results for Sierpinski gasket given in [10] to this class. Under each invariant operator for renormalization, the Laplace operator, Green function, Dirichlet form, and Neumann derivatives are explicitly constructed as the natural limits of those on finite pre-self-similar sets which approximate the p.c.f. self-similar sets. Also harmonic functions are shown to be finite dimensional, and they are characterized by the solution of an infinite system of finite difference equations.
Modular forms of weight $\frac12$ defined on products of $p$-adic upper half-planes
Anne
Schwartz
757-773
Abstract: We continue Stark's study of modular forms defined on products of $ p$-adic upper half-planes. Specifically, we restrict to the case of the number field $\mathbb{Q}$ and one finite prime. In this setting we develop a multiplier system for modular forms of weight $ \frac{1} {2}$, and provide an example of such a form.
Pseudocompact and countably compact abelian groups: Cartesian products and minimality
Dikran N.
Dikranjan;
Dmitrii B.
Shakhmatov
775-790
Abstract: Denote by $\mathcal{G}$ the class of all Abelian Hausdorff topological groups. A group $G \in \mathcal{G}$ is minimal (totally minimal) if every continuous group isomorphism (homomorphism) $i:G \to H$ of $G$ onto $H \in \mathcal{G}$ is open. For $G \in \mathcal{G}$ let $\kappa (G)$ be the smallest cardinal $\tau \geq 1$ such that the minimality of $ {G^\tau }$ implies the minimality of all powers of $G$. For $\mathcal{Q} \subset \mathcal{G}$, $\mathcal{Q} \ne \emptyset$, we set $\kappa (\mathcal{Q}) = \sup \{ \kappa (G):G \in \mathcal{G}\}$ and denote by $\alpha (\mathcal{Q})$ the smallest cardinal $\tau \geq 1$ having the following property: If $ \{ {G_i}:i \in I\} \subset \mathcal{Q}$, $ I \ne \emptyset$, and each subproduct $\prod {\{ {G_i}:i \in J\} }$, with $J \subset I$, $ J \ne \emptyset$, and $\vert J\vert \leq \tau $, is minimal, then the whole product $ \prod {\{ {G_i}:i \in I\} }$ is minimal. These definitions are correct, and $ \kappa (G) \leq {2^\omega }$ and $\kappa (\mathcal{Q}) \leq \alpha (\mathcal{Q}) \leq {2^\omega }$ for all $G \in \mathcal{G}$ and any $\mathcal{Q} \subset \mathcal{G}$, $\mathcal{Q} \ne \emptyset$, while it can happen that $\kappa (\mathcal{Q}) < \alpha (\mathcal{Q})$ for some $ \mathcal{Q} \subset \mathcal{G}$. Let $\mathcal{C} = \{ G \in \mathcal{G}:G\;{\text{is}}\;{\text{countably}}\;{\text{compact}}\} $ and $\mathcal{P} = \{ G \in \mathcal{G}:G\;{\text{is}}\;{\text{pseudocompact}}\} $. If $G \in \mathcal{C}$ is minimal, then $G \times H$ is minimal for each minimal (not necessarily Abelian) group $H$; in particular, ${G^n}$ is minimal for every natural number $ n$. We show that $\alpha (\mathcal{C}) = \omega$, and so either $\kappa (\mathcal{C}) = 1$ or $\kappa (\mathcal{C}) = \omega$. Under Lusin's Hypothesis ${2^{{\omega _1}}} = {2^\omega }$ we construct $\{ {G_n}:n \in \mathbb{N}\} \subset \mathcal{P}$ and $ H \in \mathcal{P}$ such that: (i) whenever $ n \in \mathbb{N}$, $ G_n^n$ is totally minimal, but $G_n^{n + 1}$ is not even minimal, so $\kappa ({G_n}) = n + 1$; and (ii) $ {H^n}$ is totally minimal for each natural number $n$, but $ {H^\omega}$ is not even minimal, so $ \kappa (H) = \omega$. Under $ {\text{MA}} + \neg {\text{CH}}$, conjunction of Martin's Axiom with the negation of the Continuum Hypothesis, we construct $G \in \mathcal{P}$ such that $ {G^\tau }$ is totally minimal for each $ \tau < {2^\omega }$, while ${G^{{2^\omega }}}$ is not minimal, so $ \kappa (G) = {2^\omega }$. This yields $\alpha (\mathcal{P}) = \kappa (\mathcal{P}) = {2^\omega }$ under ${\text{MA}} + \neg {\text{CH}}$. We also present an example of a noncompact minimal group $G \in \mathcal{C}$, which should be compared with the following result obtained by the authors quite recently: Totally minimal groups $G \in \mathcal{C}$ are compact.
The Lagrangian Gauss image of a surface in Euclidean $3$-space
Marek
Kossowski
791-803
Abstract: We describe a correspondence between special nonimmersed surfaces in Euclidean $3$-space and exact Lagrangian immersions in the cotangent bundle of the unit sphere. This provides several invariants for such nonimmersive maps: the degree of the Gauss map, the Gauss-Maslov class, and the polarization index. The objectives of this paper are to compare these invariants in the cases where the underlying map immerses or fails to immerse and to describe the extend to which these invariants can be prescribed.
Weak topologies for the closed subsets of a metrizable space
Gerald
Beer;
Roberto
Lucchetti
805-822
Abstract: The purpose of this article is to propose a unified theory for topologies on the closed subsets of a metrizable space. It can be shown that all of the standard hyperspace topologies--including the Hausdorff metric topology, the Vietoris topology, the Attouch-Wets topology, the Fell topology, the locally finite topology, and the topology of Mosco convergence--arise as weak topologies generated by families of geometric functionals defined on closed sets. A key ingredient is the simple yet beautiful interplay between topologies determined by families of gap functionals and those determined by families of Hausdorff excess functionals.
The geometric structure of skew lattices
Jonathan
Leech
823-842
Abstract: A skew lattice is a noncommutative associative analogue of a lattice and as such may be viewed both as an algebraic object and as a geometric object. Whereas recent papers on skew lattices primarily treated algebraic aspects of skew lattices, this article investigates their intrinsic geometry. This geometry is obtained by considering how the coset geometries of the maximal primitive subalgebras combine to form a global geometry on the skew lattice. While this geometry is derived from the algebraic operations, it can be given a description that is independent of these operations, but which in turn induces them. Various aspects of this geometry are investigated including: its general properties; algebraic and numerical consequences of these properties; connectedness; the geometry of skew lattices in rings; connections between primitive skew lattices and completely simple semigroups; and finally, this geometry is used to help classify symmetric skew lattices on two generators.
Besov spaces on domains in ${\bf R}\sp d$
Ronald A.
DeVore;
Robert C.
Sharpley
843-864
Abstract: We study Besov spaces $ B_q^\alpha ({L_p}(\Omega ))$, $0 < p,q,\alpha < \infty$, on domains $\Omega$ in $ {\mathbb{R}^d}$ . We show that there is an extension operator $\mathcal{E}$ which is a bounded mapping from $B_q^\alpha ({L_p}(\Omega ))$ onto $B_q^\alpha ({L_p}({\mathbb{R}^d}))$. This is then used to derive various properties of the Besov spaces such as interpolation theorems for a pair of $B_q^\alpha ({L_p}(\Omega ))$, atomic decompositions for the elements of $B_q^\alpha ({L_p}(\Omega ))$, and a description of the Besov spaces by means of spline approximation.
The limit sets of some infinitely generated Schottky groups
Richard
Schwartz
865-875
Abstract: Let $P$ be a packing of balls in Euclidean space ${E^n}$ having the property that the radius of every ball of $P$ lies in the interval $[1/k,k]$. If $G$ is a Schottky group associated to $P$, then the Hausdorff dimension of the topological limit set of $G$ is less than a uniform constant $C(k,n) < n$. In particular, this limit set has zero volume.
Support theorems for Radon transforms on real analytic line complexes in three-space
Jan
Boman;
Eric Todd
Quinto
877-890
Abstract: In this article we prove support theorems for Radon transforms with arbitrary nonzero real analytic measures on line complexes (three-dimensional sets of lines) in ${\mathbb{R}^3}$. Let $f$ be a distribution of compact support on ${\mathbb{R}^3}$. Assume $Y$ is a real analytic admissible line complex and ${Y_0}$ is an open connected subset of $ Y$ with one line in $ {Y_0}$ disjoint from supp$\;f$. Under weak geometric assumptions, if the Radon transform of $f$ is zero for all lines in ${Y_0}$, then supp$ \;f$ intersects no line in $ {Y_0}$. These theorems are more general than previous results, even for the classical transform. We also prove a support theorem for the Radon transform on a nonadmissible line complex. Our proofs use analytic microlocal analysis and information about the analytic wave front set of a distribution at the boundary of its support.
The Weil-Petersson symplectic structure at Thurston's boundary
A.
Papadopoulos;
R. C.
Penner
891-904
Abstract: The Weil-Petersson Kähler structure on the Teichmüller space $\mathcal{T}$ of a punctured surface is shown to extend, in an appropriate sense, to Thurston's symplectic structure on the space $ \mathcal{M}{\mathcal{F}_0}$ of measured foliations of compact support on the surface. We introduce a space $ {\widetilde{\mathcal{M}\mathcal{F}}_0}$ of decorated measured foliations whose relationship to $ \mathcal{M}{\mathcal{F}_0}$ is analogous to the relationship between the decorated Teichmüller space $\tilde{\mathcal{T}}$ and $ \mathcal{T}$. $ \widetilde{\mathcal{M}{\mathcal{F}_0}}$ is parametrized by a vector space, and there is a natural piecewise-linear embedding of $ \mathcal{M}{\mathcal{F}_0}$ in $ \widetilde{\mathcal{M}{\mathcal{F}_0}}$ which pulls back a global differential form to Thurston's symplectic form. We exhibit a homeomorphism between $ \tilde{\mathcal{T}}$ and $ {\widetilde{\mathcal{M}\mathcal{F}}_0}$ which preserves the natural two-forms on these spaces. Following Thurston, we finally consider the space $ \mathcal{Y}$ of all suitable classes of metrics of constant Gaussian curvature on the surface, form a natural completion $ \overline{\mathcal{Y}}$ of $\mathcal{Y}$, and identify $ \overline{\mathcal{Y}} - \mathcal{Y}$ with $ \mathcal{M}{\mathcal{F}_0}$. An extension of the Weil-Petersson Kähler form to $ \mathcal{Y}$ is found to extend continuously by Thurston's symplectic pairing on $ \mathcal{M}{\mathcal{F}_0}$ to a two-form on $ \overline{\mathcal{Y}}$ itself.
On mapping class groups of contractible open $3$-manifolds
Robert
Myers
1-46
Abstract: Let $W$ be an irreducible, eventually end-irreducible contractible open $3$-manifold other than ${{\mathbf{R}}^3}$, and let $V$ be a "good" exhaustion of $ W$. Let $\mathcal{H}(W;V)$ be the subgroup of the mapping class group $ \mathcal{H}(W)$ which is "eventually carried by $V$." This paper shows how to compute $\mathcal{H}(W;V)$ in terms of the mapping class groups of certain compact $3$-manifolds associated to $V$. The computation is carried out for a genus two example and for the classical genus one example of Whitehead. For these examples $\mathcal{H}(W) = \mathcal{H}(W;V)$.
Subgroups of Bianchi groups and arithmetic quotients of hyperbolic $3$-space
Fritz
Grunewald;
Joachim
Schwermer
47-78
Abstract: Let $\mathcal{O}$ be the ring of integers in an imaginary quadratic number-field. The group $ {\text{PSL}}_2(\mathcal{O})$ acts discontinuously on hyperbolic $3$-space $H$. If $\Gamma \leq {\text{PSL}}_2(\mathcal{O})$ is a torsionfree subgroup of finite index then the manifold $ \Gamma \backslash H$ can be compactified to a manifold $ {M_\Gamma }$ so that the inclusion $\Gamma \backslash H \leq {M_\Gamma }$ is a homotopy equivalence. $ {M_\Gamma }$ is a compact with boundary. The boundary being a union of finitely many $2$-tori. This paper contains a computer-aided study of subgroups of low index in $ {\text{PSL}}_2(\mathcal{O})$ for various $ \mathcal{O}$. The explicit description of these subgroups leads to a study of the homeomorphism types of the ${M_\Gamma }$.
Pseudobases in direct powers of an algebra
Paul
Bankston
79-90
Abstract: A subset $ P$ of an abstract algebra $ A$ is a pseudobasis if every function from $P$ into $A$ extends uniquely to an endomorphism on $ A$. $A$ is called $\kappa$-free has a pseudobasis of cardinality $ \kappa$; $A$ is minimally free if $ A$ has a pseudobasis. (The 0-free algebras are "rigid" in the strong sense; the $ 1$-free groups are always abelian, and are precisely the additive groups of $ E$-rings.) Our interest here is in the existence of pseudobases in direct powers $ {A^I}$ of an algebra $ A$. On the positive side, if $A$ is a rigid division ring, $\kappa$ is a cardinal, and there is no measurable cardinal $\mu$ with $ \vert A\vert < \mu \leq \kappa$, then ${A^I}$ is $\kappa$-free whenever $\vert I\vert = \vert{A^\kappa }\vert$. On the negative side, if $A$ is a rigid division ring and there is a measurable cardinal $\mu$ with $ \vert A\vert < \mu \leq \vert I\vert$, then ${A^I}$ is not minimally free.
$K$-theory of Eilenberg-Mac Lane spaces and cell-like mapping problem
A. N.
Dranishnikov
91-103
Abstract: There exist cell-like dimension raising maps of $6$-dimensional manifolds. The existence of such maps is proved by using $K$-theory of Eilenberg-Mac Lane complexes.
Toral actions on $4$-manifolds and their classifications
M. Ho
Kim
105-130
Abstract: The existence of a cross-section is proved for some nonorientable $ 4$-manifolds with a $ {T^2}$-action. Two $ 4$-manifolds with a $ {T^2}$-action, which have the same previously known invariants, are constructed. By using a new homotopy invariant, they are proved to be homotopy inequivalent. Finally a stable diffeomorphism theorem is proved.
The semigroup property of value functions in Lagrange problems
Peter R.
Wolenski
131-154
Abstract: The Lagrange problem in the calculus of variations exhibits the principle of optimality in a particularly simple form. The binary operation of inf-composition applied to the value functions of a Lagrange problem equates the principle of optimality with a semigroup property. This paper finds the infinitesimal generator of the semigroup by differentiating at $t = 0$. The type of limit is epigraphical convergence in a uniform sense. Moreover, the extent to which a semigroup is uniquely determined by its infinitesimal generator is addressed. The main results provide a new approach to existence and uniqueness questions in Hamilton-Jacobi theory. When $ L$ is in addition finite-valued, the results are given in terms of pointwise convergence.
Extensions of \'etale by connected group spaces
David B.
Jaffe
155-173
Abstract: The main theorem, in rough terms, asserts the following. Let $ K$ and $D$ be group spaces over a scheme $ S$. Assume that $ K$ has connected fibers and that $D$ is finite and étale over $ S$ . Assume that there exists a single finite, surjective, étale, Galois morphism $ \overline S \to S$ which decomposes (scheme-theoretically) every extension of $D$ by $K$. Let $\pi = \operatorname{Aut}(\overline S /S)$. Then group space extensions of $D$ with kernel $K$ are in bijective correspondence with pairs $(\xi ,\chi )$ consisting of a $ \pi$-group extension $\displaystyle \xi :1 \to K(\overline S) \to X \to D(\overline S ) \to 1$ and a $\pi$-group homomorphism $\chi :X \to \operatorname{Aut}(\overline K )$ which lifts the conjugation map $X \to \operatorname{Aut}(K(\overline S ))$ and which agrees with the conjugation map $K(\bar S) \to \operatorname{Aut}(\overline K )$. In this way, the calculation of group space extensions is reduced to a purely group-theoretic calculation.
Obstructions and hypersurface sections (minimally elliptic singularities)
Kurt
Behnke;
Jan Arthur
Christophersen
175-193
Abstract: We study the obstruction space ${T^2}$ for minimally elliptic surface singularities. We apply the main lemma of our previous paper [3] which relates ${T^2}$ to deformations of hypersurface sections. To use this we classify general hypersurface sections of minimally elliptic singularities. As in the rational singularity case there is a simple formula for the minimal number of generators for ${T^2}$ as a module over the local ring. This number is in many cases (e.g. for cusps of Hilbert modular surfaces) equal to the vector space dimension of ${T^2}$.
Intersection theory of linear embeddings
Sean
Keel
195-212
Abstract: We study intersection theoretic properties of subschemes defined by ideal sheaves of linear type in particular their behavior with respect to blowups and segre classes.
Functorial construction of Le Barz's triangle space with applications
Sean
Keel
213-229
Abstract: We give a new functorial construction of the space of triangles introduced by Le Barz. This description is used to exhibit the space as a composition of smooth blowups, to obtain a space of unordered triangles, and to study how the space varies in a family.
The first two obstructions to the freeness of arrangements
Sergey
Yuzvinsky
231-244
Abstract: In his previous paper the author characterized free arrangements by the vanishing of cohomology modules of a certain sheaf of graded modules over a polynomial ring. Thus the homogeneous components of these cohomology modules can be viewed as obstructions to the freeness of an arrangement. In this paper the first two obstructions are studied in detail. In particular the component of degree zero of the first nontrivial cohomology module has a close relation to formal arrangements and to the operation of truncation. This enables us to prove that in dimension greater than two every free arrangement is formal and not a proper truncation of an essential arrangement.
A dynamical proof of the multiplicative ergodic theorem
Peter
Walters
245-257
Abstract: We shall give a proof of the following result of Oseledec, in which $ GL(d)$ denotes the space of invertible $d \times d$ real matrices, $\vert\vert \bullet \vert\vert$ denotes any norm on the space of $d \times d$ matrices, and $ {\log ^+ }(t) = \max (0,\log (t))$ for $t \in [0,\infty )$.
Inverse monoids, trees and context-free languages
Stuart W.
Margolis;
John C.
Meakin
259-276
Abstract: This paper is concerned with a study of inverse monoids presented by a set $ X$ subject to relations of the form ${e_i} = {f_i}$, $i \in I$, where ${e_i}$ and ${f_i}$ are Dyck words, i.e. idempotents of the free inverse monoid on $X$. Some general results of Stephen are used to reduce the word problem for such a presentation to the membership problem for a certain subtree of the Cayley graph of the free group on $X$. In the finitely presented case the word problem is solved by using Rabin's theorem on the second order monadic logic of the infinite binary tree. Some connections with the theory of rational subsets of the free group and the theory of context-free languages are explored.
Quadratic transformation formulas for basic hypergeometric series
Mizan
Rahman;
Arun
Verma
277-302
Abstract: Starting with some of the known transformation formulas for well-poised $_2{\phi _1}$ and very-well-poised $_8{\phi _7}$ basic hypergeometric series we obtain $q$-analogues of $36$ quadratic transformation formulas given in $\S2.11$ of Higher transcendental functions, Vol. 1, edited by Erdélyi et al. We also derive some new quadratic transformation formulas that give rise to identities connecting very-well-poised but unbalanced $ _{10}{\phi _9}$ series in base $q$ with very-well-poised and balanced $_{12}{\phi _{11}}$ series in base $ {q^2}$. A Rogers-Ramanujan type identity is also found as a limiting case.
Harmonic volume, symmetric products, and the Abel-Jacobi map
William M.
Faucette
303-327
Abstract: The author generalizes B. Harris' definition of harmonic volume to the algebraic cycle ${W_k} - W_k^- $ for $k > 1$ in the Jacobian of a nonsingular algebraic curve $X$ . We define harmonic volume, determine its domain, and show that it is related to the image $ \nu$ of ${W_k} - W_k^-$ in the Griffiths intermediate Jacobian. We derive a formula expressing harmonic volume as a sum of integrals over a nested sequence of submanifolds of the $k$-fold symmetric product of $X$ . We show that $\nu$ , when applied to a certain class of forms, takes values in a discrete subgroup of $ {\mathbf{R}}/{\mathbf{Z}}$ and hence, when suitably extended to complexvalued forms, is identically zero modulo periods on primitive forms if $k \geq 2$. This implies that the image of ${W_k} - W_k^-$ is identically zero in the Griffiths intermediate Jacobian if $k \geq 2$. We introduce a new type of intermediate Jacobian which, like the Griffiths intermediate Jacobian, varies holomorphically with moduli, and we consider a holomorphic torus bundle on Torelli space with this fiber. We use the relationship mentioned above between $\nu$ and harmonic volume to compute the variation of $\nu$ when considered as a section of this bundle. This variational formula allows us to show that the image of ${W_k} - W_k^- $ in this intermediate Jacobian is nondegenerate.
A short proof of Zheludev's theorem
F.
Gesztesy;
B.
Simon
329-340
Abstract: We give a short proof of Zheludev's theorem that states the existence of precisely one eigenvalue in sufficiently distant spectral gaps of a Hill operator subject to certain short-range perturbations. As a by-product we simultaneously recover Rofe-Beketov's result about the finiteness of the number of eigenvalues in essential spectral gaps of the perturbed Hill operator. Our methods are operator theoretic in nature and extend to other one-dimensional systems such as perturbed periodic Dirac operators and weakly perturbed second order finite difference operators. We employ the trick of using a selfadjoint Birman-Schwinger operator (even in cases where the perturbation changes sign), a method that has already been successfully applied in different contexts and appears to have further potential in the study of point spectra in essential spectral gaps.
Number of orbits of branch points of ${\bf R}$-trees
Renfang
Jiang
341-368
Abstract: An $R$-tree is a metric space in which any two points are joined by a unique arc. Every arc is isometric to a closed interval of $R$ . When a group $G$ acts on a tree ($Z$-tree) $X$ without inversion, then $X/G$ is a graph. One gets a presentation of $G$ from the quotient graph $X/G$ with vertex and edge stabilizers attached. For a general $R$-tree $X$, there is no such combinatorial structure on $ X/G$. But we can still ask what the maximum number of orbits of branch points of free actions on $R$-trees is. We prove the finiteness of the maximum number for a family of groups, which contains free products of free abelian groups and surface groups, and this family is closed under taking free products with amalgamation.
On Dehn functions and products of groups
Stephen G.
Brick
369-384
Abstract: If $G$ is a finitely presented group then its Dehn function--or its isoperimetric inequality--is of interest. For example, $G$ satisfies a linear isoperimetric inequality iff $ G$ is negatively curved (or hyperbolic in the sense of Gromov). Also, if $ G$ possesses an automatic structure then $G$ satisfies a quadratic isoperimetric inequality. We investigate the effect of certain natural operations on the Dehn function. We consider direct products, taking subgroups of finite index, free products, amalgamations, and HNN extensions.
Subvarieties of moduli space determined by finite groups acting on surfaces
John F. X.
Ries
385-406
Abstract: Suppose the finite group $G$ acts as orientation preserving homeomorphisms of the oriented surface $S$ of genus $g$. This determines an irreducible subvariety $ \mathcal{M}_g^{[G]}$ of the moduli space $ {\mathcal{M}_g}$ of Riemann surfaces of genus $g$ consisting of all surfaces with a group $ {G_1}$ of holomorphic homeomorphisms of the same topological type as $ G$. This family is determined by an equivalence class of epimorphisms $ \psi$ from a Fuchsian group $\Gamma$ to $G$ whose kernel is isomorphic to the fundamental group of $S$. To examine the singularity of ${\mathcal{M}_g}$ along this family one needs to know the full automorphism group of a generic element of $ \mathcal{M}_g^{[G]}$. In $ \S2$ we show how to compute this from $\psi$. Let $ \mathcal{M}_g^G$ denote the locus of all Riemann surfaces of genus $ g$ whose automorphism group contains a subgroup isomorphic to $G$. In $\S3$ we show that the irreducible components of this subvariety do not necessarily correspond to the families above, that is, the components cannot be put into a one-to-one correspondence with the topological actions of $G$. In $\S4$ we examine the actions of $G$ on the spaces of holomorphic $ k$-differentials and on the first homology. We show that when $G$ is not cyclic, the characters of these actions do not necessarily determine the topological type of the action of $G$ on $S$.
On the theory of Frobenius extensions and its application to Lie superalgebras
Allen D.
Bell;
Rolf
Farnsteiner
407-424
Abstract: By using an approach to the theory of Frobenius extensions that emphasizes notions related to associative forms, we obtain results concerning the trace and corestriction mappings and transitivity. These are employed to show that the extension of enveloping algebras determined by a subalgebra of a Lie superalgebra is a Frobenius extension, and to study certain questions in representation theory.
Actions of linearly reductive groups on PI-algebras
Nikolaus
Vonessen
425-442
Abstract: Let $G$ be a linearly reductive group acting rationally on a $ {\text{PI}}$-algebra $ R$. We study the relationship between $R$ and the fixed ring ${R^G}$ , generalizing earlier results obtained under the additional hypothesis that $R$ is affine.
A classification of the finite extensions of a multidimensional Bernoulli shift
Janet Whalen
Kammeyer
443-457
Abstract: The finite extensions of a multidimensional Bernoulli shift are classified completely, up to factor isomorphism, and up to isomorphism. If such an extension is weakly mixing then it must be Bernoulli; otherwise, it has a finite rotation factor, which has a Bernoulli complementary algebra. This result is extended to multidimensional Bernoulli flows and Bernoulli shifts of infinite entropy.
Characteristic matrices and spectral properties of evolutionary systems
M. A.
Kaashoek;
S. M.
Verduyn Lunel
479-517
Abstract: In this paper we introduce the notion of a characteristic matrix for a large class of unbounded operators and study the precise connection between characteristic matrices and spectral properties of evolutionary systems. In particular, we study so-called multiplicity theorems. Several examples will illustrate our results.
Generalized group presentation and formal deformations of CW complexes
Richard A.
Brown
519-549
Abstract: A Peiffer-Whitehead word system $ \mathcal{W}$, or generalized group presentation, consists of generators, relators (words of order $2$), and words of higher order $n$ that represent elements of a free crossed module $(n = 3)$ or a free module $(n > 3)$. The ${P_n}$-equivalence relation on word systems generalizes the extended Nielsen equivalence relation on ordinary group presentations. Word systems, called homotopy readings, can be associated with any connected $ {\text{CW}}$ complex $ K$ by removing a maximal tree and selecting one generator or word per cell, via relative homotopy. Given homotopy readings ${\mathcal{W}_1}$ and ${\mathcal{W}_2}$ of finite ${\text{CW}}$ complexes ${K_1}$ and ${K_2}$ respectively, we show that ${\mathcal{W}_1}$ is ${P_n}$-equivalent to ${\mathcal{W}_2}$ if and only if ${K_1}$ formally $(n + 1)$-deforms to $ {K_2}$. This extends results of P. Wright (1975) and W. Metzler (1982) for the case $n = 2$. For $n \geq 3$, it follows that ${\mathcal{W}_1}$ is ${P_n}$-equivalent to ${\mathcal{W}_2}$ if and only if ${K_1}$ and ${K_2}$ have the same simple homotopy type.
Estimates of the Caccioppoli-Schauder type in weighted function spaces
Giovanni Maria
Troianiello
551-573
Abstract: We deal with imbeddings of certain weighted function spaces as well as with the corresponding norm estimates for solutions to second order elliptic problems. We redemonstrate some results of Gilbarg and Hörmander by a technique, entirely different from theirs, which enables us to cover a range of parameters excluded by them.
Cohomology of the symplectic group ${\rm Sp}\sb 4({\bf Z})$. I. The odd torsion case
Alan
Brownstein;
Ronnie
Lee
575-596
Abstract: Let ${h_2}$ be the degree two Siegel space and $ Sp(4,\mathbb{Z})$ the symplectic group. The quotient $Sp(4,\mathbb{Z})\backslash {h_2}$ can be interpreted as the moduli space of stable Riemann surfaces of genus $2$. This moduli space can be decomposed into two pieces corresponding to the moduli of degenerate and nondegenerate surfaces of genus $2$. The decomposition leads to a Mayer-Vietoris sequence in cohomology relating the cohomology of $ Sp(4,\mathbb{Z})$ to the cohomology of the genus two mapping class group $\Gamma _2^0$. Using this tool, the $3$- and $5$-primary pieces of the integral cohomology of $Sp(4,\mathbb{Z})$ are computed.
Hausdorff dimension of wild fractals
T. B.
Rushing
597-613
Abstract: We show that for every $s \in [n - 2,n]$ there exists a homogeneously embedded wild Cantor set ${C^s}$ in $\mathbb{R}^n, n \geq 3$, of (local) Hausdorff dimension $s$. Also, it is shown that for every $s \in [n - 2,n]$ and for any integer $k \ne n$ such that $1 \leq k \leq s$, there exist everywhere wild $ k$-spheres and $ k$-cells, in $\mathbb{R}^n, n \geq 3$, of (local) Hausdorff dimension $s$.
Hypoellipticity on Cauchy-Riemann manifolds
Johannes A.
Petersen
615-639
Abstract: Using a recent homotopy formula by Trèves, it is shown that the existence of $(q + 1)$-dimensional holomorphic supporting manifolds is a sufficient condition for the hypoellipticity on level $q$ and $n - q$ of a tangential Cauchy-Riemann complex of ${\text{CR}}$-dimension $n$. In the hypersurface case, this result is given microlocally.
The Mizohata structure on the sphere
Jorge
Hounie
641-649
Abstract: We prove that a compact surface that admits a Mizohata structure is (homeomorphic to) a sphere and that there exists exactly one such structure $ \mathcal{L}$ up to conjugation by diffeomorphisms. We also characterize the range and the kernel of the operator ${\delta _0}$ induced by $ \mathcal{L}$ , i.e., obtained from the exterior derivative on functions by passing to the quotient modulo ${\mathcal{L}^\bot }$ .
The monotonicity of the entropy for a family of degree one circle maps
Lluís
Alsedà;
Francesc
Mañosas
651-684
Abstract: For the natural biparametric family of piecewise linear circle maps with two pieces we show that the entropy increases when any of the two parameters increases. We also describe the regions of the parameter space where the monotonicity is strict.
On the structure of twisted group $C\sp *$-algebras
Judith A.
Packer;
Iain
Raeburn
685-718
Abstract: We first give general structural results for the twisted group algebras $ {C^{\ast} }(G,\sigma )$ of a locally compact group $G$ with large abelian subgroups. In particular, we use a theorem of Williams to realise ${C^{\ast}}(G,\sigma )$ as the sections of a ${C^{\ast}}$-bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras. Next we use a device of Rosenberg to show that, when $\Gamma$ is a discrete subgroup of a solvable Lie group $G$, the $K$-groups ${K_ {\ast} }({C^{\ast} }(\Gamma ,\sigma ))$ are isomorphic to certain twisted $K$-groups ${K^{\ast} }(G/\Gamma ,\delta (\sigma ))$ of the homogeneous space $G/\Gamma$, and we discuss how the twisting class $\delta (\sigma ) \in {H^3}(G/\Gamma ,\mathbb{Z})$ depends on the cocycle $\sigma$. For many particular groups, such as $ {\mathbb{Z}^n}$ or the integer Heisenberg group, $ \delta (\sigma )$ always vanishes, so that ${K_ {\ast} }({C^{\ast} }(\Gamma ,\sigma ))$ is independent of $\sigma$, but a detailed analysis of examples of the form ${\mathbb{Z}^n} \rtimes \mathbb{Z}$ shows this is not in general the case.
$H\sp p$- and $L\sp p$-variants of multiparameter Calder\'on-Zygmund theory
Anthony
Carbery;
Andreas
Seeger
719-747
Abstract: We consider Calderón-Zygmund operators on product domains. Under certain weak conditions on the kernel a singular integral operator can be proved to be bounded on ${H^p}(\mathbb{R} \times \mathbb{R} \times \cdots \times \mathbb{R}), 0 < p \leq 1$, if its behaviour on ${L^2}$ and on certain scalar-valued and vector-valued rectangle atoms is known. Another result concerns an extension of the authors' results on $ {L^p}$-variants of Calderón-Zygmund theory [1,23] to the product-domain-setting. As an application, one obtains estimates for Fourier multipliers and pseudo-differential operators.
Anomalies associated to the polar decomposition of ${\rm GL}(n,{\bf C})$
Steven
Rosenberg
749-760
Abstract: Let $D$ be a selfadjoint elliptic differential operator on a hermitian bundle over a compact manifold. For positive $D$, the variation of the functional determinant of $ D$ under positive definite hermitian gauge transformations is calculated. This corresponds to computing a gauge anomaly in the nonunitary directions of the polar decomposition of the frame bundle $ {\text{GL}}(E)$. The variation of the eta invariant for general $D$ is also calculated. If $ D$ is not selfadjoint, the integrand in the heat equation proof of the Atiyah-Singer Index Theorem is interpreted as an anomaly for ${D^{\ast} }D$ . In particular, the gauge anomaly for semiclassical Yang-Mills theory is computed.
Eta invariants of Dirac operators on foliated manifolds
Goran
Perić
761-782
Abstract: We define the eta function of Dirac operators on foliated manifolds. We show that the eta functions are regular at the origin thus defining corresponding eta invariants of foliated manifolds. Under the hypothesis of invertibility of the operator in question, we prove the Atiyah-Singer relative index theorem for Dirac operators on foliated manifolds. Then we discuss the homotopy invariance of the index with respect to secondary data.
A PL geometric study of algebraic $K$ theory
Bi Zhong
Hu
783-808
Abstract: This paper manages to apply the Farrell-Jones theory on algebraic $ K$-groups of closed negatively curved riemannian manifolds to Gromov's hyperbolic group theory. The paper reaches the conclusion that for any finite polyhedron $K$ with negative curvature, $ \operatorname{Wh}({\pi _1}K) = 0$ .
On the generalized Ramanujan-Nagell equation $x\sp 2-D=2\sp {n+2}$
Mao Hua
Le
809-825
Abstract: Let $D$ be a positive integer which is odd. In this paper we prove that the equation ${x^2} - D = {2^{n + 2}}$ has at most three positive integer solutions $(x,n)$ except when $D = {2^{2m}} - 3 \cdot {2^{m + 1}} + 1$ , where $m$ is a positive integer with $m \geq 3$ .
Entropy for canonical shifts
Marie
Choda
827-849
Abstract: For a $ ^{\ast}$-endomorphism $ \sigma$ of an injective finite von Neumann algebra $A$ , we investigate the relations among the entropy $H(\sigma )$ for $\sigma$ , the relative entropy $H(A\vert\sigma (A))$ of $ \sigma (A)$ for $ A$ , the generalized index $ \lambda (A,\sigma (A))$, and the index for subfactors. As an application, we have the following relations for the canonical shift $ \Gamma$ for the inclusion $N \subset M$ of type II$_{1}$ factors with the finite index $[M:N]$, $\displaystyle H(A\vert\Gamma (A)) \leq 2H(\Gamma ) \leq \log \lambda {(A,\Gamma (A))^{ - 1}} = 2\log [M:N],$ where $A$ is the von Neumann algebra generated by the two of the relative commutants of $M$. In the case of that $ N \subset M$ has finite depth, then all of them coincide.
On the self-intersections of foliation cycles
Yoshihiko
Mitsumatsu
851-860
Abstract: The existence of a transverse invariant measure imposes a strong restriction on the transverse complexity of a foliated manifold. The homological self-intersection of the corresponding foliation cycle measures the complexity around its support. In the present paper, the vanishing of the self-intersection is proven under some regularity condition on the measure.
An inverse problem for circle packing and conformal mapping
Ithiel
Carter;
Burt
Rodin
861-875
Abstract: Figure 1.1 illustrates the fact that if a region is almost packed with circles of radius $ \varepsilon$ in the hexagonal pattern and if the unit disk is packed in an isomorphic pattern with circles of varying radii then, after suitable normalization, the correspondence of circles converges to the Riemann mapping function as $\varepsilon \to 0$ (see [15]). In the present paper an inverse of this result is obtained as illustrated by Figure 1.2; namely, if the unit disk is almost packed with $\varepsilon $-circles there is an isomorphic circle packing almost filling the region such that, after suitable normalization, the circle correspondence converges to the conformal map of the disk onto the region as $ \varepsilon \to 0$. Note that this set up yields an approximate triangulation of the region by joining the centers of triples of mutually tangent circles. Since this triangulation is intimately related to the Riemann mapping it may be useful for grid generation [18].
Generation and propagation of interfaces in reaction-diffusion systems
Xinfu
Chen
877-913
Abstract: This paper is concerned with the asymptotic behavior, as $\varepsilon \searrow 0$, of the solution $ ({u^\varepsilon },{v^\varepsilon })$ of the second initial-boundary value problem of the reaction-diffusion system: $\displaystyle \left\{ {\begin{array}{*{20}{c}} {u_t^\varepsilon - \varepsilon \... ...varepsilon } - \gamma {\upsilon ^\varepsilon }} \end{array} } \right.$ where $\gamma > 0$ is a constant. When $v \in ( - 2\sqrt 3 /9,2\sqrt 3 /9)$, $f$ is bistable in the sense that the ordinary differential equation $ {u_t} = f(u,v)$ has two stable solutions $ u = {h_ - }(v)$ and $u = {h_ + }(v)$ and one unstable solution $u = {h_0}(v)$, where ${h_ - }(v), {h_0}(v)$, and ${h_ + }(v)$ are the three solutions of the algebraic equation $ f(u,v) = 0$. We show that, when the initial data of $v$ is in the interval $( - 2\sqrt 3 /9,2\sqrt 3 /9)$, the solution $ ({u^\varepsilon },{v^\varepsilon })$ of the system tends to a limit $ (u,v)$ which is a solution of a free boundary problem, as long as the free boundary problem has a unique classical solution. The function $u$ is a "phase" function in the sense that it coincides with ${h_ + }(v)$ in one region ${\Omega _ + }$ and with $ {h_ - }(v)$ in another region ${\Omega _ - }$. The common boundary (free boundary or interface) of the two regions ${\Omega _ - }$ and $ {\Omega _ + }$ moves with a normal velocity equal to $\mathcal{V}(v)$, where $\mathcal{V}( \bullet )$ is a function that can be calculated. The local (in time) existence of a unique classical solution to the free boundary problem is also established. Further we show that if initially $ u( \bullet, 0) - {h_0}(v( \bullet, 0))$ takes both positive and negative values, then an interface will develop in a short time $ O(\varepsilon \vert\ln \varepsilon \vert)$ near the hypersurface where $u(x,0) - {h_0}(v(x,0)) = 0$.
Brauer-Hilbertian fields
Burton
Fein;
David J.
Saltman;
Murray
Schacher
915-928
Abstract: Let $F$ be a field of characteristic $ p$ ($p = 0$ allowed), and let $ F(t)$ be the rational function field in one variable over $F$. We say $F$ is Brauer-Hilbertian if the following holds. For every $\alpha$ in the Brauer group $\operatorname{Br}(F(t))$ of exponent prime to $ p$, there are infinitely many specializations $ t \to a \in F$ such that the specialization $\bar \alpha \in \operatorname{Br}(F)$ is defined and has exponent equal to that of $ \alpha$. We show every global field is Brauer-Hilbertian, and if $ K$ is Hilbertian and $ F$ is finite separable over $K(t)$, $F$ is Brauer-Hilbertian.
Single loop space decompositions
David J.
Anick
929-940
Abstract: The method of single loop space decompositions, in which $\Omega X$ is factored into minimal factors, is an important one for understanding the unstable homotopy of many simply-connected spaces $ X$. This paper begins with a survey of the major known theorems along these lines. We then give a necessary and sufficient condition for $ \Omega X$ to be decomposable as a product of spaces belonging to a certain list. We conclude with a nontrivial instance of an application of this condition.
Transfinite graphs and electrical networks
A. H.
Zemanian
1-36
Abstract: All prior theories of infinite electrical networks assume that such networks are finitely connected, that is, between any two nodes of the network there is a finite path. This work establishes a theory for transfinite electrical networks wherein some nodes are not connected by finite paths but are connected by transfinite paths. Moreover, the voltages at those nodes may influence each other. The main difficulty to surmount for this extension is the construction of an appropriate generalization of the concept of connectedness. This is accomplished by extending the idea of a node to encompass infinite extremities of a graph. The construction appears to be novel and leads to a hierarchy of transfinite graphs indexed by the finite and infinite ordinals. Two equivalent existence and uniqueness theorems are established for transfinite resistive electrical networks based upon Tellegen's equation, one using currents and the other using voltages as the fundamental quantities. Kirchhoff's laws do not suffice for this purpose and indeed need not hold everywhere in infinite networks. Although transfinite countable electrical networks have in general an uncountable infinity of extremities, called "tips," the number of different tip voltages may be radically constrained by both the graph of the network and its resistance values. Conditions are established herein under which various tip voltages are compelled to be the same. Furthermore, a theorem of Shannon-Hagelbarger on the concavity of resistance functions is extended to the driving-point resistance between any two extremities of arbitrary ranks. This is based upon an extension of Thomson's least power principle to transfinite networks.
Isotropy groups of homotopy classes of maps
G.
Triantafillou
37-48
Abstract: Let $\operatorname{aut}(X)$ be the group of homotopy classes of self-homotopy equivalences of a space $ X$ and let $[f] \in [X,Y]$ be a homotopy class of maps from $ X$ to $Y$ . The aim of this paper is to prove that under certain nilpotency and finiteness conditions the isotropy group $ \operatorname{aut}{(X)_{[f]}}$ of $[f]$ under the action of $\operatorname{aut}(X)$ on $[X,Y]$ is commensurable to an arithmetic group. Therefore $ \operatorname{aut}{(X)_{[f]}}$ is a finitely presented group by a result of Borel and Harish-Chandra.
The classification of complete minimal surfaces with total curvature greater than $-12\pi$
Francisco J.
López
49-74
Abstract: We classify complete orientable minimal surfaces with finite total curvature $- 8\pi$.
Subcriticality and gaugeability of the Schr\"odinger operator
Z.
Zhao
75-96
Abstract: We investigate a Schrödinger operator $- \Delta /2 + V$ in ${R^d}\;(d \geq 3)$ with a potential $V$ in the class ${K_d}$ satisfying a similar Kato condition at infinity, and prove an equivalence theorem connecting various conditions on subcriticality, strong positivity and gaugeability of the operator.
QED domains and NED sets in $\overline{\bf R}{}\sp n$
Shan Shuang
Yang
97-120
Abstract: This paper contributes to the theory of quasiextremal distance (or ${\text{QED}}$) domains. We associate with every ${\text{QED}}$ domain $D$ two $ {\text{QED}}$ constants $ M(D)$ and ${M^{\ast} }(D)$ and exhibit how these constants reflect the geometry of $D$. For example, we give a geometric characterization for $ {\text{QED}}$ domains $ D$ with ${M^{\ast}}(D) = 2$ and obtain some sharp estimates of ${\text{QED}}$ constants $M(D)$ and ${M^{\ast} }(D)$ for different kinds of domains.
Mixed groups
Paul
Hill;
Charles
Megibben
121-142
Abstract: As the culmination of a series of several papers, we establish here a combinatorial characterization of Warfield groups (that is, direct summands of simply presented abelian groups) in terms of knice subgroups--a refinement of the concept of nice subgroup appropriate to the study of groups containing elements of infinite order. Central to this theory is the class of $k$-groups, those in which 0 is a knice subgroup, and the proof that this class is closed under the formation of knice isotype subgroups. In particular, a direct summand of a $k$-group is a $k$-group. As an application of our Axiom $ 3$ characterization of Warfield groups, we prove that $k$-groups of cardinality $ {\aleph _1}$ have sequentially pure projective dimension $\leq 1$; or equivalently, if $ H$ is a knice isotype sub-group of the Warfield group $G$ with $ \vert G/H\vert = {\aleph _1}$, then $H$ is itself a Warfield group.
Central extensions of current algebras
Paul
Zusmanovich
143-152
Abstract: The second cohomology group of Lie algebras of kind $L \otimes U$ with trivial coefficients is investigated, where $L$ admits a decomposition with one-dimensional root spaces and $U$ is an arbitrary associative commutative algebra with unit. This paper gives a unification of some recent results of C. Kassel and A. Haddi and provides a determination of central extensions of certain modular semisimple Lie algebras.
Subgroup conditions for groups acting freely on products of spheres
Judith H.
Silverman
153-181
Abstract: Let $d$ and $h$ be integers such that either $d \geq 2$ and $h = {2^d} - 1$, or $d = 4$ and $h = 5$. Suppose that the group $\mathcal{G}$ contains an elementary-abelian $ 2$-subgroup ${E_d}$ of rank $d$ with an element $\sigma$ of order $h$ in its normalizer. We show that if $\mathcal{G}$ admits a free and $ {{\mathbf{F}}_2}$-cohomologically trivial action on $ {({S^n})^d}$, then some nontrivial power of $\sigma$ centralizes ${E_d}$. The cohomology ring $ {H^{\ast} }({E_d};{{\mathbf{F}}_2}) \simeq {{\mathbf{F}}_2}[{y_1}, \ldots ,{y_d}]$ is a module over the Steenrod algebra $\mathcal{A}(2)$. Let $\theta \in {{\mathbf{F}}_2}[{y_1}, \ldots ,{y_d}]$, and let $ c \geq d - 2$ be an integer. We show that $\theta$ divides $S{q^{{2^i}}}(\theta )$ in the polynomial ring for $0 \leq i \leq c \Leftrightarrow \theta = {\tau ^{{2^{c - d + 3}}}}\pi $ , where $\tau$ divides $S{q^{{2^i}}}(\tau )$ for $0 \leq i \leq d - 3$ and $\pi$ is a product of linear forms.
Meromorphic extension of analytic continued fractions across their divergence line with applications to orthogonal polynomials
Hans-J.
Runckel
183-212
Abstract: For the limit periodic $J$-fraction $K( - {a_n}/(\lambda + {b_n}))$, ${a_n}$, ${b_n} \in \mathbb{C}$, $n \in \mathbb{N}$, which is normalized such that it converges and represents a meromorphic function $f(\lambda )$ on $ {\mathbb{C}^{\ast} }: = \mathbb{C}\backslash [ - 1,1]$, the numerators $ {A_n}$ and denominators $ {B_n}$ of its $ n$th approximant are explicitly determined for all $n \in \mathbb{N}$. Under natural conditions on the speed of convergence of ${a_n}$, ${b_n}$, $n \to \infty $, the asymptotic behaviour of the orthogonal polynomials ${B_n}$, $ {A_{n + 1}}$ (of first and second kind) is investigated on ${\mathbb{C}^{\ast} }$ and $[ - 1,1]$. An explicit representation for $f(\lambda )$ yields continuous extension of $ f$ from ${\mathbb{C}^{\ast} }$ onto upper and lower boundary of the cut $( - 1,1)$. Using this and a determinant relation, which asymptotically connects both sequences $ {A_n}$, ${B_n}$, one obtains nontrivial explicit formulas for the absolutely continuous part (weight function) of the distribution functions for the orthogonal polynomial sequences ${B_n}$, ${A_{n + 1}}$, $n \in \mathbb{N}$. This leads to short proofs of results which generalize and supplement results obtained by P. G. Nevai [7]. Under a stronger condition the explicit representation for $ f(\lambda )$ yields meromorphic extension of $f$ from $ {\mathbb{C}^{\ast} }$ across $( - 1,1)$ onto a region of a second copy of $\mathbb{C}$ which there is bounded by an ellipse, whose focal points $\pm 1$ are first order algebraic branch points for $ f$. Then, by substitution, analogous results on continuous and meromorphic extension are obtained for limit periodic continued fractions $K( - {a_n}(z)/(\lambda (z) + {b_n}(z)))$, where $ {a_n}(z)$, ${b_n}(z)$, $ \lambda (z)$ are holomorphic on a region in $ \mathbb{C}$. Finally, for $ T$-fractions $ T(z) = K( - {c_n}z/(1 + {d_n}z))$ with $ {c_n} \to c$, ${d_n} \to d$, $ n \to \infty$, the exact convergence regions are determined for all $ c$, $d \in \mathbb{C}$. Again, explicit representations for $T(z)$ yield continuous and meromorphic extension results. For all $c$, $ d \in \mathbb{C}$ the regions (on Riemann surfaces) onto which $T(z)$ can be extended meromorphically, are described explicitly.
Characteristic cycles of holomorphic discrete series
Jen-Tseh
Chang
213-227
Abstract: The characteristic cycles of standard modules arising from closed $ K$-orbits in a flag variety are studied and those of holomorphic discrete series are determined explicitly. Also the asymptotic expansion of the distribution characters of holomorphic discrete series are computed; the result verifies a conjecture of D. Vogan in this special case.
On conjugacy separability of fundamental groups of graphs of groups
M.
Shirvani
229-243
Abstract: A complete determination of when the elements of a fundamental group of a (countable) graph of profinite groups are conjugacy distinguished is given. By embedding an arbitrary fundamental group $G$ into one with profinite vertex groups and making use of the above result, questions on conjugacy separability of $G$ can be reduced to the solution of equations in the vertex groups of $G$.
Fundamental solutions for hypoelliptic differential operators depending analytically on a parameter
Frank
Mantlik
245-257
Abstract: Let $ P(\lambda ,D) = \sum\nolimits_{\vert\alpha \vert \leq m} {{a_\alpha }(\lambda ){D^\alpha }}$ be a differential operator with constant coefficients ${a_\alpha }$ depending analytically on a parameter $\lambda$. Assume that each $P(\lambda ,D)$ is hypoelliptic and that the strength of $ P(\lambda ,D)$ is independent of $\lambda$. Under this condition we show that there exists a regular fundamental solution of $P(\lambda ,D)$ which also depends analytically on $ \lambda$.
Delta methods in enveloping algebras of Lie superalgebras
Jeffrey
Bergen;
D. S.
Passman
259-280
Abstract: Let $L$ be a Lie superalgebra over a field $ K$ of characteristic $ \ne 2$ . We define $\displaystyle \Delta (L) = \{ l \in L\vert{\dim _K}[L,l] < \infty \}.$ Then $ \Delta (L)$ is a Lie ideal of $L$ and is restricted if $L$ is restricted. $ \Delta (L)$ is the Lie superalgebra analog of the Lie delta ideal, used by the authors in the study of enveloping rings, and also of the finite conjugate center of a group, used in the study of group algebras and crossed products. In this paper we examine $U(L)$, where depending upon $\operatorname{char}K$, $U(L)$ denotes either the enveloping algebra or the restricted enveloping algebra of $L$. We show that $ \Delta (L)$ controls certain properties of $U(L)$. Specifically, we consider semiprimeness, primeness, almost constants, almost centralizers, central closures, and the Artinian condition.
Generalized second-order derivatives of convex functions in reflexive Banach spaces
Chi Ngoc
Do
281-301
Abstract: Generalized second-order derivatives introduced by Rockafellar in finite-dimensional spaces are extended to convex functions in reflexive Banach spaces. Parallel results are shown in the infinite-dimensional case. A result that plays an important role in applications is that the generalized second-order differentiability is preserved under the integral sign.
Nonnegative curvature on piecewise constant curvature spaces
Robert
Peszek
303-315
Abstract: We consider piecewise flat and piecewise spherical spaces. We prove that every piecewise flat cobordism which is a product near the boundary and has nonnegative curvature must be trivial in the metric sense. We also obtain several restrictions for piecewise $p$-spherical manifolds with nonnegative curvature and diameter equal to $\pi p$. We prove that such a manifold must be homeomorphic to a sphere and that it is a disjoint union of minimal paths connecting two points, which have length $\pi p$.
Strong Tutte functions of matroids and graphs
Thomas
Zaslavsky
317-347
Abstract: A strong Tutte function of matroids is a function of finite matroids which satisfies $F({M_1} \oplus {M_2}) = F({M_1})F({M_2})$ and $F(M) = {a_e}F(M\backslash e) + {b_e}F(M/e)$ for $ e$ not a loop or coloop of $ M$, where ${a_e}$, ${b_e}$ are scalar parameters depending only on $ e$. We classify strong Tutte functions of all matroids into seven types, generalizing Brylawski's classification of Tutte-Grothendieck invariants. One type is, like Tutte-Grothendieck invariants, an evaluation of a rank polynomial; all types are given by a Tutte polynomial. The classification remains valid if the domain is any minor-closed class of matroids containing all three-point matroids. Similar classifications hold for strong Tutte functions of colored matroids, where the parameters depend on the color of $e$, and for strong Tutte functions of graphs and edge-colored graphs whose values do not depend on the attachments of loops. The latter classification implies new characterizations of Kauffman's bracket polynomials of signed graphs and link diagrams.
$\Sigma\sb n$ definable sets without $\Sigma\sb n$ induction
C. T.
Chong;
K. J.
Mourad
349-363
Abstract: We prove that the Friedberg-Muchnik Theorem holds in all models of ${\Sigma _1}$ collection under the base theory ${P^- } + I{\Sigma _0}$. Generalizations to higher dimensional analogs are discussed. We also study the splitting of r.e. sets in these weak models of arithmetic.
Cohomological aspects of hypergraphs
F. R. K.
Chung;
R. L.
Graham
365-388
Abstract: By a $k$-graph we will mean a collection of $ k$-element subsets of some fixed set $V$. A $k$-graph can be regarded as a $(k - 1)$-chain on $ {2^V}$, the simplicial complex of all subsets of $V$, over the coefficient group $\mathbb{Z}/2$, the additive group of integers modulo $2$. The induced group structure on the $ (k - 1)$-chains leads to natural definitions of the coboundary $\delta$ of a chain, the cochain complex of $ C = \{ {C^k},\delta \}$ and the usual cohomology groups ${H^k}(C;\mathbb{Z}/2)$. In particular, it is possible to construct what could be called "higher-order" coboundary operators $ {\delta ^{(i)}}$, where ${\delta ^{(i)}}$ increases dimension by $i$ (rather than just $1$). In this paper we will develop various properties of these $ {\delta ^{(i)}}$, and in particular, compute the corresponding cohomology groups for ${2^V}$ over $ \mathbb{Z}/2$. It turns out that these groups depend in a rather subtle way on the arithmetic properties of $i$.
Algebra for Heckoid groups
Robert
Riley
389-409
Abstract: We introduce an infinite collection of (Laurent) polynomials associated with a $2$-bridge knot or link normal form $K = (\alpha ,\beta )$. Experimental evidence suggests that these "Heckoid polynomials" define the affine representation variety of certain groups, the Heckoid groups, for $K$ . We discuss relations which hold in the image of the generic representation for each polynomial. We show that, with a certain change of variable, each Heckoid polynomial divides the nonabelian representation polynomial of $L$ , where $L$ belongs to an infinite collection of $2$-bridge knots/links determined by $ K$ and the Heckoid polynomial. Finally, we introduce a "precusp polynomial" for each $2$-bridge knot normal form, and show it is the product of two (possibly reducible) non-constant polynomials. We are preparing a sequel on the Heckoid groups and the evidence for some of the geometrical assertions stated in the introduction.
Universal spaces for ${\bf R}$-trees
John C.
Mayer;
Jacek
Nikiel;
Lex G.
Oversteegen
411-432
Abstract: ${\mathbf{R}}$-trees arise naturally in the study of groups of isometries of hyperbolic space. An ${\mathbf{R}}$-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. It follows that an ${\mathbf{R}}$-tree is locally arcwise connected, contractible, and one-dimensional. Unique and local arcwise connectivity characterize ${\mathbf{R}}$-trees among metric spaces. A universal ${\mathbf{R}}$-tree would be of interest in attempting to classify the actions of groups of isometries on ${\mathbf{R}}$-trees. It is easy to see that there is no universal $ {\mathbf{R}}$-tree. However, we show that there is a universal separable ${\mathbf{R}}$-tree $ {T_{{\aleph _0}}}$ . Moreover, for each cardinal $\alpha ,3 \leq \alpha \leq {\aleph _0}$ , there is a space ${T_\alpha } \subset {T_{{\aleph _0}}}$ , universal for separable $ {\mathbf{R}}$-trees, whose order of ramification is at most $\alpha$ . We construct a universal smooth dendroid $D$ such that each separable ${\mathbf{R}}$-tree embeds in $D$ ; thus, has a smooth dendroid compactification. For nonseparable $ {\mathbf{R}}$-trees, we show that there is an $ {\mathbf{R}}$-tree ${X_\alpha }$ , such that each ${\mathbf{R}}$-tree of order of ramification at most $\alpha$ embeds isometrically into ${X_\alpha }$ . We also show that each ${\mathbf{R}}$-tree has a compactification into a smooth arboroid (a nonmetric dendroid). We conclude with several examples that show that the characterization of $ {\mathbf{R}}$-trees among metric spaces, rather than, say, among first countable spaces, is the best that can be expected.
Geometric aspects of reduction of order
James
Sherring;
Geoff
Prince
433-453
Abstract: Using the differential geometry of vectorfields and forms we reinterpret and extend the traditional idea of an integrating factor for a first order differential equation with symmetry. In particular, we provide a simple and manifestly geometric approach to reduction of order via symmetry for ordinary differential equations which largely obviates the necessity for canonical coordinates and the associated quotient manifolds. In so doing, some new results which generalise the class of Lie group actions which can be used to solve ordinary differential equations are developed.
Stable-homotopy and homology invariants of boundary links
Michael
Farber
455-477
Abstract: An $n$-dimensional $ (n \geq 5)$ link in the $ (n + 2)$-dimensional sphere is stable if the $i$th homotopy group of its complement $ X$ vanishes for $2 \leq i \leq (n + 1)/3$ and ${\pi _1}(X)$ is freely generated by meridians. In this paper a classification of stable links in terms of stable homotopy theory is given. For simple links this classification gives a complete algebraic description. We also study Poincaré duality in the space of the free covering of the complement of a boundary link. The explicit computation of the corresponding Ext-functors gives a construction of new homology pairings, generalizing the Blanchfield and the torsion pairings for knots.
Differentials of complex interpolation processes for K\"othe function spaces
N. J.
Kalton
479-529
Abstract: We continue the study of centralizers on Köthe function spaces and the commutator estimates they generate (see [29]). Our main result is that if $X$ is a super-reflexive Köthe function space then for every real centralizer $\Omega$ on $X$ there is a complex interpolation scale of Köthe function spaces through $X$ inducing $\Omega$ as a derivative, up to equivalence and a scalar multiple. Thus, in a loose sense, all real centralizers can be identified with derivatives of complex interpolation processes. We apply our ideas in an appendix to show, for example, that there is a twisted sum of two Hilbert spaces which fails to be a $ ({\text{UMD}})$-space.
On automorphisms of Markov chains
Wolfgang
Krieger;
Brian
Marcus;
Selim
Tuncel
531-565
Abstract: We prove several theorems about automorphisms of Markov chains, using the weight-per-symbol polytope.
Equivariant cohomology and lower bounds for chromatic numbers
Igor
Kříž
567-577
Abstract: We introduce a general topological method for obtaining a lower bound of the chromatic number of an $n$-graph. We present numerical lower bounds for intersection $n$-graphs.
Lannes' $T$ functor on summands of $H\sp *(B({\bf Z}/p)\sp s)$
John C.
Harris;
R. James
Shank
579-606
Abstract: Let $H$ be the $\bmod$-$p$ cohomology of the classifying space $B({\mathbf{Z}}/p)$ thought of as an object in the category, $ \mathcal{U}$, of unstable modules over the Steenrod algebra. Lannes constructed a functor $T:\mathcal{U} \to \mathcal{U}$ which is left adjoint to the functor $A \mapsto A \otimes H$. In this paper we evaluate $ T$ on the indecomposable $\mathcal{U}$-summands of ${H^{ \otimes s}}$, the tensor product of $ s$ copies of $ H$. Our formula involves the composition factors of certain tensor products of irreducible representations of the semigroup ring $ {{\mathbf{F}}_p}[{{\mathbf{M}}_{s,}}_s({\mathbf{Z}}/p)]$. The main application is to determine the homotopy type of the space of maps from $ B({\mathbf{Z}}/p)]$ to $ X$ when $X$ is a wedge summand of the space $\Sigma (B{({\mathbf{Z}}/p)^s})$.
A Fatou theorem for the solution of the heat equation at the corner points of a cylinder
Kin Ming
Hui
607-642
Abstract: In this paper the author proves existence and uniqueness of the initial-Dirichlet problem for the heat equation in a cylindrical domain $D \times (0,\infty )$ where $ D$ is a bounded smooth domain in ${R^n}$ with zero lateral values. A unique representation of the strong solution is given in terms of measures $\mu$ on $D$ and $\lambda$ on $ \partial D$. We also show that the strong solution $u(x,t)$ of the heat equation in a cylinder converges a.e. ${x_0} \in \partial D \times \{ 0\}$ as $(x,t)$ converges to points on $\partial D \times \{ 0\}$ along certain nontangential paths.
On the $\Theta$-function of a Riemannian manifold with boundary
Pei
Hsu
643-671
Abstract: Let $\Omega$ be a compact Riemannian manifold of dimension $n$ with smooth boundary. Let ${\lambda _1} < {\lambda _2} \leq \cdots$ be the eigenvalues of the Laplace-Beltrami operator with the boundary condition $ [\partial /\partial n + \gamma ]\phi = 0$ . The associated $\Theta $-function ${\Theta _\gamma }(t) = \sum\nolimits_{n = 1}^\infty {\exp [ - {\lambda _n}t]}$ has an asymptotic expansion of the form $\displaystyle {(4\pi t)^{n/2}}{\Theta _\gamma }(t) = {a_0} + {a_1}{t^{1/2}} + {a_2}t + {a_3}{t^{3/2}} + {a_4}{t^2} + \cdots .$ The values of $ {a_0}$ , ${a_1}$ are well known. We compute the coefficients ${a_2}$ and ${a_3}$ in terms of geometric invariants associated with the manifold by studying the parametrix expansion of the heat kernel $p(t,x,y)$ near the boundary. Our method is a significant refinement and improvement of the method used in [McKean-Singer, J. Differential Geometry 1 (1969), 43-69].
Liouvillian first integrals of differential equations
Michael F.
Singer
673-688
Abstract: Liouvillian functions are functions that are built up from rational functions using exponentiation, integration, and algebraic functions. We show that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a liouvillian function of several variables vanishing on the curve defined by this solution, then the system has a liouvillian first integral, that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set. We can refine this result in special cases to show that the first integral must be of a very special form. For example, we can show that if the system $dx/dz = P(x,y)$, $ dy/dz = Q(x,y)$ has a solution $(x(z),y(z))$ satisfying a liouvillian relation then either $x(z)$ and $y(z)$ are algebraically dependent or the system has a liouvillian first integral of the form $F(x,y) = \smallint RQ\,dx - RP\,dy$ where $R = \exp (\smallint U\,dx + V\,dy)$ and $U$ and $V$ rational functions of $x$ and $y$ . We can also reprove an old result of Ritt stating that a second order linear differential equation has a nonconstant solution satisfying a liouvillian relation if and only if all of its solutions are liouvillian.
Complemented ideals in the Fourier algebra and the Radon Nikod\'ym property
Brian
Forrest
689-700
Abstract: Necessary and sufficient conditions are given for an ideal $I(H)$ of the Fourier algebra to be complemented when $H$ is a closed subgroup of $G$ . Using the Radon Nikodym property, an example of a group $G$ with a normal abelian subgroup $H$ for which $I(H)$ is not complemented is presented.
Comparing periodic orbits of maps of the interval
C.
Bernhardt;
E.
Coven;
M.
Misiurewicz;
I.
Mulvey
701-707
Abstract: Let $\pi$ and $\theta$ be cyclic permutations of finite ordered sets. We say that $\pi$ forces $\theta$ if every continuous map of the interval which has a representative of $ \pi$ also has one of $ \theta$. We give a geometric version of Jungreis' combinatorial algorithm for deciding in certain cases whether $\pi$ forces $\theta$ .
A topological method for bounded solutions of nonautonomous ordinary differential equations
James R.
Ward
709-720
Abstract: The existence of bounded solutions to nonlinear nonautonomous ordinary differential equations is studied. This is done by associating the given equation to nonlinear autonomous ones by means of a family of skew-product flows related by homotopy. The existence of a bounded solution to the original differential equation is then related to the nontriviality of a certain Conley index associated with the autonomous differential equations. The existence of nontrivial bounded solutions is also considered. The differential equations studied are perturbations of homogeneous ones.
Galois groups of maximal $p$-extensions
Roger
Ware
721-728
Abstract: Let $p$ be an odd prime and $F$ a field of characteristic different from $p$ containing a primitive $p$th root of unity. Assume that the Galois group $G$ of the maximal $p$-extension of $F$ has a finite normal series with abelian factor groups. Then the commutator subgroup of $G$ is abelian. Moreover, $ G$ has a normal abelian subgroup with pro-cyclic factor group. If, in addition, $F$ contains a primitive ${p^2}$th root of unity then $G$ has generators ${\{ x,{y_i}\} _{i \in I}}$ with relations $ {y_i}{y_j} = {y_j}{y_i}$ and $ x{y_i}{x^{ - 1}} = y_i^{q + 1}$ where $q = 0$ or $q = {p^n}$ for some $n \geq 1$. This is used to calculate the cohomology ring of $G$, when $G$ has finite rank. The field $F$ is characterized in terms of the behavior of cyclic algebras (of degree $ p$) over finite $ p$-extensions.
The index of a Brauer class on a Brauer-Severi variety
Aidan
Schofield;
Michel
Van den Bergh
729-739
Abstract: Let $D$ and $E$ be central division algebras over $k$; let $K$ be the generic splitting field of $ E$; we show that the index of $D{ \otimes _k}K$ is the minimum of the indices of $D \otimes {E^{ \otimes i}}$ as $i$ varies. We use this to calculate the index of $D$ under related central extensions and to construct division algebras with special properties.
A spectral sequence for pseudogroups on ${\bf R}$
Solomon M.
Jekel
741-749
Abstract: Consider a pseudogroup $P$ of local homeomorphisms of $\mathbb{R}$ satisfying the following property: given points $ {x_0} < \cdots < {x_p}$ and $ {y_0} < \cdots < {y_p}$ in $ \mathbb{R}$ , there is an element of $P$, with domain an interval containing $[{x_0},{x_p}]$, taking each ${x_i}$ to ${y_i}$. The pseudogroup ${P^r}$ of local ${C^r}$ homeomorphisms, $0 \leq r \leq \infty$ , is of this type as is the pseudogroup $ {P^\omega }$ of local real-analytic homeomorphisms. Let $\Gamma$ be the topological groupoid of germs of elements of $P$. We construct a spectral sequence which involves the homology of a sequence of discrete groups $\{ {G_p}\}$. Consider the set $\{ f \in P\vert f(i) = i,i = 0,1, \ldots ,p\}$,; define $f\sim g$ if $f$ and $g$ agree on a neighborhood of $[0,p] \subset \mathbb{R}$. The equivalence classes under composition form the group $ {G_p}$. Theorem: There is a spectral sequence with $E_{p,q}^1 = {H_q}(B{G_p})$ which converges to $ {H_{p + q}}(B\Gamma )$. Our spectral sequence can be considered to be a version which covers the realanalytic case of some well-known theorems of J. Mather and G. Segal. The article includes some observations about how the spectral sequence applies to $ B\Gamma _1^\omega$. Further applications will appear separately.
Fourier inequalities with nonradial weights
C.
Carton-Lebrun
751-767
Abstract: Let $\mathcal{F}\;f(\gamma ) = {\smallint _{{\mathbb{R}^n}}}({e^{ - 2i\pi \gamma \bullet x}} - 1)f(x)\,dx,n > 1$, and $u$, $v$ be nonnegative functions. Sufficient conditions are found in order that $\left\Vert \mathcal{F}\;f\right\Vert _{q,u} \leq C\left\Vert f\right\Vert _{p,v}$ for all $f \in L_v^p({\mathbb{R}^n})$. Pointwise and norm approximations of $\mathcal{F}\;f$ are derived. Similar results are obtained when $u$ is replaced by a measure weight. In the case $v(x) = \vert x{\vert^{n(p - 1)}}$, a counterexample is given which shows that no Fourier inequality can hold for all $f$ in $ L_{c,0}^\infty$. Spherical restriction theorems are established. Further conditions for the boundedness of $ \mathcal{F}$ are discussed.
Radii of convergence and index for $p$-adic differential operators
Paul Thomas
Young
769-785
Abstract: We study the radii of $p$-adic convergence of solutions at a generic point of homogeneous linear differential operators whose coefficients are analytic elements. As an application we prove a conjecture of P. Robba (for a certain class of operators) concerning the relation between radii of convergence and index on analytic elements. We also give an explicit factorization theorem for $ p$-adic differential operators, based on the radii of generic convergence and the slopes of the associated Newton polygon.
The set of all iterates is nowhere dense in $C([0,1],[0,1])$
A. M.
Blokh
787-798
Abstract: We prove that if a mixing map $ f:[0,1] \to [0,1]$ belongs to the ${C^0}$-closure of the set of iterates and $f(0) \ne 0$, $ f(1) \ne 1$ then $ f$ is an iterate itself. Together with some one-dimensional techniques it implies that the set of all iterates is nowhere dense in $ C([0,1],[0,1])$ giving the final answer to the question of A. Bruckner, P. Humke and M. Laczkovich.
Bounded analytic functions on two sheeted discs
Mikihiro
Hayashi;
Mitsuru
Nakai;
Shigeo
Segawa
799-819
Abstract: Results of qualitative nature of both positive and negative directions on the point separation by bounded analytic functions of smooth subregions of two sheeted discs are given when two sheeted discs themselves are not separated by bounded analytic functions. We are, in particular, concerned about roles of branch points in two sheeted discs played in the point separation by bounded analytic functions.
Uniqueness in Cauchy problems for hyperbolic differential operators
Christopher D.
Sogge
821-833
Abstract: In this paper we prove a unique continuation theorem for second order strictly hyperbolic differential operators. Results also hold for higher order operators if the hyperbolic cones are strictly convex. These results are proved via certain Carleman inequalities. Unlike [6], the parametrices involved only have real phase functions, but they also have Gaussian factors. We estimate the parametrices and associated remainders using sharp $ {L^p}$ estimates for Fourier integral operators which are due to Brenner [1] and Seeger, Stein, and the author [5].
Unknotted solid tori and genus one Whitehead manifolds
Edward M.
Brown
835-847
Abstract: In this paper we study contractible open $3$-manifolds which are monotone unions of solid tori and which embed in a compact $3$-manifold. We show that the tori are unknotted in later tori. We then study pairs of unknotted solid tori, and prove a unique prime decomposition theorem. This is applied to the open $3$-manifolds above to get an essentially unique prime decomposition. A number of examples in the literature are analyzed, and some new examples are constructed.
Generalized Toda brackets and equivariant Moore spectra
Steven R.
Costenoble;
Stefan
Waner
849-863
Abstract: In this paper we develop a general theory of obstructions to the existence of equivariant Moore spectra. The obstructions we obtain coincide with higher order Toda brackets as defined by Spanier. We then apply the theory to show the existence of equivariant Moore spectra in various special cases.
A recurrent nonrotational homeomorphism on the annulus
Robbert J.
Fokkink;
Lex G.
Oversteegen
865-875
Abstract: We construct an area- and orientation-preserving recurrent diffeomorphism on the annulus without periodic points, which is not conjugate to a rotation. The mapping is, however, semiconjugate to an irrational rotation of a circle. Our example is a counterexample to the Birkhoff Conjecture.
The minimal degree of a finite inverse semigroup
Boris M.
Schein
877-888
Abstract: The minimal degree of an inverse semigroup $S$ is the minimal cardinality of a set $ A$ such that $ S$ is isomorphic to an inverse semigroup of one-to-one partial transformations of $ A$. The main result is a formula that expresses the minimal degree of a finite inverse semigroup $S$ in terms of certain subgroups and the ordered structure of $S$. In fact, a representation of $ S$ by one-to-one partial transformations of the smallest possible set $ A$ is explicitly constructed in the proof of the formula. All known and some new results on the minimal degree follow as easy corollaries.
Complete nonorientable minimal surfaces in ${\bf R}\sp 3$
Tōru
Ishihara
889-901
Abstract: We will study complete minimal immersions of nonorientable surfaces into $ {R^3}$. Especially, we construct a nonorientable surface ${P_2}$ which is homeomorphic to a Klein bottle and show that for any integer $m \geq 4$, there are complete minimal immersion of $ M = {P_2} - \{ q\}$, $q \in {P_2}$ in ${R^3}$ with one end and total curvature $C(M) = - 4m\pi$.
Nilpotence and torsion in the cohomology of the Steenrod algebra
Kenneth G.
Monks
903-912
Abstract: In this paper we prove the existence of global nilpotence and global torsion bounds for the cohomology of any finite Hopf subalgebra of the Steenrod algebra for the prime $2$. An explicit formula for computing such bounds is then obtained. This is used to compute bounds for ${H^{\ast} }({\mathcal{A}_n})$ for $n \leq 6$.
The uniqueness and stability of the solution of the Riemann problem of a system of conservation laws of mixed type
Hai Tao
Fan
913-938
Abstract: We establish the uniqueness and stability of the similarity solution of the Riemann problem for a $2 \times 2$ system of conservation laws of mixed type, with initial data separated by the elliptic region, which satisfies the viscosity-capillarity travelling wave admissibility criterion.
Corrigendum to: ``Characterizations of turbulent one-dimensional mappings via $\omega$-limit sets'' [Trans. Amer. Math. Soc. {\bf 326} (1991), no. 1, 261--280; MR1010884 (91j:58133)]
Michael J.
Evans;
Paul D.
Humke;
Cheng Ming
Lee;
Richard J.
O’Malley
939-940
Cosmic approximate limits and fixed points
J.
Segal;
T.
Watanabe
1-61
Abstract: We define a condition for approximate inverse systems which implies that the limit of the system has the fixed point property. Moreover, this condition is categorical in the approximate shape category. We investigate the class of complex projective $n$-space like continua with respect to the fixed point property by means of this condition. As a further application we show that the hyperspace $ C(X)$ of nonempty subcontinua of an arc-like or circle-like Hausdorff continuum $X$ has the fixed point property. We also prove that $ {2^X}$ and $C(X)$ have the fixed point property for any locally connected Hausdorff continuum $ X$.
On the K\"unneth formula for intersection cohomology
Daniel C.
Cohen;
Mark
Goresky;
Lizhen
Ji
63-69
Abstract: We find the natural perversity functions for which intersection cohomology satisfies the Künneth formula.
Homotopy invariants of nonorientable $4$-manifolds
Myung Ho
Kim;
Sadayoshi
Kojima;
Frank
Raymond
71-81
Abstract: We define a $ {{\mathbf{Z}}_4}$-quadratic function on ${\pi _2}$ for nonorientable $4$-manifolds and show that it is a homotopy invariant. We then use it to distinguish homotopy types of certain manifolds that arose from an analysis of toral action on nonorientable $4$-manifolds.
The level manifold of a generalized Toda equation hierarchy
Yoshimasa
Nakamura
83-94
Abstract: The finite nonperiodic Toda lattice equation induces a linear one-parameter flow on a space of rational functions. The level manifold of the Toda equation hierarchy is shown to be a product of lines. Our main results establish a generalization of this Toda hierarchy which will be called the cyclic-Toda hierarchy. It is proved that the cyclic-Toda hierarchy is completely integrable and its level manifold is diffeomorphic to a disjoint union of cylinders.
The classification of spinors under ${\rm GSpin}\sb {14}$ over finite fields
Xiao-Wei
Zhu
95-114
Abstract: The spinors of a $ 14$-dimensional vector space $V$ are studied with respect to the group $ \operatorname{GSpin}_{14}$ of the $14$-dimensional vector space $V$ over finite fields ${{\mathbf{F}}_q}$. Results are given as follows: (1) the decomposition of the space of spinors into $ \operatorname{GSpin}_{14}$-equivalence classes or "orbits" over ${{\mathbf{F}}_q}$, (2) the structure of the fixer of $ \operatorname{GSpin}_{14}$ for each orbit as an $ {{\mathbf{F}}_q}$-group.
$v\sb 1$-periodic homotopy groups of exceptional Lie groups: torsion-free cases
Martin
Bendersky;
Donald M.
Davis;
Mamoru
Mimura
115-135
Abstract: The $ {v_1}$-periodic homotopy groups $ v_1^{ - 1}{\pi _ {\ast} }(X;p)$ are computed explicitly for all pairs $ (X,p)$, where $ X$ is an exceptional Lie group whose integral homology has no $p$-torsion. This yields new lower bounds for $p$-exponents of actual homotopy groups of these spaces. Delicate calculations with the unstable Novikov spectral sequence are required in the proof.
Noncommutative matrix Jordan algebras
Robert B.
Brown;
Nora C.
Hopkins
137-155
Abstract: We consider noncommutative degree two Jordan algebras $\mathcal{J}$ of two by two matrices whose off diagonal entries are from an anticommutative algebra $\mathcal{S}$ . We give generators and relations for the automorphism group of $ \mathcal{J}$ and determine the derivation algebra Der $ \mathcal{J}$ in terms of mappings on $ \mathcal{S}$ . We also give an explicit construction of all $\mathcal{S}$ for which Der $\mathcal{J}$ does not kill the diagonal idempotents and give conditions for nonisomorphic $\mathcal{S}$ 's to give isomorphic $\mathcal{J}$ 's.
Fourier multipliers on Lipschitz curves
Alan
McIntosh;
Tao
Qian
157-176
Abstract: We develop the theory of Fourier multipliers acting on ${L_p}(\gamma )$ where $\gamma$ is a Lipschitz curve of the form $\gamma = \{ x + ig(x)\}$ with $\left\Vert g\right\Vert _\infty < \infty$ and $\left\Vert g\prime\right\Vert _\infty < \infty$ . The aim is to better understand convolution singular integrals $B$ defined naturally on such curves by $\displaystyle Bu(z) = {\text{p.v.}}\int_\gamma {\varphi (z - \zeta )u(\zeta )d\zeta }$ for almost all $ z \in \gamma$ .
Boundaries of Markov partitions
Jonathan
Ashley;
Bruce
Kitchens;
Matthew
Stafford
177-201
Abstract: The core of a Markov partition is the nonwandering set of the map restricted to the boundary of the partition. We show that the core of a Markov partition is always a finitely presented system. Then we show that every one sided sofic system occurs as the core of a Markov partition for an $n$-fold covering map on the circle and every two sided sofic system occurs as the core of a Markov partition for a hyperbolic automorphism of the two dimensional torus.
Removing point singularities of Riemannian manifolds
P. D.
Smith;
Deane
Yang
203-219
Abstract: We study the behavior of geodesics passing through a point singularity of a Riemannian manifold. In particular, we show that if the curvature does not blow up too rapidly near the singularity, then the singularity is at worst an orbifold singularity. The idea is to construct the exponential map centered at a singularity. Since there is no tangent space at the singularity, a surrogate is needed. We show that the vector space of radially parallel vector fields is well defined and that there is a correspondence between unit radially parallel vector fields and geodesics emanating from the singular point.
Capacitability theorem in measurable gambling theory
A.
Maitra;
R.
Purves;
W.
Sudderth
221-249
Abstract: A player in a measurable gambling house $\Gamma$ defined on a Polish state space $ X$ has available, for each $x \in X$, the collection $\Sigma (x)$ of possible distributions $\sigma$ for the stochastic process ${x_1},{x_2}, \ldots$ of future states. If the object is to control the process so that it will lie in an analytic subset $A$ of $H = X \times X \times \cdots$, then the player's optimal reward is $\displaystyle M(A)(x) = \sup \{ \sigma (A):\sigma \in \Sigma (x)\}.$ The operator $M( \bullet )(x)$ is shown to be regular in the sense that $\displaystyle M(A)(x) = \inf M(\{ \tau < \infty \} )(x),$ where the infimum is over Borel stopping times $ \tau$ such that $ A \subseteq \{ \tau < \infty \}$. A consequence of this regularity property is that the value of $M(A)(x)$ is unchanged if, as in the gambling theory of Dubins and Savage, the player is allowed to use nonmeasurable strategies. This last result is seen to hold for bounded, Borel measurable payoff functions including that of Dubins and Savage.
The regular module problem. I
T. R.
Berger;
B. B.
Hargraves;
C.
Shelton
251-274
Abstract: In the study of induced representations the following problem arises: Let $H = AG$ be a finite solvable group and ${\mathbf{k}}$ a field with $ \operatorname{char}{\mathbf{k}}\nmid\; \vert A\vert$. Let $V$ be an irreducible, faithful, primitive $ {\mathbf{k}}[AG]$-module. Suppose $H$ contains a normal extraspecial $r$-subgroup $R$ with $ Z(R) \leq Z(H)$ and that $ A$ acts faithfully on $ R$. Under what conditions does $A$ have a regular direct summand in $V$? In this paper we consider this question under the hypotheses that $ G = MR$, where $ 1 \ne M$ is normal abelian in $AM$, $A$ is nilpotent, $(\vert A\vert,\vert MR\vert) = (\vert M\vert,\vert R\vert) = 1$ , and $R/Z(R)$ is a faithful, irreducible $AM$-module. We show that $A$ has at least three regular direct summands in $V$ unless $\vert A\vert$, $\exp (M)$, and $r$ satisfy certain very restrictive conditions.
Hilbert's tenth problem for rings of algebraic functions in one variable over fields of constants of positive characteristic
Alexandra
Shlapentokh
275-298
Abstract: The author builds an undecidable model of integers with certain relations and operations in the rings of $S$-integers of algebraic function fields in one variable over fields of constants of positive characteristic, in order to show that Hilbert's Tenth Problem has no solution there.
R\'esolvant g\'en\'eralis\'e et s\'eparation des points singuliers quasi-Fredholm
J.-Ph.
Labrousse;
M.
Mbekhta
299-313
Abstract: C. Apostol et K. Clancey (Trans. Amer. Math. Soc. 215 (1976), 293-300), ont introduit la notion de "projection spectrale généralisée". Cette notion permet, en particulier, de séparer les ensembles finis de points singuliers dans le domaine semi-Fredholm $({\rho _{s\phi }}(A))$ d'un opérateur $ A$ borné dans un Hilbert H. Dans ce travail, on se propose de généraliser ce résultat au domaine quasi-Fredholm de $A({\rho _{q\phi }}(A))$, et pour cela, nous donnons une nouvelle représentation triangulaire du type d'Apostol. D'autre part on construit, pour un opérateur fermé à domaine dense dans $H$ , un résolvant généralisé vérifiant l'identité de la résolvante et analytique dans le domaine régulier de Fredholm de $A(\rho _\phi ^r(A))$ sauf éventuellement sur un ensemble au plus dénombrable de points situés prés de la frontière de ${\rho _\phi }(A)$.
Adjoint transform, overconvexity and sets of constant width
François
Bavaud
315-324
Abstract: The properties of the adjoint transform (associating to a set the intersection of all disks of given radius centered in the set) are systematically investigated, in particular its relationship with the overconvex, the parallelisation and completion of sets. Sets conjugate by the transform can be characterised in a new way as the union or the intersection of all completions of the reference body. New relationships satisfied by their areas and perimeters are derived. Two applications in problems of random intersection of disks are finally treated.
Distortion theorems for Bloch functions
Xiang Yang
Liu;
David
Minda
325-338
Abstract: We establish various distortion theorems for both normalized locally schlicht Bloch functions and normalized Bloch function with branch points. These distortion theorems give lower bounds on either $ \vert f\prime(z)\vert$ or $ \operatorname{Re} f\prime(z)$; most of our distortion theorems are sharp and all extremal functions identified. The main tools used in establishing these distortion theorems are the classical form of Julia's Lemma and a new version of Julia's Lemma that applies to certain multiple-valued analytic functions. As applications of these distortion theorems, we obtain known lower bounds for various Bloch constants and also establish improved lower bounds on a number of Marden constants for Bloch, normal and Yosida functions.
Uniqueness of radial solutions of semilinear elliptic equations
Man Kam
Kwong;
Yi
Li
339-363
Abstract: E. Yanagida recently proved that the classical Matukuma equation with a given exponent has only one finite mass solution. We show how similar ideas can be exploited to obtain uniqueness results for other classes of equations as well as Matukuma equations with more general coefficients. One particular example covered is $\Delta u + {u^p} \pm u = 0$, with $p > 1$. The key ingredients of the method are energy functions and suitable transformations. We also study general boundary conditions, using an extension of a recent result by Bandle and Kwong. Yanagida's proof does not extend to solutions of Matukuma's equation satisfying other boundary conditions. We treat these with a completely different method of Kwong and Zhang.
Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem
Tzong-Yow
Lee;
Wei-Ming
Ni
365-378
Abstract: We investigate the behavior of the solution $u(x,t)$ of $\displaystyle \left\{ {\begin{array}{*{20}{c}} {\frac{{\partial u}} {{\partial ... ...hi (x)} & {{\text{in}}\;{\mathbb{R}^n},} \end{array} } \right.$ where $\Delta = \sum\nolimits_{i = 1}^n {{\partial ^2}/\partial _{{x_i}}^2}$ is the Laplace operator, $p > 1$ is a constant, $T > 0$, and $ \varphi$ is a nonnegative bounded continuous function in ${\mathbb{R}^n}$. The main results are for the case when the initial value $\varphi$ has polynomial decay near $x = \infty$. Assuming $\varphi \sim \lambda {(1 + \vert x\vert)^{ - a}}$ with $\lambda$, $a > 0$, various questions of global (in time) existence and nonexistence, large time behavior or life span of the solution $u(x,t)$ are answered in terms of simple conditions on $\lambda$, $a$, $p$ and the space dimension $n$.
Examples of capacity for some elliptic operators
Jang-Mei
Wu
379-395
Abstract: We study $ L$-capacities for uniformly elliptic operators of nondivergence form $\displaystyle L = \sum\limits_{i,j} {{a_{ij}}(x)\frac{{{\partial ^2}}} {{\parti... ...rtial {x_j}}} + } \sum\limits_j {{a_j}(x)\frac{\partial } {{\partial {x_j}}};}$ and construct examples of large sets having zero $L$-capacity for some $L$ , and small sets having positive $ L$-capacity. The relations between ellipticity constants of the coefficients and the sizes of these sets are also considered.
Connected simple systems, transition matrices, and heteroclinic bifurcations
Christopher
McCord;
Konstantin
Mischaikow
397-422
Abstract: Given invariant sets $A$, $B$ , and $C$ , and connecting orbits $A \to B$ and $B \to C$, we state very general conditions under which they bifurcate to produce an $A \to C$ connecting orbit. In particular, our theorem is applicable in settings for which one has an admissible semiflow on an isolating neighborhood of the invariant sets and the connecting orbits, and for which the Conley index of the invariant sets is the same as that of a hyperbolic critical point. Our proof depends on the connected simple system associated with the Conley index for isolated invariant sets. Furthermore, we show how this change in connected simple systems can be associated with transition matrices, and hence, connection matrices. This leads to some simple examples in which the nonuniqueness of the connection matrix can be explained by changes in the connected simple system.
Some integrable subalgebras of the Lie algebras of infinite-dimensional Lie groups
J.
Leslie
423-443
Abstract: This paper gives a proof of Lie's second fundamental theorem in the context of infinite dimensional Lie groups; that is, we define a class of Lie subalgebras of the Lie algebra of a large class of infinite dimensional Lie groups, say $G$ , which can be realized as the Lie algebras of Lie subgroups of $G$ .
On $q$-analogues of the Fourier and Hankel transforms
Tom H.
Koornwinder;
René F.
Swarttouw
445-461
Abstract: For H. Exton's $ q$-analogue of the Bessel function (going back to W. Hahn in a special case, but different from F. H. Jackson's $q$-Bessel functions) we derive Hansen-Lommel type orthogonality relations, which, by a symmetry, turn out to be equivalent to orthogonality relations which are $q$-analogues of the Hankel integral transform pair. These results are implicit, in the context of quantum groups, in a paper by Vaksman and Korogodskiĭ. As a specialization we get ($q$-cosines and $q$-sines which admit $q$-analogues of the Fourier-cosine and Fourier-sine transforms. We also get a formula which is both an analogue of Graf's addition formula and of the Weber-Schafheitlin discontinuous integral.
An analogue of Siegel's $\phi$-operator for automorphic forms for ${\rm GL}\sb n({\bf Z})$
Douglas
Grenier
463-477
Abstract: If $\mathcal{S}{P_n}$ is the symmetric space of $n \times n$ positive matrices, $Y \in \mathcal{S}{P_n}$ can be decomposed into $\displaystyle Y = \left( {\begin{array}{*{20}{c}} 1 & 0 x & I \end{array}... ...)\left( {\begin{array}{*{20}{c}} 1 & {{T_x}} 0 & I \end{array} } \right),$ where $W \in \mathcal{S}{P_{n - 1}}$ . By letting $v \to \infty$ we obtain the $\phi $-operator that attaches to every automorphic form for $G{L_n}(\mathbb{Z})$, $f(Y)$, an automorphic form for $G{L_{n - 1}}(\mathbb{Z})$, $f\vert\phi (W)$.
Groups of piecewise linear homeomorphisms
Melanie
Stein
477-514
Abstract: In this paper we study a class of groups which may be described as groups of piecewise linear bijections of a circle or of compact intervals of the real line. We use the action of these groups on simplicial complexes to obtain homological and combinatorial information about them. We also identify large simple subgroups in all of them, providing examples of finitely presented infinite simple groups.
The Gauss map for K\"ahlerian submanifolds of ${\bf R}\sp n$
Marco
Rigoli;
Renato
Tribuzy
515-528
Abstract: We introduce a Gauss map for Kähler submanifolds of Euclidean space and study its geometry in relation to that of the given immersion. In particular we generalize a number of results of the classical theory of minimal surfaces in Euclidean space.
Hessenberg varieties
F.
De Mari;
C.
Procesi;
M. A.
Shayman
529-534
Abstract: Numerical algorithms involving Hessenberg matrices correspond to dynamical systems which evolve on the subvariety of complete flags ${S_1} \subset {S_2} \subset \cdots \subset {S_{n - 1}}$ in $ {\mathbb{C}^n}$ satisfying the condition $s({S_i}) \subset {S_{i + 1}}$, $\forall i$, where $s$ is an endomorphism of ${\mathbb{C}^n}$. This paper describes the basic topological features of the generalization to subvarieties of $G/B$, $G$ a complex semisimple algebraic group, which are indexed by certain subsets of negative roots. In the special case where the subset consists of the negative simple roots, the variety coincides with the torus embedding associated to the decomposition into Weyl chambers.
The structure of solutions of a semilinear elliptic equation
Kuo-Shung
Cheng;
Tai Chia
Lin
535-554
Abstract: We give a complete classification of solutions of the elliptic equation $ \Delta u + K(x){e^{2u}} = 0$ in $ \mathbb{R}^n, n \geq 3$, for some interesting cases of $K$.
Automorphisms of the lattice of recursively enumerable sets: promptly simple sets
P.
Cholak;
R.
Downey;
M.
Stob
555-570
Abstract: We show that for every coinfinite r.e. set $A$ there is a complete r.e. set $B$ such that ${\mathcal{L}^{\ast} }(A){ \approx _{{\text{eff}}}}{\mathcal{L}^{\ast} }(B)$ and that every promptly simple set is automorphic (in ${\mathcal{E}^{\ast} }$) to a complete set.
$N$-body observables in the Calkin algebra
Jan
Dereziński
571-582
Abstract: The commutators of many operators which are used in the phase space analysis of the $N$-body scattering are compact. This fact makes it possible to give a description of certain classes of such operators in terms of commutative ${C^{\ast} }$-algebras inside the Calkin algebra.
Admissible boundary values of bounded holomorphic functions in wedges
Franc
Forstnerič
583-593
Abstract: If $M \subset {\mathbb{C}^N}$ is a generic Cauchy-Riemann manifold and $\mathcal{W} \subset {\mathbb{C}^N}$ is a wedge domain with edge $M$, then every bounded holomorphic function on $\mathcal{W}$ has an admissible limit at almost every point of $M$. Moreover, if a holomorphic function $ f$ on $\mathcal{W}$ has a distribution boundary value $ (\operatorname{bv}\;f)$ on $ M$ that is a bounded measurable function, then $f$ is bounded on every finer wedge near $ M$ , and its admissible limit equals $ (\operatorname{bv}\;f)(p)$ at almost every point $p \in M$.
Isometries of CSL algebras
Baruch
Solel
595-606
Abstract: We show that every Jordan isomorphism of CSL algebras, whose restriction to the diagonal of the algebra is a selfadjoint map, is the sum of an isomorphism and an anti-isomorphism. It follows that every surjective linear isometry of CSL algebras is the sum of an isomorphism and an anti-isomorphism, followed by a unitary multiplication.
Harmonic maps into hyperbolic $3$-manifolds
Yair N.
Minsky
607-632
Abstract: High-energy degeneration of harmonic maps of Riemann surfaces into a hyperbolic $3$-manifold target is studied, generalizing results of [M1] in which the target is two-dimensional. The Hopf foliation of a high-energy map is mapped to an approximation of its geodesic representative in the target, and the ratio of the squared length of that representative to the extremal length of the foliation in the domain gives an estimate for the energy. The images of harmonic maps obtained when the domain degenerates along a Teichmüller ray are shown to converge generically to pleated surfaces in the geometric topology or to leave every compact set of the target when the limiting foliation is unrealizable.
Automorphisms of torsion-free nilpotent groups of class two
Manfred
Dugas;
Rüdiger
Göbel
633-646
Abstract: We construct $ 2$-divisible, torsion-free abelian groups $G$ admitting an alternating bilinear map. We use these groups $G$ to find nilpotent groups $N$ of class $2$ such that $ \operatorname{Aut}(N)$ modulo a natural normal subgroup is a prescribed group.
Finite semilattices whose monoids of endomorphisms are regular
M. E.
Adams;
Matthew
Gould
647-665
Abstract: A classification is obtained for the finite semilattices $S$ such that the monoid of endomorphisms of $S$ is regular in the semigroup-theoretic sense.
Homological theory of idempotent ideals
M.
Auslander;
M. I.
Platzeck;
G.
Todorov
667-692
Abstract: Let $\Lambda$ be an artin algebra $\mathfrak{A}$ and a two-sided ideal of $ \Lambda$. Then $\mathfrak{A}$ is the trace of a projective $ \Lambda$-module $ P$ in $\Lambda$. We study how the homological properties of the categories of finitely generated modules over the three rings $\Lambda /\mathfrak{A}$, $\Lambda$ and the endomorphism ring of $ P$ are related. We give some applications of the ideas developed in the paper to the study of quasi-hereditary algebras.
Classification of finite-dimensional universal pseudo-boundaries and pseudo-interiors
J. J.
Dijkstra;
J.
van Mill;
J.
Mogilski
693-709
Abstract: Let $n$ and $k$ be fixed integers such that $n \geq 1$ and $0 \leq k \leq n$. Let $B_k^n$ and $s_k^n$ denote the $k$-dimensional universal pseudo-boundary and the $ k$-dimensional universal pseudo-interior in $ {{\mathbf{R}}^n}$, respectively. The aim of this paper is to prove that $ B_k^n$ is homeomorphic to $ B_k^m$ if and only if $ s_k^n$ is homeomorphic to $ s_k^m$ if and only if $ n = m$ or $n$, $ m \geq 2k + 1$.
Accessible points of hereditarily decomposable chainable continua
Piotr
Minc;
W. R. R.
Transue
711-727
Abstract: In this paper it is proven that a chainable continuum $X$ can be embedded in the plane in such a way that every point is accessible from its complement if and only if it is Suslinean. An example is shown of an hereditarily decomposable chainable continuum which cannot be embedded in the plane in such a way that each endpoint is accessible.
Expansions of chromatic polynomials and log-concavity
Francesco
Brenti
729-756
Abstract: In this paper we present several results and open problems about logconcavity properties of sequences associated with graph colorings. Five polynomials intimately related to the chromatic polynomial of a graph are introduced and their zeros, combinatorial and log-concavity properties are studied. Four of these polynomials have never been considered before in the literature and some yield new expansions for the chromatic polynomial.
Quasidiagonality of direct sums of weighted shifts
Sivaram K.
Narayan
757-774
Abstract: Let $\mathcal{H}$ be a separable Hilbert space. A bounded linear operator $A$ defined on $ \mathcal{H}$ is said to be quasidiagonal if there exists a sequence $\{ {P_n}\}$ of projections of finite rank such that ${P_n} \to I$ strongly and $\left\Vert A{P_n} - {P_n}A\right\Vert \to 0$ as $n \to \infty $. We give a necessary and sufficient condition for a finite direct sum of weighted shifts to be quasidiagonal. The condition is stated using a marked graph (a graph with a $ (0)$, $( + )$ or $( - )$ attached to its vertices) that can be associated with the direct sum.
Existence of positive nonradial solutions for nonlinear elliptic equations in annular domains
Song-Sun
Lin
775-791
Abstract: We study the existence of positive nonradial solutions of equation $\Delta u + f(u) = 0$ in ${\Omega _a}$, $u = 0$ on $ \partial {\Omega _a}$, where ${\Omega _a} = \{ x \in {\mathbb{R}^n}:a < \vert x\vert < 1\}$ is an annulus in ${\mathbb{R}^n}$, $n \geq 2$, and $f$ is positive and superlinear at both 0 and $ \infty$. We use a bifurcation method to show that there is a nonradial bifurcation with mode $k$ at $ {a_k} \in (0,1)$ for any positive integer $k$ if $f$ is subcritical and for large $k$ if $f$ is supercritical. When $f$ is subcritical, then a Nehari-type variational method can be used to prove that there exists $ {a^{\ast} } \in (0,1)$ such that for any $ a \in ({a^{\ast} },1)$, the equation has a nonradial solution on ${\Omega _a}$.
$L\sp p$ estimates for the X-ray transform restricted to line complexes of Kirillov type
Hann Tzong
Wang
793-821
Abstract: Let there be given a piecewise continuous rectifiable curve $ \phi :{\mathbf{R}} \to {{\mathbf{R}}^n}$. Let ${G_{1,n}}({M_{1,n}})$ be the usual Grassmannian (bundle) in $ {{\mathbf{R}}^n}$. Define an $n$-dimensional submanifold ${M_\phi }({{\mathbf{R}}^n})$ of $ {M_{1,n}}$ as the set of all copies of ${G_{1,n}}$ along the curve $\phi$. Following Kirillov, we know that a nice function $f(x)$ can be recovered from its X-ray transform ${R_{1,n}}f$ on ${M_\phi }({{\mathbf{R}}^n})$ if and only if the curve $\phi$ intersects almost every affine hyperplane. Define a measure on ${M_\phi }({{\mathbf{R}}^n})$ by $ d\mu = d{\mu _x}(\pi )d\lambda (x)$, where $d{\mu _x}$ is the probability measure on $ {M_{1,n}}$ carried by the set of lines passing through the point $x$ and invariant under the stabilizer of $x$ in $O(n)$ and $d\lambda$ is the usual measure on $\phi$. We show that, if $n > 2$ and $\phi$ is unbounded, then $\left\Vert {R_{1,n}}f\right\Vert _{{L^q}({M_\phi }({{\mathbf{R}}^n}),d\mu )} \leq C\left\Vert f\right\Vert _{{L^p}({{\mathbf{R}}^n})}$ if and only if $p = q = n - 1$ and $\phi$ is line-like, that is, $\lambda (\phi \cap B(0;R)) = O(R)$. This result gives a classification of Kirillov line complexes in terms of ${L^p}$ estimates.
Harmonic localization of algebraic $K$-theory spectra
Stephen A.
Mitchell
823-837
Abstract: The Lichtenbaum-Quillen conjectures hold for the harmonic localization of the $K$-theory spectrum of a nice scheme. Various consequences of this fact are explored; for example, the harmonic localization of the $K$-theory of the integers at a regular prime is explicitly identified.
Existence of smooth solutions to the classical moment problems
Palle E. T.
Jorgensen
839-848
Abstract: Let $s(0),s(1), \ldots$ be a given sequence, and define $ s(n) = \overline {s( - n)}$ for $n < 0$. If $\Sigma \Sigma {\overline \xi _n}{\xi _m}s(m - n) \geq 0$ holds for all finite sequences $ {({\xi _n})_{n \in \mathbb{Z}}}$, then it is known that there is a positive Borel measure $\mu$ on the circle $ \mathbb{T}$ such that $s(n) = \smallint_{ - \pi }^\pi {{e^{int}}d\mu (t)}$, and conversely. Our main theorem provides a necessary and sufficient condition on the sequence $(s(n))$ that the measure $\mu$ may be chosen to be smooth. A measure $\mu$ is said to be smooth if it has the same spectral type as the operator $id/dt$ acting on $ {L^2}(\mathbb{T})$ with respect to Haar measure $dt$ on $ \mathbb{T}$: Equivalently, $ \mu$ is a superposition (possibly infinite) of measures of the form $\vert w(t){\vert^2}dt$ with $w \in {L^2}(\mathbb{T})$ such that $dw/dt \in {L^2}(\mathbb{T})$. The condition is stated purely in terms of the initially given sequence $(s(n))$: We show that a smooth representation exists if and only if, for some $\varepsilon \in {\mathbb{R}_ + }$, the a priori estimate $\displaystyle \sum {\sum {s(m - n){{\overline \xi }_n}{\xi _m} \geq \varepsilon {{\left\vert {\sum {ns(n){\xi _n}} } \right\vert}^2}} } $ is valid for all finite double sequences $({\xi _n})$. An analogous result is proved for the determinate (Hamburger) moment problem on the line. But the corresponding result does not hold for the indeterminate moment problem.
Orbits in unimodular Hermitian lattices
Donald G.
James
849-860
Abstract: Let $L$ be a unimodular indefinite hermitian lattice over the integers $\mathfrak{o}$ of an algebraic number field, and $ N(L,c)$ the number of primitive representations of $c \in \mathfrak{o}$ by $L$ that are inequalivant modulo the action of the integral special unitary group $SU(L)$ on $L$. The value of $N(L,c)$ is determined from the local representations via a product formula.
An extension of Attouch's theorem and its application to second-order epi-differentiation of convexly composite functions
René A.
Poliquin
861-874
Abstract: In 1977, Hedy Attouch established that a sequence of (closed proper) convex functions epi-converges to a convex function if and only if the graphs of the subdifferentials converge (in the Mosco sense) to the subdifferential of the limiting function and (roughly speaking) there is a condition that fixes the constant of integration. We show that the theorem is valid if instead one considers functions that are the composition of a closed proper convex function with a twice continuously differentiable mapping (in addition a constraint qualification is imposed). Using Attouch's Theorem, Rockafellar showed that second-order epi-differentiation of a convex function and proto-differentiability of the subdifferential set-valued mapping are equivalent, moreover the subdifferential of one-half the second-order epi-derivative is the proto-derivative of the subdifferential mapping; we will extend this result to the convexly composite setting.
Spectral multiplicity for ${\rm Gl}\sb n({\bf R})$
Jonathan
Huntley
875-888
Abstract: We study the behavior of the cuspidal spectrum of $\Gamma \backslash \mathcal{H}$, where $\mathcal{H}$ is associated to $\operatorname{Gl}_n(R)$ and $\Gamma$ is cofinite but not compact. By a technique that modifies the Lax-Phillips technique and uses ideas from wave equation techniques, if $ r$ is the dimension of $\mathcal{H}$, $ {N_\alpha }(\lambda )$ is the counting function for the Laplacian attached to a Hilbert space $ {H_\alpha }$, ${M_\alpha }(\lambda )$ is the multiplicity function, and ${H_0}$ is the space of cusp forms, we obtain the following results: Theorem 1. There exists a space of functions ${H^1}$, containing all cusp forms, such that $\displaystyle N\prime(\lambda ) = {C_r}({\text{Vol}}\;X){\lambda ^{\frac{r} {2}... ...{\frac{{r - 1}} {2}}}{\lambda ^{\frac{1} {{n + 1}}}}{(\log \lambda )^{n - 1}}).$ Theorem 2. $\displaystyle {M_0}(\lambda ) = O({\lambda ^{\frac{{r - 1}} {2}}}{\lambda ^{\frac{1} {{n + 1}}}}{(\log \lambda )^{n - 1}}).$
Nonsingular affine $k\sp *$-surfaces
Jean
Rynes
889-921
Abstract: Nonsingular affine ${k^{\ast} }$-surfaces are classified as certain invariant open subsets of projective ${k^{\ast}}$-surfaces. A graph is defined which is an equivariant isomorphism invariant of an affine ${k^{\ast}}$-surface. Over the complex numbers, it is proved that the only acyclic affine surface which admits an effective action of the group ${{\mathbf{C}}^{\ast} }$ is ${{\mathbf{C}}^2}$ which admits only linear actions of $ {{\mathbf{C}}^{\ast}}$.
Elementary proofs of the abstract prime number theorem for algebraic function fields
Wen-Bin
Zhang
923-937
Abstract: Elementary proofs of the abstract prime number theorem of the form $ \Lambda (m) = {q^m} + O({q^m}{m^{ - 1}})$ for algebraic function fields are given. The proofs use a refinement of a tauberian theorem of Bombieri.
Corrigendum to: ``Embedding graphs into colored graphs''
A.
Hajnal;
P.
Komjáth
A law of large numbers for fast price adjustment
H. Jerome
Keisler
1-51
Abstract: The purpose of this paper is to prove a law of large numbers for certain Markov processes involving large sets of weakly interacting particles. Consider a large finite set $ A$ of "particles" which move about in $m$-dimensional Euclidean space ${R^m}$. The particles interact with each other indirectly by means of an auxiliary quantity $ p$ in $d$-dimensional Euclidean space $ {R^d}$. At each time $ t$, a particle $a \in A$ is randomly selected and randomly jumps to a new location in $ {R^m}$ with a distribution depending on $p$ and its old location. At the same time, the value of $p$ changes to a new value depending on these same arguments. The parameter $p$ moves by a small amount at each time but moves fast compared to the average position of the particles. Under appropriate hypotheses on the rules of motion, we shall prove the following law of large numbers. For sufficiently large $A$, the value of $p$ will be close to its expected value with large probability, and the average position of the particles will be close to its expected value with large probability. The work was motivated by the problem of modelling the adjustment of prices in mathematical economics, where the particles $a \in A$ are agents in an exchange economy, the position of $a$ at time $t$ is the commodity vector held by agent $ a$ at time $t$, and $p$ is the price vector at time $t$.
Pseudo-isotopies of irreducible $3$-manifolds
Jeff
Kiralis
53-78
Abstract: It is shown that a certain subspace of the space of all pseudo-isotopies of any irreducible $3$-manifold is connected. This subspace consists of those pseudo-isotopies corresponding to $ 1$-parameter families of functions which have nondegenerate critical points of index $1$ and $2$ only and which contain no slides among the $ 2$-handles. Some of the techniques developed are used to prove a weak four-dimensional $h$-cobordism theorem.
Rational fibrations in differential homological algebra
Aniceto
Murillo
79-91
Abstract: In this paper, a result of [6] is generalized as follows: Given a fibration $F \to E\xrightarrow{p}B$ of simply connected spaces in which either, the fibre has finite dimensional rational cohomology, or, it has finite dimensional rational homotopy and $\rho$ induces a surjection in rational homotopy, we construct an explicit isomorphism, \begin{displaymath}\begin{array}{*{20}{c}} {\varphi :\operatorname{Ext}_{{C^\ast... ...(E;{\mathbf{Q}})}(Q,{C^\ast}(E;{\mathbf{Q}})).} \end{array} \end{displaymath} This is deduced from its "algebraic translation," a more general result in the environment of graded differential homological algebra.
Grothendieck groups of quotient singularities
Eduardo do Nascimento
Marcos
93-119
Abstract: Given a quotient singularity $R = {S^G}$ where $S = {\mathbf{C}}[[{x_1}, \ldots ,{x_n}]]$ is the formal power series ring in $n$-variables over the complex numbers ${\mathbf{C}}$, there is an epimorphism of Grothendieck groups $\psi :{G_0}(S[G]) \to {G_0}(R)$, where $ S[G]$ is the skew group ring and $\psi$ is induced by the fixed point functor. The Grothendieck group of $S[G]$ carries a natural structure of a ring, isomorphic to $ {G_0}({\mathbf{C}}[G])$. We show how the structure of ${G_0}(R)$ is related to the structure of the ramification locus of $V$ over $V/G$, and the action of $G$ on it. The first connection is given by showing that $\operatorname{Ker}\;\psi $ is the ideal generated by $ [{\mathbf{C}}]$ if and only if $G$ acts freely on $V$. That this is sufficient has been proved by Auslander and Reiten in [4]. To prove the necessity we show the following: Let $U$ be an integrally closed domain and $ T$ the integral closure of $ U$ in a finite Galois extension of the field of quotients of $U$ with Galois group $G$. Suppose that $\vert G\vert$ is invertible in $ U$, the inclusion of $ U$ in $T$ is unramified at height one prime ideals and $T$ is regular. Then ${G_0}(T[G]) \cong Z$ if and only if $U$ is regular. We analyze the situation $V = {V_1}{\coprod} _{\mathbf{C}[G]}{V_2}$ where $G$ acts freely on $ {V_1},{V_1} \ne 0$. We prove that for a quotient singularity $R,{G_0}(R) \cong {G_0}(R[[t]])$. We also study the structure of ${G_0}(R)$ for some cases with $\dim R = 3$.
Sharp estimate of the Laplacian of a polyharmonic function and applications
Ognyan Iv.
Kounchev
121-133
Abstract: The classical sharp inequality of Markov estimates the values of the derivative of the polynomial of degree $n$ in the interval $[a,b]$ through the uniform norm of the polynomial in the same interval multiplied by $2{n^2}/(b - a)$. In the present paper we provide an exact estimate for the values of the Laplacian of a polyharmonic function of degree $m$ by the uniform norm of the polyharmonic function multiplied by $2{(m - 1)^2}/{R^2}(x)$ where $R(x)$ is the distance from the point $ x$ to the boundary of the domain. The inequality of Markov (and the similar inequality of Bernstein about trigonometric polynomials) finds many applications in approximation theory for functions of one variable. We prove analogues to some of these results in the multivariate case.
Number of solutions with a norm bounded by a given constant of a semilinear elliptic PDE with a generic right-hand side
Alexander
Nabutovsky
135-166
Abstract: We consider a semilinear boundary value problem $- \Delta u + f(u,x) = 0$ in $\Omega \subset {\mathbb{R}^N}$ and $u = 0$ on $\partial \Omega$. We assume that $f$ is a $ {C^\infty }$-smooth function and $\Omega$ is a bounded domain with a smooth boundary. For any $ {C^\alpha }$-smooth perturbation $h(x)$ of the right-hand side of the equation we consider the function ${N_h}(S)$ defined as the number of ${C^{2 + \alpha }}$-smooth solutions $ u$ such that $\left\Vert u\right\Vert _{{C^0}(\Omega )} \leq S$ of the perturbed problem. How "small" $ {N_h}(S)$ can be made by a perturbation $h(x)$ such that $ \left\Vert h\right\Vert _{{C^0}(\Omega )} \leq \varepsilon ?$ We present here an explicit upper bound in terms of $\varepsilon$ , $S$ and $\displaystyle \mathop {\max }\limits_{\vert u\vert \leq S,x \in \bar \Omega } \left\Vert D_u^i f(u,x)\right\Vert \quad (i \in \{ 0,1,2\} ).$ If $S$ is fixed then $h$ can be chosen by such a way that the upper bound persists under small in ${C^0}$-topology perturbations of $ h$ . We present an explicit lower bound for the radius of the ball of such admissible perturbations.
The center of $\mathbb{Z}[S^{n+1}]$ is the set of symmetric polynomials in $n$ commuting transposition-sums
Gadi
Moran
167-180
Abstract: Let ${S_{n + 1}}$ be the symmetric group on the $n + 1$ symbols $0,1,2, \ldots ,n$. We show that the center of the group-ring $\mathbb{Z}[{S_{n + 1}}]$ coincides with the set of symmetric polynomials with integral coefficients in the $ n$ elements ${s_1}, \ldots ,{s_n} \in \mathbb{Z}[{S_{n + 1}}]$, where ${s_k} = \sum\nolimits_{0 \leq i < k} {(i,k)}$ is a sum of $k$ transpositions, $ k = 1, \ldots ,n$. In particular, every conjugacy-class-sum of ${S_{n + 1}}$ is a symmetric polynomial in ${s_1}, \ldots ,{s_n}$.
Families of sets of positive measure
Grzegorz
Plebanek
181-191
Abstract: We present a combinatorial description of those families $\mathcal{P}$ of sets, for which there is a finite measure $\mu$ such that $\inf \{ \mu (P):P \in \mathcal{P}\} > 0$. This result yields a topological characterization of measure-compactness and Borel measure-compactness. It is also applied to a problem on the existence of regular measure extensions.
L'espace des arcs d'une surface
Robert
Cauty
193-209
Abstract: We prove that, for any surface $M$, the space of arcs contained in $M$, with the topology induced by the Hausdorff distance, is homeomorphic to $M \times {\sum}^\infty$, where $ \sum = \{ ({x_i}) \in {l^2}/\sum\nolimits_{i = 1}^\infty {{{(i{x_i})}^2} < \infty \} }$.
Excluding subdivisions of infinite cliques
Neil
Robertson;
P. D.
Seymour;
Robin
Thomas
211-223
Abstract: For every infinite cardinal $k$ we characterize graphs not containing a subdivision of ${K_k}$.
Iterating maps on cellular complexes
Stephen J.
Willson
225-240
Abstract: Let $K$ be a finite simplicial complex and $ f:K \to K$ be a "skeletal" map. A digraph $D$ is defined whose vertices correspond to the simplexes of $K$ and whose arcs give information about the behavior of $f$ on the simplexes. For every walk in $D$ there exists a point of $ K$ whose iterates under $ f$ mimic the walk. Periodic walks are mimicked by a periodic point. Digraphs with uncountably many infinite walks are characterized; the corresponding maps $f$ exhibit complicated behavior.
The structure of the space of coadjoint orbits of an exponential solvable Lie group
Bradley N.
Currey
241-269
Abstract: In this paper we address the problem of describing in explicit algebraic terms the collective structure of the coadjoint orbits of a connected, simply connected exponential solvable Lie group $G$. We construct a partition $\wp$ of the dual ${\mathfrak{g}^{\ast} }$ of the Lie algebra $\mathfrak{g}$ of $G$ into finitely many $\operatorname{Ad}^{\ast} (G)$-invariant algebraic sets with the following properties. For each $\Omega \in \wp$, there is a subset $\Sigma$ of $\Omega$ which is a cross-section for the coadjoint orbits in $\Omega$ and such that the natural mapping $\Omega /\operatorname{Ad}^{\ast} (G) \to \Sigma$ is bicontinuous. Each $ \Sigma$ is the image of an analytic $ \operatorname{Ad}^{\ast}(G)$-invariant function $P$ on $\Omega$ and is an algebraic subset of $ {\mathfrak{g}^{\ast}}$. The partition $\wp$ has a total ordering such that for each $\Omega \in \wp $, $ \cup \{ \Omega \prime:\Omega \prime \leq \Omega \}$ is Zariski open. For each $ \Omega$ there is a cone $ W \subset {\mathfrak{g}^{\ast} }$, such that $\Omega$ is naturally a fiber bundle over $ \Sigma$ with fiber $ W$ and projection $ P$. There is a covering of $ \Sigma$ by finitely many Zariski open subsets $O$ such that in each $O$, there is an explicit local trivialization $ \Theta :{P^{ - 1}}(O) \to W \times O$. Finally, we show that if $\Omega$ is the minimal element of $ \wp$ (containing the generic orbits), then its cross-section $\Sigma$ is a differentiable submanifold of $ {\mathfrak{g}^{\ast} }$. It follows that there is a dense open subset $ U$ of $G\hat \emptyset$ such that $U$ has the structure of a differentiable manifold and $G\widehat\emptyset \sim U$ has Plancherel measure zero.
Stability for an inverse problem in potential theory
Hamid
Bellout;
Avner
Friedman;
Victor
Isakov
271-296
Abstract: Let $D$ be a subdomain of a bounded domain $\Omega$ in $ {\mathbb{R}^n}$ . The conductivity coefficient of $D$ is a positive constant $k \ne 1$ and the conductivity of $\Omega \backslash D$ is equal to $1$. For a given current density $ g$ on $\partial \Omega$ , we compute the resulting potential $u$ and denote by $f$ the value of $u$ on $ \partial \Omega$. The general inverse problem is to estimate the location of $ D$ from the known measurements of the voltage $f$. If ${D_h}$ is a family of domains for which the Hausdorff distance $ d(D,{D_h})$ equal to $ O(h)$ ($h$ small), then the corresponding measurements ${f_h}$ are $O(h)$ close to $f$. This paper is concerned with proving the inverse, that is, $d(D,{D_h}) \leq \frac{1}{c}\left\Vert {f_h} - f\right\Vert$ , $c > 0$ ; the domains $D$ and ${D_h}$ are assumed to be piecewise smooth. If $ n \geq 3$ , we assume in proving the above result, that ${D_h} \supset D$ (or ${D_h} \subset D$) for all small $h$ . For $n = 2$ this monotonicity condition is dropped, provided $g$ is appropriately chosen. The above stability estimate provides quantitative information on the location of ${D_h}$ by means of ${f_h}$ .
Algorithmic procedures
Harvey
Friedman;
Richard
Mansfield
297-312
Abstract: We consider the state of elementary recursion theory when the familiar $0, 1, +, \times, =, <$ of ordinary arithmetic are replaced by constants, functions, and relations from an arbitrary model.
Free actions on $\mathbb{R}$-trees
Frank
Rimlinger
313-329
Abstract: We characterize the free minimal actions of finitely generated groups on $\mathbb{R}$-trees in terms of certain equivalence relations on compact metric graphs.
Boundary tangential convergence on spaces of homogeneous type
Patricio
Cifuentes;
José R.
Dorronsoro;
Juan
Sueiro
331-350
Abstract: We study tangential convergence of convolutions with approximate identities of functions defined on a homogeneous type space and having a certain regularity. Our results contain those already known for the Euclidean case and give new ones for stratified nilpotent Lie groups and for solutions of the Dirichlet problem on Lipschitz domains.
Nonnegatively curved submanifolds in codimension two
Maria Helena
Noronha
351-364
Abstract: Let $M$ be a complete noncompact manifold with nonnegative sectional curvatures isometrically immersed in Euclidean spaces with codimension two. We show that $M$ is a product over its soul, except when the soul is the circle ${S^1}$ or $M$ is $3$-dimensional and the soul is the Real Projective Plane. We also give a rather complete description of the immersion, including the exceptional cases.
Convolution and hypergroup structures associated with a class of Sturm-Liouville systems
William C.
Connett;
Clemens
Markett;
Alan L.
Schwartz
365-390
Abstract: Product formulas of the type $\displaystyle {u_k}(\theta ){u_k}(\phi ) = \int_0^\pi {{u_k}(\xi )D(} \xi ,\theta ,\phi )\;d\xi$ are obtained for the eigenfunctions of a class of second order regular and regular singular Sturm-Liouville problems on $ [0,\pi ]$ by using the Riemann integration method to solve a Cauchy problem for an associated hyperbolic differential equation. When $D(\xi ,\theta ,\phi )$ is nonnegative (which can be guaranteed by a simple restriction on the differential operator of the Sturm-Liouville problem), it is possible to define a convolution with respect to which $M[0,\pi ]$ becomes a Banach algebra with the functions $ {u_k}(\xi )/{u_0}(\xi )$ as its characters. In fact this measure algebra is a Jacobi type hypergroup. It is possible to completely describe the maximal ideal space and idempotents of this measure algebra.
Loewy series of certain indecomposable modules for Frobenius subgroups
Zong Zhu
Lin
391-409
Abstract: We imitate some approaches in infinite dimensional representation theory of complex semisimple Lie algebras by using the truncated category method in the categories of modules for certain Frobenius subgroups of a semisimple algebraic group over an algebraically closed field of characteristic $p > 0$. By studying the translation functors from $p$-singular weights to $p$-regular weights, we obtain some results on Loewy series of certain indecomposable modules.
Degree one maps between geometric $3$-manifolds
Yong Wu
Rong
411-436
Abstract: Let $M$ and $N$ be two compact orientable $3$-manifolds, we say that $M \geq N$, if there is a degree one map from $M$ to $N$. This gives a way to measure the complexity of $ 3$-manifolds. The main purpose of this paper is to give a positive answer to the following conjecture: if there is an infinite sequence of degree one maps between Haken manifolds, then eventually all the manifolds are homeomorphic to each other. More generally, we prove a theorem which says that any infinite sequence of degree one maps between the so-called "geometric $3$-manifolds" must eventually become homotopy equivalences.
Free $\alpha$-extensions of an Archimedean vector lattice and their topological duals
Anthony J.
Macula
437-448
Abstract: Arch denotes the category of Archimedean vector lattices with vector lattice homomorphisms, and $\alpha$ denotes an uncountable cardinal number or the symbol $\infty$. $\operatorname{Arch}(\alpha )$ denotes the category of Arch objects with $\alpha$-complete Arch morphisms.
Nonexistence of nodal solutions of elliptic equations with critical growth in $\mathbb{R}^2$
Adimurthi;
S. L.
Yadava
449-458
Abstract: Let $f(t) = h(t){e^{b{t^2}}}$ be a function of critical growth. Under a suitable assumption on $h$, we prove that \begin{displaymath}\begin{array}{*{20}{c}} { - \Delta u = f(u)} & {{\text... ...} & {{\text{on}}\;\partial B(R),} \end{array} \end{displaymath} does not admit a radial solution which changes sign for sufficiently small $ R$.
Global convexity properties of some families of three-dimensional compact Levi-flat hypersurfaces
David E.
Barrett
459-474
Abstract: We consider various examples of compact Levi-flat hypersurfaces in two-dimensional complex manifolds, exploring the interplay between geometric properties of the induced foliation, behavior of the tangential Cauchy-Riemann equations along the hypersurface, and pseudoconvexity properties of small neighborhoods of the hypersurface.
Commutator theory for relatively modular quasivarieties
Keith
Kearnes;
Ralph
McKenzie
465-502
Abstract: We develop a commutator theory for relatively modular quasivarieties that extends the theory for modular varieties. We characterize relatively modular quasivarieties, prove that they have an almost-equational axiomatization and we investigate the lattice of subquasivarieties. We derive the result that every finitely generated, relatively modular quasivariety of semigroups is finitely based.
On the positive solutions of semilinear equations $\Delta u+\lambda u-hu\sp p=0$ on the compact manifolds
Tiancheng
Ouyang
503-527
Abstract: In this paper, we study the existence, nonexistence, and uniqueness of positive solutions of semilinear equations $ \Delta u + \lambda u - h{u^p}= 0$ on compact Riemannian manifolds as well as on bounded smooth domains in ${R^n}$ with homogeneous Dirichlet or Neumann boundary conditions.
The Bergman projection on Hartogs domains in ${\bf C}\sp 2$
Harold P.
Boas;
Emil J.
Straube
529-540
Abstract: Estimates in $ {L^2}$ Sobolev norms are proved for the Bergman projection in certain smooth bounded Hartogs domains in ${{\mathbf{C}}^2}$. In particular, (1) if the domain is pseudoconvex and "nonwormlike" (the normal vector does not wind on a critical set in the boundary), then the Bergman projection is regular; and (2) Barrett's counterexample domains with irregular Bergman projection nevertheless admit a priori estimates.
Invariant affine connections on Lie groups
H. Turner
Laquer
541-551
Abstract: The space of bi-invariant affine connections is determined for arbitrary compact Lie groups. In particular, there is a surprising new family of such connections on $ SU(n)$.
Finite determination on algebraic sets
L.
Kushner
553-561
Abstract: The concept of finite relative determination was introduced by Porto and Loibel $ [$P-L$]$ in 1978 and it deals with subspaces of $ {{\mathbf{R}}^n}$. In this paper we generalize this concept for algebraic sets, and relate it with finite determination on the right. We finish with an observation between Lojasiewicz ideals and finite relative determination.
Galois groups and the multiplicative structure of field extensions
Robert
Guralnick;
Roger
Wiegand
563-584
Abstract: Let $K/k$ be a finite Galois field extension, and assume $k$ is not an algebraic extension of a finite field. Let ${K^{\ast} }$ be the multiplicative group of $K$, and let $ \Theta (K/k)$ be the product of the multiplicative groups of the proper intermediate fields. The condition that the quotient group $ \Gamma = {K^{\ast} }/\Theta (K/k)$ be torsion is shown to depend only on the Galois group $G$. For algebraic number fields and function fields, we give a complete classification of those $ G$ for which $ \Gamma$ is nontrivial.
Approximation of Jensen measures by image measures under holomorphic functions and applications
Shang Quan
Bu;
Walter
Schachermayer
585-608
Abstract: We show that Jensen measures defined on $ {\mathbb{C}^n}$ or more generally on a complex Banach space $X$ can be approximated by the image of Lebesgue measure on the torus under $X$-valued polynomials defined on $\mathbb{C}$. We give similar characterizations for Jensen measures in terms of analytic martingales and Hardy martingales. The results are applied to approximate plurisubharmonic martingales by Hardy martingales, which enables us to give a characterization of the analytic Radon-Nikodym property of Banach spaces in terms of convergence of plurisubharmonic martingales, thus solving a problem of G. A. Edgar.
Isoparametric submanifolds of hyperbolic spaces
Bingle
Wu
609-626
Abstract: In this paper we prove a decomposition theorem for isoparametric submanifolds of hyperbolic spaces. And as a consequence we obtain all polar actions on hyperbolic spaces. We also prove that any isoparametric submanifold of infinite dimensional hyperbolic space is either totally geodesic, or finite dimensional.
On the analyticity of solutions of first-order nonlinear PDE
Nicholas
Hanges;
François
Trèves
627-638
Abstract: Let $(x,t) \in {R^m} \times R$ and $u \in {C^2}\,({R^m} \times R)$. We discuss local and microlocal analyticity for solutions $u$ to the nonlinear equation $\displaystyle {u_t}= f(x,t,u,{u_x})$ . Here $f(x,t,{\zeta _0},\zeta)$ is complex valued and analytic in all arguments. We also assume $f$ to be holomorphic in $ ({\zeta _0},\zeta) \in C \times {C^m}$. In particular we show that $\displaystyle {\text{WF}}_A\,u \subset \operatorname{Char}({L^u})$ where $ {\text{WF}}_A$ denotes the analytic wave-front set and $ \operatorname{Char}({L^u})$ is the characteristic set of the linearized operator $\displaystyle {L^u}= \partial /\partial t - \sum \partial \,f/\partial \,{\zeta _j}(x,t,u,{u_x})\;\partial /\partial \,{x_j}$ . If we assume $u \in {C^3}\;({R^m} \times R)$ then we show that the analyticity of $u$ propagates along the elliptic submanifolds of $ {L^u}$.
On local structures of the singularities $A\sb k\;D\sb k$ and $E\sb k$ of smooth maps
Yoshifumi
Ando
639-651
Abstract: In studying the singularities of type ${A_k}$ of smooth maps between manifolds $ N$ and $P$ the Boardman manifold ${\sum ^{i,1, \ldots,10}}$ in ${J^\infty }\,(N,P)$ has been very useful. We will construct the submanifolds $ \sum {D_k}$ and $\Sigma {E_k}$ in ${J^\infty }\,(N,P)$ playing the similar role for singularities ${D_k}$ and ${E_k}$ and study their properties in its process.
On twistor spaces of anti-self-dual Hermitian surfaces
Massimiliano
Pontecorvo
653-661
Abstract: We consider a complex surface $M$ with anti-self-dual hermitian metric $ h$ and study the holomorphic properties of its twistor space $Z$. We show that the naturally defined divisor line bundle $[X]$ is isomorphic to the $- \frac{1} {2}$ power of the canonical bundle of $ Z$, if and only if there is a Kähler metric of zero scalar curvature in the conformal class of $h$. This has strong consequences on the geometry of $M$, which were also found by C. Boyer $ [3]$ using completely different methods. We also prove the existence of a very close relation between holomorphic vector fields on $ M$ and $Z$ in the case that $M$ is compact and Kähler.
Products of commutative rings and zero-dimensionality
Robert
Gilmer;
William
Heinzer
663-680
Abstract: If $R$ is a Noetherian ring and $ n$ is a positive integer, then there are only finitely many ideals $I$ of $R$ such that the residue class ring $R/I$ has cardinality $\leq n$. If $R$ has Noetherian spectrum, then the preceding statement holds for prime ideals of $R$. Motivated by this, we consider the dimension of an infinite product of zero-dimensional commutative rings. Such a product must be either zero-dimensional or infinite-dimensional. We consider the structure of rings for which each subring is zero-dimensional and properties of rings that are directed union of Artinian subrings. Necessary and sufficient conditions are given in order that an infinite product of zero-dimensional rings be a directed union of Artinian subrings.
A bounded mountain pass lemma without the (PS) condition and applications
Martin
Schechter
681-703
Abstract: We present a version of the mountain pass lemma which does not require the ( $\mathbf{PS}$) condition. We apply this version to problems where the ( $ \mathbf{PS}$) condition is not satisfied.
A restriction theorem for modules having a spherical submodule
Nicolás
Andruskiewitsch;
Juan A.
Tirao
705-725
Abstract: We introduce the following notion: a finite dimensional representation $ V$ of a complex reductive algebraic group $G$ is called spherical of rank one if the generic stabilizer $M$ is reductive, the pair $(G,M)$ is spherical and $\dim \;{V^M}= 1$. Let $U$ be another finite dimensional representation of $G$; we denote by $ S^{\prime}(U)\;(S^{\prime}{(U)^G})$ the ring of polynomial functions on $ U$ (the ring of $ G$-invariant polynomial functions on $U$). We characterize the image of $S^{\prime}{(U \oplus V)^G}$ under the restriction map into $S^{\prime}\,(U \oplus {V^M})$ as the $W= {N_G}(M)/M$ invariants in the Rees ring associated to an ascending filtration of $S^{\prime}{(U)^M}$. Furthermore, under some additional hypothesis, we give an isomorphism between the graded ring associated to that filtration and $S^{\prime}{(U)^P}$, where $P$ is the stabilizer of an unstable point whose $G$-orbit has maximal dimension.
The Gauss map of a genus three theta divisor
Clint
McCrory;
Theodore
Shifrin;
Robert
Varley
727-750
Abstract: A smooth complex curve is determined by the Gauss map of the theta divisor of the Jacobian variety of the curve. The Gauss map is invariant with respect to the $(- 1)$-map of the Jacobian. We show that for a generic genus three curve the Gauss map is locally $ {\mathbf{Z}}/2$-stable. One method of proof is to analyze the first-order $ {\mathbf{Z}}/2$-deformations of the Gauss map of a hyperelliptic theta divisor.
Growth series of some wreath products
Walter
Parry
751-759
Abstract: The growth series of certain finitely generated groups which are wreath products are investigated. These growth series are intimately related to the traveling salesman problem on certain graphs. A large class of these growth series is shown to consist of irrational algebraic functions.
$3$-manifold groups with the finitely generated intersection property
Teruhiko
Soma
761-769
Abstract: In this paper, first we consider whether the fundamental groups of certain geometric $3$-manifolds have FGIP or not. Next we give the sufficient conditions that FGIP for $3$-manifold groups is preserved under torus sums or annulus sums and connect this result with a conjecture by Hempel $[4]$.
Exactly $k$-to-$1$ maps between graphs
Jo
Heath;
A. J. W.
Hilton
771-785
Abstract: Suppose $ k$ is a positive integer, $ G$ and $H$ are graphs, and $f$ is a $ k{\text{-to-}}1$ correspondence from a vertex set of $G$ onto a vertex set of $H$. Conditions on the adjacency matrices are given that are necessary and sufficient for $ f$ to extend to a continuous $ k{\text{-to-}}1$ map from $ G$ onto $H$.
Parametrization of a singular Lagrangian variety
Goo
Ishikawa
787-798
Abstract: We give stabilization and parametrization theorems for a class of singular varieties in the space of polynomials of one variable and generalize the results of Arnol'd and Givental'. The class contains the open swallowtails and the open Whitney umbrella. The parametrization is associated with the singularity of a stable mapping (in the sense of Thom and Mather) of kernel rank one.
Schubert calculus in complex cobordism
Paul
Bressler;
Sam
Evens
799-813
Abstract: We study the structure of the complex cobordism ring of the flag variety of a compact connected Lie group. An explicit procedure for determining products of basis elements is obtained, generalizing the work of Bernstein-Gel'fand-Gel'fand on ordinary cohomology and of Kostant-Kumar on $ K$-theory. Bott-Samelson resolutions are used to replace the classical basis of Schubert cells.
Weakly compact homomorphisms
J. E.
Galé;
T. J.
Ransford;
M. C.
White
815-824
Abstract: We study the structure of weakly compact homomorphisms between Banach algebras. In particular, it is shown that between many pairs of algebras, the only weakly compact homomorphisms are those of finite rank.
Compact actions commuting with ergodic actions and applications to crossed products
C.
Peligrad
825-836
Abstract: Let $(A,K,\beta)$ be a $ {C^{\ast}}$-dynamical system with $K$ compact. In this paper we prove a duality result for saturated actions (Theorem 3.3). The proof of this result can also be considered as an alternate proof of the corresponding result for von Neumann algebras due to Araki, Haag, Kastler and Takesaki $ [14]$. We also obtain results concerning the simplicity and the primeness of the crossed product $ A \times _\beta K$ in terms of the ergodicity of the commutant of $\beta$ (Propositions 5.3 and 5.4).
Subsequence ergodic theorems for $L\sp p$ contractions
Roger L.
Jones;
James
Olsen;
Máté
Wierdl
837-850
Abstract: In this paper certain subsequence ergodic theorems which have previously been known in the case of measure preserving point transformations, or Dunford Schwartz operators, are extended to operators which are positive contractions on $ {L^p}$ for $p$ fixed.
Maximal triads and prime decompositions of surfaces embedded in $3$-manifolds
Michael
Motto
851-867
Abstract: In 1975, Suzuki proved that prime decompositions of connected surfaces in ${S^3}$ are unique up to stable equivalence of the factors. This paper extends his result to a large class of $3$-manifolds, and demonstrates that this result does not apply to all $3$-manifolds. It also answers a question he raised by showing that it is possible for inequivalent surfaces in ${S^3}$ of the same genus to be stably equivalent. The techniques used involve the notion of Heegaard splittings of $3$-manifold triads.
The union of compact subgroups of an analytic group
Ta Sun
Wu
869-879
Abstract: Let $G$ be an analytic group. Let $\Omega (G)$ be the union of all compact subgroups of $ G$. We give a necessary and sufficient condition for $ \Omega (G)$ to be dense in $ G$ in terms of the action of a maximal compact torus $T$ of $G$ on the nilradical $N$ of $G$.
Rotation sets for homeomorphisms and homology
Mark
Pollicott
881-894
Abstract: In this article we propose a definition of rotation sets for homeomorphisms of arbitrary compact manifolds. This approach is based on taking the suspended flow and using ideas of Schwartzmann on homology and winding cycles for flows. Our main application is to give a generalisation of a theorem of Llibre and MacKay for tori to the context of surfaces of higher genus.
Growth rates, $Z\sb p$-homology, and volumes of hyperbolic $3$-manifolds
Peter B.
Shalen;
Philip
Wagreich
895-917
Abstract: It is shown that if $ M$ is a closed orientable irreducible $3$-manifold and $n$ is a nonnegative integer, and if ${H_1}(M,{\mathbb{Z}_p})$ has rank $\geq n + 2$ for some prime $p$, then every $n$-generator subgroup of ${\pi _1}\,(M)$ has infinite index in ${\pi _1}\,(M)$, and is in fact contained in infinitely many finite-index subgroups of ${\pi _1}\,(M)$. This result is used to estimate the growth rates of the fundamental group of a $ 3$-manifold in terms of the rank of the $ {\mathbb{Z}_p}$-homology. In particular it is used to show that the fundamental group of any closed hyperbolic $3$-manifold has uniformly exponential growth, in the sense that there is a lower bound for the exponential growth rate that depends only on the manifold and not on the choice of a finite generating set. The result also gives volume estimates for hyperbolic $ 3$-manifolds with enough $ {\mathbb{Z}_p}$-homology, and a sufficient condition for an irreducible $ 3$-manifold to be almost sufficiently large.
The strong maximal function on a nilpotent group
Michael
Christ
1-13
Abstract: An analogue of the strong maximal function of Jessen, Marcinkiewicz, and Zygmund is shown to be bounded on ${L^p}$, for all $p > 1$, on a nilpotent Lie group.
The transverse homoclinic dynamics and their bifurcations at nonhyperbolic fixed points
Bo
Deng
15-53
Abstract: The complete description of the dynamics of diffeomorphisms in a neighborhood of a transverse homoclinic orbit to a hyperbolic fixed point is obtained. It is topologically conjugate to a non-Bernoulli shift called $\{ {\sum,\sigma } \}$. We also obtain a more or less complete picture, referred to as the net weaving bifurcation, when the fixed point of such a system is undergoing the generic saddle-node bifurcation. The idea of homotopy conjugacy is naturally introduced to show that systems whose fixed points undergo the pitchfork, transcritical, periodic doubling, and Hopf bifurcations are all homotopically conjugate to our shift dynamics $ \{ {\sum,\sigma } \}$ in a neighborhood of a transverse homoclinic orbit. These bifurcations are also examined in the context of the spectral decomposition with respect to the maximal indecomposable nonwandering sets.
Tensor products and Grothendieck type inequalities of operators in $L\sb p$-spaces
Bernd
Carl;
Andreas
Defant
55-76
Abstract: Several results in the theory of $(p,q)$-summing operators are improved by a unified but elementary tensor product concept.
Stable splittings of the dual spectrum of the classifying space of a compact Lie group
Chun-Nip
Lee
77-111
Abstract: For a compact Lie group $G$, there is a map from the $G$-equivariant fixed point spectrum of the zero sphere to the dual spectrum of the classifying space of $G, DB{G_ + }$. When $G$ is finite, the affirmative solution to Segal's conjecture states that this map is an equivalence upon appropriate completion of the source. In the case of a compact Lie group, we obtain splitting results of $DB{G_ + }$ via this map upon taking $p$-adic completions.
Nests of subspaces in Banach space and their order types
Alvaro
Arias;
Jeff
Farmer
113-130
Abstract: This paper addresses some questions which arise naturally in the theory of nests of subspaces in Banach space. The order topology on the index set of a nest is discussed, as well as the method of spatial indexing by a vector; sufficient geometric conditions for the existence of such a vector are found. It is then shown that a continuous nest exists in any Banach space. Applications and examples follow; in particular, an extension of the Volterra nest in $ {L^\infty }[ {0,1} ]$ to a continuous one, a continuous nest in a Banach space having no two elements isomorphic to one another, and a characterization of separable ${\mathcal{L}_p}$-spaces in terms of nests.
The helical transform as a connection between ergodic theory and harmonic analysis
Idris
Assani;
Karl
Petersen
131-142
Abstract: Direct proofs are given for the formal equivalence of the $ {L^2}$ boundedness of the maximal operators corresponding to the partial sums of Fourier series, the range of a discrete helical walk, partial Fourier coefficients, and the discrete helical transform. Strong $(2, 2)$ for the double maximal (ergodic) helical transform is extended to actions of ${\mathbb{R}^d}$ and $ {\mathbb{Z}^d}$. It is also noted that the spectral measure of a measure-preserving flow has a continuity property at $\infty$, the Local Ergodic Theorem satisfies a Wiener-Wintner property, and the maximal helical transform is not weak $(1, 1)$.
A simple proof of the fundamental theorem of Kirby calculus on links
Ning
Lu
143-156
Abstract: In this paper, we relate surgeries on links and Heegaard decompositions, relate framed links and surface mapping classes, and give a simple proof of the fundamental theorem of Kirby calculus on links by the presentation of the surface mapping class groups.
A Chern character in cyclic homology
Luca Quardo
Zamboni
157-163
Abstract: We show that inner derivations act trivially on the cyclic cohomology of the normalized cyclic complex $ \mathcal{C}(\Omega)/\mathcal{D}(\Omega)$ where $\Omega$ is a differential graded algebra. This is then used to establish the fact that the map introduced in $ [$ GJ$ ]$ defines a Chern character in $K$ theory.
The structure of rings in some varieties with definable principal congruences
G. E.
Simons
165-179
Abstract: We study varieties of rings with identity that satisfy an identity of the form $xy = yp(x,y)$, where every term of the polynomial $p$ has degree greater than one. These varieties are interesting because they have definable principal congruences and are residually small. Let $\mathcal{V}$ be such a variety. The subdirectly irreducible rings in $ \mathcal{V}$ are shown to be finite local rings and are completely described. This results in structure theorems for the rings in $\mathcal{V}$ and new examples of noncommutative rings in varieties with definable principal congruences. A standard form for the defining identity is given and is used to show that $ \mathcal{V}$ also satisfies an identity of the form $xy = q(x,y)x$. Analogous results are shown to hold for varieties satisfying $ xy = q(x,y)x$.
On the resolution of a curve lying on a smooth cubic surface in ${\bf P}\sp 3$
Salvatore
Giuffrida;
Renato
Maggioni
181-201
Abstract: Let $C$ be any reduced and irreducible curve lying on a smooth cubic surface in ${\mathbb{P}^3}$. In this paper we determine the graded Betti numbers of the ideal sheaf ${\mathcal{J}_C}$.
On the genus of smooth $4$-manifolds
Alberto
Cavicchioli
203-214
Abstract: The projective complex plane and the "twisted" ${S^3}$ bundle over ${S^1}$ are proved to be the unique closed prime connected (smooth or PL) $4$-manifolds of genus two. Then the classification of the nonorientable $4$-manifolds of genus $4$ is given. Finally the genus of a manifold $ M$ is shown to be related with the $2$nd Betti number of $M$ and some applications are proved in the general (resp. simply-connected) case.
A quasiregular analogue of a theorem of Hardy and Littlewood
Craig A.
Nolder
215-226
Abstract: Suppose that $ f$ is analytic in the unit disk. A theorem of Hardy and Littlewood relates the Hölder continuity of $f$ over the unit disk to the growth of the derivative. We prove here a quasiregular analogue of this result in certain domains in $n$-dimensional space. We replace values of the derivative with a local integral average. In the process we generalize a result on the continuity of quasiconformal mappings due to Nakki and Palka. We also present another proof of the relationship between the growth of the derivative and quasiregular mappings in BMO.
An open collar theorem for $4$-manifolds
Craig R.
Guilbault
227-245
Abstract: Let ${M^4}$ be an open $4$-manifold with boundary. Conditions are given under which ${M^4}$ is homeomorphic to $\partial M \times [0,1)$. Applications include a $4$-dimensional weak $h$-cobordism theorem and a classification of weakly flat embeddings of $2$-spheres in ${S^4}$. Specific examples of $(n - 2)$-spheres embedded in $ {S^n}$ (including $ n = 4$) are also discussed.
L'espace des pseudo-arcs d'une surface
Robert
Cauty
247-263
Abstract: We prove that, for any surface $M$, the space of pseudo-arcs contained in $ M$ is homeomorphic to $M \times {l^2}$.
Frames associated with an abelian $l$-group
James J.
Madden
265-279
Abstract: Every archimedean $ l$-group (lattice-ordered group) with weak unit is shown to be isomorphic to a sub-$l$-group of the $l$-group of continuous realvalued functions on a Tychonoff locale canonically associated with the $ l$-group. This strengthens the classical Yosida representation theorem in a useful way. The proof uses methods from universal algebra and is constructive.
Two-dimensional Cremona groups acting on simplicial complexes
David
Wright
281-300
Abstract: We show that the $ 2$-dimensional Cremona group $\displaystyle \operatorname{Cr}_2 = \operatorname{Aut}_k\;k(X,Y)$ acts on a $ 2$-dimensional simplicial complex $C$, which has as vertices certain models in the function field $k(X,Y)$. The fundamental domain consists of one face $ F$. This yields a structural description of $ \operatorname{Cr}_2$ as an amalgamation of three subgroups along pairwise intersections. The subgroup ${\text{GA}}_2 = \operatorname{Aut}_k\;k[X,Y]$ (integral Cremona group) acts on $C$ by restriction. The face $ F$ has an edge $ E$ such that the ${\text{GA}}_2$ translates of $E$ form a tree $T$. The action of ${\text{GA}}_2$ on $T$ yields the well-known structure theory for ${\text{GA}}_2$ as an amalgamated free product, using Serre's theory of groups acting on trees.
A Haar-type theory of best $L\sb 1$-approximation with constraints
András
Kroó;
Darrell
Schmidt
301-319
Abstract: A general setting for constrained ${L^1}$-approximation is presented. Let $ {U_n}$ be a finite dimensional subspace of $C[a,b]$ and $L$ be a linear operator from ${U_n}$ to ${C^r}(K)\;(r = 0,1)$ where $K$ is a finite union of disjoint, closed, bounded intervals. For $\upsilon,u \in {C^r}(K)$ with $\upsilon < u$, the approximating set is $ {\tilde U_n}(\upsilon,u) = \{ p \in {U_n}:\upsilon \leq Lp \leq u\;{\text{on}}\;K\}$ and the norm is $\Vert f\Vert _w = \int_a^b {\vert f\vert w\,dx}$ where $w$ a positive continuous function on $[a,b]$. We obtain necessary and sufficient conditions for ${\tilde U_n}(\upsilon,u)$ to admit unique best $ \Vert\;\cdot\;\Vert _w$-approximations to all $ f \in C[a,b]$ for all positive continuous $w$ and all $\upsilon,u \in {C^r}(K)\;(r = 0,1)$ satisfying a nonempty interior condition. These results are applied to several ${L^1}$-approximation problems including polynomial and spline approximation with restricted derivatives, lacunary polynomial approximation with restricted derivatives, and others.
Gauge invariant quantization on Riemannian manifolds
Zhang Ju
Liu;
Min
Qian
321-333
Abstract: For every pointwise polynomial function on each fiber of the cotangent bundle of a Riemannian manifold $M$, a family of differential operators is given, which acts on the space of smooth sections of a vector bundle on $M$. Such a correspondence may be considered as a rule to quantize classical systems moving in a Riemannian manifold or in a gauge field. Some applications of our construction are also given in this paper.
On the $p$-adic completions of nonnilpotent spaces
A. K.
Bousfield
335-359
Abstract: This paper deals with the $p$-adic completion $ {F_{p\infty }}X$ developed by Bousfield-Kan for a space $X$ and prime $p$. A space $X$ is called ${F_p}$-good when the map $X \to {F_{p\infty }}X$ is a $\bmod$-$p$ homology equivalence, and called ${F_p}$-bad otherwise. General examples of ${F_p}$-good spaces are established beyond the usual nilpotent or virtually nilpotent ones. These include the polycyclic-by-finite spaces. However, the wedge of a circle with a sphere of positive dimension is shown to be ${F_p}$-bad. This provides the first example of an ${F_p}$-bad space of finite type and implies that the $p$-profinite completion of a free group on two generators must have nontrivial higher $\bmod$-$p$ homology as a discrete group. A major part of the paper is devoted to showing that the desirable properties of nilpotent spaces under the $ p$-adic completion can be extended to the wider class of $p$-seminilpotent spaces.
Normal form and linearization for quasiperiodic systems
Shui-Nee
Chow;
Kening
Lu;
Yun Qiu
Shen
361-376
Abstract: In this paper, we consider the following system of differential equations: $\displaystyle \dot \theta = \omega + \Theta (\theta,z), \quad \dot z = Az + f(\theta,z),$ where $ \theta \in {C^m}$, $\omega = ({\omega _1}, \ldots,{\omega _m}) \in {R^m}$, $z \in {C^n}$, $A$ is a diagonalizable matrix, $f$ and $\Theta$ are analytic functions in both variables and $2\pi$-periodic in each component of the vector $\theta,\Theta = O(\vert z\vert)$ and $f = O(\vert z{\vert^2})$ as $z \to 0$. We study the normal form of this system of the equations and prove that this system can be transformed to a system of linear equations $\displaystyle \dot \theta = \omega, \quad \dot z = Az$ by an analytic transformation provided that the eigenvalues of $A$ and the frequency $\omega$ satisfy certain small-divisor conditions.
Multipliers of families of Cauchy-Stieltjes transforms
R. A.
Hibschweiler;
T. H.
MacGregor
377-394
Abstract: For $\alpha > 0$ let ${\mathcal{F}_\alpha }$ denote the class of functions defined for $\vert z\vert < 1$ by integrating $1/{(1 - xz)^\alpha }$ against a complex measure on $ \vert x\vert= 1$. A function $g$ holomorphic in $\vert z\vert < 1$ is a multiplier of ${\mathcal{F}_\alpha }$ if $f \in {\mathcal{F}_\alpha }$ implies $gf \in {\mathcal{F}_\alpha }$. The class of all such multipliers is denoted by $ {\mathcal{M}_\alpha }$. Various properties of ${\mathcal{M}_\alpha }$ are studied in this paper. For example, it is proven that $\alpha < \beta$ implies $ {\mathcal{M}_\alpha } \subset {\mathcal{M}_\beta }$, and also that $ {\mathcal{M}_\alpha } \subset {H^\infty }$. Examples are given of bounded functions which are not multipliers. A new proof is given of a theorem of Vinogradov which asserts that if $f^{\prime}$ is in the Hardy class $ {H^1}$, then $f \in {\mathcal{M}_1}$. Also the theorem is improved to $ f^{\prime} \in {H^1}$ implies $ f \in {\mathcal{M}_\alpha }$, for all $ \alpha > 0$. Finally, let $\alpha > 0$ and let $f$ be holomorphic in $ \vert z\vert < 1$. It is known that $f$ is bounded if and only if its Cesàro sums are uniformly bounded in $ \vert z\vert \leq 1$. This result is generalized using suitable polynomials defined for $\alpha > 0$.
An asymptotic estimate for heights of algebraic subspaces
Jeffrey Lin
Thunder
395-424
Abstract: We count the number of subspaces of affine space with a given dimension defined over an algebraic number field with height less than or equal to $B$. We give an explicit asymptotic estimate for the number of such subspaces as $B$ goes to infinity, where the constants involved depend on the classical invariants of the number field (degree, discriminant, class number, etc.). The problem is reformulated as an estimate for the number of lattice points in a certain bounded domain.
A simplified trace formula for Hecke operators for $\Gamma\sb 0(N)$
Shepley L.
Ross
425-447
Abstract: Let $N$ and $n$ be relatively prime positive integers, let $ \chi$ be a Dirichlet character modulo $N$, and let $k$ be a positive integer. Denote by ${S_k}(N,\chi)$ the space of cusp forms on ${\Gamma _0}(N)$ of weight $k$ and character $\chi$, a space denoted simply $ {S_k}(N)$ when $ \chi$ is the trivial character. Beginning with Hijikata's formula for the trace of ${T_n}$ acting on $ {S_k}(N,\chi)$, we develop a formula which essentially reduces the computation of this trace to looking up values in a table. From this formula we develop very simple formulas for (1) the dimension of $ {S_k}(N,\chi)$ and (2) the trace of ${T_n}$ acting on ${S_k}(N)$.
Noetherian ring extensions with trace conditions
Robert B.
Warfield
449-463
Abstract: Finite ring extensions of Noetherian rings with certain restrictions on the corresponding trace ideals are studied. This setting includes finite free extensions and extensions arising from actions of finite groups when the order of the group is invertible. In this setting we establish the following results which were previously obtained (for finite extensions without trace conditions) only under strong restrictions on the rings involved. Let $R \subset S$ be an extension of Noetherian rings such that $S$ is finitely generated as a left $R$-module and such that the left trace ideal of $S$ in $R$ is equal to $R$. If $S$ is right fully bounded, or is a Jacobson ring, then $R$ has the same property; furthermore, $R$ and $S$ have the same classical Krull dimension. If $S$ is finitely generated as both a right and a left $ R$-module, if both trace ideals of $S$ in $R$ are equal to $R$, and if $S$ satisfies the strong second layer condition, then this condition also holds in $R$. Finally, we compare the link graphs of $ R$ and $S$
Band sums of links which yield composite links. The cabling conjecture for strongly invertible knots
Mario
Eudave Muñoz
463-501
Abstract: We consider composite links obtained by bandings of another link. It is shown that if a banding of a split link yields a composite knot then there is a decomposing sphere crossing the band in one arc, unless there is such a sphere disjoint from the band. We also prove that if a banding of the trivial knot yields a composite knot or link then there is a decomposing sphere crossing the band in one arc. The last theorem implies, via double branched covers, that the only way we can get a reducible manifold by surgery on a strongly invertible knot is when the knot is cabled and the surgery is via the slope of the cabling annulus.
Finite codimensional subalgebras of Stein algebras and semiglobally Stein algebras
Hà Huy Khoái;
Nguyen Văn Khuê
503-508
Abstract: The following theorem is proved: For each finite codimensional subalgebra $A$ of a Stein algebra $B$ there exists a natural number $n$ such that $B$ is algebraically isomorphic to $A \oplus {{\mathbf{C}}^n}$.
A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis in $L\sp p(0,1)$
K. P.
Choi
509-529
Abstract: Let $1 < p < \infty$. Let $d = ({d_1},{d_2}, \ldots)$ be a real-valued martingale difference sequence, $\theta = ({\theta _1},{\theta _2}, \ldots)$ is a predictable sequence taking values in $ [0,1]$. We show that the best constant of the inequality, $\displaystyle {\left\Vert {\sum\limits_{k = 1}^n {{\theta _k}{d_k}} } \right\Ve... ...p}{\left\Vert {\sum\limits_{k = 1}^n {{d_k}} } \right\Vert _p}, \quad n \geq 1,$ satisfies $\displaystyle {c_p} = \frac{p}{2} + \frac{1}{2}\;\log \;\left({\frac{{1 + y}}{2}} \right) + \frac{{{\alpha _2}}}{p} + \cdots,$ where $\gamma = {e^{ - 2}}$ and ${\alpha _2} = {\left[ {\frac{1}{2}\;\log \;\frac{{1 + \gamma }}{2}} \right]^2}... ... \;\frac{{1 + \gamma }}{2} - 2{\left({\frac{\gamma }{{1 + \gamma }}} \right)^2}$. The best constant equals the unconditional basis constant of a monotone basis of $ {L^p}(0,1)$.
Invariant subspaces with finite codimension in Bergman spaces
Alexandru
Aleman
531-544
Abstract: For an arbitrary bounded domain in $ \mathbb{C}$ there are described those finite codimensional subspaces of the Bergman space that are invariant under multiplication by $z$.
Intersection theory of moduli space of stable $n$-pointed curves of genus zero
Sean
Keel
545-574
Abstract: We give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the Chow ring.
Lattice-ordered groups whose lattices determine their additions
Paul F.
Conrad;
Michael R.
Darnel
575-598
Abstract: In this paper it is shown that several large and important classes of lattice-ordered groups, including the free abelian lattice-ordered groups, have their group operations completely determined by the underlying lattices, or determined up to $l$-isomorphism.
Justification of multidimensional single phase semilinear geometric optics
Jean-Luc
Joly;
Jeffrey
Rauch
599-623
Abstract: For semilinear strictly hyperbolic systems $ Lu= f(x,u)$, we construct and justify high frequency nonlinear asymptotic expansions of the form $\displaystyle {u^\varepsilon }(x)\sim\sum\limits_{j\, \geq \,0} {{\varepsilon ^... ...phi \,(x)/\varepsilon}, \quad L{u^\varepsilon } - f(x,{u^\varepsilon })\sim 0 .$ The study of the principal term of such expansions is called nonlinear geometric optics in the applied literature. We show (i) formal expansions with periodic profiles ${U_j}$ can be computed to all orders, (ii) the equations for the profiles from (i) are solvable, and (iii) there are solutions of the exact equations which have the formal series as high frequency asymptotic expansion.
Superharmonic functions on foliations
S. R.
Adams
625-635
Abstract: We use techniques from geometric analysis to prove that any positive, leafwise superharmonic, measurable function on a Riemannian measurable foliation with transverse invariant measure, finite total volume and complete leaves is, in fact, constant on a.e. leaf.
Generalized Szeg\H o theorems and asymptotics of cumulants by graphical methods
Florin
Avram
637-649
Abstract: We obtain some general asymptotics results about a class of deterministic sums called "sums with dependent indices," which generalize a classical theorem of Szegö. The above type of sums is encountered when establishing convergence to the Gaussian distribution of sums of Wick products by the method of cumulants. Our asymptotic results reduce in this situation the proof of the central limit theorem to the study of the connectivity of a family of associated graphs.
Central limit theorems for sums of Wick products of stationary sequences
Florin
Avram;
Robert
Fox
651-663
Abstract: We show, by the method of cumulants, that checking whether the central limit theorem for sums of Wick powers of a stationary sequence holds can be reduced to the study of an associated graph problem (see Corollary 1). We obtain thus central limit theorems under various integrability conditions on the cumulant spectral functions (Theorems 2, 3).
Symmetry of knots and cyclic surgery
Shi Cheng
Wang;
Qing
Zhou
665-676
Abstract: If a nontorus knot $ K$ admits a symmetry which is not a strong inversion, then there exists no nontrivial cyclic surgery on $K$. No surgery on a symmetric knot can produce a fake lens space or a $3$-manifold $M$ with $ \vert{\pi _1}(M)\vert= 2$. This generalizes the result of Culler-Gordon-Luecke-Shalen-Bleiler-Scharlemann and supports the conjecture that no nontrivial surgery on a nontrivial knot yields a $ 3$-manifold $M$ with $\vert{\pi _1}(M)\vert < 5$.
Stable and uniformly stable unit balls in Banach spaces
Antonio
Suárez Granero
677-695
Abstract: Let $X$ be a Banach space with closed unit ball $ {B_X}$ and, for $x \in X$, $r \geq 0$, put $B(x;r)= \{ u \in X:\vert\vert u - x\vert\vert \leq r\}$ and $V(x,r)= {B_X} \cap B(x;r)$. We say that $ {B_X}$ (or in general a convex set) is stable if the midpoint map $ {\Phi _{1/2}}:{B_X} \times {B_X} \to {B_X}$, with $ {\Phi _{1/2}}(u,\upsilon)= \frac{1}{2}(u + \upsilon)$, is open. We say that $ {B_X}$ is uniformly stable (US) if there is a map $\alpha :(0,2] \to (0,2]$, called a modulus of uniform stability, such that, for each $x,y \in {B_X}$ and $r \in (0,2],V(\frac{1} {2}(x + y);\alpha (r)) \subseteq \frac{1} {2}(V(x;r) + V(y;r))$. Among other things, we see: (i) if $ \dim X \geq 3$, then $ X$ admits an equivalent norm such that ${B_X}$ is not stable; (ii) if $\dim X < \infty$, ${B_X}$ is stable iff ${B_x}$ is US; (iii) if $X$ is rotund, $X$ is uniformly rotund iff ${B_X}$ is US; (iv) if $X$ is $ 3.2.{\text{I.P}}$, $ {B_X}$ is US and $\alpha (r)= r/2$ is a modulus of US; (v) $ {B_X}$ is US iff ${B_{{X^{ \ast \ast }}}}$ is US and $ X$, ${X^{ \ast \ast}}$ have (almost) the same modulus of US; (vi) ${B_X}$ is stable (resp. US) iff ${B_{C(K,X)}}$ is stable (resp. US) for each compact $ K$ iff ${B_{A(K,X)}}$ is stable (resp. US) for each Choquet simplex $K$; (vii) ${B_X}$ is stable iff ${B_{{L_p}(\mu,X)}}$ is stable for each measure $ \mu$ and $1 \leq p < \infty $.
Analytic operator valued function space integrals as an ${\scr L}(L\sb p,L\sb {p'})$ theory
Kun Soo
Chang;
Kun Sik
Ryu
697-709
Abstract: The existence of an analytic operator-valued function space integral as an $ \mathcal{S}({L_p},{L_{p^{\prime}}})$ theory $ (1 \leq p \leq 2)$ has been established for certain functionals involving the Lebesgue measure. Recently, Johnson and Lapidus proved the existence of the integral as an operator on $ {L_2}$ for certain functionals involving any Borel measure. We establish the existence of the integral as an operator from $ {L_p}$ to $ {L_{p^{\prime}}}\;({1 < p < 2} )$ for certain functionals involving some Borel measures.
Isomorphism invariants for abelian groups
D. M.
Arnold;
C. I.
Vinsonhaler
711-724
Abstract: Let $A= ({A_1},\ldots,{A_n})$ be an $n$-tuple of subgroups of the additive group, $Q$, of rational numbers and let $G(A)$ be the kernel of the summation map ${A_1} \oplus \cdots \oplus {A_n} \to \sum \;{A_i}$ and $G[A]$ the cokernel of the diagonal embedding $\cap \,{A_1} \to {A_1} \oplus \cdots \oplus {A_n}$. A complete set of isomorphism invariants for all strongly indecomposable abelian groups of the form $ G(A)$, respectively, $ G[A]$, is given. These invariants are then extended to complete sets of isomorphism invariants for direct sums of such groups and for a class of mixed abelian groups properly containing the class of Warfield groups.
A general condition for lifting theorems
E. Arthur
Robinson
725-755
Abstract: We define a general condition, called stability on extensions $T$ of measure preserving transformations $ S$. Stability is defined in terms of relative unique ergodicity, and as a joining property. Ergodic compact group extensions are stable, and moreover stable extensions satisfy lifting theorems similar to those satisfied by group extensions. In general, stable extensions have relative entropy zero. In the class of continuous flow extensions over strictly ergodic homeomorphisms, stable extensions are generic.
The complete integral closure of $R[X]$
Thomas G.
Lucas
757-768
Abstract: For a reduced ring $ R$ that is completely integrally closed it is not always the case that the corresponding polynomial ring $R[X]$ is completely integrally closed. In this paper the question of when $R[X]$ is completely integrally closed is shown to be related to the question of when $R$ is completely integrally closed in $T(R[X])$ the total quotient ring of $ R[X]$. A characterization of the complete integral closure of $R[X]$ is given in the main theorem and this result is used to characterize the complete integral closure of the semigroup ring $ R[S]$ when $S$ is a torsion-free cancellative monoid.
Trace functions in the ring of fractions of polycyclic group rings
A. I.
Lichtman
769-781
Abstract: Let $KG$ be the group ring of a polycyclic-by-finite group $G$ over a field $K$ of characteristic zero, $R$ be the Goldie ring of fractions of $ KG$, $S$ be an arbitrary subring of ${R_{n \times n}}$. We prove that the intersection of the commutator subring $[S,S]$ with the center $Z(S)$ is nilpotent. This implies the existence of a nontrivial trace function in ${R_{n \times n}}$.
Harnack estimates and extinction profile for weak solutions of certain singular parabolic equations
E.
DiBenedetto;
Y. C.
Kwong
783-811
Abstract: We establish an intrinsic Harnack estimate for nonnegative weak solutions of the singular equation $(1.1)$ below, for $m$ in the optimal range $((N - 2)_+/N,1)$. Intrinsic means that, due to the singularity, the space-time dimensions in the parabolic geometry must be rescaled by a factor determined by the solution itself. Consequences are, sharp supestimates on the solutions and decay rates as $ t$ approaches the extinction time. Analogous results are shown for $p$-laplacian type equations.
A phenomenon of reciprocity in the universal Steenrod algebra
Luciano
Lomonaco
813-821
Abstract: In this paper we compute the cohomology algebra of certain subalgebras $ {L_r}$ and certain quotients ${K_s}$ of the $\bmod\, 2$ universal Steenrod algebra $ Q$, the algebra of cohomology operations for $ {H_\infty }$-ring spectra (see $[$M$]$). We prove that $\displaystyle \operatorname{Ext}_{{L_r}}({F_2},{F_2}) \cong {K_{ - k + 1}}, \qquad \operatorname{Ext}_{{K_s}}({F_2},{F_2}) \cong {L_{ - s + 1}}$ with $r$, $s$ integers and $r \leq 1$, $s \geq 0$. We also observe that some of the algebras ${L_r}$, ${K_s}$ are well known objects in stable homotopy theory and in fact our computation generalizes the fact that ${H^{\ast} }({A_L}) \cong \Lambda ^{{\text{opp}}}$ and ${H^{\ast} }({\Lambda ^{{\text{opp}}}}) \cong {A_L}$ (see, for instance, $ [$P$]$). Here ${A_L}$ is the Steenrod algebra for simplicial restricted Lie algebras and $\Lambda$ is the ${E_1}$-term of the Adams spectral sequence discovered in $[$B-S$]$.
The ``Defektsatz'' for central simple algebras
Joachim
Gräter
823-843
Abstract: Let $Q$ be a central simple algebra finite-dimensional over its center $F$ and let $V$ be a valuation ring of $F$. Then $V$ has an extension to $Q$, i.e., there exists a Dubrovin valuation ring $ B$ of $Q$ satisfying $ V= F \cap B$. Generally, the number of extensions of $V$ to $Q$ is not finite and therefore the so-called intersection property of Dubrovin valuation rings ${B_1}, \ldots,{B_n}$ is introduced. This property is defined in terms of the prime ideals and the valuation overrings of the intersection ${B_1} \cap \cdots \; \cap {B_n}$. It is shown that there exists a uniquely determined natural number $n$ depending only on $V$ and having the following property: If ${B_1}, \ldots,{B_k}$ are extensions of $V$ having the intersection property then $ k \leq n$ and $ k= n$ holds if and only if $ {B_1} \cap \cdots \cap {B_k}$ is integral over $V$. Let $n$ be the extension number of $V$ to $Q$. There exist extensions ${B_1}, \cdots,{B_n}$ of $V$ having the intersection property and if $ {R_1}, \ldots,{R_n}$ are also extensions of $V$ having the intersection property then $ {B_1} \cap \cdots \cap {B_n}$ and $ {R_1} \cap \cdots \cap {R_n}$ are conjugate. The main result regarding the extension number is the Defektsatz: $[Q:F]= {f_B}(Q/F){e_B}(Q/F){n^2}{p^d}$, where $ {f_B}(Q/F)$ is the residue degree, $ {e_B}(Q/F)$ the ramification index, $n$ the extension number, $p = \operatorname{char}(V/J(V))$, and $d$ a natural number.
A deformation of tori with constant mean curvature in ${\bf R}\sp 3$ to those in other space forms
Masaaki
Umehara;
Kotaro
Yamada
845-857
Abstract: It is shown that tori with constant mean curvature in ${\mathbb{R}^3}$ constructed by Wente $ [7]$ can be deformed to tori with constant mean curvature in the hyperbolic $ 3$-space or the $ 3$-sphere.
Functors on the category of finite sets
Randall
Dougherty
859-886
Abstract: Given a covariant or contravariant functor from the category of finite sets to itself, one can define a function from natural numbers to natural numbers by seeing how the functor maps cardinalities. In this paper we answer the question: what numerical functions arise in this way from functors? The sufficiency of the conditions we give is shown by simple constructions of functors. In order to show the necessity, we analyze the way in which functions in the domain category act on members of objects in the range category, and define combinatorial objects describing this action; the permutation groups in the domain category act on these combinatorial objects, and the possible sizes of orbits under this action restrict the values of the numerical function. Most of the arguments are purely combinatorial, but one case is reduced to a statement about permutation groups which is proved by group-theoretic methods.
Affine $3$-spheres with constant affine curvature
Martin A.
Magid;
Patrick J.
Ryan
887-901
Abstract: We classify the affine hyperspheres in ${R^4}$ which have constant curvature in the affine metric $h$ and whose Pick invariant is nonzero. In particular, the metric $h$ must be flat.
On compactly supported spline wavelets and a duality principle
Charles K.
Chui;
Jian-zhong
Wang
903-915
Abstract: Let $\cdots \subset{V_{ - 1}} \subset{V_0} \subset{V_1} \subset \cdots$ be a multiresolution analysis of $ {L^2}$ generated by the $ m$th order $B$-spline ${N_m}(x)$. In this paper, we exhibit a compactly supported basic wavelet $ {\psi _m}(x)$ that generates the corresponding orthogonal complementary wavelet subspaces $\cdots,{W_{ - 1}},{W_0},{W_1}, \ldots$. Consequently, the two finite sequences that describe the two-scale relations of ${N_m}(x)$ and $ {\psi _m}(x)$ in terms of $ {N_m}(2x - j),j \in \mathbb{Z}$, yield an efficient reconstruction algorithm. To give an efficient wavelet decomposition algorithm based on these two finite sequences, we derive a duality principle, which also happens to yield the dual bases $ \{ {\tilde N_m}(x - j)\}$ and $ \{ {\tilde \psi _m}(x - j)\}$, relative to $ \{ {N_m}(x - j)\}$ and $\{ {\psi _m}(x - j)\}$, respectively.
The kernel-trace approach to right congruences on an inverse semigroup
Mario
Petrich;
Stuart
Rankin
917-932
Abstract: A kernel-trace description of right congruences on an inverse semigroup is developed. It is shown that the trace mapping is a complete $\cap$homomorphism but not a $ \vee$-homomorphism. However, the trace classes are intervals in the complete lattice of right congruences. In contrast, each kernel class has a maximum element, namely the principal right congruence on the kernel, but in general there is no minimum element in a kernel class. The kernel mapping preserves neither intersections nor joins. The set of axioms presented in [7] for right kernel systems is reviewed. A new set of axioms is obtained as a consequence of the fact that a right congruence is the intersection of the principal right congruences on the idempotent classes. Finally, it is shown that even though a congruence on a regular semigroup is the intersection of the principal congruences on the idempotent classes, the situation is not the same for right congruences on a regular semigroup. Right congruences on a regular, even orthodox, semigroup are not, in general, determined by their idempotent classes.
Analytic geometry of complex superspaces
H.
Flenner;
D.
Sundararaman
1-40
Abstract: A detailed account of the analytic geometry of complex superspaces is given in this paper. Several representability criteria and representability theorems are proved. In particular, the existence of a versal family of deformations (parametrized by a complex superspace) for any compact complex superspace is proved.
The conormal derivative problem for equations of variational type in nonsmooth domains
Gary M.
Lieberman
41-67
Abstract: It is well known that elliptic boundary value problems in smooth domains have smooth solutions, but if the domain is, say, $ {C^1}$, the solutions need not be Lipschitz. Recently Korevaar has identified a class of Lipschitz domains, in which solutions of the capillary problem are Lipschitz assuming the contact angle relates correctly to the geometry of the domain. Lipschitz bounds for more general boundary value problems in the same class of domains are proved. Applications to variational inequalities are also considered.
Earthquakes on Riemann surfaces and on measured geodesic laminations
Francis
Bonahon
69-95
Abstract: Let $S$ be a closed orientable surface of genus at least $2$. We study properties of its Teichmüller space $ \mathcal{T}(S)$, namely of the space of isotopy classes of conformal structures on $S$. W. P. Thurston introduced a certain compactification of $ \mathcal{T}(S)$ by what he called the space of projective measured geodesic laminations. He also introduced some transformations of Teichmüller space, called earthquakes, which are intimately related to the geometry of $\mathcal{T}(S)$. A general problem is to understand which geometric properties of Teichmüller space subsist at infinity, on Thurston's boundary. In particular, it is natural to ask whether earthquakes continuously extend at certain points of Thurston's boundary, and at precisely which points they do so. This is the principal question addressed in this paper.
A local Weyl's law, the angular distribution and multiplicity of cusp forms on product spaces
Jonathan
Huntley;
David
Tepper
97-110
Abstract: Let $\Gamma /\mathcal{H}$ be a finite volume symmetric space with $ \mathcal{H}$ the product of half planes. Let $ {\Delta _i}$ be the Laplacian on the $i$th half plane, and assume that we have a cusp form $\phi$, so we have ${\Delta _i}\phi = {\lambda _i}\phi$ for $i = 1,2, \ldots,n$. Let $\vec \lambda = ({\lambda _1}, \ldots,{\lambda _n})$ and let $\displaystyle R = \sqrt {r_1^2 + \cdots + r_n^2} $ with $ r_i^2 + \frac{1} {4} = {\lambda _i}$. Letting $\vec r = ({r_1}, \ldots,{r_n})$, we let $ M(\vec r)$ denote the dimension of the space of cusp forms with eigenvalue $\vec \lambda$. More generally, let $M(\vec r,a)$ denote the number of independent eigenfunctions such that the $\vec r$ associated to an eigenfunction is inside the ball of radius $a$, centered at $\vec r$. We will define a function $f(\vec r)$, which is generally equal to a linear sum of products of the ${r_i}$. We prove the following theorems. Theorem 1. $\displaystyle M(\vec r) = O\left(\frac{f(\vec r)} {(\log R)^n} \right).$ Theorem 2. $\displaystyle M (\vec{r}, A) = 2^n f(\vec{r})+O\left(\frac{f(\vec r)}{\log R} \right).$
On the symmetric square: definitions and lemmas
Yuval Z.
Flicker
111-124
Abstract: We define the symmetric square lifting for admissible and automorphic representations, from the group $H = {H_0} = {\text{SL}}(2)$, to the group $ G = {\text{PGL}}(3)$, and derive its basic properties. This lifting is defined by means of Shintani character relations. The definition is suggested by the computation of orbital integrals (stable and unstable) in our On the symmetric square: Orbital integrals, Math. Ann. 279 (1987), 173-193. It is compatible with dual group homomorphisms ${\lambda _0}:\widehat{H} \to \widehat{G}$ and ${\lambda _1}:{\widehat{H}_1} \to \widehat{G}$, where ${H_1} = {\text{PGL}}(2)$. The lifting is proven for induced, trivial and special representations, and both spherical functions and orthogonality relations of characters are studied.
On the symmetric square: applications of a trace formula
Yuval Z.
Flicker
125-152
Abstract: In this paper we prove the existence of the symmetric-square lifting of admissible and of automorphic representations from the group $ {\text{SL}}(2)$ to the group $ {\text{PGL}}(3)$. Complete local results are obtained, relating the character of an $ {\text{SL}}(2)$-packet with the twisted character of self-contragredient $ {\text{PGL}}(3)$-modules. Our global results relate packets of cuspidal representations of $ {\text{SL}}(2)$ with a square-integrable component, and self-contragredient automorphic $ {\text{PGL}}(3)$-modules with a component coming from a square-integrable one. The sharp results, which concern ${\text{SL}}(2)$ rather than ${\text{GL}}(2)$, are afforded by the usage of the trace formula. The surjectivity and injectivity of the correspondence implies that any self-contragredient automorphic $ {\text{PGL}}(3)$-module as above is a lift, and that the space of cuspidal ${\text{SL}}(2)$-modules with a square-integrable component admits multiplicity one theorem and rigidity ("strong multiplicity one") theorem for packets (and not for individual representations). The techniques of this paper, based on the usage of regular functions to simplify the trace formula, are pursued in the sequel [VI] to extend our results to all cuspidal ${\text{SL}}(2)$-modules and self-contragredient $ {\text{PGL}}(3)$-modules
Homotopie de l'espace des \'equivalences d'homotopie
Geneviève
Didierjean
153-163
Abstract: The spectral sequence of the self-fiber-homotopy-equivalences of a fibration provides a method to compute the homotopy groups of the space of self-equivalences of a space.
On Auslander-Reiten components of blocks and self-injective biserial algebras
Karin
Erdmann;
Andrzej
Skowroński
165-189
Abstract: We investigate the existence of Auslander-Reiten components of Euclidean type for special biserial self-injective algebras and for blocks of group algebras. In particular we obtain a complete description of stable Auslander-Reiten quivers for the tame self-injective algebras considered here.
Positive solutions of semilinear equations in cones
Henrik
Egnell
191-201
Abstract: In this paper we consider the problem of finding a positive solution of the equation $\Delta u + \vert x{\vert^\nu }{u^{(n + 2 + 2\nu)/(n - 2)}} = 0$ in a cone $\mathcal{C}$, with zero boundary data. We are only interested in solutions that are regular at infinity (i.e. such that $ u(x) = o(\vert x{\vert^{2 - n}})$, as $\mathcal{C} \ni x \to \infty$). We will always assume that $\nu > - 2$. We show that the existence of a solution depends on the sign of $\nu$ and also on the shape of the cone $\mathcal{C}$.
Characterization for the solvability of nonlinear partial differential equations
Elemer E.
Rosinger
203-225
Abstract: Within the nonlinear theory of generalized functions introduced earlier by the author a number of existence and regularity results have been obtained. One of them has been the first global version of the Cauchy-Kovalevskaia theorem, which proves the existence of generalized solutions on the whole of the domain of analyticity of arbitrary analytic nonlinear PDEs. These generalized solutions are analytic everywhere, except for closed, nowhere dense subsets which can be chosen to have zero Lebesgue measure. This paper gives a certain extension of that result by establishing an algebraic necessary and sufficient condition for the existence of generalized solutions for arbitrary polynomial nonlinear PDEs with continuous coefficients. This algebraic characterization, given by the so-called neutrix or off diagonal condition, is proved to be equivalent to certain densely vanishing conditions, useful in the study of the solutions of general nonlinear PDEs.
${\rm II}\sb 1$ factors, their bimodules and hypergroups
V. S.
Sunder
227-256
Abstract: In this paper, we introduce a notion that we call a hypergroup; this notion captures the natural algebraic structure possessed by the set of equivalence classes of irreducible bifinite bimodules over a II$ _{1}$ factor. After developing some basic facts concerning bimodules over II$_{1}$ factors, we discuss abstract hypergroups. To make contact with the problem of what numbers can arise as index-values of subfactors of a given II$_{1}$ factor with trivial relative commutant, we define the notion of a dimension function on a hypergroup, and prove that every finite hypergroup admits a unique dimension function, we then give some nontrivial examples of hypergroups, some of which are related to the Jones subfactors of index $4{\cos ^2}\pi /(2n + 1)$. In the last section, we study the hypergroup invariant corresponding to a bifinite module, which is used, among other things, to obtain a transparent proof of a strengthened version of what Ocneanu terms 'the crossed-product remembering the group.'
Th\'eorie de Sullivan pour la cohomologie \`a coefficients locaux
Antonio
Gómez-Tato
235-305
Abstract: The classical moment map of symplectic geometry is used to canonically associate to a unitary representation of a Lie group $G$ a $G$-invariant subset of the dual of the Lie algebra. This correspondence is in some sense dual to geometric quantization. The nature and convexity of this subset is investigated for $G$ compact semisimple.
The moment map of a Lie group representation
N. J.
Wildberger
257-268
Abstract: Given an $m \times m$ Hadamard matrix one can extract $ {m^2}$ symmetric designs on $m - 1$ points each of which extends uniquely to a $ 3$-design. Further, when $ m$ is a square, certain Hadamard matrices yield symmetric designs on $ m$ points. We study these, and other classes of designs associated with Hadamard matrices, using the tools of algebraic coding theory and the customary association of linear codes with designs. This leads naturally to the notion, defined for any prime $p$, of $p$-equivalence for Hadamard matrices for which the standard equivalence of Hadamard matrices is, in general, a refinement: for example, the sixty $24 \times 24$ matrices fall into only six $ 2$-equivalence classes. In the $16 \times 16$ case, $ 2$-equivalence is identical to the standard equivalence, but our results illuminate this case also, explaining why only the Sylvester matrix can be obtained from a difference set in an elementary abelian $2$-group, why two of the matrices cannot be obtained from a symmetric design on $16$ points, and how the various designs may be viewed through the lens of the four-dimensional affine space over the two-element field.
Hadamard matrices and their designs: a coding-theoretic approach
E. F.
Assmus;
J. D.
Key
269-293
Abstract: To every finite dimensional algebraic coefficient system (defined below) $ (\Theta,V)$ over the De Rham algebra $\Omega (M)$ of a manifold $M$, Sullivan builds a local system $ {\rho _\Theta }:{\pi _1}(M) \to V$, in the topological sense, such that the two cohomologies $H_{{\rho _\Theta }}^{\ast}(M;V)$ and $H_\Theta ^{\ast}(\Omega (M);V)$ are isomorphic. In this paper, if ${\mathbf{K}}$ is a simplicial set and $(\Theta,V)$ an algebraic system over the ${C^\infty }$ forms ${A_\infty }({\mathbf{K}})$, we prove a similar result. We use it to extend the Hirsch lemma to the case of fibration whose fiber is an Eilenberg-Mac Lane space with certain non nilpotent action of the fundamental group of the basis. We apply this to a model of the hyperbolic torus; different from the nilpotent one, this new model is a better mirror of the topology.
Equivalence of families of functions on the natural numbers
Claude
Laflamme
307-319
Abstract: We present some consequences of the inequality $\mathfrak{u} < \mathfrak{g}$ among cardinal invariants of the continuum, which has previously been shown to be consistent relative to ZFC. We are interested in its effect on two orderings of families of functions on the natural numbers; in particular we show that, under $\mathfrak{u} < \mathfrak{g}$, there are exactly five equivalence classes for both orderings (excluding the families bounded by a fixed constant function). This implies, under the same hypothesis, the existence of exactly four classes of rarefaction of measure zero sets.
Motion of level sets by mean curvature. II
L. C.
Evans;
J.
Spruck
321-332
Abstract: We give a new proof of short time existence for the classical motion by mean curvature of a smooth hypersurface. Our method consists in studying a fully nonlinear uniformly parabolic equation satisfied by the signed distance function to the surface
Examples of pseudo-Anosov homeomorphisms
Max
Bauer
333-359
Abstract: We generalize a construction in knot theory to construct a large family $\mathcal{G}\mathcal{R} = \cup \,GR(\mathcal{P})$ of mapping classes of a surface of genus $ g$ and one boundary component, where $ \mathcal{P}$ runs over some finite index set. We exhibit explicitly the set $ \mathcal{G}{\mathcal{R}^{\ast}} \subset \mathcal{G}\mathcal{R}$ that consists of pseudo-Anosov maps, find the map that realizes the smallest dilatation in $ \mathcal{G}{\mathcal{R}^{\ast}}$, and for every $ \mathcal{P}$, we give a set of defining relations for $GR(\mathcal{P})$.
An upper bound for the least dilatation
Max
Bauer
361-370
Abstract: We given an upper bound for the least dilatation arising from a pseudo-Anosov map of a closed surface of genus greater or equal to three.
Hyperfinite transversal theory
Boško
Živaljević
371-399
Abstract: A measure theoretic version of a well-known P. Hall's theorem, about the existence of a system of distinct representatives of a finite family of finite sets, has been proved for the case of the Loeb space of an internal, uniformly distributed, hyperfinite counting space. We first prove Hall's theorem for $ \Pi _1^0(\kappa)$ graphs after which we develop the version of discrete Transversal Theory. We then prove a new version of Hall's theorem in the case of $ \Sigma _1^0(\kappa)$ monotone graphs and give an example of a $\Sigma _1^0$ graph which satisfies Hall's condition and which does not possess an internal a.e. matching.
Pettis integrability
Gunnar F.
Stefánsson
401-418
Abstract: A weakly measurable function $ f:\Omega \to X$ is said to be determined by a subspace $D$ of $X$ if for each ${x^{\ast} } \in {X^{\ast} }$, ${x^{\ast}}{\vert _D} = 0$ implies that ${x^{\ast}}\;f= 0$ a.e. For a given Dunford integrable function $ f:\Omega \to X$ with a countably additive indefinite integral we show that $ f$ is Pettis integrable if and only if $f$ is determined by a weakly compactly generated subspace of $X$ if and only if $f$ is determined by a subspace which has Mazur's property. We show that if $f:\Omega \to X$ is Pettis integrable then there exists a sequence ( $ {\varphi _n}$) of $ X$ valued simple functions such that for all ${x^{\ast}} \in {X^{\ast}}$, ${x^{\ast}}f= {\lim _n}{x^{\ast}}\,{\varphi _n}$ a.e. if and only if $f$ is determined by a separable subspace of $ X$. For a bounded weakly measurable function $f:\Omega \to {X^{\ast} }$ into a dual of a weakly compactly generated space, we show that $f$ is Pettis integrable if and only if $ f$ is determined by a separable subspace of $ {X^{\ast}}$ if and only if $ f$ is weakly equivalent to a Pettis integrable function that takes its range in ${\text{cor}}_f^{\ast} (\Omega)$.
A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three
J. W.
Cannon;
Daryl
Cooper
419-431
Abstract: We show that a cocompact hyperbolic group in dimension $3$ is characterized by certain properties of its word metric which depend only on the group structure and not on any action on hyperbolic space. We prove a similar theorem for finite-volume hyperbolic groups in dimension $3$.
Ramsey theory in noncommutative semigroups
Vitaly
Bergelson;
Neil
Hindman
433-446
Abstract: By utilizing ultrafilters we give a general version of the Central Sets Theorem [$6$, Proposition 8.21]. This enables us to derive noncommutative versions of van der Waerden's Theorem and several of its generalizations. We also derive some standard results, including the Hales-Jewett Theorem.
The Gaussian map for rational ruled surfaces
Jeanne
Duflot;
Rick
Miranda
447-459
Abstract: In this paper the Gaussian map $\Phi :{ \wedge ^2}{H^0}(C,K) \to {H^0}(C,3K)$ of a smooth curve $C$ lying on a minimal rational ruled surface is computed. It is shown that the corank of $\Phi$ is determined for almost all such curves by the rational surface in which it lies. Hence, except for some special cases, a curve cannot lie on two nonisomorphic minimal rational ruled surfaces.
The equivariant Hurewicz map
L. Gaunce
Lewis
433-472
Abstract: Let $G$ be a compact Lie group, $ Y$ be a based $ G$-space, and $ V$ be a $ G$-representation. If $\pi _V^G(Y)$ is the equivariant homotopy group of $Y$ in dimension $V$ and $H_V^G(Y)$ is the equivariant ordinary homology group of $Y$ with Burnside ring coefficients in dimension $ V$, then there is an equivariant Hurewicz map $\displaystyle h:\pi _V^G(Y) \to H_V^G(Y).$ One should not expect this map to be an isomorphism, since $ H_V^G(Y)$ must be a module over the Burnside ring, but $ \pi _V^G(Y)$ need not be. However, here it is shown that, under the obvious connectivity conditions on $Y$, this map induces an isomorphism between $ H_V^G(Y)$ and an algebraically defined modification of $ \pi _V^G(Y)$. The equivariant Freudenthal Suspension Theorem contains a technical hypothesis that has no nonequivariant analog. Our results shed some light on the behavior of the suspension map when this rather undesirable technical hypothesis is not satisfied.
Global regularity on $3$-dimensional solvmanifolds
Jacek M.
Cygan;
Leonard F.
Richardson
473-488
Abstract: Let $M$ be any $3$-dimensional (nonabelian) compact solvmanifold. We apply the methods of representation theory to study the convergence of Fourier series of smooth global solutions to first order invariant partial differential equations $Df = g$ in $ {C^\infty }(M)$. We show that smooth infinite-dimensional irreducible solutions, when they exist, satisfy estimates strong enough to guarantee uniform convergence of the irreducible (or primary) Fourier series to a smooth global solution.
Characterizations of algebras arising from locally compact groups
Paul L.
Patterson
489-506
Abstract: Two Banach $ ^{\ast}$-algebras are naturally associated with a locally compact group, $ G$: the group algebra, $ {L^1}(G)$, and the measure algebra, $M(G)$. Either of these Banach algebras is a complete set of invariants for $G$. In any Banach $^{\ast}$-algebra, $A$, the norm one unitary elements form a group, $ S$. Using $S$ we characterize those Banach $ ^{\ast}$-algebras, $ A$, which are isometrically $^{\ast}$-isomorphic to $M(G)$. Our characterization assumes that $A$ is the dual of some Banach space and that its operations are continuous in the resulting weak $ ^{\ast}$ topology. The other most important condition is that the convex hull of $ S$ must be weak$ ^{\ast}$ dense in the unit ball of $A$. We characterize Banach $ ^{\ast}$-algebras which are isomerically isomorphic to ${L^1}(G)$ for some $G$ as those algebras, $A$, whose double centralizer algebra, $ D(A)$, satisfies our characterization for $M(G)$. In addition we require $A$ to consist of those elements of $ D(A)$ on which $ S$ (defined relative to $ D(A)$) acts continuously with its weak$^{\ast}$ topology. Using another characterization of $ {L^1}(G)$ we explicitly calculate the above isomorphism between $ A$ and ${L^1}(G)$.
On the singular cardinal hypothesis
W. J.
Mitchell
507-530
Abstract: We use core model theory to obtain the following lower bounds to the consistency strength for the failure of the Singular Cardinal Hypothesis: Suppose that $\kappa$ is a singular strong limit cardinal such that ${2^\kappa } > {\kappa ^ + }$. Then there is an inner model $K$ such that $o(\kappa ) = {\kappa ^{ + + }}$ in $K$ if $\kappa$ has uncountable cofinality, and $\forall \alpha < \kappa \exists \nu < \kappa o(\kappa ) \geqslant \nu$ in $K$ otherwise.
Adams' cobar equivalence
Yves
Félix;
Stephen
Halperin;
Jean-Claude
Thomas
531-549
Abstract: Let $F$ be the homotopy fibre of a continuous map $Y\xrightarrow{\omega }X$, with $X$ simply connected. We modify and extend a construction of Adams to obtain equivalences of DGA's and DGA modules, $\displaystyle \Omega {C_{\ast}}(X)\xrightarrow{ \simeq }C{U_{\ast}}(\Omega X),$ and $\displaystyle \Omega (C_{\ast}^\omega (Y);{C_{\ast}}(X))\xrightarrow{ \simeq }C{U_{\ast}}(F),$ where on the left-hand side $\Omega ( - )$ denotes the cobar construction. Our equivalences are natural in $X$ and $\omega$. Using this result we show how to read off the algebra ${H_{\ast}}(\Omega X;R)$ and the ${H_{\ast}}(\Omega X;R)$ module, ${H_{\ast}}(F;R)$, from free models for the singular cochain algebras $ C{S^{\ast}}(X)$ and $C{S^{\ast}}(Y)$; here we assume $R$ is a principal ideal domain and $ X$ and $Y$ are of finite $R$ type.
Polynomial solutions to constant coefficient differential equations
S. Paul
Smith
551-569
Abstract: Let $ {D_1}, \ldots ,{D_r} \in \mathbb{C}[\partial /\partial {x_1}, \ldots ,\partial /\partial {x_n}]$ be constant coefficient differential operators with zero constant term. Let $\displaystyle S = \{ f \in \mathbb{C}[{x_1}, \ldots ,{x_n}]\vert{D_j}(f) = 0\;{\text{for all }}1 \leqslant j \leqslant r\}$ be the space of polynomial solutions to the system of simultaneous differential equations ${D_j}(f) = 0$. It is proved that $S$ is a module over $\mathcal{D}(V)$, the ring of differential operators on the affine scheme $V$ with coordinate ring $ \mathbb{C}[\partial /\partial {x_1}, \ldots ,\partial /\partial {x_n}]/\left\langle {{D_1}, \ldots ,{D_r}} \right\rangle$. If $ V$ is smooth and irreducible, then $S$ is a simple $ \mathcal{D}(V)$-module, $ S = 1.\mathcal{D}(V)$, and the generators for $ \mathcal{D}(V)$ yield an algorithm for obtaining a basis for $S$. If $V$ is singular, then $S$ need not be simple. However, $S$ is still a simple $\mathcal{D}(V)$-module for certain curves $ V$, and certain homogeneous spaces $V$, and this allows one to obtain a basis for $ S$, through knowledge of $ \mathcal{D}(V)$.
A direct geometric proof of the Lefschetz fixed point formulas
John D.
Lafferty;
Yan Lin
Yu;
Zhang
Weiping
571-583
Abstract: In this paper we prove the Lefschetz fixed point formulas of Atiyah, Singer, Segal, and Bott for isometries by using the direct geometric method initiated by Patodi.
Studying links via closed braids. V. The unlink
Joan S.
Birman;
William W.
Menasco
585-606
Abstract: The main result is a version of Markov's Theorem which does not involve stabilization, in the special case of the $ r$-component link. As a corollary, it is proved that the stabilization index of a closed braid representative of the unlink is at most $ 1$. To state the result, we need the concept of an "exchange move", which modifies a closed braid without changing its link type or its braid index. For generic closed braids exchange moves change conjugacy class. Theorem $1$ shows that exchange moves are the only obstruction to reducing a closed $n$-braid representative of the $ r$-component unlink to the standard closed $r$-braid representative, through a sequence of braids of nonincreasing braid index.
Locally constant cohomology
E.
Spanier
607-624
Abstract: In this paper we study locally constant cohomology theories on a space $X$. We prove that for cohomology theories on a category of paracompact spaces the homotopy axiom of Eilenberg-Steenrod is a consequence of the other Eilenberg-Steenrod axioms together with continuity and either additivity or weak additivity. We also prove that if $H$ is a cohomology theory on the space of a simplicial complex $K$ which is locally constant on every open simplex of $K$ there is a spectral sequence converging to $ H(\vert K\vert)$ whose $ {E_2}$-term is the usual simplicial cohomology of $K$ with coefficients in various stacks on $ K$ defined by $ H$. This generalizes some known spectral sequences.
An intrinsic description of shape
José M. R.
Sanjurjo
625-636
Abstract: We give in this paper a description of the shape category of compacta in terms of multivalued maps. We introduce the notion of a multi-net and prove that the shape category of compacta is isomorphic to the category $HN$ whose objects are metric compacta and whose morphisms are homotopy classes of multi-nets. This description is intrinsic in the sense that it does not make use of external elements such as ANR-expansions or embeddings in appropriate AR-spaces. We present many applications of this new formulation of shape.
The connectivity of multicurves determined by integral weight train tracks
Andrew
Haas;
Perry
Susskind
637-652
Abstract: An integral weighted train track on a surface determines the isotopy class of an embedded closed $1$-manifold. We are interested in the connectivity of the resulting $1$-manifold. In general there is an algorithm for determining connectivity, and in the simplest case of a $ 2$-parameter train track on a surface of genus one there is an explicit formula. We derive a formula for the connectivity of the closed $1$-manifold determined by a $4$-parameter train track on a surface of genus two which is computable in polynomial time. We also give necessary and sufficient conditions on the parameters for the resulting $1$-manifold to be connected.
The duality between subsemigroups of Lie groups and monotone functions
Karl-Hermann
Neeb
653-677
Abstract: In this paper we give a characterization of those convex cones $ W$ in the Lie algebra ${\mathbf{L}}(G)$ of a connected Lie group $ G$ which are global in $G$, i.e. for which there exists a closed subsemigroup $ S$ in $G$ having $W$ as its tangent wedge ${\mathbf{L}}(S)$. The main result is the Characterization Theorem II.12. We also prove in Corollary II.6 that each germ of a strictly $W$-positive function belongs to a global function if there exists at least one strictly $ W$-positive function. We apply the Characterization Theorem to obtain some general conditions for globality and to give a complete description of the global cones in compact Lie algebras.
Almost periodic potentials in higher dimensions
Vassilis G.
Papanicolaou
679-696
Abstract: This work was motivated by the paper of R. Johnson and J. Moser (see [J-M] in the references) on the one-dimensional almost periodic potentials. Here we study the operator $L = - \Delta /2 - q$, where $q$ is an almost periodic function in $ {R^d}$. It is shown that some of the results of [J-M] extend to the multidimensional case (our approach includes the one-dimensional case as well). We start with the kernel $ k(t,x,y)$ of the semigroup ${e^{ - tL}}$. For fixed $t > 0$ and $u \in {R^d}$, it is known (we review the proof) that $k(t,x,x + u)$ is almost periodic in $x$ with frequency module not bigger than the one of $q$. We show that $k(t,x,y)$ is, also, uniformly continuous on $ [a,b] \times {R^d} \times {R^d}$. These results imply that, if we set $ y = x + u$ in the kernel ${G^m}(x,y;z)$ of $ {(L - z)^{ - m}}$ it becomes almost periodic in $x$ (for the case $u = 0$ we must assume that $m > d/2$), which is a generalization of an old one-dimensional result of Scharf (see [S.G]). After this, we are able to define ${w_m}(z) = {M_x}[{G^m}(x,x;z)]$, and, by integrating this $m$ times, an analog of the complex rotation number $ w(z)$ of [J-M]. We also show that, if $ e(x,y;\lambda )$ is the kernel of the projection operator ${E_\lambda }$ associated to $L$, then the mean value $\alpha (\lambda ) = {M_x}[e(x,x;\lambda )]$ exists. In one dimension, this (times $\pi$) is the rotation number. In higher dimensions ($d = 1$ included), we show that $d\alpha (\lambda )$ is the density of states measure of [A-S] and it is related to ${w_m}(z)$ in a nice way. Finally, we derive a formula for the functional derivative of ${w_m}(z;q)$ with respect to $q$, which extends a result of [J-M].
Rigidity of $p$-completed classifying spaces of alternating groups and classical groups over a finite field
Kenshi
Ishiguro
697-713
Abstract: A $p$-adic rigid structure of the classifying spaces of certain finite groups $\pi$, including alternating groups $ {A_n}$ and finite classical groups, is shown in terms of the maps into the $ p$-completed classifying spaces of compact Lie groups. The spaces $(B\pi )_p^ \wedge$ have no nontrivial retracts. As an application, it is shown that $ (B{A_n})_p^ \wedge \simeq (B{\Sigma _n})_p^ \wedge$ if and only if $n\not \equiv 0,1,\;\bmod \,p$. It is also shown that $(BSL(n,{\mathbb{F}_q}))_p^ \wedge \simeq (BGL(n,{\mathbb{F}_q}))_p^ \wedge$ where $q$ is a power of $p$ if and only if $ (n,q - 1) = 1$.
Symmetric local algebras with $5$-dimensional center
M.
Chlebowitz;
B.
Külshammer
715-731
Abstract: We prove that a symmetric split local algebra whose center is $ 5$-dimensional has dimension $5$ or $8$. This implies that the defect groups of a block of a finite group containing exactly five irreducible Frobenius characters and exactly one irreducible Brauer character have order $5$ or are nonabelian of order $8$.
Prinjective modules, reflection functors, quadratic forms, and Auslander-Reiten sequences
J. A.
de la Peña;
D.
Simson
733-753
Abstract: Let $A,\;B$ be artinian rings and let $ _A{M_B}$ be an $ (A - B)$-bimodule which is a finitely generated left $A$-module and a finitely generated right $ B$-module. A right $ _A{M_B}$-prinjective module is a finitely generated module ${X_R} = (X_A', X_B'', \varphi :X_A' \otimes_A M_B \to X''_B)$ over the triangular matrix ring $\displaystyle R = \left( {\begin{array}{*{20}{c}} A & {_A{M_B}} 0 & B \end{array} } \right)$ such that $X_A'$ is a projective $A$-module, $X''_B$ is an injective $B$-module, and $\varphi$ is a $B$-homomorphism. We study the category $ \operatorname{prin} (R)_B^A$ of right $_A{M_B}$-prinjective modules. It is an additive Krull-Schmidt subcategory of $\bmod (R)$ closed under extensions. For every $ X,\;Y$ in $\operatorname{prin} (R)_B^A,\;\operatorname{Ext} _R^2(X,\,Y) = 0$. When $R$ is an Artin algebra, the category $\operatorname{prin} (R)_B^A$ has Auslander-Reiten sequences and they can be computed in terms of reflection functors. In the case that $R$ is an algebra over an algebraically closed field we give conditions for $ \operatorname{prin} (R)_B^A$ to be representation-finite or representation-tame in terms of a Tits form. In some cases we calculate the coordinates of the Auslander-Reiten translation of a module using a Coxeter linear transformation.
Determinant expression of Selberg zeta functions. II
Shin-ya
Koyama
755-772
Abstract: This paper is the $ \operatorname{PSL} (2,{\mathbf{C}})$-version of Part I. We show that for $ \operatorname{PSL} (2,{\mathbf{C}})$ and its subgroup $\operatorname{PSL} (2,O)$, the Selberg zeta function with its gamma factors is expressed as the determinant of the Laplacians, where $ O$ is the integer ring of an imaginary quadratic field. All the gamma factors are calculated explicitly. We also give an explicit computation to the contribution of the continuous spectrum to the determinant of the Laplacian.
Toeplitz and Hankel operators on Bergman spaces
Karel
Stroethoff;
De Chao
Zheng
773-794
Abstract: In this paper we consider Toeplitz and Hankel operators on the Bergman spaces of the unit ball and the polydisk in ${\mathbb{C}^n}$ whose symbols are bounded measurable functions. We give necessary and sufficient conditions on the symbols for these operators to be compact. We study the Fredholm theory of Toeplitz operators for which the corresponding Hankel operator is compact. For these Toeplitz operators the essential spectrum is computed and shown to be connected. We also consider symbols that extend to continuous functions on the maximal ideal space of ${H^\infty }(\Omega )$; for these symbols we describe when the Toeplitz or Hankel operators are compact.
Log concave sequences of symmetric functions and analogs of the Jacobi-Trudi determinants
Bruce E.
Sagan
795-811
Abstract: We prove that various sequences of elementary and complete homogeneous symmetric functions are log concave or PF. As corollaries we show that certain sequences of $q$-binomial coefficients and $ q$-Stirling numbers have these properties. The principal technique used is a combinatorial interpretation of determinants using lattice paths due to Gessel and Viennot [G-V 85].
A global \L ojasiewicz inequality for algebraic varieties
Shanyu
Ji;
János
Kollár;
Bernard
Shiffman
813-818
Abstract: Let $X$ be the locus of common zeros of polynomials $ {f_1}, \ldots ,{f_k}$ in $ n$ complex variables. A global upper bound for the distance to $X$ is given in the form of a Lojasiewicz inequality. The exponent in this inequality is bounded by ${d^{\min (n,k)}}$ where $d = \max (3,\deg {f_i})$. The estimates are also valid over an algebraically closed field of any characteristic.
On explosions of solutions to a system of partial differential equations modelling chemotaxis
W.
Jäger;
S.
Luckhaus
819-824
Abstract: A system of partial differential equations modelling chemotactic aggregation is analysed (Keller-Segel model). Conditions on the system of parameters are given implying global existence of smooth solutions. In two space dimensions and radially symmetric situations, explosion of the bacteria concentration in finite time is shown for a class of initial values.
Holomorphic flows in ${\bf C}\sp 3,0$ with resonances
Júlio Cesar
Canille Martins
825-837
Abstract: The topological classification, by conjugacy, of the germs of holomorphic diffeomorphisms $f:\,{{\mathbf{C}}^2},0 \to {{\mathbf{C}}^2},0$ with $df(0) = \operatorname{diag} ({\lambda _1},{\lambda _2})$, where $ {\lambda _1}$ is a root of unity and $ \vert{\lambda _2}\vert \ne 1$ is given. This type of diffeomorphism appears as holonomies of singular foliations ${\mathcal{F}_X}$ induced by holomorphic vector fields $X:{{\mathbf{C}}^3},0 \to {{\mathbf{C}}^3},0$ normally hyperbolic and resonant. An explicit example of a such vector field without holomorphic invariant center manifold is presented. We prove that there are no obstructions in the holonomies for ${\mathcal{F}_X}$ to be topologically equivalent to a product type foliation.
A model in which GCH holds at successors but fails at limits
James
Cummings
1-39
Abstract: Starting with GCH and a $ {\mathcal{P}_3}\kappa$-hypermeasurable cardinal, a model is produced in which $ {2^\lambda } = {\lambda ^ + }$ if $\lambda$ is a successor cardinal and ${2^\lambda } = {\lambda ^{ + + }}$ if $ \lambda$ is a limit cardinal. The proof uses a Reverse Easton extension followed by a modified Radin forcing.
Continuation theorems for periodic perturbations of autonomous systems
Anna
Capietto;
Jean
Mawhin;
Fabio
Zanolin
41-72
Abstract: It is first shown in this paper that, whenever it exists, the coincidence degree of the left-hand member of an autonomous differential equation $ {\text{g}}$. This result provides efficient continuation theorems specially for $\omega$-periodic perturbations of autonomous systems. Extensions to differential equations in flow-invariant ENR's are also given.
Two characteristic numbers for smooth plane curves of any degree
Paolo
Aluffi
73-96
Abstract: We use a sequence of blow-ups over the projective space parametrizing plane curves of degree $d$ to obtain some enumerative results concerning smooth plane curves of arbitrary degree. For $d = 4$, this gives a first modern verification of results of H. G. Zeuthen.
Compactifications of locally compact groups and closed subgroups
A. T.
Lau;
P.
Milnes;
J. S.
Pym
97-115
Abstract: Let $G$ be a locally compact group with closed normal subgroup $N$ such that $G/N$ is compact. In this paper, we construct various semigroup compactifications of $G$ from compactifications of $N$ of the same type. This enables us to obtain specific information about the structure of the compactifi cation of $G$ from the structure of the compactification of $ N$. Our results seem to be interesting and new even when $G$ is the additive group of real numbers and $N$ is the integers. Applications and other examples are given.
On the distribution of extremal points of general Chebyshev polynomials
András
Kroó;
Franz
Peherstorfer
117-130
Abstract: For a linear subspace $ {\mathcal{U}_n} = {\operatorname{span}}[{\varphi _1}, \ldots ,{\varphi _n}]$ in $ C[a,b]$ we introduce general Chebyshev polynomials as solutions of the minimization problem $ {\operatorname{min}_{{a_i}}}{\left\Vert {{\varphi _n} - \sum\nolimits_{i = 1}^{n - 1} {{a_i}{\varphi _i}} } \right\Vert _C}$. For such a Chebyshev polynomial we study the distribution of its extremal points (maximum and minimum points) in terms of structural and approximative properties of $ {\mathcal{U}_n}$.
Ces\`aro summability of double Walsh-Fourier series
F.
Móricz;
F.
Schipp;
W. R.
Wade
131-140
Abstract: We introduce quasi-local operators (these include operators of Calderón-Zygmund type), a hybrid Hardy space $ {{\mathbf{H}}^\sharp }$ of functions of two variables, and we obtain sufficient conditions for a quasi-local maximal operator to be of weak type $(\sharp ,1)$. As an application, we show that Cesàro means of the double Walsh-Fourier series of a function $f$ converge a.e. when $f$ belongs to $ {{\mathbf{H}}^\sharp }$. We also obtain the dyadic analogue of a summability result of Marcienkiewicz and Zygmund valid for all $f \in {L^1}$ provided summability takes place in some positive cone.
Extending cellular cohomology to $C\sp *$-algebras
Ruy
Exel;
Terry A.
Loring
141-160
Abstract: A filtration on the $ K$-theory of $ {C^*}$-algebras is introduced. The relative quotients define groups ${H_n}(A),n \geq 0$, for any $ {C^*}$-algebra $ A$, which we call the spherical homology of $A$. This extends cellular cohomology in the sense that $\displaystyle {H_n}(C(X)) \otimes {\mathbf{Q}} \cong {H^n}(X;{\mathbf{Q}})$ for $ X$ a finite CW-complex. While no extension of cellular cohomology which is derived from a filtration on $K$-theory can be additive, Morita-invariant, and continuous, ${H_n}$ is shown to be infinitely additive, Morita invariant for unital ${C^*}$-algebras, and continuous in limited cases.
Maximal entropy odd orbit types
William
Geller;
Juán
Tolosa
161-171
Abstract: A periodic orbit of a continuous map of an interval induces in a natural way a cyclic permutation, called its type. We consider a family of orbit types of period $n$ congruent to $1$ ( $ \operatorname{mod} 4$) introduced recently by Misiurewicz and Nitecki. We prove that the Misiurewicz-Nitecki orbit types and their natural generalizations to the remaining odd periods $n$ have maximal entropy among all orbit types of period $n$, and even among all $n$-permutations.
A decomposition theorem for the spectral sequence of Lie foliations
Jesús A.
Alvarez López
173-184
Abstract: For a Lie $\mathfrak{g}$-foliation $ \mathcal{F}$ on a closed manifold $M$, there is an "infinitesimal action of $\mathfrak{g}$ on $M$ up to homotopy along the leaves", in general it is not an action but defines an action of the corresponding connected simply connected Lie group $\mathfrak{S}$ on the term ${E_1}$ of the spectral sequence associated to $\mathcal{F}$. Even though ${E_1}$ in general is infinite-dimensional and non-Hausdorff (with the topology induced by the $ {\mathcal{C}^\infty }$-topology), it is proved that this action can be averaged when $ \mathfrak{S}$ is compact, obtaining a tensor decomposition theorem of $ {E_2}$. It implies duality in the whole term ${E_2}$ for Riemannian foliations on closed oriented manifolds with compact semisimple structural Lie algebra.
A damped hyperbolic equation on thin domains
Jack K.
Hale;
Geneviève
Raugel
185-219
Abstract: For a damped hyperbolic equation in a thin domain in ${{\mathbf{R}}^3}$ over a bounded smooth domain in $ {{\mathbf{R}}^2}$, it is proved that the global attractors are upper semicontinuous. It is shown also that a global attractor exists in the case of the critical Sobolev exponent.
On the linear independence of certain cohomology classes in the classifying space for subfoliations
Demetrio
Domínguez
221-232
Abstract: The purpose of this paper is to establish the linear independence of certain cohomology classes in the Haefliger classifying space $ B{\Gamma _{({q_1},{q_2})}}$ for sub-foliations of codimension $({q_{1,}}{q_2})$. The classes considered are of secondary type, not belonging to the subalgebra of $ H(B{\Gamma _{({q_1},{q_2})}},R)$ generated by the union of the universal characteristic classes for foliations of codimension $ {q_1}$ and ${q_2}$ respectively, and are elements of the kernel of the canonical homomorphism $ H(B{\Gamma _{({q_1},{q_2})}},R) \to H(B{\Gamma _{{q_1}}} \times B{\Gamma _d},R)$ with $ d = {q_2} - {q_1} > 0$.
Poincar\'e-Lefschetz duality for the homology Conley index
Christopher
McCord
233-252
Abstract: The Conley index for continuous dynamical systems is defined for (one-sided) semiflows. For (two-sided) flows, there are two indices defined: one for the forward flow; and one for the reverse flow. In general, the two indices give different information about the flow; but for flows on orientable manifolds, there is a duality isomorphism between the homology Conley indices of the forward and reverse flows. This duality preserves the algebraic structure of many of the constructions of the Conley index theory: sums and products; continuation; attractor-repeller sequences and connection matrices.
The modular representation theory of $q$-Schur algebras
Jie
Du
253-271
Abstract: We developed some basic theory of characteristic zero modular representations of $q$-Schur algebras. We described a basis of the $ q$-Schur algebra in terms of the relative norm which was first introduced by P. Hoefsmit and L. Scott, and studied the product of two such basis elements. We also defined the defect group of a primitive idempotent in a $q$-Schur algebra and showed that such a defect group is just the vertex of the corresponding indecomposable $ {\mathcal{H}_F}$-module.
The Green correspondence for the representations of Hecke algebras of type $A\sb {r-1}$
Jie
Du
273-287
Abstract: We first prove the conjecture mentioned by Leonard K. Jones in his thesis. By applying this conjecture, we obtain that the vertex of an indecomposable ${\mathcal{H}_F}$-module is an $l$-parabolic subgroup. Finally, we establish the Green correspondence for the representations of Hecke algebras of type $ {A_{r - 1}}$.
The determination of minimal projections and extensions in $L\sp 1$
B. L.
Chalmers;
F. T.
Metcalf
289-305
Abstract: Equations are derived which are shown to be necessary and sufficient for finite rank projections in ${L^1}$ to be minimal. More generally, these equations are also necessary and sufficient to determine operators of minimal norm which extend a fixed linear action on a given finite-dimensional subspace of ${L^1}$ and thus may be viewed as an extension of the Hahn-Banach theorem to higher dimensions in the $ {L^1}$ setting. These equations are solved in terms of an ${L^1}$ best approximation problem and the required orthogonality conditions. Moreover, this solution has a simple geometric interpretation. Questions of uniqueness are considered and a number of examples are given to illustrate the usefulness of these equations in determining minimal projections and extensions, including the minimal $ {L^1}$ projection onto the quadratics.
A relationship between the Jones and Kauffman polynomials
Christopher
King
307-323
Abstract: A simple relationship is presented between the Kauffman polynomial of a framed link $L$ and the Jones polynomial of a derived link $ \tilde L$. The link is $ \tilde L$ obtained by splitting each component of $L$ into two parallel strands, using the framing to determine the linking number of the strands. The relation is checked in several nontrivial examples, and a proof of the general result is given.
A modification of Shelah's oracle-c.c. with applications
Winfried
Just
325-356
Abstract: A method of constructing iterated forcing notions that has a scope of applications similar to Shelah's oracle-c.c. is presented. This method yields a consistency result on homomorphisms of quotient algebras of the Boolean algebra $ \mathcal{P}(\omega )$. Also, it is shown to be relatively consistent with ZFC that the Boolean algebra of Lebesgue measurable subsets of the unit interval has no projective lifting.
Moderate deviations and associated Laplace approximations for sums of independent random vectors
A.
de Acosta
357-375
Abstract: Let $\{ {X_j}\}$ be an i.i.d. sequence of Banach space valued r.v.'s and let ${S_n} = \sum\nolimits_{j = 1}^n {{X_j}}$. For certain positive sequences $ {b_n} \to \infty$, we determine the exact asymptotic behavior of $E{\operatorname{exp}}\{ (b_n^2/n)\Phi ({S_n}/{b_n})\}$, where $\Phi$ is a smooth function. We also prove a large deviation principle for $\{ \mathcal{L}({S_n}/{b_n})\}$.
Young measures and an application of compensated compactness to one-dimensional nonlinear elastodynamics
Peixiong
Lin
377-413
Abstract: We study the existence problem for the equations of $1$-dimensional nonlinear elastodynamics. We obtain the convergence of ${L^p}(p < \infty )$ bounded approximating sequences generated by the method of vanishing viscosity and the Lax-Friedrichs scheme. The analysis uses Young measures, Lax entropies, and the method of compensated compactness.
Generalized potentials and obstacle scattering
Richard L.
Ford
415-431
Abstract: Potential scattering theory is a very well-developed and understood subject. Scattering for Schràdinger operators represented formally by $- \Delta + V$, where $ V$ is a generalized function such as a $\delta$-function, is less understood and requires form perturbation techniques. A general scattering theory for a large class of such singular perturbations of the Laplacian is developed. The theory has application to obstacle scattering. One considers an alternative mathematical model of an obstacle in ${{\mathbf{R}}^n}$. Instead of representing the obstacle by deleting the region inhabited by the obstacle from $ {{\mathbf{R}}^n}$, the surface of the obstacle is treated as impenetrable. The impenetrable surface is understood to be the limiting case of a sequence of highly spiked potentials whose support converges to the surface of the obstacle and whose peaks grow without bound. The limiting case is identified as a $\delta$-function acting on the surface of the obstacle. Hamiltonians for the limiting case are constructed and the conditions governing the existence and completeness of the associated wave operators are determined through application of the general theory.
The Stefan problem with small surface tension
Avner
Friedman;
Fernando
Reitich
465-515
Abstract: The Stefan problem with small surface tension $ \varepsilon$ is considered. Assuming that the classical Stefan problem (with $\varepsilon = 0$) has a smooth free boundary $\Gamma$, we denote the temperature of the solution by ${\theta _0}$ and consider an approximate solution $ {\theta _0} + \varepsilon u$ for the case where $ \varepsilon \ne 0$, $\varepsilon$ small. We first establish the existence and uniqueness of $u$, and then investigate the effect of $u$ on the free boundary $\Gamma$. It is shown that small surface tension affects the free boundary $\Gamma$ radically differently in the two-phase problem than in the one-phase problem.
Iteration of a composition of exponential functions
Xiaoying
Dong
517-526
Abstract: We show that for certain complex parameters ${\lambda _1},\ldots,{\lambda _{n - 1}}$ and ${\lambda _n}$ the Julia set of the function $\displaystyle {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}}$ is the whole plane $\mathbb{C}$. We denote by $\Lambda$ the set of $n$-tuples $({\lambda _1},\ldots,{\lambda _n}),{\lambda _1},\ldots,{\lambda _n} \in \mathbb{R}$ for which the equation $\displaystyle {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}} - z= 0$ has exact two real solutions. In fact, one of them is an attracting fixed point of $\displaystyle {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}},$ which is denoted by $q$. We also show that when $({\lambda _1},\ldots,{\lambda _n})\, \in \Lambda$, the Julia set of $\displaystyle {e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}}$ is the complement of the basin of attraction of $q$. The ideas used in this note may also be applicable to more general functions.
Complete coinductive theories. II
A. H.
Lachlan
527-562
Abstract: Let $T$ be a complete theory over a relational language which has an axiomatization by $\exists \forall $-sentences. The properties of models of $T$ are studied. It is shown that existential formulas are stable. A theory of forking and independence based on Boolean combinations of existential formulas in $\exists \forall $-saturated models of $ T$ is developed for which the independence relation is shown to satisfy a very strong triviality condition. It follows that $T$ is tree-decomposable in the sense of Baldwin and Shelah. It is also shown that if the language is finite, then $T$ has a prime model.
On the existence of conformal measures
Manfred
Denker;
Mariusz
Urbański
563-587
Abstract: A general notion of conformal measure is introduced and some basic properties are studied. Sufficient conditions for the existence of these measures are obtained, using a general construction principle. The geometric properties of conformal measures relate equilibrium states and Hausdorff measures. This is shown for invariant subsets of ${S^1}$ under expanding maps.
Continuity of translation in the dual of $L\sp \infty(G)$ and related spaces
Colin C.
Graham;
Anthony T. M.
Lau;
Michael
Leinert
589-618
Abstract: Let $X$ be a Banach space and $ G$ a locally compact Hausdorff group that acts as a group of isometric linear operators on $X$. The operation of $x \in G$ on $X$ will be denoted by ${L_x}$. We study the set ${X_c}$ of elements $\mu \in X$ such that $x \mapsto {L_x}\mu$ is continuous with respect to the topology on $G$ and the norm-topology on $X$. The spaces $X$ studied include $M{(G)^{\ast} },{\text{LUC}}{(G)^{\ast} },{L^\infty }{(G)^{\ast} },{\text{VN}}(G)$, and ${\text{VN}}{(G)^{\ast} }$. In most cases, characterizations of ${X_c}$ do not appear to be possible, and we give constructions that illustrate this. We relate properties of ${X_c}$ to properties of $G$. For example, if ${X_c}$ is sufficiently small, then $G$ is compact, or even finite, depending on the case. We give related results and open problems.
Inequalities for the integral means of holomorphic functions and their derivatives in the unit ball of $\bold C\sp n$
Ji Huai
Shi
619-637
Abstract: In this paper, the following two inequalities are proved: $\displaystyle \int_0^1 {{(1 - r)}^{a\vert\alpha\vert + b}}M_p^a(r,D^{\alpha} f)... ...alpha \vert = m} \int_0^1 {(1 - r)}^{am + b}M_p^a(r,D^{\alpha }f)\,dr \right\}$ where $\alpha = ({\alpha _1}, \ldots,{\alpha _n})$ is multi-index, $0 < p < \infty,0 < a < \infty$ and $- 1 < b < \infty$. These are a generalization of some classical results of Hardy and Littlewood. Using these two inequalities, we generalize a theorem in $[9]$. The methods used in the proof of Theorem 1 lead us to obtain Theorem 7, which enables us to generalize some theorems about the pluriharmonic conjugates in $ [8]$ and $[2]$.
Change of variable results for $A\sb p$- and reverse H\"older ${\rm RH}\sb r$-classes
R.
Johnson;
C. J.
Neugebauer
639-666
Abstract: We study conditions under which the map $\displaystyle {T_{h,\gamma }}w(x)= w(h(x))\vert h^{\prime}(x){\vert^\gamma }$ maps the Muckenhoupt class ${A_p}$ into ${A_q}$ and the reverse Hölder class $R{H_{{r_1}}}$ into $ R{H_{{r_2}}}$.
Cardinal representations for closures and preclosures
F.
Galvin;
E. C.
Milner;
M.
Pouzet
667-693
Abstract: A cardinal representation of a preclosure $\varphi$ on a set $E$ is a family $\mathcal{A} \subseteq \mathcal{P}(E)$ such that for any set $X \subseteq \cup \mathcal{A},\varphi (X) = E$ holds if and only if $ \vert X \cap A\vert= \vert A\vert$ for every $A \in \mathcal{A}$. We show, for example (Theorem 2.3) that any topological closure has such a representation, but there are closures which have no cardinal representation (Theorem 11.2). We prove that, if $k$ is finite and a closure has no independent set of size $k + 1$, then it has a cardinal representation, $\mathcal{A}$, of size $\vert\mathcal{A}\vert \leq k$ (Theorem 2.4). This result is used to give a new proof of a theorem of D. Duffus and M. Pouzet [4] about gaps in a lattice of finite breadth. We do not know if a closure which has no infinite independent set necessarily has a cardinal representation, but we do prove this is so for the special case of a closure on a countable set (Theorem 2.5). Even in this special case, nothing can be said about the size of the cardinal representation; however, if the closure is algebraic, then there is a finite cardinal representation (Theorem 2.6). These results do not hold for preclosures in general, but if a preclosure on a countable set has no independent set of size $ k + 1$ ($k$ finite), then it has a cardinal representation $ \mathcal{A}$ of size $ \vert\mathcal{A}\vert \leq k$ (Theorem 2.7).
Quantitative stability of variational systems. I. The epigraphical distance
Hédy
Attouch;
Roger J.-B.
Wets
695-729
Abstract: This paper proposes a global measure for the distance between the elements of a variational system (parametrized families of optimization problems).
Equivariant fixed point index and fixed point transfer in nonzero dimensions
Carlos
Prieto;
Hanno
Ulrich
731-745
Abstract: Dold's fixed point index and fixed point transfer are generalized for certain coincidence situations, namely maps which change the "equivariant dimension." Those invariants change the dimension correspondingly. A formula for the index of a situation over a space with trivial group action is exhibited. For the transfer, a generalization of Dold's Lefschetz-Hopf trace formula is proved.
Regular points for ergodic Sina\u\i measures
Masato
Tsujii
747-766
Abstract: Ergodic properties of smooth dynamical systems are considered. A point is called regular for an ergodic measure $\mu$ if it is generic for $\mu$ and the Lyapunov exponents at it coincide with those of $\mu$. We show that an ergodic measure with no zero Lyapunov exponent is absolutely continuous with respect to unstable foliation $ [$L$]$ if and only if the set of all points which are regular for it has positive Lebesgue measure.
On weak convergence in dynamical systems to self-similar processes with spectral representation
Michael T.
Lacey
767-778
Abstract: Let $(X,\mu,T)$ be an aperiodic dynamical system. Set ${S_m}f = f + \cdots + f \circ {T^{m - 1}}$, where $ f$ is a measurable function on $X$. Let $Y(t)$ be one of a class of self-similar process with a "nice" spectral representation, for instance, either a fractional Brownian motion, a Hermite process, or a harmonizable fractional stable motion. We show that there is an $f$ on $X$, and constants ${A_m} \to + \infty$ so that $\displaystyle A_m^{ - 1}{S_{[mt]}}f\mathop \Rightarrow \limits^d Y(t),$ the convergence being understood in the sense of weak convergence of all finite dimensional distributions in $t$.
Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator
R. J.
Beerends
779-814
Abstract: Chebyshev polynomials of the first and the second kind in $ n$ variables ${z_1},{z_2}, \ldots,{z_n}$ are introduced. The variables ${z_1},{z_2}, \ldots,{z_n}$ are the characters of the representations of $SL(n + 1,{\mathbf{C}})$ corresponding to the fundamental weights. The Chebyshev polynomials are eigenpolynomials of a second order linear partial differential operator which is in fact the radial part of the Laplace-Beltrami operator on certain symmetric spaces. We give an explicit expression of this operator in the coordinates ${z_1},{z_2}, \ldots,{z_n}$ and then show how many results in the literature on differential equations satisfied by Chebyshev polynomials in several variables follow immediately from well-known results on the radial part of the Laplace-Beltrami operator. Related topics like orthogonality, symmetry relations, generating functions and recurrence relations are also discussed. Finally we note that the Chebyshev polynomials are a special case of a more general class of orthogonal polynomials in several variables.
Regularity properties of commutators and layer potentials associated to the heat equation
John L.
Lewis;
Margaret A. M.
Murray
815-842
Abstract: In recent years there has been renewed interest in the solution of parabolic boundary value problems by the method of layer potentials. In this paper we consider graph domains $D = \{ (x,t):x > f(t)\}$ in ${\mathcal{R}^2}$, where the boundary function $ f$ is in ${I_{1/2}}({\text{BMO}})$. This class of domains would appear to be the minimal smoothness class for the solvability of the Dirichlet problem for the heat equation by the method of layer potentials. We show that, for $1 < p < \infty$, the boundary single-layer potential operator for $D$ maps ${L^p}$ into the homogeneous Sobolev space $ {I_{1/2}}({L^p})$. This regularity result is obtained by studying the regularity properties of a related family of commutators. Along the way, we prove ${L^p}$ estimates for a class of singular integral operators to which the $ {\text{T1}}$ Theorem of David and Journé does not apply. The necessary estimates are obtained by a variety of real-variable methods.
Multibasic Eulerian polynomials
Dominique
Foata;
Doron
Zeilberger
843-862
Abstract: Eulerian polynomials with several bases are defined. Their combinatorial interpretations are given as well as congruence properties modulo some ideals generated by cyclotomic polynomials.
Heegaard diagrams of $3$-manifolds
Mitsuyuki
Ochiai
863-879
Abstract: For a $ 3$-manifold $M(L)$ obtained by an integral Dehn surgery along an $n$-bridge link $L$ with $n$-components we define a concept of planar Heegaard diagrams of $M(L)$ using a link diagram of $L$. Then by using Homma-Ochiai-Takahashi's theorem and a planar Heegaard diagram of $ M(L)$ we will completely determine if $M(L)$ is the standard $3$-sphere in the case when $L$ is a $2$-bridge link with $2$-components.
On lifting Hecke eigenforms
Lynne H.
Walling
881-896
Abstract: A classical Hilbert modular form $f \in {\mathcal{M}_k}({\Gamma _0}(\mathcal{N},\mathcal{I}),{\chi _\mathcal{N}})$ cannot be an eigenform for the full Hecke algebra. We develop a means of lifting a classical form to a modular form $F \in { \oplus _\lambda }{\mathcal{M}_k}({\Gamma _0}(\mathcal{N},{\mathcal{I}_\lambda }),{\chi _\mathcal{N}})$ which is an eigenform for the full Hecke algebra. Using this lift, we develop the newform theory for a space of cusp forms ${\mathcal{S}_k}({\Gamma _0}(\mathcal{N},\mathcal{I}),{\chi _\mathcal{N}})$; we also use theta series to construct eigenforms for the full Hecke algebra.
Cohomology of metacyclic groups
Johannes
Huebschmann
1-72
Abstract: Let ${\mathbf{e}}:1 \to N \to G \to K \to 1$ be an extension of a finite cyclic group $N$ by a finite cyclic group $ K$. Using homological perturbation theory, we introduce the beginning of a free resolution of the integers ${\mathbf{Z}}$ over the group ring ${\mathbf{Z}}G$ of $G$ in such a way that the resolution reflects the structure of $G$ as an extension of $N$ by $K$, and we use this resolution to compute the additive structure of the integral cohomology of $ G$ in many cases. We proceed by first establishing a number of special cases, thereafter constructing suitable cohomology classes thereby obtaining a lower bound, then computing characteristic classes introduced in an earlier paper, and, finally, exploiting these classes, obtaining upper bounds for the cohomology via the integral cohomology spectral sequence of the extension ${\mathbf{e}}$. The calculation is then completed by comparing the two bounds.
$L\sp p$ bounds for spectral multipliers on nilpotent groups
Michael
Christ
73-81
Abstract: A criterion is given for the ${L^p}$ boundedness of a class of spectral multiplier operators associated to left-invariant, homogeneous subelliptic second-order differential operators on nilpotent Lie groups, generalizing a theorem of Hörmander for radial Fourier multipliers on Euclidean space. The order of differentiability required is half the homogeneous dimension of the group, improving previous results in the same direction.
Pair-dense relation algebras
Roger D.
Maddux
83-131
Abstract: The central result of this paper is that every pair-dense relation algebra is completely representable. A relation algebra is said to be pair-dense if every nonzero element below the identity contains a "pair". A pair is the relation algebraic analogue of a relation of the form $ \{ \langle {a,a} \rangle,\langle {b,b} \rangle \}$ (with $a= b$ allowed). In a simple pair-dense relation algebra, every pair is either a "point" (an algebraic analogue of $\{ \langle {a,a} \rangle \}$) or a "twin" (a pair which contains no point). In fact, every simple pair-dense relation algebra $ \mathfrak{A}$ is completely representable over a set $U$ iff $ \vert U\vert= \kappa + 2\lambda$, where $\kappa$ is the number of points of $\mathfrak{A}$ and $\lambda$ is the number of twins of $\mathfrak{A}$. A relation algebra is said to be point-dense if every nonzero element below the identity contains a point. In a point-dense relation algebra every pair is a point, so a simple point-dense relation algebra $ \mathfrak{A}$ is completely representable over $U$ iff $\vert U\vert= \kappa$, where $ \kappa$ is the number of points of $ \mathfrak{A}$. This last result actually holds for semiassociative relation algebras, a class of algebras strictly containing the class of relation algebras. It follows that the relation algebra of all binary relations on a set $ U$ may be characterized as a simple complete point-dense semiassociative relation algebra whose set of points has the same cardinality as $U$. Semiassociative relation algebras may not be associative, so the equation $(x;y);z= x;(y;z)$ may fail, but it does hold if any one of $x,y$, or $z$ is $1$. In fact, any rearrangement of parentheses is possible in a term of the form ${x_0}; \ldots;{x_{\alpha - 1}}$, in case one of the
An infinite family in ${}\sb 2\pi\sp {\rm s}\sb *$ at Adams filtration seven
Wên Hsiung
Lin
133-149
Abstract: We prove the family $ \{ h_i^2{h_3}{d_1}\}$ in $ \operatorname{Ext}_A^{7,\ast}({\mathbb{Z}_2},{\mathbb{Z}_2})$ detects homotopy elements in the $2$-adic stable homotopy of spheres $_2\pi_{\ast}^S$ where $A$ is the $\bmod \;2$ Steenrod algebra.
Waldhausen's classification theorem for finitely uniformizable $3$-orbifolds
Yoshihiro
Takeuchi
151-200
Abstract: We define a map between two orbifolds. With respect to this map, we generalize $3$-manifold theory to $3$-orbifolds. As the main goal, we generalize the Waldhausen's classification theorem of Haken $ 3$-manifolds to finitely uniformizable $3$-orbifolds. For applications of the developed theory, we introduce an invariant for links and tangles by using the orbifold fundamental group. With the invariant, we classify a class of links and show the untangling theorem.
The associated Askey-Wilson polynomials
Mourad E. H.
Ismail;
Mizan
Rahman
201-237
Abstract: We derive some contiguous relations for very well-poised $_8{\phi _7}$ series and use them to construct two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials. We then use these solutions to find explicit representations of two families of associated Askey-Wilson polynomials. We identify the corresponding continued fractions as quotients of two very well-poised $_8{\phi _7}$ series and find the weight functions.
Twisted calibrations
Timothy A.
Murdoch
239-257
Abstract: The methods of calibrated geometry are extended to include nonorientable submanifolds which can be oriented by some real Euclidean line bundle. Specifically, if there exists a line bundle-valued differential form of comass one which restricts to a submanifold to be a density, then the submanifold satisfies a minimizing property. The results are applied to show that the cone on the Veronese surface minimizes among a general class of comparison $3$-folds.
Grothendieck's existence theorem in analytic geometry and related results
Siegmund
Kosarew
259-306
Abstract: We state and prove several kinds of analytification theorems of formal objects (such as coherent sheaves and formal complex spaces) which are in the spirit of Grothendieck's algebraization theorem in [EGA, III]. The formulation of the results was derived from deformation theory and especially M. Artin's work on representability of functors. The methods of proof depend heavily on a deeper study of cotangent complexes and resolvants. As applications one can deduce the convergence of formal versal deformations in diverse situations.
On topological classification of function spaces $C\sb p(X)$ of low Borel complexity
T.
Dobrowolski;
W.
Marciszewski;
J.
Mogilski
307-324
Abstract: We prove that if $ X$ is a countable nondiscrete completely regular space such that the function space $ {C_p}(X)$ is an absolute $ {F_{\sigma \,\delta }}$-set, then ${C_p}(X)$ is homeomorphic to ${\sigma ^\infty }$, where $\sigma = \{ ({x_i}) \in {{\mathbf{R}}^\infty }:{x_i}= 0$ for all but finitely many $i\}$. As an application we answer in the negative some problems of A. V. Arhangel'skii by giving examples of countable completely regular spaces $ X$ and $Y$ such that $X$ fails to be a ${b_R}$-space and a $k$-space (and hence $X$ is not a $ {k_\omega }$-space and not a sequential space) and $Y$ fails to be an $ {\aleph _0}$-space while the function spaces ${C_p}(X)$ and ${C_p}(Y)$ are homeomorphic to ${C_p}(\mathfrak{X})$ for the compact metric space $\mathfrak{X}= \{ 0\} \cup \{ {n^{ - 1}}:n= 1,2, \ldots \}$.
A representation theorem for cyclic analytic two-isometries
Stefan
Richter
325-349
Abstract: A bounded linear operator $T$ on a complex separable Hilbert space $\mathcal{H}$ is called a $2$-isometry if ${T^{\ast 2}}{T^2} - 2{T^{ \ast }}T + I = 0$. We say that $T$ is analytic if ${ \cap _{n> 0}}\,{T^n}\,\mathcal{H}= (0)$. In this paper we show that every cyclic analytic $ 2$-isometry can be represented as multiplication by $z$ on a Dirichlet-type space $D(\mu)$. Here $\mu$ denotes a finite positive Borel measure on the unit circle. For two measures $\mu$ and $\nu$ the $2$-isometries obtained as multiplication by $ z$ on $D(\mu)$ and $D(\nu)$ are unitarily equivalent if and only if $ \mu = \nu$. We also investigate similarity and quasisimilarity of these $ 2$-isometries, and we apply our results to the invariant subspaces of the Dirichlet shift.
Homogeneous distributions on the Heisenberg group and representations of ${\rm SU}(2,1)$
R. C.
Fabec
351-391
Abstract: A 'Fourier' transform of tempered distributions on the Heisenberg group is defined to analyze homogeneous distributions relative the group of dilations $(z,t) \mapsto (r\,z,{r^2}t)$, $r \in {\mathbf{R}}$. An inversion formula is derived for the abelian central Fourier transform of the distribution. These formulas are applied to the family of homogeneous distributions defining the intertwining operators for the group $ {\text{SU}}(2,1)$. Explicit unitary structures are determined on subquotient representations and their spectral decompositions on the minimal parabolic subgroup are obtained.
Sharp square-function inequalities for conditionally symmetric martingales
Gang
Wang
393-419
Abstract: Let $f$ be a conditionally symmetric martingale taking values in a Hilbert space $\mathbb{H}$ and let $S(f)$ be its square function. If ${\nu _p}$ is the smallest positive zero of the confluent hypergeometric function and $ {\mu _p}$ is the largest positive zero of the parabolic cylinder function of parameter $p$, then the following inequalities are sharp: $\displaystyle \Vert f \Vert _{p} \leq \nu_{p}\Vert S(f)\Vert _{p}$ if$\displaystyle \;0 < p \leq 2,$ $\displaystyle \Vert f \Vert _{p} \leq \mu_{p} \Vert S(f)\Vert _{p}$ if$\displaystyle \;p \geq 3,$ $\displaystyle \nu_{p}\Vert S(f)\Vert _{p}\; \leq\; \Vert f\Vert _p$ if$\displaystyle \; p \geq 2.$ Moreover, the constants $\nu_p$ and $\mu_p$ for the cases mentioned above are also best possible for the Marcinkiewicz-Paley inequalities for Haar functions.
Szeg\H o's theorem on a bidisc
Takahiko
Nakazi
421-432
Abstract: G. Szegö showed that $\displaystyle \inf \;\int_0^{2\pi } {\vert 1 - f{\vert^2}w\,d\theta /2\pi = \exp \;\int_0^{2\pi } {\log \,w\,d\theta /2\pi } }$ where $ f$ ranges over analytic polynomials with mean value zeros. We study extensions of the Szegö's theorem on the disc to the bidisc. We show that the quantity is a mixed form of an arithmetic mean and a geometric one of $w$ in some special cases.
$2$-to-$1$ maps on hereditarily indecomposable continua
Jo
Heath
433-444
Abstract: Suppose $ f$ is a $2{\text{-to-}}1$ continuous map from the hereditarily indecomposable continuum $X$ onto a continuum $Y$. In order for it to be the case that each proper subcontinuum $C$ in $Y$ has as its preimage two disjoint continua each of which $f$ maps homeomorphically onto $C$, it is obviously necessary that $ f$ satisfy the condition that each nondense connected subset of $Y$ has disconnected preimage. We show that this condition is also sufficient, and thus any $ 2{\text{-to-}}1$ continuous map from a hereditarily indecomposable continuum satisfying this condition must be confluent and have an image that is hereditarily indecomposable.
Polarized surfaces of $\Delta$-genus $3$
Maria Lucia
Fania;
Elvira Laura
Livorni
445-463
Abstract: Let $X$ be a smooth, complex, algebraic, projective surface and let $L$ be an ample line bundle on it. Let $\Delta = \Delta \,(X,L)= {c_1}{(L)^2} + 2 - {h^0}(L)$ denote the $\Delta$-genus of the pair $(X,L)$. The purpose of this paper is to classify such pairs under the assumption that $\Delta = 3$ and the complete linear system $ \vert L \vert$ contains a smooth curve. If $d \geq 7$ and $ g \geq \Delta$, Fujita has shown that $L$ is very ample and $g= \Delta$. If $d \geq 7$ and $ g < \Delta = 3$, then $ g= 2$ and those pairs have been studied by Fujita and Beltrametti, Lanteri, and Palleschi. To study the remaining cases we have examined the two possibilities of $L + tK$ being nef or not, for $t= 1,2$. In the cases in which $ L + 2K$ is nef it turned out to be very useful to iterate the adjunction mapping for ample line bundles as it was done by Biancofiore and Livorni in the very ample case. If $g > \Delta$ there are still open cases to solve in which completely different methods are needed.
Stability of travelling wave solutions of diffusive predator-prey systems
R.
Gardner;
C. K. R. T.
Jones
465-524
Abstract: The stability of travelling wave solutions of singularly perturbed, diffusive predator-prey systems is proved by showing that the linearized operator about such a solution has no unstable spectrum and that the translation eigenvalue at $\lambda = 0$ is simple. The proof illustrates the application of some recently developed geometric and topological methods for counting eigenvalues.
The structure of open continuous mappings having two valences
A. K.
Lyzzaik;
Kenneth
Stephenson
525-566
Abstract: The authors study open continuous functions which map the unit disc to compact Riemann surfaces and which assume each value in the range space (with a finite number of exceptions) either $p$ or $q$ times for some positive integers $ p$, $q$. Although the questions here originated in efforts to understand mapping properties of locally univalent analytic functions, the authors remove analyticity assumptions and show that the underlying issues are topological and combinatoric in nature. The mappings are studied by embedding their image surfaces in compact covering spaces, a setting which allows the consideration of fairly general ranges and which accommodates branch and exceptional points. Known results are generalized and extended; several open questions are posed, particularly regarding the higher dimensional analogues of the results.
A formal Mellin transform in the arithmetic of function fields
David
Goss
567-582
Abstract: The Mellin transform is a fundamental tool of classical arithmetic. We would also like such a tool in the arithmetic of function fields based on Drinfeld modules, although a construction has not yet been found. One formal approach to finding Mellin transforms in classical theory is through $ p$-adic measures. It turns out that this approach also works for function fields. Thus this paper is devoted to exploring what can be learned this way. We will establish some very enticing connections with gamma functions and the Kummer-Vandiver conjecture for function fields.
Linear topological classifications of certain function spaces
Vesko M.
Valov
583-600
Abstract: Some linear classification results for the spaces ${C_P}(X)$ and $C_P^{\ast}\,(X)$ are proved.
On the range of the Radon $d$-plane transform and its dual
Fulton B.
Gonzalez
601-619
Abstract: We present direct, group-theoretic proofs of the range theorem for the Radon $d$-plane transform $f \to \hat f$ on $ \mathcal{S}({\mathbb{R}^n})$. (The original proof, by Richter, involves extensive use of local coordinate calculations on $ G(d,n)$, the Grassmann manifold of affine $d$-planes in $ {\mathbb{R}^n}$.) We show that moment conditions are not sufficient to describe this range when $d < n - 1$, in contrast to the compactly supported case. Finally, we show that the dual $ d$-plane transform maps $ \mathcal{E}(G(d,n))$ surjectively onto $ \mathcal{E}({\mathbb{R}^n})$.
Mod\`ele minimal \'equivariant et formalit\'e
Thierry
Lambre
621-639
Abstract: We study the rational equivariant homotopy type of a topological space $ X$ equipped with an action of the group of integers modulo $n$. For $n= {p^k}$ ($p$ prime, $k$ a positive integer), we build an algebraic model which gives the rational equivariant homotopy type of $ X$. The homotopical fixed-point set appears in the construction of a model of the fixed-points set. In general, this model is different from $ {\text{G}}$. Triantafillou's model $ [{\text{T1}}]$. For $ n= p$ ($p$ prime), we then give a notion of equivariant formality. We prove that this notion is equivalent to the formalizability of the inclusion of fixed-points set $i:{X^{{\mathbb{Z}_p}}} \to X$. Examples and counterexamples of $ {\mathbb{Z}_p}$-formal spaces are given.
$k$-cobordism for links in $S\sp 3$
Tim D.
Cochran
641-654
Abstract: We give an explicit finite set of (based) links which generates, under connected sum, the $k$-cobordism classes of links. We show that the union of these generating sets, $2 \leq k < \infty$, is not a generating set for $\omega$-cobordism classes or even $ \infty$-cobordism classes. For $2$-component links in ${S^3}$ we define $(2,k)$-corbordism and show that the concordance invariants ${\beta ^i},i \in {\mathbb{Z}^+}$, previously defined by the author, are invariants under $ (2,i + 1)$-cobordism. Moreover we show that the $(2,k)$-cobordism classes of links (with linking number 0) is a free abelian group of rank $k - 1$, detected precisely by ${\beta ^1} \times \cdots \times {\beta^{k - 1}}$. We write down a basis. The union of these bases $(2 \leq k < \infty)$ is not a generating set for $ (2,\infty)$ or $(2,\omega)$-cobordism classes. However, we can show that $\prod _{i = 1}^\infty {\beta ^i}(\;)$ is an isomorphism from the group of $(2,\infty)$-cobordism classes to the subgroup $\mathcal{R} \subset \prod _{i = 1}^\infty \mathbb{Z}$ of linearly recurrent sequences, so a basis exists by work of T. Jin.
Isotopy invariants of graphs
D.
Jonish;
K. C.
Millett
655-702
Abstract: The development of oriented and semioriented algebraic invariants associated to a class of embeddings of regular four valent graphs is given. These generalize the analogous invariants for classical knots and links, can be determined from them by means of a weighted averaging process, and define them by means of a new state model. This development includes the elucidation of the elementary spatial equivalences (generalizations of the classical Reidemeister moves), and the extension of fundamental concepts in classical knot theory, such as the linking number, to this class spatial graphs.
Derived functors of unitary highest weight modules at reduction points
Pierluigi Möseneder
Frajria
703-738
Abstract: The derived functors introduced by Zuckerman are applied to the unitary highest weight modules of the Hermitian symmetric pairs of classical type. The construction yields "small" unitary representations which do not have a highest weight. The infinitesimal character parameter of the modules we consider is such that their derived functors are nontrivial in more than one degree; at the extreme degrees where the cohomology is nonvanishing, it is possible to determine the ${\mathbf{K}}$-spectrum of the resulting representations completely. Using this information it is shown that, in most cases, the derived functor modules are unitary, irreducible, and not of highest weight type. Their infinitesimal character and lowest ${\mathbf{K}}$-type are also easily computed.
Reflected Brownian motion in a cone with radially homogeneous reflection field
Y.
Kwon;
R. J.
Williams
739-780
Abstract: This work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties. (i) The state space is a cone in $d$-dimensions $ (d \geq 3)$, and the process behaves in the interior of the cone like ordinary Brownian motion. (ii) The process reflects instantaneously at the boundary of the cone, the direction of reflection being fixed on each radial line emanating from the vertex of the cone. (iii) The amount of time that the process spends at the vertex of the cone is zero (i.e., the set of times for which the process is at the vertex has zero Lebesgue measure). The question of existence and uniqueness is cast in precise mathematical terms as a submartingale problem in the style used by Stroock and Varadhan for diffusions on smooth domains with smooth boundary conditions. The question is resolved in terms of a real parameter $\alpha$ which in general depends in a rather complicated way on the geometric data of the problem, i.e., on the cone and the directions of reflection. However, a criterion is given for determining whether $\alpha > 0$. It is shown that there is a unique continuous strong Markov process satisfying (i)-(iii) above if and only if $ \alpha < 2$, and that starting away from the vertex, this process does not reach the vertex if $ \alpha \leq 0$ and does reach the vertex almost surely if $0 < \alpha < 2$. If $\alpha \geq 2$, there is a unique continuous strong Markov process satisfying (i) and (ii) above; it reaches the vertex of the cone almost surely and remains there. These results are illustrated in concrete terms for some special cases. The process considered here serves as a model for comparison with a reflected Brownian motion in a cone having a nonradially homogeneous reflection field. This is discussed in a subsequent work by Kwon.
Terms in the Selberg trace formula for ${\rm SL}(3,{\scr Z})\backslash{\rm SL}(3,{\scr R})/{\rm SO}(3,{\scr R})$ associated to Eisenstein series coming from a minimal parabolic subgroup
D. I.
Wallace
781-793
Abstract: In this paper we compute the contribution to the trace formula for $SL(3,\mathcal{Z})$ of the integrals associated to inner products of Eisenstein series. We show these reduce to corresponding integrals for a lower rank trace formula plus a few residual terms.
Convex optimization and the epi-distance topology
Gerald
Beer;
Roberto
Lucchetti
795-813
Abstract: Let $\Gamma (X)$ denote the proper, lower semicontinuous, convex functions on a Banach space $X$, equipped with the completely metrizable topology $\tau$ of uniform convergence of distance functions on bounded sets. A function $f$ in $ \Gamma (X)$ is called well-posed provided it has a unique minimizer, and each minimizing sequence converges to this minimizer. We show that well-posedness of $f \in \Gamma (X)$ is the minimal condition that guarantees strong convergence of approximate minima of $\tau$-approximating functions to the minimum of $ f$. Moreover, we show that most functions in $\langle \Gamma (X),{\tau _{aw}}\rangle$ are well-posed, and that this fails if $ \Gamma (X)$ is topologized by the weaker topology of Mosco convergence, whenever $ X$ is infinite dimensional. Applications to metric projections are also given, including a fundamental characterization of approximative compactness.
Stable rank and approximation theorems in $H\sp \infty$
Leonardo A.
Laroco
815-832
Abstract: It is conjectured that for ${H^\infty }$ the Bass stable rank $($bsr$)$ is $1$ and the topological stable rank $($tsr$)$ is $2$. ${\text{bsr}}({H^\infty })= 1$ if and only if for every $({f_1},{f_2})\; \in {H^\infty } \times {H^\infty }$ which is a corona pair (i.e., there exist $ {g_{1}}$, ${g_2} \in {H^\infty }$ such that ${f_1}{g_1} + {f_2}{g_2}= 1$) there exists a $g \in {H^\infty }$ such that ${f_1} + {f_2}g \in {({H^\infty })^{ - 1}}$, the invertibles in $ {H^\infty }$; however, it suffices to consider corona pairs $({f_1},{f_2})$ where ${f_1}$ is a Blaschke product. It is also shown that there exists a $ g \in {H^\infty }$ such that $ {f_1} + {f_2}g \in \exp ({H^\infty })$ if and only if $ \log {f_1}$ can be boundedly, analytically defined on $\{ {z \in \mathbb{D}:\vert\,{f_2}(z)\vert < \delta } \}$, for some $\delta > 0$. ${\text{tsr}}({H^\infty })= 2$ if and only if the corona pairs are uniformly dense in ${H^\infty } \times {H^\infty }$; however, it suffices to show that the corona pairs are uniformly dense in pairs of Blaschke products. This condition would be satisfied if the interpolating Blaschke products were uniformly dense in the Blaschke products. For $b$ an inner function, $K= {H^2} \ominus b\,{H^2}$ is an ${H^\infty }$-module via the compressed Toeplitz operators $ {C_f}= {P_K}{T_f}{\vert _K}$, for $ f \in {H^\infty }$, where $ {T_f}$ is the Toeplitz operator ${T_f}g= f\,g$, for $g \in {H^2}$. Some stable rank questions can be recast as lifting questions: for $({f_i})_1^n \subset{H^\infty }$, there exist $ ({g_i})_1^n$, $({h_i})_1^n \subset{H^\infty }$ such that $ \sum\nolimits_{i = 1}^n {({f_i} + b\,{g_i}){h_i}= 1}$ if and only if the compressed Toeplitz operators $ ({C_{{f_i}}})_1^n$ may be lifted to Toeplitz operators $({T_{{F_i}}})_1^n$ which generate $B({H^2})$ as an ideal.
A multidimensional Wiener-Wintner theorem and spectrum estimation
John J.
Benedetto
833-852
Abstract: Sufficient conditions are given for a bounded positive measure $ \mu$ on ${\mathbb{R}^d}$ to be the power spectrum of a function $ \varphi$. Applications to spectrum estimation are made for the cases in which a signal $\varphi$ is known or its autocorrelation $ {P_\phi }$ is known. In the first case, it is shown that $\displaystyle \int {\vert\hat f(\gamma)\vert^2}d{\mu _\phi}(\gamma)= \mathop {\... ...}\,\frac{1}{\vert B(R )\vert}\,\int_{B(R)} \vert f \ast \varphi (t)\vert^2\;dt,$ where ${\hat P}_{\varphi }= {\mu _\varphi }$, $ B(R)$ is the $ d$-dimensional ball of radius $R$, and $f$ ranges through a prescribed function space. In the second case, an example, which is a variant of the Tomas-Stein restriction theorem, is $\displaystyle \forall f \in {L^1}({\mathbb{R}^d})\, \cap \,{L^p}({\mathbb{R}^d}... ...ght)\;\left(\parallel f {\parallel _{1}} + \parallel f{\parallel _{p}} \right),$ where $1 \leq p < 2d/(d + 1)$ and the power spectrum ${\mu _{d - 1}}$ is the compactly supported restriction of surface measure to the unit sphere $\sum\nolimits_{d - 1} { \subseteq } \;{{\hat{\mathbb{R}}}^d}$.
Asymptotic integrations of nonoscillatory second order differential equations
Shao Zhu
Chen
853-865
Abstract: The linear differential equation (1) $(r(t)x^{\prime})^{\prime} + (f(t) + q(t))x= 0$ is viewed as a perturbation of the equation (2) $ (r(t)y^{\prime})^{\prime} + (f(t)y = 0$, where $r > 0$, $f$ and $q$ are real-valued continuous functions. Suppose that (2) is nonoscillatory at infinity and $ {y_1}$, ${y_2}$ are principal, nonprincipal solutions of (2), respectively. Adapted Riccati techniques are used to obtain an asymptotic integration for the principal solution ${x_1}$ of (1). Under some mild assumptions, we characterize that (1) has a principal solution $ {x_1}$ satisfying ${x_1}= {y_1}(1 + o(1))$. Sufficient (sometimes necessary) conditions under which the nonprincipal solution ${x_2}$ of (1) behaves, in three different degrees, like ${y_2}$ as $ t \to \infty$ are also established.
On subordinated holomorphic semigroups
Alfred S.
Carasso;
Tosio
Kato
867-878
Abstract: If $[{e^{ - tA}}]$ is a uniformly bounded $ {C_0}$ semigroup on a complex Banach space $X$, then $- {A^\alpha },$, $0 < \alpha < 1$, generates a holomorphic semigroup on $X$, and $ [{e^{ - t{A^\alpha }}}]$ is subordinated to $ [{e^{ - tA}}]$ through the Lévy stable density function. This was proved by Yosida in 1960, by suitably deforming the contour in an inverse Laplace transform representation. Using other methods, we exhibit a large class of probability measures such that the subordinated semigroups are always holomorphic, and obtain a necessary condition on the measure's Laplace transform for that to be the case. We then construct probability measures that do not have this property.
Radon-Nikod\'ym properties associated with subsets of countable discrete abelian groups
Patrick N.
Dowling
879-890
Abstract: With any subset of a countable discrete abelian we associate with it three Banach space properties. These properties are Radon-Nikodym type properties. The relationship between these properties is investigated. The results are applied to give new characterizations of Riesz subsets and Rosenthal subsets of countable discrete abelian groups.
Shadows of convex bodies
Keith
Ball
891-901
Abstract: It is proved that if $C$ is a convex body in ${\mathbb{R}^n}$ then $C$ has an affine image $\tilde C$ (of nonzero volume) so that if $ P$ is any $ 1$-codimensional orthogonal projection, $\displaystyle \vert P\tilde C\vert \geq \,\vert\tilde C{\vert^{(n - 1)\,/\,n}}.$ It is also shown that there is a pathological body, $ K$, all of whose orthogonal projections have volume about $\sqrt n$ times as large as $\vert K{\vert^{(n - 1)\,/\,n}}$.
Area integral estimates for the biharmonic operator in Lipschitz domains
Jill
Pipher;
Gregory
Verchota
903-917
Abstract: Let $D \subseteq {{\mathbf{R}}^n}$ be a Lipschitz domain and let $u$ be a function biharmonic in $D$, i.e., $\Delta \Delta u= 0$ in $D$. We prove that the nontangential maximal function and the square function of the gradient of $ u$ have equivalent ${L^p}(d\mu)$ norms, where $d\mu \in {A^\infty }\,(d\sigma)$ and $ d\sigma$ is surface measure on $\partial D$.
Erratum to: ``Tiled orders of finite global dimension'' [Trans. Amer. Math. Soc. {\bf 322} (1990), no. 1, 329--342; MR0968884 (91b:16016)]
Hisaaki
Fujita
919-920
The Schwartz space of a general semisimple Lie group. II. Wave packets associated to Schwartz functions
Rebecca A.
Herb
1-69
Abstract: Let $G$ be a connected semisimple Lie group. If $G$ has finite center, Harish-Chandra used Eisenstein integrals to construct Schwartz class wave packets of matrix coefficients and showed that every $ K$-finite function in the Schwartz space is a finite sum of such wave packets. This paper is the second in a series which generalizes these results of Harish-Chandra to include the case that $ G$ has infinite center. In this paper, the Plancherel theorem is used to decompose $ K$-compact Schwartz class functions (those with $K$-types in a compact set) as finite sums of wave packets. A new feature of the infinite center case is that the individual wave packets occurring in the decomposition of a Schwartz class function need not be Schwartz class. These wave packets are studied to obtain necessary conditions for a wave packet of Eisenstein integrals to occur in the decomposition of a Schwartz class function. Applied to the case that $f$ itself is a single wave packet, the results of this paper yield a complete characterization of Schwartz class wave packets.
The construction of analytic diffeomorphisms for exact robot navigation on star worlds
Elon
Rimon;
Daniel E.
Koditschek
71-116
Abstract: A Euclidean Sphere World is a compact connected submanifold of Euclidean $n$-space whose boundary is the disjoint union of a finite number of $(n - 1)$ dimensional Euclidean spheres. A Star World is a homeomorph of a Euclidean Sphere World, each of whose boundary components forms the boundary of a star shaped set. We construct a family of analytic diffeomorphisms from any analytic Star World to an appropriate Euclidean Sphere World "model." Since our construction is expressed in closed form using elementary algebraic operations, the family is effectively computable. The need for such a family of diffeomorphisms arises in the setting of robot navigation and control. We conclude by mentioning a topological classification problem whose resolution is critical to the eventual practicability of these results.
Linear series with an $N$-fold point on a general curve
David
Schubert
117-124
Abstract: A linear series $(V,\mathcal{L})$ on a curve $X$ has an $N$-fold point along a divisor $D$ of degree $N$ if $\dim (V \cap {H^0}\;(X,\mathcal{L}\,(- D))) \geq \dim \;V - 1$. The dimensions of the families of linear series with an $N$-fold point are determined for general curves.
Weighted Sobolev-Poincar\'e inequalities and pointwise estimates for a class of degenerate elliptic equations
Bruno
Franchi
125-158
Abstract: In this paper we prove a Sobolev-Poincaré inequality for a class of function spaces associated with some degenerate elliptic equations. These estimates provide us with the basic tool to prove an invariant Harnack inequality for weak positive solutions. In addition, Hölder regularity of the weak solutions follows in a standard way.
A dense set of operators with tiny commutants
Domingo A.
Herrero
159-183
Abstract: For a (bounded linear) operator $T$ on a complex, separable, infinite-dimensional Hilbert space $ \mathcal{H}$, let $\mathcal{A}\,(T)$ and ${\mathcal{A}^a}(T)$ denote the weak closure of the polynomials in $T$ and, respectively, the weak closure of the rational functions with poles outside the spectrum of $ T$. Let $\mathcal{A}^{\prime}(T)$ and $\mathcal{A}''(T)$ denote the commutant and, respectively, the double commutant of $T$. We say that $T$ has a tiny commutant if $ \mathcal{A}^{\prime}(T)= {\mathcal{A}^a}(T)$. By constructing a large family of "models" and by using standard techniques of approximation, it is shown that $T \in \mathcal{L}\,(\mathcal{H}):T$ has a tiny commutant is norm-dense in the algebra $ \mathcal{L}\,(\mathcal{H})$ of all operators acting on $ \mathcal{H}$. Other related results: Let $ \operatorname{Lat}\;\mathcal{B}$ denote the invariant subspace lattice of a subalgebra $\mathcal{B}$ of $ \mathcal{L}(\mathcal{H})$. For a Jordan curve $\gamma \subset {\mathbf{C}}$, let $\hat \gamma$ denote the union of $\gamma$ and its interior; for $T \in \mathcal{L}\;(\mathcal{H})$, let ${\rho _{s - F}}\,(T)= \{ \lambda \in {\mathbf{C}}:\lambda - T$ is a semi-Fredholm operator, and let $ \rho _{s - F}^ + (T)(\rho _{s - F}^ - (T))= \{ \lambda \in {\rho _{s - F}}(T):{\text{ind}}(\lambda - T) > 0\;(< 0,{\text{resp.)\} }}$. With this notation in mind, it is shown that ${\{ T \in \mathcal{L}(\mathcal{H}):\mathcal{A}(T)= {\mathcal{A}^a}(T)\} ^ - } ... ...peratorname{Lat}\;{\mathcal{A}^a}(T)\} ^ - }= \{ A \in \mathcal{L}(\mathcal{H})$ if $\gamma$ (Jordan curve) $\subset\rho _{s - F}^ \pm (A)$, then $\hat \gamma \subset \sigma (A)\}$; moreover, $\{ A \in \mathcal{L}(\mathcal{H})$: if $\gamma$ (Jordan curve) $\subset\rho _{s - F}^ \pm (A)$, then ${\text{ind}}(\lambda - A)$ is constant on $\hat \gamma \cap {\rho _{s - F}}(A)\} \subset {\{ T \in \mathcal{L}(\mathcal{H... ...orname{Lat}\;\mathcal{A}^{\prime}(T)\} \subset\{ A \in \mathcal{L}(\mathcal{H})$: if $\gamma$ (Jordan curve) $\subset\rho _{s - F}^ \pm (A)$, then $ \hat \gamma \cap {\rho _{s - F}}(A) \subset\rho _{s - F}^ \pm (A)\} \subset \{ T \in \mathcal{L}(\mathcal{H}):\mathcal{A}(T)= {\mathcal{A}^a}(T)\}$. (The first and the last inclusions are proper.) The results also include a partial analysis of $ \operatorname{Lat}\;\mathcal{A}''(T)$.
Automorphisms and twisted forms of generalized Witt Lie algebras
William C.
Waterhouse
185-200
Abstract: We prove that the automorphisms of the generalized Witt Lie algebras $ W(m,{\mathbf{n}})$ over arbitrary commutative rings of characteristic $p \geq 3$ all come from automorphisms of the algebras on which they are defined as derivations. By descent theory, this result then implies that if a Lie algebra over a field becomes isomorphic to $ W(m,{\mathbf{n}})$ over the algebraic closure, it is a derivation algebra of the type studied long ago by Ree. Furthermore, all isomorphisms of those derivation algebras are induced by isomorphisms of their underlying associative algebras.
Alexander duality and Hurewicz fibrations
Steven C.
Ferry
201-219
Abstract: We explore conditions under which the restriction of the projection map $ p:{S^n} \times B \to B$ to an open subset $ U \subset S^n \times B$ is a Hurewicz fibration. As a consequence, we exhibit Hurewicz fibrations $p:E \to I$ such that: (i) $p:E \to I$ is not a locally trivial bundle, (ii) $p^{ - 1}(t)$ is an open $n$-manifold for each $t$, and (iii) $p\; \circ \;{\text{proj:E}} \times {R^1} \to I$ is a locally trivial bundle. The fibers in our examples are distinguished by having nonisomorphic fundamental groups at infinity. We also show that when the fibers of a Hurewicz fibration with open $ n$-manifold fibers have finitely generated $ (n - 1){\text{st}}$ homology, then all fibers have the same finite number of ends. This last shows that the punctured torus and the thrice punctured two-sphere cannot both be fibers of a Hurewicz fibration $p:E \to I$ with open $2$-manifold fibers.
On the relative reflexivity of finitely generated modules of operators
Bojan
Magajna
221-249
Abstract: Let $\mathcal{R}$ be a von Neumann algebra on a Hilbert space $ \mathcal{H}$ with commutant $ \mathcal{R}^{\prime}$ and centre $ \mathcal{C}$. For each subspace $ \mathcal{S}$ of $\mathcal{R}$ let $ \operatorname{ref}_\mathcal{R}\,(\mathcal{S})$ be the space of all $B \in \mathcal{R}$ such that $XBY= 0$ for all $X,Y \in \mathcal{R}$ satisfying $X\,\mathcal{S}\,Y = 0$. If $ \operatorname{ref}_\mathcal{R}\,(\mathcal{S})= \mathcal{S}$, the space $\mathcal{S}$ is called $ \mathcal{R}$-reflexive. (If $\mathcal{R}= \mathcal{B}(\mathcal{H})$ and $\mathcal{S}$ is an algebra containing the identity operator, $ \mathcal{R}$-reflexivity reduces to the usual reflexivity in operator theory.) The main result of the paper is the following: if $\mathcal{S}$ is one-dimensional, or if $\mathcal{S}$ is arbitrary finite-dimensional but $\mathcal{R}$ has no central portions of type $ {{\text{I}}_n}$ for $ n > 1$, then the space $\overline {\mathcal{C}\mathcal{S}}$ is $ \mathcal{R}$-reflexive and the space $\overline {\mathcal{R}^{\prime}\,\mathcal{S}}$ is $ \mathcal{B}(\mathcal{H})$-reflexive, where the bar denotes the closure in the ultraweak operator topology. If $\mathcal{R}$ is a factor, then $ \mathcal{R}^{\prime}\,\mathcal{S}$ is closed in the weak operator topology for each finite-dimensional subspace $\mathcal{S}$ of $ \mathcal{R}$.
$q$-tensor space and $q$-Weyl modules
Richard
Dipper;
Gordon
James
251-282
Abstract: We obtain the irreducible representations of the $q$-Schur algebra, motivated by the fact that these representations give all the irreducible representations of $G{L_n}(q)$ in the nondescribing characteristic. The irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction.
A modified Schur algorithm and an extended Hamburger moment problem
Olav
Njåstad
283-311
Abstract: An algorithm for a Pick-Nevanlinna problem where the interpolation points coalesce into a finite set of points on the real line is introduced, its connection with certain multipoint Padé approximation problems is discussed, and the results are used to obtain the solutions of an extended Hamburger moment problem.
Almost tangent and cotangent structures in the large
G.
Thompson;
U.
Schwardmann
313-328
Abstract: We examine some global properties of integrable almost tangent and cotangent manifolds. In particular, we extend several results which essentially characterize tangent and cotangent bundles as, respectively, regular almost tangent and cotangent structures.
Solving ordinary differential equations in terms of series with real exponents
D. Yu.
GrigorЬev;
M. F.
Singer
329-351
Abstract: We generalize the Newton polygon procedure for algebraic equations to generate solutions of polynomial differential equations of the form $\sum\nolimits_{i = 0}^\infty {{\alpha _i}{x^{{\beta _i}}}}$ where the $ {\alpha _i}$ are complex numbers and the $ {\beta _i}$ are real numbers with ${\beta _0} > {\beta _1} > \cdots$. Using the differential version of the Newton polygon process, we show that any such a series solution is finitely determined and show how one can enumerate all such solutions of a given polynomial differential equation. We also show that the question of deciding if a system of polynomial differential equations has such a power series solution is undecidable.
First steps in descriptive theory of locales
John
Isbell
353-371
Abstract: F. Hausdorff and D. Montgomery showed that a subspace of a completely metrizable space is developable if and only if it is ${F_\sigma }$ and $ {G_\delta }$. This extends to arbitrary metrizable locales when " ${F_\sigma }$" and " $ {G_\delta }$" are taken in the localic sense (countable join of closed, resp. meet of open, sublocales). In any locale, the developable sublocales are exactly the complemented elements of the lattice of sublocales. The main further results of this paper concern the strictly pointless relative theory, which exists because--always in metrizable locales-- there exist nonzero pointless-absolute $ {G_\delta}^{\prime}{\text{s}}$, ${G_\delta }$ in every pointless extension. For instance, the pointless part $ {\text{pl}}({\mathbf{R}})$ of the real line is characterized as the only nonzero zero-dimensional separable metrizable pointless-absolute $ {G_\delta }$. There is no nonzero pointless-absolute $ {F_\sigma }$. The pointless part of any metrizable space is, if not zero, second category, i.e. not a countable join of nowhere dense sublocales.
Coefficient ideals
Kishor
Shah
373-384
Abstract: Let $R$ be a $d$-dimensional Noetherian quasi-unmixed local ring with maximal ideal $M$ and an $M$-primary ideal $I$ with integral closure $ \overline I$. We prove that there exist unique largest ideals $ {I_k}$ for $1 \leq k \leq d$ lying between $I$ and $ \overline I$ such that the first $k + 1$ Hilbert coefficients of $ I$ and ${I_k}$ coincide. These coefficient ideals clarify some classical results related to $\overline I$. We determine their structure and immediately apply the structure theorem to study the associated primes of the associated graded ring of $ I$.
On complete congruence lattices of complete lattices
G.
Grätzer;
H.
Lakser
385-405
Abstract: The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In this paper, we characterize this lattice as a complete lattice. In other words, for a complete lattice $L$, we construct a complete lattice $ K$ such that $ L$ is isomorphic to the lattice of complete congruence relations of $K$. Regarding $K$ as an infinitary algebra, this result strengthens the characterization theorem of congruence lattices of infinitary algebras of G. Grätzer and W. A. Lampe. In addition, we show how to construct $ K$ so that it will also have a prescribed automorphism group.
Definable singularity
William J.
Mitchell
407-426
Abstract: The main result of this paper is a characterization of singular cardinals in terms of the core model, assuming that there is no model of $\exists \kappa \,o(\kappa)= {\kappa ^{ + + }}$. This characterization is used to prove a result in infinitary Ramsey theory. In the course of the proof we develop a simplified statement of the covering lemma for sequences of measures which avoids the use of mice. We believe that this development will be capable of isolating almost all applications of the covering lemma from the detailed structure of the core model.
Control of degenerate diffusions in ${\bf R}\sp d$
Omar
Hijab
427-448
Abstract: An optimal regularity result is established for the viscosity solution of the degenerate elliptic equation $\displaystyle - Av + F(x,\upsilon ,D\upsilon )= 0,$ $A= \frac{1}{2}\sum {{a_{ij}}(x){\partial ^2}/\partial \,x{_i}\,\partial \,{x_j}}, x \in {{\mathbf{R}}^d}$. We assume the equation is of Bellman type, i.e. $F(x,\upsilon ,p)= {\sup _{u \in U}}[b(x,u) \cdot p + c(x,u)\upsilon - f(x,u)]$, $U \subset{{\mathbf{R}}^d}$. If we set $\lambda \equiv {\inf _{x,u}}c(x,u)$, then there exists $ {\lambda _0} \geq 0$ such that $0 < \lambda < {\lambda _0}$ implies $\upsilon$ is Hölder, while $\lambda > {\lambda _0}$ implies $ \upsilon$ is Lipschitz. The following is established: Suppose the equation is also of Lipschitz type, i.e. suppose there is a Lipschitz function $ u(x,\upsilon ,p)$ such that the supremum in $ F\,(x,\upsilon ,p)$ is uniquely attained at $ u= u\,(x,\upsilon ,p)$; then there exists ${\lambda _1} > {\lambda _0}$ such that $\lambda > {\lambda _1}$ implies $ \upsilon$ is ${C^{1,1}},$ i.e. $D\upsilon$ exists and is Lipschitz.
Flowbox manifolds
J. M.
Aarts;
L. G.
Oversteegen
449-463
Abstract: A separable and metrizable space $X$ is called a flowbox manifold if there exists a base for the open sets each of whose elements has a product structure with the reals $\operatorname{Re}$ as a factor such that a natural consistency condition is met. We show how flowbox manifolds can be divided into orientable and nonorientable ones. We prove that a space $X$ is an orientable flowbox manifold if and only if $X$ can be endowed with the structure of a flow without restpoints. In this way we generalize Whitney's theory of regular families of curves so as to include self-entwined curves in general separable metric spaces. All spaces under consideration are separable and metrizable.
Plane autonomous systems with rational vector fields
Harold E.
Benzinger
465-483
Abstract: The differential equation $\dot z= R(z)$ is studied, where $ R$ is an arbitrary rational function. It is shown that the Riemann sphere is decomposed into finitely many open sets, on each of which the flow is analytic and, in each time direction, there is common long-term behavior. The boundaries of the open sets consist of those points for which the flow fails to be analytic in at least one time direction. The main idea is to express the differential equation as a continuous Newton method $\dot z = - f(z)/f^{\prime}\;(z)$, where $f$ is an analytic function which can have branch points and essential singularities. A method is also given for the computer generation of phase plane portraits which shows the correct time parametrization and which is noniterative, thereby avoiding the problems associated with the iteration of rational functions.
Fixed set systems of equivariant infinite loop spaces
Steven R.
Costenoble;
Stefan
Waner
485-505
Abstract: We develop machinery enabling us to show that suitable $G$-spaces, including the equivariant version of $BF$, are equivariant infinite loop spaces. This involves a "recognition principle" for systems of spaces which behave formally like the system of fixed sets of a $G$-space; that is, we give a necessary and sufficient condition for such a system to be equivalent to the fixed set system of an equivariant infinite loop space. The advantage of using the language of fixed set systems is that one can frequently replace the system of fixed sets of an actual $G$-space by an equivalent formal system which is considerably simpler, and which admits the requisite geometry necessary for delooping. We also apply this machinery to construct equivariant Eilenberg-Mac Lane spaces corresponding to Mackey functors.
Blow-up of straightening-closed ideals in ordinal Hodge algebras
Winfried
Bruns;
Aron
Simis;
Ngô Viêt
Trung
507-528
Abstract: We study a class of ideals $I$ in graded ordinal Hodge algebras $ A$. These ideals are distinguished by the fact that their powers have a canonical standard basis. This leads to Hodge algebra structures on the Rees ring and the associated graded ring. Furthermore, from a natural standard filtration one obtains a depth bound for $A/{I^n}$ which, under certain conditions, is sharp for $n$ large. Frequently one observes that ${I^n}= {I^{(n)}}$. Under suitable hypotheses it is possible to calculate the divisor class group of the Rees algebra. Our main examples are ideals of "virtual" maximal minors and ideals of maximal minors "fixing a submatrix".
Lie flows of codimension $3$
E.
Gallego;
A.
Reventós
529-541
Abstract: We study the following realization problem: given a Lie algebra of dimension $3$ and an integer $q,0 \leq q \leq 3$, is there a compact manifold endowed with a Lie flow transversely modeled on $\mathcal{G}$ and with structural Lie algebra of dimension $q$? We give here a quite complete answer to this problem but some questions remain still open $({\text{cf.}}\;\S2$.
Towards a functional calculus for subnormal tuples: the minimal normal extension
John B.
Conway
543-567
Abstract: In this paper the study of a functional calculus for subnormal $ n$-tuples is initiated and the minimal normal extension problem for this functional calculus is explored. This problem is shown to be equivalent to a mean approximation problem in several complex variables which is solved. An analogous uniform approximation problem is also explored. In addition these general results are applied together with The Area and the The Coarea Formula from Geometric Measure Theory to operators on Bergman spaces and to the tensor product of two subnormal operators. The minimal normal extension of the tensor product of the Bergman shift with itself is completely determined.
Vanishing of $H\sp 2\sb w(M,K(H))$ for certain finite von Neumann algebras
Florin
Rădulescu
569-584
Abstract: We prove the vanishing of the second Hochschild cohomology group $H_w^2\,(M,K(H))$, whenever $M \subset B(H)$ is a finite countably decomposable von Neumann algebra not containing a non $ \Gamma$-factor or a factor without Cartan subalgebra as a direct summand. Here $ H$ is a Hubert space, and $ K(H)$ the compact operators.
On the distance between unitary orbits of weighted shifts
Laurent
Marcoux
585-612
Abstract: In this paper, we consider invertible bilateral weighted shift operators acting on a complex separable Hilbert space $\mathcal{H}$. They have the property that there exist a constant $\tau > 0$ and an orthonormal basis $ {\{ {{e_i}} \}_{i \in \mathbb{Z}}}$ for $ \mathcal{H}$ with respect to which a shift $V$ acts by $W{e_i}= {w_i}{e_{i + 1}},i \in \mathbb{Z}$ and $ {\mathbf{\vert}}{w_i}{\mathbf{\vert}} \geq \tau$. The equivalence class $\mathcal{U}(W)= \{ {U^{\ast}}\;WU:U \in \mathcal{B}(\mathcal{H}),U\;{\text{unitary}}\}$ of weighted shifts with weight sequence (with respect to the basis $ {\{ {U^{\ast}}{e_i}\} _{i \in \mathbb{Z}}}$ for $ \mathcal{H})$ identical to that of $W$ forms the unitary orbit of $W$. Given two shifts $W$ and $V$, one can define a distance $ d(\mathcal{U}(V),\mathcal{U}(W))= \inf \{\parallel \,X - Y\parallel :X \in \mathcal{U}(V),Y \in \mathcal{U}(W)\} $ between the unitary orbits of $W$ and $V$. We establish numerical estimates for upper and lower bounds on this distance which depend upon information drawn from finite dimensional restrictions of these operators.
Monogenic differential calculus
F.
Sommen
613-632
Abstract: In this paper we study differential forms satisfying a Dirac type equation and taking values in a Clifford algebra. For them we establish a Cauchy representation formula and we compute winding numbers for pairs of nonintersecting cycles in $ {\mathbb{R}^m}$ as residues of special differential forms. Next we prove that the cohomology spaces for the complex of monogenic differential forms split as direct sums of de Rham cohomology spaces. We also study duals of spaces of monogenic differential forms, leading to a general residue theory in Euclidean space. Our theory includes the one established in our paper [11] and is strongly related to certain differential forms introduced by Habetha in [4].
Harmonic measure versus Hausdorff measures on repellers for holomorphic maps
Anna
Zdunik
633-652
Abstract: This paper is a continuation of a joint paper of the author with F. Przytycki and M. Urbański. We study a harmonic measure on a boundary of so-called repelling boundary domain; an important example is a basin of a sink for a rational map. Using the results of the above-mentioned paper we prove that either the boundary of the domain is an analytically embedded circle or interval, or else the harmonic measure is singular with respect to the Hausdorff measure corresponding to the function ${\phi _c}(t)= t\;\exp \,\left(c\sqrt {\log \frac{1} {t}\,\log \,\log \,\log } \frac{1} {t}\right)$ for some $c > 0$.
Recursive linear orders with incomplete successivities
Rodney G.
Downey;
Michael F.
Moses
653-668
Abstract: A recursive linear order is said to have intrinsically complete successivities if, in every recursive copy, the successivities form a complete set. We show (Theorem 1) that there is a recursive linear order with intrinsically complete successivities but (Theorem 2) that this cannot be a discrete linear oder. We investigate the related issues of intrinsically non-low and non-semilow successivities in discrete linear orders. We show also (Theorem 3) that no recursive linear order has intrinsically $wtt$-complete successivities.
The two-sided Stefan problem with a spatially dependent latent heat
Terry R.
McConnell
669-699
Abstract: We prove existence and uniqueness of solutions to a problem which generalizes the two-sided Stefan problem. The initial temperature distribution and variable latent heat may be given by positive measures rather than point functions, and the free boundaries which result are essentially arbitrary increasing functions which need not exhibit any degree of smoothness in general. Nevertheless, the solutions are "classical" in the sense that all derivatives and boundary values have the classical interpretation. We also study connections with the Skorohod embedding problem of probability theory and with a general class of optimal stopping problems.
Varieties of periodic attractor in cellular automata
Mike
Hurley
701-726
Abstract: We apply three alternate definitions of "attractor" to cellular automata. Examples are given to show that using the different definitions can give different answers to the question "Does this cellular automaton have a periodic attractor?" The three definitions are the topological notion of attractor as used by C. Conley, a more measure-theoretic version given by J. Milnor, and a variant of Milnor's definition that is based on the concept of the "center of attraction" of an orbit. Restrictions on the types of periodic orbits that can be periodic attractors for cellular automata are described. With any of these definitions, a cellular automaton has at most one periodic attractor. Additionally, if Conley's definition is used, then a periodic attractor must be a fixed point. Using Milnor's definition, each point on a periodic attractor must be fixed by all shifts, so the number of symbols used is an upper bound on the period; whether the actual upper bound is $1$ is unknown. With the third definition this restriction is removed, and examples are given of onedimensional cellular automata on three symbols that have finite "attractors" of arbitrarily large size (with the third definition, a finite attractor is not necessarily a single periodic orbit).
Classification of balanced sets and critical points of even functions on spheres
Charles V.
Coffman
727-747
Abstract: The Lyusternik-Schnirelman approach to the study of critical points of even functionals on the sphere ${S^N}$ employs min-max or max-min principles whose formulation uses a numerical invariant that is defined for compact balanced subsets of ${S^N}$. The Krasnosel'skii genus is an example. Here we study a general class of such invariants (which is quite large) with particular attention to the following questions: formulation of dual variational principles, multiplicity results for critical points, and determination of the Morse index of nondegenerate critical points.
Invariant arcs, Whitney levels, and Kelley continua
M.
van de Vel
749-771
Abstract: As an application of convexity in spaces of arcs, three results of a somewhat different nature have been obtained. The first one gives some simple conditions under which an arc of a semilattice is mapped back into itself by an order-preserving function. The second result states that certain Whitney levels are absolute retracts. Finally, Kelley continua are characterized by what we call approximating coselections.
A transitive homeomorphism on the pseudoarc which is semiconjugate to the tent map
Judy
Kennedy
773-793
Abstract: A powerful theorem and construction of Wayne Lewis are used to build two homeomorphisms on the pseudoarc, each of which is semiconjugate to the tent map on the unit interval. The first homeomorphism is transitive, thus answering a question of Marcy Barge as to whether such homeomorphisms exist. The second homeomorphism admits wandering points. Also, it is proven that any homeomorphism on the pseudoarc that is semiconjugate to the tent map and is irreducible with respect to the semiconjugacy must either be transitive or admit wandering points.
Representations of knot groups in ${\rm SU}(2)$
Eric Paul
Klassen
795-828
Abstract: This paper is a study of the structure of the space $R(K)$ of representations of classical knot groups into $ {\text{SU}}(2)$. Let $ \hat R(K)$ equal the set of conjugacy classes of irreducible representations. In $\S I$, we interpret the relations in a presentation of the knot group in terms of the geometry of ${\text{SU}}(2)$; using this technique we calculate $ \hat R(K)$ for $ K$ equal to the torus knots, twist knots, and the Whitehead link. We also determine a formula for the number of binary dihedral representations of an arbitrary knot group. We prove, using techniques introduced by Culler and Shalen, that if the dimension of $\hat R(K)$ is greater than $1$, then the complement in ${S^3}$ of a tubular neighborhood of $ K$ contains closed, nonboundary parallel, incompressible surfaces. We also show how, for certain nonprime and doubled knots, $ \hat R(K)$ has dimension greater than one. In $\S II$, we calculate the Zariski tangent space, ${T_\rho }(R(K))$, for an arbitrary knot $ K$, at a reducible representation $\rho$, using a technique due to Weil. We prove that for all but a finite number of the reducible representations, $\dim {T_\rho }(R(K))= 3$. These nonexceptional representations possess neighborhoods in $ R(K)$ containing only reducible representations. At the exceptional representations, which correspond to real roots of the Alexander polynomial, $\dim {T_\rho }(R(K)) = 3 + 2k$ for a positive integer $ k$. In those examples analyzed in this paper, these exceptional representations can be expressed as limits of arcs of irreducible representations. We also give an interpretation of these "extra" tangent vectors as representations in the group of Euclidean isometries of the plane.
On the existence and uniqueness of positive solutions for competing species models with diffusion
E. N.
Dancer
829-859
Abstract: In this paper, we consider strictly positive solutions of competing species systems with diffusion under Dirichlet boundary conditions. We obtain a good understanding of when strictly positive solutions exist, obtain new nonuniqueness results and a number of other results, showing how complicated these equations can be. In particular, we consider how the shape of the underlying domain affects the behaviour of the equations.
Actions of loop groups on harmonic maps
M. J.
Bergvelt;
M. A.
Guest
861-886
Abstract: We describe a general framework in which subgroups of the loop group $ \Lambda G{l_n}\mathbb{C}$ act on the space of harmonic maps from ${S^2}$ to $ G{l_n}\mathbb{C}$. This represents a simplification of the action considered by Zakharov-Mikhailov-Shabat [ZM, ZS] in that we take the contour for the Riemann-Hilbert problem to be a union of circles; however, it reduces the basic ingredient to the well-known Birkhoff decomposition of $\Lambda G{l_n}\mathbb{C}$, and this facilitates a rigorous treatment. We give various concrete examples of the action, and use these to investigate a suggestion of Uhlenbeck [Uh] that a limiting version of such an action ("completion") gives rise to her fundamental process of "adding a uniton". It turns out that this does not occur, because completion preserves the energy of harmonic maps. However, in the special case of harmonic maps from ${S^2}$ to complex projective space, we describe a modification of this completion procedure which does indeed reproduce "adding a uniton".
An equivariant torus theorem for involutions
W. H.
Holzmann
887-906
Abstract: A complete classification is given for equivariant surgery on incompressible tori with respect to involutions with possible $ 1$- or $ 2$-dimensional fixed sets.
Weighted inequalities for maximal functions associated with general measures
Kenneth F.
Andersen
907-920
Abstract: For certain positive Borel measures $\mu$ on $ {\mathbf{R}}$ and for $ {T_\mu }$ any of three naturally associated maximal function operators of Hardy-Littlewood type, the weight pairs $(u,\upsilon)$ for which ${T_\mu }$ is of weak type $(p,p),1 \leq p < \infty $, and of strong type $(p,p),1 < p < \infty$, are characterized. Only minimal assumptions are placed on $ \mu$; in particular, $ \mu$ need not satisfy a doubling condition nor need it be continuous.
Hyperbolic invariants of knots and links
Colin
Adams;
Martin
Hildebrand;
Jeffrey
Weeks
1-56
Abstract: Tables of values for the hyperbolic volume, number of symmetries, cusp volume and conformal invariants of the cusps are given for hyperbolic knots through ten crossings and hyperbolic links of $2, 3$ and $4$ components through $9$ crossings. The horoball patterns and the canonical triangulations are displayed for knots through eight crossings and for particularly interesting additional examples of knots and links.
Crossed simplicial groups and their associated homology
Zbigniew
Fiedorowicz;
Jean-Louis
Loday
57-87
Abstract: We introduce a notion of crossed simplicial group, which generalizes Connes' notion of the cyclic category. We show that this concept has several equivalent descriptions and give a complete classification of these structures. We also show how many of Connes' results can be generalized and simplified in this framework.
Separating points from closed convex sets over ordered fields and a metric for $\tilde{R}^n$
Robert O.
Robson
89-99
Abstract: Let $R$ be an arbitrary ordered field, let $ \bar R$ be a real closure, and let $\tilde R$ and $ {\tilde R^n}$ denote the real spectra of $\bar R[X]$ and $\bar R[{X_1}, \ldots,{X_n}]$. We prove that a closed convex subset in ${R^n}$ may be separated from a point not in it via a continuous "linear" functional taking values in $ \tilde R$ and that there is a $\tilde R$-valued metric on ${\tilde R^n}$. The methods rely on the ultrafilter interpretation of points in ${\tilde R^n}$ and on the existence of suprema and infima of sets in $\tilde R$.
Quadratic models for generic local $3$-parameter bifurcations on the plane
Freddy
Dumortier;
Peter
Fiddelaers
101-126
Abstract: The first chapter deals with singularities occurring in quadratic planar vector fields. We make distinction between singularities which as a general system are of finite codimension and singularities which are of infinite codimension in the sense that they are nonisolated, or Hamiltonian, or integrable, or that they have an axis of symmetry after a linear coordinate change or that they can be approximated by centers. In the second chapter we provide quadratic models for all the known versal $ k$-parameter unfoldings with $k = 1,2,3$, except for the nilpotent focus which cannot occur as a quadratic system. We finally show that a certain type of elliptic points of codimension $ 4$ does not have a quadratic versal unfolding.
Action on Grassmannians associated with a field extension
Patrick
Rabau
127-155
Abstract: We examine the action of the general linear group ${\text{GL}}_L(V)$ on the set of all $K$-subspaces of $V$, where $L/K$ is a finite field extension and $V$ is a finite-dimensional vector space over $L$. The orbits are completely classified in the case of quadratic and cubic extensions; for infinite fields, the number of orbits is shown to be infinite if the degree of the extension is at least four. As an application we obtain $q$-analogues of tranformation and evaluation formulas for hypergeometric functions due to Gessel and Stanton.
Action on Grassmannians associated with commutative semisimple algebras
Dae San
Kim;
Patrick
Rabau
157-178
Abstract: Let $A$ be a finite-dimensional commutative semisimple algebra over a field $k$ and let $V$ be a finitely generated faithful $ A$-module. We study the action of the general linear group ${\text{GL}}_A(V)$ on the set of all $k$-subspaces of $V$ and show that, if the field $k$ is infinite, there are infinitely many orbits as soon as $A$ has dimension at least four. If $ A$ has dimension two or three, the number of orbits is finite and independent of the field; in each such case we completely classify the orbits by means of a certain number of integer parameters and determine the structure of the quotient poset obtained from the action of $ {\text{GL}}_A(V)$ on the poset of $k$-subspaces of $V$.
Unit groups and class numbers of real cyclic octic fields
Yuan Yuan
Shen
179-209
Abstract: The generating polynomials of D. Shanks' simplest quadratic and cubic fields and M.-N. Gras' simplest quartic and sextic fields can be obtained by working in the group $ {\mathbf{PG}}{{\mathbf{L}}_2}({\mathbf{Q}})$. Following this procedure and working in the group $ {\mathbf{PG}}{{\mathbf{L}}_2}({\mathbf{Q}}(\sqrt 2))$, we obtain a family of octic polynomials and hence a family of real cyclic octic fields. We find a system of independent units which is close to being a system of fundamental units in the sense that the index has a uniform upper bound. To do this, we use a group theoretic argument along with a method similar to one used by T. W. Cusick to find a lower bound for the regulator and hence an upper bound for the index. Via Brauer-Siegel's theorem, we can estimate how large the class numbers of our octic fields are. After working out the first three examples in $ \S5$, we make a conjecture that the index is $8$. We succeed in getting a system of fundamental units for the quartic subfield. For the octic field we obtain a set of units which we conjecture to be fundamental. Finally, there is a very natural way to generalize the octic polynomials to get a family of real $ {2^n}$-tic number fields. However, to select a subfamily so that the fields become Galois over $ {\mathbf{Q}}$ is not easy and still a lot of work on these remains to be done.
Extensions of measures invariant under countable groups of transformations
Adam
Krawczyk;
Piotr
Zakrzewski
211-226
Abstract: We consider countably additive, nonnegative, extended real-valued measures vanishing on singletons. Given a group $G$ of bijections of a set $ X$ and a $G$-invariant measure $m$ on $X$ we ask whether there exists a proper $ G$-invariant extension of $ m$. We prove, among others, that if $ \mathbb{Q}$ is the group of rational translations of the reals, then there is no maximal $ \mathbb{Q}$-invariant extension of the Lebesgue measure on $\mathbb{R}$. On the other hand, if ${2^\omega }$ is real-valued measurable, then there exists a maximal $\sigma$-finite $ \mathbb{Q}$-invariant measure defined on a proper $\sigma$-algebra of subsets of $\mathbb{R}$.
Brownian motion in a wedge with variable skew reflection
L. C. G.
Rogers
227-236
Abstract: Does planar Brownian motion confined to a wedge by skew reflection on the sides approach the vertex of the wedge? This question has been answered by Varadhan and Williams in the case where the direction of reflection is constant on each of the sides, but here we address the question when the direction reflected is allowed to vary. A necessary condition, and a sufficient condition, are obtained for the vertex to be reached. The conditions are of a geometric nature, and the gap between them is quite small.
On the braid index of alternating links
Kunio
Murasugi
237-260
Abstract: We show that, at least for an alternating fibered link or $ 2$-bridge link $ L$, there is an exact formula which expresses the braid index ${\mathbf{b}}(L)$ of $L$ as a function of the $2$-variable generalization ${P_L}(l,m)$ of the Jones polynomial.
Characterizations of turbulent one-dimensional mappings via $\omega$-limit sets
Michael J.
Evans;
Paul D.
Humke;
Cheng Ming
Lee;
Richard J.
O’Malley
261-280
Abstract: The structure of $ \omega$-limit sets for nonturbulent functions is studied, and various characterizations for turbulent and chaotic functions are obtained. In particular, it is proved that a continuous function mapping a compact interval into itself is turbulent if and only if there exists an $\omega$-limit set which is a unilaterally convergent sequence
Parity and generalized multiplicity
P. M.
Fitzpatrick;
Jacobo
Pejsachowicz
281-305
Abstract: Assuming that $ X$ and $Y$ are Banach spaces and $ \alpha :[a,b] \to \mathcal{L}(X,Y)$ is a path of linear Fredholm operators with invertible endpoints, in $[{\text{F}} - \text{P}1]$ we defined a homotopy invariant of $\alpha,\sigma (\alpha,I) \in {{\mathbf{Z}}_2}$, the parity of $\alpha$ on $I$. The parity plays a fundamental role in bifurcation problems, and in degree theory for nonlinear Fredholm-type mappings. Here we prove (a) that, generically, the parity is a $\bmod\, 2$ count of the number of transversal intersections of $\alpha (I)$ with the set of singular operators, (b) that if $ {\lambda _0}$ is an isolated singular point of $\alpha$, then the local parity $\displaystyle \sigma (\alpha,{\lambda _0}) \equiv \mathop {\lim }\limits_{\vare... ...\to 0} \sigma (\alpha,[{\lambda _0} - \varepsilon,{\lambda _0} + \varepsilon ])$ remains invariant under Lyapunov-Schmidt reduction, and (c) that $\sigma (\alpha,{\lambda _0}) = {(- 1)^{{M_G}({\lambda _0})}}$, where ${M_G}({\lambda _0})$ is any one of the various concepts of generalized multiplicity which have been defined in the context of linearized bifurcation data.
Cyclic Galois extensions and normal bases
C.
Greither
307-343
Abstract: A Kummer theory is presented which does not need roots of unity in the ground ring. For $R$ commutative with $ {p^{ - 1}} \in R$ we study the group of cyclic Galois extensions of fixed degree $ {p^n}$ in detail. Our theory is well suited for dealing with cyclic $ {p^n}$-extensions of a number field $K$ which are unramified outside $p$. We then consider the group $ \operatorname{Gal}({\mathcal{O}_K}[{p^{ - 1}}],{C_{{p^n}}})$ of all such extensions, and its subgroup $ {\text{NB}}({\mathcal{O}_K}[{p^{ - 1}}],{C_{{p^n}}})$ of extensions with integral normal basis outside $p$. For the size of the latter we get a simple asymptotic formula $ (n \to \infty)$, and the discrepancy between the two groups is in some way measured by the defect $\delta$ in Leopoldt's conjecture.
The solution of length four equations over groups
Martin
Edjvet;
James
Howie
345-369
Abstract: Let $G$ be a group, $F$ the free group generated by $t$ and let $r(t) \in G \ast F$. The equation $r(t) = 1$ is said to have a solution over $ G$ if it has a solution in some group that contains $G$. This is equivalent to saying that the natural map $G \to \langle G \ast F\vert r(t)\rangle$ is injective. There is a conjecture (attributed to M. Kervaire and F. Laudenbach) that injectivity fails only if the exponent sum of $t$ in $r(t)$ is zero. In this paper we verify this conjecture in the case when the sum of the absolute values of the exponent of $t$ in $r(t)$ is equal to four.
Algebraic hulls and smooth orbit equivalence
Alessandra
Iozzi
371-384
Abstract: For $i = 1,2,$ let ${\mathcal{F}_i}$ be foliations on smooth manifolds ${M_i}$ determined by the actions of connected Lie groups ${H_i}$; we describe here some results which provide an obstruction, in terms of a geometric invariant of the actions, to the existence of a diffeomorphism between the $ \mathcal{F}_i^{\prime}{\text{s}}$.
The holomorphic discrete series of an affine symmetric space and representations with reproducing kernels
G.
Ólafsson;
B.
Ørsted
385-405
Abstract: Consider a semisimple connected Lie group $G$ with an affine symmetric space $ X$. We study abstractly the intertwining operators from the discrete series of $ X$ into representations with reproducing kernel and, in particular, into the discrete series of $G$; each such is given by a convolution with an analytic function. For $X$ of Hermitian type, we consider the holomorphic discrete series of $X$ and here derive very explicit formulas for the intertwining operators. As a corollary we get a multiplicity one result for the series in question.
A measure of smoothness related to the Laplacian
Z.
Ditzian
407-422
Abstract: A $K$-functional on $f \in C\,({R^d})$ given by $\displaystyle \tilde K\,(f,{t^2})= \inf (\vert\vert f - g\vert\vert + {t^2}\vert\vert\Delta g\vert\vert;g \in {C^2}\,({R^d}))$ will be shown to be equivalent to the modulus of smoothness $\displaystyle \tilde w\,(f,t)= \mathop {\sup }\limits_{0 < h \leq t} \,\left\Ve... ...,df(x) - \sum\limits_{i = 1}^d {[f(x + h{e_i}) + f(x - h{e_i})]} } \right\Vert.$ The situation for other Banach spaces of functions on ${R^d}$ will also be resolved.
The cohomology of certain function spaces
Martin
Bendersky;
Sam
Gitler
423-440
Abstract: We study a spectral sequence converging to the cohomology of the configuration space of $n$ ordered points in a manifold. A chain complex is constructed with homology equal to the ${E_2}$ term. If the field is the rationals and the manifold is formal then the spectral sequence is shown to collapse. The results are applied to compute the Anderson spectral sequence converging to the cohomology of a function space.
Remarks on forced equations of the double pendulum type
Gabriella
Tarantello
441-452
Abstract: Motivated by the double pendulum equation we consider Lagrangian systems with potential $V = V(t,q)$ periodic in each of the variables $t,q = ({q_1}, \ldots,{q_N})$. We study periodic solutions for the corresponding equation of motion subject to a periodic force $f = f(t)$. If $f$ has mean value zero, the corresponding variational problem admits a $ {{\mathbf{Z}}^N}$ symmetry which yields $N + 1$ distinct periodic solutions (see [9]). Here we consider the case where the average of $f$, though bounded, is no longer required to be zero. We show how this situation becomes more delicate, and in general it is only possible to claim no more than two periodic solutions.
Some model theory of compact Lie groups
Ali
Nesin;
Anand
Pillay
453-463
Abstract: We consider questions of first order definability in a compact Lie group $G$. Our main result is that if such $G$ is simple (and centerless) then the Lie group structure of $G$ is first order definable from the abstract group structure. Along the way we also show (i) if $ G$ is non-Abelian and connected then a copy of the field $\mathbb{R}$ is interpretable. in $(G, \cdot)$, and (ii) any "$ 1$-dimensional" field interpretable in $ (\mathbb{R}, +, \cdot)$ is definably (i.e., semialgebraically) isomorphic to the ground field $ \mathbb{R}$.
Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture
Michel L.
Lapidus
465-529
Abstract: Let $\Omega$ be a bounded open set of $ {\mathbb{R}^n}\;(n \geq 1)$ with "fractal" boundary $\Gamma$. We extend Hermann Weyl's classical theorem by establishing a precise remainder estimate for the asymptotics of the eigenvalues of positive elliptic operators of order $ 2m\;(m \geq 1)$ on $ \Omega$. We consider both Dirichlet and Neumann boundary conditions. Our estimate--which is expressed in terms of the Minkowski rather than the Hausdorff dimension of $\Gamma $--specifies and partially solves the Weyl-Berry conjecture for the eigenvalues of the Laplacian. Berry's conjecture--which extends to "fractals" Weyl's conjecture--is closely related to Kac's question "Can one hear the shape of a drum?"; further, it has significant physical applications, for example to the scattering of waves by "fractal" surfaces or the study of porous media. We also deduce from our results new remainder estimates for the asymptotics of the associated "partition function" (or trace of the heat semigroup). In addition, we provide examples showing that our remainder estimates are sharp in every possible "fractal" (i.e., Minkowski) dimension. The techniques used in this paper belong to the theory of partial differential equations, the calculus of variations, approximation theory and--to a lesser extent--geometric measure theory. An interesting aspect of this work is that it establishes new connections between spectral and "fractal" geometry.
Partially acyclic manifold decompositions yielding generalized manifolds
David F.
Snyder
531-571
Abstract: Let $G$ be an upper semicontinuous decomposition (used) of the $(n + k)$-manifold $M$ into subcontinua having the shape of closed orientable $n$-manifolds $(2 < n,k)$. We define $G$ to be $j$-acyclic if for every element $g$ of $G$ the reduced Čech homology of $g$ vanishes up through dimension $ j$. The primary objective of this investigation is to determine the local connectivity properties of the decomposition space $ B = M/G$ if $G$ is $(k - 2)$-acyclic and $B$ is finite dimensional. The Leray-Grothendieck spectral sequence of the decomposition map $ p$ is analyzed, which relegates the principal part of the investigation to studying the structure of the Leray sheaf of $p$ and its relation to the local cohomology of $B$. Let $E$ denote the subset of $B$ over which the Leray sheaf is not locally constant, $K$ the subset of $E$ over which the Leray sheaf is not locally Hausdorff, and $D = E - K$. Then we get as our main result, which extends work of R. J. Daverman and J. J. Walsh, and generalizes a result of D. S. Coram and P. Duvall as well, Theorem. Let $G$ be a $(k - 2)$-acyclic decomposition of the $ (n + k)$-manifold $ M$ such that $ k < n + 2$, $ B = M/G$ is finite dimensional, and the set $E$ does not locally separate $B$. Then $B$ is a generalized $ k$-manifold, if either $k = n + 1$, or $k < n + 1$ and $M$ is orientable.
On surfaces and Heegaard surfaces
Klaus
Johannson
573-591
Abstract: This paper is concerned with the intersection of surfaces and Heegaard surfaces in closed orientable $3$-manifolds $M$. Given a Heegaard decomposition $(M,{V_1},{V_2})$ it will be shown that any surface (orientable or not) in $M$ is equivalent to a surface which intersects $ {V_1}$ in discs whose total number is limited from above by some function in the genus of $ \partial {V_1}$ alone. The equivalence relation in question is generated by disc- and annulus-compressions.
Recursive functionals and quantifiers of finite types revisited. V
S. C.
Kleene
593-630
Abstract: This is the last in a sequence of papers that redoes the theory of recursion in finite types. A key feature of the theory is that a computation can succeed (or finish) even if some of its subcomputations do not, if these turn out to be irrelevant to the total computation. I give a detailed description of computations involving oracles for type $3$ functionals. The computation may be viewed formally as a transfinite sequence of symbolic expressions, but I also describe a semantics in which each expression is given a concrete realization.
$C\sp \infty$ loop algebras and noncommutative Bott periodicity
N. Christopher
Phillips
631-659
Abstract: We construct the noncommutative analogs ${\Omega _\infty }A$ and ${\Omega _{{\text{lip}}}}A$ of the ${C^\infty }$ and Lipschitz loop spaces for a pro-${C^\ast}$-algebra $A$ equipped with a suitable dense subalgebra. With $ {U_{{\text{nc}}}}$ and $ P$ being the classifying algebras for $K$-theory earlier introduced by the author, we then prove that there are homotopy equivalences ${\Omega _\infty }{U_{{\text{nc}}}} \simeq P$ and ${\Omega _\infty }P \simeq {U_{{\text{nc}}}}$. This result is a noncommutative analog of Bott periodicity in the form $\Omega U \simeq {\mathbf{Z}} \times BU$ and $\Omega ({\mathbf{Z}} \times BU) \simeq U$.
Absolute continuity results for superprocesses with some applications
Steven N.
Evans;
Edwin
Perkins
661-681
Abstract: Let ${X^1}$ and ${X^2}$ be instances of a measure-valued Dawson-Watanabe $\xi$-super process where the underlying spatial motions are given by a Borel right process, $ \xi$, and where the branching mechanism has finite variance. A necessary and sufficient condition on $X_0^1$ and $X_0^2$ is found for the law of $X_s^1$ to be absolutely continuous with respect to the law of $X_t^2$. The conditions are the natural absolute continuity conditions on $\xi$, but some care must be taken with the set of times $s$, $t$ being considered. The result is used to study the closed support of super-Brownian motion and give sufficient conditions for the existence of a nontrivial "collision measure" for a pair of independent super-Lévy processes or, more generally, for a super-Lévy process and a fixed measure. The collision measure gauges the extent of overlap of the two measures. As a final application, we give an elementary proof of the instantaneous propagation of a super-Lévy process to all points to which the underlying Lévy process can jump. This result is then extended to a much larger class of superprocesses using different techniques.
Second-order elliptic operators and heat kernels on Lie groups
Ola
Bratteli;
Derek W.
Robinson
683-713
Abstract: Arendt, Batty, and Robinson proved that each second-order strongly elliptic operator $C$ associated with left translations on the $ {L_p}$-spaces of a Lie group $G$ generates an interpolating family of semigroups $T$, whenever the coefficients of $ C$ are sufficiently smooth. We establish that $T$ has an integral kernel $K$ satisfying the bounds $ {{\mathbf{R}}^d}$.
Quasi-projective and relative cohomological dimension of groups
Juan M.
Alonso
715-739
Abstract: We give a geometric interpretation of the quasi-projective dimension of groups, a notion introduced by Howie and Schneebeli [H-S1] as a generalization of the Identity Property.
A spectral commutant lifting theorem
Hari
Bercovici;
Ciprian
Foias;
Allen
Tannenbaum
741-763
Abstract: The commutant lifting theorem of [24] may be regarded as a very general interpolation theorem from which a number of classical interpolation results may be deduced. In this paper we prove a spectral version of the commutant lifting theorem in which one bounds the spectral radius of the interpolant and not the norm. We relate this to a spectral analogue of classical matricial Nevanlinna-Pick interpolation.
Alternating sequences and induced operators
M. A.
Akcoglu;
R. E.
Bradley
765-791
Abstract: We show that when a positive ${L_p}$ contraction is equipped with a norming function having full support, then it is related in a natural way to an operator on any other ${L_p}$ space, $1 < p < \infty$. This construction is used to generalize a theorem of Rota concerning the convergence of alternating sequences.
$K\sb 1$-groups, quasidiagonality, and interpolation by multiplier projections
Shuang
Zhang
793-818
Abstract: We relate the following conditions on a $\sigma$-unital ${C^\ast}$-algebra $A$ with the " $ {\text{FS}}$ property": (a) ${K_1}(A) = 0$; (b) every projection in $M(A)/A$ lifts; (c) the general Weyl-von Neumann theorem holds in $M(A)$: Any selfadjoint element $h$ in $M(A)$ can be written as $ h = \sum\nolimits_{i = 1}^\infty {{\lambda _i}{p_i} + a} $ for some selfadjoint element $a$ in $A$, some bounded real sequence $\{ {\lambda _i}\}$, and some mutually orthogonal projections $\{ {p_i}\}$ in $A$ with $\sum\nolimits_{i = 1}^\infty {{p_i} = 1}$; (d) $M(A)$ has $ {\text{FS}}$; and (e) interpolation by multiplier projections holds: For any closed projections $p$ and $q$ in $ {A^{\ast \ast}}$ with $ pq = 0$, there is a projection $r$ in $M(A)$ such that $p \leq r \leq 1 - q$. We prove various equivalent versions of (a)-(e), and show that (e) $\Leftrightarrow$ (d) $\Leftrightarrow$ (c) $ \Rightarrow$ (b) $\Leftarrow$ (a), and that (a) $\Leftrightarrow$ (b) if, in addition, $ A$ is stable. Combining the above results, we obtain counterexamples to the conjecture of G. K. Pedersen "$A$ has $ FS \Rightarrow M(A)$ has ${\text{FS}}$" (for example the stabilized Bunce-Deddens algebras). Hence the generalized Weyl-von Neumann theorem does not generally hold in $ L({H_A})$ for $ \sigma$-unital $ {C^\ast}$-algebras with ${\text{FS}}$.
The symbolic representation of billiards without boundary condition
Takehiko
Morita
819-828
Abstract: We consider a dynamical system with elastic reflections in the whole plane and show that such a dynamical system can be represented as a symbolic flow over a mixing subshift of finite type. This fact enables us to prove an analogue of the prime number theorem for the closed orbits of such a dynamical system.
Support algebras of $\sigma$-unital $C\sp *$-algebras and their quasi-multipliers
Hua Xin
Lin
829-854
Abstract: We study certain dense hereditary $^\ast$-subalgebras of $\sigma$-unital ${C^\ast}$-algebras and their relations with the Pedersen ideals. The quasi-multipliers of the dense hereditary $^\ast$-subalgebras are also studied.
Groups of prime power order as Frobenius-Wielandt complements
Carlo M.
Scoppola
855-874
Abstract: It is known that the Sylow subgroups of a Frobenius complement are cyclic or generalized quaternion. In this paper it is shown that there are no restrictions at all on the structure of the Sylow subgroups of the Frobenius-Wielandt complements that appear in the well-known Wielandt's generalization of Frobenius' Theorem. Some examples of explicit constructions are also given.
Partitions, irreducible characters, and inequalities for generalized matrix functions
Thomas H.
Pate
875-894
Abstract: Given a partition $\alpha = \{ {\alpha _1},{\alpha _2}, \ldots ,{\alpha _s}\}$, ${\alpha _1} \geq {\alpha _2} \geq \cdots \geq {\alpha _s}$, of $n$ we let $ {X_\alpha }$ denote the derived irreducible character of ${S_n}$, and we associate with $\alpha$ a derived partition $\displaystyle \alpha\prime = \{ {\alpha _1} - 1,{\alpha _2} - 1, \ldots ,{\alpha _t} - 1,{\alpha _{t + 1}}, \ldots ,{\alpha _s},{1^t}\}$ where $ t$ denotes the smallest positive integer such that $ {\alpha _t} > {\alpha _{t + 1}}\;({\alpha _{s + 1}} = 0)$. We show that if $ Y$ is a decomposable $\mathbb{C}$-valued $n$-linear function on ${\mathbb{C}^m} \times {\mathbb{C}^m} \times \cdots \times {\mathbb{C}^m}$ ($n$-copies) then $\left\langle {{X_\alpha }Y,Y} \right\rangle \geq \left\langle {{X_\alpha },Y,Y} \right\rangle$. Translating into the notation of matrix theory we obtain an inequality involving the generalized matrix functions ${d_{{X_\alpha }}}$ and $ {d_{{X_{\alpha\prime}}}}$, namely that $\displaystyle {({X_\alpha }(e))^{ - 1}}{d_{{X_\alpha }}}(B) \geq {({X_{\alpha\prime}}(e))^{ - 1}}{d_{{X_{\alpha\prime}}}}(B)$ for each $ n \times n$ positive semidefinite Hermitian matrix $B$. This result generalizes a classical result of I. Schur and includes many other known inequalities as special cases.
A Picard theorem with an application to minimal surfaces. II
Peter
Hall
895-902
Abstract: Let $ f:{\mathbf{C}} \to {{\mathbf{R}}^n}$ be a parabolic minimal surface such that the normals to $f$ omit $n + k$ directions in general position, $k \geq 0$. We obtain sharp bounds on the dimension of the affine span of $ f$ and of the linear span of the Gauss map of $f$.
Covering moves
R.
Piergallini
903-920
Abstract: In this paper we give a positive answer to a long standing question posed by Montesinos, by introducing new covering moves, in order to relate any two colored diagrams representing the same $3$-manifold as simple branched $3$-covering of ${S^3}$.
The generalized Dowling lattices
Phil
Hanlon
1-37
Abstract: In this paper we study a new class of lattices called the generalized Dowling lattices. These lattices are parametrized by a positive integer $n$, a finite group $G$, and a meet sublattice $K$ of the lattice of subgroups of $G$. For an appropriate choice of $ K$ the generalized Dowling lattice $ {D_n}(G,K)$ agrees with the ordinary Dowling lattice ${D_n}(G)$. For a different choice of $ K$, the generalized Dowling lattices are the lattice of intersections of a set of subspaces in complex space. The set of subspaces, defined in terms of a representation of $ G$, generalizes the thick diagonal in $ {\mathbb{C}^n}$. We compute the Möbius function and characteristic polynomial of the lattice $ {D_n}(G,K)$ along with the homology of $ {D_n}(G,K)$ in terms of the homology of $K$. We go on to compute the character of $ G$ wr ${S_n}$ acting on the homology of ${D_n}(G,K)$. This computation provides a nontrivial generalization of a result due to Stanley concerning the character of ${S_n}$ acting on the top homology of the partition lattice.
The adjoint arc in nonsmooth optimization
Philip D.
Loewen;
R. T.
Rockafellar
39-72
Abstract: We extend the theory of necessary conditions for nonsmooth problems of Bolza in three ways: first, we incorporate state constraints of the intrinsic type $ x(t) \in X(t)$ for all $ t$; second, we make no assumption of calmness or normality; and third, we show that a single adjoint function of bounded variation simultaneously satisfies the Hamiltonian inclusion, the Euler-Lagrange inclusion, and the Weierstrass-Pontryagin maximum condition, along with the usual transversality relations.
Strictly cyclic operator algebras
John
Froelich
73-86
Abstract: We prove several results about the lattice of invariant subspaces of general strictly cyclic and strongly strictly cyclic operator algebras. A reflexive operator algebra $ A$ with a commutative subspace lattice is strictly cyclic iff $ \operatorname{Lat}{(A)^ \bot }$ contains a finite number of atoms and each nonzero element of $\operatorname{Lat}{(A)^ \bot }$ contains an atom. This leads to a characterization of the $ n$-strictly cyclic reflexive algebras with a commutative subspace lattice as well as an extensive generalization of D. A. Herrero's result that there are no triangular strictly cyclic operators. A reflexive operator algebra $ A$ with a commutative subspace lattice is strongly strictly cyclic iff $ \operatorname{Lat}(A)$ satisfies A.C.C. The distributive lattices which are attainable by strongly strictly cyclic reflexive algebras are the complete sublattices of $\{ 0,1] \times \{ 0,1\} \times \cdots$ which satisfy A.C.C. We also show that if $ \operatorname{Alg}(\mathcal{L})$ is strictly cyclic and $\mathcal{L} \subseteq$ atomic m.a.s.a. then $ \operatorname{Alg}(\mathcal{L})$ contains a strictly cyclic operator.
Random products of contractions in Banach spaces
J.
Dye;
M. A.
Khamsi;
S.
Reich
87-99
Abstract: We show that the random product of a finite number of $(W)$ contractions converges weakly in all smooth reflexive Banach spaces. If one of the contractions is compact, then the convergence is uniform.
Minimal submanifolds of $E\sp {2n+1}$ arising from degenerate ${\rm SO} (3)$ orbits on the Grassmannian
J. M.
Landsberg
101-117
Abstract: We give new examples of minimal submanifolds of ${{\mathbf{E}}^{2n + 1}}$ characterised by having their Gauss map's image lie in degenerate $SO(3)$ orbits of ${G_{p,2n + 1}}$, the Grassmannian of $ p$-planes in ${{\mathbf{E}}^{2n + 1}}$ (where the action on ${G_{p,2n + 1}}$ is induced from the irreducible $ SO(3)$ action on $ {{\mathbf{R}}^{2n + 1}}$). These submanifolds are all given explicitly in terms of holomorphic data and are linearly full in ${{\mathbf{E}}^{2n + 1}}$.
How porous is the graph of Brownian motion?
J. T.
Cox;
Philip S.
Griffin
119-140
Abstract: We prove that the graph of Brownian motion is almost surely porous, and determine the Hausdorff dimension of sets with a given porosity index. In particular we show that the porosity index of the graph is ${\gamma _0} \doteq 0.6948$.
Smooth dynamics on Weierstrass nowhere differentiable curves
Brian R.
Hunt;
James A.
Yorke
141-154
Abstract: We consider a family of smooth maps on an infinite cylinder which have invariant curves that are nowhere smooth. Most points on such a curve are buried deep within its spiked structure, and the outermost exposed points of the curve constitute an invariant subset which we call the "facade" of the curve. We find that for surprisingly many of the maps in the family, all points in the facades of their invariant curves are eventually periodic.
Inner amenable locally compact groups
Anthony To Ming
Lau;
Alan L. T.
Paterson
155-169
Abstract: In this paper we study the relationship between amenability, inner amenability and property $P$ of a von Neumann algebra. We give necessary conditions on a locally compact group $G$ to have an inner invariant mean $ m$ such that $ m(V) = 0$ for some compact neighborhood $V$ of $G$ invariant under the inner automorphisms. We also give a sufficient condition on $G$ (satisfied by the free group on two generators or an I.C.C. discrete group with Kazhdan's property $ T$, e.g., $ {\text{SL}}(n,\mathbb{Z})$, $n \geq 3$) such that each linear form on $ {L^2}(G)$ which is invariant under the inner automorphisms is continuous. A characterization of inner amenability in terms of a fixed point property for left Banach $G$-modules is also obtained.
Subelliptic estimates for the $\overline \partial$-Neumann problem for $n-1$ forms
Lop-Hing
Ho
171-185
Abstract: In this note we deal with the problem of the subelliptic estimates of the $\bar \partial $-Neumann problem on nonpseudoconvex domains. In the first part we give a necessary condition for $n - 1$ forms in a class of domains. In the second part we give the exact estimate for a class of domains where the Levi form of a vector field $L$ is bounded below by a certain function.
Utility functions which ensure the adequacy of stationary strategies
Michael G.
Monticino
187-204
Abstract: Within a Dubins and Savage gambling framework, a stationary strategy is a strategy which selects a gamble at each time based solely on the gambler's present fortune. We determine conditions upon the gambler's utility function under which stationary strategies allow the gambler to maximize his return. The class of utility functions which satisfies these conditions, termed nearly leavable shift invariant functions, is large and contains many of the common gambling utility functions. Moreover, this class is closed under uniform limits. These results are obtained with the setting of an analytic gambling house.
Hamilton-Jacobi equations with singular boundary conditions on a free boundary and applications to differential games
Martino
Bardi;
Pierpaolo
Soravia
205-229
Abstract: A class of Hamilton-Jacobi equations arising in generalized timeoptimal control problems and differential games is considered. The natural global boundary value problem for these equations has a singular boundary condition on a free boundary. The uniqueness of the continuous solution and of the free boundary is proved in the framework of viscosity solutions. A local uniqueness theorem is also given, as well as some existence results and several applications to control and game theory. In particular a relaxation theorem (weak form of the bang-bang principle) is proved for a class of nonlinear differential games.
On completing unimodular polynomial vectors of length three
Ravi A.
Rao
231-239
Abstract: It is shown that if $ R$ is a local ring of dimension three, with $ \frac{1} {2} \in R$, then a polynomial three vector $ ({v_0}(X),{v_1}(X),{v_2}(X))$ over $R[X]$ can be completed to an invertible matrix if and only if it is unimodular. In particular, if $1/3! \in R$, then every stably free projective $R[{X_1}, \ldots ,{X_n}]$-module is free.
The Maslov class of the Lagrange surfaces and Gromov's pseudo-holomorphic curves
L. V.
Polterovich
241-248
Abstract: For an immersed Lagrange submanifold $W \subset {T^\ast }X$, one can define a nonnegative integer topologic invariant $ m(W)$ such that the image of $ {H_1}(W;{\mathbf{Z}})$ under the Maslov class is equal to $m(W) \cdot {\mathbf{Z}}$. In this paper, the value of $ m(W)$ is calculated for the case of a two-dimensional oriented manifold $ X$ with the universal cover homeomorphic to $ {{\mathbf{R}}^2}$ and an embedded Lagrange torus $W$. It is proved that if $X = {{\mathbf{T}}^2}$ and $W$ is homologic to the zero section, then $ m(W) = 0$. In all the other cases $m(W) = 2$. The last result is true also for a wide class of oriented properly embedded Lagrange surfaces in ${T^\ast }{{\mathbf{R}}^2}$. The proof is based on the Gromov's theory of pseudo-holomorphic curves. Some applications to the hamiltonian mechanics are mentioned.
On the convergence of moment problems
J. M.
Borwein;
A. S.
Lewis
249-271
Abstract: We study the problem of estimating a nonnegative density, given a finite number of moments. Such problems arise in numerous practical applications. As the number of moments increases, the estimates will always converge weak$ ^\ast$ as measures, but need not converge weakly in ${L_1}$. This is related to the existence of functions on a compact metric space which are not essentially Riemann integrable (in some suitable sense). We characterize the type of weak convergence we can expect in terms of Riemann integrability, and in some cases give error bounds. When the estimates are chosen to minimize an objective function with weakly compact level sets (such as the Bolzmann-Shannon entropy) they will converge weakly in ${L_1}$. When an ${L_p}$ norm $ (1 < p < \infty)$ is used as the objective, the estimates actually converge in norm. These results provide theoretical support to the growing popularity of such methods in practice.
New results on the Pompeiu problem
Nicola
Garofalo;
Fausto
Segàla
273-286
Abstract: Let $ {p_N}(w) = \sum\nolimits_{k = 0}^N {{a_k}{w^k}}$, $w \in \mathbb{C}$, $N \in \mathbb{N}$, be a polynomial with complex coefficients. In this paper we prove that if $D \subset {\mathbb{R}^2}$ is a simply-connected bounded open set whose boundary is a closed, simple curve parametrized by $x(s) = {x_1}(s) + i{x_2}(s) = {p_N}({e^{is}})$, $s \in [ - \pi ,\pi ]$, then $ D$ has the Pompeiu property unless $N = 1$ and ${p_1}(w) = {a_1}w + {a_2}$ in which case $ D$ is a disk. This result supports the conjecture that modulo sets of zero two-dimensional Lebesgue measure, the disk is the only simply-connected, bounded open set which fails to have the Pompeiu property.
Hyperholomorphic functions and second order partial differential equations in ${\bf R}\sp n$
R. Z.
Yeh
287-318
Abstract: Hyperholomorphic functions in ${R^n}$ with $n \geq 2$ are introduced, extending the hitherto considered hyperholomorphic functions in ${R^2}$. A Taylor formula is derived, and with it a unique representation theorem is proved for hyperholomorphic functions that are real analytic at the origin. Hyperanalyticity is seen to be generally a consequence of hyperholomorphy and real analyticity combined. Results for hyperholomorphic functions are applied to gradients of solutions of second order homogeneous partial differential equations with constant coefficients. Polynomial solutions of such a second order equation are obtained by a matrix algorithm. These polynomials are modified and combined to form polynomial bases for real analytic solutions. It is calculated that in such a basis there are $ (m + n - 3)!(2m + n - 2)/m!(n - 2)!$ homogeneous polynomials of degree $ m$.
Some results on the \v Sarkovski\u\i partial ordering of permutations
Irwin
Jungreis
319-344
Abstract: If $\pi$ is a cyclic permutation and $ x$ is a periodic point of a continuous function $f:{\mathbf{R}} \mapsto {\mathbf{R}}$ with ${\text{period}}(x) = {\text{order}}(\pi) = n$, then we say that $x$ has type $\pi$ if the orbit of $x$ consists of points ${x_1} < {x_2} < \cdots < {x_n}$ with $f({x_i}) = {x_{\pi (i)}}$. In analogy with Sarkovskii's Theorem, we define a partial ordering on cyclic permutations by $\theta \to \pi$ if every continuous function with a periodic point of type $\theta$ also has a point of type $\pi$. In this paper we examine this partial order form the point of view of critical points, itineraries, and kneading sequences. We show that $\theta \to \pi$ if and only if the maxima of $ \theta$ are "higher" and the minima "lower" than those of $\pi$, where "higher" and "lower" are precisely defined in terms of itineraries. We use this to obtain several results about $\to$: there are no minimal upper bounds; if $ \pi$ and $\theta$ have the same number of critical points (or if they differ by $1$ or sometimes $2$), then $ \theta \to \pi$ if and only if $ \theta \to {\pi_\ast}$ for some period double $ {\pi_\ast}$ of $ \pi$; and finally, we prove a conjecture of Baldwin that maximal permutations of size $n$ have $n - 2$ critical points, and obtain necessary and sufficient conditions for such a permutation to be maximal.
Adapted sets of measures and invariant functionals on $L\sp p(G)$
Rodney
Nillsen
345-362
Abstract: Let $G$ be a locally compact group. If $ G$ is compact, let $ L_0^p(G)$ denote the functions in ${L^p}(G)$ having zero Haar integral. Let $ {M^1}(G)$ denote the probability measures on $G$ and let ${\mathcal{P}^1}(G) = {M^1}(G) \cap {L^1}(G)$. If $ S \subseteq {M^1}(G)$, let $ \Delta ({L^p}(G),S)$ denote the subspace of ${L^p}(G)$ generated by functions of the form $f - \mu\ast f$, $ f \in {L^p}(G)$, $ \mu \in S$. If $ G$ is compact, $ \Delta ({L^p}(G),S) \subseteq L_0^p(G)$ . When $G$ is compact, conditions are given on $ S$ which ensure that for some finite subset $F$ of $S$, $\Delta ({L^p}(G),F) = L_0^p(G)$ for all $1 < p < \infty$. The finite subset $ F$ will then have the property that every $F$-invariant linear functional on ${L^p}(G)$ is a multiple of Haar measure. Some results of a contrary nature are presented for noncompact groups. For example, if $1 \leq p \leq \infty$, conditions are given upon $ G$, and upon subsets $ S$ of ${M^1}(G)$ whose elements satisfy certain growth conditions, which ensure that ${L^p}(G)$ has discontinuous, $ S$-invariant linear functionals. The results are applied to show that for $1 \leq p \leq \infty$, ${L^p}(\mathbb{R})$ has an infinite, independent family of discontinuous translation invariant functionals which are not $ {\mathcal{P}^1}(\mathbb{R})$-invariant.
Resonance and the second BVP
Victor L.
Shapiro
363-387
Abstract: Let $\Omega \subset {\mathbb{R}^N}$ be a bounded open connected set with the cone property, and let $1 < p < \infty$ . Also, let $ Qu$ be the $2m$th order quasilinear differential operator in generalized divergence form: $\displaystyle Qu = \sum\limits_{1 \leq \vert\alpha \vert \leq m} {{{(- 1)}^{\vert\alpha \vert}}{D^\alpha }{A_\alpha }(x,{\xi _m}(u))},$ where for $u \in {W^{m,p}}$, ${\xi _m}(u) = \{ {D^\alpha }u:\vert\alpha \vert \leq m\}$. (For $m = 1$, $Qu = - \sum\nolimits_{i = 1}^N {{A_i}(x,u,Du)}$.) Under four assumptions on $ {A_\alpha }$--Carathéodory, growth, monotonicity for $\vert\alpha \vert = m$, and ellipticity--results at resonance are established for the equation $Qu = G + f(x,u)$, where $G \in {[{W^{m,p}}(\Omega)]^\ast }$ and $f(x,u)$ satisfies a one-sided condition (plus others). For the case $m = 1$ , these results are tantamount to generalized solutions of the second BVP.
On the topology and geometric construction of oriented matroids and convex polytopes
Jürgen
Richter;
Bernd
Sturmfels
389-412
Abstract: This paper develops new combinatorial and geometric techniques for studying the topology of the real semialgebraic variety $ \mathcal{R}(M)$ of all realizations of an oriented matroid $M$ . We focus our attention on point configurations in general position, and as the main result we prove that the realization space of every uniform rank $3$ oriented matroid with up to eight points is contractible. For these special classes our theorem implies the isotopy property which states the spaces $\mathcal{R}(M)$ are path-connected. We further apply our methods to several related problems on convex polytopes and line arrangements. A geometric construction and the isotopy property are obtained for a large class of neighborly polytopes. We improve a result of M. Las Vergnas by constructing a smallest counterexample to a conjecture of G. Ringel, and, finally, we discuss the solution to a problem of R. Cordovil and P. Duchet on the realizability of cyclic matroid polytopes.
On the structure of certain locally compact topological groups
Ta Sun
Wu
413-434
Abstract: A locally compact topological group $G$ is called an $ ({\text{H}})$ group if $ G$ has a maximal compact normal subgroup with Lie factor. In this note, we study the problem when a locally compact group is an $({\text{H}})$ group.
Weak type estimates for a singular convolution operator on the Heisenberg group
Loukas
Grafakos
435-452
Abstract: On the Heisenberg group $ {\mathbb{H}^n}$ with coordinates $(z,t) \in {\mathbb{C}^n} \times \mathbb{R}$, define the distribution $K(z,t) = L(z)\delta (t)$, where $L(z)$ is a homogeneous distribution on $ {\mathbb{C}^n}$ of degree $ - 2n$ , smooth away from the origin and $ \delta (t)$ is the Dirac mass in the $t$ variable. We prove that the operator given by convolution with $K$ maps $ {H^1}({\mathbb{H}^n})$ to weak $ {L^1}({\mathbb{H}^n})$.
Effective lower bounds for the norm of the Poincar\'e $\Theta$-operator
Mark
Sheingorn
453-463
Abstract: Motivated by McMullen's proof of Kra's conjecture that the norm of the Poincaré theta operator ${\Theta _{q,\Gamma }}$ is less than $1$ for every $q$ and $\Gamma$ of finite volume, this paper provides explicit lower bounds for this norm. These bounds are sufficient to show that $\left\Vert {{\Theta _{q,\Gamma }}} \right\Vert \to 1$ for fixed $\Gamma$ as $ q \to \infty$. Here the difference from $1$ is less than $ O(\frac{{{{(2\pi e)}^{q - 2}}}}{{{q^{q - 2}}}})$. For $\Gamma (N) \subseteq \Gamma \subseteq {\Gamma _0}(N)$, $\left\Vert {{\Theta _{q,\Gamma }}} \right\Vert \to 1$ for fixed $q$ as $N \to \infty $. Here the difference from $1$ is $ O({N^{35 - q}})$. We prove these results by estimating $\frac{{{{\left\Vert {{\Theta _{q,\Gamma }}({f_p})} \right\Vert}_{{A_q}(\Gamma)}}}} {{{{\left\Vert {{f_p}} \right\Vert}_{{A_q}}}}}$ where the ${f_p}$ are cusp forms of weight $p \leq q - 2$. (Thus such functions may in general tend to optimize $ {\Theta _{q,\Gamma }}$.) In the case of the congruence subgroups they are taken to be products of $\Delta$ and Eisenstein series. Effective formulae are presented for all implied constants.
Commutation methods applied to the mKdV-equation
F.
Gesztesy;
W.
Schweiger;
B.
Simon
465-525
Abstract: An explicit construction of solutions of the modified Korteweg-de Vries equation given a solution of the (ordinary) Korteweg-de Vries equation is provided. Our theory is based on commutation methods (i.e., $N = 1$ supersymmetry) underlying Miura's transformation that links solutions of the two evolution equations.
Ramsey theorems for knots, links and spatial graphs
Seiya
Negami
527-541
Abstract: An embedding $ f:G \to {{\mathbf{R}}^3}$ of a graph $G$ into $ {{\mathbf{R}}^3}$ is said to be linear if each edge $f(e)\quad (e \in E(G))$ is a straight line segment. It will be shown that for any knot or link type $ k$, there is a finite number $R(k)$ such that every linear embedding of the complete graph ${K_n}$ with at least $R(k)$ vertices $ (n \geqslant R(k))$ in $ {{\mathbf{R}}^3}$ contains a knot or link equivalent to $k$.
Ultra-irreducibility of induced representations of semidirect products
Henrik
Stetkær
543-554
Abstract: Let the Lie group $ G$ be a semidirect product, $G = SK$, of a connected, closed, normal subgroup $ S$ and a closed subgroup $ K$. Let $\Lambda$ be a nonunitary character of $ S$, and let ${K_\Lambda }$ be its stability subgroup in $ K$. Let ${I^{\Lambda \mu }}$, for any irreducible representation $\mu$ of $ {K_\Lambda }$, denote the representation $ {I^{\Lambda \mu }}$ of $ G$ induced by the representation $\Lambda \mu$ of $S{K_\Lambda }$. The representation spaces are subspaces of the distributions. We show that ${I^{\Lambda \mu }}$ is ultra-irreducible when the corresponding Poisson transform is injective, and find a sufficient condition for this injectivity.
Local singularities such that all deformations are tangentially flat
Bernd
Herzog
555-601
Abstract: We give a criterion for a local ring $({B_0},{\mathfrak{n}_0})$ containing a field to have only tangentially flat deformations. Various examples of such local rings are constructed.
Construction of units in integral group rings of finite nilpotent groups
Jürgen
Ritter;
Sudarshan K.
Sehgal
603-621
Abstract: Let $U$ be the group of units of the integral group ring of a finite group $G$. We give a set of generators of a subgroup $B$ of $U$. This subgroup is of finite index in $ U$ if $G$ is an odd nilpotent group. We also give an example of a $2$-group such that $B$ is of infinite index in $U$.
The Selberg trace formula. VIII. Contribution from the continuous spectrum
M. Scott
Osborne;
Garth
Warner
623-653
Abstract: The purpose of this paper is to isolate the contribution from the continuous spectrum to the Selberg trace formula.
A generalized Berele-Schensted algorithm and conjectured Young tableaux for intermediate symplectic groups
Robert A.
Proctor
655-692
Abstract: The Schensted and Berele algorithms combinatorially mimic the decompositions of ${ \otimes ^k}V$ with respect to $ {\operatorname{GL} _N}$ and ${\operatorname{Sp} _{2n}}$. Here we present an algorithm which is a common generalization of these two algorithms. "Intermediate symplectic groups" ${\operatorname{Sp} _{2n,m}}$ are defined. These groups interpolate between ${\operatorname{GL} _N}$ and ${\operatorname{Sp} _N}$. We conjecture that there is a decomposition of $ { \otimes ^k}V$ with respect to ${\operatorname{Sp} _{2n,m}}$ which is described by the output of the new algorithm.
On the growth of solutions of $f''+gf'+hf=0$
Simon
Hellerstein;
Joseph
Miles;
John
Rossi
693-706
Abstract: Suppose $ g$ and $h$ are entire functions with the order of $ h$ less than the order of $ g$. If the order of $ g$ does not exceed $\tfrac{1} {2}$, it is shown that every (necessarily entire) nonconstant solution $f$ of the differential equation
Hankel operators on the Bergman space of bounded symmetric domains
Ke He
Zhu
707-730
Abstract: Let $\Omega$ be a bounded symmetric domain in $ {\mathbb{C}^n}$ with normalized volume measure $dV$. Let $P$ be the orthogonal projection from ${L^2}(\Omega ,dV)$ onto the Bergman space $L_a^2(\Omega )$ of holomorphic functions in ${L^2}(\Omega ,dV)$. Let $\overline P$ be the orthogonal projection from ${L^2}(\Omega ,dV)$ onto the closed subspace of antiholomorphic functions in ${L^2}(\Omega ,dV)$. The "little" Hankel operator $ {h_f}$ with symbol $ f$ is the operator from $L_a^2(\Omega )$ into ${L^2}(\Omega ,dV)$ defined by ${h_f}g = \overline P (fg)$. We characterize the boundedness, compactness, and membership in the Schatten classes of the Hankel operators ${h_f}$ in terms of a certain integral transform of the symbol $f$. These characterizations are further studied in the special cases of the open unit ball and the poly-disc in $ {\mathbb{C}^n}$.
The Frobenius-Perron operator on spaces of curves
P.
Góra;
A.
Boyarsky
731-746
Abstract: Let $\tau :{R^2} \to {R^2}$ be a diffeomorphism which leaves a compact set $A$ invariant. Let $B \subset A$ be such that $ \tau$ can map out of $ B$. Assume that $ \tau$ has a hyperbolic fixed point $p$ in $B$. Let $ \mathcal{C}$ be a space of smooth curves in $B$. We define a normalized Frobenius-Perron operator on the vector bundle of Lipschitz continuous functions labelled by the curves in $\mathcal{C}$, and use it to prove the existence of a unique, smooth conditionally invariant measure $ \mu$ on a segment $ {V^u}$ of the unstable manifold of $p$. A formula for the computation of ${f^{\ast}}$, the density of $\mu$, is derived, and $\mu ({\tau ^{ - 1}}{V^u})$ is shown to be equal to the reciprocal of the maximal modulus eigenvalue of the Jacobian of $\tau$ at $p$.
A cohomological approach to the Brauer-Long group and the groups of Galois extensions and strongly graded rings
S.
Caenepeel;
M.
Beattie
747-775
Abstract: Let $G$ be a finite abelian group, and $ R$ a commutative ring. The Brauer-Long group $\operatorname{BD} (R,G)$ is described by an exact sequence $\displaystyle 1 \to {\operatorname{BD} ^s}(R,G) \to \operatorname{BD} (R,G)\xrightarrow{\beta }\operatorname{Aut} (G \times {G^{\ast}})(R)$ where $ {\operatorname{BD} ^s}(R,G)$ is a product of étale cohomology groups, and Im $\beta$ is a kind of orthogonal subgroup of $ \operatorname{Aut} (G \times {G^{\ast}})(R)$. This sequence generalizes some other well-known exact sequences, and restricts to two split exact sequences describing Galois extensions and strongly graded rings.
Relative Frobenius of plane singularities
D.
Daigle
777-791
Abstract: In view of the well-known conjecture concerning the classification of lines in the affine plane in characteristic $p > 0$, it is desirable to understand how the characteristic pairs of an irreducible algebroid plane curve are affected by the relative Frobenius. This paper determines the relation between the characteristic sequences $[x,y]$ and $[x,{y^p}]$, where $x$ and $y$ are formal power series in one variable with coefficients in a field of characteristic $p > 0$.
Prescribing curvature on compact surfaces with conical singularities
Marc
Troyanov
793-821
Abstract: We study the Berger-Nirenberg problem on surfaces with conical singularities, i.e. we discuss conditions under which a function on a Riemann surface is the Gaussian curvature of some conformal metric with a prescribed set of singularities of conical types.
Univalence criteria and quasiconformal extensions
J. M.
Anderson;
A.
Hinkkanen
823-842
Abstract: Let $f$ be a locally univalent meromorphic function in the unit disk $\Delta$. Recently, Epstein obtained a differential geometric proof for the fact that if $f$ satisfies an inequality involving a suitable real-valued function $\sigma$, then $f$ is univalent in $\Delta$ and has a quasiconformal extension to the sphere. We give a more classical proof for this result by means of an explicit quasiconformal extension, and obtain generalizations of the result under suitable conditions even if $\sigma$ is allowed to be complex-valued and $ \Delta$ is replaced by a quasidisk.
On pseudo-differentiability
Roberto
Cominetti
843-865
Abstract: We present some new relations between the pseudo-derivatives and parabolic epiderivatives recently introduced by Rockafellar, and also its infinite dimensional counterparts. Significant extensions of the most important known results are proven, which further clarify the range of applicability of this new theory.
Linearization of bounded holomorphic mappings on Banach spaces
Jorge
Mujica
867-887
Abstract: The main result in this paper is the following linearization theorem. For each open set $U$ in a complex Banach space $E$, there is a complex Banach space ${G^\infty }(U)$ and a bounded holomorphic mapping $ {g_U}:U \to {G^\infty }(U)$ with the following universal property: For each complex Banach space $F$ and each bounded holomorphic mapping $ f:U \to F$, there is a unique continuous linear operator ${T_f}:{G^\infty }(U) \to F$ such that ${T_f} \circ {g_U} = f$. The correspondence $f \to {T_f}$ is an isometric isomorphism between the space $ {H^\infty }(U;F)$ of all bounded holomorphic mappings from $U$ into $F$, and the space $L({G^\infty }(U);F)$ of all continuous linear operators from $ {G^\infty }(U)$ into $ F$. These properties characterize ${G^\infty }(U)$ uniquely up to an isometric isomorphism. The rest of the paper is devoted to the study of some aspects of the interplay between the spaces $ {H^\infty }(U;F)$ and $L({G^\infty }(U);F)$.
On certain partial differential operators of finite odd type
A. Alexandrou
Himonas
889-900
Abstract: Let $P$ be a linear partial differential operator of order $m \geqslant 1$ with real-analytic coefficients defined in $\Omega$, an open set of ${\mathbb{R}^n}$, and let $\gamma$ be in the cotangent space of $ \Omega$ minus the zero section. If $P$ is of odd finite type $k$ and if the Hörmander numbers are $ 1 = {k_1} < {k_2},{k_2}$ odd, then $P$ is analytic hypoelliptic at $ \gamma$. These operators are not semirigid.
The existence of generalized isothermal coordinates for higher-dimensional Riemannian manifolds
Jian Guo
Cao
901-920
Abstract: We shall show that, for any given point $p$ on a Riemannian manifold $(M,{g^0})$, there is a pointwise conformal metric $g = \Phi {g^0}$ in which the $ g$-geodesic sphere centered at $p$ with radius $r$ has constant mean curvature $1/r$ for all sufficiently small $ r$. Furthermore, the exponential map of $g$ at $p$ is a measure preserving map in a small ball around $p$.
Generalized iteration of forcing
M.
Groszek;
T.
Jech
1-26
Abstract: Generalized iteration extends the usual notion of iterated forcing from iterating along an ordinal to iterating along any partially ordered set. We consider a class of forcings called perfect tree forcing. The class includes Axiom A forcings with a finite splitting property, such as Cohen, Laver, Mathias, Miller, Prikry-Silver, and Sacks forcings. If $ \mathcal{P}$ is a perfect tree forcing, there is a decomposition $\mathcal{Q} * \mathcal{R}$ such that $\mathcal{Q}$ is countably closed, $\mathcal{R}$ has the countable chain condition, and $ \mathcal{Q} * \mathcal{R}$ adds a $ \mathcal{P}$-generic set. Theorem. The mixed-support generalized iteration of perfect tree forcing decompositions along any well-founded partial order preserves ${\omega _1}$. Theorem. If ${\text{ZFC}}$ is consistent, so is $ {\text{ZFC + }}{{\text{2}}^\omega }$ is arbitrarily large + whenever $\mathcal{P}$ is a perfect tree forcing and $\mathcal{D}$ is a collection of ${\omega _1}$ dense subsets of $\mathcal{P}$, there is a $\mathcal{D}$-generic filter on $\mathcal{P}$.
Zeros of solutions and of Wronskians for the differential equation $L\sb ny+p(x)y=0$
Uri
Elias
27-40
Abstract: The equation which is studied here is ${L_n}y + p(x)y = 0,a \leq x \leq b$, where $ {L_n}$ is a disconjugate differential operator and $p(x)$ is of a fixed sign. We prove that certain solutions of the equation and corresponding odd-order minors of the Wronskian have an equal number of zeros, and we apply this property to oscillation problems.
Relative cohomology and projective twistor diagrams
S. A.
Huggett;
M. A.
Singer
41-57
Abstract: The use of relative cohomology in the investigation of functionals on tensor products of twistor cohomology groups is considered and yields a significant reduction in the problem of looking for contours for the evaluation of (projective) twistor diagrams. The method is applied to some simple twistor diagrams and is used to show that the standard twistor kernel for the first order massless scalar ${\phi^4}$ vertex admits a (cohomological) contour for only one of the physical channels. A new kernel is constructed for the ${\phi^4}$ vertex which admits contours for all channels.
Box-spaces and random partial orders
Béla
Bollobás;
Graham
Brightwell
59-72
Abstract: Winkler [2] studied random partially ordered sets, defined by taking $ n$ points at random in $ {[0,1]^d}$, with the order on these points given by the restriction of the order on ${[0,1]^d}$. Bollobás and Winkler [1] gave several results on the height of such a random partial order. In this paper, we extend these results to a more general setting. We define a box-space to be, roughly speaking, a partially ordered measure space such that every two intervals of nonzero measure are isomorphic up to a scale factor. We give some examples of box-spaces, including (i) ${[0,1]^d}$ with the usual measure and order, and (ii) Lorentzian space-time with the order given by causality. We show that, for every box-space, there is a constant $d$ which behaves like the dimension of the space. In the second half of the paper, we study random partial orders defined by taking a Poisson distribution on a box-space. (This is of course essentially the same as taking $n$ random points in a box-space.) We extend the results of Bollobás and Winkler to these random posets. In particular we show that, for a box-space $ X$ of dimension $ d$, there is a constant $ {m_X}$ such that the length of a longest chain tends to ${m_X}{n^{1/d}}$ in probability.
Strong Bertini theorems
Steven
Diaz;
David
Harbater
73-86
Abstract: We show that the singular locus of the general member of a linear system has dimension less than that predicted by Bertini's theorem, provided that the base locus is scheme-theoretically smooth. As corollaries, we obtain a result about complete intersection varieties containing a given subvariety and a result concerning liaison.
Nonsingular algebraic curves in ${\bf R}{\rm P}\sp 1\times{\bf R}{\rm P}\sp 1$
Sachiko
Matsuoka
87-107
Abstract: We give some restrictions for the mutual position of the connected components of a nonsingular algebraic curve in the product space ${\mathbf{R}}{P^1} \times {\mathbf{R}}{P^1}$ of two real projective lines. We obtain our main theorem by calculating the Brown invariant of a certain quadratic form determined by the algebraic curve. Moreover, we consider a double covering of $ {\mathbf{C}}{P^1} \times {\mathbf{C}}{P^1}$ branched along the complexification of our curve and antiholomorphic involutions that are the lifts of the complex conjugation.
Boundary value problems for degenerate elliptic-parabolic equations of the fourth order
Robert G.
Root
109-134
Abstract: We consider boundary value problems for the fourth-order linear equation $\displaystyle {A^{ijrs}}{u_{ijrs}} + {A^{ijr}}{u_{ijr}} + {A^{ij}}{u_{ij}} - \gamma {({a^{ij}}{u_i})_j} + {A^i}{u_i} + Fu = f\quad {\text{in}}\overline \Omega$ with smooth coefficients. The fourth-order part may degenerate on arbitrary subsets of $ \overline \Omega$ i.e., $ {A^{ijrs}}(x){m_{ij}}{m_{rs}} \geq 0$ for all $n \times n$ matrices $ M$, with no restriction on where equality occurs. We assume the ${a^{ij}}$ part of the operator is uniformly elliptic (of second order) on $\Omega$ while $\gamma$ is a parameter allowing us to increase modulus of ellipticity as much as needed. As in Fichera's second-order elliptic-parabolic equations [see, for example, Sulle equazioni differenziali lineari ellitico-paraboliche del secondo ordine, Atti Accad. Naz. Lincei Mem. (8) 5 (1956), 1-30], because of the degeneracy, there may be characteristic portions of the boundary; however, we restrict our attention to the noncharacteristic case. We define a weak solution to the Dirichlet problem and obtain existence and uniqueness results. The question of regularity is addressed; elliptic regularization is used to obtain a Sobolev-type global regularity result. The equation models an anisotropic, inhomogeneous plate under tension that can lose stiffness at any point and in any direction. The regularity result has the satisfying physical interpretation that sufficient tension results in a smooth solution.
On vector bundles on $3$-folds with sectional genus $1$
Edoardo
Ballico
135-147
Abstract: Here we give a classification (in characteristic zero) of pairs $(V,E)$ with $V$ being a smooth, connected, complete $ 3$-fold and $E$ a rank-$2$ spanned ample vector bundle on $ V$ with sectional genus $ 1$. The proof uses the partial classification of Fano $3$-folds and Mori theory.
Determinant expression of Selberg zeta functions. I
Shin-ya
Koyama
149-168
Abstract: We show that for $ {\text{PSL}}(2,{\mathbf{R}})$ and its congruence subgroup, the Selberg zeta function with its gamma factors is expressed as the determinant of the Laplacian. All the gamma factors are calculated explicitly. We also give an explicit computation to the contribution of the continuous spectrum to the determinant of the Laplacian.
On the interior of subsemigroups of Lie groups
K. H.
Hofmann;
W. A. F.
Ruppert
169-179
Abstract: Let $G$ denote a Lie group with Lie algebra $\mathfrak{g}$ and with a subsemigroup $S$ whose infinitesimal generators generate $ \mathfrak{g}$. We construct real analytic curves $\gamma :{{\mathbf{R}}^ + } \to S$ such that $\dot \gamma (0)$ is a preassigned tangent vector of $S$ at the origin and that $\gamma(t)$ is in the interior of $S$ for all positive $t$. Among the consequences, we find that the boundary of $S$ has to be reasonably well behaved. Our procedure involves the construction of certain linear generating sets from a given Lie algebra generating set, and this may be of independent interest.
On the characteristic classes of actions of lattices in higher rank Lie groups
Garrett
Stuck
181-200
Abstract: We show that under certain assumptions, the measurable cohomology class of the linear holonomy cocycle of a foliation yields information about the characteristic classes of the foliation. Combined with the results of a previous paper, this yields vanishing theorems for characteristic classes of certain actions of lattices in higher rank semisimple Lie groups.
Similarity orbits and the range of the generalized derivation $X\to MX-XN$
Allen
Schweinsberg
201-211
Abstract: If $M$ and $N$ are bounded operators on infinite dimensional complex Hilbert spaces $ \mathcal{H}$ and $\mathcal{K}$, let $\tau (X) = MX - XN$ for $X$ in $ \mathcal{L}(\mathcal{K},\mathcal{H})$. The closure of the range of $\tau$ is characterized when $ M$ and $N$ are normal. There is a close connection between the range of $\tau$ and operators $C$ for which $ [\begin{array}{*{20}{c}} M \& C 0 \& N \end{array} ]$ is in the closure of the similarity orbit of $ [\begin{array}{*{20}{c}} M \& 0 0 ^ N \end{array} ]$. This latter set is characterized and compared with the closure of the range of $\tau$.
Optimal H\"older and $L\sp p$ estimates for $\overline\partial\sb b$ on the boundaries of real ellipsoids in ${\bf C}\sp n$
Mei-Chi
Shaw
213-234
Abstract: Let $D$ be a real ellipsoid in ${{\mathbf{C}}^n},n \geq 3$, with defining function $\rho (z) = \sum\nolimits_{k = 1}^n {(x_k^{2{n_k}} + y_k^{2{m_k}})} - 1$, ${z_k} = {x_k} + i{y_k}$, where ${n_k},{m_k} \in N$. In this paper we study the sharp Hàlder and ${L^p}$ estimates for the solutions of the tangential Cauchy-Riemann equations ${\overline \partial _b}$ on the boundary $\partial D$ of $D$ using the integral kernel method. In particular, we proved that if $\alpha \in L_{(0,1)}^\infty (\partial D)$ such that ${\overline \partial _b}\alpha = 0$ on $\partial D$ in the distribution sense, then there exists a $u \in {\Lambda _{1/2m}}(\partial D)$ satisfying $ {\overline \partial _b}u = \alpha$ and ${\left\Vert u \right\Vert _{{\Lambda _{1/2m}}(\partial D)}} \leq c{\left\Vert \alpha \right\Vert _{{L^\infty }(\partial D)}}$ for some constant $ c > 0$ independent of $ \alpha$, where ${\Lambda _{1/2m}}(\partial D)$ is the Lipschitz space with exponent $\frac{1} {{2m}}$ and $ 2m = {\max _{1 \leq k \leq n}}\min (2{n_k},2{m_k})$ is the type of the domain $ D$.
Hausdorff dimension of divergent Teichm\"uller geodesics
Howard
Masur
235-254
Abstract: Let $g > 1$ be given and let $k = ({k_1}, \ldots ,{k_n})$ be an $ n$-tuple of positive integers whose sum is $4g - 4$. Denote by ${Q_k}$ the set of all holomorphic quadratic differentials on compact Riemann surfaces of genus $ g$ whose zeros have orders ${k_1}, \ldots, {k_n}$. $Q_k$ is called a stratum inside the cotangent space of all holomorphic quadratic differentials over the Teichmüller space of genus $g$. Let ${Q_k}/\operatorname{Mod} (g)$ be the moduli space where $ \operatorname{Mod} (g)$ is the mapping class group. Each $q \in {Q_k}$ defines a Teichmüller geodesic. Theorem. There exists $\delta > 0$ so that for almost all $q \in {Q_k}$, the set of $\theta$, such that the geodesic defined by $ {e^{i\theta }}q$ eventually leaves every compact set in ${Q_k}/\operatorname{Mod} (g)$, has Hausdorff dimension $\theta$.
The decompositions of Schur complexes
Hyoung J.
Ko
255-270
Abstract: This paper presents a method for finding the characteristic-free Pieri type decompositions of Schur modules, Weyl modules, and Schur complexes. We also introduce several new combinatorial rules for computing the Littlewood-Richardson coefficients.
A convergent framework for the multicomponent KP-hierarchy
G. F.
Helminck;
G. F.
Post
271-292
Abstract: In this paper we describe how to construct convergent solutions of the multicomponent KP-hierarchy, starting from a certain open subset of the Grassmann manifold of a special kind of Banach space, and derive an expression of its solutions in terms of Fredholm determinants. Further we show that the simplest nonscalar reduction of the present hierarchy leads to the AKNS-hierarchy.
Extending discrete-valued functions
John
Kulesza;
Ronnie
Levy;
Peter
Nyikos
293-302
Abstract: In this paper, we show that for a separable metric space $X$, every continuous function from a subset $S$ of $X$ into a finite discrete space extends to a continuous function on $X$ if and only if every continuous function from $ S$ into any discrete space extends to a continuous function on $X$. We also show that if there is no inner model having a measurable cardinal, then there is a metric space $X$ with a subspace $S$ such that every $2$-valued continuous function from $S$ extends to a continuous function on all of $X$, but not every discrete-valued continuous function on $S$ extends to such a map on $X$. Furthermore, if Martin's Axiom is assumed, such a space can be constructed so that not even $ \omega$-valued functions on $S$ need extend. This last result uses a version of the Isbell-Mrowka space $\Psi$ having a ${C^ * }$-embedded infinite discrete subset. On the other hand, assuming the Product Measure Extension Axiom, no such $\Psi$ exists.
Periodicity and decomposability of basin boundaries with irrational maps on prime ends
Russell B.
Walker
303-317
Abstract: Planar basin boundaries of iterated homeomorphisms induce homeomorphisms on prime ends. When the basin is connected, simply connected, and has a compact connected boundary, the space of prime ends is a topological circle. If the induced homeomorphism on prime ends has rational rotation number, the basin boundary contains periodic orbits. Several questions as to basin boundary periodics, decomposability, and minimality, when the induced map on prime ends has irrational rotation number, are answered by construction of both homeomorphisms and diffeomorphisms. Examples in the literature of basin boundaries with interesting prime end dynamics have been sparse. Prime end dynamics has drawn recent interest as a natural tool for the study of strange attractors.
On the distance of subspaces of $l\sp n\sb p$ to $l\sp k\sb p$
William B.
Johnson;
Gideon
Schechtman
319-329
Abstract: It is proved that if $l_p^n$ is well-isomorphic to $X \oplus Y$ and $X$ either has small dimension or is a Euclidean space, then $Y$ is well-isomorphic to $l_p^k$, $ k = \operatorname{dim} Y$. The proofs use new forms of the finite dimensional decomposition method. It is shown that the constant of equivalence between a normalized $K$-unconditional basic sequence in $l_p^n$ and a subsequence of the unit vector basis of $l_p^n$ is greatest, up to a constant depending on $ K$, when the sequence spans a $2$-Euclidean space.
Stable patterns in a viscous diffusion equation
A.
Novick-Cohen;
R. L.
Pego
331-351
Abstract: We consider a pseudoparabolic regularization of a forward-backward nonlinear diffusion equation ${u_t} = \Delta (f(u) + \nu {u_t})$, motivated by the problem of phase separation in a viscous binary mixture. The function $f$ is non-monotone, so there are discontinuous steady state solutions corresponding to arbitrary arrangements of phases. We find that any bounded measurable steady state solution $u(x)$ satisfying $f(u) = {\text{constant}}$,
Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture
Alberto
Albano;
Sheldon
Katz
353-368
Abstract: We study the deformation theory of lines on the Fermat quintic threefold. We formulate an infinitesimal version of the generalized Hodge conjecture, and use our analysis of lines to prove it in a special case.
Noether-Lefschetz locus for surfaces
Sung-Ock
Kim
369-384
Abstract: We generalize M. Green's Explicit Noether-Lefschetz Theorem to the family of smooth complete intersection surfaces in the higher dimensional projective spaces. Moreover, we give a new proof of the Density Theorem due to C. Ciliberto, J. Harris, and R. Miranda [5].
Markov partitions for expanding maps of the circle
Matthew
Stafford
385-403
Abstract: We study Markov partitions for orientation-preserving expanding maps of the circle whose rectangles are connected. Up to a reordering of basis elements, the class of induced matrices arising for such partitions is characterized. Then the study focuses on the subclass of partitions for which each boundary set is a periodic orbit. We show that, if the boundary orbit of a partition is well-distributed, the partition and its symmetries can be constructed. An accompanying result is concerned with double covers of the circle only. It says that, for a given period, all partitions bounded by ill-distributed orbits have the same induced matrix.
Maximal representations of surface groups in bounded symmetric domains
Luis
Hernández
405-420
Abstract: Let $\Gamma$ be the fundamental group of a hyperbolic surface of genus $g$; for $1 \le p \le q,PSU(p,q)$ is the group of isometries of a certain Hermitian symmetric space ${D_{p,q}}$ of rank $p$. There exists a characteristic number $c:\operatorname{Hom} (\Gamma ,PSU(p,q)) \to \mathbb{R}$, which is constant on each connected component and such that $\vert c(\rho )\vert \leq 4p\pi (g - 1)$ for every representation $\rho$. We show that representations with maximal characteristic number (plus some nondegeneracy condition if $p > 2$ leave invariant a totally geodesic subspace of ${D_{p,q}}$ isometric to ${D_{p,p}}$.
Infinitesimal rigidity for the action of ${\rm SL}(n,{\bf Z})$ on ${\bf T}\sp n$
James W.
Lewis
421-445
Abstract: Let $\Gamma = {\mathbf{SL}}(n,\mathbb{Z})$ or any subgroup of finite index. Then the action of $ \Gamma$ on ${\mathbb{T}^n}$ by automorphisms is infinitesimally rigid for $n \ge 7$, i.e., the cohomology ${\text{H}^1}(\Gamma ,\operatorname{Vec} ({\mathbb{T}^n})) = 0$, where $\operatorname{Vec} ({\mathbb{T}^n})$ denotes the module of $ {C^\infty }$ vector fields on $ {\mathbb{T}^n}$.
Symplectic double groupoids over Poisson $(ax+b)$-groups
Kentaro
Mikami
447-463
Abstract: First, we classify all the multiplicative Poisson structures on the $ (ax + b)$-group and determine their dual Poisson Lie groups. Next, we show the existence of symplectic groupoid over the Poisson $ (ax + b)$-group. Finally, by the Hamilton-Jacobi method we construct nontrivial symplectic double groupoids and conclude that for each pair of nondegenerate multiplicative Poisson structures of the $(ax + b)$-group there exists a symplectic double groupoid.
Maxwell's equations in a periodic structure
Xinfu
Chen;
Avner
Friedman
465-507
Abstract: Consider a diffraction of a beam of particles in ${\mathbb{R}^3}$ when the dielectric coefficient is a constant $ {\varepsilon _1}$ above a surface $S$ and a constant $ {\varepsilon _2}$ below a surface $S$, and the magnetic permeability is constant throughout $ {\mathbb{R}^3}$. $ S$ is assumed to be periodic in the ${x_1}$ direction and of the form $ {x_1} = {f_1}(s),\,{x_3} = {f_3}(s),\,{x_2}$ arbitrary. We prove that there exists a unique solution to the time-harmonic Maxwell equations in $ {\mathbb{R}^3}$ having the form of refracted waves for $ {x_3} \ll 1$ and of transmitted waves for $ - {x_3} \gg 1$ if and only if there exists a unique solution to a certain system of two coupled Fredholm equations. Thus, in particular, for all the $ \varepsilon$'s, except for a discrete number, there exists a unique solution to the Maxwell equations.
Cavitational flows and global injectivity of conformal maps
Massimo
Lanza de Cristoforis
509-527
Abstract: This paper treats some new mathematical aspects of the two-dimensional cavitational problem of the flow of a perfect fluid past an obstacle. Natural regularity conditions of very general type are found to ensure the global injectivity of the complex-potential and the presence of at most one zero of its derivative on the boundary of the flow. This derivative is the complex velocity. Previous authors have hypothesized the properties obtained here. The same regularity conditions are then shown to be satisfied by the classical solutions found via Villat's integral equation. A simple counterexample in $ \S4$ shows that the global injectivity of a holomorphic map defined on an unbounded Jordan domain cannot be deduced solely from its injectivity on the boundary. This simple fact raises new questions on the relation between cavitational flows and Villat's integral equation, which are discussed in $\S3$.
On the homology of ${\rm SU}(n)$ instantons
Charles P.
Boyer;
Benjamin M.
Mann;
Daniel
Waggoner
529-561
Abstract: In this paper we study the homology of the moduli spaces of instantons associated to principal $ {\mathbf{SU}}(n)$ bundles over the four-sphere. This is accomplished by exploiting an "iterated loop space" structure implicit in the disjoint union of all moduli spaces associated to a fixed $ {\mathbf{SU}}(n)$ with arbitrary instanton number and relating these spaces to the known homology structure of the four-fold loop space on $ B{\mathbf{SU}}(n)$.
Complex representations of matrix semigroups
Jan
Okniński;
Mohan S.
Putcha
563-581
Abstract: Let $M$ be a finite monoid of Lie type (these are the finite analogues of linear algebraic monoids) with group of units $G$. The multiplicative semigroup ${\mathcal{M}_n}(F)$, where $F$ is a finite field, is a particular example. Using Harish-Chandra's theory of cuspidal representations of finite groups of Lie type, we show that every complex representation of $M$ is completely reducible. Using this we characterize the representations of $ G$ extending to irreducible representations of $M$ as being those induced from the irreducible representations of certain parabolic subgroups of $ G$. We go on to show that if $F$ is any field and $S$ any multiplicative subsemigroup of ${\mathcal{M}_n}(F)$, then the semigroup algebra of $S$ over any field of characteristic zero has nilpotent Jacobson radical. If $S = {\mathcal{M}_n}(F)$, then this algebra is Jacobson semisimple. Finally we show that the semigroup algebra of $ {\mathcal{M}_n}(F)$ over a field of characteristic zero is regular if and only if $\operatorname{ch} (F) = p > 0$ and $F$ is algebraic over its prime field.
An $L\sp 2$-cohomology construction of unitary highest weight modules for ${\rm U}(p,q)$
Lisa A.
Mantini
583-603
Abstract: In this paper a geometric construction is given of all unitary highest weight modules of $ G = \operatorname{U} (p,q)$. The construction is based on the unitary model of the $ k$th tensor power of the metaplectic representation in a Bargmann-Segal-Fock space of square-integrable differential forms. The representations are constructed as holomorphic sections of certain vector bundles over $G/K$, and the construction is implemented via an integral transform analogous to the Penrose transform of mathematical physics.
Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface
Carsten
Thomassen
605-635
Abstract: We describe all regular tilings of the torus and the Klein bottle. We apply this to describe, for each orientable (respectively nonorientable) surface $S$, all (but finitely many) vertex-transitive graphs which can be drawn on $S$ but not on any surface of smaller genus (respectively crosscap number). In particular, we prove the conjecture of Babai that, for each $g \geqslant 3$, there are only finitely many vertex-transitive graphs of genus $g$. In fact, they all have order $< {10^{10}}g$. The weaker conjecture for Cayley graphs was made by Gross and Tucker and extends Hurwitz' theorem that, for each $g \geqslant 2$, there are only finitely many groups that act on the surface of genus $g$. We also derive a nonorientable version of Hurwitz' theorem.
Weak type $(1,1)$ estimates for heat kernel maximal functions on Lie groups
Michael
Cowling;
Garth
Gaudry;
Saverio
Giulini;
Giancarlo
Mauceri
637-649
Abstract: For a Lie group $ G$ with left-invariant Haar measure and associated Lebesgue spaces $ {L^p}(G)$, we consider the heat kernels $ {\{ {p_t}\} _{t > 0}}$ arising from a right-invariant Laplacian $ \Delta$ on $G$: that is, $u(t, \cdot ) = {p_t}{\ast}f$ solves the heat equation $(\partial /\partial t - \Delta )u = 0$ with initial condition $ u(0, \cdot ) = f( \cdot )$. We establish weak-type $(1,1)$ estimates for the maximal operator $\mathcal{M}(\mathcal{M}\;f = {\sup _{t > 0}}\vert{p_t}{\ast}f\vert)$ and for related Hardy-Littlewood maximal operators in a variety of contexts, namely for groups of polynomial growth and for a number of classes of Iwasawa $AN$ groups. We also study the "local" maximal operator $ {\mathcal{M}_0}({\mathcal{M}_0}f = {\sup _{0 < t < 1}}\vert{p_t}{\ast}f\vert)$ and related Hardy-Littlewood operators for all Lie groups.
Characteristic numbers for unoriented ${\bf Z}$-homology manifolds
Sandro
Buoncristiano;
Derek
Hacon
651-663
Abstract: It is shown that the analogue of Thom's theorem on Stiefel-Whitney numbers holds for $ {\mathbf{Z}}$-homology manifolds
Concentrated cyclic actions of high periodicity
Daniel
Berend;
Gabriel
Katz
665-689
Abstract: The class of concentrated periodic diffeomorphisms $ g:M \to M$ is introduced. A diffeomorphism is called concentrated if, roughly speaking, its normal eigenvalues range in a small (with respect to the period of $g$ and the dimension of $M$) arc on the circle. In many ways, the cyclic action generated by such a $g$ behaves on the one hand as a circle action and on the other hand as a generic prime power order cyclic action. For example, as for circle actions, $\operatorname{Sign} (g,M) = \operatorname{Sign} ({M^g})$, provided that the left-hand side is an integer; as for prime power order actions, $g$ cannot have a single fixed point if $ M$ is closed. A variety of integrality results, relating to the usual signatures of certain characteristic submanifolds of the regular neighbourhood of ${M^g}$ in $M$ to $\operatorname{Sign} (g,M)$ via the normal $ g$-representations, is established.
A cubic counterpart of Jacobi's identity and the AGM
J. M.
Borwein;
P. B.
Borwein
691-701
Abstract: We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is $\displaystyle {a_{n + 1}}: = \frac{{{a_n} + 2{b_n}}} {3}\quad {\text{and}}\quad... ...}: = \sqrt[3]{{{b_n}\left( {\frac{{a_n^2 + {a_n}{b_n} + b_n^2}} {3}} \right).}}$ The limit of this iteration is identified in terms of the hypergeometric function $ {}_2{F_1}(1/3,2/3;1; \cdot )$, which supports a particularly simple cubic transformation.
Distal functions and unique ergodicity
Ebrahim
Salehi
703-713
Abstract: A. Knapp [5] has shown that the set, $D(S)$, of all distal functions on a group $ S$ is a norm closed subalgebra of ${l^\infty }(S)$ that contains the constants and is closed under the complex conjugation and left translation by elements of $S$. Also it is proved that [7] for any $k \in \mathbb{N}$ and any $\lambda \in \mathbb{R}$ the function $f:\mathbb{Z} \to \mathbb{C}$ defined by $f(n) = {e^{i\lambda {n^k}}}$ is distal on $ \mathbb{Z}$. Now let ${\mathbf{W}}$ be the norm closure of the algebra generated by the set of functions $\displaystyle \{ n \mapsto {e^{i\lambda {n^k}}}:k \in \mathbb{N},\;\lambda \in \mathbb{R}\} ,$ which will be called the Weyl algebra. According to the facts mentioned above, all members of the Weyl Algebra are distal functions on $\mathbb{Z}$. In this paper, we will show that any element of $ {\mathbf{W}}$ is uniquely ergodic (Theorem 2.13) and that the set ${\mathbf{W}}$ does not exhaust all the distal functions on $ \mathbb{Z}$ (Theorem 2.14). The latter will answer the question that has been asked (to the best of my knowledge) by P. Milnes [6]. The term Weyl algebra is suggested by S. Glasner. I would like to express my warmest gratitude to S. Glasner for his helpful advise, and to my advisor Professor Namioka for his enormous helps and contributions.
Roots of unity and the Adams-Novikov spectral sequence for formal $A$-modules
Keith
Johnson
715-726
Abstract: The cohomology of a Hopf algebroid related to the Adams-Novikov spectral sequence for formal $A$-modules is studied in the special case in which $ A$ is the ring of integers in the field obtained by adjoining $p$th roots of unity to $ {\widehat{\mathbb{Q}}_p}$, the $p$-adic numbers. Information about these cohomology groups is used to give new proofs of results about the ${E_2}$ term of the Adams spectral sequence based on $ 2$-local complex $ K$-theory, and about the odd primary Kervaire invariant elements in the usual Adams-Novikov spectral sequence.
Infinitely many co-existing sinks from degenerate homoclinic tangencies
Gregory J.
Davis
727-748
Abstract: The evolution of a horseshoe is an interesting and important phenomenon in Dynamical Systems as it represents a change from a nonchaotic state to a state of chaos. As we are interested in determining how this transition takes place, we are studying certain families of diffeomorphisms. We restrict our attention to certain one-parameter families $\{ {F_t}\}$ of diffeomorphisms in two dimensions. It is assumed that each family has a curve of dissipative periodic saddle points, ${P_t};\;F_t^n({P_t}) = {P_t}$, and $\vert\det DF_t^n({P_t})\vert < 1$. We also require the stable and unstable manifolds of $ {P_t}$ to form homoclinic tangencies as the parameter $t$ varies through ${t_0}$. Our emphasis is the exploration of the behavior of families of diffeomorphisms for parameter values $t$ near ${t_0}$. We show that there are parameter values $ t$ near ${t_0}$ at which ${F_t}$ has infinitely many co-existing periodic sinks.
Generalizations of Picard's theorem for Riemann surfaces
Pentti
Järvi
749-763
Abstract: Let $D$ be a plane domain, $E \subset D$ a compact set of capacity zero, and $ f$ a holomorphic mapping of $D\backslash E$ into a hyperbolic Riemann surface $W$. Then there is a Riemann surface $ W'$ containing $ W$ such that $ f$ extends to a holomorphic mapping of $D$ into $W'$. The same conclusion holds if hyperbolicity is replaced by the assumption that the genus of $ W$ be at least two. Furthermore, there is quite a general class of sets of positive capacity which are removable in the above sense for holomorphic mappings into Riemann surfaces of positive genus, except for tori.
Strong shape for topological spaces
Jerzy
Dydak;
Sławomir
Nowak
765-796
Abstract: Strong shape equivalences for topological spaces are introduced in a way which generalizes easily to inverse systems of topological spaces. Each space is then mapped via a strong shape equivalence into a fibrant inverse system of ANRs. This leads naturally to defining the strong shape category SSh for topological spaces. Other descriptions of SSh are also provided.
Conformal automorphisms and conformally flat manifolds
William M.
Goldman;
Yoshinobu
Kamishima
797-810
Abstract: A geometric structure on a smooth $n$-manifold $M$ is a maximal collection of distinguished charts modeled on a $1$-connected $n$-dimensional homogeneous space $ X$ of a Lie group $ G$ where coordinate changes are restrictions of transformations from $ G$. There exists a developing map $dev:wm \to X$ which is always locally a diffeomorphism. It is in general far from globally being a diffeomorphism. We study the rigid property of developing maps of $(G,X)$-manifolds. As an application we shall classify closed conformally flat manifolds $M$ when the universal covering space $ \tilde M$ supports a one parameter group of conformal transformations.
A countably compact topological group $H$ such that $H\times H$ is not countably compact
Klaas Pieter
Hart;
Jan
van Mill
811-821
Abstract: Using $ {\mathbf{M}}{{\mathbf{A}}_{{\text{countable}}}}$ we construct a topological group with the properties mentioned in the title.
M\"obius invariant Hilbert spaces of holomorphic functions in the unit ball of ${\bf C}\sp n$
Ke He
Zhu
823-842
Abstract: We prove that there exists a unique Hilbert space of holomorphic functions in the open unit ball of ${\mathbb{C}^n}$ whose (semi-) inner product is invariant under Möbius transformations.
Generating modules efficiently over noncommutative Noetherian rings
S. C.
Coutinho
843-856
Abstract: The Forster-Swan Theorem gives an upper bound on the number of generators of a module over a commutative ring in terms of local data. Stafford showed that this theorem could be generalized to arbitrary right and left noetherian rings. In this paper a similar result is proved for right noetherian rings with finite Krull dimension. A new dimension function--the basic dimension--is the main tool used in the proof of this result.
A nonstandard resonance problem for ordinary differential equations
Shair
Ahmad
857-875
Abstract: Necessary and sufficient conditions are established for the existence of bounded solutions for a class of second order differential equations.
Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term
J.
García Azorero;
I.
Peral Alonso
877-895
Abstract: We study the existence of solutions for the following nonlinear degenerate elliptic problems in a bounded domain $ \Omega \subset {{\mathbf{R}}^N}$ $\displaystyle - \operatorname{div} (\vert\nabla u{\vert^{p - 2}}\nabla u) = \ve... ...{\vert^{{p^{\ast}} - 2}}u + \lambda \vert u{\vert^{q - 2}}u,\qquad \lambda > 0,$ where ${p^{\ast}}$ is the critical Sobolev exponent, and $ u{\vert _{\delta \Omega }} \equiv 0$. By using critical point methods we obtain the existence of solutions in the following cases: If $ p < q < {p^{\ast}}$, there exists $ {\lambda _0} > 0$ such that for all $ \lambda > {\lambda _0}$ there exists a nontrivial solution. If $ \max (p,{p^{\ast}} - p/(p - 1)) < q < {p^{\ast}}$, there exists nontrivial solution for all $ \lambda > 0$. If $ 1 < q < p$ there exists ${\lambda _1}$ such that, for $0 < \lambda < {\lambda _1}$, there exist infinitely many solutions. Finally, we obtain a multiplicity result in a noncritical problem when the associated functional is not symmetric.
On the spectral character of Toeplitz operators on multiply connected domains
Kevin F.
Clancey
897-910
Abstract: An explicit resolvent formula is given for selfadjoint Toeplitz operators acting on the least harmonic majorant Hardy spaces of a multiply connected planar domain. This formula is obtained by using theta functions associated with the double of the domain. Several consequences concerning the spectral resolutions of selfadjoint Toeplitz operators are deduced.
Locally flat $2$-knots in $S\sp 2\times S\sp 2$ with the same fundamental group
Yoshihisa
Sato
911-920
Abstract: We consider a locally flat $2$-sphere in $ {S^2} \times {S^2}$ representing a primitive homology class $\xi$, which is referred to as a $ 2$-knot in ${S^2} \times {S^2}$ representing $\xi$. Then for any given primitive class $\xi$, there exists a $2$-knot in $ {S^2} \times {S^2}$ representing $\xi$ with simply-connected complement. In this paper, we consider the classification of $ 2$-knots in ${S^2} \times {S^2}$ whose complements have a fixed fundamental group. We show that if the complement of a $2$-knot $S$ in $ {S^2} \times {S^2}$ is simply connected, then the ambient isotopy type of $ S$ is determined. In the case of nontrivial ${\pi _1}$, however, we show that the ambient isotopy type of a $2$-knot in $ {S^2} \times {S^2}$ with nontrivial ${\pi _1}$ is not always determined by ${\pi _1}$.
Closed convex hulls of unitary orbits in von Neumann algebras
Fumio
Hiai;
Yoshihiro
Nakamura
1-38
Abstract: Let $\mathcal{M}$ be a von Neumann algebra. The distance $ \operatorname{dist} (x,\operatorname{co} \mathcal{U}(y))$ between $ x$ and $ \operatorname{co} \mathcal{U}(y)$ for selfadjoint operators $x$, $ y \in \mathcal{M}$ and the distance $ \operatorname{dist} (\varphi ,\operatorname{co} \mathcal{U}(\psi ))$ between $\varphi$ and $\operatorname{co} \mathcal{U}(\psi )$ for selfadjoint elements $\varphi$, $\psi \in {\mathcal {M}_*}$ are exactly estimated, where $\operatorname{co} \mathcal{U}(y)$ and $ \operatorname{co} \mathcal{U}(\psi )$ are the convex hulls of the unitary orbits of $y$ and $\psi$, respectively. This is done separately in the finite factor case, in the infinite semifinite factor case, and in the type III factor case. Simple formulas of distances between two convex hulls of unitary orbits are also given. When $ \mathcal{M}$ is a von Neumann algebra on a separable Hilbert space, the above cases altogether are combined under the direct integral decomposition of $ \mathcal{M}$ into factors. As a result, it is known that if $\mathcal{M}$ is $\sigma$-finite and $ x \in \mathcal{M}$ is selfadjoint, then $\overline {\operatorname{co} } \mathcal{U}(x) = {\overline {\operatorname{co} } ^{\mathbf{w}}}\mathcal{U}(x)$ where $ \overline {\operatorname{co} } \mathcal{U}(x)$ and ${\overline {\operatorname{co} } ^{\mathbf{w}}}\mathcal{U}(x)$ are the closures of $ \operatorname{co} \mathcal{U}(x)$ in norm and in the weak operator topology, respectively.
Computing the topological entropy of general one-dimensional maps
P.
Góra;
A.
Boyarsky
39-49
Abstract: A matrix-theoretic method for computing the topological entropy of continuous, piecewise monotonic maps of the interval is presented. The method results in a constructive procedure which is easily implemented on the computer. Examples for families of unimodal, nonunimodal and discontinuous maps are presented.
Microlocal Holmgren's theorem for a class of hypo-analytic structures
S.
Berhanu
51-64
Abstract: A microlocal version of Holmgren's Theorem is proved for a certain class of the hypo-analytic structures of Baouendi, Chang, and Treves.
A discrete approach to monotonicity of zeros of orthogonal polynomials
Mourad E. H.
Ismail;
Martin E.
Muldoon
65-78
Abstract: We study the monotonicity with respect to a parameter of zeros of orthogonal polynomials. Our method uses the tridiagonal (Jacobi) matrices arising from the three-term recurrence relation for the polynomials. We obtain new results on monotonicity of zeros of associated Laguerre, Al-Salam-Carlitz, Meixner and Pollaczek polynomials. We also derive inequalities for the zeros of the Al-Salam-Carlitz and Meixner polynomials.
Commuting fully invariant congruences on free completely regular semigroups
F.
Pastijn
79-92
Abstract: We show that "almost all" fully invariant congruences on every free completely regular semigroup commute. From this it is shown that the lattice of completely regular semigroup varieties is arguesian.
On Hermite-Fej\'er interpolation in a Jordan domain
Charles K.
Chui;
Xie Chang
Shen
93-109
Abstract: The Hermite-Fejér interpolation problem on a Jordan domain is studied. Under certain mild conditions on the smoothness of the boundary curve, we give both uniform and $ {L^p}$, $0 < p < \infty$, estimates on the rate of convergence. Our estimates are sharp even for the unit disk setting.
Genericity of nontrivial $H$-superrecurrent $H$-cocycles
Karma
Dajani
111-132
Abstract: We prove that most ${\text{H}}$-cocycles for a nonsingular ergodic transformation of type $ {\text{II}}{{\text{I}}_\lambda }$, $0 < \lambda < 1$, are $ {\text{H}}$-superrecurrent. This is done by showing that the set of nontrivial $ {\text{H}}$-superrecurrent ${\text{H}}$-cocycles form a dense ${G_\delta}$ set with respect to the topology of convergence in measure.
Stability of Newton boundaries of a family of real analytic singularities
Masahiko
Suzuki
133-150
Abstract: Let ${f_t}(x,y)$ be a real analytic $t$-parameter family of real analytic functions defined in a neighborhood of the origin in $ {\mathbb{R}^2}$. Suppose that ${f_t}(x,y)$ admits a blow analytic trivilaization along the parameter $t$ (see the definition in $\S1$ of this paper). Under this condition, we prove that there is a real analytic $t$-parameter family ${\sigma _t}(x,y)$ with ${\sigma _0}(x,y)=(x,y)$ and ${\sigma _t}(0,0)=(0,0)$ of local coordinates in which the Newton boundaries of $ {f_t}(x,y)$ are stable. This fact claims that the blow analytic equivalence among real analytic singularities is a fruitful relationship since the Newton boundaries of singularities contains a lot of informations on them.
Lie supergroup actions on supermanifolds
Charles P.
Boyer;
O. A.
Sánchez-Valenzuela
151-175
Abstract: Lie supergroups are here understood as group objects in the category of supermanifolds (as in [$2$, $5$, and $15$]). Actions of Lie supergroups in supermanifolds are defined by means of diagrams of supermanifold morphisms. Examples of such actions are given. Among them emerge the linear actions discussed in [$2$, $5$, and $12$] and the natural actions on the Grassmannian supermanifolds studied in [$6$-$9$ and $13$]. The nature of the isotropy subsupergroup associated to an action is fully elucidated; it is exhibited as an embedded subsupergroup in the spirit of the theory of smooth manifolds and Lie groups and with no need for the Lie-Hopf algebraic approach of Kostant in [$3$]. The notion of orbit is also discussed. Explicit calculations of isotropy subsupergroups are included. Also, an alternative proof of the fact that the structural sheaf of a Lie supergroup is isomorphic to the sheaf of sections of a trivial exterior algebra bundle is given, based on the triviality of its supertangent bundle.
Hypersurface variations are maximal. II
James A.
Carlson
177-196
Abstract: We show that certain variations of Hodge structure defined by sufficiently ample hypersurfaces are maximal integral manifolds of Griffiths' horizontal distribution.
A characterization of the complemented translation-invariant subspaces of $H\sp 1({\bf R})$
Dale E.
Alspach
197-207
Abstract: The purpose of this paper is to characterize the complemented translation-invariant subspaces of ${H^1}({\mathbf{R}})$ in terms of the zero set of the Fourier transform. It is shown that if $X$ is such a subspace then $X = I(A)$ where $A$ is in the ring generated by arithmetic progressions and lacunary sequences and $A$ is $ \varepsilon$-separated for some $\varepsilon > 0$. This proves a conjecture of the author and D. Ullrich.
The hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type
A.
Boggess;
R.
Dwilewicz;
A.
Nagel
209-232
Abstract: We show that the hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type in ${C^n}$ contains an open set in ${C^n}$ which emanates from the hypersurface a distance which is proportional to the length of the minor axis of the nonisotropic ball. In addition, we prove a maximal function estimate for plurisubharmonic functions which is important in the study of boundary values of holomorphic functions.
Finite group actions on the moduli space of self-dual connections. I
Yong Seung
Cho
233-261
Abstract: Let $M$ be a smooth simply connected closed $4$-manifold with positive definite intersection form. Suppose a finite group $G$ acts smoothly on $M$. Let $ \pi :E \to M$ be the instanton number one quaternion line bundle over $ M$ with a smooth $ G$-action such that $ \pi$ is an equivariant map. We first show that there exists a Baire set in the $ G$-invariant metrics on $ M$ such that the moduli space $\mathcal{M}_ * ^G$ of $G$-invariant irreducible self-dual connections is a manifold. By utilizing the $ G$-transversality theory of T. Petrie, we then identify cohomology obstructions to globally perturbing the full space ${\mathcal{M}_ * }$ of irreducible self-dual connections to a $G$-manifold when $G = {{\mathbf{Z}}_2}$ and the fixed point set of the $ {\mathbf{Z}}_2$ action on $ M$ is a nonempty collection of isolated points and Riemann surfaces.
Sobolev interpolation inequalities with weights
Cristian E.
Gutiérrez;
Richard L.
Wheeden
263-281
Abstract: We study weighted local Sobolev interpolation inequalities of the form \begin{displaymath}\begin{gathered}\frac{1} {{{w_2}(B)}}{\int\limits_B {\vert u(... ...\vert u(x){\vert^p}v(x)dx} } \right), \end{gathered} \end{displaymath} , where $1 < p < \infty,h > 1, B$ is a ball in $ {{\mathbf{R}}^n}$, and $ v$ ,${w_1}$, and ${w_2}$ are weight functions. The case $p = 2$ is of special importance in deriving regularity results for solutions of degenerate parabolic equations. We also study the analogous inequality without the second summand on the right in the case $u$ has compact support in $B$, and we derive global Landau inequalities ${\left\Vert {\nabla u} \right\Vert _{L_w^q}} \leq c\left\Vert {\nabla u} \righ... ...eft\Vert {{\nabla ^2}u} \right\Vert _{L_v^p}^a,0 < a < 1,1 < p \leq q < \infty $, when $u$ has compact support.
Puret\'e, rigidit\'e, et morphismes entiers
Gabriel
Picavet
283-313
Abstract: Bousfield and Kan have shown that a ring morphism with domain ${\mathbf{Z}}$ is rigid; we say that a ring morphism is rigid if it admits a factorization by an epimorphism, followed by a pure morphism. A ring $ A$ is said to be rigid if every morphism with domain $A$ is a rigid one. Our principal results are: the rigid domains are the Prüferian rings $ A$, with $\operatorname{Dim} (A) \leq 1$, and the Noetherian rigid rings are the Z.P.I. rings. The quasi-compact open sets of an affine rigid scheme, having as underlying ring a domain or a Noetherian ring, are affine and schematically dense if they contain the assassin of the ring. Every injective integral ring morphism with rigid domain is a pure morphism. We give two criteria of purity for integral injective morphisms. As a consequence of these results we obtain the following properties: if $A$ is a normal ring, containing the field of rationals, or is a regular ring, containing a field, every injective integral morphism with domain $A$ is a pure one. For a reduced ring, we define the category of reduced modules and show that any injective integral morphism is pure with respect to the category of the reduced modules.
Structure locale de l'espace des r\'etractions d'une surface
Robert
Cauty
315-334
Abstract: Let $\Sigma$ be a compact connected $ 2$-manifold, and $\mathcal{R}(\Sigma )$ the space of retractions of $ \Sigma$. We prove that $ \mathcal{R}(\Sigma )$ is an ${l^2}$-manifold if the boundary of $\Sigma$ is not empty, and is the union of an ${l^2}$-manifold and an isolated point $ {\text{i}}{{\text{d}}_\Sigma }$ if $\Sigma$ is closed.
Theorems of Hardy and Paley for vector-valued analytic functions and related classes of Banach spaces
O.
Blasco;
A.
Pełczyński
335-367
Abstract: We investigate the classes of Banach spaces where analogues of the classical Hardy inequality and the Paley gap theorem hold for vector-valued functions. We show that the vector-valued Paley theorem is valid for a large class of Banach spaces (necessarily of cotype $2$) which includes all Banach lattices of cotype $2$, all Banach spaces whose dual is of type $ 2$ and also the preduals of ${C^ * }$-algebras. For the trace class $ {S_1}$ and the dual of the algebra of all bounded operators on a Hilbert space a stronger result holds; namely, the vector-valued analogue of the Fefferman theorem on multipliers from $ {H^1}$ into ${l^1}$; in particular for the latter spaces the vector-valued Hardy inequality holds. This inequality is also true for every Banach space of type $ > 1$ (Bourgain).
Indecomposable Cohen-Macaulay modules and their multiplicities
Dorin
Popescu
369-387
Abstract: The main aim of this paper is to find a large class of rings for which there are indecomposable maximal Cohen-Macaulay modules of arbitrary high multiplicity (or rank in the case of domains).
On collectionwise normality of locally compact, normal spaces
Zoltán T.
Balogh
389-411
Abstract: We prove that by adjoining supercompact many Cohen or random reals to a model of ZFC set theory, in the resulting model, every normal locally compact space is collectionwise normal. In the same models, countably paracompact, locally compact $ {T_3}$-spaces are expandable. Local compactness in the above theorems can be weakened to being of point-countable type, a condition that is implied by both Čech-completeness and first countability.
Univalent functions which map onto regions of given transfinite diameter
P. L.
Duren;
M. M.
Schiffer
413-428
Abstract: By a variational method, the sharp upper bound is obtained for the second coefficients of normalized univalent functions which map the unit disk onto regions of prescribed transfinite diameter, or logarithmic capacity.
Inner functions and cyclic vectors in the Bloch space
J. M.
Anderson;
J. L.
Fernández;
A. L.
Shields
429-448
Abstract: In this paper we construct a singular inner function whose polynomial multiples are dense in the little Bloch space ${\mathcal{B}_0}$. To do this we construct a singular measure on the unit circle with "best possible" control of both the first and second differences.
Circuit partitions and the homfly polynomial of closed braids
François
Jaeger
449-463
Abstract: We present an expansion of the homfly polynomial $P(D,z,a)$ of a braid diagram $D$ in terms of its circuit partitions. Another aspect of this result is an expression of $ P(D,z,a)$ as the trace of a matrix associated to $D$ in a simple way. We show how certain degree properties of the homfly polynomial can be derived easily from this model. In particular we obtain that if $ D$ is a positive braid diagram on $n$ strings with $w$ crossings, the maximum degree of $P(D,z,a)$ in the variable $a$ equals $n - 1 - w$. Nous présentons une expansion pour le polynôme homfly $P(D,z,a)$ d'un diagramme de tresse $ D$ en termes de ses partitions en circuits. Un autre aspect de ce résultat consiste en une expression de $P(D,z,a)$ comme trace d'une matrice associee de façon simple à $D$. Nous montrons comment certaines propriétés de degré du polynôme homfly dérivent simplement de ce modèle. En particulier nous obtenons que pour un diagramme de tresse positif $ D$ à $n$ brins et $w$ croisements, le degré maximum de $ P(D,z,a)$ en la variable $ a$ est égal à $ n - 1- w$.
On sets that are uniquely determined by a restricted set of integrals
J. H. B.
Kemperman
417-458
Abstract: In many applied areas, such as tomography and crystallography, one is confronted by an unknown subset $S$ of a measure space $ (X,\lambda )$ such as ${{\mathbf{R}}^n}$, or an unknown function $0 \leqslant \phi \leqslant 1$ on $ X$, having known moments (integrals) relative to a specified class $ F$ of functions $f:X \to {\mathbf{R}}$. Usually, these $ F$-moments do not fully determine the object $S$ or function $\phi$. We will say that $S$ is a set of uniqueness if no other function $ 0 \leqslant \psi \leqslant 1$ has the same $F$-moments as $S$ in so far as the latter moments exist. Here, $ S$ is identified with its indicator function. Within this general setting and with no further assumptions, we develop a powerful sufficient condition for uniqueness, called generalized additivity. When $F$ is a finite class, this condition of generalized additivity is shown to be also necessary for uniqueness. For each $\phi$, which is not the indicator function of a set of uniqueness, there exist infinitely many sets having the same $F$-moments as $\phi$, provided $ (X,\lambda ,F)$ is nonatomic or regular and, moreover, `strongly rich', a condition which is satisfied in many applications. Using such general results, we also study the uniqueness problem for measures with given marginals relative to a given system of projections ${\pi _j}:X \to {Y_j}(j \in J)$. Here, one likes to know, for instance, what subsets $S$ of $X$ are uniquely determined by the corresponding set of projections (X-ray pictures). It is allowed that $ \lambda (S) = \infty$. Our results are also relevant to a wide class of optimization problems.
Epicomplete Archimedean $l$-groups and vector lattices
Richard N.
Ball;
Anthony W.
Hager
459-478
Abstract: An object $ G$ in a category is epicomplete provided that the only morphisms out of $ G$ which are simultaneously epi and mono are the isomorphisms. We characterize the epicomplete objects in the category ${\mathbf{Arch}}$, whose objects are the archimedean lattice-ordered groups (archimedean $\ell $-groups) and whose morphisms are the maps preserving both group and lattice structure ($\ell$-homomorphisms). Recall that a space is basically disconnected if the closure of each cozero subset is open. Theorem. The following are equivalent for $G \in {\mathbf{Arch}}$. (a) $G$ is ${\mathbf{Arch}}$ epicomplete. (b) $ G$ is an ${\mathbf{Arch}}$ extremal suboject of $ D(Y)$ for some basically disconnected compact Hausdorff space $ Y$. Here $ D(Y)$ denotes the continuous extended real-valued functions on $ Y$ which are finite on a dense subset. (c) $G$ is conditionally and laterally $ \sigma$-complete (meaning each countable subset of positive elements of $G$ which is either bounded or pairwise disjoint has a supremum), and $G$ is divisible. The analysis of $ {\mathbf{Arch}}$ rests on an analysis of the closely related category ${\mathbf{W}}$, whose objects are of the form $ (G,u)$, where $G \in {\mathbf{Arch}}$ and $u$ is a weak unit (meaning $g \wedge u = 0$ implies $g = 0$ for all $g \in G$), and whose morphisms are the $ \ell$-homomorphism preserving the weak unit. Theorem. The following are equivalent for $(G,u) \in {\mathbf{W}}$. (a) $(G,u)$ is ${\mathbf{W}}$ epicomplete. (b) $ (G,u)$ is ${\mathbf{W}}$ isomorphic to $ (D(Y),1)$. (c) $ (G,u)$ is conditionally and laterally $\sigma$-complete, and $G$ is divisible.
The Arf and Sato link concordance invariants
Rachel
Sturm Beiss
479-491
Abstract: The Kervaire-Arf invariant is a $Z/2$ valued concordance invariant of knots and proper links. The $\beta$ invariant (or Sato's invariant) is a $ Z$ valued concordance invariant of two component links of linking number zero discovered by J. Levine and studied by Sato, Cochran, and Daniel Ruberman. Cochran has found a sequence of invariants $\{ {\beta _i}\} $ associated with a two component link of linking number zero where each ${\beta _i}$ is a $Z$ valued concordance invariant and ${\beta _0} = \beta$. In this paper we demonstrate a formula for the Arf invariant of a two component link $L = X \cup Y$ of linking number zero in terms of the $\beta$ invariant of the link: $\displaystyle \operatorname{arf} (X \cup Y) = \operatorname{arf} (X) + \operatorname{arf} (Y) + \beta (X \cup Y)\quad (\bmod 2).$ This leads to the result that the Arf invariant of a link of linking number zero is independent of the orientation of the link's components. We then establish a formula for $\vert\beta \vert$ in terms of the link's Alexander polynomial $\Delta (x,y) = (x - 1)(y - 1)f(x,y)$: $\displaystyle \vert\beta (L)\vert = \vert f(1,1)\vert.$ Finally we find a relationship between the ${\beta _i}$ invariants and linking numbers of lifts of $X$ and $Y$ in a $Z/2$ cover of the compliment of $X \cup Y$.
Multipliers, linear functionals and the Fr\'echet envelope of the Smirnov class $N\sb *({\bf U}\sp n)$
Marek
Nawrocki
493-506
Abstract: Linear topological properties of the Smirnov class $ {N_{\ast}}({\mathbb{U}^n})$ of the unit polydisk $ {\mathbb{U}^n}$ in ${\mathbb{C}^n}$ are studied. All multipliers of $ {N_{\ast}}({\mathbb{U}^n})$ into the Hardy spaces $ {H_p}({\mathbb{U}^n}),\;0 < p \leqslant \infty$, are described. A representation of the continuous linear functionals on $ {N_{\ast}}({\mathbb{U}^n})$ is obtained. The Fréchet envelope of $ {N_{\ast}}({\mathbb{U}^n})$ is constructed. It is proved that if $ n > 1$, then $ {N_{\ast}}({\mathbb{U}^n})$ is not isomorphic to $ {N_{\ast}}(\mathbb{U}{^1})$.
Varieties of group representations and Casson's invariant for rational homology $3$-spheres
S.
Boyer;
A.
Nicas
507-522
Abstract: Andrew Casson's ${\mathbf{Z}}$-valued invariant for ${\mathbf{Z}}$-homology $3$-spheres is shown to extend to a ${\mathbf{Q}}$-valued invariant for ${\mathbf{Q}}$-homology $3$-spheres which is additive with respect to connected sums. We analyze conditions under which the set of abelian ${\operatorname{SL} _2}({\mathbf{C}})$ and $ \operatorname{SU} (2)$ representations of a finitely generated group is isolated. A formula for the dimension of the Zariski tangent space to an abelian ${\operatorname{SL} _2}({\mathbf{C}})$ or $ \operatorname{SU} (2)$ representation is obtained. We also derive a sum theorem for Casson's invariant with respect to toroidal splittings of a $ {\mathbf{Z}}$-homology $ 3$-sphere.
The connection matrix in Morse-Smale flows
James F.
Reineck
523-545
Abstract: In a Morse-Smale flow with no periodic orbits, it is shown that the connection matrix is unique. In the case of periodic orbits, nonuniqueness can occur. We show that on $2$-manifolds, with some technical assumptions, given a connection matrix for the flow, one can replace the periodic orbits with doubly-connected rest points and obtain a new flow with no periodic orbits having the given connection matrix.
Baer modules over domains
Paul C.
Eklof;
László
Fuchs;
Saharon
Shelah
547-560
Abstract: For a commutative domain $R$ with $1$, an $R$-module $B$ is called a Baer module if $ \operatorname{Ext} _R^1(B,T) = 0$ for all torsion $R$-modules $T$. The structure of Baer modules over arbitrary domains is investigated, and the problem is reduced to the case of countably generated Baer modules. This requires a general version of the singular compactness theorem. As an application we show that over $ h$-local Prüfer domains, Baer modules are necessarily projective. In addition, we establish an independence result for a weaker version of Baer modules.
Dualizing complexes of affine semigroup rings
Uwe
Schäfer;
Peter
Schenzel
561-582
Abstract: For an affine semigroup ring we construct the dualizing complex in terms of the semigroup and the homology of the face lattice of the polyhedral cone spanned by the semigroup. As a consequence there are characterizations of locally Cohen-Macaulay rings, Buchsbaum rings, and Cohen-Macaulay rings as well as Serre's condition ${\mathcal{S}_l}$.
Weak Chebyshev subspaces and $A$-subspaces of $C[a,b]$
Wu
Li
583-591
Abstract: In this paper we show some very interesting properties of weak Chebyshev subspaces and use them to simplify Pinkus's characterization of $A$subspaces of $C[a,b]$. As a consequence we obtain that if the metric projection ${P_G}$ from $C[a,b]$ onto a finite-dimensional subspace $ G$ has a continuous selection and elements of $G$ have no common zeros on $(a,b)$, then $G$ is an $A$-subspace.
$N$-body Schr\"odinger operators with finitely many bound states
W. D.
Evans;
Roger T.
Lewis
593-626
Abstract: In this paper we consider a class of second-order elliptic operators which includes atomic-type $N$-body operators for $N > 2$. Our concern is the problem of predicting the existence of only a finite number of bound states corresponding to eigenvalues below the essential spectrum. We obtain a criterion which is natural for the problem and easy to apply as is demonstrated with various examples. While the criterion applies to general second-order elliptic operators, sharp results are obtained when the Hamiltonian of an atom with an infinitely heavy nucleus of charge $Z$ and $N$ electrons of charge $1$ and mass $ \tfrac{1} {2}$ is considered.
Global families of limit cycles of planar analytic systems
L. M.
Perko
627-656
Abstract: The global behavior of any one-parameter family of limit cycles of a planar analytic system $\dot x = f(x,\lambda )$ depending on a parameter $\lambda \in R$ is determined. It is shown that any one-parameter family of limit cycles belongs to a maximal one-parameter family which is either open or cyclic. If the family is open, then it terminates as the parameter or the orbits become unbounded, or it terminates at a critical point or on a (compound) separatrix cycle of the system. This implies that the periods in a one-parameter family of limit cycles can become unbounded only if the orbits become unbounded or if they approach a degenerate critical point or (compound) separatrix cycle of the system. This is a more specific result for planar analytic systems than Wintner's principle of natural termination for $n$-dimensional systems where the periods can become unbounded in strange ways. This work generalizes Duffs results for one-parameter families of limit cycles generated by a one-parameter family of rotated vector fields. In particular, it is shown that the behavior at a nonsingular, multiple limit cycle of any one-parameter family of limit cycles is exactly the same as the behavior at a multiple limit cycle of a one-parameter family of limit cycles generated by a one-parameter family of rotated vector fields.
Solving Beltrami equations by circle packing
Zheng-Xu
He
657-670
Abstract: We use Andreev-Thurston's theorem on the existence of circle packings to construct approximating solutions to the Beltrami equations on Riemann surfaces. The convergence of the approximating solutions on compact subsets will be shown. This gives a constructive proof of the existence theorem for Beltrami equations.
Mean summability methods for Laguerre series
Krzysztof
Stempak
671-690
Abstract: We apply a construction of generalized convolution in $\displaystyle {L^1}({\mathbb{R}_ + } \times \mathbb{R},{x^{2\alpha - 1}}dxdt),\qquad \alpha \geqslant 1,$ cf. [8], to investigate the mean convergence of expansions in Laguerre series. Following ideas of [4, 5] we construct a functional calculus for the operator $\displaystyle L = - \left( {\frac{{{\partial ^2}}} {{\partial {x^2}}} + \frac{{... ... {t^2}}}} \right),\qquad x > 0,\quad t \in \mathbb{R},\quad \alpha \geqslant 1.$ Then, arguing as in [3], we prove results concerning the mean convergence of some summability methods for Laguerre series. In particular, the classical Abel-Poisson and Bochner-Riesz summability methods are included.
On the smoothness of convex envelopes
A.
Griewank;
P. J.
Rabier
691-709
Abstract: We examine differentiability properties of the convex envelope $\operatorname{conv} E$ of a given function $ E:{{\mathbf{R}}^n} \to ( - \infty ,\infty ]$ in terms of properties of $ E$. It is shown that $ {C^1}$ as well as optimal ${C^{1,\alpha }}$ regularity results, $0 < \alpha \leqslant 1$, can be obtained under general conditions.
An asymptotic formula for hypo-analytic pseudodifferential operators
S.
Berhanu
711-729
Abstract: An asymptotic expansion formula for hypo-analytic pseudodifferential operators is proved and applications are given.
Random Blaschke products
W. George
Cochran
731-755
Abstract: Let $\{ {\theta _n}(\omega )\} $ be a sequence of independent random variables uniformly distributed on $ [0,2\pi ]$, and let ${z_n}(\omega ) = {r_n}{e^{i{\theta _n}(\omega )}}$ for a fixed but arbitrary sequence of radii $ {r_n}$ satisfying the Blaschke condition $\sum {(1 - {r_n}) < \infty }$. We show that the random Blaschke product with zeros ${z_n}(\omega )$ is almost surely not in the little Bloch space, and we describe necessary conditions and sufficient conditions on the radii ${r_n}$ so that $\{ {z_n}(\omega )\}$ is almost surely an interpolating sequence.
Lie algebra modules with finite-dimensional weight spaces. I
S. L.
Fernando
757-781
Abstract: Let $\mathfrak{g}$ denote a reductive Lie algebra over an algebraically closed field of characteristic zero, and let $\mathfrak{h}$ denote a Cartan subalgebra of $\mathfrak{g}$. In this paper we study finitely generated $ \mathfrak{g}$-modules that decompose into direct sums of finite dimensional $\mathfrak{h}$-weight spaces. We show that the classification of irreducible modules in this category can be reduced to the classification of a certain class of irreducible modules, those we call torsion free modules. We also show that if $\mathfrak{g}$ is a simple Lie algebra that admits a torsion free module, then $\mathfrak{g}$ is of type $A$ or $C$.
Tauberian theorems for the Laplace-Stieltjes transform
C. J. K.
Batty
783-804
Abstract: Let $ \alpha :[0,\infty ) \to {\mathbf{C}}$ be a function of locally bounded variation, with $\alpha (0) = 0$, whose Laplace-Stieltjes transform $g(z) = \int_0^\infty {{e^{ - zt}}d\alpha (t)}$ is absolutely convergent for $\operatorname{Re} z > 0$. Let $E$ be the singular set of $ g$ in $i{\mathbf{R}}$, and suppose that $0 \notin E$. Various estimates for $ \lim {\sup _{t \to \infty }}\vert\alpha (t) - g(0)\vert$ are obtained. In particular, $ \alpha (t) \to g(0)$ as $t \to \infty$ if \begin{displaymath}\begin{gathered}({\text{i)}}\quad E\,{\text{is null,}} ... ...vert\alpha (s) - \alpha (t)\vert = 0. \end{gathered} \end{displaymath} This result contains Tauberian theorems for Laplace transforms, power series, and Dirichlet series.
Stability of individual elements under one-parameter semigroups
Charles J. K.
Batty;
Vù Quôc Phóng
805-818
Abstract: Let $\{ T(t):t \geqslant 0\}$ be a $ {C_0}$-semigroup on a Banach space $X$ with generator $A$, and let $x \in X$. If $\sigma (A) \cap i{\mathbf{R}}$ is empty and $t \mapsto T(t)x$ is uniformly continuous, then $ \vert\vert T(t)x\vert\vert \to 0$ as $t \to \infty $. If the semigroup is sun-reflexive, $\sigma (A) \cap i{\mathbf{R}}$ is countable, $ P\sigma (A) \cap i{\mathbf{R}}$ is empty, and $ t \mapsto T(t)x$ is uniformly weakly continuous, then $ T(t)x \to 0$ weakly as $t \to \infty$. Questions of almost periodicity and of stabilization of contraction semigroups on Hilbert space are also discussed.
Mean-square approximation by polynomials on the unit disk
Thomas L.
Kriete;
Barbara D.
MacCluer
1-34
Abstract: We investigate ${P^2}(\mu )$, the closure in ${L^2}(\mu )$ of the complex polynomials, for certain measures $\mu$ on the closed unit disk in the complex plane. Specifically, we present a condition on $\mu$ which guarantees that ${P^2}(\mu )$ decomposes into a natural direct sum.
The fixed point theorem in equivariant cohomology
J. D. S.
Jones;
S. B.
Petrack
35-49
Abstract: In this paper we study the ${S^1}$-equivariant de Rham cohomology of infinite dimensional ${S^1}$-manifolds. Our main example is the free loop space $LX$ where $X$ is a finite dimensional manifold with the circle acting by rotating loops. We construct a new form of equivariant cohomology $h_T^*$ which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces $LX$. This gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of Witten [16], Atiyah [2], and Bismut [5]. In particular we show, following Atiyah in [2], that the $\hat A$-polynomial of $X$ arises as an equivariant characteristic class, in the theory $h_T^*$, of the normal bundle to $X$, considered as the space of constant loops, in $LX$.
Generalized second derivatives of convex functions and saddle functions
R. T.
Rockafellar
51-77
Abstract: The theory of second-order epi-derivatives of extended-real-valued functions is applied to convex functions on ${\mathbb{R}^n}$ and shown to be closely tied to proto-differentiation of the corresponding subgradient multifunctions, as well as to second-order epi-differentiation of conjugate functions. An extension is then made to saddle functions, which by definition are convex in one argument and concave in another. For this case a concept of epi-hypo-differentiability is introduced. The saddle function results provide a foundation for the sensitivity analysis of primal and dual optimal solutions to general finite-dimensional problems in convex optimization, since such solutions are characterized as saddlepoints of a convex-concave Lagrangian function, or equivalently as subgradients of the saddle function conjugate to the Lagrangian.
The crank of partitions mod $8,\;9$ and $10$
Frank G.
Garvan
79-94
Abstract: Recently new combinatorial interpretations of Ramanujan's partition congruences modulo $5$, $7$ and $11$ were found. These were in terms of the crank. A refinement of the congruence modulo $ 5$ is proved. The number of partitions of $5n + 4$ with even crank is congruent to 0 modulo $ 5$. The residue of the even crank modulo $10$ divides these partitions into five equal classes. Other relations for the crank modulo $ 8$, $9$ and $10$ are also proved. The dissections of certain generating functions associated with these results are calculated. All of the results are proved by elementary methods.
Homotopy theory of minimal simplicial spaces
Mark D.
Pritt
95-119
Abstract: Various aspects of homotopy theory in the category of minimal simplicial spaces are studied. It is shown that the usual results of homotopy theory hold in this category, and necessary and sufficient conditions are given under which a simplicial space has the homotopy type of a minimal simplicial space. Continuous cohomology in this category is also studied.
Existence and uniqueness of the Riemann problem for a nonlinear system of conservation laws of mixed type
L.
Hsiao;
P.
de Mottoni
121-158
Abstract: We study the system of conservation laws given by $\displaystyle \left\{ {_{{\upsilon _t} + {{[\upsilon (a + u)]}_x} = 0\quad (a >... ...{\text{a}}{\text{constant}}),}^{{u_t} + {{[u(1 - \upsilon )]}_x} = 0,}} \right.$ with any Riemann initial data $({u_ \mp },{\upsilon _ \mp })$. The system is elliptic in the domain where $ {(\upsilon - u + a - 1)^2} + 4(a - 1)u < 0$ and strictly hyperbolic when $ {(\upsilon - u + a - 1)^2} + 4(a - 1)u > 0$. We combine and generalize Lax criterion and Oleinik-Liu criterion to introduce the generalized entropy condition (G.E.C.) by which we can show that the Riemann problem always has a weak solution (any discontinuity satisfies the G.E.C.) for any initial data, however not necessarily unique. We introduce the minimum principle then in the definition of an admissible weak solution for the Riemann problem and the existence and uniqueness of the solution for any Riemann data.
The diffeotopy group of the twisted $2$-sphere bundle over the circle
M. Ho
Kim;
Frank
Raymond
159-168
Abstract: The diffeotopy group of the nontrivial $2$-sphere bundle over the circle is shown to be isomorphic to ${\mathbb{Z}_2} \oplus {\mathbb{Z}_2}$. The first generator is induced by a reflection across the base circle, while a second generator comes from rotating the $2$-sphere fiber as one travels around the base circle. The technique employed also shows that homotopic diffeomorphisms are diffeotopic.
Rotation and winding numbers for planar polygons and curves
Branko
Grünbaum;
G. C.
Shephard
169-187
Abstract: The winding and rotation numbers for closed plane polygons and curves appear in various contexts. Here alternative definitions are presented, and relations between these characteristics and several other integer-valued functions are investigated. In particular, a point-dependent "tangent number" is defined, and it is shown that the sum of the winding and tangent numbers is independent of the point with respect to which they are taken, and equals the rotation number.
Basic dual homotopy invariants of Riemannian foliations
Peter Y. H.
Pang
189-199
Abstract: In this paper, we use the Sullivan minimal model construction to produce invariants for Riemannian foliations. Existence and vanishing results are proved for these invariants.
Unusually large gaps between consecutive primes
Helmut
Maier;
Carl
Pomerance
201-237
Abstract: Let $G(x)$ denote the largest gap between consecutive primes below $x$. In a series of papers from 1935 to 1963, Erdàs, Rankin, and Schànhage showed that $\displaystyle G(x) \geq (c + o(1)){\operatorname{log}}x{\operatorname{loglog}}x{\operatorname{loglogloglog}}x{({\operatorname{logloglog}}x)^{ - 2}}$ , where $c = {e^\gamma }$ and $\gamma$ is Euler's constant. Here, this result is shown with $c = {c_0}{e^\gamma }$ where ${c_0} = 1.31256 \ldots$ is the solution of the equation $ 4/{c_0} - {e^{ - 4/{c_0}}} = 3$. The principal new tool used is a result of independent interest, namely, a mean value theorem for generalized twin primes lying in a residue class with a large modulus.
$L\sp 1$-approximation with constraints
Allan
Pinkus;
Hans
Strauss
239-261
Abstract: In this paper we study problems of best ${L^1}$-approximation to continuous functions from finite-dimensional subspaces under a variety of constraints. Included are problems of bounded coefficient approximation, approximation with interpolation, restricted range approximation, and restricted range and derivative approximation. Emphasis is placed on problems of uniqueness.
Boundary behavior of the fast diffusion equation
Y. C.
Kwong
263-283
Abstract: The fast diffusion equation $\Delta {\upsilon ^m} = {\upsilon _t}$, $0 < m < 1$, is a degenerate nonlinear parabolic equation of which the existence of a unique continuous weak solution has been established. In this paper we are going to obtain a Lipschitz growth rate of the solution at the boundary of $ \Omega$ and estimate that in terms of the various data.
A class of algebras similar to the enveloping algebra of ${\rm sl}(2)$
S. P.
Smith
285-314
Abstract: Fix $f \in {\mathbf{C}}[X]$. Define $R = {\mathbf{C}}[A,B,H]$ subject to the relations $\displaystyle HA - AH = A,\quad HB - BH = - B,\quad AB - BA = f(H)$ . We study these algebras (for different $ f$) and in particular show how they are similar to (and different from) $U({\text{sl}}(2))$, the enveloping algebra of $ {\text{sl}}(2,{\mathbf{C}})$. There is a notion of highest weight modules and a category $ \mathcal{O}$ for such $ R$. For each $ n > 0$, if $ f(x) = {(x + 1)^{n + 1}} - {x^{n + 1}}$, then $R$ has precisely $n$ simple modules in each finite dimension, and every finite-dimensional $R$-module is semisimple.
Interpretations of Euclidean geometry
S.
Świerczkowski
315-328
Abstract: Following Tarski, we view $n$-dimensional Euclidean geometry as a first-order theory ${E_n}$ with an infinite set of axioms about the relations of betweenness (among points on a line) and equidistance (among pairs of points). We show that for $k < n$, ${E_n}$ does not admit a $k$-dimensional interpretation in the theory RCF of real closed fields, and we deduce that $ {E_n}$ cannot be interpreted $r$-dimensionally in ${E_s}$, when $r \cdot s < n$.
Tiled orders of finite global dimension
Hisaaki
Fujita
329-341
Abstract: We define a projective link between maximal ideals, with respect to which an idealizer preserves being of finite global dimension. Let $D$ be a local Dedekind domain with the quotient ring $K$. We show that for $2 \leq n \leq 5$, every tiled $D$-order of finite global dimension in ${(K)_n}$ is obtained by iterating idealizers w.r.t. projective links from a hereditary order. For $n \geq 6$, we give a tiled $ D$-order in ${(K)_n}$ without this property, which is also a counterexample to Tarsy's conjecture, saying that the maximum finite global dimension of such an order is $n - 1$.
A character-theoretic approach to embeddings of rooted maps in an orientable surface of given genus
D. M.
Jackson;
T. I.
Visentin
343-363
Abstract: The group algebra of the symmetric group and properties of the irreducible characters are used to derive combinatorial properties of embeddings of rooted maps in orientable surfaces of arbitrary genus. In particular, we show that there exists, for each genus, a correspondence between the set of rooted quadrangulations and a set of rooted maps of all lower genera with a distinguished subset of vertices.
Character theory and rooted maps in an orientable surface of given genus: face-colored maps
D. M.
Jackson;
T. I.
Visentin
365-376
Abstract: The character theoretic approach [5] to the enumeration of rooted maps in an orientable surface of arbitrary genus is extended to $2$-face-colorable rooted maps. In particular, we show that there exists, for each genus, a correspondence between the set of $2$-colored triangulations and a set of $2$-colored rooted maps of all lower genera with a distinguished subset of vertices.
On a multiplier conjecture for univalent functions
V.
Gruenberg;
F.
Rønning;
St.
Ruscheweyh
377-393
Abstract: Let $\mathcal{S}$ be the set of normalized univalent functions, and let $ \mathcal{D}$ be the subset of $\mathcal{S}$ containing functions with the property:
Time-delay operators in semiclassical limit. II. Short-range potentials
Xue Ping
Wang
395-416
Abstract: This work is a continuation of [27]. We prove that quantum time-delay operator localized in a nontrapping energy interval is in fact an $h$-pseudodifferential operator with the $ h$-principal symbol given by the classical time-delay function composed with the incoming wave operator in classical mechanics. The classical limit of time-delay operator is also given.
The influence of boundary data on the number of solutions of boundary value problems with jumping nonlinearities
Greg A.
Harris
417-464
Abstract: This paper contains results concerning the number of solutions, as a function of the boundary data, for boundary value problems with jumping nonlinearities. An example seems to indicate that boundary data has a different influence on this number than does forcing data. Through approximating techniques this example leads to lower bounds on solution numbers for the more general case
Cardinal conditions for strong Fubini theorems
Joseph
Shipman
465-481
Abstract: If $ {\kappa _1},{\kappa _2}, \ldots ,{\kappa _n}$ are cardinals with ${\kappa _1}$ the cardinality of a nonmeasurable set, and for $i = 2,3, \ldots ,n$ ${\kappa _i}$ is the cardinality of a set of reals which is not the union of ${\kappa _{i - 1}}$ measure-0 sets, then for any nonnegative function $f:{{\mathbf{R}}^n} \to {\mathbf{R}}$ all of the iterated integrals $\displaystyle {I_\sigma } = \iint \cdots \int {f({x_1},{x_2}, \ldots ,{x_n})d{x_{\sigma (1)}}d{x_{\sigma (2)}} \cdots d{x_{\sigma (n)}},\quad \sigma \in {S_n}}$ , which exist are equal. If all $n!$ of the integrals exist, then the weaker condition of the case $n = 2$ implies they are equal. These cardinal conditions are consistent with and independent of ZFC, and follow from the existence of a real-valued measure on the continuum. Other necessary conditions and sufficient conditions for the existence and equality of iterated integrals are also treated.
Realization of the level one standard $\tilde{C}_{2k+1}$-modules
Kailash C.
Misra
483-504
Abstract: In this paper we study the level one standard (or irreducible integrable highest weight) modules for the affine symplectic Lie algebras. In particular, we give concrete realizations of all level one standard modules for the affine symplectic Lie algebras of even rank.
Large deviations in dynamical systems and stochastic processes
Yuri
Kifer
505-524
Abstract: The paper exhibits a unified approach to large deviations of dynamical systems and stochastic processes based on the existence of a pressure functional and on the uniqueness of equilibrium states for certain dense sets of functions. This enables us to generalize recent results from [OP, Y, and D] on large deviations for dynamical systems, as well, as to recover Donsker-Varadhan's [DV2] large deviation estimates for Markov processes.
The generalized Lusternik-Schnirelmann category of a product space
Mónica
Clapp;
Dieter
Puppe
525-532
Abstract: We continue to study the notions of $ \mathcal{A}$-category and strong $ \mathcal{A}$-category which we introduced in [2]. We give a characterization of them in terms of homotopy colimits and then use it to prove some product theorems in this context.
Zeta functions of formal languages
Jean
Berstel;
Christophe
Reutenauer
533-546
Abstract: Motivated by symbolic dynamics and algebraic geometry over finite fields, we define cyclic languages and the zeta function of a language. The main result is that the zeta function of a cyclic language which is recognizable by a finite automation is rational.
A short proof of principal kinematic formula and extensions
W.
Rother;
M.
Zähle
547-558
Abstract: Federer's extension of the classical principal kinematic formula of integral geometry to sets with positive reach is proved in a direct way by means of generalised unit normal bundles, associated currents, and the coarea theorem. This enables us to extend the relation to more general sets. At the same time we get a short proof for the well-known variants from convex geometry and differential geometry.
Curves on $K$-theory and the de Rham homology of associative algebras
John G.
Ryan
559-582
Abstract: This paper describes the generalization to arbitrary associative algebras of the complex of "typical curves on algebraic $ K$-theory" and shows, in particular, that for certain $ {\mathbf{Q}}$-algebras, $ A$, the complex is isomorphic to the "generalized de Rham complex," $(H{H_*}(A),B)$, in which $B$ is Connes' operator acting on the Hochschild homology groups of $A$.
On the integrals of a singular real analytic differential form in ${\bf R}\sp n$
A.
Meziani
583-594
Abstract: In this paper, we study the constancy on the fibers for the continuous integrals of a complex-valued real analytic differential form in ${R^n}$. Then we prove an isomorphism result between the space of smooth integrals and a space built from spaces of Whitney functions.
On the integrability of singular differential forms in two complex variables
A.
Meziani
595-620
Abstract: In this work we study the integrability of a germ at $0 \in {{\mathbf{C}}^2}$ of a singular differential form for which the closure of the integral curves are analytic varieties that pass through 0. The focii of this paper are the existence of pure meromorphic integrals, the linearization and the nonexistence of a topological criterion for transcendental integrability.
Classifying sets of measure zero with respect to their open covers
Winfried
Just;
Claude
Laflamme
621-645
Abstract: Developing ideas of Borel and Fréchet, we define a partial preorder which classifies measure zero sets of reals according to their open covers and study the induced partial order on the equivalence classes. The more "rarefied" a set of measure zero, the higher it will range in our partial order. Main results: The sets of strong measure zero form one equivalence class that is the maximum element of our order. There is a second highest class that contains all uncountable closed sets of measure zero. There is a minimum class that contains all dense ${G_\delta }$-subsets of the real line of measure zero. There exist at least four classes, and if Martin's axiom holds, then there are as many classes as subsets of the real line. It is also consistent with ZFC that there is a second lowest class.
A topological persistence theorem for normally hyperbolic manifolds via the Conley index
Andreas
Floer
647-657
Abstract: We prove that the cohomology ring of a normally hyperbolic manifold of a diffeomorphism $f$ persists under perturbation of $ f$. We do not make any quantitative assumptions on the expansion and contraction rates of $Df$ on the normal and the tangent bundles of $ N$.
On solvable groups of finite Morley rank
Ali
Nesin
659-690
Abstract: We investigate solvable groups of finite Morley rank. We find conditions on $ G$ for $G'$ to split in $G$. In particular, if $G'$ is abelian and $Z(G) = 1$ we prove that
Families of rational surfaces preserving a cusp singularity
Lee J.
McEwan
691-716
Abstract: Families of rational surfaces containing resolutions of cusp singularities are explicitly constructed. It is proved that the families constructed are universal deformations at each point. Two different monodromy formulas are established; one of these is shown to be connected to automorphisms of Inoue-Hirzebruch surfaces. Some evidence (but no proof) is offered for the conjecture that finite base changes of the families we construct are the versal-deformation spaces for singular Inoue-Hirzebruch surfaces.
Local orders whose lattices are direct sums of ideals
Jeremy
Haefner
717-740
Abstract: Let $R$ be a complete local Dedekind domain with quotient field $K$ and let $\Lambda$ be a local $R$-order in a separable $K$-algebra. This paper classifies those orders $ \Lambda$ such that every indecomposable $R$-torsionfree $\Lambda$-module is isomorphic to an ideal of $ \Lambda$. These results extend to the noncommutative case some results for commutative rings found jointly by this author and L. Levy.
A spanning set for ${\scr C}(I\sp n)$
Thomas
Bloom
741-759
Abstract: $\mathcal{C}({I^n})$ denotes the Banach space of continuous functions on the unit $n$-cube, ${I^n}$, in $ {{\mathbf{R}}^n}$. Let $\{ {a^i}\}$, $ i = 0,1,2, \ldots ,$, be a countable collection of $n$-tuples of positive real numbers satisfying $ {\operatorname{lim}_i}a_j^i = + \infty$ for $ j = 1, \ldots ,n$. We canonically enlarge the family of monomials $\{ {x^{{a^i}}}\}$ to a family of functions $\mathcal{F}(A)$. Conjecture. The linear span of $ \mathcal{F}(A)$ is dense in $ \mathcal{C}({I^n})$ if and only if $\Sigma _{i = 0}^\infty 1/\left\vert {{a^i}} \right\vert = + \infty$. For $n = 1$ this is equivalent to the Müntz-Szasz theorem. For $n > 1$ we prove the necessity in general and the sufficiency under the additional hypothesis that there exist constants $G$, $N > 1$ such that $\left\vert {{a^i}} \right\vert \leq G{\operatorname{exp}}({i^N})$ for all $i$.
Some global results on extension of CR-objects in complex manifolds
Guido
Lupacciolu
761-774
Abstract: We prove some results concerning the holomorphic extendability of CR-objects defined on real hypersurfaces of a complex manifold. After a preliminary generalization of the classic theorem on the extendability from the boundary of a relatively compact domain, we discuss the extendability from a part of the boundary of such a domain, the one side extendability from a piece of hypersurface and the extendability from the boundary of an unbounded domain.
Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type
Nicola
Garofalo;
Ermanno
Lanconelli
775-792
Abstract: In this paper we establish a uniform Harnack inequality for a class of degenerate equations whose prototype is Kolmogorov's equations in $ {{\mathbf{R}}^3}:{D_{{\text{yy}}}}u - {\text{y}}{D_z}u - {D_t}u = 0$. Our approach is based on mean value formulas for solutions of the equation under consideration on the level sets of the fundamental solution.
Hyperbolicity properties of $C\sp 2$ multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions
Tomasz
Nowicki;
Sebastian
van Strien
793-810
Abstract: In this paper we study the dynamical properties of general ${C^2}$ maps $f:[0,1] \to [0,1]$ with quadratic critical points (and not necessarily unimodal). We will show that if such maps satisfy the well-known Collet-Eckmann conditions then one has (a) hyperbolicity on the set of periodic points; (b) nonexistence of wandering intervals; (c) sensitivity on initial conditions; and (d) exponential decay of branches (intervals of monotonicity) of ${f^n}$ as $ n \to \infty ;$ For these results we will not make any assumptions on the Schwarzian derivative $f$. We will also give an estimate of the return-time of points that start near critical points.
Existence and uniqueness of algebraic curvature tensors with prescribed properties and an application to the sphere theorem
Walter
Seaman
811-823
Abstract: An existence and uniqueness theorem is proved for algebraic curvature tensors and then applied to yield a global geometric theorem for locally weakly quarter pinched Riemannian manifolds whose second Betti number is nonzero.
Abstract functional-differential equations and reaction-diffusion systems
R. H.
Martin;
H. L.
Smith
1-44
Abstract: Several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a Banach space setting. The ideas are applied to reaction-diffusion systems that have time delays in the nonlinear reaction terms. The techniques presented here include differential inequalities, invariant sets, and Lyapunov functions, and therefore they provide for a wide range of applicability. The results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays.
The $27$-dimensional module for $E\sb 6$. III
Michael
Aschbacher
45-84
Abstract: This is the third in a series of five papers investigating the subgroup structure of the universal Chevalley group $G = {E_6}(F)$ of type ${E_6}$ over a field $F$ and the geometry induced on the $ 27$-dimensional $ FG$-module $V$ by the symmetric trilinear form $ f$ preserved by $ G$. The series uses the geometry on $V$ to describe and enumerate (up to a small list of ambiguities) all closed maximal subgroups of $ G$ when $F$ is finite or algebraically closed.
On Gel\cprime fand pairs associated with solvable Lie groups
Chal
Benson;
Joe
Jenkins;
Gail
Ratcliff
85-116
Abstract: Let $G$ be a locally compact group, and let $K$ be a compact subgroup of ${\operatorname{Aut}}(G)$, the group of automorphisms of $G$. There is a natural action of $K$ on the convolution algebra $ {L^1}(G)$, and we denote by $L_K^1(G)$ the subalgebra of those elements in $ {L^1}(G)$ that are invariant under this action. The pair $(K,G)$ is called a Gelfand pair if $ L_K^1(G)$ is commutative. In this paper we consider the case where $ G$ is a connected, simply connected solvable Lie group and $K \subseteq {\operatorname{Aut}}(G)$ is a compact, connected group. We characterize such Gelfand pairs $(K,G)$, and determine a moduli space for the associated $K$-spherical functions.
On the existence of central sequences in subfactors
Dietmar H.
Bisch
117-128
Abstract: We prove a relative version of [Co1, Theorem 2.1] for a pair of type $ {\text{I}}{{\text{I}}_1}$-factors $N \subset M$. This gives a list of necessary and sufficient conditions for the existence of nontrivial central sequences of $M$ contained in the subfactor $N$. As an immediate application we obtain a result by Bédos [Be, Theorem A], showing that if $N$ has property $\Gamma$ and $G$ is an amenable group acting freely on $ N$ via some action $ \sigma$, then the crossed product $ N{ \times _\sigma }G$ has property $\Gamma$. We also include a proof of a relative Mc Duff-type theorem (see [McD, Theorems $ 1$, $2$ and $3$]), which gives necessary and sufficient conditions implying that the pair $N \subset M$ is stable.
On the bihomogeneity problem of Knaster
Krystyna
Kuperberg
129-143
Abstract: The author constructs a locally connected, homogeneous, finitedimensional, compact, metric space which is not bihomogeneous, thus providing a compact counterexample to a problem posed by B. Knaster around 1921.
Iterated spinning and homology spheres
Alexander I.
Suciu
145-157
Abstract: Given a closed $ n$-manifold ${M^n}$ and a tuple of positive integers $ P$, let ${\sigma _P}M$ be the $P$-spin of $M$. If $ {M^n} \not\backsimeq{S^n}$ and $P \ne Q$ (as unordered tuples), it is shown that ${\sigma _P}M\not\backsimeq{\sigma _Q}M$ if either (1) $ {H_*}({M^n})\not\cong{H_*}({S^n})$, (2)${\pi _1}M$ finite, (3) $M$ aspherical, or (4) $n = 3$. Applications to the homotopy classification of homology spheres and knot exteriors are given.
Identities on quadratic Gauss sums
Paul
Gérardin;
Wen-Ch’ing Winnie
Li
159-182
Abstract: Given a local field $ F$, each multiplicative character $\theta$ of the split algebra $F \times F$ or of a separable quadratic extension of $F$ has an associated generalized Gauss sum $\gamma _\theta ^F$. It is a complex valued function on the character group of ${F^ \times } \times F$, meromorphic in the first variable. We define a pairing between such Gauss sums and study its properties when $F$ is a nonarchimedean local field. This has important applications to the representation theory of $ GL(2,F)$ and correspondences $ [{\text{GL}}3]$.
The Jacobian module of a Lie algebra
J. P.
Brennan;
M. V.
Pinto;
W. V.
Vasconcelos
183-196
Abstract: There is a natural way to associate to the commuting variety $ C(A)$ of an algebra $ A$ a module over a polynomial ring. It serves as a vehicle to study the arithmetical properties of $C(A)$, particularly Cohen-Macaulayness. The focus here is on Lie algebras and some of their representations.
Dichromatic link invariants
Jim
Hoste;
Mark E.
Kidwell
197-229
Abstract: We investigate the skein theory of oriented dichromatic links in $ {S^3}$. We define a new chromatic skein invariant for a special class of dichromatic links. This invariant generalizes both the two-variable Alexander polynomial and the twisted Alexander polynomial. Alternatively, one may view this new invariant as an invariant of oriented monochromatic links in ${S^1} \times {D^2}$, and as such it is the exact analog of the twisted Alexander polynomial. We discuss basic properties of this new invariant and applications to link interchangeability. For the full class of dichromatic links we show that there does not exist a chromatic skein invariant which is a mutual extension of both the two-variable Alexander polynomial and the twisted Alexander polynomial.
Composite ribbon number one knots have two-bridge summands
Steven A.
Bleiler;
Mario
Eudave Muñoz
231-243
Abstract: A composite ribbon knot which can be sliced with a single band move has a two-bridge summand.
Gel\cprime fer functions, integral means, bounded mean oscillation, and univalency
Shinji
Yamashita
245-259
Abstract: A Gelfer function $ f$ is a holomorphic function in $ D = \{ \left\vert z \right\vert < 1\}$ such that $f(0) = 1$ and $ f(z) \ne - f(w)$ for all $ z$, $w$ in $D$. The family $G$ of Gelfer functions contains the family $ P$ of holomorphic functions $f$ in $D$ with $f(0) = 1$ and Re $f > 0$ in $D$. If $f$ is holomorphic in $D$ and if the ${L^2}$ mean of $f'$ on the circle $\{ \left\vert z \right\vert = r\}$ is dominated by that of a function of $G$ as $r \to 1 - 0$, then $f \in BMOA$. This has two recent and seemingly different results as corollaries. A core of the proof is the fact that ${\operatorname{log}}f \in BMOA$ if $f \in G$. Besides the properties obtained concerning $f \in G$ itself, we shall investigate some families of functions where the roles played by $ P$ in Univalent Function Theory are replaced by those of $G$. Some exact estimates are obtained.
Nonmonomial characters and Artin's conjecture
Richard
Foote
261-272
Abstract: If $E/F$ is a Galois extension of number fields with solvable Galois group $G$, the main result of this paper proves that if the Dedekind zeta-function of $ E$ has a zero of order less than $ {\mathcal{M}_G}$ at the complex point ${s_0} \ne 1$, then all Artin $ L$-series for $ G$ are holomorphic at $ {s_0}$ -- here ${\mathcal{M}_G}$ is the smallest degree of a nonmonomial character of any subgroup of $G$. The proof relies only on certain properties of $L$-functions which are axiomatized to give a purely character-theoretic statement of this result.
Isometric isomorphisms between Banach algebras related to locally compact groups
F.
Ghahramani;
A. T.
Lau;
V.
Losert
273-283
Abstract: Let ${G_1}$, ${G_2}$ be locally compact groups. We prove in this paper that if $T$ is an isometric isomorphism from the Banach algebra ${\text{LUC}}{({G_1})^*}$ (the continuous dual of the Banach space of left uniformly continuous functions on ${G_1}$, equipped with Arens multiplication) onto $ {\text{LUC}}{({G_2})^*}$, then $T$ maps $M({G_1})$ onto $M({G_2})$ and $ {L^1}({G_1})$ onto ${L^1}({G_2})$. We also prove that any isometric isomorphism from $ {L^1}{({G_1})^{**}}$ (second conjugate algebra of $ {L^1}({G_1})$) onto $ {L^1}{({G_2})^{**}}$ maps ${L^1}({G_1})$ onto $ {L^1}({G_2})$.
Maximal polynomials and the Ilieff-Sendov conjecture
Michael J.
Miller
285-303
Abstract: In this paper, we consider those complex polynomials which have all their roots in the unit disk, one fixed root, and all the roots of their first derivatives as far as possible from a fixed point. We conjecture that any such polynomial has all the roots of its derivative on a circle centered at the fixed point, and as many of its own roots as possible on the unit circle. We prove a part of this conjecture, and use it to define an algorithm for constructing some of these polynomials. With this algorithm, we investigate the 1962 conjecture of Sendov and the 1969 conjecture of Goodman, Rahman and Ratti and (independently) Schmeisser, obtaining counterexamples of degrees $6$, $8$, $10$, and $12$ for the latter.
Complex interpolation for normed and quasi-normed spaces in several dimensions. III. Regularity results for harmonic interpolation
Zbigniew
Slodkowski
305-332
Abstract: The paper continues the study of one of the complex interpolation methods for families of finite-dimensional normed spaces ${\{ {{\mathbf{C}}^n},\vert\vert \cdot \vert{\vert _z}\} _{z \in G}}$, where $ G$ is open and bounded in $ {{\mathbf{C}}^k}$. The main result asserts that (under a mild assumption on the datum) the norm function $(z,w) \to \vert\vert w\vert\vert _z^2$ belongs to some anisotropic Sobolew class and is characterized by a nonlinear PDE of second order. The proof uses the duality theorem for the harmonic interpolation method (obtained earlier by the author). A new, simpler proof of this duality relation is also presented in the paper.
Weakly almost periodic functions and thin sets in discrete groups
Ching
Chou
333-346
Abstract: A subset $ E$ of an infinite discrete group $G$ is called (i) an ${R_W}$-set if any bounded function on $G$ supported by $E$ is weakly almost periodic, (ii) a weak $ p$-Sidon set $(1 \leq p < 2)$ if on ${l^1}(E)$ the ${l^p}$-norm is bounded by a constant times the maximal ${C^*}$-norm of ${l^1}(G)$, (iii) a $T$-set if $xE \cap E$ and $Ex \cap E$ are finite whenever $x \ne e$, and (iv) an $FT$-set if it is a finite union of $ T$-sets. In this paper, we study relationships among these four classes of thin sets. We show, among other results, that (a) every infinite group $G$ contains an ${R_W}$-set which is not an $FT$-set; (b) countable weak $p$-Sidon sets, $1 \leq p < 4/3$ are $FT$-sets.
Continuous spatial semigroups of $*$-endomorphisms of ${\germ B}({\germ H})$
Robert T.
Powers;
Geoffrey
Price
347-361
Abstract: To each continuous semigroup of $*$-endomorphisms $\alpha$ of $\mathfrak{B}\left( \mathfrak{H} \right)$ with an intertwining semigroup of isometries there is associated a $*$-representation $\pi$ of the domain $\mathfrak{O}(\delta )$ of the generator of $ \alpha$. It is shown that the Arveson index $ {d_ * }(\alpha )$ is the number of times the representation $ \pi$ contains the identity representation of $\mathfrak{O}(\delta )$. This result is obtained from an analysis of the relation between two semigroups of isometries, $U$ and $S$, satisfying the condition $S{(t)^*}U(t) = {e^{ - \lambda t}}I$ for $ t \geq 0$ and $\lambda > 0$.
Jacobi polynomials as generalized Faber polynomials
Ahmed I.
Zayed
363-378
Abstract: Let ${\mathbf{B}}$ be an open bounded subset of the complex $z$-plane with closure $\overline {\mathbf{B}}$ whose complement ${\overline {\mathbf{B}} ^c}$ is a simply connected domain on the Riemann sphere. $z = \psi (w)$ map the domain $\left\vert w \right\vert > \rho \quad (\rho > 0)$ one-to-one conformally onto the domain $ {\overline {\mathbf{B}} ^c}$ such that $ \psi (\infty ) = \infty$. Let $R(w) = \sum\nolimits_{n = 0}^\infty {{c_n}{w^{ - n}}}$, ${c_0} \ne 0$ be analytic in the domain $\left\vert w \right\vert > \rho$ with $R(w) \ne 0$. Let $ F(z) = \sum\nolimits_{n = 0}^\infty {{b_n}} {z^n}$, $F*(z) = \sum\nolimits_{n = 0}^\infty {\frac{1} {{{b_n}}}} {z^n}$ be analytic in $\left\vert z \right\vert < 1$ and analytically continuable to any point outside $\left\vert z \right\vert < 1$ along any path not passing through the points $z = 0,1,\infty$. The generalized Faber polynomials $\{ {P_n}(z)\} _{n = 0}^\infty$ of ${\mathbf{B}}$ are defined by $ \{ P_n^{(\alpha ,\beta )}(z)\} _{n = 0}^\infty$ are generalized Faber polynomials of any region $ {\mathbf{B}}$, then it must be the elliptic region $ \{ z:\vert z + 1\vert + \vert z - 1\vert < \rho + \frac{1}{\rho },\rho > 1\} ;$ (2) the only Jacobi polynomials that can be classified as generalized Faber polynomials are the Tchebycheff polynomials of the first kind, some normalized Gegenbauer polynomials, some normalized Jacobi polynomials of type $\{ P_n^{(\alpha ,\alpha + 1)}(z)\} _{n = 0}^\infty$, $\{ P_n^{(\beta + 1,\beta )}(z)\} _{n = 0}^\infty$ and there are no others, no matter how one normalizes them; (3) the Hermite and Laguerre polynomials cannot be generalized Faber polynomials of any region.
The nonstandard treatment of Hilbert's fifth problem
Joram
Hirschfeld
379-400
Abstract: We give a nonstandard proof that every locally Euclidean group is a Lie group. The heart of the proof is a strong nonstandard variant of Gleason's lemma for a class of groups that includes all locally Euclidean groups.
Generalized local Fatou theorems and area integrals
B. A.
Mair;
Stan
Philipp;
David
Singman
401-413
Abstract: Let $X$ be a space of homogeneous type and $ W$ a subset of $X \times (0,\infty )$. Then, under minimal conditions on $W$, we obtain a relationship between two modes of convergence at the boundary $X$ for functions defined on $ W$. This result gives new local Fatou theorems of the Carleson-type for solutions of Laplace, parabolic and Laplace-Beltrami equations as immediate consequences of the classical results. Lusin area integral characterizations for the existence of limits within these more general approach regions are also obtained.
The structure of Legendre foliations
Myung-Yull
Pang
417-455
Abstract: The local and global structure of Legendre foliations of contact manifolds is analysed. The main invariant of a Legendre foliation is shown to be a quadratic form on the tangent bundle to the foliation--the fundamental quadratic form. The equivalence problem is solved in the case when the fundamental quadratic form is nondegenerate and a generalization of Chern's solution to the equivalence problem for Finsler manifolds is obtained. A normal form for Legendre foliations is given which is closely related to Weinstein's structure theorem for Lagrangian foliations. It is shown that every compact, simply connected leaf of a Legendre foliation is diffeomorphic to a sphere.
Second order theta functions and vector bundles over Jacobi varieties
David S.
Yuen
457-492
Abstract: We consider the Picard vector bundles defined over Jacobi varieties. The rank $g + 1$ Picard bundle imbeds in the rank $ {2^g}$ Clifford bundle, so the second order theta functions, viewed appropriately, span the dual of the Picard bundle over each fiber. We prove a result on the minimum number of such second order theta functions required to span the whole bundle at each point. We give an application of using these functions to describe subvarieties of the Jacobian. There follow comments on which functions we could use, and generalizations to higher order theta functions.
Polynomial flows on ${\bf C}\sp n$
Brian A.
Coomes
493-506
Abstract: We show that polynomial flows on $ {\mathbb{R}^n}$ extend to functions holomorphic on ${\mathbb{C}^{n + 1}}$ and that the group property holds after this extension. Then we give some methods, based on power series, for determining when a vector field has a polynomial flow.
Smooth great circle fibrations and an application to the topological Blaschke conjecture
C. T.
Yang
507-524
Abstract: We study great smooth circle fibrations of round spheres and Blaschke manifolds of the homotopy type of complex projective spaces.
Matroid representations and free arrangements
Günter M.
Ziegler
525-541
Abstract: We show that Terao's Conjecture ("Freeness of the module of logarithmic forms at a hyperplane arrangement is determined by its abstract matroid") holds over fields with at most four elements. However, an example demonstrates that the field characteristic has to be fixed for this.
Leavable gambling problems with unbounded utilities
A.
Maitra;
R.
Purves;
W.
Sudderth
543-567
Abstract: The optimal return function $U$ of a Borel measurable gambling problem with a positive utility function is known to be universally measurable. With a negative utility function, however, $ U$ may not be so measurable. As shown here, the measurability of $ U$ for all Borel gambling problems with negative utility functions is equivalent to the measurability of all PCA sets, a property of such sets known to be independent of the usual axioms of set theory. If the utility function is further required to satisfy certain uniform integrability conditions, or if the gambling problem corresponds to an optimal stopping problem, the optimal return function is measurable. Another return function $W$ is introduced as an alternative to $U$. It is shown that $W$ is always measurable and coincides with $ U$ when the utility function is positive.
Boundedness versus periodicity over commutative local rings
Vesselin N.
Gasharov;
Irena V.
Peeva
569-580
Abstract: Over commutative graded local artinian rings, examples are constructed of periodic modules of arbitrary minimal period and modules with bounded Betti numbers, which are not eventually periodic. They provide counterexamples to a conjecture of D. Eisenbud, that every module with bounded Betti numbers over a commutative local ring is eventually periodic of period $2$. It is proved however, that the conjecture holds over rings of small length.
Semialgebraic expansions of ${\bf C}$
David
Marker
581-592
Abstract: We prove no nontrivial expansion of the field of complex numbers can be obtained from a reduct of the field of real numbers.
The Dirichlet problem for radially homogeneous elliptic operators
Richard F.
Bass
593-614
Abstract: The Dirichlet problem in the unit ball is considered for the strictly elliptic operator $L = \sum {{a_{ij}}{D_{ij}}} $, where the $ {a_{ij}}$, are smooth away from the origin and radially homogeneous: ${a_{ij}}(rx) = {a_{ij}}(x),\;r > 0,\;x \ne 0$. Existence and uniqueness are proved for solutions in a certain space of functions. Necessary and sufficient conditions are given for an extended maximum principle to hold.
Endomorphism rings of formal $A\sb 0$-modules
Shuji
Yamagata
615-623
Abstract: Let ${A_0}$ be the valuation ring of a finite extension ${K_0}$ of ${Q_p}$ and $ A \supset {A_0}$ be a complete discrete valuation ring with the perfect residue field. We consider the endomorphism rings of $ n$-dimensional formal $ {A_0}$-modules $ \Gamma$ over $ A$ of finite $ {A_0}$-height with reduction absolutely simple up to isogeny. Especially we prove commutativity of ${\operatorname{End} _{A,{A_0}}}(\Gamma )$. Given an arbitrary finite unramified extension $ {K_1}$ of ${K_0}$, a variety of examples (different dimensions and different ${A_0}$-heights) is constructed whose absolute endomorphism rings are isomorphic to the valuation ring of ${K_1}$.
On a theorem of Stein
Steven G.
Krantz
625-642
Abstract: In this paper the Kobayashi metric on a domain in ${{\mathbf{C}}^n}$ is used to define a new function space. Elements of this space belong to a nonisotropic Lipschitz class. It is proved that if $f$ is holomorphic on the domain and in the classical Lipschitz space ${\Lambda _\alpha }$ then in fact $f$ is in the new function space. The result contains the original result of Stein on this subject and provides the optimal result adapted to any domain. In particular, it recovers the Hartogs extension phenomenon in the category of Lipschitz spaces.
Butler groups of infinite rank. II
Manfred
Dugas;
Paul
Hill;
K. M.
Rangaswamy
643-664
Abstract: A torsion-free abelian group $G$ is called a Butler group if $\operatorname{Bext} (G,T) = 0.$ for any torsion group $ T$. We show that every Butler group $G$ of cardinality $ {\aleph _1}$ is a $ {B_2}$-group; i.e., $ G$ is a union of a smooth ascending chain of pure subgroups ${G_\alpha }$ where $ {G_{\alpha + 1}} = {G_\alpha } + {B_\alpha },{B_\alpha }$ a Butler group of finite rank. Assuming the validity of the continuum hypothesis (CH), we show that every Butler group of cardinality not exceeding $ {\aleph _\omega }$ is a $ {B_2}$-group. Moreover, we are able to prove that the derived functor $ {\operatorname{Bext} ^2}(A,T) = 0$ for any torsion group $T$ and any torsion-free $A$ with $\vert A\vert \leqslant {\aleph _\omega }$. This implies that under CH all balanced subgroups of completely decomposable groups of cardinality $\leqslant {\aleph _\omega }$ are $ {B_2}$-groups.
Symmetry properties of the solutions to Thomas-Fermi-Dirac-von Weizs\"acker type equations
Rafael D.
Benguria;
Cecilia
Yarur
665-675
Abstract: We consider a semilinear elliptic equation with a spherically symmetric potential (specifically, Thomas-Fermi-Dirac-von Weizsäcker type equations without electronic repulsion). Assuming some regularity properties of the solutions at the origin and at infinity, we prove that the solutions have spherical symmetry.
Comparison of commuting one-parameter groups of isometries
Ola
Bratteli;
Hideki
Kurose;
Derek W.
Robinson
677-694
Abstract: Let $\alpha ,\;\beta$ be two commuting strongly continuous one-parameter groups of isometries on a Banach space $\mathcal{A}$ with generators ${\delta _\alpha }$ and ${\delta _\beta }$, and analytic elements $\mathcal{A}_\omega ^\alpha ,\;\mathcal{A}_\omega ^\beta$, respectively. Then it is easy to show that if ${\delta _\alpha }$ is relatively bounded by ${\delta _\beta }$, then $\mathcal{A}_\omega ^\beta \subseteq \mathcal{A}_\omega ^\alpha$, and in this paper we establish the inverse implication for unitary one-parameter groups on Hilbert spaces and for one-parameter groups of $ ^{\ast}$-automorphisms of abelian $ {C^{\ast}}$-algebras. It is not known in general whether the inverse implication holds or not, but it does not hold for one-parameter semigroups of contractions.
On the sparsity of representations of rings of pure global dimension zero
Birge
Zimmermann-Huisgen;
Wolfgang
Zimmermann
695-711
Abstract: It is shown that the rings $R$ all of whose left modules are direct sums of finitely generated modules satisfy the following finiteness condition: For each positive integer $ n$ there are only finitely many isomorphism types of (a) indecomposable left $ R$-modules of length $ n$; (b) finitely presented indecomposable right $R$-modules of length $n$; (c) indecomposable right $R$-modules having minimal projective resolutions with $n$ relations. Moreover, our techniques yield a very elementary proof for the fact that the presence of the above decomposability hypothesis for both left and right $R$-modules entails finite representation type.
The orderability and closed images of scattered spaces
S.
Purisch
713-725
Abstract: A (totally) orderable scattered space and a space homeomorphic to a subspace of an ordinal space are characterized in terms of a neighborhood subbase for each of their points plus what corresponds to a neighborhood base for each of their non-$Q$-gaps. These generalize the characterizations in [ P$_{1}$] of an orderable compact scattered space and in [B] of a space homeomorphic to a compact ordinal space. Generalizing a result in [M] it is shown that a space is orderable and scattered iff it is the $ 2$ to $1$ image under a closed map of a subspace of an ordinal space. In response to a question of Telgarsky [T] a simple description is given of a closed map with discrete fibers from an orderable scattered space onto an orderable perfect space. Maps that preserve length conditions on a scattered space are touched upon.
Maximal functions on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions
Miguel A.
Ariño;
Benjamin
Muckenhoupt
727-735
Abstract: A characterization is given of a class of classical Lorentz spaces on which the Hardy Littlewood maximal operator is bounded. This is done by determining the weights for which Hardy's inequality holds for nonincreasing functions. An alternate characterization, valid for nondecreasing weights, is also derived.
A new proof of the strong partition relation on $\omega\sb 1$
Steve
Jackson
737-745
Abstract: Assuming the axiom of determinacy, we give a new proof of the strong partition relation on $ {\omega _1}$. The proof is direct and avoids appeal to complicated set-theoretic machinery.
A canonical extension for analytic functions on Banach spaces
Ignacio
Zalduendo
747-763
Abstract: Given Banach spaces $ E$ and $F$, a Banach space ${G_{EF}}$ is presented in which $ E$ is embedded and which seems a natural space to which extend $F$-valued analytic functions. Any $ F$-valued analytic function defined on a subset $U$ of $E$ may be extended to an open neighborhood of $ U$ in ${G_{EF}}$. This extension generalizes that of Aron and Berner. It is also related to the Arens product in Banach algebras, to the functional calculus for bounded linear operators, and to an old problem of duality in spaces of analytic functions. A characterization of the Aron-Berner extension is given in terms of continuity properties of first-order differentials.
Parametrization of domains in $\hat{\bf C}$: the logarithmic domains
Johannes
Michaliček;
Rodolfo
Wehrhahn
765-777
Abstract: We prove a generalization of Riemann's mapping theorem: Every $ n$-fold connected domain in $ \widehat{\mathbf{C}}$, whose boundary does not contain isolated points, is conformal equivalent to a logarithmic domain. The logarithmic domains are characterized by a Green's function consisting of a finite sum of logarithms.
Discontinuous ``viscosity'' solutions of a degenerate parabolic equation
Michiel
Bertsch;
Roberta
Dal Passo;
Maura
Ughi
779-798
Abstract: We study a nonlinear degenerate parabolic equation of the second order. Regularizing the equation by adding some artificial viscosity, we construct a generalized solution. We show that this solution is not necessarily continuous at all points.
Bounds for projection constants and $1$-summing norms
Hermann
König;
Nicole
Tomczak-Jaegermann
799-823
Abstract: It is shown that projection constants $ \lambda ({X_n})$ of $ n$-dimensional normed spaces ${X_n}$ satisfy $\lambda ({X_n}) \leqslant \sqrt n - c/\sqrt n$ where $c > 0$ is a numerical constant. Similarly, the $ 1$-summing norms of (the identity of) ${X_n}$ can be estimated by $ {\pi _1}({X_n}) \geqslant \sqrt n + c/\sqrt n$. These estimates are the best possible: for prime $n$, translation-invariant $ n$-dimensional spaces $ {X_n}$ such that $ \lambda ({X_n}) \geqslant \sqrt n - 2/\sqrt n$ and ${\pi _1}({X_n}) \leqslant \sqrt n + 2/\sqrt n$ can be constructed. For these spaces Gordon-Lewis constants and distances to Hilbert spaces are large as well: $\operatorname{gl} ({X_n}) \geqslant \tfrac{1} {3}\sqrt n ,d({X_n},l_2^n) = \sqrt n $.
The initial-Neumann problem for the heat equation in Lipschitz cylinders
Russell M.
Brown
1-52
Abstract: We prove existence and uniqueness for solutions of the initial-Neumann problem for the heat equation in Lipschitz cylinders when the lateral data is in ${L^p}$, $1 < p < 2+\varepsilon$, with respect to surface measure. For convenience, we assume that the initial data is zero. Estimates are given for the parabolic maximal function of the spatial gradient. An endpoint result is established when the data lies in the atomic Hardy space ${H^1}$. Similar results are obtained for the initial-Dirichlet problem when the data lies in a space of potentials having one spatial derivative and half of a time derivative in ${L^p}$, $1 < p < 2+\varepsilon$, with a corresponding Hardy space result when $p = 1$. Using these results, we show that our solutions may be represented as single-layer heat potentials. By duality, it follows that solutions of the initial-Dirichlet problem with data in $ {L^q}$,
Layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains
Jeff E.
Lewis
53-76
Abstract: The symbolic calculus of pseudodifferential operators of Mellin type is applied to study layer potentials on a plane domain ${\Omega ^ + }$ whose boundary ${\partial\Omega ^ + }$ is a curvilinear polygon. A "singularity type" is a zero of the determinant of the matrix of symbols of the Mellin operators and can be used to calculate the "bad values" of $p$ for which the system is not Fredholm on ${L^p}(\partial {\Omega ^ + })$. Using the method of layer potentials we study the singularity types of the system of elastostatics $\displaystyle L{\mathbf{u}} = \mu \Delta {\mathbf{u}} + (\lambda + \mu )\nabla \operatorname{div} {\mathbf{u}} = 0.$ in a plane domain ${\Omega ^ + }$ whose boundary ${\partial\Omega ^ + }$ is a curvilinear polygon. Here $ \mu > 0$ and $-\mu \le \lambda \le +\infty$. When $\lambda = +\infty$, the system is the Stokes system of hydrostatics. For the traction double layer potential, we show that all singularity types in the strip $0 < \operatorname{Re} z < 1$ lie in the interval $\left( {\frac{1} {2},1} \right)$ so that the system of integral equations is a Fredholm operator of index 0 on ${L^p}(\partial {\Omega ^ + })$ for all $p$, $ 2 \le p < \infty$. The explicit dependence of the singularity types on $ \lambda$ and the interior angles $\theta$ of $ {\partial\Omega ^ + }$ is calculated; the singularity type of each corner is independent of $\lambda$ iff the corner is nonconvex.
A continuous localization and completion
Norio
Iwase
77-90
Abstract: The main goal of this paper is to construct a localization and completion of Bousfield-Kan type as a continuous functor for a virtually nilpotent CW-complex. Then the localization and completion of an ${A_n}$-space is given to be an $ {A_n}$-homomorphism between ${A_n}$-spaces. For any general compact Lie group, this gives a continuous equivariant localization and completion for a virtually nilpotent $ G$-CW-complex. More generally, we have a continuous localization with respect to a system of core rings for a virtually nilpotent $\mathbf{D}$-CW-complex for a polyhedral category $\mathbf{D}$.
$L\sp p$ inequalities for entire functions of exponential type
Qazi I.
Rahman;
G.
Schmeisser
91-103
Abstract: Let $f$ be an entire function of exponential type $\tau$ belonging to ${L^p}$ on the real line. It has been known since a long time that
Classification of crossed-product $C\sp *$-algebras associated with characters on free groups
Hong Sheng
Yin
105-143
Abstract: We study the classification problem of crossed-product $ {C^ * }$-algebras of the form $C_r^ * (G){ \times _{{\alpha _\chi }}}{\mathbf{Z}}$, where $G$ is a discrete group, $\chi$ is a one-dimensional character of $ G$, and ${\alpha_\chi}$ is the unique $ *$-automorphism of $C_r^ * (G)$ such that if $U$ is the left regular representation of $ G$, then $ {\alpha_{\chi}(U_{g})=\chi(g)U_{g}}$, $g \in G$. When $ {G = F_{n}}$, the free group on $n$ generators, we have a complete classification of these crossed products up to $*$-isomorphism which generalizes the classification of irrational and rational rotation $ {C^ * }$-algebras. We show that these crossed products are determined by two $ K$-theoretic invariants, that these two invariants correspond to two orbit invariants of the characters under the natural $ \operatorname{Aut} ({F_n})$-action, and that these two orbit invariants completely classify the characters up to automorphisms of $ {F_n}$. The classification of crossed products follows from these results. We also consider the same problem for $G$ some other groups.
Local rigidity of symmetric spaces
V.
Schroeder;
W.
Ziller
145-160
Abstract: We show that on a symmetric space of noncompact or compact type the metric is locally rigid in the sense that if one changes the metric locally but preserves the curvature bounds, then the new metric is isometric to the old one. We also prove an analytic continuation property for symmetric spaces of rank $\ge 3$.
Weight strings in nonstandard representations of Kac-Moody algebras
Meighan I.
Dillon
161-169
Abstract: We consider the weights which occur in arbitrary irreducible highest weight representations of Kac-Moody algebras and determine conditions under which certain weights may or may not occur.
Minimal identities of symmetric matrices
Wen Xin
Ma;
Michel L.
Racine
171-192
Abstract: Let ${H_n}(F)$ denote the subspace of symmetric matrices of ${M_n}(F)$, the full matrix algebra with coefficients in a field $F$. The subspace ${H_n}(F)\subset {M_n}(F)$ does not have any polynomial identity of degree less than $2n$. Let $\displaystyle T_k^i({x_1}, \ldots ,{x_k}) = \sum\limits_{\begin{array}{*{20}{c}... ... {{{( - 1)}^\sigma }{x_{\sigma (1)}}} {x_{\sigma (2)}} \cdots {x_{\sigma (k)}},$ , and $e(n) = n$ if $n$ is even, $n + 1$ if $n$ is odd. For all $ n \geq 1,T_{2n}^i$ is an identity of ${H_n}(F)$. If the characteristic of $ F$ does not divide $ e(n)!$ and if $ n \ne 3$, then any homogeneous polynomial identity of ${H_n}(F)$ of degree $2n$ is a consequence of $T_{2n}^i$. The case $n = 3$ is also dealt with. The proofs are algebraic, but an equivalent formulation of the first result in graph-theoretical terms is given.
The nonlinear geometry of linear programming. III. Projective Legendre transform coordinates and Hilbert geometry
J. C.
Lagarias
193-225
Abstract: This paper studies projective scaling trajectories, which are the trajectories obtained by following the infinitesimal version of Karmarkar's linear programming algorithm. A nonlinear change of variables, projective Legendre transform coordinates, is introduced to study these trajectories. The projective Legendre transform mapping has a coordinate-free geometric interpretation in terms of the notion of "centering by a projective transformation." Let ${\mathsf{H}}$ be a set of linear programming constraints $ \{ \langle {{\mathbf{a}}_j},{\mathbf{x}}\rangle \geq {b_j}:1 \leq j \leq m\}$ on $ {{\mathbf{R}}^n}$ such that its polytope of feasible solutions ${P_{\mathsf{H}}}$ is bounded and contains ${\mathbf{0}}$ in its interior. The projective Legendre transform mapping ${\psi _{\mathsf{H}}}$ is given by $\displaystyle {\psi _{\mathsf{H}}}({\mathbf{x}}) = \frac{{{\phi _{\mathsf{H}}}(... ...{{{\mathbf{a}}_j}}} {{\langle {{\mathbf{a}}_j},{\mathbf{x}}\rangle - {b_j}}}.}$ Here $ {\phi _{\mathsf{H}}}(x)$ is the Legendre transform coordinate mapping introduced in part II. ${\psi _{\mathsf{H}}}({\mathbf{x}})$ is a one-to-one and onto mapping of the interior of the feasible solution polytope $ \operatorname{Int} ({P_{\mathsf{H}}})$ to the interior of its polar polytope $ \operatorname{Int} (P_{\mathsf{H}}^\circ )$. The set of projective scaling trajectories with objective function $\langle {\mathbf{c}},{\mathbf{x}}\rangle - {c_0}$ are mapped under ${\psi _{\mathsf{H}}}$ to the set of straight line segments in $\operatorname{Int} (P_{\mathsf{H}} ^\circ )$ passing through the boundary point $- {\mathbf{c}}/{c_0}$ of $P_{\mathsf{H}} ^\circ$. As a consequence the projective scaling trajectories (for all objective functions) can be interpreted as the complete set of "geodesics" (actually distinguished chords) of a projectively invariant metric geometry on $\operatorname{Int} ({P_{\mathsf{H}}})$, which is isometric to Hilbert geometry on the interior of the polar polytope $ P_{\mathsf{H}}^\circ$.
A regularity theory for variational problems with higher order derivatives
F. H.
Clarke;
R. B.
Vinter
227-251
Abstract: We consider problems in the calculus of variations in one independent variable and where the Lagrangian involves derivatives up to order $N$, $N \ge 1$. Existence theory supplies mild hypotheses under which there are minimizers for such problems, but they need to be strengthened for standard necessary conditions to apply. For problems with $N > 1$, this paper initiates investigation of regularity properties, and associated necessary conditions, which obtain strictly under the hypotheses of existence theory. It is shown that the $N$th derivative of a minimizer is locally essentially bounded off a closed set of zero measure, the set of "points of bad behaviour". Additional hypotheses are shown to exclude occurrence of points of bad behaviour. Finally a counter example suggests respects in which problems with $N > 1$ exhibit pathologies not present in the $N = 1$ case.
Elliptic problems involving an indefinite weight
M.
Faierman
253-279
Abstract: We consider a selfadjoint elliptic eigenvalue problem, which is derived formally from a variational problem, of the form $Lu = \lambda \omega (x)u$ in $ \Omega$, ${B_j}u = 0$ on $\Gamma$, $j = 1, \ldots ,m$, where $ L$ is a linear elliptic operator of order $2m$ defined in a bounded open set $ \Omega \subset {{\mathbf{R}}^n}\quad (n \geq 2)$ with boundary $\Gamma$, the ${B_j}$ are linear differential operators defined on $\Gamma$, and $\omega$ is a real-valued function assuming both positive and negative values. For our problem we prove the completeness of the eigenvectors and associated vectors in two function spaces which arise naturally in such an indefinite problem. We also establish some results concerning the eigenvalues of the problem which complement the known results and investigate the structure of the principal subspaces.
Using subnormality to show the simple connectivity at infinity of a finitely presented group
Joseph S.
Profio
281-292
Abstract: A CW-complex $ X$ is simply connected at infinity if for each compact $C$ in $X$ there exists a compact $D$ in $X$ such that loops in $X - D$ are homotopically trivial in $X - C$. Let $G$ be a finitely presented group and $ X$ a finite CW-complex with fundamental group $G$. $G$ is said to be simply connected at infinity if the universal cover of $X$ is simply connected at infinity. B. Jackson and C. M. Houghton have independently shown that if $ G$ and a normal subgroup $ H$ are infinite finitely presented groups with $G/H$ infinite and either $H$ or $G/H$ $1$-ended, then $G$ is simply connected at infinity. In the case where $H$ is $1$-ended, we exhibit a class of groups showing that the "finitely presented" hypothesis on $H$ cannot be reduced to "finitely generated." We address the question: if $N$ is normal in $H$ and $H$ is normal in $G$ and these are infinite groups with $ N$ and $G$ finitely presented and either $ N$ or $G/H$ is $1$-ended, is $G$ simply connected at infinity? In the case that $ N$ is $1$-ended, the answer is shown to be yes. In the case that $G/H$ is $1$-ended, we exhibit a class of such groups that are not simply connected at infinity.
Nonmetrizable topological dynamics and Ramsey theory
Vitaly
Bergelson;
Neil
Hindman
293-320
Abstract: Applying ideas from topological dynamics in compact metric spaces to the Stone-Cěch compactification of a discrete semigroup, several new proofs of old results and some new results in Ramsey Theory are obtained. In particular, two ultrafilter proofs of van der Waerden's Theorem are given. An ultrafilter approach to "central" sets (sets which are combinatorially rich) is developed. This enables us to show that for any partition of the positive integers one cell is both additively and multiplicatively central. Also, a fortuitous answer to a question of Ellis is obtained.
The Schr\"odinger equation with a quasi-periodic potential
Steve
Surace
321-370
Abstract: We consider the Schràdinger equation $\displaystyle - \frac{{{d^2}}} {{d{x^2}}}\psi + \varepsilon (\cos x + \cos (\alpha x + \vartheta ))\psi = E\psi$ where $\varepsilon$ is small and $\sigma$ satisfies the Diophantine inequality $\displaystyle \vert p + q\alpha \vert \geq C/{q^2}{\text{for}}p{\text{,}}q \in {\mathbf{Z}},q \ne 0.$ . We look for solutions of the form $\displaystyle \psi (x) = {e^{iKx}}q(x) = {e^{iKx}}\sum {{\psi _{mn}}{e^{inx}}} {e^{im(\alpha x + \vartheta )}}$ . If we try to solve for $\psi = {\psi _{mn}}$ we are led to the Schràdinger equation on the lattice ${{\mathbf{Z}}^2}$ $\displaystyle H(K)\psi = (\varepsilon \Delta + V(K))\psi = E\psi$ where $ \Delta$ is the discrete Laplacian (without diagonal terms) and $V(K)$ is some potential on ${{\mathbf{Z}}^2}$ . We have two main results: (1) For $ \varepsilon$ sufficiently small, $H(K)$ has pure point spectrum for almost every $ K$. (2) For $\varepsilon$ sufficiently small, the operator $\displaystyle - {d^2}/d{x^2} + \varepsilon (\cos x + \cos (\alpha x + \vartheta ))$ has no point spectrum. To prove our results, we must get decay estimates on the Green's function ${(E - H)^{ - 1}}$. The decay of the eigenfunction follows from this. In general, we must keep track of small divisors which can make the Green's function large. This is accomplished by a KAM (Kolmogorov, Arnold, Moser) type of multiscale perturbation analysis.
Analysis of a class of probability preserving measure algebras on compact intervals
William C.
Connett;
Alan L.
Schwartz
371-393
Abstract: The measure algebras of the title are those which are also hypergroups with some regularity conditions. Examples include the convolutions associated with Jacobi polynomial series and Fourier Bessel series. It is shown here that there is a one-to-one correspondence between these hypergroups and a class of Sturm-Liouville problems which have the characters of the hypergroup as eigenfunctions. The interplay between these two characterizations allows a detailed analysis which includes a Hilb-type formula for the characters and asymptotic estimates for the Plancherel measure and the eigenvalues of the associated Sturm-Liouville problem.
A topological characterization of ${\bf R}$-trees
John C.
Mayer;
Lex G.
Oversteegen
395-415
Abstract: $\mathbf{R}$-trees arise naturally in the study of groups of isometries of hyperbolic space. An $\mathbf{R}$-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals $ \mathbf{R}$. Actions on $\mathbf{R}$-trees can be viewed as ideal points in the compactification of groups of isometries. As such they have applications to the study of hyperbolic manifolds. Our concern in this paper, however, is with the topological characterization of $\mathbf{R}$-trees. Our main theorem is the following: Let $(X,p)$ be a metric space. Then $ X$ is uniquely arcwise connected and locally arcwise connected if, and only if, $X$ admits a compatible metric $ d$ such that $ (X,d)$ is an $\mathbf{R}$-tree. Essentially, we show how to put a convex metric on a uniquely arcwise connected, locally arcwise connected, metrizable space.
Modular representation theory of finite groups with T.I. Sylow $p$-subgroups
H. I.
Blau;
G. O.
Michler
417-468
Abstract: Let $p$ be a fixed prime, and let $ G$ be a finite group with a T.I. Sylow $p$-subgroup $P$. Let $ N = {N_G}(P)$ and let $ k(G)$ be the number of conjugacy classes of $G$. If $z(G)$ denotes the number of $p$-blocks of defect zero, then we show in this article that $ z(G) = k(G) - k(N)$. This result confirms a conjecture of J. L. Alperin. Its proof depends on the classification of the finite simple groups. Brauer's height zero conjecture and the Alperin-McKay conjecture are also verified for finite groups with a T.I. Sylow $p$-subgroup.
Hall-Littlewood functions, plane partitions, and the Rogers-Ramanujan identities
John R.
Stembridge
469-498
Abstract: We apply the theory of Hall-Littlewood functions to prove several multiple basic hypergeometric series identities, including some previously known generalizations of the Rogers-Ramanujan identities due to G. E. Andrews and D. M. Bressoud. The techniques involve the adaptation of a method due to I. G. Macdonald for calculating partial fraction expansions of certain types of symmetric formal power series. Macdonald originally used this method to prove a pair of generating function identities for plane partitions conjectured by MacMahon and Bender-Knuth. We show that this method can also be used to prove another pair of plane partition identities recently obtained by R. A. Proctor.
Defect relations for degenerate meromorphic maps
Wan Xi
Chen
499-515
Abstract: Using a concept called subgeneral position and adapting a weight function created by E. I. Nochka, this work proves the Cartan's conjecture on defect relations for a degenerate meromorphic map from a parabolic manifold into a projective space.
Weighted inequalities for one-sided maximal functions
F. J.
Martín-Reyes;
P.
Ortega Salvador;
A.
de la Torre
517-534
Abstract: Let $M_g^ +$ be the maximal operator defined by $\displaystyle M_g^ + f(x) = \mathop {\sup }\limits_{h > 0} \left( {\int_x^{x + ... ... f(t)\vert g(t)dt} } \right){\left( {\int_x^{x + h} {g(t)dt} } \right)^{ - 1}},$ where $g$ is a positive locally integrable function on ${\mathbf{R}}$. We characterize the pairs of nonnegative functions $(u,v)$ for which $M_g^ +$ applies ${L^p}(v)$ in ${L^p}(u)$ or in weak- ${L^p}(u)$. Our results generalize Sawyer's (case $ g = 1$) but our proofs are different and we do not use Hardy's inequalities, which makes the proofs of the inequalities self-contained.
The $v\sb 1$-periodic homotopy groups of an unstable sphere at odd primes
Robert D.
Thompson
535-559
Abstract: The $\bmod \;p$ ${v_1}$-periodic homotopy groups of a space $ X$ are defined by considering the homotopy classes of maps of a Moore space into $ X$ and then inverting the Adams self map. In this paper we compute the $ p$ $ {v_1}$-periodic homotopy groups of an odd dimensional sphere, localized at an odd prime. This is done by showing that these groups are isomorphic to the stable $\bmod \;p$ ${v_1}$-periodic homotopy groups of $B\Sigma _p^{2(p - 1)n}$, the $2(p - 1)n$ skeleton of the classifying space for the symmetric group ${\Sigma _p}$. There is a map ${\Omega ^{2n + 1}}{S^{2n + 1}} \to {\Omega ^\infty }(J \wedge B\Sigma _p^{2(p - 1)n})$, where $ J$ is a spectrum constructed from connective $K$-theory, and the image in homotopy is studied.
Algebraic distance graphs and rigidity
M.
Homma;
H.
Maehara
561-572
Abstract: An algebraic distance graph is defined to be a graph with vertices in $ {E^n}$ in which two vertices are adjacent if and only if the distance between them is an algebraic number. It is proved that an algebraic distance graph with finite vertex set is complete if and only if the graph is "rigid". Applying this result, we prove that (1) if all the sides of a convex polygon $\Gamma$ which is inscribed in a circle are algebraic numbers, then the circumradius and all diagonals of $\Gamma$ are also algebraic numbers, (2) the chromatic number of the algebraic distance graph on a circle of radius $r$ is $\infty$ or $2$ accordingly as $r$ is algebraic or not. We also prove that for any $ n > 0$, there exists a graph $G$ which cannot be represented as an algebraic distance graph in ${E^n}$.
Jumps of orderings
C. J.
Ash;
C. G.
Jockusch;
J. F.
Knight
573-599
Abstract: Here it is shown that for each recursive ordinal $\alpha \geqslant 2$ and each Turing degree ${\mathbf{d}} > {{\mathbf{0}}^{(\alpha )}}$, there is a linear ordering ${\mathbf{A}}$ such that $ {\mathbf{d}}$ is least among the $\alpha$th jumps of degrees of (open diagrams of) isomorphic copies of $ {\mathbf{A}}$ and for $\beta < \alpha$, the set of $\beta$th jumps of degrees of copies of ${\mathbf{A}}$ has no least element.
Topological entropy of fixed-point free flows
Romeo F.
Thomas
601-618
Abstract: Topological entropy was introduced as an invariant of topological conjugacy and also as an analogue of measure theoretic entropy. Topological entropy for one parameter flows on a compact metric spaces is defined by Bowen. General statements are proved about this entropy, but it is not easy to calculate the topological entropy, and to show it is invariant under conjugacy. For all this I would like to try to pose a new direction and study a definition for the topological entropy that involves handling the technical difficulties that arise from allowing reparametrizations of orbits. Some well-known results are proved as well using this definition. These results enable us to prove some results which seem difficult to prove using Bowen's definition. Also we show here that this definition is equivalent to Bowen's definition for any flow without fixed points on a compact metric space. Finally, it is shown that the topological entropy of an expansive flow can be defined globally on a local cross sections.
The heat equation for Riemannian foliations
Seiki
Nishikawa;
Mohan
Ramachandran;
Philippe
Tondeur
619-630
Abstract: Let $\mathcal{F}$ be a Riemannian foliation on a closed oriented manifold $M$, with the transversal Laplacian ${\Delta _B}$ acting on the basic forms $\Omega _B^r(\mathcal{F})$ of degree $r \geqslant 0$. We construct the fundamental solution $ e_B^r(x,y,t)$ for the basic heat operator $\partial /\partial t + {\Delta _B}$, and prove existence and uniqueness for the solution of the heat equation on $ \Omega _B^r(\mathcal{F})$. As an application we give a new proof for the deRham-Hodge decomposition theorem for $ {\Delta _B}$ in $\Omega _B^r(\mathcal{F})$, generalizing the approach to the classical deRham-Hodge theorem pioneered by Milgram and Rosenbloom.
Dirac manifolds
Theodore James
Courant
631-661
Abstract: A Dirac structure on a vector space $V$ is a subspace of $V$ with a skew form on it. It is shown that these structures correspond to subspaces of $V \oplus {V^{\ast}}$ satisfying a maximality condition, and having the property that a certain symmetric form on $V \oplus {V^{\ast}}$ vanishes when restricted to them. Dirac structures on a vector space are analyzed in terms of bases, and a generalized Cayley transformation is defined which takes a Dirac structure to an element of $O(V)$. Finally a method is given for passing a Dirac structure on a vector space to a Dirac structure on any subspace. Dirac structures on vector spaces are generalized to smooth Dirac structures on a manifold $P$, which are defined to be smooth subbundles of the bundle $ TP \oplus {T^{\ast}}P$ satisfying pointwise the properties of the linear case. If a bundle $L \subset TP \oplus {T^{\ast}}P$ defines a Dirac structure on $P$, then we call $L$ a Dirac bundle over $P$. A $3$-tensor is defined on Dirac bundles whose vanishing is the integrability condition of the Dirac structure. The basic examples of integrable Dirac structures are Poisson and presymplectic manifolds; in these cases the Dirac bundle is the graph of a bundle map, and the integrability tensors are $[B,B]$ and $d\Omega$ respectively. A function $f$ on a Dirac manifold is called admissible if there is a vector field $X$ such that the pair $(X,df)$ is a section of the Dirac bundle $ L$; the pair $ (X,df)$ is called an admissible section. The set of admissible functions is shown to be a Poisson algebra. A process is given for passing Dirac structures to a submanifold $Q$ of a Dirac manifold $P$. The induced bracket on admissible functions on $Q$ is in fact the Dirac bracket as defined by Dirac for constrained submanifolds.
Link homotopy with one codimension two component
Paul A.
Kirk
663-688
Abstract: Link maps with one codimension two component are studied and an invariant of link maps modulo link homotopy is constructed using ideas from knot theory and immersion theory. This invariant is used to give examples of nontrivial link homotopy classes and to show that there are infinitely many distinct link homotopy classes in many dimensions. A link map with the codimension two component embedded is shown to be nullhomotopic. These ideas are applied to the special case of $2$-spheres in ${S^4}$ to give simple examples of the failure of the Whitney trick in dimension $4$.
Lie algebra representations of dimension $<p\sp 2$
Helmut
Strade
689-709
Abstract: Various methods of representation theory of modular Lie algebras are improved. As an application the structure of the Lie algebras having a faithful irreducible module of dimension $< {p^2}$ is determined. Applications to the classification theory of modular simple Lie algebras are given.
On the Dirichlet space for finitely connected regions
Kit Chak
Chan
711-728
Abstract: This paper is devoted to the study of the Dirichlet space $\operatorname{Dir} (G)$ for finitely connected regions $G$; we are particularly interested in the algebra of bounded multiplication operators on this space. Results in different directions are obtained. One direction deals with the structure of closed subspaces invariant under all bounded multiplication operators. In particular, we show that each such subspace contains a bounded function. For regions with circular boundaries we prove that a finite codimensional closed subspace invariant under multiplication by $ z$ must be invariant under all bounded multiplication operators, and furthermore it is of the form $p\operatorname{Dir} (G)$, where $p$ is a polynomial with all its roots lying in $G$. Another direction is to study cyclic and noncyclic vectors for the algebra of all bounded multiplication operators. Typical results are: if $f \in \operatorname{Dir} (G)$ and $f$ is bounded away from zero then $ f$ is cyclic; on the other hand, if the zero set of the radial limit function of $ f$ on the boundary has positive logarithmic capacity, then $f$ is not cyclic. Also, some other sufficient conditions for a function to be cyclic are given. Lastly, we study transitive operator algebras containing all bounded multiplication operators; we prove that they are dense in the algebra of all bounded operators in the strong operator topology.
The method of negative curvature: the Kobayashi metric on ${\bf P}\sb 2$ minus $4$ lines
Michael J.
Cowen
729-745
Abstract: Bloch, and later H. Cartan, showed that if ${H_1}, \ldots ,{H_{n + 2}}$ are $n + 2$ hyperplanes in general position in complex projective space ${{\mathbf{P}}_n}$, then $ {{\mathbf{P}}_n} - {H_1} \cup \cdots \cup {H_{n + 2}}$ is (in current terminology) hyperbolic modulo $\Delta$, where $\Delta$ is the union of the hyperplanes $({H_{^1}} \cap \cdots \cap {H_k}) \oplus ({H_{k + 1}} \cap \cdots \cap {H_{n + 2}})$ for $2 \leqslant k \leqslant n$ and all permutations of the ${H_i}$. Their results were purely qualitative. For $ n = 1$, the thrice-punctured sphere, it is possible to estimate the Kobayashi metric, but no estimates were known for $n \geqslant 2$. Using the method of negative curvature, we give an explicit model for the Kobayashi metric when $n = 2$.
Local behavior of solutions of quasilinear elliptic equations with general structure
J.-M.
Rakotoson;
William P.
Ziemer
747-764
Abstract: This paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form $ \operatorname{div} A(x,u,\nabla u) + B(x,u,\nabla u)$, where $A$ and $B$ are Borel measurable, are solutions to the equation $ \operatorname{div} A(x,u,\nabla u) + B(x,u,\nabla u) = \mu$ for some nonnegative Radon measure $\mu$. Among other things, it is shown that if $ u$ is a Hölder continuous solution to this equation, then the measure $ \mu$ satisfies the growth property $\mu [B(x,r)] \leqslant M{r^{n - p + \varepsilon }}$ for all balls $B(x,r)$ in $ {{\mathbf{R}}^n}$. Here $\varepsilon$ depends on the Hölder exponent of $ u$ while $p > 1$ is given by the structure of the differential operator. Conversely, if $ \mu$ is assumed to satisfy this growth condition, then it is shown that $ u$ satisfies a Harnack-type inequality, thus proving that $u$ is locally bounded. Under the additional assumption that $A$ is strongly monotonic, it is shown that $ u$ is Hölder continuous.
Amenability of weighted convolution algebras on locally compact groups
Niels
Grønbæk
765-775
Abstract: We give a direct transition from the existence of a bounded right approximate identity in the diagonal ideal for a weighted convolution algebra on a locally compact group to the existence of translation invariant means on an associated weighted ${L^\infty }$-space, thus giving a characterization of amenability for such an algebra.
$\Lambda(q)$ processes
Ron C.
Blei
777-786
Abstract: Motivated by some classical notions in harmonic analysis, $\Lambda (q)$ processes are introduced in the context of a study of stochastic interdependencies. An extension of a classical theorem of Salem and Zygmund regarding random Fourier series is obtained. The Littlewood exponent of $\Lambda (q)$ processes is estimated and, in some archetypical cases, computed.
Holomorphic maps which preserve intrinsic metrics or measures
Ian
Graham
787-803
Abstract: Suppose that $ M$ is a domain in a taut complex manifold $M'$, and that $\Omega$ is a strictly convex bounded domain in $ {{\mathbf{C}}^n}$. We consider the following question: given a holomorphic map $F:M \to \Omega$ which is an isometry for the infinitesimal Kobayashi metric at one point, must $ F$ be biholomorphic? With an additional technical assumption on the behavior of the Kobayashi distance near points of $\partial M$, we show that $F$ gives a biholomorphism of $ M$ with an open dense subset of $\Omega$. Moreover, $F$ extends as a homeomorphism from a larger domain $\tilde M$ to $\Omega$. We also give some related results--refinements of theorems of Bland and Graham and Fornaess and Sibony, and the answer to a question of Graham and Wu.
Asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions
Zhou Ping
Xin
805-820
Abstract: This paper concerns the large time behavior toward planar rarefaction waves of the solutions for scalar viscous conservation laws in several dimensions. It is shown that a planar rarefaction wave is nonlinearly stable in the sense that it is an asymptotic attractor for the viscous conservation law. This is proved by using a stability result of rarefaction wave for scalar viscous conservation laws in one dimension and an elementary $ {L^2}$-energy method.
Remarks and corrections for: ``Groups acting on affine algebras'' [Trans. Amer. Math. Soc. {\bf 310} (1988), no. 2, 485--497; MR0940913 (89i:16029)]
Daniel R.
Farkas
821-823
Bundle theories for topological manifolds
C. B.
Hughes;
L. R.
Taylor;
E. B.
Williams
1-65
Abstract: Manifold approximate fibrations arise in the geometric topology of manifolds and group actions on topological manifolds. The primary purpose of this paper is to classify manifold approximate fibrations in terms of the lifting problem for a certain bundle. Our classification meshes well with the classical classifications of fibrations and bundles and, hence, we are able to attack questions such as the following. When is a fibration controlled homotopy equivalent to a manifold approximate fibration? When is a manifold approximate fibration controlled homeomorphic to a bundle?
H\"older domains and Poincar\'e domains
Wayne
Smith;
David A.
Stegenga
67-100
Abstract: A domain $D \subset {R^d}$ of finite volume is said to be a $p$-Poincaré domain if there is a constant $ {M_p}(D)$ so that $\displaystyle {\int\limits_D {\vert u - {u_D}\vert} ^p}dx \leq M_p^p(D){\int\limits_D {\vert\nabla u\vert} ^p}dx$ for all functions $u \in {C^1}(D)$. Here ${u_D}$ denotes the mean value of $u$ over $D$. Techniques involving the quasi-hyperbolic metric on $D$ are used to establish that various geometric conditions on $D$ are sufficient for $D$ to be a $p$-Poincaré domain. Domains considered include starshaped domains, generalizations of John domains and Hàlder domains. $D$ is a Hàlder domain provided that the quasi-hyperbolic distance from a fixed point ${x_0} \in D$ to $x$ is bounded by a constant multiple of the logarithm of the euclidean distance of $x$ to the boundary of $D$. The terminology is derived from the fact that in the plane, a simply connected Hàlder domain has a Hàlder continuous Riemann mapping function from the unit disk onto $D$. We prove that if $D$ is a Hàlder domain and $ p \ge d$, then $ D$ is a $ p$-Poincaré domain. This answers a question of Axler and Shields regarding the image of the unit disk under a Hàlder continuous conformal mapping. We also consider geometric conditions which imply that the imbedding of the Sobolev space $ {W^{1,p}}(D) \to {L^p}(D)$ is compact, and prove that this is the case for a Hàlder domain $D$.
Upper bounds for ergodic sums of infinite measure preserving transformations
Jon
Aaronson;
Manfred
Denker
101-138
Abstract: For certain conservative, ergodic, infinite measure preserving transformations $T$ we identify increasing functions $A$, for which $\displaystyle \limsup \limits_{n \to \infty } \frac{1} {{A(n)}}\sum\limits_{k = 1}^n {f \circ } {T^k} = \int_X {fd\mu } \quad {\text{a}}{\text{.e}}{\text{.}}$ holds for any nonnegative integrable function $f$. In particular the results apply to some Markov shifts and number-theoretic transformations, and include the other law of the iterated logarithm.
Complete localization of domains with noncompact automorphism groups
Kang-Tae
Kim
139-153
Abstract: We prove a characterization of the domains in ${{\mathbf{C}}^n}$ with an automorphism orbit accumulating at a boundary point at which the boundary is real analytic and convex up to a biholomorphic change of local coordinates. This result generalizes the well-known Wong-Rosay theorem on strongly pseudoconvex domains to the case of locally convex domains with real analytic boundaries.
A complete classification of the piecewise monotone functions on the interval
Stewart
Baldwin
155-178
Abstract: We define two functions $f$ and $g$ on the unit interval $[0,1]$ to be strongly conjugate iff there is an order-preserving homeomorphism $h$ of $[0,1]$ such that $g = {h^{ - 1}}fh$ (a minor variation of the more common term "conjugate", in which $ h$ need not be order-preserving). We provide a complete set of invariants for each continuous (strictly) piecewise monotone function such that two such functions have the same invariants if and only if they are strongly conjugate, thus providing a complete classification of all such strong conjugacy classes. In addition, we provide a criterion which decides whether or not a potential invariant is actually realized by some piecewise monotone continuous function.
Uniqueness of the critical point of the solutions to some semilinear elliptic boundary value problems in ${\bf R}\sp 2$
Shigeru
Sakaguchi
179-190
Abstract: We consider some two-dimensional semilinear elliptic boundary value problems over a bounded convex domain in ${{\mathbf{R}}^2}$ and show the uniqueness of the critical point of the solutions.
Power series space representations of nuclear Fr\'echet spaces
Dietmar
Vogt
191-208
Abstract: Let $E$ be a nuclear graded Fréchet space such that the norms satisfy inequalities $\vert\vert\vert\vert _k^2 \leq {C_k}\vert\vert\vert{\vert _{k - 1}}\vert\vert\vert{\vert _{k - 1}}$ for all $k$, let $F$ be a graded Fréchet space such that the dual (extended real valued) norms satisfy inequalities $\vert\vert\vert\vert _k^{*2} \leq {D_k}\vert\vert\vert\vert _{k - 1}^*\vert\vert\vert\vert _{k + 1}^*$ for all $k$, and let $A$ be a tame (resp. linearly tame) linear map from $F$ to $E$. Then there exists a tame (resp. linearly tame) factorization of $A$ through a power series space $\Lambda _\infty ^2(\alpha )$. In the case of a tame quotient map, $E$ is tamely equivalent to a power series space of infinite type. This applies in particular to the range of a tame (resp. linearly tame) projection in a power series space $\Lambda _\infty ^2(\alpha )$. In this case one does not need nuclearity. It also applies to the tame spaces in the sense of the various implicit function theorems. If they are nuclear, they are tamely equivalent to power series spaces ${\Lambda _\infty }(\alpha )$.
Complete coinductive theories. I
A. H.
Lachlan
209-241
Abstract: Let $T$ be a complete theory over a relational language which has an axiomatization by $\exists \forall $-sentences. The properties of models of $T$ are studied. It is shown that quantifier-free formulas are stable. This limited stability is used to show that in $ \exists \forall$-saturated models the elementary types of tuples are determined by their $\exists$-types and algebraicity is determined by existential formulas. As an application, under the additional assumption that no quantifier-free formula has the FCP, the models $ \mathcal{M}$ of $ T$ are completely characterized in terms of certain 0-definable equivalence relations on cartesian powers of $M$. This characterization yields a result similar to that of Schmerl for the case in which $T$ is ${\aleph _0}$-categorical.
A Plancherel formula for parabolic subgroups
Mie
Nakata
243-256
Abstract: We obtain explicit Plancherel formulas for the parabolic subgroups $ P$ of $p$-adic unitary groups which fix one dimensional isotropic subspaces. By means of certain limits of difference operators (called strong derivatives), we construct a Dixmier-Pukanszky operator which compensates for the nonunimodularity of the group $P$. Then, we compute the Plancherel formula of $N \cdot A$, where $ N$ is the nilradical of $ P$ and
Intrinsic formality and certain types of algebras
Gregory
Lupton
257-283
Abstract: In this paper, a type of algebra is introduced and studied from a rational homotopy point of view, using differential graded Lie algebras. The main aim of the paper is to establish whether or not such an algebra is the rational cohomology algebra of a unique rational homotopy type of spaces. That is, in the language of rational homotopy, whether or not such an algebra is intrinsically formal. Examples are given which show that, in general, this is not so--7.8 and 7.9. However, whilst it is true that not all such algebras are intrinsically formal, some of them are. The main results of this paper show a certain class of these algebras to be intrinsically formal--Theorem $2$ (6.1); and a second, different type of algebra also to be intrinsically formal--Theorem $ 1$ (5.2), which type of algebra overlaps with the first type in many examples of interest. Examples are given in $\S7$.
Almost split sequences and Zariski differentials
Alex
Martsinkovsky
285-307
Abstract: Let $R$ be a complete two-dimensional integrally closed analytic $k$-algebra. Associated with $R$ is the Auslander module $ A$ from the fundamental sequence $0 \to {\omega _R} \to A \to R \to k \to 0$ and the module of Zariski differentials ${D_k}{(R)^{ * * }}$. We conjecture that these modules are isomorphic if and only if $R$ is graded. We prove this conjecture for (a) hypersurfaces $f = X_3^n + {\text{g}}({X_1},{X_2})$, (b) quotient singularities, and (c) $R$ graded Gorenstein.
Isomorphism universal varieties of Heyting algebras
M. E.
Adams;
V.
Koubek;
J.
Sichler
309-328
Abstract: A variety $\mathbf{V}$ is group universal if every group $ G$ is isomorphic to the automorphism group ${\operatorname{Aut}}(A)$ of an algebra $A \in \mathbf{V}$; if, in addition, all finite groups are thus representable by finite algebras from $\mathbf{V}$, the variety $ \mathbf{V}$ is said to be finitely group universal. We show that finitely group universal varieties of Heyting algebras are exactly the varieties which are not generated by chains, and that a chain-generated variety $\mathbf{V}$ is group universal just when it contains a four-element chain. Furthermore, we show that a variety $ \mathbf{V}$ of Heyting algebras is group universal whenever the cyclic group of order three occurs as the automorphism group of some $A \in \mathbf{V}$. The results are sharp in the sense that, for every group universal variety and for every group $G$, there is a proper class of pairwise nonisomorphic Heyting algebras $ A \in \mathbf{V}$ for which ${\operatorname{Aut}}(A) \cong G$.
Surfaces of $E\sp 4$ satisfying certain restrictions on their normal bundle
Th.
Hasanis;
D.
Koutroufiotis;
P.
Pamfilos
329-347
Abstract: We consider smooth surfaces in ${E^4}$ whose normal bundles satisfy certain geometric conditions that entail the vanishing of the normal curvature, and prove that their Gauss curvatures cannot be bounded from above by a negative number. We also give some results towards a classification of flat surfaces with flat normal bundle in ${E^4}$.
A notion of rank for unitary representations of general linear groups
Roberto
Scaramuzzi
349-379
Abstract: A notion of rank for unitary representations of general linear groups over a locally compact, nondiscrete field is defined. Rank measures how singular a representation is, when restricted to the unipotent radical of a maximal parabolic subgroup. Irreducible representations of small rank are classified. It is shown how rank determines to a large extent the asymptotic behavior of matrix coefficients of the representations.
Local approximation by certain spaces of exponential polynomials, approximation order of exponential box splines, and related interpolation problems
N.
Dyn;
A.
Ron
381-403
Abstract: Local approximation order to smooth complex valued functions by a finite dimensional space $ \mathcal{H}$, spanned by certain products of exponentials by polynomials, is investigated. The results obtained, together with a suitable quasi-interpolation scheme, are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline. The analysis of a typical space $\mathcal{H}$ is based here on the identification of its dual with a certain space $\mathcal{P}$ of multivariate polynomials. This point of view allows us to solve a class of multivariate interpolation problems by the polynomials from $\mathcal{P}$, with interpolation data characterized by the structure of $ \mathcal{H}$, and to construct bases of $ \mathcal{P}$ corresponding to the interpolation problem.
Curvatures and similarity of operators with holomorphic eigenvectors
Mitsuru
Uchiyama
405-415
Abstract: The curvature of the holomorphic vector bundle generated by eigenvectors of operators is estimated, and the necessary and sufficient conditions for contractions to be similar or quasi-similar with unilateral shifts are given.
Erratum to: ``Remarks on classical invariant theory''
Roger
Howe
An invariant of regular isotopy
Louis H.
Kauffman
417-471
Abstract: This paper studies a two-variable Laurent polynomial invariant of regular isotopy for classical unoriented knots and links. This invariant is denoted ${L_K}$ for a link $K$, and it satisfies the axioms: 1. Regularly isotopic links receive the same polynomial. 2. $ {L_{[{\text{unk}}]}} = 1$. 3. $ {L_{[{\text{unk}}]}} = aL,\qquad {L_{[{\text{unk}}]}} = {a^{ - 1}}L$. 4. $ {L_{[{\text{unk}}]}} + {L_{[{\text{unk]}}}} = z({L_{[{\text{unk]}}}} + {L_{[{\text{unk]}}}})$. Small diagrams indicate otherwise identical parts of larger diagrams. Regular isotopy is the equivalence relation generated by the Reidemeister moves of type II and type III. Invariants of ambient isotopy are obtained from $L$ by writhe-normalization.
Generalized balanced tournament designs
E. R.
Lamken
473-490
Abstract: A generalized balanced tournament design, $GBTD(n,k)$, defined on a $kn$-set $V$, is an arrangement of the blocks of a $(kn,k,k - 1)$-$BIBD$ defined on $V$ into an $ n \times (kn - 1)$ array such that (1) every element of $V$ is contained in precisely one cell of each column, and (2) every element of $V$ is contained in at most $k$ cells of each row. In this paper, we introduce $ GBTD(n,k)s$ and describe connections between these designs and several other types of combinatorial designs. We also show how to use $GBTDs$ to construct resolvable, near resolvable, doubly resolvable and doubly near resolvable $ BIBDs$.
Two differential-difference equations arising in number theory
Ferrell S.
Wheeler
491-523
Abstract: We survey many old and new results on solutions of the following pair of adjoint differential-difference equations: (1) $\displaystyle \sum\limits_{\begin{array}{*{20}{c}} {1 < n \leqslant x} {{P_2... ... x)}^\alpha }} \qquad (x \to \infty ,\;u \geqslant 1,\;\alpha \in {\mathbf{R}})$ where $ {P_1}(n)$ and $ {P_2}(n)$ are the first and second largest prime divisors of $n$ and $f(u)$ satisfies (2) with $(a,b) = (1 - \alpha , - 1)$.
Large deviations in dynamical systems
Lai-Sang
Young
525-543
Abstract: We prove some large deviation estimates for continuous maps of compact metric spaces and apply them to attractors in differentiable dynamics, rate of escape problems, and to shift spaces.
On infinite-dimensional manifold triples
Katsuro
Sakai;
Raymond Y.
Wong
545-555
Abstract: Let $Q$ denote the Hilbert cube $ {[ - 1,1]^\omega },\;s = {( - 1,1)^\omega }$ the pseudo-interior of $Q,\;\Sigma = \{ ({x_i}) \in s\vert\sup \vert{x_i}\vert < 1\}$ and $\sigma = \{ ({x_i}) \in s\vert{x_i} = 0\;{\text{except for finitely many}}\;i\} $. A triple $(X,M,N)$ of separable metrizable spaces is called a $ (Q,\Sigma ,\sigma )$- (or $(s,\Sigma ,\sigma )$-)manifold triple if it is locally homeomorphic to $(Q,\Sigma ,\sigma )$ (or $(s,\Sigma ,\sigma )$). In this paper, we study such manifold triples and give some characterizations.
Bounded polynomial vector fields
Anna
Cima;
Jaume
Llibre
557-579
Abstract: We prove that, for generic bounded polynomial vector fields in ${{\mathbf{R}}^n}$ with isolated critical points, the sum of the indices at all their critical points is ${( - 1)^n}$. We characterize the local phase portrait of the isolated critical points at infinity for any bounded polynomial vector field in ${{\mathbf{R}}^2}$. We apply this characterization to show that there are exactly seventeen different behaviours at infinity for bounded cubic polynomial vector fields in the plane.
Invariant Radon transforms on a symmetric space
Jeremy
Orloff
581-600
Abstract: Injectivity and support theorems are proved for a class of Radon transforms, $ {R_\mu }$, for $ \mu$ a smooth family of measures defined on a certain space of affine planes in ${\mathbb{X}_0}$, where ${\mathbb{X}_0}$ is the tangent space, of a Riemannian symmetric space of rank one. The transforms are defined by integrating against $\mu$ over these planes. We show that if $ {R_\mu }f$ is supported inside a ball of radius $R$ then so is $f$. This is true for $f \in L_c^2({\mathbb{X}_0})$ or
Ergodic and mixing properties of equilibrium measures for Markov processes
Enrique D.
Andjel
601-614
Abstract: Let $S(t)$ be the semigroup corresponding to a Markov process on a metric space $X$. Suppose $S(t)$ commutes with a homeomorphism $T$ of $X$. We prove that under certain conditions, an equilibrium measure for the process is ergodic under $ T$. We also show that, under stronger conditions this measure must be mixing under $ T$. Several applications of these results are given.
Domain-independent upper bounds for eigenvalues of elliptic operators
Stephen M.
Hook
615-642
Abstract: Let $\Omega \subseteq {\mathbb{R}^m}$ be a bounded open set, $ \partial \Omega$ its boundary and $\Delta$ the Laplacian on ${\mathbb{R}^m}$. Consider the elliptic differential equation: (1) $\displaystyle - \Delta u = \lambda u\quad {\text{in}}\;\Omega ;\qquad u = 0\quad {\text{on}}\;\partial \Omega .$ It is known that the eigenvalues, ${\lambda _i}$, of (1) satisfy (2) $\displaystyle \sum\limits_{i = 1}^n {\frac{{{\lambda _i}}} {{{\lambda _{n + 1}} - {\lambda _i}}}} \geqslant \frac{{mn}} {4}$ provided that ${\lambda _{n + 1}} > {\lambda _n}$. In this paper we abstract the method used by Hile and Protter [2] to establish (2) and apply the method to a variety of second-order elliptic problems, in particular, to all constant coefficient problems. We then consider a variety of higher-order problems and establish an extension of (2) for problem (1) where the Laplacian is replaced by a more general operator in a Hilbert space.
Hamilton-Jacobi equations with state constraints
I.
Capuzzo-Dolcetta;
P.-L.
Lions
643-683
Abstract: In the present paper we consider Hamilton-Jacobi equations of the form $H(x,u,\nabla u) = 0,\;x \in \Omega$, where $ \Omega$ is a bounded open subset of ${R^n},H$ is a given continuous real-valued function of $(x,s,p) \in \Omega \times R \times {R^n}$ and $ \nabla u$ is the gradient of the unknown function $u$. We are interested in particular solutions of the above equation which are required to be supersolutions, in a suitable weak sense, of the same equation up to the boundary of $\Omega$. This requirement plays the role of a boundary condition. The main motivation for this kind of solution comes from deterministic optimal control and differential games problems with constraints on the state of the system, as well from related questions in constrained geodesics.
Hausdorff dimension of harmonic measures on negatively curved manifolds
Yuri
Kifer;
François
Ledrappier
685-704
Abstract: We show by probabilistic means that harmonic measures on manifolds, whose curvature is sandwiched between two negative constants have positive Hausdorff dimensions. A lower bound for harmonic measures of open sets is derived, as well. We end with the results concerning the Hausdorff dimension of harmonic measures on universal covers of compact negatively curved manifolds.
Symmetric derivates, scattered, and semi-scattered sets
Chris
Freiling
705-720
Abstract: We call a set right scattered (left scattered) if every nonempty subset contains a point isolated on the right (left). We establish the following monotonicity theorem for the symmetric derivative. If a real function $f$ has a nonnegative lower symmetric derivate on an open interval $I$, then there is a nondecreasing function $ g$ such that $f(x) > g(x)$ on a right scattered set and $f(x) < g(x)$ on a left scattered set. Furthermore, if $R$ is any right scattered set and $L$ is any left scattered set disjoint with $R$, then there is a function which is positive on $R$, negative on $L$, zero otherwise, and which has a zero lower symmetric derivate everywhere. We obtain some consequence including an analogue of the Mean Value Theorem and a new proof of an old theorem of Charzynski.
A conformal inequality related to the conditional gauge theorem
Terry R.
McConnell
721-733
Abstract: We prove the inequality $h{(x)^{ - 1}}G(x,y)h(y) \leqslant cG(x,y) + c$, where $G$ is the Green function of a plane domain $ D,\;h$ is positive and harmonic on $D$, and $c$ is a constant whose value depends on the topological nature of the domain. In particular, for the class of proper simply connected domains $c$ may be taken to be an absolute constant. As an application, we prove the Conditional Gauge Theorem for plane domains of finite area for which the constant $c$ in the above inequality is finite.
Ergodicity of finite-energy diffusions
Timothy C.
Wallstrom
735-747
Abstract: Recently, the existence of a class of diffusion processes with highly singular drift coefficients has been established under the assumption of "finite energy." The drift singularities of these diffusions greatly complicate their ergodicity properties; indeed, they can render the diffusion nonergodic. In this paper, a method is given for estimating the relaxation time of a finite-energy diffusion, when it is ergodic. These results are applied to show that the set of $\operatorname{spin} - \tfrac{1} {2}$ diffusions of stochastic mechanics is uniformly ergodic.
The stochastic mechanics of the Pauli equation
Timothy C.
Wallstrom
749-762
Abstract: In stochastic mechanics, the Bopp-Haag-Dankel diffusions on $ {\mathbb{R}^3} \times \operatorname{SO} (3)$ are used to represent particles with spin. Bopp and Haag showed that in the limit as the particle's moment of inertia $I$ goes to zero, the solutions of the Bopp-Haag equations converge to that of the regular Pauli equation. Nelson has conjectured that in the same limit, the projections of the Bopp-Haag-Dankel diffusions onto $ {\mathbb{R}^3}$ converge to a Markovian limit process. In this paper, we prove this conjecture for spin $\operatorname{spin} \;\tfrac{1} {2}$ and regular potentials, and identify the limit process as the diffusion naturally associated with the solution to the regular Pauli equation.
Metrizable spaces where the inductive dimensions disagree
John
Kulesza
763-781
Abstract: A method for constructing zero-dimensional metrizable spaces is given. Using generalizations of Roy's technique, these spaces can often be shown to have positive large inductive dimension. Examples of $ {\mathbf{N}}$-compact, complete metrizable spaces with $\operatorname{ind} = 0$ and $\operatorname{Ind} = 1$ are provided, answering questions of Mrowka and Roy. An example with weight $\mathfrak{c}$ and positive Ind such that subspaces with smaller weight have $\operatorname{Ind} = 0$ is produced in ZFC. Assuming an additional axiom, for each cardinal $\lambda$ a space of positive Ind with all subspaces with weight less than $\lambda$ strongly zero-dimensional is constructed.
Simple Lie algebras of characteristic $p$ with dependent roots
Georgia
Benkart;
J. Marshall
Osborn
783-807
Abstract: We investigate finite dimensional simple Lie algebras over an algebraically closed field $ {\mathbf{F}}$ of characteristic $p \geqslant 7$ having a Cartan subalgebra $ H$ whose roots are dependent over $ {\mathbf{F}}$. We show that $H$ must be one-dimensional or for some root $\alpha$ relative to $H$ there is a $1$-section $ {L^{(\alpha )}}$ such that the core of $ {L^{(\alpha )}}$ is a simple Lie algebra of Cartan type $H{(2:\underline m :\Phi )^{(2)}}$ or $W(1:\underline n )$ for some $n > 1$. The results we obtain have applications to studying the local behavior of simple Lie algebras and to classifying simple Lie algebras which have a Cartan subalgebra of dimension less than $ p - 2$.
Immersions of positively curved manifolds into manifolds with curvature bounded above
Nadine L.
Menninga
809-821
Abstract: Let $M$ be a compact, connected, orientable Riemannian manifold of dimension $n - 1 \geqslant 2$, and let $x$ be an isometric immersion of $ M$ into an $n$-dimensional Riemannian manifold $ N$. Let $K$ denote sectional curvature and $ i$ denote the injectivity radius. Assume, for some constant positive constant $ c$, that $ K(N) \leqslant 1/(4{c^2}),\quad 1/{c^2} \leqslant K(M)$, and $\pi c \leqslant i(N)$. Then the radius of the smallest $N$-ball containing $x(M)$ is less than $\tfrac{1} {2}\pi c$ and $x$ is in fact an imbedding of $M$ into $N$, whose image bounds a convex body.
Sums of linked ideals
Bernd
Ulrich
1-42
Abstract: It is shown that the sum of two geometrically linked ideals in the linkage class of a complete intersection is again an ideal in the linkage class of a complete intersection. Conversely, every Gorenstein ideal (of height at least two) in the linkage class of a complete intersection can be obtained as a "generalized localization" of a sum of two geometrically linked ideals in the linkage class of a complete intersection. We also investigate sums of doubly linked Gorenstein ideals. As an application, we construct a perfect prime ideal which is strongly nonobstructed, but not strongly Cohen-Macaulay, and a perfect prime ideal which is not strongly nonobstructed, but whose entire linkage class is strongly Cohen-Macaulay.
Unknotted homology classes on unknotted surfaces in $S\sp 3$
Bruce
Trace
43-56
Abstract: Suppose $ F$ is a closed, genus $ g$ surface which is standardly embedded in ${S^3}$. Let $\gamma$ denote a primitive element in $ {H_1}(F)$ which satisfies ${\theta _F}(\gamma ,\gamma ) = 0$ where ${\theta _F}$ is the Seifert pairing on $ F$. We obtain a number theoretic condition which is equivalent to $\gamma$ being realizable by a curve (in $F$) which is unknotted in ${S^3}$. Various related observations are included.
The minimal normal extension for $M\sb z$ on the Hardy space of a planar region
John
Spraker
57-67
Abstract: Multiplication by the independent variable on ${H^2}(R)$ for $R$ a bounded open region in the complex plane $\mathbb{C}$ is a subnormal operator. This paper characterizes its minimal normal extension $ N$. Any normal operator is determined by a scalar-valued spectral measure and a multiplicity function. It is a consequence of some standard operator theory that a scalar-valued spectral measure for $N$ is harmonic measure for $R$, $\omega$. This paper investigates the multiplicity function $m$ for $N$. It is shown that $m$ is bounded above by two $\omega $-a.e., and necessary and sufficient conditions are given for $m$ to attain this upper bound on a set of positive harmonic measure. Examples are given which indicate the relationship between $N$ and the boundary of $R$.
Some weighted inequalities on product domains
Henry
Lin
69-85
Abstract: We extend the results of R. Fefferman [3] on the bidisc to higher product domains via induction. As an application, we extend the weighted inequality for Calderon-Zygmund operators on the bidisc to higher product domains, and we also extend the result of the Littlewood-Paley operator corresponding to the arbitrary disjoint rectangles to the weighted case.
Abelian and nondiscrete convergence groups on the circle
A.
Hinkkanen
87-121
Abstract: A group $ G$ of homeomorphisms of the unit circle onto itself is a convergence group if every sequence of elements of $G$ contains a subsequence, say ${{\text{g}}_n}$, such that either (i) $ {{\text{g}}_n} \to {\text{g}}$ and ${\text{g}}_n^{ - 1} \to {{\text{g}}^{ - 1}}$ uniformly on the circle where $ {\text{g}}$ is a homeomorphism, or (ii) ${{\text{g}}_n} \to {{\text{x}}_0}$ and $ {\text{g}}_n^{ - 1} \to {{\text{y}}_0}$ uniformly on compact subsets of the complements of $ \{ {{\text{y}}_0}\}$ and $ \{ {{\text{x}}_0}\}$, respectively, for some points ${{\text{x}}_0}$ and $ {{\text{y}}_0}$ of the circle (possibly ${{\text{x}}_0}{\text{ = }}{{\text{y}}_0}$). For example, a group of $K$-quasisymmetric maps, for a fixed $K$, is a convergence group. We show that if $G$ is an abelian or nondiscrete convergence group, then there is a homeomorphism $f$ such that $f \circ G \circ {f^{ - 1}}$ is a group of Màbius transformations.
Outlet points and homogeneous continua
Paweł
Krupski;
Janusz R.
Prajs
123-141
Abstract: (1) A proof is presented for Bing's conjecture that homogeneous, treelike continua are hereditarily indecomposable. As a consequence, each homogeneous curve admits the continuous decomposition into the maximal terminal, homeomorphic, homogeneous, hereditarily indecomposable, treelike subcontinua. (2) A homogeneous, hereditarily unicoherent continuum contains either an arc or arbitrarily small, nondegenerate, indecomposable subcontinua. (3) A treelike continuum with property $K$ which is homogeneous with respect to confluent light mappings contains no two nondegenerate subcontinua with the one-point intersection.
Homogeneous continua in Euclidean $(n+1)$-space which contain an $n$-cube are $n$-manifolds
Janusz R.
Prajs
143-148
Abstract: Let $X$ be a homogeneous continuum and let ${E^n}$ be Euclidean $n$-space. We prove that if $X$ is properly contained in a connected $ (n + 1)$-manifold, then $ X$ contains no $ n$-dimensional umbrella (i.e. a set homeomorphic to the set $\{ ({x_1}, \ldots ,{x_{n + 1}}) \in {E^{n + 1}}:x_1^2 + \cdots + x_{n + 1}^2 \leq 1$ and ${x_{n + 1}} \leq 0$ and either ${x_1} = \cdots = {x_n} = 0$ or ${x_{n + 1}} = 0\}$). Combining this fact with an earlier result of the author we conclude that if $ X$ lies in ${E^{n + 1}}$ and topologically contains $ {E^n}$, then $ X$ is an $ n$-manifold.
Leray functor and cohomological Conley index for discrete dynamical systems
Marian
Mrozek
149-178
Abstract: We introduce the Leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological Conley index of an isolated invariant set of a homeomorphism on a locally compact metric space. We prove the homotopy and additivity properties for this index and compute the index in some examples. As one of applications we prove the existence of nonconstant, bounded solutions of the Euler approximation of a certain system of ordinary differential equations.
Existence of weak solutions for the Navier-Stokes equations with initial data in $L\sp p$
Calixto P.
Calderón
179-200
Abstract: The existence of weak solutions for the Navier-Stokes equations for the infinite cylinder with initial data in $ {L^p}$ is considered in this paper. We study the case of initial data in ${L^p}({R^n})$, $2 < p < n$, and $n = 3,4$. An existence theorem is proved covering these important cases and therefore, the "gap" between the Hopf-Leray theory $(p = 2)$ and that of Fabes-Jones-Riviere $ (p > n)$ is bridged. The existence theorem gives a new method of constructing global solutions. The cases $p = n$ are treated at the end of the paper.
Addendum to the paper: ``Existence of weak solutions for the Navier-Stokes equations with initial data in $L\sp p$'' [Trans. Amer. Math. Soc. {\bf 318} (1990), no. 1, 179--200; MR0968416 (90k:35199)]
Calixto P.
Calderón
201-207
Abstract: This paper considers the existence of global weak solutions for the Navier-Stokes equations in the infinite cylinder $ {{\mathbf{R}}^n} \times {{\mathbf{R}}_ + }$ with initial data in $ {L^r}$, $n \geq 3$, $1 < r < \infty$. An imbedding theorem as well as related initial value problems are also studied, thus completing results in [2].
A classification of Baire class $1$ functions
A. S.
Kechris;
A.
Louveau
209-236
Abstract: We study in this paper various ordinal ranks of (bounded) Baire class $ 1$ functions and we show their essential equivalence. This leads to a natural classification of the class of bounded Baire class $ 1$ functions ${\mathcal{B}_1}$ in a transfinite hierarchy $ \mathcal{B}_1^\xi (\xi < {\omega _1})$ of "small" Baire classes, for which (for example) an analysis similar to the Hausdorff-Kuratowski analysis of $ \Delta _2^0$ sets via transfinite differences of closed sets can be carried out. The notions of pseudouniform convergence of a sequence of functions and optimal convergence of a sequence of continuous functions to a Baire class $ 1$ function $f$ are introduced and used in this study.
Inequalities for eigenvalues of selfadjoint operators
Stephen M.
Hook
237-259
Abstract: We establish several inequalities for eigenvalues of selfadjoint operators in Hilbert space. The results are quite general. In particular, let $\Omega$ be a region in ${{\mathbf{R}}^n},\partial \Omega$ its boundary and $ \Delta$ the Laplace operator in $ {{\mathbf{R}}^n}$. Let $ p(x)$ be a polynomial of degree $m$ having nonnegative real coefficients. We show that if the problems (1) $- \Delta u = \lambda u$ in $\Omega ;u = 0$ on $ \partial \Omega$; (2) $p( - \Delta )\upsilon = \mu \upsilon$ in $\Omega ;\upsilon$ and its first $m - 1$ derivatives$ =0$ on$ \partial \Omega$; and (3) $ {( - \Delta )^m}w = vw$ in $ \Omega ;w$ and its first $m - 1$ derivatives$ =0$ on$ \partial \Omega$ are selfadjoint with discrete spectra of finite multiplicity ${\lambda _1} \leq {\lambda _2} \leq \cdots$ etc. then (4) $p(\Gamma _i^{1/m}) \geq {\mu _i} \geq p({\lambda _i})$ for each index $i$. The set of problems (1), (2), (3) and the result (4) is only one example of our more general result. The above problems (1), (2), and (3) can be thought of as related through the single operator given by the Laplacian. We also establish results for eigenvalues for unrelated operators. Let $ A$, $B$ and $A + B$ be selfadjoint on domains ${D_A},{D_B}$, and $ {D_{A + B}}$ with $ {D_{A + B}} \subseteq {D_A} \cap {D_B}$. If $A$, $B$, and $A + B$ have discrete spectra $\{ {\lambda _i}\} _{i = 1}^\infty ,\{ {\mu _i}\} _{i = 1}^\infty$ and $\{ {\Gamma _i}\} _{i = 1}^\infty$ arranged in ascending order, as above, then inequality (5) $\sum\nolimits_{i = 1}^n {{\Gamma _i}} \geq \sum\nolimits_{i = 1}^n {({\lambda _i} + {v_i})}$ is established for each positive integer $n$.
Algebraically invariant extensions of $\sigma$-finite measures on Euclidean space
Krzysztof
Ciesielski
261-273
Abstract: Let $G$ be a group of algebraic transformations of $ {{\mathbf{R}}^n}$, i,e., the group of functions generated by bijections of $ {{\mathbf{R}}^n}$ of the form $ ({f_1}, \ldots ,{f_n})$ where each ${f_i}$ is a rational function with coefficients in ${\mathbf{R}}$ in $n$-variables. For a function $\gamma :G \to (0,\infty )$ we say that a measure $ \mu$ on ${{\mathbf{R}}^n}$ is $\gamma$-invariant when $\mu (g[A]) = \gamma (g)\cdot\mu (A)$ for every $ g \in G$ and every $ \mu$-measurable set $ A$. We will examine the question: "Does there exist a proper $\gamma $-invariant extension of $ \mu ?$ We prove that if $ \mu$ is $\sigma $-finite then such an extension exists whenever $G$ contains an uncountable subset of rational functions $H \subset {({\mathbf{R}}({X_1}, \ldots ,{X_n}))^n}$ such that $\mu (\{ x:{h_1}(x) = {h_2}(x)\} ) = 0$ for all $ {h_1},{h_2} \in H,{h_1} \ne {h_2}$. In particular if $G$ is any uncountable subgroup of affine transformations of ${{\bf {R}}^n},\gamma (g{\text{)}}$ is the absolute value of the Jacobian of $g \in G$ and $\mu$ is a $\gamma$-invariant extension of the $ n$-dimensional Lebesgue measure then $\mu$ has a proper $\gamma$-invariant extension. The conclusion remains true for any $\sigma$-finite measure if $G$ is a transitive group of isometries of ${{\mathbf{R}}^n}$. An easy strengthening of this last corollary gives also an answer to a problem of Harazisvili.
Unique continuation for $\Delta+v$ and the C. Fefferman-Phong class
Sagun
Chanillo;
Eric
Sawyer
275-300
Abstract: We show that the strong unique continuation property holds for the inequality $\left\vert {\Delta u} \right\vert \leq \left\vert \upsilon \right\vert\left\vert u \right\vert$, where the potential $\upsilon (x)$ satisfies the C. Fefferman-Phong condition in a certain range of $p$ values. We also deal with the situation of $ u(x)$ vanishing at infinity. These are all consequences of appropriate Carleman inequalities.
Massey products in the cohomology of groups with applications to link theory
David
Stein
301-325
Abstract: Invariants of links in ${S^3}$ are developed using a modification of the Massey product of one-dimensional classes in the cohomology of certain groups. The theory yields two types of invariants, invariants which depend upon a collection of meridians, or basing, of a link, and invariants which do not. The invariants, which are independent of the basing, are compared with John Milnor's $\overline \mu $-invariants. For two component links, a collection of ostensibly based invariants is shown to be independent of the basing. If the linking number of the components of such a link is zero, the resulting invariants may be equivalent to the Sato-Levine-Cochran invariants.
On the nonimmersion of products of real projective spaces
Hyun-Jong
Song;
W. Stephen
Wilson
327-334
Abstract: In this paper we utilize $B{P^*}(\;)$, a generalized cohomology theory associated with the Brown-Peterson spectrum to prove a nonimmersion theorem for products of real projective spaces.
A Hurewicz spectral sequence for homology
David A.
Blanc
335-354
Abstract: For any connected space $ {\mathbf{X}}$ and ring $ R$, we describe a first-quadrant spectral sequence converging to ${\tilde H_*}({\bf {X}};R)$, whose $ {E^2}$-term depends only on the homotopy groups of $ {\mathbf{X}}$ and the action of the primary homotopy operations on them. We show that (for simply connected $ {\mathbf{X}}$) the $ {E^2}$-term vanishes below a line of slope $1/2$; computing part of the ${E^2}$-term just above this line, we find a certain periodicity, which shows, in particular, that this vanishing line is best possible. We also show how the differentials in this spectral sequence can be used to compute certain Toda brackets.
Weighted norm estimates for the Fourier transform with a pair of weights
Jan-Olov
Strömberg;
Richard L.
Wheeden
355-372
Abstract: We prove weighted norm inequalities of the form $\displaystyle {\left\Vert {\hat f} \right\Vert _{L_u^q}} \leq C{\left\Vert f \right\Vert _{H_\upsilon ^p}},\quad 0 < p \leq q < \infty ,$ for the Fourier transform on ${{\mathbf{R}}^n}$. For some weight functions $ \upsilon$, the Hardy space $H_\upsilon ^p$ on the right can be replaced by $L_\upsilon ^p$. The proof depends on making an atomic decomposition of $ f$ and using cancellation properties of the atoms.
Cells and the reflection representation of Weyl groups and Hecke algebras
J. Matthew
Douglass
373-399
Abstract: Let $\mathcal{H}$ be the generic algebra of the finite crystallographic Coxeter group $W$, defined over the ring $ \mathbb{Q}[{u^{1/2}},{u^{ - 1/2}}]$. First, the two-sided cell corresponding to the reflection representation of $\mathcal{H}$ is shown to consist of the nonidentity elements of $W$ having a unique reduced expression. Next, the matrix entries of this representation are computed in terms of certain Kazhdan-Lusztig polynomials. Finally, the Kazhdan-Lusztig polynomials just mentioned are described in case $W$ is of type ${{\text{A}}_{l - 1}}$ or ${{\text{B}}_l}$.
A one-phase hyperbolic Stefan problem in multi-dimensional space
De Ning
Li
401-415
Abstract: The hyperbolic heat transfer model is obtained by replacing the classical Fourier's law with the relaxation relation $ \tau \vec qt + \vec q = - k\nabla T$. The sufficient and necessary conditions are derived for the local existence and uniqueness of classical solutions for multi- ${\text{D}}$ Stefan problem of hyperbolic heat transfer model where phase change is accompanied with delay of latent heat storage.
Real analytic desingularization and subanalytic sets: an elementary approach
H. J.
Sussmann
417-461
Abstract: We give a proof of a theorem on desingularization of real-analytic functions which is a weaker version of H. Hironaka's result, but has the advantage of being completely self-contained and elementary, and not involving any machinery from algebraic geometry. We show that the basic facts about subanalytic sets can be proved from this result.
On linear topological properties of $H\sp 1$ on spaces of homogeneous type
Paul F. X.
Müller
463-484
Abstract: Let $(X,d,\mu )$ be a space of homogeneous type. Let $B = \{ x \in X:\mu \{ x\} = 0\}$, then $\mu (B) > 0$ implies that ${H^1}(X,d,\mu )$ contains a complemented copy of ${H^1}(\delta )$. This applies to Hardy spaces $ {H^1}(\partial \Omega ,d,\omega )$ associated to weak solutions of uniformly elliptic operators in divergence form. Under smoothness assumptions of the coefficients of the elliptic operators, we obtain that ${H^1}(\partial \Omega ,d,\omega )$ is isomorphic to ${H^1}(\delta )$.
Equivariant BP-cohomology for finite groups
N.
Yagita
485-499
Abstract: The Brown-Peterson cohomology rings of classifying spaces of finite groups are studied, considering relations to the other generalized cohomology theories. In particular, ${\operatorname{BP} ^{\ast}}(M)$ are computed for minimal nonabelian $p$-groups $M$. As an application, we give a necessary condition for the existence of nonabelian $p$-subgroups of compact Lie groups.
The enumerative geometry of plane cubics. I. Smooth cubics
Paolo
Aluffi
501-539
Abstract: We construct a variety of complete plane cubics by a sequence of five blow-ups over $ {\mathbb{P}^9}$. This enables us to translate the problem of computing characteristic numbers for a family of plane cubics into one of computing five Segre classes, and to recover classic enumerative results of Zeuthen and Maillard.
Geometrical implications of certain infinite-dimensional decompositions
N.
Ghoussoub;
B.
Maurey;
W.
Schachermayer
541-584
Abstract: We investigate the connections between the "global" structure of a Banach space (i.e. the existence of certain finite and infinite dimensional decompositions) and the geometrical properties of the closed convex bounded subsets of such a space (i.e. the existence of extremal and other topologically distinguished points). The global structures of various--supposedly pathological-- examples of Banach spaces constructed by R. C. James turn out to be more "universal" than expected. For instance James-tree-type (resp. James-matrix-type) decompositions characterize Banach spaces with the Point of Continuity Property (resp. the Radon-Nikodým Property). Moreover, the Convex Point of Continuity Property is stable under the formation of James-infinitely branching tree-type "sums" of infinite dimensional factors. We also give several counterexamples to various questions relating some topological and geometrical concepts in Banach spaces.
Semisimple representations of quivers
Lieven
Le Bruyn;
Claudio
Procesi
585-598
Abstract: We discuss the invariant theory of the variety of representations of a quiver and present generators and relations. We connect this theory of algebras with a trace satisfying a formal Cayley-Hamilton identity
The domain spaces of quasilogarithmic operators
M.
Cwikel;
B.
Jawerth;
M.
Milman
599-609
Abstract: The construction of intermediate Banach spaces in interpolation theory and the study of commutator inequalities in this context are closely related to certain nonlinear operators $ \Omega$. Here an explicit characterization of the domain spaces of these operators $\Omega$ is obtained, and the characterization is related to logarithmic Sobolev inequalities.
The extreme projections of the regular simplex
P.
Filliman
611-629
Abstract: The largest and smallest projections of the regular $n$-dimensional simplex into a $ k$-dimensional subspace are determined for certain values of $n$ and $k$. These results suggest that the smallest $ k$-dimensional projection and the largest $(n - k)$-dimensional projection occur in orthogonal subspaces of ${R^n}$.
The spectra and commutants of some weighted composition operators
James W.
Carlson
631-654
Abstract: An operator $ {T_{ug}}$ on a Hilbert space $H$ of functions on a set $X$ defined by ${T_{ug}}(f) = u(f \circ g)$, where $f$ is in $H,\;u:X \to {\mathbf{C}}$ and $ g:X \to X$, is called a weighted composition operator. In this paper $X$ is the set of integers and $H = {L^2}({\mathbf{Z}},\mu )$, where $\mu$ is a measure whose sigma-algebra is the power set of $ {\mathbf{Z}}$. One distinguished space is ${l^2} = {L^2}({\mathbf{Z}},\mu )$, where $\mu$ is counting measure. The most important results given here are the determination of the spectrum of ${T_{ug}}$ on ${l^2}$ and a characterization of the commutant of ${T_g}$ on ${L^2}({\mathbf{Z}},\mu )$. To obtain many of the results it was necessary to assume the function $ g$ to be one-to-one except on a finite subset of the integers.
Rational approximations to L-S category and a conjecture of Ganea
Barry
Jessup
655-660
Abstract: The rational version of Ganea's conjecture for L-S category, namely that $\operatorname{cat} (S \times {\Sigma ^k}) = \operatorname{cat} (S) + 1$, if $S$ is a rational space and ${\Sigma ^k}$ denotes the $k$-sphere, is still open. Recently, a module type approximation to $ \operatorname{cat} (S)$, was introduced by Halperin and Lemaire. We have previously shown that $ M\operatorname{cat}$ satisfies Ganea's conjecture. Here we show that for $ (r - 1)$ connected $ S$, if $M\operatorname{cat} (S)$ is at least $\dim S/2r$, then $ M\operatorname{cat} (S) = \operatorname{cat} (S)$. This yields Ganea's conjecture for these spaces. We also extend other properties of $ M\operatorname{cat}$, previously unknown for cat, to these spaces.
Metric transforms and Euclidean embeddings
M.
Deza;
H.
Maehara
661-671
Abstract: It is proved that if $0 \leqslant c \leqslant 0.72/n$ then for any $n$-point metric space $(X,d)$, the metric space $(X,{d^c})$ is isometrically embeddable into a Euclidean space. For $6$-point metric space, $c = \tfrac{1} {2}{\log _2}\tfrac{3} {2}$ is the largest exponent that guarantees the existence of isometric embeddings into a Euclidean space. Such largest exponent is also determined for all $ n$-point graphs with "truncated distance".
Weak stability in the global $L\sp 1$-norm for systems of hyperbolic conservation laws
Blake
Temple
673-685
Abstract: We prove that solutions for systems of two conservation laws which are generated by Glimm's method are weakly stable in the global ${L^1}$-norm. The method relies on a previous decay result of the author, together with a new estimate for the ${L^1}$ Lipschitz constant that relates solutions at different times. The estimate shows that this constant can be bounded by the supnorm of the solution, and is proved for any number of equations. The techniques do not rely on the existence of a family of entropies, and moreover the results would generalize immediately to more than two equations if one were to establish the stability of solutions in the supnorm for more than two equations.
Enlacements du mouvement brownien autour des courbes de l'espace
Jean-François
Le Gall;
Marc
Yor
687-722
Abstract: Limit theorems are proved for the winding numbers of a three-dimensional Brownian motion around certain curves in space. In particular, the joint asymptotic distribution of the winding numbers around two curves is obtained. This joint distribution generalizes the asymptotic law of the winding numbers of a planar Brownian motion around two points, which has recently been given by Pitman and Yor. The limiting distributions are closely related to the time spent by a linear Brownian motion above and below a multiple of its maximum process. Proofs rely on stochastic calculus for continuous semi-martingales.
Perturbed dynamical systems with an attracting singularity and weak viscosity limits in Hamilton-Jacobi equations
B.
Perthame
723-748
Abstract: We give a new PDE proof of the Wentzell-Freidlin theorem concerning small perturbations of a dynamical system \begin{displaymath}\begin{gathered}{L_\varepsilon }{u_\varepsilon } = - \tfrac{\... ... \quad {\text{on}}\;\partial \Omega . \end{gathered} \end{displaymath} We prove that, if $b$ has a single attractive singular point, ${u_\varepsilon }$ converges uniformly on compact subsets of $\Omega$, and with an exponential decay, to a constant $\mu$, and we determine $\mu$. We also treat the case of Neumann boundary condition. In order to do so, we perform the asymptotic analysis for some ergodic measure which leads to a study of the viscosity limit of a Hamilton-Jacobi equation. This is achieved under very general assumptions by using a weak formulation of the viscosity limits of these equations. Résumé. Nous donnons une nouvelle preuve, par des méthodes EDP, du théorème de Wentzell-Freidlin concernant les petites perturbations d'un système dynamique: \begin{displaymath}\begin{gathered}{L_\varepsilon }{u_\varepsilon } = - \tfrac{\... ...\quad {\text{sur}}\;\partial \Omega . \end{gathered} \end{displaymath} Nous prouvons que, si $b$ a un seul point singulier attractif, alors ${u_\varepsilon }$ converge vers une constant $\mu$, uniformément sur tout compact, et avec une vitesse exponentielle. Nous déterminons $ \mu$. Nous traitons aussi le cas de conditions aux limites de Neuman. Pour cela, nous faisons l'analyse asymptotique d'une mesure ergodique intervenant naturellement dans le problème, ce qui revient à étudier la limite par viscosité évanescente dans une équation de Hamilton-Jacobi. Ceci est réalisé sous des hypothèses très générales gâce à un passage à la limite faible dans cette équation.
Prime ideals in differential operator rings. Catenarity
K. A.
Brown;
K. R.
Goodearl;
T. H.
Lenagan
749-772
Abstract: Let $R$ be a commutative algebra over the commutative ring $k$, and let $\Delta = \{ {\delta _1}, \ldots ,{\delta _n}\}$ be a finite set of commuting $k$-linear derivations from $R$ to $R$. Let $T = R[{\theta _1}, \ldots ,{\theta _n};{\delta _1}, \ldots ,{\delta _n}]$ be the corresponding ring of differential operators. We define and study an isomorphism of left $ R$-modules between $ T$ and its associated graded ring $ R[{x_1}, \ldots ,{x_n}]$, a polynomial ring over $R$. This isomorphism is used to study the prime ideals of $T$, with emphasis on the question of catenarity. We prove that $T$ is catenary when $R$ is a commutative noetherian universally catenary $k$-algebra and one of the following cases occurs: (A) $k$ is a field of characteristic zero and $ \Delta$ acts locally finitely; (B) $k$ is a field of positive characteristic; (C) $ k$ is the ring of integers, $R$ is affine over $k$, and $\Delta$ acts locally finitely.
The Mourre estimate for dispersive $N$-body Schr\"odinger operators
Jan
Dereziński
773-798
Abstract: We prove the Mourre estimate for a certain class of dispersive $ N$-body Schrödinger operators. Using this estimate we derive some properties of those operators such as the local finiteness of the finite spectrum and the absence of the singular continuous spectrum.
The Schubert calculus, braid relations, and generalized cohomology
Paul
Bressler;
Sam
Evens
799-811
Abstract: Let $X$ be the flag variety of a compact Lie group and let $ {h^{\ast}}$ be a complex-oriented generalized cohomology theory. We introduce operators on $ {h^{\ast}}(X)$ which generalize operators introduced by Bernstein, Gel'fand, and Gel'fand for rational cohomology and by Demazure for $K$-theory. Using the Becker-Gottlieb transfer, we give a formula for these operators, which enables us to prove that they satisfy braid relations only for the two classical cases, thereby giving a topological interpretation of a theorem proved by the authors and extended by Gutkin.
Construction by isotopy. II
Daniel S.
Silver
813-823
Abstract: Construction by isotopy is a technique introduced by Iain R. Aitchison for obtaining doubly slice fibered knots in any dimension. We show that if $k$ is any doubly slice fibered $(n - 2)$-knot, $n \geqslant 5$, such that ${\pi _1}({S^n} - k) \cong Z$, then $k$ is constructible by isotopy. We also prove that the $m$-twist-spin of any doubly slice knot is constructible by isotopy. Consequently, there exists a double slice knot constructible by isotopy that is not the double of any disk knot. We also give an example of a doubly slice fibered $6$-knot that is not constructible by isotopy.
Application of the generalized Weierstrass preparation theorem to the study of homogeneous ideals
Mutsumi
Amasaki
1-43
Abstract: The system of Weierstrass polynomials, defined originally for ideals in convergent power series rings, together with its sequence of degrees allows us to analyze a homogeneous ideal directly. Making use of it, we study local cohomology modules, syzygies, and then graded Buchsbaum rings. Our results give a formula which to some extent clarifies the connection among the matrices appearing in the free resolution starting from a system of Weierstrass polynomials, a rough classification of graded Buchsbaum rings in the general case and a complete classification of graded Buchsbaum integral domains of codimension two.
Invariant tori for the billiard ball map
Valery
Kovachev;
Georgi
Popov
45-81
Abstract: For an $ n$-dimensional domain $\Omega (n \geq 3)$ with a smooth boundary which is strictly convex in a neighborhood of an elliptic closed geodesic $ \mathcal{O}$, the existence of a family of invariant tori for the billiard ball map with a positive measure is proved under the assumptions of nondegeneracy and $N$-elementarity, $N \geq 5$, of the corresponding to $\mathcal{O}$ Poincaré map. Moreover, the conjugating diffeomorphism constructed is symplectic. An analogous result is obtained in the case $n=2$. It is shown that the lengths of the periodic geodesics determine uniquely the invariant curves near the boundary and the billiard ball map on them up to a symplectic diffeomorphism.
Replacing homotopy actions by topological actions. II
Larry
Smith
83-90
Abstract: A homotopy action of a group $G$ on a space $X$ is a homomorphism from $G$ to the group ${\operatorname{HAUT}}(X)$ of homotopy classes of homotopy equivalences of $X$. George Cooke developed an obstruction theory to determine if a homotopy action is equivalent up to homotopy to a topological action. The question studied in this paper is: Given a diagram of spaces with homotopy actions of $G$ and maps between them that are equivariant up to homotopy, when can the diagram be replaced by a homotopy equivalent diagram of $G$-spaces and $G$-equivariant maps? We find that the obstruction theory of Cooke has a natural extension to this context.
Representing sets of ordinals as countable unions of sets in the core model
Menachem
Magidor
91-126
Abstract: We prove the following theorems. Theorem 1 $(\neg {0^\char93 })$. Every set of ordinals which is closed under primitive recursive set functions is a countable union of sets in $L$. Theorem 2. (No inner model with an Erdàs cardinal, i.e. $ \kappa \to {({\omega _1})^{ < \omega }}$.) For every ordinal $ \beta$, there is in $K$ an algebra on $\beta$ with countably many operations such that every subset of $\beta$ closed under the operations of the algebra is a countable union of sets in $K$.
Laws of trigonometry on ${\rm SU}(3)$
Helmer
Aslaksen
127-142
Abstract: The orbit space of congruence classes of triangles in $SU(3)$ has dimension $8$. Each corner is given by a pair of tangent vectors $(X,Y)$, and we consider the $8$ functions $ {\text{tr}}{X^2},i{\text{tr}}{X^3},{\text{tr}}{Y^2},i{\text{tr}}{Y^3},{\text{tr}}XY,i{\text{tr}}{X^2}Y,i{\text{tr}}X{Y^2}$ and ${\text{tr}}{X^2}{Y^2}$ which are invariant under the full isometry group of $SU(3)$. We show that these $8$ corner invariants determine the isometry class of the triangle. We give relations (laws of trigonometry) between the invariants at the different corners, enabling us to determine the invariants at the remaining corners, including the values of the remaining side and angles, if we know one set of corner invariants. The invariants that only depend on one tangent vector we will call side invariants, while those that depend on two tangent vectors will be called angular invariants. For each triangle we then have $ 6$ side invariants and $ 12$ angular invariants. Hence we need $18 - 8 = 10$ laws of trigonometry. If we restrict to $SU(2)$, we get the cosine laws of spherical trigonometry. The basic tool for deriving these laws is a formula expressing $ {\text{tr}}({\operatorname{exp}}X{\operatorname{exp}}Y)$ in terms of the corner invariants.
Topological equivalence of foliations of homogeneous spaces
Dave
Witte
143-166
Abstract: For $i = 1,2$, let $ {\Gamma _i}$ be a lattice in a connected Lie group ${G_i}$, and let ${X_i}$ be a connected Lie subgroup of ${G_i}$. The double cosets ${\Gamma _i}g{X_i}$ provide a foliation ${\mathcal{F}_i}$ of the homogeneous space ${\Gamma _i}\backslash {G_i}$. Assume that ${X_1}$ and ${X_2}$ are unimodular and that ${\mathcal{F}_1}$ has a dense leaf. If $ {G_1}$ and ${G_2}$ are semisimple groups to which the Mostow Rigidity Theorem applies, or are simply connected nilpotent groups (or are certain more general solvable groups), we use an idea of D. Benardete to show that any topological equivalence of ${\mathcal{F}_1}$ and ${\mathcal{F}_2}$ must be the composition of two very elementary maps: an affine map and a map that takes each leaf to itself.
Rapidly decreasing functions in reduced $C\sp *$-algebras of groups
Paul
Jolissaint
167-196
Abstract: Let $\Gamma$ be a group. We associate to any length-function $L$ on $\Gamma$ the space $H_L^\infty (\Gamma )$ of rapidly decreasing functions on $\Gamma$ (with respect to $L$), which coincides with the space of smooth functions on the $k$-dimensional torus when $\Gamma = {{\bf {Z}}^k}$. We say that $\Gamma$ has property (RD) if there exists a length-function $L$ on $\Gamma$ such that $H_L^\infty (\Gamma )$ is contained in the reduced $ {C^*}$-algebra $C_r^*(\Gamma )$ of $\Gamma$. We study the stability of property (RD) with respect to some constructions of groups such as subgroups, over-groups of finite index, semidirect and amalgamated products. Finally, we show that the following groups have property (RD): (1) Finitely generated groups of polynomial growth; (2) Discrete cocompact subgroups of the group of all isometries of any hyperbolic space.
The number of solutions of norm form equations
Wolfgang M.
Schmidt
197-227
Abstract: A norm form is a form $ F({X_1}, \ldots ,{X_n})$ with rational coefficients which factors into linear forms over $ {\mathbf{C}}$ but is irreducible or a power of an irreducible form over ${\mathbf{Q}}$. It is known that a nondegenerate norm form equation $F({x_1}, \ldots ,{x_n}) = m$ has only finitely many solutions $({x_1}, \ldots ,{x_n}) \in {{\mathbf{Z}}^n}$. We derive explicit bounds for the number of solutions. When $ F$ has coefficients in ${\mathbf{Z}}$, these bounds depend only on $ n$, $m$ and the degree of $F$, but are independent of the size of the coefficients of $F$.
Almost periodic operators in ${\rm VN}(G)$
Ching
Chou
229-253
Abstract: Let $G$ be a locally compact group, $ A(G)$ the Fourier algebra of $G$, $B(G)$ the Fourier-Stieltjes algebra of $ G$ and ${\text{VN}}(G)$ the von Neumann algebra generated by the left regular representation $ \lambda$ of $ G$. Then $A(G)$ is the predual of ${\text{VN}}(G)$; $ {\text{VN}}(G)$ is a $ B(G)$-module and $ A(G)$ is a closed ideal of $ B(G)$. Let ${\text{AP}}(\hat G) = \{ T \in {\text{VN}}(G):u \mapsto u \cdot T$ is a compact operator from $A(G)$ into ${\text{VN}}(G)\}$, the space of almost periodic operators in $ {\text{VN}}(G)$. Let $C_\delta ^*(G)$ be the ${C^*}$-algebra generated by $\{ \lambda (x):x \in G\} $. Then $ C_\delta ^*(G) \subset {\text{AP}}(\hat G)$. For a compact $G$, let $E$ be the rank one operator on ${L^2}(G)$ that sends $h \in {L^2}(G)$ to the constant function $\int {h(x)dx}$. We have the following results: (1) There exists a compact group $G$ such that $E \in$ AP$(\hat G)\backslash C_\delta ^*(G)$. (2) For a compact Lie group $G$, $E \in {\text{AP(}}\hat G{\text{)}} \Leftrightarrow E \in C_\delta ^*(G) \Leftrightarrow {L^\infty }(G)$ has a unique left invariant mean $\Leftrightarrow G$ is semisimple. (3) If $ G$ is an extension of a locally compact abelian group by an amenable discrete group then ${\text{AP}}(\hat G) = C_\delta ^*(G)$. (4) Let $ G = {{\mathbf{F}}_r}$, the free group with $r$ generators, $ 1 < r < \infty$. If $ T \in {\text{VN}}(G)$ and $u \mapsto u \cdot T$ is a compact operator from $B(G)$ into $ {\text{VN}}(G)$ then $T \in C_\delta ^*(G)$.
Complex interpolation of normed and quasinormed spaces in several dimensions. II. Properties of harmonic interpolation
Zbigniew
Slodkowski
255-285
Abstract: This paper is a continuation of the study of harmonic interpolation families of normed or quasinormed spaces parametrized by points of a domain in $ {{\mathbf{C}}^k}$. It is shown, among other things, that each of the following properties holds for all the intermediate quasinormed spaces, if it holds for all given boundary spaces: (1) being a normed space; (2) being a Hilbert space; (3) satisfying the triangle inequality by the $ r$th power of the quasinorm; (4) being uniformly convex; and (5) being uniformly smooth. As a principal tool, the notion of a harmonic set valued function (a generalization of analytic multifunction) is introduced and studied.
Extensions of valuation rings in central simple algebras
H.-H.
Brungs;
J.
Gräter
287-302
Abstract: Certain subrings $ R$ of simple algebras $ Q$, finite dimensional over their center $K$, are studied. These rings are called $ Q$-valuation rings since they share many properties with commutative valuation rings. Let $V$ be a valuation ring of $K$, the center of $Q$, and let $ \mathcal{R}$ be the set of $ Q$-valuation rings $ R$ in $Q$ with $ R \cap K = V$, then $ \left\vert \mathcal{R} \right\vert \geq 1$. This extension theorem, which does not hold if one considers only total valuation rings, was proved by N. I. Dubrovin. Here, first a somewhat different proof of this result is given and then information about the set $ \mathcal{R}$ is obtained. Theorem. The elements in $\mathcal{R}$ are conjugate if $ V$ has finite rank. Theorem. The elements in $\mathcal{R}$ are total valuation rings if $\mathcal{R}$ contains one total valuation ring. In this case $Q$ is a division ring. Theorem. $\mathcal{R}$ if $ \mathcal{R}$ contains an invariant total valuation ring.
A fixed point theorem for weakly chainable plane continua
Piotr
Minc
303-312
Abstract: In this paper the fixed point theorem is proven for every plane acyclic continuum $X$ with the property that every indecomposable continuum in the boundary of $X$ is contained in a weakly chainable subcontinuum of $X$.
Zero integrals on circles and characterizations of harmonic and analytic functions
Josip
Globevnik
313-330
Abstract: We determine the kernels of two circular Radon transforms of continuous functions on an annulus and use this to obtain a characterization of harmonic functions in the open unit disc which involves Poisson averages over circles computed at only one point of the disc and to obtain a version of Morera's theorem which involves only the circles which surround the origin.
Eventual finite order generation for the kernel of the dimension group representation
J. B.
Wagoner
331-350
Abstract: The finite order generation problem (FOG) in symbolic dynamics asks whether every element in the kernel of the dimension group representation of a subshift of finite type $({X_A},{\sigma _A})$ is a product of elements of finite order in the group $ {\operatorname{Aut}}({\sigma _A})$ of homeomorphisms of ${X_A}$ commuting with $ {\sigma _A}$. We study the space of strong shift equivalences over the nonnegative integers, and the first application is to prove Eventual FOG which says that every inert symmetry of ${\sigma _A}$ is a product of finite order homeomorphisms of ${X _A}$ commuting with sufficiently high powers of ${\sigma _A}$. Then we discuss the relation of FOG to Williams' lifting problem (LIFT) for symmetries of fixed points. In particular, either FOG or LIFT is false. Finally, we also discuss $p$-adic convergence and other implications of Eventual FOG for gyration numbers.
Extending $H\sp p$ functions from subvarieties to real ellipsoids
Kenzō
Adachi
351-359
Abstract: Let $\Omega$ be a domain in ${C^n}$ which is a somewhat generalized type of the real ellipsoid. Let $V$ be a subvariety in $\Omega$ which intersects $\partial \Omega$ transversally. Then there exists an operator $E:{H^p}(V) \to {H^p}(\Omega )$ satisfying $Ef{\vert _\nu } = f$.
Centers of generic Hecke algebras
Lenny K.
Jones
361-392
Abstract: Let $W$ be a Weyl group and let $W'$ be a parabolic subgroup of $ W$. Define $A$ as follows: $\displaystyle A = R{ \otimes _{{\mathbf{Q}}[u]}}\mathcal{A}(W)$ where $\mathcal{A}(W)$ is the generic algebra of type $ {A_n}$ over ${\mathbf{Q}}[u]$ an indeterminate, associated with the group $W$, and $R$ is a $ {\mathbf{Q}}[u]$-algebra, possibly of infinite rank, in which $u$ is invertible. Similarly, we define $A'$ associated with $W'$. Let $M$ be an $A - A$ bimodule, and let $b \in M$. Define the relative norm [14] $ b \in {Z_M}(A') = \{ m \in M\vert ma' = a'm\quad \forall a' \in A'\}$, then $\alpha = ({k_1},{k_2}, \ldots ,{k_z})$ be a partition of $n$ and let ${S_\alpha } = \Pi _{i = 1}^Z{S_{{k_i}}}$ be a "left-justified" parabolic subgroup of ${S_n}$ of shape $\alpha$. Define $\displaystyle {b_\alpha } = {N_{{S_n},{S_\alpha }}}({\mathcal{N}_\alpha })$ , where $\displaystyle {\mathcal{N}_\alpha } = \prod\limits_{i = 1}^z {{N_{{S_{{k_i} - 1}},{S_1}}}({a_{{w_i}}})}$ with $ {w_i}$ a ${k_i}$-cycle of length ${k_i} - 1$ in $ {S_{{k_i}}}$. Then the main result of this paper is Theorem. The set $\{ {b_\alpha }\vert\alpha \vdash n\}$ is a basis for $ {Z_{A({S_n})}}(A({S_n}))$ over ${\mathbf{Q}}[u,{u^{ - 1}}]$. Remark. The norms $ {b_\alpha }$ in $ {Z_{A({S_n})}}(A({S_n}))$ are analogs of conjugacy class sums in the center of $ {\mathbf{Q}}{S_n}$ and, in fact, specialization of these norms at $ u = 1$ gives the standard conjugacy class sum basis of the center of ${\mathbf{Q}}{S_n}$ up to coefficients from ${\mathbf{Q}}$.
Topological spaces whose Baire measure admits a regular Borel extension
Haruto
Ohta;
Ken-ichi
Tamano
393-415
Abstract: A completely regular, Hausdorff space $X$ is called a Măík space if every Baire measure on $X$ admits an extension to a regular Borel measure. We answer the questions about Măík spaces asked by Wheeler [29] and study their topological properties. In particular, we give examples of the following spaces: A locally compact, measure compact space which is not weakly Bairedominated; i.e., it has a sequence $ {F_n} \downarrow \emptyset$ of regular closed sets such that $ { \cap _{n \in \omega }}{B_n} \ne \emptyset$ whenever ${B_n}$'s are Baire sets with ${F_n} \subset {B_n}$; a countably paracompact, non-Măík space; a locally compact, non-Măík space $X$ such that the absolute $E(X)$ is a Măík space; and a locally compact, Măík space $ X$ for which $ E(X)$ is not. It is also proved that Michael's product space is not weakly Baire-dominated.
On the Kummer congruences and the stable homotopy of $B$U
Andrew
Baker;
Francis
Clarke;
Nigel
Ray;
Lionel
Schwartz
385-432
Abstract: We study the torsion-free part of the stable homotopy groups of the space $ BU$, by considering upper and lower bounds. The upper bound is furnished by the ring $ P{K_{\ast}}(BU)$ of coaction primitives into which $\pi _{\ast}^S(BU)$ is mapped by the complex $ K$-theoretic Hurewicz homomorphism $\displaystyle \pi _{\ast}^S(BU) \to P{K_{\ast}}(BU).$ We characterize $P{K_{\ast}}(BU)$ in terms of symmetric numerical polynomials and describe systematic families of elements by utilizing the classical Kummer congruences among the Bernoulli numbers. For a lower bound we choose the ring of those framed bordism classes which may be represented by singular hypersurfaces in $ BU$. From among these we define families of classes constructed from regular neighborhoods of embeddings of iterated Thom complexes in Euclidean space. Employing techniques of duality theory, we deduce that these two families correspond, except possibly in the lowest dimensions, under the Hurewicz homomorphism, which thus provides a link between the algebra and the geometry. In the course of this work we greatly extend certain $e$-invariant calculations of J. F. Adams.
Negative scalar curvature metrics on noncompact manifolds
John
Bland;
Morris
Kalka
433-446
Abstract: In this paper we prove that every noncompact smooth manifold admits a complete metric of constant negative scalar curvature.
Multiplicity of the adjoint representation in simple quotients of the enveloping algebra of a simple Lie algebra
Anthony
Joseph
447-491
Abstract: Let $\mathfrak{g}$ be a complex simple Lie algebra, $\mathfrak{h}$ a Cartan subalgebra and $U(\mathfrak{g})$ the enveloping algebra of $\mathfrak{g}$. We calculate for each maximal two-sided ideal $ {J_{\max }}(\lambda ):\lambda \in {\mathfrak{h}^{\ast}}$ of $ U(\mathfrak{g})$ the number of times the adjoint representation occurs in $U(\mathfrak{g})/{J_{\max }}(\lambda )$. This is achieved by reduction via the Kazhdan-Lusztig polynomials to the case when $\lambda$ lies on a corner, i.e. is a multiple of a fundamental weight. Remarkably in this case one can always present $U(\mathfrak{g})/{J_{\max }}(\lambda )$ as a (generalized) principal series module and here we also calculate its Goldie rank as a ring which is a question of independent interest. For some of the more intransigent cases it was necessary to use recent very precise results of Lusztig on left cells. The results are used to show how a recent theorem of Gupta established for "nonspecial" $\lambda$ can fail if $\lambda$ is singular. Finally we give a quite efficient procedure for testing if an induced ideal is maximal.
Conjugacy classes whose square is an infinite symmetric group
Gadi
Moran
493-522
Abstract: Let ${X_\nu }$ be the set of all permutations $ \xi$ of an infinite set $ A$ of cardinality ${\aleph _\nu }$ with the property: every permutation of $A$ is a product of two conjugates of $\xi$. The set ${X_0}$ is shown to be the set of permutations $ \xi$ satisfying one of the following three conditions: (1) $ \xi$ has at least two infinite orbits. (2) $\xi$ has at least one infinite orbit and infinitely many orbits of a fixed finite size $n$. (3) $\xi$ has: no infinite orbit; infinitely many finite orbits of size $k,l$ and $k + l$ for some positive integers $k,l$; and infinitely many orbits of size $> 2$. It follows that $\xi \in {X_0}$ iff some transposition is a product of two conjugates of $\xi$, and $\xi$ is not a product $\sigma i$, where $\sigma$ has a finite support and $i$ is an involution. For $\nu > 0,\;\xi \in {X_\nu }$ iff $ \xi$ moves ${\aleph _\nu }$ elements, and satisfies (1), (2) or $(3')$, where $(3')$ is obtained from (3) by omitting the requirement that $\xi$ has infinitely many orbits of size $ > 2$. It follows that for $ \nu > 0,\;\xi \in {X_\nu }\;$ iff $\xi$ moves $ {\aleph _\nu }$ elements and some transposition is the product of two conjugates of $ \xi$. The covering number of a subset $X$ of a group $G$ is the smallest power of $X$ (if any) that equals $G$ [AH]. These results complete the classification of conjugacy classes in infinite symmetric groups with respect to their covering number.
Isometric dilations for infinite sequences of noncommuting operators
Gelu
Popescu
523-536
Abstract: This paper develops a dilation theory for $\{ {T_n}\} _{n = 1}^\infty $ an infinite sequence of noncommuting operators on a Hilbert space, when the matrix $[{T_1},{T_2}, \ldots ]$ is a contraction. A Wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for $\{ {T_n}\} _{n = 1}^\infty $ are obtained. Some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given. This results are used to extend the Sz.-Nagy-Foiaş lifting theorem to this noncommutative setting.
A new algebraic approach to microlocalization of filtered rings
Maria Jesus
Asensio;
Michel
Van den Bergh;
Freddy
Van Oystaeyen
537-553
Abstract: Using the construction of the Rees ring associated to a filtered ring we provide a description of the microlocalization of the filtered ring by using only purely algebraic techniques. The method yields an easy approach towards the study of exactness properties of the microlocalization functor. Every microlocalization at a regular multiplicative Ore set in the associated graded ring can be obtained as the completion of a localization at an Ore set of the filtered ring.
A note on local change of diffeomorphism
Mikiya
Masuda
555-566
Abstract: Let $D(M)$ be the group of pseudo-isotopy classes of orientation preserving diffeomorphisms of a compact manifold $M$ which restrict to the identity on $\partial M$. If a compact manifold $N$ of the same dimension as $M$ is embedded in $M$, then extending maps in $ D(N)$ as the identity on the exterior of $N$ defines a homomorphism $E:D(N) \to D(M)$. We ask if the kernel of $ E$ is finite and show that this is the case for special cases.
Behavior of polynomials of best uniform approximation
E. B.
Saff;
V.
Totik
567-593
Abstract: We investigate the asymptotic behavior of the polynomials $\{ {P_n}(f)\} _0^\infty$ of best uniform approximation to a function $f$ that is continuous on a compact set $K$ of the complex plane ${\mathbf{C}}$ and analytic in the interior of $ K$, where $K$ has connected complement. For example, we show that for "most" functions $ f$, the error $f - {P_n}(f)$ does not decrease faster at interior points of $K$ than on $K$ itself. We also describe the possible limit functions for the normalized error $(f - {P_n}(f))/{E_n}$, where ${E_n}: = \vert\vert f - {P_n}(f)\vert{\vert _K}$, and the possible limit distributions of the extreme points for the error. In contrast to these results, we show that "near best" polynomial approximants to $ f$ on $K$ exist that converge more rapidly at the interior points of $K$.
On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains
Catherine
Bandle;
Howard A.
Levine
595-622
Abstract: In this paper we study the first initial-boundary value problem for $ {u_t} = \Delta u + {u^p}$ in conical domains $D = (0,\infty ) \times \Omega \subset {R^N}$ where $ \Omega \subset {S^{N - 1}}$ is an open connected manifold with boundary. We obtain some extensions of some old results of Fujita, who considered the case $D = {R^N}$. Let $\lambda = - {\gamma _ - }$ where ${\gamma _ - }$ is the negative root of $ \gamma (\gamma + N - 2) = {\omega _1}$ and where $ {\omega _1}$ is the smallest Dirichlet eigenvalue of the Laplace-Beltrami operator on $\Omega$. We prove: If $1 < p < 1 + 2/(2 + \lambda )$, there are no nontrivial global solutions. If $1 < p < 1 + 2/\lambda$, there are no stationary solutions in $ D - \{ 0\}$ except $u \equiv 0$. If $1 + 2/\lambda < p < (N + 1)/(N - 3)$ (if $ N > 3$, arbitrary otherwise) there are singular stationary solutions $ {u_s}$. If $u(x,0) \leqslant {u_s}(x)$, the solutions are global. If $1 + 2/\lambda < p < (N + 2)/(N - 2)$ and $u(x,0) \leqslant {u_s}$, with $ u(x,0) \in C(\overline D )$, the solutions decay to zero. If $1 + 2/N < p$, there are global solutions. For $1 < p < \infty$, there are ${L^\infty }$ data of arbitrarily small norm, decaying exponentially fast at $ r = \infty$, for which the solution is not global. We show that if $ D$ is the exterior of a bounded region, there are no global, nontrivial, positive solutions if $ 1 < p < 1 + 2/N$ and that there are such if $ p > 1 + 2/N$. We obtain some related results for ${u_t} = \Delta u + \vert x{\vert^\sigma }{u^p}$ in the cone.
Regular transition functions and regular superprocesses
E. B.
Dynkin
623-634
Abstract: The class of regular Markov processes is very close to the class of right processes studied by Meyer, Getoor and others. We say that a transition function $p$ is regular if it is the transition function of a well-defined regular Markov process. A characterization of regular transition functions is given which implies that, if $p$ is regular, then the Dawson-Watanabe and the Fleming-Viot supertransition functions over $ p$ belong to the same class.
Complex approximation of real functions by reciprocals of polynomials
Daniel
Wulbert
635-652
Abstract: Characterizations are given for local and global best rational approximations to a real function. The characterizations are specialized to reciprocals of polynomials, where they are used to settle some conjectures and questions.
Interpolation between Sobolev and between Lipschitz spaces of analytic functions on starshaped domains
Emil J.
Straube
653-671
Abstract: We show that on a starshaped domain $\Omega$ in $ {\operatorname{C} ^n}$ (actually on a somewhat larger, biholomorphically invariant class) the $ {\mathcal{L}^p}$-Sobolev spaces of analytic functions form an interpolation scale for both the real and complex methods, for each $p,\;0 < p \leqslant \infty$. The case $p = \infty$ gives the Lipschitz scale; here the functor $ {(,)^{[\theta ]}}$ has to be considered (rather than ${(,)_{[\theta ]}}$).
The complex bordism of groups with periodic cohomology
Anthony
Bahri;
Martin
Bendersky;
Donald M.
Davis;
Peter B.
Gilkey
673-687
Abstract: Is is proved that if $BG$ is the classifying space of a group $ G$ with periodic cohomology, then the complex bordism groups $M{U_{\ast}}(BG)$ are obtained from the connective $ K$-theory groups $k{u_{\ast}}(BG)$ by just tensoring up with the generators of $ M{U_{\ast}}$ as a polynomial algebra over $ k{u_{\ast}}$. The explicit abelian group structure is also given. The bulk of the work is the verification when $G$ is a generalized quaternionic group.
Processes disjoint from weak mixing
S.
Glasner;
B.
Weiss
689-703
Abstract: We show that the family $ {\mathcal{W}^ \bot }$ of ergodic measure preserving transformations which are disjoint from every weakly mixing m.p.t. properly contains the family $ \mathcal{D}$ of distal ergodic m.p.t. In the topological case we show that $ \mathcal{P}\mathcal{I}$, the family of proximally isometric flows is properly contained in the family $\mathcal{M}({\mathcal{W}^ \bot })$ of multipliers for ${\mathcal{W}^ \bot }$.
The structure of an even liaison class
Giorgio
Bolondi;
Juan C.
Migliore
1-37
Abstract: We describe a structure called the Lazarsfeld-Rao property for even liaison classes in projective space. This property holds for many even liaison classes of curves in $ {{\mathbf{P}}^3}$. We give a procedure for showing that an even liaison class in codimension $2$ possesses this property, and we prove it for a family of even liaison classes in codimension $ 2$ in any $ {{\mathbf{P}}^n},\;n \geqslant 3$. However, we conjecture that it in fact holds for every even liaison class in codimension $ 2$, so we want to give consequences for an even liaison class that possesses this property. The main element in describing this structure is the notion of a basic double link. The Lazarsfeld-Rao property says that there exist minimal elements of the even liaison class and that any element of the even liaison class can be deformed to a curve obtained by a sequence of basic double links beginning with any minimal element. We show that there is a unique standard type of sequence for any given element of the even liaison class. As a result, we can express the even liaison class as a disjoint union of irreducible nonempty families parameterized by certain finite sequences of integers. The standard numerical invariants of the elements of any family can be computed from the associated sequence of integers. We apply this to surfaces in ${{\mathbf{P}}^4}$. Our main tool for these results is a deformation technique related to liaison in codimension $2$. We also study Schwartau's procedure of Liaison Addition in codimension $2$ from the point of view of vector bundles. Using this, we give a different sort of structure for an even liaison class with the Lazarsfeld-Rao property.
Generators for the bordism algebra of immersions
M. A.
Aguilar
39-51
Abstract: Let us denote by $ I(n,k)$ the group of bordism classes of immersions of closed smooth $n$-manifolds in closed smooth $ (n + k)$-manifolds $ (k > 0)$. We can make $I({\ast},k)$ into a graded algebra over the unoriented bordism ring. This algebra is polynomial. In this paper we give two sets of immersions which are polynomial generators.
On some limit theorems for continued fractions
Jorge D.
Samur
53-79
Abstract: As a consequence of previous results on mixing random variables, some functional limit theorems for quantities related to the continued fraction expansion of a random number in $ (0,1)$ are given.
Harmonic measure and radial projection
Donald E.
Marshall;
Carl
Sundberg
81-95
Abstract: Among all curves in the closed unit disk that meet every radius, there is one, ${\gamma _0}$, whose harmonic measure at the origin is minimal. We give an explicit description of ${\gamma _0}$ and compute its harmonic measure. We also give a quadratically convergent algorithm to compute the harmonic measure of one side of a rectangle at its center.
Remarks on approximation methods in degree theory
W.
Kryszewski;
B.
Przeradzki;
S.
Wereński
97-114
Abstract: An approximative approach to a generalized theory of the topological mapping degree is presented. Some new wide classes of operators acting in Banach spaces, which include $ A$-proper mappings of Petryshyn, are introduced and studied from the viewpoint of the homotopic properties of the topological degree. The results are applied in some existence aspects of abstract nonlinear equations.
The generalized Borel conjecture and strongly proper orders
Paul
Corazza
115-140
Abstract: The Borel Conjecture is the statement that $C = {[\mathbb{R}]^{ < {\omega _1}}}$, where $ C$ is the class of strong measure zero sets; it is known to be independent of ZFC. The Generalized Borel Conjecture is the statement that $C = {[\mathbb{R}]^{ < {\mathbf{c}}}}$. We show that this statement is also independent. The construction involves forcing with an $ {\omega _2}$-stage iteration of strongly proper orders; this latter class of orders is shown to include several well-known orders, such as Sacks and Silver forcing, and to be properly contained in the class of $\omega$-proper, $ {\omega ^\omega }$-bounding orders. The central lemma is the observation that A. W. Miller's proof that the statement $({\ast})$ "Every set of reals of power c can be mapped (uniformly) continuously onto $ [0,1]$" holds in the iterated Sacks model actually holds in several other models as well. As a result, we show for example that $ ({\ast})$ is not restricted by the presence of large universal measure zero $({{\text{U}}_0})$ sets (as it is by the presence of large $C$ sets). We also investigate the $ \sigma$-ideal $\mathcal{J} = \{ X \subset \mathbb{R}:X\;{\text{cannot be mapped uniformly continuously onto }}[0,1]\}$ and prove various consistency results concerning the relationships between $ \mathcal{J},\;{{\text{U}}_0}$, and AFC (where $\operatorname{AFC} = \{ X \subset \mathbb{R}:X\;{\text{is always first category\} }}$). These latter results partially answer two questions of J. Brown.
A generalisation of the matroid lift construction
Geoff
Whittle
141-159
Abstract: This paper introduces a general matroid-theoretic construction which includes, as special cases, elementary lifts of matroids and bias matroids of biased graphs. To perform the construction on a matroid $M$, it is necessary (but not sufficient) to have a submodular function inducing $ M$. Elementary lifts are obtained when the submodular function chosen is the rank function of $M$. We define what is meant by a $ k$-induced matroid. These matroids simultaneously generalise matroids of graphs, transversal matroids and Dilworth truncations. They are induced by a particularly natural class of submodular functions. The effect of the above construction on $k$-induced matroids using these natural submodular functions is studied. Results on minors of $k$-induced matroids and the matroids obtained from them using the construction are given.
On a problem of S. Mazur
László
Székelyhidi
161-164
Abstract: In this work a generalization of Mazur's problem concerning the continuity of linear functionals is given.
On the variety of plane curves of degree $d$ with $\delta$ nodes and $\kappa$ cusps
Pyung-Lyun
Kang
165-192
Abstract: Let ${{\mathbf{P}}^N}$ be the projective space which parametrizes all plane curves of degree $d$ and $V(d,\delta ,\kappa )$ the subvariety of ${{\mathbf{P}}^N}$ consisting of all reduced and irreducible plane curves of degree $d$ with $\delta$ nodes and $\kappa$ cusps as their only singularities. In this paper we prove that $V(d,\delta ,\kappa )$ is irreducible if $\kappa \leqslant 3$, except possibly when $\kappa = 3$ and $d = 5$ or $6$.
Quantization of curvature of harmonic two-spheres in Grassmann manifolds
Yunbo
Zheng
193-214
Abstract: Various pinching theorems for curvature of minimal two-spheres in Grassmann manifolds have been proved. In particular, we show that when the curvature is large, then the minimal map from ${S^2}$ into $G(m,N)$ must be either holomorphic or antiholomorphic. Also, minimal two-spheres of curvature $\kappa \geqslant 2$ in $ G(2,4)$ have been classified.
Complemented subspaces of products of Banach spaces
Paweł
Domański;
Augustyn
Ortyński
215-231
Abstract: It is proved that: (i) every complemented subspace in an infinite product of ${L_1}$-predual Banach spaces $\prod\nolimits_{i \in I} {{X_i}}$ is isomorphic to $Z \times {{\mathbf{K}}^\mathfrak{m}}$, where $ \dim {\mathbf{K}} = 1,\;\mathfrak{m} \leqslant \operatorname{card} I$ and $ Z$ is isomorphic to a complemented subspace of $ \prod\nolimits_{i \in J} {{X_i},\;J \subseteq I,\;Z}$ contains an isomorphic cop[ill] of $c_0^{\operatorname{card} J}$; (ii) every injective lcs (in particular, Fréchet) is of the form $Z \times {{\mathbf{K}}^\mathfrak{m}},\;\dim {\mathbf{K}} = 1$, where $Z$ has a fundamental family of seminorms of the cardinality $\tau$ and $Z$ contains an isomorphic copy of $l_\infty ^\mathfrak{r}$ (this is a generalization of Lindenstrauss' theorem on injective Banach spaces); (iii) whenever $ X \simeq {l_p},\;1 \leqslant p \leqslant \infty$, or $X \simeq {c_0}$, then every complemented subspace in a power $ {X^\mathfrak{m}}$ ( $\mathfrak{m}$ is an arbitrary cardinal number) is isomorphic to $ {X^\mathfrak{r}} \times {{\mathbf{K}}^\mathfrak{s}},\;\mathfrak{r} + s \leqslant \mathfrak{m}$ (a generalization of the results due to Lindenstrauss and Pełczyński for $ \mathfrak{m} = 1$).
Double shock fronts for hyperbolic systems of conservation laws in multidimensional space
An Ton
Bui;
De Ning
Li
233-250
Abstract: The existence of a unique double shock front for hyperbolic systems of conservation laws in several space variables is established, extending an earlier result of Metivier. An example of a double shock wave arising from physical applications is given.
$*$-differential identities of prime rings with involution
Chen-Lian
Chuang
251-279
Abstract: Main Theorem. Let $R$ be a prime ring with involution $ ^{\ast}$. Suppose that $\phi (x_i^{{\Delta _j}},{(x_i^{{\Delta _j}})^{\ast}}) = 0$ is a $ {\ast}$-differential identity for $R$, where $ {\Delta _j}$ are distinct regular words of derivations in a basis $ M$ with respect to a linear order $<$ on $M$. Then $\phi ({z_{ij}},z_{ij}^{\ast}) = 0$ is a ${\ast}$-generalized identity for $ R$, where $ {z_{ij}}$ are distinct indeterminates. Along with the Main Theorem above, we also prove the following: Proposition 1. Suppose that $^{\ast}$ is of the second kind and that $ C$ is infinite. Then $R$ is special. Proposition 2. Suppose that ${S_W}(V) \subseteq R \subseteq {L_W}(V)$. Then $Q$, the two-sided quotient ring of $ R$, is equal to $ {L_W}(V)$. Proposition 3 (Density theorem). Suppose that $ {}_DV$ and $ {W_D}$ are dual spaces with respect to the nondegenerate bilinear form $(,)$. Let $ {v_1}, \ldots ,{v_s},\;v_s^\prime , \ldots ,v_s^\prime \in V$ and ${u_1}, \ldots ,{u_t},\;u_1^\prime , \ldots ,u_t^\prime \in W$ be such that $\{ {v_1}, \ldots ,{v_s}\} $ is $ D$-independent in $ V$ and $\{ {u_1}, \ldots ,{u_t}\}$ is $D$-independent in $W$. Then there exists $a \in {S_W}(V)$ such that ${v_i}a = v_i^\prime \,(i = 1, \ldots ,s)$ and ${a^{\ast}}{u_j} = u_j^\prime \,(j = 1, \ldots ,t)$ if and only if $(v_i',{u_j}) = ({v_i},u_j')$ for $i = 1, \ldots ,s$ and $j = 1, \ldots ,t$. Proposition 4. Suppose that $R$ is a prime ring with involution $ ^{\ast}$ and that $ f$ is a $ {\ast}$-generalized polynomial. If $f$ vanishes on a nonzero ideal of $ R$, than $ f$ vanishes on $ Q$, the two-sided quotient ring of $R$.
Some explicit cases of the Selberg trace formula for vector valued functions
Jeffrey
Stopple
281-293
Abstract: The trace formula for $ SL(2,{\mathbf{Z}})$ can be developed for vector-valued functions which satisfy an automorphic condition involving a group representation $\pi$. This paper makes this version explicit for the class of representations which can be realized as representations of the finite group $PSL(2,{\mathbf{Z}}/q)$ for some prime $ q$. The body of the paper is devoted to computing, for the singular representations $ \pi$, the determinant of the scattering matrix $ \Phi (s,\pi )$ on which the applications depend. The first application is a version of the Roelcke-Selberg conjecture. This follows from known results once the scattering matrix is given. The study of representations of $SL(2,{\mathbf{Z}})$ in finite-dimensional vector spaces of (scalar-valued) holomorphic forms dates back to Hecke. Similar problems can be studied for vector spaces of Maass wave forms, with fixed level $ q$ and eigenvalue $ \lambda$. One would like to decompose the natural representation of $SL(2,{\mathbf{Z}})$ in this space, and count the multiplicities of its irreducible components. The eigenvalue estimate obtained for vector-valued forms is equivalent to an asymptotic count, as $\lambda \to \infty$, of these multiplicities.
Realization of the level two standard ${\rm sl}(2k+1,{\bf C})\sp \sim$-modules
Kailash C.
Misra
295-309
Abstract: In this paper we study the level two standard modules for the affine special linear Lie algebras. In particular, we give the vertex operator realizations of all level two standard modules for the affine special linear Lie algebras of odd rank.
Additive cohomology operations
Jeanne
Duflot
311-325
Abstract: The bigraded group $ \{ {H_i}({\Sigma _n},{\mathbf{Z}}/p)\}$ becomes a Hopf algebra, if multiplication is induced by restriction, and comultiplication is induced by transfer. Using Steenrod's method of considering elements of this bigraded group as $ \bmod {\text{ - }}p$ cohomology operations, the primitives of this Hopf algebra correspond to additive cohomology operations. In this paper we use the results known about the homology and cohomology of the symmetric groups and the operations they induce in $ \bmod {\text{ - }}p$ cohomology to write down two (additive) bases of the bigraded vector space of primitives of the above Hopf algebra.
On the dissipative evolution equations associated with the Zakharov-Shabat system with a quadratic spectral parameter
Jyh-Hao
Lee
327-336
Abstract: In this paper we derive some results for the Zakharov-Shabat system of the form $ dm/dx = {z^2}[J,m] + (zQ + P)m$; $J$ is diagonal and skew-Hermitian $ [8,10,12]$. Following the idea of R. Beals and R. R. Coifman, we estimate the wedge products of the columns of $m$ by ${L^2}$-norm of the potential $(Q,P)\,[4]$. By this result we have the global existence of the dissipative evolution equations associated with this spectral problem if the generic initial data $(Q(x,0),\,P(x,0)) = ({Q_0},{P_0})$ is of Schwartz class.
Extendible sets in Peano arithmetic
Stuart T.
Smith
337-367
Abstract: Let $\mathcal{A}$ be a structure and let $ U$ be a subset of $\vert\mathcal{A}\vert$. We say $U$ is extendible if whenever $\mathcal{B}$ is an elementary extension of $\mathcal{A}$, there is a $V \subseteq \vert\mathcal{B}\vert$ such that $(\mathcal{A},U) \prec (\mathcal{B},V)$. Our main results are: If $ \mathcal{M}$ is a countable model of Peano arithmetic and $U$ is a subset of $\vert\mathcal{M}\vert$, then $U$ is extendible iff $ U$ is parametrically definable in $ \mathcal{M}$. Also, the cofinally extendible subsets of $\vert\mathcal{M}\vert$ are exactly the inductive subsets of $ \vert\mathcal{M}\vert$. The end extendible subsets of $\vert\mathcal{M}\vert$ are not completely characterized, but we show that if $ \mathcal{N}$ is a model of Peano arithmetic of arbitrary cardinality and $ U$ is any bounded subset of $\mathcal{N}$, then $U$ is end extendible.
The intrinsic conformal structure and Gauss map of a light-like hypersurface in Minkowski space
Marek
Kossowski
369-383
Abstract: We begin by pointing out two subtleties in the global properties of hypersurfaces in Minkowski space which inherit a uniformly degenerate metric (i.e., the existence of global space-like sections and the notion of an icon; see Appendices 1 and 2). We then construct a Gauss map for such hypersurfaces and an intrinsic invariant. This leads us to results concerning light-like hypersurfaces which parallel known results concerning surfaces in Euclidean space.
Degree theory for equivariant maps. I
J.
Ize;
I.
Massabò;
A.
Vignoli
433-510
Abstract: A degree theory for equivariant maps is constructed in a simple geometrical way. This degree has all the basic properties of the usual degree theories and takes its values in the equivariant homotopy groups of spheres. For the case of a semifree ${S^1}$-action, a complete computation of these groups is given, the range of the equivariant degree is determined, and the general ${S^1}$-action is reduced to that special case. Among the applications one recovers and unifies both the degree for autonomous differential equations defined by Fuller [F] and the ${S^1}$-degree for gradient maps introduced by Dancer [Da]. Also, a simple but very useful formula of Nirenberg [N] is generalized (see Theorem 4.4(ii)).
Geometric quantization and the universal enveloping algebra of a nilpotent Lie group
Niels Vigand
Pedersen
511-563
Abstract: We study geometric quantization in connection with connected nilpotent Lie groups. First it is proved that the quantization map associated with a (real) polarized coadjoint orbit establishes an isomorphism between the space of polynomial quantizable functions and the space of polynomial quantized operators. Our methods allow noninductive proofs of certain basic facts from Kirillov theory. It is then shown how the quantization map connects with the universal enveloping algebra. This is the main result of the paper. Finally we show how one can explicitly compute global canonical coordinates on coadjoint orbits, and that this can be done simultaneously on all orbits contained in a given stratum of what we call "the fine $ \mathcal{F}$-stratification of the dual of the Lie algebra". This is a generalization of a result of M. Vergne about simultaneous canonical coodinates for orbits in general position.
Area integral estimates for caloric functions
Russell M.
Brown
565-589
Abstract: We study the relationship between the area integral and the parabolic maximal function of solutions to the heat equation in domains whose boundary satisfies a $ \left({\frac{1}{2},1}\right)$ mixed Lipschitz condition. Our main result states that the area integral and the parabolic maximal function are equivalent in $ {L^p}(\mu)$, $0 < p < \infty$. The measure $\mu$ must satisfy Muckenhoupt's ${A_\infty }$-condition with respect to caloric measure. We also give a Fatou theorem which shows that the existence of parabolic limits is a.e. (with respect to caloric measure) equivalent to the finiteness of the area integral.
The universal von Staudt theorems
Francis
Clarke
591-603
Abstract: We prove general forms of von Staudt's theorems on the Bernoulli numbers. As a consequence we are able to deduce strong versions of a number of congruences involving various generalisations of the Bernoulli numbers. For example we obtain an improved form of a congruence due to Hurwitz involving the Laurent series coefficients of the Weierstrass elliptic function associated with a square lattice.
Value functions on central simple algebras
Patrick J.
Morandi
605-622
Abstract: In this paper we study noncommutative valuation rings as defined by Dubrovin. While there is in general no valuation associated to a Dubrovin valuation ring, we show that there is a value function associated to any Dubrovin valuation ring integral over its center. By using value functions we obtain information on Dubrovin valuation rings in a tensor product, both generalizing and giving a much simpler proof of a result about valued division algebras. By being able to work directly with central simple algebras we gain new information about division algebras over Henselian fields.
Integral Dubrovin valuation rings
Patrick J.
Morandi;
Adrian R.
Wadsworth
623-640
Abstract: In the preceding paper, Dubrovin valuation rings integral over their centers in central simple algebras were characterized by value functions. Here, these value functions are used to give a method for extending integral Dubrovin valuation rings in generalized crossed product algebras. Several applications of this extension theorem are given, including new and more natural proofs of some theorems on valued division algebras over Henselian fields.
A Mandelbrot set whose boundary is piecewise smooth
M. F.
Barnsley;
D. P.
Hardin
641-659
Abstract: It is proved that the Mandelbrot set associated with the pair of maps $ {w_{1,2}}:{\mathbf{C}} \to {\mathbf{C}}, {w_1}(z) = sz + 1, {w_2}(z) = {s^\ast }z - 1$, with parameter $s \in {\mathbf{C}}$, is connected and has piecewise smooth boundary.
On infinite root systems
R. V.
Moody;
A.
Pianzola
661-696
Abstract: We define in an axiomatic fashion the concept of a set of root data that generalizes the usual concept of root system of a Kac-Moody Lie algebra. We study these objects from a purely formal and geometrical point of view as well as in relation to their associated Lie algebras. This leads to a coherent theory of root systems, bases, subroot systems, Lie algebras defined by root data, and subalgebras.
Pixley-Roy hyperspaces of $\omega$-graphs
J. D.
Mashburn
697-709
Abstract: The techniques developed by Wage and Norden are used to show that the Pixley-Roy hyperspaces of any two $\omega$-graphs are homeomorphic. The Pixley-Roy hyperspaces of several subsets of ${{\mathbf{R}}^n}$ are also shown to be homeomorphic.
Isolated singularities of the Schr\"odinger equation with a good potential
Juan Luis
Vázquez;
Cecilia
Yarur
711-720
Abstract: We study the behaviour near an isolated singularity, say 0, of nonnegative solutions of the Schrödinger equation $- \Delta u + Vu = 0$ defined in a punctured ball $0 < \vert x\vert < R$. We prove that whenever the potential $V$ belongs to the Kato class ${K_n}$ the following alternative, well known in the case of harmonic functions, holds: either $\vert x{\vert^{n - 2}}u(x)$ has a positive limit as $\vert x\vert \to 0$ or $u$ is continuous at 0. In the first case $ u$ solves the equation $- \Delta u + Vu = a\delta$ in $\{ \vert x\vert < R\} $. We discuss the optimality of the class ${K_n}$ and extend the result to solutions $ u \ngeq 0$ of $- \Delta u + Vu = f$.
Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities
A. C.
Lazer;
P. J.
McKenna
721-739
Abstract: We give conditions for the existence, uniqueness, and asymptotic stability of periodic solutions of a second-order differential equation with piecewise linear restoring and $2\pi$-periodic forcing where the range of the derivative of the restoring term possibly contains the square of an integer. With suitable restrictions on the restoring and forcing in the undamped case, we give a necessary and sufficient condition.
Deviations of trajectory averages and the defect in Pesin's formula for Anosov diffeomorphisms
Steven
Orey;
Stephan
Pelikan
741-753
Abstract: A large deviation theorem at the Donsker Varadhan level three is obtained for the convergence of trajectory averages of Anosov diffeomorphisms. It is possible to provide an explicit description of the rate function.
Hyperconvexity and nonexpansive multifunctions
Robert
Sine
755-767
Abstract: It is shown that a ball intersection valued nonexpansive multifunction on a hyperconvex space admits a nonexpansive point valued selection. This implies fixed point theorems for such multifunctions and to certain point valued nonexpansive maps. The result is used to study best approximation and to show the space of all nonexpansive maps of a bounded hyperconvex space is hyperconvex.
Relations between $H\sp p\sb u$ and $L\sp p\sb u$ in a product space
Jan-Olov
Strömberg;
Richard L.
Wheeden
769-797
Abstract: Relations between $ L_u^p$ and $H_u^p$ are studied for the product space ${{\mathbf{R}}^1} \times {{\mathbf{R}}^1}$ in the case $1 < p < \infty$ and $u({x_1},{x_2}) = \vert{Q_1}({x_1}){\vert^p}\vert{Q_2}({x_2}){\vert^p}w({x_1},{x_2})$, where ${Q_1}$ and ${Q_2}$ are polynomials and $w$ satisfies the ${A_p}$ condition for rectangles. A description of the distributions in $ H_u^p$ is given. Questions about boundary values and about the existence of dense subsets of smooth functions satisfying appropriate moment conditions are also considered.
Mappings of trees and the fixed point property
M. M.
Marsh
799-810
Abstract: We investigate weakly confluent, universal, and related mappings of trees and their relationships to the fixed point property for tree-like continua. This investigation leads to some new results, to generalizations of some known results, and to a partial solution of a question of H. Cook.
On a maximal function on compact Lie groups
Michael
Cowling;
Christopher
Meaney
811-822
Abstract: Suppose that $ G$ is a compact Lie group with finite centre. For each positive number $ s$ we consider the $ \operatorname{Ad}(G)$-invariant probability measure ${\mu _s}$ carried on the conjugacy class of $\exp (s{H_\rho })$ in $G$. This one-parameter family of measures is used to define a maximal function $\mathcal{M}\,f$, for each continuous function $ f$ on $G$. Our theorem states that there is an index ${p_0}$ in $(1,2)$, depending on $G$, such that the maximal operator $\mathcal{M}$ is bounded on ${L^p}(G)$ when $p$ is greater than ${p_0}$. When the rank of $G$ is greater than one, this provides an example of a controllable maximal operator coming from averages over a family of submanifolds, each of codimension greater than one.
Scattering for the Yang-Mills equations
John C.
Baez
823-832
Abstract: We construct wave and scattering operators for the Yang-Mills equations on Minkowski space, ${{\mathbf{M}}_0} \cong {{\mathbf{R}}^4}$. Sufficiently regular solutions of the Yang-Mills equations on $ {{\mathbf{M}}_0}$ are known to extend uniquely to solutions of the corresponding equations on the universal cover of its conformal compactification, $ \tilde{\mathbf{M}} \cong {\mathbf{R}} \times {S^3}$. Moreover, the boundary of $ {{\mathbf{M}}_0}$ as embedded in $ \tilde{\mathbf{M}}$ is the union of "lightcones at future and past infinity", $ {C_ \pm }$. We construct wave operators ${W_ \pm }$ as continuous maps from a space ${\mathbf{X}}$ of time-zero Cauchy data for the Yang-Mills equations to Hilbert spaces ${\mathbf{H}}({C_ \pm })$ of Goursat data on $ {C_ \pm }$. The scattering operator is then a homeomorphism $ S:{\mathbf{X}} \to {\mathbf{X}}$ such that $ U{W_ + } = {W_ - }S$, where $U:{\mathbf{H}}({C_ + }) \to {\mathbf{H}}({C_ - })$ is the linear isomorphism arising from a conformal transformation of $ \tilde{\mathbf{M}}$ mapping ${C_ - }$ onto ${C_ + }$. The maps ${W_ \pm }$ and $S$ are shown to be smooth in a certain sense.
Regularity of the metric entropy for expanding maps
Marek
Rychlik
833-847
Abstract: The main result of the current paper is an estimate of the radius of the nonperipheral part of the spectrum of the Perron-Frobenius operator for expanding mappings. As a consequence, we are able to show that the metric entropy of an expanding map has modulus of continuity $x\log (1/x)$ on the space of ${C^2}$-expandings. We also give an explicit estimate of the rate of mixing for ${C^1}$-functions in terms of natural constants. It seems that the method we present can be generalized to other classes of dynamical systems, which have a distinguished invariant measure, like $ \operatorname{Axiom}$ A diffeomorphisms. It also can be adopted to show that the entropy of the quadratic family ${f_\mu }(x) = 1 - \mu {x^2}$ computed with respect to the absolutely continuous invariant measure found in Jakobson's Theorem varies continuously (the last result is going to appear somewhere else).
Infima of convex functions
Gerald
Beer
849-859
Abstract: Let $\Gamma (X)$ be the lower semicontinuous, proper, convex functions on a real normed linear space $ X$. We produce a simple description of what is, essentially, the weakest topology on $\Gamma (X)$ such that the value functional $f \to \inf f$ is continuous on $\Gamma (X)$. When $X$ is reflexive, convergence of a sequence in this topology is equivalent to Mosco plus pointwise convergence of the corresponding sequence of conjugate convex functions.
Higher-order Sugawara operators for affine Lie algebras
Roe
Goodman;
Nolan R.
Wallach
1-55
Abstract: Let $\hat{\mathfrak{g}}$ be the affine Lie algebra associated to a simple Lie algebra $\mathfrak{g}$. Representations of $\hat{\mathfrak{g}}$ are described by current fields $ X(\zeta)$ on the circle $ {\mathbf{T}}\;(X \in \mathfrak{g}$ and $\zeta \in {\mathbf{T}})$. In this paper a linear map $ \sigma$ from the symmetric algebra $ S(\mathfrak{g})$ to (formal) operator fields on a suitable category of $\hat{\mathfrak{g}}$ modules is constructed. The operator fields corresponding to $\mathfrak{g}$-invariant elements of $S(\mathfrak{g})$ are called Sugawara fields. It is proved that they satisfy commutation relations of the form $(\ast)$ $\displaystyle [\sigma (u)(\zeta),X(\eta)] = {c_\infty }D\delta (\zeta /\eta)\sigma ({\nabla _X}u)(\zeta) + {\text{higher-order}}\;{\text{terms}}$ with the current fields, where $ {c_\infty }$ is a renormalization of the central element in $\hat{\mathfrak{g}}$ and $D\delta$ is the derivative of the Dirac delta function. The higher-order terms in $(\ast)$ are studied using results from invariant theory and finite-dimensional representation theory of $ \mathfrak{g}$. For suitably normalized invariants $u$ of degree $4$ or less, these terms are shown to be zero. This vanishing is also proved for $\mathfrak{g} = {\text{sl}}(n,{\mathbf{C}})$ and $u$ running over a particular choice of generators for the symmetric invariants. The Sugawara fields defined by such invariants commute with the current fields whenever $ {c_\infty }$ is represented by zero. This property is used to obtain the commuting ring, composition series, and character formulas for a class of highest-weight modules for $\hat{\mathfrak{g}}$.
Functional equations, tempered distributions and Fourier transforms
John A.
Baker
57-68
Abstract: This paper introduces a method for solving functional equations based on the Fourier transform of tempered distributions.
Multiresolution approximations and wavelet orthonormal bases of $L\sp 2({\bf R})$
Stephane G.
Mallat
69-87
Abstract: A multiresolution approximation is a sequence of embedded vector spaces ${({{\mathbf{V}}_j})_{j \in {\text{z}}}}$ for approximating $ {{\mathbf{L}}^2}({\mathbf{R}})$ functions. We study the properties of a multiresolution approximation and prove that it is characterized by a $2\pi$-periodic function which is further described. From any multiresolution approximation, we can derive a function $\psi (x)$ called a wavelet such that ${(\sqrt {{2^j}} \psi ({2^j}x - k))_{(k,j) \in {{\text{z}}^2}}}$ is an orthonormal basis of $ {{\mathbf{L}}^2}({\mathbf{R}})$. This provides a new approach for understanding and computing wavelet orthonormal bases. Finally, we characterize the asymptotic decay rate of multiresolution approximation errors for functions in a Sobolev space $ {{\mathbf{H}}^s}$.
Injectivity of operator spaces
Zhong-Jin
Ruan
89-104
Abstract: We study the structure of injective operator spaces and the existence and uniqueness of the injective envelopes of operator spaces. We give an easy example of an injective operator space which is not completely isometric to any $ {C^\ast }$-algebra. This answers a question of Wittstock [23]. Furthermore, we show that an operator space $E$ is injective if and only if there exists an injective ${C^\ast }$-algebra $A$ and two projections $p$ and $q$ in $A$ such that $E$ is completely isometric to $pAq$.
Invariant measures and equilibrium states for piecewise $C\sp {1+\alpha}$ endomorphisms of the unit interval
Christopher J.
Bose
105-125
Abstract: A differentiable function is said to be $ {C^{1 + \alpha }}$ if its derivative is a Hölder continuous function with exponent $\alpha > 0$. We show that three well-known results about invariant measures for piecewise monotonic and ${C^2}$ endomorphisms of the unit interval are in fact true for piecewise monotonic and ${C^{1 + \alpha }}$ maps. We show the existence of unique, ergodic measures equivalent to Lebesgue measure for $ {C^{1 + \alpha }}$ Markov maps, extending a result of Bowen and Series for the $ {C^2}$ case. We present a generalization of Adler's Folklore Theorem for maps which satisfy a restricted mixing condition, and we show that these $ {C^{1 + \alpha }}$ mixing endomorphisms possess unique equilibrium states, a result which was shown for the ${C^2}$ case by P. Walters.
On Cournot-Nash equilibrium distributions for games with a nonmetrizable action space and upper semi-continuous payoffs
M. Ali
Khan
127-146
Abstract: We report results on the existence of a Cournot-Nash equilibrium distribution for games in which the action space is not necessarily metrizable and separable and the payoff functions are not necessarily continuous. Our work relies on the theory of Radon measures as developed by Schwartz-Topsoe and on the epitopology as developed by Dolecki-Salinetti-Wets
The structure of quasimultipliers of $C\sp *$-algebras
Hua Xin
Lin
147-172
Abstract: Let $A$ be a ${C^\ast }$-algebra and ${A^{\ast\ast}}$ its enveloping ${W^\ast }$-algebra. Let ${\text{LM}}(A)$ be the left multipliers of $ A$, ${\text{RM}}(A)$ the right multipliers of $ A$ and ${\text{QM}}(A)$ the quasi-multipliers of $ A$. A question was raised by Akemann and Pedersen [1] whether $ {\text{QM}}(A) = {\text{LM}}(A) + {\text{RM}}(A)$. McKennon [20] gave a nonseparable counterexample. L. Brown [6] shows the answer is negative for stable (separable) ${C^\ast }$-algebras also. In this paper, we mainly consider $\sigma $-unitial ${C^\ast }$-algebras. We give a criterion for ${\text{QM}}(A) = {\text{LM}}(A) + {\text{RM}}(A)$. In the case that $A$ is stable, we give a necessary and sufficient condition for ${\text{QM}}(A) = {\text{LM}}(A) + {\text{RM}}(A)$. We also give answers for other ${C^\ast }$-algebras.
Sarkovski\u\i's theorem for hereditarily decomposable chainable continua
Piotr
Minc;
W. R. R.
Transue
173-188
Abstract: Sarkovskii's theorem, which fails to hold for chainable continua, is shown to hold for hereditarily decomposable chainable continua.
Hamburger moment problems and orthogonal polynomials
T. S.
Chihara
189-203
Abstract: We consider a sequence of orthogonal polynomials given by the classical three term recurrence relation. We address the problem of deciding the determinacy or indeterminacy of the associated Hamburger moment problem on the basis of the behavior of the coefficients in the three term recurrence relation. Comparisons are made with other criteria in the literature. The efficacy of the criteria obtained is illustrated by application to many specific examples of orthogonal polynomials.
A generalization of the Levine-Tristram link invariant
Lawrence
Smolinsky
205-217
Abstract: Invariants to $ m$-component links are defined and are shown to be link cobordism invariants under certain conditions. Examples are given.
A new $3$-dimensional shrinking criterion
Robert J.
Daverman;
Dušan
Repovš
219-230
Abstract: We introduce a new shrinking criterion for cell-like upper semicontinuous decompositions $G$ of topological $3$-manifolds, such that the embedding dimension (in the sense of Štan'ko) of the nondegeneracy set of $ G$ is at most one. As an immediate application, we prove a recognition theorem for $3$-manifolds based on a new disjoint disks property.
The structure of some equivariant Thom spectra
Steven R.
Costenoble
231-254
Abstract: We show that the equivariant Thom spectra $M{O_{{{\text{Z}}_2}}}$ and $m{O_{{{\text{Z}}_2}}}$ do not split as wedges of equivariant Eilenberg-Mac Lane spectra, as they do nonequivariantly. This is done by finding two-stage Postnikov towers giving these spectra, and determining the nontrivial $k$-invariants. We also consider the question: In what sense is the spectrum $m{O_{{{\text{Z}}_2}}}$ representing unoriented bordism unique?
Compact abelian prime actions on von Neumann algebras
Klaus
Thomsen
255-272
Abstract: We classify the compact abelian actions on semifinite injective von Neumann algebras with factor fixed point algebra. The method uses that the (nonzero) eigenspaces of such an action contain unitaries which give rise to a classifying invariant.
Optimal $L\sp p$ and H\"older estimates for the Kohn solution of the $\overline\partial$-equation on strongly pseudoconvex domains
Der-Chen E.
Chang
273-304
Abstract: Let $\Omega$ be an open, relatively compact subset in $ {{\mathbf{C}}^{n + 1}}$, and assume the boundary of $\Omega$, $ \partial \Omega$, is smooth and strongly pseudoconvex. Let $\operatorname{Op}(K)$ be an integral operator with mixed type homogeneities defined on $\overline \Omega$: i.e., $K$ has the form as follows: $\displaystyle \sum\limits_{k,l \geq 0} {{E_k}{H_l},}$ where ${E_k}$ is a homogeneous kernel of degree $ - k$ in the Euclidean sense and ${H_l}$ is homogeneous of degree $- l$ in the Heisenberg sense. In this paper, we study the optimal ${L^p}$ and Hölder estimates for the kernel $ K$. We also use Lieb-Range's method to construct the integral kernel for the Kohn solution $\overline {{\partial^\ast}} {\mathbf{N}}$ of the Cauchy-Riemann equation on the Siegel upper-half space and then apply our results to $ \overline {{\partial^\ast}} {\mathbf{N}}$. On the other hand, we prove Lieb-Range's kernel gains $1$ in "good" directions (hence gains $1/2$ in all directions) via Phong-Stein's theory. We also discuss the transferred kernel from the Siegel upper-half space to $\Omega$.
Nonlinearly equivalent representations of quaternionic $2$-groups
Washington
Mio
305-321
Abstract: We construct new examples of nonlinearly equivalent finite-dimensional real linear representations of quaternionic $2$-groups, which cannot be obtained from equivalent representations of cyclic groups by induction and composition techniques.
Coupled points in the calculus of variations and applications to periodic problems
Vera
Zeidan;
Pier Luigi
Zezza
323-335
Abstract: The aim of this paper is to introduce the definition of coupled points for the problems of the calculus of variations with general boundary conditions, and to develop second order necessary conditions for optimality. When one of the end points is fixed, our necessary conditions reduce to the known ones involving conjugate points. We also apply our results to the periodic problems of the calculus of variations.
Cauchy integral equalities and applications
Boo Rim
Choe
337-352
Abstract: We study bounded holomorphic functions $\pi$ on the unit ball ${B_n}$ of $ {\mathbb{C}^n}$ satisfying the following so-called Cauchy integral equalities: \begin{displaymath}\begin{array}{*{20}{c}} {C[{\pi ^{m + 1}}\bar \pi ] = {\gamma _m}{\pi ^m}} & {(m = 0,1,2, \ldots)} \end{array} \end{displaymath} for some sequence ${\gamma _m}$ depending on $\pi$. Among the applications are the Ahern-Rudin problem concerning the composition property of holomorphic functions on ${B_n}$, a projection theorem about the orthogonal projection of $ {H^2}({B_n})$ onto the closed subspace generated by holomorphic polynomials in $ \pi$, and some new information about the inner functions. In particular, it is shown that if we interpret ${\text{BMOA}}({B_n})$ as the dual of ${H^1}({B_n})$, then the map $g \to g \circ \pi$ is a linear isometry of ${\text{BMOA}}({B_1})$ into ${\text{BMOA}}({B_n})$ for every inner function $ \pi$ on ${B_n}$ such that $\pi (0) = 0$.
Radially symmetric solutions to a superlinear Dirichlet problem in a ball with jumping nonlinearities
Alfonso
Castro;
Alexandra
Kurepa
353-372
Abstract: Let $p,\varphi :[0,T] \to R$ be bounded functions with $\varphi > 0$. Let $g:{\mathbf{R}} \to {\mathbf{R}}$ be a locally Lipschitzian function satisfying the superlinear jumping condition: (i) $ {\lim _{u \to - \infty }}(g(u)/u) \in {\mathbf{R}}$ (ii) $ {\lim _{u \to \infty }}(g(u)/{u^{1 + \rho }}) = \infty $ for some $\rho > 0$, and (iii) $ {\lim _{u \to \infty }}{(u/g(u))^{N/2}}(NG(\kappa u) - ((N - 2)/2)u \cdot g(u)) = \infty$ for some $ \kappa \in (0,1]$ where $ G$ is the primitive of $ g$. Here we prove that the number of solutions of the boundary value problem $\Delta u + g(u) = p(\left\Vert x\right\Vert) + c\varphi (\left\Vert x\right\Vert)$ for $x \in {{\mathbf{R}}^N}$ with $ \left\Vert x\right\Vert < T,u(x) = 0$ for $ \left\Vert x\right\Vert = T$ tends to $+ \infty$ when $c$ tends to $+ \infty$. The proofs are based on the "energy" and "phase plane" analysis.
A higher order invariant of differential manifolds
Gregory A.
Fredricks;
Peter B.
Gilkey;
Phillip E.
Parker
373-388
Abstract: We discuss conditions under which a lens space is $s$th order flat.
$\overline\partial\sb b$-equations on certain unbounded weakly pseudo-convex domains
Hyeonbae
Kang
389-413
Abstract: We found an explicit closed formula for the relative fundamental solution of $ {\bar \partial _b}$ on the surface ${H_k} = \{ ({z_1},{z_2}) \in {\mathbb{C}^2}:\operatorname{Im} {z_2} = \vert{z_1}{\vert^{2k}}\}$ . We then make estimates of the relative fundamental solution in terms of the nonisotropic metric associated with the surface. The estimates lead us to the regularity results. We also study the problem of finding weights $\omega$ so that ${\bar \partial _b}$ as an operator from $L_\omega ^2$ to ${L^2}$ has a closed range. We find the best possible weight among radial weights.
The transfer ideal of quadratic forms and a Hasse norm theorem mod squares
David B.
Leep;
Adrian R.
Wadsworth
415-432
Abstract: Any finite degree field extension $K/F$ determines an ideal ${\mathcal{T}_{K/F}}$ of the Witt ring $WF$ of $F$, called the transfer ideal, which is the image of any nonzero transfer map $WK \to WF$. The ideal ${\mathcal{T}_{K/F}}$ is computed for certain field extensions, concentrating on the case where $ K$ has the form $F\left({\sqrt {{a_1}} , \ldots ,\sqrt {{a_n}} } \right)$, ${a_i} \in F$. When $F$ and $K$ are global fields, we investigate whether there is a local global principle for membership in ${\mathcal{T}_{K/F}}$. This is shown to be equivalent to the existence of a "Hasse norm theorem mod squares," i.e., a local global principle for the image of the norm map $ {N_{K/F}}: {K^\ast}/{K^{\ast2}} \to {F^\ast}/{F^{\ast2}}$. It is shown that such a Hasse norm theorem holds whenever $K = F(\sqrt{a_1},\ldots,\sqrt{a_n})$, although it does not always hold for more general extensions of global fields, even some Galois extensions with group $ \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$.
Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems
G.
Keller
433-497
Abstract: Transfer operators and zeta functions of piecewise monotonic and of more general piecewise invertible dynamical systems are studied. To this end we construct Markov extensions of given systems, develop a kind of Fredholm theory for them, and carry the results back to the original systems. This yields e.g. bounds on the number of ergodic maximal measures or equilibrium states.
The nonlinear geometry of linear programming. I. Affine and projective scaling trajectories
D. A.
Bayer;
J. C.
Lagarias
499-526
Abstract: This series of papers studies a geometric structure underlying Karmarkar's projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure studied is the set of trajectories obtained by integrating this vector field, which we call $P$-trajectories. We also study a related vector field, the affine scaling vector field, and its associated trajectories, called $A$-trajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. Affine and projective scaling vector fields are each defined for linear programs of a special form, called strict standard form and canonical form, respectively. This paper derives basic properties of $P$-trajectories and $A$-trajectories. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for $P$-trajectories and $A$-trajectories. It shows that projective transformations map $P$-trajectories into $P$-trajectories. It presents Karmarkar's interpretation of $A$-trajectories as steepest descent paths of the objective function $ \left\langle {{\mathbf{c}},{\mathbf{x}}} \right\rangle $ with respect to the Riemannian geometry $d{s^2} = \sum\nolimits_{i = 1}^n {d{x_i}d{x_i}/x_i^2}$ restricted to the relative interior of the polytope of feasible solutions. $ P$-trajectories of a canonical form linear program are radial projections of $ A$-trajectories of an associated standard form linear program. As a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program: This algorithm is essentially Karmarkar's algorithm. These trajectories are studied in subsequent papers by two nonlinear changes of variables called Legendre transform coordinates and projective Legendre transform coordinates, respectively. It will be shown that $P$-trajectories have an algebraic and a geometric interpretation. They are algebraic curves, and they are geodesics (actually distinguished chords) of a geometry isometric to a Hubert geometry on a polytope combinatorially dual to the polytope of feasible solutions. The $A$-trajectories of strict standard form linear programs have similar interpretations: They are algebraic curves, and are geodesics of a geometry isometric to Euclidean geometry.
The nonlinear geometry of linear programming. II. Legendre transform coordinates and central trajectories
D. A.
Bayer;
J. C.
Lagarias
527-581
Abstract: Karmarkar's projective scaling algorithm for solving linear programming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem. This paper studies the set of trajectories obtained by integrating this vector field, called $P$-trajectories, as well as a related set of trajectories, called $A$-trajectories. The $ A$-trajectories arise from another linear programming algorithm, the affine scaling algorithm. The affine and projective scaling vector fields are each defined for linear programs of a special form, called standard form and canonical form, respectively. These trajectories are studied using a nonlinear change of variables called Legendre transform coordinates, which is a projection of the gradient of a logarithmic barrier function. The Legendre transform coordinate mapping is given by rational functions, and its inverse mapping is algebraic. It depends only on the constraints of the linear program, and is a one-to-one mapping for canonical form linear programs. When the polytope of feasible solutions is bounded, there is a unique point mapping to zero, called the center. The $ A$-trajectories of standard form linear programs are linearized by the Legendre transform coordinate mapping. When the polytope of feasible solutions is bounded, they are the complete set of geodesics of a Riemannian geometry isometric to Euclidean geometry. Each $A$-trajectory is part of a real algebraic curve. Each $P$-trajectory for a canonical form linear program lies in a plane in Legendre transform coordinates. The $P$-trajectory through $ {\mathbf{0}}$ in Legendre transform coordinates, called the central $ P$-trajectory, is part of a straight line, and is contained in the $ A$-trajectory through ${\mathbf{0}}$, called the central $ A$-trajectory. Each $ P$-trajectory is part of a real algebraic curve. The central $A$-trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form $\langle {\mathbf{c}},{\mathbf{x}}\rangle = \mu$. It is also the set of minima of a parametrized family of logarithmic barrier functions. Power-series expansions are derived for the central $ A$-trajectory, which is also the central $P$-trajectory. These power-series have a simple recursive form and are useful in developing "higher-order" analogues of Karmarkar's algorithm. $ A$-trajectories are defined for a general linear program. Using this definition, it is shown that the limit point ${{\mathbf{x}}_\infty }$ of a central $A$-trajectory on the boundary of the feasible solution polytope $P$ is the center of the unique face of $ P$ containing ${{\mathbf{x}}_\infty }$ in its relative interior. The central trajectory of a combined primal-dual linear program has a simple set of polynomial relations determining it as an algebraic curve. These relations are a relaxed form of the complementary slackness conditions. This central trajectory algebraically projects onto the central trajectories of both the primal and dual linear programs, and this gives an algebraic correspondence between points on the positive parts of the central trajectories of the primal and dual linear programs. Two Lagrangian dynamical systems with simple Lagrangians are shown to have $ A$-trajectories as ${\mathbf{\dot q}}$-trajectories. The Hamiltonian dynamical systems associated to these Lagrangian systems are completely integrable.
Lorentz spaces that are isomorphic to subspaces of $L\sp 1$
Carsten
Schütt
583-595
Abstract: We show which Lorentz spaces are isomorphic to subspaces of ${L^1}$ and which are not.
A Picard theorem with an application to minimal surfaces
Peter
Hall
597-603
Abstract: We prove a Picard theorem for holomorphic maps from ${\mathbf{C}}$ to a quadric hypersurface. This implies a theorem on the number of directions in general position omitted by the normals to a minimal surface of the conformal type of $ {\mathbf{C}}$.
Some properties of the curve straightening flow in the plane
Anders
Linnér
605-618
Abstract: We will explicitly compute the gradient of the total squared curvature functional on a space of closed curves. An example shows that the flow along the gradient trajectory may cause curves to develop self-intersections. We prove the existence of strictly convex curves that momentarily turn nonconvex. In conclusion we use computer graphics to illustrate how self-intersections come about.
Stability in the isoperimetric problem for convex or nearly spherical domains in ${\bf R}\sp n$
Bent
Fuglede
619-638
Abstract: For convex bodies $ D$ in ${{\mathbf{R}}^n}$ the deviation $d$ from spherical shape is estimated from above in terms of the (dimensionless) isoperimetric deficiency $\Delta$ of $D$ as follows: $d \leq f(\Delta)$ (for $ \Delta$ sufficiently small). Here $f$ is an explicit elementary function vanishing continuously at 0. The estimate is sharp as regards the order of magnitude of $f$. The dimensions $n = 2$ and $3$ present anomalies as to the form of $ f$. In the planar case $ n = 2$ the result is contained in an inequality due to T. Bonnesen. A qualitative consequence of the present result is that there is stability in the classical isoperimetric problem for convex bodies $D$ in $ {{\mathbf{R}}^n}$ in the sense that, as $D$ varies, $d \to 0$ for $ \Delta \to 0$. The proof of the estimate $ d \leq f(\Delta)$ is based on a related estimate in the case of domains (not necessarily convex) that are supposed a priori to be nearly spherical in a certain sense.
A uniform algebra of analytic functions on a Banach space
T. K.
Carne;
B.
Cole;
T. W.
Gamelin
639-659
Abstract: Let $A(B)$ be the uniform algebra on the unit ball of a dual Banach space $\mathcal{Z} = {\mathcal{Y}^\ast}$ generated by the weak-star continuous linear functionals. We focus on three related problems: (i) to determine when $A(B)$ is a tight uniform algebra; (ii) to describe which functions in $ {H^\infty }(B)$ are approximable pointwise on $B$ by bounded nets in $A(B)$; and (iii) to describe the weak topology of $B$ regarded as a subset of the dual of $ A(B)$. With respect to the second problem, we show that any polynomial in elements of $ {\mathcal{Y}^{\ast\ast}}$ can be approximated pointwise on $B$ by functions in $A(B)$ of the same norm. This can be viewed as a generalization of Goldstine's theorem. In connection with the third problem, we introduce a class of Banach spaces, called $\Lambda$-spaces, with the property that if $\{ {x_j}\}$ is a bounded sequence in $\mathcal{X}$ such that $P({x_j}) \to 0$ for any $m$-homogeneous analytic function $P$ on $\mathcal{X}, m \geq 1$, then ${x_j} \to 0$ in norm. We show for instance that a Banach space has the Schur property if and only if it is a $\Lambda$-space with the Dunford-Pettis property.
Weighted norm inequalities for the continuous square function
J. Michael
Wilson
661-692
Abstract: We prove new weighted norm inequalities for real-variable analogues of the Lusin area function. We apply our results to obtain new: (i) weighted norm inequalities for singular integral operators; (ii) weighted Sobolev inequalities; (iii) eigenvalue estimates for degenerate Schrödinger operators.
On two-cardinal properties of ideals
Jacek
Cichoń
693-708
Abstract: We investigate two-cardinal properties of ideals. These properties involve notions such as Luzin sets, special coverings, etc. We apply our results to the ideals of meagre sets and of negligible sets in the real line. In case of the negligible sets, we relate these properties to caliber and precalibers of the measure algebra.
Boundary layers in approximate solutions
K. T.
Joseph
709-726
Abstract: In this paper we study the development of boundary layers in parabolic approximate solutions of the initial boundary value problem for linear strictly hyperbolic systems of equations in one space variable. We also analyse the boundary layer behaviour.
Sobolev inequalities for products of powers
A. Eduardo
Gatto;
Richard L.
Wheeden
727-743
Abstract: We derive weighted Sobolev inequalities of the form ${\left\Vert f \right\Vert _{L_u^q}} \leq C{\left\Vert {\nabla f} \right\Vert _{L_v^p}}$, $f \in C_0^\infty ({{\mathbf{R}}^n})$, $1 < p \leq q < \infty$, for classes of weight functions $u$, $v$ which include $v$'s that are a finite product of certain power weights times an ${A_p}$ function.
The graded Witt ring and Galois cohomology. II
Jón Kr.
Arason;
Richard
Elman;
Bill
Jacob
745-780
Abstract: A primary problem in the theory of quadratic forms over a field $ F$ of characteristic different from two is to prove that the rings $H_q^\ast F$ and $GWF$ are isomorphic. Here $H_q^\ast F = {H^\ast }(\operatorname{Gal}({F_q}/F),{\mathbf{Z}}/2{\mathbf{Z}}))$, where ${F_q}$ is the quadratic closure of $ F$, and $GWF$ is the graded Witt ring associated to the fundamental ideal of even dimensional forms in the Witt ring $WF$ of $F$. In this paper, we assume we are given a field extension $K$ of $F$ such that $WK$ is 'close' to $WF$ or $ H_q^{\ast} K$ is 'close' to $H_q^\ast F$. A method is developed to obtain information about these graded rings over $F$ and its $2$-extensions from information about the corresponding graded ring of $K$. This relative theory extends and includes the previously developed absolute case where $K = {F_q}$. Applications are also given to show that $ H_q^\ast F$ and $ GWF$ are isomorphic for a collection of fields arising naturally from the theory of abstract Witt rings.
On certain integrals associated to CR-functions
Telemachos
Hatziafratis
781-802
Abstract: We construct explicit $\bar \partial $-primitives of Cauchy-Fantappiè type kernels on analytic subvarieties of domains in $ {{\mathbf{C}}^n}$, outside the zero set of a holomorphic map defined on the variety. Then we use these primitives to derive, by means of a process of passing to a residue, integral formulas for $ {\text{CR}}$-functions defined on subsets of the boundary of a variety. Certain geometric restrictions on the varieties are imposed. These primitives apply in the particular case of the Bochner-Martinelli kernel in domains in ${{\mathbf{C}}^n}$ and we use these primitives, in this case, to prove a criterion for holomorphic extendability of $ {\text{CR}}$-functions defined on certain subsets of the boundary of a domain in $ {{\mathbf{C}}^n}$.
Topologically principal part of analytic functions
Etsuo
Yoshinaga
803-814
Abstract: The problem of $ {C^0}$-sufficiency of jets is one of the most interesting problems in the theory of functions or singularities. Roughly speaking, it is the problem of determining a topologically principal part of the Taylor expansion of a given function $f(x)$ at the origin of Euclidean space. Here, the topologically principal part should satisfy the properties that it is as small as possible a part of the Taylor expansion of $f(x)$ and that the local topological type of $ f(x)$ at the origin is determined by it. If a function $f(x)$ is an isolated singularity at the origin or has a nondegenerate Newton principal part (see (1.2)), then we know some answers to this problem (see (1.1), (1.3)). The purpose of this paper is to give some results for this problem for any analytic function. The main results are formulated in (1.5), (1.6), and (1.7).
Isospectral potentials on a discrete lattice. III
Thomas
Kappeler
815-824
Abstract: Continuing prior work of the author, spectral problems for a discretized version of the Schrödinger equation 'Laplacian plus potential' on the torus are considered. Spectral invariants are constructed and with their help isospectral sets of certain nongeneric potentials are determined.
Projective $n$-folds of log-general type. I
M.
Beltrametti;
A.
Biancofiore;
A. J.
Sommese
825-849
Abstract: Let $X$ be a normal $n$-dimensional Gorenstein irreducible projective subvariety of a complex projective space with codimension $3$ singularities. This article introduces new projective invariants of $X$ and shows that they satisfy certain inequalities and relations making them useful tools in the projective classification of varieties. The main tools used are the recent precise adjunction theoretic results on classification of projective varieties.
The group of automorphisms of $L\sp 1(0,1)$ is connected
F.
Ghahramani
851-859
Abstract: It is shown that the group of the automorphisms of the radical convolution algebra $ {L^1}(0,1)$ is connected in the operator norm topology, and thus every automorphism is of the form ${e^{\lambda d}}{e^q}$, where $\lambda$ is a complex number, $ d$ is the derivation $df(x) = xf(x)$ and $q$ is a quasinilpotent derivation.
On invariants of graphs with applications to knot theory
Kunio
Murasugi
1-49
Abstract: To each weighted graph $\Gamma$, two invariants, a polynomial ${P_\Gamma }(x,y,z)$ and the signature $\sigma (\Gamma)$, are defined. The various partial degress of $ {P_\Gamma }(x,y,z)$ and $\sigma (\Gamma)$ are expressed in terms of maximal spanning graphs of $\Gamma$. Furthermore, one unexpected property of Tutte's dichromate is proved. These results are applied to knots or links in ${S^3}$.
Propagation of $L\sp q\sb k$-smoothness for solutions of the Euler equation
Gustavo
Ponce
51-61
Abstract: The motion of an ideal incompressible fluid is described by a system of partial differential equations known as the Euler equation. Considering the initial value problem for this equation, we prove that in a classical solution the $ L_k^q$-regularity of the data propagates along the fluid lines. Our method consists of combining properties of the $\varepsilon$-approximate solution with ${L^q}$-energy estimates and simple results of classical singular integral operators. In particular, for the two-dimensional case we present an elementary proof.
Mel\cprime nikov transforms, Bernoulli bundles, and almost periodic perturbations
Kenneth R.
Meyer;
George R.
Sell
63-105
Abstract: In this paper we study nonlinear time-varying perturbations of an autonomous vector field in the plane ${R^2}$ . We assume that the unperturbed equation, i.e. the given vector field has a homoclinic orbit and we present a generalization of the Melnikov method which allows us to show that the perturbed equation has a transversal homoclinic trajectory. The key to our generalization is the concept of the Melnikov transform, which is a linear transformation on the space of perturbation functions. The appropriate dynamical setting for studying these perturbation is the concept of a skew product flow. The concept of transversality we require is best understood in this context. Under conditions whereby the perturbed equation admits a transversal homoclinic trajectory, we also study the dynamics of the perturbed vector field in the vicinity of this trajectory in the skew product flow. We show the dynamics near this trajectory can have the exotic behavior of the Bernoulli shift. The exact description of this dynamical phenomenon is in terms of a flow on a fiber bundle, which we call, the Bernoulli bundle. We allow all perturbations which are bounded and uniformly continuous in time. Thus our theory includes the classical periodic perturbations studied by Melnikov, quasi periodic and almost periodic perturbations, as well as toroidal perturbations which are close to quasi periodic perturbations.
Global solvability of the derivative nonlinear Schr\"odinger equation
Jyh-Hao
Lee
107-118
Abstract: The derivative nonlinear Schrödinger equation $($DNLS$)$ \begin{displaymath}\begin{array}{*{20}{c}} {i{q_t} = {q_{xx}} \pm {{({q^\ast }{q... ...sqrt { - 1} ,{q^\ast }(z) = \overline {q(z)} ,} \end{array} \end{displaymath} was first derived by plasma physicists [9,10]. This equation was used to interpret the propagation of circular polarized nonlinear Alfvén waves in plasma. Kaup and Newell obtained the soliton solutions of DNLS in 1978 [5]. The author obtained the local solvability of DNLS in his dissertation [6]. In this paper we obtain global existence (in time $ t$) of Schwartz class solutions of DNLS if the ${L^2}$-norm of the generic initial data $ q(x,0)$ is bounded.
Summability of Hermite expansions. I
S.
Thangavelu
119-142
Abstract: We study the summability of one-dimensional Hermite expansions. We prove that the critical index for the Riesz summability is $ 1/6$. We also prove analogues of the Fejér-Lebesgue theorem and Riemann's localisation principle.
Summability of Hermite expansions. II
S.
Thangavelu
143-170
Abstract: We study the summability of $n$-dimensional Hermite expansions where $n > 1$. We prove that the critical index for the Riesz summability is $ (n - 1)/2$. We also prove analogues of the Fejér-Lebesgue theorem and Riemann's localisation principle when the index $ \alpha$ of the Riesz means is $> (3n - 2)/6$ .
Equivalent conditions to the spectral decomposition property for closed operators
I.
Erdélyi;
Sheng Wang
Wang
171-186
Abstract: The spectral decomposition property has been instrumental in developing a local spectral theory for closed operators acting on a complex Banach space. This paper gives some necessary and sufficient conditions for a closed operator to possess the spectral decomposition property.
On the Cauchy problem and initial traces for a degenerate parabolic equation
E.
DiBenedetto;
M. A.
Herrero
187-224
Abstract: We consider the Cauchy problem (f) $\displaystyle \left\{ {\begin{array}{*{20}{c}} {{u_t} - \operatorname{div}(\ver... ...= {u_0}(x),} & {x \in {{\mathbf{R}}^N},} \end{array} } \right.$ and discuss existence of solutions in some strip ${S_T} \equiv {{\mathbf{R}}^N} \times (0,T)$, $0 < T \leq \infty$, in terms of the behavior of $x \to {u_0}(x)$ as $\vert x\vert \to \infty $. The results obtained are optimal in the class of nonnegative locally bounded solutions, for which a Harnack-type inequality holds. Uniqueness is shown under the assumption that the initial values are taken in the sense of $ L_{{\text{loc}}}^1({{\mathbf{R}}^N})$.
Similarity, quasisimilarity, and operator factorizations
Raúl E.
Curto;
Lawrence A.
Fialkow
225-254
Abstract: We introduce and illustrate an operator factorization technique to study similarity and quasisimilarity of Hilbert space operators. The technique allows one to generate, in a systematic way, families of "test" operators, and to check for similarity and quasisimilarity with a given model. In the case of the unilateral shift ${U_ + }$, we obtain a one-parameter family of nonhyponormal, noncontractive, shift-like operators in the similarity orbit of ${U_ + }$. We also obtain new characterizations of quasisimilarity and similarity in terms of invariant operator ranges, and conditions for spectral and essential spectral inclusions.
Superprocesses and their linear additive functionals
E. B.
Dynkin
255-282
Abstract: Let $X = ({X_t},P)$ be a measure-valued stochastic process. Linear functionals of $X$ are the elements of the minimal closed subspace $ L$ of ${L^2}(P)$ which contains all $ {X_t}(B)$ with $\smallint {{X_t}{{(B)}^2}\;dP\; < \infty }$. Various classes of $L$-valued additive functionals are investigated for measure-valued Markov processes introduced by Watanabe and Dawson. We represent such functionals in terms of stochastic integrals and we derive integral and differential equations for their Laplace transforms. For an important particular case--"weighted occupation times"--such equations have been established earlier by Iscoe. We consider Markov processes with nonstationary transition functions to reveal better the principal role of the backward equations. This is especially helpful when we derive the formula for the Laplace transforms.
Index formulas for elliptic boundary value problems in plane domains with corners
Gregory
Eskin
283-348
Abstract: We derive the conditions for the operator corresponding to a general elliptic boundary value problem in a plane domain with corners to be Fredholm and give an explicit formula for the index of this operator.
Variational problems on contact Riemannian manifolds
Shukichi
Tanno
349-379
Abstract: We define the generalized Tanaka connection for contact Riemannian manifolds generalizing one for nondegenerate, integrable ${\text{CR}}$ manifolds. Then the torsion and the generalized Tanaka-Webster scalar curvature are defined properly. Furthermore, we define gauge transformations of contact Riemannian structure, and obtain an invariant under such transformations. Concerning the integral related to the invariant, we define a functional and study its first and second variational formulas. As an example, we study this functional on the unit sphere as a standard contact manifold.
Singular relaxation moduli and smoothing in three-dimensional viscoelasticity
Wolfgang
Desch;
Ronald
Grimmer
381-404
Abstract: We develop a semigroup setting for linear viscoelasticity in three-dimensional space with tensor-valued relaxation modulus and give a criterion on the relaxation kernel for differentiability and analyticity of the solutions. The method is also extended to a simple problem in thermoviscoelasticity.
On $H\sb *(\Omega\sp {n+2}S\sp {n+1};{\bf F}\sb 2)$
Thomas J.
Hunter
405-420
Abstract: In this paper, we study ${H_\ast}{\Omega ^{n + 2}}{S^{n + 1}}$ . Here $ \Omega X$ denotes the space of pointed maps $ {S^1} \to X$, and $ {H_\ast}$ represents homology modulo $2$. We show that the Eilenberg-Moore spectral sequence $\operatorname{Tor}_{{H^\ast }\Omega _0^{n + 1}{S^{n + 1}}}^{\ast\ast}({F_{2,}}{F_2}) \Rightarrow {H^\ast }{\Omega ^{n + 2}}{S^{n + 1}}$ collapses, and we identify the kernel of the Whitehead product map ${\Omega ^{n + 1}}{p_\ast}:{H_\ast}{\Omega ^{n + 3}}{S^{2n + 1}} \to {H_\ast}{\Omega ^{n + 1}}{S^n}$ . These observations yield two different descriptions of ${H_\ast}{\Omega ^{n + 2}}{S^{n + 1}}$ up to extension.
Limitation topologies on function spaces
Philip L.
Bowers
421-431
Abstract: Four competing definitions for limitation topologies on the set of continuous functions $C(X,Y)$ are compared.
Orbital parameters for induced and restricted representations
Ronald L.
Lipsman
433-473
Abstract: General formulas for the spectral decomposition of both induced and restricted representations are laid out for the case of connected Lie groups $H \subset G$. The formulas--which detail the actual spectrum, the multiplicits, and the spectral measure--are in terms of the usual parameters in the so-called orbit method. A proof of these formulas is given in the nilpotent situation. The proof is much simpler than a previously obtained proof using nilpotent algebraic geometry. It is also capable of generalization to nonnilpotent groups. With that in mind, many new examples are presented for semisimple and symmetric homogeneous spaces. Also, a start is made in the case of exponential solvable homogeneous spaces with the treatment of both normal and conormal subgroups.
Periodic orbits of maps of $Y$
Lluís
Alsedà;
Jaume
Llibre;
Michał
Misiurewicz
475-538
Abstract: We introduce some notions that are useful for studying the behavior of periodic orbits of maps of one-dimensional spaces. We use them to characterize the set of periods of periodic orbits for continuous maps of $Y = \{ z \in {\mathbf{C}}:{z^3} \in [0,1]\}$ into itself having zero as a fixed point. We also obtain new proofs of some known results for maps of an interval into itself.
Remarks on classical invariant theory
Roger
Howe
539-570
Abstract: A uniform formulation, applying to all classical groups simultaneously, of the First Fundamental Theory of Classical Invariant Theory is given in terms of the Weyl algebra. The formulation also allows skew-symmetric as well as symmetric variables. Examples illustrate the scope of this formulation.
Projections onto translation-invariant subspaces of $H\sp 1({\bf R})$
Dale E.
Alspach;
David C.
Ullrich
571-588
Abstract: Recently I. Klemes has characterized the complemented translation-invariant subspaces of $ {H^1}(\mathbb{T})$. In this paper we investigate the case of ${H^1}(\mathbb{R})$. The main results are that the hull of a complemented translation-invariant subspace is $ \varepsilon$-separated for some $ \varepsilon > 0$, and that an $ \varepsilon$-separated subset of $ {\mathbb{R}^ + }$ which is in the ring generated by cosets of closed subgroups of $\mathbb{R}$ (intersected with ${\mathbb{R}^ + }$) and lacunary sequences is the hull of a complemented ideal.
Accessory parameters for punctured spheres
Irwin
Kra
589-617
Abstract: This paper contains some qualitative results about the accessory parameters for punctured spheres with signature. We show that the Fuchsian uniformizing connection, and hence also the accessory parameters, for the surface depends real analytically on moduli. We also show that the important invariants of a uniformization of a punctured sphere such as the accessory parameters, Fuchsian groups, Poincaré metrics, and covering maps vary continuously under degenerations such as coalescing of punctures.
A calculus approach to hyperfunctions. II
Tadato
Matsuzawa
619-654
Abstract: We consider any hyperfunctions with the compact support as initial values of the solutions of the heat equation. The main aim of this paper is to unify the theory of distributions and hyperfunctions as well as simplify proofs of some important results via heat kernel.
Band-limited functions: $L\sp p$-convergence
Juan A.
Barceló;
Antonio
Córdoba
655-669
Abstract: We consider the set ${B_p}(\Omega)$ (functions of ${L^p}({\mathbf{R}})$ whose Fourier spectrum lies in $[ - \Omega , + \Omega ]$). We prove that the prolate spheroidal wave functions constitute a basis of this space if and only if $4/3 < p < 4$. The result is obtained as a consequence of the analogous problem for the spherical Bessel functions. The proof rely on a weighted inequality for the Hilbert transform.
Small zeros of quadratic forms over number fields. II
Jeffrey D.
Vaaler
671-686
Abstract: Let $F$ be a nontrivial quadratic form in $ N$ variables with coefficients in a number field $k$ and let $ \mathcal{Z}$ be a subspace of ${k^N}$ of dimension $M,1 \leq M \leq N$. If $F$ restricted to $ \mathcal{Z}$ vanishes on a subspace of dimension $ L,1 \leq L < M$, and if the rank of $F$ restricted to $ \mathcal{Z}$ is greater than $M - L$, then we show that $F$ must vanish on $M - L + 1$ distinct subspaces $ {\mathcal{X}_0},{\mathcal{X}_1}, \ldots ,{\mathcal{X}_{M - L}}$ in $\mathcal{Z}$ each of which has dimension $ L$. Moreover, we show that for each pair $ {\mathcal{X}_0},{\mathcal{X}_1},1 \leq l \leq M - L$, the product of their heights $ H({\mathcal{X}_0})H({\mathcal{X}_1})$ is relatively small. Our results generalize recent work of Schlickewei and Schmidt.
A characterization of nonchaotic continuous maps of the interval stable under small perturbations
D.
Preiss;
J.
Smítal
687-696
Abstract: Recent results of the second author show that every continuous map of the interval to itself either has every trajectory approximable by cycles (sometimes this is possible even in the case when the trajectory is not asymptotically periodic) or is $ \varepsilon$-chaotic for some $\varepsilon > 0$. In certain cases, the first property is stable under small perturbations. This means that a perturbed map can be chaotic but the chaos must be small whenever the perturbation is small. In other words, there are nonchaotic maps without "chaos explosions". In the paper we give a characterization of these maps along with some consequences. Using the main result it is possible to prove that generically the nonchaotic maps are stable.
Optimal stopping of two-parameter processes on nonstandard probability spaces
Robert C.
Dalang
697-719
Abstract: We prove the existence of optimal stopping points for upper semicontinuous two-parameter processes defined on filtered nonstandard (Loeb) probability spaces that satisfy a classical conditional independence hypothesis. The proof is obtained via a lifting theorem for elements of the convex set of randomized stopping points, which shows in particular that extremal elements of this set are ordinary stopping points.
Lie groups that are closed at infinity
Harry F.
Hoke
721-735
Abstract: A noncompact Riemannian manifold $M$ is said to be closed at infinity if no bounded volume form which is also bounded away from zero can be written as the exterior derivative of a bounded form on $M$ . The isoperimetric constant of $M$ is defined by $ h(M) = \inf \{ {\text{vol}}(\partial S)/{\text{vol}}(S)\}$ where $S$ ranges over compact domains with boundary in $ M$. It is shown that a Lie group $G$ with left invariant metric is closed at infinity if and only if $h(G) = 0$ if and only if $G$ is amenable and unimodular. This result relates these geometric invariants of $G$ to the algebraic structure of $ G$ since the conditions amenable and unimodular have algebraic characterizations for Lie groups. $G$ is amenable if and only if $G$ is a compact extension of a solvable group and $G$ is unimodular if and only if $ \operatorname{Tr}({\text{ad}}\,X) = 0$ for all $X$ in the Lie algebra of $G$. An application is the clarification of relationships between several conditions for the existence of transversal invariant measures for a foliation of a compact manifold by the orbits of a Lie group action.
Characterizations of normal quintic $K$-$3$ surfaces
Jin Gen
Yang
737-751
Abstract: If a normal quintic surface is birational to a $K$-$3$ surface then it must contain from one to three triple points as its only essential singularities. A $ K$-$3$ surface is the minimal model of a normal quintic surface with only one triple point if and only if it contains a nonsingular curve of genus two and a nonsingular rational curve crossing each other transversally. The minimal models of normal quintic $K$-$3$ surfaces with several triple points can also be characterized by the existence of some special divisors.
Extending homeomorphisms and applications to metric linear spaces without completeness
Tadeusz
Dobrowolski
753-784
Abstract: A method of extending homeomorphisms between compacta metric spaces is presented. The main application is that homeomorphisms between compacta of an infinite-dimensional locally convex metric linear space extend to the whole space. A lemma used in the proof of this fact together with the method of absorbing sets is employed to show that every $\sigma$ -compact normed linear space is homeomorphic to a dense linear subspace of a Hilbert space. A discussion of the relative topological equivalence of absorbing sets in noncomplete spaces is included. The paper is concluded with some controlled versions of an isotopy extension theorem.
Conjugation and the prime decomposition of knots in closed, oriented $3$-manifolds
Katura
Miyazaki
785-804
Abstract: In this paper we consider the prime decomposition of knots in closed, oriented $3$-manifolds. (For classical knots one can easily prove the uniqueness of prime decomposition by using a standard innermost disk argument.) We define a new relation, conjugation, between oriented knots in closed, oriented $3$-manifolds and prove the following results. (1) The prime decomposition is, roughly speaking, uniquely determined up to conjugation, (2) there is a prime knot $\mathcal{R}$ in $ {S^1} \times {S^2}$ such that $\mathcal{R}\char93 {\mathcal{K}_1} = \mathcal{R}\char93 {\mathcal{K}_2}$ if ${\mathcal{K}_1}$ is a conjugation of ${\mathcal{K}_2}$, and (3) if a knot $\mathcal{K}$ has a prime decomposition which does not contain $ \mathcal{R}$, then it is the unique prime decomposition of $\mathcal{K}$ .
Stable processes with drift on the line
Sidney C.
Port
805-841
Abstract: The stable processes on the line having a drift are investigated. Except for the symmetric Cauchy processes with drift these are all transient and points are nonpolar sets. Explicit information about the potential kernel is obtained and this is used to obtain specific results about hitting times and places for various sets.
Cyclic extensions of $K(\sqrt{-1})/K$
Jón Kr.
Arason;
Burton
Fein;
Murray
Schacher;
Jack
Sonn
843-851
Abstract: In this paper the height $ {\text{ht}}(L/K)$ of a cyclic $2$-extension of a field $K$ of characteristic $\ne 2$ is studied. Here ${\text{ht}}(L/K) \geq n$ means that there is a cyclic extension $E$ of $ K,E \supset L$, with $[E:L] = {2^n}$. Necessary and sufficient conditions are given for ${\text{ht}}(L/K) \geq n$ provided $K(\sqrt { - 1})$ contains a primitive $ {2^n}$th root of unity. Primary emphasis is placed on the case $L = K(\sqrt { - 1})$. Suppose $ {\text{ht}}(K(\sqrt { - 1})/K) \geq 1$. It is shown that $ {\text{ht}}(K(\sqrt { - 1})/K) \geq 2$ and if $K$ is a number field then $ {\text{ht}}(K(\sqrt { - 1})/K) \geq n$ for all $n$. For each $n \geq 2$ an example is given of a field $ K$ such that $ {\text{ht}}(K(\sqrt { - 1})/K) \geq n$ but ${\text{ht}}(K(\sqrt { - 1})/K) \ngeq n + 1$.
On the Hopf index and the Conley index
Christopher K.
McCord
853-860
Abstract: The following generalization of the Poincaré-Hopf index theorem is proved: If $S$ is an isolated invariant set of a flow on a manifold $M$, then the sum of the Hopf indices on $ S$ is equal (up to a sign) to the Euler characteristic of the homology Conley index of $S$.
Local smooth isometric embeddings of low-dimensional Riemannian manifolds into Euclidean spaces
Gen
Nakamura;
Yoshiaki
Maeda
1-51
Abstract: Local smooth isometric embedding problems of low dimensional Riemannian manifolds into Euclidean spaces are studied. Namely, we prove the existence of a local smooth isometric embedding of a smooth Riemannian $3$-manifold with nonvanishing curvature into Euclidean $6$-space. For proving this, we give a local solvability theorem for a system of a nonlinear PDE of real principal type. To obtain the local solvability theorem, we need a tame estimate for the linearized equation corresponding to the given PDE, which is presented by two methods. The first is based on the result of Duistermaat-Hörmander which constructed the exact right inverse for linear PDEs of real principal type by using Fourier integral operators. The second method uses more various properties of Fourier integral operators given by Kumano-go, which seems to be a simpler proof than the above.
Cell-like mappings and nonmetrizable compacta of finite cohomological dimension
Sibe
Mardešić;
Leonard R.
Rubin
53-79
Abstract: Compact Hausdorff spaces $X$ of cohomological dimension ${\dim _Z}X \leq n$ are characterized as cell-like images of compact Hausdorff spaces $Z$ with covering dimension $Z \leq n$. The proof essentially uses the newly developed techniques of approximate inverse systems.
The space of harmonic maps of $S\sp 2$ into $S\sp 4$
Bonaventure
Loo
81-102
Abstract: Every branched superminimal surface of area $4\pi d$ in ${S^4}$ is shown to arise from a pair of meromorphic functions $ ({f_1},{f_2})$ of bidegree $ (d,d)$ such that $ {f_1}$ and ${f_2}$ have the same ramification divisor. Conditions under which branched superminimal surfaces can be generated from such pairs of functions are derived. For each $d \geq 1$ the space of harmonic maps (i.e branched superminimal immersions) of ${S^2}$ into ${S^4}$ of harmonic degree $d$ is shown to be a connected space of complex dimension $2d + 4$ .
Weakly almost periodic flows
R.
Ellis;
M.
Nerurkar
103-119
Abstract: The notion of the enveloping semigroup of a flow is applied to some situations in ergodic theory. In particular, weakly almost periodic functions on groups are studied and Moore's ergodic theorem is proved.
The spectral measure and Hilbert transform of a measure-preserving transformation
James
Campbell;
Karl
Petersen
121-129
Abstract: V. F. Gaposhkin gave a condition on the spectral measure of a normal contraction on ${L^2}$ sufficient to imply that the operator satisfies the pointwise ergodic theorem. We prove that unitary operators which come from measure-preserving transformations satisfy a stronger version of this condition. This follows from the fact that the rotated ergodic Hubert transform is a continuous function of its parameter. The maximal inequality on which the proof depends follows from an analytic inequality related to the Carleson-Hunt Theorem on the a.e. convergence of Fourier series.
Totally categorical structures
Ehud
Hrushovski
131-159
Abstract: A first order theory is totally categorical if it has exactly one model in each infinite power. We prove here that every such theory admits a finite language, and is finitely axiomatizable in that language, modulo axioms stating that the structure is infinite. This was conjectured by Vaught. We also show that every ${\aleph _0}$-stable, $ {\aleph _0}$-categorical structure is a reduct of one that has finitely many models in small uncountable powers. In the case of structures of disintegrated type we nearly find an explicit structure theorem, and show that the remaining obstacle resides in certain nilpotent automorphism groups.
Cauchy-Szeg\H o maps, invariant differential operators and some representations of ${\rm SU}(n+1,1)$
Christopher
Meaney
161-186
Abstract: Fix an integer $n > 1$. Let $ G$ be the semisimple Lie group $ {\text{SU}}(n + 1,1)$ and $ K$ be the subgroup $ {\text{S(U}}(n + 1) \times {\text{U}}(1))$. For each finite dimensional representation $ (\tau ,{\mathcal{H}_\tau })$ of $K$ there is the space of smooth $\tau $-covariant functions on $ G$, denoted by ${C^\infty }(G,\tau)$ and equipped with the action of $ G$ by right translation. Now take $ (\tau ,{\mathcal{H}_\tau })$ to be $({\tau _{p,p}},{\mathcal{H}_{p,p}})$, the representation of $K$ on the space of harmonic polynomials on $ {{\mathbf{C}}^{n + 1}}$ which are bihomogeneous of degree $(p,p)$. For a real number $\nu$ there is the corresponding spherical principal series representation of $ G$, denoted by $({\pi _\nu },{{\mathbf{I}}_{1,\nu }})$. In this paper we show that, as a $ (\mathfrak{g},K)$-module, the irreducible quotient of ${{\mathbf{I}}_{1,1 - n - 2p}}$ can be realized as the space of the $K$-finite elements of the kernel of a certain invariant first order differential operator acting on $ {C^\infty }(G,{\tau _{p,p}})$. Johnson and Wallach had shown that these representations are not square-integrable. Thus, some exceptional representations of $ G$ are realized in a manner similar to Schmid's realization of the discrete series. The kernels of the differential operators which we use here are the intersection of kernels of some Schmid operators and quotient maps, which we call Cauchy-Szegö maps, a generalization the Szegö maps used by Knapp and Wallach. We also identify this representation of $G$ with an end of complementary series representation.
Local properties of secant varieties in symmetric products. I
Mark E.
Huibregtse;
Trygve
Johnsen
187-204
Abstract: Let $L$ be a line bundle on an abstract nonsingular curve $C$, let $V \subset {H^0}(C,L)$ be a linear system, and denote by ${C^{(d)}}$ the symmetric product of $d$ copies of $C$. There exists a canonically defined $ {C^{(d)}}$-bundle map: $\displaystyle \sigma :V \otimes {\mathcal{O}_{{C^{(d)}}}} \to {E_L},$ where ${E_L}$ is a bundle of rank $ d$ obtained from $ L$ by a so-called symmetrization process. The various degenerary loci of $ \sigma$ can be considered as subsecant schemes of ${C^{(d)}}$. Our main result, Theorem 4.2, is given in $\S4$, where we obtain a local matrix description of $ \sigma$ valid (also) at points on the diagonal in ${C^{(d)}}$, and thereby we can determine the completions of the local rings of the secant schemes at arbitrary points. In $\S5$ we handle the special case of giving a local scheme structure to the zero set of $\sigma$.
Local properties of secant varieties in symmetric products. II
Trygve
Johnsen
205-220
Abstract: Let $V$ be a linear system on a curve $ C$. In Part I we described a method for studying the secant varieties $ V_d^r$ locally. The varieties $V_d^r$ are contained in the $d$-fold symmetric product $ {C^{(d)}}$. In this paper (Part II) we apply the methods from Part I. We give a formula for local tangent space dimensions of the varieties $V_d^1$ valid in all characteristics (Theorem 2.4). Assume $\operatorname{rk}\;V = n + 1$ and $\operatorname{char} K = 0$. In $\S\S3$ and $4$ we describe in detail the projectivized tangent cones of the varieties $V_n^1$ for a large class of points. The description is a generalization of earlier work on trisecants for a space curve. In $\S5$ we study the curve in ${C^{(2)}}$ consisting of divisors $D$ such that $ 2D \in V_4^1$ . We give multiplicity formulas for all points on this curve in $ {C^{(2)}}$ in terms of local geometrical invariants of $C$. We assume $\operatorname{char} K = 0$.
$\Delta$-closures of ideals and rings
Louis J.
Ratliff
221-247
Abstract: It is shown that if $ R$ is a commutative ring with identity and $\Delta$ is a multiplicatively closed set of finitely generated nonzero ideals of $ R$, then the operation $I \to {I_\Delta } = { \cup _{K \in \Delta }}(IK:K)$ is a closure operation on the set of ideals $I$ of $R$ that satisfies a partial cancellation law, and it is a prime operation if and only if $R$ is $\Delta$-closed. Also, if none of the ideals in $ \Delta$ is contained in a minimal prime ideal, then ${I_\Delta } \subseteq {I_a}$, the integral closure of $I$ in $R$, and if $\Delta$ is the set of all such finitely generated ideals and $I$ contains an ideal in $\Delta$, then $ {I_\Delta } = {I_a}$. Further, $R$ has a natural $\Delta$-closure ${R^\Delta },A \to {A^\Delta }$ is a closure operation on a large set of rings $A$ that contain $R$ as a subring, $A \to {A^\Delta }$ behaves nicely under certain types of ring extension, and every integral extension overring of $R$ is $ {R^\Delta }$ for an appropriate set $\Delta$. Finally, if $R$ is Noetherian, then the associated primes of ${I_\Delta }$ are also associated primes of ${I_\Delta }K$ and $ {(IK)_\Delta }$ for all $K \in \Delta$.
Braids, link polynomials and a new algebra
Joan S.
Birman;
Hans
Wenzl
249-273
Abstract: A class function on the braid group is derived from the Kauffman link invariant. This function is used to construct representations of the braid groups depending on $2$ parameters. The decomposition of the corresponding algebras into irreducible components is given and it is shown how they are related to Jones' algebras and to Brauer's centralizer algebras.
Integral representations of positive definite matrix-valued distributions on cylinders
Jürgen
Friedrich
275-299
Abstract: The notion of a $ G$-continuous matrix-valued positive definite distribution on $\displaystyle {S_N}(2a) \times {{\mathbf{R}}^M} \times G$ is introduced, where $ G$ is an abelian separable locally compact group and where ${S_N}(2a)$ is an open ball around zero in $ {\mathbf{R}^N}$ with radius $2a > 0$. This notion generalizes that one of strongly continuous positive definite operator-valued functions. For these objects, a Bochner-type theorem gives a suitable integral representation if $N = 1$ or if the matrix-valued distribution is invariant w.r.t. rotations in ${\mathbf{R}^N}$. As a consequence, appropriate extensions to the whole group are obtained. In particular, we show that a positive definite function on a certain cylinder in a separable real Hilbert space $ H$ may be extended to a characteristic function of a finite positive measure on $ H$, if it is invariant w.r.t. rotations and continuous w.r.t. a suitable topology.
Cosmicity of cometrizable spaces
Gary
Gruenhage
301-315
Abstract: A space $ X$ is cometrizable if $ X$ has a coarser metric topology such that each point of $X$ has a neighborhood base of metric closed sets. Most examples in the literature of spaces obtained by modifying the topology of the plane or some other metric space are cometrizable. Assuming the Proper Forcing Axiom (PFA) we show that the following statements are equivalent for a cometrizable space $ X$ : (a) $X$ is the continuous image of a separable metric space; (b) $ {X^\omega }$ is hereditarily separable and hereditarily Lindelöf, (c) ${X^2}$ has no uncountable discrete subspaces; (d) $X$ is a Lindelöf semimetric space; (e) $ X$ has the pointed ${\text{ccc}}$. This result is a corollary to our main result which states that, under PFA, if $ X$ is a cometrizable space with no uncountable discrete subspaces, then either $X$ is the continuous image of a separable metric space or $X$ contains a copy of an uncountable subspace of the Sorgenfrey line.
On the regularity up to the boundary in the Dirichlet problem for degenerate elliptic equations
Adalberto P.
Bergamasco;
Jorge A.
Gerszonowicz;
Gerson
Petronilho
317-329
Abstract: We give a simple proof of the regularity up to the boundary of solutions of the Dirichlet problem for a class of second-order degenerate elliptic equations in the plane. We show that the method of transfer to the boundary via the associated heat equations, can be used to reduce the problem to proving the ellipticity or hypoellipticity of a pseudodifferential operator on the boundary.
An HNN-extension with cyclic associated subgroups and with unsolvable conjugacy problem
Jody Meyer
Lockhart
331-345
Abstract: In this paper, we consider the conjugacy problem for ${\text{HNN}}$-extensions of groups with solvable conjugacy problem for which the associated subgroups are cyclic. An example of such a group with unsolvable conjugacy problem is constructed. A similar construction is given for free products with amalgamation.
The lifting problem for affine structures in nilpotent Lie groups
Nguiffo B.
Boyom
347-379
Abstract: Affine manifolds occur in several situations in pure and applied mathematics, (e.g. leaves of Lagrangian foliations, completely integrable Hamiltonian systems, linear representations of virtually polycyclic groups, geometric quantization and so on). This work is devoted to left invariant affinely flat structures in Lie groups. We are mainly concerned with the following situation. Let $ G$ and ${G_0}$ be nilpotent Lie groups of dimension $n + 1$ and $n$ , respectively and let $h:G \to {G_0}$ be a continuous homomorphism from $ G$ onto ${G_0}$ . Given a left invariant affinely flat structure $ ({G_0},{\nabla _0})$ the lifting problem is to discover whether $ G$ has a left invariant affinely flat structure $ (G,\nabla)$ such that $ h$ becomes an affine morphism. In the present work we answer positively when $({G_0},{\nabla _0})$ is "normal". Therefore the existence problem for a left invariant complete affinely flat structure in nilpotent Lie groups is solved by applying the following subsequent results. Let $ \mathfrak{A}f({G_0})$ be the set of left invariant affinely flat structures in the nilpotent Lie group $ {G_0},({1^ \circ })\;\mathfrak{A}f({G_0}) \ne \emptyset $ implies the existence of normal structure $ ({G_0},{\nabla _0}) \in \mathfrak{A}f({G_0});({2^ \circ })\;h:G \to {G_0}$ being as above every normal structure $({G_0},{\nabla _0})$ has a normal lifted in $\mathfrak{A}f(G)$.
The connectedness of symmetric and skew-symmetric degeneracy loci: even ranks
Loring W.
Tu
381-392
Abstract: A degeneracy locus is the set of points where a vector-bundle map has rank at most a given integer. Such a set is symmetric or skew-symmetric according as whether the vector-bundle map is symmetric or skew-symmetric. We prove a connectedness result, first conjectured by Fulton and Lazarsfeld, for skew-symmetric degeneracy loci and for symmetric degeneracy loci of even ranks.
Rigidity of pseudo-holomorphic curves of constant curvature in Grassmann manifolds
Quo-Shin
Chi;
Yunbo
Zheng
393-406
Abstract: Rigidity of minimal immersions of constant curvature in harmonic sequences generated by holomorphic curves in Grassmann manifolds is studied in this paper by lifting them to holomorphic curves in certain projective spaces. We prove that for such curves the curvature must be positive, and that all such simply connected curves in $ C{P^n}$ are generated by Veronese curves, thus generalizing Calabi's counterpart for holomorphic curves in $C{P^n}$. We also classify all holomorphic curves from the Riemann sphere into $G(2,4)$ whose curvature is equal to $2$ into two families, which illustrates pseudo-holomorphic curves of positive constant curvature in $G(m,N)$ are in general not unitarily equivalent, constracting to the fact that generic isometric complex submanifolds in a Kaehler manifold are congruent.
Prescribing zeros of functions in the Nevanlinna class on weakly pseudo-convex domains in ${\bf C}\sp 2$
Mei-Chi
Shaw
407-418
Abstract: Let $D$ be a bounded weakly pseudo-convex domain in $ {{\mathbf{C}}^2}$ of uniform strict type. For any positive divisor $ M$ of $D$ with finite area, there exists a holomorphic function $f$ in the Nevanlinna class such that $ M$ is the zero set of $ f$. The proof is to study the solutions of $ \bar \partial$ with ${L^1}(\partial D)$ boundary values.
On hypersurfaces of hyperbolic space infinitesimally supported by horospheres
Robert J.
Currier
419-431
Abstract: This paper is concerned with complete, smooth immersed hypersurfaces of hyperbolic space that are infinitesimally supported by horospheres. This latter condition may be restated as requiring that all eigenvalues of the second fundamental form, with respect to a particular unit normal field, be at least one. The following alternative must hold: either there is a point where all the eigenvalues of the second fundamental form are strictly greater than one, in which case the hypersurface is compact, imbedded and diffeomorphic to a sphere; or, the second fundamental form at every point has $1$ as an eigenvalue, in which case the hypersurface is isometric to Euclidean space and is imbedded in hyperbolic space as a horosphere.
Bifurcation of critical periods for plane vector fields
Carmen
Chicone;
Marc
Jacobs
433-486
Abstract: A bifurcation problem in families of plane analytic vector fields which have a nondegenerate center at the origin for all values of a parameter $\lambda \in {{\mathbf{R}}^N}$ is studied. In particular, for such a family, the period function $(\xi ,\lambda) \mapsto P(\xi ,\lambda)$ is defined; it assigns the minimum period to each member of the continuous band of periodic orbits (parametrized by $\xi \in {\mathbf{R}}$) surrounding the origin. The bifurcation problem is to determine the critical points of this function near the center with $\lambda$ as bifurcation parameter. Generally, if the function $\rho$, given by $ \xi \mapsto P(\xi ,{\lambda_\ast}) - P(0,{\lambda_\ast})$, vanishes to order $2k$ at the origin, then it is shown that the period function, after a perturbation of ${\lambda_\ast}$, has at most $k$ critical points near the origin. If $ \rho$ vanishes to infinite order, i.e., the center is isochronous, it is shown that the number of critical points of $P$ for perturbations of ${\lambda_\ast}$ depends on the number of generators of the ideal of all Taylor coefficients of $\rho (\xi ,\lambda)$, where the coefficients are considered elements of the ring of convergent power series in $\lambda$. Specifically, if the ideal is generated by the first $2k$ Taylor coefficients, then a perturbation of ${\lambda_\ast}$ produces at most $k$ critical points of $P$ near the origin. These theorems are applied to the quadratic systems with Bautin centers and to one degree of freedom "kinetic+potential" Hamiltonian systems with polynomial potentials. For the quadratic systems a complete solution of the bifurcation problem is obtained. For the Hamiltonian systems a number of results are proved independent of the degree of the potential and a complete solution is obtained for potentials of degree less than seven. Aside from their intrinsic interest, monotonicity properties of the period function are important in the question of existence and uniqueness of autonomous boundary value problems, in the study of subharmonic bifurcation of periodic oscillations, and in the analysis of the problem of linearization. In this regard it is shown that a Hamiltonian system with a polynomial potential of degree larger than two cannot be linearized. However, while these topics are the subject of a large literature, the spirit of this paper is more akin to that of N. Bautin's treatment of the multiple Hopf bifurcation for quadratic systems and the work on various forms of the weakened Hilbert's 16th problem first posed by V. Arnold.
On the reconstruction of topological spaces from their groups of homeomorphisms
Matatyahu
Rubin
487-538
Abstract: For various classes $ K$ of topological spaces we prove that if $ {X_1},{X_2} \in K$ and ${X_1},{X_2}$ have isomorphic homeomorphism groups, then ${X_1}$ and ${X_2}$ are homeomorphic. Let $G$ denote a subgroup of the group of homeomorphisms $H(X)$ of a topological space $X$. A class $K$ of $ \langle X,G\rangle$'s is faithful if for every $\langle {X_1},{G_1}\rangle ,\langle {X_2},{G_2}\rangle \in K$, if $\varphi :{G_1} \to {G_2}$ is a group isomorphism, then there is a homeomorphism $h$ between ${X_1}$ and ${X_2}$ such that for every $g \in {G_1}\;\varphi (g) = hg{h^{ - 1}}$. Theorem 1: The following class is faithful: $ \{ \langle X,H(X)\rangle \vert(X$ is a locally finite-dimensional polyhedron in the metric or coherent topology or $X$ is a Euclidean manifold with boundary) and for every $ x \in X\;x$ is an accumulation point of $\{ g(x)\vert g \in H(X)\} \} \cup \{ \langle X,G\rangle \vert X$ is a differentiable or a $ PL$-manifold and $ G$ contains the group of differentiable or piecewise linear homeomorphisms$ \}$ $\cup \{ \langle X,H(X)\rangle \vert X$ is a manifold over a normed vector space over an ordered field$\}$. This answers a question of Whittaker $[{\text{W}}]$, who asked about the faithfulness of the class of Banach manifolds. Theorem 2: The following class is faithful: $\{ \langle X,G\rangle \vert X$ is a locally compact Hausdorff space and for every open $T \subseteq X$ and $x \in T\;\{ g(x)\vert g \in H(X)$ and $ g \upharpoonright (X - T) = \operatorname{Id}\}$ is somewhere dense$ \}$. Note that this class includes Euclidean manifolds as well as products of compact connected Euclidean manifolds. Theorem 3: The following class is faithful: $\{ \langle X,H(X)\rangle \vert$ (1) $X$ is a 0-dimensional Hausdorff space; (2) for every $x \in X$ there is a regular open set whose boundary is $\{ x\}$; (3) for every $x \in X$ there are ${g_{1,}}{g_2} \in G$ such that $x \ne {g_1}(x) \ne {g_2}(x) \ne x$, and (4) for every nonempty open $V \subseteq X$ there is $g \in H(X) - \{ \operatorname{Id}\}$ such that $g \upharpoonright (X - V) = \operatorname{Id}\}$. Note that (2) is satisfied by 0-dimensional first countable spaces, by order topologies of linear orderings, and by normed vector spaces over fields different from $ {\mathbf{R}}$. Theorem 4: We prove (Theorem 2.23.1) that for an appropriate class ${K^T}$ of trees $\{ \langle \operatorname{Aut}(T),T; \leq , \circ ,\operatorname{Op}\rangle \vert T \in {K^T}\}$ is first-order interpretable in $\{ \operatorname{Aut}(T)\vert T \in {K^T}\}$.
Bifurcation of a unique stable periodic orbit from a homoclinic orbit in infinite-dimensional systems
Shui-Nee
Chow;
Bo
Deng
539-587
Abstract: Under some generic conditions, we show how a unique stable periodic orbit can bifurcate from a homoclinic orbit for semilinear parabolic equations and retarded functional differential equations. This is a generalization of a result of Šil'nikov for ordinary differential equations.
Two-dimensional Riemann problem for a single conservation law
Tong
Zhang;
Yu Xi
Zheng
589-619
Abstract: The entropy solutions to the partial differential equation $\displaystyle (\partial /\partial t)u(t,x,y) + (\partial /\partial x)f(u(t,x,y)) + (\partial /\partial y)g(u(t,x,y)) = 0,$ with initial data constant in each quadrant of the $(x,y)$ plane, have been constructed and are piecewise smooth under the condition $f''(u) \ne 0, g''(u) \ne 0, (f''(u)/g''(u))\prime \ne 0$. This problem generalizes to several space dimensions the important Riemann problem for equations in one-space dimension. Although existence and uniqueness of solutions are well known, little is known about the qualitative behavior of solutions. It is this with which we are concerned here.
Topological types of finitely-$C\sp 0$-$K$-determined map-germs
Takashi
Nishimura
621-639
Abstract: In this article, we investigate the following two problems Problem 1. Is finite- $ {C^0}{\text{-}}K$-determinacy a topological invariant among analytic map-germs? Problem 2. Do the topological types of all finitely- $ {C^0}{\text{-}}K$-determined map-germs have topological moduli, i.e. do they have infinitely many topological types with the cardinal number of continuum? Problem $ 1$ is solved affirmatively in the complex case. Problem $2$ is solved negatively in the complex case; and affirmatively in the real case.
Sharp estimates for the nontangential maximal function and the Lusin area function in Lipschitz domains
Rodrigo
Bañuelos;
Charles N.
Moore
641-662
Abstract: Let $u$ be a harmonic function on a domain of the form $D = \{ (x,y):x \in {{\mathbf{R}}^n},y \in {\mathbf{R}},y > \phi (x)\}$ where $\phi :{{\mathbf{R}}^n} \to {\mathbf{R}}$ is a Lipschitz function. The authors show a good-$\lambda$ inequality between $ Au$, the Lusin area function of $u$, and $Nu$, the nontangential maximal function of $ u$. This leads to an $ {L^p}$ inequality of the form $\left\Vert Au\right\Vert _p \leq C_p\left\Vert Nu\right\Vert _p$ which is sharp in the sense that $ {C_p}$ is of the smallest possible order in $p$ as $p \to \infty $. For $P \in \partial D$ and $t > 0$ we also consider the functions $Au(P + (0,t))$ and $ Nu(P + (0,t))$ and show that a corollary of the good-$\lambda$ inequality is a law of the iterated logarithm involving these two functions as $ t \to 0$. If $ n = 1$ and $\phi$ has a small Lipschitz constant the above results are shown valid with the roles of $ Nu$ and $Au$ interchanged.
Piecewise linearization of real-valued subanalytic functions
Masahiro
Shiota
663-679
Abstract: We show that for a subanalytic function $f$ on a locally compact subanalytic set $ X$ there exists a unique subanalytic triangulation (a simplicial complex $ K$, a subanalytic homeomorphism $\pi :\vert K\vert \to X$) such that $f \circ \pi {\vert _\sigma }, \sigma \in K$, are linear.
A unified theory for real vs. complex rational Chebyshev approximation on an interval
Arden
Ruttan;
Richard S.
Varga
681-697
Abstract: A unified approach is presented for determining all the constants $ {\gamma _{m,n}}\;(m \geq 0,n \geq 0)$ which occur in the study of real vs. complex rational Chebyshev approximation on an interval. In particular, it is shown that ${\gamma _{m,m + 2}} = 1/3\;(m \geq 0)$, a problem which had remained open.
Hypergraphs with finitely many isomorphism subtypes
Henry A.
Kierstead;
Peter J.
Nyikos
699-718
Abstract: Let $\mathcal{H} = (H,E)$ be an $n$-uniform infinite hypergraph such that the number of isomorphism types of induced subgraphs of $\mathcal{H}$ of cardinality $\lambda$ is finite for some infinite $ \lambda$. We solve a problem due independently to Jamison and Pouzet, by showing that there is a finite subset $K$ of $H$ such that the induced subgraph on $H - K$ is either empty or complete. We also characterize such hypergraphs in terms of finite (not necessarily uniform) hypergraphs.
A Poisson-Plancherel formula for the universal covering group with Lie algebra of type $B\sb n$
Peter
Dourmashkin
719-738
Abstract: A proof is given for the Poisson-Plancherel formula for Lie groups of type ${B_n}$ using the recurrence relations for the Plancherel function on adjacent Cartan subalgebras given in [12] and the recurrence relations for the discrete series constants which determine a $ G$-invariant generalized function on $ {{\mathbf{g}}^\ast }$ appearing in the formula.
Higher-dimensional analogues of the modular and Picard groups
C.
Maclachlan;
P. L.
Waterman;
N. J.
Wielenberg
739-753
Abstract: Clifford algebras are used to describe arithmetic groups which are generalizations of the modular and Picard groups.
Cohomology equations and commutators of germs of contact diffeomorphisms
Augustin
Banyaga;
Rafael
de la Llave;
C. Eugene
Wayne
755-778
Abstract: We study the group of germs of contact diffeomorphisms at a fixed point. We prove that the abelianization of this group is isomorphic to the multiplicative group of real positive numbers. The principal ingredient in this proof is a version of the Sternberg linearization theorem in which the conjugating diffeomorphism preserves the contact structure.
Uniform analyticity of orthogonal projections
R. R.
Coifman;
Margaret A. M.
Murray
779-817
Abstract: Let $X$ denote the circle $T$ or the interval $[ - 1,1]$, and let $d\mu$ denote a nonnegative, absolutely continuous measure on $X$ . Under what conditions does the Gram-Schmidt procedure in the weighted space ${L^2}(X,{\omega ^2}\;d\mu)$ depend analytically on the logarithm of the weight function $ \omega$? In this paper, we show that, in numerous examples of interest, $\log \omega \in BMO$ is a sufficient (often necessary!) condition for analyticity of the Gram-Schmidt procedure. These results are then applied to establish the local analyticity of certain infinite-dimensional Toda flows.
Nonuniqueness for solutions of the Korteweg-de Vries equation
Amy
Cohen;
Thomas
Kappeler
819-840
Abstract: Variants of the inverse scattering method give examples of nonuniqueness for the Cauchy problem for $ {\text{KdV}}$. One example gives a nontrivial $ {C^\infty }$ solution $ u$ in a domain $\{ (x,t):0 < t < H(x)\}$ for a positive nondecreasing function $H$ , such that $u$ vanishes to all orders as $t \downarrow 0$ . This solution decays rapidly as $x \to + \infty $ , but cannot be well behaved as $x$ moves left. A different example of nonuniqueness is given in the quadrant $x \geq 0,t \geq 0$, with nonzero initial data.
Finitely presented lattices: canonical forms and the covering relation
Ralph
Freese
841-860
Abstract: A canonical form for elements of a lattice freely generated by a partial lattice is given. This form agrees with Whitman's canonical form for free lattices when the partial lattice is an antichain. The connection between this canonical form and the arithmetic of the lattice is given. For example, it is shown that every element of a finitely presented lattice has only finitely many minimal join representations and that every join representation can be refined to one of these. An algorithm is given which decides if a given element of a finitely presented lattice has a cover and finds them if it does. An example is given of a nontrivial, finitely presented lattice with no cover at all.
Ergodic attractors
Charles
Pugh;
Michael
Shub
1-54
Abstract: Using the graph transform method, we give a geometric treatment of Pesin's invariant manifold theory. Beyond deriving the existence, uniqueness, and smoothness results by Fathi, Herman, and Yoccoz our method allows us to do four things: optimally conserve smoothness, deal with endomorphisms, prove absolute continuity of the Pesin laminations, and produce ergodic attractors.
Semicharacteristics, bordism, and free group actions
James F.
Davis;
R. James
Milgram
55-83
Abstract: In this paper we give characteristic class formulae for all semicharacteristic classes of all compact, closed manifolds with finite fundamental groups. These invariants are identified with elements in certain odd $L$-groups, and exactly which elements occur is specified. An appendix calculates the cohomology of the model groups needed. A second appendix determines the structure of the $L$-groups needed.
Determinacy of sufficiently differentiable maps
Alan M.
Selby
85-113
Abstract: Variants of the algebraic conditions of Mather are shown to be sufficient for the $k$-determinacy of ${C^u}$ maps with respect to $j$-flat, contact (or right) ${C^r}$ equivalence relations where $ u - k \leq r \leq u - k + j + 1$ and $ 0 \leq j < k \leq u$. The required changes of coordinates and matrix-valued functions are constructed from the variation of coefficients in polynomials. The main result follows from a finite-dimensional, polynomial pertubation argument which employs a parameter-dependent polynomial representation of functions based on Taylor's formula. For $r > k$, the algebraic conditions are seen to be necessary.
The spectrum of the Schr\"odinger operator
Martin
Schechter
115-128
Abstract: We describe the negative spectrum of the Schrödinger operator with a singular potential. We determine the exact value of the bottom of the spectrum and estimate it from above and below. We describe the dependence of a crucial constant on the eigenvalue parameter and discuss some of its properties. We show how recent results of others are simple consequences of a theorem proved by the author in 1972.
Rigidity for complete Weingarten hypersurfaces
M.
Dajczer;
K.
Tenenblat
129-140
Abstract: We classify, locally and globally, the ruled Weingarten hypersurfaces of the Euclidean space. As a consequence of the local classification and a rigidity theorem of Dajczer and Gromoll, it follows that a complete Weingarten hypersurface which does not contain an open subset of the form $ {L^3} \times {{\mathbf{R}}^{n - 3}}$, where ${L^3}$ is unbounded and $n \geq 3$, is rigid.
Quadrature and harmonic $L\sp 1$-approximation in annuli
D. H.
Armitage;
M.
Goldstein
141-154
Abstract: Open sets $ D$ in ${R^N}\;(N \geq 3)$ with the property that $ \bar D$ is a closed annulus $\{ x:{r_1} \leq \;\left\Vert x\right\Vert \; \leq {r_2}\}$ are characterized by quadrature formulae involving mean values of certain harmonic functions. One such characterization is used to give a criterion for the existence of a best harmonic $ {L^1}$ approximant to a function which is subharmonic (and satisfies some other conditions) in an annulus.
Deforming a PL submanifold of Euclidean space into a hyperplane
Jože
Vrabec
155-178
Abstract: Let $M$ be a closed, $k$-connected, $m$-dimensional $ {\text{PL}}$ submanifold of ${\mathbb{R}^{2m - k - 1}}\;(1 \leq k \leq m - 4)$. The main result of this paper states that if $ m - k$ is even, then every embedding of $M$ into $ {\mathbb{R}^{2m - k}}$ can be isotopically deformed into ${\mathbb{R}^{2m - k - 1}}$, and specifies which embeddings of $M$ into $ {\mathbb{R}^{2m - k}}$ can be deformed into ${\mathbb{R}^{2m - k - 1}}$ in case $ m - k$ is odd.
Holomorphic foliations in ruled surfaces
Xavier
Gómez-Mont
179-201
Abstract: We analyse the universal families of holomorphic foliations with singularities in a ruled surface. In terms of Chern classes we determine the general and the special families. We also classify all nonsingular foliations.
Singularities of the scattering kernel and scattering invariants for several strictly convex obstacles
Vesselin M.
Petkov;
Luchezar N.
Stojanov
203-235
Abstract: Let $\Omega \subset {{\mathbf{R}}^n}$ be a domain such that ${{\mathbf{R}}^n}\backslash \Omega$ is a disjoint union of a finite number of compact strictly convex obstacles with $ {C^\infty }$ smooth boundaries. In this paper the singularities of the scattering kernel $ s(t,\theta ,\omega)$, related to the wave equation in ${\mathbf{R}} \times \Omega $ with Dirichlet boundary condition, are studied. It is proved that for every $\omega \in {S^{n - 1}}$ there exists a residual subset $ \mathcal{R}(\omega)$ of ${S^{n - 1}}$ such that for each $\theta \in \mathcal{R}(\omega),\theta \ne \omega$ $\displaystyle {\text{singsupp}}\,s(t,\theta ,\omega) = {\{ - {T_\gamma }\} _\gamma },$ where $ \gamma$ runs over the scattering rays in $\Omega$ with incoming direction $\omega$ and with outgoing direction $ \theta$ having no segments tangent to $ \partial \Omega$, and ${T_\gamma }$ is the sojourn time of $ \gamma$. Under some condition on $\Omega$, introduced by M. Ikawa, the asymptotic behavior of the sojourn times of the scattering rays related to a given configuration, as well as the precise rate of the decay of the coefficients of the main singularity of $ s(t,\theta ,\omega)$, is examined.
Some families of isoparametric hypersurfaces and rigidity in a complex hyperbolic space
Micheal H.
Vernon
237-256
Abstract: The geometric notion of equivalence for submanifolds in a chosen ambient space is that of congruence. In this study, a certain type of isoparametric hypersurface of a complex hyperbolic space form is shown to have a rigid immersion by utilizing the congruences of a Lorentzian hyperbolic space form that lies as an ${S^1}$-fiber bundle over the complex hyperbolic space. Several families of isoparametric hypersurfaces (namely tubes and horospheres) are constructed whose immersions are rigid.
Summation, transformation, and expansion formulas for bibasic series
George
Gasper
257-277
Abstract: An indefinite bibasic sum containing three parameters is evaluated and used to derive bibasic extensions of Euler's transformation formula and of a Fields and Wimp expansion formula. It is also used to derive a transformation formula involving four independent bases, a $ q$-Lagrange inversion formula, and some quadratic, cubic and quartic summation formulas.
Well-quasi-ordering infinite graphs with forbidden finite planar minor
Robin
Thomas
279-313
Abstract: We prove that given any sequence $ {G_1},{G_2}, \ldots$ of graphs, where ${G_1}$ is finite planar and all other ${G_i}$ are possibly infinite, there are indices $i,j$ such that $i < j$ and ${G_i}$ is isomorphic to a minor of ${G_j}$ . This generalizes results of Robertson and Seymour to infinite graphs. The restriction on $ {G_1}$ cannot be omitted by our earlier result. The proof is complex and makes use of an excluded minor theorem of Robertson and Seymour, its extension to infinite graphs, Nash-Williams' theory of better-quasi-ordering, especially his infinite tree theorem, and its extension to something we call tree-structures over ${\text{QO}}$-categories, which includes infinitary version of a well-quasi-ordering theorem of Friedman.
Nearly representable operators
R.
Kaufman;
Minos
Petrakis;
Lawrence H.
Riddle;
J. J.
Uhl
315-333
Abstract: Among Bourgain's many remarkable theorems is one from 1980 which states that if $T$ is a non-Dunford-Pettis operator from ${L_1}[0,1]$ into an arbitrary Banach space, then there is a Dunford-Pettis operator $ D$ from ${L_1}[0,1]$ into $ {L_1}[0,1]$ such that the composition $T \circ D$ is not Bochner representable. This theorem sets up the following question: What are the operators $T$ from $ {L_1}[0,1]$ into a Banach space $X$ such that $T \circ D$ is Bochner representable for all Dunford-Pettis operators $D:{L_1}[0,1] \to {L_1}[0,1]$ ? We call such an operator nearly representable. In view of Bourgain's theorem, all nearly representable operators are Dunford-Pettis. If $X$ is a Banach space such that all nearly representable operators are, in addition, Bochner representable, then we say $X$ has the near Radon-Nikodym property (NRNP) and ask which Banach spaces have the NRNP? This paper is an attempt to provide at least partial answers to these questions. The first section collects terminology, gives the introductory results and shows that the NRNP is a three space property. The second section studies a continuity property that implies near representability. Finally, the third section contains the main result of the paper, Theorem 15, which states that if $ T:{L_1}[0,1] \to {L_1}[0,1]$ is a nonrepresentable operator, there exists a Dunford-Pettis operator $D:{L_1}[0,1] \to {L_1}[0,1]$ such that $ T \circ D$ is also nonrepresentable. This implies that the ${\text{NRNP}}$ is shared by ${L_1}[0,1]$, lattices not containing $ {c_0}$, and $ {L_1}({\mathbf{T}})/H_0^1$.
A separable space with no remote points
Alan
Dow
335-353
Abstract: In the model obtained by adding $ {\omega _2}$ side-by-side Sacks reals to a model of ${\mathbf{CH}}$, there is a separable nonpseudocompact space with no remote points. To prove this it is also shown that in this model the countable box product of Cantor sets contains a subspace of size ${\omega _2}$ such that every uncountable subset has density ${\omega _1}$. Furthermore assuming the existence of a measurable cardinal $\kappa$ with ${2^\kappa } = {\kappa ^ + }$, a space $X$ is produced with no isolated points but with remote points in $\upsilon X - X$. It is also shown that a pseudocompact space does not have remote points.
Digital representations using the greatest integer function
Bruce
Reznick
355-375
Abstract: Let ${S_d}(\alpha)$ denote the set of all integers which can be expressed in the form $\sum {{\varepsilon _i}[{\alpha ^i}]}$, with ${\varepsilon _i} \in \{ 0, \ldots ,d - 1\}$, where $d \geq 2$ is an integer and $\alpha \geq 1$ is real, and let $ {I_d}$ denote the set of $ \alpha$ so that $ {S_d}(\alpha) = {{\mathbf{Z}}^ + }$. We show that ${I_d} = [1,{r_d}) \cup \{ d\}$, where $ {r_2} = {13^{1/4}},{r_3} = {22^{1/3}}$ and ${r_2} = {({d^2} - d - 2)^{1/2}}$ for $ d \geq 4$. If $\alpha \notin {I_d}$ we show that ${T_d}(\alpha)$, the complement of ${S_d}(\alpha)$, is infinite, and discuss the density of $ {T_d}(\alpha)$ when $\alpha < d$. For $d \geq 4$ and a particular quadratic irrational $\beta = \beta (d) < d$, we describe ${T_d}(\beta)$ explicitly and show that $\vert{T_d}(\beta) \cap [0,n]\vert$ is of order ${n^{e(d)}}$, where $e(d) < 1$.
Generalizations of the stacked bases theorem
Paul
Hill;
Charles
Megibben
377-402
Abstract: Let $H$ be a subgroup of the free abelian group $G$. In order for there to exist a basis ${\{ {x_i}\} _{i \in I}}$ of $G$ for which $H = { \oplus _{i \in I}}\langle {n_i}{x_i}\rangle$ for suitable nonnegative integers $ {n_i}$, it is obviously necessary for $G/H$ to be a direct sum of cyclic groups. In the 1950's, Kaplansky raised the question of whether this condition on $G/H$ is sufficient for the existence of such a basis. J. Cohen and H. Gluck demonstrated in 1970 that the answer is "yes"; their result is known as the stacked bases theorem, and it extends the classical and well-known invariant factor theorem for finitely generated abelian groups. In this paper, we develop a theory that contains and, in fact, generalizes in several directions the stacked bases theorem. Our work includes a complete classification, using numerical invariants, of the various free resolutions of any abelian group.
Harnack's inequality for degenerate Schr\"odinger operators
Cristian E.
Gutiérrez
403-419
Abstract: We prove a Harnack inequality for nonnegative weak solutions of certain Schrödinger equations of the form $Lu - Vu = 0$ where $L$ is a second order degenerate elliptic operator in divergence form and $V$ is a potential in certain class.
The uniform bound problem for local birational nonsingular morphisms
Bernard
Johnston
421-431
Abstract: It is known that any factorization of a local birational morphism $f:\operatorname{Spec}\;S \to \operatorname{Spec}\;R$ of nonsingular (affine) schemes of arbitrary dimension via other nonsingular schemes must be finite in length. This fact generalizes the classical Local Factorization Theorem of Zariski and Abhyankar, which states that there is a unique such factorization, that given by quadratic transformations, in the surface case. A much stronger generalization is given here, namely, that there exists a uniform bound on the lengths of all such factorizations, provided that $R$ is excellent. This bound is explicitly calculated for some concrete extensions and examples are given to show that this is the strongest generalization possible in some sense.
The error in spatial truncation for systems of parabolic conservation laws
Hung Ju
Kuo
433-465
Abstract: In this paper we investigate the behavior of the solution of \begin{displaymath}\begin{array}{*{20}{c}} {{u_t} = D{u_{xx}} - f{{(u)}_x},} ... ...in {L^\infty },\qquad u(t, \pm L) = {u^ \pm },} \end{array} \end{displaymath} where $t \geqslant 0$ and $x \in [ - L,L]$. Solutions of this equation are considered to be approximations to the solutions of the corresponding parabolic conservation laws. We obtain decay results on the norms of the difference between the solution for $L$ infinite and the solution when $L$ is finite.
Regular coverings of homology $3$-spheres by homology $3$-spheres
E.
Luft;
D.
Sjerve
467-481
Abstract: We study $ 3$-manifolds that are homology $3$-spheres and which admit nontrivial regular coverings by homology $3$-spheres. Our main theorem establishes a relationship between such coverings and the canonical covering of the $3$-sphere ${S^3}$ onto the dodecahedral space $ {D^3}$. We also give methods for constructing irreducible sufficiently large homology $3$-spheres $ \tilde M,\;M$ together with a degree $1$ map $ h:M \to {D^3}$ such that $ \tilde M$ is the covering space of $M$ induced from the universal covering ${S^3} \to {D^3}$ by means of the degree $ 1$ map $h:M \to {D^3}$. Finally, we show that if $p:\tilde M \to M$ is a nontrivial regular covering and $\tilde M,\;M$ are homology spheres with $ M$ Seifert fibered, then $\tilde M = {S^3}$ and $M = {D^3}$.
Unramified class field theory for orders
Peter
Stevenhagen
483-500
Abstract: The main theorem of unramified class field theory, which states that the class group of the ring of integers of a number field $ K$, is canonically isomorphic to the Galois group of the maximal totally unramified abelian extension of $K$ over $K$, is generalized and proved for all infinite commutative rings with unit that, like rings of integers, are connected and finitely generated as a module over ${\mathbf{Z}}$. Modulo their nilradical, these rings are exactly the connected orders in products of number fields.
A criterion for the boundedness of singular integrals on hypersurfaces
Stephen W.
Semmes
501-513
Abstract: This paper gives geometric conditions on a hypersurface in ${{\mathbf{R}}^n}$ so that certain singular integrals on that hypersurface define bounded operators on $ {L^2}$. These singular integrals include the Cauchy integral operator in the sense of Clifford analysis and in particular the double layer potential. For curves in the plane, this condition is more general than the chord-arc condition but less general than the Ahlfors-David condition. The main tool is the $T(b)$ theorem [DJS].
A $K$-theoretic invariant for dynamical systems
Yiu Tung
Poon
515-533
Abstract: Let $(X,T)$ be a zero-dimensional dynamical system. We consider the quotient group $G = C(X,Z)/B(X,T)$, where $C(X,Z)$ is the group of continuous integer-valued functions on $X$ and $B(X,T)$ is the subgroup of functions of the form $f - f \circ T$. We show that if $(X,T)$ is topologically transitive, then there is a natural order on $G$ which makes $G$ an ordered group. This order structure gives a new invariant for the classification of dynamical systems. We prove that for each $n$, the number of fixed points of $ {T^n}$ is an invariant of the ordered group $G$. Then we show how $G$ can be computed as a direct limit of finite rank ordered groups. This is used to study the conditions under which $\lq G$ is a dimension group. Finally we discuss the relation between $G$ and the ${K_0}$-group of the crossed product ${C^{\ast}}$-algebra associated to the system $ (X,T)$.
On the linear representation of braid groups
D. D.
Long
535-560
Abstract: We give a new derivative of the Burau and Gassner representations of the braid and pure braid groups. Various applications are explored.
The connection matrix theory for Morse decompositions
Robert D.
Franzosa
561-592
Abstract: The connection matrix theory for Morse decompositions is introduced. The connection matrices are matrices of maps between the homology indices of the sets in the Morse decomposition. The connection matrices cover, in a natural way, the homology index braid of the Morse decomposition and provide information about the structure of the Morse decomposition. The existence of connection matrices of Morse decompositions is established, and examples illustrating applications of the connection matrix are provided.
Characterization of nonlinear semigroups associated with semilinear evolution equations
Shinnosuke
Oharu;
Tadayasu
Takahashi
593-619
Abstract: Nonlinear continuous perturbations of linear dissipative operators are considered from the point of view of the nonlinear semigroup theory. A general class of nonlinear perturbations of linear contraction semigroups in a Banach space $ X$ is introduced by means of a lower semicontinuous convex functional $ [{\text{unk}}]:X \to [0,\infty ]$ and two notions of semilinear infinitesimal generators of the associated nonlinear semigroups are formulated. Four types of necessary and sufficient conditions are given for a semilinear operator $A + B$ of the class to be the infinitesimal generator of a nonlinear semigroup $\{ S(t):t \geqslant 0\}$ on the domain $C$ of $B$ such that for $x \in C$ the $C$-valued function $ S( \cdot )x$ on $[0,\infty )$ provides a unique mild solution of the semilinear evolution equation $[{\text{unk]}}(u( \cdot ))$. It turns out that various types of characterizations of nonlinear semigroups associated with semilinear evolution equations are obtained and, in particular, a semilinear version of the Hille-Yosida theorem is established in a considerably general form.
Equivariant Morse theory for starshaped Hamiltonian systems
Claude
Viterbo
621-655
Abstract: Let $\Sigma$ be a starshaped hypersurface in $ {R^{2n}}$; the problem of finding closed characteristics of $ \Sigma$ can be classically reduced to a variational problem. This leads to studying an ${S^1}$-equivariant functional on a Hilbert space. The equivariant Morse theory of this functional, together with the assumption that $\Sigma$ only has finitely many geometrically distinct characteristics, leads to a remarkable formula relating the average indices of the characteristics. Using this formula one can prove, at least for $n$ even, that genetically there are infinitely many characteristics (cf. [E1] for the convex case).
On the monoid of tame extensions
Cornelius
Greither;
D. K.
Harrison
657-682
Abstract: This paper deals with not necessarily maximal orders in abelian extensions of number fields. We restrict our attention to orders invariant under the Galois group $G$. Based on recent work of Childs and Hurley [CH], we introduce a notion of tameness for such orders (actually this is done in a slightly more general setting). The maximal order is tame in this sense if and only if the field extension is tamely ramified.
Stability of viscous scalar shock fronts in several dimensions
Jonathan
Goodman
683-695
Abstract: We prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions. The proof uses the "integrated equation" and an effective equation for the motion of the front itself. We derive energy estimates that balance terms from the integrated equation with terms from the front motion equation.
Invariants of graphs in three-space
Louis H.
Kauffman
697-710
Abstract: By associating a collection of knots and links to a graph in three-dimensional space, we obtain computable invariants of the embedding type of the graph. Two types of isotopy are considered: topological and rigid-vertex isotopy. Rigid-vertex graphs are a category mixing topological flexibility with mechanical rigidity. Both categories constitute steps toward models for chemical and biological networks. We discuss chirality in both rigid and topological contexts.
Nonlinear second order elliptic partial differential equations at resonance
R.
Iannacci;
M. N.
Nkashama;
J. R.
Ward
711-726
Abstract: In this paper we study the solvability of boundary value problems for semilinear second order elliptic partial differential equations of resonance type in which the nonlinear perturbation is not (necessarily) required to satisfy the Landesman-Lazer condition or the monotonicity assumption. The nonlinearity may be unbounded and some crossing of eigenvalues is allowed. Selfadjoint and nonselfadjoint resonance problems are considered.
Infix congruences on a free monoid
C. M.
Reis
727-737
Abstract: A congruence $ \rho$ on a free monoid ${X^{\ast}}$ is said to be infix if each class $ C$ of $\rho$ satisfies $u \in C$ and $xuy \in C$ imply $xy = 1$. The main purpose of this paper is a characterization of commutative maximal infix congruences. These turn out to be congruences induced by homomorphisms $\tau$ from $ {X^{\ast}}$ to ${{\mathbf{N}}^0}$, the monoid of nonnegative integers under addition, with ${\tau ^{ - 1}}(0) = 1$.
Codimension two complete noncompact submanifolds with nonnegative curvature
Maria Helena
Noronha
739-748
Abstract: We study the topology of complete noncompact manifolds with non-negative sectional curvatures isometrically immersed in Euclidean spaces with codimension two. We investigate some conditions which imply that such a manifold is a topological product of a soul by a Euclidean space and this gives a complete topological description of this manifold.
Multiple solutions of perturbed superquadratic second order Hamiltonian systems
Yi Ming
Long
749-780
Abstract: In this paper we prove the existence of infinitely many distinct $ T$-periodic solutions for the perturbed second order Hamiltonian system $V:{{\mathbf{R}}^N} \to {\mathbf{R}}$ is continuously differentiable and superquadratic, and that $ f$ is square integrable and $T$-periodic. In the proof we use the minimax method of the calculus of variation combining with a priori estimates on minimax values of the corresponding functionals.
Approximating continuous functions by holomorphic and harmonic functions
Christopher J.
Bishop
781-811
Abstract: If $\Omega$ is a Widom domain in the plane (e.g., finitely connected) and $f$ is any bounded harmonic function on $ \Omega$ which is not holomorphic, then we prove the algebra ${H^\infty }(\Omega )[f]$ contains all the uniformly continuous functions on $\Omega$. The basic tools are the solution of the $\overline \partial$ equation with ${L^\infty }$ estimates and some estimates on the level sets of functions in BMOA.
Factorization of diffusions on fibre bundles
Ming
Liao
813-827
Abstract: Let $\pi :M \to N$ be a fibre bundle with a $G$-structure and a connection. A $ G$-invariant operator $ A$ on the standard fibre $ F$ is "shifted" to an operator ${A^{\ast}}$ on $M$ and a semielliptic operator $B$ on $N$ is "lifted" to an operator $\tilde B$ on $M$. Let ${X_t}$ be an $A$-diffusion on $F$, let ${Y_t}$ be a $B$-diffusion on $N$ which is independent of ${X_t}$ and let ${\Psi _t}$ be its horizontal lift in the associated principal bundle. Then ${Z_t} = {\Psi _t}({X_t})$ is a diffusion on $M$ with generator ${A^{\ast}} + \tilde B$. Conversely, such a factorization is possible only if the fibre bundle has a proper $ G$-structure. In the case of a Riemannian submersion, $X,\;Y$ and $Z$ can be taken to be Brownian motions and the existence of a $G$-structure then means that the fibres are totally geodesic.
Determination of all imaginary cyclic quartic fields with class number $2$
Kenneth
Hardy;
Richard H.
Hudson;
David
Richman;
Kenneth S.
Williams
1-55
Abstract: It is proved that there are exactly $8$ imaginary cyclic quartic fields with class number $2$.
Continuous cohomology and real homotopy type
Edgar H.
Brown;
Robert H.
Szczarba
57-106
Abstract: Various aspects of homotopy theory in the category of simplicial spaces are developed. Topics covered include continuous cohomology, continuous de Rham cohomology, the Kan extension condition, the homotopy relation, fibrations, the Serre spectral sequence, real homotopy type and its relation to graded commutative differential algebras over the reals.
Realizing rotation vectors for torus homeomorphisms
John
Franks
107-115
Abstract: We consider the rotation set $\rho (F)$ for a lift $F$ of a homeomorphism $f:{T^2} \to {T^2}$, which is homotopic to the identity. Our main result is that if a vector $v$ lies in the interior of $\rho (F)$ and has both coordinates rational, then there is a periodic point $x \in {T^2}$ with the property that $\displaystyle \frac{{{F^q}({x_0}) - {x_0}}}{q} = v$ where ${x_0} \in {R^2}$ is any lift of $x$ and $q$ is the least period of $x$.
Lie sphere transformations and the focal sets of certain taut immersions
Steven G.
Buyske
117-133
Abstract: We study the images of certain taut or Dupin hypersurfaces, including their focal sets, under Lie sphere transformations (generalizations of conformal transformations of euclidean or spherical space). After the introduction, the method of studying hypersurfaces as Lie sphere objects is developed. In two recent papers, Cecil and Chern use submanifolds of the space of lines on the Lie quadric. Here we use submanifolds of the Lie quadric itself instead. The third section extends the concepts of tightness and tautness to semi-euclidean space. The final section shows that if a hypersurface is the Lie sphere image of certain standard constructions (tubes, cylinders, and rotations) over a taut immersion, the resulting family of curvature spheres is taut in the Lie quadric. The sheet of the focal set will be tight in euclidean space if it is compact. In particular, if a hypersurface in euclidean space is the Lie sphere image of an isoparametric hypersurface each compact sheet of the focal set will be tight.
Peripherally specified homomorphs of knot groups
Dennis
Johnson;
Charles
Livingston
135-146
Abstract: Let $G$ be a group and let $\mu$ and $\lambda$ be elements of $G$. Necessary and sufficient conditions are presented for the solution of the following problem: Is there a knot $K$ in ${S^3}$ and a representation $\rho :{\pi _1}({S^3} - K) \to G$ such that $\rho (m) = \mu $ and $\rho (l) = \lambda$, where $m$ and $l$ are the meridian and longitude of $K$?
Spectral decompositions of one-parameter groups of isometries on Hardy spaces
Dimitri
Karayannakis
147-166
Abstract: Spectral decompositions of strongly continuous one-parameter groups of surjective isometries on Hardy spaces of the disk ${\mathbf{D}}$ and the torus ${{\mathbf{T}}^2}$ are examined; a concrete description of the (pointwise) action of these decompositions is presented, mainly in the parabolic case, leading to a complete description of the action of the partial sum-operators of M. Riesz when carried from ${L^p}({\mathbf{R}})$ to ${H^p}({\mathbf{D}})$, $ 1 < p \leq 2$. The (pointwise) action of the spectral decompositions of these isometric groups on $ {H^p}({{\mathbf{T}}^2})$, $1 < p < \infty$ is also examined and concrete descriptions are derived, mainly in the parabolic case.
Differential-difference operators associated to reflection groups
Charles F.
Dunkl
167-183
Abstract: There is a theory of spherical harmonics for measures invariant under a finite reflection group. The measures are products of powers of linear functions, whose zero-sets are the mirrors of the reflections in the group, times the rotation-invariant measure on the unit sphere in ${{\mathbf{R}}^n}$. A commutative set of differential-difference operators, each homogeneous of degree $ -1$, is the analogue of the set of first-order partial derivatives in the ordinary theory of spherical harmonics. In the case of $ {{\mathbf{R}}^2}$ and dihedral groups there are analogues of the Cauchy-Riemann equations which apply to Gegenbauer and Jacobi polynomial expansions.
Torsion points on abelian \'etale coverings of ${\bf P}\sp 1-\{0,1,\infty\}$
Robert F.
Coleman
185-208
Abstract: Let $X \to {{\mathbf{P}}^1}$ be an Abelian covering of degree $m$ over $ {\mathbf{Q}}({\mu _m})$ unbranched outside 0, $1$ and $\infty$. If the genus of $X$ is greater than $1$ embed $X$ in its Jacobian $J$ in such a way that one of the points above 0, $ 1$ or $\infty$ is mapped to the origin. We study the set of torsion points of $J$ which lie on $X$. In particular, we prove that this set is defined over an extension of $ {\mathbf{Q}}$ unramified outside $6m$. We also obtain information about the orders of these torsion points.
A converse to the mean value property on homogeneous trees
Massimo A.
Picardello;
Wolfgang
Woess
209-225
Abstract: The homogeneous tree ${\mathbf{T}}$ of degree $q + 1\quad (q \geq 2)$ may be considered as a discrete analogue of the open unit disc ${\mathbf{D}}$. On $ {\mathbf{D}}$, every harmonic function satisfies the mean value property (MVP) at every point. Conversely, positive functions on ${\mathbf{D}}$ having the MVP with respect to a ball with specified radius at each point of ${\mathbf{D}}$ are harmonic under certain assumptions concerning the radius function: results of this type are due to J. R. Baxter, W. Veech and others. Here we consider harmonic functions on ${\mathbf{T}}$ with respect to a natural choice of a discrete Laplacian: the analogous MVP is true in this setting. We present a Lipschitz-type condition on the radius function (which now has integer values and refers to the discrete metric of ${\mathbf{T}}$) under which harmonicity holds for positive functions whose value at each point is the mean of its values over the ball of the radius assigned to this point. The method is based upon our previous results concerning the geometrical realization of Martin boundaries of certain transition operators as the space of ends of the underlying graph.
Proper holomorphic mappings from the two-ball to the three-ball
J. A.
Cima;
T. J.
Suffridge
227-239
Abstract: We prove that a proper mapping of the two ball in ${\mathbf{C}^{n}}$ into the three ball, which is $ {C^2}$ on the closed two ball is equivalent to one of four normalized polynomial mappings. This improves the known result of Faran. The proof is basic using Taylor expansions.
Torsion points of generic formal groups
Michael
Rosen;
Karl
Zimmermann
241-253
Abstract: Let $F$ be a generic formal group of height $h$ defined over $A = {{\mathbf{Z}}_p}[[{t_1}, \ldots ,{t_{h - 1}}]]$. Let $K$ be the quotient field of $A$. We show the natural map ${\rho _n}:{\text{Gal}}(K(\operatorname{ker} [{p^n}])/K) \to G{L_h}({\mathbf{Z}}/{p^n}{\mathbf{Z}})$ isomorphisms for all $n \ge 1$ provided $p \ne 2$.
Topological entropy of homoclinic closures
Leonardo
Mendoza
255-266
Abstract: In this paper we study the topological entropy of certain invariant sets of diffeomorphisms, namely the closure of the set of transverse homoclinic points associated with a hyperbolic periodic point, in terms of the growth rate of homoclinic orbits. First we study homoclinic closures which are hyperbolic in $n$-dimensional compact manifolds. Using the pseudo-orbit shadowing property of basic sets we prove a formula similar to Bowen's one on the growth of periodic points. For the nonuniformly hyperbolic case we restrict our attention to compact surfaces.
Excessive measures and the existence of right semigroups and processes
J.
Steffens
267-290
Abstract: Given a resolvent $({U_\alpha })$ on a Lusin space $(E,\mathfrak{E})$, the paper gives necessary and sufficient conditions in terms of the excessive measures that ensure the existence of a right process, resp. a right continuous semigroup, on $(E,\mathfrak{E})$ with resolvent $({U_\alpha })$. Furthermore, a notion of nonbranch points with respect to $({U_\alpha })$ is introduced--also in terms of the excessive measures--and various characterizations are given. They show, in particular, the equivalence of this definition with those introduced and discussed by Engelbert and Wittmann.
Spreading of singularities at the boundary in semilinear hyperbolic mixed problems. II. Crossing and self-spreading
Mark
Williams
291-321
Abstract: The creation of anomalous singularities in solutions to nonlinear hyperbolic equations due to crossing or self-spreading in free space is by now rather well understood. In this paper we study how anomalous singularities are produced in mixed problems for semilinear wave equations $\square u = f(u)$ on the half-space ${\mathbf{R}}_ + ^{n + 1},u \in H_{{\operatorname{loc}}}^s,s > (n + 1)/2$, due to crossing and self-spreading at boundary points. Several phenomena appear in the problems considered here which distinguish spreading at the boundary from spreading in free space: (1) Anomalous singularities of strength $\sim 2s - n/2$ can arise when incoming singularity-bearing rays cross or self-spread at a point on the boundary. A consequence of this, announced in [14], is that the analogue of Beals's $3s$ theorem fails for reflection in second-order mixed problems. Although ${H^r}$ regularity for $r < \sim 3s -n$ propagates along null bicharacteristics in free space, for $r < \sim 2s -n/2$ it does not in general reflect. (2) For nonlinear wave equations in free space, anomalous singular support is never produced by the interaction of fewer than three bicharacteristics, unless self-spreading occurs. However, anomalous singularities can arise when a pair of rays cross at a boundary point. (3) Suppose $ \square u = {u^2}$ and $u \in {C^\infty }$ on the boundary. For certain choices of initial data, anomalous singularities of strength $ \sim 2s - n/2$ arise at the boundary from three sources: interactions of incoming rays with incoming rays, incoming rays with reflected rays, and reflected rays with reflected rays. Singularities produced by the incoming-reflected interactions differ in sign from and are strictly weaker than the other two types, so some cancellations occur. As the incoming rays approach being gliding rays, the difference in strength decreases and hence the cancellations become increasingly significant.
Primary cycles on the circle
Louis
Block;
Ethan M.
Coven;
Leo
Jonker;
Michał
Misiurewicz
323-335
Abstract: We consider cycles, i.e., periodic orbits, of continuous degree one maps of the circle. One cycle forces another if every such map that has a cycle which looks like the first also has a cycle which looks like the second. We characterize primary cycles, i.e., those which force no other cycle of the same period.
Conjugating homeomorphisms to uniform homeomorphisms
Katsuro
Sakai;
Raymond Y.
Wong
337-356
Abstract: Let $H(X)$ denote the group of homeomorphisms of a metric space $X$ onto itself. We say that $h \in H(X)$ is conjugate to $g \in H(X)$ if $ {g = fhf^{-1}}$ for some $f \in H(X)$. In this paper, we study the questions: When is $h \in H(X)$ conjugate to $g \in H(X)$ which is a uniform homeomorphism or can be extended to a homeomorphism $\tilde{g}$ on the metric completion of $ X$ Typically for a complete metric space $X$, we prove that $ h \in H(X)$ is conjugate to a uniform homeomorphism if $H$ is uniformly approximated by uniform homeomorphisms. In case $ X = \mathbf{R}$, we obtain a stronger result showing that every homeomorphism on $\mathbf{R}$ is, in fact, conjugate to a smooth Lipschitz homeomorphis. For a noncomplete metric space $ X$, we provide answers to the existence of $\tilde{g}$ under several different settings. Our results are concerned mainly with infinite-dimensional manifolds.
Translation semigroups and their linearizations on spaces of integrable functions
Annette
Grabosch
357-390
Abstract: Of concern is the unbounded operator $D({A_\Phi }) = \{ f \in {W^{1,1}}:f(0) = \Phi (f)\}$ which is considered on the Banach space $E$ of Bochner integrable functions on an interval with values in a Banach space $F$. Under the assumption that $\Phi$ is a Lipschitz continuous operator from $E$ to $F$, it is shown that $ {A_{\Phi}}$ generates a strongly continuous translation semigroup ${({T_\Phi }(t))_{t \geq 0}}$. For linear operators $\Phi$ several properties such as essential-compactness, positivity, and irreducibility of the semigroup $ {({T_\Phi }(t))_{t \geq 0}}$ depending on the operator $\Phi$ are studied. It is shown that if $F$ is a Banach lattice with order continuous norm, then ${({T_\Phi }(t))_{t \geq 0}}$ is the modulus semigroup of $ {({T_\Phi }(t))_{t \geq 0}}$. Finally spectral properties of ${A_\Phi}$ are studied and the spectral bound $s({A_\Phi })$ is determined. This leads to a result on the global asymptotic behavior in the case where $\Phi$ is linear and to a local stability result in the case where $\Phi$ is Fréchet differentiable.
Variations of Hodge structure, Legendre submanifolds, and accessibility
James A.
Carlson;
Domingo
Toledo
391-411
Abstract: Variations of Hodge structure of weight two are integral manifolds for a distribution in the tangent bundle of a period domain. This distribution has dimension ${h^{2,0}}{h^{1,1}}$ and is nonintegrable for ${h^{2,0}} > 1$. In this case it is known that the dimension of an integral manifold does not exceed $ \frac{1} {2}{h^{2,0}}{h^{1,1}}$. Here we give a new proof, based on an analogy between Griffiths' horizontal differential system of algebraic geometry and the contact system of classical mechanics. We show also that any two points in such a domain can be joined by a horizontal curve which is piecewise holomorphic.
Specializations of finitely generated subgroups of abelian varieties
D. W.
Masser
413-424
Abstract: Given a generic Mordell-Weil group over a function field, we can specialize it down to a number field. It has been known for some time that the resulting homomorphism of groups is injective "infinitely often". We prove that this is in fact true "almost always", in a sense that is quantitatively nearly best possible.
Correction to: ``Harmonically immersed surfaces in ${\bf R}\sp n$'' [Trans. Amer. Math. Soc. {\bf 307} (1988), no. 1, 363--372; MR0936822 (89g:53006)]
Gary R.
Jensen;
Marco
Rigoli
425-428
Monotone reducibility over the Cantor space
Randall
Dougherty
433-484
Abstract: Define the partial ordering $\leqslant$ on the Cantor space ${}^\omega 2$ by $x \leqslant y$ iff $\forall n\,x(n) \leqslant y(n)$ (this corresponds to the subset relation on the power set of $ \omega$). A set $A \subseteq {}^\omega 2$ is monotone reducible to a set $B \subseteq {}^\omega 2$ iff there is a monotone (i.e., $x \leqslant y \Rightarrow f(x) \leqslant f(y)$) continuous function $f:{}^\omega 2 \to {}^\omega 2$ such that $ x \in A$ iff $f(x) \in B$. In this paper, we study the relation of monotone reducibility, with emphasis on two topics: (1) the similarities and differences between monotone reducibility on monotone sets (i.e., sets closed upward under $\leqslant$) and Wadge reducibility on arbitrary sets; and (2) the distinction (or lack thereof) between `monotone' and `positive,' where `positive' means roughly `a priori monotone' but is only defined in certain specific cases. (For example, a $\Sigma _2^0$-positive set is a countable union of countable intersections of monotone clopen sets.) Among the main results are the following: Each of the six lowest Wadge degrees contains one or two monotone degrees (of monotone sets), while each of the remaining Wadge degrees contains uncountably many monotone degrees (including uncountable antichains and descending chains); and, although `monotone' and `positive' coincide in a number of cases, there are classes of monotone sets which do not match any notion of `positive.'
Groups acting on affine algebras
Daniel R.
Farkas
485-497
Abstract: General actions of groups on commutative affine domains are studied. We prove a finiteness theorem for orbits of ideals and an ergodic theorem inspired by results from the theories of group algebras and universal enveloping algebras.
Finite basis theorems for relatively congruence-distributive quasivarieties
Don
Pigozzi
499-533
Abstract: $\mathcal{Q}$ is any quasivariety. A congruence relation $\Theta$ on a member $ {\mathbf{A}}$ of $\mathcal{Q}$ is a $ \mathcal{Q}$-congruence if ${\mathbf{A}}/\Theta \in \mathcal{Q}$. The set $ Co{n_\mathcal{Q}}{\mathbf{A}}$ of all $ \mathcal{Q}$-congruences is closed under arbitrary intersection and hence forms a complete lattice $ {\mathbf{Co}}{{\mathbf{n}}_\mathcal{Q}}{\mathbf{A}}$. $ \mathcal{Q}$ is relatively congruence-distributive if $ {\mathbf{Co}}{{\mathbf{n}}_\mathcal{Q}}{\mathbf{A}}$ is distributive for every $ {\mathbf{A}} \in \mathcal{Q}$. Relatively congruence-distributive quasivarieties occur naturally in the theory of abstract data types. $ \mathcal{Q}$ is finitely generated if it is generated by a finite set of finite algebras. The following generalization of Baker's finite basis theorem is proved. Theorem I. Every finitely generated and relatively congruence-distributive quasivariety is finitely based. A subquasivariety $ \mathcal{R}$ of an arbitrary quasivariety $ \mathcal{Q}$ is called a relative subvariety of $ \mathcal{Q}$ if it is of the form $\mathcal{V} \cap \mathcal{Q}$ for some variety $\mathcal{V}$, i.e., a base for $\mathcal{R}$ can be obtained by adjoining only identities to a base for $ \mathcal{Q}$. Theorem II. Every finitely generated relative subvariety of a relatively congruence-distributive quasivariety is finitely based. The quasivariety of generalized equality-test algebras is defined and the structure of its members studied. This gives rise to a finite algebra whose quasi-identities are finitely based while its identities are not. Connections with logic and the algebraic theory of data types are discussed.
The $q$-Selberg polynomials for $n=2$
Kevin W. J.
Kadell
535-553
Abstract: We have conjectured that Selberg's integral has a plethora of extensions involving the Selberg polynomials and proved that these are the Schur functions for $k = 1$. We prove this conjecture for $ n = 2$ and show that the polynomials are, in a formal sense, Jacobi polynomials. We conjecture an orthogonality relation for the Selberg polynomials which combines orthogonality relations for the Schur functions and Jacobi polynomials. We extend a basic Schur function identity. We give a $ q$-analogue of the Selberg polynomials for $n = 2$ using the little $q$-Jacobi polynomials.
Differential delay equations that have periodic solutions of long period
Steven A.
Chapin
555-566
Abstract: If $f:{\mathbf{R}} \to {\mathbf{R}}$ is a continuous odd function satisfying $xf(x) > 0$, $x \ne 0$, and $ f(x) = o({x^{ - 2}})$ as $x \to \infty$, then so-called periodic solutions of long period seem to play a prominent role in the dynamics of $({\ast})$ $xf(x) \geqslant 0$. These solutions have quite different qualitative features than in the odd case.
Factoring operators satisfying $p$-estimates
Stan
Byrd
567-582
Abstract: Necessary and sufficient conditions for a positive operator to factor through a Banach lattice satisfying upper and lower estimates are presented. These conditions are then combined to give a necessary condition for a positive operator to factor through a super-reflexive Banach lattice. An example is given to show that, in spite of the name given by Beauzamy, uniformly convexifying operators need not factor through any uniformly convex lattice
The resolvent parametrix of the general elliptic linear differential operator: a closed form for the intrinsic symbol
S. A.
Fulling;
G.
Kennedy
583-617
Abstract: Nonrecursive, explicit expressions are obtained for the term of arbitrary order in the asymptotic expansion of the intrinsic symbol of a resolvent parametrix of an elliptic linear differential operator, of arbitrary order and algebraic structure, which acts on sections of a vector bundle over a manifold. Results for the conventional symbol are included as a special case.
Cobordism classes of manifolds with category four
Harpreet
Singh
619-628
Abstract: The Lusternik-Schnirelmann category of a manifold $M$ is the smallest integer $ k$ such that $ M$ can be covered by $ k$ open sets each of which is contractible in $M$. The classification up to cobordism of manifolds with category $3$ was completed by the author in 1985. The object of this paper is to attempt a similar classification of manifolds with category $4$.
The classifying topos of a continuous groupoid. I
Ieke
Moerdijk
629-668
Abstract: We investigate some properties of the functor $B$ which associates to any continuous groupoid $ G$ its classifying topos $ BG$ of equivariant $ G$-sheaves. In particular, it will be shown that the category of toposes can be obtained as a localization of a category of continuous groupoids.
Decidability and invariant classes for degree structures
Manuel
Lerman;
Richard A.
Shore
669-692
Abstract: We present a decision procedure for the $ \forall \exists$-theory of $\mathcal{D}$. The decision procedure follows easily from these results. As a corollary to the $\forall \exists $-decision procedure for $\mathcal{D}$, we show that no degree ${\mathbf{a}} > {\mathbf{0}}$ is definable by any $ \exists \forall$-formula of degree theory. As a start on restricting the formulas which could possibly define the various jump classes we classify the generalized jump classes which are invariant for any $\forall$ or $\exists$-formula. The analysis again uses the decision procedure for the $ \forall \exists$-theory of $\mathcal{D}$. A similar analysis is carried out for the high/low hierarchy using the decision procedure for the $ \forall \exists$-theory of $\mathcal{C}$ is $\sigma $-invariant if $ \sigma ({\mathbf{a}})$ holds for every $ {\mathbf{a}}$ in $\mathcal{C}$.)
Markov-Duffin-Schaeffer inequality for polynomials with a circular majorant
Q. I.
Rahman;
G.
Schmeisser
693-702
Abstract: If $p$ is a polynomial of degree at most $ n$ such that $\vert p(x)\vert \leqslant \sqrt {1 - {x^2}}$ for $- 1 \leqslant x \leqslant 1$, then for each $ k$, $\max \vert{p^{(k)}}(x)\vert$ on $[ - 1,\,1]$ is maximized by the polynomial $({x^2} - 1){U_{n - 2}}(x)$ where $ {U_m}$ is the $ m$th Chebyshev polynomial of the second kind. The purpose of this paper is to investigate if it is enough to assume $\vert p(x)\vert \leqslant \sqrt {1 - {x^2}}$ at some appropriately chosen set of $n + 1$ points in $[ - 1,\,1]$. The problem is inspired by a well-known extension of Markov's inequality due to Duffin and Schaeffer.
On inductive limits of certain $C\sp *$-algebras of the form $C(X)\otimes F$
Cornel
Pasnicu
703-714
Abstract: A certain class of $ {\ast}$-homomorphisms $ C(X) \otimes A \to C(Y) \otimes B$, called compatible with a map defined on $ Y$ with values in the set of all closed nonempty subsets of $X$, is studied. A local description of ${\ast}$-homomorphisms $C(X) \otimes A \to C(Y) \otimes B$ is given considering separately the cases $X = {\text{point}}$ and $A = {\mathbf{C}}$; this is done in terms of continuous "quasifields" of $ {C^{\ast}}$-algebras. Conditions under which an inductive limit $\underrightarrow {\lim }(C({X_k}) \otimes {A_k},\,{\Phi _k})$, where each ${\Phi _k}$ is of the above type, is $ {\ast}$-isomorphic with the tensor product of a commutative ${C^{\ast}}$-algebra with an AF algebra are given. For such inductive limits the isomorphism problem is considered.
On James' type spaces
Abderrazzak
Sersouri
715-745
Abstract: We study the spaces $ E$ which are isometric to their biduals $ {E^{{\ast}{\ast}}}$, and satisfy $\dim ({E^{{\ast}{\ast}}}/E) < \infty$. We show that these spaces have several common points with the usual James' space. Our study leads to a kind of classification of these spaces and we show that there are essentially four different basic structures for such spaces in the complex case, and five in the real case.
Ordinal rankings on measures annihilating thin sets
Alexander S.
Kechris;
Russell
Lyons
747-758
Abstract: We assign a countable ordinal number to each probability measure which annihilates all $H$-sets. The descriptive-set theoretic structure of this assignment allows us to show that this class of measures is coanalytic non-Borel. In addition, it allows us to quantify the failure of Rajchman's conjecture. Similar results are obtained for measures annihilating Dirichlet sets.
Electrophoretic traveling waves
P. C.
Fife;
O. A.
Palusinski;
Y.
Su
759-780
Abstract: An existence-uniqueness-approximability theory is given for a prototypical mathematical model for the separation of ions in solution by an imposed electric field. The separation is accomplished during the formation of a traveling wave, and the mathematical problem consists in finding a traveling wave solution of a set of diffusion-advection equations coupled to a Poisson equation. A basic small parameter $ \varepsilon$ appears in an apparently singular manner, in that when $\varepsilon = 0$ (which amounts to assuming the solution is everywhere electrically neutral), the last (Poisson) equation loses its derivative, and becomes an algebraic relation among the concentrations. Since this relation does not involve the function whose derivative is lost, the type of "singular" perturbation represented here is nonstandard. Nevertheless, the traveling wave solution depends in a regular manner on $\varepsilon$, even at $\varepsilon = 0$; and one of the principal aims of the paper is to show this regular dependence.
The continuation theory for Morse decompositions and connection matrices
Robert D.
Franzosa
781-803
Abstract: The continuation theory for ($<$-ordered) Morse decompositions and the indices defined on them--the homology index braid and the connection matrices--is established. The equivalence between $<$-ordered Morse decompositions and $ <$-consistent attractor filtrations is displayed. The spaces of ($ <$-ordered) Morse decompositions for a product parametrization of a local flow are introduced, and the local continuation of ($ <$-ordered) Morse decompositions is obtained via the above-described equivalence and the local continuation of attractors. The homology index braid and the connection matrices of an admissible ordering of a Morse decomposition are shown to be invariant on path components of the corresponding space of $<$-ordered Morse decompositions. This invariance is used to prove that the collection of connection matrices of a Morse decomposition is upper semicontinuous over the space of Morse decompositions (and over the parameter space) under local continuation.
Generalizations of Cauchy's summation theorem for Schur functions
G. E.
Andrews;
I. P.
Goulden;
D. M.
Jackson
805-820
Abstract: Cauchy's summation theorem for Schur functions is generalized, and a number of related results are given. The result is applied to a combinatorial problem involving products of pairs of permuations, by appeal to properties of the group algebra of the symmetric group.
Multilinear convolutions defined by measures on spheres
Daniel M.
Oberlin
821-835
Abstract: Let $\sigma$ be Lebesgue measure on ${\Sigma _{n - 1}}$ and write $\sigma = ({\sigma _1}, \ldots ,{\sigma _n})$ for an element of ${\Sigma _{n - 1}}$. For functions ${f_1}, \ldots ,{f_n}$ on $ {\mathbf{R}}$, define $\displaystyle T({f_1}, \ldots ,{f_n})(x) = \int_{{\Sigma _{n - 1}}} {{f_1}(x - {\sigma _1}) \cdots {f_n}(x - {\sigma _n})\,d\sigma ,\qquad x \in {\mathbf{R}}.}$ This paper partially answers the question: for which values of $p$ and $q$ is there an inequality $\displaystyle \vert\vert T({f_1}, \ldots ,{f_n})\vert{\vert _q} \leqslant C\vert\vert{f_1}\vert{\vert _p} \cdots \vert\vert{f_n}\vert{\vert _p}?$
Approximation properties for orderings on $*$-fields
Thomas C.
Craven
837-850
Abstract: The goal of this paper is to extend the main theorems on approximation properties of the topological space of orderings from formally real fields to skew fields with an involution $ ^{\ast}$. To accomplish this, the concept of $^{\ast}$-semiordering is developed and new theorems are obtained for lifting $^{\ast}$orderings from the residue class field of a real valuation.
Spherical functions on Cartan motion groups
Cary
Rader
1-45
Abstract: This paper gives a reasonably complete treatment of harmonic analysis on Cartan motion groups. Included is an explicit parameterization of irreducible spherical functions of general $K$-type, and of the nonunitary dual (and its topology). Also included is the explicit Plancherel measure, the Paley Wiener theorem, and an asymptotic expansion of general matrix entries. (These are generalized Bessel functions.) However the main result is Theorem 19, a technical result which measures the size of the centralizer of $K$ in the universal enveloping algebra of the corresponding reductive group.
Holomorphic maps from ${\bf C}\sp n$ to ${\bf C}\sp n$
Jean-Pierre
Rosay;
Walter
Rudin
47-86
Abstract: We study holomorphic mappings from $ {{\mathbf{C}}^n}$ to ${{\mathbf{C}}^n}$, and especially their action on countable sets. Several classes of countable sets are considered. Some new examples of Fatou-Bieberbach maps are given, and a nondegenerate map is constructed so that the volume of the image of ${{\mathbf{C}}^n}$ is finite. An Appendix is devoted to the question of linearization of contractions.
The existence of least area surfaces in $3$-manifolds
Joel
Hass;
Peter
Scott
87-114
Abstract: This paper presents a new and unified approach to the existence theorems for least area surfaces in $3$-manifolds.
Decay rates of Fourier transforms of curves
B. P.
Marshall
115-126
Abstract: Let $d\mu$ be a smooth measure on a nondegenerate curve in $ {{\mathbf{R}}^n}$. This paper examines the decay rate of spherical averages of its Fourier transform $ \widehat{d\mu }$. Thus estimates of the following form are considered: $\displaystyle {\left( {\int_{{\sum _r}} {\vert\widehat{d\mu }(\xi {\vert^p}d\xi } } \right)^{1/p}} \leqslant C{r^{ - \sigma }}\vert\vert f\vert\vert$ where ${\sum _r} = \{ \xi \in {{\mathbf{R}}^n}:\vert\xi \vert = r\}$.
Iterating the basic construction
Mihai
Pimsner;
Sorin
Popa
127-133
Abstract: Let $N \subset M$ be a pair of type II$_{1}$ factors with finite Jones' index and $N \subset M \subset {M_1} \subset {M_2} \subset \cdots \subset {M_n} \subset \cdots \subset {M_{2n + 1}}$ be the associated tower of type II$_{1}$ factors obtained by iterating Jones' basic construction. We give an explicit formula of a projection in ${M_{2n + 1}}$ which implements the conditional expectation of ${M_n}$ onto $N$, thus showing that $ {M_{2n + 1}}$ comes naturally from the basic construction associated to the pair $N \subset {M_n}$. From this we deduce several properties of the relative commutant
The cohomology of presheaves of algebras. I. Presheaves over a partially ordered set
Murray
Gerstenhaber;
Samuel D.
Schack
135-165
Abstract: To each presheaf (over a poset) of associative algebras $\mathbb{A}$ we associate an algebra $\mathbb{A}!$. We define a full exact embedding of the category of (presheaf) $ \mathbb{A}$-bimodules in that of $ \mathbb{A}!$-bimodules. We show that this embedding preserves neither enough (relative) injectives nor enough (relative) projectives, but nonetheless preserves (relative) Yoneda cohomology. The cohomology isomorphism links the deformations of manifolds, algebraic presheaves, and algebras. It also implies that the cohomology of any triangulable space is isomorphic to the Hochschild cohomology of an associative algebra. (The latter isomorphism preserves all known cohomology operations.) We conclude the paper by exhibiting for each associative algebra and triangulable space a "product" which is again an associative algebra.
Stable extensions of homeomorphisms on the pseudo-arc
Judy
Kennedy
167-178
Abstract: We prove the following: Theorem. If $P'$ is a proper subcontinuum of the pseudoarc $P,\,h'$ is a homeomorphism from $ P'$ onto itself, and $\Theta$ is an open set in $P$ that contains $P'$, then there is a homeomorphism $ h$ from $ P$ onto itself such that $h\vert P' = h'$ and $h(x) = x$ for $x \notin \Theta$.
A construction of pseudo-Anosov homeomorphisms
Robert C.
Penner
179-197
Abstract: We describe a generalization of Thurston's original construction of pseudo-Anosov maps on a surface $F$ of negative Euler characteristic. In fact, we construct whole semigroups of pseudo-Anosov maps by taking appropriate compositions of Dehn twists along certain families of curves; our arguments furthermore apply to give examples of pseudo-Anosov maps on nonorientable surfaces. For each self-map $f:F \to F$ arising from our recipe, we construct an invariant "bigon track" (a slight generalization of train track) whose incidence matrix is Perron-Frobenius. Standard arguments produce a projective measured foliation invariant by $f$. To finally prove that $f$ is pseudo-Anosov, we directly produce a transverse invariant projective measured foliation using tangential measures on bigon tracks. As a consequence of our argument, we derive a simple criterion for a surface automorphism to be pseudo-Anosov.
Stable maps into free $G$-spaces
J. P. C.
Greenlees
199-215
Abstract: In this paper we introduce a systematic method for calculating the group of stable equivariant maps $ {[X,\,Y]^G}$ [3, 18] into a $G$-free space or spectrum $Y$. In fact the method applies without restriction on $X$ whenever $G$ is a $p$-group and $Y$ is $p$-complete and satisfies standard finiteness assumptions. The method is an Adams spectral sequence based on a new equivariant cohomology theory ${c^{\ast}}(X)$ which we introduce in $\S1$. This spectral sequence is quite calculable and provides a natural generalisation of the classical Adams spectral sequence based on ordinary $ \bmod p$ cohomology. It also geometrically realises certain inverse limits of nonequivariant Adams spectral sequences which have been useful in the study of the Segal conjecture [19, 5, 21, 9].
Moduli spaces of Riemann surfaces of genus two with level structures. I
Ronnie
Lee;
Steven H.
Weintraub
217-237
Abstract: The cohomology of modular varieties defined by congruence subgroups of $ {\operatorname{Sp} _4}({\mathbf{Z}})$ whose levels lie between $2$ and $4$ is studied. Using a counting argument and the techniques of zeta functions, the authors completely determine the cohomology of a particular variety of this type.
On the Wiener criterion and quasilinear obstacle problems
Juha
Heinonen;
Tero
Kilpeläinen
239-255
Abstract: We study the Wiener criterion and variational inequalities with irregular obstacles for quasilinear elliptic operators $ A$, $ A(x,\,\nabla u) \cdot \nabla u \approx \vert\nabla u{\vert^p}$, in ${{\mathbf{R}}^n}$. Local solutions are continuous at Wiener points of the obstacle function; if $ p > n - 1$, the converse is also shown to be true. If $p > n - 1$, then a characterization of the thinness of a set at a point is given in terms of $A$-superharmonic functions.
An Erd\H os-Wintner theorem for differences of additive functions
Adolf
Hildebrand
257-276
Abstract: An Erdös-Wintner type criterion is given for the convergence of the distributions ${D_x}(z) = {[x]^{ - 1}}\char93 \{ 1 \leqslant n \leqslant x:\,f(n + 1) - f(n) \leqslant z\}$, where $f$ is a real-valued additive function. A corollary of this result is that an additive function $ f$, for which $f(n + 1) - f(n)$ tends to zero on a set of density one, must be of the form $f = \lambda \log$ for some constant $\lambda$. This had been conjectured by Erdős.
Decompositions of continua over the hyperbolic plane
James T.
Rogers
277-291
Abstract: The following theorem is proved. Theorem. Let $X$ be a homogeneous continuum such that ${H^1}(X) \ne 0$. If $\mathcal{G}$ is the collection of maximal terminal proper subcontinua of $X$, then (1) The collection $\mathcal{G}$ is a monotone, continuous, terminal decomposition of $X$, (2) The nondegenerate elements of $\mathcal{G}$ are mutually homeomorphic, indecomposable, cell-like, terminal, homogeneous continua of the same dimension as $X$, (3) The quotient space is a homogeneous continuum, and (4) The quotient space does not contain any proper, nondegenerate, terminal subcontinuum. This theorem is related to the Jones' Aposyndetic Decomposition Theorem. The proof involves the hyperbolic plane and a subset of the circle at $\infty$, called the set of ends of a component of the universal cover of $X$.
Borel orderings
Leo
Harrington;
David
Marker;
Saharon
Shelah
293-302
Abstract: We show that any Borel linear order can be embedded in an order preserving way into $ {2^\alpha }$ for some countable ordinal $\alpha$ and that any thin Borel partial order can be written as a union of countably many Borel chains.
On a class of functionals invariant under a ${\bf Z}\sp n$ action
Paul H.
Rabinowitz
303-311
Abstract: Consider a system of ordinary differential equations of the form $ ({\ast})$ $\displaystyle \ddot q + {V_q}(t,\,q) = f(t)$ where $f$ and $V$ are periodic in $t$, $V$ is periodic in the components of $q = ({q_1}, \ldots ,{q_n})$, and the mean value of $f$ vanishes. By showing that a corresponding functional is invariant under a natural ${{\mathbf{Z}}^n}$ action, a simple variational argument yields at least $n + 1$ distinct periodic solutions of (*). More general versions of (*) are also treated as is a class of Neumann problems for semilinear elliptic partial differential equations.
Amalgamation for inverse and generalized inverse semigroups
T. E.
Hall
313-323
Abstract: For any amalgam $(S,\,T;\,U)$ of inverse semigroups, it is shown that the natural partial order on $S{{\ast}_U}T$, the (inverse semigroup) free product of $S$ and $T$ amalgamating $U$, has a simple form on $S \cup T$. In particular, it follows that the semilattice of $ S{{\ast}_U}T$ is a bundled semilattice of the corresponding semilattice amalgam $ (E(S),\,E(T);\,E(U))$; taken jointly with a result of Teruo Imaoka, this gives that the class of generalized inverse semigroups has the strong amalgamation property. Preserving finiteness is also considered.
Homology of smooth splines: generic triangulations and a conjecture of Strang
Louis J.
Billera
325-340
Abstract: For $\Delta$ a triangulated $ d$-dimensional region in $ {{\mathbf{R}}^d}$, let $S_m^r(\Delta )$ denote the vector space of all $ {C^r}$ functions $ F$ on $\Delta$ that, restricted to any simplex in $\Delta$, are given by polynomials of degree at most $m$. We consider the problem of computing the dimension of such spaces. We develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds $\Delta$ in the plane, getting lower bounds on the dimension of $ S{}_m^r(\Delta )$ for all $ r$. For $r = 1$, we prove a conjecture of Strang concerning the generic dimension of the space of $ {C^1}$ splines over a triangulated manifold in $ {{\mathbf{R}}^2}$. Finally, we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region.
Normal structure in dual Banach spaces associated with a locally compact group
Anthony To Ming
Lau;
Peter F.
Mah
341-353
Abstract: In this paper we investigated when the dual of a certain function space defined on a locally compact group has certain geometric properties. More particularly, we asked when weak$^{*}$ compact convex subsets in these spaces have normal structure, and when the norm of these spaces satisfies one of several types of Kadec-Klee property. As samples of the results we have obtained, we have proved, among other things, the following two results: (1) The measure algebra of a locally compact group has weak$^{*}$-normal structure iff it has property SUKK$ ^{*}$ iff it has property SKK$^{*}$ iff the group is discrete; (2) Among amenable locally compact groups, the Fourier-Stieltjes algebra has property SUKK$^{*}$ iff it has property SKK$^{*}$ iff the group is compact. Consequently the Fourier-Stieltjes algebra has weak$ ^{*}$-normal structure when the group is compact.
Extremal analytic discs with prescribed boundary data
Chin-Huei
Chang;
M. C.
Hu;
Hsuan-Pei
Lee
355-369
Abstract: This paper concerns the existence and uniqueness of extremal analytic discs with prescribed boundary data in a bounded strictly linearly convex domain $D$ in $ {{\mathbf{C}}^n}$. We prove that for any two distinct points $p$, $q$ in $\partial D$ (respectively, $p \in \partial D$ and a vector $v$ such that $\sqrt { - 1} v \in {T_p}(\partial D)$ and $\langle v,\,\overline \nu (p)\rangle = \sum\nolimits_1^n {{v_j}{{\overline \nu }_j}(p) > 0}$ where $ \nu (p)$ is the outward normal to $\partial D$ at $p$) there exists an extremal analytic disc $ f$ passing through $ p$, $q$ if $\partial D \in {C^k}$, $k \geqslant 3$ (respectively, $ f(1) = p$, $ \partial D \in {C^k}$, $k \geqslant 14$). Consequently, we can foliate $\overline D$ with these extremal analytic discs.
Banach spaces with separable duals
M.
Zippin
371-379
Abstract: It is proved that every Banach space with a separable dual embeds into a space with a shrinking basis. It follows that every separable reflexive space can be embedded in a reflexive space with a basis.
Finite covers of $3$-manifolds containing essential tori
John
Luecke
381-391
Abstract: It is shown in this paper that if a Haken $3$-manifold contains an incompressible torus that is not boundary-parallel then either it has a finite cover that is a torus-bundle over the circle or it has finite covers with arbitrarily large first Betti number.
Nonconvex variational problems with general singular perturbations
Nicholas C.
Owen
393-404
Abstract: We study the effect of a general singular perturbation on a nonconvex variational problem with infinitely many solutions. Using a scaling argument and the theory of $\Gamma $-convergence of nonlinear functionals, we show that if the solutions of the perturbed problem converge in ${L^1}$ as the perturbation parameter goes to zero, then the limit function satisfies a classical minimal surface problem.
A characterization of the weakly continuous polynomials in the method of compensated compactness
Robert C.
Rogers;
Blake
Temple
405-417
Abstract: We present a sufficient condition for weak continuity in the method of compensated compactness. The condition links weak continuity to the structure of the wave cone and the characteristic set for polynomials of degree greater than two. The condition applies to all the classical examples of weakly continuous functions and generalizes the Quadratic Theorem and the Wedge Product Theorem. In fact, the condition reduces to the Legendre-Hadamard Necessary Condition when the polynomial is quadratic, and also whenever a certain orthogonality condition is satisfied. The condition is derived by isolating conditions under which the quadratic theorem can be iterated.
Bifurcation phenomena associated to the $p$-Laplace operator
Mohammed
Guedda;
Laurent
Véron
419-431
Abstract: We determine the structure of the set of the solutions $u$ of $- {(\vert{u_x}{\vert^{p - 2}}{u_x})_x} + f(u) = \lambda \vert u{\vert^{p - 2}}u$ on $(0,\,1)$ such that $ u(0) = u(1) = 0$, where $p > 1$ and $\lambda \in {\mathbf{R}}$. We prove that the solutions with $k$ zeros are unique when $1 < p \leqslant 2$ but may not be so when $ p > 2$.
Ideals associated to deformations of singular plane curves
Steven
Diaz;
Joe
Harris
433-468
Abstract: We consider in this paper the geometry of certain loci in deformation spaces of plane curve singularities. These loci are the equisingular locus $ES$ which parametrizes equisingular or topologically trivial deformations, the equigeneric locus $ EG$ which parametrizes deformations of constant geometric genus, and the equiclassical locus $EC$ which parametrizes deformations of constant geometric genus and class. (The class of a reduced plane curve is the degree of its dual.) It was previously known that the tangent space to $ES$ corresponds to an ideal called the equisingular ideal and that the support of the tangent cone to $EG$ corresponds to the conductor ideal. We show that the support of the tangent cone to $EC$ corresponds to an ideal which we call the equiclassical ideal. By studying these ideals we are able to obtain information about the geometry and dimensions of $ES$, $EC$, and $EG$. This allows us to prove some theorems about the dimensions of families of plane curves with certain specified singularities.
Definable sets in ordered structures. III
Anand
Pillay;
Charles
Steinhorn
469-476
Abstract: We show that any $ o$-minimal structure has a strongly $o$-minimal theory.
Positive quadratic differential forms and foliations with singularities on surfaces
Víctor
Guíñez
477-502
Abstract: To every positive $ {C^r}$-quadratic differential form defined on an oriented two manifold is associated a pair of transversal one-dimensional $ {C^r}$-foliations with common singularities. An open set of positive $ {C^r}$-quadratic differential forms with structural stable associated foliations is characterized and it is proved that this set is dense in the space of positive $ {C^\infty }$-quadratic differential forms with ${C^2}$-topology. Also a realization theorem is established.
$v\sb 1$-periodic ${\rm Ext}$ over the Steenrod algebra
Donald M.
Davis;
Mark
Mahowald
503-516
Abstract: For a large family of modules $M$ over the $\bmod 2$ Steenrod algebra $A$, $\operatorname{Ext} _A^{s,t}(M,\,{{\mathbf{Z}}_2})$ is periodic for $t < 4s$ with respect to operators $v_1^{2n}$ of period $({2^n},\,3 \cdot {2^n})$ for varying $n$. $v_1^{ - 1}\operatorname{Ext} _A^{s,t}(M,\,{{\mathbf{Z}}_2})$ can be defined by extending this periodic behavior outside this range. We calculate this completely when $M = {H^{\ast}}(Y)$, where $Y$ is the suspension spectrum of ${\mathbf{R}}{P^2} \wedge {\mathbf{C}}{P^2}$.
Mixed norm estimates for certain means
Lennart
Börjeson
517-541
Abstract: We obtain estimates of the mean $\displaystyle F_x^\gamma (t) = {C_\gamma }\int_{\vert y\vert < 1} {{{(1 - \vert y{\vert^2})}^\gamma }f(x - ty)\,dy}$ in mixed Lebesgue and Sobolev spaces. They generalize earlier estimates of the spherical mean $F_x^{ - 1}(t) = C\;\int_{{S^{n - 1}}} {f(x - ty)\,dS(y)}$ and of solutions of the wave equation ${\Delta _x}u = {\partial ^2}u/\partial {t^2}$.
The minimal model of the complement of an arrangement of hyperplanes
Michael
Falk
543-556
Abstract: In this paper the methods of rational homotopy theory are applied to a family of examples from singularity theory. Let ${\mathbf{A}}$ be a finite collection of hyperplanes in $ {{\mathbf{C}}^l}$, and let $M = {{\mathbf{C}}^l} - \bigcup\nolimits_{H \in {\mathbf{A}}} H$. We say ${\mathbf{A}}$ is a rational $K(\pi ,\,1)$ arrangement if the rational completion of $M$ is aspherical. For these arrangements an identity (the LCS formula) is established relating the lower central series of $ {\pi _1}(M)$ to the cohomology of $M$. This identity was established by group-theoretic means for the class of fiber-type arrangements in previous work. We reproduce this result by showing that the class of rational $K(\pi ,\,1)$ arrangements contains all fiber-type arrangements. This class includes the reflection arrangements of types ${A_l}$ and ${B_l}$. There is much interest in arrangements for which $M$ is a $K(\pi ,\,1)$ space. The methods developed here do not apply directly because $M$ is rarely a nilpotent space. We give examples of $K(\pi ,\,1)$ arrangements which are not rational $K(\pi ,\,1)$ for which the LCS formula fails, and $K(\pi ,\,1)$ arrangements which are not rational $K(\pi ,\,1)$ where the LCS formula holds. It remains an open question whether rational $K(\pi ,\,1)$ arrangements are necessarily $K(\pi ,\,1)$.
Bounds on the $L\sp 2$ spectrum for Markov chains and Markov processes: a generalization of Cheeger's inequality
Gregory F.
Lawler;
Alan D.
Sokal
557-580
Abstract: We prove a general version of Cheeger's inequality for discrete-time Markov chains and continuous-time Markovian jump processes, both reversible and nonreversible, with general state space. We also prove a version of Cheeger's inequality for Markov chains and processes with killing. As an application, we prove $ {L^2}$ exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs.
Trace identities and $\bf {Z}/2\bf {Z}$-graded invariants
Allan
Berele
581-589
Abstract: We prove Razmyslov's theorem on trace identities for ${M_{k,\,l}}$ using the invariant theory of $ \operatorname{pl} (k,\,l)$.
On the nonlinear eigenvalue problem $\Delta u+\lambda e\sp u=0$
Takashi
Suzuki;
Ken’ichi
Nagasaki
591-608
Abstract: The structure of the set $ \mathcal{C}$ of solutions of the nonlinear eigenvalue problem $\Delta u + \lambda {e^u} = 0$ under Dirichlet condition in a simply connected bounded domain $ \Omega$ is studied. Through the idea of parametrizing the solutions $(u,\,\lambda )$ in terms of $s = \lambda \,\int_\Omega {{e^u}\,dx}$, some profile of $\mathcal{C}$ is illustrated when $ \Omega$ is star-shaped. Finally, the connectivity of the branch of Weston-Moseley's large solutions to that of minimal ones is discussed.
Locally bounded sets of holomorphic mappings
José
Bonet;
Pablo
Galindo;
Domingo
García;
Manuel
Maestre
609-620
Abstract: Several results and examples about locally bounded sets of holomorphic mappings defined on certain classes of locally convex spaces (Baire spaces, $(DF)$-spaces, $C(X)$-spaces) are presented. Their relation with the classification of locally convex spaces according to holomorphic analogues of barrelled and bornological properties of the linear theory is considered.
Stochastic perturbations to conservative dynamical systems on the plane. I. Convergence of invariant distributions
G.
Wolansky
621-639
Abstract: We consider a nonlinear system on the plane, given by an oscillator with homoclinic orbits. The above system is subjected to a perturbation, composed of a deterministic part and a random (white noise) part. Assuming the existence of a finite, invariant measure to the perturbed system, we deal with the convergence of the measures to a limit measure, as the perturbation parameter tends to zero. The limit measure is constructed in terms of the action function of the unperturbed oscillator, and the strong local ${L_2}$ convergence of the associated densities is proved.
Stochastic perturbations to conservative dynamical systems on the plane. II. Recurrency conditions
G.
Wolansky
641-657
Abstract: We consider a conservative system on the plane, subjected to a perturbation. The above perturbation is composed of a deterministic part and a random (white noise) part. We discuss the conditions under which there exists a unique, finite invariant measure to the perturbed system, and the weak compactness of the above measures for small enough perturbation's parameter.
A canonical subspace of $H\sp *(B{\rm O})$ and its application to bordism
Errol
Pomerance
659-670
Abstract: A particularly nice canonical subspace of $ {H^{\ast}}(BO)$ is defined. The bordism class of a map $f:X \to Y$, where $X$ and $Y$ are compact, closed manifolds, can be determined by the characteristic numbers corresponding to elements of this subspace, and these numbers can be easily calculated. As an application, we study the "fixed-point manifold" of a parameter family of self-maps $F:M \times X \to X$, thus refining to bordism the usual homological analysis of the diagonal which is the basis of the standard Lefschetz fixed point theorem.
A Stone-type representation theorem for algebras of relations of higher rank
H.
Andréka;
R. J.
Thompson
671-682
Abstract: The Stone representation theorem for Boolean algebras gives us a finite set of equations axiomatizing the class of Boolean set algebras. Boolean set algebras can be considered to be algebras of unary relations. As a contrast here we investigate algebras of $n$-ary relations (originating with Tarski). The new algebras have more operations since there are more natural set theoretic operations on $n$-ary relations than on unary ones. E.g. the identity relation appears as a new constant. The Resek-Thompson theorem we prove here gives a finite set of equations axiomatizing the class of algebras of $ n$-ary relations (for every ordinal $n$).
Translates of exponential box splines and their related spaces
Asher
Ben-Artzi;
Amos
Ron
683-710
Abstract: Exponential box splines ($EB$-splines) are multivariate compactly supported functions on a regular mesh which are piecewise in a space $ \mathcal{H}$ spanned by exponential polynomials. This space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients. The main part of this paper is devoted to algebraic analysis of the space $ {\mathbf{H}}$ of all entire functions spanned by the integer translates of an $ EB$-spline. This investigation relies on a detailed description of $\mathcal{H}$ and its discrete analog $\mathcal{S}$. The approach taken here is based on the observation that the structure of $\mathcal{H}$ is relatively simple when $\mathcal{H}$ is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process. Also, we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of Fourier analysis. Thus, while generalizing the known theory of polynomial box splines, the results here offer a simpler approach and a new insight towards this important special case. We also identify and study in detail several types of singularities which occur only for complex $EB$-splines. The first is when the Fourier transform of the $EB$-spline vanishes at some critical points, the second is when $ \mathcal{H}$ cannot be embedded in $ \mathcal{S}$ and the third is when $ {\mathbf{H}}$ is a proper subspace of $ \mathcal{H}$. We show, among others, that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh.
Algebras on the disk and doubly commuting multiplication operators
Sheldon
Axler;
Pamela
Gorkin
711-723
Abstract: We prove that a bounded analytic function $f$ on the unit disk is in the little Bloch space if and only if the uniformly closed algebra on the disk generated by $ {H^\infty }$ and $\overline f$ does not contain the complex conjugate of any interpolating Blaschke product. A version of this result is then used to prove that if $ f$ and $g$ are bounded analytic functions on the unit disk such that the commutator $ {T_f}T_g^{\ast} - T_g^{\ast}{T_f}$ (here ${T_f}$ denotes the operator of multiplication by $f$ on the Bergman space of the disk) is compact, then $(1 - \vert z{\vert^2})\min \{ \vert f' (z)\vert,\;\vert g' (z)\vert\} \to 0$ as $\vert z\vert \uparrow 1$.
Field theories in the modern calculus of variations
Andrzej
Nowakowski
725-752
Abstract: Two methods of construction of fields of extremals ("geodesic coverings") in the generalized problem of Bolza are given and, as a consequence, sufficient conditions for optimality in a form similar to Weierstrass' are formulated. The first field theory is an extension of Young's field theory-- "concourse of flights" for our problem; the other describes a nonclassical treatment of field theory which allows one to reject the "self-multiplier restriction".
Volumes of small balls on open manifolds: lower bounds and examples
Christopher B.
Croke;
Hermann
Karcher
753-762
Abstract: Question: "Under what curvature assumptions on a complete open manifold is the volume of balls of a fixed radius bounded below independent of the center point?" Two theorems establish such assumptions and two examples sharply limit their weakening. In particular we give an example of a metric on $ {{\mathbf{R}}^4}$ (extending to higher dimensions) of positive Ricci curvature, whose sectional curvatures decay to 0, and such that the volume of balls goes uniformly to 0 as the center goes to infinity.
Regularity of weak solutions of parabolic variational inequalities
William P.
Ziemer
763-786
Abstract: In this paper, parabolic operators of the form $\displaystyle {u_t} - \operatorname{div} A(x,\,t,\,u,\,Du) - B(x,\,t,\,u,\,Du)$ are considered where $A$ and $B$ are Borel measurable and subject to linear growth conditions. Let $\psi :\,\Omega \to {R^1}$ be a Borel function bounded above (an obstacle) where $\Omega \subset {R^{n + 1}}$. Let $u \in {W^{1,2}}(\Omega )$ be a weak solution of the variational inequality in the following sense: assume that $u \geqslant \psi $ q.e. and $\displaystyle \int_\Omega {{u_t}\varphi + A \cdot D\varphi - B\varphi \geqslant 0}$ whenever $\varphi \in W_0^{1,2}(\Omega )$ and $\varphi \geqslant u - \psi$ q.e. Here q.e. means everywhere except for a set of classical parabolic capacity. It is shown that $u$ is continuous even though the obstacle may be discontinuous. A mild condition on $\psi$ which can be expressed in terms of the fine topology is sufficient to ensure the continuity of $ u$. A modulus of continuity is obtained for $u$ in terms of the data given for $\psi$.
The box product of countably many copies of the rationals is consistently paracompact
L. Brian
Lawrence
787-796
Abstract: By proving the theorem stated in the title, we show that local compactness in the factor spaces is not necessary for paracompactness in the box product.
Fast algorithms for multiple evaluations of the Riemann zeta function
A. M.
Odlyzko;
A.
Schönhage
797-809
Abstract: The best previously known algorithm for evaluating the Riemann zeta function, $ \zeta (\sigma + it)$, with $ \sigma$ bounded and $ t$ large to moderate accuracy (within $ \pm {t^{ - c}}$ for some $ c > 0$, say) was based on the Riemann-Siegel formula and required on the order of ${t^{1/2}}$ operations for each value that was computed. New algorithms are presented in this paper which enable one to compute any single value of $\zeta (\sigma + it)$ with $\sigma$ fixed and $ T \leqslant t \leqslant T + {T^{1/2}}$ to within $ \pm {t^{ - c}}$ in $O({t^\varepsilon })$ operations on numbers of $ O(\log t)$ bits for any $\varepsilon > 0$, for example, provided a precomputation involving $O({T^{1/2 + \varepsilon }})$ operations and $O({T^{1/2 + \varepsilon }})$ bits of storage is carried out beforehand. These algorithms lead to methods for numerically verifying the Riemann hypothesis for the first $n$ zeros in what is expected to be $O({n^{1 + \varepsilon }})$ operations (as opposed to about ${n^{3/2}}$ operations for the previous method), as well as improved algorithms for the computation of various arithmetic functions, such as $\pi (x)$. The new zeta function algorithms use the fast Fourier transform and a new method for the evaluation of certain rational functions. They can also be applied to the evaluation of $L$-functions, Epstein zeta functions, and other Dirichlet series.
Hausdorff dimension in graph directed constructions
R. Daniel
Mauldin;
S. C.
Williams
811-829
Abstract: We introduce the notion of geometric constructions in ${{\mathbf{R}}^m}$ governed by a directed graph $ G$ and by similarity ratios which are labelled with the edges of this graph. For each such construction, we calculate a number $ \alpha$ which is the Hausdorff dimension of the object constructed from a realization of the construction. The measure of the object with respect to ${\mathcal{H}^\alpha }$ is always positive and $ \sigma$-finite. Whether the $ {\mathcal{H}^\alpha }$-measure of the object is finite depends on the order structure of the strongly connected components of $G$. Some applications are given.
On classical Clifford theory
Morton E.
Harris
831-842
Abstract: Let $k$ be a field, let $N$ be a normal subgroup of a finite group $ H$ and let $M$ be a completely reducible $ k[N]$-module. We give sufficient conditions for a finite dimensional (finite) group crossed product $k$-algebra to be a Frobenius or symmetric $ k$-algebra. These results imply that $ k[H]/(J(k[N])k[H])$ and the endomorphism $k$-algebra, ${\operatorname{End} _{k[H]}}({M^H})$, of the induced module ${M^H}$ are symmetric $k$-algebras. We also completely describe the $ k[H]$-indecomposable decomposition of ${M^H}$. It follows that the head and socle of an indecomposable component of ${M^H}$ are irreducible isomorphic $ k[H]$-modules.
Seifert matrices and $6$-knots
J. A.
Hillman;
C.
Kearton
843-855
Abstract: A new classification of simple $ {\mathbf{Z}}$-torsion-free $ 2q$-knots, $q \geqslant 3$, is given in terms of Seifert matrices modulo an equivalence relation. As a result the classification of such $2q$-knots, $q \geqslant 4$, in terms of $ F$-forms is extended to the case $q = 3$.
Correction to: ``Differential identities in prime rings with involution'' [Trans. Amer. Math. Soc. {\bf 291} (1985), no. 2, 765--787; MR0800262 (87f:16013)]
Charles
Lanski
857-859
Geometry of the Severi variety
Steven
Diaz;
Joe
Harris
1-34
Abstract: This paper is concerned with the geometry of the Severi variety $ W$ parametrizing plane curves of given degree and genus, and specifically with the relations among various divisor classes on $ W$. Two types of divisor classes on $W$ are described: those that come from the intrinsic geometry of the curves parametrized, and those characterized by extrinsic properties such as the presence of cusps, tacnodes, hyperflexes, etc. The goal of the paper is to express the classes of the extrinsically defined divisors in terms of the intrinsic ones; this, along with other calculations such as the determination of the canonical class of $W$, is carried out by using various enumerative techniques. One corollary is that the variety of nodal curves of given degree and genus in the plane is affine.
Generating combinatorial complexes of polyhedral type
Egon
Schulte
35-50
Abstract: The paper describes a method for generating combinatorial complexes of polyhedral type. Building blocks ${\mathbf{B}}$ are implanted into the maximal simplices of a simplicial complex ${\mathbf{C}}$, on which a group operates as a combinatorial reflection group. Of particular interest is the case where $ {\mathbf{B}}$ is a polyhedral block and $ {\mathbf{C}}$ the barycentric subdivision of a regular incidence-polytope ${\mathbf{K}}$ together with the action of the automorphism group of $ {\mathbf{K}}$.
A random graph with a subcritical number of edges
B.
Pittel
51-75
Abstract: A random graph ${G_n}(\operatorname{prob} (\operatorname{edge} ) = p)\;(p = c/n,\,0 < c < 1)$ on $n$ labelled vertices is studied. There are obtained limiting distributions of the following characteristics: the lengths of the longest cycle and the longest path, the total size of unicyclic components, the number of cyclic vertices, the number of distinct component sizes, and the middle terms of the component-size order sequence. For instance, it is proved that, with probability approaching ${(1 - c)^{1/2}}\exp (\sum\nolimits_{j = 1}^l {{c^j}/2j)}$ as $n \to \infty$, the random graph does not have a cycle of length $> l$. Another result is that, with probability approaching $1$, the size of the $\nu$th largest component either equals an integer closest to $a\;\log (bn/\nu \,{\log ^{5/2}}n)$, $a = a(c)$, $b = b(c)$, or is one less than this integer, provided that $\nu \to \infty $ and $\nu = o(n/{\log ^{5/2}}n)$.
Odd primary periodic phenomena in the classical Adams spectral sequence
Paul
Shick
77-86
Abstract: We study certain periodic phenomena in the cohomology of the $ \bmod \;p$ Steenrod algebra which are related to the polynomial generators $ {v_n} \in {\pi _{\ast}}BP$. A chromatic resolution of the ${E_2}$ term of the classical Adams spectral sequence is constructed.
Carleson measures and multipliers of Dirichlet-type spaces
Ron
Kerman;
Eric
Sawyer
87-98
Abstract: A function $ \rho$ from $[0,\,1]$ onto itself is a Dirichlet weight if it is increasing, $\rho '' \leqslant 0$ and ${\lim _{x \to 0 + }}x/\rho (x) = 0$. The corresponding Dirichlet-type space, ${D_\rho }$, consists of those bounded holomorphic functions on $U = \{ z \in {\mathbf{C}}:\,\vert z\vert < 1\}$ such that $M({D_\rho }) = \{ g:\,U \to {\mathbf{C}}:gf \in {D_\rho },\forall f \in {D_\rho }\}$.
Zelevinski algebras related to projective representations
M.
Bean;
P.
Hoffman
99-111
Abstract: We define $ L$- $\operatorname{PSH} $-algebras, and prove a classification theorem for such objects. The letters refer respectively to a ground ring $L$ and to the positivity, selfadjointness and Hopf structures on an algebra, the basic example of which occurred in the study of projective representations of ${S_n}$. This is analogous to an idea over ${\mathbf{Z}}$ due to Zelevinski in connection with linear representations.
A sparse Graham-Rothschild theorem
Hans Jürgen
Prömel;
Bernd
Voigt
113-137
Abstract: The main result of this paper is a sparse version of the Graham-Rothschild partition theorem for $n$-parameter sets [R. L. Graham and B. L. Rothschild, Ramsey's theorem for $n$-parameter sets, Trans. Amer. Math. Soc. 159 (1971), 257-292]. In particular, a sparse version of Hales-Jewett's theorem is proved. We give several applications, e.g., for arithmetic progressions and finite sums of integers, confirming conjectures of J. Spencer and of J. Nešetřil and V. Rödl. We also consider graphs defined on parameter sets and prove a sparse and restricted induced partition theorem for such graphs, extending results from [H. J. Prömel, Induced partition properties of combinatorial cubes, J. Combin. Theory Ser. A 39 (1985), 177-208] and [P. Frankl, R. L. Graham, and V. Rödl, Induced restricted Ramsey theorems for spaces, J. Combin. Theory Ser. A 44 (1987), 120-128].
Function spaces generated by blocks associated with spheres, Lie groups and spaces of homogeneous type
Aleš
Založnik
139-164
Abstract: Functions generated by blocks were introduced by M. Taibleson and G. Weiss in the setting of the one-dimensional torus $ T$ [TW1]. They showed that these functions formed a space "close" to the class of integrable functions for which we have almost everywhere convergence of Fourier series. Together with S. Lu [LTW] they extended the theory to the $n$-dimensional torus where this convergence result (for Bochner-Riesz means at the critical index) is valid provided we also restrict ourselves to $ L\log L$. In this paper we show that this restriction is not needed if the underlying domain is a compact semisimple Lie group (or certain more general spaces of a homogeneous type). Other considerations (for example, these spaces form an interesting family of quasi-Banach spaces; they are connected with the notion of entropy) guide one in their study. We show how this point of view can be exploited in the setting of more general underlying domains.
Pseudoconvex classes of functions. III. Characterization of dual pseudoconvex classes on complex homogeneous spaces
Zbigniew
Slodkowski
165-189
Abstract: Invariant classes of functions on complex homogeneous spaces, with properties similar to those of the class of plurisubharmonic functions, are studied. The main tool is a regularization method for these classes, and the main theorem characterizes dual classes of functions (where duality is defined in terms of the local maximum property). These results are crucial in proving a duality theorem for complex interpolation of normed spaces, which is given elsewhere.
Umbral calculus, binomial enumeration and chromatic polynomials
Nigel
Ray
191-213
Abstract: We develop the concept of partition categories, in order to extend the Mullin-Rota theory of binomial enumeration, and simultaneously to provide a natural setting for recent applications of the Roman-Rota umbral calculus to computations in algebraic topology. As a further application, we describe a generalisation of the chromatic polynomial of a graph.
Trace Paley-Wiener theorem in the twisted case
J. D.
Rogawski
215-229
Abstract: A version of the trace Paley-Wiener theorem for a reductive $p$-adic group in the context of twisted harmonic analysis with respect to an outer automorphism is proved.
Valuations on meromorphic functions of bounded type
Mitsuru
Nakai
231-252
Abstract: The primary purpose of this paper is to show that every valuation on the field of meromorphic functions of bounded type on a finitely sheeted unlimited covering Riemann surface is a point valuation if and only if the same is true on its base Riemann surface. The result is then applied to concrete examples and some related results are obtained.
The Bergman spaces, the Bloch space, and Gleason's problem
Ke He
Zhu
253-268
Abstract: Suppose $ f$ is a holomorphic function on the open unit ball ${B_n}$ of $ {{\mathbf{C}}^n}$. For $1 \leqslant p < \infty$ and $m > 0$ an integer, we show that $f$ is in $ {L^p}({B_n},\,dV)$ (with $ dV$ the volume measure) iff all the functions $ {\partial ^m}f/\partial {z^{\alpha \,}}\;(\vert\alpha \vert\, = m)$ are in ${L^p}({B_n},\,dV)$. We also prove that $ f$ is in the Bloch space of ${B_n}$ iff all the functions $ {\partial ^m}f/\partial {z^\alpha }\;(\vert\alpha \vert\, = m)$ are bounded on ${B_n}$. The corresponding result for the little Bloch space of ${B_n}$ is established as well. We will solve Gleason's problem for the Bergman spaces and the Bloch space of ${B_n}$ before proving the results stated above. The approach here is functional analytic. We make extensive use of the reproducing kernels of $ {B_n}$. The corresponding results for the polydisc in ${{\mathbf{C}}^n}$ are indicated without detailed proof.
Representations of Hecke algebras
Eugene
Gutkin
269-277
Abstract: We find all operators of a certain type that satisfy the braid relations corresponding to any generalized Cartan matrix.
Kazhdan-Lusztig polynomials for Hermitian symmetric spaces
Brian D.
Boe
279-294
Abstract: A nonrecursive scheme is presented to compute the Kazhdan-Lusztig polynomials associated to a classical Hermitian symmetric space, extending a result of Lascoux-Schützenberger for grassmannians. The polynomials for the exceptional Hermitian domains are also tabulated. All the Kazhdan-Lusztig polynomials considered are shown to be monic.
On the dual of an exponential solvable Lie group
Bradley N.
Currey
295-307
Abstract: Let $G$ be a connected, simply connected exponential solvable Lie group with Lie algebra $\mathfrak{g}$. The Kirillov mapping $\eta :\,\,\mathfrak{g}{\ast}/\operatorname{Ad} {\ast}(G) \to \hat G$ gives a natural parametrization of $\hat G$ by co-adjoint orbits and is known to be continuous. In this paper a finite partition of $ \mathfrak{g}{\ast}/\operatorname{Ad} {\ast}(G)$ is defined by means of an explicit construction which gives the partition a natural total ordering, such that the minimal element is open and dense. Given $ \pi \in \hat G$, elements in the enveloping algebra of ${\mathfrak{g}_c}$ are constructed whose images under $\pi$ are scalar and give crucial information about the associated orbit. This information is then used to show that the restriction of $\eta$ to each element of the above-mentioned partition is a homeomorphism.
On the canonical rings of some Horikawa surfaces. I
Valentin
Iliev
309-323
Abstract: This paper is devoted to finding necessary and sufficient conditions for a graded ring to be the canonical ring of a minimal surface of general type with ${K^2} = 2{p_g} - 3$, $ {p_g} \geqslant 3$, and such that its canonical linear system has one base point.
A Diophantine problem on elliptic curves
Robert
Tubbs
325-338
Abstract: This paper examines simultaneous diophantine approximations to coordinates of certain points on a product of elliptic curves. Specifically, let $\wp (z)$ be a Weierstrass elliptic function with algebraic invariants and complex multiplication. Suppose that $ \beta$ is cubic over the "field of multiplications" of $\wp (z)$ and that $u \in \mathbb{C}$ such that $\zeta = (\wp (u),\,\wp (\beta u),\,\wp ({\beta ^2}u))$ is defined. We study approximations to $ \zeta$ by points which lie on curves defined over $ \mathbb{Z}$.
Boundary behavior of invariant Green's potentials on the unit ball in ${\bf C}\sp n$
K. T.
Hahn;
David
Singman
339-354
Abstract: Let $p(z) = \int_B {G(z,\,w)\,d\mu (w)}$ be an invariant Green's potential on the unit ball $ B$ in $ {{\mathbf{C}}^n}\;(n \geqslant 1)$, where $G$ is the invariant Green's function and $ \mu$ is a positive measure with $\int_B {{{(1 - \vert w{\vert^2})}^n}\,d\mu (w) < \infty }$. In this paper, a necessary and sufficient condition on a subset $E$ of $B$ such that for every invariant Green's potential $ p$, $\displaystyle \mathop {\lim }\limits_{z \to e} \,\inf {(1 - \vert z{\vert^2})^n}p(z) = 0,\qquad e = (1,\,0,\, \ldots ,\,0)\; \in \partial B,\;z \in E,$ is given. The condition is that the capacity of the sets $ E \cap \{ z \in B\vert\;\vert z - e\vert < \varepsilon \}$, $\varepsilon > 0$, is bounded away from 0. The result obtained here generalizes Luecking's result, see [L], on the unit disc in ${\mathbf{C}}$.
Symmetry diffeomorphism group of a manifold of nonpositive curvature
Patrick
Eberlein
355-374
Abstract: Let $\tilde M$ denote a complete simply connected manifold of nonpositive sectional curvature. For each point $p \in \tilde M$ let $ {s_p}$ denote the diffeomorphism of $\tilde M$ that fixes $p$ and reverses all geodesics through $ p$. The symmetry diffeomorphism group $ {G^{\ast}}$ generated by all diffeomorphisms $\{ {s_p}:\,p \in \tilde M\} $ extends naturally to group of homeomorphisms of the boundary sphere $\tilde M(\infty )$. A subset $X$ of $ \tilde M(\infty )$ is called involutive if it is invariant under ${G^{\ast}}$. Theorem. Let $X \subseteq \tilde M(\infty )$ be a proper, closed involutive subset. For each point $p \in \tilde M$ let $ N(p)$ denote the linear span in $ {T_p}\tilde M$ of those vectors at $p$ that are tangent to a geodesic $ \gamma$ whose asymptotic equivalence class $\gamma (\infty )$ belongs to $ X$. If $ N(p)$ is a proper subspace of $ {T_p}\tilde M$ for some point $p \in \tilde M$, then $ \tilde M$ splits as a Riemannian product ${\tilde M_1} \times {\tilde M_2}$ such that $N$ is the distribution of $ \tilde M$ induced by ${\tilde M_1}$. This result has several applications that include new results as well as great simplifications in the proofs of some known results. In a sequel to this paper it is shown that if $ \tilde M$ is irreducible and $ \tilde M(\infty )$ admits a proper, closed involutive subset $X$, then $\tilde M$ is isometric to a symmetric space of noncompact type and rank $ k \geqslant 2$.
On zeros of a system of polynomials and application to sojourn time distributions of birth-and-death processes
Ken-iti
Sato
375-390
Abstract: Zeros of the following system of polynomials are considered: $\displaystyle \left\{ \begin{gathered}{P_0}(x) = 1, {P_1}(x) = {B_0} ... ..._{n - 1}}(x)\quad {\text{for}}\;n \geqslant 1. \end{gathered} \right.$ Numbers of positive and negative zeros are determined and a separation property of the zeros of $ {P_m}(x)$ and $ {P_n}(x)$ is proved under the condition that ${C_n} > 0$ and $ {P_n}(0) > 0$ for every $ n$. No condition is imposed on ${A_n}$. These results are applied to determination of the distribution of a sojourn time with general (not necessarily positive) weight function for a birth-and-death process up to a first passage time. Unimodality and infinite divisibility of the distribution follow.
Local $H$-maps of $B{\rm U}$ and applications to smoothing theory
Timothy
Lance
391-424
Abstract: When localized at an odd prime $p$, the classifying space $PL/O$ for smoothing theory splits as an infinite loop space into the product $C \times N$ where $C = {\text{Cokernel}}\,(J)$ and $N$ is the fiber of a $p$-local $H$-map $BU \to BU$. This paper studies spaces which arise in this latter fashion, computing the cohomology of their Postnikov towers and relating their $ k$-invariants to properties of the defining self-maps of $BU$. If $Y$ is a smooth manifold, the set of homotopy classes $ [Y,\,N]$ is a certain subgroup of resmoothings of $Y$, and the $k$-invariants of $N$ generate obstructions to computing that subgroup. These obstructions can be directly related to the geometry of $Y$ and frequently vanish.
The fundamental module of a normal local domain of dimension $2$
Yuji
Yoshino;
Takuji
Kawamoto
425-431
Abstract: The fundamental module $E$ of a normal local domain $(R,\,\mathfrak{m})$ of dimension $2$ is defined by the nonsplit exact sequence $0 \to K \to E \to \mathfrak{m} \to 0$, where $K$ is the canonical module of $R$. We prove that, if $R$ is complete with $R/\mathfrak{m} \simeq \mathbb{C}$, then $ E$ is decomposable if and only if $R$ is a cyclic quotient singularity. Various other properties of fundamental modules will be discussed.
Some P.V.-equivalences and a classification of $2$-simple prehomogeneous vector spaces of type ${\rm II}$
Tatsuo
Kimura;
Shin-ichi
Kasai;
Masanobu
Taguchi;
Masaaki
Inuzuka
433-494
Abstract: A classification of $ 2$-simple prehomogeneous vector spaces is completed by using some P.V.-equivalences together with [3]. Some part is very different from the previous classification of the irreducible or simple cases [1, 2], and some new method is necessary. This result shows the difficult point of a classification problem of reductive prehomogeneous vector spaces.
Hardy spaces of vector-valued functions: duality
Oscar
Blasco
495-507
Abstract: We prove here that the Hardy space of $B$-valued functions ${H^1}(B)$ defined by using the conjugate function and the one defined in terms of $B$-valued atoms do not coincide for a general Banach space. The condition for them to coincide is the UMD property on $B$. We also characterize the dual space of both spaces, the first one by using $ {B^{\ast}}$-valued distributions and the second one in terms of a new space of vector-valued measures, denoted $ \mathcal{B}\mathcal{M}\mathcal{O}({B^{\ast}})$, which coincides with the classical $\operatorname{BMO} ({B^{\ast}})$ of functions when ${B^{\ast}}$ has the RNP.
The number of solutions to linear Diophantine equations and multivariate splines
Wolfgang
Dahmen;
Charles A.
Micchelli
509-532
Abstract: In this paper we study how the number of nonnegative integer solutions of $s$ integer linear equations in $n \geqslant s$ unknowns varies as a function of the inhomogeneous terms. Aside from deriving various recurrence relations for this function, we establish some of its detailed structural properties. In particular, we show that on certain subsets of lattice points it is a polynomial. The univariate case ($s = 1$) yields E. T. Bell's description of Sylvester's denumerants. Our approach to this problem relies upon the use of polyhedral splines. As an example of this method we obtain results of R. Stanley on the problem of counting the number of magic squares.
A characterization of two weight norm inequalities for fractional and Poisson integrals
Eric T.
Sawyer
533-545
Abstract: For $1 < p \leqslant q < \infty$ and $ w(x)$, $v(x)$ nonnegative functions on $ {{\mathbf{R}}^n}$, we show that the weighted inequality $\displaystyle {\left( {\int {\vert Tf{\vert^q}w} } \right)^{1/q}} \leqslant C{\left( {\int {{f^p}v} } \right)^{1/p}}$ holds for all $f \geqslant 0$ if and only if both $\displaystyle \int {{{[T({\chi _Q}{v^{1 - p'}})]}^q}w \leqslant {C_1}{{\left( {\int_Q {{v^{1 - p'}}} } \right)}^{q/p}} < \infty }$ and $\displaystyle {\int {{{[T({\chi _Q}w)]}^{p'}}{v^{1 - p'}} \leqslant {C_2}\left( {\int_Q w } \right)} ^{p'/q'}} < \infty $ hold for all dyadic cubes $Q$. Here $T$ denotes a fractional integral or, more generally, a convolution operator whose kernel $K$ is a positive lower semicontinuous radial function decreasing in $\vert x\vert$ and satisfying $K(x) \leqslant CK(2x)$, $x \in {{\mathbf{R}}^n}$. Applications to degenerate elliptic differential operators are indicated. In addition, a corresponding characterization of those weights $v$ on $ {{\mathbf{R}}^n}$ and $ w$ on ${\mathbf{R}}_ + ^{n + 1}$ for which the Poisson operator is bounded from ${L^p}(v)$ to ${L^q}(w)$ is given.
Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators
K. F.
Andersen;
E. T.
Sawyer
547-558
Abstract: The weight functions $u(x)$ for which $ {R_\alpha }$, the Riemann-Liouville fractional integral operator of order $\alpha > 0$, is bounded from ${L^p}({u^p}\,dx)$ to $ {L^q}({u^q}\,dx)$, $1 < p < 1/\alpha$, $1/q = 1/p - \alpha$, are characterized. Further, given $p$,$q$ with $ 1/q \geqslant 1/p - \alpha$, the weight functions $u > 0$ a.e. (resp. $ v < \infty$ a.e.) for which there is $v < \infty $ a.e. (resp. $ u > 0$ a.e.) so that ${R_\alpha }$ is bounded from ${L^p}({v^p}\,dx)$ to ${L^q}({u^q}\,dx)$ are characterized. Analogous results are obtained for the Weyl fractional integral. The method involves the use of complex interpolation of analytic families of operators to obtain similar results for fractional "one-sided" maximal function operators which are of independent interest.
Orthogonal polynomials on several intervals via a polynomial mapping
J. S.
Geronimo;
W.
Van Assche
559-581
Abstract: Starting from a sequence $ \{ {p_n}(x;\,{\mu _0})\}$ of orthogonal polynomials with an orthogonality measure ${\mu _0}$ supported on ${E_0} \subset [ - 1,\,1]$, we construct a new sequence $ \{ {p_n}(x;\,\mu )\}$ of orthogonal polynomials on $E = {T^{ - 1}}({E_0})$ ($T$ is a polynomial of degree $N$) with an orthogonality measure $ \mu$ that is related to $ {\mu _0}$. If ${E_0} = [ - 1,\,1]$, then $E = {T^{ - 1}}([ - 1,\,1])$ will in general consist of $N$ intervals. We give explicit formulas relating $\{ {p_n}(x;\,\mu )\}$ and $\{ {p_n}(x;\,{\mu _0})\}$ and show how the recurrence coefficients in the three-term recurrence formulas for these orthogonal polynomials are related. If one chooses $T$ to be a Chebyshev polynomial of the first kind, then one gets sieved orthogonal polynomials.
Deficient values and angular distribution of entire functions
Lo
Yang
583-601
Abstract: Let $f(z)$ be an entire function of positive and finite order $\mu$. If $f(z)$ has a finite number of Borel directions of order $\geqslant \mu $, then the sum of numbers of finite nonzero deficient values of $f(z)$ and all its primitives does not exceed $2\mu$. The proof is based on several lemmas and application of harmonic measure.
The central limit theorem for empirical processes under local conditions: the case of Radon infinitely divisible limits without Gaussian component
Niels T.
Andersen;
Evarist
Giné;
Joel
Zinn
603-635
Abstract: Weak convergence results are obtained for empirical processes indexed by classes $ \mathcal{F}$ of functions in the case of infinitely divisible purely Poisson (in particular, stable) Radon limits, under conditions on the local modulus of the processes $\{ f(X):\,f \in \mathcal{F}\}$ ("bracketing" conditions). They extend (and slightly improve upon) a central limit theorem of Marcus and Pisier (1984) for Lipschitzian processes. The law of the iterated logarithm is also considered. The examples include Marcinkiewicz type laws of large numbers for weighted empirical processes and for the dual-bounded-Lipschitz distance between a probability in $ {\mathbf{R}}$ and its associated empirical measures.
Eigensharp graphs: decomposition into complete bipartite subgraphs
Thomas
Kratzke;
Bruce
Reznick;
Douglas
West
637-653
Abstract: Let $\tau (G)$ be the minimum number of complete bipartite subgraphs needed to partition the edges of $G$, and let $r(G)$ be the larger of the number of positive and number of negative eigenvalues of $ G$. It is known that $\tau (G) \geqslant r(G)$; graphs with $\tau (G) = r(G)$ are called eigensharp. Eigensharp graphs include graphs, trees, cycles ${C_n}$ with $n = 4$ or $n \ne 4k$, prisms ${C_n}\square {K_2}$ with $n \ne 3k$, "twisted prisms" (also called "Möbius ladders") ${M_n}$ with $n = 3$ or $n \ne 3k$, and some Cartesian products of cycles. Under some conditions, the weak (Kronecker) product of eigensharp graphs is eigensharp. For example, the class of eigensharp graphs with the same number of positive and negative eigenvalues is closed under weak products. If each graph in a finite weak product is eigensharp, has no zero eigenvalues, and has a decomposition into $\tau (G)$ stars, then the product is eigensharp. The hypotheses in this last result can be weakened. Finally, not all weak products of eigensharp graphs are eigensharp.
Differentiation theorem for Gaussian measures on Hilbert space
Jaroslav
Tišer
655-666
Abstract: It is shown that the differentiation theorem is valid in infinitely dimensional Hilbert space with certain Gaussian measures. The proof uses result from harmonic analysis concerning the behavior of Hardy-Littlewood maximal operator in highly dimensional space.
Explicit formula for weighted scalar nonlinear hyperbolic conservation laws
Philippe
LeFloch;
Jean-Claude
Nédélec
667-683
Abstract: We prove a uniqueness and existence theorem for the entropy weak solution of nonlinear hyperbolic conservation laws of the form $\displaystyle \frac{\partial } {{\partial t}}(ru) + \frac{\partial } {{\partial x}}(rf(u)) = 0,$ with initial data and boundary condition. The scalar function $u = u(x,\,t)$, $x > 0$, $t > 0$, is the unknown, the function $f = f(u)$ is assumed to be strictly convex with inf $ f( \cdot ) = 0$ and the weight function $r = r(x)$, $x > 0$, to be positive (for example, $r(x) = {x^\alpha }$, with an arbitrary real $ \alpha$). We give an explicit formula, which generalizes a result of P. D. Lax. In particular, a free boundary problem for the flux $ r( \cdot )f(u( \cdot , \cdot ))$ at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to B. L. Keyfitz.
Complex interpolation of normed and quasinormed spaces in several dimensions. I
Zbigniew
Slodkowski
685-711
Abstract: A variety of complex interpolation methods for families of normed or quasi-normed spaces, parametrized by points of domains in complex homogeneous spaces, parametrized by points of domains in complex homogeneous spaces, is developed. Results on existence, continuity, uniqueness, reiteration and duality for interpolation are proved, as well as on interpolation of operators. A minimum principle for plurisubharmonic functions is obtained and used as a tool for the duality theorem.
Construction of an inner function in the little Bloch space
Kenneth
Stephenson
713-720
Abstract: An explicit construction using Riemann surfaces and Brownian motion is given for an inner function in the unit disc which is not a finite Blaschke product yet belongs to the little Bloch space $ {\mathcal{B}_0}$. In addition to showing how an inner function can meet the geometric conditions for $ {\mathcal{B}_0}$, this example settles an open question concerning the finite ranges of inner functions: the values which it takes only finitely often are dense in the disc.
Equivalence and strong equivalence of actions on handlebodies
John
Kalliongis;
Andy
Miller
721-745
Abstract: An algebraic characterization is given for the equivalence and strong equivalence classes of finite group actions on $ 3$-dimensional handlebodies. As one application it is shown that each handlebody whose genus is bigger than one admits only finitely many finite group actions up to equivalence. In another direction, the algebraic characterization is used as a basis for deriving an explicit combinatorial description of the equivalence and strong equivalence classes of the cyclic group actions of prime order on handlebodies with genus larger than one. This combinatorial description is used to give a complete closed-formula enumeration of the prime order cyclic group actions on such handlebodies.
${\rm SO}(2)$-equivariant vector fields on $3$-manifolds: moduli of stability and genericity
Genesio Lima
dos Reis;
Geovan Tavares
dos Santos
747-763
Abstract: An open and dense class of vector fields on $3$-dimensional compact manifolds equivariant under the action of $ \operatorname{SO} (2)$ is defined. Each such vector field has finite moduli of stability. We also exhibit an open and dense subset of the $ \operatorname{SO} (2)$-equivariant gradient vector fields which are structurally stable.
Templates and train tracks
George
Frank
765-784
Abstract: Within the context of Smale flows on compact manifolds, this article deals with a relationship between abstract templates, branched $1$-manifolds (train tracks), and laminations representing unstable separatrices of basic sets. We show that an abstract template, the richest in information of the above three entities, determines a member of each of the remaining two groups, and partial determinations in other directions are developed. As a result of this relationship, an obstruction to the realization of certain abstract templates in nonsingular Smale flows on homology $3$-spheres is raised.
Normal derivative for bounded domains with general boundary
Guang Lu
Gong;
Min Ping
Qian;
Martin L.
Silverstein
785-809
Abstract: Let $D$ be a general bounded domain in the Euclidean space ${R^n}$. A Brownian motion which enters from and returns to the boundary symmetrically is used to define the normal derivative as a functional for $ f$ with $f$, $\nabla f$ and $\Delta f$ all in ${L^2}$ on $D$. The corresponding Neumann condition (normal derivative $= 0$) is an honest boundary condition for the $ {L^2}$ generator of reflected Brownian notion on $D$. A conditioning argument shows that for $ D$ and $f$ sufficiently smooth this general definition of the normal derivative agrees with the usual one.
Wiener's criterion for parabolic equations with variable coefficients and its consequences
Nicola
Garofalo;
Ermanno
Lanconelli
811-836
Abstract: In a bounded set in ${{\mathbf{R}}^{n + 1}}$ we study the problem of the regularity of boundary points for the Dirichlet problem for a parabolic operator with smooth coefficients. We give a geometric characterization, modelled on Wiener's criterion for Laplace's equation, of those boundary points that are regular. We also present some important consequences. Here is the main one: a point is regular for a variable coefficient operator if and only if it is regular for the constant coefficient operator obtained by freezing the coefficients at that point.
Rigidity theorems for right angled reflection groups
Ennis
Rosas
837-848
Abstract: Let $\Gamma$ be a right angled reflection group. Let $M$ and $M'$ be Coxeter manifolds. Then any $ \Gamma$-map
Engulfing and subgroup separability for hyperbolic groups
D. D.
Long
849-859
Abstract: If a group is subgroup separable, otherwise known as locally extended residually finite or LERF, one can pass from immersions to embeddings in some finite covering space. We show that a certain 'engulfing' property gives subgroup separability for a large and useful class of subgroups of hyperbolic $3$-manifold groups.
Generalized quotients in Coxeter groups
Anders
Björner;
Michelle L.
Wachs
1-37
Abstract: For ($W$, $S$) a Coxeter group, we study sets of the form $\displaystyle W/V = \{ w \in W\vert l(wv) = l(w) + l(v)\;{\text{for all}}\;v \in V\} ,$ , where $V \subseteq W$. Such sets $W/V$, here called generalized quotients, are shown to have much of the rich combinatorial structure under Bruhat order that has previously been known only for the case when $ V \subseteq S$ (i.e., for minimal coset representatives modulo a parabolic subgroup). We show that Bruhat intervals in $ W/V$, for general $V \subseteq W$, are lexicographically shellable. The Möbius function on $W/V$ under Bruhat order takes values in $\{ - 1,\,0,\, + 1\}$. For finite groups $ W$, generalized quotients are the same thing as lower intervals in the weak order. This is, however, in general not true. Connections with the weak order are explored and it is shown that $W/V$ is always a complete meet-semilattice and a convex order ideal as a subset of $W$ under weak order. Descent classes $ {D_I} = \{ w \in W\vert l(ws) < l(w) \Leftrightarrow s \in I,\;{\text{for all}}\;s \in S\}$, $ I \subseteq S$, are also analyzed using generalized quotients. It is shown that each descent class, as a poset under Bruhat order or weak order, is isomorphic to a generalized quotient under the corresponding ordering. The latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics. For instance, the set of standard Young tableaux of a fixed shape or the set of linear extensions of a rooted forest, suitably interpreted, form generalized quotients. We prove a factorization result for the quotients that come from rooted forests, which shows that algebraically these quotients behave as a system of minimal "coset" representatives of a subset which is in general not a subgroup. We also study the rank generating function for certain quotients in the symmetric group.
On the local boundedness of singular integral operators
Mark
Leckband
39-56
Abstract: The class of singular integral operators whose kernels satisfy the usual smoothness conditions is studied. Let such an operator be denoted by $K$. We establish necessary conditions that imply $K$ has local (weighted) ${L^p}$ norm inequalities. The underlying principle is as follows. If ${\chi _Q}$ is the characteristic function of a fixed cube $Q$ of ${R^n}$, or all of ${R^n}$, then $ K{\chi _Q}$ and (the adjoint of $K$) $ {K^{\ast}}{\chi _Q}$ determine the boundedness properties of $K$ for functions supported in a proper fraction of $Q$.
Weighted norm inequalities for potential operators
Martin
Schechter
57-68
Abstract: We give sufficient conditions for inequalities of the form $\displaystyle {\left( {\int {{{\left( {\int {G(x - y)f(y)\,d\mu (y)} } \right)}... ...1/q}}\, \leqslant C{\left( {\int {\vert f(y){\vert^p}d\nu (y)} } \right)^{1/p}}$ to hold for measurable functions $f$. We determine the dependence of the constant $C$ on the measures $\mu$, $\nu$, $\omega$ and give some applications.
Conformal distortion of boundary sets
D. H.
Hamilton
69-81
Abstract: Conformal maps $ f$ of the disk into itself have the property that $\dim {f^{ - 1}}(F) \leqslant \dim F$ for any set $F$ on the unit circle.
Subordination families and extreme points
Yusuf
Abu-Muhanna;
D. J.
Hallenbeck
83-89
Abstract: Let $s(F)$ denote the set of functions subordinate to a univalent function $F$ in $\Delta$ the unit disk. Let ${B_0}$ denote the set of functions $ \phi (z)$ analytic in $ \Delta$ satisfying $\vert\phi (z)\vert < 1$ and $\phi (0) = 0$. We prove that if $f = F \circ \phi$ is an extreme point of $ s(F)$, then $\phi$ is an extreme point of $ {B_0}$. Let $D = F(s)$ and $\lambda (w,\,\partial D)$ denote the distance between $w$ and $ \partial D$ (boundary of $ D$). We also prove that if $ \phi$ is an extreme point of ${B_0}$ and $ \vert\phi ({e^{it}})\vert < 1$ for almost all $t$, then $\int_0^{2\pi } {\log \lambda (F(\phi ({e^{it}}){e^{i\theta }}),\,\partial D)\,dt = - \infty }$ for almost all $\theta$.
Generalized Chebyshev polynomials associated with affine Weyl groups
Michael E.
Hoffman;
William Douglas
Withers
91-104
Abstract: We begin with a compact figure that can be folded into smaller replicas of itself, such as the interval or equilateral triangle. Such figures are in one-to-one correspondence with affine Weyl groups. For each such figure in $ n$-dimensional Euclidean space, we construct a sequence of polynomials ${P_k}:{{\mathbf{R}}^n} \to {{\mathbf{R}}^n}$ so that the mapping ${P_k}$ is conjugate to stretching the figure by a factor $k$ and folding it back onto itself. If $n = 1$ and the figure is the interval, this construction yields the Chebyshev polynomials (up to conjugation). The polynomials $ {P_k}$ are orthogonal with respect to a suitable measure and can be extended in a natural way to a complete set of orthogonal polynomials.
Local uncertainty inequalities for locally compact groups
John F.
Price;
Alladi
Sitaram
105-114
Abstract: Let $G$ be a locally compact unimodular group equipped with Haar measure $m$, $\hat G$ its unitary dual and $\mu$ the Plancherel measure (or something closely akin to it) on $\hat G$. When $G$ is a euclidean motion group, a non-compact semisimple Lie group or one of the Heisenberg groups we prove local uncertainty inequalities of the following type: given $\theta \in \left[ {0,\tfrac{1} {2}} \right.)$ there exists a constant $ {K_\theta }$ such that for all $f$ in a certain class of functions on $G$ and all measurable $E \subseteq \hat G$, $\displaystyle {\left( {\int_E {\operatorname{Tr} (\pi {{(f)}^{\ast}}\pi (f))\,d... ...\leqslant {K_\theta }\mu {(E)^\theta }\vert\vert{\phi _\theta }f\vert{\vert _2}$ where ${\phi _\theta }$ is a certain weight function on $ G$ (for which an explicit formula is given). When $G = {{\mathbf{R}}^k}$ the inequality has been established with ${\phi _\theta }(x) = \vert x{\vert^{k\theta }}$.
The Brauer group of graded continuous trace $C\sp *$-algebras
Ellen Maycock
Parker
115-132
Abstract: Let $X$ be a locally compact Hausdorff space. The graded Morita equivalence classes of separable, $ {{\mathbf{Z}}_2}$-graded, continuous trace $ {C^{\ast}}$-algebras which have spectrum $X$ form a group, ${\operatorname{GBr} ^\infty }(X)$, the infinite-dimensional graded Brauer group of $X$. Techniques from algebraic topology are used to prove that ${\operatorname{GBr} ^\infty }(X)$ is isomorphic via an isomorphism $w$ to the direct sum $\check{H}^1(X; \underline{\mathbf{Z}}_2) \oplus \check{H}^3 (X; \underline{\mathbf{Z}})$. The group ${\operatorname{GBr} ^\infty }(X)$ includes as a subgroup the ungraded continuous trace ${C^{\ast}}$-algebras, and the Dixmier-Douady invariant of such an ungraded $ {C^{\ast}}$-algebra is its image in $\check{H}^3 (X; \underline{\mathbf{Z}})$ under $w$.
Proper knot theory in open $3$-manifolds
Peter
Churchyard;
David
Spring
133-142
Abstract: This paper introduces a theory of proper knots, i.e., smooth proper embeddings of $ {{\mathbf{R}}^1}$ into open $3$-manifolds. Proper knot theory is distinguished by the fact that proper isotopies of knots are not ambient in general. A uniqueness theorem for proper knots is proved in case the target manifold is the interior of a one-dimensional handlebody.
The closing lemma for generalized recurrence in the plane
Maria Lúcia Alvarenga
Peixoto
143-158
Abstract: We prove a version of the Closing Lemma for ${C^r}$ vector fields on the plane, $r \geqslant 1$, and for a kind of recurrence obtained using the concept of prolongational limit sets. We call it generalized recurrence. Given a nonperiodic point $p$ in the generalized recurrent set we perturb the vector field in order to get a new vector field arbitrarily close to it, with a closed orbit through $ p$.
Stable manifolds in the method of averaging
Stephen
Schecter
159-176
Abstract: Consider the differential equation $\dot z = \varepsilon f(z,\,t,\,\varepsilon )$, where $f$ is $T$periodic in $t$ and $ \varepsilon > 0$ is a small parameter, and the averaged equation $\dot z = \overline f (z): = (1/T)\,\int_0^T {\,f(z,\,t,\,0)\,dt}$. Suppose the averaged equation has a hyperbolic equilibrium at $z = 0$ with stable manifold $\overline W$. Let ${\beta _\varepsilon }(t)$ denote the hyperbolic $ T$-periodic solution of $\dot z = \varepsilon f(z,\,t,\,\varepsilon )$ near $z \equiv 0$. We prove a result about smooth convergence of the stable manifold of ${\beta _\varepsilon }(t)$ to $\overline W \times {\mathbf{R}}$ as $\varepsilon \to 0$. The proof uses ideas of Vanderbauwhede and van Gils about contractions on a scale of Banach spaces.
Uniformly fat sets
John L.
Lewis
177-196
Abstract: In this paper we study closed sets $E$ which are "locally uniformly fat" with respect to a certain nonlinear Riesz capacity. We show that $ E$ is actually "locally uniformly fat" with respect to a weaker Riesz capacity. Two applications of this result are given. The first application is concerned with proving Sobolev-type inequalities in domains whose complements are uniformly fat. The second application is concerned with the Fekete points of $E$.
Euler spaces of analytic functions
James
Rovnyak
197-208
Abstract: A formula due to Euler and Legendre is used to construct finite-difference counterparts to the Dirichlet space. The spaces have integral representations and characterizations in terms of area integrals. Their reproducing kernels are logarithms of the reproducing kernels of the Newton spaces, which are counterparts to the Hardy class. A Hilbert space with reproducing kernel $\displaystyle \log [(1/\overline w z)\,\log \;1/(1 - \overline w z)]$ is also shown to exist and to be related to Bernoulli numbers and combinatorial theory.
On the complete ${\rm GL}(n,{\bf C})$-decomposition of the stable cohomology of ${\rm gl}\sb n(A)$
Phil
Hanlon
209-225
Abstract: Let $A$ be a graded, associative ${\mathbf{C}}$-algebra. For each $n$ let $g{l_n}(A)$ denote the Lie algebra of $n \times n$ matrices with entries from $ A$. In this paper we extend the Loday-Quillen theorem to nontrivial isotypic components of $ GL(n,\,{\mathbf{C}})$ acting on the Lie algebra cohomology of $g{l_n}(A)$. For $\alpha$ and $\beta$ partitions of some nonnegative integer $ m$ let $ {[\alpha ,\,\beta ]_n} \in {{\mathbf{Z}}^n}$ denote the maximal $ GL(n,\,{\mathbf{C}})$-weight given by $\displaystyle {[\alpha ,\,\beta ]_n} = \sum\limits_i {{\alpha _i}{e_i}} - \sum\limits_j {{\beta _j}{e_{n + 1 - j}}.}$ We show that the $ {[\alpha ,\,\beta ]_n}$-isotypic component of the Lie algebra cohomology of $ g{l_n}(A)$ stabilizes when $n \to \infty $ and is equal to $\displaystyle HR{C^{\ast}}(A) \otimes ({\tilde H^{\ast}}{(A;\,{\mathbf{C}})^{ \otimes m}} \otimes {S^\alpha } \otimes {S^\beta }){s_m}$ where ${\tilde H^{\ast}}(A;\,{\mathbf{C}})$ is the reduced Hochschild cohomology of $A$ with trivial coefficients, where $ HR{C^{\ast}}(A)$ is the graded exterior algebra generated by the cyclic cohomology of $A$, where $ {S^\alpha }$ and ${S^\beta }$ are the irreducible $ {S_m}$-modules indexed by $ \alpha$ and $ \beta$ and where the action of ${S_m}$ on $\tilde H{(A;\,{\mathbf{C}})^{ \otimes m}}$ is the exterior action.
Pseudo-orbit shadowing in the family of tent maps
Ethan M.
Coven;
Ittai
Kan;
James A.
Yorke
227-241
Abstract: We study the family of tent maps--continuous, unimodal, piecewise linear maps of the interval with slopes $\pm s$, $\sqrt 2 \leqslant s \leqslant 2$. We show that tent maps have the shadowing property (every pseudo-orbit can be approximated by an actual orbit) for almost all parameters $s$, although they fail to have the shadowing property for an uncountable, dense set of parameters. We also show that for any tent map, every pseudo-orbit can be approximated by an actual orbit of a tent map with a perhaps slightly larger slope.
Smoothness up to the boundary for solutions of the nonlinear and nonelliptic Dirichlet problem
C. J.
Xu;
C.
Zuily
243-257
Abstract: For the Dirichlet problem associated with a general real second order p.d.e. $F(x,\,u,\,\nabla u,\,{\nabla ^2}u) = 0$ in a smooth open set $\Omega$ of $ {{\mathbf{R}}^n}$, we prove smoothness up to the boundary of the solution $ u$ for which the linearized characteristic form is nonnegative and satisfies Hörmander's brackets condition, the boundary of $ \Omega$ being noncharacteristic.
Some inequalities for singular convolution operators in $L\sp p$-spaces
Andreas
Seeger
259-272
Abstract: Suppose that a bounded function $m$ satisfies a localized multiplier condition ${\sup _{t > 0}}\vert\vert\phi m({t^P} \cdot )\vert{\vert _{{M_p}}} < \infty$, for some bump function $\phi$. We show that under mild smoothness assumptions $m$ is a Fourier multiplier in ${L^p}$. The approach uses the sharp maximal operator and Littlewood-Paley-theory. The method gives new results for lacunary maximal functions and for multipliers in Triebel-Lizorkin-spaces.
A Riemannian geometric invariant and its applications to a problem of Borel and Serre
Bang-Yen
Chen;
Tadashi
Nagano
273-297
Abstract: A new geometric invariant will be introduced, studied and determined on compact symmetric spaces.
Variational principles for Hill's spherical vortex and nearly spherical vortices
Yieh Hei
Wan
299-312
Abstract: In this paper, vortex rings are regarded as axisymmetric motions without swirl of an incompressible inviscid fluid in space, with vorticity confined to their finite cores. The main results of this paper are (H) Hill's spherical vortex is a "nondegenerate" local maximum of the energy function subject to a fixed impulse, among vortex rings. (N) Norbury's nearly spherical vortex is a "nondegenerate" local maximum of the energy function subject to a fixed impulse, and a fixed circulation. Estimates are established to overcome the discontinuity of vorticity distributions, and the singular behavior of Stoke's stream functions near the axis of symmetry. The spectral analysis involves the use of Legendre's functions.
Iwasawa's $\lambda\sp -$-invariant and a supplementary factor in an algebraic class number formula
Kuniaki
Horie
313-328
Abstract: Let $l$ be a prime number and $k$ an imaginary abelian field. Sinnott [12] has shown that the relative class number of $k$ is expressed by the so-called index of the Stickelberger ideal of $k$, with a "supplementary factor" $ {c^ - }$ in $ \mathbb{N}/2 = \{ n/2\vert n \in \mathbb{N}\}$, and that if $k$ varies through the layers of the basic $ {\mathbb{Z}_l}$-extension over an imaginary abelian field, then ${c^ - }$ becomes eventually constant. On the other hand, ${c^ - }$ can take any value in $\mathbb{N}/2$ as $k$ ranges over the imaginary abelian fields (cf. [10]). In this paper, we shall study relations between the supplementary factor $ {c^ - }$ and Iwasawa's ${\lambda ^ - }$-invariant for the basic $ {\mathbb{Z}_l}$-extension over $k$, our discussion being based upon some formulas of Kida [8, 9], those of Sinnott [12], and fundamental results concerning a finite abelian $ l$-group acted on by a cyclic group. As a consequence, we shall see that the ${\lambda ^ - }$-invariant goes to infinity whenever $k$ ranges over a sequence of imaginary abelian fields such that the $l$-part of ${c^ - }$ goes to infinity.
On the zero set of a holomorphic one-form on a compact complex manifold
Michael J.
Spurr
329-339
Abstract: On any compact complex surface $M$, divisors of nonnegative self-intersection which are contained in the zero set (or in the integral set) of a holomorphic $1$-form are shown to induce a fibration of $ M$ onto a Riemann surface. This result is extended to higher dimensions for $ M$ projective. Applications to zero sets of holomorphic $1$-forms on surfaces are given.
The Morse index theorem where the ends are submanifolds
Diane
Kalish
341-348
Abstract: In this paper the Morse Index Theorem is proven in the case where submanifolds $P$ and $Q$ are at the endpoints of a geodesic, $ \gamma$. At $ \gamma$, the index of the Hessian of the energy function defined on paths joining $P$ and $Q$ is computed using $P$-focal points, and a calculation at the endpoint of $\gamma$, involving the second fundamental form of $ Q$.
The blow-up surface for nonlinear wave equations with small spatial velocity
Avner
Friedman;
Luc
Oswald
349-367
Abstract: Consider the Cauchy problem for ${u_{tt}} - {\varepsilon ^2}\Delta u = f(u)$ in space dimension $ \leqslant 3$ where $ f(u)$ is superlinear and nonnegative. The solution blows up on a surface $t = {\phi _\varepsilon }(x)$. Denote by $t = \phi (x)$ the blow-up surface corresponding to $v'' = f(v)$. It is proved that $\vert{\phi _\varepsilon }(x) - \phi (x)\vert \leqslant C{\varepsilon ^2}$, $\vert\nabla ({\phi _\varepsilon }(x) - \phi (x))\vert \leqslant C{\varepsilon ^2}$ in a neighborhood of any point ${x_0}$ where $ \phi ({x_0}) < \infty$.
Traveling wave solutions of a gradient system: solutions with a prescribed winding number. I
David
Terman
369-389
Abstract: Consideration is given to a system of equations of the form ${u_t} = {u_{xx}} + \nabla F(u)$, $u \in {{\mathbf{R}}^2}$. In a previous paper [6], conditions of $F$ were given which guarantee that the system possesses infinitely many traveling wave solutions. The solutions are now characterized by how many times they wind around in phase space. A winding number for solutions is defined. It is demonstrated that for each positive integer $K$, there exists at least two traveling wave solutions, each with winding number $K$ or $K + 1$.
Traveling wave solutions of a gradient system: solutions with a prescribed winding number. II
David
Terman
391-412
Abstract: This paper completes the analysis begun in [2] concerning the existence of traveling wave solutions of a system of the form $ {u_t} = {u_{xx}} + \nabla F(u)$, $u \in {{\mathbf{R}}^2}$. In [2] a notion of winding number for solutions was defined, and the proof that there exists a traveling wave solution with a prescribed winding number was reduced to a purely algebraic problem. In this paper the algebraic problem is solved.
Spectrum reducing extension for one operator on a Banach space
C. J.
Read
413-429
Abstract: In this paper we show that, given an operator $T$ on a Banach space $X$, there is an extension $Y$ of $X$ such that $T$ extends in a natural way to an operator ${T^ \sim }$ on $Y$, and the spectrum of $ {T^ \sim }$ is the approximate point spectrum of $T$. This answers a question posed by Bollobás, and contributes to a theory investigated by Shilov, Arens, Bollobás, etc. The unusual transfinite construction is similar to that which we used earlier to find an inverse producing extension for a commutative unital Banach algebra which eliminates the residual spectrum of one element. We also give a counterexample, consisting of a Banach algebra $L$ containing elements ${g_1}$ and ${g_2}$ such that in no extension $L'$ of $L$ are the residual spectra of ${g_1}$ and ${g_{_2}}$ eliminated simultaneously.
Errata to: ``Where does the $L\sp p$-norm of a weighted polynomial live?''
H. N.
Mhaskar;
E. B.
Saff
431
A fractal-like algebraic splitting of the classifying space for vector bundles
V.
Giambalvo;
David J.
Pengelley;
Douglas C.
Ravenel
433-455
Abstract: The connected covers of the classifying space $BO$ induce a decreasing filtration $\{ {B_n}\}$ of ${H_{\ast}}(BO;\,Z/2)$ by sub-Hopf algebras over the Steenrod algebra $A$. We describe a multiplicative grading on $ {H_{\ast}}(BO;\,Z/2)$ inducing a direct sum splitting of ${B_n}$ over ${A_n}$, where $ \{ {A_n}\}$ is the usual (increasing) filtration of $A$. The pieces in the splittings are finite, and the grading extends that of ${H_{\ast}}{\Omega ^2}{S^3}$ which splits it into Brown-Gitler modules. We also apply the grading to the Thomifications $ \{ {M_n}\}$ of $\{ {B_n}\}$, where it induces splittings of the corresponding cobordism modules over the entire Steenrod algebra. These generalize algebraically the previously known topological splittings of the connective cobordism spectra $MO$, $MSO$ and $M\,Spin$.
Applications of nonstandard models and Lebesgue measure to sequences of natural numbers
Steven C.
Leth
457-468
Abstract: By use of a nonstandard model, sequences of natural numbers are associated with a collection of closed subsets of reals in a natural way. The topological and measure-theoretic properties of the associated closed sets are used to prove standard theorems and define new density functions on sequences.
Asymptotic periodicity of the iterates of positivity preserving operators
M.
Miklavčič
469-479
Abstract: Assume that (A1) $X$ is a real Banach space. (A2) $ {X^ + }$ is a closed subset of $X$ with the following properties: (i) if $x \in {X^ + }$, $y \in {X^ + }$, $\alpha \in [0,\,\infty )$ then $x + y \in {X^ + }$ and $\alpha x \in {X^ + }$; (ii) there exists ${M_0} \in (0,\,\infty )$ such that for each $x \in X$ there exist ${x_ + } \in {X^ + }$ and ${x_ - } \in {X^ + }$ which satisfy $\displaystyle x = {x_ + } - {x_ - },\qquad \vert\vert{x_ + }\vert\vert \leqslan... ...t\vert,\qquad \vert\vert{x_ - }\vert\vert \leqslant {M_0}\vert\vert x\vert\vert$ and if $x = {y_ + } - {y_ - }$ for some ${y_ + } \in {X^ + }$, ${y_ - } \in {X^ + }$ then ${y_ + } - {x_ + } \in {X^ + }$; (iii) if $x \in {X^ + }$, $ y \in {X^ + }$ then $\vert\vert x\vert\vert \leqslant \vert\vert x + y\vert\vert$. (A3) $B$ is a bounded linear operator on $ X$. (A4) $B{X^ + } \subset {X^ + }$. (A5) $ {F_0}$ is a nonempty compact subset of $X$ and ${\lim _{n \to \infty }}\operatorname{dist} ({B^n}x,\,{F_0}) = 0$ whenever $x \in {X^ + }$ and $\vert\vert x\vert\vert = 1$. Then $ {B^n}x$ is asymptotically periodic for every $x \in X$. This, and other properties of $ B$, are proven in the paper.
A classification of a class of $3$-branchfolds
Yoshihiro
Takeuchi
481-502
Abstract: An $n$-orbifold is a topological space provided with a local modelling on (an open set in ${{\mathbf{R}}^n}$)/(a finite group action). Mainly, we deal with $3$-branchfolds (i.e. $3$-orbifolds with $1$-dimensional singular locus). We define a map between two $3$-branchfolds. With respect to this map, we prove some facts parallel to $3$-manifold theorems. Using the facts, we classify a class of $3$-branchfolds, analogous to Waldhausen's classification theorem of Haken manifolds.
Total linking number modules
Oziride Manzoli
Neto
503-533
Abstract: Given a codimension two link $L$ in a sphere ${S^k}$ with complement $X = {S^k} - L$, the total linking number covering of $ L$ is the covering $\hat X \to X$ associated to the surjection ${\pi _1}(X) \to Z$ defined by sending the meridians to $1$. The homology ${H_{\ast}}(\hat X)$ define weaker invariants than the homology of the universal abelian covering of $L$. The groups $ {H_i}(\hat X)$ are modules over $Z\left[ {t,\,{t^{ - 1}}} \right]$ and this work gives an algebraic characterization of these modules for $ k \geqslant 4$ except for the pseudo null part of $ {H_1}(\hat X)$.
Travelling wave solutions to a gradient system
James F.
Reineck
535-544
Abstract: Given a system of reaction-diffusion equations where the nonlinearity is derived from a potential with certain restrictions, we use the Conley index and the connection matrix to show that there is a travelling wave solution connecting the maxima of the potential.
The dynamics of rotating waves in scalar reaction diffusion equations
S. B.
Angenent;
B.
Fiedler
545-568
Abstract: The maximal compact attractor for the RDE ${u_t} = {u_{xx}} + f(u,\,{u_x})$ with periodic boundary conditions is studied. It is shown that any $\omega$-limit set contains a rotating wave, i.e., a solution of the form $U(x - ct)$. A number of heteroclinic orbits from one rotating wave to another are constructed. Our main tool is the Nickel-Matano-Henry zero number. The heteroclinic orbits are obtained via a shooting argument, which relies on a generalized Borsuk-Ulam theorem.
Linear supergroup actions. I. On the defining properties
Oscar Adolfo
Sánchez-Valenzuela
569-595
Abstract: This paper studies the notions of linearity and bilinearity in the category of supermanifolds. Following the work begun by [OASV2], we deal with supermanifoldifications of supervector spaces. The ${{\mathbf{R}}^{1\vert 1}}$-module operations are defined componentwise. The linearity and bilinearity properties are stated by requiring commutativity of some appropriate diagrams of supermanifold morphisms. It is proved that both linear and bilinear supermanifold morphisms are completely determined by their underlying continuous maps, which in turn have to be linear (resp., bilinear) in the usual sense. It is observed that whereas linear supermanifold morphisms are vector bundle maps, bilinear supermanifold morphisms are not. A natural generalization of the bilinear evaluation map $\operatorname{Hom} (V,\,W) \times V \to W\;((F,\,v) \mapsto F(v))$ is given and some applications pointing toward the notions of linear supergroup actions and adjoint and coadjoint actions are briefly discussed.
Remarks on Grassmannian supermanifolds
Oscar Adolfo
Sánchez-Valenzuela
597-614
Abstract: This paper studies some aspects of a particular class of examples of supermanifolds; the supergrassmannians, introduced in [Manin]. Their definition, in terms of local data and glueing isomorphisms, is reviewed. Explicit formulas in local coordinates are given for the Lie group action they come equipped with. It is proved that, for those supergrassmannians whose underlying manifold is an ordinary grassmannian, their structural sheaf can be realized as the sheaf of sections of the exterior algebra bundle of some canonical vector bundle. This realization holds true equivariantly for the Lie group action in question, thus making natural in these cases the identification given in [Batchelor]. The proof depends on the computation of the transition functions of the supercotangent bundle as defined in a previous work [OASV 2]. Finally, it is shown that there is a natural supergroup action involved (in the sense of [OASV 3]) and hence, the supergrassmannians may be regarded as examples of superhomogeneous spaces--a notion first introduced in [Kostant]. The corresponding Lie superalgebra action can be realized as superderivations of the structural sheaf; explicit formulas are included for those supergrassmannians identifiable with exterior algebra vector bundles.
Infinitely many periodic solutions for the equation: $u\sb {tt}-u\sb {xx}\pm \vert u\vert \sp {p-1}u=f(x,t)$. II
Kazunaga
Tanaka
615-645
Abstract: Existence of forced vibrations of nonlinear wave equation: \begin{displaymath}\begin{array}{*{20}{c}} {{u_{tt}} - {u_{xx}} \pm \vert u{\ver... ...,t) \in (0,\,\pi ) \times {\mathbf{R}},} \end{array} \end{displaymath} is considered. For all $p \in (1,\,\infty )$ and $f(x,\,t) \in {L^{(p + 1)/p}}$, existence of infinitely many periodic solutions is proved. This improves the results of the author [29, 30]. We use variational methods to show the existence result. Minimax arguments and energy estimates for the corresponding functional play an essential role in the proof.
Tangent cones to discriminant loci for families of hypersurfaces
Roy
Smith;
Robert
Varley
647-674
Abstract: A deformation of a variety with (nonisolated) hypersurface singularities, such as a projective hypersurface or a theta divisor of an abelian variety, determines a rational map of the singular locus to projective space and the resulting projective geometry of the singular locus describes how the singularities propagate in the deformation. The basic principle is that the projective model of the singular locus is dual to the tangent cone to the discriminant of the deformation. A detailed study of the method, which emerged from interpreting Andreotti-Mayer's work on theta divisors in terms of Schlessinger's deformation theory of singularities, is given along with examples, applications, and multiplicity formulas.
Automorphisms and isomorphisms of real Henselian fields
Ron
Brown
675-703
Abstract: Let $K$ and $L$ be ordered algebraic extensions of an ordered field $F$. Suppose $K$ and $L$ are Henselian with Archimedean real closed residue class fields. Then $K$ and $L$ are shown to be $F$-isomorphic as ordered fields if they have the same value group. Analogues to this result are proved involving orderings of higher level, unordered extensions, and, when $K$ and $L$ are maximal valued fields, transcendental extensions. As a corollary, generalized real closures at orderings of higher level are shown to be determined up to isomorphism by their value groups. The results on isomorphisms are applied to the computation of automorphism groups of $K$ and to the study of the fixed fields of groups of automorphisms of $K$. If $K$ is real closed and maximal with respect to its canonical valuation, then these fixed fields are shown to be exactly those real closed subfields of $ K$ which are topologically closed in $K$. Generalizations of this fact are proved. An example is given to illustrate several aspects of the problems considered here.
The differential operator ring of an affine curve
Jerry L.
Muhasky
705-723
Abstract: The purpose of this paper is to investigate the structure of the ring $ D(R)$ of all linear differential operators on the coordinate ring of an affine algebraic variety $X$ (possibly reducible) over a field $k$ (not necessarily algebraically closed) of characteristic zero, concentrating on the case that dim $X \leqslant 1$. In this case, it is proved that $D(R)$ is a (left and right) noetherian ring with (left and right) Krull dimension equal to dim $ X$, that the endomorphism ring of any simple (left or right) $D(R)$-module is finite dimensional over $ k$, that $D(R)$ has a unique smallest ideal $ L$ essential as a left or right ideal, and that $D(R)/L$ is finite dimensional over $ k$. The following ring-theoretic tool is developed for use in deriving the above results. Let $D$ be a subalgebra of a left noetherian $ k$-algebra $E$ such that $E$ is finitely generated as a left $D$-module and all simple left $ E$-modules have finite dimensional endomorphism rings (over $k$), and assume that $D$ contains a left ideal $I$ of $E$ such that $E/I$ has finite length. Then it is proved that $ D$ is left noetherian and that the endomorphism ring of any simple left $ D$-module is finite dimensional over $k$.
Strong homology is not additive
S.
Mardešić;
A. V.
Prasolov
725-744
Abstract: Using the continuum hypothesis (CH) we show that strong homology groups $\overline {{H_p}} (X)$ do not satisfy Milnor's additivity axiom. Moreover, CH implies that strong homology does not have compact supports and that $\overline {{H_p}} (X)$ need not vanish for $p < 0$.
Products of involution classes in infinite symmetric groups
Gadi
Moran
745-762
Abstract: Let $A$ be an infinite set. Denote by $ {S_A}$ the group of all permutations of $A$, and let ${R_i}$, denote the class of involutions of $ A$ moving $\vert A\vert$ elements and fixing $ i$ elements $(0 \leqslant i \leqslant \vert A\vert)$. The products $ {R_i}{R_j}$ were determined in [M1]. In this article we treat the products $ {R_{{i_1}}} \cdots {R_{{i_n}}}$ for $ n \geqslant 3$. Let INF denote the set of permutations in ${S_A}$ moving infinitely many elements. We show: (1) ${R_{{i_1}}} \cdots {R_{{i_n}}} = {S_A}$ for $n \geqslant 4$. (2)(a) ${R_i}{R_j}{R_k} = \operatorname{INF}$ if $\{ i,\,j,\,k\} $ contains two integers of different parity; (b) ${R_i}{R_j}{R_k} = {S_A}$ if $i + j + k > 0$ and all integers in $\{ i,\,j,\,k\}$ have the same parity. (3) $ R_0^3 = {S_A}\backslash E$, where $\theta \in E$ iff $\theta$ satisfies one of the following three conditions: (i) $\theta$ moves precisely three elements. (ii) $ \theta$ moves precisely five elements. (iii) $\theta$ moves precisely seven elements and has order $ 12$. These results were announced in 1973 in [MO]. (1) and part of (2)(a) were generalized recently by Droste [D1, D2].
Borel measures and Hausdorff distance
Gerald
Beer;
Luzviminda
Villar
763-772
Abstract: In this article we study the restriction of Borel measures defined on a metric space $X$ to the nonempty closed subsets $\operatorname{CL} (X)$ of $X$, topologized by Hausdorff distance. We show that a $\sigma$-finite Radon measure is a Borel function on $ \operatorname{CL} (X)$, and characterize those $X$ for which each outer regular Radon measure on $ X$ is semicontinuous when restricted to $ \operatorname{CL} (X)$. A number of density theorems for Radon measures are also presented.
Automorphisms of hyperbolic dynamical systems and $K\sb 2$
Frank
Zizza
773-797
Abstract: Let $\sigma :\Sigma \to \Sigma $ be a subshift of finite type and $\operatorname{Aut} (\sigma )$ be the group of homeomorphisms of $\Sigma$ which commute with $\sigma$. In [Wl], Wagoner constructs an invariant for the group $\operatorname{Aut} (\sigma )$ using $K$-theoretic methods. Smooth hyperbolic dynamical systems can be modeled by subshifts of finite type over the nonwandering sets. In this paper we extend Wagoner's construction to produce an invariant on the group of homeomorphisms of a smooth manifold which commute with a fixed hyperbolic diffeomorphism. We then proceed to show that this dynamical invariant can be calculated (at least $\bmod 2$) from the homology groups of the manifold and the action of the diffeomorphism and the homeomorphisms on the homology groups.
Cauchy problem of hyperbolic conservation laws in multidimensional space with intersecting jump initial data
De Ning
Li
799-812
Abstract: Cauchy problem of hyperbolic conservation laws in multidimensional space is considered, where the initial data have several jump discontinuity surfaces which develop into shock fronts intersecting at a common submanifold. Local existence is proved, assuming compatible conditions and uniform stability. For isentropic flow in $ 2$-dimensional space, the interaction of two shock fronts and the nonexistence of three intersecting shock fronts are discussed.
A very singular solution of a quasilinear degenerate diffusion equation with absorption
L. A.
Peletier;
Jun Yu
Wang
813-826
Abstract: The object of this paper is to study the existence of a nonnegative solution of the Cauchy problem $\displaystyle {u_t} = \operatorname{div} (\vert\nabla u{\vert^{p - 2}}\nabla u) - {u^q},\qquad u(x,\,0) = 0\quad {\text{if}}\;x \ne 0,$ which is more singular at $ (0,\,0)$ than the fundamental solution of the corresponding equation without the absorption term.
Chains on CR manifolds and Lorentz geometry
Lisa K.
Koch
827-841
Abstract: We show that two nearby points of a strictly pseudoconvex CR manifold are joined by a chain. The proof uses techniques of Lorentzian geometry via a correspondence of Fefferman. The arguments also apply to more general systems of chain-like curves on CR manifolds.
Deformations of finite-dimensional algebras and their idempotents
M.
Schaps
843-856
Abstract: Let $B$ be a finite dimensional algebra over an algebraically closed field $K$. If we represent primitive idempotents by points and basis vectors in $ {e_i}B{e_j}$ by "arrows" from ${e_j}$ to ${e_i}$, then any specialization of the algebra acts on this directed graph by coalescing points. This implies that the number of irreducible components in the scheme parametrizing $n$-dimensional algebras is no less than the number of loopless directed graphs with a total of $ n$ vertices and arrows. We also show that the condition of having a distributive ideal lattice is open.
A remark on a theorem of Vo Van Tan
Mihnea
Colţoiu
857-859
Abstract: In this paper we consider the following problem: Let $(X,\,S)$ be a $1$-convex manifold with $1$-dimensional exceptional set $ S$. Does it follow that $ X$ is a Kähler manifold? Although this was answered in the affirmative by Vo Van Tan in two papers, we show that his proofs are wrong. It is also shown that the Kähler condition implies that any strongly pseudoconvex domain $D \Subset X$ is embeddable, i.e. can be realized as a closed analytic submanifold in some $ {{\mathbf{C}}^N} \times {{\mathbf{P}}_M}$. On the other hand it is known that under some additional assumptions on $ S$ ($S$ is not rational or $S \simeq {{\mathbf{P}}^1}$ and $ \operatorname{dim} X \ne 3$) it follows that $X$ is embeddable, in particular it is Kählerian.
Calibrations on ${\bf R}\sp 8$
J.
Dadok;
R.
Harvey;
F.
Morgan
1-40
Abstract: Calibrations are a powerful tool for constructing minimal surfaces. In this paper we are concerned with $ 4$-manifolds $M \subset {{\mathbf{R}}^8}$. If a differential form $\varphi \in { \bigwedge ^4}{{\mathbf{R}}^8}$ calibrates all tangent planes of $M$ then $M$ is area minimizing. For $\varphi$ in one of several large subspaces of $ { \bigwedge ^4}{{\mathbf{R}}^8}$ we compute its comass, that is the maximal value of $\varphi$ on the Grassmannian of oriented $ 4$-planes. We then determine the set $G(\varphi ) \subset G(4,\,{{\mathbf{R}}^8})$ on which this maximum is attained. These are the planes calibrated by $\varphi$, the possible tangent planes to a manifold calibrated by $\varphi$. The families of calibrations obtained include the well-known examples: special Lagrangian, Kähler, and Cayley.
Curves of genus $2$ with split Jacobian
Robert M.
Kuhn
41-49
Abstract: We say that an algebraic curve has split jacobian if its jacobian is isogenous to a product of elliptic curves. If $ X$ is a curve of genus $ 2$, and $f:X \to E$ a map from $X$ to an elliptic curve, then $ X$ has split jacobian. It is not true that a complement to $E$ in the jacobian of $X$ is uniquely determined, but, under certain conditions, there is a canonical choice of elliptic curve $E'$ and algebraic
Sieving the positive integers by small primes
D. A.
Goldston;
Kevin S.
McCurley
51-62
Abstract: Let $Q$ be a set of primes that has relative density $\delta$ among the primes, and let $\phi (x,\,y,\,Q)$ be the number of positive integers $\leqslant x$ that have no prime factor $\leqslant y$ from the set $Q$. Standard sieve methods do not seem to give an asymptotic formula for $\phi (x,\,y,\,Q)$ in the case that $\tfrac{1}{2} \leqslant \delta < 1$. We use a method of Hildebrand to prove that $\displaystyle \phi (x,y,Q)\tilde{x}f(u)\prod\limits_{\mathop {p < y}\limits_{p \in Q} } {\left( {1 - \frac{1}{p}} \right)} $ as $x \to \infty$, where $u = \frac{{\log x}}{{\log y}}$ and $f(u)$ is defined by $\displaystyle {u^\delta }f(u) = \left\{ {\begin{array}{*{20}{c}} {\frac{{{e^{{\... ...1 + t)}^{\delta - 1}}\;dt,} } & {u > 1.} \end{array} } \right.$ This may also be viewed as a generalization of work by Buchstab and de Bruijn, who considered the case where $ Q$ consisted of all primes.
Construction of manifolds of positive scalar curvature
Rodney
Carr
63-74
Abstract: We prove that a regular neighborhood of a codimension $\geqslant 3$ subcomplex of a manifold can be chosen so that the induced metric on its boundary has positive scalar curvature. A number of useful facts concerning manifolds of positive scalar curvature follow from this construction. For example, we see that any finitely presented group can appear as the fundamental group of a compact $4$-manifold with such a metric.
First- and second-order epi-differentiability in nonlinear programming
R. T.
Rockafellar
75-108
Abstract: Problems are considered in which an objective function expressible as a max of finitely many ${C^2}$ functions, or more generally as the composition of a piecewise linear-quadratic function with a ${C^2}$ mapping, is minimized subject to finitely many ${C^2}$ constraints. The essential objective function in such a problem, which is the sum of the given objective and the indicator of the constraints, is shown to be twice epi-differentiable at any point where the active constraints (if any) satisfy the Mangasarian-Fromovitz qualification. The epi-derivatives are defined by taking epigraphical limits of classical first-and second-order difference quotients instead of pointwise limits, and they reveal properties of local geometric approximation that have not previously been observed.
Exact bounds for the stochastic upward matching problem
WanSoo T.
Rhee;
Michel
Talagrand
109-125
Abstract: We draw at random independently and according to the uniform distribution two sets of $n$ points of the unit square. We consider a maximum matching of points of the first set with points of the second set with the restriction that a point can be matched only with a point located at its upper right. Then with probability close to one, the number of unmatched points is of order ${n^{1/2}}{(\log n)^{3/4}}$.
Characteristic multipliers and stability of symmetric periodic solutions of $\dot x(t)=g(x(t-1))$
Shui-Nee
Chow;
Hans-Otto
Walther
127-142
Abstract: We study the scalar delay differential equation $\dot x(t) = g(x(t - 1))$ with negative feedback. We assume that the nonlinear function $g$ is odd and monotone. We prove that periodic solutions $x(t)$ of slowly oscillating type satisfying the symmetry condition $x(t) = - x(t - 2)$, $t \in {\mathbf{R}}$, are nondegenerate and have all nontrivial Floquet multipliers strictly inside the unit circle. This says that the periodic orbit $ \{ {x_t}:t \in {\mathbf{R}}\}$ in the phase space $ C[ - 1,\,0]$ is orbitally exponentially asymptotically stable.
Functions that preserve the uniform distribution of sequences
William
Bosch
143-152
Abstract: In this paper, necessary and sufficient conditions are given for certain functions to preserve the uniform distribution of sequences. An analytic condition allows the construction of examples. An application is also given.
A strong contractivity property for semigroups generated by differential operators
Robert M.
Kauffman
153-169
Abstract: Frequently, nonconservative semigroups generated by partial differential operators in $ {L_{2,\rho }}({R^k})$ have the property that initial conditions which are large at $\vert x\vert = \infty$ become immediately small at infinity for all $t > 0$. This property is related to the rate of decay of eigenfunctions of the differential operator. In this paper this phenomenon is investigated for a large class of differential operators of second and higher order. New estimates on the rate of decay of the eigenfunctions are included, which are related in special cases to those of Agmon.
Conditional gauge and potential theory for the Schr\"odinger operator
M.
Cranston;
E.
Fabes;
Z.
Zhao
171-194
Abstract: This paper extends the Conditional Gauge Theorem to more general operators and less regular domains than in previous works. We use this to obtain potential-theoretic results for the Schrödinger equation.
The connection map for attractor-repeller pairs
Christopher
McCord
195-203
Abstract: In the Conley index theory, the connection map of the homology attractor-repeller sequence provides a means of detecting connecting orbits between a repeller and attractor in an isolated invariant set. In this work, the connection map is shown to be additive: under suitable decompositions of the connecting orbit set, the connection map of the invariant set equals the sum of the connection maps of the decomposition elements. This refines the information provided by the homology attractor-repeller sequence. In particular, the properties of the connection map lead to a characterization of isolated invariant sets with hyperbolic critical points as an attractor-repeller pair.
Multiple fibers on rational elliptic surfaces
Brian
Harbourne;
William E.
Lang
205-223
Abstract: Our main result, Theorem (0.1), classifies multiple fibers on rational elliptic surfaces over algebraically closed fields of arbitrary characteristic. One result of this is the existence in positive characteristics of tame multiple fibers of additive type for several of the Kodaira fiber-types for which no examples were previously known.
Folds and cusps in Banach spaces with applications to nonlinear partial differential equations. II
M. S.
Berger;
P. T.
Church;
J. G.
Timourian
225-244
Abstract: Earlier the authors have given abstract properties characterizing the fold and cusp maps on Banach spaces, and these results are applied here to the study of specific nonlinear elliptic boundary value problems. Functional analysis methods are used, specifically, weak solutions in Sobolev spaces. One problem studied is the inhomogeneous nonlinear Dirichlet problem $\displaystyle \Delta u + \lambda u - {u^3} = g\quad {\text{on}}\;\Omega ,\qquad u\vert\partial \Omega = 0,$ where $\Omega \subset {{\mathbf{R}}^n}(n \leqslant 4)$ is a bounded domain. Another is a nonlinear elliptic system, the von Kármán equations for the buckling of a thin planar elastic plate when compressive forces are applied to its edge.
Isometry groups of Riemannian solvmanifolds
Carolyn S.
Gordon;
Edward N.
Wilson
245-269
Abstract: A simply connected solvable Lie group $R$ together with a left-invariant Riemannian metric $g$ is called a (simply connected) Riemannian solvmanifold. Two Riemannian solvmanifolds $(R,\,g)$ and $ (R' ,\,g' )$ may be isometric even when $R$ and $R'$ are not isomorphic. This article addresses the problems of (i) finding the "nicest" realization $ (R,\,g)$ of a given solvmanifold, (ii) describing the embedding of $R$ in the full isometry group $ I(R,\,g)$, and (iii) testing whether two given solvmanifolds are isometric. The paper also classifies all connected transitive groups of isometries of symmetric spaces of noncompact type and partially describes the transitive subgroups of $I(R,\,g)$ for arbitrary solvmanifolds $(R,\,g)$.
Regularity of solutions of two-dimensional Monge-Amp\`ere equations
Friedmar
Schulz;
Liang Yuan
Liao
271-277
Abstract: In the paper we investigate the regularity of solutions $z(x,\,y) \in {C^{1,1}}(\Omega )$, resp. ${C^{1,1}}(\overline \Omega )$ of elliptic Monge-Ampére equations of the form $\displaystyle Ar + 2Bs + Ct + (rt - {s^2}) = E.$ It is shown that $z(x,\,y) \in {C^{2,\alpha }}(\Omega )$, resp. $ {C^{2,\alpha }}(\overline \Omega )$, with corresponding a priori estimates, if $ A,\,B,\,C,\,E \in {C^\alpha }(\Omega \times {{\mathbf{R}}^3})$. The results are deduced via the Campanato technique for equations of variational structure invoking a Legendre-like transformation.
Some sharp inequalities for martingale transforms
K. P.
Choi
279-300
Abstract: Two sharp inequalities for martingale transforms are proved. These results extend some earlier work of Burkholder. The inequalities are then extended to stochastic integrals and differentially subordinate martingales.
Spinor bundles on quadrics
Giorgio
Ottaviani
301-316
Abstract: We define some stable vector bundles on the complex quadric hypersurface $ {Q_n}$ of dimension $ n$ as the natural generalization of the universal bundle and the dual of the quotient bundle on ${Q_4} \simeq \operatorname{Gr} (1,\,3)$. We call them spinor bundles. When $n = 2k - 1$ there is one spinor bundle of rank ${2^{k - 1}}$. When $n = 2k$ there are two spinor bundles of rank ${2^{k - 1}}$. Their behavior is slightly different according as $ n \equiv 0\;(\bmod 4)$ or $ n \equiv 2\;(\bmod 4)$. As an application, we describe some moduli spaces of rank $ 3$ vector bundles on $ {Q_5}$ and ${Q_6}$.
Constant isotropic submanifolds with $4$-planar geodesics
Jin Suk
Pak;
Kunio
Sakamoto
317-333
Abstract: Let $f$ be an isometric immersion of a Riemannian manifold $M$ into $ \overline M$. We prove that if $f$ is constant isotropic, $4$-planar geodesic and $\overline M$ is a Euclidean sphere, then $ M$ is isometric to one of compact symmetric spaces of rank equal to one and $ f$ is congruent to a direct sum of standard minimal immersions. We also determine constant isotropic, $4$-planar geodesic, totally real immersions into a complex projective space of constant holomorphic sectional curvature.
Pseudodifferential operators with coefficients in Sobolev spaces
Jürgen
Marschall
335-361
Abstract: Pseudo-differential operators with coefficients in Sobolev spaces $ {H^{r,q}},1 \leqslant q \leqslant \infty$, and their adjoints are studied on Hardy-Sobolev spaces ${H^{s,p}},\;0 < p \leqslant \infty$. A symbolic calculus for these operators is developed, and the microlocal properties are studied. Finally, the invariance under coordinate transformations is proved.
Harmonically immersed surfaces of ${\bf R}\sp n$
Gary R.
Jensen;
Marco
Rigoli
363-372
Abstract: Some generalizations of classical results in the theory of minimal surfaces $ f:M \to {{\mathbf{R}}^n}$ are shown to hold in the more general case of harmonically immersed surfaces.
Bonnesen-style inequalities for Minkowski relative geometry
J. R.
Sangwine-Yager
373-382
Abstract: Two Bonnesen-style inequalities are obtained for the relative inradius of one convex body with respect to another in $ n$-dimensional space. Both reduce to the known planar inequality; one sharpens the relative isoperimetric inequality, the other states that a quadratic polynomial is negative at the inradius. Circumradius inequalities follow.
Homogeneous continua in Euclidean $(n+1)$-space which contain an $n$-cube are locally connected
Janusz R.
Prajs
383-394
Abstract: We prove that each homogeneous continuum which topologically contains an $ n$-dimensional unit cube and lies in $(n + 1)$-dimensional Euclidean space is locally connected.
Embedding graphs into colored graphs
A.
Hajnal;
P.
Komjáth
395-409
Abstract: If $X$ is a graph, $\kappa$ a cardinal, then there is a graph $ Y$ such that if the vertex set of $Y$ is $\kappa$-colored, then there exists a monocolored induced copy of $X$; moreover, if $X$ does not contain a complete graph on $ \alpha$ vertices, neither does $Y$. This may not be true, if we exclude noncomplete graphs as subgraphs. It is consistent that there exists a graph $X$ such that for every graph $Y$ there is a two-coloring of the edges of $ Y$ such that there is no monocolored induced copy of $X$. Similarly, a triangle-free $X$ may exist such that every $ Y$ must contain an infinite complete graph, assuming that coloring $Y$'s edges with countably many colors a monocolored copy of $X$ always exists.
Coordinatization in superstable theories. II
Steven
Buechler
411-417
Abstract: In this paper we prove Theorem A. Suppose that $ T$ is superstable and $ U(a/A) = \alpha + 1$, for some $\alpha$. Then in ${T^{{\text{eq}}}}$ there is a $c \in \operatorname{acl} (Aa)\backslash \operatorname{acl} (A)$ such that one of the following holds. (i) $ U(c/A) = 1$. (ii) $ \operatorname{stp} (c/A)$ has finite Morley rank. In fact, this strong type is semiminimal with respect to a strongly minimal set.
Area-minimizing integral currents with boundaries invariant under polar actions
Julian C.
Lander
419-429
Abstract: Let $G$ be a compact, connected subgroup of $SO(n)$ acting on $ {{\mathbf{R}}^n}$, and let the action of $G$ be polar. (Polar actions include the adjoint action of a Lie group $H$ on the tangent space to the symmetric space $ G/H$ at the identity coset.) Let $B$ be an $(m - 1)$-dimensional submanifold without boundary, invariant under the action of $G$, and lying in the union of the principal orbits of $G$. It is shown that, if $S$ is an area-minimizing integral current with boundary $B$, then $S$ is invariant under the action of $G$. This result is extended to a larger class of boundaries, and to a class of parametric integrands including the area integrand.
Errata to: ``Nonsingular quadratic differential equations in the plane'' [Trans. Amer. Math. Soc. {\bf 301} (1987), no. 2, 845--859; MR0882718 (88b:58109)]
M. I. T.
Camacho;
C. F. B.
Palmeira
431
$k$-flat structures and exotic characteristic classes
Lisa R.
Goldberg
433-453
Abstract: We generalize the concept of "foliation" and define $k$-flat structures; these are smooth vector bundles with affine connections whose characteristic forms vanish above a certain dimension. Using semisimplicial techniques we construct a classifying space for $k$-flat structures, and prove a classification theorem for these structures on smooth manifolds. Techniques from rational homotopy theory are used to relate the exotic characteristic classes of foliations to the rational homotopy groups and cohomology of the classifying space.
On the theory of biorthogonal polynomials
A.
Iserles;
S. P.
Nørsett
455-474
Abstract: Let $\varphi (x,\,\mu )$ be a distribution in $x \in {\mathbf{R}}$ for every $\mu$ in a real parameter set $ \Omega$. Subject to additional technical conditions, we study $m$th degree monic polynomials $ {p_m}$ that satisfy the biorthogonality conditions $\displaystyle \int_{ - \infty }^\infty {{p_m}(x)\,d\varphi (x,{\mu _l}) = 0,} \qquad l = 1,\,2, \ldots ,\,m,\;m \geqslant 1$ , for a distinct sequence ${\mu _1},\,{\mu _2},\, \ldots \; \in \Omega \,$. Necessary and sufficient conditions for existence and uniqueness are established, as well as explicit determinantal and integral representations. We also consider loci of zeros, existence of Rodrigues-type formulae and reducibility to standard orthogonality. The paper is accompanied by several examples of biorthogonal systems.
Classifying $1$-handles attached to knotted surfaces
Jeffrey
Boyle
475-487
Abstract: We study a method of obtaining knotted surfaces in the $4$-sphere ${S^4}$ by attaching embedded $ 2$-dimensional $ 1$-handles to a given knot. The main result is there is a one-to-one correspondence between the $1$-handles that can be attached to a knot and the double cosets of the peripheral subgroup in the group of the knot. Many examples and applications are given.
Geodesics and conformal transformations of Heisenberg-Reiter spaces
J. F.
Torres Lopera
489-498
Abstract: Generalized Heisenberg groups, in the sense of Reiter, can be endowed with left-invariant metrics whose geodesies and curvature are obtained. Using these curvature data it is also proved that on their nilmanifolds (compact or not), every conformal transformation is in fact an isometry. A large family of nonisometric examples is given.
Topological equivalence of flows on homogeneous spaces, and divergence of one-parameter subgroups of Lie groups
Diego
Benardete
499-527
Abstract: Let $\Gamma$ and $\Gamma '$ be lattices, and $\phi$ and $\phi '$ one-parameter subgroups of the connected Lie groups $G$ and $G'$. If one of the following conditions (a), (b), or (c) hold, Theorem A states that if the induced flows on the homogeneous spaces $G/\Gamma$ and $G' /\Gamma '$ are topologically equivalent, then they are topologically equivalent by an affine map. (a) $G$ and $G'$ are one-connected and nilpotent. (b) $ G$ and $G'$ are one-connected and solvable, and for all $X$ in $L(G)$ and $X'$ in $L(G' )$, $ \operatorname{ad} (x)$ and
On certain fibred ribbon disc pairs
Iain R.
Aitchison;
Daniel S.
Silver
529-551
Abstract: We prove that for any free group automorphism ${\phi ^{\ast}}$ having a specified form there exists an invertible ribbon disc pair $({B^4},\,{D^2})$ such that the closure of ${B_4} - \operatorname{nbd} ({D^2})$ fibres over the circle with fibre a handlebody and monodromy equal to ${\phi ^{\ast}}$. We apply this to obtain results about ribbon $1$- and $2$-knots.
Classification of continuous $JBW\sp *$-triples
G.
Horn;
E.
Neher
553-578
Abstract: We show that every $JB{W^{\ast}}$-triple without a direct summand of type I is isometrically isomorphic to an ${l^\infty }$-sum $\mathcal{R}{ \otimes ^\infty }H(A,\,\alpha )$ where $\mathcal{R}$ is a ${w^{\ast}}$-closed right ideal in a ${W^{\ast}}$-algebra $B$ and $ H(A,\,\alpha )$ are the elements of a $ {W^{\ast}}$-algebra $ A$ which are symmetric under a C-linear involution $\alpha$ of $A$. Both $A$ and $B$ do not have a direct ( $ {W^{\ast}}$-algebra) summand of type I. In order to refine the decomposition $\mathcal{R}{ \otimes ^\infty }H(A,\,\alpha )$ we define and characterize types of $ JB{W^{\ast}}$-triples.
The Casson-Gordon invariants in high-dimensional knot theory
Daniel
Ruberman
579-595
Abstract: The Casson-Gordon invariants of knots in all dimensions are interpreted in terms of surgery theory. Applications are given to finding non-doubly slice knots, and doubly slice knots which are not the double of a disk knot. In even dimensions, the property of being doubly slice is shown to be largely homotopy theoretic, while in odd dimensions the surgery-theoretic method shows such properties to depend on more than the homotopy type.
Alexander modules of links with all linking numbers zero
M. L.
Platt
597-605
Abstract: In this paper we characterize the Alexander modules of links resulting from a surgical modification on the trivial link of any number of components. Using the presentation matrix obtained, we derive some properties of the Alexander polynomials of such links.
Exceptional boundary sets for solutions of parabolic partial differential inequalities
G. N.
Hile;
R. Z.
Yeh
607-621
Abstract: Let $\mathcal{M}$ be a second order, linear, parabolic partial differential operator with coefficients defined in a domain $\mathcal{D} = \Omega \times (0,\,T)$ in $ {{\mathbf{R}}^n} \times {\mathbf{R}}$, with $\Omega$ a domain in $ {{\mathbf{R}}^n}$. Let $ u$ be a suitably regular real function in $ \mathcal{D}$ such that $ u$ is bounded below and $\mathcal{M}u$ is bounded above in $\mathcal{D}$. If $ u \geqslant 0$ on $\Omega \times \{ 0\}$ except on a set $\Gamma \times \{ 0\}$, with $\Gamma$ a subset of $\Omega$ of suitably restricted Hausdorff dimension, then necessarily $u \geqslant 0$ also on $\Gamma \times \{ 0\}$. The allowable Hausdorff dimension of $\Gamma$ depends on the coefficients of $\mathcal{M}$. For example, if $\mathcal{M}$ is the heat operator $\Delta - \partial /\partial t$, the Hausdorff dimension of $\Gamma$ needs to be smaller than the number of space dimensions $n$. Analogous results are valid for exceptional boundary sets on the lateral boundary, $\partial \Omega \times (0,\,T)$, of $\mathcal{D}$.
Flows on vector bundles and hyperbolic sets
Dietmar
Salamon;
Eduard
Zehnder
623-649
Abstract: This note deals with C. Conley's topological approach to hyperbolic invariant sets for continuous flows. It is based on the notions of isolated invariant sets and Morse decompositions and it leads to the concept of weak hyperbolicity.
On the vanishing of homology and cohomology groups of associative algebras
Rolf
Farnsteiner
651-665
Abstract: This paper establishes sufficient conditions for the vanishing of the homology and cohomology groups of an associative algebra with coefficients in a two-sided module.
Tensor products of composition algebras, Albert forms and some exceptional simple Lie algebras
B. N.
Allison
667-695
Abstract: In this paper, we study algebras with involution that are isomorphic after base field extension to the tensor product of two composition algebras. To any such algebra $(\mathcal{A},\, - )$, we associate a quadratic form $Q$ called the Albert form of $(\mathcal{A},\, - )$. The Albert form is used to give necessary and sufficient conditions for two such algebras to be isotopic. Using a Lie algebra construction of Kantor, we are then able to give a description of the isomorphism classes of Lie algebras of index $F_{4,1}^{21}$, $ {}^2E_{6,1}^{29}$, $E_{7,1}^{48}$ and $ E_{8,1}^{91}$. That description is used to obtain a classification of the indicated Lie algebras over $ {\mathbf{R}}(({T_1}, \ldots ,{T_n})),\;n \leqslant 3$.
Outer functions in function algebras on the bidisc
Håkan
Hedenmalm
697-714
Abstract: Let $f$ be a function in the bidisc algebra $ A({{\mathbf{D}}^2})$ whose zero set $Z(f)$ is contained in $\{ 1\} \times \overline {\mathbf{D}}$. We show that the closure of the ideal generated by $f$ coincides with the ideal of functions vanishing on $Z(f)$ if and only if $f( \cdot ,\,\alpha )$ is an outer function for all $ \alpha \in \overline {\mathbf{D}}$, and $ f(1,\, \cdot )$ either vanishes identically or is an outer function. Similar results are obtained for a few other function algebras on $ {{\mathbf{D}}^2}$ as well.
Riemannian $4$-symmetric spaces
J. A.
Jiménez
715-734
Abstract: The main purpose of this paper is to classify the compact simply connected Riemannian $4$-symmetric spaces. As homogeneous manifolds, these spaces are of the form $G/L$ where $G$ is a connected compact semisimple Lie group with an automorphism $\sigma$ of order four whose fixed point set is (essentially) $L$. Geometrically, they can be regarded as fiber bundles over Riemannian $2$-symmetric spaces with totally geodesic fibers isometric to a Riemannian $2$-symmetric space. A detailed description of these fibrations is also given. A compact simply connected Riemannian $4$-symmetric space decomposes as a product $ {M_1} \times \ldots \times {M_r}$ where each irreducible factor is: (i) a Riemannian $2$-symmetric space, (ii) a space of the form $ \{ U \times U \times U \times U\} /\Delta U$ with $U$ a compact simply connected simple Lie group, $\Delta U =$ diagonal inclusion of $U$, (iii) $\{ U \times U\} /\Delta {U^\theta }$ with $ U$ as in (ii) and ${U^\theta }$ the fixed point set of an involution $ \theta$ of $U$, and (iv) $U/K$ with $U$ as in (ii) and $K$ the fixed point set of an automorphism of order four of $U$. The core of the paper is the classification of the spaces in (iv). This is accomplished by first classifying the pairs $(\mathfrak{g},\,\sigma )$ with $\mathfrak{g}$ a compact simple Lie algebra and $ \sigma$ an automorphism of order four of $ \mathfrak{g}$. Tables are drawn listing all the possibilities for both the Lie algebras and the corresponding spaces. For $ U$ "classical," the automorphisms $\sigma$ are explicitly constructed using their matrix representations. The idea of duality for $ 2$-symmetric spaces is extended to $4$-symmetric spaces and the duals are determined. Finally, those spaces that admit invariant almost complex structures are also determined: they are the spaces whose factors belong to the class (iv) with $ K$ the centralizer of a torus.
Construction of cohomology of discrete groups
Y. L.
Tong;
S. P.
Wang
735-763
Abstract: A correspondence between Hermitian modular forms and vector valued harmonic forms in locally symmetric spaces associated to $U(p,\,q)$ is constructed and also shown in general to be nonzero. The construction utilizes Rallis-Schiffmann type theta functions and simplified arguments to circumvent differential geometric calculations used previously in related problems.
Derivatives of meromorphic functions of finite order
Werner P.
Kohs;
Jack
Williamson
765-772
Abstract: Let $F$ be a nonentire, meromorphic function of finite order with only real zeros and real poles such that $F'$ has no zeros. We classify all such real $ F$ and all such strictly nonreal $F$ whose poles are of bounded multiplicities. We also give examples of such $F$ which are strictly nonreal and whose poles are of unbounded multiplicities.
Primeness and sums of tangles
Mario
Eudave Muñoz
773-790
Abstract: We consider knots and links obtained by summing a rational tangle and a prime tangle. For a given prime tangle, we show that there are at most three rational tangles that will induce a composite or splittable link. In fact, we show that there is at most one rational tangle that will give a splittable link. These results extend Scharlemann's work.
Representations of anisotropic unitary groups
Donald G.
James
791-804
Abstract: Let $SU(f)$ be the special unitary group of an anisotropic hermitian form $f$ over a field $k$. Assume $f$ represents only one norm class in $k$. The representations $ \alpha :\,SU(f) \to SL(n,\,R)$ are characterized when $R$ is a commutative ring with $2$ not a zero divisor and $n = \dim f \geqslant 3$ with $n \ne 4,\,6$. In particular, a partial classification of the normal subgroups of $SU(f)$ is given when $k$ is the function field ${\mathbf{C}}(X)$.
Finite-codimensional invariant subspaces of Bergman spaces
Sheldon
Axler;
Paul
Bourdon
805-817
Abstract: For a large class of bounded domains in $ \mathbb{C}$, we describe those finite codimensional subspaces of the Bergman space that are invariant under multiplication by $ z$. Using different techniques for certain domains in ${\mathbb{C}^N}$, we describe those finite codimensional subspaces of the Bergman space that are invariant under multiplication by all the coordinate functions.
General gauge theorem for multiplicative functionals
K. L.
Chung;
K. M.
Rao
819-836
Abstract: We generalize our previous work on the gauge theorem and its various consequences and complements, initiated in [8] and somewhat extended by subsequent investigations (see [6]). The generalization here is two-fold. First, instead of the Brownian motion, the underlying process is now a fairly broad class of Markov processes, not necessarily having continuous paths. Second, instead of the Feynman-Kac functional, the exponential of a general class of additive functionals is treated. The case of Schrödinger operator $\Delta /2 + \nu $, where $\nu$ is a suitable measure, is a simple special case. The most general operator, not necessarily a differential one, which may arise from our potential equations is briefly discussed toward the end of the paper. Concrete instances of applications in this case should be of great interest.
Tauberian theorems and stability of one-parameter semigroups
W.
Arendt;
C. J. K.
Batty
837-852
Abstract: The main result is the following stability theorem: Let $ \mathcal{T} = {(T(t))_{t \geqslant 0}}$ be a bounded ${C_0}$-semigroup on a reflexive space $ X$. Denote by $ A$ the generator of $\mathcal{T}$ and by $ \sigma (A)$ the spectrum of $A$. If $ \sigma (A) \cap i{\mathbf{R}}$ is countable and no eigenvalue of $A$ lies on the imaginary axis, then $ {\lim _{t \to \infty }}T(t)x = 0$ for all $x \in X$.
A class of nonlinear Sturm-Liouville problems with infinitely many solutions
Renate
Schaaf;
Klaus
Schmitt
853-859
Abstract: This paper is concerned with the existence of solutions of nonlinear Sturm-Liouville problems whose linear part is at resonance. It is shown that such problems may have infinitely many solutions if the nonlinear perturbations are periodic.
Tiling the projective foliation space of a punctured surface
Lee
Mosher
1-70
Abstract: There is a natural way to associate, to each ideal triangulation of a punctured surface a cell decomposition of the projective foliation space of the punctured surface.
The automorphism group of a shift of finite type
Mike
Boyle;
Douglas
Lind;
Daniel
Rudolph
71-114
Abstract: Let $({X_T},{\sigma _T})$ be a shift of finite type, and $G = \operatorname{aut} ({\sigma _T})$ denote the group of homeomorphisms of ${X_T}$ commuting with $ {\sigma _T}$. We investigate the algebraic properties of the countable group $ G$ and the dynamics of its action on ${X_T}$ and associated spaces. Using "marker" constructions, we show $G$ contains many groups, such as the free group on two generators. However, $G$ is residually finite, so does not contain divisible groups or the infinite symmetric group. The doubly exponential growth rate of the number of automorphisms depending on $n$ coordinates leads to a new and nontrivial topological invariant of $ {\sigma _T}$ whose exact value is not known. We prove that, modulo a few points of low period, $G$ acts transitively on the set of points with least ${\sigma _T}$-period $n$. Using $p$-adic analysis, we generalize to most finite type shifts a result of Boyle and Krieger that the gyration function of a full shift has infinite order. The action of $G$ on the dimension group of ${\sigma _T}$ is investigated. We show there are no proper infinite compact $G$-invariant sets. We give a complete characterization of the $G$-orbit closure of a continuous probability measure, and deduce that the only continuous $G$-invariant measure is that of maximal entropy. Examples, questions, and problems complement our analysis, and we conclude with a brief survey of some remaining open problems.
Infinitesimally rigid polyhedra. II. Modified spherical frameworks
Walter
Whiteley
115-139
Abstract: In the first paper, Alexandrov's Theorem was studied, and extended, to show that convex polyhedra form statically rigid frameworks in space, when built with plane-rigid faces. This second paper studies two modifications of these polyhedral frameworks: (i) block polyhedral frameworks, with some discs as open holes, other discs as space-rigid blocks, and the remaining faces plane-rigid; and (ii) extended polyhedral frameworks, with individually added bars (shafts) and selected edges removed. Inductive methods are developed to show the static rigidity of particular patterns of holes and blocks and of extensions, in general realizations of the polyhedron. The methods are based on proof techniques for Steinitz's Theorem, and a related coordinatization of the proper realizations of a $3$-connected spherical polyhedron. Sample results show that: (a) a single $k$-gonal block and a $k$-gonal hole yield static rigidity if and only if the block and hole are $k$-connected in a vertex sense; and (b) a $ 4$-connected triangulated sphere, with one added bar, is a statically rigid circuit (removing any one bar leaves a minimal statically rigid framework). The results are also interpreted as a description of which dihedral angles in a triangulated sphere will flex when one bar is removed.
On the second fundamental theorem of Nevanlinna
Arturo
Fernández Arias
141-163
Abstract: It is shown that a condition on the size of the exceptional set in the second fundamental theorem of Nevanlinna cannot be improved. The method is based on a construction of Hayman and also makes use of a quantitative version of a result of F. Nevanlinna about the growth of the characteristic function of a meromorophic function omitting a finite number of points
Weyl groups and the regularity properties of certain compact Lie group actions
Eldar
Straume
165-190
Abstract: The geometric weight system of a $G$-manifold $X$ (acyclic or spherical) is the nonlinear analogue of the weight system of a linear representation. We study the possible realization of a given $G$-weight pattern, via the interaction between roots, weights and the Weyl group, together with various fixed point results of P. A. Smith type. If the orbit structure is reasonably simple, then the $ G$-weight pattern must in fact coincide with that of a simple representation. This in turn implies that $X$ is (orthogonally) modeled on the linear $ G$-space, e.g., with the same orbit types. In particular, complete results in this direction are obtained for a certain family of $G$-manifolds, $G$ a classical group. In this family the weight patterns are of $2$-parametric type, and it includes essentially all cases where the principal isotropy type is nontrivial. This family also covers many cases with trivial principal isotropy type.
Estimates for $(\overline\partial-\mu\partial)\sp {-1}$ and Calder\'on's theorem on the Cauchy integral
Stephen W.
Semmes
191-232
Abstract: One can view the Cauchy integral operator as giving the solution to a certain $\overline \partial$ problem. If one has a quasiconformal mapping on the plane that takes the real line to the curve, then this $\bar \partial$ problem on the curve can be pulled back to a $ \bar \partial - \mu \partial$ problem on the line. In the case of Lipschitz graphs (or chordarc curves) with small constant, we show how a judicial choice of q.c. mapping and suitable estimates for $ \bar \partial - \mu \partial$ gives a new approach to the boundedness of the Cauchy integral. This approach has the advantage that it is better suited to related problems concerning ${H^\infty }$ than the usual singular integral methods. Also, these estimates for the Beltrami equation have application to quasiconformal and conformal mappings, taken up in a companion paper.
Quasiconformal mappings and chord-arc curves
Stephen W.
Semmes
233-263
Abstract: Given a quasiconformal mapping $\rho$ on the plane, what conditions on its dilatation $ \mu$ guarantee that $\rho ({\mathbf{R}})$ is rectifiable and $ \rho {\vert _{\mathbf{R}}}$ is locally absolutely continuous? We show in this paper that if $\mu$ satisfies certain quadratic Carleson measure conditions, with small norm, then $\rho ({\mathbf{R}})$ is a chord-arc curve with small constant, and $\rho (x) = \rho (0) + \int_0^x {{e^{a(t)}}dt}$ for $ x \in {\mathbf{R}}$, with $ a \in \operatorname{BMO}$ having small norm. Conversely, given any such map from ${\mathbf{R}} \to {\mathbf{C}}$, we show that it has an extension to $ {\mathbf{C}}$ with the right kind of dilatation. Similar results hold with ${\mathbf{R}}$ replaced by a chord-arc curve. Examples are given that show that it would be hard to improve these results. Applications are given to conformal welding and the theorem of Coifman and Meyer on the real analyticity of the Riemann mapping on the manifold of chord-arc curves.
Elliptic and parabolic BMO and Harnack's inequality
Hugo
Aimar
265-276
Abstract: We give a generalization of the John-Nirenberg lemma which can be applied to prove ${A_2}$ type conditions for small powers of positive solutions of elliptic and parabolic, degenerate and nondegenerate operators.
A Brouwer translation theorem for free homeomorphisms
Edward E.
Slaminka
277-291
Abstract: We prove a generalization of the Brouwer Translation Theorem which applies to a class of homeomorphisms (free homeomorphisms) which admit fixed points, but retain a dynamical property of fixed point free orientation preserving homeomorphsims. That is, if $h:{M^2} \to {M^2}$ is a free homeomorphism where $ {M^2}$ is a surface, then whenever $D$ is a disc and $h(D) \cap D = \emptyset $, we have that $ {h^n}(D) \cap D = \emptyset$ for all $n \ne 0$. Theorem. Let $h$ be a free homeomorphism of $ {S^2}$, the two-sphere, with finite fixed point set $F$. Then each $p \in {S^2} - F$ lies in the image of an embedding ${\phi _p}:({R^2},\,0) \to ({S^2} - F,\,p)$ such that: (i) $h{\phi _p} = {\phi _p}\tau $, where $\tau (z) = z + 1$ is the canonical translation of the plane, and (ii) the image of each vertical line under ${\phi _p}$ is closed in ${S^2} - F$.
There is no exactly $k$-to-$1$ function from any continuum onto $[0,1]$, or any dendrite, with only finitely many discontinuities
Jo W.
Heath
293-305
Abstract: Katsuura and Kellum recently proved [8] that any (exactly) $ k$-to$1$ function from $[0,\,1]$ onto $[0,\,1]$ must have infinitely many discontinuities, and they asked if the theorem remains true if the domain is any (compact metric) continuum. The result in this paper, that any (exactly) $k$-to-$1$ function from a continuum onto any dendrite has finitely many discontinuities, answers their question in the affirmative.
A truncated Gauss-Kuz\cprime min law
Doug
Hensley
307-327
Abstract: The transformations $ {T_n}$ which map $x \in [0,\,1)$ onto 0 (if $x \leqslant 1/(n + 1)$), and to $\{ 1/x\}$ otherwise, are truncated versions of the continued fraction transformation $T:x \to \{ 1/x\}$ (but $0 \to 0$). An analog to the Gauss-Kuzmin result is obtained for these ${T_n}$, and is used to show that the Lebesgue measure of $T_n^{ - k}\{ 0\} $ approaches $ 1$ exponentially. From this fact is obtained a new proof that the ratios $ \nu /k$, where $ \nu$ denotes any solution of $ {\nu ^2} \equiv - 1\bmod k$, are uniformly distributed $\bmod 1$ in the sense of Weyl.
The first case of Fermat's last theorem is true for all prime exponents up to $714,591,416,091,389$
Andrew
Granville;
Michael B.
Monagan
329-359
Abstract: We show that if the first case of Fermat's Last Theorem is false for prime exponent $p$ then ${p^2}$ divides ${q^p} - q$ for all primes $q \leqslant 8q$. As a corollary we state the theorem of the title.
An approach to homotopy classification of links
J. P.
Levine
361-387
Abstract: A reformulation and refinement of the $ \overline \mu$-invariants of Milnor are used to give a homotopy classification of $ 4$ component links and suggest a possible general homotopy classification. The main idea is to use the (reduced) group of a link and its "geometric" automorphisms to define the precise indeterminacy of these invariants.
The cohomology representation of an action of $C\sb p$ on a surface
Peter
Symonds
389-400
Abstract: When a finite group $ G$ acts on a surface $ S$, then ${H^1}(S;\,{\mathbf{Z}})$ posseses naturally the structure of a $ {\mathbf{Z}}G$-module with invariant symplectic inner product. In the case of a cyclic group of odd prime order we describe explicitly this symplectic inner product space in terms of the fixed-point data of the action.
Testing analyticity on rotation invariant families of curves
Josip
Globevnik
401-410
Abstract: Let $\Gamma \subset C$ be a piecewise smooth Jordan curve, symmetric with respect to the real axis, which contains the origin in its interior and which is not a circle centered at the origin. Let $\Omega$ be the annulus obtained by rotating $ \Gamma$ around the origin. We characterize the curves $\Gamma$ with the property that if $f \in C(\Omega )$ is analytic on $s\Gamma$ for every $s$, $\vert s\vert = 1$, then $f$ is analytic in Int $\Omega$.
Fixed points of arc-component-preserving maps
Charles L.
Hagopian
411-420
Abstract: The following classical problem remains unsolved: If $ M$ is a plane continuum that does not separate the plane and $f$ is a map of $M$ into $M$, must $f$ have a fixed point? We prove that the answer is yes if $f$ maps each arc-component of $ M$ into itself. Since every deformation of a space preserves its arc-components, this result establishes the fixed-point property for deformations of nonseparating plane continua. It also generalizes the author's theorem [10] that every arcwise connected nonseparating plane continuum has the fixed-point property. Our proof shows that every arc-component-preserving map of an indecomposable plane continuum has a fixed point. We also prove that every tree-like continuum that does not contain uncountably many disjoint triods has the fixed-point property for arc-component-preserving maps.
Fonctions sph\'eriques des espaces sym\'etriques compacts
Jean-Louis
Clerc
421-431
Abstract: An integral formula, similar to Harish-Chandra's formula for spherical functions on a noncompact Riemannian symmetric space $G/K$ is given for the spherical functions of the compact dual $U/K$. As a consequence, an asymptotic expansion, as the parameter tends to infinity, is obtained, by using the (complex) stationary phase method. RÉSUMÉ. On démontre une formule intégrale pour les fonctions sphériques d'un espace symétrique de type compact $U/K$, analogue de la formule d'Harish-Chandra pour le dual non-compact $G/K$. En conséquence on obtient un équivalent asymptotique lorsque le paramètre tend vers l'infini, en utilisant la méthode de la phase stationnaire complexe.
Operator methods and Lagrange inversion: a unified approach to Lagrange formulas
Ch.
Krattenthaler
431-465
Abstract: We present a general method of proving Lagrange inversion formulas and give new proofs of the $s$-variable Lagrange-Good formula [13] and the $q$-Lagrange formulas of Garsia [7], Gessel [10], Gessel and Stanton [11, 12] and the author [18]. We also give some $q$-analogues of the Lagrange formula in several variables.
Paracommutators---boundedness and Schatten-von Neumann properties
Svante
Janson;
Jaak
Peetre
467-504
Abstract: A very general class of operators, acting on functions in $ {L^2}({{\mathbf{R}}^d})$, is introduced. The name "paracommutator" has been chosen because of the similarity with the paramultiplication of Bony and also because paracommutators comprise as a special case commutators of Calderón-Zygmund operators, as well as many other interesting examples (Hankel and Toeplitz operators etc.). The main results, extending previous results by Peller and others, express boundedness and Schatten-von Neumann properties of a paracommutator in terms of its symbol.
On the mean value property of harmonic functions and best harmonic $L\sp 1$-approximation
Myron
Goldstein;
Werner
Haussmann;
Lothar
Rogge
505-515
Abstract: The present paper deals with the inverse mean value property of harmonic functions and with the existence, uniqueness, and characterization of a best harmonic $ {L^1}$-approximant to strictly subharmonic functions. The main theorem concerning the inverse mean value property of harmonic functions is based on a generalization of a theorem due to Ü. Kuran as well as on an approximation theorem proved by J. C. Polking and also by L. I. Hedberg. The inverse mean value property will be applied in order to prove necessary and sufficient conditions for the existence of a best harmonic $ {L^1}$-approximant to a subharmonic function $s$ satisfying $ \Delta s > 0$ a.e. in the unit ball.
Minimal $K$-types for $G\sb 2$ over a $p$-adic field
Allen
Moy
517-529
Abstract: We single out certain representations of compact open subgroups of $ {G_2}$ over a $ p$-adic field and show they play a role in the representation theory of $ {G_2}$ similar to minimal $ K$-types in the theory of real groups.
Measured laminations in $3$-manifolds
Ulrich
Oertel
531-573
Abstract: An essential measured lamination embedded in an irreducible, orientable $ 3$-manifold $M$ is a codimension $1$ lamination with a transverse measure, carried by an incompressible branched surface satisfying further technical conditions. Weighted incompressible surfaces are examples of essential measured laminations, and the inclusion of a leaf of an essential measured lamination into $M$ is injective on ${\pi _1}$. There is a space $ \mathcal{P}\mathcal{L}(M)$ whose points are projective classes of essential measured laminations. Projective classes of weighted incompressible surfaces are dense in $ \mathcal{P}\mathcal{L}(M)$. The space $ \mathcal{P}\mathcal{L}(M)$ is contained in a finite union of cells (of different dimensions) embedded in an infinite-dimensional projective space, and contains the interiors of these cells. Most of the properties of the incompressible branched surfaces carrying measured laminations are preserved under the operations of splitting or passing to sub-branched surfaces.
The space of incompressible surfaces in a $2$-bridge link complement
W.
Floyd;
A.
Hatcher
575-599
Abstract: Projective lamination spaces for $2$-bridge link complements are computed explicitly.
Complex algebraic geometry and calculation of multiplicities for induced representations of nilpotent Lie groups
L.
Corwin;
F. P.
Greenleaf
601-622
Abstract: Let $G$ be a connected, simply connected nilpotent Lie group, $H$ a Lie subgroup, and $\sigma$ an irreducible unitary representation of $ H$. In a previous paper, the authors and G. Grelaud gave an explicit direct integral decomposition (with multiplicities) of $ \operatorname{Ind} (H \uparrow G,\,\sigma )$. One consequence of that work was that the multiplicity function was either a.e. infinite or a.e. bounded. In this paper, it is proved that if the multiplicity function is bounded, its parity is a.e. constant. The proof is algebraic-geometric in nature and amounts to an extension of the familiar fact that for almost all polynomials over $ R$ of fixed degree, the parity of the number of roots is a.e. constant. One consequence of the methods is that if $G$ is a complex nilpotent Lie group and $ H$ a complex Lie subgroup, then the multiplicity is a.e. constant.
Arens regularity of the algebra $A\hat\otimes B$
A.
Ülger
623-639
Abstract: Let $A$ and $B$ be two Banach algebras. On the projective tensor product $ A\hat \otimes \,B$ of $ A$ and $B$ there exists a natural algebra structure. In this note we study Arens regularity of the Banach algebra $ A\hat \otimes \,B$.
On Aitchison's construction by isotopy
Daniel
Silver
641-652
Abstract: We describe a method introduced by I. Aitchison for constructing doubly slice fibered $n$-knots. We prove that all high-dimensional simple doubly slice fibered $n$-knots can be obtained by this construction. (Even-dimensional $n$-knots are required to be $Z$-torsion-free.) We also show that any possible rational Seifert form can be realized by a doubly slice fibered classical knot.
The continuous $(\alpha, \beta)$-Jacobi transform and its inverse when $\alpha+\beta+1$ is a positive integer
G. G.
Walter;
A. I.
Zayed
653-664
Abstract: The continuous $(\alpha ,\,\beta )$-Jacobi transform is introduced as an extension of the discrete Jacobi transform by replacing the polynomial kernel by a continuous one. An inverse transform is found for both the standard and a modified normalization and applied to a version of the sampling theorem. An orthogonal system forming a basis for the range is shown to have some unusual properties, and is used to obtain the inverse.
Random perturbations of reaction-diffusion equations: the quasideterministic approximation
Mark I.
Freidlin
665-697
Abstract: Random fields ${u^\varepsilon }(t,\,x) = (u_1^\varepsilon (t,\,x), \ldots ,u_n^\varepsilon (t,\,x))$, defined as the solutions of a system of the PDE due. $\displaystyle \frac{{\partial u_k^\varepsilon }} {{\partial t}} = {L_k}u_k^\var... ...x;\,u_1^\varepsilon , \ldots ,u_n^\varepsilon ) + \varepsilon {\zeta _k}(t,\,x)$ are considered. Here ${L_k}$ are second-order linear elliptic operators, ${\zeta _k}$ are Gaussian white-noise fields, independent for different $k$, and $ \varepsilon$ is a small parameter. The most attention is given to the problem of determining the behavior of the invariant measure ${\mu ^\varepsilon }$ of the Markov process $u_t^\varepsilon = (u_1^\varepsilon (t,\, \cdot ), \ldots ,u_n^\varepsilon (t,\, \cdot ))$ in the space of continuous functions as $\varepsilon \to 0$, and also of describing transitions of $ u_t^\varepsilon$ between stable stationary solutions of nonperturbed systems of PDE. The behavior of ${\mu ^\varepsilon }$ and the transitions are defined by large deviations for the field ${u^\varepsilon }(t,\,x)$.
The density manifold and configuration space quantization
John D.
Lafferty
699-741
Abstract: The differential geometric structure of a Fréchet manifold of densities is developed, providing a geometrical framework for quantization related to Nelson's stochastic mechanics. The Riemannian and symplectic structures of the density manifold are studied, and the Schrödinger equation is derived from a variational principle. By a theorem of Moser, the density manifold is an infinite dimensional homogeneous space, being the quotient of the group of diffeomorphisms of the underlying base manifold modulo the group of diffeomorphisms which preserve the Riemannian volume. From this structure and symplectic reduction, the quantization procedure is equivalent to Lie-Poisson equations on the dual of a semidirect product Lie algebra. A Poisson map is obtained between the dual of this Lie algebra and the underlying projective Hilbert space.
A space-time property of a class of measure-valued branching diffusions
Edwin A.
Perkins
743-795
Abstract: If $d > \alpha$, it is shown that the $ d$-dimensional branching diffusion of index $\alpha$, studied by Dawson and others, distributes its mass over a random support in a uniform manner with respect to the Hausdorff ${\phi _\alpha }$-measure, where $ {\phi _\alpha }(x) = {x^\alpha }\log \log 1/x$. More surprisingly, it does so for all positive times simultaneously. Slightly less precise results are obtained in the critical case $d = \alpha$. In particular, the process is singular at all positive times a.s. for $d \geqslant \alpha$.
Some applications of tree-limits to groups. I
Kenneth
Hickin
797-839
Abstract: Sharper applications to group theory are given of an elegant construction -- the "tree-limit"--which S. Shelah circulated as a preprint in 1977 and used to obtain $\infty$-$\omega$-enlargements to power ${2^\omega }$ of certain countable homogeneous groups and skew fields. In this paper we enlarge the class of groups to which this construction can be interestingly applied and we obtain permutation representations of countable degree of the tree-limit groups; we obtain uncountable subgroup-incomparability for enlargements of countable existentially closed groups and even in nonhomogeneous cases we obtain the very strong "archetypal direct limit property" (which implies $ \infty$-$\omega $-equivalence (see (1.0)) of the permutation representations). We are able to control the permutation representations which get stretched by the tree-limit by varying the point-stabilizer subgroups (see (5.5)). In particular we can archetypally stretch in $ {2^\omega }$ subgroup-incomparable ways any homogeneous permutation representation of a countable locally finite group in which every finite subgroup has infinitely many regular orbits (Theorem 4). We discuss cases where tree-limits are subgroups of inverse limits.
Univalent functions in $H\cdot \overline H(D)$
Z.
Abdulhadi;
D.
Bshouty
841-849
Abstract: Functions in $H \cdot \overline H (D)$ are sense-preserving of the form $f = h \cdot \overline g$ where $h$ and $g$ are in $H(D)$. Such functions are solutions of an elliptic nonlinear P.D.E. that is studied in detail especially for its univalent solutions.
Correction to: ``Cartan subalgebras of simple Lie algebras'' [Trans. Amer. Math. Soc. {\bf 234} (1977), no. 2, 435--446; MR0480650 (58 \#806)]
Robert Lee
Wilson
851-855
Corrigendum to: ``Affine semigroups and Cohen-Macaulay rings generated by monomials'' [Trans. Amer. Math. Soc. {\bf 298} (1986), no. 1, 145--167; MR0857437 (87j:13032)]
Lê Tuân
Hoa;
Ngô Viêt
Trung
857
Crossed products of continuous-trace $C\sp \ast$-algebras by smooth actions
Iain
Raeburn;
Jonathan
Rosenberg
1-45
Abstract: We study in detail the structure of $ {C^{\ast}}$-crossed products of the form $ A \rtimes {}_\alpha G$, where $A$ is a continuous-trace algebra and $\alpha$ is an action of a locally compact abelian group $G$ on $A$, especially in the case where the action of $ G$ on $\hat A$ has a Hausdorff quotient and only one orbit type. Under mild conditions, the crossed product has continuous trace, and we are often able to compute its spectrum and Dixmier-Douady class. The formulae for these are remarkably interesting even when $G$ is the real line.
New combinatorial interpretations of Ramanujan's partition congruences mod $5,7$ and $11$
F. G.
Garvan
47-77
Abstract: Let $p(n)$ denote the number of unrestricted partitions of $n$. The congruences referred to in the title are $ p(5n + 4)$, $p(7n + 5)$ and $p(11n + 6) \equiv 0$ ($\bmod 5$, $7$ and $11$, respectively). Dyson conjectured and Atkin and Swinnerton-Dyer proved combinatorial results which imply the congruences $\bmod 5$ and $7$. These are in terms of the rank of partitions. Dyson also conjectured the existence of a "crank" which would likewise imply the congruence $\bmod 11$. In this paper we give a crank which not only gives a combinatorial interpretation of the congruence $\bmod 11$ but also gives new combinatorial interpretations of the congruences $\bmod 5$ and $7$. However, our crank is not quite what Dyson asked for; it is in terms of certain restricted triples of partitions, rather than in terms of ordinary partitions alone. Our results and those of Dyson, Atkin and Swinnerton-Dyer are closely related to two unproved identities that appear in Ramanujan's "lost" notebook. We prove the first identity and show how the second is equivalent to the main theorem in Atkin and Swinnerton-Dyer's paper. We note that all of Dyson's conjectures $\bmod 5$ are encapsulated in this second identity. We give a number of relations for the crank of vector partitions $\bmod 5$ and $7$, as well as some new inequalities for the rank of ordinary partitions $\bmod 5$ and $7$. Our methods are elementary relying for the most part on classical identities of Euler and Jacobi.
A lifting theorem and uniform algebras
Takahiko
Nakazi;
Takanori
Yamamoto
79-94
Abstract: In this paper we discuss the possible generalizations of a lifting theorem of a $2 \times 2$ matrix to uniform algebras. These have applications to Hankel operators, weighted norm inequalities for conjugation operators and Toeplitz operators on uniform algebras. For example, the Helson-Szegö theorems for general uniform algebras follow.
Sharp distortion theorems for quasiconformal mappings
G. D.
Anderson;
M. K.
Vamanamurthy;
M.
Vuorinen
95-111
Abstract: Continuing their earlier work on distortion theory, the authors prove some dimension-free distortion theorems for $ K$-quasiconformal mappings in ${R^n}$. For example, one of the present results is the following sharp variant of the Schwarz lemma: If $ f$ is a $ K$-quasiconformal self-mapping of the unit ball ${B^n}$, $ n \geqslant 2$, with $ f(0) = 0$, then ${4^{1 - {K^2}}}\vert x{\vert^K} \leqslant \vert f(x)\vert \leqslant {4^{1 - 1/{K^2}}}\vert x{\vert^{1/K}}$ for all $x$ in ${B^n}$.
Euler-Poincar\'e characteristic and higher order sectional curvature. I
Chuan-Chih
Hsiung;
Kenneth Michael
Shiskowski
113-128
Abstract: The following long-standing conjecture of H. Hopf is well known. Let $ M$ be a compact orientable Riemannian manifold of even dimension $n \geqslant 2$. If $M$ has nonnegative sectional curvature, then the Euler-Poincaré characteristic $ \chi (M)$ is nonnegative. If $M$ has nonpositive sectional curvature, then $ \chi (M)$ is nonnegative or nonpositive according as $n \equiv 0$ or $ 2\bmod 4$. This conjecture for $n = 4$ was proved first by J. W. Milnor and then by S. S. Chern by a different method. The main object of this paper is to prove this conjecture for a general $ n$ under an extra condition on higher order sectional curvature, which holds automatically for $n = 4$. Similar results are obtained for Kähler manifolds by using holomorphic sectional curvature, and F. Schur's theorem about the constancy of sectional curvature on a Riemannian manifold is extended.
Infinite rank Butler groups
Manfred
Dugas;
K. M.
Rangaswamy
129-142
Abstract: A torsion-free abelian group $G$ is said to be a Butler group if $\operatorname{Bext} (G,\,T)$ for all torsion groups $T$. It is shown that Butler groups of finite rank satisfy what we call the torsion extension property (T.E.P.). A crucial result is that a countable Butler group $G$ satisfies the T.E.P. over a pure subgroup $ H$ if and only if $ H$ is decent in $ G$ in the sense of Albrecht and Hill. A subclass of the Butler groups are the so-called ${B_2}$-groups. An important question left open by Arnold, Bican, Salce, and others is whether every Butler group is a ${B_2}$-group. We show under $(V = L)$ that this is indeed the case for Butler groups of rank $ {\aleph _1}$. On the other hand it is shown that, under ZFC, it is undecidable whether a group $B$ for which $\operatorname{Bext} (B,\,T) = 0$ for all countable torsion groups $T$ is indeed a ${B_2}$-group.
Coexistence theorems of steady states for predator-prey interacting systems
Lige
Li
143-166
Abstract: In this paper we give necessary and sufficient conditions for the existence of positive solutions of steady states for predator-prey systems under Dirichlet boundary conditions on $ \Omega \Subset {{\mathbf{R}}^n}$. We show that the positive coexistence of predatorprey densities is completely determined by the "marginal density," the unique density of prey or predator while the other one is absent, i.e. the $({u_0},\,0)$ or $ (0,\,{\nu _0})$. More specifically, the situation of coexistence is determined by the spectral behavior of certain operators related to these marginal densities and is also completely determined by the stability properties of these marginal densities. The main results are Theorems 1 and 4.2.
Ensembles de Riesz
Valérie
Tardivel
167-174
Abstract: Let $G$ be an abelian countable discrete group. We show that there exists no positive characterization of Riesz subsets of $G$, by proving that the Riesz subsets of $ G$ form a coanalytic non-Borel subset of ${2^G}$.
Longtime dynamics of a conductive fluid in the presence of a strong magnetic field
C.
Bardos;
C.
Sulem;
P.-L.
Sulem
175-191
Abstract: We prove existence in the large of localized solutions to the MHD equations for an ideal conducting fluid subject to a strong magnetic field. We show that, for large time, the dynamics may reduce to linear Alfven waves.
Isometries between function spaces
Krzysztof
Jarosz;
Vijay D.
Pathak
193-206
Abstract: Surjective isometries between some classical function spaces are investigated. We give a simple technical scheme which verifies whether any such isometry is given by a homeomorphism between corresponding Hausdorff compact spaces. In particular the answer is positive for the ${C^1}(X)$, $\operatorname{AC} [0,1]$, ${\operatorname{Lip} _\alpha }(X)$ and $ {\operatorname{lip} _\alpha }(X)$ spaces provided with various natural norms.
On the behavior of harmonic functions near a boundary point
Wade
Ramey;
David
Ullrich
207-220
Abstract: Several results on the behavior of harmonic functions at an individual boundary point are obtained. The results apply to positive harmonic functions as well as to Poisson integrals of functions in BMO.
Local projective resolutions and translation functors for Kac-Moody algebras
Wayne
Neidhardt
221-245
Abstract: Let $\mathfrak{g}$ be a Kac-Moody algebra defined by a not necessarily symmetrizable generalized Cartan matrix. We define translation functors and use them to show that the multiplicities $(M({w_1} \cdot \lambda ):L({w_2} \cdot \lambda ))$ are independent of the dominant integral weight $ \lambda$, depending only on the elements of the Weyl group. In order to define the translation functors, we introduce the notion of local projective resolutions and use them to develop the machinery of homological algebra in certain categories of $\mathfrak{g}$-modules.
Representations of hyperharmonic cones
Sirkka-Liisa
Eriksson
247-262
Abstract: Hyperharmonic cones are ordered convex cones possessing order properties similar to those of hyperharmonic functions on harmonic spaces. The dual of a hyperharmonic cone is defined to be the set of extended real-valued additive and left order-continuous mappings $(\not \equiv \infty )$. The second dual gives a representation of certain hyperharmonic cones in which suprema of upward directed families are pointwise suprema, although infima of pairs of functions are not generally pointwise infima. We obtain necessary and sufficient conditions for the existence of a representation of a hyperharmonic cone in which suprema of upward directed families are pointwise suprema and infima of pairs of functions are pointwise infima.
Convergence acceleration for generalized continued fractions
Paul
Levrie;
Lisa
Jacobsen
263-275
Abstract: The main result in this paper is the proof of convergence acceleration for a suitable modification (as defined by de Bruin and Jacobsen) in the case of an $n$-fraction for which the underlying recurrence relation is of Perron-Kreuser type. It is assumed that the characteristic equations for this recurrence relation have only simple roots with differing absolute values.
Complementation in Kre\u\i n spaces
Louis
de Branges
277-291
Abstract: A generalization of the concept of orthogonal complement is introduced in complete and decomposable complex vector spaces with scalar product.
$k$-dimensional regularity classifications for $s$-fractals
Miguel Ángel
Martín;
Pertti
Mattila
293-315
Abstract: We study subsets $ E$ of ${{\mathbf{R}}^n}$ which are ${H^s}$ measurable and have $0 < {H^s}(E) < \infty$, where $ {H^s}$ is the $ s$-dimensional Hausdorff measure. Given an integer $k$, $ s \leqslant k \leqslant n$, we consider six ($s$, $k$) regularity definitions for $ E$ in terms of $ k$-dimensional subspaces or surfaces of $ {{\mathbf{R}}^n}$. If $ s = k$, they all agree with the (${H^k}$, $k$) rectifiability in the sense of Federer, but in the case $s < k$ we show that only two of them are equivalent. We also study sets with positive lower density, and projection properties in connection with these regularity definitions.
Counting semiregular permutations which are products of a full cycle and an involution
D. M.
Jackson
317-331
Abstract: Character theoretic methods and the group algebra of the symmetric group are used to derive properties of the number of permutations, with only $p$-cycles, for an arbitrary but fixed $ p$, which are expressible as the product of a full cycle and a fixed point free involution. This problem has application to single face embeddings of $p$-regular graphs on surfaces of given genus.
The ideal structure of certain nonselfadjoint operator algebras
Justin
Peters
333-352
Abstract: Let $(X,\,\phi )$ be a locally compact dynamical system, and ${{\mathbf{Z}}^ + }{ \times _\phi }\,{C_0}(X)$ the norm-closed subalgebra of the crossed product $Z{ \times _\phi }{C_0}(X)$ generated by the nonnegative powers of $\phi$ in case $\phi$ is a homeomorphism. If $ \phi$ is just a continuous map, ${{\mathbf{Z}}^ + }{ \times _\phi }{C_0}$ can still be defined by a crossed product type construction. The ideal structure of these algebras is determined in case $\phi$ acts freely. A class of strictly transitive Banach modules is described, indicating that for the nonselfadjoint operator algebras considered here, not all irreducible representations are on Hilbert space. Finally in a special case, the family of all invariant maximal right ideals is given.
Operator theoretical realization of some geometric notions
Qing
Lin
353-367
Abstract: This paper studies the realization of certain geometric constructions in Cowen-Douglas operator class. Through this realization, some operator theoretical phenomena are easily seen from the corresponding geometric phenomena. In particular, we use this technique to solve the first-order equivalence problem and introduce a new operation among certain operators.
The action of a solvable group on an infinite set never has a unique invariant mean
Stefan
Krasa
369-376
Abstract: Theorem 1 of the paper proves a conjecture of J. Rosenblatt on nonuniqueness of invariant means for the action of a solvable group $G$ on an infinite set $X$. The same methods used in this proof yield even a more general result: Nonuniqueness still holds if $ G$ is an amenable group containing a solvable subgroup $H$ such that $ \operatorname{card} (G/H) \leqslant \operatorname{card} (H)$.
A problem in convexity leading to the analysis of two functional equations
John V.
Ryff
377-396
Abstract: Transformation semigroups can often be studied effectively by examining their orbit structure. If the class of transformations has a special quality, such as convexity, it is generally reflected in the orbits. This work is concerned with such a circumstance. The goal is to examine the behavior of transformations on extreme points of orbits through the construction of a class of extreme operators. The construction leads naturally to the study of two functional equations which are analyzed in detail. Information about solutions is obtained through different $ {L^2}$-methods depending on whether or not two basic parameters are rational or irrational. In two cases all solutions are classified. In a third an example of a spanning set of solutions is obtained. Techniques of harmonic analysis and ergodic theory are used to study the functional equations.
Interpolation of Besov spaces
Ronald A.
DeVore;
Vasil A.
Popov
397-414
Abstract: We investigate Besov spaces and their connection with dyadic spline approximation in $ {L_p}(\Omega )$, $0 < p \leqslant \infty$. Our main results are: the determination of the interpolation spaces between a pair of Besov spaces; an atomic decomposition for functions in a Besov space; the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines.
Finite order solutions of second order linear differential equations
Gary G.
Gundersen
415-429
Abstract: We consider the differential equation $f\not \equiv 0$ of the equation will have infinite order. We will also find conditions on $ A(z)$ and $B(z)$ which will guarantee that any finite order solution $ f\not \equiv 0$ of the equation will not have zero as a Borel exceptional value. We will also show that if $A(z)$ and $B(z)$ satisfy certain growth conditions, then any finite order solution of the equation will satisfy certain other growth conditions. Related results are also proven. Several examples are given to complement the theory.
Borel classes and closed games: Wadge-type and Hurewicz-type results
A.
Louveau;
J.
Saint-Raymond
431-467
Abstract: For each countable ordinal $\xi$ and pair $ ({A_0},\,{A_1})$ of disjoint analytic subsets of $ {2^\omega }$, we define a closed game ${J_\xi }({A_0},\,{A_1})$ and a complete $\Pi _\xi ^0$ subset ${H_\xi }$ of $ {2^\omega }$ such that (i) a winning strategy for player I constructs a $\sum _\xi ^0$ set separating ${A_0}$ from ${A_1}$; and (ii) a winning strategy for player II constructs a continuous map $\varphi :{2^\omega } \to {A_0} \cup {A_1}$ with ${\varphi ^{ - 1}}({A_0}) = {H_\xi }$. Applications of this construction include: A proof in second order arithmetics of the statement "every $\Pi _\xi ^0$ non $ \sum _\xi ^0$ set is $\Pi _\xi ^0$-complete"; an extension to all levels of a theorem of Hurewicz about $\sum _2^0$ sets; a new proof of results of Kunugui, Novikov, Bourgain and the authors on Borel sets with sections of given class; extensions of results of Stern and Kechris. Our results are valid in arbitrary Polish spaces, and for the classes in Lavrentieff's and Wadge's hierarchies.
Longer than average intervals containing no primes
A. Y.
Cheer;
D. A.
Goldston
469-486
Abstract: We present two methods for proving that there is a positive proportion of intervals which contain no primes and are longer than the average distance between consecutive primes. The first method is based on an argument of Erdös which uses a sieve upper bound for prime twins to bound the density function for gaps between primes. The second method uses known results about the first three moments for the distribution of intervals with a given number of primes. Better results are obtained by assuming that the first $n$ moments are Poisson. The related problem of longer than average gaps between primes is also considered.
Manifolds on which only tori can act
Kyung Bai
Lee;
Frank
Raymond
487-499
Abstract: A list of various types of connected, closed oriented manifolds are given. Each of the manifolds support some of the well-known compact transformation group properties enjoyed by aspherical manifolds. We list and describe these classes and their transformation group properties in increasing generality. We show by various examples that these implications can never be reversed. This establishes a hierarchy in terms of spaces in one direction and the properties they enjoy in the opposite direction.
Comparison between analytic capacity and the Buffon needle probability
Takafumi
Murai
501-514
Abstract: We show that analytic capacity and the Buffon needle probability are not comparable.
Weak limits of projections and compactness of subspace lattices
Bruce H.
Wagner
515-535
Abstract: A strongly closed lattice of projections on a Hilbert space is compact if the associated algebra of operators has a weakly dense subset of compact operators. If the lattice is commutative, there are necessary and sufficient conditions for compactness, one in terms of the structure of the lattice, and the other in terms of a measure on the lattice. There are many examples of compact lattices, and two main types of examples of noncompact lattices. Compactness is also related to the study of weak limits of certain projections.
Dense morphisms in commutative Banach algebras
Gustavo
Corach;
Fernando Daniel
Suárez
537-547
Abstract: Using a new notion of stability we compute exactly the stable rank of the polydisc algebra, extend Oka's extension theorem to $ n$-tuples of functions without common zeros and give an estimation for a question raised by Swan concerning the stable rank of a dense subalgebra of a given Banach algebra.
Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups
L.
Corwin;
F. P.
Greenleaf;
G.
Grélaud
549-583
Abstract: Let $K$ be a Lie subgroup of the connected, simply connected nilpotent Lie group $G$, and let $ \mathfrak{k}$, $\mathfrak{g}$ be the corresponding Lie algebras. Suppose that $\sigma$ is an irreducible unitary representation of $K$. We give an explicit direct integral decomposition of ${\operatorname{Ind} _{k \to G}}\sigma$ into irreducibles. The description uses the Kirillov orbit picture, which gives a bijection between $G^\wedge$ and the coadjoint orbits in $ {\mathfrak{g}^{\ast}}$ (and similarly for $ K^\wedge,\,{\mathfrak{k}^{\ast}}$). Let $P:{\mathfrak{k}^{\ast}} \to {\mathfrak{g}^{\ast}}$ be the canonical projection, let $ {\mathcal{O}_\sigma } \subset {\mathfrak{k}^{\ast}}$ be the orbit corresponding to $\sigma$, and, for $ \pi \in G^\wedge$, let ${\mathcal{O}_\pi } \subset {\mathfrak{g}^{\ast}}$ be the corresponding orbit. The main result of the paper says essentially that $ \pi \in G^\wedge$ appears in the direct integral iff ${P^{ - 1}}({\mathcal{O}_\sigma })$ meets $ {\mathcal{O}_\pi }$; the multiplicity of $\pi$ is the number of ${\operatorname{Ad} ^{\ast}}(K)$-orbits in ${\mathcal{O}_\pi } \cap {P^{ - 1}}({\mathcal{O}_\sigma })$. There is also a natural description of the measure class in the integral.
Invariant subspaces in Banach spaces of analytic functions
Stefan
Richter
585-616
Abstract: We study the invariant subspace structure of the operator of multiplication by $z$, ${M_z}$, on a class of Banach spaces of analytic functions. For operators on Hilbert spaces our class coincides with the adjoints of the operators in the Cowen-Douglas class ${\mathcal{B}_1}(\overline \Omega )$. We say that an invariant subspace $ \mathcal{M}$ satisfies $ \operatorname{cod} \mathcal{M} = 1$ if $ z\mathcal{M}$ has codimension one in $ \mathcal{M}$. We give various conditions on invariant subspaces which imply that $ \operatorname{cod} \mathcal{M} = 1$. In particular, we give a necessary and sufficient condition on two invariant subspaces $\mathcal{M}$, $ \mathcal{N}$ with $\operatorname{cod} \mathcal{M} = \operatorname{cod} \mathcal{N} = 1$ so that their span again satisfies $ \operatorname{cod} (\mathcal{M} \vee \mathcal{N}) = 1$. This result will be used to show that any invariant subspace of the Bergman space $ L_a^p,\,p \geqslant 1$, which is generated by functions in $L_a^{2p}$, must satisfy $ \operatorname{cod} \mathcal{M} = 1$. For an invariant subspace $\mathcal{M}$ we then consider the operator $S = M_z^{\ast}\vert{\mathcal{M}^ \bot }$. Under some extra assumption on the domain of holomorphy we show that the spectrum of $S$ coincides with the approximate point spectrum iff $\operatorname{cod} \mathcal{M} = 1$. Finally, in the last section we obtain a structure theorem for invariant subspaces with $\operatorname{cod} \mathcal{M} = 1$. This theorem applies to Dirichlet-type spaces.
Nonlinear stability of vortex patches
Yun
Tang
617-638
Abstract: To establish the nonlinear (Liapunov) stability of both circular and elliptical vortex patches in the plane for the nonlinear dynamical system generated by the two-dimensional Euler equations of incompressible, inviscid hydrodynamics. This is accomplished by using a relative variational principle in terms of energy function. A counterexample shows that our result in the case of an elliptical vortex patch is the best one that can be attained by applying the energy estimate.
On the elliptic equations $\Delta u=K(x)u\sp \sigma$ and $\Delta u=K(x)e\sp {2u}$
Kuo-Shung
Cheng;
Jenn-Tsann
Lin
639-668
Abstract: We give some nonexistence results for the equations $\Delta u = K(x){u^\sigma }$ and $\Delta u = K(x){e^{2u}}$ for $K(x) \geqslant 0$.
Weighted norm estimates for Sobolev spaces
Martin
Schechter
669-687
Abstract: We give sufficient conditions for estimates of the form $\displaystyle {\int {\left\vert {u(x)} \right\vert} ^q}d\mu (x) \leqslant C\left\Vert u \right\Vert _{s,p}^1,\qquad u \in {H^{s,p}},$ to hold, where $ \mu (x)$ is a measure and $ {\left\Vert u \right\Vert _{s,p}}$ is the norm of the Sobolev space ${H^{s,p}}$. If $d\mu = dx$, this reduces to the usual Sobolev inequality. The general form has much wider applications in both linear and nonlinear partial differential equations. An application is given in the last section.
Linear series with cusps and $n$-fold points
David
Schubert
689-703
Abstract: A linear series $ (V,\,\mathcal{L})$ on a curve $X$ has an $n$-fold point along a divisor $D$ of degree $n$ if $\dim (V \cap {H^0}(X,\,\mathcal{L}( - D))) \geqslant \dim (V) - 1$. The linear series has a cusp of order $e$ at a point $P$ if $\dim (V \cap {H^0}(X,\,\mathcal{L}( - (e + 1)P))) \geqslant \dim (V) - 1$. Linear series with cusps and $n$-fold points are shown to exist if certain inequalities are satisfied. The dimensions of the families of linear series with cusps are determined for general curves.
Chaotic maps with rational zeta function
H. E.
Nusse
705-719
Abstract: Fix a nontrivial interval $X \subset {\mathbf{R}}$ and let $f \in {C^1}(X,\,X)$ be a chaotic mapping. We denote by ${A_\infty }(f)$ the set of points whose orbits do not converge to a (one-sided) asymptotically stable periodic orbit of $f$ or to a subset of the absorbing boundary of $ X$ for $f$. A. We assume that $f$ satisfies the following conditions: (1) the set of asymptotically stable periodic points for $ f$ is compact (an empty set is allowed), and (2) $A{\,_\infty }(f)\,$ is compact, $f$ is expanding on ${A_\infty }(f)$. Then we can associate a matrix $ {A_f}$ with entries either zero or one to the mapping $f$ such that the number of periodic points for $ f$ with period $ n$ is equal to the trace of the matrix ${\left[ {{A_f}} \right]^n}$; furthermore the zeta function of $f$ is rational having the eigenvalues of $ {A_f}$ as poles. B. We assume that $ f \in {C^3}(X,\,X)$ such that: (1) the Schwarzian derivative of $f$ is negative, and (2) the closure of ${A_\infty }(f)$ is compact and ${A_\infty }(f)$. Then we obtain the same result as in A.
Spaces of geodesic triangulations of the sphere
Marwan
Awartani;
David W.
Henderson
721-732
Abstract: We study questions concerning the homotopy-type of the space $\operatorname{GT} (K)$ of geodesic triangulations of the standard $n$-sphere which are (orientation-preserving) isomorphic to $K$. We find conditions which reduce this question to analogous questions concerning spaces of simplexwise linear embeddings of triangulated $n$-cells into $n$-space. These conditions are then applied to the $2$-sphere. We show that, for each triangulation $ K$ of the $2$-sphere, certain large subspaces of $ \operatorname{GT} (K)$ are deformable (in $ \operatorname{GT} (K)$) into a subsapce homeomorphic to $\operatorname{SO} (3)$. It is conjectured that (for $ n = 2$) $\operatorname{GT} (K)$ has the homotopy of $ \operatorname{SO} (3)$. In a later paper the authors hope to use these same conditions to study the homotopy type of spaces of geodesic triangulations of the $n$-sphere, $n > 2$.
On maximal functions associated to hypersurfaces and the Cauchy problem for strictly hyperbolic operators
Christopher D.
Sogge
733-749
Abstract: In this paper we prove a maximal Fourier integral theorem for the types of operators which arise in the study of maximal functions associated to averaging over hypersurfaces and also the Cauchy problem for hyperbolic operators. We apply the Fourier integral theorem to generalize Stein's spherical maximal theorem (see [8]) and also to prove a sharp theorem for the almost everywhere convergence to ${L^p}$ initial data of solutions to the Cauchy problem for second order strictly hyperbolic operators. Our results improve those of Greenleaf [3] and Ruiz [6]. We also can prove almost everywhere convergence to ${L^2}$ initial data for operators of order $m \geqslant 3$.
On the initial-boundary value problem for a Bingham fluid in a three-dimensional domain
Jong Uhn
Kim
751-770
Abstract: The initial-boundary value problem associated with the motion of a Bingham fluid is considered. The existence and uniqueness of strong solution is proved under a certain assumption on the data. It is also shown that the solution exists globally in time when the data are small and that the solution converges to a periodic solution if the external force is time-periodic.
Decompositions of Banach lattices into direct sums
P. G.
Casazza;
N. J.
Kalton;
L.
Tzafriri
771-800
Abstract: We consider the problem of decomposing a Banach lattice $Z$ as a direct sum $Z = X \oplus Y$ where $X$ and $Y$ are complemented subspaces satisfying a condition of incomparability (e.g. every operator from $ Y$ to $X$ is strictly singular). We treat both the atomic and nonatomic cases. In particular we answer a question of Wojtaszczyk by showing that ${L_1} \oplus {L_2}$ has unique structure as a nonatomic Banach lattice.
The equivariant Conner-Floyd isomorphism
Steven R.
Costenoble
801-818
Abstract: This paper proves two equivariant generalizations of the Conner-Floyd isomorphism relating unitary cobordism and $ K$-theory. It extends a previous result of Okonek for abelian groups to all compact Lie groups. We also show that the result for finite groups is true using either the geometric or homotopical versions of cobordism.
Spectral measures, boundedly $\sigma$-complete Boolean algebras and applications to operator theory
Werner J.
Ricker
819-838
Abstract: A systematic study is made of spectral measures in locally convex spaces which are countably additive for the topology of uniform convergence on bounded sets, briefly, the bounded convergence topology. Even though this topology is not compatible for the duality with respect to the pointwise convergence topology it turns out, somewhat surprisingly, that the corresponding ${L^1}$-spaces for the spectral measure are isomorphic as vector spaces. This fact, together with I. Kluvanek's notion of closed vector measure (suitably developed in our particular setting) makes it possible to extend to the setting of locally convex spaces a classical result of W. Bade. Namely, it is shown that if $ B$ is a Boolean algebra which is complete (with respect to the bounded convergence topology) in Bade's sense, then the closed operator algebras generated by $B$ with respect to the bounded convergence topology and the pointwise convergence topology coincide.
Symmetry breaking for a class of semilinear elliptic problems
Mythily
Ramaswamy;
P. N.
Srikanth
839-845
Abstract: We study positive solutions of the Dirichlet problem for $- \Delta u = {u^p} - \lambda$, $p > 1$, $ \lambda > 0$, on the unit ball $\Omega$. We show that there exists a positive solution $ ({u_0},\,{\lambda _0})$ of this problem which satisfies in addition $\partial {u_0}/\partial n = 0$ on $\partial \Omega $. We prove also that at $ ({u_0},\,{\lambda _0})$, the symmetry breaks, i.e. asymmetric solutions bifurcate from the positive radial solutions.
Correction to: ``Meromorphic functions that share four values'' [Trans. Amer. Math. Soc. {\bf 277} (1983), no. 2, 545--567; MR0694375 (84g:30028)]
Gary G.
Gundersen
847-850
Invariance results for delay and Volterra equations in fractional order Sobolev spaces
F.
Kappel;
K.
Kunisch
1-51
Abstract: Invariance of the trajectories of infinite delay- and Volterra-type equations in fractional order Sobolev spaces are derived under minimal assumptions on the problem data. Properties of fractional order Sobolev spaces defined over intervals are summarized.
Oscillatory integrals and Fourier transforms of surface carried measures
Michael
Cowling;
Giancarlo
Mauceri
53-68
Abstract: We suppose that $ S$ is a smooth hypersurface in $ {{\mathbf{R}}^{n + 1}}$ with Gaussian curvature $\kappa$ and surface measure $dS$, $w$ is a compactly supported cut-off function, and we let $ {\mu _\alpha }$ be the surface measure with $d{\mu _\alpha } = w{\kappa ^\alpha }\,dS$. In this paper we consider the case where $S$ is the graph of a suitably convex function, homogeneous of degree $d$, and estimate the Fourier transform $ {\hat \mu _\alpha }$. We also show that if $S$ is convex, with no tangent lines of infinite order, then $ {\hat \mu _\alpha }(\xi )$ decays as $ \vert\xi {\vert^{ - n / 2}}$ provided $\alpha \geqslant [(n + 3)/2]$. The techniques involved are the estimation of oscillatory integrals; we give applications involving maximal functions.
Application of group cohomology to space constructions
Paul
Igodt;
Kyung Bai
Lee
69-82
Abstract: From a short exact sequence of crossed modules $1 \to K \to H \to \bar H \to 1$ and a $ 2$-cocycle $(\phi ,\,h) \in {Z^2}(G;\,H)$, a $ 4$-term cohomology exact sequence $H_{ab}^1(G;\,Z) \to H_{(\bar \phi ,\,\bar h)}^1(G;\,\bar H,\bar Z)\mathop \to ... ...(G;\,K):{\psi _{{\text{out}}}} = {\phi _{{\text{out}}}}\} \to H_{ab}^2(G;\,Z)}$ is obtained. Here the first and the last term are the ordinary (=abelian) cohomology groups, and $Z$ is the center of the crossed module $ H$. The second term is shown to be in one-to-one correspondence with certain geometric constructions, called Seifert fiber space construction. Therefore, it follows that, if both the end terms vanish, the geometric construction exists and is unique.
On strongly summable ultrafilters and union ultrafilters
Andreas
Blass;
Neil
Hindman
83-97
Abstract: We prove that union ultrafilters are essentially the same as strongly summable ultrafilters but ordered-union ultrafilters are not. We also prove that the existence of ultrafilters of these sorts implies the existence of $ P$-points and therefore cannot be established in ZFC.
Holomorphic kernels and commuting operators
Ameer
Athavale
101-110
Abstract: Necessary and sufficient conditions in terms of operator polynomials are obtained for an $m$-tuple $T = ({T_1}, \ldots ,{T_m})$ of commuting bounded linear operators on a separable Hilbert space $ \mathcal{H}$ to extend to an $\dot m$-tuple $S = ({S_1}, \ldots ,{S_m})$ of operators on some Hilbert space $ \mathcal{K}$, where each $ {S_i}$ is realized as a $ {\ast}$-representation of the adjoint of a multiplication operator on the tensor product of a special type of functional Hilbert spaces. Also, necessary and sufficient conditions in terms of operator polynomials are obtained for $ T$ to have a commuting normal extension.
Taming wild extensions with Hopf algebras
Lindsay N.
Childs
111-140
Abstract: Let $K \subset L$ be a Galois extension of number fields with abelian Galois group $G$ and rings of integers $R \subset S$, and let $\mathcal{A}$ be the order of $S$ in $KG$. If $ \mathcal{A}$ is a Hopf $ R$-algebra with operations induced from $KG$, then $S$ is locally isomorphic to $\mathcal{A}$ as $ \mathcal{A}$-module. Criteria are found for $ \mathcal{A}$ to be a Hopf algebra when $ K = {\mathbf{Q}}$ or when $ L/K$ is a Kummer extension of prime degree. In the latter case we also obtain a complete classification of orders over $R$ in $L$ which are tame or Galois $ H$-extensions, $ H$ a Hopf order in $ KG$, using a generalization of the discriminant.
The Szeg\H o kernel as a singular integral kernel on a family of weakly pseudoconvex domains
Katharine Perkins
Diaz
141-170
Abstract: The Szegö kernels on the weakly pseudoconvex domains $\{ \operatorname{Im} {z_2} > \vert{z_1}{\vert^{2k}}\}$, $k \in {Z^ + }$, have been computed by Greiner and Stein. After constructing a global, nonisotropic pseudometric suitable for Calderón-Zygmund singular integral theory on the boundaries of the domains, we study principal value operators associated to these Szegö kernels. We prove that the principal value operators are bounded on ${L^p}$ for $1 < p < \infty$, and that they preserve certain nonisotropic Lipschitz classes. We then derive a Plemelj formula that relates the principal value operators to the Szegö projections. From this formula we deduce that the Szegö projections are also bounded on ${L^p}$, for $ 1 < p < \infty$, and that they preserve the same nonisotropic Lipschitz classes.
A strong generalization of Helgason's theorem
Kenneth D.
Johnson
171-192
Abstract: Let $G$ be a simple Lie group with $ KAN$ an Iwasawa decomposition of $G$, and let $M$ be the centralizer of $A$ in $K$. Suppose ${K_1}$ is a fixed, closed, normal, analytic subgroup of $K$, and set $ {\mathbf{P}}({K_1})$ equal to the set of all parabolic subgroups $P$ of $G$ which contain $MAN$ such that $ {K_1}P = G$ and ${K_1} \cap P$ is normal in the reductive part of $ P$. Suppose $\pi :G \to GL(V)$ is an irreducible representation of $G$. Then, if ${\mathbf{P}}({K_1}) \ne \emptyset$, we obtain necessary and sufficient conditions for ${V^{{K_1}}}$, the space of ${K_1}$-fixed vectors, to be $\ne (0)$. Moreover, reciprocity formulas are obtained which determine $ \dim {V^{{K_1}}}$.
The Morava $K$-theories of some classifying spaces
Nicholas J.
Kuhn
193-205
Abstract: Let $P$ be a finite abelian $ p$-group with classifying space $BP$. We compute, in representation theoretic terms, the Morava $K$-theories of the stable wedge summands of $ BP$. In particular, we obtain a simple, and purely group theoretic, description of the rank of $ K{(s)^{\ast}}(BG)$ for any finite group $G$ with an abelian $p$-Sylow subgroup. A minimal amount of topology quickly reduces the problem to an algebraic one of analyzing truncated polynomial algebras as modular representations of the semigroup ${M_n}({\mathbf{Z}} / p)$.
Equivariant geometry and Kervaire spheres
Allen
Back;
Wu-Yi
Hsiang
207-227
Abstract: The intrinsic geometry of metrics on the Kervaire sphere which are invariant under a large transformation group (cohomogeneity one) is studied. Invariant theory is used to describe the behavior of these metrics near the singular orbits. Nice expressions for the Ricci and sectional curvatures are obtained. The nonexistence of invariant metrics of positive sectional curvature is proven, and Cheeger's construction of metrics of positive Ricci curvature is discussed.
Universal Loeb-measurability of sets and of the standard part map with applications
D.
Landers;
L.
Rogge
229-243
Abstract: It is shown in this paper that for $K$-saturated models many important external sets of nonstandard analysis--such as monadic sets or the set of all near-standard points or all pre-near-standard points or all compact points--are universally Loeb-measurable, i.e. Loeb-measurable with respect to every internal content. We furthermore obtain universal Loeb-measurability of the standard part map for topological spaces which are not covered by previous results in this direction. Moreover, the standard part map can be used as a measure preserving transformation for all $ \tau$-smooth measures, and not only for Radon-measures as known up to now. Applications of our results lead to simple new proofs for theorems of classical measure theory. We obtain e.g. the extension of $\tau$-smooth Baire-measures to $ \tau$-smooth Borel-measures, the decomposition theorems for $\tau$-smooth Baire-measures and $ \tau$-smooth Borel-measures and Kakutani's theorem for product measures.
Contributions to the theory of set valued functions and set valued measures
Nikolaos S.
Papageorgiou
245-265
Abstract: Measurable multifunctions and multimeasures with values in a Banach space are studied. We start by proving a variation of the known Dunford theorem for weak compactness in $ {L^1}(X)$. With a similar technique we prove that the range of certain vector valued integrals that appear in applications is $ w$-compact and convex. Also we obtain Dunford-Pettis type theorems for sequences of integrably bounded multifunctions. Some pointwise $w$-compactness theorems are also obtained for certain families of measurable multifunctions. Then we prove a representation theorem for additive, set valued operators defined on ${L^1}(X)$. Finally, in the last section, a detailed study of transition multimeasures is conducted and several representation theorems are proved.
Riemann problems for nonstrictly hyperbolic $2\times 2$ systems of conservation laws
David G.
Schaeffer;
Michael
Shearer
267-306
Abstract: The Riemann problem is solved for $2 \times 2$ systems of hyperbolic conservation laws having quadratic flux functions. Equations with quadratic flux functions arise from neglecting higher order nonlinear terms in hyperbolic systems that fail to be strictly hyperbolic everywhere. Such equations divide into four classes, three of which are considered in this paper. The solution of the Riemann problem is complicated, with new types of shock waves, and new singularities in the dependence of the solution on the initial data. Several ideas are introduced to help organize and clarify the new phenomena.
Visibility and rank one in homogeneous spaces of $K\leq 0$
María J.
Druetta
307-321
Abstract: In this paper we study relationships between the visibility axiom and rank one in homogeneous spaces of nonpositive curvature. We obtain a complete classification (in terms of rank) of simply connected homogeneous spaces of nonpositive curvature and dimension $\leqslant 4$. We provide examples, in every $\dim \geqslant 4$, of simply connected, irreducible homogeneous spaces $ (K \leqslant 0)$ which are neither visibility manifolds nor symmetric spaces.
Ideals of holomorphic functions with $C\sp \infty$ boundary values on a pseudoconvex domain
Edward
Bierstone;
Pierre D.
Milman
323-342
Abstract: We give natural sufficient conditions for the solution of several problems concerning division in the space ${\mathcal{A}^\infty }(\Omega )$ of holomorphic functions with $ {\mathcal{C}^\infty }$ boundary values on a pseudoconvex domain $ \Omega$ with smooth boundary. The sufficient conditions come from upper semicontinuity with respect to the analytic Zariski topology of a local invariant of coherent analytic sheaves (the "invariant diagram of initial exponents"), and apply to division in the space of ${\mathcal{C}^\infty }$ Whitney functions on an arbitrary closed set. Our theorem on division in $ {\mathcal{A}^\infty }(\Omega )$ follows using Kohn's theorem on global regularity in the $\bar \partial $-Neumann problem.
Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations
Gary M.
Lieberman
343-353
Abstract: We consider solutions (and subsolutions and supersolutions) of the boundary value problem \begin{displaymath}\begin{array}{*{20}{c}} {{a^{ij}}(x,\,u,\,Du){D_{ij}}u + a(x,... ...(x)u = g(x)\quad {\text{on}}\;\partial \Omega } \end{array} \end{displaymath} for a Lipschitz domain $\Omega$, a positive-definite matrix-valued function $ [{a^{ij}}]$, and a vector field $\beta$ which points uniformly into $ \Omega$. Without making any continuity assumptions on the known functions, we prove Harnack and Hölder estimates for $u$ near $ \partial \Omega$. In addition we bound the $ {L^\infty }$ norm of $ u$ near $\partial \Omega$ in terms of an appropriate $ {L^p}$ norm and the known functions. Our approach is based on that for the corresponding interior estimates of Trudinger.
All infinite groups are Galois groups over any field
Manfred
Dugas;
Rüdiger
Göbel
355-384
Abstract: Let $G$ be an arbitrary monoid with $ 1$ and right cancellation, and $K$ be a given field. We will construct extension fields $F \supseteq K$ with endomorphism monoid End $F$ isomorphic to $G$ modulo Frobenius homomorphisms. If $ G$ is a group, then Aut $ F = G$. Let ${F^G}$ denote the fixed elements of $ F$ under the action of $ G$. In the case that $ G$ is an infinite group, also ${F^G} = K$ and $G$ is the Galois group of $F$ over $K$. If $G$ is an arbitrary group, and $G = 1$, respectively, this answers an open problem (R. Baer 1967, E. Fried, C. U. Jensen, J. Thompson) and if $G$ is infinite, the result is an infinite analogue of the still unsolved Hilbert-Noether conjecture inverting Galois theory. Observe that our extensions $K \subset F$ are not algebraic. We also suggest to consider the case $K = {\mathbf{C}}$ and $ G = \{ 1\}$.
Existence domains of holomorphic functions of restricted growth
M.
Jarnicki;
P.
Pflug
385-404
Abstract: The paper presents a large class of domains $G$ of holomorphy in $ {{\mathbf{C}}^n}$ such that, for any $N > 0$, there exists a nonextendable holomorphic function $f$ on $G$ with $ \vert f\vert\delta _G^N$ bounded where ${\delta _G}(z): = \min ({(1 + \vert z{\vert^2})^{ - 1 / 2}},\,\operatorname{dist} (z,\,\partial G))$. Any fat Reinhardt domain of holomorphy belongs to this class. On the other hand we characterize those Reinhardt domains of holomorphy which are existence domains of bounded holomorphic functions.
Traveling waves in combustion processes with complex chemical networks
Steffen
Heinze
405-416
Abstract: The existence of traveling waves for laminar flames with complex chemistry is proved. The crucial assumptions are that all reactions have to be exothermic and that no cycles occur in the graph of the reaction network. The method is to solve the equations first in a bounded interval by a degree argument and then taking the infinite domain limit.
Classifying spaces for foliations with isolated singularities
Peter
Greenberg
417-429
Abstract: Let ${\Gamma ^a} \subset \Gamma $ be transitive pseudogroups on $ {{\mathbf{R}}^n}$, such that, for any element $ g:\,U \to V$ of $ \Gamma$, there is a locally finite subset $S \subset U$, such that $g{\vert _{U - S}}$ is an element of ${\Gamma ^a}$. We construct $B\Gamma$, up to weak homotopy type, from $B{\Gamma ^a}$ and the classifying spaces of certain groups of germs. As an application, the classifying space of the pseudogroup of orientation-preserving, piecewise linear homeomorphisms between open subsets of ${\mathbf{R}}$ is weakly homotopy equivalent to $ B{\mathbf{R}}{\ast}B{\mathbf{R}}$.
Homogeneous random measures and a weak order for the excessive measures of a Markov process
P. J.
Fitzsimmons
431-478
Abstract: Let $X = ({X_t},\,{P^x})$ be a right Markov process and let $ m$ be an excessive measure for $X$. Associated with the pair $(X,\,m)$ is a stationary strong Markov process $ ({Y_t},\,{Q_m})$ with random times of birth and death, with the same transition function as $X$, and with $m$ as one dimensional distribution. We use $({Y_t},\,{Q_m})$ to study the cone of excessive measures for $X$. A "weak order" is defined on this cone: an excessive measure $\xi$ is weakly dominated by $m$ if and only if there is a suitable homogeneous random measure $\kappa$ such that $({Y_t},\,{Q_\xi })$ is obtained by "birthing" $({Y_t},\,{Q_m})$, birth in $[t,\,t + dt]$ occurring at rate $\kappa (dt)$. Random measures such as $ \kappa$ are studied through the use of Palm measures. We also develop aspects of the "general theory of processes" over $({Y_t},\,{Q_m})$, including the moderate Markov property of $ ({Y_t},\,{Q_m})$ when the arrow of time is reversed. Applications to balayage and capacity are suggested.
Semistability at $\infty$, $\infty$-ended groups and group cohomology
Michael L.
Mihalik
479-485
Abstract: A finitely presented group $G$, is semistable at $\infty$ if for some (equivalently any) finite complex $X$, with $ {\pi _1}(X) = G$, any two proper maps $r,\,s:[0,\,\infty ) \to \tilde X$ ($\equiv$ the universal cover of $ X$) that determine the same end of $\tilde X$ are properly homotopic in $\tilde X$. If $G$ is semistable at $\infty$, then ${H^2}(G;\,ZG)$ is free abelian. 0- and $ 2$-ended groups are all semistable at $\infty$. Theorem. If $G = A{{\ast}_C}B$ where $C$ is finite and $ A$ and $ B$ are finitely presented, semistable at $\infty$ groups, then $G$ is semistable at $ \infty$. Theorem. If $\alpha :C \to D$ is an isomorphism between finite subgroups of the finitely presented semistable at $\infty$ group $H$, then the resulting $HNN$ extension is semistable at $\infty$. Combining these results with the accessibility theorem of M. Dunwoody gives Theorem. If all finitely presented $1$-ended groups are semistable at $ \infty$, then all finitely presented groups are semistable at $ \infty$.
The asymptotic behavior of the solutions of degenerate parabolic equations
Catherine
Bandle;
M. A.
Pozio;
Alberto
Tesei
487-501
Abstract: Existence of stationary states is established by means of the method of upper and lower solutions. The structure of the solution set is discussed and a uniqueness property for certain classes is proved by a generalized maximum principle. It is then shown that all solutions of the parabolic equation converge to a stationary state.
Squares of conjugacy classes in the infinite symmetric groups
Manfred
Droste
503-515
Abstract: Using combinatorial methods, we will examine squares of conjugacy classes in the symmetric groups ${S_\nu }$ of all permutations of an infinite set of cardinality $ {\aleph _\nu }$. For arbitrary permutations $p \in {S_\nu }$, we will characterize when each element $s \in {S_\nu }$ with finite support can be written as a product of two conjugates of $p$, and if $p$ has infinitely many fixed points, we determine when all elements of ${S_\nu }$ are products of two conjugates of $ p$. Classical group-theoretical theorems are obtained from similar results.
A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions
J. M.
Borwein;
D.
Preiss
517-527
Abstract: We show that, typically, lower semicontinuous functions on a Banach space densely inherit lower subderivatives of the same degree of smoothness as the norm. In particular every continuous convex function on a space with a Gâteaux (weak Hadamard, Fréchet) smooth renorm is densely Gâteaux (weak Hadamard, Fréchet) differentiable. Our technique relies on a more powerful analogue of Ekeland's variational principle in which the function is perturbed by a quadratic-like function. This "smooth" variational principle has very broad applicability in problems of nonsmooth analysis.
Conjugacy classes in algebraic monoids
Mohan S.
Putcha
529-540
Abstract: Let $M$ be a connected linear algebraic monoid with zero and a reductive group of units $ G$. The following theorem is established. Theorem. There exist affine subsets $ {M_1}, \ldots ,{M_k}$ of $M$, reductive groups ${G_1}, \ldots ,{G_k}$ with antiautomorphisms $^{\ast}$, surjective morphisms ${\theta _i}:{M_i} \to {G_i}$, such that: (1) Every element of $ M$ is conjugate to an element of some ${M_i}$, and (2) Two elements $ a$, $b$ in ${M_i}$ are conjugate in $M$ if and only if there exists $x \in {G_i}$ such that $x{\theta _i}(a){x^{\ast}} = {\theta _i}(b)$. As a consequence, it is shown that $M$ is a union of its inverse submonoids.
Nil $K$-theory maps to cyclic homology
Charles A.
Weibel
541-558
Abstract: Algebraic $ K$-theory breaks into two pieces: nil $K$-theory and Karoubi-Villamayor $ K$-theory. Karoubi has constructed Chern classes from the latter groups into cyclic homology. We construct maps from nil $K$-theory to cyclic homology which are compatible with Karoubi's maps, but with a degree shift. Several recent results show that in characteristic zero our map is often an isomorphism.
Hyperarithmetical index sets in recursion theory
Steffen
Lempp
559-583
Abstract: We define a family of properties on hyperhypersimple sets and show that they yield index sets at each level of the hyperarithmetical hierarchy. An extension yields a $ \Pi _1^1$-complete index set. We also classify the index set of quasimaximal sets, of coinfinite r.e. sets not having an atomless superset, and of r.e. sets major in a fixed nonrecursive r.e. set.
Singularly perturbed Dirichlet problems with subquadratic nonlinearities
Albert J.
DeSanti
585-593
Abstract: Boundary and interior layer theory is provided for a class of singularly perturbed Dirichlet problems with subquadratic nonlinearities in the derivative terms. The results obtained generalize and extend well-known results on the semilinear problem.
$p$-ranks and automorphism groups of algebraic curves
Shōichi
Nakajima
595-607
Abstract: Let $X$ be an irreducible complete nonsingular curve of genus $g$ over an algebraically closed field $k$ of positive characteristic $ p$. If $g \geqslant 2$, the automorphism group $ \operatorname{Aut} (X)$ of $ X$ is known to be a finite group, and moreover its order is bounded from above by a polynomial in $g$ of degree four (Stichtenoth). In this paper we consider the $p$-rank $\gamma$ of $X$ and investigate relations between $ \gamma$ and $\operatorname{Aut} (X)$. We show that $\gamma$ affects the order of a Sylow $ p$-subgroup of $ \operatorname{Aut} (X)\;(\S3)$ and that an inequality $ \vert\operatorname{Aut} (X)\vert \leqslant 84(g - 1)g$ holds for an ordinary (i.e. $\gamma = g$) curve $X\,(\S4)$.
On weak continuity and the Hodge decomposition
Joel W.
Robbin;
Robert C.
Rogers;
Blake
Temple
609-618
Abstract: We address the problem of determining the weakly continuous polynomials for sequences of functions that satisfy general linear first-order differential constraints. We prove that wedge products are weakly continuous when the differential constraints are given by exterior derivatives. This is sufficient for reproducing the Div-Curl Lemma of Murat and Tartar, the null Lagrangians in the calculus of variations and the weakly continuous polynomials for Maxwell's equations. This result was derived independently by Tartar who stated it in a recent survey article [7]. Our proof is explicit and uses the Hodge decomposition.
Towers and injective cohomology algebras
Paul
Goerss;
Larry
Smith
619-636
Abstract: Let $Y$ be a space of finite type such that ${H^{\ast}}Y$ is injective as an unstable algebra over the Steenrod algebra $A$ and such that ${\overline H ^{\ast}}Y$ is $A$-unbounded. Let $X$ be a simply connected $p$-complete space. Then any map of $A$-algebras $f:{H^{\ast}}\Omega X \to {H^{\ast}}Y$ can be realized as a map of spaces.
$K\sb {l+1}$-free graphs: asymptotic structure and a $0$-$1$ law
Ph. G.
Kolaitis;
H. J.
Prömel;
B. L.
Rothschild
637-671
Abstract: The structure of labeled ${K_{l + 1}}$-free graphs is investigated asymptotically. Through a series of stages of successive refinement the structure of "almost all" such graphs is found sufficiently precisely to prove that they are in fact $l$-colorable ($l$-partite). With the asymptotic information obtained it is shown also that in the class of ${K_{l + 1}}$-free graphs there is a first-order labeled 0-$1$ law. With this result, and those cases already known, we can say that any infinite class of finite undirected graphs with amalgamations, induced subgraphs and isomorphisms has a 0-$1$ law.
The Radon-Nikod\'ym property and the Kre\u\i n-Milman property are equivalent for strongly regular sets
Walter
Schachermayer
673-687
Abstract: The result announced in the title is proved. As corollaries we obtain that RNP and KMP are equivalent for subsets of spaces with an unconditional basis and for $K$-convex Banach spaces. We also obtain a sharpening of a result of R. Huff and P. Morris: A dual space has the RNP iff all separable subspaces have the KMP.
Branched coverings of $2$-complexes and diagrammatic reducibility
S. M.
Gersten
689-706
Abstract: The condition that all spherical diagrams in a $2$-complex be reducible is shown to be equivalent to the condition that all finite branched covers be aspherical. This result is related to the study of equations over groups. Furthermore large classes of $ 2$-complexes are shown to be diagrammatically reducible in the above sense; in particular, every $2$-complex has a subdivision which admits a finite branched cover which is diagrammatically reducible.
Group actions on the complex projective plane
Dariusz M.
Wilczyński
707-731
Abstract: Let $G$ be a finite or compact Lie group. It is shown that $G$ acts on the complex projective plane (resp. on the Chern manifold) if and only if $G$ is isomorphic to a subgroup (resp. a pseudofree subgroup) of $PU(3)$. All actions considered are effective, locally smooth, and trivial on homology.
Nilpotent spaces of sections
Jesper Michael
Møller
733-741
Abstract: The space of sections of a fibration is nilpotent provided the base is finite $CW$-complex and the fiber is nilpotent. Moreover, localization commutes with the formation of section spaces.
A finiteness theorem in the Galois cohomology of algebraic number fields
Wayne
Raskind
743-749
Abstract: In this note we show that if $k$ is an algebraic number field with algebraic closure $\overline k$ and $M$ is a finitely generated, free ${{\mathbf{Z}}_l}$-module with continuous $ \operatorname{Gal} (\overline k /k)$-action, then the continuous Galois cohomology group $ {H^1}(k,\,M)$ is a finitely generated $ {{\mathbf{Z}}_l}$-module under certain conditions on $M$ (see Theorem 1 below). Also, we present a simpler construction of a mapping due to S. Bloch which relates torsion algebraic cycles and étale cohomology.
An operator-theoretic formulation of asynchronous exponential growth
G. F.
Webb
751-763
Abstract: A strongly continuous semigroup of bounded linear operators $ T(t)$, $t \geqslant 0$, in the Banach space $X$ has asynchronous exponential growth with intrinsic growth constant ${\lambda _0}$ provided that there is a nonzero finite rank operator ${P_0}$ in $X$ such that ${\lim _{t \to \infty }}{e^{ - {\lambda _0}t}}T(t) = {P_0}$. Necessary and sufficient conditions are established for $T(t)$, $ t \geqslant 0$, to have asynchronous exponential growth. Applications are made to a maturity-time model of cell population growth and a transition probability model of cell population growth.
Pure subgroups of torsion-free groups
Paul
Hill;
Charles
Megibben
765-778
Abstract: In this paper, we show that certain new notions of purity stronger than the classical concept are relevant to the study of torsion-free abelian groups. In particular, implications of $ {\ast}$-purity, a concept introduced in one of our recent papers, are investigated. We settle an open question (posed by Nongxa) by proving that the union of an ascending countable sequence of ${\ast}$-pure subgroups is completely decomposable provided the subgroups are. This result is false for ordinary purity. The principal result of the paper, however, deals with $\Sigma$-purity, a concept stronger than $ {\ast}$-purity but weaker than the usual notion of strong purity. Our main theorem, which has a number of corollaries including the recent result of Nongxa that strongly pure subgroups of separable groups are again separable, states that a $ \Sigma$-pure subgroup of a $k$-group is itself a $k$-group. Among other results is the negative resolution of the conjecture (valid in the countable case) that a strongly pure subgroup of a completely decomposable group is again completely decomposable.
Quasi $F$-covers of Tychonoff spaces
M.
Henriksen;
J.
Vermeer;
R. G.
Woods
779-803
Abstract: A Tychonoff topological space is called a quasi $F$-space if each dense cozero-set of $X$ is $ {C^{\ast}}$-embedded in $ X$. In Canad. J. Math. 32 (1980), 657-685 Dashiell, Hager, and Henriksen construct the "minimal quasi $F$-cover" $QF(X)$ of a compact space $X$ as an inverse limit space, and identify the ring $C(QF(X))$ as the order-Cauchy completion of the ring $ {C^{\ast}}(X)$. In On perfect irreducible preimages, Topology Proc. 9 (1984), 173-189, Vermeer constructed the minimal quasi $F$-cover of an arbitrary Tychonoff space. In this paper the minimal quasi $F$-cover of a compact space $X$ is constructed as the space of ultrafilters on a certain sublattice of the Boolean algebra of regular closed subsets of $X$. The relationship between $ QF(X)$ and $QF(\beta X)$ is studied in detail, and broad conditions under which $\beta (QF(X)) = QF(\beta X)$ are obtained, together with examples of spaces for which the relationship fails. (Here $\beta X$ denotes the Stone-Čech compactification of $X$.) The role of $QF(X)$ as a "projective object" in certain topological categories is investigated.
Rings of differential operators on invariant rings of tori
Ian M.
Musson
805-827
Abstract: Let $k$ be an algebraically closed field of characteristic zero and $G$ a torus acting diagonally on ${k^s}$. For a subset $\beta$ of ${\mathbf{s}} = \{ 1,\,2, \ldots ,\,s\}$, set ${U_\beta } = \{ u \in {k^s}\vert{u_j} \ne 0\;{\text{if}}\;j \in \beta \}$. Then $G$ acts on $\mathcal{O}({U_\beta })$, the ring of regular functions on ${U_\beta }$, and we study the ring $ D(\mathcal{O}{({U_\beta })^G})$ of all differential operators on the invariant ring. More generally suppose that $\Delta$ is a set of subsets of s, such that each invariant ring $\mathcal{O}{({U_\beta })^G}$, $\beta \in \Delta$, has the same quotient field. We prove that ${ \cap _{\beta \in \Delta }}D(\mathcal{O}{({U_\beta })^G})$ is Noetherian and finitely generated as a $ k$-algebra. Now $ G$ acts on each $ D(\mathcal{O}({U_\beta }))$ and there is a natural map $\displaystyle \theta :\bigcap\limits_{\beta \in \Delta } {D{{(\mathcal{O}({U_\b... ..._{\beta \in \Delta } {D(\mathcal{O}{{({U_\beta })}^G}) = D({Y_\Delta } / G)} }$ obtained by restriction of the differential operators. We find necessary and sufficient conditions for $ \theta$ to be surjective and describe the kernel of $\theta$. The algebras ${ \cap _{\beta \in \Delta }}D{(\mathcal{O}({U_\beta }))^G}$ and ${ \cap _{\beta \in \Delta }}D(\mathcal{O}{({U_\beta })^G})$ carry a natural filtration given by the order of the differential operators. We show that the associated graded rings are finitely generated commutative algebras and are Gorensetin rings. We also determine the centers of ${ \cap _{\beta \in \Delta }}D{(\mathcal{O}({U_\beta }))^G}$ and ${ \cap _{\beta \in \Delta }}D(\mathcal{O}{({U_\beta })^G})$.
Stopping times and $\Gamma$-convergence
J.
Baxter;
G.
Dal Maso;
U.
Mosco
1-38
Abstract: The equation $ \partial u/\partial t = \Delta u - \mu u$ represents diffusion with killing. The strength of the killing is described by the measure $ \mu$, which is not assumed to be finite or even $\sigma$-finite (to illustrate the effect of infinite values for $\mu$, it may be noted that the diffusion is completely absorbed on any set $A$ such that $ \mu (B) = \infty$ for every nonpolar subset $B$ of $A$). In order to give rigorous mathematical meaning to this general diffusion equation with killing, one may interpret the solution $u$ as arising from a variational problem, via the resolvent, or one may construct a semigroup probabilistically, using a multiplicative functional. Both constructions are carried out here, shown to be consistent, and applied to the study of the diffusion equation, as well as to the study of the related Dirichlet problem for the equation $\Delta u - \mu u = 0$. The class of diffusions studied here is closed with respect to limits when the domain is allowed to vary. Two appropriate forms of convergence are considered, the first being $\gamma $-convergence of the measures $\mu$, which is defined in terms of the variational problem, and the second being stable convergence in distribution of the multiplicative functionals associated with the measures $\mu$. These two forms of convergence are shown to be equivalent.
Character table and blocks of finite simple triality groups $\sp 3D\sb 4(q)$
D. I.
Deriziotis;
G. O.
Michler
39-70
Abstract: Based on recent work of Spaltenstein [14] and the Deligne-Lusztig theory of irreducible characters of finite groups of Lie type, in this paper the character table of the finite simple groups ${}^3{D_4}(q)$ is given. As an application we obtain a classification of the irreducible characters of ${}^3{D_4}(q)$ into $r$-blocks for all primes $r > 0$. This enables us to verify Brauer's height zero conjecture, his conjecture on the bound of irreducible characters belonging to a give block, and the Alperin-McKay conjecture for the simple triality groups $ {}^3{D_4}(q)$. It also follows that for every prime $r$ there are blocks of defect zero in ${}^3{D_4}(q)$.
An \'etale cohomology duality theorem for number fields with a real embedding
Mel
Bienenfeld
71-96
Abstract: The restriction on $ 2$-primary components in the Artin-Verdier duality theorem [2] has been eliminated by Zink [9], who has shown that the sheaf of units for the étale topology over the ring of integers of any number field acts as a dualizing sheaf for a modified cohomology of sheaves. The present paper provides an alternate means of removing the $2$-primary restriction. Like Zink's, it involves a topology which includes infinite primes, but it avoids modified cohomology and will be more directly applicable in the proof of a theorem of Lichtenbaum regarding zeta- and $L$-functions [4, 5]. Related results--including the cohomology of units sheaves, the norm theorem, and punctual duality theorem of Mazur [6]--are also affected by the use of a topology including the infinite primes. The corresponding results in the new setting are included here.
Braids and the Jones polynomial
John
Franks;
R. F.
Williams
97-108
Abstract: An important new invariant of knots and links is the Jones polynomial, and the subsequent generalized Jones polynomial or two-variable polynomial. We prove inequalities relating the number of strands and the crossing number of a braid with the exponents of the variables in the generalized Jones polynomial which is associated to the link formed from the braid by connecting the bottom ends to the top ends. We also relate an exponent in the polynomial to the number of components of this link.
Where does the $L\sp p$-norm of a weighted polynomial live?
H. N.
Mhaskar;
E. B.
Saff
109-124
Abstract: For a general class of nonnegative weight functions $w(x)$ having bounded or unbounded support $ \Sigma \subset {\mathbf{R}}$, the authors have previously characterized the smallest compact set $ {\mathfrak{S}_w}$, having the property that for every $n = 1,\,2, \ldots$ and every polynomial $ P$ of degree $\leqslant n$, $\displaystyle \vert\vert{[w(x)]^n}P(x)\vert{\vert _{{L^\infty }(\Sigma )}} = \vert\vert{[w(x)]^n}P(x)\vert{\vert _{{L^\infty }({\mathfrak{S}_w})}}$ . In the present paper we prove that, under mild conditions on $w$, the ${L^p}$-norms $ (0 < p < \infty )$ of such weighted polynomials also "live" on ${\mathfrak{S}_w}$ in the sense that for each $ \eta > 0$ there exist a compact set $\Delta$ with Lebesgue measure $m(\Delta ) < \eta$ and positive constants $ {c_1}$, ${c_2}$ such that $\displaystyle \vert\vert{w^n}P\vert{\vert _{{L^p}(\Sigma )}} \leqslant (1 + {c_... ... - {c_2}n))\vert\vert{w^n}P\vert{\vert _{{L^p}({\mathfrak{S}_w} \cup \Delta )}}$ . As applications we deduce asymptotic properties of certain extremal polynomials that include polynomials orthogonal with respect to a fixed weight over an unbounded interval. Our proofs utilize potential theoretic arguments along with Nikolskii-type inequalities.
The Milne problem for the radiative transfer equations (with frequency dependence)
François
Golse
125-143
Abstract: We study the following stationary frequency dependent transport equation: \begin{displaymath}\begin{array}{*{20}{c}} {\mu {\partial _x}f + \sigma (\nu ,\,... ... (\mu ,\,\nu ),\qquad \nu > 0,\;u \in ]0;\,1[,} \end{array} \end{displaymath} where ${B_\nu }$ is the well-known Planck function appearing in astrophysics. We are able to describe the asymptotic behavior of $f$ and $T$ for $x$ large, when $ \sigma (\nu ,\,T)$ is of the special form $\sigma (\nu ,\,T) = \sigma (\nu )k(T)$. Our method relies mainly on the monotonicity of the nonlinearity. The proof does not use any linearization of the equation; in particular, no smallness assumption on the data $\varphi$ (in any sense) is required. Résumé. Nous étudions l'équation de transport stationnaire avec dépendance en fréquence: \begin{displaymath}\begin{array}{*{20}{c}} {\mu {\partial _x}f + \sigma (\nu ,\,... ...\mu ,\,\nu );\qquad \nu > 0,\;\mu \in ]0;\,1[.} \end{array} \end{displaymath} Lorsque $ \sigma (\nu ,\,T)$ est de la forme particulière $\sigma (\nu ,\,T) = \sigma (\nu )k(T)$, nous savons décrire le comportement asymptotique de $ f$ et $T$ pour $x$ grand. Notre méthode repose principalement sur la monotonie de la non-linéarité. La preuve n'utilise aucune linéarisation de l'équation; en particulier, nous n'avons besoin d'aucune hypothèse de petitesse (d'aucune sorte) sur la donnée $\varphi$.
The structure of groups which are almost the direct sum of countable abelian groups
Alan H.
Mekler
145-160
Abstract: The notion of being in standard form is defined for the groups described in the title of the paper which are of cardinality ${\omega _1}$. Being in "standard form" is a structural description of the group. The consequences of being in standard form are explored, sometimes with the use of additional set-theoretic axioms. It is shown that it is consistent that a large class of these groups, including every weakly ${\omega _1}$-separable $ {\omega _1}$-$ \Sigma$-cyclic group of cardinality $ {\omega _1}$, can be put in standard form.
Scalar curvature and warped products of Riemann manifolds
F.
Dobarro;
E.
Lami Dozo
161-168
Abstract: We establish the relationship between the scalar curvature of a warped product $M \times {}_fN$ of Riemann manifolds and those ones of $M$ and $N$. Then we search for weights $f$ to obtain constant scalar curvature on $M \times {}_fN$ when $M$ is compact.
Global existence for $1$D, compressible, isentropic Navier-Stokes equations with large initial data
David
Hoff
169-181
Abstract: We prove the global existence of weak solutions of the Cauchy problem for the Navier-Stokes equations of compressible, isentropic flow of a polytropic gas in one space dimension. The initial velocity and density are assumed to be in $ {L^2}$ and ${L^2} \cap BV$ respectively, modulo additive constants. In particular, no smallness assumptions are made about the intial data. In addition, we prove a result concerning the asymptotic decay of discontinuities in the solution when the adiabatic constant exceeds $ 3/2$.
Strong Ramsey theorems for Steiner systems
Jaroslav
Nešetřil;
Vojtěch
Rödl
183-192
Abstract: It is shown that the class of partial Steiner $(k,\,l)$-systems has the edge Ramsey property, i.e., we prove that for every partial Steiner $ (k,\,l)$-system $\mathcal{G}$ there exists a partial Steiner $ (k,\,l)$-system $\mathcal{H}$ such that for every partition of the edges of $ \mathcal{H}$ into two classes one can find an induced monochromatic copy of $\mathcal{G}$. As an application we get that the class of all graphs without cycles of lengths $ 3$ and $4$ has the edge Ramsey property. This solves a longstanding problem in the area.
Polynomial approximation in the mean with respect to harmonic measure on crescents
John
Akeroyd
193-199
Abstract: For $1 \leqslant s < \infty $ and "nice" crescents $ G$, this paper gives a necessary condition (Theorem 2.6) and a sufficient condition (Theorem 2.5) for density of the polynomials in the generalized Hardy space ${H^s}(G)$. These conditions are easily tested and almost equivalent.
On a property of Castelnuovo varieties
Ciro
Ciliberto
201-210
Abstract: Castelnuovo varieties are those irreducible complete varieties in a projective space whose geometric genus is maximal according to their dimension, degree and embedding dimension. In this paper, extending results by Severi and Accola, we prove that, under suitable conditions, such varieties are birational if and only if they are projectively equivalent.
Bounded weak solutions of an elliptic-parabolic Neumann problem
J.
Hulshof
211-227
Abstract: In this paper we establish existence and uniqueness for bounded weak solutions of an elliptic-parabolic Neumann problem. We also describe the asymptotic behavior as $t \to \infty$.
On the differentiability of Lipschitz-Besov functions
José R.
Dorronsoro
229-240
Abstract: ${L^r}$ and ordinary differentiability is proved for functions in the Lipschitz-Besov spaces $ B_a^{p,q},\;1 \leqslant p < \infty ,\;1 \leqslant q \leqslant \infty ,\;a > 0$, using certain maximal operators measuring smoothness. These techniques allow also the study of lacunary directional differentiability and of tangential convergence of Poisson integrals.
Thue equations with few coefficients
Wolfgang M.
Schmidt
241-255
Abstract: Let $F(x,\,y)$ be a binary form of degree $r \geqslant 3$ with integer coefficients, and irreducible over the rationals. Suppose that only $s + 1$ of the $r + 1$ coefficients of $ F$ are nonzero. Then the Thue equations $ F(x,\,y) = 1$ has $\ll {(rs)^{1/2}}$ solutions. More generally, the inequality $\vert F(x,\,y)\vert \leqslant h$ has $\ll {(rs)^{1/2}}{h^{2/r}}(1 + \log {h^{1/r}})$ solutions.
A new proof that Teichm\"uller space is a cell
A. E.
Fischer;
A. J.
Tromba
257-262
Abstract: A new proof is given, using the energy of a harmonic map, that Teichmüller space is a cell.
Riccati techniques and variational principles in oscillation theory for linear systems
G. J.
Butler;
L. H.
Erbe;
A. B.
Mingarelli
263-282
Abstract: We consider the seond order differential system $(1)\,Y'' + Q(t)Y = 0$, where $Q$, $Y$ are $n \times n$ matrices with $Q = Q(t)$ a continuous symmetric matrix-valued function, $t \in [a,\, + \infty ]$. We obtain a number of sufficient conditions in order that all prepared solutions $ Y(t)$ of $(1)$ are oscillatory. Two approaches are considered, one based on Riccati techniques and the other on variational techniques, and involve assumptions on the behavior of the eigenvalues of $ Q(t)$ (or of its integral). These results extend some well-known averaging techniques for scalar equations to system $(1)$.
Smale flows on the three-sphere
Ketty
de Rezende
283-310
Abstract: In this paper, a complete classification of Smale flows on $ {S^3}$ is obtained. This classification is presented by means of establishing a concise set of properties that must be satisfied by an (abstract) Lyapunov graph associated to a Smale flow and a Lyapunov function. We show that these properties are necessary, that is, given a Smale flow and a Lyapunov function, its Lyapunov graph satisfies this set of properties. We also show that these properties are sufficient, that is, given an abstract Lyapunov graph $ L'$ satisfying this set of properties, it is possible to realize a Smale flow on $ {S^3}$ that has a graph $ L$ as its Lyapunov graph where $L$ is equal to $L'$ up to topological equivalence. The techniques employed in proving that the conditions imposed on the graph are necessary involve some use of homology theory. Geometrical methods are used to construct the flow on ${S^3}$ associated to the given graph and therefore establish the sufficiency of the above conditions. The main theorem in this paper generalizes a result of Franks [8] who classified nonsingular Smale flows on $ {S^3}$.
Sporadic and irrelevant prime divisors
Stephen
McAdam;
L. J.
Ratliff
311-324
Abstract: Let $I$ represent a regular ideal in a Noetherian ring $R$. If $W$ is a finite set of prime ideals in $ R$, some conditions on $ W$ are given assuring that an $I$ can be found such that $W$ is exactly the set of primes which are in $ \operatorname{Ass} R/I$ but not in $\operatorname{Ass} R/{I^n}$ for all large $ n$. Furthermore, if $ I$ is fixed, and if $ P$ is a prime ideal containing $I$, some conditions are given assuring that in the Rees ring ${\mathbf{R}} = R[u,\,It],\,(u,\,P,\,It){\mathbf{R}}$ is a prime divisor of $u{\mathbf{R}}$.
A formula for the resolvent of $(-\Delta)\sp m+M\sp {2m}\sb q$ with applications to trace class
Peter
Takáč
325-344
Abstract: We derive a formula for the resolvent of the elliptic operator $H = {( - \Delta )^m} + M_q^{2m}$ on $ {L_2}({\mathbb{R}^N})$ in terms of bounded integral operators ${S_\lambda }$ and $ {T_\lambda }$ whose kernels we know explicitly. We use this formula to specify the domain of the operator ${A_\lambda } = (H + \lambda I){M_p}$ on ${L_2}({\mathbb{R}^N})$, and to estimate the Hilbert-Schmidt norm of its inverse $A_\lambda ^{ - 1}$, for $\lambda \geqslant 0$. Finally we exploit the last two results to prove a trace class criterion for an integral operator $K$ on $ {L_2}({\mathbb{R}^N})$.
On the generalized spectrum for second-order elliptic systems
Robert Stephen
Cantrell;
Chris
Cosner
345-363
Abstract: We consider the system of homogeneous Dirichlet boundary value problems $ ({\ast})$ $\displaystyle {L_1}u = \lambda [{a_{11}}(x)u + {a_{12}}(x)v],\quad {L_2}v = \mu [{a_{12}}(x)u + {a_{22}}(x)v]$ in a smooth bounded domain $ \Omega \subseteq {{\mathbf{R}}^N}$, where ${L_1}$ and ${L_2}$ are formally self-adjoint second-order strongly uniformly elliptic operators. Using linear perturbation theory, continuation methods, and the Courant-Hilbert variational eigenvalue characterization, we give a detailed qualitative and quantitative description of the real generalized spectrum of $ ({\ast})$, i.e., the set $(\lambda ,\,\mu ) \in {{\mathbf{R}}^2}:\,({\ast})$ has a nontrivial solution. The generalized spectrum, a term introduced by Protter in 1979, is of considerable interest in the theory of linear partial differential equations and also in bifurcation theory, as it is the set of potential bifurcation points for associated semilinear systems.
Scalar curvatures on $S\sp 2$
Wen Xiong
Chen;
Wei Yue
Ding
365-382
Abstract: A theorem for the existence of solutions of the nonlinear elliptic equation $- \Delta u + 2 = R(x){e^u},\;x \in {S^2}$, is proved by using a "mass center" analysis technique and by applying a continuous "flow" in ${H^1}({S^2})$ controlled by $\nabla R$.
A GCH example of an ordinal graph with no infinite path
Jean A.
Larson
383-393
Abstract: It is hard to find nontrivial positive partition relations which hold for many ordinals in ordinary set theory, or even ordinary set theory with the additional assumption of the Generalized Continuum Hypothesis. Erdös, Hajnal and Milner have proved that limit ordinals $ \alpha < \omega _1^{\omega + 2}$ satisfy a positive partition relation that can be expressed in graph theoretic terms. In symbols one writes $\alpha \to {(\alpha ,\,\operatorname{infinite} \operatorname{path} )^2}$ to mean that every graph on an ordinal $\alpha$ either has a subset order isomorphic to $ \alpha$ in which no two points are joined by an edge or has an infinite path. This positive result generalizes to ordinals of cardinality $ {\aleph _m}$ for $ m$ a natural number. However, the argument, based on a set mapping theorem, works only on the initial segment of the limit ordinals of cardinality ${\aleph _m}$ for which the set mapping theorem is true. In this paper, the Generalized Continuum Hypothesis is used to construct counterexamples for a cofinal set of ordinals of cardinality ${\aleph _m}$, where $m$ is a natural number at least two.
Convergence in distribution of products of random matrices: a semigroup approach
Arunava
Mukherjea
395-411
Abstract: The problem of weak convergence of the sequence of convolution powers of a probability measure has been considered in this paper in the general context of a noncompact semigroup and in particular, in the semigroup of nonnegative and real matrices. Semigroup methods have been used to give simple proofs of some recent results of Kesten and Spitzer in nonnegative matrices. It has been also shown that these methods often lead to similar results in the more general context of real matrices.
Homological stability for ${\rm O}\sb {n,n}$ over a local ring
Stanisław
Betley
413-429
Abstract: Let $R$ be a local ring, ${V^{2n}}$ a free module over $ R$ of rank $2n$ and $q$ a bilinear form on ${V^{2n}}$ which has in some basis the matrix $ \left\vert {\begin{array}{*{20}{c}} 0 & 1 1 & 0 \end{array} } \right\vert\,$. Let ${O_{n,n}}$ be the group of automorphisms of $ {V^{2n}}$ which preserve $ q$. We prove the following theorem: if $n$ is big enough with respect to $k$ then the inclusion homomorphism $ i:{O_{n,n}} \to {O_{n + 1,n + 1}}$ induces an isomorphism $ {i_{\ast}}:{H_k}({O_{n,n}};\,Z) \to {H_k}({O_{n + 1,n + 1}};Z)$.
Stationary configurations of point vortices
Kevin Anthony
O’Neil
383-425
Abstract: The motion of point vortices in a plane of fluid is an old problem of fluid mechanics, which was given a Hamiltonian formulation by Kirchhoff. Stationary configurations are those which remain self-similar throughout the motion. Results of two types are presented. Configurations which are in equilibrium or which translate uniformly are counted using methods of algebraic geometry, which establish necessary and sufficient conditions for existence. Relative equilibria (rigidly rotating configurations) which lie on a line are studied using a topological construction applicable to other power-law systems. Upper and lower bounds for such configurations are found for vortices with mixed circulations. Arrangements of three vortices which collide in finite time are well known. One-dimensional families of such configurations are shown to exist for more than three vortices. Stationary configurations of four vortices are examined in detail.
The Fraser-Horn and Apple properties
Joel
Berman;
W. J.
Blok
427-465
Abstract: We consider varieties $\mathcal{V}$ in which finite direct products are skew-free and in which the congruence lattices of finite directly indecomposables have a unique coatom. We associate with $ \mathcal{V}$ a family of derived varieties, $ d(\mathcal{V})$: a variety in $ d(\mathcal{V})$ is generated by algebras $ {\mathbf{A}}$ where the universe of $ {\mathbf{A}}$ consists of a congruence class of the coatomic congruence of a finite directly indecomposable algebra ${\mathbf{B}} \in \mathcal{V}$ and the operations of $ {\mathbf{A}}$ are those of ${\mathbf{B}}$ that preserve this congruence class. We also consider the prime variety of $\mathcal{V}$, denoted ${\mathcal{V}_0}$, generated by all finite simple algebras in $ \mathcal{V}$. We show how the structure of finite algebras in $\mathcal{V}$ is determined to a considerable extent by $ {\mathcal{V}_0}$ and $d(\mathcal{V})$. In particular, the free $\mathcal{V}$-algebra on $n$ generators, $ {{\mathbf{F}}_\mathcal{V}}(n)$, has as many directly indecomposable factors as $ {{\mathbf{F}}_{{\mathcal{V}_0}}}(n)$ and the structure of these factors is determined by the varieties $ d(\mathcal{V})$. This allows us to produce in many cases explicit formulas for the cardinality of $ {{\mathbf{F}}_\mathcal{V}}(n)$. Our work generalizes the structure theory of discriminator varieties and, more generally, that of arithmetical semisimple varieties. The paper contains many examples of algebraic systems that have been investigated in different contexts; we show how these all fit into a general scheme.
Graded Lie algebras of the second kind
Jih Hsin
Chêng
467-488
Abstract: The associated Lie algebra of the Cartan connection for an abstract CR-hypersurface admits a gradation of the second kind. In this article, we give two ways to characterize this kind of graded Lie algebras, namely, geometric characterization in terms of symmetric spaces and algebraic characterization in terms of root systems. A complete list of this class of Lie algebras is given.
The multiplicity of isolated two-dimensional hypersurface singularities
Henry B.
Laufer
489-496
Abstract: Consider an isolated two-dimensional complex analytic hypersurface singularity $(V,p)$. A relation is given between the abstract topology of $(V,p)$ and the multiplicity of $ (V,p)$, yielding an upper bound for the multiplicity. This relation is a necessary condition for a Gorenstein singularity to be a hypersurface.
The topology of resolution towers
Selman
Akbulut;
Henry
King
497-521
Abstract: An obstruction theory is given to determine when a space has a resolution tower. This can be used to decide whether or not the space is homeomorphic to a real algebraic set.
Produced representations of Lie algebras and Harish-Chandra modules
Michael J.
Heumos
523-534
Abstract: The comultiplication of the universal enveloping algebra of a Lie algebra is used to give modules produced from a subalgebra, an additional compatible structure of a module over an algebra of formal power series. When only the $ \mathfrak{k}$-finite elements of this algebra act on a module, conditions are given that insure that it is the Harish-Chandra module of a produced module. The results are then applied to Zuckerman derived functor modules for reductive Lie algebras. The main application describes a setting where the Zuckerman functors and production from a subalgebra commute.
Prime ideals in enveloping rings
D. S.
Passman
535-560
Abstract: Let $L$ be a Lie algebra over the field $ K$ of characteristic 0 and let $U(L)$ denote its universal enveloping algebra. If $R$ is a $K$-algebra and $L$ acts on $R$ as derivations, then there is a natural ring generated by $R$ and $U(L)$ which is denoted by $R\char93 U(L)$ and called the smash product of $ R$ by $U(L)$. The aim of this paper is to describe the prime ideals of this algebra when it is Noetherian. Specifically we show that there exists a twisted enveloping algebra $U(X)$ on which $L$ acts and a precisely defined one-to-one correspondence between the primes $P$ of $R\char93 U(L)$ with $ P \cap R = 0$ and the $ L$-stable primes of $ U(X)$. Here $X$ is a Lie algebra over some field $C \supseteq K$.
Strong multiplicity theorems for ${\rm GL}(n)$
George T.
Gilbert
561-576
Abstract: Let $\pi = \otimes {\pi _\upsilon }$ be a cuspidal automorphic representation of $GL(n,{F_A})$, where ${F_A}$ denotes the adeles of a number field $ F$. Let $E$ be a Galois extension of $ F$ and let $\{ g\}$ denote a conjugacy class of the Galois group. The author considers those cuspidal automorphic representations which have local components ${\pi _\upsilon }$ whenever the Frobenius of the prime $\upsilon$ is $\{ g\}$, showing that such representations are often easily described and finite in number. This generalizes a result of Moreno [Bull. Amer. Math. Soc. 11 (1984), pp. 180-182].
Proof of a conjecture of Kostant
Dragomir Ž.
Đoković
577-585
Abstract: Let ${\mathfrak{g}_0} = {\mathfrak{k}_0} + {\mathfrak{p}_0}$ be a Cartan decomposition of a semisimple real Lie algebra and $\mathfrak{g} = \mathfrak{k} + \mathfrak{p}$ its complexification. Denote by $G$ the adjoint group of $ \mathfrak{g}$ and by ${G_0},K,{K_0}$ the connected subgroups of $ G$ with respective Lie algebras $ {\mathfrak{g}_0},\mathfrak{k},{\mathfrak{k}_0}$. A conjecture of Kostant asserts that there is a bijection between the $ {G_0}$-conjugacy classes of nilpotent elements in $ {\mathfrak{g}_0}$ and the $ K$-orbits of nilpotent elements in $ \mathfrak{p}$ which is given explicitly by the so-called Cayley transformation. This conjecture is proved in the paper.
Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations
Klaus W.
Schaaf
587-615
Abstract: This paper is a generalization of the theory of the KPP and bistable nonlinear diffusion equations. It is shown that traveling wave solutions exist for nonlinear parabolic functional differential equations (FDEs) which behave very much like the well-known solutions of the classical KPP and bistable equations. Among the techniques used are maximum principles, sub- and supersolutions, phase plane techniques for FDEs and perturbation of linear operators.
VMO, ESV, and Toeplitz operators on the Bergman space
Ke He
Zhu
617-646
Abstract: This paper studies the largest ${C^*}$-subalgebra $Q$ of ${L^\infty }({\mathbf{D}})$ such that the Toeplitz operators ${T_f}$ on the Bergman space $L_a^2({\mathbf{D}})$ with symbols $f$ in $Q$ have a symbol calculus modulo the compact operators. $Q$ is characterized by a condition of vanishing mean oscillation near the boundary. I also give several other necessary and sufficient conditions for a bounded function to be in $Q$. After decomposing $Q$ in a "nice" way, I study the Fredholm theory of Toeplitz operators with symbols in $Q$. The essential spectrum of ${T_f}(f \in Q)$ is shown to be connected and computable in terms of the Stone-Cěch compactification of $ {\mathbf{D}}$. The results in this article partially answer a question posed in [3] and give several new necessary and sufficient conditions for a bounded analytic function on the open unit disc to be in the little Bloch space ${\mathcal{B}_0}$.
The connectedness of the group of automorphisms of $L\sp 1(0,1)$
F.
Ghahramani
647-659
Abstract: For each of the radical Banach algebras $ {L^1}(0,1)$ and $ {L^1}(w)$ an integral representation for the automorphisms is given. This is used to show that the groups of the automorphisms of ${L^1}(0,1)$ and ${L^1}(w)$ endowed with bounded strong operator topology (BSO) are arcwise connected. Also it is shown that if $ \vert\vert\vert \cdot \vert\vert{\vert _p}$ denotes the norm of $B({L^p}(0,1)$, $ {L^1}(0,1))$, $1 < p \leq \infty$, then the group of automorphisms of $ {L^1}(0,1)$ topologized by $\vert\vert\vert \cdot \vert\vert{\vert _p}$ is arcwise connected. It is shown that every automorphism $\theta$ of $ {L^1}(0,1)$ is of the form $\theta = {e^{\lambda d}}{\operatorname{lim}}{e^{qn}}({\text{BSO}})$, where each ${q_n}$ is a quasinilpotent derivation. It is shown that the group of principal automorphisms of $ {l^1}(w)$ under operator norm topology is arcwise connected, and every automorphism has the form $ {e^{i\alpha d}}{({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }$, where $\alpha \in {\mathbf{R}}$, $\lambda > 0$, and $D$ is a derivation, and where $ {({e^{\lambda d}}{e^D}{e^{ - \lambda d}})^ - }$ denotes the extension by continuity of ${e^{\lambda d}}{e^D}{e^{ - \lambda d}}$ from a dense subalgebra of ${l^1}(w)$ to ${l^1}(w)$.
Convex subcones of the contingent cone in nonsmooth calculus and optimization
Doug
Ward
661-682
Abstract: The tangential approximants most useful in nonsmooth analysis and optimization are those which lie "between" the Clarke tangent cone and the Bouligand contigent cone. A study of this class of tangent cones is undertaken here. It is shown that although no convex subcone of the contingent cone has the isotonicity property of the contingent cone, there are such convex subcones which are more "accurate" approximants than the Clarke tangent cone and possess an associated subdifferential calculus that is equally strong. In addition, a large class of convex subcones of the contingent cone can replace the Clarke tangent cone in necessary optimality conditions for a nonsmooth mathematical program. However, the Clarke tangent cone plays an essential role in the hypotheses under which these calculus rules and optimality conditions are proven. Overall, the results obtained here suggest that the most complete theory of nonsmooth analysis combines a number of different tangent cones.
Degrees of splittings and bases of recursively enumerable subspaces
R. G.
Downey;
J. B.
Remmel;
L. V.
Welch
683-714
Abstract: This paper analyzes the interrelationships between the (Turing) of r.e. bases and of r.e. splittings of r.e. vector spaces together with the relationship of the degrees of bases and the degrees of the vector spaces they generate. For an r.e. subspace $V$ of $ {V_\infty }$, we show that $ \alpha$ is the degree of an r.e. basis of $V$ iff $\alpha$ is the degree of an r.e. summand of $ V$ iff $\alpha$ is the degree and dependence degree of an r.e. summand of $V$. This result naturally leads to explore several questions regarding the degree theoretic properties of pairs of summands and the ways in which bases may arise.
On the central limit theorem for dynamical systems
Robert
Burton;
Manfred
Denker
715-726
Abstract: Given an aperiodic dynamical system $ (X,T,\mu )$ then there is an $f \in {L^2}(\mu )$ with $\smallint fd\mu = 0$ satisfying the Central Limit Theorem, i.e. if $ {S_m}f = f + f \circ T + \cdots + f \circ {T^{m - 1}}$ and $ {\sigma _m} = {\left\Vert {{S_m}f} \right\Vert _2}$ then $\displaystyle \mu \left\{ {x\vert\frac{{{S_m}f(x)}}{{{\sigma _m}}} < u} \right\... ...fty }^u {{\text{exp}}} \left[ {\frac{{ - {\upsilon ^2}}}{2}} \right]d\upsilon .$ The analogous result also holds for flows.
On the Stickelberger ideal and the relative class number
Tatsuo
Kimura;
Kuniaki
Horie
727-739
Abstract: Let $k$ be any imaginary abelian field, $ R$ the integral group ring of $G = {\text{Gal}}(k/\mathbb{Q})$, and $S$ the Stickelberger ideal of $k$. Roughly speaking, the relative class number ${h^ - }$ of $k$ is expressed as the index of $S$ in a certain ideal $A$ of $R$ described by means of $G$ and the complex conjugation of $k;{c^ - }{h^ - } = [A:S]$, with a rational number ${c^ - }$ in $\frac{1} {2}\mathbb{N} = \{ n/2;n \in \mathbb{N}\}$, which can be described without ${h^ - }$ and is of lower than $ {h^ - }$ if the conductor of $k$ is sufficiently large (cf. [6, 9, 10]; see also [5]). We shall prove that $2{c^ - }$, a natural number, divides $ 2{([k:\mathbb{Q}]/2)^{[k:\mathbb{Q}]/2}}$. In particular, if $ k$ varies through a sequence of imaginary abelian fields of degrees bounded, then ${c^ - }$ takes only a finite number of values. On the other hand, it will be shown that ${c^ - }$ can take any value in $\frac{1} {2}\mathbb{N}$ when $k$ ranges over all imaginary abelian fields. In this connection, we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields.
Une minoration de la norme de l'op\'erateur de Cauchy sur les graphes lipschitziens
Guy
David
741-750
Abstract: It was shown by T. Murai that the norm of the operator defined by the Cauchy kernel on the graph of a Lipschitz function $ A$ is less than
$K$-theory and right ideal class groups for HNP rings
Timothy J.
Hodges
751-767
Abstract: Let $R$ be an hereditary Noetherian prime ring, let $S$ be a "Dedekind closure" of $R$ and let $ \mathcal{T}$ be the category of finitely generated $S$-torsion $R$-modules. It is shown that for all $i \geq 0$, there is an exact sequence $0 \to {K_i}(\mathcal{T}) \to {K_i}(R) \to {K_i}(S) \to 0$. If $i = 0$, or $R$ has finitely many idempotent ideals then this sequence splits. A notion of "right ideal class group" is then introduced for hereditary Noetherian prime rings which generalizes the standard definition of class group for hereditary orders over Dedekind domains. It is shown that there is a decomposition $ {K_0}(R) \cong {\text{Cl}}(R) \oplus F$ where $F$ is a free abelian group whose rank depends on the number of idempotent maximal ideals of $ R$. Moreover there is a natural isomorphism ${\text{Cl}}(R) \cong {\text{Cl}}(S)$ and this decomposition corresponds closely to the splitting of the above exact sequence for ${K_0}$.
The normal subgroup structure of the Picard group
Benjamin
Fine;
Morris
Newman
769-786
Abstract: The Picard group $ \Gamma$ is $PS{L_2}(Z[i])$, the group of linear fractional transformations with Gaussian integer coefficients. We examine the structure of the normal subgroups of $ \Gamma$. In particular we give a complete classification of the normal subgroups for indices less than $60$ and show that beyond this there are large gaps in the possible indices. This classification depends on the structure of the derived series. Finally we give examples of normal noncongruence subgroups.
An infinite-dimensional Hamiltonian system on projective Hilbert space
Anthony M.
Bloch
787-796
Abstract: We consider here the explicit integration of a Hamiltonian system on infinite-dimensional complex projective space. The Hamiltonian, which is the restriction of a linear functional to this projective space, arises in the problem of line fitting in complex Hilbert space (or, equivalently, the problem of functional approximation) or as the expectation value of a model quantum mechanical system. We formulate the system here as a Lax system with parameter, showing how this leads to an infinite set of conserved integrals associated with the problem and to an explicit formulation of the flow in action-angle form via an extension of some work of J. Moser. In addition, we find the algebraic curve naturally associated with the system.
Holomorphic mappings on $l\sb 1$
Raymond A.
Ryan
797-811
Abstract: We describe the holomorphic mappings of bounded type, and the arbitrary holomorphic mappings from the complex Banach space $ {l_1}$ into a complex Banach space $X$. It is shown that these mappings have monomial expansions and the growth of the norms of the coefficients is characterized in each case. This characterization is used to give new descriptions of the compact open topology and the Nachbin ported topology on the space $ \mathcal{H}({l_1};X)$ of holomorphic mappings, and to prove a lifting property for holomorphic mappings on ${l_1}$. We also show that the monomials form an equicontinuous unconditional Schauder basis for the space $(\mathcal{H}({l_1}),{\tau _0})$ of holomorphic functions on ${l_1}$ with the topology of uniform convergence on compact sets.
Reflection, removable singularities, and approximation for partial differential equations. II
Leon
Ehrenpreis
1-45
Abstract: Let ${\Omega ^j}$ be domains in ${R^n}$. For each $j$ we are given a system ${{\mathbf{D}}^j}$ of linear constant coefficient operators and a function ${f^j}$ on $ {\Omega ^j}$ satisfying ${{\mathbf{D}}^j}{f^j} = 0$. When the $ {f^j}$ satisfy certain compatibility conditions on the intersections ${{\mathbf{D}}^j}$ on larger domains. As a consequence of our methods we are able to sharpen Hartogs' theorems to allow for continuation of solutions of overdetermined systems over noncompact sets.
On the invariance of $q$-convexity and hyperconvexity under finite holomorphic surjections
Nguyẽn Văn
Khuê;
Lê Văn
Thành
47-54
Abstract: In this note we have proved that 0-convexity and hyperconvexity are invariant under finite holomorphic surjections. Invariance of cohomological $q$-convexity for the case of finite dimension also has been established.
The $L\sp 2$-boundedness of pseudodifferential operators
I. L.
Hwang
55-76
Abstract: We give a new proof of the Calderon-Vaillancourt theorem. We also obtain the ${L^2}$-continuity of $a(x,D)$ if its symbol $a(x,\xi )$ satisfies some suitable conditions.
Variations on Lusin's theorem
Jack B.
Brown;
Karel
Prikry
77-86
Abstract: We prove a theorem about continuous restrictions of Marczewski measurable functions to large sets. This theorem is closely related to the theorem of Lusin about continuous restrictions of Lebesgue measurable functions to sets of positive measure and the theorem of Nikodym and Kuratowski about continuous restrictions of functions with the Baire property (in the wide sense) to residual sets. This theorem is used to establish Lusin-type theorems for universally measurable functions and functions which have the Baire property in the restricted sense. The theorems are shown (under assumption of the Continuum Hypothesis) to be "best possible" within a certain context.
Almost split sequences for rational double points
Maurice
Auslander;
Idun
Reiten
87-97
Abstract: Let $R$ be a complete local two-dimensional integrally closed noetherian nonregular Gorenstein domain with maximal ideal $m$ over an algebraically closed field $ k$, such that $R/m \simeq k$. Then the category of finitely generated reflexive modules is known to have almost split sequences, and our main result is that the $ AR$-quiver is an extended Dynkin quiver $ {\tilde A_n}$, ${\tilde D_n}$, $ {\tilde E_6}$, ${\tilde E_7}$, or $ {\tilde E_8}$.
Half-canonical series on algebraic curves
Montserrat
Teixidor i Bigas
99-115
Abstract: Denote by $\mathcal{M}_g^r$ the locus in the moduli space of curves of genus $g$ of those curves which have a theta-characteristic of (projective) dimension at least $r$. We give an upper bound for the dimension of $ \mathcal{M}_g^r$ and we determine this dimension completely for $r \leqslant 4$. For $r \leqslant 4$, we prove also that a generic point in every component of $\mathcal{M}_g^r$ has a single theta-characteristic of this dimension.
On certain $3$-generator Artin groups
Craig C.
Squier
117-124
Abstract: We describe the three $3$-generator Artin groups that correspond to the three sets $\{ p,q,r\}$ of positive integer solutions of ${p^{ - 1}} + {q^{ - 1}} + {r^{ - 1}} = 1$. In each case, we show that the Artin group is a free product with amalgamation or HNN extension involving finitely generated free groups and subgroups of finite index.
Periodic points and automorphisms of the shift
Mike
Boyle;
Wolfgang
Krieger
125-149
Abstract: The automorphism group of a topological Markov shift is studied by way of periodic points and unstable sets. A new invariant for automorphisms of dynamical systems, the gyration function, is used to characterize those automorphisms of finite subsystems of the full shift on $n$ symbols which can be extended to a composition of involutions of the shift. It is found that for any automorphism $U$ of a subshift of finite type $S$, for all large integers $ M$ the map $U{S^M}$ is a topological Markov shift whose unstable sets equal those of $S$. This fact yields, by way of canonical measures and dimension groups, information about dynamical properties of $U{S^k}$ such as the zeta function and entropy.
Complete groups with nonabelian composition factors
Jay
Zimmerman
151-159
Abstract: A finite group is said to be complete if it has trivial center and if every automorphism is an inner automorphism. A finite group with nonabelian composition factors has a unique completely reducible radical (CR radical). We consider finite groups with nonabelian composition factors whose CR radical consists of complete simple groups and we give necessary and sufficient conditions for such a group to be complete. This involves finding group theoretic conditions which are necessary and sufficient for a finite centerless group to occur as a self-normalizing subgroup of a direct product of symmetric groups.
Interpolating sequences in the polydisc
Bo
Berndtsson;
Sun-Yung A.
Chang;
Kai-Ching
Lin
161-169
Abstract: Let ${H^\infty }({D^n})$ denote the set of all bounded analytic functions defined on the polydisc $ {D^n}$ of ${{\mathbf{C}}^n}$. In this note, we give a sufficient condition for sequences of points in ${D^n}$ to be interpolating sequences for ${H^\infty }({D^n})$. We also discuss some conditions for interpolation of general domains.
Special points on first order partial differential equations and the deformations of solutions
Marek
Kossowski
171-184
Abstract: The object of this paper is to identify four cases of special behavior in a class of first order PDE for a real valued function. (The class of PDE may be thought of as perturbation of PDE with singular solutions.) In each case we show how invariants of the PDE determine properties of solutions. The properties of solutions examined here are the structure of critical points and singularities induced by cotangent projection. These properties are described in the sense of constructing local models for solutions and characterizing their behavior under small deformations. We will find two cases where deformations exhibit bifurcation phenomena, and describe generic deformations.
Martin boundaries of random walks: ends of trees and groups
Massimo A.
Picardello;
Wolfgang
Woess
185-205
Abstract: Consider a transient random walk ${X_n}$ on an infinite tree $T$ whose nonzero transition probabilities are bounded below. Suppose that ${X_n}$ is uniformly irreducible and has bounded step-length. (Alternatively, $ {X_n}$ can be regarded as a random walk on a graph whose metric is equivalent to the metric of $T$.) The Martin boundary of ${X_n}$ is shown to coincide with the space $ \Omega$ of all ends of $ T$ (or, equivalently, of the graph). This yields a boundary representation theorem on $\Omega$ for all positive eigenfunctions of the transition operator, and a nontangential Fatou theorem which describes their boundary behavior. These results apply, in particular, to finitely supported random walks on groups whose Cayley graphs admit a uniformly spanning tree. A class of groups of this type is constructed.
The set of continuous functions with everywhere convergent Fourier series
M.
Ajtai;
A. S.
Kechris
207-221
Abstract: This paper deals with the descriptive set theoretic properties of the class $ \operatorname{EC}$ of continuous functions with everywhere convergent Fourier series. It is shown that this set is a complete coanalytic set in $C(T)$. A natural coanalytic rank function on $ \operatorname{EC}$ is studied that assigns to each $f \in \operatorname{EC}$ a countable ordinal number, which measures the "complexity" of the convergence of the Fourier series of $f$. It is shown that there exist functions in $ \operatorname{EC}$ (in fact even differentiable ones) which have arbitrarily large countable rank, so that this provides a proper hierarchy on $ \operatorname{EC}$ with ${\omega _1}$ distinct levels.
Totally ramified valuations on finite-dimensional division algebras
J.-P.
Tignol;
A. R.
Wadsworth
223-250
Abstract: Division algebras $ D$ with valuation $ v$ are studied, where $ D$ is finite-dimensional and totally ramified over its center $F$ (i.e., the ramification index of $ v$ over $v{\vert _F}$ equals $[D:F]$). Such division algebras have appeared in some important constructions, but the structure of these algebras has not been systematically analyzed before. When $v{\vert _F}$ is Henselian a full classification of the $F$-subalgebras of $D$ is given. When $F$ has a Henselian valuation $v$ with separably closed residue field and $A$ is any tame central simple $F$-algebra, an algorithm is given for computing the underlying division algebra of $ A$ from a suitable subgroup of $ {A^{\ast}}/{F^{\ast}}$. Some examples are constructed using this valuation theory, including the first example of finite-dimensional $ F$-central division algebras ${D_1}$ and ${D_2}$ with ${D_1}{ \otimes _F}{D_2}$ not a division ring, but $ {D_1}$ and ${D_2}$ having no common subfield $K \supsetneqq F$.
On a problem concerning permutation polynomials
Gerhard
Turnwald
251-267
Abstract: Let $S(f)$ denote the set of integral ideals $ I$ such that $ f$ is a permutation polynomial modulo $I$, where $f$ is a polynomial over the ring of integers of an algebraic number field. We obtain a classification for the sets $S$ which may be written in the form $S(f)$.
On the distribution of the number of prime factors of sums $a+b$
P.
Erdős;
H.
Maier;
A.
Sárközy
269-280
Abstract: We continue a series of investigations by A. Balog and two of the authors (P. Erdös and A. Sárközy) on the arithmetic properties of the elements $a + b$, where $a \in {\mathbf{A}}$, $b \in {\mathbf{B}}$, $ {\mathbf{A}}$ and ${\mathbf{B}}$ "dense sequences." The present paper transfers the famous Erdös-Kac theorem on the normal distribution of the number of distinct prime factors of integers to such "sum sequences."
Small zeros of quadratic forms over number fields
Jeffrey D.
Vaaler
281-296
Abstract: Let $F$ be a nontrivial quadratic form in $ N$ variables with coefficients in a number field $k$ and let $A$ be a $K \times N$ matrix over $k$. We show that if the simultaneous equations $ F({\mathbf{x}}) = 0$ and $A{\mathbf{x}} = 0$ hold on a subspace $\mathfrak{X}$ of dimension $L$ and $L$ is maximal, then such a subspace $\mathfrak{X}$ can be found with the height of $\mathfrak{X}$ relatively small. In particular, the height of $ \mathfrak{X}$ can be explicitly bounded by an expression depending on the height of $F$ and the height of $A$. We use methods from geometry of numbers over adèle spaces and local to global techniques which generalize recent work of H. P. Schlickewei.
Approximation by equivariant homeomorphisms. I
Mark
Steinberger;
James
West
297-317
Abstract: Locally linear (= locally smoothable) actions of finite groups on finite dimensional manifolds are considered in which two incident components of fixed point sets of subgroups either coincide or one has codimension at least three in the other. For these actions, an equivariant $ \alpha$-approximation theorem is proved using engulfing techniques. As corollaries are obtained equivariant "fibrations are bundles" and "controlled $h$-cobordism" theorems, as well as an equivariant version of Edwards' cell-like mapping theorem and the vanishing of the set of transfer-invariant $ G$-homotopy topological structures, rel boundary, on ${T^n} \times {D_\rho }$ (when ${T^n}$ is the $n$-torus with trivial $G$ action and ${D_\rho }$ is a representation disc).
The Conner-Floyd map for formal $A$-modules
Keith
Johnson
319-332
Abstract: A generalization of the Conner-Floyd map from complex cobordism to complex $ K$-theory is constructed for formal $A$-modules when $A$ is the ring of algebraic integers in a number field or its $p$-adic completion. This map is employed to study the Adams-Novikov spectral sequence for formal $ A$-modules and to confirm a conjecture of D. Ravenel.
A residual property of certain linear groups
Peter F.
Stebe
333-340
Abstract: An extension of residual finiteness, residual finiteness with respect to nests, is demonstrated for certain subgroups of $ GL(n,Z)$, the polycyclic by finite groups. It is also shown that groups containing a free subgroup of rank greater than $1$ cannot have the property. It is not settled whether or not there are other solvable by finite groups, subgroups allowed by Tits' theorem, that are residually finite with respect to nests.
Splitting of closed ideals in $({\rm DFN})$-algebras of entire functions and the property $({\rm DN})$
Reinhold
Meise;
B. Alan
Taylor
341-370
Abstract: For a plurisubharmonic weight function $p$ on $ {{\mathbf{C}}^n}$ let ${A_p}({{\mathbf{C}}^n})$ denote the (DFN)-algebra of all entire functions on $ {{\mathbf{C}}^n}$ which do not grow faster than a power of $\exp (p)$. We prove that the splitting of many finitely generated closed ideals of a certain type in $ {A_p}({{\mathbf{C}}^n})$, the splitting of a weighted $\overline \partial $-complex related with $ p$, and the linear topological invariant (DN) of the strong dual of ${A_p}({{\mathbf{C}}^n})$ are equivalent. Moreover, we show that these equivalences can be characterized by convexity properties of $p$, phrased in terms of greatest plurisubharmonic minorants. For radial weight functions $ p$, this characterization reduces to a covexity property of the inverse of $ p$. Using these criteria, we present a wide range of examples of weights $ p$ for which the equivalences stated above hold and also where they fail.
The proximal normal formula in Banach space
J. M.
Borwein;
J. R.
Giles
371-381
Abstract: Approximation by proximal normals to the Clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis. We give simple general versions of such formulae in infinite dimensional Banach spaces which satisfy different geometrical conditions. Our first class, of spaces with uniformly Gâteaux differentiable norm includes the Hilbert space case and the formulae is attained through dense subsets. Our second class, of reflexive Kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space. Our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of Ekeland and Lebourg. Resume. L'approximation par les normales proximales au sous-différentiel généralisé de Clarke pour une fonction de distance produit d'un ensemble non-vide fermé et le cône normal à l'ensemble produit des normales proximales sont objets d'importance pour l'analyse non-régulière. Nous donnons deux versions simples et générales de telles formules dans les espaces de Banach de dimensions infinies. Premièrement, nous examinons la classe des espaces avec norme uniformément Gâteaux-dérivable qui comprend les espaces de Hilbert. Deuxièmement, nous examinons la classe espaces réflexifs Kadec et lisses. Enfin, notre méthode produit l'existence des solutions pour une classe de problèmes à l'optimisation.
The space of framed functions
Kiyoshi
Igusa
431-477
Abstract: We define the notion of a ``framed function'' on a compact smooth manifold $ N$ and we show that the space of all framed functions on $N$ is $(\operatorname{dim} \,N - 1)$-connected. A framed function on $N$ is essentially a smooth function $N \to \mathbf{R}$ with only Morse and birth-death singularities together with certain additional structure.
Bordism of semifree circle actions on Spin manifolds
Lucília Daruiz
Borsari
479-487
Abstract: Using traditional methods in bordism theory, an almost complete description of the rational bordism groups of semifree circle actions on Spin manifolds is given. The single remaining problem, to describe the ideal of $\Omega _ \ast ^{{\operatorname{Spin}}}\, \otimes \,\mathbf{Q}$, generated by bordism classes of Spin manifolds admitting a semifree action of odd type, has been recently solved by S. Ochanine $[\mathbf{O}]$.
Induced group actions, representations and fibered skew product extensions
R. C.
Fabec
489-513
Abstract: Let $G$ be a locally compact group acting ergodically on $Y$. We introduce the notion of an action of this group action and study the notions of induced group actions, ergodicity, and fibered product extensions in this context. We also characterize fibered skew product actions built over a cocycle.
Lattice embeddings in the recursively enumerable truth table degrees
Christine Ann
Haught
515-535
Abstract: It is shown that every finite lattice, and in fact every recursively presentable lattice, can be embedded in the r.e. tt-degrees by a map preserving least and greatest elements. The decidability of the $1$-quantifier theory of the r.e. tt-degrees in the language with $\leqslant ,\, \vee ,\, \wedge ,\,0$, and 1 is obtained as a corollary.
Infinitely many traveling wave solutions of a gradient system
David
Terman
537-556
Abstract: We consider a system of equations of the form ${u_t} = {u_{xx}} + \nabla F(u)$. A traveling wave solution of this system is one of the form $u(x,\,t) = U(z),\,z = x + \theta t$. Sufficient conditions on $F(u)$ are given to guarantee the existence of infinitely many traveling wave solutions.
Stability of harmonic maps and eigenvalues of the Laplacian
Hajime
Urakawa
557-589
Abstract: The index and nullity of the Hessian of the energy for every harmonic map are estimated above by a geometric quantity. The stability theory of harmonic maps is developed and as an application, the Kähler version of the Lichnerowicz-Obata theorem about the first eigenvalue of the Laplacian is proved.
Koszul homology and the structure of low codimension Cohen-Macaulay ideals
Wolmer V.
Vasconcelos
591-613
Abstract: The relationship between the properties of the Koszul homology modules of two ideals connected by linkage is studied. If the ideal $I$ is either (i) a Cohen-Macaulay ideal of codimension 3, or (ii) a Gorenstein ideal of codimension 4, the one-dimensional Koszul module carries considerable information on the structural nature of the linkage class of $I$ in case (i), or on the conormal module of $ I$ in case (ii). Emphasis is given to the verification of the properties by computation.
Supersymmetry, twistors, and the Yang-Mills equations
Michael
Eastwood
615-635
Abstract: This article investigates a supersymmetric proof due to Witten of the twistor description of general Yang-Mills fields due to Green, Isenberg, and Yasskin. In particular, some rigor is added and the rather complicated calculations are given in detail.
Balanced subgroups of finite rank completely decomposable abelian groups
Loyiso G.
Nongxa
637-648
Abstract: It is proved that, if a finite rank completely decomposable group has extractable typeset of cardinality at most 5, all its balanced subgroups are also completely decomposable. Balanced Butler groups with extractable typeset of size at most 3 are almost completely decomposable and decompose into rank 1 and/or rank 3 indecomposable summands. We also construct an indecomposable balanced Butler group whose extractable typeset is of size 4 which fails to be almost completely decomposable.
On the M\"obius function
Helmut
Maier
649-664
Abstract: We investigate incomplete convolutions of the Möbius function of the form $\sum\nolimits_{d\vert n;d \leq z} {\mu (d)}$. It is shown that for almost all integers $n$ one can find $z$ for which this sum is large.
Pseudo-Chern classes of an almost pseudo-Hermitian manifold
Yasuo
Matsushita
665-677
Abstract: For an almost pseudo-Hermitian manifold, pseudo-Chern classes are defined on its complexified tangent bundle with the pseudo-Hermitian structure as represented by certain $ {\text{ad}}(U(p,\,q))$-invariant forms on the manifold. It is shown that such a manifold always admits an almost Hermitian structure, and hence that Chern classes are also defined on the complexified tangent bundle with such an almost Hermitian structure. A relation between the pseudo-Chern classes and the Chern classes is established. From the relation, the pseudo-Chern classes are considered as the characteristic classes which measure how the almost pseudo-Hermitian structure deviates from an almost Hermitian structure.
Quadratic geometry of numbers
Hans Peter
Schlickewei;
Wolfgang M.
Schmidt
679-690
Abstract: We give upper bounds for zeros of quadratic forms. For example we prove that for any nondegenerate quadratic form $ \mathfrak{F}({x_1}, \ldots ,\,{x_n})$ with rational integer coefficients which vanishes on a $d$-dimensional rational subspace $(d > 0)$ there exist sublattices ${\Gamma _0},\,{\Gamma _1},\, \ldots \,,{\Gamma _{n - d}}$ of $ {\mathbf{Z}^n}$ of rank $ d$, on which $\mathfrak{F}$ vanishes, with the following properties: $\displaystyle {\text{rank}}({\Gamma _0} \cap {\Gamma _i}) = d - 1,\quad {\text{rank}}({\Gamma _0} \cup {\Gamma _1} \cup \cdots \cup {\Gamma _{n - d}}) = n$ and $\displaystyle {(\det \,{\Gamma _0})^{n - d}}\det \,{\Gamma _1} \cdots \det \,{\Gamma _{n - d}} \ll {F^{{{(n - d)}^2}}}$ , where $F$ is the maximum modulus of the coefficients of $ \mathfrak{F}$.
On linear Volterra equations of parabolic type in Banach spaces
Jan
Prüss
691-721
Abstract: Linear integrodifferential equations of Volterra type in a Banach space are studied in case the main part of the equation generates an analytic ${C_0}$-semigroup. Under very general assumptions it is shown that a resolvent operator exists and that many of the solution properties of parabolic evolution equations are inherited. The results are then applied to integro-partial differential equations of parabolic type.
Scalar curvature functions in a conformal class of metrics and conformal transformations
Jean-Pierre
Bourguignon;
Jean-Pierre
Ezin
723-736
Abstract: This article addresses the problem of prescribing the scalar curvature in a conformal class. (For the standard conformal class on the $2$-sphere, this is usually referred to as the Nirenberg problem.) Thanks to the action of the conformal group, integrability conditions due to J. L. Kazdan and F. W. Warner are extended, and shown to be universal. A counterexample to a conjecture by J. L. Kazdan on the role of first spherical harmonics in these integrability conditions on the standard sphere is given. Using the action of the conformal groups, some existence results are also given.
Prime ideals in polycyclic crossed products
D. S.
Passman
737-759
Abstract: In this paper, we describe the prime ideals $P$ in crossed products $R \ast G$ with $R$ a right Noetherian ring and with $G$ a polycyclic-by-finite group. This is achieved through a series of reductions. To start with, we may assume that $P \cap R = 0$ so that $ R$ is a $G$-prime ring. The first step uses a technique of M. Lorenz and the author to reduce to a prime ring and a subgroup of finite index in $ G$. Next if $R$ is prime, then we show that the prime ideals of $R \ast G$ disjoint from $R$ are explicitly determined by the primes of a certain twisted group algebra of a normal subgroup of $G$. Finally the prime ideals in twisted group algebras of polycyclic-by-finite groups are studied by lifting the situation to ordinary group algebras where the results of J. E. Roseblade can be applied.
Positive forms and dilations
Wacław
Szymański
761-780
Abstract: By using the quadratic form and unbounded operator theory a new approach to the general dilation theory is presented. The boundedness condition is explained in terms of the Friedrichs extension of symmetric operators. Unbounded dilations are introduced and discussed. Applications are given to various problems involving positive definite functions.
Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions
Kenneth I.
Gross;
Donald St. P.
Richards
781-811
Abstract: Hypergeometric functions of matrix argument arise in a diverse range of applications in harmonic analysis, multivariate statistics, quantum physics, molecular chemistry, and number theory. This paper presents a general theory of such functions for real division algebras. These functions, which generalize the classical hypergeometric functions, are defined by infinite series on the space $S = S(n,\,\mathbf{F})$ of all $n \times n$ Hermitian matrices over the division algebra $ \mathbf{F}$. The theory depends intrinsically upon the representation theory of the general linear group $G = GL(n,\,\mathbf{F})$ of invertible $n \times n$ matrices over $ \mathbf{F}$, and the theme of this work is the full exploitation of the inherent group theory. The main technique is the use of the method of ``algebraic induction'' to realize explicitly the appropriate representations of $ G$, to decompose the space of polynomial functions on $S$, and to describe the ``zonal polynomials'' from which the hypergeometric functions are constructed. Detailed descriptions of the convergence properties of the series expansions are given, and integral representations are provided. Future papers in this series will develop the fine structure of these functions.
Toeplitz operators on the Segal-Bargmann space
C. A.
Berger;
L. A.
Coburn
813-829
Abstract: In this paper, we give a complete characterization of those functions on $2n$-dimensional Euclidean space for which the Berezin-Toeplitz quantizations admit a symbol calculus modulo the compact operators. The functions in question are characterized by a condition of ``small oscillation at infinity'' .
Hardy spaces of heat functions
H. S.
Bear
831-844
Abstract: We consider spaces of solutions of the one-dimensional heat equation on appropriate bounded domains in the $ (x,\,t)$-plane. The domains we consider have the property that they are parabolically star-shaped at some point; i.e., each downward half-parabola from some center point intersects the boundary exactly once. We introduce parabolic coordinates $(r,\,\theta)$ in such a way that the curves $ \theta =$constant are the half-parabolas, and dilation by multiplying by $ r$ preserves heat functions. An integral kernel is introduced by specializing to this situation the very general kernel developed by Gleason and the author for abstract harmonic functions. The combination of parabolic coordinates and kernel function provides a close analogy with the Poisson kernel and polar coordinates for harmonic functions on the disc, and many of the Hardy space theorems for harmonic functions generalize to this setting. Moreover, because of the generality of the Bear-Gleason kernel, much of this theory extends nearly verbatim to other situations where there are polar-type coordinates (such that the given space of functions is preserved by the ``radial'' expansion) and the maximum principle holds. For example, most of these theorems hold unchanged for harmonic functions on a radial star in $ {\mathbf{R}^n}$. As ancillary results we give a simple condition that a boundary point of a plane domain be regular, and give a new local Phragmén-Lindelöf theorem for heat functions.
Nonsingular quadratic differential equations in the plane
M. I. T.
Camacho;
C. F. B.
Palmeira
845-859
Abstract: We consider the problem of determining the number of inseparable leaves of nonsingular polynomial differential equations of degree two. As a corollary of a classification theorem for the foliation defined by these equations, we prove that this number is at most 2.
Fredholm, Hodge and Liouville theorems on noncompact manifolds
Robert
Lockhart
1-35
Abstract: Fredholm, Liouville, Hodge, and ${L^2}$-cohomology theorems are proved for Laplacians associated with a class of metrics defined on manifolds that have finitely many ends. The metrics are conformal to ones that are asymptotically translation invariant. They are not necessarily complete. The Fredholm results are, of necessity, with respect to weighted Sobolev spaces. Embedding and compact embedding theorems are also proved for these spaces.
Uniform distribution of two-term recurrence sequences
William Yslas
Vélez
37-45
Abstract: Let ${u_0},\,{u_1},\,A,\,B$ be rational integers and for $n \geqslant 2$ define ${u_n} = A{u_{n - 1}} + B{u_{n - 2}}$. The sequence $({u_n})$ is clearly periodic modulo $ m$ and we say that $ ({u_n})$ is uniformly distributed modulo $m$ if for every $s$, every residue modulo $m$ occurs the same number of times in the sequence of residues ${u_s},\,{u_{s + 1}},\, \ldots ,\,{u_{s + N - 1}}$, where $N$ is the period of $({u_n})$ modulo $m$. If $({u_n})$ is uniformly distributed modulo $ m$ then $m$ divides $N$, so we write $N = mf$. Several authors have characterized those $ m$ for which $ ({u_n})$ is uniformly distributed modulo $m$. In fact in this paper we will show that a much stronger property holds when $m = {p^k},\,p$, a prime. Namely, if $({u_n})$ is uniformly distributed modulo $ {p^k}$ with period $ {p^k}f$, then every residue modulo ${p^k}$ appears exactly once in the sequence ${u_s},\,{u_{s + f}},\, \ldots ,\,{u_{s + ({p^k} - 1)f}}$, for every $s$. We also characterize those composite $ m$ for which this more stringent property holds.
$F$-purity and rational singularity in graded complete intersection rings
Richard
Fedder
47-62
Abstract: A simple criterion is given for determining ``almost completely'' whether the positively graded complete intersection ring $R = K[{X_1},\, \ldots ,\,{X_{n + t}}]/({G_i},\, \ldots ,\,{G_t})$, of dimension $n$, has an $F$-pure type singularity at $m = ({X_1},\, \ldots ,\,{X_{n + t}})$. Specifically, if $\operatorname{deg} ({X_i}) = {\alpha _i} > 0$ for $1 \leq i \leq n + t$ and $ \operatorname{deg} ({G_i}) = {d_i} > 0$ for $ i \leq i \leq t$, then there exists an integer $\delta$ determined by the singular locus of $ R$ such that: (1) $ R$ has $F$-pure type if $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} < \delta$. (2) $R$ does not have $F$-pure type if $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} > 0$. The characterization given by this theorem is particularly effective if the singularity of $R$ at $m$ is isolated. In that case, $\delta = 0$ so that only the condition $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0$ is not solved by the above result. In particular, it follows from work of Kei-ichi Watanabe that if $ R$ has an isolated rational singularity, then $R$ has $F$-pure type. The converse is also ``almost true'' with the only exception being the case where $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0$. In proving this criterion, a weak but more stable form of $F$-purity, called $F$-contractedness, is defined and explored. $ R$ is $F$-contracted (in characteristic $ p > 0$) if every system of parameters for $m$ is contracted with respect to the Frobenius map $F:\,R \to R$. Just as for $F$-purity, the notion of $F$-contracted type is defined in characteristic 0 by reduction to characteristic $ p$. The two notions of $ F$-pure (type) and $ F$-contracted (type) coincide when $R$ is Gorenstein; whence, in particular, when $ R$ is a complete intersection ring.
The binary matroids with no $4$-wheel minor
James G.
Oxley
63-75
Abstract: The cycle matroids of wheels are the fundamental building blocks for the class of binary matroids. Brylawski has shown that a binary matroid has no minor isomorphic to the rank-3 wheel $ M({\mathcal{W}_3})$ if and only if it is a series-parallel network. In this paper we characterize the binary matroids with no minor isomorphic to $M({\mathcal{W}_4})$. This characterization is used to solve the critical problem for this class of matroids and to extend results of Kung and Walton and Welsh for related classes of binary matroids.
Hypergeometric functions over finite fields
John
Greene
77-101
Abstract: In this paper the analogy between the character sum expansion of a complex-valued function over $ {\text{GF}}(p)$ and the power series expansion of an analytic function is exploited in order to develop an analogue for hypergeometric series over finite fields. It is shown that such functions satisfy many summation and transformation formulas analogous to their classical counterparts.
T-degrees, jump classes, and strong reducibilities
R. G.
Downey;
C. G.
Jockusch
103-136
Abstract: It is shown that there exist r.e. degrees other than 0 and $\mathbf{0}^{\prime}$ which have a greatest r.e. $ 1$-degree. This solves an old question of Rogers and Jockusch. We call such degrees $1$-topped. We show that there exist incomplete $1$-topped degrees above any low r.e. degree, but also show that no nonzero low degree is $1$-topped. It then follows by known results that all incomplete $1$-topped degrees are low$_{2}$ but not low. We also construct cappable nonzero $1$-topped r.e. degrees and examine the relationships between $1$-topped r.e. degrees and high r.e. degrees. Finally, we give an analysis of the ``local'' relationships of r.e. sets under various strong reducibilities. In particular, we analyze the structure of r.e. ${\text{wtt-}}$ and $ {\text{tt}}$-degrees within a single r.e. $ {\text{T}}$-degree. We show, for instance, that there is an r.e. degree which contains a greatest r.e. $ {\text{wtt-}}$-degree and a least r.e. $ {\text{tt}}$-degree yet does not consist of a single r.e. ${\text{wtt}}$-degree. This depends on a new construction of a nonzero r.e. $ {\text{T}}$-degree with a least $ {\text{tt}}$-degree, which proves to have several further applications.
A regularity result for viscosity solutions of Hamilton-Jacobi equations in one space dimension
R.
Jensen;
P. E.
Souganidis
137-147
Abstract: Viscosity solutions of Hamilton-Jacobi equations need only to be continuous. Here we prove that, in the special case of a one-dimensional stationary problem, under quite general assumptions, Lipschitz continuous viscosity solutions have right and left derivatives at every point. Moreover, these derivatives have some kind of continuity properties.
Cohomology theories on spaces
E.
Spanier
149-161
Abstract: In this paper a previously proven uniqueness theorem for nonnegative cohomology theories on the same space is extended to cohomology theories on the same finite-dimensional space. In this form it is applicable to extraordinary cohomology theories. An example is given to show that the theorem does not hold without finite dimensionality.
The \'etale cohomology of $p$-torsion sheaves. I
William Anthony
Hawkins
163-188
Abstract: This paper generalizes a formula of Grothendieck, Ogg, and Shafarevich that expresses the Euler-Poincaré characteristic of a constructible sheaf of ${F_l}$-modules on a smooth, proper curve, over an algebraically closed field $k$ of characteristic $p > 0$, as a sum of local and global terms, where $l \ne p$. The primary focus is on removing the restriction on $l$. We begin with calculations for $ p$-torsion sheaves trivialized by $p$-extensions, but using etale cohomology to give a unified proof for all primes $l$. In the remainder of this work, only $ p$-torsion sheaves are considered. We show the existence on ${X_{{\text{et}}}}$, $X$ a scheme of characteristic $ p$, of a short exact sequence of sheaves, involving the tangent space at the identity of a finite, flat, height 1, commutative group scheme, and the subsheaf fixed by the $p$th power endomorphism; the latter turns out to be an etale group scheme. A corollary gives complete results on the Euler-Poincaré characteristic of a constructible sheaf of $ {F_p}$-modules on a smooth, proper curve, over an algebraically closed field $ k$ of characteristic $ p > 0$, when the generic stalk has rank $p$. Explicit computations are given for the Euler characteristics of such $p$-torsion sheaves on ${P^1}$ and a result on elliptic surfaces is included. A study is made of the comparison of the $ p$-ranks of abelian extensions of curves. Several examples of $p$-ranks for nonhyperelliptic curves are discussed. The paper concludes with a brief sketch of results on certain constructible sheaves of $ {F_q}$-modules, $q={p^r},\,r \ge 1$.
The zero dispersion limit of the Korteweg-de Vries equation with periodic initial data
Stephanos
Venakides
189-226
Abstract: We study the initial value problem for the Korteweg-de Vries equation $\displaystyle ({\text{i}})\quad {u_t} - 6u{u_x} + {\varepsilon ^2}{u_{xxx}} = 0$ in the limit of small dispersion, i.e., $\varepsilon \to 0$. When the unperturbed equation $\displaystyle ({\text{ii}})\quad {u_t} - 6u{u_x} = 0$ develops a shock, rapid oscillations arise in the solution of the perturbed equation (i) In our study: a. We compute the weak limit of the solution of (i) for periodic initial data as $ \varepsilon \to 0$. b. We show that in the neighborhood of a point $ (x,\,t)$ the solution $ u(x,\,t,\,\varepsilon)$ can be approximated either by a constant or by a periodic or by a quasiperiodic solution of equation (i). In the latter case the associated wavenumbers and frequencies are of order $ O(1/\varepsilon )$. c. We compute the number of phases and the wave parameters associated with each phase of the approximating solution as functions of $x$ and $t$. d. We explain the mechanism of the generation of oscillatory phases. Our computations in a and c are subject to the solution of the Lax-Levermore evolution equations (7.7). Our results in b-d rest on a plausible averaging assumption.
On root invariants of periodic classes in ${\rm Ext}\sb A({\bf Z}/2,{\bf Z}/2)$
Paul
Shick
227-237
Abstract: We prove that if a class in the cohomology of the mod 2 Steenrod algebra is $ \operatorname{mod}\,2$-periodic in the sense of [10], then its root invariant must be $ {\upsilon _{n + 1}}$-periodic, where $ {\upsilon _{n}}$ denotes the $n$th generator of ${\pi _ \ast }({\text{BP}})$.
Sets of uniqueness in compact, $0$-dimensional metric groups
D. J.
Grubb
239-249
Abstract: An investigation is made of sets of uniqueness in a compact 0-dimensional space. Such sets are defined by pointwise convergence of sequences of functions that generalize partial sums of trigonometric series on Vilenkin groups. Several analogs of classical uniqueness theorems are proved, including a version of N. Bary's theorem on countable unions of closed sets of uniqueness.
A cohomological pairing of half-forms
P. L.
Robinson
251-261
Abstract: Blattner and Rawnsley have constructed half-forms for regular polarizations of arbitrary index. We show how to pair these half-forms into a line bundle fashioned purely from the symplectic data, with no assumption on the intersection of the polarizations. Our pairing agrees with the regular BKS pairing when the polarizations are positive.
The structure of $\sigma$-ideals of compact sets
A. S.
Kechris;
A.
Louveau;
W. H.
Woodin
263-288
Abstract: Motivated by problems in certain areas of analysis, like measure theory and harmonic analysis, where $\sigma$-ideals of compact sets are encountered very often as notions of small or exceptional sets, we undertake in this paper a descriptive set theoretic study of $\sigma$-ideals of compact sets in compact metrizable spaces. In the first part we study the complexity of such ideals, showing that the structural condition of being a $\sigma$-ideal imposes severe definability restrictions. A typical instance is the dichotomy theorem, which states that $\sigma$-ideals which are analytic or coanalytic must be actually either complete coanalytic or else ${G_\delta}$. In the second part we discuss (generators or as we call them here) bases for $ \sigma$-ideals and in particular the problem of existence of Borel bases for coanalytic non-Borel $\sigma$-ideals. We derive here a criterion for the nonexistence of such bases which has several applications. Finally in the third part we develop the connections of the definability properties of $ \sigma$-ideals with other structural properties, like the countable chain condition, etc.
On scrambled sets for chaotic functions
A. M.
Bruckner;
Thakyin
Hu
289-297
Abstract: Some recent research has raised questions concerning the possible sizes of scrambled sets for chaotic functions. We answer these questions by showing that a scrambled set can have full measure, but cannot be residual although a scrambled set can be second category in every interval. We also indicate relationships that exist between chaotic functions and transitive functions.
The moduli of compact continuations of an open Riemann surface of genus one
M.
Shiba
299-311
Abstract: Let $(R,\,\{ A,\,B\} )$ be a marked open Riemann surface of genus one. Denote by $(T,\,\{ {A_T},\,{B_T}\} ,i)$ a pair of a marked torus $ (T,\,\{ {A_T},\,{B_T}\} )$ and a conformal embedding $i$ of $R$ into $T$ with $i(A)$ and $i(B)$ homotopic respectively to $ {A_T}$ and ${B_T}$. We say that $(T,\,\{ {A_T},\,{B_T}\} ,i)$ and $(T^{\prime},\,\{ {A_T^{\prime}},\,{B_T^{\prime}}\} ,i^{\prime})$ are equivalent if $i^{\prime} \circ {i^{ - 1}}$ extends to a conformal mapping of $T$ onto $ {T^\prime}$. The equivalence classes are called compact continuations of $(R,\,\{ A,\,B\} )$ and the set of moduli of compact continuations of $(R,\,\{ A,\,B\} )$ is denoted by $M = M(R,\,\{ A,\,B\} )$. Then $M$ is a closed disk in the upper half plane. The radius of $M$ represents the size of the ideal boundary of $ R$ and gives a generalization of Schiffer's span for planar domains; in particular, it vanishes if and only if $R$ belongs to the class ${O_{AD}}$. On the other hand, any holomorphic differential on $R$ with distinguished imaginary part produces in a canonical manner a compact continuation of $(R,\,\{ A,\,B\} )$. Such a compact continuation is referred to as a hydrodynamic continuation of $(R,\,\{ A,\,B\} )$. The boundary of $M$ parametrizes in a natural way the space of hydrodynamic continuations; i.e., the hydrodynamic continuations have extremal properties.
Isotype submodules of $p$-local balanced projective groups
Mark
Lane
313-325
Abstract: By giving necessary and sufficient conditions for two isotype submodules of a $p$-local balanced projective group to be equivalent, we are able to introduce a general theory of isotype submodules of $p$-local balanced projective groups (or $ IB$ modules). Numerous applications of the above result are available particularly for the special class of $IB$ modules introduced by Wick (known as SKT modules). We first show that the class of SKT modules is closed under direct summands, and then we are able to show that if $H$ appears as an isotype submodule of the $ p$-local balanced projective group $G$ such that $G/H$ is the coproduct of countably generated torsion groups, then $H$ is an SKT module. Finally we show that $ IB$ modules satisfy general structural properties such as transitivity, full transitivity, and the equivalence of ${p^\alpha }$-high submodules.
Positive solutions of systems of semilinear elliptic equations: the pendulum method
Joseph
Glover
327-342
Abstract: Conditions are formulated which guarantee the existence of positive solutions for systems of the form \begin{displaymath}\begin{gathered}- \Delta {u_1} + {f_1}({u_1}, \ldots ,\,{u_n}... ...}({u_1}, \ldots ,\,{u_n}) = {\mu _n}, \end{gathered} \end{displaymath} , where $\Delta$ is the Laplacian (with Dirichlet boundary conditions) on an open domain in ${\mathbf{R}^d}$, and where each ${\mu_i}$ is a positive measure. The main tools used are probabilistic potential theory, Markov processes, and an iterative scheme which is not a generalization of the one used for quasimonotone systems. Quasimonotonicity is not assumed and new results are obtained even for the case where $\partial {f_k}/\partial {x_j} > 0$ for every $ k$ and $j$.
Global solvability on compact nilmanifolds of three or more steps
Jacek M.
Cygan;
Leonard F.
Richardson
343-373
Abstract: We apply the methods of representation theory of nilpotent Lie groups to study the convergence of Fourier series of smooth global solutions to first order invariant partial differential equations $Df = g$ in $ {C^\infty}$ of a compact nilmanifold of three or more steps. We investigate which algebraically well-defined conditions on $D$ in the complexified Lie algebra imply that smooth infinite-dimensional irreducible solutions, when they exist, satisfy estimates strong enough to guarantee uniform convergence of the irreducible (or primary) Fourier series to a smooth global solution. This extends and improves the results of an earlier two step paper.
Convergence of series of scalar- and vector-valued random variables and a subsequence principle in $L\sb 2$
S. J.
Dilworth
375-384
Abstract: Let $({d_n})_{n = 1}^\infty$ be a martingale difference sequence in ${L_0}(X)$, where $X$ is a uniformly convex Banach space. We investigate a necessary condition for convergence of the series $ \sum {_{n = 1}^\infty {a_n}{d_n}}$. We also prove a related subsequence principle for the convergence of a series of square-integrable scalar random variables.
Hamiltonian analysis of the generalized problem of Bolza
F. H.
Clarke
385-400
Abstract: On étudie le problème généralisé de Bolza en calcul des variations. Presented at the International Conference on the Calculus of Variations held to honour the memory of Leonida Tonelli, Scuola Normale Superiore, Pisa, March 1986. On obtient des conditions nécessaires en forme hamiltonienne, sous des hypothèses moins exigeantes qu'antérieurement, en particulier sans qualification sur les contraintes. Le lien avec les problèmes de contrôle optimal est développé, ainsi que l'apport de ces conditions à la théorie de la régularité de la solution. We obtain necessary conditions in Hamiltonian form for the generalized problem of Bolza in the calculus of variations. These are proven in part by an extension to Hamiltonians of Tonelli's method of auxiliary Lagrangians. One version of the conditions is of a new character since it is obtained in the absence of any constraint qualification on the data. A new regularity theorem is shown to be a consequence of the necessary conditions.
New results on automorphic integrals and their period functions
Richard A.
Cavaliere
401-412
Abstract: Automorphic integrals, being generalizations of automorphic forms on discrete subgroups of $ SL(2,\,\mathbf{R})$, share properties similar to those of forms. In this article I obtain a natural boundary result for integrals which is similar to that which holds for forms. If an automorphic integral on a given group behaves like a form on a subgroup of finite index (i.e., the period functions are identically zero), then in fact the integral must be a form on the whole group. Specializing to modular integrals with integer dimension I obtain a lower bound on the number of poles of the period functions which, of necessity, lie in quadratic extensions of the rationals.
The isometry groups of manifolds and the automorphism groups of domains
Rita
Saerens;
William R.
Zame
413-429
Abstract: We prove that every compact Lie group can be realized as the (full) automorphism group of a strictly pseudoconvex domain and as the (full) isometry group of a compact, connected, smooth Riemannian manifold.
Infinite-dimensional linear systems with unbounded control and observation: a functional analytic approach
Dietmar
Salamon
383-431
Abstract: The object of this paper is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators. On the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem. The implications of the theory for large classes of functional and partial differential equations are discussed in detail.
Sufficiency conditions for $L\sp p$-multipliers with power weights
Benjamin
Muckenhoupt;
Richard L.
Wheeden;
Wo-Sang
Young
433-461
Abstract: Weighted norm inequalities in ${R^1}$ are proved for multiplier operators with the multiplier function of Hörmander type. The operators are initially defined on the space ${\mathcal{S}_{0,0}}$ of Schwartz functions whose Fourier transforms have compact support not including 0. This restriction on the domain of definition makes it possible to use weight functions of the form ${\left\vert x \right\vert^\alpha }$ for $ \alpha$ larger than usually considered. For these weight functions, if $(\alpha + 1)/p$ is not an integer, a strict inequality on $\alpha$ is shown to be sufficient for a norm inequality to hold. A sequel to this paper shows that the weak version of this inequality is necessary.
Sufficiency conditions for $L\sp p$ multipliers with general weights
Benjamin
Muckenhoupt;
Richard L.
Wheeden;
Wo-Sang
Young
463-502
Abstract: Weighted norm inequalities in ${R^1}$ are proved for multiplier operators with the multiplier function satisfying Hörmander type conditions. The operators are initially defined on the space $ {\mathcal{S}_{0,0}}$ of Schwartz functions whose Fourier transforms have compact support not including 0. This restriction on the domain of definition makes it possible to use a larger class of weight functions than usually considered; weight functions used here are of the form ${\left\vert {g(x)} \right\vert^p}V(x)$ where $ g(x)$) is a polynomial of arbitrarily high degree and $V(x)$ is in ${A_p}$. For weight functions in ${A_p}$, the results hold for all Schwartz functions. The periodic case is also considered.
Necessity conditions for $L\sp p$ multipliers with power weights
Benjamin
Muckenhoupt
503-520
Abstract: It is shown that if multiplier operators are bounded on with weight $ {\left\vert x \right\vert^\alpha }$ for all functions in the space ${\mathcal{S}_{0,0}}$ of Schwartz functions whose Fourier transforms have compact support not including 0 and all multiplier functions in a standard Hörmander type multiplier class, then $\alpha$ must satisfy certain inequalities. This is a sequel to a previous paper in which conditions on $ \alpha$ that were almost the same were shown to be sufficient for the norm inequality to hold.
Some weighted norm inequalities for the Fourier transform of functions with vanishing moments
Cora
Sadosky;
Richard L.
Wheeden
521-533
Abstract: Weighted $ L^p$ norm inequalities are derived between a function and its Fourier transform in case the function has vanishing moments up to some order. For weights of the form ${\left\vert x \right\vert^\gamma }$, the results concern values of $\gamma$ which are outside the range which is normally considered.
Polynomial algebras have polynomial growth
David R.
Finston
535-556
Abstract: The definitions and basic properties of Gelfand-Kirillov dimension are extended to algebras over a field which are not necessarily associative. The results are applied to the algebra of polynomial functions on an arbitrary finite dimensional algebra to obtain polynomial growth (i.e. integral G-K dimension) for these algebras. The G-K dimension of the polynomial algebra in one indeterminate is shown to be constant on the category of all finite dimensional nomial extensions of an associative algebra.
Near coherence of filters. II. Applications to operator ideals, the Stone-\v Cech remainder of a half-line, order ideals of sequences, and slenderness of groups
Andreas
Blass
557-581
Abstract: The set-theoretic principle of near coherence of filters (NCF) is known to be neither provable nor refutable from the usual axioms of set theory. We show that NCF is equivalent to the following statements, among others: (1) The ideal of compact operators on Hilbert space is not the sum of two smaller ideals. (2) The Stone-Čech remainder of a half-line has only one composant. (This was first proved by J. Mioduszewski.) (3) The partial ordering of slenderness classes of abelian groups, minus its top element, is directed upward (and in fact has a top element). Thus, all these statements are also consistent and independent.
The MacRae invariant and the first local Chern character
Paul
Roberts
583-591
Abstract: The first local Chern character of a bounded complex of locally free sheaves on a scheme $Y$ is given by intersection with a Cartier divisor. In the case of the resolution of a module of finite projective dimension, this is the invariant defined by MacRae.
The $b{\rm o}$-Adams spectral sequence
Wolfgang
Lellmann;
Mark
Mahowald
593-623
Abstract: Due to its relation to the image of the $J$-homomorphism and first order periodicity (Bott periodicity), connective real $K$-theory is well suited for problems in $ 2$-local stable homotopy that arise geometrically. On the other hand the use of generalized homology theories in the construction of Adams type spectral sequences has proved to be quite fruitful provided one is able to get a hold on the respective $ {E_2}$-terms. In this paper we make a first attempt to construct an algebraic and computational theory of the ${E_2}$-term of the bo-Adams spectral sequence. This allows for some concrete computations which are then used to give a proof of the bounded torsion theorem of [8] as used in the geometric application of [2]. The final table of the $ {E_2}$-term for $\pi _ \ast ^S$ in ${\operatorname{dim}}\, \leq 20$ shows that the statement of this theorem cannot be improved. No higher differentials appear in this range of the bo-Adams spectral sequence. We observe, however, that such a differential has to exist in dim 30.
A bilinear form for Spin manifolds
Peter S.
Landweber;
Robert E.
Stong
625-640
Abstract: This paper studies the bilinear form on $ {H^j}(M;{Z_2})$ defined by $\left[ {x,\,y} \right] = x\,{\text{S}}{{\text{q}}^2}y[M]$ when $M$ is a closed Spin manifold of dimension $ 2j + 2$. In analogy with the work of Lusztig, Milnor, and Peterson for oriented manifolds, the rank of this form on integral classes gives rise to a cobordism invariant.
Modules and stability theory
Anand
Pillay;
Mike
Prest
641-662
Abstract: Modules are now widely recognized as important examples of stable structures. In fact, in the light of results and conjectures of Zilber [Zi] ( $ {\aleph _1}$-categorical structures are ``field-like'', ``module-like'' or ``trivial''), we may consider modules as one of the typical examples of stable structures. Our aim here is both to prove some new results in the model theory of modules and to highlight the particularly clear form of, and the algebraic content of, the concepts of stability theory when applied to modules. One of the main themes of this paper is the connection between stability-theoretic notions, such as ranks, and algebraic decomposition of models. We will usually work with $T$, a complete theory of $R$-modules, for some ring $R$. In $\S2$ we show that the various stability-theoretic ranks, when defined, are the same. In $\S3$ we show that $T$ (not necessarily superstable) is nonmultidimensional (in the sence of Shelah [Sh1]). In $\S4$ we consider the algebraic content of saturation and we show, for example, that if $ M$ is a superstable module then $M$ is $ F_{{\aleph _0}}^a$-saturated just if $M$ is pure-injective and realizes all types in finitely many free variables over $\phi$. In $\S5$ we use our methods to reprove Ziegler's theorem on the possible spectrum functions. In $\S6$ we show the profusion (in a variety of senses) of regular types. In $\S7$ we give a structure theorem for the models of $ T$ in the case where $ T$ has $U$-rank 1.
On nonbinary $3$-connected matroids
James G.
Oxley
663-679
Abstract: It is well known that a matroid is binary if and only if it has no minor isomorphic to ${U_{2,4}}$, the $4$-point line. Extending this result, Bixby proved that every element in a nonbinary connected matroid is in a ${U_{2,4}}$-minor. The result was further extended by Seymour who showed that every pair of elements in a nonbinary $3$-connected matroid is in a $ {U_{2,4}}$-minor. This paper extends Seymour's theorem by proving that if $\left\{ {x,\,y,\,z} \right\}$ is contained in a nonbinary $3$-connected matroid $M$, then either $M$ has a ${U_{2,4}}$-minor using $\left\{ {x,\,y,\,z} \right\}$, or $ M$ has a minor isomorphic to the rank-$3$ whirl that uses $\left\{ {x,\,y,\,z} \right\}$ as its rim or its spokes.
Analytic functions with prescribed cluster sets
L. W.
Brinn
681-693
Abstract: Suppose that $0 < R \leq + \infty$. A monotonic boundary path (mb-path) in $\left\vert z \right\vert < R$ is a simple continuous curve $z = z(s)$, $0 \leq s < 1$, in $\left\vert z \right\vert < R$ such that $ \left\vert {z(s)} \right\vert \to R$ strictly monotonically as $s \to 1$. Suppose that $ f$ is a complex valued function, defined in $ \left\vert z \right\vert < R$, and that $t$ is any mb-path in $\left\vert z \right\vert < R$. The cluster set of $f$ on $t$ is the set of those points $w$ on the Riemann sphere for which there exists a sequence $ \{ {z_n}\}$ of points of $ t$ with ${\operatorname{lim}_{n \to \infty }}\left\vert {{z_n}} \right\vert = R$ and $ {\operatorname{lim}_{n \to \infty }}f({z_n}) = w$. The cluster set is denoted by $ {C_t}(f)$. If the cluster set is a single point set, that point is called the asymptotic value of $f$ on $t$. If the function $f$ is continuous, then ${C_t}(f)$ is a continuum on the Riemann sphere. It is a conjecture of F. Bagemihl and W. Seidel that if $\mathcal{T}$ is a family of mb-paths in $\left\vert z \right\vert < R$ satisfying certain conditions, and if $\mathcal{K}$ is an analytic set of continua on the Riemann sphere, then there exists a function $ f$, analytic in $\left\vert z \right\vert < R$, such that $\left\{ {{C_t}(f)\vert t \in \mathcal{T}} \right\} = \mathcal{K}$. A restricted form of the conjecture is mentioned in [3, p. 100]. Our principal results show the correctness of the conjecture in the case that $\mathcal{K}$ is the collection of all continua on the Riemann sphere and $\mathcal{T}$ is a tress of a certain type. The results are generalized in several directions. In particular, our technique for constructing the analytic function $f$ extends immediately to the case in which $\mathcal{K}$ is any closed set of continua on the sphere. Specific examples of closed sets lead to several corollaries.
A characterization of the kernel of the Poincar\'e series operator
Makoto
Masumoto
695-704
Abstract: Let $\Gamma$ be a finitely generated Fuchsian group of the first kind acting on the unit disk $ \Delta$. The kernel of the Poincaré series operator of the Hardy space ${H^p},\,1 < p < \infty$, onto the Bers space $ {A_q}(\Gamma )$ of integrable holomorphic automorphic forms of weight $- 2q,\,q \in {\mathbf{Z}},\,q \geq 2$, on $\Delta$ for $\Gamma$ is characterized in terms of Eichler integrals of order $1 - q$ on $\Delta$ for $\Gamma$.
A generalized Fatou theorem
B. A.
Mair;
David
Singman
705-719
Abstract: In this paper, a general Fatou theorem is obtained for functions which are integrals of kernels against measures on ${{\mathbf{R}}^n}$. These include solutions of Laplace's equation on an upper half-space, parabolic equations on an infinite slab and the heat equation on a right half-space. Lebesgue almost everywhere boundary limits are obtained within regions which contain sequences approaching the boundary with any prescribed degree of tangency.
On bounded analytic functions in finitely connected domains
Zbigniew
Slodkowski
721-736
Abstract: A new proof of the corona theorem for finitely connected domains is given. It is based on a result on the existence of a meromorphic selection from an analytic set-valued function. The latter fact is also applied to the study of finitely generated ideals of $ {H^\infty }$ over multiply connected domains.
Eichler-Shimura homology, intersection numbers and rational structures on spaces of modular forms
Svetlana
Katok;
John J.
Millson
737-757
Abstract: In this paper we reinterpret the main results of [8] using the intersection theory of cycles with coefficients. To this end we give a functorial interpretation of Eichler-Schimura periods.
A renewal theorem for random walks in multidimensional time
J.
Galambos;
K.-H.
Indlekofer;
I.
Kátai
759-769
Abstract: Let $X,\,{X_1},\,{X_2}, \ldots $ be a family of integer valued, independent and identically distributed random variables with positive mean and finite (positive) variance. Let ${S_n} = {X_1} + \,{X_2} + \cdots + {X_n}$. The asymptotic behavior of the weighted sum $R(k) = \sum {a_n}P({S_n} = k)$, with summation over $n \geq 1$, is investigated as $k \to + \infty$. In the special case ${a_n} = {d_r}(n)$, the number of solutions of the equation $n = {n_1}{n_2} \cdots {n_r}$ in positive integers $ {n_j},\,1 \leq j \leq r,\,R(k)$ becomes the renewal function $Q(k)$ for a random walk in $ r$-dimensional time whose terms are distributed as $X$. Under some assumptions on the magnitude of ${a_n}$ and of $A(x) = \sum\nolimits_{n \leq x} {{a_n}}$, (i) it is shown that $R(k)$ is asymptotically distribution free as $k \to + \infty$, (ii) the proper order of magnitude of $R(k)$ is determined, and under some further restrictions on $A(x)$, (iii) a simple asymptotic formula is given for $R(k)$. From (i), the known asymptotic formula for $ Q(k)$ with $r = 2$ or 3 is deduced under the sole assumption of finite variance. The relaxation of previous moment assumptions requires a new inequality for the sum of the divisor function ${d_r}(n),\,1 \leq n \leq x$, which by itself is of interest.
A general theory of canonical forms
Richard S.
Palais;
Chuu-Lian
Terng
771-789
Abstract: If $G$ is a compact Lie group and $ M$ a Riemannian $ G$-manifold with principal orbits of codimension $k$ then a section or canonical form for $ M$ is a closed, smooth $ k$-dimensional submanifold of $M$ which meets all orbits of $M$ orthogonally. We discuss some of the remarkable properties of $G$-manifolds that admit sections, develop methods for constructing sections, and consider several applications.
${\bf Z}/p{\bf Z}$ actions on $(S\sp n)\sp k$
Alejandro
Adem
791-809
Abstract: Let ${\mathbf{Z}}/p$ act on a finitistic space $ X$ with integral cohomology isomorphic to that of $ {({S^n})^k}$ as a ring. We show a direct relationship between the ${\mathbf{Z}}/p$-module structure of ${H^n}(X;{\mathbf{Z}})$ and the nature of the fixed-point set. In particular, we obtain a significant restriction on $ {H^n}(X;{\mathbf{Z}})$ for free actions.
Addendum to: ``Group-graded rings, smash products, and group actions'' [Trans. Amer. Math. Soc. {\bf 282} (1984), no. 1, 237--258; MR0728711 (85i:16002)]
M.
Cohen;
S.
Montgomery
810-811
Eisenstein series and the Selberg trace formula. II
H.
Jacquet;
D.
Zagier
1-48
Abstract: The integral of the kernel of the trace formula against an Eisenstein series is investigated. The analytic properties of this integral imply the divisibility of the convolution $L$-function attached to a form by the zeta function of the field.
Knapp-Wallach Szeg\H o integrals. II. The higher parabolic rank case
B. E.
Blank
49-59
Abstract: Let $G$ be a connected reductive linear Lie group with compact center and real rank $l$. For each integer $k(1 \leqslant k \leqslant l)$ and each discrete series representation $\pi$ of $G$, an explicit embedding of $\pi$ into a generalized principal series representation induced from a parabolic subgroup of rank $ k$ is given. The existence of such embeddings was proved by W. Schmid. In this paper an explicit integral formula with Szegö kernel is given which provides these mappings.
A dimension formula for Hermitian modular cusp forms of degree two
Min King
Eie
61-72
Abstract: An explicit dimension formula for the vector space of Hermitian modular cusp forms of degree two with respect to the modular group $ {\Gamma _2}({\mathbf{Z}}[i]) = \operatorname{SU} (2,2) \cap {M_4}({\mathbf{Z}}[i])$ is obtained via the Selberg trace formula and its arithmetic properties. Also, a generating function for the graded ring of Hermitian cusp forms of degree two is given.
Complex geometry and the asymptotics of Harish-Chandra modules for real reductive Lie groups. I
Luis G.
Casian;
David H.
Collingwood
73-107
Abstract: Let $G$ be a connected semisimple real matrix group. It is now apparent that the representation theory of $G$ is intimately connected with the complex geometry of the flag variety $ \mathcal{B}$. By studying appropriate orbit structures on $\mathcal{B}$, we are naturally led to representation theory in the category of Harish-Chandra modules $ \mathcal{H}\mathcal{C}$, or the representation theory of category $\mathcal{B}$, associated to $\mathcal{H}\mathcal{C}$ and
The Szeg\H o projection: Sobolev estimates in regular domains
Harold P.
Boas
109-132
Abstract: The Szegö projection preserves global smoothness in weakly pseudoconvex domains that are regular in the sense of Diederich, Fornæss, and Catlin. It preserves local smoothness near boundary points of finite type.
Wrappings of permutations
Saul
Stahl
133-152
Abstract: A theory of wrappings of permutations is constructed which is analogous to the well-known concept of branched coverings of Riemann surfaces. It is shown that this theory is strong enough to contain combinatorial definitions of such well-known groups as Fuchsian groups of the first kind and triangle groups.
The homology and higher representations of the automorphism group of a Riemann surface
S. A.
Broughton
153-158
Abstract: The representations of the automorphism group of a compact Riemann surface on the first homology group and the spaces of $ q$-differentials are decomposed into irreducibles. As an application it is shown that ${M_{24}}$ is not a Hurwitz group.
A global approach to the Rankin-Selberg convolution for ${\rm GL}(3,{\bf Z})$
Solomon
Friedberg
159-174
Abstract: We discuss the Rankin-Selberg convolution on $\operatorname{GL} (3,{\mathbf{Z}})$ in the `classical' language of symmetric spaces and automorphic forms.
Orthogonal polynomials, measures and recurrences on the unit circle
Paul
Nevai
175-189
Abstract: New characterizations are given for orthogonal polynomials on the unit circle and the associated measures in terms of the reflection coefficients in the recurrence equation satisfied by the polynomials.
Periodic phenomena in the classical Adams spectral sequence
Mark
Mahowald;
Paul
Shick
191-206
Abstract: We investigate certain periodic phenomena in the classical Adams sepctral sequence which are related to the polynomial generators $ {\nu _n}$ in ${\pi _{\ast}}(\operatorname{BP} )$. We define the notion of a class $a$ in ${\operatorname{Ext} _A}({\mathbf{Z}}/2,{\mathbf{Z}}/2)$ being ${\nu _n}$-periodic or ${\nu _n}$-torsion and prove that classes that are $ {\nu _n}$-torsion are also $ {\nu _k}$-torsion for all $ k$ such that $0 \leqslant k \leqslant n$. This allows us to define a chromatic filtration of ${\operatorname{Ext} _A}({\mathbf{Z}}/2,{\mathbf{Z}}/2)$ paralleling the chromatic filtration of the Novikov spectral sequence ${E_2}$-term given in [13].
Oscillatory phenomena associated to semilinear wave equations in one spatial dimension
T.
Cazenave;
A.
Haraux
207-233
Abstract: Let $g$ be a nonincreasing, odd $ {C^1}$ function and $l > 0$. We establish that for any solution $u \in C({\mathbf{R}};H_0^1(0,l))$ of the equation ${u_{tt}} - {u_{xx}} + g(u) = 0$ and any ${x_0} \in ]0,l[$, the function $t \mapsto u(t,{x_0})$ satisfies the following alternative: either $u(t,{x_0}) = 0,\forall t \in {\mathbf{R}}$, or $ \forall a \in {\mathbf{R}}$, there exist ${t_1}$ and ${t_2}$ in $ [a,a + 2l]$ such that $u({t_1},{x_0}) > 0$ and $u({t_2},{x_0}) < 0$. We study the structure of the set of points satisfying the first possibility. We give analogous results for ${u_x}$ and for some other homogeneous boundary conditions.
Stability results for a diffusion equation with functional drift approximating a chemotaxis model
James M.
Greenberg;
Wolfgang
Alt
235-258
Abstract: A hyperbolic-parabolic "chemotaxis" system modelling aggregation of motile cells by production of a diffusible chemoattractant, is approximated by a scalar diffusion equation for the cell density, where the drift term is an explicit functional of the current density profile. We prove the unique existence and, using the Hopf-Cole transformation, the local stability of an equilibrium, i.e. a steady aggregation state. We also discuss the limiting hyperbolic case of vanishing random motility with the formation of shocks describing cell clumps.
Forbidden intersections
Peter
Frankl;
Vojtěch
Rödl
259-286
Abstract: About ten years ago P. Erdös conjectured that if $\mathcal{F}$ is a family of subsets of $\{ 1,2, \ldots ,n\}$ without $F$, $ \vert\mathcal{F}\vert < {(2 - \varepsilon )^n}$ holds for some positive absolute constant $ \varepsilon$. Here this conjecture is proved in a stronger form (Theorem 1.1), which solves a 250 problem of Erdös. Suppose $\mathcal{C}$ is a code (i.e., a collection of sequences of length $n$) over an alphabet of $q$ elements, where $\tfrac{1} {2} > \delta > 0$ is arbitrary. Suppose further that there are no two codewords at Hamming distance $d$ where $d$ is a fixed integer, $\delta n < d < (1 - \delta )n$, and $ d$ is even if $ q = 2$. Then $ \vert\mathcal{C}\vert < {(q - \varepsilon )^n}$, where $\varepsilon > 0$ depends only on $ q$ and $\delta$. The following conjecture of Erdös and Szemerédi is also proved: If $ \mathcal{F}$ is a family of subsets of $ \{ 1,2, \ldots ,n\}$ not containing a weak $\Delta$-system of size $r$ (cf. Definition 1.8), then $ \vert\mathcal{F}\vert < {(2 - {\varepsilon _r})^n}$, ${\varepsilon _r} > 0$ holds. An old conjecture of Larman and Rogers is established in the following stronger form: Let $ \mathcal{A}$ be a collection of $4n$-dimensional $( \pm 1)$-vectors, $ r \geqslant 2$ is a fixed integer. Suppose that $A$ does not contain $r$ pairwise orthogonal vectors. Then $ \vert\mathcal{A}\vert < {(2 - \varepsilon )^{4n}}$. All these results can be deduced from our most general result (Theorem 1.16) which concerns the intersection pattern of families of partitions. This result has further implications in Euclidean Ramsey theory as well as for isometric embeddings into the Hamming space $H(n,q)$ (cf. Theorem 9.1).
Two-dimensional nonlinear boundary value problems for elliptic equations
Gary M.
Lieberman
287-295
Abstract: Boundary regularity of solutions of the fully nonlinear boundary value problem $\displaystyle F(x,u,Du,{D^2}u) = 0\quad {\text{in}}\;\Omega ,\qquad G(x,u,Du) = 0\quad {\text{on}}\;\partial \Omega$ is discussed for two-dimensional domains $\Omega$. The function $F$ is assumed uniformly elliptic and $G$ is assumed to depend (in a nonvacuous manner) on $Du$. Continuity estimates are proved for first and second derivatives of $u$ under weak hypotheses for smoothness of $ F$, $G$, and $\Omega$.
Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval
Louis
Block;
Ethan M.
Coven
297-306
Abstract: We say that a continuous map $f$ of a compact interval to itself is linear Markov if it is piecewise linear, and the set of all $ {f^k}(x)$, where $k \geqslant 0$ and $x$ is an endpoint of a linear piece, is finite. We provide an effective classification, up to topological conjugacy, for linear Markov maps and an effective procedure for determining whether such a map is transitive. We also consider expanding Markov maps, partly to motivate the proof of the more complicated linear Markov case.
On weakly countably determined Banach spaces
Sophocles
Mercourakis
307-327
Abstract: For a topological space $X$, let ${C_1}(X)$ denote the Banach space of all bounded functions $ f:X \to {\mathbf{R}}$ such that for every $ \varepsilon > 0$ the set $\{ x \in X:\vert f(x)\vert \geqslant \varepsilon \}$ is closed and discrete in $X$, endowed with the supremum norm. The main theorem is the following: Let $L$ be a weakly countably determined subset of a Banach space; then there exist a subset $\Sigma '$ of the Baire space $ \Sigma$, a compact space $ K$, and a bounded linear one-to-one operator ${C_1}(\Sigma ' \times K)$ (resp. ${C_1}(\Sigma \times K)$) (a compact space $ \Omega$ is called Gulko or Talagrand compact if $ C(\Omega )$ is WCD or a weakly $K$-analytic Banach space); the characterization of WCD (resp. weakly $K$-analytic) Banach spaces $E$, using one-to-one operators from ${E^{\ast}}$ into ${C_1}(\Sigma \times K)$); and the existence of equivalent "good" norms on $E$ and $ {E^{\ast}}$ simultaneously.
Area and Hausdorff dimension of Julia sets of entire functions
Curt
McMullen
329-342
Abstract: We show the Julia set of $\lambda \sin (z)$ has positive area and the action of $ \lambda \sin (z)$ on its Julia set is not ergodic; the Julia set of $\lambda \exp (z)$ has Hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero.
Matrices whose powers are $M$-matrices or $Z$-matrices
Shmuel
Friedland;
Daniel
Hershkowitz;
Hans
Schneider
343-366
Abstract: A matrix $ A$ all of whose (positive) powers are $Z$-matrices is called here a $ ZM$-matrix. A matrix is called a $ZMA$-matrix if all powers of $A$ are irreducible $Z$-matrices. We prove that the spectrum of a $ZMA$-matrix is real and only the eigenvalue minimal in absolute value may be negative. By means of an operation called inflation which generalizes the Kronecker product of two matrices, we determine the class of $ZMA$-matrices of order $n$ in terms of the classes of $ZMA$-matrices of smaller orders. We use this result to show that a $ZMA$-matrix is positively diagonally similar to a symmetric matrix. Similar results hold for $ MMA$-matrices which are defined in analogy with $ZMA$-matrices in terms of $M$-matrices, and for $ZMO$-matrices which are defined to be $ZM$-matrices such that all odd powers are irreducible and all even powers reducible. We also prove that a matrix is a $ZMA$-, $ZMO$- or $MMA$-matrix under apparently weaker conditions. If $A$ is a real matrix such that all sufficiently large powers of $A$ are $Z$-matrices, then $A$ is a $ZMA$-matrix if ${A^2}$ is irreducible, $A$ is a $ZMO$-matrix if $A$ is irreducible and ${A^2}$ is reducible, and $A$ is an $MMA$-matrix if $A$ is an irreducible $Z$-matrix and some odd power of $A$ is an $M$-matrix.
Whitney continua of curves
Hisao
Kato
367-381
Abstract: In this paper, we prove several theorems relating shape properties of Whitney continua of curves. In particular, we investigate the fundamental dimension and the shape type of Whitney continua of curves.
Isoperimetric inequalities for the least harmonic majorant of $\vert x\vert \sp p$
Makoto
Sakai
431-472
Abstract: Let $D$ be an open set in the $d$-dimensional Euclidean space ${{\mathbf{R}}^d}$ containing the origin 0 and let $ {h^{(p)}}(x,D)$ be the least harmonic majorant of $\vert x{\vert^p}$ in $D$, where $ 0 < p < \infty$ if $d \geqslant 2$ and $1 \leqslant p < \infty $ if $d = 1$. We shall be concerned with the following isoperimetric inequalities $ {h^{(p)}}{(0,D)^{1/p}} \leqslant cr(D)$, where $r(D)$ denotes the volume radius of $D$, namely, a ball with radius $ r(D)$ has the same volume as $D$ has and $c$ is a constant dependent on $d$ and $p$ but independent of $D$. We fix $d$ and denote by $c(p)$ the infimum of such constants $ c$. As a function of $ p$, $c(p)$ is nondecreasing and satisfies $c(p) \geqslant 1$. We shall show (1) there are positive constants ${C_1}$ and ${C_2}$ such that $ {C_1}{p^{(d - 1)/d}} \leqslant c(p) \leqslant {C_2}{p^{(d - 1)/d}}$ for $p \geqslant 1$, (2) $c(p) = 1$ if $p \leqslant d + {2^{1 - d}}$. Many estimations of $ {h^{(p)}}(0,D)$ and their applications are also given.
Some results on locally finitely presentable categories
M.
Makkai;
A. M.
Pitts
473-496
Abstract: We prove that any full subcategory of a locally finitely presentable (l.f.p.) category having small limits and filtered colimits preserved by the inclusion functor is itself l.f.p. Here "full" may be weakened to "full with respect to isomorphisms." Further, we characterize those left exact functors $I:{\mathbf{C}} \to {\mathbf{D}}$ between small categories with finite limits for which the functor $ {I^{\ast}}:{\mathbf{LEX}}({\mathbf{D}},{\text{Set)}} \to {\mathbf{LEX}}{\text{(}}{\mathbf{C}}{\text{,Set)}}$ induced by composition is full and faithful. As an application, we prove a theorem on sheaf representations, a consequence of which is that, for any site $\mathcal{C} = ({\mathbf{C}},J)$ on a category $ {\mathbf{C}}$ with finite limits, defined by a subcanonical Grothendieck topology $J$, the closure in $ {\mathbf{LEX}}({\mathbf{C}},{\text{Set)}}$ under small limits and filtered colimits of the models of $ \mathcal{C}$ is the whole of $ {\mathbf{LEX}}({\mathbf{C}},{\text{Set)}}$.
On the theory of fundamental norming bounded biorthogonal systems in Banach spaces
Paolo
Terenzi
497-511
Abstract: Let $X$ and $Y$ be quasi complementary subspaces of a separable Banach space $B$ and let $({z_n})$ be a sequence complete in $X$. Then (a) there exists a uniformly minimal norming $M$-basis $({x_n})$ of $X$ with ${x_m} \in \operatorname{span} {({z_n})_{n \geqslant {q_m}}}$ for every $m$, $ {q_m} \to \infty$; (b) if $({x_n})$ is a uniformly minimal norming $ M$-basis of $X$, there exists a uniformly minimal norming $M$-basis of $B$ which is an extension of $({x_n})$; (c) there exists a uniformly minimal norming $M$-basis $ ({x_n}) \cup ({y_n})$ of $ B$ with $({x_n}) \subset X$ and $({y_n}) \subset Y$.
A finiteness condition on regular local overrings of a local domain
Bernard
Johnston
513-524
Abstract: The local factorization theorem of Zariski and Abhyankar implies that between a given pair of $2$-dimensional regular local rings, $S \supseteq R$, having the same quotient field, every chain of regular local rings must be finite. It is shown in this paper that this property extends to every such pair of regular local rings, regardless of dimension. An example is given to show that this does not hold if "regular" is replaced by "Cohen-Macaulay," by "normal," or by "rational singularity." More generally, it is shown that the set $\mathcal{R}(R)$ of $n$dimensional regular local rings birationally dominating a given $n$-dimensional local domain, $R$, and ordered by containment, satisfies the descending chain condition. An example is given to show that if $R$ is regular the two examples of minimal elements of $ \mathcal{R}(R)$ given by J. Sally do not exhaust the set of minimal elements of $ \mathcal{R}(R)$.
Porous sets and quasisymmetric maps
Jussi
Väisälä
525-533
Abstract: A set $ A$ in ${R^n}$ is called porous if there is $\alpha > 0$ such that every ball $\overline B (x,r)$ contains a point whose distance from $A$ is at least $\alpha r$. We show that porosity is preserved by quasisymmetric maps, in particular, by bilipschitz maps. Local versions are also given.
The variation of the de Rham zeta function
Steven
Rosenberg
535-557
Abstract: Special values of the zeta function $\zeta (s)$ for the Laplacian on forms $ \Delta$ on a compact Riemannian manifold are known to have geometric significance. We compute the variation of these special values with respect to the variation of the metric and write down the Euler-Lagrange equation for conformal variations. The invariant metric on a locally symmetric space is shown to be critical for every local Lagrangian. We also compute the variation of
Parallel translation of curvature along geodesics
James J.
Hebda
559-572
Abstract: According to the Cartan-Ambrose-Hicks Theorem, two simply-connected, complete Riemannian manifolds are isometric if, given a certain correspondence between all the broken geodesics emanating from a point in one manifold, and all those emanating from a point in the other, the parallel translates of the curvature tensor agree along corresponding broken geodesics. For generic metrics on a surface, the hypothesis can be refined so that it is enough to compare curvature along corresponding unbroken geodesics in order to obtain the isometry.
Norms of Hankel operators and uniform algebras
Takahiko
Nakazi
573-580
Abstract: Two generalizations of the classical Hankel operators are defined on an abstract Hardy space that is associated with a uniform algebra. In this paper the norms of Hankel operators are studied. This has applications to weighted norm inequalities for conjugation operators, and invertible Topelitz operators. The results in this paper have applications to concrete uniform algebras, for example, a polydisc algebra and a uniform algebra which consists of rational functions.
Application of a theorem of M. G. Kre\u\i n to singular integrals
Rainer
Wittmann
581-599
Abstract: We give Hölder and ${L^2}$ estimates for singular integrals on homogeneous spaces in the sense of Coifman and Weiss. The fundamental tool which allows us to pass from Hölder to $ {L^2}$ estimates, is a theorem of M. G. Krein.
Polynomial invariants of graphs
Seiya
Negami
601-622
Abstract: We define two polynomials $f(G)$ and $ {f^{\ast}}(G)$ for a graph $ G$ by a recursive formula with respect to deformation of graphs. Analyzing their various properties, we shall discuss when two graphs have the same polynomials.
Endomorphisms of right ideals of the Weyl algebra
J. T.
Stafford
623-639
Abstract: Let $A = A(k)$ be the first Weyl algebra over an infinite field $k$, let $P$ be any noncyclic, projective right ideal of $ A$ and set $S = \operatorname{End} (P)$. We prove that, as $ k$-algebras, $S\not \cong A$. In contrast, there exists a noncyclic, projective right ideal $Q$ of $S$ such that $S \cong \operatorname{End} (Q)$. Thus, despite the fact that they are Morita equivalent, $S$ and $A$ have surprisingly different properties. For example, under the canonical maps, $ {\operatorname{Aut} _k}(A) \cong {\operatorname{Pic} _k}(A) \cong {\operatorname{Pic} _k}(S)$. In contrast, ${\operatorname{Aut} _k}(S)$ has infinite index in ${\operatorname{Pic} _k}(S)$.
A closed separable subspace of $\beta{\bf N}$ which is not a retract
Petr
Simon
641-655
Abstract: We shall exhibit a countable subset, $X$, of $ {{\mathbf{N}}^{\ast}}$ whose closure is not a retract of $\beta {\mathbf{N}}$. The points of $X$ are constructed in $ c$ steps with the aid of an independent matrix of subsets of $\omega$.
Seifert matrices and boundary link cobordisms
Ki Hyoung
Ko
657-681
Abstract: To an $ m$-component boundary link of odd dimension, a matrix is associated by taking the Seifert pairing on a Seifert surface of the link. An algebraic description of the set of boundary link cobordism classes of boundary links is obtained by using this matrix invariant.
A classification of simple Lie modules having a $1$-dimensional weight space
D. J.
Britten;
F. W.
Lemire
683-697
Abstract: Let $L$ denote a simple Lie algebra over the complex numbers. In this paper, we classify and construct all simple $L$ modules which may be infinite dimensional but have at least one $1$-dimensional weight space. This completes the study begun earlier by the authors for the case of $ L = {A_n}$. The approach presented here relies heavily on the results of Suren Fernando whose dissertation dealt with simple weight modules and their weight systems.
Some moduli spaces for rank $2$ stable reflexive sheaves on ${\bf P}\sp 3$
Rosa M.
Miró-Roig
699-717
Abstract: In [Ma], Maruyama proved that the set $M({c_1},{c_2},{c_3})$ of isomorphism classes of rank $ 2$ stable reflexive sheaves on $ {{\mathbf{P}}^3}$ with Chern classes $ ({c_1},{c_2},{c_3})$ has a natural structure as an algebraic scheme. Until now, there are no general results about these schemes concerning dimension, irreducibility, rationality, etc. and only in a few cases a precise description of them is known. This paper is devoted to the following cases: (i) $M( - 1,{c_2},c_2^2 - 2r{c_2} + 2r(r + 1))$ with ${c_2} \geqslant 4$, $ 1 \leqslant r \leqslant ( - 1 + \sqrt {4{c_2} - 7} )/2$; and (ii) $ M( - 1,{c_2},c_2^2 - 2(r - 1){c_2})$ with $ {c_2} \geqslant 8$, $2 \leqslant r \leqslant ( - 1 + \sqrt {4{c_2} - 7} )/2$.
The adelic zeta function associated with the space of binary cubic forms with coefficients in a function field
Boris A.
Datskovsky
719-745
Abstract: In this paper we study the adelic zeta function associated with the prehomogeneous vector space of binary cubic forms, defined over a function field. We establish its rationality, find its poles and residues and a simple functional equation that this zeta function satisfies.
Representations of crossed products by coactions and principal bundles
M. B.
Landstad;
J.
Phillips;
I.
Raeburn;
C. E.
Sutherland
747-784
Abstract: The main purpose of this paper is to establish a covariant representation theory for coactions of locally compact groups on ${C^{\ast}}$-algebras (including a notion of exterior equivalence), to show how these results extend the usual notions for actions of groups on ${C^{\ast}}$-algebras, and to apply these ideas to classes of coactions termed pointwise unitary and locally unitary to obtain a complete realization of the isomorphism theory of locally trivial principal $ G$-bundles in this context. We are also able to obtain all (Cartan) principal $ G$-bundles in this context, but their isomorphism theory remains elusive.
Counting cycles in permutations by group characters, with an application to a topological problem
D. M.
Jackson
785-801
Abstract: The character theory of the symmetric group is used to derive properties of the number of permutations, with $k$ cycles, which are expressible as the product of a full cycle with an element of an arbitrary, but fixed, conjugacy class. For the conjugacy class of fixed point free involutions, this problem has application to the analysis of singularities in surfaces.
On the canonical element conjecture
Sankar P.
Dutta
803-811
Abstract: The canonical element conjecture is proved in the following two cases: (i) depth $A = \dim A - 1$, $H_m^{n - 1}(A)$ is decomposable; (ii) depth $A = \dim A - 1$, $H_m^{n - 1}{(A)^ \vee }$ is cyclic. The equivalence of the C.E.C. and the improved new intersection theorem is also established.
Univalent harmonic functions
W.
Hengartner;
G.
Schober
1-31
Abstract: Several families of complex-valued, univalent, harmonic functions are studied from the point of view of geometric function theory. One class consists of mappings of a simply-connected domain onto an infinite horizontal strip with a normalization at the origin. Extreme points and support points are determined, as well as sharp estimates for Fourier coefficients and distortion theorems. Next, mappings in $ \left\vert z \right\vert > 1$ are considered that leave infinity fixed. Some coefficient estimates, distortion theorems, and covering properties are obtained. For such mappings with real boundary values, many extremal problems are solved explicitly.
Completely reducible operators that commute with compact operators
Shlomo
Rosenoer
33-40
Abstract: It is shown that if $ T$ is a completely reducible operator on a Banach space and $TK = KT$, where $K$ is an injective compact operator with a dense range, then $T$ is a scalar type spectral operator. Other related results are also obtained.
Applications of the covering lemma for sequences of measures
W.
Mitchell
41-58
Abstract: We present several applications of the covering lemma for the core model for sequences of measures, including characterizations of the large cardinal strength necessary to make the filter of closed, unbounded subsets of ${\omega _1}$ an ultrafilter or to change the cofinality of a regular cardinal, and a characterization of the minimal inner model containing an arbitrary elementary embedding.
On the values at integers of the Dedekind zeta function of a real quadratic field
David
Kramer
59-79
Abstract: In 1976 Shintani gave a decomposition of the Dedekind zeta function, $\zeta \kappa (s)$, of a totally real number field into a finite sum of functions, each given by a Dirichlet series whose meromorphic continuation assumes rational values at negative integers. He obtained a formula for these values, thereby giving an expression for $ \zeta \kappa ( - n),$, $n = 0,\,1,\,2, \ldots$. Earlier, Zagier had studied the special case of $\zeta (A,\,s)$, the narrow ideal class zeta function for a real quadratic field. He decomposes $\zeta (A,\,s)$ into ${\Sigma _A}{Z_Q}(s)$, where ${Z_Q}(s)$ is given as a Dirichlet series associated to a binary quadratic form $Q(x,\,y) = a{x^2} + bxy + c{y^2}$, and the summation is over a canonically given finite cycle of ``reduced'' quadratic forms associated to a narrow ideal class $A$. He then obtains a formula for ${Z_Q}( - n)$ as a rational function in the coefficients of the form $Q$. Since the denominator of $\zeta (A,\, - n)$ is known not to depend on the class $A$, whereas the coefficients of reduced forms attain arbitrarily large values, it is natural to ask whether the rational function in Zagier's formula might be replaced by a polynomial. In this paper such a result is obtained. For example, Zagier gives $\displaystyle 15120\zeta (A, - 2) = \sum\limits_A {\frac{{{b^5} - 10a{b^3}c + 3... ...}} + \frac{{{b^5} - 10a{b^3}c + 30{a^2}b{c^2}}}{{{c^3}}} - 21b(a + c){\text{ }}$ while our result is $\displaystyle 15120\zeta (A,\, - 2) = \frac{1} {2}\left( {\mathop \sum \limits_... ...mits_{\theta A} } \right)\,(60{a^2} - 117ab + 76ac + 38{b^2} - 117bc + 60{c^2})$ , where $ \theta$ is the narrow ideal class consisting of principal ideals generated by elements of negative norm. Starting with a representation of ${Z_Q}(1 + n)$ due to Shanks and Zagier for $n = 1,\,2,\,3, \ldots$ as a certain transcendental function of the coefficients of $ Q$, we also obtain the result that $ \zeta (A,\,1 + n)$ is given as the same sum of reduced quadratic forms as in the formula for $ \zeta (A,\, - n)$, times the appropriate ``gamma factor.'' This gives a new proof of the functional equation of $\zeta (A,\,s)$ at integer values of $ s$, and suggests the possibility that one might be able to prove the functional equation for all $s$ by finding some relation between $ {Z_Q}(s)$ and ${Z_Q}(1 - s)$. So far we have not found such a relation.
Concavity of solutions of the porous medium equation
Philippe
Bénilan;
Juan Luis
Vázquez
81-93
Abstract: We consider the problem $\displaystyle \left( {\text{P}} \right)\quad \quad \left\{ {\begin{array}{*{20}... ...\in {\mathbf{R}},\,t > 0} {{\text{for}}\,x \in {\mathbf{R}}} \end{array}$ where $m > 1$ and ${u_0}$ is a continuous, nonnegative function that vanishes outside an interval $(a,\,b)$ and such that $(u_0^{m - 1})'' \leq - C \leq 0$ in $ (a,\,b)$. Using a Trotter-Kato formula we show that the solution conserves the concavity in time: for every $t > 0,\,u(x,t)$ vanishes outside an interval $\Omega (t) = ({}_{\zeta 1}(t),\,{}_{\zeta 2}(t))$ and $\displaystyle {({u^{m - 1}})_{xx}} \leq - \frac{C} {{1 + C(m(m + 1)/(m - 1))t}}$ in $\Omega (t)$. Consequently the interfaces $x{ = _{\zeta i}}(t)$, $i = 1,\,2$, are concave curves. These results also give precise information about the large time behavior of solutions and interfaces.
Dimension de Hausdorff des ensembles de z\'eros et d'interpolation pour $A\sp \infty(D)$
Jacques
Chaumat;
Anne-Marie
Chollet
95-114
Abstract: Soit $D$ un domaine borné strictement pseudoconvexe dans $ {{\mathbf{C}}^n}$ à frontière régulière $\partial D$ et soit $ {A^\infty }(D)$ la classe des fonctions holomorphes dans $D$, indéfiniment dérivables dans $ \overline D$. Un sous-ensemble compact $E$ de $ \partial D$ est un ensemble de zéros pour $ {A^\infty }(D)$ s'il existe une fonction de $ {A^\infty }(D)$ s'annulant seulement sur $E$. C'est un ensemble d'interpolation d'ordre infini pour ${A^\infty }(D)$ si, pour toute fonction $f$ de classe $ {C^\infty }$ dans ${{\mathbf{C}}^n}$ telle que $\overline \partial f$ soit plate sur $ E$, il existe une fonction $ F$ de ${A^\infty }(D)$ telle que $F - f$ soit plate sur $E$. On construit ici des ensembles de dimension de Hausdorff $n$. Ce résultat est le meilleur possible dans le cas d'ensembles totalement réels. Le point de vue utilisé pour montrer qu'un sous-ensemble $ E$ de $\partial D$ est d'interpolation d'ordre infini pour ${A^\infty }(D)$ est de vérifier qu'il a la propriété de division par ${A^\infty }(D)$, c'est-á-dire, que, pour toute famille de fonctions ${({f_i})_{i \in {\text{N}}}}$ de ${C^\infty }(\overline D )$, plates sur $E$, il existe une fonction $F$ de $ {A^\infty }(D)$, plate sur $ E$ et nulle seulement sur $ E$ et une famille de fonctions $ {({k_i})_{i \in {\text{N}}}}$ de $ {C^\infty }(\overline D )$, plates sur $E$, telles que l'on ait, pour tout $i$ dans $ {\mathbf{N}}$, ${f_i} = F{k_i}$.
The structure of the critical set in the mountain pass theorem
Patrizia
Pucci;
James
Serrin
115-132
Abstract: We show that the critical set generated by the Mountain Pass Theorem of Ambrosetti and Rabinowitz must have a well-defined structure. In particular, if the underlying Banach space is infinite dimensional then either the critical set contains a saddle point of mountain-pass type, or the set of local minima intersects at least two components of the set of saddle points. Related conclusions are also established for the finite dimensional case, and when other special conditions are assumed. Throughout the paper, no hypotheses of nondegeneracy are required on the critical set.
Persistence of form and the value group of reducible cubics
P. D. T. A.
Elliott
133-143
Abstract: It is proved that the values of $ x({x^2} + c)$, $c \ne 0$, at positive integers, multiplicatively generate the positive rationals. Analogs in rational function fields are obtained.
Regressive partition relations for infinite cardinals
András
Hajnal;
Akihiro
Kanamori;
Saharon
Shelah
145-154
Abstract: The regressive partition relation, which turns out to be important in incompleteness phenomena, is completely characterized in the transfinite case. This work is related to Schmerl $ \left[ {\mathbf{S}} \right]$, whose characterizations we complete.
Construction of group actions on four-manifolds
Allan L.
Edmonds
155-170
Abstract: It is shown that any cyclic group of odd prime order acts on any closed, simply connected topological $4$-manifold, inducing the identity on integral homology. The action is locally linear except perhaps at one isolated fixed point. In the case of primes greater than three a more careful argument is used to show that the action can be constructed to be locally linear.
Countably generated Douglas algebras
Keiji
Izuchi
171-192
Abstract: Under a certain assumption of $f$ and $g$ in $ {L^\infty }$ which is considered by Sarason, a strong separation theorem is proved. This is available to study a Douglas algebra $[{H^\infty },\,f]$ generated by ${H^\infty }$ and $f$. It is proved that (1) ball $(B/{H^\infty } + C)$ does not have exposed points for every Douglas algebra $B$, (2) Sarason's three functions problem is solved affirmatively, (3) some characterization of $ f$ for which $[{H^\infty },\,f]$ is singly generated, and (4) the $M$-ideal conjecture for Douglas algebras is not true.
A remark on the blowing-up of solutions to the Cauchy problem for nonlinear Schr\"odinger equations
O.
Kavian
193-203
Abstract: We consider solutions to $i{u_t} = \Delta u + {\left\vert u \right\vert^{p - 1}}u$, $u(0) = {u_0}$, where $x$ belongs to a smooth domain $ \Omega \subset {{\mathbf{R}}^N}$, and we prove that under suitable conditions on $ p$, $N$ and ${u_0} \in {H^2}(\Omega ) \cap H_0^1(\Omega )$, $ {\left\Vert {\nabla u(t)} \right\Vert _{{L^2}}}$ blows up in finite time. The range of $p$'s for which blowing-up occurs depends on whether $ \Omega$ is starshaped or not. Examples of blowing-up under Neuman or periodic boundary conditions are given. On considère des solutions de $i{u_t} = \Delta u + {\left\vert u \right\vert^{p - 1}}u$, $u(0) = {u_0}$, où la variable d'espace $x$ appartient à un domaine régulier $ \Omega \subset {{\mathbf{R}}^N}$, et on prouve que sous des conditions adéquates sur $p$, $N$ et ${u_0} \in {H^2}(\Omega ) \cap H_0^1(\Omega )$, $ {\left\Vert {\nabla u(t)} \right\Vert _{{L^2}}}$ explose en temps fini. Les valeurs de $p$ pour lesquelles l'explosion a lieu dépend de la forme de l'ouvert $\Omega$ (en fait $\Omega$ étoilé ou non). On donne également des exemples d'explosion sous des conditions de Neuman ou périodiques au bord.
Category and group rings in homotopy theory
William J.
Ralph
205-223
Abstract: It frequently arises in algebraic topology that a function $\beta :G \to H$, between two groups, is not a homomorphism. We show that in many standard situations $ \beta$ induces a group homomorphism $\overline \beta :{\mathbf{Z}}(G)/{\mathcal{A}^d} \to H$, where $ {\mathcal{A}^d}$ is a power of the augumentation ideal in the group ring ${\mathbf{Z}}(G)$. A typical example is $ \beta :[X,\,Y] \to [{S^2}X,\,{S^2}Y]$ where $Y$ is some $H$-group, in which case $d$ can be taken to be $1 + {\text{cat}}\,X$.
Geometric theory of extremals in optimal control problems. I. The fold and Maxwell case
I.
Kupka
225-243
Abstract: The behavior of the extremal curves in optimal control theory is much more complex than that of their namesakes in the classical calculus of variations. Here we analyze the simplest instances of singular behavior of these extremals. Among others, in sharp contrast to the classical case, a $ {C^0}$-limit of extremals may not be an extremal. In the simplest cases (elliptic fold and Maxwell points) of this occurrence, the limits are trajectories of a new vector field. A special case of this field showed up in some work of V. I. Arnold. Results related to ours have been obtained in low dimension by I. Ekeland.
Nonharmonic Fourier series and spectral theory
Harold E.
Benzinger
245-259
Abstract: We consider the problem of using functions ${g_n}(x): = exp(i{\lambda _n}x)$ to form biorthogonal expansions in the spaces ${L^p}( - \pi ,\,\pi )$, for various values of $ p$. The work of Paley and Wiener and of Levinson considered conditions of the form $\left\vert {{\lambda _n} - n} \right\vert \leq \Delta (p)$ which insure that $ \{ {g_n}\}$ is part of a biorthogonal system and the resulting biorthogonal expansions are pointwise equiconvergent with ordinary Fourier series. Norm convergence is obtained for $p = 2$. In this paper, rather than imposing an explicit growth condition, we assume that $ \{ {\lambda _n} - n\}$ is a multiplier sequence on ${L^p}( - \pi ,\,\pi )$. Conditions are given insuring that $\{ {g_n}\}$ inherits both norm and pointwise convergence properties of ordinary Fourier series. Further, ${\lambda _n}$ and ${g_n}$ are shown to be the eigenvalues and eigenfunctions of an unbounded operator $ \Lambda$ which is closely related to a differential operator, $i\Lambda$ generates a strongly continuous group and $ - {\Lambda ^2}$ generates a strongly continuous semigroup. Half-range expansions, involving ${\text{cos}}{\lambda _n}x$ or ${\text{sin}}{\lambda _n}x$ on $(0,\,\pi )$ are also shown to arise from linear operators which generate semigroups. Many of these results are obtained using the functional calculus for well-bounded operators.
Growth properties of functions in Hardy fields
Maxwell
Rosenlicht
261-272
Abstract: This paper continues the author's earlier work on the notion of rank in a Hardy field. Further results are given on functions in Hardy fields of finite rank, including extensions of Hardy's results on the rates of growth of his logarithmico-exponential functions.
On the behavior near the crest of waves of extreme form
C. J.
Amick;
L. E.
Fraenkel
273-298
Abstract: The angle $ \phi$ which the free boundary of an extreme wave makes with the horizontal is the solution of a singular, nonlinear integral equation that does not fit (as far as we know) into the theory of compact operators on Banach spaces. It has been proved only recently that solutions exist and that (as Stokes suggested in 1880) these solutions represent waves with sharp crests of included angle $2\pi /3$. In this paper we use the integral equation, known properties of solutions and the technique of the Mellin transform to obtain the asymptotic expansion $\displaystyle ( * )\qquad \phi (s) = \frac{\pi } {6} + \sum\limits_{n = 1}^k {{a_n}{s^{{\mu _n}}} + o({s^{{\mu _k}}})} \quad {\text{as}}\,s \downarrow 0$ , to arbitrary order; the coordinate $s$ is related to distance from the crest as measured by the velocity potential rather than by length. The first few (and probably all) of the exponents $ {\mu _n}$ are transcendental numbers. We are unable to evaluate the coefficients $ {a_n}$ explicitly, but define some in terms of global properties of $\phi$, and the others in terms of earlier coefficients. It is proved in [8] that $ {a_1} < 0$, and follows here that ${a_2} > 0$. The derivation of (*) includes an assumption about a question in number theory; if that assumption should be false, logarithmic terms would enter the series at very large values of $ n$.
The asymptotic behavior near the crest of waves of extreme form
J. B.
McLeod
299-302
Abstract: The angle which the free boundary of an extreme wave makes with the horizontal is the solution of a singular, nonlinear integral equation. It has been proved only recently that solutions exist and that (as Stokes suggested in 1880) these solutions represent waves with sharp crests of included angle $ \frac{2} {3}\pi$. Amick and Fraenkel have investigated the asymptotic behavior of the free surface near the crest and obtained an asymptotic expansion for this behavior, but are unable to say whether the leading term in this expansion has a nonzero coefficient (and so whether it is in fact the leading term or not). The present paper shows that the coefficient is nonzero and determines its sign.
On inductive limits of matrix algebras of holomorphic functions
Justin
Peters
303-318
Abstract: Let $\mathfrak{A}$ be a UHF algebra and $ \mathcal{A}({\mathbf{D}})$ the disk algebra. If $\mathfrak{A} = {\left[ {{ \cup _{n \geq 1}}{\mathfrak{A}_n}} \right]^ - }$ and $\alpha$ is a product-type automorphism of $\mathfrak{A}$ which leaves each ${\mathfrak{A}_n}$ invariant, then $ \alpha$ defines an embedding $\displaystyle \mathfrak{A}_n \otimes \mathcal{A}({\mathbf{D}}) \stackrel{\imath_n}{\hookrightarrow} {\mathfrak{A}_{n + 1}} \otimes \mathcal{A}({\mathbf{D}})$ . The inductive limit of this system is a Banach algebra whose maximal ideal space is closely related to that of the disk algebra if the Connes spectrum $ \Gamma (\alpha )$ is finite.
First layer formulas for characters of ${\rm SL}(n,{\bf C})$
John R.
Stembridge
319-350
Abstract: Some problems concerning the decomposition of certain characters of $ SL(n,\,{\mathbf{C}})$ are studied from a combinatorial point of view. The specific characters considered include those of the exterior and symmetric algebras of the adjoint representation and the Euler characteristic of Hanlon's so-called ``Macdonald complex.'' A general recursion is given for computing the irreducible decomposition of these characters. The recursion is explicitly solved for the first layer representations, which are the irreducible representations corresponding to partitions of $ n$. In the case of the exterior algebra, this settles a conjecture of Gupta and Hanlon. A further application of the recursion is used to give a family of formal Laurent series identities that generalize the (equal parameter) $q$-Dyson Theorem.
Best constants in norm inequalities for the difference operator
Hans G.
Kaper;
Beth E.
Spellman
351-372
Abstract: Let $ \xi = {({\xi _m})_{m \in {\mathbf{Z}}}}$ be an arbitrary element of the sequence space ${l^\infty }({\mathbf{Z}})$, and let $ \Delta$ be the difference operator on ${l^\infty }({\mathbf{Z}}):\Delta \xi = {({\xi _{m + 1}} - {\xi _m})_{m \in {\mathbf{Z}}}}$. The object of this investigation is the best possible value $\displaystyle C(n,\,k) = {\operatorname{sup}}\{ {Q_{n,k}}(\xi ):\xi \in {l^\infty }({\mathbf{Z}}),\,{\Delta ^n}\xi \ne 0\}$ of the quotient $\displaystyle {Q_{n,k}}(\xi ) = \frac{{\left\Vert {{\Delta ^k}\xi } \right\Vert... ...i \right\Vert}^{(n - k)/n}}{{\left\Vert {{\Delta ^n}\xi } \right\Vert}^{k/n}}}}$ , where $n = 2,\:3, \ldots$; $k = 1, \ldots ,\,n - 1$. It is shown that $ C(n,\,k)$ is at least equal to the corresponding constant $K(n,\,k)$, determined by Kolmogorov [Moscov. Gos. Univ. Uchen. Zap. Mat. 30 (1939), 3-13; Amer. Math. Soc. Transl. (1) 2 (1962), 233-243] for the differential operator $D$ on ${L^\infty }({\mathbf{R}})$, and exactly equal to $K(n,\,k)$ if $k = n - 1$. Lower bounds for $C(n,\,k)$ are derived that show that $ C(n,\,k)$ is generally greater than $K(n,\,k)$. The values of $C(n,\,k)$, $ k = 1, \ldots ,\,n - 1$, are computed for $ n = 2, \ldots ,5$.
On embedding of group rings of residually torsion free nilpotent groups into skew fields
A.
Eizenbud;
A. I.
Lichtman
373-386
Abstract: It is proven that the group ring of an amalgamated free product of residually torsion free nilpotent groups is a domain and can be embedded in a skew field. This is a generalization of J. Lewin's theorem, proven for the case of free groups. Our proof is based on the study of the Malcev-Neumann power series ring $K\left\langle G \right\rangle$ of a residually torsion free nilpotent group $G$. It is shown that its subfield $ D$, generated by the group ring $KG$, does not depend on the order of $G$ for many kinds of orders and the study of $D$ can be reduced in some sense to the case when $ G$ is nilpotent.
Integration on noncompact supermanifolds
Mitchell J.
Rothstein
387-396
Abstract: We note that the Berezin integral, which is ill-defined for noncompact supermanifolds, is a distribution with support on the underlying manifold. This leads to the discovery of correction terms in the Berezinian transformation law and thereby eliminates the boundary ambiguities.
$K$-theory and multipliers of stable $C\sp \ast$-algebras
J. A.
Mingo
397-411
Abstract: The main theorem is that if $A$ is a $C^{\ast}$-algebra with a countable approximate identity consisting of projections, then the unitary group of $ M(A \otimes K)$ is contractible. This gives a realization, via the index map, of ${K_0}(A)$ as components in the set of Fredholm operators on ${H_A}$.
The dimension of closed sets in the Stone-\v Cech compactification
James
Keesling
413-428
Abstract: In this paper properties of compacta $K$ in $ \beta X\backslash X$ are studied for Lindelöf spaces $X$. If ${\operatorname{dim}}\,K = \infty$, then there is a mapping $f:K \to {T^c}$ such that $f$ is onto and every mapping homotopic to $ f$ is onto. This implies that there is an essential family for $K$ consisting of $c$ disjoint pairs of closed sets. It also implies that if $K = \cup \left\{ {{K_\alpha }\vert\alpha < c} \right\}$ with each $ {K_\alpha }$ closed, then there is a $\beta$ such that $ {\operatorname{dim}}\,{K_\beta } = \infty$. Assume $K$ is a compactum in $\beta X\backslash X$ as above. Then if $ {\operatorname{dim}}\,K = n$, there is a closed set $K'$ in $K$ such that ${G_\delta }$-set in $K'$ contains an $n$-dimensional compactum. This holds for $ n$ finite or infinite. If ${\operatorname{dim}}\,K = n$ and $K = \cup \left\{ {{K_\alpha }\vert\alpha < {\omega _1}} \right\}$ with each ${K_\alpha }$ closed, then there must be a $ \beta$ such that $ {\operatorname{dim}}\,{K_\beta } = n$.
Erratum to: ``Bounds on the dimension of variations of Hodge structure''
James A.
Carlson
429
A global theory of internal solitary waves in two-fluid systems
C. J.
Amick;
R. E. L.
Turner
431-484
Abstract: The problem analyzed is that of two-dimensional wave motion in a heterogeneous, inviscid fluid confined between two rigid horizontal planes and subject to gravity $g$. It is assumed that a fluid of constant density $ {\rho _ + }$ lies above a fluid of constant density ${\rho _ - } > {\rho _ + } > 0$ and that the system is nondiffusive. Progressing solitary waves, viewed in a moving coordinate system, can be described by a pair $ (\lambda ,w)$, where the constant $\lambda = g/{c^2}$, $c$ being the wave speed, and where $w(x,\eta ) + \eta$ is the height at a horizontal position $x$ of the streamline which has height $ \eta$ at $x = \pm \infty$. It is shown that among the nontrivial solutions of a quasilinear elliptic eigenvalue problem for $ (\lambda ,w)$ is an unbounded connected set in ${\mathbf{R}} \times (H_0^1 \cap {C^{0,1}})$. Various properties of the solution are shown, and the behavior of large amplitude solutions is analyzed, leading to the alternative that internal surges must occur or streamlines with vertical tangents must occur.
Hypoelliptic convolution equations in the space ${\scr K}'\sb e$
Dae Hyeon
Pahk
485-495
Abstract: We consider convolution equations in the space $ \exp ({e^{k\vert x\vert}})$ for some constant $k$. Our main results are to find conditions for convolution operators to be hypoelliptic in
Recursive labelling systems and stability of recursive structures in hyperarithmetical degrees
C. J.
Ash
497-514
Abstract: We show that, under certain assumptions of recursiveness in $\mathfrak{A}$, the recursive structure $\mathfrak{A}$ is $ \Delta _\alpha ^0$-stable for $ \alpha < \omega _1^{CK}$ if and only if there is an enumeration of $\mathfrak{A}$ using a $\Sigma _\alpha ^0$ set of recursive ${\Sigma _\alpha }$ infinitary formulae and finitely many parameters from $ \mathfrak{A}$. This extends the results of [1]. To do this, we first obtain results concerning $ \Delta _\alpha ^0$ paths in recursive labelling systems, also extending results of [1]. We show, more generally, that a path and a labelling can simultaneously be defined, when each node of the path is to be obtained by a $\Delta _\alpha ^0$ function from the previous node and its label.
Explosion problems for symmetric diffusion processes
Kanji
Ichihara
515-536
Abstract: We discuss the explosion problem for a symmetric diffusion process. Hasminskii's idea cannot be applied to this case. Instead, the theory of Dirichlet forms is employed to obtain criteria for conservativeness and explosion of the process. The fundamental criteria are given in terms of the $\alpha$-equilibrium potentials and $ \alpha$-capacities of the unit ball centered at the origin. They are applied to obtain sufficient conditions on the coefficients of the infinitesimal generator for conservativeness and explosion.
Vector bundles on complex projective spaces and systems of partial differential equations. I
Peter F.
Stiller
537-548
Abstract: This paper establishes and investigates a relationship between the space of solutions of a system of constant coefficient partial differential equations and the cohomology ($ {H^1}$ in particular) of an associated vector bundle/reflexive sheaf on complex projective space. Using results of Grothendieck and Shatz on vector bundles over projective one-space, the case of partial differential equations in two variables is completely analyzed. The final section applies results about vector bundles on higher-dimensional projective spaces to the case of three or more variables.
The Euler characteristic as an obstruction to compact Lie group actions
Volker
Hauschild
549-578
Abstract: Actions of compact Lie groups on spaces $X$ with $ {H^{\ast}}(X,{\mathbf{Q}}) \cong {\mathbf{Q}}[{x_1}, \ldots ,{x_n}]/{I_0}$, $Q \in {I_0}$ a definite quadratic form, $\deg {x_i} = 2$, are considered. It is shown that the existence of an effective action of a compact Lie group $G$ on such an $X$ implies $\chi (X) \equiv O(\vert WG\vert)$, where $ \chi (X)$ is the Euler characteristic of $X$ and $\vert WG\vert$ means the order of the Weyl group of $ G$. Moreover the diverse symmetry degrees of such spaces are estimated in terms of simple cohomological data. As an application it is shown that the symmetry degree ${N_t}(G/T)$ is equal to $\dim G$ if $G$ is a compact connected Lie group and $T \subset G$ its maximal torus. Effective actions of compact connected Lie groups $K$ on $G/T$ with $ \dim K = \dim G$ are completely classified.
Around Effros' theorem
J. J.
Charatonik;
T.
Maćkowiak
579-602
Abstract: Some stronger versions of the Effros theorem are discussed in the paper, not only for homeomorphisms but also for some other mappings, e.g. for open ones. Equivalent formulations of the theorem are presented as the $\varepsilon$-push property and the existence of a so-called Effros' metric.
Invariants of the Lusternik-Schnirelmann type and the topology of critical sets
Mónica
Clapp;
Dieter
Puppe
603-620
Abstract: We introduce and study in detail generalizations of the notion of Lusternik-Schnirelmann category which give information about the topology of the critical set of a differentiable function. We also improve a result of T. Ganea about the equality of the strong category and the category (even in the classical case).
A characterization and another construction of Janko's group $J\sb 3$
Richard
Weiss
621-633
Abstract: Graphs $ \Gamma$ with the following properties are classified: (i) $\Gamma$ is $(G,s)$-transitive for some $s \geqslant 4$ and some group $G \leqslant \operatorname{aut} (\Gamma )$ such that each vertex stabilizer in $G$ is finite, (ii) $s \geqslant (g - 1)/2$, where $g$ is the girth of $\Gamma$, and (ii) $\Gamma$ is connected. A new construction of ${J_3}$ is given.
The Pontryagin maximum principle from dynamic programming and viscosity solutions to first-order partial differential equations
Emmanuel Nicholas
Barron;
Robert
Jensen
635-641
Abstract: We prove the Pontryagin Maximum Principle for the Lagrange problem of optimal control using the fact that the value function of the problem is the viscosity solution of the associated Hamilton-Jacobi-Bellman equation. The proof here makes rigorous the formal proof of Pontryagin's principle known for at least three decades.
On the zeros of successive derivatives of even Laguerre-P\'olya functions
Li-Chien
Shen
643-652
Abstract: Using "method of steepest descent", we prove that the final set (in the sense of Polya) of a class of even Laguerre-Polya functions is the entire real axis.
On maximal functions and Poisson-Szeg\H o integrals
Juan
Sueiro
653-669
Abstract: We study a class of maximal functions of Hardy-Littlewood type defined on spaces of homogeneous type and we give necessary and sufficient conditions for the corresponding maximal operators to be of weak type $(1,1)$. As a consequence we show that Poisson-Szegö integrals of ${L^p}$ functions possess certain boundary limits which are not implied by Korányi's theorem.
Crossed products and inner actions of Hopf algebras
Robert J.
Blattner;
Miriam
Cohen;
Susan
Montgomery
671-711
Abstract: This paper develops a theory of crossed products and inner (weak) actions of arbitrary Hopf algebras on noncommutative algebras. The theory covers the usual examples of inner automorphisms and derivations, and in addition is general enough to include "inner" group gradings of algebras. We prove that if $\pi :H \to \overline H$ is a Hopf algebra epimorphism which is split as a coalgebra map, then $ H$ is algebra isomorphic to $A{\char93 _\sigma }H$, a crossed product of $H$ with the left Hopf kernel $A$ of $\pi$. Given any crossed product $A{\char93 _\sigma }H$ with $H$ (weakly) inner on $A$, then $ A{\char93 _\sigma }H$ is isomorphic to a twisted product ${A_\tau }[H]$ with trivial action. Finally, if $ H$ is a finite dimensional semisimple Hopf algebra, we consider when semisimplicity or semiprimeness of $A$ implies that of $ A{\char93 _\sigma }H$; in particular this is true if the (weak) action of $ H$ is inner.
Construction of high-dimensional knot groups from classical knot groups
Magnhild
Lien
713-722
Abstract: In this paper we study constructions of high dimensional knot groups from classical knot groups. We study certain homomorphic images of classical knot groups. Specifically, let $ K$ be a classical knot group and $w$ any element in $K$. We are interested in the quotient groups $ G$ obtained by centralizing $w$, i.e. $G = K/[K,w]$, and ask whether $ G$ is itself a knot group. For certain $K$ and $w$ we show that $G$ can be realized as the group of a knotted $ 3$-sphere in $ 5$-space, but $ G$ is not realizable by a $ 2$-sphere in $ 4$-space. By varying $ w$, we also obtain quotients that are groups of knotted $2$-spheres in $4$-space, but they cannot be realized as the groups of classical knots. We have examples of quotients $K/[K,w]$ that have nontrivial second homology. Hence these groups cannot be realized as knot groups of spheres in any dimension. However, we show that these groups are groups of knotted tori in ${S^4}$.
Doubly sliced knots which are not the double of a disk
L.
Smolinsky
723-732
Abstract: In this paper we show that double disk knots can be distinguished from general doubly sliced knots in dimensions $4n + 1$.
Singularly perturbed quadratically nonlinear Dirichlet problems
Albert J.
DeSanti
733-746
Abstract: The Dirichlet problem for singularly perturbed elliptic equations of the form $ \varepsilon \Delta u = A({\mathbf{x}},u)\nabla u \cdot \nabla u + {\mathbf{B}}({\mathbf{x}},u) \cdot \nabla u + C({\mathbf{x}},u)$ in $\Omega \in {E^n}$ is studied. Under explicit and easily checked conditions, solutions are shown to exist for $\varepsilon$ sufficiently small and to exhibit specified asymptotic behavior as $\varepsilon \to 0$. The results are obtained using a method based on the theory of partial differential inequalities.
Diffuse sequences and perfect $C\sp \ast$-algebras
Charles A.
Akemann;
Joel
Anderson;
Gert K.
Pedersen
747-762
Abstract: The concept of a diffuse sequence in a $ {C^{\ast}}$-algebra is introduced and exploited to complete the classification of separable, perfect $ {C^{\ast}}$-algebras. A ${C^{\ast}}$-algebra is separable and perfect exactly when the closure of the pure state space consists entirely of atomic states.
Tameness and local normal bases for objects of finite Hopf algebras
Lindsay N.
Childs;
Susan
Hurley
763-778
Abstract: Let $R$ be a commutative ring, $ S$ an $R$-algebra, $H$ a Hopf $R$algebra, both finitely generated and projective as $ R$-modules, and suppose $ S$ is an $H$-object, so that $ {H^{\ast}} = {\operatorname{Hom} _R}(H,R)$ acts on $S$ via a measuring. Let $I$ be the space of left integrals of ${H^{\ast}}$. We say $S$ has normal basis if $S \cong H$ as $ {H^{\ast}}$modules, and $ S$ has local normal bases if ${S_p} \cong {H_p}$ as $H_p^{\ast}$-modules for all prime ideals $ p$ of $R$. When $R$ is a perfect field, $H$ is commutative and cocommutative, and certain obvious necessary conditions on $S$ hold, then $S$ has normal basis if and only if $IS = R = {S^{{H^{\ast}}}}$. If $ R$ is a domain with quotient field $K$, $H$ is cocommutative, and $L = S \otimes {}_RK$ has normal basis as $({H^{\ast}} \otimes K)$-module, then $ S$ has local normal bases if and only if $ IS = R = {S^{{H^{\ast}}}}$.
Minimal submanifolds of a sphere with bounded second fundamental form
Hillel
Gauchman
779-791
Abstract: Let $h$ be the second fundamental form of an $n$-dimensional minimal submanifold $M$ of a unit sphere ${S^{n + p}}(p \geqslant 2)$, $S$ be the square of the length of $ h$, and $\sigma (u) = \vert\vert h(u,u)\vert{\vert^2}$ for any unit vector $u \in TM$. Simons proved that if $S \leqslant n/(2 - 1/p)$ on $M$, then either $ S \equiv 0$, or $S \equiv n/(2 - 1/p)$. Chern, do Carmo, and Kobayashi determined all minimal submanifolds satisfying $S \equiv n/(2 - 1/p)$. In this paper the analogous results for $ \sigma (u)$ are obtained. It is proved that if $\sigma (u) \leqslant \tfrac{1} {3}$, then either $\sigma (u) \equiv 0$, or $\sigma (u) \equiv \tfrac{1} {3}$. All minimal submanifolds satisfying $\sigma (u)$ are determined. A stronger result is obtained if $M$ is odd-dimensional.
On the Hausdorff dimension of some graphs
R. Daniel
Mauldin;
S. C.
Williams
793-803
Abstract: Consider the functions $\displaystyle {W_b}(x) = \sum\limits_{n = - \infty }^\infty {{b^{ - \alpha n}}[\Phi ({b^n}x + {\theta _n}) - \Phi ({\theta _n})],} $ where $b > 1$, $0 < \alpha < 1$, each ${\theta _n}$ is an arbitrary number, and $ \Phi$ has period one. We show that there is a constant $C > 0$ such that if $b$ is large enough, then the Hausdorff dimension of the graph of ${W_b}$ is bounded below by $2 - \alpha - (C/\ln b)$. We also show that if a function $f$ is convex Lipschitz of order $\alpha$, then the graph of $ f$ has $\sigma $-finite measure with respect to Hausdorff's measure in dimension $2 - \alpha$. The convex Lipschitz functions of order $ \alpha$ include Zygmund's class $ {\Lambda _\alpha }$. Our analysis shows that the graph of the classical van der Waerden-Tagaki nowhere differentiable function has $ \sigma$-finite measure with respect to $ h(t) = t/\ln (1/t)$.
The diagonal entries in the formula ``quasitriangular $-$ compact $=$ triangular'' and restrictions of quasitriangularity
Domingo A.
Herrero
1-42
Abstract: A (bounded linear) Hilbert space operator $T$ is called quasitriangular if there exists an increasing sequence $\{ {P_n}\} _{n = 0}^\infty $ of finite-rank orthogonal projections, converging strongly to 1, such that $\left\Vert {(1 - {P_n})T{P_n}} \right\Vert \to 0\,(n \to \infty )$. This definition, due to P. R. Halmos, plays a very important role in operator theory. The core of this article is a concrete answer to the following problem: Suppose $T$ is a quasitriangular operator and $\Gamma = \{ {\lambda _j}\} _{j = 1}^\infty$ is a sequence of complex numbers. Find necessary and sufficient conditions for the existence of a compact operator $K$ (of arbitrarily small norm) so that $T - K$ is triangular with respect to some orthonormal basis, and the sequence of diagonal entries of $T - K$ coincides with $\Gamma$. For instance, if no restrictions are put on the norm of $K$, then $T$ and $\Gamma$ must be related as follows: (a) if ${\lambda _0}$ is a limit point of $\Gamma$ and ${\lambda _0} - T$ is semi-Fredholm, then $ {\operatorname{ind}}({\lambda _0} - T) > 0$; and (b) if $\Omega$ is an open set intersecting the Weyl spectrum of $T$, whose boundary does not intersect this set, then $ \{ j:{\lambda _j} \in \Omega \}$ is a denumerable set of indices. Particularly important is the case when $\Gamma = \{ 0,0,0, \ldots \}$. The following are equivalent for an operator $T$: (1) there is an integral sequence $\{ {P_n}\} _{n = 0}^\infty$ of orthogonal projections, with rank ${P_n} = n$ for all $n$, converging strongly to 1, such that $\left\Vert {(1 - {P_n})T{P_{n + 1}}} \right\Vert \to 0\,(n \to \infty )$; (2) from some compact $ K,\,T - K$ is triangular, with diagonal entries equal to 0; (3) $T$ is quasitriangular, and the Weyl spectrum of $T$ is connected and contains the origin. The family ${({\text{StrQT}})_{ - 1}}$ of all operators satisfying (1) (and hence (2) and (3)) is a (norm) closed subset of the algebra of all operators; moreover, $ {({\text{StrQT}})_{ - 1}}$ is invariant under similarity and compact perturbations and behaves in many senses as an analog of Halmos's class of quasitriangular operators, or an analog of the class of extended quasitriangular operators $ {({\text{StrQT}})_{ - 1}}$, introduced by the author in a previous article. If $ \{ {P_n}\} _{n = 0}^\infty$ is as in (1), but condition $\left\Vert {(1 - {P_n})T{P_{n + 1}}} \right\Vert \to 0\,(n \to \infty )$ is replaced by (1') $ \left\Vert {(1 - {P_{{n_k}}})T{P_{{n_k} + 1}}} \right\Vert \to 0\,(k \to \infty )$ for some subsequence $\{ {n_k}\} _{k = 1}^\infty $, then (1') is equivalent to (3'), $T$ is quasitriangular, and its Weyl spectrum contains the origin. The family ${({\text{QT}})_{ - 1}}$ of all operators satisfying (1') (and hence (3')) is also a closed subset, invariant under similarity and compact perturbations, and provides a different analog to Halmos's class of quasitriangular operators. Both classes have ``$ m$-versions'' ( $ {({\text{StrQT}})_{ - m}}$ and, respectively, ${({\text{QT}})_{ - m}}$, $m = 1,2,3, \ldots$) with similar properties. ( $ {({\text{StrQT}})_{ - m}}$ is the class naturally associated with triangular operators $A$ such that the main diagonal and the first $ (m - 1)$ superdiagonals are identically zero, etc.) The article also includes some applications of the main result to certain nest algebras ``generated by orthonormal bases.''
Decay with a rate for noncompactly supported solutions of conservation laws
Blake
Temple
43-82
Abstract: We show that solutions of the Cauchy problem for systems of two conservation laws decay in the supnorm at a rate that depends only on the ${L^1}$ norm of the initial data. This implies that the dissipation due to increasing entropy dominates the nonlinearities in the problem at a rate depending only on the ${L^1}$ norm of the initial data. Our results apply to any BV initial data satisfying ${u_0}( \pm \infty ) = 0$ and $ {\operatorname{Sup}}\{ {u_0}( \cdot )\} \ll 1$. The problem of decay with a rate independent of the support of the initial data is central to the issue of continuous dependence in systems of conservation laws because of the scale invariance of the equations. Indeed, our result implies that the constant state is stable with respect to perturbations in $ L_{{\operatorname{loc}}}^1$. This is the first stability result in an $ {L^p}$ norm for systems of conservation laws. It is crucial that we estimate decay in the supnorm since the total variation does not decay at a rate independent of the support of the initial data. The main estimate requires an analysis of approximate characteristics for its proof. A general framework is developed for the study of approximate characteristics, and the main estimate is obtained for an arbitrary number of equations.
Surgery on codimension one immersions in ${\bf R}\sp {n+1}$: removing $n$-tuple points
J. Scott
Carter
83-101
Abstract: The self-intersection sets of immersed $n$-manifolds in $(n + 1)$-space provide invariants of the $ n$th stable stem and the $ (n + 1)$st stable homotopy of infinite real projective space. Theorems of Eccles [5] and others [1, 8, 14, 19] relate these invariants to classically defined homotopy theoretic invariants. In this paper a surgery theory of immersions is developed; the given surgeries affect the self-intersection sets in specific ways. Using such operations a given immersion may be surgered to remove $ (n + 1)$-tuple and $ n$-tuple points, provided the $ {\mathbf{Z}}/2$-valued $ (n + 1)$-tuple point invariant vanishes $(n \geq 5)$. This invariant agrees with the Kervaire invariant for $ n = 4k + 1$. These results first appeared in my dissertation [2]; a summary was presented in [3]. Some results and methods have been improved since these works were written. In particular, the proof of Theorem 14 has been simplified.
On generalizing Boy's surface: constructing a generator of the third stable stem
J. Scott
Carter
103-122
Abstract: An analysis of Boy's immersion of the projective plane in $ 3$-space is given via a collection of planar figures. An analogous construction yields an immersion of the $3$-sphere in $4$-space which represents a generator of the third stable stem. This immersion has one quadruple point and a closed curve of triple points whose normal matrix is a $ 3$-cycle. Thus the corresponding multiple point invariants do not vanish. The construction is given by way of a family of three dimensional cross sections.
Frobenius reciprocity and extensions of nilpotent Lie groups
Jeffrey
Fox
123-144
Abstract: In $\S1$ we use $ {C^\infty }$-vector methods, essentially Frobenius reciprocity, to derive the Howe-Richardson multiplicity formula for compact nilmanifolds. In $\S2$ we use Frobenius reciprocity to generalize and considerably simplify a reduction procedure developed by Howe for solvable groups to general extensions of nilpotent Lie groups. In $\S3$ we give an application of the previous results to obtain a reduction formula for solvable Lie groups.
Affine semigroups and Cohen-Macaulay rings generated by monomials
Ngô Viêt
Trung;
Lê Tuân
Hoa
145-167
Abstract: We give a criterion for an arbitrary ring generated by monomials to be Cohen-Macaulay in terms of certain numerical and topological properties of the additive semigroup generated by the exponents of the monomials. As a consequence, the Cohen-Macaulayness of such a ring is dependent upon the characteristic of the ground field.
Polar classes and Segre classes on singular projective varieties
Shoji
Yokura
169-191
Abstract: We investigate the relation between polar classes of complex varieties and the Segre class of $K$. Johnson [Jo]. Results are obtained for hypersurfaces of projective spaces and for certain varieties with isolated singularities.
Index filtrations and the homology index braid for partially ordered Morse decompositions
Robert
Franzosa
193-213
Abstract: On a Morse decomposition of an invariant set in a flow there are partial orderings defined by the flow. These are called admissible orderings of the Morse decomposition. The index filtrations for a total ordering of a Morse decomposition are generalized in this paper with the definition and proof of existence of index filtrations for admissible partial orderings of a Morse decomposition. It is shown that associated to an index filtration there is a collection of chain complexes and chain maps called the chain complex braid of the index filtration. The homology index braid of the corresponding admissible ordering of the Morse decomposition is obtained by passing to homology in the chain complex braid.
Simplexwise linear untangling
David W.
Henderson
215-226
Abstract: In this paper we show how to canonically untangle simplexwise linear spanning arcs of a convex $2$-cell. Specifically, we show that the space of such arcs is contractible. The main step in the contraction is a flow along the gradient field of an energy function. A $3$-dimensional version of this result would imply the Smale Conjecture--Hatcher Theorem.
Morse theory for codimension-one foliations
Steven C.
Ferry;
Arthur G.
Wasserman
227-240
Abstract: It is shown that a smooth codimension-one foliation on a compact simply-connected manifold has a compact leaf if and only if every smooth real-valued function on the manifold has a cusp singularity.
Dense imbedding of test functions in certain function spaces
Michael
Renardy
241-243
Abstract: In a recent paper [1], J. U. Kim studies the Cauchy problem for the motion of a Bingham fluid in ${R^2}$. He points out that the extension of his results to three dimensions depends on proving the denseness of ${C^\infty }$-functions with compact support in certain spaces. In this note, such a result is proved.
Rational Moore $G$-spaces
Peter J.
Kahn
245-271
Abstract: This paper obtains some existence and uniqueness results for Moore spaces in the context of the equivariant homotopy theory of Bredon. This theory incorporates fixed-point-set data as part of the structure and so is a refinement of the classical equivariant homotopy theory. To avoid counterexamples to existence in the classical case and to focus on new phenomena involving the fixed-point-set structure, most of the results involve rational spaces. In this setting, there are no obstacles to existence, but a notion of projective dimension presents an obstacle to uniqueness: uniqueness is proved, subject to constraint on the projective dimension, and an example shows that this constraint is sharp. Various related existence results are proved and computations are given of certain equivariant mapping sets $ [X,\,Y]$, $X$ an equivariant Moore space.
Equivariant homology decompositions
Peter J.
Kahn
273-287
Abstract: This paper presents some results on the existence of homology decompositions in the context of the equivariant homotopy theory of Bredon. To avoid certain obstructions to the existence of equivariant Moore spaces occurring already in classical equivariant homotopy theory, most of the work of this paper is done ``over the rationals.'' The standard construction of homology decompositions by Eckmann and Hilton can be followed in the present equivariant context until it is necessary to produce appropriate $k'$-invariants. For these, the Eckmann-Hilton construction uses a certain Universal Coefficient Theorem for homotopy sets. The relevant extension of this to the equivariant situation is an equivariant Federer spectral sequence, which is developed in $\S2$. Using this, we can formulate conditions which imply the existence of the desired $ k'$-invariants, and hence the existence of the homology decomposition. The conditions involve a certain notion of projective dimension. For one application, equivariant homology decompositions always exist when the group has prime order.
On Brownian excursions in Lipschitz domains. I. Local path properties
Krzysztof
Burdzy;
Ruth J.
Williams
289-306
Abstract: A necessary and sufficient condition is given for a Brownian excursion law in a Lipschitz domain to share the local path properties with an excursion law in a halfspace. This condition is satisfied for all boundary points of every ${C^{1,\alpha }}$-domain, $ \alpha > 0$. There exists a ${C^1}$-domain such that the condition is satisfied almost nowhere on the boundary. A probabilistic interpretation and applications to minimal thinness and boundary behavior of Green functions are given.
Locally geodesically quasiconvex functions on complete Riemannian manifolds
Takao
Yamaguchi
307-330
Abstract: In this article, we investigate the topological structure of complete Riemannian manifolds admitting locally geodesically quasiconvex functions, whose family includes all geodesically convex functions. The existence of a locally geodesically quasiconvex function is equivalent to the existence of a certain filtration by locally convex sets. Our argument contains Morse theory for the lower limit function of a given locally geodesically quasiconvex function.
Simply-connected $4$-manifolds with a given boundary
Steven
Boyer
331-357
Abstract: Let $M$ be a closed, oriented, connected $ 3$-manifold. For each bilinear, symmetric pairing $({{\mathbf{Z}}^n},\,L)$, our goal is to calculate the set $ {\mathcal{V}_L}(M)$ of all oriented homeomorphism types of compact, $ 1$-connected, oriented $ 4$-manifolds with boundary $ M$ and intersection pairing isomorphic to $ ({{\mathbf{Z}}^n},\,L)$. For each pair $ ({{\mathbf{Z}}^n},\,L)$ which presents $ {H_ \ast }(M)$, we construct a double coset space $B_L^t(M)$ and a function $c_L^t:{\mathcal{V}_L}(M) \to B_L^t(M)$. The set $ B_L^t(M)$ is the quotient of the group of all link-pairing preserving isomorphisms of the torsion subgroup of ${H_1}(M)$ by two naturally occuring subgroups. When $ ({{\mathbf{Z}}^n},\,L)$ is an even pairing, we construct another double coset space ${\hat B_L}(M)$, a function ${\hat c_L}:{\mathcal{V}_L}(M) \to {\hat B_L}(M)$ and a projection ${p_2}:{\hat B_L}(M) \to B_L^t(M)$ such that $ {p_2} \cdot {\hat c_L} = c_L^t$. Our main result states that when $({{\mathbf{Z}}^n},\,L)$ is even the function ${\hat c_L}$ is injective, as is the function $ c_L^t \times \Delta :{\mathcal{V}_L}(M) \to B_L^t(M) \times {\mathbf{Z}}/2$ when $ ({{\mathbf{Z}}^n},\,L)$ is odd. Here $\Delta$ is a Kirby-Siebenmann obstruction to smoothing. It follows that the sets ${\mathcal{V}_L}(M)$ are finite and of an order bounded above by a constant depending only on $ {H_1}(M)$. We also show that when ${H_1}(M;{\mathbf{Q}}) \cong 0$ and $({{\mathbf{Z}}^n},\,L)$ is even, $c_L^t = {p_2} \cdot {\hat c_L}$ is injective. It seems likely that via the functions $c_L^t \times \Delta$ and ${\hat c_L}$, the sets $B_L^t(M) \times {\mathbf{Z}}/2$ and ${\hat B_L}(M)$ calculate ${\mathcal{V}_L}(M)$ when $({{\mathbf{Z}}^n},\,L)$ is respectively odd and even. We verify this in several cases, most notably when $ {H_1}(M)$ is free abelian. The results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two $1$-connected $4$-manifolds extending a given homeomorphism of their boundaries. The theory developed is then applied to show that there is an $m > 0$, depending only on ${H_1}(M)$, such that for any self-homeomorphism $f$ of $M$, ${f^m}$ extends to a self-homeomorphism of any $ 1$-connected, compact $ 4$-manifold with boundary $ M$.
The chromatic number of Kneser hypergraphs
N.
Alon;
P.
Frankl;
L.
Lovász
359-370
Abstract: Suppose the $ r$-subsets of an $ n$-element set are colored by $t$ colors. THEOREM 1.1. If $n \geq (t - 1)(k - 1) + k \cdot r$, then there are $k$ pairwise disjoint $r$-sets having the same color. This was conjectured by Erdös $[{\mathbf{E}}]$ in 1973. Let $T(n,\,r,\,s)$ denote the Turán number for $ s$-uniform hypergraphs (see $\S1$). THEOREM 1.3. If $\varepsilon > 0,\,t \leq (1 - \varepsilon )T(n,\,r,\,s)/(k - 1)$, and $ n > {n_0}(\varepsilon ,\,r,\,s,\,k)$, then there are $k$ $r$-sets ${A_1},{A_2}, \ldots ,{A_k}$ having the same color such that $\left\vert {{A_i} \cap {A_j}} \right\vert < s$ for all $ 1 \leq i < j \leq k$. If $ s = 2,\,\varepsilon$ can be omitted. Theorem 1.1 is best possible. Its proof generalizes Lovász' topological proof of the Kneser conjecture (which is the case $ k = 2$). The proof uses a generalization, due to Bárány, Shlosman, and Szücs of the Borsuk-Ulam theorem. Theorem 1.3 is best possible up to the $\varepsilon$-term (for large $n$). Its proof is purely combinatorial, and employs results on kernels of sunflowers.
On the Cauchy problem associated with the motion of a Bingham fluid in the plane
Jong Uhn
Kim
371-400
Abstract: This paper discusses an initial value problem for the variational inequality which describes the motion of a Bingham fluid in the plane. The existence of strong solution is established.
Large-time behavior of solutions to a scalar conservation law in several space dimensions
Patricia
Bauman;
Daniel
Phillips
401-419
Abstract: We consider solutions of the Cauchy problem in ${\mathbf{R}}_ + ^{n + 1}$ for the equation ${u_t} + {\operatorname{div}_x}f(u) = 0$. The initial data is assumed to be a compact perturbation of a function of the form, $\varphi (x) = a$ for $\left\langle {x,\,\mu } \right\rangle > 0$, $\varphi (x) = b$ for $\left\langle {x,\,\mu } \right\rangle < 0$, where $a$ and $b$ are constants and $\mu$ is a given unit vector. The Cauchy problem together with an entropy condition on $u$ is known to be well posed. The solution with unperturbed initial data, $\varphi (x)$, is a traveling shock, $\varphi (x - \overrightarrow k t)$, provided that $\varphi (x - \overrightarrow k t)$ satisfies the entropy condition (an inequality on $a,\,b,\,\mu$, and $f$). Assuming this type of condition on $ \varphi$, we study the large-time behavior of $u$. In particular, we show that $u$ converges to a traveling shock whose profile agrees with $\left\langle {x,\,\mu } \right\rangle = 0$ outside of a compact set.
Fixed sets of framed $G$-manifolds
Stefan
Waner
421-429
Abstract: This note describes restrictions on the framed bordism class of a framed manifold $Y$ in order that it be the fixed set of some framed $ G$-manifold $M$ with $G$ a finite group. These results follow from a recently proved generalization of the Segal conjecture, and imply, in particular, that if $M$ is a framed $G$-manifold of sufficiently high dimension, and if $G$ is a $p$-group, then the number of ``noncancelling'' fixed points is either zero or approaches infinity as the dimension of $M$ goes to infinity. Conversely, we give sufficient conditions on the framed bordism class of a manifold $ Y$ that it be the fixed set of some framed $G$-manifold $M$ of arbitrarily high dimension.
Some product formulae for nonsimply connected surgery problems
R. J.
Milgram;
Andrew
Ranicki
383-413
Abstract: For an $ n$-dimensional normal map $f :{M^n} \to {N^n}$ with finite fundamental group $ {\pi _1}(N) = \pi$ and PL $ 1$ torsion kernel $ Z[\pi ]$-modules ${K_{\ast}}(M)$ the surgery obstruction $ {\sigma _{\ast}}(f) \in L_n^h(Z[\pi ])$ is expressed in terms of the projective classes $[{K_{\ast}}(M)] \in {\tilde K_0}(Z[\pi ])$, assuming ${K_i}(M) = 0$ if $n = 2i$. This expression is used to evaluate in certain cases the surgery obstruction ${\sigma _ {\ast} }(g) \in L_{m + n}^h(Z[{\pi _1} \times \pi ])$ of the $(m + n)$-dimensional normal map $ g = 1 \times f:{M_1} \times M \to {M_1} \times N$ defined by product with an $ m$-dimensional manifold $ {M_1}$, where ${\pi _1} = {\pi _1}({M_1})$.
Bounds for prime solutions of some diagonal equations. II
Ming Chit
Liu
415-426
Abstract: Let ${b_j}$ and $m$ be certain integers. In this paper we obtain a bound for prime solutions ${p_j}$ of the diagonal equations of order $ k,\;{b_1}p_1^k + \cdots + {b_s}p_s^k = m$. The bound obtained is $ {C^{{{(\log B)}^2}}} + C\vert m{\vert^{1/k}}$ where $B = {\max _j}\{ e,\vert{b_j}\vert\}$ and $ C$ are positive constants depending at most on $k$.
Groups presented by finite two-monadic Church-Rosser Thue systems
J.
Avenhaus;
K.
Madlener;
F.
Otto
427-443
Abstract: It is shown that a group $G$ can be defined by a monoid-presentation of the form $(\Sigma ;T)$, where $T$ is a finite two-monadic Church-Rosser Thue system over $\Sigma$, if and only if $G$ is isomorphic to the free product of a finitely generated free group with a finite number of finite groups.
Homologie de l'espace des sections d'un fibr\'e
Claude
Legrand
445-459
Abstract: For a fiber bundle with a finite cohomology dimension and $1$-connected base $B$ and $1$-connected fiber $F$, we obtain the homology of the section space by an ${E^1}$-spectral sequence. In the "stable" range the $ {E^1}$-terms are the homology of a product of Eilenberg-Mac Lane space of type $ K({H^{p - i}}(B;{\pi _p}F),i)$ (the same as those of the $ {E^1}$-spectral sequences which converges to the homology of the functional space $\operatorname{Hom} (B,F)$ [10]). The differential is the product of two operations: one appears in the ${E^1}$-spectral sequence, which converges to the homology of $\operatorname{Hom} (B,F)$; the second one is a "cup-product" determined by the fiber structure of the bundle. This spectral sequence is obtained by a Moore-Postnikov tower of the fiber, which generalizes Kahn's methods [9].
Random power series generated by ergodic transformations
Judy
Halchin;
Karl
Petersen
461-485
Abstract: Generalizing classical studies of power series with sequences of independent random variables as coefficients, we study series of the forms $\displaystyle {g_{x,\phi }}(z) = \sum\limits_{n = 0}^\infty {\phi ({T^n}x){z^n}... ...m\limits_{n = 1}^\infty {\phi (x)\phi (Tx) \cdots \phi ({T^{n - 1}}x){z^n},} }$ where $T$ is an ergodic measure-preserving transformation on a probability space $(X,\mathcal{B},\mu )$ and $\phi$ is a measurable complex-valued function which is a.e. nonzero. When ${f_{x,\phi }}$ is entire, its order of growth at infinity measures the speed of divergence of the ergodic averages of $\log \vert\phi \vert$. We give examples to show that any order is possible for any $ T$ and that different orders are possible for fixed $\phi$. For fixed $T$, the set of $\phi$ which produce infinite order is residual in the subset of ${L^1}(X)$ consisting of those $\phi$ which are a.e. nonzero and produce entire $ {f_{x,\phi }}$. As in a theorem of Pólya for gap series, if ${f_{x,\phi }}$ is entire and has finite order, then it assumes every value infinitely many times. The functions $ \phi \in {L^1}(X)$ for which ${g_{x,\phi }}$ is rational a.e. are exactly the finite sums of eigenfunctions of $ T$; their poles are all simple and are the inverses of the corresponding eigenvalues. By combining this result with a skew product construction, we can also characterize when ${f_{x,\phi }}$ is rational, provided that $ \phi$ takes one of several particular forms.
The BGG resolution, character and denominator formulas, and related results for Kac-Moody algebras
Wayne
Neidhardt
487-504
Abstract: Let $\mathfrak{g}$ be a Kac-Moody algebra defined by a (not necessarily symmetrizable) generalized Cartan matrix. We construct a BGG-type resolution of the irreducible module $ L(\lambda )$ with dominant integral highest weight $\lambda$, and we use this to obtain character and denominator formulas analogous to those of Weyl. We also determine a condition on the algebra which is sufficient for these formulas to take their classical form, and which implies that the set of defining relations is complete.
Construction of a family of non-self-dual gauge fields
Ignacio
Sols
505-508
Abstract: Using the generalization of vector bundles by reflexive sheaves recently introduced by R. Hartshorne in [2] we construct a $ 15$-dimensional family of nontrivial complex gauge fields $(U,E,\nabla )$ which are not self-dual nor anti-self-dual. ($U$ is an affine neighborhood in $ {Q_4} = \operatorname{Gr} (2,{{\mathbf{C}}^4})$ of the stereographic compactification ${S^4}$ of $ {\mathbb{R}^4}$, $ E$ is a vector bundle on $ U$ and $\nabla$ is a connection on it whose curvature $\phi$ satisfies the inequalities ${}^{\ast}\phi \ne \phi$ and ${}^{\ast}\phi \ne - \phi $.)
The intersection topology w.r.t. the real line and the countable ordinals
G. M.
Reed
509-520
Abstract: If ${\Upsilon _1}$ and $ {\Upsilon _2}$ are topologies defined on the set $X$, then the intersection topology w.r.t. ${\Upsilon _1}$ and ${\Upsilon _2}$ is the topology $\Upsilon$ on $X$ such that $\{ {U_1} \cap {U_2}\vert{U_1} \in {\Upsilon _1}\;{\text{and}}\;{U_2} \in {\Upsilon _2}\}$ is a basis for $ (X,\Upsilon )$. In this paper, the author considers spaces in the class $\mathcal{C}$, where $(X,\Upsilon ) \in \mathcal{C}$ iff $X = \{ {x_\alpha }\vert\alpha < {\omega _1}\} \subseteq {\mathbf{R}}$, ${\Upsilon _{\mathbf{R}}}$ is the inherited real line topology on $X$, $ {\Upsilon _{{\omega _1}}}$ is the order topology on $X$ of type $ {\omega _1}$, and $ \Upsilon$ is the intersection topology w.r.t. ${\Upsilon _{\mathbf{R}}}$ and ${\Upsilon _{{\omega _1}}}$. This class is shown to be a surprisingly useful tool in the study of abstract spaces. In particular, it is shown that: (1) If $ X \in \mathcal{C}$, then $ X$ is a completely regular, submetrizable, pseudo-normal, collectionwise Hausdorff, countably metacompact, first countable, locally countable space with a base of countable order that is neither subparacompact, metalindelöf, cometrizable, nor locally compact. (2) $(\operatorname{MA} + \neg \operatorname{CH} )$ If $X \in \mathcal{C}$, then $ X$ is perfect. (3) There exists in ZFC an $ X \in \mathcal{C}$ such that $X$ is not normal. (4) $(\operatorname{CH} )$ There exists $X \in \mathcal{C}$ such that $X$ is collectionwise normal and ${\omega _1}$-compact but not perfect.
On the rational homotopy Lie algebra of a fixed point set of a torus action
Christopher
Allday;
Volker
Puppe
521-528
Abstract: Let $X$ be a simply connected topological space, and let ${\mathcal{L}_{\ast}}(X)$ be its rational homotopy Lie algebra. Suppose that a torus acts on $X$ with fixed points, and suppose that $ F$ is a simply connected component of the fixed point set. If ${\mathcal{L}_{\ast}}(X)$ is finitely presented and if $ F$ is full, then it is shown that ${\mathcal{L}_{\ast}}(F)$ is finitely presented, and that the numbers of generators and relations in a minimal presentation of ${\mathcal{L}_{\ast}}(F)$ do not exceed the numbers of generators and relations (respectively) in a minimal presentation of ${\mathcal{L}_{\ast}}(X)$. Various other related results are given.
Shape properties of Whitney maps for hyperspaces
Hisao
Kato
529-546
Abstract: In this paper, some shape properties of Whitney maps for hyperspaces are investigated. In particular, the following are proved: (1) Let $X$ be a continuum and let $ \mathfrak{H}$ be the hyperspace ${2^X}$ or $C(X)$ of $X$ with the Hausdorff metric. Then if $ \omega$ is any Whitney map for $ \mathfrak{H}$, for any $0 \leqslant s \leqslant t \leqslant \omega (X){\omega ^{ - 1}}(t)$ is an approximate strong deformation retract of ${\omega ^{ - 1}}([s,t])$. In particular, $\operatorname{Sh} ({\omega ^{ - 1}}(t)) = \operatorname{Sh} ({\omega ^{ - 1}}([s,t]))$. (2) Pointed $1$-movability is a Whitney property. (3) For any given $ {\text{n}} < \infty$, the property of (cohomological) dimension $\leqslant n$ is a sequential strong Whitney-reversible property. (4) The property of being chainable or circle-like is a sequential strong Whitney-reversible property. (5) The property of being an FAR is a Whitney property for $1$-dimensional continua. Property (2) is an affirmative answer to a problem of J. T. Rogers [16, 112]. Properties (3) and (4) are affirmative answers to problems of S. B. Nadler [20, (14.57) and 21].
A formula for Casson's invariant
Jim
Hoste
547-562
Abstract: Suppose $ H$ is a homology $ 3$-sphere obtained by Dehn surgery on a link $L$ in a homology $3$-sphere $M$. If every pair of components of $L$ has zero linking number in $ M$, then we give a formula for the Casson invariant, $ \lambda (H)$, in terms of $\lambda (M)$, the surgery coefficients of $ L$, and a certain coefficient from each of the Conway polynomials of $ L$ and all its sublinks. A few consequences of this formula are given.
On ${\bf R}\sp \infty\;(Q\sp \infty)$-manifold bundles over CW complexes
Vo Thanh
Liem
563-585
Abstract: Let $\Lambda \in \mathcal{C}\mathcal{W}(\mathcal{C}) \cup \mathcal{C}\mathcal{W}(\mathcal{M})$ be a pseudo CW complex generated either by Hausdorff compact spaces or by metric spaces. In the theory of manifolds modeled on $ {R^\infty }$ or ${Q^\infty }$, we will prove the $\Lambda $-fiber-preserving versions of the following: Equivalences among the notions of $D$-sets, $ {D^{\ast}}$-sets and infinite deficient sets; relative stability theorems; relative deformation of homotopy equivalences to homeomorphisms; strong unknotting theorem for $ D$-embeddings; and $ \alpha$-approximation theorems.
Stratification of continuous maps of an interval
L. S.
Block;
W. A.
Coppel
587-604
Abstract: We define the motion of turbulence for a continuous map of an interval into the line and study its relation with periodic and homoclinic points. We define also strongly simple orbits and show, in particular, that they represent periodic orbits with minimum entropy. Further results are obtained for unimodal maps with negative Schwarzian, which sharpen recent results of Block and Hart.
A topological proof of the equivariant Dehn lemma
Allan L.
Edmonds
605-615
Abstract: An elementary topological proof is given for a completely general version of the Equivariant Dehn Lemma, in the spirit of the original proof of the nonequivariant version due to C. D. Papakyriakopolous in 1957.
Smooth maps, pullback path spaces, connections, and torsions
Kuo Tsai
Chen
617-627
Abstract: By generalizing the local version of the usual differential geometric notion of connections and that of torsions, a model for the pullback path space of a smooth map is constructed from the induced map of the de Rham complexes. The pullback path space serves not only as a homotopy fiber but also as a device reflecting differentiable properties of the smooth map. Applications are discussed.
Range transformations on a Banach function algebra
Osamu
Hatori
629-643
Abstract: We study the range transformations $\operatorname{Op} ({A_{D,}}\operatorname{Re} B)$ and $\operatorname{Op} ({A_D},B)$ for Banach function algebras $A$ and $B$. As a special instance, the harmonicity of functions in $\operatorname{Op} ({A_D},\operatorname{Re} A)$ for a nontrivial function algebra $A$ is established and is compared with previous investigations of $\operatorname{Op} ({A_D},A)$ and $\operatorname{Op} ({(\operatorname{Re} A)_I},(\operatorname{Re} A))$ for an interval $I$. In $\S2$ we present some results on $\operatorname{Op} ({A_D},B)$ and use them to show that functions in ${\operatorname{Op} ^C}({A_D},B)$ are analytic for certain Banach function algebras.
Boundary behavior of a nonparametric surface of prescribed mean curvature near a reentrant corner
Alan R.
Elcrat;
Kirk E.
Lancaster
645-650
Abstract: Let $\Omega$ be an open set in ${{\mathbf{R}}^2}$ which is locally convex at each point of its boundary except one, say $(0,0)$. Under certain mild assumptions, the solution of a prescribed mean curvature equation on $ \Omega$ behaves as follows: All radial limits of the solution from directions in $ \Omega$ exist at $ (0,0)$, these limits are not identical, and the limits from a certain half-space $ (H)$ are identical. In particular, the restriction of the solution to $\Omega \cap H$ is the solution of an appropriate Dirichlet problem.
Cauchy problem for nonlinear hyperbolic systems of partial differential equations
Victoria
Yasinovskaya
651-668
Abstract: We proved the sharp Sobolev estimate for Cauchy data for the general type of hyperbolic systems of nonlinear partial differential equations, which leads to a local existence and uniqueness theorem for solutions of the Cauchy problem in Sobolev spaces.
Poisson integrals of regular functions
José R.
Dorronsoro
669-685
Abstract: Tangential convergence of Poisson integrals is proved for certain spaces of regular functions which contain the spaces of Bessel potentials of ${L^p}$ functions, $1 < p < \infty$, and of functions in the local Hardy space ${h^1}$, and the corresponding tangential maximal functions are shown to be of strong $ p$ type, $p \geqslant 1$.
Dimension-free quasiconformal distortion in $n$-space
G. D.
Anderson;
M. K.
Vamanamurthy;
M.
Vuorinen
687-706
Abstract: Most distortion theorems for $K$-quasiconformal mappings in ${{\mathbf{R}}^n}$, $n \geqslant 2$, depend on both $n$ and $K$ in an essential way, with bounds that become infinite as $n$ tends to $\infty$. The present authors obtain dimension-free versions of four well-known distortion theorems for quasiconformal mappings--namely, bounds for the linear dilatation, the Schwarz lemma, the $ \Theta$-distortion theorem, and the $\eta$-quasisymmetry property of these mappings. They show that the upper estimates they have obtained in each of these four main results remain bounded as $ n$ tends to $\infty$ with $K$ fixed. The proofs are based on a "dimensioncancellation" property of the function $ t \mapsto {\tau ^{ - 1}}(\tau (t)/K),\,t > 0,\,K > 0$, where $\tau (t)$ is the capacity of a Teichmüller extremal ring in $ {{\mathbf{R}}^n}$. The authors also prove a dimension-free distortion theorem for the absolute (cross) ratio under $ K$-quasiconformal mappings of $ {\overline {\mathbf{R}} ^n}$, from which several other distortion theorems follow as special cases.
Fine structure of the integral exponential functions below $2\sp {2\sp x}$
Bernd I.
Dahn
707-716
Abstract: Integral exponential functions are the members of the least class of real functions containing $1$, the identity function, and closed under addition, multiplication, and binary exponentiation sending $f$ and $g$ to ${f^g}$. This class is known to be wellordered by the relation of eventual dominance. It is shown that for each natural number $n$ the order type of the integral exponential functions below $ {2^{{x^n}}}$ (below ${x^{{x^n}}}$) is exactly ${\omega ^{{\omega ^{2n - 1}}}}$ ( ${\omega ^{{\omega ^{2n}}}}$ respectively). The proof, using iterated asymptotic expansions, contains also a new proof that integral exponential functions below $ {2^{{2^x}}}$ are wellordered.
On the factorizations of ordinary linear differential operators
G. J.
Etgen;
G. D.
Jones;
W. E.
Taylor
717-728
Abstract: Relations are found between the nonvanishing of certain Wronskians and disconjugacy properties of $ {L_n}y + py = 0$, where $ {L_n}y$ is a disconjugate operator and $p$ is sign definite. The results are then used to show ways in which $ {L_n}y + py$ can be factored.
On the local behavior of $\Psi(x,y)$
Adolf
Hildebrand
729-751
Abstract: $\Psi (x,y)$ denotes the number of positive integers $\leqslant x$ and free of prime factors $> y$. In the range $y \geqslant \exp ({(\log \log x)^{5/3 + \varepsilon }})$, $\Psi (x,y)$ can be well approximated by a "smooth" function, but for $y \leqslant {(\log x)^{2 - \varepsilon }}$, this is no longer the case, since then the influence of irregularities in the distribution of primes becomes apparent. We show that $ \Psi (x,y)$ behaves "locally" more regular by giving a sharp estimate for $\Psi (cx,y)/\Psi (x,y)$, valid in the range $x \geqslant y \geqslant 4\log x$, $1 \leqslant c \leqslant y$.
A strong containment property for discrete amenable groups of automorphisms on $W\sp \ast$ algebras
Edmond E.
Granirer
753-761
Abstract: Let $G$ be a countable group of automorphisms on a $ {W^{\ast}}$ algebra $\mathcal{M}$ and let ${\phi _0}$ be a ${w^{\ast}}{G_\delta }$ point of the set of $ G$ invariant states on $\mathcal{M}$ which belong to $ {w^{\ast}}\operatorname{cl} \operatorname{Co} E$, where $E$ is a set of (possibly pure) states on $\mathcal{M}$. If $G$ is amenable, then the cyclic representation ${\pi _{{\phi _0}}}$ corresponding to $ {\phi _0}$ is contained in $ \{ \oplus {\pi _\phi };\phi \in E\}$. This property characterizes amenable groups. Related results are obtained.
On the a.e. convergence of the arithmetic means of double orthogonal series
F.
Móricz
763-776
Abstract: The extension of the coefficient test of Menšov and Kaczmarz ensuring the a.e. $(C,1,1)$-summability of double orthogonal series has been stated by two authors. Unfortunately, their proofs turned out to be deficient. Now we present a general theory, in the framework of which a complete proof of this test can also be obtained. Besides, we extend the relevant theorems of Kolmogorov and Kaczmarz from single orthogonal series to double ones, establishing the a.e. equiconvergence of the lacunary subsequences of the rectangular partial sums and of the entire sequence of the arithmetic means. The corresponding tests ensuring the a.e. $(C,1,0)$ and $(C,0,1)$-summability are also treated.
All triangles are Ramsey
Peter
Frankl;
Vojtěch
Rödl
777-779
Abstract: Given a triangle $ ABC$ and an integer $ r$, $r \geqslant 2$, it is shown that for $ n$ sufficiently large and an arbitrary $r$-coloring of ${R^n}$ one can find a monochromatic copy of $ ABC$.
Above and below subgroups of a lattice-ordered group
Richard N.
Ball;
Paul
Conrad;
Michael
Darnel
1-40
Abstract: In an $ l$-group $G$, this paper defines an $l$-subgroup $A$ to be above an $l$-subgroup $B$ (or $B$ to be below $A$) if for every integer $n$, $a \in A$, and $b \in B$, $n(\vert a\vert \wedge \vert b\vert) \leqslant \vert a\vert$. It is shown that for every $ l$-subgroup $A$, there exists an $l$-subgroup $B$ maximal below $A$ which is closed, convex, and, if the $ l$-group $G$ is normal-valued, unique, and that for every $l$-subgroup $B$ there exists an $l$-subgroup $A$ maximal above $B$ which is saturated: if $0 = x \wedge y$ and $ x + y \in A$, then $ x \in A$. Given $ l$-groups $A$ and $B$, it is shown that every lattice ordering of the splitting extension $ G = A \times B$, which extends the lattice orders of $A$ and $B$ and makes $A$ lie above $B$, is uniquely determined by a lattice homomorphism $\pi$ from the lattice of principal convex $ l$-subgroups of $ A$ into the cardinal summands of $B$. This extension is sufficiently general to encompass the cardinal sum of two $l$-groups, the lex extension of an $ l$-group by an $ o$-group, and the permutation wreath product of two $l$-groups. Finally, a characterization is given of those abelian $l$-groups $G$ that split off below: whenever $ G$ is a convex $ l$-subgroup of an $ l$-group $H$, $H$ is then a splitting extension of $G$ by $A$ for any $l$-subgroup $A$ maximal above $G$ in $H$.
Equivariant Morse theory for flows and an application to the $N$-body problem
Filomena
Pacella
41-52
Abstract: In this paper, using Conley's index and equivariant cohomology, some Morse type inequalities are deduced for a flow equivariant with respect to the action of a compact topological group. In the case of a gradient flow induced by a nondegenerate smooth function these inequalities coincide with those described by R. Bott. The theory is applied to the study of the central configurations of $N$-bodies.
Weighted inequalities for the one-sided Hardy-Littlewood maximal functions
E.
Sawyer
53-61
Abstract: Let ${M^ + }f(x) = {\sup _{h > 0}}(1/h)\int_x^{x + h} {\vert f(t)\vert\,dt}$ denote the one-sided maximal function of Hardy and Littlewood. For $w(x) \geqslant 0$ on $R$ and $1 < p < \infty$, we show that $ {M^ + }$ is bounded on $ {L^p}(w)$ if and only if $ w$ satisfies the one-sided $ {A_p}$ condition: $\displaystyle \left( {A_p^ + } \right)\qquad \left[ {\frac{1} {h}\int_{a - h}^a... ...1} {h}\int_a^{a + h} {w{{(x)}^{ - 1/(p - 1)}}dx} } \right]^{p - 1}} \leqslant C$ for all real $a$ and positive $h$. If in addition $v(x) \geqslant 0$ and $\sigma = {v^{ - 1/(p - 1)}}$,then ${M^ + }$ is bounded from ${L^p}(v)$ to ${L^p}(w)$ if and only if $\displaystyle \int_I {{{[{M^ + }({\chi _I}\sigma )]}^p}w \leqslant C\int_I {\sigma < \infty } }$ for all intervals $I = (a,b)$ such that $\int_{ - \infty }^a {w > 0}$. The corresponding weak type inequality is also characterized. Further properties of $A_p^ +$ weights, such as $A_p^ + \Rightarrow A_{p - \varepsilon }^ +$ and $A_p^ + = (A_1^ + ){(A_1^ - )^{1 - p}}$, are established.
\`A propos de ``wedges'' et d'``edges'', et de prolongements holomorphes
Jean-Pierre
Rosay
63-72
Abstract: Holomorphic extensions in wedges of continuous functions defined on edges, which are extensions in the distributional sense, are shown to be genuine continuous extensions, and a ${\mathcal{C}^1}$ version of the edge of the wedge theorem is proved.
Weighted nonlinear potential theory
David R.
Adams
73-94
Abstract: The potential theoretic idea of the "thinness of a set at a given point" is extended to the weighted nonlinear potential theoretic setting--the weights representing in general singularities/degeneracies--and conditions on these weights are given that guarantee when two such notions are equivalent at the given point. When applied to questions of boundary regularity for solutions to (degenerate) elliptic second-order partial differential equations in bounded domains, this result relates the boundary Wiener criterion for one operator to that of another, and in the linear case gives conditions for boundary regular points to be the same for various operators. The methods also yield two weight norm inequalities for Riesz potentials $\displaystyle {\left( {\int {{{({I_\alpha }{\ast}f)}^q}v\,dx} } \right)^{1/q}} \leqslant {\left( {\int {{f^p}w\,dx} } \right)^{1/p}},$ $1 < p \leqslant q < \infty$, which at least in the first-order case $(\alpha = 1)$ have found some use in a number of places in analysis.
Isometries on $L\sb {p,1}$
N. L.
Carothers;
B.
Turett
95-103
Abstract: The extreme points of the sphere of the Lorentz function space ${L_{p,1}}[0,1]$ are computed. As an application, the linear isometries from ${L_{p,1}}$ into itself are completely described.
Equivariant minimal immersions of $S\sp 2$ into $S\sp {2m}(1)$
Norio
Ejiri
105-124
Abstract: We classify the directrix curves associated with equivariant minimal immersions of ${S^2}$ into $ {S^{2m}}(1)$ and obtain some applications.
Unknotting information from $4$-manifolds
T. D.
Cochran;
W. B. R.
Lickorish
125-142
Abstract: Results of S. K. Donaldson, and others, concerning the intersection forms of smooth $4$-manifolds are used to give new information on the unknotting numbers of certain classical knots. This information is particularly sensitive to the signs of the knot crossings changed in an unknotting process.
Attracting orbits in Newton's method
Mike
Hurley
143-158
Abstract: It is well known that the dynamical system generated by Newton's Method applied to a real polynomial with all of its roots real has no periodic attractors other than the fixed points at the roots of the polynomial. This paper studies the effect on Newton's Method of roots of a polynomial "going complex". More generally, we consider Newton's Method for smooth real-valued functions of the form ${f_\mu }(x) = g(x) + \mu $, $\mu$ a parameter. If ${\mu _0}$ is a point of discontinuity of the map $\mu \to$ (the number of roots of ${f_\mu }$), then, in the presence of certain nondegeneracy conditions, we show that there are values of $\mu$ near ${\mu _0}$ for which the Newton function of $ {f_\mu }$ has nontrivial periodic attractors.
The axioms of supermanifolds and a new structure arising from them
Mitchell J.
Rothstein
159-180
Abstract: An analysis of supermanifolds over an arbitrary graded-commmutative algebra is given, proceeding from a set of axioms the first of which is that the derivations of the structure sheaf of a supermanifold are locally free. These axioms are satisfied not by the sheaf of $ {G^\infty }$ functions, as has been asserted elsewhere, but by an extension of this sheaf. A given $ {G^\infty }$ manifold may admit many supermanifold extensions, and it is unknown at present whether there are ${G^\infty }$ manifolds that admit no such extension. When the underlying graded-commutative algebra is commutative, the axioms reduce to the Berezin-Kostant supermanifold theory.
Absolute subretracts and weak injectives in congruence modular varieties
Brian A.
Davey;
L. G.
Kovács
181-196
Abstract: Absolute subretracts and weak injectives in congruence modular varieties of universal algebras are investigated by focusing attention on the directly indecomposibles. The proofs rely on a congruence modular version of generalized direct products (direct products with amalgamation) and on the generalized Jónsson Lemma for congruence modular varieties. The results have immediate application to varieties of groups or rings.
Eigenvalues below the essential spectra of singular elliptic operators
W. D.
Evans;
Roger T.
Lewis
197-222
Abstract: A new technique is developed for determining if the number of eigenvalues below the essential spectrum of a singular elliptic differential operator is finite. A method is given for establishing lower bounds for the least point of the essential spectrum in terms of the behavior of the coefficients and weight near the singularities. Higher-order operators are included in these results as well as second-order Schrödinger operators.
The blow-up boundary for nonlinear wave equations
Luis A.
Caffarelli;
Avner
Friedman
223-241
Abstract: Consider the Cauchy problem for a nonlinear wave equation $\square u = F(u)$ in $N$ space dimensions, $N \leqslant 3$, with $ F$ superlinear and nonnegative. It is well known that, in general, the solution blows up in finite time. In this paper it is shown, under some assumptions on the Cauchy data, that the blow-up set is a space-like surface $t = \phi (x)$ with $\phi (x)$ continuously differentiable.
A regularity theorem for minimizing hypersurfaces modulo $\nu$
Frank
Morgan
243-253
Abstract: It is proved that an $(n - 1)$-dimensional, area-minimizing flat chain modulo $\nu$ in $ {{\mathbf{R}}^n}$, with smooth extremal boundary of at most $\nu /2$ components, has an interior singular set of Hausdorff dimension at most $ n - 8$. Similar results hold for more general integrands.
Countable-dimensional universal sets
Roman
Pol
255-268
Abstract: The main results of this paper are a construction of a countable union of zero dimensional sets in the Hilbert cube whose complement does not contain any subset of finite dimension $n \geqslant 1$ (Theorem 2.1, Corollary 2.3) and a construction of universal sets for the transfinite extension of the Menger-Urysohn inductive dimension (Theorem 2.2, Corollary 2.4).
Chaotic functions with zero topological entropy
J.
Smítal
269-282
Abstract: Recently Li and Yorke introduced the notion of chaos for mappings from the class ${C^0}(I,I)$, where $I$ is a compact real interval. In the present paper we give a characterization of the class $M \subset {C^0}(I,I)$ of mappings chaotic in this sense. As is well known, $ M$ contains the mappings of positive topological entropy. We show that $ M$ contains also certain (but not all) mappings that have both zero topological entropy and infinite attractors. Moreover, we show that the complement of $M$ consists of maps that have only trajectories approximate by cycles. Finally, it turns out that the original Li and Yorke notion of chaos can be replaced by (an equivalent notion of) $\delta$-chaos, distinguishable on a certain level $\delta > 0$.
BMO rational approximation and one-dimensional Hausdorff content
Joan
Verdera
283-304
Abstract: Let $X \subset {\mathbf{C}}$ be compact and let $f \in \operatorname{VMO} ({\mathbf{C}})$. We give necessary and sufficient conditions on $f$ and $X$ for ${f_{\vert X}}$ to be the limit of a sequence of rational functions without poles on $X$ in the norm of $\operatorname{BMO} (X)$, the space of functions of bounded mean oscillation on $X$. We also characterize those compact $X \subset {\mathbf{C}}$ with the property that the restriction to $X$ of each function in $\operatorname{VMO} ({\mathbf{C}})$, which is holomorphic on $\mathop X\limits^ \circ $, is the limit of a sequence of rational functions with poles off $ X$. Our conditions involve the notion of one-dimensional Hausdorff content. As an application, a result related to the inner boundary conjecture is proven.
The Radon transform on ${\rm SL}(2,{\bf R})/{\rm SO}(2,{\bf R})$
D. I.
Wallace;
Ryuji
Yamaguchi
305-318
Abstract: Let $G$ be $SL(2,{\mathbf{R}})$. $G$ acts on the upper half-plane $\mathcal{H}$ by the Möbius transformation, providing $ \mathcal{H}$ with the Riemannian metric structure along with the Laplacian, $ \Delta$. We study the integral transform along each geodesic. $G$ acts on $ \mathcal{P}$, the space of all geodesics, in a natural way, providing $\mathcal{P}$ with its invariant measure and its own Laplacian. ( $ \mathcal{P}$ actually is a coset space of $G$.) Therefore the above transform can be viewed as a map from a suitable function space on $\mathcal{H}$ to a suitable function space on $\mathcal{P}$. We prove a number of properties of this transform, including the intertwining properties with its Laplacians and its relation to the Fourier transforms.
Analytic perturbation of the Taylor spectrum
Zbigniew
Slodkowski
319-336
Abstract: Let ${T_1}(z), \ldots ,{T_m}(z)$, $z \in G \subset {{\mathbf{C}}^k}$, be analytic families of bounded operators in a complex Banach space $X$, such that for each $z \in G$ the operators ${T_i}(z)$ and ${T_j}(z)$, $ i,j = 1, \ldots ,n$, commute. Main result: If $K(z)$ denotes the Taylor spectrum of the tuple $ ({T_1}(z), \ldots ,{T_m}(z))$, then the set-valued function $K:G \to {2^{{\mathbf{C}}m}}$ is analytic. Analyticity of such set-valued functions is defined here by a simultaneous local maximum property of $ k$-tuples of complex polynomials on the graph of $K$.
Regularity results for an elliptic-parabolic free boundary problem
M.
Bertsch;
J.
Hulshof
337-350
Abstract: We study an elliptic-parabolic free boundary problem in one space dimension. We give several regularity results for both the weak solution and the free boundary. In particular conditions are given which ensure that the free boundary is a ${C^1}$-curve.
On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type
Wei-Ming
Ni;
Izumi
Takagi
351-368
Abstract: We derive a priori estimates for positive solutions of the Neumann problem for some semilinear elliptic systems (i.e., activator-inhibitor systems in biological pattern formation theory), as well as for semilinear single equations related to such systems. By making use of these a priori estimates, we show that under certain assumptions, there is no positive nonconstant solutions for single equations or for activator-inhibitor systems when the diffusion coefficient (of the activator, in the case of systems) is sufficiently large; we also study the existence of nonconstant solutions for specific domains.
Vector fields in the vicinity of a circle of critical points
J.-F.
Mattei;
M. A.
Teixeira
369-381
Abstract: In this paper the $ {C^S}$-conjugacy between vector fields on $ {{\mathbf{R}}^2}$ having a circle of critical points is studied.
Interaction de deux chocs pour un syst\`eme de deux lois de conservation, en dimension deux d'espace
Guy
Métivier
431-479
Abstract: The existence of shock front solutions to a system of conservation laws in several space variables has been proved by A. Majda, solving a Cauchy problem, with a suitable discontinuous Cauchy data. But, in general, the solution to such a Cauchy problem will present $N$ singularities, $ N$ being the number of laws. In this paper we solve (locally) this Cauchy problem, with a Cauchy data which is piecewise smooth, in the case where all the singularities are expected to be shock waves. Actually the construction is written for a system of two laws, with two space variables and similarly, for such a system, the same method enables us to study the interaction of two shock waves. The key point, in the construction below, is the study of a nonlinear, free boundary Goursat problem.
Peano arithmetic and hyper-Ramsey logic
James H.
Schmerl
481-505
Abstract: It is known that $ {\text{PA}}({Q^2})$, Peano arithmetic in a language with the Ramsey quantifier, is complete and compact and that its first-order consequences are the same as those of $\Pi _1^1{\text{-CA}_0}$. A logic $ \mathcal{H}{\mathcal{R}_\omega }$, called hyper-Ramsey logic, is defined; it is the union of an increasing sequence $\mathcal{H}{\mathcal{R}_1} \subseteq {\mathcal{H}_{\mathcal{R}2}} \subseteq \mathcal{H}{\mathcal{R}_3} \subseteq \cdots$ of sublogics, and $ \mathcal{H}{\mathcal{R}_1}$ contains $L({Q^2})$. It is proved that $ {\text{PA}}(\mathcal{H}{\mathcal{R}_n})$, which is Peano arithmetic in the context of $ \mathcal{H}{\mathcal{R}_n}$, has the same first-order consequences as $\Pi _n^1{\text{-CA}_0}$. A by-product and ingredient of the proof is, for example, the existence of a model of $ {\text{CA}}$ having the form $(\mathcal{N}, {\text{Class}}(\mathcal{N}))$.
Hypothesis testing in integral geometry
Peter
Waksman
507-520
Abstract: Probability distributions are defined relative to a fixed plane domain and are calculated explicitly when the domain is a union of coordinate rectangles. The theory of approximating step functions by the resulting special functions gives an interpretation of the problem of guessing a domain given a random sample of observations.
The Riemann hypothesis and the Tur\'an inequalities
George
Csordas;
Timothy S.
Norfolk;
Richard S.
Varga
521-541
Abstract: A solution is given to a fifty-eight year-old open problem of G Pólya, involving the Taylor coefficients of the Riemann $ \xi$-function.
Equivariant intersection forms, knots in $S\sp 4$, and rotations in $2$-spheres
Steven P.
Plotnick
543-575
Abstract: We consider the problem of distinguishing the homotopy types of certain pairs of nonsimply-connected four-manifolds, which have identical three-skeleta and intersection pairings, by the equivariant isometry classes of the intersection pairings on their universal covers. As applications of our calculations, we: (i) construct distinct homology four-spheres with the same three-skeleta, (ii) generalize a theorem of Gordon to show that any nontrivial fibered knot in ${S^4}$ with odd order monodromy is not determined by its complement, and (iii) give a more constructive proof of a theorem of Hendriks concerning rotations in two-spheres embedded in threemanifolds.
On the depth of the symmetric algebra
J.
Herzog;
M. E.
Rossi;
G.
Valla
577-606
Abstract: Let $(R,\mathfrak{m})$ be a local ring. Assume that $R = A/I$, where $(A,\mathfrak{n})$ is a regular local ring and $I \subseteq {\mathfrak{n}^2}$ is an ideal. The depth of the symmetric algebra $S(\mathfrak{m})$ of $ \mathfrak{m}$ over $ R$ is computed in terms of the depth of the associated graded module $ {\text{gr}_\mathfrak{n}}(I)$ and the so-called "strong socle condition." Explicit results are obtained, for instance, if $I$ is generated by a super-regular sequence, if $I$ has a linear resolution or if $ I$ has projective dimension one.
The dual of the Bergman space $A\sp 1$ in symmetric Siegel domains of type ${\rm II}$
David
Békollé
607-619
Abstract: An affirmative answer is given to the following conjecture of R. Coifman and R. Rochberg: in any symmetric Siegel domain of type II, the dual of the Bergman space ${A^1}$ coincides with the Bloch space of holomorphic functions and can be realized as the Bergman projection of $ {L^\infty }$.
The Bergman projection of $L\sp \infty$ in tubes over cones of real, symmetric, positive-definite matrices
David
Békollé
621-639
Abstract: We determine a defining kernel for the Bergman projection of ${L^\infty }$ in tubes over cones of real, symmetric, positive-definite matrices.
On the homology of associative algebras
David J.
Anick
641-659
Abstract: We present a new free resolution for $k$ as an $G$-module, where $G$ is an associative augmented algebra over a field $k$. The resolution reflects the combinatorial properties of $G$.
On secondary bifurcations for some nonlinear convolution equations
F.
Comets;
Th.
Eisele;
M.
Schatzman
661-702
Abstract: On the $ d$-dimensional torus ${{\mathbf{T}}^d} = {({\mathbf{R}}/{\mathbf{Z}})^d}$, we study the nonlinear convolution equation $\displaystyle u(t) = g\{ \lambda \cdot w \ast u(t)\} , \quad t \in {{\mathbf{T}}^d}, \lambda > 0.$ where $ \ast$ is the convolution on $ {{\mathbf{T}}^d}$, $ w$ is an integrable function which is not assumed to be even, and $g$ is bounded, odd, increasing, and concave on $ {{\mathbf{R}}^ + }$. A typical example is $ g = {\text{th}}$. For a general function $w$, we show by the standard theory of local bifurcation that, if the eigenspace of the linearized problem is of dimension $2$, a branch of solutions bifurcates at $\lambda = {(g\prime(0)\hat w(p))^{ - 1}}$ from the zero solution, and we show that it can be extended to infinity. For special simple forms of $ w$, we show that the first bifurcating branch has no secondary bifurcation, but the other branches can. These results are related to the theory of spin models on ${{\mathbf{T}}^d}$ in statistical mechanics, where they allow one to show the existence of a secondary phase transition of first order, and to some models of periodic structures in the brain in neurophysiology.
Optimal-partitioning inequalities for nonatomic probability measures
John
Elton;
Theodore P.
Hill;
Robert P.
Kertz
703-725
Abstract: Suppose ${\mu _1}, \ldots ,{\mu _n}$ are nonatomic probability measures on the same measurable space $(S,\mathcal{B})$. Then there exists a measurable partition $ \{ {S_i}\} _{i = 1}^n$ of $ S$ such that $ {\mu _i}({S_i}) \geq {(n + 1 - M)^{ - 1}}$ for all $i = 1, \ldots ,n$, where $M$ is the total mass of $\vee _{i = 1}^n\,{\mu _i}$ (the smallest measure majorizing each ${\mu _i}$). This inequality is the best possible for the functional $M$, and sharpens and quantifies a well-known cake-cutting theorem of Urbanik and of Dubins and Spanier. Applications are made to ${L_1}$-functions, discrete allocation problems, statistical decision theory, and a dual problem.
Playful Boolean algebras
Boban
Veličković
727-740
Abstract: We show that for an atomless complete Boolean algebra $\mathcal{B}$ of density $\leq {2^{{\aleph _0}}}$, the Banach-Mazur, the split and choose, and the Ulam game on $\mathcal{B}$ are equivalent. Moreover, one of the players has a winning strategy just in trivial cases: Empty wins iff $ \mathcal{B}$ adds a real; Nonempty wins iff $ \mathcal{B}$ has a $ \sigma$-closed dense set. This extends some previous results of Foreman, Jech, and Vojtáš
Brownian motion at a slow point
Martin T.
Barlow;
Edwin A.
Perkins
741-775
Abstract: If $c > 1$ there are points $ T(\omega)$ such that the piece of a Brownian path $B,X(t) = B(T + t) - B(T)$, lies within the square root boundaries $ \pm c\sqrt t$. We study probabilistic and sample path properties of $X$. In particular, we show that $ X$ is an inhomogeneous Markov process satisfying a certain stochastic differential equation, and we analyze the local behaviour of its local time at zero.
On the covering dimension of the set of solutions of some nonlinear equations
P. M.
Fitzpatrick;
I.
Massabò;
J.
Pejsachowicz
777-798
Abstract: We prove an abstract theorem whose sole hypothesis is that the degree of a certain map is nonzero and whose conclusions imply sharp, multidimensional continuation results. Applications are given to nonlinear partial differential equations.
Distant future and analytic measures
Jun-ichi
Tanaka
799-814
Abstract: Using a representation of analytic measures in terms of a flow built under a function, it is shown that a positive measure is the total variation measure of an analytic measure if and only if the distant future is the zero subspace. This settles a problem posed by Forelli in connection with his generalization of F. and M. Riesz theorems. We also provide another version of Helson's existence theorem.
Stable finitely homogeneous structures
G.
Cherlin;
A. H.
Lachlan
815-850
Abstract: Let $L$ be a finite relational language and $ \operatorname{Hom}(L,\omega)$ denote the class of countable $L$-structures which are stable and homogeneous. The main result of the paper is that there exists a natural number $c(L)$ such that for any transitive $\mathcal{M} \in \operatorname{Hom}(L;\omega)$, if $E$ is a maximal 0-definable equivalence relation on $ \mathcal{M}$, then either $ \vert\mathcal{M}/E\vert < c(L)$, or $ \mathcal{M}/E$ is coordinatizable. In an earlier paper the second author analyzed certain subclasses $\operatorname{Hom}(L, r) (r < \omega)$ of $ \operatorname{Hom}(L,\omega)$ for all sufficiently small $r$. Thus the earlier analysis now applies to $ \operatorname{Hom}(L,\omega)$.
Word maps, isotopy and entropy
David
Fried
851-859
Abstract: We find diffeomorphisms of low entropy in each isotopy class on ${S^3} \times {S^3}$. These arise as word maps, a nonabelian analogue of toral automorphisms. Hyperbolic examples of equal entropy are also found. The group ${\pi _0}\;\operatorname{Diff}\,({S^3} \times {S^3})$ is computed.
Relative pressure, relative equilibrium states, compensation functions and many-to-one codes between subshifts
Peter
Walters
1-31
Abstract: Let $S:X \to X,T:Y \to Y$ be continuous maps of compact metrizable spaces, and let $\pi :X \to Y$ be a continuous surjection with $ \pi \circ S = T \circ \pi$. We investigate the notion of relative pressure, which was introduced by Ledrappier and Walters, and study some maximal relative pressure functions that tie in with relative equilibrium states. These ideas are connected with the notion of compensation function, first considered by Boyle and Tuncel, and we show that a compensation function always exists when $S$ and $T$ are subshifts. A function $F \in C(X)$ is a compensation function if $P(S,F + \phi \circ \pi) = P(T,\phi)\forall \phi \in C(Y)$. When $S$ and $T$ are topologically mixing subshifts of finite type, we relate compensation functions to lifting $ T$-invariant measures to $ S$-invariant measures, obtaining some results of Boyle and Tuncel. We use compensation functions to describe different types of quotient maps $\pi$. An example is given where no compensation function exists.
Isometries for the Legendre-Fenchel transform
Hédy
Attouch;
Roger J.-B.
Wets
33-60
Abstract: It is shown that on the space of lower semicontinuous convex functions defined on ${R^n}$, the conjugation map--the Legendre-Fenchel transform--is an isometry with respect to some metrics consistent with the epi-topology. We also obtain isometries for the infinite dimensional case (Hilbert space and reflexive Banach space), but this time they correspond to topologies finer than the Moscoepi-topology.
Inequalities for the ergodic maximal function and convergence of the averages in weighted $L\sp p$-spaces
F. J.
Martín-Reyes
61-82
Abstract: This paper is concerned with the characterization of those positive functions $w$ such that Hopf's averages associated to an invertible measure preserving transformation $ T$ and a positive function $ g$ converge almost everywhere for every $ f \in {L^p}(w\,d\mu)$. We also study mean convergence when $g$ satisfies a "doubling condition" over orbits. In order to do this, we first characterize the pairs of positive functions $(u,w)$ such that the ergodic maximal operator associated to $T$ and $g$ is of weak or strong type with respect to the measures $w\,d\mu$ and $u\,d\mu$.
Desingularizations of plane vector fields
F.
Cano
83-93
Abstract: The singularities of a plane vector field can be reduced under quadratic blowing ups. We describe a control method for the singularities of the vector field which works for ground fields of any characteristic and which has no essential obstruction for generalizing to higher dimensional cases.
The spectrum $(P\wedge{\rm BP}\langle 2\rangle)\sb {-\infty}$
Donald M.
Davis;
David C.
Johnson;
John
Klippenstein;
Mark
Mahowald;
Steven
Wegmann
95-110
Abstract: The spectrum ${(P \wedge {\text{BP}}\langle {\text{2}}\rangle)_{ - \infty }}$ is defined to be the homotopy inverse limit of spectra $ {P_{ - k}} \wedge {\text{BP}}\langle {\text{2}}\rangle $, where ${P_{ - k}}$ is closely related to stunted real projective spaces, and ${\text{BP}}\langle {\text{2}}\rangle$ is formed from the Brown-Peterson spectrum. It is proved that this spectrum is equivalent to the infinite product of odd suspensions of the $2$-adic completion of the spectrum of connective $ K$-theory. An odd-primary analogue is also proved.
Factorial property of a ring of automorphic forms
Shigeaki
Tsuyumine
111-123
Abstract: A ring of automorphic forms is shown to be factorial under some conditions on the domain and on the Picard group. As an application, we show that any divisor on the moduli space $ {\mathfrak{M}_g}$ of curves of genus $g \geqslant 3$ is defined by a single element, and that the Satake compactification of $ {\mathfrak{M}_g}$ is written as a projective spectrum of a factorial graded ring. We find a single element which defines the closure of $ {\mathfrak{M}\prime_4}$ in $ {\mathfrak{M}_4}$ where $ {\mathfrak{M}\prime_4}$ is the moduli of curves of genus four whose canonical curves are exhibited as complete intersections of quadric cones and of cubics in ${\mathbb{P}^3}$.
Insufficiency of Torres' conditions for two-component classical links
M. L.
Platt
125-136
Abstract: Torres has given necessary conditions for a polynomial to be the Alexander polynomial of a two component link. For certain links, additional conditions are necessary. Hillman gave one example for linking number $6$. Here we give examples for all other linking numbers except $0, \pm 1$, and $\pm 2$.
Real hypersurfaces and complex submanifolds in complex projective space
Makoto
Kimura
137-149
Abstract: Let $M$ be a real hypersurface in ${P^n}({\mathbf{C}})$ be the complex structure and $ \xi$ denote a unit normal vector field on $M$. We show that $M$ is (an open subset of) a homogeneous hypersurface if and only if $M$ has constant principal curvatures and $ J\xi$ is principal. We also obtain a characterization of certain complex submanifolds in a complex projective space. Specifically, ${P^m}({\mathbf{C}})$ (totally geodesic), $ {Q^n},{P^1}({\mathbf{C}}) \times {P^n}({\mathbf{C}}),SU(5)/S(U(2) \times U(3))$ and $ SO(10)/U(5)$ are the only complex submanifolds whose principal curvatures are constant in the sense that they depend neither on the point of the submanifold nor on the normal vector.
Spanier-Whitehead duality in \'etale homotopy
Roy
Joshua
151-166
Abstract: We construct a $ (\bmod{\text{-}}l)$ Spanier-Whitehead dual for the etale homotopy type of any geometrically unibranched and projective variety over an algebraically closed field of arbitrary characteristic. The Thom space of the normal bundle to imbedding any compact complex manifold in a large sphere as a real submanifold provides a Spanier-Whitehead dual for the disjoint union of the manifold and a base point. Our construction generalises this to any characteristic. We also observe various consequences of the existence of a $ (\bmod {\text{-}}l)$ Spanier-Whitehead dual.
Paths and cycles in tournaments
Andrew
Thomason
167-180
Abstract: Sufficient conditions are given for the existence of an oriented path with given end vertices in a tournament. As a consequence a conjecture of Rosenfeld is established. This states that if $n$ is large enough, then every non-strongly oriented cycle of order $n$ is contained in every tournament of order $ n$.
Equivariant bundles and cohomology
A.
Kozlowski
181-190
Abstract: Let $G$ be a topological group, $ A$ an abelian topological group on which $G$ acts continuously and $X$ a $G$-space. We define "equivariant cohomology groups" of $X$ with coefficients in $A$, $ H_G^i(X;A)$, for $ i \geq 0$ which generalize Graeme Segal's continuous cohomology of the topological group $G$ with coefficients in $A$. In particular we have $H_G^1(X;A) \simeq$ equivalence classes of principal $(G,A)$-bundles over $X$. We show that when $G$ is a compact Lie group and $A$ an abelian Lie group we have for $i > 1\;H_G^i(X;A) \simeq {H^i}(EG{ \times _G}X;\tau A)$ where $\tau A$ is the sheaf of germs of sections of the bundle $(X \times EG \times A)/G \to (X \times EG)/G$. For $ i = 1$ and the trivial action of $G$ on $A$ this is a theorem of Lashof, May and Segal.
A parametrix for step-two hypoelliptic diffusion equations
Thomas
Taylor
191-215
Abstract: In this paper I construct a parametrix for the hypoelliptic diffusion equations $ (\partial /\partial t - L)u = 0$, where $L = \sum\nolimits_{a = 1}^n {g_a^2}$ and where the $ {g_a}$ are vector fields which satisfy the property that they, together with all of the commutators $ [{g_{a,}}{g_b}]$ for $a < b$, are at each point linearly independent and span the tangent space.
Automorphic images of commutative subspace lattices
K. J.
Harrison;
W. E.
Longstaff
217-228
Abstract: Let $C(H)$ denote the lattice of all (closed) subspaces of a complex, separable Hilbert space $ H$. Let $({\text{AC)}}$ be the following condition that a subspace lattice $\mathcal{F} \subseteq C(H)$ may or may not satisfy: (AC) \begin{displaymath}\begin{array}{*{20}{c}} {\mathcal{F} = \phi (\mathcal{L})\;{\... ...;{\text{lattice}}\;\mathcal{L} \subseteq C(H).} \end{array} \end{displaymath} Then $ \mathcal{F}$ satisfies $({\text{AC}})$ if and only if $\mathcal{A} \subseteq \mathcal{B}$ for some Boolean algebra subspace lattice $\mathcal{B} \subseteq C(H)$ with the property that, for every $K,L \in \mathcal{B}$, the vector sum $ K + L$ is closed. If $\mathcal{F}$ is finite, then $\mathcal{F}$ satisfies $({\text{AC}})$ if and only if $\mathcal{F}$ is distributive and $ K + L$ is closed for every $ K,L \in \mathcal{F}$. In finite dimensions $ \mathcal{F}$ satisfies $({\text{AC}})$ if and only if $\mathcal{F}$ is distributive. Every $\mathcal{F}$ satisfying $({\text{AC}})$ is reflexive. For such $\mathcal{F}$, given vectors $x,y \in H$, the solvability of the equation $Tx = y$ for $T \in \operatorname{Alg}\,\mathcal{F}$ is investigated.
``Almost'' implies ``near''
Robert M.
Anderson
229-237
Abstract: We formulate a formal language in which it is meaningful to say that an object almost satisfies a property. We then show that any object which almost satisfies a property is near an object which exactly satisfies the property. We show how this principle can be used to prove existence theorems. We give an example showing how one may strengthen the statement to give information about the relationship between the amount by which the object fails to satisfy the property and the distance to the nearest object which satisfies the property. Examples are given concerning commuting matrices, additive sequences, Brouwer fixed points, competitive equilibria, and differential equations.
On excursions of reflecting Brownian motion
Pei
Hsu
239-264
Abstract: We discuss the properties of excursions of reflecting Brownian motion on a bounded smooth domain in ${R^d}$ and give a procedure for constructing the process from the excursions and the boundary process. Our method is computational and can be applied to general diffusion processes with reflecting type boundary conditions on compact manifolds.
On integers free of large prime factors
Adolf
Hildebrand;
Gérald
Tenenbaum
265-290
Abstract: The number $\Psi (x,y)$ of integers $\leq x$ and free of prime factors $> y$ has been given satisfactory estimates in the regions $y \leq {(\log x)^{3/4 - \varepsilon }}$ and $y > \exp \{ {(\log \log x)^{5/3 + \varepsilon }}\}$. In the intermediate range, only very crude estimates have been obtained so far. We close this "gap" and give an expression which approximates $\Psi (x,y)$ uniformly for $x \geq y \geq 2$ within a factor $1 + O((\log y)/(\log x) + (\log y)/y)$. As an application, we derive a simple formula for $\Psi (cx,y)/\Psi (x,y)$, where $1 \leq c \leq y$. We also prove a short interval estimate for $ \Psi (x,y)$.
On a conormal module of smooth set theoretic complete intersections
M.
Boratyński
291-300
Abstract: We prove that $V \subset {\mathbf{A}}_k^n$ ($ V$-smooth) is a set-theoretic complete intersection (stci) if and only if $ V$ imbedded as a zero section of its normal bundle is a stci, we give a characterization of smooth codimension $2$ stci of index $\leq 4$ in terms of their conormal modules.
Chebyshev rank in $L\sb 1$-approximation
András
Kroó
301-313
Abstract: Let ${C_\omega }(K)$ denote the space of continuous functions endowed with the norm ${\smallint _K}\omega \left\vert f \right\vert = {\left\Vert f \right\Vert _\omega },\omega > 0$. In this paper we characterize the subspaces $ {U_n} \subset {C_\omega }(K)$ having Chebyshev rank at most $k\;(0 \leq k \leq n - 1)$ with respect to all bounded positive weights $\omega$. Various applications of main results are also presented.
Blow up near higher modes of nonlinear wave equations
Natalia
Sternberg
315-325
Abstract: This paper is concerned with the instability properties of higher modes of the nonlinear wave equation ${u_{tt}} - \Delta u - f(u) = 0$ defined on a smoothly bounded domain with Dirichlet boundary conditions. It is shown that they are unstable in the sense that in any neighborhood of a higher mode there exists a solution of the given equation which blows up in finite time.
Orbits of the pseudocircle
Judy
Kennedy;
James T.
Rogers
327-340
Abstract: The following theorem is proved. Theorem. The pseudocircle has uncountably many orbits under the action of its homeomorphism group. Each orbit is the union of uncountably many composants.
Inequalities for some maximal functions. II
M.
Cowling;
G.
Mauceri
341-365
Abstract: Let $S$ be a smooth compact hypersurface in ${{\mathbf{R}}^n}$, and let $\mu$ be a measure on $S$, absolutely continuous with respect to surface measure. For $t$ in ${{\mathbf{R}}^ + },{\mu _t}$ denotes the dilate of $ \mu$ by $t$, normalised to have the same total variation as $\mu$: for $f$ in $ \mathcal{S}({{\mathbf{R}}^n}),{\mu ^\char93 }f$ denotes the maximal function $ {\sup _{t > 0}}\vert{\mu _t}\ast f\vert$. We seek conditions on $\mu$ which guarantee that the a priori estimate $\displaystyle \left\Vert \mu^\char93 f\right\Vert _p \leq C\left\Vert f \right\Vert _p, \quad f \in S(\mathbf{R}^n),$ holds; this estimate entails that the sublinear operator ${\mu ^\char93 }$ extends to a bounded operator on the Lebesgue space $ {L^p}({{\mathbf{R}}^n})$. Our methods generalise E. M. Stein's treatment of the "spherical maximal function" [5]: a study of "Riesz operators", $g$-functions, and analytic families of measures reduces the problem to that of obtaining decay estimates for the Fourier transform of $\mu$. These depend on the geometry of $ S$ and the relation between $\mu$ and surface measure on $S$. In particular, we find that there are natural geometric maximal operators limited on ${L^p}({{\mathbf{R}}^n})$ if and only if $p \in (q,\infty ];q$ is some number in $(1,\infty)$, and may be greater than $2$. This answers a question of S. Wainger posed by Stein [6]>.
A critical set with nonnull image has large Hausdorff dimension
Alec
Norton
367-376
Abstract: The question of how complicated a critical set must be to have a nonnull image is answered by relating its Hausdorff dimension to the (Hölder) differentiability of the map. This leads to a new extension of the Morse-Sard Theorem. The main tool is an extended version of Morse's Lemma.
On starshaped rearrangement and applications
Bernhard
Kawohl
377-386
Abstract: A radial symmetrization technique is investigated and new properties are proven. The method transforms functions $ u$ into new functions $ {u^\ast}$ with starshaped level sets and is therefore called starshaped rearrangement. This rearrangement is in general not equimeasurable, a circumstance with some surprising consequences. The method is then applied to certain variational and free boundary problems and yields new results on the geometrical properties of solutions to these problems. In particular, the Lipschitz continuity of free boundaries can now be easily obtained in a new fashion.
Harmonic analysis on Grassmannian bundles
Robert S.
Strichartz
387-409
Abstract: The harmonic analysis of the Grassmannian bundle of $k$-dimensional affine subspaces of ${{\mathbf{R}}^n}$, as a homogeneous space of the Euclidean motion group, is given explicitly. This is used to obtain the diagonalization of various generalizations of the Radon transform between such bundles. In abstract form, the same technique gives the Plancherel formula for any unitary representation of a semidirect product $G \times V$ ($V$ a normal abelian subgroup) induced from an irreducible unitary representation of a subgroup of the form $H \times W$.
The Fefferman metric and pseudo-Hermitian invariants
John M.
Lee
411-429
Abstract: C. Fefferman has shown that a real strictly pseudoconvex hypersurface in complex $n$-space carries a natural conformal Lorentz metric on a circle bundle over the manifold. This paper presents two intrinsic constructions of the metric, valid on an abstract $ {\text{CR}}$ manifold. One is in terms of tautologous differential forms on a natural circle bundle; the other is in terms of Webster's pseudohermitian invariants. These results are applied to compute the connection and curvature forms of the Fefferman metric explicitly.
On quintic surfaces of general type
Jin Gen
Yang
431-473
Abstract: The study of quintic surfaces is of special interest because $ 5$ is the lowest degree of surfaces of general type. The aim of this paper is to give a classification of the quintic surfaces of general type over the complex number field ${\mathbf{C}}$. We show that if $S$ is an irreducible quintic surface of general type; then it must be normal, and it has only elliptic double or triple points as essential singularities. Then we classify all such surfaces in terms of the classification of the elliptic double and triple points. We give many examples in order to verify the existence of various types of quintic surfaces of general type. We also make a study of the double or triple covering of a quintic surface over ${{\mathbf{P}}^2}$ obtained by the projection from a triple or double point on the surface. This reduces the classification of the surfaces to the classification of branch loci satisfying certain conditions. Finally we derive some properties of the Hilbert schemes of some types of quintic surfaces.
Deformations of complete minimal surfaces
Harold
Rosenberg
475-489
Abstract: A notion of deformation is defined and studied for complete minimal surfaces in ${R^3}$ and ${R^3}/G,G$ a group of translations. The catenoid, Enneper's surface, and the surface of Meeks-Jorge, modelled on a $3$-punctured sphere, are shown to be isolated. Minimal surfaces of total curvature $4\pi$ in ${R^3}/Z$ and $ {R^3}/{Z^2}$ are studied. It is proved that the helicoid and Scherk's surface are isolated under periodic perturbations.
Some remarks on deformations of minimal surfaces
Harold
Rosenberg;
Éric
Toubiana
491-499
Abstract: We consider complete minimal surfaces (c.m.s.'s) in ${R^3}$ and their deformations. $ {M_1}$ is an $\varepsilon$-deformation of ${M_0}$ if ${M_1}$ is a graph over ${M_0}$ in an $ \varepsilon$ tubular neighborhood of ${M_0}$ and ${M_1}$ is $ \varepsilon \;{C^1}$-close to ${M_0}$. A minimal surface $M$ is isolated if all c.m.s.'s which are sufficiently small deformations of $ M$ are congruent to $ M$. In this paper we construct an example of a nonisolated c.m.s. It is modelled on a $4$-punctured sphere and is of finite total curvature. On the other hand, we prove that a c.m.s. discovered by Meeks and Jorge, modelled on the sphere punctured at the fourth roots of unity, is isolated.
Unicity of a holomorphic functional calculus in infinite dimensions
José E.
Galé
501-508
Abstract: L. Waelbroeck gives a holomorphic functional calculus for Banach algebras and analytic functions on Banach spaces. The properties of this calculus extend the well-known properties for the case of several complex variables. In this last situation, W. Zame has obtained a theorem of unicity where the famous condition of compatibility is dropped. We obtain a theorem analogous to Zame's for Waelbroeck's calculus restricted to a certain algebra of germs of functions. We consider Banach spaces whose topological duals have the bounded approximation property. Also, results of the same kind as above are given for bornological algebras.
Nonlinear oblique boundary value problems for nonlinear elliptic equations
Gary M.
Lieberman;
Neil S.
Trudinger
509-546
Abstract: We consider the nonlinear oblique derivative boundary value problem for quasilinear and fully nonlinear uniformly elliptic partial differential equations of second order. The elliptic operators satisfy natural structure conditions as introduced by Trudinger in the study of the Dirichlet problem while for the boundary operators we formulate general structure conditions which embrace previously considered special cases such as the capillarity condition. The resultant existence theorems include previous work such as that of Lieberman on quasilinear equations and Lions and Trudinger on Neumann boundary conditions.
Weakly definable types
L. A. S.
Kirby;
A.
Pillay
547-563
Abstract: We study some generalizations of the notion of a definable type, first in an abstract setting in terms of ultrafilters on certain Boolean algebras, and then as applied to model theory.
Definable sets in ordered structures. I
Anand
Pillay;
Charles
Steinhorn
565-592
Abstract: This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the $ \mathcal{O}$-minimal structures. The definition of this class and the corresponding class of theories, the strongly $\mathcal{O}$-minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizations of $\mathcal{O}$-minimal ordered groups and rings. Several other simple results are collected in $ \S3$. The primary tool in the analysis of $ \mathcal{O}$-minimal structures is a strong analogue of "forking symmetry," given by Theorem 4.2. This result states that any (parametrically) definable unary function in an $\mathcal{O}$-minimal structure is piecewise either constant or an order-preserving or reversing bijection of intervals. The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all ${\aleph _0}$-categorical $ \mathcal{O}$-minimal structures (Theorem 6.1).
Definable sets in ordered structures. II
Julia F.
Knight;
Anand
Pillay;
Charles
Steinhorn
593-605
Abstract: It is proved that any 0-minimal structure $M$ (in which the underlying order is dense) is strongly 0-minimal (namely, every $ N$ elementarily equivalent to $M$ is 0-minimal). It is simultaneously proved that if $M$ is 0-minimal, then every definable set of $ n$-tuples of $ M$ has finitely many "definably connected components."
Congruences on regular semigroups
Francis
Pastijn;
Mario
Petrich
607-633
Abstract: Let $S$ be a regular semigroup and let $ \rho$ be a congruence relation on $S$. The kernel of $\rho$, in notation $\ker \rho$, is the union of the idempotent $ \rho$-classes. The trace of $\rho$, in notation $\operatorname{tr}\,\rho $, is the restriction of $ \rho$ to the set of idempotents of $S$. The pair $(\ker \rho ,\operatorname{tr}\,\rho )$ is said to be the congruence pair associated with $\rho$. Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple $((\rho \vee \mathcal{L})/\mathcal{L},\ker \rho ,(\rho \vee \mathcal{R})/\mathcal{R})$ is said to be the congruence triple associated with $\rho$. Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple. The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of $S$. For congruence relations $\rho$ and $\theta$, put $\rho {T_l}\theta \;[\rho {T_r}\theta ,\rho T\theta ]$ if and only if $\rho \vee \mathcal{L} = \theta \vee \mathcal{L}\;[\rho \vee \mathcal{R} = \theta \vee \mathcal{R},\operatorname{tr}\rho = \operatorname{tr}\theta ]$. Then $ {T_l},{T_r}$ and $ T$ are complete congruences on the congruence lattice of $S$ and $ T = {T_l} \cap {T_r}$.
Closed geodesics on a Riemann surface with application to the Markov spectrum
A. F.
Beardon;
J.
Lehner;
M.
Sheingorn
635-647
Abstract: This paper determines those Riemann surfaces on which each nonsimple closed geodesic has a parabolic intersection--that is, an intersection in the form of a loop enclosing a puncture or a deleted disk. An application is made characterizing the simple closed geodesic on $H/\Gamma (3)$ in terms of the Markov spectrum. The thrust of the situation is this: If we call loops about punctures or deleted disks boundary curves, then if the surface has "little" topology, each nonsimple closed geodesic must contain a boundary curve. But if there is "enough" topology, there are nonsimple closed geodesics not containing boundary curves.
Total stability of sets for nonautonomous differential systems
Zhivko S.
Athanassov
649-663
Abstract: The principal purpose of this paper is to present sufficient conditions for total stability, or stability under constantly acting perturbations, of sets of a sufficiently general kind for nonautonomous ordinary differential equations. To do this, two Liapunov-like functions with specific properties are used. The obtained results include and considerably improve the classical results on total stability of isolated equilibrium points. Applications are presented to study the stability of nonautonomous Lurie-type nonlinear equations.
The divergence theorem
W. F.
Pfeffer
665-685
Abstract: We define a well-behaved multidimensional Riemann type integral such that the divergence of any vector field continuous in a compact interval and differentiable in its interior is integrable, and the integral equals the flux of the vector field out of the interval.
Boundary behavior of positive solutions of the heat equation on a semi-infinite slab
B. A.
Mair
687-697
Abstract: In this paper, the abstract Fatou-Naim-Doob theorem is used to investigate the boundary behavior of positive solutions of the heat equation on the semi-infinite slab $X = {{\mathbf{R}}^{n - 1}} \times {{\mathbf{R}}_ + } \times (0,T)$. The concept of semifine limit is introduced, and relationships are obtained between fine, semifine, parabolic, one-sided parabolic and two-sided parabolic limits at points on the parabolic boundary of $ X$. A Carleson-Calderón-type local Fatou theorem is also obtained for solutions on a union of two-sided parabolic regions.
An extremal problem for analytic functions with prescribed zeros and $r$th derivative in $H\sp \infty$
A.
Horwitz;
D. J.
Newman
699-713
Abstract: Let $({\alpha _1}, \ldots ,{\alpha _n})$ be $ n$ points in the unit disc $ U$. Suppose $g$ is analytic in $U$, $g({\alpha _1}) = \cdots = g({\alpha _n}) = 0$ (multiplicities included), and $\Vert g\prime\Vert _{\infty } \leq 1$. Then we prove that $\vert g(z)\vert \leq \vert\phi (z)\vert$ for all $z \in U$, where $\phi ({\alpha _1}) = \cdots = \phi ({\alpha _n}) = 0$ and $ \phi\prime(z)$ is a Blaschke product of order $n - 1$. We extend this result in a natural way to convex domains $D$ with analytic boundary. For $D$ not convex we show that there is no extremal function $\phi$.
Axiom $3$ modules
Paul
Hill;
Charles
Megibben
715-734
Abstract: By introducing the concept of a knice submodule, a refinement of the notion of nice subgroup, we are able to formulate a version of the third axiom of countability appropriate to the study of $p$-local mixed groups in the spirit of the well-known characterization of totally projective $p$-groups. Our Axiom $3$ modules, in fact, form a class of $ {{\mathbf{Z}}_{\mathbf{p}}}$-modules, encompassing the totally projectives in the torsion case, for which we prove a uniqueness theorem and establish closure under direct summands. Indeed Axiom $3$ modules turn out to be precisely the previously classified Warfield modules. But with the added power of the third axiom of countability characterization, we derive numerous new results, including the resolution of a long-standing problem of Warfield and theorems in the vein of familiar criteria due to Kulikov and Pontryagin.
Torsion free groups
Paul
Hill;
Charles
Megibben
735-751
Abstract: In this paper we introduce the class of torsion free $k$-groups and the notion of a knice subgroup. Torsion free $k$-groups form a class of groups more extensive than the separable groups of Baer, but they enjoy many of the same closure properties. We establish a role for knice subgroups of torsion free groups analogous to that played by nice subgroups in the study of torsion groups. For example, among the torsion free groups, the balanced projectives are characterized by the fact that they satisfy the third axiom of countability with respect to knice subgroups. Separable groups are characterized as those torsion free $k$-groups with the property that all finite rank, pure knice subgroups are direct summands. The introduction of these new classes of groups and subgroups is based on a preliminary study of the interplay between primitive elements and $\ast$-valuated coproducts. As a by-product of our investigation, new proofs are obtained for many classical results on separable groups. Our techniques lead naturally to the discovery that a balanced subgroup of a completely decomposable group is itself completely decomposable provided the corresponding quotient is a separable group of cardinality not exceeding ${\aleph _1}$; that is, separable groups of cardinality ${\aleph _1}$ have balanced projective dimension $\leq 1$.
Algebraic relations among solutions of linear differential equations
Michael F.
Singer
753-763
Abstract: Using power series methods, Harris and Sibuya [3,4] recently showed that if $k$ is an ordinary differential field of characteristic zero and $y \ne 0$ is an element of a differential extension of $ k$ such that $ y$ and $1/y$ satisfy linear differential equations with coefficients in $k$, then $y\prime/y$ is algebraic over $k$. Using differential galois theory, we generalize this and characterize those polynomial relations among solutions of linear differential equations that force these solutions to have algebraic logarithmic derivatives. We also show that if $ f$ is an algebraic function of genus $\geq 1$ and if $y$ and $f(y)$ or $y$ and $ {e^{\int y}}$ satisfy linear differential equations, then $y$ is an algebraic function.
$L\sp p$ inequalities for stopping times of diffusions
R. Dante
DeBlassie
765-782
Abstract: Let ${X_t}$ be a solution to a stochastic differential equation. Easily verified conditions on the coefficients of the equation give ${L^p}$ inequalities for stopping times of ${X_t}$ and the maximal function. An application to Brownian motion with radial drift is also discussed.
On algebras with convolution structures for Laguerre polynomials
Yūichi
Kanjin
783-794
Abstract: In this paper we treat the convolution algebra connected with Laguerre polynomials which was constructed by Askey and Gasper [1]. For this algebra, we study the maximal ideal space, Wiener's general Tauberian theorem, spectral synthesis and Helson sets. We also study Sidon sets and idempotent measures for the algebras with dual convolution structures.
Generic dynamics and monotone complete $C\sp \ast$-algebras
Dennis
Sullivan;
B.
Weiss;
J. D. Maitland
Wright
795-809
Abstract: Let $R$ be any ergodic, countable generic equivalence relation on a perfect Polish space $ X$. It follows from the main theorem of $\S1$ that, modulo a meagre subset of $ X,R$ may be identified with the relation of orbit equivalence ensuing from a canonical action of $ {\mathbf{Z}}$. Answering a longstanding problem of Kaplansky, Takenouchi and Dyer independently gave cross-product constructions of Type III $A{W^\ast}$-factors which were not von Neumann algebras. As a specialization of a much more general result, obtained in $\S3$, we show that the Dyer factor is isomorphic to the Takenouchi factor.
Quantization and Hamiltonian $G$-foliations
L.
Pukanszky
811-847
Abstract: As it was recognized twenty five years ago by A. A. Kirillov, in the unitary representation theory of nilpotent Lie groups a crucial role is played by orbits of the coadjoint representation. B. Kostant noted that, for any connected Lie group, these orbits admit a symplectic structure and lend themselves to an intrinsic characterization. The present author later observed, that already for the purposes of unitary representation theory of solvable Lie groups, this concept has to be enlarged and replaced by that of a generalized orbit. One objective of this paper is their intrinsic characterization. Other results prepare the way for the geometric construction of the corresponding unitary representations, to be developed later.
Connected locally connected toposes are path-connected
I.
Moerdijk;
G. C.
Wraith
849-859
Abstract: A conjecture of A. Joyal is proved, which states that, in contrast to topological spaces, toposes which are connected and locally connected are also path-connected. The reason for this phenomenon is the triviality of cardinality considerations in the topos-theoretic setting; any inhabited object pulls back to an enumerable object under some open surjective geometric morphism. This result points towards a homotopy theory for toposes.
Long-range potential scattering by Enss's method in two Hilbert spaces
Denis A. W.
White
1-33
Abstract: Existence and completeness of wave operators is established by a straightforward transposition of the original short range result of Enss into an appropriate two-Hilbert space setting. Applied to long range quantum mechanical potential scattering, this result in conjunction with recent work of Isozaki and Kitada reduces the problem of proving existence and completeness of wave operators to that of approximating solutions of certain partial differential equations on cones in phase space. As an application existence and completeness of wave operators is established for Schrödinger operators with a long range multiplicative and possibly rapidly oscillating potential.
Brownian excursions from hyperplanes and smooth surfaces
Krzysztof
Burdzy
35-57
Abstract: A skew-product decomposition of the $n$-dimensional $ (n \geq 2)$ Brownian excursion law from a hyperplane is obtained. This is related to a Kolmogorov-type test for excursions from hyperplanes. Several results concerning existence, uniqueness and form of Brownian excursion laws from sufficiently "flat" surfaces are given. Some of these theorems are potential-theoretic in spirit. An extension of the results concerning excursion laws to an exit system in a Lipschitz domain is supplied.
Irreducibility of moduli spaces of cyclic unramified covers of genus $g$ curves
R.
Biggers;
M.
Fried
59-70
Abstract: Let $ ({C_1}, \ldots ,{C_r}G) = ({\mathbf{C}},G)$ be an $r$-tuple consisting of a transitive subgroup $ G$ of ${S_m}$ and $r$ conjugacy classes ${C_1}, \ldots ,{C_r}$ of $G$. We consider the concept of the moduli space $ \mathcal{H}({\mathbf{C}},G)$ of compact Riemann surface covers of the Riemann sphere of Nielsen class $({\mathbf{C}},G)$. The irreducibility of $ \mathcal{H}({\mathbf{C}},G)$ is equivalent to the transitivity of a specific permutation representation of the Hurwitz monodromy group $(\S1)$, but there are few general tools to decide questions about this representation. Theorem 2 gives a class of examples of $({\mathbf{C}},G)$ for which $ \mathcal{H}({\mathbf{C}},G)$ is irreducible. As an immediate corollary this gives an elementary proof and generalization of the irreduciblity of the moduli space of cyclic unramified covers of genus $g$ curves (for which Deligne and Mumford [ ${\mathbf{DM}}$, Theorem 5.15] applied Teichmüller theory and Dehn's theorem). This contrasts with the examples of $ ({\mathbf{C}},G)$ in $[{\mathbf{BFr}}]$ for which $ \mathcal{H}({\mathbf{C}},G)$ is reducible. These kinds of questions combined with the study of the existence of rational subvarieties of $ \mathcal{H}({\mathbf{C}},G)$ have application to the realization of a group $ G$ as the Galois group of a regular extension of $ \mathbb{Q}(t)\;[{\mathbf{Fr3}},\S4]$.
Defining equations for real analytic real hypersurfaces in ${\bf C}\sp n$
John P.
D’Angelo
71-84
Abstract: A defining function for a real analytic real hypersurface can be uniquely written as $2\operatorname{Re} (H) + E$, where $H$ is holomorphic and $ E$ contains no pure terms. We study how $H$ and $E$ change when we perform a local biholomorphic change of coordinates, or multiply by a unit. One of the main results is necesary and sufficient conditions on the first nonvanishing homogeneous part of $ E$ (expanded in terms of $ H$) beyond ${E_{00}}$ that serve as obstructions to writing a defining equation as $2\operatorname{Re} (h) + e$, where $e$ is independent of $ h$. We also find necessary pluriharmonic obstructions to doing this, which arise from the easier case of attempting to straighten the hypersurface.
Lyapunov exponents for a stochastic analogue of the geodesic flow
A. P.
Carverhill;
K. D.
Elworthy
85-105
Abstract: New invariants for a Riemannian manifold are defined as Lyapunov exponents of a stochastic analogue of the geodesic flow. A lower bound is given reminiscent of corresponding results for the geodesic flow, and an upper bound is given for surfaces of positive curvature. For surfaces of constant negative curvature a direct method via the Doob $ h$-transform is used to determine the full Lyapunov structure relating the stable manifolds to the horocycles.
The singularities of the $3$-secant curve associated to a space curve
Trygve
Johnsen
107-118
Abstract: Let $C$ be a curve in ${P^3}$ over an algebraically closed field of characteristic zero. We assume that $C$ is nonsingular and contains no plane component except possibly an irreducible conic. In [ $ {\mathbf{GP}}$] one defines closed $r$-secant varieties to $C$, $r \in N$. These varieties are embedded in $ G$, the Grassmannian of lines in ${P^3}$. Denote by $T$ the $3$-secant variety (curve), and assume that the set of $4$-secants is finite. Let $\tilde T$ be the curve obtained by blowing up the ideal of $4$-secants in $T$. The curve $\tilde T$ is in general not in $G$. We study the local geometry of $ \tilde T$ at any point whose fibre of the blowing-up map is reduced at the point. The multiplicity of $\tilde T$ at such a point is determined in terms of the local geometry of $C$ at certain chosen secant points. Furthermore we give a geometrical interpretation of the tangential directions of $\tilde T$ at a singular point. We also give a criterion for whether all the tangential directions are distinct or not.
Regions of variability for univalent functions
Peter
Duren;
Ayşenur
Ünal
119-126
Abstract: Let $S$ be the standard class of univalent functions in the unit disk, and let ${S_0}$ be the class of nonvanishing univalent functions $g$ with $g(0) = 1$. It is shown that the regions of variability $\{ g(r):g \in {S_0}\}$ and $\{ (1 - {r^2})f\prime(r):f \in S\}$ are very closely related but are not quite identical.
Weighted weak $(1,1)$ and weighted $L\sp p$ estimates for oscillating kernels
Sagun
Chanillo;
Douglas S.
Kurtz;
Gary
Sampson
127-145
Abstract: Weak type $ (1,1)$ and strong type $ (p,p)$ inequalities are proved for operators defined by oscillating kernels. The techniques are sufficiently general to derive versions of these inequalities using weighted norms.
Jones polynomials of alternating links
Kunio
Murasugi
147-174
Abstract: Let $ {J_K}(t) = {a_r}{t^r} + \cdots + {a_s}{t^s},r > s$, be the Jones polynomial of a knot $K$ in ${S^3}$. For an alternating knot, it is proved that $r - s$ is bounded by the number of double points in any alternating projection of $K$. This upper bound is attained by many alternating knots, including $2$-bridge knots, and therefore, for these knots, $r - s$ gives the minimum number of double points among all alternating projections of $K$. If $K$ is a special alternating knot, it is also proved that ${a_s} = 1$ and $s$ is equal to the genus of $K$. Similar results hold for links.
Affine manifolds and orbits of algebraic groups
William M.
Goldman;
Morris W.
Hirsch
175-198
Abstract: This paper is the sequel to The radiance obstruction and parallel forms on affine manifolds (Trans. Amer. Math. Soc. 286 (1984), 629-649) which introduced a new family of secondary characteristic classes for affine structures on manifolds. The present paper utilizes the representation of these classes in Lie algebra cohomology and algebraic group cohomology to deduce new results relating the geometric properties of a compact affine manifold ${M^n}$ to the action on ${{\mathbf{R}}^n}$ of the algebraic hull ${\mathbf{A}}(\Gamma )$ of the affine holonomy group $\Gamma \subseteq \operatorname{Aff}({{\mathbf{R}}^n})$. A main technical result of the paper is that if $M$ has a nonzero cohomology class represented by a parallel $k$-form, then every orbit of ${\mathbf{A}}(\Gamma )$ has dimension $\geq k$. When $M$ is compact, then ${\mathbf{A}}(\Gamma )$ acts transitively provided that $M$ is complete or has parallel volume; the converse holds when $\Gamma$ is nilpotent. A $4$-dimensional subgroup of $ \operatorname{Aff}({{\mathbf{R}}^3})$ is exhibited which does not contain the holonomy group of any compact affine $3$-manifold. When $M$ has solvable holonomy and is complete, then $M$ must have parallel volume. Conversely, if $ M$ has parallel volume and is of the homotopy type of a solvmanifold, then $ M$ is complete. If $ M$ is a compact homogeneous affine manifold or if $M$ possesses a rational Riemannian metric, then it is shown that the conditions of parallel volume and completeness are equivalent.
Approximation order from certain spaces of smooth bivariate splines on a three-direction mesh
Rong Qing
Jia
199-212
Abstract: Let $\Delta$ be the mesh in the plane obtained from a uniform square mesh by drawing in the north-east diagonal in each square. Let $\pi _{k,\Delta }^\rho$ be the space of bivariate piecewise polynomial functions in ${C^\rho }$, of total degree $\leq k$, on the mesh $\Delta$. Let $ m(k,\rho )$ denote the approximation order of $\pi _{k,\Delta }^\rho$. In this paper, an upper bound for $m(k,\rho )$ is given. In the space $3 \leq 2k - 3\rho \leq 7$, the exact values of $m(k,\rho )$ are obtained: \begin{displaymath}\begin{array}{*{20}{c}} {m(k,\rho ) = 2k - 2\rho - 1} ... ...or}}\;2k - 3\rho = 5,6\;{\text{or}}\;7.} \end{array} \end{displaymath} In particular, this result answers negatively a conjecture of de Boor and Höllig.
Weighted and vector-valued inequalities for potential operators
Francisco J.
Ruiz Blasco;
José L.
Torrea Hernández
213-232
Abstract: In this paper we develop some aspect of a general theory parallel to the Calderón-Zygmund theory for operator valued kernels, where the operators considered map functions defined on ${R^n}$ into functions defined on $ R_ + ^{n + 1} = {R^n} \times [0,\infty )$. In particular, we apply the obtained results to get vector-valued inequalities for the Poisson integral and fractional integrals. Some weighted norm inequalities are also considered for fractional integrals.
$S\sp 1$-equivariant function spaces and characteristic classes
Benjamin M.
Mann;
Edward Y.
Miller;
Haynes R.
Miller
233-256
Abstract: We determine the structure of the homology of the Becker-Schultz space $SG({S^1}) \simeq Q({\mathbf{C}}P_ + ^\infty \wedge {S^1})$ of stable ${S^1}$-equivariant self-maps of spheres (with standard free ${S^1}$-action) as a Hopf algebra over the Dyer-Lashof algebra. We use this to compute the homology of $BSG({S^1})$. Along the way, we give a fresh account of the partially framed transfer construction and the Becker-Schultz homotopy equivalence. We compute the effect in homology of the "${S^1}$-transfers" ${\mathbf{C}}P_ + ^\infty \wedge {S^1} \to Q((B{{\mathbf{Z}}_{{p^n}}})_+ ),n \geq 0$, and of the equivariant $ J$-homomorphisms $ SO \to Q({\mathbf{R}}P_ + ^\infty )$ and $U \to Q({\mathbf{C}}P_ + ^\infty \wedge {S^1})$. By composing, we obtain $ U \to Q{S^0}$ in homology, answering a question of J. P. May.
Well-posedness of higher order abstract Cauchy problems
Frank
Neubrander
257-290
Abstract: The paper is concerned with differential equations of the type $\displaystyle {u^{(n + 1)}}(t) - A{u^{(n)}}(t) - {B_1}{u^{(n - 1)}}(t) - \cdots - {B_n}u(t) = 0$ ($\ast$) in a Banach space $E$ where $A$ is a linear operator with dense domain $ D(A)$ and ${B_1}, \ldots ,{B_n}$ are closed linear operators with $ D(A) \subset D({B_k})$ for $1 \leq k \leq n$. The main result is the equivalence of the following two statements: (a) $A$ has nonempty resolvent set and for every initial value $({x_0}, \ldots ,{x_n}) \in {(D(A))^{n + 1}}$ the equation $( \ast )$ has a unique solution in ${C^{n + 1}}({{\mathbf{R}}^ + },E) \cap {C^n}({{\mathbf{R}}^n},[D(A)])([D(A)]$ denotes the Banach space $ D(A)$ endowed with the graph norm); (b) $A$ is the generator of a strongly continuous semigroup. Under additional assumptions on the operators $ {B_k}$, which are frequently fulfilled in applications, we obtain continuous dependence of the solutions on the initial data; i.e., well-posedness of $( \ast )$. Using Laplace transform methods, we give explicit expressions for the solutions in terms of the operators $A$, ${B_k}$. The results are then used to discuss strongly damped semilinear second order equations.
The problem of embedding $S\sp n$ into ${\bf R}\sp {n+1}$ with prescribed Gauss curvature and its solution by variational methods
V. I.
Oliker
291-303
Abstract: A way to recover a closed convex hypersurface from its Gauss curvature is to find a positive function over ${S^n}$ whose graph would represent the hypersurface in question. Then one is led to a nonlinear elliptic problem of Monge-Ampère type on $ {S^n}$. Usually, geometric problems involving operators of this type are too complicated to be suggestive for a natural functional whose critical points are candidates for solutions of such problems. It turns out that for the problem indicated in the title, such a functional exists and has interesting geometric properties. With the use of this functional, we obtain new existence results for hypersurfaces with prescribed curvature as well as strengthen some that are already known.
Eigenvalues of elliptic boundary value problems with an indefinite weight function
Jacqueline
Fleckinger;
Michel L.
Lapidus
305-324
Abstract: We consider general selfadjoint elliptic eigenvalue problems (P) $\displaystyle \mathcal{A}u = \lambda r(x)u,$ in an open set $\Omega \subset {{\mathbf{R}}^k}$. Here, the operator $ \mathcal{A}$ is positive and of order $2m$ and the "weight" $r$ is a function which changes sign in $ \Omega$ and is allowed to be discontinuous. A scalar $\lambda$ is said to be an eigenvalue of $({\text{P}})$ if $\mathcal{A}u = \lambda ru$--in the variational sense--for some nonzero $u$ satisfying the appropriate growth and boundary conditions. We determine the asymptotic behavior of the eigenvalues of $ ({\text{P}})$, under suitable assumptions. In the case when $\Omega$ is bounded, we assumed Dirichlet or Neumann boundary conditions. When $ \Omega$ is unbounded, we work with operators of "Schrödinger type"; if we set $ r \pm = \max ( \pm r,0)$, two cases appear naturally: First, if $\Omega$ is of "weighted finite measure" (i.e., $ \int_\Omega {{{({r_ + })}^{k/2m}} < + \infty \;} {\text{or}}\;\int_\Omega {{{({r_ - })}^{k/2m}} < + \infty }$), we obtain an extension of the well-known Weyl asymptotic formula; secondly, if $\Omega$ is of "weighted infinite measure" (i.e., $\int_\Omega {{{({r_ + })}^{k/2m}} = + \infty \;{\text{or}}\;\int_\Omega {{{({r_ - })}^{k/2m}} = + \infty } }$), our results extend the de Wet-Mandl formula (which is classical for Schrödinger operators with weight $r \equiv 1$). When $\Omega$ is bounded, we also give lower bounds for the eigenvalues of the Dirichlet problem for the Laplacian.
Random recursive constructions: asymptotic geometric and topological properties
R. Daniel
Mauldin;
S. C.
Williams
325-346
Abstract: We study some notions of "random recursive constructions" in Euclidean $ m$-space which lead almost surely to a particular type of topological object; e.g., Cantor set, Sierpiński curve or Menger curve. We demonstrate that associated with each such construction is a "universal" number $\alpha$ such that almost surely the random object has Hausdorff dimension $\alpha$. This number is the expected value of the sum of some ratios which in the deterministic case yields Moran's formula.
Embedding strictly pseudoconvex domains into balls
Franc
Forstnerič
347-368
Abstract: Every relatively compact strictly pseudoconvex domain $D$ with $ {{\mathbf{C}}^2}$ boundary in a Stein manifold can be embedded as a closed complex submanifold of a finite dimensional ball. However, for each $n \geq 2$ there exist bounded strictly pseudoconvex domains $D$ in $ {\mathbb{C}^n}$ with real-analytic boundary such that no proper holomorphic map from $D$ into any finite dimensional ball extends smoothly to $ \overline D$.
Splitting strongly almost disjoint families
A.
Hajnal;
I.
Juhász;
S.
Shelah
369-387
Abstract: We say that a family $\mathcal{A} \subset {[\lambda ]^\kappa }$ is strongly almost disjoint if something more than just $\vert A \cap B\vert < \kappa$, e.g. that $ \vert A \cap B\vert < \sigma < \kappa$, is assumed for $A$, $B \in \mathcal{A}$. We formulate conditions under which every such strongly a.d. family is "essentially disjoint", i.e. for each $A \in \mathcal{A}$ there is $ F(A) \in {[A]^{ < \kappa }}$ so that $\{ A\backslash F(A):A \in \mathcal{A}\}$ is disjoint. On the other hand, we get from a supercompact cardinal the consistency of $ {\text{GCH}}$ plus the existence of a family $\mathcal{A} \subset {[{\omega _{\omega + 1}}]^{{\omega _1}}}$ whose elements have pairwise finite intersections and such that it does not even have property $B$. This solves an old problem raised in [4]. The same example is also used to produce a graph of chromatic number $ {\omega _2}$ on ${\omega _{\omega + 1}}$ that does not contain $[\omega ,\omega ]$, answering a problem from [5]. We also have applications of our results to "splitting" certain families of closed subsets of a topological space. These improve results from [ $ {\mathbf{3}},{\mathbf{12}}$ and $ {\mathbf{13}}$].
Uniqueness results for homeomorphism groups
Robert R.
Kallman
389-396
Abstract: Let $X$ be a separable metric manifold and let $ \mathcal{H}(X)$ be the homeomorphism group of $X$. Then $ \mathcal{H}(X)$ has a unique topology in which it is a complete separable metric group. Similar results are demonstrated for a much wider class of spaces, $X$, and for many subgroups of the homeomorphism group.
On the generic structure of cohomology modules for semisimple algebraic groups
Henning Haahr
Andersen
397-415
Abstract: Let $G$ be a connected semisimple algebraic group over a field of positive characteristic. Denote by $B$ a Borel subgroup. Our main result says that generically the cohomology modules for line bundles on $ G/B$ have simple socles and simple heads, and we identify the corresponding highest weights. As one of the consequences we discover a certain symmetry among extensions of simple modules for $G$.
The associative forms of the graded Cartan type Lie algebras
Rolf
Farnsteiner
417-427
Abstract: This paper determines the Cartan type Lie algebras that possess a nonsingular associative form.
Erratum to: ``Generic algebras'' [Trans. Amer. Math. Soc. {\bf 275} (1983), no. 2, 497--510; MR0682715 (84h:18010)]
John
Isbell
429
A convergent series expansion for hyperbolic systems of conservation laws
Eduard
Harabetian
383-424
Abstract: We consider the discontinuous piecewise analytic initial value problem for a wide class of conservation laws that includes the full three-dimensional Euler equations. The initial interaction at an arbitrary curved surface is resolved in time by a convergent series. Among other features the solution exhibits shock, contact, and expansion waves as well as sound waves propagating on characteristic surfaces. The expansion waves correspond to the one-dimensional rarefactions but have a more complicated structure. The sound waves are generated in place of zero strength shocks, and they are caused by mismatches in derivatives.
A rigidity property for the set of all characters induced by valuations
Robert
Bieri;
John R. J.
Groves
425-434
Abstract: If $K$ is a field and $G$ a finitely generated multiplicative subgroup of $K$ then every real valuation on $K$ induces a character $G \to {\mathbf{R}}$. It is known that the set $ \Delta (G) \subseteq {{\mathbf{R}}^n}$ of all characters induced by valuations is polyhedral. We prove that $\Delta (G)$ satisfies a certain rigidity property and apply this to give a new and conceptual proof of the Brewster-Roseblade result [4] on the group of automorphisms of $K$ stabilizing $G$.
Spectral theory of the linearized Vlasov-Poisson equation
Pierre
Degond
435-453
Abstract: We study the spectral theory of the linearized Vlasov-Poisson equation, in order to prove that its solution behaves, for large times, like a sum of plane waves. To obtain such an expansion involving damped waves, we must find an analytical extension of the resolvent of the equation. Then, the poles of this extension are no longer eigenvalues and must be interpreted as eigenmodes, associated to ``generalized eigenfunctions'' which actually are linear functionals on a Banach space of analytic functions.
The Auslander-Reiten quiver of a simple curve singularity
Ernst
Dieterich;
Alfred
Wiedemann
455-475
Abstract: With any simple curve singularity (plane, complex, affine-algebraic) of Dynkin-type $\Delta$ we associate the category of all finitely generated torsionfree modules over its complete local ring. For each of these module categories we calculate the Auslander-Reiten quiver. We suggest the construction of the ``twisted quiver'' of a quiver with involution and valuation of arrows which gives rise to a (purely combinatorial) one-to-one correspondence between the Auslander-Reiten quiver and the Dynkin diagram $ \Delta$.
General convergence of continued fractions
Lisa
Jacobsen
477-485
Abstract: We introduce a new concept of convergence of continued fractions--general convergence. Moreover, we compare it to the ordinary convergence concept and to strong convergence. Finally, we prove some properties of general convergence.
Volume of mixed bodies
Erwin
Lutwak
487-500
Abstract: By using inequalities obtained for the volume of mixed bodies and the Petty Projection Inequality, (sharp) isoperimetric inequalities are derived for the projection measures (Quermassintegrale) of a convex body. These projection measure inequalities, which involve mixed projection bodies (zonoids), are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body. The inequality obtained for the volume of mixed bodies is also used to derive a form of the Brunn-Minkowski inequality involving mixed bodies. As an application, inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies.
Martingale transforms and complex uniform convexity
J.
Bourgain;
W. J.
Davis
501-515
Abstract: Martingale transforms and Calderon-Zygmund singular integral operators are bounded as operators from ${L_2}({L_1})$ to $ {L_2}({L_q})$ when $0 < q < 1$. If $ Y$ is a reflexive subspace of ${L_1}$ then ${L_1}/Y$ can be renormed to be $2$-complex uniformly convex. A new proof of the cotype 2 property of ${L_1}/{H_1}$ is given.
On $K\sb 3$ of truncated polynomial rings
Janet
Aisbett
517-536
Abstract: Group homology spectral sequences are used to investigate ${K_3}$ of truncated polynomial rings. If $F$ is a finite field of odd characteristic, we show that relative ${K_2}$ of the pair $(F\left[ t \right]/({t^q}),\,({t^k}))$, which has been identified by van der Kallen and Stienstra, is isomorphic to ${K_3}(F\left[ t \right]/({t^k}),\,(t))$ when $q$ is sufficiently large. We also show that $ {H_3}({\text{SL}}\,{\mathbf{Z}}\left[ t \right]/({t^k});{\mathbf{Z}}) = {{\mathbf{Z}}^{k - 1}} \oplus {\mathbf{Z}}/24$ and is isomorphic to the associated ${K_3}$ group modulo an elementary abelian $ 2$-group.
Stability of minimal orbits
John E.
Brothers
537-552
Abstract: Let $G$ be a compact Lie group of isometries of a riemannian manifold $M$. It is well known that the minimal principal orbits are those on which the volume function ${\mathbf{v}}$, which assigns to $p \in M$ the volume of the orbit of $ p$, is critical. It is shown that stability of a minimal orbit on which the hessian of $ {\mathbf{v}}$ is nonnegative is determined by the degree of involutivity of the distribution of normal planes to the orbits. Specifically, if the lengths of the tangential components of Lie brackets of pairs of orthonormal normal vector fields are sufficiently small relative to the hessian of ${\mathbf{v}}$, then the minimal orbit is stable, and conversely. Computable lower bounds are obtained for the values of these parameters at which stability turns to instability. These lower bounds are positive even in the case where $ {\mathbf{v}}$ is constant, and are finite unless the normal distribution is involutive. Several examples in which $M$ is a compact classical Lie group and $G$ is a subgroup of $M$ are discussed, showing in particular that the above estimates are sharp.
Unitary quasilifting: applications
Yuval Z.
Flicker
553-565
Abstract: Let $U(3)$ be the quasi-split unitary group in three variables defined using a quadratic extension $ E/F$ of number fields. Complete local and global results are obtained for the $ \sigma$-endo-(unstable) lifting from $U(2)$ to $ {\text{GL}}(3,\,E)$. This is used to establish quasi-(endo-)lifting for automorphic forms from $U(2)$ to $U(3)$ by means of base change from $U(3)$ to ${\text{GL}}(3,\,E)$. Base change quasi-lifting is also proven. Continuing the work of $\left[ {\mathbf{I}} \right]$, the exposition is elementary, and uses only a simple form of an identity of trace formulas, and base change transfer of orbital integrals of spherical functions.
Resonance and quasilinear ellipticity
Victor L.
Shapiro
567-584
Abstract: Two resonance-type existence theorems for periodic solutions of second order quasilinear elliptic partial differential equations are established. The first theorem is a best possible result, and the second theorem presents conditions which are both necessary and sufficient.
On the ampleness of homogeneous vector bundles
Dennis M.
Snow
585-594
Abstract: A formula is proved which expresses the ampleness of a homogeneous vector bundle over $G/P$ in terms of the distance of the weights of the representation of $P$ to certain dominant weights of $G$.
Spherical polynomials and the periods of a certain modular form
David
Kramer
595-605
Abstract: The space of cusp forms on $ {\text{S}}{{\text{L}}_2}({\mathbf{Z}})$ of weight $2k$ is spanned by certain modular forms with rational periods.
Conformally flat manifolds whose development maps are not surjective. I
Yoshinobu
Kamishima
607-623
Abstract: Let $M$ be an $n$-dimensional conformally flat manifold. A universal covering of $ M,\,\tilde M$ admits a conformal development map into ${S^n}$. When a development map is not surjective, we can relate the boundary of the development image with the limit set of the holonomy group of $ M$. In this paper, we study properties of closed conformally flat manifolds whose development maps are not surjective.
Fixed points of topologically stable flows
Mike
Hurley
625-633
Abstract: This paper concerns certain necessary conditions for a flow to be topologically stable (in the sense of P. Walters). In particular, it is shown that under fairly general conditions one can conclude that a topologically stable flow has a finite number of fixed points, and each of these is isolated in the chain recurrent set of the flow.
Contributions from conjugacy classes of regular elliptic elements in Hermitian modular groups to the dimension formula of Hermitian modular cusp forms
Min King
Eie
635-645
Abstract: The dimension of the vector space of hermitian modular cusp forms on the hermitian upper half plane can be obtained from the Selberg trace formula; in this paper we shall compute the contributions from conjugacy classes of regular elliptic elements in hermitian modular groups by constructing an orthonomal basis in a certain Hilbert space of holomorphic functions. A generalization of the main Theorem can be applied to the dimension formula of cusp forms of $SU(p,\,q)$. A similar theorem was given for the case of regular elliptic elements of $ {\text{Sp}}(n,\,{\mathbf{Z}})$ in [5] via a different method.
Functional equations for character series associated with $n\times n$ matrices
Edward
Formanek
647-663
Abstract: Let $A$ be either the ring of invariants or the trace ring of $r$ generic $n \times n$ matrices. Then $ A$ has a character series $ \chi (A)$ which is a symmetric rational function of commuting variables ${x_1}, \ldots ,{x_r}$. The main result is that if $r \geq {n^2}$, then $\chi (A)$ satisfies the functional equation $\displaystyle \chi (A)(x_1^{ - 1}, \ldots ,x_r^{ - 1}) = {( - 1)^d}{({x_1} \cdots {x_r})^{{n^2}}}\chi (A)({x_1}, \ldots ,{x_r})$ , where $d$ is the Krull dimension of $A$.
Left separated spaces with point-countable bases
William G.
Fleissner
665-677
Abstract: Theorem 2.2 lists properties equivalent to left separated spaces in the class of ${T_1}$ with point-countable bases, with examples preventing plausible additions to this list. For example, $X$ is left iff $X$ is $\sigma$-weakly separated or $X$ has a closure preserving cover by countable closed sets, but $X$ is left separated does not imply that $ X$ is $\sigma $-discrete. Theorem 2.2 is used to show that the following reflection property holds after properly collapsing a supercompact cardinal to ${\omega _2}$: If $X$ is a not $\sigma$-discrete metric space, then $X$ has a not $\sigma$-discrete subspace of cardinality less than ${\omega _2}$. Similar reflection properties are shown true in some models and false in others.
On the generalized Nakayama conjecture and the Cartan determinant problem
K. R.
Fuller;
B.
Zimmermann-Huisgen
679-691
Abstract: For Artin algebras allowing certain filtered module categories, the Generalized Nakayama Conjecture is shown to be true; our result covers all positively graded Artin algebras and those whose radical cube is zero. For the corresponding class of left artinian rings we prove that finite global dimension forces the determinant of the Cartan matrix to be 1.
Weak type estimates for Bochner-Riesz spherical summation multipliers
Sagun
Chanillo;
Benjamin
Muckenhoupt
693-703
Abstract: We consider the Bochner-Riesz multiplier $\displaystyle \widehat{{T_\delta }f}(\xi ) = {(1 - {\left\vert \xi \right\vert^2})^\delta } + \hat f(\xi ),\qquad \delta > 0,$ where $\widehat{}$ denotes the Fourier transform. It is shown that the multiplier operator ${T_\delta }$ is weak type $({p_0},\,{p_0})$ acting on $ {L^{p0}}({{\mathbf{R}}^n})$ radial functions, where ${p_0}$ is the critical value $2n/(n + 1 + 2\delta )$.
The complex equilibrium measure of a symmetric convex set in ${\bf R}\sp n$
Eric
Bedford;
B. A.
Taylor
705-717
Abstract: We give a formula for the measure on a convex symmetric set $K$ in $ {{\mathbf{R}}^n}$ which is the Monge-Ampere operator applied to the extremal plurisubharmonic function ${L_K}$ for the convex set. The measure is concentrated on the set $K$ and is absolutely continuous with respect to Lebesgue measure with a density which behaves at the boundary like the reciprocal of the square root of the distance to the boundary. The precise asymptotic formula for $x \in K$ near a boundary point ${x_0}$ of $K$ is shown to be of the form $ c({x_0})/{[{\operatorname{dist}}(x,\,\partial K)]^{ - 1/2}}$, where the constant $ c({x_0})$ depends both on the curvature of $K$ at ${x_0}$ and on the global structure of $K$.
Trellises formed by stable and unstable manifolds in the plane
Robert W.
Easton
719-732
Abstract: A trellis is the figure formed by the stable and unstable manifolds of a hyperbolic periodic point of a diffeomorphism of a $2$-manifold. This paper describes and classifies some trellises. The set of homoclinic points is linearly ordered as a subset of the stable manifold and again as a subset of the unstable manifold. Each homoclinic point is assigned a type number which is constant on its orbit. Combinatorial properties of trellises are studied using type numbers and the pair of linear orderings. Trellises are important because their closures in some cases are strange attractors and in other cases are ergodic zones.
Algebraic meridians of knot groups
Chichen M.
Tsau
733-747
Abstract: We propose the conjecture that every automorphism of a knot group preserves the meridian up to inverse and conjugation. We establish the conjecture for all composite knots, all torus knots, most cable knots, and at most one exception for hyperbolic knots; moreover we prove that the Property P Conjecture implies our conjecture. We also investigate hyperbolic knots in more detail, and give an example of figure-eight knot group and its automorphisms.
Vector bundles and projective modules
Leonid N.
Vaserstein
749-755
Abstract: Serre and Swan showed that the category of vector bundles over a compact space $X$ is equivalent to the category of finitely generated projective modules over the ring of continuous functions on $X$. In this paper, titled after the famous paper by Swan, this result is extended to an arbitrary topological space $X$. Also the well-known homotopy classification of the vector bundles over compact $X$ up to isomorphism is extended to arbitrary $X$. It is shown that the ${K_0}$-functor and the Witt group of the ring of continuous functions on $X$ coincide, and they are homotopy-type invariants of $ X$.
Brownian motions of ellipsoids
J. R.
Norris;
L. C. G.
Rogers;
David
Williams
757-765
Abstract: The object of this paper is to provide an elementary treatment (involving no differential geometry) of Brownian motions of ellipsoids, and, in particular, of some remarkable results first obtained by Dynkin. The canonical right-invariant Brownian motion $G = \{ G(t)\}$ on $ {\text{GL}}(n)$ induces processes $X = G{G^T}$ and $Y = {G^T}G$ on the space of positive-definite symmetric matrices. The motion of the common eigenvalues of $ X$ and $Y$ is analysed. It is further shown that the orthonormal frame of eigenvectors of $ X$ ultimately behaves like Brownian motion on $ {\text{O}}(n)$, while that of $Y$ converges to a limiting value. The $ Y$ process is that studied by Dynkin and Orihara. From a naive standpoint, the $ X$ process would seem to provide a more natural model.
Global boundedness for a delay-differential equation
Stephan
Luckhaus
767-774
Abstract: The inequality $ ({\partial _t}u - \Delta u)(t,\,x)\qquad \leq \qquad u(t,\,x)(1 - u(t - \tau ,\,x))$ is investigated. It is shown that nonnegative solutions of the Dirichlet problem in a bounded interval remain bounded as time goes to infinity, whereas in a more dimensional domain, in general, this holds only if the delay is not too large.
$\Omega$-stable limit set explosions
S. E.
Patterson
775-798
Abstract: Certain diffeomorphisms of two-dimensional manifolds are considered. These diffeomorphisms have a finite hyperbolic limit set which contains a limit set cycle. The only nontransverse cycle connection in these cycles is a complete coincidence of one component of the unstable manifold of one periodic point with one component of the stable manifold of some other periodic point. A one-parameter family of diffeomorphisms containing the original diffeomorphism is described. It is shown that for parameter values arbitrarily near the parameter value corresponding to the original map these diffeomorphisms have a much enlarged limit set and are $\Omega$-stable.
$G$-deformations and some generalizations of H. Weyl's tube theorem
Oldřich
Kowalski;
Lieven
Vanhecke
799-811
Abstract: We prove an invariance theorem for the volumes of tubes about submanifolds in arbitrary analytic Riemannian manifolds under $ G$-deformations of the second order. For locally symmetric spaces or two-point homogeneous spaces we give stronger invariance theorems using only $G$-deformations of the first order. All these results can be viewed as generalizations of the result of H. Weyl about isometric deformations and the volumes of tubes in spaces of constant curvature. They are derived from a new formula for the volume of a tube about a submanifold.
Potent axioms
Matthew
Foreman
1-28
Abstract: This paper suggests alternatives to the ordinary large cardinal axioms of set theory. These axioms can be viewed as generalizations of large cardinals and exhibit many of the same phenomena. They are shown to imply the G.C.H., every set of reals in $L({\mathbf{R}})$ is Lebesgue measurable, and various results in combinatorics, algebra and model theory.
Extensions of Verma modules
Kevin J.
Carlin
29-43
Abstract: A spectral sequence is introduced which computes extensions in category $\mathcal{O}$ in terms of derived functors associated to coherent translation functors. This is applied to the problem of computing extensions of one Verma module by another when the highest weights are integral and regular. Some results are obtained which are consistent with the Gabber-Joseph conjecture. The main result is that the highest-degree nonzero extension is one-dimensional. The spectral sequence is also applied to the Kazhdan-Lusztig conjecture and related to the work of Vogan in this area.
Bounds on the dimension of variations of Hodge structure
James A.
Carlson
45-64
Abstract: We derive upper bounds on the dimension of a variation of Hodge structure of weight two and show that these bounds are sharp. Using them we exhibit maximal geometric variations of Hodge structure. Analogous results for higher weight are obtained in the presence of a nondegeneracy hypothesis, and variations coming from hypersurfaces are shown to be nondegenerate. Maximal geometric variations of higher weight are also constructed.
Clarke's gradients and epsilon-subgradients in Banach spaces
Jay S.
Treiman
65-78
Abstract: A new characterization of Clarke's normal cone to a closed set in a Banach space is given. The normal cone is characterized in terms of weak-star limits of epsilon normals. A similar characterization of Clarke's generalized gradients is also presented. Restrictions must be placed on the Banach spaces to make the formulas valid.
Hyperreflexivity and a dual product construction
David R.
Larson
79-88
Abstract: We show that an example of a nonhyperreflexive CSL algebra recently constructed by Davidson and Power is a special case of a general and natural reflexive subspace construction. Completely different techniques of proof are needed because of absence of symmetry. It is proven that if $\mathcal{S}$ and $ \mathcal{I}$ are reflexive proper linear subspaces of operators acting on a separable Hilbert space, then the hyperreflexivity constant of ${({\mathcal{S}_ \bot } \otimes {\mathcal{I}_ \bot })^ \bot }$ is at least as great as the product of the constants of $ \mathcal{S}$ and $\mathcal{I}$.
On sieved orthogonal polynomials. III. Orthogonality on several intervals
Mourad E. H.
Ismail
89-111
Abstract: We introduce two generalizations of Chebyshev polynomials. The continuous spectrum of either is $\{ x: - 2\sqrt c /(1 + c) \leqslant {T_k}(x) \leqslant 2\sqrt c /(1 + c)\}$, where $c$ is a positive parameter. The weight function of the polynomials of the second kind is ${\{ 1 - ({(1 + c)^2}/4\operatorname{c} )T_k^2(x)\} ^{1/2}}/\vert{U_{k - 1}}(x)\vert$ when $c \geqslant 1$. When $c < 1$ we pick up discrete masses located at the zeros of ${U_{k - 1}}(x)$. The weight function of the polynomials of the first kind is also included. Sieved generalizations of the symmetric Pollaczek polynomials and their $q$-analogues are also treated. Their continuous spectra are also the above mentioned set. The $ q$-analogues include a sieved version of the Rogers $q$-ultraspherical polynomials and another set of $q$-ultraspherical polynomials discovered by Askey and Ismail. Generating functions and explicit formulas are also derived.
Some generalized Brown-Gitler spectra
Paul G.
Goerss;
John D. S.
Jones;
Mark E.
Mahowald
113-132
Abstract: Brown-Gitler spectra for the homology theories associated with the spectra $K{{\mathbf{Z}}_p}^ \wedge $ , $bo$, and $bu$ are constructed. Complexes adapted to the new Brown-Gitler spectra are produced and a spectral sequence converging to stable maps into these spectra is constructed and examined.
$H\sp p$-classes on rank one symmetric spaces of noncompact type. I. Nontangential and probabilistic maximal functions
Patricio
Cifuentes
133-149
Abstract: Two kinds of $ {H^p}$-classes of harmonic functions are defined on a general rank one symmetric space of noncompact type. The first one is introduced by using a nontangential maximal function. The second is related to the diffusion generated by the Laplace-Beltrami operator. The equivalence of the two classes is proven for $0 < p < \infty$.
Classification of metabelian $p$-groups
Wu Nan
Chou
151-176
Abstract: Let $G$ be a two-generator metabelian group of exponent $p$ with class of nilpotence $c$, where $c \leqslant p - 1$ and $ p$ is an odd prime. In this paper, we shall consider the classification problem when $ \vert{G_2}/{G_3}\vert = p$, $ \vert{G_3}/{G_4}\vert = {p^2}$and $\vert{G_4}/{G_5}\vert \leqslant {p^2}$.
On linking double lines
Juan
Migliore
177-185
Abstract: A double line is a nonreduced locally Cohen-Macaulay scheme of degree two supported on a line in projective three-space. The heart of this work is to compute the associated Hartshorne-Rao module for such a curve. We can then say exactly when two such curves are in the same liaison class and in fact when they are directly linked. In particular, we find that $C$ is only self-linked in characteristic two.
Joint spectra and analytic set-valued functions
M.
Klimek
187-196
Abstract: We investigate analyticity of joint spectra of ${A^m}$-valued holomorphic mappings, where $ A$ denotes a complex Banach algebra. We show also that if $K$ is an analytic set-valued function whose values are compact subsets of ${{\mathbf{C}}^n}$ and $d$ is the transfinite diameter in ${{\mathbf{C}}^n}$, then the upper-semicontinuous regularization of $\log d(K)$ is plurisubharmonic. Moreover, we give higher dimensional extensions of Aupetit's Scarcity Theorem.
Anosov diffeomorphisms and expanding immersions. II
Lowell
Jones
197-216
Abstract: This paper continues the study of hyperbolic attractors, expanding immersions, and quotient solenoids which was begun in a previous paper of the same title. The main result states that certain hyperbolic attractors are topologically conjugate to an Anosov diffeomorphism.
Existence and nonoscillation theorems for an Emden-Fowler equation with deviating argument
William F.
Trench
217-231
Abstract: Sufficient conditions are given for a generalized Emden-Fowler equation with deviating argument to have nonoscillatory solutions with prescribed asymptotic behavior as $t \to \infty $. The integrability condition on the nonlinear term requires only conditional convergence, supplemented by a condition on the order of convergence, which is automatically satisfied in some important special cases. The exponent in the nonlinear term may be any real number. The deviating argument is not assumed to be purely advanced or retarded, and, in some cases, need not tend to infinity. Some of the results are global, in that the desired solution is shown to exist on a given interval, rather than only for sufficiently large $t$.
A bilaterally deterministic $\rho$-mixing stationary random sequence
Richard C.
Bradley
233-241
Abstract: A (nondegenerate) strictly stationary sequence $({X_k},\;k \in {\mathbf{Z}})$ of random variables is constructed such that the $\rho$-mixing (maximal correlation mixing) condition is satisfied and each ${X_k}$ is measurable with respect to the double tail $\sigma$-field.
Properties of relatively free inverse semigroups
N. R.
Reilly;
P. G.
Trotter
243-262
Abstract: The objective of this paper is to study structural properties of relatively free inverse semigroups in varieties of inverse semigroups. It is shown, for example, that if $ S$ is combinatorial (i.e., $\mathcal{H}$ is trivial), completely semisimple (i.e., every principal factor is a Brandt semigroup or, equivalently, $S$ does not contain a copy of the bicyclic semigroup) or $E$-unitary (i.e., $E(S)$ is the kernel of the minimum group congruence) then the relatively free inverse semigroup $F{\mathcal{V}_X}$ on the set $X$ in the variety $\mathcal{V}$ generated by $S$ is also combinatorial, completely semisimple or $E$-unitary, respectively. If $ S$ is a fundamental (i.e., the only congruence contained in $\mathcal{H}$ is the identity congruence) and $ \vert X\vert \geqslant {\aleph _0}$, then $ F{\mathcal{V}_X}$ is also fundamental. $ F{\mathcal{V}_X}$ may not be fundamental if $ \vert X\vert < {\aleph _0}$. It is also shown that for any variety of groups $\mathcal{U}$ and for $\vert X\vert \geqslant {\aleph _0}$, there exists a variety of inverse semigroups $\mathcal{V}$ which is minimal with respect to the properties (i) $ F{\mathcal{V}_X}$ is fundamental and (ii) $\mathcal{V} \cap \mathcal{G} = \mathcal{U}$, where $ \mathcal{G}$ is the variety of groups. In the main result of the paper it is shown that there exists a variety $\mathcal{V}$ for which $F{\mathcal{V}_X}$ is not completely semisimple, thereby refuting a long standing conjecture.
On coupled multiparameter nonlinear elliptic systems
Robert Stephen
Cantrell
263-285
Abstract: This paper considers the system of nonlinear Dirichlet boundary value problems $\displaystyle \left\{ \begin{gathered}Lu(x) = \lambda f(u(x),v(x)) Lv(x) = \mu g(u(x),v(x)) \end{gathered} \right\},\qquad x \in \Omega ,$ a bounded domain in $ {{\mathbf{R}}^n}$. Here $ L$ is a strongly, uniformly elliptic linear partial differential operator, $ \lambda$, $\mu$ are real parameters, and $ f$, $g:{{\mathbf{R}}^2} \to R$ are smooth with $\displaystyle f(0,0) = 0 = g(0,0).$ A detailed analysis of the solution set to the system is given from the point of view of several parameter bifurcation theory.
Dominated permutations of subsequences of random variables
Aaron
Meyerowitz;
Mark
Schwartz
287-294
Abstract: The generalized strong law of large numbers of Komlós may be extended to include certain dominated permutations of the random variables. A further extension to larger classes of permutations is obtained through decompositions of sequences of positive integers.
On type of metric spaces
J.
Bourgain;
V.
Milman;
H.
Wolfson
295-317
Abstract: Families of finite metric spaces are investigated. A notion of metric type is introduced and it is shown that for Banach spaces it is consistent with the standard notion of type. A theorem parallel to the Maurey-Pisier Theorem in Local Theory is proved. Embeddings of ${l_p}$-cubes into the ${l_1}$-cube (Hamming cube) are discussed.
Nonexistence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of Euclidean space
Ralph
Howard;
S. Walter
Wei
319-331
Abstract: Call a compact Riemannian manifold $M$ a strongly unstable manifold if it is not the range or domain of a nonconstant stable harmonic map and also the homotopy class of any map to or from $ M$ contains elements of arbitrarily small energy. If $M$ is isometrically immersed in Euclidean space, then a condition on the second fundamental form of $ M$ is given which implies $ M$ is strongly unstable. As compact isotropy irreducible homogeneous spaces have "standard" immersions into Euclidean space this allows a complete list of the strongly unstable compact irreducible symmetric spaces to be made.
Boundary uniqueness theorems in ${\bf C}\sp n$
Joseph A.
Cima;
Emil J.
Straube
333-339
Abstract: Let $n$-dimensional manifolds ${\Gamma _k},\,k = 1,2, \ldots$, be given in a smoothly bounded domain $\Omega \subset {{\mathbf{C}}^n}$. Assume that the ${\Gamma _k}$ "converge" to an $ n$-dimensional, totally real manifold $\Gamma \subseteq \partial \Omega$ and that a function $f$ analytic in $\Omega$ has the property that its traces $ {f_k}$ on ${\Gamma _k}$ have distributional limit zero as $k \to \infty $ (or assume that ${f_k} \to 0$ pointwise). Then under the assumption that $f$ is polynomially bounded near $P \in \Gamma$ by $ {(\operatorname{dist} (z,\partial \Omega ))^{ - 1}}$ we conclude that $ f$ is identically zero.
Lower bounds of the gap between the first and second eigenvalues of the Schr\"odinger operator
Qi Huang
Yu;
Jia Qing
Zhong
341-349
Abstract: In this paper the authors prove the following theorem: Let $ \Omega$ be a smooth strictly convex bounded domain in ${R^n}$ and $V:\Omega \to R$ a nonnegative convex function. Suppose $ {\lambda _1}$ and ${\lambda _2}$ are the first and second nonzero eigenvalues of the equation $\displaystyle - \Delta f + Vf = \lambda f,\qquad f{\vert _{\partial \Omega }} \equiv 0.$ Then ${\lambda _2} - {\lambda _1} \geqslant {\pi ^2}/{d^2}$, where $d$ is the diameter of $\Omega$.
The law of the iterated logarithm in uniformly convex Banach spaces
Michel
Ledoux
351-365
Abstract: We give necessary and sufficient conditions for a random variable $ X$ with values in a uniformly convex Banach space $B$ to satisfy the law of the iterated logarithm. Precisely, we show that a $B$-valued random variable $X$ satisfies the (compact) law of the iterated logarithm if and only if $E\{ \vert\vert X\vert{\vert^2}/{L_2}\vert\vert X\vert\vert\} < \infty$, the family $\{ \vert{x^{\ast}}(X){\vert^2};\,{x^{\ast}} \in {B^{\ast}},\,\vert\vert{x^{\ast}}\vert\vert = 1\}$ is uniformly integrable and ${S_n}/{a_n} \to 0$ in probability.
An analytic set-valued selection and its applications to the corona theorem, to polynomial hulls and joint spectra
Zbigniew
Slodkowski
367-377
Abstract: It is shown that for every annulus $P = \{ z \in {{\mathbf{C}}^n}:\delta < \vert z\vert < r\}$, $ \delta > 0$, there exists a set-valued correspondence $z \to K(z):P \to {2^{{{\mathbf{C}}^n}}}$, whose graph is a bounded relatively closed subset of the manifold $\{ (z,w) \in P \times {{\mathbf{C}}^n}:{z_1}{w_1} + \cdots + {z_n}{w_n} = 1\} $ which can be covered by $ n$-dimensional analytic manifolds. The analytic set-valued selection $ K$ obtained thereby is then applied to several problems in complex analysis and spectral theory which involve solving the equation ${a_1}{x_1} + \cdots + {a_n}{x_n} = y$. For example, an elementary proof is given of the following special case of a theorem due to Oka: every bounded pseudoconvex domain in $ {{\mathbf{C}}^2}$ is a domain of holomorphy.
Corrigendum to: ``On isometric embeddings of graphs'' [Trans. Amer. Math. Soc. {\bf 288} (1985), no. 2, 527--536; MR0776391 (86f:05055b)]
R. L.
Graham;
P. M.
Winkler
379
Correction to: ``Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions'' [Trans. Amer. Math. Soc. {\bf 282} (1984), no. 1, 1--32; MR0728700 (85h:03015)]
C. Ward
Henson;
Lee A.
Rubel
381
The Gauss map for surfaces. I. The affine case
Joel L.
Weiner
431-446
Abstract: Let $M$ be a connected oriented surface and let $G_2^c$ be the Grassmannian of oriented $ 2$-planes in Euclidean $ (2 + c)$-space, ${{\mathbf{E}}^{2 + c}}$. Smooth maps $t:M \to G_2^c$ are studied to determine whether or not they are Gauss maps. Both local and global results are obtained. If $t$ is a Gauss map of an immersion $X:\;M \to {{\mathbf{E}}^{2 + c}}$, we study the extent to which $t$ uniquely determines $X$ under certain circumstances.
The Gauss map for surfaces. II. The Euclidean case
Joel L.
Weiner
447-466
Abstract: We study smooth maps $t:\;M \to G_2^c$ of a Riemann surface $M$ into the Grassmannian $G_2^c$ of oriented $2$-planes in $ {{\mathbf{E}}^{2 + c}}$ and determine necessary and sufficient conditons on $ t$ in order that it be the Gauss map of a conformal immersion $X:\;M \to {{\mathbf{E}}^{2 + c}}$. We sometimes view $t$ as an oriented riemannian vector bundle; it is a subbundle of $ {\mathbf{E}}_M^{2 + c}$, the trivial bundle over $M$ with fibre $ {{\mathbf{E}}^{2 + c}}$. The necessary and sufficient conditions obtained for simply connected $M$ involve the curvatures of $t$ and $ {t^ \bot }$, the orthogonal complement of $t$ in $ {\mathbf{E}}_M^{2 + c}$, as well as certain components of the tension of $ t$ viewed as a map $t:\;M \to {S^C}(1)$, where ${S^C}(1)$ is a unit sphere of dimension $ C$ that contains $ G_2^c$ as a submanifold in a natural fashion. If $t$ satisfies a particular necessary condition, then the results take two different forms depending on whether or not $t$ is the Gauss map of a conformal minimal immersion. The case $ t:\;M \to G_2^2$ is also studied in some additional detail.
The least $r$-free number in an arithmetic progression
Kevin S.
McCurley
467-475
Abstract: Let ${n_r}(a,q)$ be the least $r$-free number in the arithmetic progession $ a$ modulo $q$. Several results are proved that give lower bounds for $ {n_r}(a,q)$, improving on previous results due to Erdös and Warlimont. In addition, a heuristic argument is given, leading to two conjectures that would imply that the results of the paper are close to best possible.
Extensions of a theorem of Wintner on systems with asymptotically constant solutions
William F.
Trench
477-483
Abstract: A theorem of Wintner concerning sufficient conditions for a system
A simple theory of differential calculus in locally convex spaces
Richard A.
Graff
485-509
Abstract: A theory of differential calculus for nonlinear maps between general locally convex spaces is developed. All convergence notions are topological, and only familiarity with basic results from point set topology, differential calculus in Banach spaces, and locally convex space theory is assumed. The chain rule for continuous $k$th order differentiability, smoothness of inverse functions, and function space continuity properties of higher order derivatives are examined. It is shown that this theory extends the classical Fréchet theory of differential calculus for maps between Banach spaces.
Rational singularities and almost split sequences
Maurice
Auslander
511-531
Abstract: The main aim of this paper is to relate almost split sequences to singularity theory by showing that the McKay quiver built from the finite-dimensional representations of a finite subgroup $G$ of $\operatorname{GL} (2,{\mathbf{C}})$, where ${\mathbf{C}}$ is the complex numbers, is isomorphic to the $AR$ quiver of the reflexive modules of the quotient singularity associated with $G$.
Best rational approximations of entire functions whose Maclaurin series coefficients decrease rapidly and smoothly
A. L.
Levin;
D. S.
Lubinsky
533-545
Abstract: Let $f = \Sigma _{j = 0}^\infty {a_j}{z^j}$ be an entire function which satisfies $\displaystyle \vert{a_{j - 1}}a{ _{j + 1}}/a_j^2\vert \leqslant {\rho ^2},\qquad j = 1,2,3, \ldots ,$ where $0 < \rho < {\rho _0}$ and $ {\rho _0} = 0.4559 \ldots$ is the positive root of the equation $ 2\Sigma _{j = 1}^\infty {\rho ^{{j^2}}} = 1$. Let $r > 0$ be fixed. Let ${W_{LM}}$ denote the rational function of type $ (L,M)$ of best approximation to $f$ in the uniform norm on $\vert z\vert \leqslant r$. We show that for any sequence of nonnegative integers $\{ {M_L}\} _{L = 1}^\infty $ that satisfies $ {M_L} \leqslant 10L,\,L = 1,2,3, \ldots$, the rational approximations ${W_{L{M_L}}}$ converge to $f$ throughout $ {\mathbf{C}}$ as $L \to \infty$. In particular, convergence takes place for the diagonal sequence and for the row sequences of the Walsh array for $f$.
Martingale transforms and related singular integrals
Rodrigo
Bañuelos
547-563
Abstract: The operators obtained by taking conditional expectation of continuous time martingale transforms are studied, both on the circle $ T$ and on ${{\mathbf{R}}^n}$. Using a Burkholder-Gundy inequality for vector-valued martingales, it is shown that the vector formed by any number of these operators is bounded on ${L^p}({{\mathbf{R}}^n}),\,1 < p < \infty$, with constants that depend only on $p$ and the norms of the matrices involved. As a corollary we obtain a recent result of Stein on the boundedness of the Riesz transforms on $ {L^p}({{\mathbf{R}}^n}),\,1 < p < \infty$, with constants independent of $ n$.
Baer's lemma and Fuchs's problem 84a
Ulrich
Albrecht
565-582
Abstract: An indecomposable, torsion-free, reduced abelian group $A$ has the properties that (i) each subgroup $B$ of an $A$-projective group with ${S_A}(B) = B$ is $A$-projective and (ii) each subgroup $B$ of a group $G$ with $ {S_A}(G) + B = G$ and $ G/B$ $A$-projective is a direct summand if and only if $A$ is self-small and flat as a left $ E(A)$-module, and $ E(A)$ is right hereditary. Furthermore, a group-theoretic characterization is given for torsion-free, reduced abelian groups with a right and left Noetherian, hereditary endomorphism ring. This is applied to Fuchs' Problem 84a. Finally, various applications of the results of this paper are given.
Solvability of differential equations with linear coefficients of real type
Rainer
Felix
583-591
Abstract: Let $L$ be the infinitesimal generator associated with a flow on a manifold $M$. Regarding $L$ as an operator on a space of testfunctions we deal with the question if $L$ has closed range. (Questions of this kind are investigated in [4, 1, 2].) We provide conditions under which $L + \mu 1:\mathcal{S}(M) \to \mathcal{S}(M)$, $ \mu \in {\mathbf{C}}$, has closed range, where $M = {{\mathbf{R}}^n} \times K$, $K$ being a compact manifold; here $\mathcal{S}(M)$ is the Schwartz space of rapidly decreasing smooth functions. As a consequence we show that the differential operator $ {\Sigma _{i,j}}{a_{ij}}{x_j}(\partial /\partial {x_i}) + b$ defines a surjective mapping of the space $ \mathcal{S}({{\mathbf{R}}^n})$ of tempered distributions onto itself provided that all eigenvalues of the matrix $({a_{ij}})$ are real. (In the case of imaginary eigenvalues this is not true in general [3].)
Conditions suffisantes de r\'esolubilit\'e locale pour des op\'erateurs invariants \`a gauche sur des groupes nilpotents. II
Pierre
Lévy-Bruhl
593-603
Abstract: On donne des conditions suffisantes de résolubilité locale pour des opérateurs différentiels invariants à gauche sur certains groupes de Lie nilpotents gradués. Ces conditions portent sur l'image de l'opérateur par certaines représentations unitaires irréductibles du groupe.
A solution of Warner's 3rd problem for representations of holomorphic type
Floyd L.
Williams
605-612
Abstract: In response to one of ten problems posed by G. Warner, we assign (to the extent that it is possible) a geometric or cohomological interpretation-- in the sense of Langlands--to the multiplicty in ${L^2}(\Gamma \backslash G)$ of an irreducible unitary representation $\pi$ of a semisimple Lie group $G$, where $\Gamma$ is a discrete subgroup of $G$, in the case when $\pi$ has a highest weight.
Topological properties of Souslin subsets
R. W.
Hansell
613-622
Abstract: Let $X$ be a subparacompact regular space such that the projection map $p:X \times {\mathbf{P}} \to X$, where ${\mathbf{P}}$ is the space of irrational numbers, preserves collections of sets having a $\sigma $-locally finite refinement. It is shown that $p$ then preserves generalized ${F_\sigma }$-sets. It follows that, if $ X$ has any tpological property which is hereditary with respect to generalized ${F_\sigma }$-sets, then every Souslin subset of $ X$ will also have this property in the relative topology. Such topological properties include nearly all covering properties (paracompactness, metacompactness, etc.), as well as normality, collectionwise normality, and the Lindelöf property. We show that the above mapping property will hold whenever $X$ is a $ P(\omega )$-space, thus, in particular, when $X$ is any Souslin (hence any Baire) subset of a compact space crossed with a metrizable space. Additional topological properties of Souslin subsets, such as topological completeness, realcompactness, $ z$-embeddedness, and the properties of being a $P$-space or $\Sigma$-space, are also considered.
A Martin boundary in the plane
Thomas S.
Salisbury
623-642
Abstract: Let $E$ be an open connected subset of Euclidean space, with a Green function, and let $ \lambda$ be harmonic measure on the Martin boundary $\Delta$ of $E$. We will show that, except for a $\lambda \otimes \lambda $-null set of $(x,y) \in {\Delta ^2}$, $x$ is an entrance point for Brownian motion conditioned to leave $E$ at $y$. R. S. Martin gave examples in dimension $ 3$ or higher, for which there exist minimal accessible Martin boundary points $ x \ne y$ for which this condition fails. We will give a similar example in dimension $2$.
Entire functions which are infinitely integer-valued at a finite number of points
P.
Lockhart;
E. G.
Straus
643-654
Abstract: This paper determines arithmetic limits for the growth rates of entire functions which are infinitely integer valued on a finite set $S$. The characterization of such functions with growth rate less than the arithmetic limit is complete if there exist exponential polynomials which are infinitely integer valued on $S$.
Simple homotopy type of finite $2$-complexes with finite abelian fundamental group
M. Paul
Latiolais
655-662
Abstract: Theorem 1. Let $K$ be a finite $2$-dimensional $CW$-complex with $ {\pi _1}(K)$ finite and abelian. Then every element of the Whitehead group of $K$ is realizable as the torsion of a self-homotopy equivalence on $K$. Theorem 2. Homotopy equivalence and simple homotopy equivalence are the same for finite $ 2$-dimensional $ CW$-complexes with finite abelian fundamental groups.
Almost Mizohata operators
Nicholas
Hanges
663-675
Abstract: The range of every almost Mizohata operator is characterized via an integral formula.
The Stefan problem with heating: appearance and disappearance of a mushy region
M.
Bertsch;
P.
de Mottoni;
L. A.
Peletier
677-691
Abstract: We consider a material which is initially in the solid state and then, due to heating, starts to melt. We describe the appearance of a so-called mushy region, i.e., a region in which the material is in neither a solid nor a liquid state. The main result is that after a finite time the mushy region has disappeared and only the solid and the liquid phases have remained.
A new construction of noncrossed product algebras
Bill
Jacob;
Adrian R.
Wadsworth
693-721
Abstract: New examples of noncrossed product division algebras are obtained, using methods different from all previous noncrossed product constructions. The examples are division algebras over intersections of $p$-Henselian valued fields, and they have Schur index ${p^m}$ and exponent ${p^n}$ for any prime number $p$ and any integers $m \geqslant n \geqslant 2\;(n \geqslant 3\;{\text{if}}\;p = 2)$. The basic tools used in the construction are valuation theory and Galois cohomology; no generic methods are applied and there is no p.i. theory. Along the way, local-global principles are proved for central simple algebras over intersections of $ p$-Henselian valued fields.
Potentials producing maximally sharp resonances
Evans M.
Harrell;
Roman
Svirsky
723-736
Abstract: We consider quantum-mechanical potentials consisting of a fixed background plus an additional piece constrained only by having finite height and being supported in a given finite region in dimension $d \leqslant 3$. We characterize the potentials in this class that produce the sharpest resonances. In the one-dimensional or spherically symmetric specialization, a quite detailed description is possible. The maximally sharp resonances that we find are, roughly speaking, caused by barrier confinement of a metastable state, although in some situations they call for interactions in the interior of the confining barrier as well.
Equilibrium points of nonatomic games over a Banach space
M. Ali
Khan
737-749
Abstract: Schmeidler's results on the equilibrium points of nonatomic games with strategy sets in Euclidean $n$-space are generalized to nonatomic games with strategy sets in a Banach space. Our results also extend previous work of the author which assumed the Banach space to be separable and its dual to possess the Radon-Nikodým property. Our proofs use recent results in functional analysis.
Principal lines on surfaces immersed with constant mean curvature
C.
Gutiérrez;
J.
Sotomayor
751-766
Abstract: Configurations of lines of principal curvature on constant mean curvature immersed surfaces are studied. Analytical models for these configurations near general isolated umbilical points and particular types of ends are found. From the existence of transversal invariant measures for the foliations by principal lines, established here, follows that the union of recurrent lines of principal curvature is an open set. Examples illustrating all possible cases are given.
Weak limits of stopped diffusions
J. R.
Baxter;
R. V.
Chacon;
N. C.
Jain
767-792
Abstract: We consider the following homogenization problem: Let Brownian motion in $ {{\mathbf{R}}^d}$, $d \geqslant 3$, be killed on the surface of many small absorbing bodies (standard diffusion equation with Dirichlet boundary conditions). We investigate the limit as the number of bodies approaches infinity and the size of the bodies approaches 0. By taking a weak limit of stopping times we replace a convergence problem on the state space by an identification of the limit on the sample space. This technique then gives results without smoothness assumptions which were previously necessary.
On the dichotomy problem for tensor algebras
J.
Bourgain
793-798
Abstract: Let $I$, $J$ be discrete spaces and $E \subset I \times J$. Then either $E$ is a $V$-Sidon set (in the sense of $[{\mathbf{2}},\S11]$), or the restriction algebra $ A(E)$ is analytic. The proof is based on probabilistic methods, involving Slépian's lemma.
Maximal functions associated with curves and the Calder\'on-Zygmund method of rotations
Shuichi
Sato
799-806
Abstract: Let ${\delta _t}(t > 0)$ be a dilation in $ {{\mathbf{R}}^n}(n \geqslant 2)$ defined by $\displaystyle {\delta _t}x = ({t^{{\alpha _1}}}{x_1},{t^{{\alpha _2}}}{x_2}, \ldots ,{t^{{\alpha _n}}}{x_n})\qquad (x = ({x_1},{x_2}, \ldots ,{x_n})),$ where $ {\alpha _i} > 0(i = 1,2, \ldots ,n)$ and ${\alpha _i} \ne {\alpha _j}$ if $i \ne j$. For $\nu \in {{\mathbf{R}}^n}$ with $\vert\nu \vert = 1$, let $ {\Gamma _\nu }:(0,\infty ) \to {{\mathbf{R}}^n}$ be a curve defined by ${\Gamma _\nu }(t) = {\delta _t}\nu \,(0 < t < \infty )$. Using maximal functions associated with the curves $ {\Gamma _\nu }$, we define an operator $M$ which is a nonisotropic analogue of the one studied in R. Fefferman [2]. It is proved that $M$ is a bounded operator on ${L^p}({{\mathbf{R}}^n})$ for some $p$ with $1 < p < 2$. As its application we prove the $ {L^p}$ boundedness of operators of the form ${T^{\ast}}(f)(x) = {\sup _{\varepsilon > 0}}\vert{T_\varepsilon }(f)(x)\vert$, where ${T_\varepsilon }$ is an integral operator associated with a variable kernel with mixed homogeneity.
On multiplication algebras
David R.
Finston
807-818
Abstract: The basic properties of multiplication algebras of nonassociative algebras over rings are introduced, including a discussion of multiplication algebras of tensor products of algebras. A characterization of semisimple artinian multiplication algebras is given along with a discussion of the nature of the simple factors of a multiplication algebra modulo its Jacobson radical. A criterion distinguishing the multiplication algebras of certain associative algebras is proved. Examples are included to illustrate certain proved results.
The distribution of solutions to equations over finite fields
Todd
Cochrane
819-826
Abstract: Let ${\mathbb{F}_q}$ be the finite field in $q = {p^f}$ elements, $\underline F (\underline x )$ be a $ k$-tuple of polynomials in ${\mathbb{F}_q}[{x_1}, \ldots ,{x_n}]$, $ V$ be the set of points in $ \mathbb{F}_q^n$ satisfying $\underline F (\underline x ) = \underline 0$ and $ S$, $T$ be any subsets of $\mathbb{F}_q^n$. Set $\phi (V,\underline 0 ) = \vert V\vert - {q^{n - k}}$, $\displaystyle \phi (V,\underline y ) = \sum\limits_{\underline x \in V} {e\left... ...ot \underline y )} \right)\quad {\text{for}}\;\underline y \ne \underline 0 ,}$ and $\Phi (V) = {\max _{\underline y }}\vert\phi (V,\underline y )\vert$. We use finite Fourier series to show that $ (S + T) \cap V$ is nonempty if $ \vert S\vert\vert T\vert > {\Phi ^2}(V){q^{2k}}$. In case $q = p$ we deduce from this, for example, that if $C$ is a convex subset of ${\mathbb{R}^n}$ symmetric about a point in $ {\mathbb{Z}^n}$, of diameter $< 2p$ (with respect to the sup norm), and $\operatorname{Vol} (C) > {2^{2n}}\Phi (V){p^k}$, then $C$ contains a solution of $ \underline F (\underline x ) \equiv \underline 0 (\bmod p)$. We also show that if $B$ is a box of points in $\mathbb{F}_q^n$ not contained in any $ (n - 1)$-dimensional subspace and $\vert B\vert > 4 \cdot {2^{nf}}\Phi (V){q^k}$, then $B \cap V$ contains $n$ linearly independent points.
Some results concerning the boundary zero sets of general analytic functions
Robert D.
Berman
827-836
Abstract: Two results concerning the boundary zero sets of analytic functions on the unit disk $\Delta$ are proved. First we consider nonconstant analytic functions $f$ on $\Delta$ for which the radial limit function ${f^{\ast}}$ is defined at each point of the unit circumference $C$. We show that a subset $E$ of $C$ is the zero set of $ {f^{\ast}}$ for some such function $f$ if and only if it is a ${\mathcal{G}_\delta }$ that is not metrically dense in any open arc of $C$. We then give a precise version of an asymptotic radial uniqueness theorem and its converse. The constructions given in the proofs of each of these theorems employ an approximation theorem of Arakeljan.
Growth rates and critical exponents of classes of binary combinatorial geometries
Joseph P. S.
Kung
837-859
Abstract: We prove that a binary geometry of rank $n\;(n \geqslant 2)$ not containing $M({K_5})$ and ${F_7}$ (respectively, $M({K_5})$ and ${C_{10}}$) as a minor has at most $3n - 3$ (respectively, $ 4n - 5$) points. Here, $ M({K_5})$ is the cycle geometry of the complete graph on five vertices, $ {F_7}$ the Fano plane, and $ {C_{10}}$ a certain rank $ 4$ ten-point geometry containing the dual Fano plane $ F_7^{\ast}$ as a minor. Our technique is elementary and uses the notion of a bond graph. From these results, we deduce upper bounds on the critical exponents of these geometries.
Self-maps of loop spaces. I
H. E. A.
Campbell;
F. P.
Peterson;
P. S.
Selick
1-39
Abstract: In this paper we study self-maps of $ {\Omega ^k}{S^{m + 1}}$ and show that, except for the cases $m = 1,\,3,\,7$, or $p$ and $m$, if $f$ induces an isomorphism on $ {H_{m + 1 - k}}({\Omega ^k}{S^{m + 1}};\,Z/pZ)$ with $k < m$, then ${f_{(p)}}$ is a homotopy equivalence.
Self-maps of loop spaces. II
H. E. A.
Campbell;
F. R.
Cohen;
F. P.
Peterson;
P. S.
Selick
41-51
Abstract: We study under what conditions on a finite CW complex $X$ is $Q(X)$ atomic.
Derivatives of mappings with applications to nonlinear differential equations
Martin
Schechter
53-69
Abstract: We present a new definition of differentiation for mappings of sets in topological vector spaces. Complete flexibility is allowed in choosing the topology with which the derivative is taken. We determine the largest space on which the derivative can act. Our definition includes all others hitherto given, and the basic theorems of calculus hold for it. Applications are considered here and elsewhere.
A rearranged good $\lambda$ inequality
Richard J.
Bagby;
Douglas S.
Kurtz
71-81
Abstract: Let $Tf$ be a maximal Calderón-Zygmund singular integral, $Mf$ the Hardy-Littlewood maximal function, and $ w$ an ${A_\infty }$ weight. We replace the ``good $ \lambda$'' inequality $\displaystyle w\left( {\{ x:\,Tf(x) > 2\lambda \,{\text{and}}\,Mf(x) \leq \vare... ...a \} } \right) \leq C(\varepsilon )w\left( {\{ x:\,Tf(x) > \lambda \} } \right)$ by the rearrangement inequality $\displaystyle (Tf)_w^ \ast (t) \leq C(Mf)_w^ \ast (t/2) + (Tf)_w^ \ast (2t)$ and show that it gives better estimates for $Tf$. In particular, we obtain best possible weighted ${L^p}$ bounds, previously unknown exponential integrability estimates, and simplified derivations of known unweighted estimates for ${(Tf)^ \ast }$.
Conjugacy problem in ${\rm GL}\sb 2({\bf Z}[\sqrt{-1}])$ and units of quadratic extensions of ${\bf Q}(\sqrt{-1})$
Hironori
Onishi
83-98
Abstract: A highly efficient procedure for deciding if two given elements of $ {\text{G}}{{\text{L}}_2}(\mathbf{Z}[\sqrt { - 1} ])$ are conjugate or not will be presented. It makes use of a continued fraction algorithm in $\mathbf{Z}[\sqrt { - 1} ]$ and gives a fundamental unit of any given quadratic extension of $\mathbf{Q}(\sqrt { - 1} )$.
The balanced-projective dimension of abelian $p$-groups
L.
Fuchs;
P.
Hill
99-112
Abstract: The balanced-projective dimension of every abelian $p$-group is determined by means of a structural property that generalizes the third axiom of countability. As a corollary to our general structure theorem, we show for $\lambda = {\omega _n}$ that every ${p^\lambda }$-high subgroup of a $p$-group $G$ has balanced-projective dimension exactly $n$ whenever $G$ has cardinality $ {\aleph _n}$ but ${p^\lambda }G \ne 0$. Our characterization of balanced-projective dimension also leads to new classes of groups where different infinite dimensions are distinguished.
The fifth and seventh order mock theta functions
George E.
Andrews
113-134
Abstract: The theory of Bailey chains is extended to yield identities for Hecke type modular forms and related generalizations. The extended results allow us to produce Hecke type series for the fifth and seventh order mock theta functions. New results on the generating function for sums of three squares also follow, and a new proof that every integer is the sum of three triangular numbers is given.
Strange attractors of uniform flows
Ittai
Kan
135-159
Abstract: Consider orbitally stable attractors of those flows on the open solid torus ${D^2} \times {S^1}$ which have uniform velocity in the ${S^1}$ direction (uniform flows). It is found that any such attractor is the frontier of a strictly nested sequence of positively invariant open solid tori. Necessary and sufficient conditions related to these tori are derived for an arbitrary set to be an orbitally stable attractor. When the cross-section of an orbitally stable attractor is a Cantor set, the first return map is found to be conjugate to an irrational rotation on a certain compact abelian group. New examples are constructed of orbitally stable attractors of uniform ${C^\infty }$ flows whose cross-sections have uncountably many components (one of these attractors has positive $3$-dimensional Lebesgue measure).
Multivariate rational approximation
Ronald A.
DeVore;
Xiang Ming
Yu
161-169
Abstract: We estimate the error in approximating a function $f$ by rational functions of degree $ n$ in the norm of $ {L_q}(\Omega ),\,\Omega : = {[0,\,1]^d}$. Among other things, we prove that if $ f$ is in the Sobolev space $W_p^k(\Omega )$ and if $k/d - 1/p + 1/q > 0$, then $f$ can be approximated by rational functions of degree $n$ to an order $ O({n^{ - k/d}})$.
On periodic solutions of superlinear parabolic problems
Maria J.
Esteban
171-189
Abstract: In this paper we study the existence of positive nontrivial periodic solutions of semilinear parabolic problems. Most of the nonlinearities considered are of the superlinear type. Some bifurcation results are proved as well.
Quasilinear evolution equations and parabolic systems
Herbert
Amann
191-227
Abstract: It is shown that general quasilinear parabolic systems possess unique maximal classical solutions for sufficiently smooth initial values, provided the boundary conditions are ``time-independent''. Moreover it is shown that, in the autonomous case, these equations generate local semiflows on appropriate Sobolev spaces. Our results apply, in particular, to the case of prescribed boundary values (Dirichlet boundary conditions).
Regularization for $n$th-order linear boundary value problems using $m$th-order differential operators
D. A.
Kouba;
John
Locker
229-255
Abstract: Let $X$ and $Y$ denote real Hilbert spaces, and let $L:\,X \to Y$ be a closed densely-defined linear operator having closed range. Given an element $ y \in Y$, we determine least squares solutions of the linear equation $ Lx = y$ by using the method of regularization. Let $Z$ be a third Hilbert space, and let $T:\,X \to Z$ be a linear operator with $ \mathcal{D}(L) \subseteq \mathcal{D}(T)$. Under suitable conditions on $ L$ and $T$ and for each $\alpha \ne 0$, we show that there exists a unique element $ {x_\alpha } \in \mathcal{D}(L)$ which minimizes the functional ${G_\alpha }(x) = {\left\Vert {Lx - y} \right\Vert^2} + {\alpha ^2}{\left\Vert {Tx} \right\Vert^2}$, and the $ {x_\alpha }$ converge to a least squares solution ${x_0}$ of $Lx = y$ as $ \alpha \to 0$. We apply our results to the special case where $L$ is an $n$th-order differential operator in $X = {L^2}[a,b]$, and we regularize using for $ T$ an $m$th-order differential operator in ${L^2}[a,b]$ with $m \le n$. Using an approximating space of Hermite splines, we construct numerical solutions to $ Lx = y$ by the method of continuous least squares and the method of discrete least squares.
Derivation, $L\sp \Psi$-bounded martingales and covering conditions
Michel
Talagrand
257-291
Abstract: Let $(\Omega ,\,\Sigma ,\,P)$ be a complete probability space. Let ${({\Sigma _t})_{t \in J}}$ be a directed family of sub-$\sigma$-algebras of $\Sigma$. Let $ (\Phi,\,\Psi)$ be a pair of conjugate Young functions. We investigate the covering conditions that are equivalent to the essential convergence of ${L^\Psi }$-bounded martingales. We do not assume that either $\Phi$ or $\Psi$ satisfy the $ {\Delta _2}$ condition. We show that when $\Phi$ satisfies condition Exp, that is when there exists an $a > 0$ such that $\Phi (u) \leq \operatorname{exp} \,au$ for each $u \ge 0$, the essential convergence of $ {L^\Psi }$-bounded martingales is equivalent to the classical covering condition $ {V_\Phi }$. This covers in particular the classical case $\Psi (t) = t{(\operatorname{log} \,t)^ + }$. The growth condition Exp on $\Phi$ cannot be relaxed. When $ J$ contains a countable cofinite set, we show that the essential convergence of $ {L^\Psi }$-bounded martingales is equivalent to a covering condition $ {D_\Phi }$ (that is weaker than ${V_\Phi }$). When $\Phi$ fails condition Exp, condition $ {D_\Phi }$ is optimal. Roughly speaking, in the case of ${L^1 }$-bounded martingales, condition $ {D_\Phi }$ means that, locally, the Vitali condition with finite overlap holds. We also investigate the case where $J$ does not contain a countable cofinal set and $\Phi$ fails condition Exp. In this case, it seems impossible to characterize the essential convergence of $ {L^\Psi }$-bounded martingales by a covering condition. Using the Continuum Hypothesis, we also produce an example where all equi-integrable ${L^1 }$-bounded martingales, but not all $ {L^1 }$-bounded martingales, converge essentially. Similar results are also established in the derivation setting.
McKay quivers and extended Dynkin diagrams
Maurice
Auslander;
Idun
Reiten
293-301
Abstract: Let $k$ be an algebraically closed field and $G$ a finite nontrivial group whose order is not divisible by the characteristic of $k$. Associated with an $m$-dimensional representation of $ G$ is the McKay quiver, whose vertices correspond to the irreducible representations of $G$. We show that if $m = 2$, then the underlying graph of the separated McKay quiver is a finite union of extended Dynkin diagrams.
The normal closure of the coproduct of rings over a division ring
Wallace S.
Martindale
303-317
Abstract: Let $R = {R_1}\coprod {R_2}$ be the coproduct of $ \Delta$-rings $ {R_1}$ and ${R_2}$ with 1 over a division ring $\Delta ,\qquad {R_1} \ne \Delta ,\qquad {R_2} \ne \Delta$, with at least one of the dimensions ${({R_i}:\Delta )_r},\,{({R_i}:\Delta )_l},\,i = 1,\,2$, greater than 2. If ${R_1}$ and ${R_2}$ are weakly $1$-finite (i.e., one-sided inverses are two-sided) then it is shown that every $X$-inner automorphism of $ R$ (in the sense of Kharchenko) is inner, unless $ {R_1},\,{R_2}$ satisfy one of the following conditions: (I) each $ {R_i}$ is primary (i.e., $ {R_i} = \Delta + T,\,{T^2} = 0$), (II) one ${R_i}$ is primary and the other is $ 2$-dimensional, (III) char. $\Delta = 2$, one ${R_i}$ is not a domain, and one ${R_i}$ is $2$-dimensional. This generalizes a recent joint result with Lichtman (where each ${R_i}$ was a domain) and an earlier joint result with Montgomery (where each ${R_i}$ was a domain and $\Delta$ was a field).
Approximation theorems for Nash mappings and Nash manifolds
Masahiro
Shiota
319-337
Abstract: Let $0 \leq r < \infty$. A Nash function on ${\mathbf{R}^n}$ is a ${C^r}$ function whose graph is semialgebraic. It is shown that a ${C^r}$ Nash function is approximated by a ${C^\omega}$ Nash one in a strong topology defined in the same way as the usual topology on the space $\mathcal{S}$ of rapidly decreasing ${C^\infty}$ functions. A ${C^r}$ Nash manifold in ${\mathbf{R}^n}$ is a semialgebraic ${C^r}$ manifold. We also prove that a ${C^r}$ Nash manifold for $r \ge 1$ is approximated by a ${C^\omega}$ Nash manifold, from which we can classify all ${C^r}$ Nash manifolds by ${C^r}$ Nash diffeomorphisms.
On the existence and uniqueness of complex structure and spaces with ``few'' operators
Stanisław J.
Szarek
339-353
Abstract: We construct a $ 2n$-dimensional real normed space whose (Banach-Mazur) distance to the set of spaces admitting complex structure is of order $ {n^{1/2}}$, and two complex $n$-dimensional normed spaces which are isometric as real spaces, but whose complex Banach-Mazur distance is of order $n$. Both orders of magnitude are the largest possible. We also construct finite-dimensional spaces with the property that all ``well-bounded'' operators on them are ``rather small'' (in the sense of some ideal norm) perturbations of multiples of identity. We also state some ``metatheorem'', which can be used to produce spaces with various pathological properties.
Reflexivity and order properties of scalar-type spectral operators in locally convex spaces
P. G.
Dodds;
B.
de Pagter;
W.
Ricker
355-380
Abstract: One of the principal results of the paper is that each scalar-type spectral operator in the quasicomplete locally convex space $X$ is reflexive. The paper also studies in detail the relation between the theory of equicontinuous spectral measures in locally convex spaces and the order properties of equicontinuous Bade complete Boolean algebras of projections.
The trace of an action and the degree of a map
Daniel Henry
Gottlieb
381-410
Abstract: Two integer invariants of a fibration are defined: the degree, which generalizes the usual notion, and the trace. These numbers represent the smallest transfers for integral homology which can be constructed for the fibrations. Since every action gives rise to a fibration, we have the trace of an action. A list of properties of this trace is developed. This list immediately gives, in a mechanical way, new proofs and generalizations of theorems of Borsuk-Ulam, P. A. Smith, Conner and Floyd, Bredon, W. Browder, and G. Carlsson.
Long time asymptotics of the Korteweg-de Vries equation
Stephanos
Venakides
411-419
Abstract: We study the long time evolution of the solution to the Kortewegde Vries equation with initial data $\upsilon (x)$ which satisfy $\displaystyle \lim \limits_{x \to - \infty } \upsilon (x) = - 1,\qquad \lim \limits_{x \to + \infty } \upsilon (x) = 0$ We show that as $t \to \infty$ the step emits a wavetrain of solitons which asymptotically have twice the amplitude of the initial step. We derive a lower bound of the number of solitons separated at time $ t$ for $t$ large.
Symmetric semicontinuity implies continuity
Jaromir
Uher
421-429
Abstract: The main result of this paper shows that, for any function, symmetric semicontinuity on a measurable set $E$ implies continuity a.e. in $ E$ and, similarly, that symmetric semicontinuity on a set residual in $ R$ implies continuity on a set residual in $R$. These propositions are used to prove more precise versions of the fundamental connections between symmetric and ordinary differentiability.
Compensated compactness and general systems of conservation laws
Ronald J.
DiPerna
383-420
Abstract: We outline a general program and present some new results dealing with oscillations in weakly convergent solution sequences to systems of conservation laws. The analysis employs the Young measure and the Tartar-Murat theory of compensated compactness and deals with systems of hyperbolic and elliptic type.
Algorithmic solution of extremal digraph problems
W. G.
Brown;
P.
Erdős;
M.
Simonovits
421-449
Abstract: For a given family $\mathcal{L}$ of digraphs, we study the "extremal" digraphs on $n$ vertices containing no member of $\mathcal{L}$, and having the maximum number of arcs, $ \operatorname{ex} (n,\mathcal{L})$. We resolve conjectures concerning the set $\{ {\lim _{n \to \infty }}(\operatorname{ex} (n,\mathcal{L})/{n^2})\}$ as $ \mathcal{L}$ ranges over all possible families, and describe a "finite" algorithm that can determine, for any $\mathcal{L}$, all matrices $A$ for which a sequence $\{ A(n)\}$ of "matrix digraphs" is asymptotically extremal ($A(n)$ contains no member of $\mathcal{L}$ and has $ \operatorname{ex} (n,\mathcal{L}) + o({n^2})$ arcs as $ n \to \infty$.) Résumé. Pour une famille donnée, $\mathcal{L}$, de digraphes, on étudie les digraphes "extrémaux" à $n$ sommets qui ne contiennent aucun membre de $ \mathcal{L}$, et qui possèdent le nombre maximal d'arêtes, $ \operatorname{ex} (n,\mathcal{L})$. On résolue des conjectures qui concernent l'ensemble $\{ {\lim _{n \to \infty }}(\operatorname{ex} (n,\mathcal{L})/{n^2})\}$ où $\mathcal{L}$ soit une famille quelconque, et on présente un algorithme "fini" qui peut déterminer, pour chaque $ \mathcal{L}$, toute matrice $A$ pour laquelle une suite $\{ A(n)\}$ de "digraphes matriciels" est extrémale asymptotiquement ($ A(n)$ ne contient aucun membre de $ \mathcal{L}$ et possède $\operatorname{ex} (n,\mathcal{L}) + o({n^2})$ arêtes lorsque $ n \to \infty$.)
Les groupes $\omega$-stables de rang fini
Daniel
Lascar
451-462
Abstract: We prove that a group $G$ which is $\omega$-stable of finite Morley rank is nonmultidimensional. If moreover it is connected and does not have any infinite normal abelian definable subgroup, then it is isomorphic to $ \Pi {H_i}/K$, where the $ {H_i}$ are ${\omega _1}$-categorical groups and $K$ is a finite group.
Linearizing certain reductive group actions
H.
Bass;
W.
Haboush
463-482
Abstract: Is every algebraic action of a reductive algebraic group $ G$ on affine space ${{\mathbf{C}}^n}$ equivalent to a linear action? The "normal linearization theorem" proved below implies that, if each closed orbit of $G$ is a fixed point, then ${{\mathbf{C}}^n}$ is $G$-equivariantly isomorphic to $ {({{\mathbf{C}}^n})^G} \times {{\mathbf{C}}^m}$ for some linear action of $ G$ on ${{\mathbf{C}}^m}$.
Graphic flows and multiple disjointness
Joseph
Auslander;
Nelson
Markley
483-499
Abstract: A graphic flow is a totally minimal flow for which the only minimal subsets of the product flow are the graphs of the powers of the generating homeomorphism. The POD flows of Furstenberg, Keynes, and Shapiro [5] are examples of graphic flows. Graphic flows are in some ways analogous to ergodic systems with minimal self-joinings [11]. Various disjointness results concerning graphic flows and their powers are obtained, and their regular factors are determined.
Radial limits of $M$-subharmonic functions
David
Ullrich
501-518
Abstract: "$ M$-subharmonic" functions are defined in the unit ball of ${{\mathbf{C}}^n}$. Their basic properties are developed, leading to the following generalization of a theorem of Littlewood: An $M$-subharmonic function such that its restrictions to spheres centered at the origin are bounded in $ {L^1}$ must have radial limits almost everywhere on the unit sphere.
A Dowker product
Amer
Bešlagić
519-530
Abstract: $\diamondsuit$ implies that there is a (normal) countably paracompact space $X$ such that ${X^2}$ is normal and not countably paracompact.
Stationary solutions of chemotaxis systems
Renate
Schaaf
531-556
Abstract: The Keller-Segel Model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance (Chemotaxis). This paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns. Examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response.
Representation of set valued operators
Nikolaos S.
Papageorgiou
557-572
Abstract: In this paper we prove representation theorems for set valued additive operators acting on the spaces $L_X^1(X = {\text{separable Banach space)}}$, ${L^1}$ and $ {L^\infty }$. Those results generalize well-known ones for single valued operators and among them the celebrated Dunford-Pettis theorem. The properties of these representing integrals are studied. We also have a differentiability result for multifunctions analogous to the one that says that an absolutely continuous function from a closed interval into a Banach space with the Radon-Nikodým property is almost everywhere differentiable and also it is the primitive of its strong derivative. Finally we have a necessary and sufficient condition for the set of integrable selectors of a multifunction to be $ w$-compact in $ L_X^1$. This result is a new very general result about $w$-compactness in the Lebesgue-Bochner space $ L_X^1$.
Joint continuity of division of smooth functions. I. Uniform Lojasiewicz estimates
Mark Alan
Mostow;
Steven
Shnider
573-583
Abstract: In this paper we study the question of the existence of a continuous inverse to the multiplication mapping $(f,g) \to (fg,g)$ defined on pairs of ${C^\infty }$ functions on a manifold $M$. Obviously, restrictions must be imposed on the domain of such an inverse. This leads us to the study of a modified problem: Find an appropriate domain for the inverse of $(f,G) \to (f(p \circ G),G)$, where $ G$ is a ${C^\infty }$ mapping of the manifold $M$ into an analytic manifold $ N$ and $p$ is a fixed analytic function on $ N$. We prove a theorem adequate for application to the study of inverting the mapping $(A,X) \to (A,AX)$, where $ X$ is a vector valued ${C^\infty }$ function and $A$ is a square matrix valued ${C^\infty }$ function on $M$ whose determinant may vanish on a nowhere dense set.
Joint continuity of division of smooth functions. II. The distance to a Whitney stratified set from a transversal submanifold
Mark Alan
Mostow
585-594
Abstract: Let $S$ be a closed set in ${{\mathbf{R}}^m}$, and let a ${C^1}$ Whitney stratification of $ S$ be given. (Actually, only Whitney (a)-regularity is needed.) Let $ f:{{\mathbf{R}}^n} \to {{\mathbf{R}}^m}$ be a ${C^1}$ map transversal to all the strata. Assume that the image of $f$ intersects $S$. Then for each compact set $K$ in $ {{\mathbf{R}}^n}$, the Euclidean distances $\rho (x,{f^{ - 1}}(S))$ and $\rho (f(x),S)$, for $x$ in $K$, are bounded by constant multiples of each other. The bounding constants can be chosen to work for all maps $g$ which are close enough to $f$ in a ${C^1}$ sense on a neighborhood of $ K$. This result is used in part I (written jointly with S. Shnider) to prove a result on the joint continuity of the division of smooth functions [MS].
Polyhedral resolutions of algebraic varieties
James A.
Carlson
595-612
Abstract: We give a method for constructing relatively small smooth simplicial resolutions of singular projective algebraic varieties. For varieties of dimension $n$, at most $n$ applications of the basic process yields a resolution of combinatorial dimension at most $ n$. The object so obtained may be used to compute the mixed Hodge stucture of the underlying variety.
Embeddings of locally connected compacta
Gerard A.
Venema
613-627
Abstract: Let $X$ be a $k$-dimensional compactum and $f:X \to {M^n}$ a map into a piecewise linear $ n$-manifold, $n \geqslant k + 3$. The main result of this paper asserts that if $X$ is locally $(2k - n)$-connected and $f$ is $(2k - n + 1)$-connected, then $ f$ is homotopic to a CE equivalence. In particular, every $ k$-dimensional, $ r$-connected, locally $ r$-connected compactum is CE equivalent to a compact subset of ${{\mathbf{R}}^{2k - r}}$ as long as $r \leqslant k - 3$.
Nonmonotoneity of Picard principle
Mitsuru
Nakai;
Toshimasa
Tada
629-644
Abstract: Two nonnegative ${C^\infty }$ functions $P(z)$ and $Q(z)$ on the punctured unit disk $0 < \vert z\vert \leqslant 1$ are constructed such that $ Q(z) \leqslant P(z)$ and there exists only one Martin minimal boundary point for the equation $ \Delta u = Pu$ over $ z = 0$ and, nevertheless, there exist exactly two Martin minimal boundary points for the equation $ \Delta u = Qu$ over $ z = 0$.
An irreducible representation of a symmetric star algebra is bounded
Subhash J.
Bhatt
645-652
Abstract: A $ {\ast}$-algebra $ A$ is called symmetric if $ (1 + {x^{\ast}}x)$ is invertible in $A$ for each $x$ in $A$. An irreducible hermitian representation of a symmetric ${\ast}$-algebra $A$ maps $A$ onto an algebra of bounded operators.
Isometric homotopy in codimension two
John Douglas
Moore
653-663
Abstract: This article investigates the structure of the space of isometric immersions from a simply connected $n$-dimensional Riemannian manifold with positive sectional curvatures into $(n + 2)$-dimensional Euclidean space ${E^{n + 2}}$. It is proven that if $n \geqslant 4$ and ${M^n}$ is such a manifold which admits a ${C^\infty }$ isometric immersion as a hypersurface in ${E^{n + 1}}$, then any ${C^\infty }$ isometric immersion from $ {M^n}$ into ${E^{n + 2}}$ is $ {C^{2n - 4}}$ homotopic through isometric immersions to an immersion whose image lies in some hyperplane.
Local and global envelopes of holomorphy of domains in ${\bf C}\sp 2$
Eric
Bedford
665-674
Abstract: A criterion is given for a smoothly bounded domain $D \subset {{\mathbf{C}}^2}$ to be locally extendible to a neighborhood of a point ${z_0} \in \partial D$. (This result may also be formulated in terms of extension of CR functions on $ \partial D$.) This is related to the envelope of holomorphy of the semitubular domain $\displaystyle \Omega (\Phi ) = \{ (z,w) \in {{\mathbf{C}}^2}:\operatorname{Re} w + {r^k}\Phi (\theta ) < 0\} ,$ where $r = \vert z\vert$, $ \theta = \arg (z)$. Necessary and sufficient conditions are given for the envelope of holomorphy of $\Omega (\Phi )$ to be ${{\mathbf{C}}^2}$. These conditions are equivalent to the existence of a subharmonic minorant for ${r^k}\Phi (\theta )$.
Combinatorial lower bound arguments for deterministic and nondeterministic Turing machines
Wolfgang
Maass
675-693
Abstract: We introduce new techniques for proving quadratic lower bounds for deterministic and nondeterministic $ 1$-tape Turing machines (all considered Turing machines have an additional one-way input tape). In particular, we derive for the simulation of $2$-tape Turing machines by $1$-tape Turing machines an optimal quadratic lower bound in the deterministic case and a nearly optimal lower bound in the nondeterministic case. This answers the rather old question whether the computing power of the considered types of Turing machines is significantly increased when more than one tape is used (problem Nos. 1 and 7 in the list of Duris, Galil, Paul, Reischuk [3]). Further, we demonstrate a substantial superiority of nondeterminism over determinism and of co-nondeterminism over nondeterminism for $ 1$-tape Turing machines.
A dynamical meaning of fractal dimension
Steve
Pelikan
695-703
Abstract: When two attractors of a dynamical system have a common basin boundary $B$, small changes in initial conditions which lie near $B$ can result in radically different long-term behavior of the trajectory. A quantitative description of this phenomenon is obtained in terms of the fractal dimension of the basin boundary $B$.
Noncommutative algebras of dimension three over integral schemes
Rick
Miranda;
Mina
Teicher
705-712
Abstract: In this article we describe the algebraic data which is equivalent to giving an associative, noncommutative algebra ${\mathcal{O}_X}$ over an integral $k$-scheme $Y$ (where $k$ is an algebraically closed field of characteristic $\ne 3$), which is locally free of rank $3$. The description allows us to conclude that, essentially, all such are locally upper triangular $2 \times 2$ matrices, with degenerations of a restricted form allowed.
A canonical partition of the periodic orbits of chaotic maps
Kathleen T.
Alligood
713-719
Abstract: We show that the periodic orbits of an area-contracting horseshoe map can be partitioned into subsets of orbits of minimum period $k,\;2k,\;4k,\;8k \ldots $, for some positive integer $k$. This partition is natural in the following sense: for any parametrized area-contracting map which forms a horseshoe, the orbits in one subset of the partition are contained in a single component of orbits in the full parameter space. Furthermore, prior to the formation of the horseshoe, this component contains attracting orbits of minimum period ${2^m}k$, for each nonnegative integer $ m$.
Rational homotopy of spaces of maps into spheres and complex projective spaces
Jesper Michael
Møller;
Martin
Raussen
721-732
Abstract: We investigate the rational homotopy classification problem for the components of some function spaces with $ {S^n}$ or ${\mathbf{C}}{P^n}$ as target space.
Boundary behavior of ${\rm BMO}(B\sb n)$
Paula A.
Russo
733-740
Abstract: If $f$ is a holomorphic function of bounded mean oscillation in the unit ball of ${{\mathbf{C}}^n}$, then it has radial limits at almost every point of the boundary of the ball. The question remains as to how nicely one can expect this function to behave on subsets of the boundary of zero measure. For example, there is a holomorphic BMO function in the ball that has a finite radial limit at no point of the $n$-torus. We show here that this is not an isolated phenomenon; there exists at least one other $ n$-dimensional submanifold of the boundary of the ball with this same behavior.
Duality theorems in deformation theory
Hubert
Goldschmidt
1-49
Abstract: We give a unified treatment of the construction of the Calabi sequence, which is a resolution of the sheaf of Killing vector fields on a Riemannian manifold of constant curvature, and of the resolution of the sheaf of conformal Killing vector fields on a conformally flat Riemannian manifold of dimension $ \geqslant 3$ introduced in [7]. We also explain why the latter resolution is selfadjoint and associate to certain geometric structures selfadjoint resolutions of their infinitesimal automorphisms.
$(n-1)$-axial ${\rm SO}(n)$ and ${\rm SU}(n)$ actions on homotopy spheres
R. D.
Ball
51-79
Abstract: Let $G(n) = O(n)$ or $U(n)$ and $ SG(n) = SO(n)$ or $ SU(n)$. For each integer $m \geqslant 1$ a family $\{ {S_{\gamma ,\sigma }}:\gamma \in H,\sigma \in K\}$ of $(n - 1)$-axial $SG(n)$ homotopy spheres ${S_{\gamma ,\sigma }}$ is constructed. Each ${S_{\gamma ,\sigma }}$ has fixed point set of dimension $ (m - 1) \geqslant 0$ and orbit space of dimension $r = \tfrac{1} {2}n(n - 1) + (m - 1)$ (resp. $r = {(n - 1)^2} + m - 1$) if $SG(n) = SO(n)$ (resp. $SU(n)$). $H$ is ${\pi _{r - 1}}(SG(n)/G(n - 1))$. $K$ is trivial if $SG(n) = SO(n)$ and is a homotopy theoretically defined subgroup of sections of an ${S^2}$ bundle depending only on $m$ and $n$ if $SG(n) = SU(n)$. Assume that $ m$ and $n$ satisfy the mild restriction $ \S5$, (1). It is shown that the above family is universal for $(n - 1)$-axial $SG(n)$ homotopy spheres and provides a classification analogous to the classification of fibre bundles: for each $(n - 1)$-axial $SG(n)$ homotopy sphere $\Sigma$ there is a ${S_{\gamma ,\sigma }}$ and a unique equivariant stratified map $\Sigma \to {S_{\gamma ,\sigma }}$. $ \Sigma$ is equivariantly diffeomorphic to the pullback of ${S_{\gamma ,\sigma }}$ via the map $ B(\Sigma ) \to B({S_{\gamma ,\sigma }})$ of orbit spaces. If $SG(n) = SO(n)$ then $\gamma$ is unique (and $ \sigma = 1$). If $SG(n) = SU(n)$ then $\gamma$ is unique modulo the image of $\displaystyle {\pi _{r - 1}}S(U(n - 2) \times U(2))/U(k - 1) \times U(1)\quad {\text{in}}\;H.$ An example is given showing that the differentiable structure of the underlying smooth manifold of $ {S_{\gamma ,\sigma }}$ may be exotic.
${\rm Aut}(F)\to{\rm Aut}(F/F'')$ is surjective for free group $F$ of rank $\geq 4$
Seymour
Bachmuth;
Horace Y.
Mochizuki
81-101
Abstract: In this article, it is shown that the group of automorphisms of the free metabelian group $\Phi (n)$ of rank $ n \geqslant 4$ is not only finitely generated but in fact every automorphism of $ \Phi (n)$ is induced by an automorphism of the free group of the same rank $ n$. This contrasts sharply with the authors' earlier result [4] that any set of generators of the group of automorphisms of the free metabelian group $\Phi (3)$ of rank $3$ contains infinitely many automorphisms which are not induced by an automorphism of the free group of rank $3$.
A commutator theorem and weighted BMO
Steven
Bloom
103-122
Abstract: The main result of this paper is a commutator theorem: If $\mu$ and $\lambda$ are ${A_p}$ weights, then the commutator $H$, ${M_b}$ is a bounded operator from ${L^p}(\mu )$ into $ {L^p}(\lambda )$ if and only if $b \in {\operatorname{BMO} _{{{(\mu {\lambda ^{ - 1}})}^{1/p}}}}$. The proof relies heavily on a weighted sharp function theorem. Along the way, several other applications of this theorem are derived, including a doubly-weighted $ {L^p}$ estimate for BMO. Finally, the commutator theorem is used to obtain vector-valued weighted norm inequalities for the Hilbert transform.
On the decomposition numbers of the finite general linear groups. II
Richard
Dipper
123-133
Abstract: Let $q$ be a prime power, $G = {\operatorname{GL} _n}(q)$ and let $ r$ be a prime not dividing $ q$. Using representations of Hecke algebras associated with symmetric groups over arbitrary fields, the $r$-modular irreducible $G$-modules are classified. The decomposition matrix $D$ of $G$ (with respect to $r$) is partly described in terms of decomposition matrices of Hecke algebras, and it is shown that $ D$ is lower unitriangular, provided the irreducible characters and irreducible Brauer characters of $G$ are suitably ordered.
Singular integrals and approximate identities on spaces of homogeneous type
Hugo
Aimar
135-153
Abstract: In this paper we give conditions for the ${L^2}$-boundedness of singular integrals and the weak type $(1,1)$ of approximate identities on spaces of homogeneous type. Our main tools are Cotlar's lemma and an extension of a theorem of Zó.
Group-graded rings and duality
Declan
Quinn
155-167
Abstract: We give an alternative construction of the duality between finite group actions and group gradings on rings which was shown by Cohen and Montgomery in [1]. This duality is then used to extend known results on skew group rings to corresponding results for large classes of group-graded rings. Finally we modify the construction slightly to handle infinite groups.
Quasiconformal and bi-Lipschitz homeomorphisms, uniform domains and the quasihyperbolic metric
Gaven J.
Martin
169-191
Abstract: Let $D$ be a proper subdomain of ${R^n}$ and ${k_D}$ the quasihyperbolic metric defined by the conformal metric tensor $d{\overline s ^2} = \operatorname{dist} {(x,\partial D)^{ - 2}}d{s^2}$. The geodesics for this and related metrics are shown, by purely geometric methods, to exist and have Lipschitz continuous first derivatives. This is sharp for ${k_D}$; we also obtain sharp estimates for the euclidean curvature of such geodesics. We then use these results to prove a general decomposition theorem for uniform domains in ${R^n}$, in terms of embeddings of bi-Lipschitz balls. We also construct a counterexample to the higher dimensional analogue of the decomposition theorem of Gehring and Osgood.
Classification of semisimple algebraic monoids
Lex E.
Renner
193-223
Abstract: Let $X$ be a semisimple algebraic monoid with unit group $G$. Associated with $E$ is its polyhedral root system $(X,\Phi ,C)$, where $X = X(T)$ is the character group of the maximal torus $T \subseteq G$, $\Phi \subseteq X(T)$ is the set of roots, and $C = X(\overline T )$ is the character monoid of $ \overline T \subseteq E$ (Zariski closure). The correspondence $E \to (X,\Phi ,C)$ is a complete and discriminating invariant of the semisimple monoid $E$, extending the well-known classification of semisimple groups. In establishing this result, monoids with preassigned root data are first constructed from linear representations of $G$. That done, we then show that any other semisimple monoid must be isomorphic to one of those constructed. To do this we devise an extension principle based on a monoid analogue of the big cell construction of algebraic group theory. This, ultimately, yields the desired conclusions.
Brownian motion with polar drift
R. J.
Williams
225-246
Abstract: Consider a strong Markov process ${X^0}$ that has continuous sample paths in ${R^d}(d \geqslant 2)$ and the following two properties. (1) Away from the origin $ {X^0}$ behaves like Brownian motion with a polar drift given in spherical polar coordinates by $\mu (\theta )/2r$. Here $\mu$ is a bounded Borel measurable function on the unit sphere in ${R^d}$, with average value $\overline \mu$. (2) ${X^0}$ is absorbed at the origin. It is shown that $ {X^0}$ reaches the origin with probability zero or one as $\overline \mu \geqslant 2 - d$ or $< 2 - d$. Indeed, ${X^0}$ is transient to $+ \infty$ if $\overline \mu > 2 - d$ and null recurrent if $\bar \mu = 2 - d$. Furthermore, if $\bar \mu < 2 - d$ (i.e., ${X^0}$ reaches the origin), then $ {X^0}$ does not approach the origin in any particular direction. Indeed, there is a single Martin boundary point for ${X^0}$ at the origin. The question of the existence and uniqueness of a strong Markov process with continuous sample paths in ${R^d}$ that behaves like ${X^0}$ away from the origin, but spends zero time there (in the sense of Lebesgue measure), is also resolved here.
Algebraic and etale $K$-theory
William G.
Dwyer;
Eric M.
Friedlander
247-280
Abstract: We define etale $ K$-theory, interpret various conjectures of Quillen and Lichtenbaum in terms of a map from algebraic $K$-theory to etale $K$-theory, and then prove that this map is surjective in many cases of interest.
The extended module of indecomposables for mod $2$ finite $H$-spaces
R. M.
Kane
281-304
Abstract: We construct a generalization of the usual module of indecomposables for the $\bmod 2$ cohomology of a finite $H$-space. Structure theorems are obtained regarding how the Steenrod algebra acts on this module.
On the solution of certain systems of partial difference equations and linear dependence of translates of box splines
Wolfgang
Dahmen;
Charles A.
Micchelli
305-320
Abstract: This paper is concerned with solving systems of partial difference equations which arise when studying linear dependence of translates of box splines.
On the restriction of the Fourier transform to a conical surface
Bartolome Barcelo
Taberner
321-333
Abstract: Let $\Gamma$ be the surface of a circular cone in $ {{\mathbf{R}}^3}$. We show that if $1 \leqslant p < 4/3$, $1/q = 3(1 - 1/p)$ and $f \in {L^p}({{\mathbf{R}}^3})$, then the Fourier transform of $f$ belongs to ${L^q}(\Gamma ,d\sigma )$ for a certain natural measure $\sigma$ on $\Gamma$. Following P. Tomas we also establish bounds for restrictions of Fourier transforms to conic annuli at the endpoint $p = 4/3$, with logarithmic growth of the bound as the thickness of the annulus tends to zero.
Multiparameter maximal functions along dilation-invariant hypersurfaces
Hasse
Carlsson;
Peter
Sjögren;
Jan-Olov
Strömberg
335-343
Abstract: Consider the hypersurface ${x_{n + 1}} = \Pi _1^nx_i^{{\alpha _i}}$ in ${{\mathbf{R}}^{n + 1}}$. The associated maximal function operator is defined as the supremum of means taken over those parts of the surface lying above the rectangles $\{ 0 \leqslant {x_i} \leqslant {h_i},\;i = 1, \ldots ,n\}$. We prove that this operator is bounded on ${L^p}$ for $p > 1$. An analogous result is proved for a quadratic surface in $ {{\mathbf{R}}^3}$.
Saturation of the closed unbounded filter on the set of regular cardinals
Thomas J.
Jech;
W. Hugh
Woodin
345-356
Abstract: For any $\alpha < {\kappa ^ + }$, the following are equiconsistent: (a) $\kappa$ is measurable of order $\alpha$, (b) $\kappa$ is $\alpha$-Mahlo and the filter $ {\mathbf{C}}[\operatorname{Reg} ]$ is saturated.
$L\sp p$ estimates for Schr\"odinger evolution equations
M.
Balabane;
H. A.
Emamirad
357-373
Abstract: We prove that for Cauchy data in $ {L^1}({{\mathbf{R}}^n})$, the solution of a Schrödinger evolution equation with constant coefficients of order $ 2m$ is uniformly bounded for $t \ne 0$, with bound $(1 + \vert t{\vert^{ - c}})$, where $ c$ is an integer, $c > n/2m - 1$. Moreover it belongs to ${L^q}({{\mathbf{R}}^n})$ if $q > q(m,n)$, with its ${L^p}$ norm bounded by $ ({L^p},{L^q})$ estimates. On the other hand, we prove that for Cauchy data in $ {L^p}({{\mathbf{R}}^n})$, such a Cauchy problem is well posed as a distribution in the $t$-variable with values in $ {L^p}({{\mathbf{R}}^n})$, and we compute the order of the distribution. We apply these two results to the study of Schrödinger equations with potential in $ {L^p}({{\mathbf{R}}^n})$. We give an estimate of the resolvent operator in that case, and prove an asymptotic boundedness for the solution when the Cauchy data belongs to a subspace of $ {L^p}({{\mathbf{R}}^n})$.
Existence in the large for Riemann problems for systems of conservation laws
Michael
Sever
375-381
Abstract: An existence theorem in the large is obtained for the Riemann problem for nonlinear systems of conservation laws. Our principal assumptions are strict hyperbolicity, genuine nonlinearity in the strong sense, and the existence of a convex entropy function. The entropy inequality is used to obtain an a priori estimate of the strengths of the shocks and refraction waves forming a solution; existence of such a solution then follows by an application of finite-dimensional degree theory. The case of a single degenerate field is also included, with an additional assumption on the existence of Riemann invariants.
Structure and dimension of global branches of solutions to multiparameter nonlinear equations
J.
Ize;
I.
Massabò;
J.
Pejsachowicz;
A.
Vignoli
383-435
Abstract: This paper is concerned with the topological dimension of global branches of solutions appearing in different problems of Nonlinear Analysis, in particular multiparameter (including infinite dimensional) continuation and bifurcation problems. By considering an extension of the notion of essential maps defined on sets and using elementary point set topology, we are able to unify and extend, in a selfcontained fashion, most of the recent results on such problems. Our theory applies whenever any generalized degree theory with the boundary dependence property may be used, but with no need of algebraic structures. Our applications to continuation and bifurcation follow from the nontriviality of a local invariant, in the stable homotopy group of a sphere, and give information on the local dimension and behavior of the sets of solutions, of bifurcation points and of continuation points.
Volumes of tubes about complex submanifolds of complex projective space
Alfred
Gray
437-449
Abstract: Simple formulas in terms of Chern classes are given for the volume of a tube about a Kähler submanifold of a space of constant holomorphic sectional curvature. A comparison theorem which generalizes these formulas is also given. Tubes about complete intersections in complex projective space are studied in detail.
Propagation of singularities for nonstrictly hyperbolic semilinear systems in one space dimension
Lucio
Micheli
451-485
Abstract: We consider the creation and propagation of singularities in the solutions of semilinear nonstrictly hyperbolic systems in one space dimension when the initial data has jump discontinuities. We show that singularities travelling along characteristics can branch at points of degeneracy of the vector fields on all other forward characteristics. We prove a lower bound for the strength of these new singularities, and we give an example showing that our result cannot be improved in general.
On the existence and classification of extensions of actions on submanifolds of disks and spheres
Amir
Assadi;
William
Browder
487-502
Abstract: Given a $ G$-action $\psi :G \times W \to W$ and an embedding $W \subset {D^n}$, when is it possible to find a $ G$-action $\phi :G \times {D^n} \to D$ such that $ {D^n} - W$ is $ G$-free? Sufficient conditions of cohomological nature for the existence of such extensions are given and the extensions are classified. This leads to the characterization of the stationary point sets and classification of semifree actions on disks up to $G$-diffeomorphism under suitable dimension hypotheses.
A note on automorphic forms of weight one and weight three
Peter F.
Stiller
503-518
Abstract: In this paper the author develops an interesting relationship between classical automorphic forms of weights one and three, and the solutions of certain second order differential equations related to elliptic (modular) surfaces. In particular for a cusp form of weight three, it is shown that the special values of the associated Dirichlet series can be determined from the periods of an inhomogeneous differential equation, or what is the same thing, the monodromy of an associated third order differential equation. Explicit examples are provided for principal congruence subgroups $\Gamma (N)$ with $N \equiv 0\,\operatorname{mod} \,4$.
The cuspidal group and special values of $L$-functions
Glenn
Stevens
519-550
Abstract: The structure of the cuspidal divisor class group is investigated by relating this structure to arithmetic properties of special values of $L$-functions of weight two Eisenstein series. A new proof of a theorem of Kubert (Proposition 3.1) concerning the group of modular units is derived as a consequence of the method. The key lemma is a nonvanishing result (Theorem 2.1) for values of the ``$L$-function'' attached to a one-dimensional cohomology class over the relevant-congruence subgroup. Proposition 4.7 provides data regarding Eisenstein series and associated subgroups of the cuspidal divisor class group which the author hopes will simplify future calculations in the cuspidal group.
Subgraphs of random graphs
D. H.
Fremlin;
M.
Talagrand
551-582
Abstract: Let $\Delta \subseteq {[\omega ]^2}$ be an undirected graph on $\omega$, and let $u \in [0,\,1]$. Following P. Erdös and A. Hajnal, we write $(\omega ,\,2,\,u) \Rightarrow \Delta$ to mean: whenever ${E_1} \subseteq [0,\,1]$ is a measurable set of Lebesgue measure at least $u$ for every $I \in {[\omega ]^2}$, then there is some $t \in [0,\,1]$ such that $ \Delta$ appears in the graph $ {\Gamma _t} = \{ I:\,t \in {E_I}\}$ in the sense that there is a strictly increasing function $ f:\,\omega \to \omega$ such that $\{ f(i),\,f(j)\} \in {\Gamma _t}$ whenever $\{ i,\,j\} \in \Delta$. We give an algorithm for determining when $(\omega ,\,2,\,u) \Rightarrow \Delta$ for finite $\Delta$, and we show that for infinite $ \Delta ,\,(\omega ,\,2,\,u) \Rightarrow \Delta$ if there is a $\upsilon < u$ such that $ (\omega ,\,2,\,\upsilon ) \Rightarrow {\Delta ^\prime }$ for every finite $ \Delta^{\prime} \subseteq \Delta$. Our results depend on a new condition, expressed in terms of measures on $ \beta\omega$, sufficient to imply that $\Delta$ appears in $\Gamma$ (Theorem 2F), and enable us to identify the extreme points of some convex sets of measures (Theorem 5H).
A ``Tits-alternative'' for subgroups of surface mapping class groups
John
McCarthy
583-612
Abstract: It has been observed that surface mapping class groups share various properties in common with the class of linear groups (e.g., $ [\mathbf{BLM},\,\mathbf{H}]$). In this paper, the known list of such properties is extended to the ``Tits-Alternative'', a property of linear groups established by J. Tits $[\mathbf{T}]$. In fact, we establish that every subgroup of a surface mapping class group is either virtually abelian or contains a nonabelian free group. In addition, in order to establish this result, we develop a theory of attractors and repellers for the action of surface mapping classes on Thurston's projective lamination spaces $ [\mathbf{Th1}]$. This theory generalizes results known for pseudo-Anosov mapping classes $ [\mathbf{FLP}]$.
Convergence of conditional expectations and strong laws of large numbers for multivalued random variables
Fumio
Hiai
613-627
Abstract: Fatou's lemmas and Lebesgue's convergence theorems are established for multivalued conditional expectations of random variables having values in the closed subsets of a separable Banach space. Strong laws of large numbers are also given for such multivalued random variables.
Elements of finite order for finite monadic Church-Rosser Thue systems
Friedrich
Otto
629-637
Abstract: A Thue system $ T$ over $\Sigma$ is said to allow nontrivial elements of finite order, if there exist a word $u \in {\Sigma ^ \ast }$ and integers $n \ge 0$ and $k \ge 1$ such that $u \nleftrightarrow \,_T^ \ast \lambda$ and ${u^{n + k}} \leftrightarrow \,_T^ \ast {u^n}$. Here the following decision problem is shown to be decidable: Instance. A finite, monadic, Church-Rosser Thue system $T$ over $\Sigma$. Question. Does $ T$ allow nontrivial elements of finite order? By a result of Muller and Schupp this implies in particular that given a finite monadic Church-Rosser Thue system $T$ it is decidable whether the monoid presented by $T$ is a free group or not.
Completely unstable dynamical systems
Sudhir K.
Goel;
Dean A.
Neumann
639-668
Abstract: We associate with the $ {C^r}\,(r\, \ge \,1)$ dynamical system $\phi$ on an $m$-manifold $M$, the orbit space $M/\phi$, defined to be the set of orbits of $\phi$ with the quotient topology. If $\phi$ is completely unstable, $ M/\phi$ turns out to be a ${C^r}\,(m\, - \,1)$-nonseparated manifold. It is known that for a completely unstable flow $ \phi$ on a contractible manifold $ M,\,M/\phi$ is Hausdorff if and only if $\phi$ is parallelizable. In general, we place an order on the non-Hausdorff points of $M/\phi$ (essentially) by setting $\bar p < \bar q$ if and only if ${\pi ^{ - 1}}(\bar q) \subseteq {J^ + }({\pi ^{ - 1}}(\bar p))$. Our result is that $(M,\,\phi)$ is topologically equivalent to $(M^{\prime},\,\phi ^{\prime})$ if and only if $ M/\phi$ is order isomorphic to $M^{\prime}/\phi ^{\prime}$.
Maps between surfaces
Richard
Skora
669-679
Abstract: The Uniqueness Conjecture states if $\phi ,\,\psi :\,M \to N$ are $d$-fold, simple, primitive, branched coverings between closed, connected surfaces, then $ \phi$ and $\psi$ are equivalent. The Uniqueness Conjecture is proved in the case that $M$ and $N$ are nonorientable and $N = \mathbf{R}{P^2}$ or Klein bottle. It is also proved in the case that $M$ and $N$ are nonorientable and $d/2 < d\chi (N) - \chi (M)$. As an application it is shown that two $d$-fold, branched coverings $\phi :{M_1} \to N,\,\psi :{M_2} \to N$ between closed, connected surfaces are branched cobordant.
Arbitrarily large continuous algebras on one generator
Jiří
Adámek;
Václav
Koubek;
Evelyn
Nelson;
Jan
Reiterman
681-699
Abstract: Generation of order-continuous algebras is investigated for various concepts of continuity. For the continuity understood as the preservation of joins of countably-directed sets, arbitrarily large infinitary continuous algebras on one generator are constructed.
Projective modules in the category ${\scr O}\sb S$: self-duality
Ronald S.
Irving
701-732
Abstract: Given a parabolic subalgebra $ {\mathfrak{p}_S}$ of a complex, semisimple Lie algebra $ \mathfrak{g}$, there is a naturally defined category ${\mathcal{O}_S}$ of $ \mathfrak{g}$-modules which includes all the $ \mathfrak{g}$-modules induced from finite-dimensional $ {\mathfrak{p}_S}$-modules. This paper treats the question of whether certain projective modules in $ {\mathcal{O}_S}$ are isomorphic to their dual modules. The projectives in question are the projective covers of those simple modules occurring in the socles of generalized Verma modules. Their self-duality is proved in a number of cases, and additional information is obtained on the generalized Verma modules.
Projective modules in the category ${\scr O}\sb S$: Loewy series
Ronald S.
Irving
733-754
Abstract: Let $\mathfrak{g}$ be a complex, semisimple Lie algebra with a parabolic subalgebra ${\mathfrak{p}_S}$. The Loewy lengths and Loewy series of generalized Verma modules and of their projective covers in $ {\mathcal{O}_S}$ are studied with primary emphasis on the case in which ${\mathfrak{p}_S}$ is a Borel subalgebra and ${\mathcal{O}_S}$ is the category $\mathcal{O}$. An examination of the change in Loewy length of modules under translation leads to the calculation of Loewy length for Verma modules and for self-dual projectives in $\mathcal{O}$, assuming the Kazhdan-Lusztig conjecture (in an equivalent formulation due to Vogan). In turn, it is shown that the Loewy length results imply Vogan's statement, and lead to the determination of Loewy length for the self-dual projectives and certain generalized Verma modules in ${\mathcal{O}_S}$. Under the stronger assumption of Jantzen's conjecture, the radical and socle series are computed for self-dual projectives in $\mathcal{O}$. An analogous result is formulated for self-dual projectives in ${\mathcal{O}_S}$ and proved in certain cases.
The Schur multiplier of a nilpotent group
Ursula Martin
Webb
755-763
Abstract: In this paper we obtain upper and lower bounds for the rank of the Schur multiplier of a nilpotent group in terms of the nilpotency class and the number of generators and rank of the derived quotient.
Differential identities in prime rings with involution
Charles
Lanski
765-787
Abstract: Let $R$ be a prime ring with involution. Using work of V. K. Kharchenko it is shown that any generalized identity for $R$ involving derivations of $R$ and the involution of $R$ is a consequence of the generalized identities with involution which $R$ satisfies. In obtaining this result, a generalization, to rings satisfying a GPI, of the classical theorem characterizing inner derivations of finite-dimensional simple algebras is required. Consequences of the main theorem are that in characteristic zero no outer derivation of $R$ can act algebraically on the set of symmetric elements of $R$, and if the images of the set of symmetric elements under the derivations of $R$ satisfy a polynomial relation, then $ R$ must satisfy a generalized polynomial identity.
Factorization of diagonally dominant operators on $L\sb 1([0,1],X)$
Kevin T.
Andrews;
Joseph D.
Ward
789-800
Abstract: Let $X$ be a separable Banach space. It is shown that every diagonally dominant invertible operator on $ {L_1}([0,\,1],\,X)$ can be factored uniquely as a product of an invertible upper triangular operator and an invertible unit lower triangular operator.
Multipliers on the space of semiperiodic sequences
Manuel Núñez
Jiménez
801-811
Abstract: Semiperiodic sequences are defined to be the uniform limit of periodic sequences. They form a space of continuous functions on a compact group $\Delta$. We study the properties of the Radon measures on $\Delta$ in order to classify the multipliers for the space of semiperiodic sequences, paying special attention to those which can be realized as transference functions of physically constructible filters.
Connected simple systems and the Conley index of isolated invariant sets
Dietmar
Salamon
1-41
Abstract: The object of this paper is to present new and simplified proofs for most of the basic results in the index theory for flows. Simple, explicit formulae are derived for the maps which play a central role in the theory. The presentation is self-contained.
Subellipticity of the $\bar \partial$-Neumann problem on nonpseudoconvex domains
Lop-Hing
Ho
43-73
Abstract: Following the work of Kohn, we give a sufficient condition for subellipticity of the $ \overline \partial$-Neumann problem for not necessarily pseudoconvex domains. We define a sequence of ideals of germs and show that if $1$ is in any of them, then there is a subelliptic estimate. In particular, we prove subellipticity under some specific conditions for $n - 1$ forms and for the case when the Levi-form is diagonalizable. For the necessary conditions, we use another method to prove that there is no subelliptic estimate for $q$ forms if the Levi-form has $n - q - 1$ positive eigenvalues and $ q$ negative eigenvalues. This was proved by Derridj. Using similar techniques, we prove a necessary condition for subellipticity for some special domains. Finally, we give a remark to Catlin's theorem concerning the hypoellipticity of the $\overline \partial$-Neumann problem in the case of nonpseudoconvex domains.
On Lipschitz homogeneity of the Hilbert cube
Aarno
Hohti
75-86
Abstract: The main contribution of this paper is to prove the conjecture of [Vä] that the Hilbert cube $Q$ is Lipschitz homogeneous for any metric $ {d_s}$, where $ s$ is a decreasing sequence of positive real numbers ${s_k}$ converging to zero, $ {d_s}(x,y) = \sup \{ {s_k}\vert{x_k} - {y_k}\vert:k \in N\}$, and $ R(s) = \sup \{ {s_k}/{s_{k + 1}}:k \in N\} < \infty $. In addition to other results, we shall show that for every Lipschitz homogeneous compact metric space $X$ there is a constant $\lambda < \infty$ such that $X$ is homogeneous with respect to Lipschitz homeomorphisms whose Lipschitz constants do not exceed $\lambda$. Finally, we prove that the hyperspace $ {2^I}$ of all nonempty closed subsets of the unit interval is not Lipschitz homogeneous with respect to the Hausdorff metric.
Small zeros of quadratic forms
Wolfgang M.
Schmidt
87-102
Abstract: We give upper and lower bounds for zeros of quadratic forms in the rational, real and $p$-adic fields. For example, given $r > 0$, $s > 0$, there are infinitely many forms $\mathfrak{F}$ with integer coefficients in $r + s$ variables of the type $ (r,s)$ (i.e., equivalent over ${\mathbf{R}}$ to $ X_1^2 + \cdots + X_r^2 - X_{r + 1}^2 - \cdots - X_{r + s}^2$ such that every nontrivial integer zero $ {\mathbf{x}}$ has $ \vert{\mathbf{x}}\vert \gg {F^{r/2s}}$, where $F$ is the maximum modulus of the coefficients of $ \mathfrak{F}$.
The traction problem for incompressible materials
Y. H.
Wan
103-119
Abstract: The traction problem for incompressible materials is treated as a bifurcation problem, where the applied loads are served as parameters. We take both the variational approach and the classical power series approach. The variational approach provides a natural, unified way of looking at this problem. We obtain a count of the number of equilibria together with the determination of their stability. In addition, it also lays down the foundation for the Signorini-Stoppelli type computations. We find second order sufficient conditions for the existence of power series solutions. As a consequence, the linearization stability follows, and it clarifies in some sense the role played by the linear elasticity in the context of the nonlinear elasticity theory. A systematic way of calculating the power series solution is also presented.
Descriptive complexity of function spaces
D.
Lutzer;
J.
van Mill;
R.
Pol
121-128
Abstract: In this paper we show that ${C_\pi }(X)$, the set of continuous, real-valued functions on $X$ topologized by the pointwise convergence topology, can have arbitrarily high Borel or projective complexity in $ {{\mathbf{R}}^X}$ even when $X$ is a countable regular space with a unique limit point. In addition we show how to construct countable regular spaces $X$ for which $ {C_\pi }(X)$ lies nowhere in the projective hierarchy of the complete separable metric space $ {{\mathbf{R}}^X}$.
Propagation estimates for Schr\"odinger-type operators
Arne
Jensen
129-144
Abstract: Propagation estimates for a Schrödinger-type operator are obtained using multiple commutator techniques. A new method is given for obtaining estimates for powers of the resolvent. As an application, micro-local propagation estimates are obtained for two-body Schrödinger operators with smooth long-range potentials.
The Cauchy problem for $u\sb t=\Delta u\sp m$ when $0<m<1$
Miguel A.
Herrero;
Michel
Pierre
145-158
Abstract: This paper deals with the Cauchy problem for the nonlinear diffusion equation $\partial u/\partial t - \Delta (u\vert u{\vert^{m - 1}}) = 0$ on $(0,\infty ) \times {{\mathbf{R}}^N},u(0, \cdot ) = {u_0}$ when $0 < m < 1$ (fast diffusion case). We prove that there exists a global time solution for any locally integrable function ${u_0}$: hence, no growth condition at infinity for $ {u_0}$ is required. Moreover the solution is shown to be unique in that class. Behavior at infinity of the solution and $ L_{\operatorname{loc} }^\infty$-regularizing effects are also examined when $ m \in (\max \{ (N - 2)/N,0\} ,1)$.
Probabilistic square functions and a priori estimates
Andrew G.
Bennett
159-166
Abstract: We obtain a priori estimates for Riesz transforms and their variants, that is, estimates with bounds independent of the dimension of the space and/or the nature of the boundary. The key to our results is to give probabilistic definitions which do not depend on the geometry of the situation for the transformations in question. We then use probabilistic square functions to prove our a priori estimates.
Symmetric positive systems with boundary characteristic of constant multiplicity
Jeffrey
Rauch
167-187
Abstract: The theory of maximal positive boundary value problems for symmetric positive systems is developed assuming that the boundary is characteristic of constant multiplicity. No such hypothesis is needed on a neighborhood of the boundary. Both regularity theorems and mixed initial boundary value problems are discussed. Many classical ideas are sharpened in the process.
Baire sets of $k$-parameter words are Ramsey
Hans Jürgen
Prömel;
Bernd
Voigt
189-201
Abstract: We show that Baire sets of $k$-parameter words are Ramsey. This extends a result of Carlson and Simpson, A dual form of Ramsey's theorem, Adv. in Math. 53 (1984), 265-290. Employing the method established therefore, we derive analogous results for Dowling lattices and for ascending $k$-parameter words.
Constant term identities extending the $q$-Dyson theorem
D. M.
Bressoud;
I. P.
Goulden
203-228
Abstract: Andrews [1] has conjectured that the constant term in a certain product is equal to a $q$-multinomial coefficient. This conjecture is a $q$-analogue of Dyson's conjecture [5], and has been proved, combinatorically, by Zeilberger and Bressoud [15]. In this paper we give a combinatorial proof of a master theorem, that the constant term in a similar product, computed over the edges of a nontransitive tournament, is zero. Many constant terms are evaluated as consequences of this master theorem including Andrews' $q$-Dyson theorem in two ways, one of which is a $q$-analogue of Good's [6] recursive proof.
Hyponormal operators quasisimilar to an isometry
Pei Yuan
Wu
229-239
Abstract: An expression for the multiplicity of an arbitrary contraction is presented. It is in terms of the isometries which can be densely intertwined to the given contraction. This is then used to obtain a generalization of a result of Sz.-Nagy and Foiaş concerning the existence of a $C{._0}$ contraction which is a quasiaffine transform of a contraction. We then consider the problem when a hyponormal operator is quasisimilar to an isometry or, more generally, when two hyponormal contractions are quasisimilar to each other. Our main results in this respect generalize previous ones obtained by Hastings and the author. For quasinormal and certain subnormal operators, quasisimilarity or similarity to an isometry may even imply unitary equivalence.
Entropy and knots
John
Franks;
R. F.
Williams
241-253
Abstract: We show that a smooth flow on ${S^3}$ with positive topological entropy must possess periodic closed orbits in infinitely many different knot type equivalence classes.
Global solvability and regularity for $\bar \partial$ on an annulus between two weakly pseudoconvex domains
Mei-Chi
Shaw
255-267
Abstract: Let ${M_1}$ and ${M_2}$ be two bounded pseudo-convex domains in $ {{\mathbf{C}}^n}$ with smooth boundaries such that ${\overline M _1} \subset {M_2}$. We consider the Cauchy-Riemann operators $\overline \partial$ on the annulus $M = {M_2}\backslash {\overline M _1}$. The main result of this paper is the following: Given a $ \overline \partial$-closed $(p,q)$ form $\alpha$, $0 < q < n$, which is ${C^\infty }$ on $ \overline M$ and which is cohomologous to zero on $M$, there exists a $(p,q - 1)$ form $u$ which is $ {C^\infty }$ on $\overline M$ such that $\overline \partial u = \alpha$.
Cobordism of $(k)$-framed manifolds
E.
Micha
269-280
Abstract: The cobordism theories that arise by considering manifolds whose stable normal bundle has category $k$ are introduced. Using these theories, we define a new filtration of the homotopy groups of spheres. We study the filtration and obtain an upper bound for the filtration of elements in the stable $n$-stem.
The homotopy theory of cyclic sets
W. G.
Dwyer;
M. J.
Hopkins;
D. M.
Kan
281-289
Abstract: The aim of this note is to show that the homotopy theory of the cyclic sets of Connes [3] is equivalent to that of $ \operatorname{SO} (2)$-spaces (i.e. spaces with a circle action) and hence to that of spaces over $K(Z,2)$.
On monomial algebras of finite global dimension
David J.
Anick
291-310
Abstract: Let $G$ be an associative monomial ${\mathbf{k}}$-algebra. If $G$ is assumed to be finitely presented, then either $G$ contains a free subalgebra on two monomials or else $G$ has polynomial growth. If instead $G$ is assumed to have finite global dimension, then either $G$ contains a free subalgebra or else $ G$ has a finite presentation and polynomial growth. Also, a graded Hopf algebra with generators in degree one and relations in degree two contains a free Hopf subalgebra if the number of relations is small enough.
Uniqueness for a forward backward diffusion equation
Alan V.
Lair
311-317
Abstract: Let $\phi$ be continuous, have at most finitely many local extrema on any bounded interval, be twice continuously differentiable on any closed interval on which there is no local extremum and be strictly decreasing on any closed interval on which it is decreasing. We show that the initial-boundary value problem for $ {u_t} = \phi {({u_x})_x}$ with Neumann boundary conditions has at most one smooth solution.
On nonlinear scalar Volterra integral equations. I
Hans
Engler
319-336
Abstract: The scalar nonlinear Volterra integral equation $\displaystyle u(t) + \int_0^t {g(t,s,u(s))\,ds = f(t)\qquad (0 \leqslant t)}$ is studied. Conditions are given under which the difference of two solutions can be estimated by the variation of the difference of the corresponding right-hand sides. Criteria for the existence of $\lim u(t)$ (as $t \to \infty$) are given, and existence and uniqueness questions are also studied.
The degrees of r.e. sets without the universal splitting property
R. G.
Downey
337-351
Abstract: It is shown that every nonzero r.e. degree contains an r.e. set without the universal splitting property. That is, if $ \delta$ is any r.e. nonzero degree, there exist r.e. sets $\emptyset < {}_TB < {}_TA$ with $\deg (A) = \delta$ such that if ${A_0} \sqcup {A_1}$ is an r.e. splitting of $ A$, then ${A_0}\not \equiv {}_TB$. Some generalizations are discussed.
Remarks on the stability of shock profiles for conservation laws with dissipation
Robert L.
Pego
353-361
Abstract: Two remarks are made. The first is to establish the stability of monotone shock profiles of the KdV-Burgers equation, based on an energy method of Goodman. The second remark illustrates, specifically in Burgers' equation, that uniform rates of decay are not to be expected for perturbations of shock profiles in typical norms.
Divisions of space by parallels
G. L.
Alexanderson;
John E.
Wetzel
363-377
Abstract: An arrangement of hyperplanes in $ {\mathbb{E}^d}$ is a "plaid" provided its hyperplanes form no multiple flats of intersection and lie in parallel families that are in general position. We develop some geometrically natural formulas for the number of $r$-faces that are formed by such an arrangement.
Dirichlet series and automorphic forms on unitary groups
Tobias
Orloff
431-456
Abstract: In a special case our unitary group takes the form $\displaystyle G = \{ g \in {\text{GL}}(p + 2,{\mathbf{C}}){\vert^t}\bar gRg = R\} .$ Here $\displaystyle R = \left( {\begin{array}{*{20}{c}} S & 0 & 0 0 & 0 & 1 0 & { - 1} & 0 \end{array} } \right)$ is a skew-Hermitian matrix with entries in an imaginary quadratic number field $ K$. We suppose that $- iR$ has signature $ (p + 1,1)$. This group acts naturally on the symmetric domain $\displaystyle D = \left\{ {w \in {{\mathbf{C}}^p},z \in {\mathbf{C}}\vert\operatorname{Im} (z) > - {{\frac{1}{2}}^t}\bar wSw} \right\}.$ If $ \Gamma = G \cap {\text{SL}}(p + 2,{\mathcal{O}_K})$ with ${\mathcal{O}_K}$ the ring of integers in $ K$, then an automorphic form $f(w,z)$ with respect to $\Gamma$ has an expansion ${\Sigma _r}{g_r}(w) \cdot {e^{2\pi irz}}$. The functions ${g_r}(w)$ are theta functions. Given another automorphic form $g(w,z)$ with an expansion ${\Sigma _s}{h_s}(w) \cdot {e^{2\pi isz}}$ we define a Dirichlet series ${\Sigma _r}\langle {g_r},{h_r}\rangle {r^{ - s}}$ . Here $\langle {g_r},{h_r}\rangle $ is a certain positive definite inner product on the space of theta functions. The series is obtained as an integral of Rankin type: $\displaystyle {\int _{{P_\Gamma }\backslash D}}f\bar g\cdot{\left( {\operatorname{Im} (z) + \frac{1} {2}{i^t}\bar wSw} \right)^s}dw\;d\bar w\;dz\;d\bar z$ with ${P_\Gamma } \subseteq \Gamma$ a subgroup of "translations". The series is analytically continued by studying the Eisenstein series arising when the above integral is transformed into an integral over $\Gamma \backslash D$. In the case $p = 1$ our results have an application to some recent work of Shintani, where the Euler product attached to an eigenfunction of the Hecke operators is obtained, up to some simple factors, as a series of the above type.
a.c. groups: extensions, maximal subgroups, and automorphisms
Kenneth
Hickin
457-481
Abstract: In $\S1$ we extend the results of [ ${\mathbf{3}}$] on centralizers to r.e. subgroups and show, e.g., that every a.c. group has an $\infty - \omega $-equivalent subgroup of the same power which is embedded maximally in itself; and we pursue a natural typology of maximal subgroups. $\S2$ shows that if $A$ is a countable group of automorphisms of a countable a.c. group $G$ such that $A \supset \operatorname{Inn}\;G$, then there exists $\tau \in \operatorname{Aut}\;G$ such that the $ {\text{HNN}}$ extension ( $ A,\tau :{\tau ^{ - 1}}g\tau = \tau (g)$ for all $g \in \operatorname{Inn}\;G$) is a subgroup of $ \operatorname{Aut}\;G$. We show in $\S3$ that every a.c. group with a countable skeleton has a proper extension to an a.c. group having any skeleton that contains the original one and any f.g. group which contains the countable a.c. group equivalent to the original one as an r.e. subset. This uses Ziegler's construction [ $ {\mathbf{7}}$]. Finally, in $\S4$, also using Ziegler's construction we show that there exists an a.c. group $A$ of any power and having any countable skeleton which has a free subgroup $M$ such that for all $x \in A - M$ and $y \in A$ there exist free generators $a,b,c \in M$ such that $y = {(ax)^b}{(ax)^c}$.
The Euler equation for functionals with linear growth
Gabriele
Anzellotti
483-501
Abstract: We give a first variation formula for functionals of the type $\int_\Omega {f(x,\mu )}$, where $f(x,p):\Omega \times {{\mathbf{R}}^k} \to {\mathbf{R}}$ is of linear growth in $p$ for large $\vert p\vert$ and $\mu$ is a $ {{\mathbf{R}}^k}$-valued measure in $\Omega$. The Euler equation for the minima of various functionals defined on spaces of ${\text{BV}}$ functions is then studied.
Free boundary regularity for surfaces minimizing ${\rm Area}(S)+c\,{\rm Area}(\partial S)$
Edith A.
Cook
503-526
Abstract: In ${{\mathbf{R}}^n}$, fix a hyperplane $Z$ and $ a\;(k - 1)$-dimensional surface $F$ lying to one side of $Z$ with boundary in $Z$. We prove the existence of $S$ and $B$ minimizing $\operatorname{Area}(S) + c\operatorname{Area}(B)$ among all $k$-dimensional $S$ having boundary $F \cup B$, where $B$ is a free boundary constrained to lie in $ Z$. We prove that except possibly on a set of Hausdorff dimension $ k - 2$, $S$ is locally a ${C^{1,\alpha }}$ manifold with ${C^{1,\alpha }}$ boundary $B$ for $ 0 < \alpha < 1/2$. If $k = n - 1$, $ {C^{1,\alpha }}$ is replaced by real analytic.
On a class of Banach spaces of functions associated with the notion of entropy
Boris
Korenblum
527-553
Abstract: A class of function spaces on the circle is introduced which contain all continuous functions of bounded variation but are included in the set of all continuous functions. The corresponding dual spaces consist of certain types of generalized measures. One application of these spaces is a new convergence test for Fourier series which includes both the Dirichlet-Jordan and the Dini-Lipschitz tests.
A geometric approach to the linear Penrose transform
I. B.
Penkov
555-575
Abstract: We show that under a certain cohomological condition the theorem of Witten, Isenberg, Yasskin and Green about the inverse Penrose transform of a (non-self-dual) connection $ \nabla$ (together with Manin's description of its curvature ${F_\nabla }$) is true in a quite general situation. We then present a (multidimensional) version of the Penrose transform of a vector bundle in the language of jets. This gives a coordinate-free interpretation of certain results of Henkin and Manin, coding a number of classical field equations in terms of obstructions to infinitesimal extension of cohomology classes.
On the spectra of compact nilmanifolds
Jeffrey S.
Fox
577-583
Abstract: We show the equivalence of the Howe-Richardson multiplicity formula for compact nilmanifolds and the formula obtained by Corwin and Greenleaf using the Selberg trace formula.
$P$-points with countably many constellations
Ned I.
Rosen
585-596
Abstract: If the continuum hypothesis $ ({\text{CH}})$ is true, then for any $P$ point ultrafilter $D$ (on the set of natural numbers) there exist initial segments of the Rudin-Keisler ordering, restricted to (isomorphism classes of) $P$ points which lie above $ D$, of order type ${\aleph _1}$. In particular, if $ D$ is an ${\text{RK}}$-minimal ultrafilter, then we have $({\text{CH}})$ that there exist $P$-points with countably many constellations.
The dual Weierstrass-Laguerre transform
Deborah Tepper
Haimo
597-613
Abstract: An inversion algorithm is derived for the dual Weierstrass-Laguerre transform $ \int_0^\infty {{g_\alpha }(x,y;1)\varphi (y){y^\alpha }{e^{ - y}}/(\alpha + 1)dy}$, where the function ${g_\alpha }(x,y,t)$ is associated with the source solution of the Laguerre differential heat equation $x{u_{xx}}(x,t) = (\alpha + 1 - x){u_x}(x,t) = {u_t}(x,t)$. Correspondingly, sufficient conditions are established for a function to be represented by a Weierstrass-Laguerre Stieltjes transform $ \int_0^\infty {{g_\alpha }(x,y;1)\;d\beta (y)}$ of a nondecreasing function $ \beta$.
Regular linear algebraic monoids
Mohan S.
Putcha
615-626
Abstract: In this paper we study connected regular linear algebraic monoids. If $ \phi :{G_0} \to {\text{GL}}(n,K)$ is a representation of a reductive group $ {G_0}$, then the Zariski closure of $K\phi ({G_0})$ in ${\mathcal{M}_n}(K)$ is a connected regular linear algebraic monoid with zero. In $\S2$ we study abstract semigroup theoretic properties of a connected regular linear algebraic monoid with zero. We show that the principal right ideals form a relatively complemented lattice, that the idempotents satisfy a certain connectedness condition, and that these monoids are $V$-regular. In $\S3$ we show that when the ideals are linearly ordered, the group of units is nearly simple of type $ {A_l},{B_l},{C_l},{F_4}\;{\text{or}}\;{G_2}$. In $\S4$, conjugacy classes are studied by first reducing the problem to nilpotent elements. It is shown that the number of conjugacy classes of minimal nilpotent elements is always finite.
On minimal surfaces in a K\"ahler manifold of constant holomorphic sectional curvature
Jon G.
Wolfson
627-646
Abstract: This paper studies minimal surfaces in Kähler manifolds of constant holomorphic sectional curvature using the technique of the moving frame. In particular, we provide a classification of the minimal two-spheres in ${\mathbf{C}}{P^n}$, complex projective $ n$-space, equipped with the Fubini-Study metric. This classification can be described as follows: To each holomorphic curve in ${\mathbf{C}}{P^n}$ classically there is associated a particular framing of ${{\mathbf{C}}^{n + 1}}$ called the Frenet frame. Each element of the Frenet frame induces a minimal surface in $ {\mathbf{C}}{P^n}$. The classification theorem states that all minimal surfaces of topological type of the two-sphere occur in this manner. The theorem is proved using holomorphic differentials that occur naturally on minimal surfaces in Kähler manifolds of constant holomorphic sectional curvature together with the Riemann-Roch Theorem.
Normal subgroups of doubly transitive automorphism groups of chains
Richard N.
Ball;
Manfred
Droste
647-664
Abstract: We characterize the structure of the normal subgroup lattice of $ 2$-transitive automorphism groups $A(\Omega )$ of infinite chains $(\Omega , \leqslant )$ by the structure of the Dedekind completion $(\bar \Omega , \leqslant )$ of the chain $(\Omega , \leqslant )$. As a consequence we obtain various group-theoretical results on the normal subgroups of $A(\Omega )$, including that any proper subnormal subgroup of $ A(\Omega )$ is indeed normal and contained in a maximal proper normal subgroup of $A(\Omega )$, and that $A(\Omega )$ has precisely $5$ normal subgroups if and only if the coterminality of the chain $(\Omega , \leqslant )$ is countable.
Solving semilinear partial differential equations with probabilistic potential theory
Joseph
Glover;
P. J.
McKenna
665-681
Abstract: Techniques of probabilistic potential theory are applied to solve $- Lu + f(u) = \mu$, where $\mu$ is a signed measure, $ f$ a (possibly discontinuous) function and $L$ a second order elliptic or parabolic operator on ${R^d}$ or, more generally, the infinitesimal generator of a Markov process. Also formulated are sufficient conditions guaranteeing existence of a solution to a countably infinite system of such equations.
The de Branges theorem on univalent functions
Carl H.
FitzGerald;
Ch.
Pommerenke
683-690
Abstract: We present a simplified version of the de Branges proof of the Lebedev-Milin conjecture which implies the Robertson and Bieberbach conjectures. As an application of an analysis of the technique, it is shown that the method could not be used directly to prove the Bieberbach conjecture.
A note on a question of Bass
Sankar P.
Dutta
691-700
Abstract: In this paper we study the following question raised by Bass: Is a local ring with a finitely generated module of finite injective dimension Cohen-Macaulay? We prove that the answer is in the affirmative when a certain local cohomology of the ring is either decomposible or cyclic. We also study the above question in some special cases and some of its implications.
Unstable ground state of nonlinear Klein-Gordon equations
Jalal
Shatah
701-710
Abstract: In this paper we prove the instability of the ground state, i.e. least energy steady-state solution of nonlinear Klein-Gordon equations with space dimension $n \geqslant 3$.
Directed sets and cofinal types
Stevo
Todorčević
711-723
Abstract: We show that $1,\omega ,{\omega _1},\omega \times {\omega _1}$ and ${[{\omega _1}]^{ < \omega }}$ are the only cofinal types of directed sets of size ${\aleph _1}$, but that there exist many cofinal types of directed sets of size continuum.
On the group ${\rm SSF}(G),\;G$ a cyclic group of prime order
M.
Maller;
J.
Whitehead
725-733
Abstract: We extend the definition of the obstruction group ${\text{SSF}}(G)$ in the case where $G$ is a cyclic group of prime order. We show that an endomorphism of a free $ZG$-module is a direct summand of a virtual permutation if its characteristic polynomial has the appropriate form. Among these endomorphisms the virtual permutations are detected by ${K_0}$. The main application is in detecting Morse-Smale isotopy classes.
Abelian rank of normal torsion-free finite index subgroups of polyhedral groups
Youn W.
Lee
735-745
Abstract: Suppose that $ P$ is a convex polyhedron in the hyperbolic $3$-space with finite volume and $P$ has integer $( > 1)$ submultiples of $\pi$ as dihedral angles. We prove that if the rank of the abelianization of a normal torsion-free finite index subgroup of the polyhedral group $ G$ associated to $ P$ is one, then $ P$ has exactly one ideal vertex of type $(2,2,2,2)$ and $G$ has an index two subgroup which does not contain any one of the four standard generators of the stabilizer of the ideal vertex.
Homology and cell structure of nilpotent spaces
Robert H.
Lewis
747-760
Abstract: Let $A$ and $X$ denote finitely dominated nilpotent ${\text{CW}}$ complexes. We are interested in questions relating the homology groups of such spaces to their cell structure and homotopy type. We solve a problem posed by Brown and Kahn, that of constructing nilpotent complexes of minimal dimension. When the fundamental group is finite, the three-dimensional complex we construct may not be finite; we then construct a finite six-dimensional complex. We investigate the set of possible cofibers of maps $A \to X$, and find a severe restriction. When it is met and the fundamental group is finite, $ X$ can be constructed from $ A$ by attaching cells in a natural way. The restriction implies that the classical notion of homology decomposition has no application to nilpotent complexes. We show that the Euler characteristic of $X$ must be zero. Several corollaries are derived to the theory of finitely dominated nilpotent complexes. Several of these results depend upon a purely algebraic theorem that we prove concerning the vanishing of homology of nilpotent modules over nilpotent groups.
Variants of the maximal double Hilbert transform
Elena
Prestini
761-771
Abstract: We prove the boundedness on $ {L_p}({T^2})$, $1 < p < \infty$, of two variants of the double Hilbert transform and maximal double Hilbert transform. They have an application to a problem of almost everywhere convergence of double Fourier series.
Changing the depth of an ordered set by decomposition
E. C.
Milner;
K.
Prikry
773-785
Abstract: The depth of a partially ordered set $ \langle P, < \rangle$ is the smallest ordinal $\gamma$ such that $\langle P, < \rangle$ does not embed ${\gamma ^\ast}$. The width of $\langle P, < \rangle$ is the smallest cardinal number $\mu$ such that there is no antichain of size $ \mu + 1$ in $P$. We show that if $\gamma > \omega$ and $\gamma$ is not an infinite successor cardinal, then any partially ordered set of depth $\gamma$ can be decomposed into $ \operatorname{cf}(\vert\gamma \vert)$ parts so that the depth of each part is strictly less than $\gamma$. If $ \gamma = \omega$ or if $ \gamma$ is an infinite successor cardinal, then for any infinite cardinal $ \lambda$ there is a linearly ordered set of depth $\gamma$ such that for any $\lambda $-decomposition one of the parts has the same depth $\gamma$. These results are used to solve an analogous problem about width. It is well known that, for any cardinal $\lambda$, there is a partial order of width $ \omega$ which cannot be split into $\lambda$ parts of finite width. We prove that, for any cardinal $\lambda$ and any infinite cardinal $ \nu$, there is a partial order of width $ {\nu ^ + }$ which cannot be split into $\lambda$ parts of smaller width.
Local vanishing properties of solutions of elliptic and parabolic quasilinear equations
J. Ildefonso
Díaz;
Laurent
Véron
787-814
Abstract: We use a local energy method to study the vanishing property of the weak solutions of the elliptic equation $- \operatorname{div}\;A(x,u,Du) + B(x,u,Du) = 0$ and of the parabolic equation $\partial \psi (u)/\partial t - \operatorname{div}\;\mathcal{A}(t,x,u,Du) + \mathcal{B}(t,x,u,Du) = 0$. The results are obtained without any assumption of monotonicity on $A$, $B$, $ \mathcal{A}$ and $\mathcal{B}$.
The determinant of the Eisenstein matrix and Hilbert class fields
I.
Efrat;
P.
Sarnak
815-824
Abstract: We compute the determinant of the Eisenstein matrix associated to the Hilbert-Blumenthal modular group $ {\text{PSL}_2}({\mathcal{O}_k})$, and express it in terms of the zeta function of the Hilbert class field of $K$.
Propagation of $C\sp \infty$ regularity for fully nonlinear second order strictly hyperbolic equations in two variables
Paul
Godin
825-830
Abstract: It is shown that if $ u$ is a ${C^3}$ solution of a fully nonlinear second order strictly hyperbolic equation in two variables, then $u$ is $ {C^\infty }$ at a point $ m$ as soon as it is ${C^\infty }$ at some point of each of the two bicharacteristic curves through $m$. For semilinear equations, such a result was obtained before by Rauch and Reed if $u \in {C^1}$
Separation in countably paracompact spaces
W. Stephen
Watson
831-842
Abstract: We study the question "Are discrete families of points separated in countably paracompact spaces?" in the class of first countable spaces and the class of separable spaces.
On derivations annihilating a maximal abelian subalgebra
Geoffrey L.
Price
843-850
Abstract: Let $\mathcal{A}$ be an $ {\text{AF}}\;{C^\ast}$-algebra, and let $\delta$ be a closed $\ast$-derivation which annihilates the maximal abelian subalgebra $ \mathcal{C}$ of diagonal elements of $ \mathcal{A}$. Then we show that $\delta$ generates an approximately inner $ {C^\ast}$-dynamics on $\mathcal{A}$, and that $\delta$ is a commutative $\ast $-derivation. Any two closed $\ast$-derivations vanishing on $\mathcal{C}$ are shown to be strongly commuting. More generally, if $\delta$ is a semiderivation on $\mathcal{A}$ which vanishes on $\mathcal{C}$, we prove that $\delta$ is a generator of a semigroup of strongly positive contractions of $\mathcal{A}$.
A quasi-invariance theorem for measures on Banach spaces
Denis
Bell
851-855
Abstract: We show that for a measure $\gamma$ on a Banach space directional differentiability implies quasi-translation invariance. This result is shown to imply the Cameron-Martin theorem. A second application is given in which $ \gamma$ is the image of a Gaussian measure under a suitably regular map.
Erratum to: ``Localization of equivariant cohomology rings''
J.
Duflot
857-858
Homogeneous Borel sets of ambiguous class two
Fons
van Engelen
1-39
Abstract: We describe and characterize all homogeneous subsets of the Cantor set which are both an $ {F_{\sigma \delta }}$ and a $ {G_{\delta \sigma }}$; it turns out that there are $ {\omega _1}$ such spaces.
Regular cardinals in models of ${\rm ZF}$
Moti
Gitik
41-68
Abstract: We prove the following Theorem. Suppose $ M$ is a countable model of $ZFC$ and $\kappa$ is an almost huge cardinal in $ M$. Let $ A$ be a subset of $ \kappa$ consisting of nonlimit ordinals. Then there is a model $ {N_A}$ of $ ZF$ such that ${\aleph _\alpha }$ is a regular cardinal in $ {N_A}$ iff $\alpha \in A$ for every $\alpha > 0$.
Free lattice-ordered groups represented as $o$-$2$ transitive $l$-permutation groups
Stephen H.
McCleary
69-79
Abstract: It is easy to pose questions about the free lattice-ordered group $ {F_\eta }$ of rank $ \eta > 1$ whose answers$ ^{2}$ are "obvious", but difficult to verify. For example: 1. What is the center of ${F_\eta }$? 2. Is ${F_\eta }$ directly indecomposable? 3. Does $ {F_\eta }$ have a basic element? 4. Is ${F_\eta }$ completely distributive? Question 1 was answered recently by Medvedev, and both $ 1$ and $2$ by Arora and McCleary, using Conrad's representation of ${F_\eta }$ via right orderings of the free group $ {G_\eta }$. Here we answer all four questions by using a completely different tool: The (faithful) representation of $ {F_\eta }$ as an $o{\text{-}}2$-transitive $l$-permutation group which is pathological (has no nonidentity element of bounded support). This representation was established by Glass for most infinite $ \eta$, and is here extended to all $\eta > 1$. Curiously, the existence of a transitive representation for ${F_\eta }$ implies (by a result of Kopytov) that in the Conrad representation there is some right ordering of ${G_\eta }$ which suffices all by itself to give a faithful representation of ${F_\eta }$. For finite $\eta$, we find that every transitive representation of ${F_\eta }$ can be made from a pathologically $o{\text{-}}2$-transitive representation by blowing up the points to $o$-blocks; and every pathologically $o{\text{-}}2$-transitive representation of $ {F_\eta }$ can be extended to a pathologically $ o{\text{-}}2$-transitive representation of $ {F_{{\omega _0}}}$.
An even better representation for free lattice-ordered groups
Stephen H.
McCleary
81-100
Abstract: The free lattice-ordered group ${F_\eta }$ (of rank $\eta$) has been studied in two ways: via the Conrad representation on the various right orderings of the free group ${G_\eta }$ (sharpened by Kopytov's observation that some one right ordering must by itself give a faithful representation), and via the Glass-McCleary representation as a pathologically $ o{\text{-}}2$-transitive $ l$-permutation group. Each kind of representation yields some results which cannot be obtained from the other. Here we construct a representation giving the best of both worlds--a right ordering $({G_\eta }, \leqslant )$ on which the action of $ {F_\eta }$ is both faithful and pathologically $ o{\text{-}}2$-transitive. This $ ({G_\eta }, \leqslant )$ has no proper convex subgroups. The construction is explicit enough that variations of it can be utilized to get a great deal of information about the root system $ {\mathcal{P}_\eta }$ of prime subgroups of ${F_\eta }$. All ${\mathcal{P}_\eta }$'s with $1 < \eta < \infty $ are $ o$-isomorphic. This common root system $ {\mathcal{P}_f}$ has only four kinds of branches (singleton, three-element, $ {\mathcal{P}_f}$ and $ {\mathcal{P}_{{\omega _0}}}$), each of which occurs ${2^{{\omega _0}}}$ times. Each finite or countable chain having a largest element occurs as the chain of covering pairs of some root of ${\mathcal{P}_f}$.
Subspaces of ${\rm BMO}({\bf R}\sp n)$
Michael
Frazier
101-125
Abstract: We consider subspaces of $ {\text{BMO}}({{\mathbf{R}}^n})$ generated by one singular integral transform. We show that the averages along ${x_j}$-lines of the $j$ th Riesz transform of $g \in {\text{BMO}} \cap {L^2}({{\mathbf{R}}^n})$ or $ g \in {L^\infty }({{\mathbf{R}}^n})$ satisfy a certain strong regularity property. One consquence of this result is that such functions satisfy a uniform doubling condition on a.e. $ {x_j}$-line. We give an example to show, however, that the restrictions to $ {x_j}$-lines of the Riesz transform of $g \in {\text{BMO}} \cap {L^2}({{\mathbf{R}}^n})$ do not necessarily have uniformly bounded ${\text{BMO}}$ norm. Also, for a Calderón-Zygmund singular integral operator $K$ with real and odd kernel, we show that $ K({\text{BMO}_c}) \subseteq \overline {{L^\infty } + K(L_c^\infty )}$, where $L_c^\infty$ and $ {\text{BMO}_c}$ are the spaces of ${L^\infty }$ or ${\text{BMO}}$ functions of compact support, respectively, and the closure is taken in ${\text{BMO}}$ norm.
A proof of Andrews' $q$-Dyson conjecture for $n=4$
Kevin W. J.
Kadell
127-144
Abstract: Andrews' $ q$-Dyson conjecture is that the constant term in a polynomial associated with the root system $ {A_{n - 1}}$ is equal to the $q$-multinomial coefficient. Good used an identity to establish the case $q = 1$, which was originally raised by Dyson. Andrews established his conjecture for $n \leqslant 3$ and Macdonald proved it when $ {a_1} = {a_2} = \cdots = {a_n} = 1,2$ or $\infty$ for all $ n \geqslant 2$. We use a $ q$-analog of Good's identity which involves a remainder term and linear algebra to establish the conjecture for $ n = 4$. The remainder term arises because of an essential problem with the $ q$-Dyson conjecture: the symmetry of the constant term. We give a number of conjectures related to the symmetry.
Good and OK ultrafilters
Alan
Dow
145-160
Abstract: In this paper we extend Kunen's construction of ${\alpha ^ + }$-good ultrafilters on $\mathcal{P}(\alpha )$ to more general algebras, as well as the construction of ${\alpha ^ + }$-OK ultrafilters. In so doing, we prove the existence of $({2^\alpha } \times {\alpha ^ + })$-independent matrices, as defined by Kunen, in these algebras. Some of the topological properties of the Stone spaces of these algebras are then investigated. We find points, for example, in $ U(\alpha )$ which can be regarded as a generalization of weak $P$-points.
On twisted lifting
Yuval Z.
Flicker
161-178
Abstract: If $\sigma$ is a generator of the galois group of a finite cyclic extension $E/F$ of local or global fields, and $\varepsilon$ is a character of $ {C_E}( = {E^ \times }\;{\text{or}}\;{E^ \times }\backslash {{\mathbf{A}}^ \times })$ whose restriction to $ {C_F}$ has order $ n$, then the irreducible admissible or automorphic representations $ \pi$ of ${\text{GL}}(n)$ over $E$ with $^\sigma \pi \cong \pi \otimes \varepsilon$ are determined.
On the relative consistency strength of determinacy hypotheses
Alexander S.
Kechris;
Robert M.
Solovay
179-211
Abstract: For any collection of sets of reals $C$, let $ C{\text{-DET}}$ be the statement that all sets of reals in $C$ are determined. In this paper we study questions of the form: For given $C \subseteq C\prime$, when is $C\prime {\text{-DET}}$ equivalent, equiconsistent or strictly stronger in consistency strength than $C {\text{-DET}}$ (modulo ${\text{ZFC}}$)? We focus especially on classes $ C$ contained in the projective sets.
Bifurcation from a heteroclinic solution in differential delay equations
Hans-Otto
Walther
213-233
Abstract: We study a class of functional differential equations $\dot x(t) = af(x(t - 1))$ with periodic nonlinearity $ f:{\mathbf{R}} \to {\mathbf{R}},0 < f$ in $(A,0)$ and $f < 0$ in $(0,B),f(A) = f(0) = f(B) = 0$ . Such equations describe a state variable on a circle with one attractive rest point (given by the argument $\xi = 0$ of $f$) and with reaction lag $a$ to deviations. We prove that for a certain critical value $a = {a_0}$ there exists a heteroclinic solution going from the equilibrium solution $t \to A$ to the equilibrium $ t \to B$. For $a - {a_0} > 0$, this heteroclinic connection is destroyed, and periodic solutions of the second kind bifurcate. These correspond to periodic rotations on the circle.
Unitary structures on cohomology
C. M.
Patton;
H.
Rossi
235-258
Abstract: Let ${{\mathbf{C}}^{p + q}}$ be endowed with a hermitian form $H$ of signature $(p,q)$. Let ${M_r}$ be the manifold of $r$-dimensional subspaces of ${{\mathbf{C}}^{p + q}}$ on which $H$ is positive-definite and let $ E$ be the determinant bundle of the tautological bundle on ${M_r}$. We show (starting from the oscillator representation of ${\text{SU}}(p,q))$ that there is an invariant subspace of ${H^{r(p - r)}}({M_r},\mathcal{O}(E(p + k)))$ which defines a unitary representation of $ {\text{SU}}(p,q)$. For $W \in {M_p},\operatorname{Gr}(r,W)$ is the subvariety of $r$-dimensional subspaces of $W$. Integration over $\operatorname{Gr}(r,W)$ associates to an $ r(p - r)$-cohomology class $ \alpha$, a function $P(\alpha )$ on ${M_p}$. We show that this map is injective and provides an intertwining operator with representations of $ {\text{SU}}(p,q)$ on spaces of holomorphic functions on Siegel space
Minimal surfaces of constant curvature in $S\sp n$
Robert L.
Bryant
259-271
Abstract: In this note, we study an overdetermined system of partial differential equations whose solutions determine the minimal surfaces in ${S^n}$ of constant Gaussian curvature. If the Gaussian curvature is positive, the solution to the global problem was found by [Calabi], while the solution to the local problem was found by [Wallach]. The case of nonpositive Gaussian curvature is more subtle and has remained open. We prove that there are no minimal surfaces in ${S^n}$ of constant negative Gaussian curvature (even locally). We also find all of the flat minimal surfaces in ${S^n}$ and give necessary and sufficient conditions that a given two-torus may be immersed minimally, conformally, and flatly into ${S^n}$.
On the boundary behaviour of generalized Poisson integrals on symmetric spaces
Henrik
Schlichtkrull
273-280
Abstract: On a Riemannian symmetric space $X$ of the noncompact type we introduce a generalized Poisson transformation from functions on the minimal boundary to functions on the maximal compactification whose restrictions to $X$ are eigenfunctions of the invariant differential operators. Some continuity- and "Fatou"-theorems are proved.
Analytic uniformly bounded representations of ${\rm SU}(1,n+1)$
Ronald J.
Stanke
281-302
Abstract: By analytically continuing suitably normalized spherical principal series, a family of uniformly bounded representations of $SU(1,n + 1)$, all of which act on the same Hilbert space ${L^2}({{\mathbf{R}}^{2n + 1}})$, is constructed which is parametrized by complex numbers $s$ lying in the strip $- 1 < \operatorname{Re} (s) < 1$. The proper normalization of the principal series representations involves the intertwining operators of equivalent principal series representations. These intertwining operators are first analyzed using Fourier analysis on the Heisenberg group.
Absolutely continuous invariant measures that are maximal
W.
Byers;
A.
Boyarsky
303-314
Abstract: Let $A$ be a certain irreducible $0{\text{-}}1$ matrix and let $\tau$ denote the family of piecewise linear Markov maps on $[0,1]$ which are consistent with $ A$. The main result of this paper characterizes those maps in $\tau$ whose (unique) absolutely continuous invariant measure is maximal, and proves that for "most" of the maps that are consistent with $ A$, the absolutely continuous invariant measure is not maximal.
On the decomposition numbers of the finite general linear groups
Richard
Dipper
315-344
Abstract: Let $G = {\text{GL}_n}(q)$, $q$ a prime power, and let $r$ be an odd prime not dividing $q$. Let $s$ be a semisimple element of $G$ of order prime to $r$ and assume that $r$ divides. ${q^{\deg (\Lambda )}} - 1$ for all elementary divisors $\Lambda$ of $s$. Relating representations of certain Hecke algebras over symmetric groups with those of $ G$, we derive a full classification of all modular irreducible modules in the $ r$-block ${B_s}$ of $G$ with semisimple part $s$. The decomposition matrix $D$ of ${B_s}$ may be partly described in terms of the decomposition matrices of the symmetric groups corresponding to the Hecke algebras above. Moreover $ D$ is lower unitriangular. This applies in particular to all $r$-blocks of $G$ if $r$ divides $q - 1$. Thus, in this case, the $ r$-decomposition matrix of $ G$ is lower unitriangular.
Existence of weak solutions to stochastic differential equations in the plane with continuous coefficients
J.
Yeh
345-361
Abstract: Let $B$ be a $2$-parameter Brownian motion on ${\mathbf{R}}_ + ^2$. Consider the nonMarkovian stochastic differential system in $2$-parameter $\displaystyle \left\{ {\begin{array}{*{20}{c}} {dX(z) = \alpha (z,X)\;dB(z) + \... ...text{for}}\;z \in \partial {\mathbf{R}}_ + ^2,} \end{array} } \right.$ i.e., $\displaystyle \left\{ {\begin{array}{*{20}{c}} {X(z) = X(0) + \int_{{R_z}} {\al... ...}}_ + ^2,} {x(0) = \xi ,} & {} \end{array} } \right.$ where $ {R_z} = [0,s] \times [0,t]$ for $z = (s,t) \in {\mathbf{R}}_ + ^2$. An existence theorem for weak solutions of the system is proved in this paper. Under the assumption that $\alpha$ and $\beta$ satisfy a continuity condition and a growth condition and ${\mathbf{E}}[{\xi ^6}] < \infty$, it is shown that there exist a $2$-parameter stochastic process $X$ and a $2$-parameter Brownian motion $B$ on some probability space satisfying the stochastic integral equation above, with $ X(0)$ having the same probability distribution as $\xi$.
Strongly pure subgroups of separable torsion-free abelian groups
Loyiso G.
Nongxa
363-373
Abstract: In this paper we prove that countable strongly pure subgroups of completely decomposable groups are completely decomposable. We also show that strongly pure subgroups of separable torsion-free groups are separable.
Harmonic functions on semidirect extensions of type $H$ nilpotent groups
Ewa
Damek
375-384
Abstract: Let $S = NA$ be a semidirect extension of a Heisenberg type nilpotent group $N$ by the one-parameter group of dilations, equipped with the Riemannian structure, which generalizes this of the symmetric space. Let ${\{ {P_a}(y)\} _{a > 0}}$ be a Poisson kernel on $N$ with respect to the Laplace-Beltrami operator. Then every bounded harmonic function $F$ on $S$ is a Poisson integral $F(yb) = f \ast {P_b}(y)$ of a function $f \in {L^\infty }(N)$. Moreover the harmonic measures $\mu _a^b$ defined by ${P_b} = {P_a} \ast \mu _a^b,b > a$, are radial and have smooth densities. This seems to be of interest also in the case of a symmetric space of rank $ 1$.
Degree theory on oriented infinite-dimensional varieties and the Morse number of minimal surfaces spanning a curve in ${\bf R}\sp n$. I. $n\geq 4$
A. J.
Tromba
385-413
Abstract: A degree theory applicable to Plateau's problem is developed and the Morse equality for minimal surfaces spanning a contour in $ {{\mathbf{R}}^n},n \geq 4$, is proved.
Spectral properties of elementary operators. II
Lawrence A.
Fialkow
415-429
Abstract: Let $A = ({A_1}, \ldots ,{A_n})$ and $B = ({B_1}, \ldots ,{B_n})$ denote commutative $n$-tuples of operators on a Hilbert space $\mathcal{H}$. Let ${R_{AB}}$ denote the elementary operator on $\mathcal{L}(\mathcal{H})$ defined by $ {R_{AB}}(X) = {A_1}X{B_1} + \cdots + {A_n}X{B_n}$. We obtain new expressions for the essential spectra of ${R_{AB}}$ and $ {R_{AB}}\vert\mathcal{J}$ (the restriction of ${R_{AB}}$ to a norm ideal $\mathcal{J}$ of $ \mathcal{L}(\mathcal{H})$). We also study isolated points of joint spectra defined in the sense of $ {\text{R}}$. Harte.
Properties of center manifolds
Jan
Sijbrand
431-469
Abstract: The center manifold has a number of puzzling properties associated with the basic questions of existence, uniqueness, differentiability and analyticity which may cloud its profitable application in e.g. bifurcation theory. This paper aims to deal with some of these subtle properties. Regarding existence and uniqueness, it is shown that the cut-off function appearing in the usual existence proof is responsible for the selection of a single center manifold, thereby hiding the inherent nonuniqueness. Conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection. This intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium. In the case of nonuniqueness the difference between any two center manifolds will be less than $ {c_1}\exp ({c_2}{x^{ - 1}})$ with ${c_1}$ and ${c_2}$ constants, which explains why the formal Taylor expansions of different center manifolds are the same, while the expansions do not converge. The differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost. Center manifolds are usually not analytic; however, an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant. Using this manifold, a new and very simple proof of the Lyapunov subcenter theorem is given.
The typical structure of the sets $\{x\colon\;f(x)=h(x)\}$ for $f$ continuous and $h$ Lipschitz
Zygmunt
Wójtowicz
471-484
Abstract: Let $R$ be the space of real numbers and $ C$ the space of continuous functions $ f:[0,1] \to R$ with the uniform norm. Bruckner and Garg prove that there exists a residual set $B$ in $C$ such that for every function $f \in B$ there exists a countable dense set ${\Lambda _f}$ in $R$ such that: for $\lambda \notin {\Lambda _f}$ the top and bottom levels in the direction $\lambda$ of $f$ are singletons, in between these levels there are countably many levels in the direction $\lambda$ of $f$ that consist of a nonempty perfect set together with a single isolated point, and the remaining levels in the direction $\lambda$ of $f$ are all perfect; for $\lambda \in {\Lambda _f}$ the level structure in the direction $\lambda$ of $f$ is the same except that one (and only one) of the levels has two isolated points instead of one. In this paper we show that the analogue of the above theorem holds: if we replace the family of straight lines $\{ \lambda x + c\}$ by a $2$-parameter family $H$ that is almost uniformly Lipschitz; and if we replace $ \{ \lambda x + c\}$ by a homeomorphical image of a certain $2$-parameter family $H$ that is almost uniformly Lipschitz.
Fixed points and conjugacy classes of regular elliptic elements in ${\rm Sp}(3,{\bf Z})$
Min King
Eie;
Chung Yuan
Lin
485-496
Abstract: In this paper, we obtain $13$ isolated fixed points (up to a $ \operatorname{Sp}(3,{\mathbf{Z}})$-equivalence) and $86$ conjugacy classes of regular elliptic elements in $ \operatorname{Sp}(3,{\mathbf{Z}})$. Hence the contributions from regular elliptic conjugacy classes in $ \operatorname{Sp}(3,{\mathbf{Z}})$ to the dimension formula computed via the Selberg trace formula can be computed explictly by the main theorem of [ $ {\mathbf{4}}$ or ${\mathbf{5}}$].
The Cauchy integral, Calder\'on commutators, and conjugations of singular integrals in ${\bf R}\sp n$
Margaret A. M.
Murray
497-518
Abstract: We consider the Cauchy integral and Hilbert transform for Lipschitz domains in the Clifford algebra based on ${R^n}$. The Hilbert transform is shown to be the generating function for the Calderón commutators in ${R^n}$. We make use of an intrinsic characterization of these commutators to obtain ${L^2}$ estimates. These estimates are used to show the analyticity of the Hilbert transform and of the conjugation of singular integral operators by bi-Lipschitz changes of variable in ${R^n}$.
Quotients by ${\bf C}\sp\ast \times{\bf C}\sp\ast$ actions
Andrzej
Białynicki-Birula;
Andrew John
Sommese
519-543
Abstract: Let $T \approx {{\mathbf{C}}^\ast} \times {{\mathbf{C}}^\ast}$ act meromorphically on a compact Kähler manifold $X$, e.g. algebraically on a projective manifold. The following is a basic question from geometric invariant theory whose answer is unknown even if $X$ is projective. PROBLEM. Classify all $T$-invariant open subsets $U$ of $X$ such that the geometric quotient $ U \to U/T$ exists with $U/T$ a compact complex space (necessarily algebraic if $X$ is). In this paper a simple to state and use solution to this problem is given. The classification of $U$ is reduced to finite combinatorics. Associated to the $T$ action on $X$ is a certain finite $2$-complex $ \mathcal{C}(X)$. Certain $ \{ 0,1\}$ valued functions, called moment measures, are defined in the set of $ 2$-cells of $\mathcal{C}(X)$. There is a natural one-to-one correspondence between the $U$ with compact quotients, $U/T$, and the moment measures.
On differential equations associated with Euler product expressions
Ian
Knowles
545-573
Abstract: A method is given by which one may associate (uniquely) certain differential equations with analytic functions defined by certain Euler product expressions. Some of the consequences of this construction include results relating the location of the zeros of the analytic functions to asymptotic properties of the solutions of the differential equations, as well as a differential equation characterization of those Dirichlet series with multiplicative coefficients.
Fractional integrals on weighted $H\sp p$ spaces
Angel E.
Gatto;
Cristian E.
Gutiérrez;
Richard L.
Wheeden
575-589
Abstract: We characterize the pairs of doubling weights $(u,v)$ on ${R^n}$ such that $\displaystyle \parallel {I_\alpha }f{\parallel _{H_u^q}} \leq c\parallel f{\parallel _{H_v^p}}, \quad 0 < p < q < \infty $ , where ${I_\alpha },\alpha > 0$, is the fractional integral operator. We also consider the behavior of an associated maximal function. Applications of the results to Sobolev inequalities in weighted $ {L^p}$ spaces are given.
Invariant regions for systems of conservation laws
David
Hoff
591-610
Abstract: We describe necessary and sufficient conditions for a region in ${{\mathbf{R}}^n}$ to be invariant for (Glimm) solutions of the system of $n$ conservation laws ${u_t} + f{(u)_x} = 0$. We also make some observations about the invariance of such regions for certain finite difference approximations of solutions of systems of conservation laws.
Error bounds for Glimm difference approximations for scalar conservation laws
David
Hoff;
Joel
Smoller
611-642
Abstract: We derive error bounds for the Glimm difference approximation to the solution of a genuinely nonlinear scalar conservation law with BV initial data. We show that the ${L^1}$ error is bounded by $O(\Delta {x^{1/6}}\vert\log \Delta x\vert)$ in the general case, and by $O(\Delta {x^{1/2}}\vert\log \Delta x\vert)$ for a generic class of piecewise constant data.
A kernel approach to the local solvability of the tangential Cauchy Riemann equations
A.
Boggess;
M.-C.
Shaw
643-658
Abstract: An integral kernel approach is given for the proof of the theorem of Andreotti and Hill which states that the $Y(q)$ condition of Kohn is a sufficient condition for local solvability of the tangential Cauchy Riemann equations on a real hypersurface in $ {{\mathbf{C}}^n}$. In addition, we provide an integral kernel approach to nonsolvability for a certain class of real hypersurfaces in the case when $Y(q)$ is not satisfied.
Convexity conditions and intersections with smooth functions
S.
Agronsky;
A. M.
Bruckner;
M.
Laczkovich;
D.
Preiss
659-677
Abstract: A continuous function that agrees with each member of a family $\mathcal{F}$ of smooth functions in a small set must itself possess certain desirable properties. We study situations that arise when $\mathcal{F}$ consists of the family of polynomials of degree at most $n$, as well as certain larger families and when the small sets of agreement are finite. The conclusions of our theorems involve convexity conditions. For example, if a continuous function $f$ agrees with each polynomial of degree at most $n$ in only a finite set, then $f$ is $(n + 1)$-convex or $(n + 1)$-concave on some interval. We consider also certain variants of this theorem, provide examples to show that certain improvements are not possible and present some applications of our results.
Six standard deviations suffice
Joel
Spencer
679-706
Abstract: Given $ n$ sets on $n$ elements it is shown that there exists a two-coloring such that all sets have discrepancy at most $K{n^{1/2}}$, $K$ an absolute constant. This improves the basic probabilistic method with which $K = c{(\ln n)^{1/2}}$. The result is extended to $ n$ finite sets of arbitrary size. Probabilistic techniques are melded with the pigeonhole principle. An alternate proof of the existence of Rudin-Shapiro functions is given, showing that they are exponential in number. Given $n$ linear forms in $n$ variables with all coefficients in $[ - 1, + 1]$ it is shown that initial values $ {p_1}, \ldots ,{p_n} \in \{ 0,1\}$ may be approximated by ${\varepsilon _1}, \ldots ,{\varepsilon _n} \in \{ 0,1\}$ so that the forms have small error.
Convolution equations in spaces of distributions with one-sided bounded support
R.
Shambayati;
Z.
Zielezny
707-713
Abstract: Let $ \mathcal{D}\prime(0,\infty )$ be the space of distributions on $ R$ with support in $[0,\infty )$ and $\mathcal{S}\prime(0,\infty )$ its subspace consisting of tempered distributions. We characterize the distributions $S \in \mathcal{D}\prime(0,\infty )$ for which $S\, \ast \mathcal{D}\prime(0,\infty ) = \mathcal{D}\prime(0,\infty )$, where $\ast$ is the convolution. We also characterize the distributions $S \in \mathcal{S}\prime(0,\infty )$ for which $S \ast \mathcal{S}\prime(0,\infty ) = \mathcal{S}\prime(0,\infty )$.
Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces; the case of finite volume
Peter D.
Lax;
Ralph S.
Phillips
715-735
Abstract: Let $\Gamma$ be a discrete subgroup of automorphisms of $ {{\mathbf{H}}^n}$, with fundamental polyhedron of finite volume, finite number of sides, and $N$ cusps. Denote by $ {\Delta _\Gamma }$ the Laplace-Beltrami operator acting on functions automorphic with respect to $\Gamma$. We give a new short proof of the fact that ${\Delta _\Gamma }$ has absolutely continuous spectrum of uniform multiplicity $N$ on $( - \infty ,{((n - 1)/2)^2})$, plus a standard discrete spectrum. We show that this property of the spectrum is unchanged under arbitrary perturbation of the metric on a compact set. Our method avoids Eisenstein series entirely and proceeds instead by constructing explicitly a translation representation for the associated wave equation.
On the proper Steenrod homotopy groups, and proper embeddings of planes into $3$-manifolds
Matthew G.
Brin;
T. L.
Thickstun
737-755
Abstract: Standard algebraic invariants of proper homotopy type are discussed. These do not naturally fit into long exact sequences. Groups of proper homotopy classes of proper maps of Euclidean spaces and open annuli which do naturally form a long exact sequence are defined, and a diagram relating these groups to the standard algebraic invariants of proper homotopy type is given. The structures defined are used to compare several notions of essentiality for proper maps. Some results and examples are given for proper maps of spaces into manifolds of dimension $ 2$ and $3$. These results are used to add information to a theorem of Brown and Feustel about properly embedding planes in $3$-manifolds.
Obstruction theory and multiparameter Hopf bifurcation
Jorge
Ize
757-792
Abstract: The Hopf bifurcation problem is treated as an example of an equivariant bifurcation. The existence of a local bifurcating solution is given by the nonvanishing of an obstruction to extending a map defined on a complex projective space and is computed using the complex Bott periodicity theorem. In the case of the classical Hopf bifurcation the results of Chow, Mallet-Paret and Yorke are recovered without using any special index as the Fuller degree: There is bifurcation if the number of exchanges of stability is nonzero. A global theorem asserts that the sum of the local invariants on a bounded component of solutions must be zero.
Equivariant diffeomorphisms with simple recurrences on two-manifolds
W.
de Melo;
G. L.
dos Reis;
P.
Mendes
793-807
Abstract: We consider the class of diffeomorphisms, on compact two-dimensional manifolds, which are invariant under the action of a compact Lie group $G$ and whose nonwandering set consists of a finite number of $G$-orbits. We describe the modulus of stability of almost all diffeomorphisms in this class.
Directed graphs and traveling waves
David
Terman
809-847
Abstract: The existence of traveling wave solutions for equations of the form $ {u_t} = {u_{xx}} + F\prime(u)$ is considered. All that is assumed about $ F$ is that it is sufficiently smooth, ${\lim _{\vert u\vert \to \infty }}F(u) = - \infty$, $F$ has only a finite number of critical points, each of which is nondegenerate, and if $ A$ and $B$ are distinct critical points of $ F$, then $F(A) \ne F(B)$. The results demonstrate that, for a given function $F$, there may exist zero, exactly one, a finite number, or an infinite number of waves which connect two fixed, stable rest points. The main technique is to identify the phase planes, which arise naturally from the problem, with an array of integers. While the phase planes may be very complicated, the arrays of integers are always quite simple to analyze. Using the arrays of integers one is able to construct a directed graph; each path in the directed graph indicates a possible ordering, starting with the fastest, of which waves must exist. For a large class of functions $ F$ one is then able to use the directed graphs in order to determine how many waves connect two stable rest points.
Characteristic classes of transversely homogeneous foliations
Chal
Benson;
David B.
Ellis
849-859
Abstract: The foliations studied in this paper have transverse geometry modeled on a homogeneous space $G/H$ with transition functions given by the left action of $G$. It is shown that the characteristic classes for such a foliation are determined by invariants of a certain flat bundle. This is used to prove that when $ G$ is semisimple, the characteristic classes are rigid under smooth deformations, extending work of Brooks, Goldman and Heitsch.
The Dirac operator on spaces with conical singularities and positive scalar curvatures
Arthur Weichung
Chou
1-40
Abstract: We study, in the spirit of Jeff Cheeger, the Dirac operator on a space with conical singularities. We obtain a Bochner-type vanishing theorem and prove an index theorem in the singular case. Also, the relationship with manifolds with boundary is explored. In the Appendix two methods of deforming the metric near the boundary are established and applied to obtain several new results on constructing complete metrics with positive scalar curvature.
Degrees of indiscernibles in decidable models
H. A.
Kierstead;
J. B.
Remmel
41-57
Abstract: We show that the problem of finding an infinite set of indiscernibles in an arbitrary decidable model of a first order theory is essentially equivalent to the problem of finding an infinite path through a recursive $\omega$-branching tree. Similarly, we show that the problem of finding an infinite set of indiscernibles in a decidable model of an $\omega$-categorical theory with decidable atoms is essentially equivalent to finding an infinite path through a recursive binary tree.
Spaces with coarser minimal Hausdorff topologies
Jack
Porter;
Johannes
Vermeer
59-71
Abstract: A technique is developed, using $H$-closed extensions, for determining when certain Hausdorff spaces are Katetov, i.e., have a coarser minimal Hausdorff topology. Our technique works for Čech-complete Lindelöf spaces, complete metrizable spaces, and many other spaces. Also, a number of interesting examples are presented; the most striking is an example of a Katetov space whose semiregularization is not Katetov.
Regularity properties of solutions to the basic problem in the calculus of variations
F. H.
Clarke;
R. B.
Vinter
73-98
Abstract: This paper concerns the basic problem in the calculus of variations: minimize a functional $J$ defined by $\displaystyle J(x) = \int_a^b {L(t,x(t),\dot x(t))\;dt}$ over a class of arcs $ x$ whose values at $ a$ and $b$ have been specified. Existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs, conditions which must be strengthened in order that the standard necessary conditions apply. The question arises: What necessary conditions hold merely under hypotheses of existence theory, say the classical Tonelli conditions? It is shown that, given a solution $x$, there exists a relatively open subset $ \Omega$ of $[a,b]$, of full measure, on which $ x$ is locally Lipschitz and satisfies a form of the Euler-Lagrange equation. The main theorem, of which this is a corollary, can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions. Three such classes are identified, and results of Bernstein, Tonelli, and Morrey are extended. One of these classes is of a novel nature, and its study implies the new result that when $L$ is independent of $t$, the solution has essentially bounded derivative.
General defect relations of holomorphic curves
Kiyoshi
Niino
99-113
Abstract: Let $ x:{\mathbf{C}} \to {P_n}{\mathbf{C}}$ be a holomorphic curve of finite lower order $ \mu$, and let $A = \{ \alpha \}$ be an arbitrary finite family of holomorphic curves $\alpha :{\mathbf{C}} \to {({P_n}{\mathbf{C}})^\ast}$ satisfying $ T(r,\alpha ) = o(T(r,x))\;(r \to \infty )$. Suppose $x$ is nondegenerate with respect to $A$, and $A$ is in general position. We show the following general defect relations: (1) $ x$ has at most $ n$ deficient curves in $ A$ if $\mu = 0$. (2) $\sum\nolimits_{\alpha \in A} {\delta (\alpha ) \leq n\;{\text{if}}\;0 < \mu \leq 1/2}$. (3) $ \sum\nolimits_{\alpha \in A} {\delta (\alpha ) \leq [2n\mu ] + 1\;{\text{if}}\;1/2 < \mu < + \infty } $.
Anosov diffeomorphisms and expanding immersions. I
Lowell
Jones
115-131
Abstract: The purpose of this paper is to develop a theory for representing Anosov diffeomorphisms by expanding immersions on compact branched manifolds. This theory was motivated by R. F. Williams' study of expanding attractors [15,17].
Abstract theory of abelian operator algebras: an application of forcing
Thomas J.
Jech
133-162
Abstract: The abstract abelian operator theory is developed from a general standpoint, using the method of forcing and Boolean-valued models.
Uniform operators
Hsiao Lan
Wang;
Joseph G.
Stampfli
163-169
Abstract: A general representation theorem for uniform operators is obtained which enables one to verify a conjecture of Cowen and Douglas in the presence of a mild additional restriction.
On the index of a number field
Enric
Nart
171-183
Abstract: Arithmetic invariants are found which determine the index $i(K)$ of a number field $K$. They are used to obtain an explicit formula under certain restrictions on $ K$. They provide also a complete explanation of a phenomenon conjectured by Ore [ ${\mathbf{8}}$] and showed by Engstrom in a particular case [ $ {\mathbf{2}}$].
On contractions of semisimple Lie groups
A. H.
Dooley;
J. W.
Rice
185-202
Abstract: A limiting formula is given for the representation theory of the Cartan motion group associated to a Riemannian symmetric pair $(G,K)$ in terms of the representation theory of $ G$.
Some structure theory for a class of triple systems
Nora C.
Hopkins
203-212
Abstract: This paper deals with a class of triple systems satisfying two generalized five linear identities and having nondegenerate bilinear forms with certain properties. If $(M,\{ ,,\} )$ is such a triple system with bilinear form $\phi (,)$, it is shown that if $M$ is semisimple, then $ M$ is the direct sum of simple ideals if $\phi$ is symmetric or symplectic or if $ M$ is completely reducible as a module for its right multiplication algebra $\mathcal{L}$. It is also shown that if $M$ is a completely reducible $\mathcal{L}$-module, $M$ is the direct sum of a semisimple ideal and the center of $M$. Such triple systems can be embedded into certain nonassociative algebras and the results on the triple systems are extended to these algebras.
Complete linear systems on rational surfaces
Brian
Harbourne
213-226
Abstract: We determine the dimension, fixed components and base points of complete linear systems on blowings-up of ${{\mathbf{P}}^2}$ having irreducible anticanonical divisor.
A nonshrinkable decomposition of $S\sp 3$ whose nondegenerate elements are contained in a cellular arc
W. H.
Row;
John J.
Walsh
227-252
Abstract: A decomposition $ G$ of ${S^3}$ is constructed with the following properties: (1) The set ${N_G}$ of all nondegenerate elements consists of a null sequence of arcs and $ J = {\text{CL}}( \cup \{ g \in {N_G}\} )$ is a simple closed curve. (2) Each arc contained in $J$ is cellular. (3) $J$ is the boundary of a disk $Q$ that is locally flat except at points of $J$. (4) The decomposition $G$ is not shrinkable; that is, the decomposition space is not homeomorphic to $ {S^3}$.
The splittability and triviality of $3$-bridge links
Seiya
Negami;
Kazuo
Okita
253-280
Abstract: A method to simplify $3$-bridge projections of links and knots, called a wave move, is discussed in general situation and it is shown what kind of properties of $3$-bridge links and knots can be recognized from their projections by wave moves. In particular, it will be proved that every $3$-bridge projection of a splittable link or a trivial knot can be transformed into a disconnected one or a hexagon, respectively, by a finite sequence of wave moves. As its translation via the concept of $2$-fold branched coverings of ${S^3}$, it follows that every genus $ 2$ Heegaard diagram of ${S^2} \times {S^2}\char93 L(p,q)$ or $ {S^3}$ can be transformed into one of specific standard forms by a finite sequence of operations also called wave moves.
On the asphericity of ribbon disc complements
James
Howie
281-302
Abstract: The complement of a ribbon $n$-disc in the $(n + 2)$-ball has a $2$-dimensional spine which shares some of the combinatorial properties of classical knot complement spines. It is an open question whether such $2$-complexes are always aspherical. To any ribbon disc we associate a labelled oriented tree, from which the homotopy type of the complement can be recovered, and we prove asphericity in certain special cases described by conditions on this tree. Our main result is that the complement is aspherical whenever the associated tree has diameter at most $ 3$.
Deductive varieties of modules and universal algebras
Leslie
Hogben;
Clifford
Bergman
303-320
Abstract: A variety of universal algebras is called deductive if every subquasivariety is a variety. The following results are obtained: (1) The variety of modules of an Artinian ring is deductive if and only if the ring is the direct sum of matrix rings over local rings, in which the maximal ideal is principal as a left and right ideal. (2) A directly representable variety of finite type is deductive if and only if either (i) it is equationally complete, or (ii) every algebra has an idempotent element, and a ring constructed from the variety is of the form (1) above.
Filtering cohomology and lifting vector bundles
E. Graham
Evans;
Phillip
Griffith
321-332
Abstract: For a module $ M$ over a local Cohen-Macaulay ring $R$ we develop a (finite) sequence of presentations of $ M$ which facilitates the study of invariants arising from the cohomology modules of $M$. As an application we use this data, in case $ R$ is regular and $ M$ represents a vector bundle on the punctured spectrum of $R$ with a vanishing cohomology module, to obtain bounds on how far $M$ can be lifted as a vector bundle.
Monge-Amp\`ere measures associated to extremal plurisubharmonic functions in ${\bf C}\sp n$
Norman
Levenberg
333-343
Abstract: We consider the extremal plurisubharmonic functions $L_E^\ast$ and $U_E^\ast$ associated to a nonpluripolar compact subset $E$ of the unit ball $B \subset {{\mathbf{C}}^n}$ and show that the corresponding Monge-Ampère measures ${(d{d^c}L_E^\ast )^n}$ and ${(d{d^c}U_E^\ast )^n}$ are mutually absolutely continuous. We then discuss the polynomial growth condition $({L^\ast})$, a generalization of Leja's polynomial condition in the plane, and study the relationship between the asymptotic behavior of the orthogonal polynomials associated to a measure on $E$ and the $ ({L^\ast})$ condition.
Points fixes d'applications holomorphes dans un domaine born\'e convexe de ${\bf C}\sp n$
Jean-Pierre
Vigué
345-353
Abstract: Let $D$ be a bounded convex domain in $ {{\mathbf{C}}^n}$. We prove that the set $V$ of fixed points of a holomorphic map $ f:D \to D$ is a complex submanifold of $D$ and, if $V$ is not empty, $V$ is a holomorphic retract of $D$.
Chaos, periodicity, and snakelike continua
Marcy
Barge;
Joe
Martin
355-365
Abstract: The results of this paper relate the dynamics of a continuous map $ f$ of the interval and the topology of the inverse limit space with bonding map $ f$. These inverse limit spaces have been studied by many authors, and are examples of what Bing has called "snakelike continua". Roughly speaking, we show that when the dynamics of $ f$ are complicated, the inverse limit space contains indecomposable subcontinua. We also establish a partial converse.
The fine structure of transitive Riemannian isometry groups. I
Carolyn S.
Gordon;
Edward N.
Wilson
367-380
Abstract: Let $M$ be a connected homogeneous Riemannian manifold, $G$ the identity component of the full isometry group of $M$ and $H$ a transitive connected subgroup of $G$. $G = HL$, where $L$ is the isotropy group at some point of $ M$. $M$ is naturally identified with the homogeneous space $ H/H \cap L$ endowed with a suitable left-invariant Riemannian metric. This paper addresses the problem: Given a realization of $ M$ as a Riemannian homogeneous space of a connected Lie group $H$, describe the structure of the full connected isometry group $G$ in terms of $H$. This problem has already been studied in case $ H$ is compact, semisimple of noncompact type, or solvable. We use the fact that every Lie group is a product of subgroups of these three types in order to study the general case.
On the ideals of a Noetherian ring
J. T.
Stafford
381-392
Abstract: We construct various examples of Noetherian rings with peculiar ideal structure. For example, there exists a Noetherian domain $ R$ with a minimal, nonzero ideal $I$, such that $R/I$ is a commutative polynomial ring in $ n$ variables, and a Noetherian domain $S$ with a (second layer) clique that is not locally finite. The key step in the construction of these rings is to idealize at a right ideal $I$ in a Noetherian domain $ T$ such that $ T/I$ is not Artinian.
Ergodic semigroups of epimorphisms
Daniel
Berend
393-407
Abstract: The conditions for ergodicity of semigroups of epimorphisms of compact groups are studied. In certain cases ergodic semigroups are shown to contain small ergodic subsemigroups. Properties related to ergodicity, such as that of admitting no infinite closed invariant proper subset of the group, are discussed for semigroups of epimorphisms and of affine transformations.
On the structure of weakly compact subsets of Hilbert spaces and applications to the geometry of Banach spaces
S.
Argyros;
V.
Farmaki
409-427
Abstract: A characterization of weakly compact subsets of a Hilbert space, when they are considered as subsets of $B$-spaces with an unconditional basis, is given. We apply this result to renorm a class of reflexive $ B$-spaces by defining a norm uniformly convex in every direction. We also prove certain results related to the factorization of operators. Finally, we investigate the structure of weakly compact subsets of $ {L^1}(\mu )$.
Erratum to: ``On wave fronts propagation in multicomponent media'' [Trans. Amer. Math. Soc. {\bf 276} (1983), no. 1, 181--191; MR0684501 (84a:35027)]
M. I.
Freidlin
429
Analysis on analytic spaces and non-self-dual Yang-Mills fields
N. P.
Buchdahl
431-469
Abstract: This paper gives a cohomological description of the Witten-Isenberg-Yasskin-Green generalization to the non-self-dual case of Ward's twistor construction for self-dual Yang-Mills fields. The groundwork for this description is presented in Part I: with a brief introduction to analytic spaces and differential forms thereon, it contains an investigation of the exactness of the holomorphic relative de Rham complex on formal neighbourhoods of submanifolds, results giving sufficient conditions for the invertibility of pull-back functors on categories of analytic objects, and a discussion of the extension problem for analytic objects in the context of the formalism earlier introduced. Part II deals with non-self-dual Yang-Mills fields: the Yang-Mills field and current are identified in terms of the Griffiths obstructions to extension, including a proof of Manin's result that "current = obstruction to third order". All higher order obstructions are identified, there being at most ${N^2}$ for a bundle of rank $N$. An ansatz for producing explicit examples of non-self-dual fields is obtained by using the correspondence. This ansatz generates $ {\text{SL}}(2,\mathbb{C})$ solutions with topological charge $1$ on ${S^4}$.
No $L\sb 1$-contractive metrics for systems of conservation laws
Blake
Temple
471-480
Abstract: Let $(\ast)$ $\displaystyle \quad {u_t} + F{(u)_x} = 0$ be any $2 \times 2$ system of conservation laws satisfying certain generic assumptions on $F$ in a neighborhood $ \mathcal{N}$ of $ u$-space. We prove that for every nondegenerate metric $D$ on $u$-space there exists states ${u_1}$ and ${u_2}$ in $ \mathcal{N}$ such that $\int_{ - \infty }^\infty {D(u(x,t),{u_1})\;dx}$ is a strictly increasing function of $t$ in a neighborhood of $ t = 0$, where $ u$ is the admissible solution of $( \ast )$ with initial data $\displaystyle u(x,0) = \left\{ {\begin{array}{*{20}{c}} {{u_1},} & {x \l... ...,} {{u_1},} & {x \geqslant 1.} \end{array} } \right.$ This contrasts with the case of a scalar equation in which $ \int_{ - \infty }^\infty {D(u(x,t),v(x,t))\;dx}$ is a decreasing function of $ t$ for all admissible solution pairs $u$ and $v$ when $D$ is taken to be the absolute value norm.
Semigroups in Lie groups, semialgebras in Lie algebras
Joachim
Hilgert;
Karl H.
Hofmann
481-504
Abstract: Consider a subsemigroup of a Lie group containing the identity and being ruled by one-parameter semigroups near the identity. We associate with it the set $W$ of its tangent vectors at the identity and obtain a subset of the Lie algebra $ L$ of the group. The set $ W$ has the following properties: (i) $W + W = W$, (ii) ${{\mathbf{R}}^ + }\,\cdot\;W \subset W$, (iii) ${W^ - } = W$, and, the crucial property, (iv) for all sufficiently small elements $ x$ and $y$ in $W$ one has $x \ast y = x + y + \frac{1} {2}[x,y] + \cdots$ (Campbell-Hausdorff!) $\in W$. We call a subset $W$ of a finite-dimensional real Lie algebra $L$ a Lie semialgebra if it satisfies these conditions, and develop a theory of Lie semialgebras. In particular, we show that a subset $W$ satisfying (i)-(iii) is a Lie semialgebra if and only if, for each point $x$ of $W$ and the (appropriately defined) tangent space ${T_x}$ to $W$ in $x$, one has $[x,{T_x}] \subset {T_x}$. (The Lie semialgebra $W$ of a subgroup is always a vector space, and for vector spaces $W$ we have ${T_x} = W$ for all $x$ in $W$, and thus the condition reduces to the old property that $W$ is a Lie algebra.) In the introduction we fully discuss all Lie semialgebras of dimension not exceeding three. Our methods include a full duality theory for closed convex wedges, basic Lie group theory, and certain aspects of ordinary differential equations.
On the structure of abelian $p$-groups
Paul
Hill
505-525
Abstract: A new kind of abelian $p$-group, called an $A$-group, is introduced. This class contains the totally projective groups and Warfield's $S$-groups as special cases. It also contains the $N$-groups recently classified by the author. These more general groups are classified by cardinal (numerical) invariants which include, but are not limited to, the Ulm-Kaplansky invariants. Thus the existing theory, as well as the classification, of certain abelian $p$-groups is once again generalized. Having classified $A$-groups (by means of a uniqueness and corresponding existence theorem) we can successfully study their structure and special properties. Such a study is initiated in the last section of the paper.
On isometric embeddings of graphs
R. L.
Graham;
P. M.
Winkler
527-536
Abstract: If $G$ is a finite connected graph with vertex set $V$ and edge set $E$, a standard way of defining a distance $ {d_G}$ on $G$ is to define ${d_G}(x,y)$ to be the number of edges in a shortest path joining $x$ and $y$ in $V$. If $(M,{d_M})$ is an arbitrary metric space, then an embedding $ \lambda :V \to M$ is said to be isometric if ${d_G}(x,y) = {d_M}(\lambda (x),\lambda (y))$ for all $x,y \in V$. In this paper we will lay the foundation for a theory of isometric embeddings of graphs into cartesian products of metric spaces.
Condensed Julia sets, with an application to a fractal lattice model Hamiltonian
M. F.
Barnsley;
J. S.
Geronimo;
A. N.
Harrington
537-561
Abstract: The Julia set for the complex rational map $z \to {z^2} - \lambda + \varepsilon /z$, where $ \lambda$ and $\varepsilon$ are complex parameters, is considered in the limit as $ \varepsilon \to 0$. The result is called the condensed Julia set for $z \to ({z^3} - \lambda z)/z$. The limit of balanced measures, associated functional equations and orthogonal polynomials are considered; it is shown, for example, that for $\lambda \geqslant 2$ the moments, orthogonal polynomials, and associated Jacobi matrix $\mathcal{J}$ can be calculated explicitly and are not those belonging to ${z^2} - \lambda$. The spectrum of $\mathcal{J}$ consists of a point spectrum $ P$ together with its derived set. The latter is the Julia set for ${z^2} - \lambda$, and carries none of the spectral mass when $ \lambda > 2$. When $\lambda = 2$, $P$ is dense in $[-2,2]$. A similar condensation in the case $\lambda = 15/4$ leads to a system which corresponds precisely to the spectrum and density of states of a two-dimensional Sierpinski gasket model Schrödinger equation. The basic ideas about condensation of Julia sets in general are described. If $R(z)$ is a rational transformation of degree greater than one, then condensation can be attached to $\displaystyle z \to R(z) + \varepsilon \sum\limits_{i = 1}^k {{{(z - {a_i})}^{ - {\gamma _i}}},}$ where the ${\gamma _i}$'s and $k$ are finite positive integers and the $ {a_i}$'s are complex numbers. If $\infty$ is an indifferent or attractive fixed point of $R(z)$, then all of the moments of the associated condensed balanced measure can be calculated explicitly, as can the orthogonal polynomials when the condensed Julia set is real. Sufficient conditions for the condensed measure $\sigma$ to be a weak limit of the balanced measures $ {\mu _\varepsilon }$ are given. Functional equations connected to the condensed measure are derived, and it is noted that their form typifies those encountered in statistical physics, in connection with partition functions for Ising hierarchical models.
Grade schemes and grade functions
Stephen
McAdam
563-590
Abstract: In recent years, two concepts similar to $R$-sequences have appeared, essential sequences and asymptotic sequences. This work explores the general nature of such sequences.
Index theory on curves
Peter
Haskell
591-604
Abstract: This paper constructs from the $\bar \partial $-operator on the smooth part of a complex projective algebraic curve a cycle in the analytically defined $K$ homology of the curve. The paper identifies the corresponding cycle in the topologically defined $ K$ homology.
Primitive group rings and Noetherian rings of quotients
Christopher J. B.
Brookes;
Kenneth A.
Brown
605-623
Abstract: Let $k$ be a field, and let $G$ be a countable nilpotent group with centre $Z$. We show that the group algebra $kG$ is primitive if and only if $ k$ is countable, $ G$ is torsion free, and there exists an abelian subgroup $A$ of $G$, of infinite rank, with $A \cap Z = 1$. Suppose now that $G$ is torsion free. Then $kG$ has a partial quotient ring $Q = kG{(kZ)^{ - 1}}$. The above characterisation of the primitivity of $kG$ is intimately connected with the question: When is $Q$ a Noetherian ring? We determine this for those groups $G$, as above, all of whose finite rank subgroups are finitely generated. In this case, $Q$ is Noetherian if and only if $ G$ has no abelian subgroup $ A$ of infinite rank with $A \cap Z = 1$.
Some remarks on the intrinsic measures of Eisenman
Ian
Graham;
H.
Wu
625-660
Abstract: This paper studies the intrinsic measures on complex manifolds first introduced by Eisenman in analogy with the intrinsic distances of Kobayashi. Some standard conjectures, together with several new ones, are considered and partial or complete answers are provided. Most of the counterexamples come from a closer examination of unbounded domains in complex euclidean space. In particular, a large class of unbounded hyperbolic domains are exhibited. Those unbounded domains of finite euclidean volume are also singled out for discussion.
Lipschitzian mappings and total mean curvature of polyhedral surfaces. I
Ralph
Alexander
661-678
Abstract: For a smooth closed surface $C$ in ${E^3}$ the classical total mean curvature is defined by $M(C) = \frac{1} {2}\int ({\kappa _1} + {\kappa _2})\;d\sigma (p)$, where ${\kappa _1},{\kappa _2}$ are the principal curvatures at $p$ on $C$. If $C$ is a polyhedral surface, there is a well known discrete version given by $M(C) = \frac{1} {2}\Sigma {l_i}(\pi - {\alpha _i})$, where ${l_i}$ represents edge length and ${\alpha _i}$ the corresponding dihedral angle along the edge. In this article formulas involving differentials of total mean curvature (closely related to the differential formula of L. Schláfli) are applied to several questions concerning Lipschitizian mappings of polyhedral surfaces. For example, the simplest formula $\Sigma {l_i}\,d{\alpha _i} = 0$ may be used to show that the remarkable flexible polyhedral spheres of R. Connelly must flex with constant total mean curvature. Related differential formulas are instrumental in showing that if $ f: {E^2} \to {E^2}$ is a distance-increasing function and $K \subset {E^2}$, then $ \operatorname{Per}(\operatorname{conv}\;K) \leqslant \operatorname{Per}(\operatorname{conv}\;f[K])$. This article (part I) is mainly concerned with problems in ${E^n}$. In the sequel (part II) related questions in ${S^n}$ and ${H^n}$, as well as ${E^n}$, will be considered.
Packing measure, and its evaluation for a Brownian path
S. James
Taylor;
Claude
Tricot
679-699
Abstract: A new measure on the subsets $E \subset {{\mathbf{R}}^d}$ is constructed by packing as many disjoint small balls as possible with centres in $E$. The basic properties of $\phi$-packing measure are obtained: many of these mirror those of $\phi$-Hausdorff measure. For $\phi (s) = {s^2}/(\log \,\log (1/s))$, it is shown that a Brownian trajectory in ${{\mathbf{R}}^d}(d \geqslant 3)$ has finite positive $\phi$-packing measure.
Simplexwise linear near-embeddings of a $2$-disk into ${\bf R}\sp 2$
Ethan D.
Bloch
701-722
Abstract: Let $K \subset {{\mathbf{R}}^2}$ be a finitely triangulated $2$-disk; a map $f:K \to {{\mathbf{R}}^2}$ is called simplexwise linear $(SL)$ if $f\vert\sigma$ is affine linear for each (closed) simplex $\sigma$ of $K$. Interest in $ {\text{SL}}$ maps originated with work of S. S. Cairns and subsequent work of R. Thom and N. H. Kuiper. Let $E(K) = \{ {\text{orientation preserving SL embeddings}}\;K \to {{\mathbf{R}}^2}\}$, $L(K) = \{ {\text{SL homeomorphism}}\;K \to K\;{\text{fixing}}\;\partial K\;{\text{pointwise}}\}$, and $\overline {E(K)} ,\overline {L(K)}$ denote their respective closures in the space of all ${\text{SL}}$ maps $K \to {{\mathbf{R}}^2}$ and the space of all ${\text{SL}}$ maps $K \to K$ fixing $ \partial K$. The main result of this paper is useful characterizations of maps in $\overline {L(K)}$ and some maps in $\overline {E(K)}$, including the relation of such maps to $ {\text{SL}}$ embeddings into the nonstandard plane.
Strictly convex simplexwise linear embeddings of a $2$-disk
Ethan D.
Bloch
723-737
Abstract: Let $K \subset {{\mathbf{R}}^2}$ be a finitely triangulated $2$-disk; a map $f:K \to {{\mathbf{R}}^2}$ is called simplexwise linear $(SL)$ if $f\vert\sigma$ is affine linear for each (closed) $ 2$-simplex $\sigma$ of $K$. Let $ E(K) = \{ {\text{orientation preserving SL embeddings}}\;K \to {{\mathbf{R}}^2}\}$, $ {E_{{\text{SC}}}}(K) = \{ f \in E(K)\vert f(K)\;{\text{is strictly convex}}\}$, and let $ \overline {E(K)}$ and $ \overline {{E_{{\text{SC}}}}(K)}$ denote their closures in the space of all ${\text{SL}}$ maps $K \to {{\mathbf{R}}^2}$. A characterization of certain elements of $ \overline {E(K)}$ is used to prove that $ {E_{{\text{SC}}}}(K)$ has the homotopy type of ${S^1}$ and to characterize those elements of $\overline {E(K)}$ which are in $ \overline {{E_{{\text{SC}}}}(K)}$, as well as to relate such maps to ${\text{SL}}$ embeddings into the nonstandard plane.
General position properties satisfied by finite products of dendrites
Philip L.
Bowers
739-753
Abstract: Let $\bar A$ be a dendrite whose endpoints are dense and let $A$ be the complement in $\bar A$ of a dense $\sigma$-compact collection of endpoints of $ \bar A$. This paper investigates various general position properties that finite products of $\bar A$ and $A$ possess. In particular, it is shown that (i) if $X$ is an $L{C^n}$-space that satisfies the disjoint $ n$-cells property, then $X \times \bar A$ satisfies the disjoint $ (n + 1)$-cells property, (ii) $ {\bar A^n} \times [ - 1,1]$ is a compact $(n + 1)$-dimensional $ {\text{AR}}$ that satisfies the disjoint $n$-cells property, (iii) ${\bar A^{n + 1}}$ is a compact $(n + 1)$-dimensional ${\text{AR}}$ that satisfies the stronger general position property that maps of $ n$-dimensional compacta into ${\bar A^{n + 1}}$ are approximable by both $Z$-maps and ${Z_n}$-embeddings, and (iv) ${A^{n + 1}}$ is a topologically complete $ (n + 1)$-dimensional ${\text{AR}}$ that satisfies the discrete $ n$-cells property and as such, maps from topologically complete separable $ n$-dimensional spaces into ${A^{n + 1}}$ are strongly approximable by closed ${Z_n}$-embeddings.
Varieties of automorphism groups of orders
W. Charles
Holland
755-763
Abstract: The group $A(\Omega )$ of automorphisms of a totally ordered set $\Omega$ must generate either the variety of all groups or the solvable variety of class $n$. In the former case, $A(\Omega )$ contains a free group of rank ${2^{{\aleph _0}}}$; in the latter case, $A(\Omega )$ contains a free solvable group of class $n - 1$ and rank ${2^{{\aleph _0}}}$.
Helical minimal immersions of compact Riemannian manifolds into a unit sphere
Kunio
Sakamoto
765-790
Abstract: An isometric immersion of a Riemannian manifold $M$ into a Riemannian manifold $\overline M$ is called helical if the image of each geodesic has constant curvatures which are independent of the choice of the particular geodesic. Suppose $ M$ is a compact Riemannian manifold which admits a minimal helical immersion of order $4$ into the unit sphere. If the Weinstein integer of $ M$ equals that of one of the projective spaces, then $M$ is isometric to that projective space with its canonical metric.
Homomorphisms between generalized Verma modules
Brian D.
Boe
791-799
Abstract: Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra and $ \mathfrak{p}$ a parabolic subalgebra. The first result is a necessary and sufficient condition, in the spirit of the Bernstein-Gelfand-Gelfand theorem on Verma modules, for Lepowsky's "standard map" between two generalized Verma modules for $\mathfrak{g}$ to be zero. The main result gives a complete description of all homomorphisms between the generalized Verma modules induced from one-dimensional $ \mathfrak{p}$-modules, in the "hermitian symmetric" situation.
Piecewise continuous almost periodic functions and mean motions
Jing Bo
Xia
801-811
Abstract: In this paper, we prove the existence of mean motion for certain noncontinuous almost periodic functions.
A relation between invariant means on Lie groups and invariant means on their discrete subgroups
John R.
Grosvenor
813-825
Abstract: Let $G$ be a Lie group, and let $ D$ be a discrete subgroup of $G$ such that the right coset space $D\backslash G$ has finite right-invariant volume. We will exhibit an injection of left-invariant means on ${l^\infty }(D)$ into left-invariant means on the left uniformly continuous bounded functions of $ G$. When $G$ is an abelian Lie group with finitely many connected components, we also show surjectivity, and when $G$ is the additive group ${{\mathbf{R}}^n}$ and $D$ is $ {{\mathbf{Z}}^n}$, the bijection will explicitly take the form of an integral over the unit cube ${[0,1]^n}$.
The bidual of the compact operators
Theodore W.
Palmer
827-839
Abstract: Let $X$ be a Banach space such that $ {X^\ast}$ has the Radon-Nikodým property. If ${X^\ast}$ also has the approximation property, then the Banach algebra $B({X^{ \ast \ast }})$ of all bounded linear operators on $ {X^{\ast\ast}}$ is isometrically isomorphic (as an algebra) to the double dual ${B_K}{(X)^{ \ast \ast }}$ of the Banach algebra of compact operators on $X$ when ${B_K}{(X)^{ \ast \ast }}$ is provided with the first Arens product. The chief result of this paper is a converse to the above statement. The converse is formulated in a strong fashion and a number of other results, including a formula for the second Arens product, are also given.
Cascade of sinks
Clark
Robinson
841-849
Abstract: In this paper it is proved that if a one-parameter family $\{ {F_t}\}$ of ${C^1}$ dissipative maps in dimension two creates a new homoclinic intersection for a fixed point $ {P_t}$ when the parameter $t = {t_0}$, then there is a cascade of quasi-sinks, i.e., there are parameter values ${t_n}$ converging to ${t_0}$ such that, for $t = {t_n}$, ${F_t}$ has a quasi-sink ${A_n}$ with each point $q$ in ${A_n}$ having period $n$. A quasi-sink ${A_n}$ for a map $F$ is a closed set such that each point $ q$ in ${A_n}$ is a periodic point and $ {A_n}$ is a quasi-attracting set (à la Conley), i.e., ${A_n}$ is the intersection of attracting sets $ A_n^j, {A_n} = { \cap _j}A_n^j$, where each $A_n^j$ has a neighborhood $U_n^j$ such that $\cap \{ {F^k}(U_n^j):k \geqslant 0\} = A_n^j$. Thus, the quasi-sinks ${A_n}$ are almost attracting sets made up entirely of points of period $n$. Gavrilov and Silnikov, and later Newhouse, proved this result when the new homoclinic intersection is created nondegenerately. In this case the sets ${A_n}$ are single, isolated (differential) sinks. In an earlier paper we proved the degenerate case when the homoclinic intersections are of finite order tangency (or the family is real analytic), again getting a cascade of sinks, not just quasi-sinks.
Restricted ramification for imaginary quadratic number fields and a multiplicator free group
Stephen B.
Watt
851-859
Abstract: Let $K$ be an imaginary quadratic number field with unit group ${E_K}$ and let $\ell$ be a rational prime such that $\ell \nmid \left\vert {{E_K}} \right\vert$. Let $S$ be any finite set of finite primes of $ K$ and let $K(\ell ,S)$ denote the maximal $\ell $-extension of $ K$ (inside a fixed algebraic closure of $K$) which is nonramified at the finite primes of $ K$ outside $S$. We show that the finitely generated pro-$\ell$-group $\Omega (\ell ,S) = \operatorname{Gal}(K(\ell ,S)/K)$ has the property that a complete set of defining relations for $\Omega (\ell ,S)$ as a pro-$\ell$-group can be obtained by lifting the nontrivial abelian or torsion relations in the maximal abelian quotient group $\Omega {(\ell ,S)^{{\text{ab}}}}$. In addition we use the key idea of the proof to derive some interesting results on towers of fields over $ K$ with restricted ramification.
Covers in free lattices
Ralph
Freese;
J. B.
Nation
1-42
Abstract: In this paper we study the covering relation $ (u \succ v)$ in finitely generated free lattices. The basic result is an algorithm which, given an element $w \in {\text{FL}}(X)$, finds all the elements which cover or are covered by $w$ (if any such elements exist). Using this, it is shown that covering chains in free lattices have at most five elements; in fact, all but finitely many covering chains in each free lattice contain at most three elements. Similarly, all finite intervals in ${\text{FL}}(X)$ are classified; again, with finitely many exceptions, they are all one-, two- or three-element chains.
Surgery in dimension four and noncompact $5$-manifolds
Daniel S.
Silver
43-50
Abstract: This paper describes a precise relationship between the problems of completing surgery in dimension four and finding boundaries for noncompact $5$-manifolds.
The determination of the Lie algebra associated to the lower central series of a group
John P.
Labute
51-57
Abstract: In this paper we determine the Lie algebra associated to the lower central series of a finitely presented group in the case where the defining relators satisfy certain independence conditions. Other central series, such as the lower $ p$-central series, are treated as well.
A reciprocity law for polynomials with Bernoulli coefficients
Willem
Fouché
59-67
Abstract: We study the zeros $ \pmod p$ of the polynomial ${\beta _p}(X) = {\Sigma _k}({B_k}/k)({X^{p - 1 - k}} - 1)$ for $p$ an odd prime, where ${B_k}$ denotes the $k$th Bernoulli number and the summation extends over $ 1 \leqslant k \leqslant p - 2$. We establish a reciprocity law which relates the congruence ${\beta _p}(r) \equiv 0\;\pmod p$ to a congruence $ {f_p}(n) \equiv 0\,\pmod r$ for $r$ a prime less than $p$ and $ n \in {\mathbf{Z}}$. The polynomial ${f_p}(x)$ is the irreducible polynomial over ${\mathbf{Q}}$ of the number $ \operatorname{Tr}_L^{{\mathbf{Q}}(\zeta )}\zeta$, where $\zeta$ is a primitive ${p^2}$ th root of unity and $ L \subset {\mathbf{Q}}(\zeta )$ is the extension of degree $p$ over $ {\mathbf{Q}}$. These congruences are closely related to the prime divisors of the indices $I(\alpha ) = (\mathcal{O}:{\mathbf{Z}}[\alpha ])$, where $ \mathcal{O}$ is the integral closure in $L$ and $ \alpha \in \mathcal{O}$ is of degree $p$ over $ {\mathbf{Q}}$. We establish congruences $\pmod p$ involving the numbers $I(\alpha )$ and show that their prime divisors $ r \ne p$ are closely related to the congruence ${r^{p - 1}} \equiv 1\,\pmod {p^2}$.
New inequalities for polynomials
C.
Frappier;
Q. I.
Rahman;
St.
Ruscheweyh
69-99
Abstract: Using a recently developed method to determine bound-preserving convolution operators in the unit disk, we derive various refinements and generalizations of the well-known inequalities of S. Bernstein and M. Riesz for polynomials. Many of these results take into account the size of one or more of the coefficients of the polynomial in question. Other results of similar nature are obtained from a new interpolation formula.
Coordinatization in superstable theories. I. Stationary types
Steven
Buechler
101-114
Abstract: Suppose $ T$ is superstable and $ P$ is a complete type over some finite set with $U(p) = \alpha + 1$ for some $\alpha$. We show how to associate with $ p$ an incidence geometry which measures the complexity of the family of extensions of $p$ of rank $\alpha$. When $p$ is stationary we give a characterization of the possible incidence geometries. As an application we prove Theorem. Suppose $M$ is superstable and has only one $ 1$-type $p \in S(\emptyset )$. Further suppose $ p$ is stationary with $U(p) = \alpha + 1$ for some $\alpha$. Then one of the following holds: (i) There is an equivalence relation $E \subset {M^2}$ with infinitely many infinite classes definable over $ \emptyset$. (ii) $ M$ is the algebraic closure of a set of Morley rank $1$. In particular, $ M$ is ${\aleph _0}$-stable of finite rank.
Stability of Godunov's method for a class of $2\times 2$ systems of conservation laws
Randall J.
LeVeque;
Blake
Temple
115-123
Abstract: We prove stability and convergence of the Godunov scheme for a special class of genuinely nonlinear $2 \times 2$ systems of conservation laws. The class of systems, which was identified and studied by Temple, is a subset of the class of systems for which the shock wave curves and rarefaction wave curves coincide. None of the equations of gas dynamics fall into this class, but equations of this type do arise, for example, in the study of multicomponent chromatography. To our knowledge this is the first time that a numerical method other than the random choice method of Glimm has been shown to be stable in the variation norm for a coupled system of nonlinear conservation laws. This implies that subsequences converge to weak solutions of the Cauchy problem, although convergence for $2 \times 2$ systems has been proved by DiPerna using the more abstract methods of compensated compactness.
Polar coordinates induced by actions of compact Lie groups
Jiri
Dadok
125-137
Abstract: Let $G$ be a connected Lie subgroup of the real orthogonal group $O(n)$. For the action of $G$ on $ {{\mathbf{R}}^n}$, we construct linear subspaces $ \mathfrak{a}$ that intersect all orbits. We determine for which $G$ there exists such an $\mathfrak{a}$ meeting all the $G$-orbits orthogonally; groups that act transitively on spheres are obvious examples. With few exceptions all possible $G$ arise as the isotropy subgroups of Riemannian symmetric spaces.
A reflexivity theorem for weakly closed subspaces of operators
Hari
Bercovici
139-146
Abstract: It was proved in [4] that the ultraweakly closed algebras generated by certain contractions on Hilbert space have a remarkable property. This property, in conjunction with the fact that these algebras are isomorphic to $ {H^\infty }$, was used in [3] to show that such ultraweakly closed algebras are reflexive. In the present paper we prove an analogous result that does not require isomorphism with ${H^\infty }$, and applies even to linear spaces of operators. Our result contains the reflexivity theorems of [3,2 and 9] as particular cases.
A bijective proof of Stanley's shuffling theorem
I. P.
Goulden
147-160
Abstract: For two permutations $\sigma$ and $\omega$ on disjoint sets of integers, consider forming a permutation on the combined sets by "shuffling" $ \sigma$ and $ \omega$ (i.e., $ \sigma$ and $ \omega$ appear as subsequences). Stanley [10], by considering $ P$-partitions and a $ q$-analogue of Saalschutz's $_3{F_2}$ summation, obtained the generating function for shuffles of $\sigma$ and $\omega$ with a given number of falls (an element larger than its successor) with respect to greater index (sum of positions of falls). It is a product of two $q$-binomial coefficients and depends only on remarkably simple parameters, namely the lengths, numbers of falls and greater indexes of $\sigma$ and $\omega$. A combinatorial proof of this result is obtained by finding bijections for lattice path representations of shuffles which reduce $ \sigma$ and $ \omega$ to canonical permutations, for which a direct evaluation of the generating function is given.
Singular integral operators of Calder\'on type and related operators on the energy spaces
Takafumi
Murai
161-178
Abstract: We show the boundedness of some singular integral operators on the energy spaces.
Best approximation and quasitriangular algebras
Timothy G.
Feeman
179-187
Abstract: If $\mathcal{P}$ is a linearly ordered set of projections on a Hilbert space and $\mathcal{K}$ is the ideal of compact operators, then $\operatorname{Alg}\, \mathcal{P} + \mathcal{K}$ is the quasitriangular algebra associated with $\mathcal{P}$. We study the problem of finding best approximants in a given quasitriangular algebra to a given operator: given $T$ and $ \mathcal{P}$, is there an $ A$ in $ \operatorname{Alg}\, \mathcal{P} + \mathcal{K}$ such that $\left\Vert {T - A} \right\Vert = \inf \{ \left\Vert {T - S} \right\Vert:S \in \operatorname{Alg}\,\mathcal{P} + \mathcal{K}\}$? We prove that if $\mathcal{A}$ is an operator subalgebra which is closed in the weak operator topology and satisfies a certain condition $\Delta$, then every operator $T$ has a best approximant in $\mathcal{A} + \mathcal{K}$. We also show that if $ \mathcal{E}$ is an increasing sequence of finite rank projections converging strongly to the identity then $ \operatorname{Alg}\,\mathcal{E}$ satisfies the condition $\Delta$. Also, we show that if $ T$ is not in $ \operatorname{Alg}\,\mathcal{E} + \mathcal{K}$ then the best approximants in $ \operatorname{Alg}\,\mathcal{E} + \mathcal{K}$ to $T$ are never unique.
Compact conformally flat hypersurfaces
Manfredo
do Carmo;
Marcos
Dajczer;
Francesco
Mercuri
189-203
Abstract: Roughly speaking, a conformal space is a differentiable manifold $ {M^n}$ in which the notion of angle of tangent vectors at a point $p \in {M^n}$ makes sense and varies differentiably with $p$; two such spaces are (locally) equivalent if they are related by an angle-preserving (local) diffeomorphism. A conformally flat space is a conformal space locally equivalent to the euclidean space $ {R^n}$. A submanifold of a conformally flat space is said to be conformally flat if so its induced conformal structure; in particular, if the codimension is one, it is called a conformally flat hypersurface. The aim of this paper is to give a description of compact conformally flat hypersurfaces of a conformally flat space. For simplicity, assume the ambient space to be $ {R^{n + 1}}$. Then, if $n \geqslant 4$, a conformally flat hypersurface $ {M^n} \subset {R^{n + 1}}$ can be described as follows. Diffeomorphically, $ {M^n}$ is a sphere $ {S^n}$ with ${b_1}(M)$ handles attached, where $ {b_1}(M)$ is the first Betti number of $M$. Geometrically, it is made up by (perhaps infinitely many) nonumbilic submanifolds of ${R^{n + 1}}$ that are foliated by complete round $ (n - 1)$-spheres and are joined through their boundaries to the following three types of umbilic submanifolds of ${R^{n + 1}}$: (a) an open piece of an $ n$-sphere or an $ n$-plane bounded by round $ (n - 1)$-sphere, (b) a round $(n - 1)$-sphere, (c) a point.
On infinite deficiency in ${\bf R}\sp \infty$-manifolds
Vo Thanh
Liem
205-226
Abstract: Using the notion of inductive proper $ q - 1 - {\text{LCC}}$ introduced in this note, we will prove the following theorems. Theorem 1. Let $M$ be an $ {R^\infty }$-manifold and let $ H:X \times I \to M$ be a homotopy such that ${H_0}$ and ${H_1}$ are $ {R^\infty }$-deficient embeddings. Then, there is a homeomorphism $ F$ of $M$ such that $F \circ {H_0} = {H_1}$. Moreover, if $ H$ is limited by an open cover $\alpha$ of $M$ and is stationary on a closed subset $ {X_0}$ of $ X$ and $ {W_0}$ is an open neighborhood of $\displaystyle H[(X - {X_0}) \times I] \quad {in\;M,}$ then we can choose $ F$ to also be $\operatorname{St}^4(\alpha )$-close to the identity and to be the identity on $\dot X_{0} \cup (M - {W_0})$. Theorem 2. Every closed, locally ${R^\infty }({Q^\infty })$-deficient subset of an $ {R^\infty }({Q^\infty })$-manifold $M$ is ${R^\infty }({Q^\infty })$-deficient in $ M$. Consequently, every closed, locally compact subset of $M$ is ${R^\infty }({Q^\infty })$-deficient in $ M$.
Sobolev space projections in strictly pseudoconvex domains
Harold P.
Boas
227-240
Abstract: The orthogonal projection from a Sobolev space ${W^s}(\Omega )$ onto the subspace of holomorphic functions is studied. This analogue of the Bergman projection is shown to satisfy regularity estimates in higher Sobolev norms when $\Omega$ is a smooth bounded strictly pseudoconvex domain in $ {{\mathbf{C}}^n}$.
Nilpotent automorphism groups of Riemann surfaces
Reza
Zomorrodian
241-255
Abstract: The action of nilpotent groups as automorphisms of compact Riemann surfaces is investigated. It is proved that the order of a nilpotent group of automorphisms of a surface of genus $g \geqslant 2$ cannot exceed $ 16(g - 1)$. Exact conditions of equality are obtained. This bound corresponds to a specific Fuchsian group given by the signature (0;2,4,8).
Invariant means on an ideal
Michel
Talagrand
257-272
Abstract: Let $G$ be a compact abelian group and $ Q$ an invariant ideal of ${L^\infty }(G)$. Let ${M_Q}$ be the set of invariant means $ \nu$ on ${L^\infty }(G)$ that are zero on $Q$, that is $\nu ({\chi _A}) = 1$ for ${\chi _A} \in Q$. We show that ${M_Q}$ is very large in the sense that a nonempty $ {G_\delta }$ subset of $ {M_Q}$ must contain a copy of $ \beta {\mathbf{N}}$. Let $ {E_Q}$ be the set of extreme points of ${M_Q}$. We show that its closure is very small in the sense that it contains no nonempty ${G_\delta }$ of ${M_Q}$. We also show that ${E_Q}$ is topologically very irregular in the sense that it contains no nonempty ${G_\delta }$ of its closure. The proofs are based on delicate constructions which rely on combinatorial type properties of abelian groups. Assume now that $ G$ is locally compact, noncompact, nondiscrete and countable at infinity. Let $ M$ be the set of invariant means on $ {L^\infty }(G)$ and $ {M_t}$, the set of topologically invariant means. We show that ${M_t}$ is very small in $M$. More precisely, each nonempty ${G_\delta }$ subset of $M$ contains a $\nu$ such that $ \nu (f) = 1$ for some $f \in C(G)$]> with <![CDATA[ $0 \leqslant f \leqslant 1$ and the support of $ f$ has a finite measure. Under continuum hypothesis, we also show that there exists points in ${M_t}$ which are extremal in $M$ (but, in general, ${M_t}$ is not a face of $M$, that is, not all the extreme points of ${M_t}$ are extremal in $M$).
Decompositions into codimension-two manifolds
R. J.
Daverman;
J. J.
Walsh
273-291
Abstract: Let $M$ denote an orientable $(n + 2)$-manifold and let $ G$ denote an upper semicontinuous decomposition of $M$ into continua having the shape of closed, orientable $n$-manifolds. The main result establishes that the decomposition space $M/G$ is a $2$-manifold.
Simply transitive groups and K\"ahler structures on homogeneous Siegel domains
Josef
Dorfmeister
293-305
Abstract: We determine the Lie algebras of all simply transitive groups of automorphisms of a homogeneous Siegel domain $D$ as modifications of standard normal $j$-algebras. We show that the Lie algebra of all automorphisms of $D$ is a "complete isometry algebra in standard position". This implies that $D$ carries a riemannian metric $ \tilde g$ with nonpositive sectional curvature satisfying Lie $ \operatorname{Iso}(D,\tilde g) = \operatorname{Lie}\; \operatorname{Aut}\,$ D. We determine all Kähler metrics $ f$ on $D$ for which the group $\operatorname{Aut}(D,f)$ of holomorphic isometries acts transitively. We prove that in this case $ \operatorname{Aut}(D,f)$ contains a simply transitive split solvable subgroup. The results of this paper are used to prove the fundamental conjecture for homogeneous Kähler manifolds admitting a solvable transitive group of automorphisms.
The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences
A.
Bellow;
V.
Losert
307-345
Abstract: In this paper on the weighted pointwise ergodic theorem we bring together a substantial amount of previous work by a number of mathematicians and integrate it into a systematic consistent scheme; we also bring some original contributions to the subject which extend its boundaries and suggest further avenues of research. The paper is divided into six sections. The most significant new results are contained in $\S\S5$ and $6$.
Multipliers on weighted $L\sb p$-spaces over certain totally disconnected groups
C. W.
Onneweer
347-362
Abstract: Let $G$ be a locally compact totally disconnected group with a suitable sequence of open compact subgroups. We prove a multiplier theorem for certain weighted ${L_p}$-spaces over $G$, which is a generalization of a Hörmander-type multiplier theorem for ${L_p}$-spaces over a local field, due to Taibleson.
Regularization of $L\sp 2$ norms of Lagrangian distributions
Steven
Izen
363-380
Abstract: Let $X$ be a compact smooth manifold, $\dim X = n$. Let $\Lambda$ be a fixed Lagrangian submanifold of $ {T^\ast}X$. The space of Lagrangian distributions $ {I^k}(X,\Lambda )$ is contained in ${L^2}(X)$ if $k < - n/4$. When $k = n/4$, $ {I^{ - n/4}}(X,\Lambda )$ just misses ${L^2}(X)$. A new inner product ${\langle u,v\rangle _R}$ is defined on $ {I^{ - n/4}}(X,\Lambda )/{I^{ - n/4 - 1}}(X,\Lambda )$ in terms of symbols. This inner product contains "${L^2}$ information" in the following sense: Slight regularizations of the Lagrangian distributions are taken, putting them in ${L^2}(X)$. The asymptotic behavior of the $ {L^2}$ inner product is examined as the regularizations approach the identity. Three different regularization schemes are presented and, in each case, ${\langle u,v\rangle _R}$ is found to regulate the growth of the ordinary ${L^2}$ inner product.
Cubes of conjugacy classes covering the infinite symmetric group
Manfred
Droste
381-393
Abstract: Using combinatorial methods, we prove the following theorem on the group $S$ of all permutations of a countably-infinte set: Whenever $p \in S$ has infinite support without being a fixed-point-free involution, then any $s \in S$ is a product of three conjugates of $p$. Furthermore, we present uncountably many new conjugacy classes $C$ of $S$ satisfying that any $s \in S$ is a product of two elements of $ C$. Similar results are shown for permutations of uncountable sets.
Functions of uniformly bounded characteristic on Riemann surfaces
Shinji
Yamashita
395-412
Abstract: A characteristic function $T(D,w,f)$ of Shimizu and Ahlfors type for a function $ f$ meromorphic in a Riemann surface $R$ is defined, where $D$ is a regular subdomain of $R$ containing a reference point $ w \in R$. Next we suppose that $R$ has the Green functions. Letting $ T(w,f) = {\lim _{D \uparrow R}}T(D,w,f)$, we define $f$ to be of uniformly bounded characteristic in $ R$, $f \in {\text{UBC}}(R)$ in notation, if ${\sup _{w \in R}}T(w,f) < \infty$. We shall propose, among other results, some criteria for $ f$ to be in ${\text{UBC}}(R)$ in various terms, namely, Green's potentials, harmonic majorants, and counting functions. They reveal that ${\text{UBC}}(\Delta )$ for the unit disk $ \Delta$ coincides precisely with that introduced in our former work. Many known facts on $ {\text{UBC}}(\Delta )$ are extended to $ {\text{UBC}}(R)$ by various methods. New proofs even for $R = \Delta$ are found. Some new facts, even for $ \Delta$, are added.
Lowest order invariants for real-analytic surfaces in ${\bf C}\sp 2$
Gary A.
Harris
413-422
Abstract: Suppose $ M$ is a general real-analytic surface in complex euclidean two-space with complex tangent space at a point $p$. Further suppose $M$ is tangent to order $k$ at $p$. This paper determines a complete set of $ k$th order local holomorphic invariants for $M$ at $p$.
An upper bound for the waiting time for nonlinear degenerate parabolic equations
Michel
Chipot;
Thomas
Sideris
423-427
Abstract: An upper bound is obtained for the time when the support of the solution of some nonlinear, degenerate parabolic equations begins to spread.
Inequalities for some maximal functions. I
Michael
Cowling;
Giancarlo
Mauceri
431-455
Abstract: This paper presents a new approach to maximal functions on ${{\mathbf{R}}^n}$. Our method is based on Fourier analysis, but is slightly sharper than the techniques based on square functions. In this paper, we reprove a theorem of E. M. Stein [16] on spherical maximal functions and improve marginally work of N. E. Aguilera [1] on the spherical maximal function in ${L^2}({{\mathbf{R}}^2})$. We prove results on the maximal function relative to rectangles of arbitrary direction and fixed eccentricity; as far as we know, these have not appeared in print for the case where $n \geqslant 3$, though they were certainly known to the experts. Finally, we obtain a best possible theorem on the pointwise convergence of singular integrals, answering a question of A. P. Calderón and A. Zygmund [3,3] to which N. E. Aguilera and E. O. Harboure [2] had provided a partial response.
Classification of semisimple rank one monoids
Lex E.
Renner
457-473
Abstract: Consider the classification problem for irreducible, normal, algebraic monoids with unit group $G$. We obtain complete results for the groups $ \operatorname{Sl}_2(K) \times {K^\ast}$, $ \operatorname{Gl}_2(K)$ and $\operatorname{PGl}_2(K) \times {K^\ast}$. If $ G$ is one of these groups let $ \mathcal{E}(G)$ denote the set of isomorphy types of normal, algebraic monoids with zero element and unit group $G$. Our main result establishes a canonical one-to-one correspondence $\mathcal{E}(G) \cong {{\mathbf{Q}}^ + }$, where ${{\mathbf{Q}}^ + }$ is the set of positive rational numbers. The classification is achieved in two steps. First, we construct a class of monoids from linear representations of $G$. That done, we show that any other $ E$ must already be one of those constructed. To do this, we devise an extension principle analogous to the big cell construction of algebraic group theory. This yields a birational comparison morphism $ \varphi :{E_r} \to E$, for some $ r \in {{\mathbf{Q}}^ + }$, which is ultimately an isomorphism because the monoid $ {E_r} \in \mathcal{E}(G)$ is regular. The relatively insignificant classification problem for normal monoids with group $ G$ and no zero element is also solved. For each $G$ there is only one such $E$ with $ G \subsetneqq E$.
Some sharp neighborhoods of univalent functions
Johnny E.
Brown
475-482
Abstract: For $\delta \geqslant 0$ and $f(z) = z + {a_2}{z^2} + \cdots$ analytic in $ \vert z\vert < 1$ let the $\delta$-neighborhood of $f$, ${N_\delta }(f)$, consist of those analytic functions $g(z) = z + {b_z}{z^2} + \cdots$ with $\sum\nolimits_{k = 2}^\infty {k\vert{a_k} - {b_k}\vert \leqslant \delta } $. We determine sufficient conditions guaranteeing which neighborhoods of certain classes of convex functions belong to certain classes of starlike functions. We extend some recent results of St. Ruscheweyh and R. Fournier and, at the same time, provide much simpler proofs. We also prove precisely how boundaries affect the value of $\delta$ for some general classes of functions.
Estimates for operators in mixed weighted $L\sp p$-spaces
Hans P.
Heinig
483-493
Abstract: A weighted Marcinkiewicz interpolation theorem is proved. If $T$ is simultaneously of weak type $ ({p_i},{q_i})$, $i = 0,1$; $1 \leqslant {p_0} < {p_1} \leqslant \infty$ and $u$, $v$ certain weight functions, then $ T$ is bounded from $ L_v^p$ to $L_u^q$ for $0 < q < p$, $ p \geqslant 1$. The result is applied to obtain weighted estimates for the Laplace and Fourier transform, as well as the Riesz potential.
Asymptotic expansions of ratios of coefficients of orthogonal polynomials with exponential weights
Attila
Máté;
Paul
Nevai;
Thomas
Zaslavsky
495-505
Abstract: Let ${p_n}(x) = {\gamma _n}{x^n} + \cdots$ denote the $n$th polynomial orthonormal with respect to the weight $ \exp ( - {x^\beta }/\beta )$ where $\beta > 0$ is an even integer. G. Freud conjectured and Al. Magnus proved that, writing $ {a_n} = {\gamma _{n - 1}}/{\gamma _n}$, the expression ${a_n}{n^{ - 1/\beta }}$ has a limit as $n \to \infty$. It is shown that this expression has an asymptotic expansion in terms of negative even powers of $n$. In the course of this, a combinatorial enumeration problem concerning one-dimensional lattice walk is solved and its relationship to a combinatorial identity of J. L. W. V. Jensen is explored.
Epimorphically closed permutative varieties
N. M.
Khan
507-528
Abstract: We show that for semigroups all permutation identities are preserved under epis and that all subvarieties of the permutative variety defined by any permutation identity $\displaystyle {x_1}{x_2} \cdots {x_n} = {x_{{i_1}}}{x_{{i_2}}} \cdots {x_{{i_n}}},$ with $n \geqslant 3$ and such that ${i_n} \ne n$ or $ {i_1} \ne 1$, are closed under epis. Finally we find some sufficient conditions that an identity be preserved under epis in conjunction with any nontrivial permutation identity.
Unstable towers in the odd primary homotopy groups of spheres
Martin
Bendersky
529-542
Abstract: The unstable elements in filtration $2$ of the unstable Novikov spectral sequence are computed. These elements are shown to survive to elements in the homotopy groups of spheres which are related to $ \operatorname{Im}\, J$. The computation is applied to determine the Hopf invariants of compositions of $\operatorname{Im}\, J$ and the exponent of certain sphere bundles over spheres.
Probabilities of first-order sentences about unary functions
James F.
Lynch
543-568
Abstract: Let $f$ be any fixed positive integer and $ \sigma$ a sentence in the first-order predicate calculus of $f$ unary functions. For positive integers $n$, an $n$-structure is a model with universe $\{ 0,1, \ldots ,n - 1\}$ and $f$ unary functions, and $\mu (n,\sigma )$ is the ratio of the number of $ n$-structures satisfying $ \sigma$ to ${n^{nf}}$, the number of $ n$-structures. We show that ${\lim _{n \to \infty }}\mu (n,\sigma )$ exists for all such $\sigma$, and its value is given by an expression consisting of integer constants and the operators $+ , - , \cdot ,/$, and ${e^x}$.
Differential operators and theta series
Solomon
Friedberg
569-589
Abstract: Let $f$ be a modular form on a congruence subgroup of $ {\text{SL}}(2,\mathbb{Z})$--not necessarily holomorphic, but an eigenfunction of the weight $k$ Casimir operator. Maass introduced differential operators (coming from the complexified universal enveloping algebra) which raise and lower by $2$ the weight of such a form and shift the eigenvalue. Here we introduce differential operators on hyperbolic $3$ space analogous to the Maass operators. These change by $2$ the weight of a modular form for an imaginary quadratic field. Theorem. The Maass operators and the hyperbolic space operators are intertwined by the imaginary quadratic Doi-Naganuma (base change) lifting. That is, the following diagram is commutative: \begin{displaymath}\begin{array}{*{20}{c}} & F & {\underset{{{\text{operators}}}... ...ext{Maass}}}}{\leftrightarrow}}} & {\tilde f} & \end{array} \end{displaymath} Using similar techniques for the dual pair $ ({\text{SL}}(2,\mathbb{R}),\;{\text{SO}}(2,1))$, we give a simple proof that the Shimura correspondences preserve holomorphicity (for weight $\geqslant 5/2$) and an explanation for this property directly in terms of the theta series (Weil representation) integral kernel. We also establish similar results for the real quadratic Doi-Naganuma lifting.
A property equivalent to the existence of scales
Howard
Becker
591-612
Abstract: Let ${\text{UNIF}}$ and $ {\text{SCALES}}$ be the propositions that every relation on ${\mathbf{R}}$ can be uniformized, and every subset of $ {\mathbf{R}}$ admits a scale, respectively. For $A \subset {\mathbf{R}}$, let $w(A)$ denote the Wadge ordinal of $ A$, and let $\delta _1^1(A)$ be the supremum of the ordinals realized in the pointclass $ {\Delta ^1}_1(A)$. Theorem $ {\text{(AD)}}$. The following are equivalent: (a) ${\text{SCALES}}$, (b) ${\text{UNIF}} +$ the set $\{ w(A):\delta _1^1(A) = {(w(A))^ + }\}$ contains an $\omega$-cub subset of $\Theta$. Using this theorem, Woodin has shown that if the theory $ {\text{(ZF}} + {\text{DC}} + {\text{AD}} + {\text{UNIF)}}$ is consistent, then the theory ${\text{(ZF}} + {\text{DC}} + {\text{AD}}_{\mathbf{R}} + {\text{SCALES)}}$ is also consistent. In this paper we give a proof of the above theorem and of a local version of it. We also study the ordinal $\delta _1^1(A)$ and give several characterizations of it.
An application of flows to time shift and time reversal in stochastic processes
E. B.
Dynkin
613-619
Abstract: A simple proposition (Theorem 1) on flows allows the investigation of random time shift and time reversal in Markov processes without assuming any regularity of paths. Theorem 5 is a generalization of Nagasawa's time reversal theorem and Theorem 4 generalizes a recent result of Getoor and Glover.
Bounded homotopy equivalences of Hilbert cube manifolds
C. Bruce
Hughes
621-643
Abstract: Let $M$ and $F$ be Hilbert cube manifolds with $ F$ compact. The purpose of this paper is to study homotopy equivalences $f:M \to {{\mathbf{R}}^m} \times F$ which have bounded control in the $ {{\mathbf{R}}^m}$-direction. Roughly, these homotopy equivalences form a semi-simplicial complex $ \mathcal{W}\mathcal{H}({{\mathbf{R}}^m} \times F)$, the controlled Whitehead space. Using results about approximate fibrations, $ \mathcal{W}\mathcal{H}({{\mathbf{R}}^m} \times F)$ is related to the semi-simplicial complex of bounded concordances on $ {{\mathbf{R}}^m} \times F$. Then the homotopy groups of $ \mathcal{W}\mathcal{H}({{\mathbf{R}}^m} \times F)$ are computed in terms of the lower algebraic $K$-theoretic functors $ {K_{ - i}}$.
Thrice-punctured spheres in hyperbolic $3$-manifolds
Colin C.
Adams
645-656
Abstract: The work of ${\text{W}}$. Thurston has stimulated much interest in the volumes of hyperbolic $3$-manifolds. In this paper, it is demonstrated that a $3$-manifold $M\prime$ obtained by cutting open an oriented finite volume hyperbolic $3$-manifold $M$ along an incompressible thrice-punctured sphere $S$ and then reidentifying the two copies of $S$ by any orientation-preserving homeomorphism of $S$ will also be a hyperbolic $3$-manifold with the same hyperbolic volume as $M$. It follows that an oriented finite volume hyperbolic $3$-manifold containing an incompressible thrice-punctured sphere shares its volume with a nonhomeomorphic hyperbolic $3$-manifold. In addition, it is shown that two orientable finite volume hyperbolic $3$-manifolds ${M_1}$ and ${M_2}$ containing incompressible thrice-punctured spheres ${S_1}$ and ${S_2}$, respectively, can be cut open along $ {S_1}$ and ${S_2}$ and then glued together along copies of ${S_1}$ and ${S_2}$ to yield a $3$-manifold which is hyperbolic with volume equal to the sum of the volumes of ${M_1}$ and ${M_2}$. Applications to link complements in $ {S^3}$ are included.
Singular behavior in nonlinear parabolic equations
Wei-Ming
Ni;
Paul
Sacks
657-671
Abstract: In this paper, we study the well-posedness of the initial-boundary value problems of some quasilinear parabolic equations, namely, nonlinear heat equations and the porous medium equation in the fast-diffusion case. We establish nonuniqueness (local in time) and/or nonregularizing effect of these equations in some critical cases. The key which leads to the resolution of these problems is to study some singular solutions of the elliptic counterparts of these parabolic problems (the so-called $ M$-solutions of the Lane-Emden equations in astrophysics).
A weighted inequality for the maximal Bochner-Riesz operator on ${\bf R}\sp 2$
Anthony
Carbery
673-680
Abstract: For $f \in \mathcal{S}({{\mathbf{R}}^2})$, let $(T_R^\alpha f)\hat \emptyset (\xi ) = (1 - \vert\xi {\vert^2}{R^2})_ + ^\alpha \hat f(\xi )$. It is a well-known theorem of Carleson and Sjölin that $T_1^\alpha$ defines a bounded operator on $ {L^4}$ if $\alpha > 0$. In this paper we obtain an explicit weighted inequality of the form $\displaystyle \int {\mathop {\sup }\limits_{0 < R < \infty } \vert T_R^\alpha f(x){\vert^2}w(x)\;dx \leqslant \int {\vert f{\vert^2}{P_\alpha }w(x)\;dx,} } $ with ${P_\alpha }$ bounded on ${L^2}$ if $ \alpha > 0$. This strengthens the above theorem of Carleson and Sjölin. The method gives information on the maximal operator associated to general suitably smooth radial Fourier multipliers of $ {{\mathbf{R}}^2}$.
Estimates of the harmonic measure of a continuum in the unit disk
Carl H.
FitzGerald;
Burton
Rodin;
Stefan E.
Warschawski
681-685
Abstract: The harmonic measure of a continuum in the unit disk is estimated from below in two ways. The first estimate is in terms of the angle subtended by the continuum as viewed from the origin. This result is a dual to the Milloux problem. The second estimate is in terms of the diameter of the continuum. This estimate was conjectured earlier as a strengthening of a theorem of D. Gaier. In preparation for the proofs several lemmas are developed. These lemmas describe some properties of the Riemann mapping function of a disk with radial incision onto a disk.
Finely harmonic functions with finite Dirichlet integral with respect to the Green measure
Bernt
Øksendal
687-700
Abstract: We consider finely harmonic functions $h$ on a fine, Greenian domain $V \subset {{\mathbf{R}}^d}$ with finite Dirichlet integral wrt $Gm$, i.e. $(\ast)$ $\displaystyle \int_V\vert\nabla h(y)\vert^2G(x,y)\,dm(y) < \infty \quad {\text{for}}\;x \in V,$ where $m$ denotes the Lebesgue measure, $G(x,y)$ the Green function. We use Brownian motion and stochastic calculus to prove that such functions $h$ always have boundary values ${h^\ast}$ along a.a. Brownian paths. This partially extends results by Doob, Brelot and Godefroid, who considered ordinary harmonic functions with finite Dirichlet integral wrt $m$ and Green lines instead of Brownian paths. As a consequence of Theorem 1 we obtain several properties equivalent to $( \ast )$, one of these being that $h$ is the harmonic extension to $ V$ of a random "boundary" function ${h^\ast}$ (of a certain type), i.e. $h(x) = {E^x}[{h^\ast}]$ for all $x \in V$. Another application is that the polar sets are removable singularity sets for finely harmonic functions satisfying $( \ast )$. This is in contrast with the situation for finely harmonic functions with finite Dirichlet integral wrt $m$.
Some estimates for nondivergence structure, second order elliptic equations
Lawrence C.
Evans
701-712
Abstract: We obtain various formal estimates for solutions of nondivergence structure, second order, uniformly elliptic ${\text{PDE}}$. These include interior lower bounds and also gradient estimates in ${L^p}$, for some $p < 0$.
The Mackey topology and complemented subspaces of Lorentz sequence spaces $d(w,p)$ for $0<p<1$
M.
Nawrocki;
A.
Ortyński
713-722
Abstract: In this paper we continue the study of Lorentz sequence spaces $ d(w,p)$, $0 < p < 1$, initiated by N. Popa [8]. First we show that the Mackey completion of $d(w,p)$ is equal to $d(v,1)$ for some sequence $v$. Next, we prove that if $d(w,p) \not\subset {l_1}$, then it contains a complemented subspace isomorphic to ${l_p}$. Finally we show that if $\lim {n^{ - 1}}\left(\sum\nolimits_{i = 1}^n {w_i}\right)^{1/p} = \infty$, then every complemented subspace of $d(w,p)$ with symmetric bases is isomorphic to $ d(w,p)$.
A kinetic approach to general first order quasilinear equations
Yoshikazu
Giga;
Tetsuro
Miyakawa;
Shinnosuke
Oharu
723-743
Abstract: This paper presents a new method for constructing entropy solutions of first order quasilinear equations of conservation type, which is illustrated in terms of the kinetic theory of gases. Regarding a quasilinear equation as a model of macroscopic conservation laws in gas dynamics, we introduce as the corresponding microscopic model an auxiliary linear equation involving a real parameter $\xi$ which plays the role of the velocity argument. Approximate solutions for the quasilinear equation are then obtained by integrating solutions of the linear equation with respect to the parameter $ \xi$. All of these equations are treated in the Fréchet space $ L_{{\text{loc}}}^1({R^n})$, and a convergence theorem for such approximate solutions to the entropy solutions is established with the aid of nonlinear semigroup theory.
On the Gauss-Bonnet theorem for complete manifolds
Steven
Rosenberg
745-753
Abstract: For a manifold diffeomorphic to the interior of a compact manifold with boundary, several classes of complete metrics are given for which the Gauss-Bonnet Theorem is valid.
Pull-backs of $C\sp \ast$-algebras and crossed products by certain diagonal actions
Iain
Raeburn;
Dana P.
Williams
755-777
Abstract: Let $G$ be a locally compact group and $p:\Omega \to T$ a principal $G$-bundle. If $A$ is a ${C^\ast}$-algebra with primitive ideal space $ T$, the pull-back $ {p^\ast}A$ of $ A$ along $p$ is the balanced tensor product $ {C_0}(\Omega ){ \otimes _{C(T)}}A$. If $\beta :G \to \operatorname{Aut}\,A$ consists of $C(T)$-module automorphisms, and $\gamma :G \to \operatorname{Aut}\,{C_0}(\Omega )$ is the natural action, then the automorphism group $ \gamma \otimes \beta$ of $ {C_0}(\Omega ) \otimes A$ respects the balancing and induces the diagonal action ${p^\ast}\beta $ of $G$ on ${p^\ast}A$. We discuss some examples of such actions and study the crossed product ${p^\ast}A{ \times _{{p^\ast}\beta }}G$. We suggest a substitute $D$ for the fixed-point algebra, prove ${p^\ast}A \times G$ is strongly Morita equivalent to $D$, and investigate the structure of $D$ in various cases. In particular, we ask when $D$ is strongly Morita equivalent to $ A$--sometimes, but by no means always--and investigate the case where $ A$ has continuous trace.
Finite codimensional ideals in function algebras
Krzysztof
Jarosz
779-785
Abstract: Assume $ S$ is a compact, metric space and let $M$ be a finite codimensional closed subspace of a complex space $C(S)$. In this paper we prove that if each element from $M$ has at least $k$ zeros in $S$, then for some ${s_1}, \ldots ,{s_k} \in S,M \subseteq \{ f \in C(S):f({s_1}) = \cdots = f({s_k}) = 0\}$.
Essential dimension lowering mappings having dense deficiency set
Mladen
Bestvina
787-798
Abstract: Two classes of surjective maps $ f:{S^m} \to {S^n}$ that are one-to-one over the image of a dense set are constructed. We show that for $ m,n \geq 3$ there is a monotone surjection $ f:{S^m} \to {S^n}$ that is one-to-one over the image of a dense set; and for $3 \leq n \leq m \leq 2n - 3$, each element of $ {\pi _m}({S^n})$ can be represented as a monotone surjection $f:{S^m} \to {S^n}$ that is one-to-one over the image of a dense set.
Analytic operator algebras (factorization and an expectation)
Baruch
Solel
799-817
Abstract: Let $M$ be a $\sigma$-finite von Neumann algebra and ${\{ {\alpha _t}\} _{t \in T}}$ a periodic flow on $M$. The algebra of analytic operators in $ M$ is $\{ a \in M:{\text{sp}_\alpha }(a) \subseteq {{\mathbf{Z}}_ + }\} $ and is denoted ${H^\infty }(\alpha )$. We prove that every invertible operator $a \in {H^\infty }(\alpha )$ can be written as $a = ub$, where $u$ is unitary in $M$ and $b \in {H^\infty }(\alpha ) \cap {H^\infty }{(\alpha )^{ - 1}}$. We also prove inner-outer factorization results for $a \in {H^\infty }(\alpha )$. Another result represents $ {H^\infty }(\alpha )$ as the image of a certain nest subalgebra (of a von Neumann algebra that contains $M$) via a conditional expectation. As corollaries we prove a distance formula and an interpolation result for the case where $M$ is an injective von Neumann algebra.
Banach spaces with the $L\sp 1$-Banach-Stone property
Peter
Greim
819-828
Abstract: It has previously been shown that separable Banach spaces $V$ with trivial $L$-structure have the $ {L^1}$-Banach-Stone property, i.e. every surjective isometry between two Bochner spaces $ {L^1}({\mu _i},V)$ induces an isomorphism of the two measure algebras. We remove the separability restriction, employing the topology of the measure algebra's Stonean space. The result is achieved via a complete description of the $L$-structure of $ {L^1}(\mu ,V)$.
Pro-Lie groups
R. W.
Bagley;
T. S.
Wu;
J. S.
Yang
829-838
Abstract: A topological group $ G$ is pro-Lie if $ G$ has small compact normal subgroups $K$ such that $G/K$ is a Lie group. A locally compact group $ G$ is an $ L$-group if, for every neighborhood $U$ of the identity and compact set $C$, there is a neighborhood $V$ of the identity such that $ gH{g^{ - 1}} \cap C \subset U$ for every $g \in G$ and every subgroup $H \subset V$. We obtain characterizations of pro-Lie groups and make several applications. For example, every compactly generated $L$-group is pro-Lie and a compactly generated group which can be embedded (by a continuous isomorphism) in a pro-Lie group is pro-Lie. We obtain related results for factor groups, nilpotent groups, maximal compact normal subgroups, and generalize a theorem of Hofmann, Liukkonen, and Mislove [4].
On Chebyshev subspaces in the space of multivariate differentiable functions
András
Kroó
839-852
Abstract: In the present paper we give a characterization of Chebyshev sub-spaces in the space of (real or complex) continuously-differentiable functions of two variables. We also discuss various applications of the characterization theorem.
On the higher Whitehead groups of a Bieberbach group
Andrew J.
Nicas
853-859
Abstract: Let $\Gamma$ be a Bieberbach group, i.e. the fundamental group of a compact flat Riemannian manifold. In this paper we show that if $p > 2$ is a prime, then the $ p$-torsion subgroup of $ {\text{Wh}_i}(\Gamma )$ vanishes for $ 0 \leq i \leq 2p - 2$, where $ {\text{Wh}_i}(\Gamma )$ is the $i$th higher Whitehead group of $\Gamma$. The proof involves Farrell and Hsiang's structure theorem for Bieberbach groups, parametrized surgery, pseudoisotopy, and Waldhausen's algebraic $K$-theory of spaces.
The role of countable dimensionality in the theory of cell-like relations
Fredric D.
Ancel
1-40
Abstract: Consider only metrizable spaces. The notion of a slice-trivial relation is introduced, and Theorem 3.2 is proved. This theorem sets forth sufficient conditions for a continuous relation with compact $ U{V^\infty }$ point images to be slice-trivial. Theorem 4.5 posits a number of necessary and sufficient conditions for a map to be a hereditary shape equivalence. Several applications of these two theorems are made, including the following. Theorem 5.1. A cell-like map $ f:X \to Y$ is a hereditary shape equivalence if there is a sequence $\{ {K_n}\}$ of closed subsets of $ Y$ such that (1) $Y - \bigcup\nolimits_{n = 1}^\infty {{K_n}}$ is countable dimensional, and (2) $ f\vert{f^{ - 1}}({K_n}):{f^{ - 1}}({K_n}) \to {K_n}$ is a hereditary shape equivalence for each $n \geq 1$. Theorem 5.9. If $ f:X \to Y$ is a proper onto map whose point inverses are $U{V^\infty }$ sets, then $Y$ is an absolute neighborhood extensor for the class of countable dimensional spaces. Furthermore, if $Y$ is countable dimensional, then $ Y$ is an absolute neighborhood retract. Theorem 5.9 is of particular interest when specialized to the identity map of a locally contractible space.
Bowen-Ruelle measures for certain piecewise hyperbolic maps
Lai-Sang
Young
41-48
Abstract: We consider a class of piecewise ${C^2}$ Lozi-like maps and prove the existence of invariant measures with absolutely continuous conditional measures on unstable manifolds
Functions of $\Phi$-bounded variation and Riemann-Stieltjes integration
Michael
Schramm
49-63
Abstract: A notion of generalized bounded variation is introduced which simultaneously generalizes many of those previously examined. It is shown that the class of functions arising from this definition is a Banach space with a suitable norm. Appropriate variation functions are defined and examined, and an analogue of Helly's theorem is estabished. The significance of this class to convergence of Fourier series is briefly discussed. A result concerning Riemann-Stieltjes integrals of functions of this class is proved.
Strong laws of large numbers for products of random matrices
Steve
Pincus
65-89
Abstract: This work, on products of random matrices, is inspired by papers of Furstenberg and Kesten (Ann. Math. Statist. 31 (1960), 457-469) and Furstenberg (Trans. Amer. Math. Soc. 108 (1963), 377-428). In particular, a formula was known for almost sure limits for normalized products of random matrices in terms of a stationary measure. However, no explicit computational techniques were known for these limits, and little was known about the stationary measures. We prove two main theorems. The first assumes that the random matrices are upper triangular and computes the almost sure limits in question. For the second, we assume the random matrices are $2 \times 2$ and Bernoulli, i.e., random matrices whose support is two points. Then the second theorem gives an asymptotic result for the almost sure limits, with rates of convergence in some cases.
Mixed projection inequalities
Erwin
Lutwak
91-105
Abstract: A number of sharp geometric inequalities for polars of mixed projection bodies (zonoids) are obtained. Among the inequalities derived is a polar projection inequality that has the projection inequality of Petty as a special case. Other special cases of this polar projection inequality are inequalities (between the volume of a convex body and that of the polar of its $i$th projection body) that are strengthened forms of the classical inequalities between the volume of a convex body and its projection measures (Quermassintegrale). The relation between the Busemann-Petty centroid inequality and the Petty projection inequality is shown to be similar to the relation that exists between the Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry. Some mixed integral inequalities are derived similar in spirit to inequalities obtained by Chakerian and others.
${\rm BMO}(\rho)$ and Carleson measures
Wayne Stewart
Smith
107-126
Abstract: This paper concerns certain generalizations of $ {\text{BMO}}$, the space of functions of bounded mean oscillation. Let $ \rho$ be a positive nondecreasing function on $ (0,\infty )$ with $\rho (0 + ) = 0$. A locally integrable function on $ {{\mathbf{R}}^m}$ is said to belong to $ {\text{BMO}}(\rho)$ if its mean oscillation over any cube $Q$ is $ O(\rho (l(Q)))$, where $ l(Q)$ is the edge length of $Q$. Carleson measures are known to be closely related to $ {\text{BMO}}$. Generalizations of these measures are shown to be similarly related to the spaces $ {\text{BMO}}(\rho)$. For a cube $Q$ in $ {{\mathbf{R}}^m},\;\vert Q\vert$ denotes its volume and $R(Q)$ is the set $ \{ (x,y) \in {\mathbf{R}}_ + ^{m + 1}:x \in Q,\;0 < y < l(Q)\}$. A measure $ \mu$ on ${\mathbf{R}}_ + ^{m + 1}$ is called a $ \rho$-Carleson measure if $ \vert\mu \vert(R(Q)) = O(\rho (l(Q))\vert Q\vert)$, for all cubes $Q$. L. Carleson proved that a compactly supported function in $ {\text{BMO}}$ can be represented as the sum of a bounded function and the balyage, or sweep, of some Carleson measure. A generalization of this theorem involving ${\text{BMO}}(\rho )$ and $\rho$-Carleson measures is proved for a broad class of growth functions, and this is used to represent $ {\text{BMO}}(\rho )$ as a dual space. The proof of the theorem is based on a proof of J. Garnett and P. Jones of Carleson's theorem. Another characterization of ${\text{BMO}}(\rho )$ using $\rho $-Carleson measures is a corollary. This result generalizes a characterization of $ {\text{BMO}}$ due to C. Fefferman. Finally, an atomic decomposition of the predual of $ {\text{BMO}}(\rho )$ is given.
Finite time analyticity for the two- and three-dimensional Rayleigh-Taylor instability
C.
Sulem;
P.-L.
Sulem
127-160
Abstract: The Rayleigh-Taylor instability refers to the dynamics of the interface between two ideal irrotational fluids of different densities superposed one over the other and in relative motion. The well-posedness of this problem is considered for two- and three-dimensional flows in the entire space and in the presence of a horizontal bottom. In the entire space, finite time analyticity of the interface is proven when the initial interface has sufficiently small gradients and is flat at infinity. In the presence of a horizontal bottom, the initial interface corrugations has also to be small initially but it is not required to vanish at infinity.
Projections on tensor product spaces
E. J.
Halton;
W. A.
Light
161-165
Abstract: $(S,\Sigma ,\mu ),(T,\Theta ,\upsilon )$ are finite, nonatomic measure spaces. $G$ and $H$ are finite-dimensional subspaces of ${L_1}(S)$ and ${L_1}(T)$ respectively. Both $G$ and $H$ contain the constant functions. It is shown that the relative projection constant of $ {L_1}(S) \otimes H + G \otimes {L_1}(T)$ in $ {L_1}(S \times T)$ is at least $3$.
On a.e. convergence of solutions of hyperbolic equations to $L\sp p$-initial data
Alberto
Ruiz
167-188
Abstract: We consider the Cauchy data problem $ u(x,0) = 0$, $\partial u(x,0)/\partial t = f(x)$, for a strongly hyperbolic second order equation in $ n$th spatial dimension, $n \geq 3$, with ${C^\infty }$ coefficients. Almost everywhere convergence of the solution of this problem to initial data, in the appropriate sense is proved for $f$ in ${L^p}$, $2n/(n + 1) < p < 2(n - 2)/(n - 3)$. The basic techniques are ${L^p}$-estimates for some maximal operators associated to the problem (see [4]), and the asymptotic expansion of the Riemann function given by D. Ludwig (see [9]).
On invariant finitely additive measures for automorphism groups acting on tori
S. G.
Dani
189-199
Abstract: Consider the natural action of a subgroup $H$ of $ {\text{GL}}(n,{\mathbf{Z}})$ on $ {{\mathbf{T}}^n}$. We relate the $H$-invariant finitely additive measures on $ ({{\mathbf{T}}^n},\mathcal{L})$ where $ \mathcal{L}$ is the class of all Lebesgue measurable sets, to invariant subtori $ C$ such that the $ H$-action on either $ C$ or ${{\mathbf{T}}^n}/C$ factors to an action of an amenable group. In particular, we conclude that if $H$ is a nonamenable group acting irreducibly on $ {{\mathbf{T}}^n}$ then the normalised Haar measure is the only $H$-invariant finitely additive probability measure on $ ({{\mathbf{T}}^n},\mathcal{L})$ such that $ \mu (R) = 0$, where $ R$ is the (countable) subgroup consisting of all elements of finite order; this answers a question raised by J. Rosenblatt. Along the way we analyse $H$-invariant finitely additive measures defined for all subsets of $ {{\mathbf{T}}^n}$ and deduce, in particular, that the Haar measure extends to an $ H$-invariant finitely additive measure defined on all sets if and only if $ H$ is amenable.
Minimal leaves in foliations
Daniel M.
Cass
201-213
Abstract: The paper defines a property of open Riemannian manifolds, called quasi-homogeneity. This property is quasi-isometry invariant and is shown to hold for any manifold which appears as a minimal leaf in a foliation. Examples are given of surfaces which are not quasi-homogeneous. One such is the well-known noncompact leaf of Reeb's foliation of ${S^3}$. These surfaces have bounded geometry.
The duration of transients
S.
Pelikan
215-221
Abstract: A transformation $ T$ defined on $X \subset {{\mathbf{R}}^n}$ for which $T(X) \supset X$ is considered. A transient in $X$ is a trajectory $x,Tx, \ldots ,{T^m}x \subset X$ so that ${T^{m + 1}}x \notin X$. In this case, $ m$ is the duration of the transient. A method for estimating the average duration of transients is given, and an example of a transformation with exceedingly long transients is described.
On the restriction of the Fourier transform to curves: endpoint results and the degenerate case
Michael
Christ
223-238
Abstract: For smooth curves $ \Gamma$ in ${{\mathbf{R}}^n}$ with certain curvature properties it is shown that the composition of the Fourier transform in $ {{\mathbf{R}}^n}$ followed by restriction to $\Gamma$ defines a bounded operator from ${L^p}({{\mathbf{R}}^n})$ to ${L^q}(\Gamma )$ for certain $p,q$. The curvature hypotheses are the weakest under which this could hold, and $ p$ is optimal for a range of $q$. In the proofs the problem is reduced to the estimation of certain multilinear operators generalizing fractional integrals, and they are treated by means of rearrangement inequalities and interpolation between simple endpoint estimates.
Periodic solutions of Hamilton's equations and local minima of the dual action
Frank H.
Clarke
239-251
Abstract: The dual action is a functional whose extremals lead to solutions of Hamilton's equations. Up to now, extremals of the dual action have been obtained either through its global minimization or through application of critical point theory. A new methodology is introduced in which local minima of the dual action are found to exist. Applications are then made to the existence of Hamiltonian trajectories having prescribed period.
The algebra of the finite Fourier transform and coding theory
R.
Tolimieri
253-273
Abstract: The role of the finite Fourier transform in the theory of error correcting codes has been explored in a recent text by Richard Blahut. In this work we study how the finite Fourier transform relates to certain polynomial identities involving weight enumerator polynomials of linear codes. These include the generalized MacWilliams identities and theorems originally due to ${\text{R}}$. Gleason concerning polynomial algebras containing weight enumerator polynomials. The Heisenberg group model of the finite Fourier transform provides certain algebras of classical theta functions which will be applied to reprove Gleason's results.
Period doubling and the Lefschetz formula
John
Franks
275-283
Abstract: This article gives an application of the Lefschetz fixed point theorem to prove, under certain hypotheses, the existence of a family of periodic orbits for a smooth map. The family has points of periods ${2^k}p$ for some $p$ and all $k \geq 0$. There is a version of the result for a parametrized family $f_t$ which shows that these orbits are "connected" in parametrized space under appropriate hypotheses.
A method for investigating geometric properties of support points and applications
Johnny E.
Brown
285-291
Abstract: A normalized univalent function $f$ is a support point of $S$ if there exists a continuous linear functional $ L$ (which is nonconstant on $S$) for which $f$ maximizes $\operatorname{Re} L(g),g \in S$. For such functions it is known that $\Gamma = {\text{C}} - f(U)$ is a single analytic arc that is part of a trajectory of a certain quadratic differential $ Q(w)\;d{w^2}$. A method is developed which is used to study geometric properties of support points. This method depends on consideration of $\operatorname{Im} \{ {w^2}Q(w)\}$ rather than the usual $\operatorname{Re} \{ {w^2}Q(w)\}$. Qualitative, as well as quantitative, applications are obtained. Results related to the Bieberbach conjecture when the extremal functions have initial real coefficients are also obtained.
Fractional integrals on weighted $H\sp p$ and $L\sp p$ spaces
Jan-Olov
Strömberg;
Richard L.
Wheeden
293-321
Abstract: We study the two weight function problem $ \parallel {I_\alpha }f{\parallel _{H_u^q}} \leqslant c\parallel f{\parallel _{H_v^p}},0 < p \leqslant q < \infty$ , for fractional integrals on Hardy spaces. If $u$ and $v$ satisfy the doubling condition and $0 < p \leqslant 1$, we obtain a necessary and sufficient condition for the norm inequality to hold. If $1 < p < \infty$ we obtain a necessary condition and a sufficient condition, and show these are the same under various additional conditions on $ u$ and $v$. We also consider the corresponding problem for $L_u^q$ and $L_v^p$, and obtain a necessary and sufficient condition in some cases.
Of planar Eulerian graphs and permutations
Gadi
Moran
323-341
Abstract: Infinite planar Eulerian graphs are used to show that for $v > 0$ the covering number of the infinite simple group $ {H_v} = S/{S^v}$ is two. Here $S$ denotes the group of all permutations of a set of cardinality $ {\aleph _v},{S^v}$ denotes its subgroup consisting of the permutations moving less than ${\aleph _v}$ elements, and the covering number of a (simple) group $G$ is the smallest positive integer $ n$ satisfying $ {C^n} = G$ for every nonunit conjugacy class $C$ in $G$.
Local operators and derivations on $C\sp \ast$-algebras
C. J. K.
Batty
343-352
Abstract: The variations on a theme of locality for a pair of operators $ (H,K)$ on a $ {C^\ast}$-algebra $\mathfrak{A}$ are expressed algebraically. If $ K$ is a $ \ast$-derivation generating an action of $ \mathbb{R}$ on $\mathfrak{A}$, and $H$ is $\ast$-linear and $K$-local, then, under certain restrictions, $ H$ is shown to be very closely related to $K$.
Fuchsian groups and algebraic number fields
P. L.
Waterman;
C.
Maclachlan
353-364
Abstract: Given the signature of a finitely-generated Fuchsian group, we find the minimal extension of the rationals for which there is a Fuchsian group having the required signature, whose matrix entries lie in this field.
Conway's field of surreal numbers
Norman L.
Alling
365-386
Abstract: Conway introduced the Field $ {\mathbf{No}}$ of numbers, which Knuth has called the surreal numbers. ${\mathbf{No}}$ is a proper class and a real-closed field, with a very high level of density, which can be described by extending Hausdorff's ${\eta _\xi }$ condition. In this paper the author applies a century of research on ordered sets, groups, and fields to the study of $ {\mathbf{No}}$. In the process, a tower of subfields, $\xi {\mathbf{No}}$, is defined, each of which is a real-closed subfield of $ {\mathbf{No}}$ that is an ${\eta _\xi }$-set. These fields all have Conway partitions. This structure allows the author to prove that every pseudo-convergent sequence in ${\mathbf{No}}$ has a unique limit in ${\mathbf{No}}$.
An idempotent completion functor in homotopy theory
Harold M.
Hastings
387-402
Abstract: We observe that Artin-Mazur style $R$-completions ($R$ is a commutative ring with identity) induce analogous idempotent completions on the weak prohomotopy category pro-Ho(Top). Because Ho(Top) is a subcategory of pro-Ho(Top) and pro-Ho(Top) is closely related to the topologized homotopy category of J. F. Adams and D. Sullivan, our construction represents the Sullivan completions as homotopy limits of idempotent functors. In addition, we show that the Sullivan completion is idempotent on those spaces (in analogy with the Bousfield and Kan ${R_\infty }$-completion on $ R$-good spaces) for which its cohomology with coefficients in $ R$ agrees with that of our Artin-Mazur style completion. Finally, we rigidify the Artin-Mazur completion to obtain an idempotent Artin-Mazur completion on a category of generalized prospaces which preserves fibration and suitably defined cofibration sequences. (Our previous results on idempotency and factorization lift to the rigid completion.) Our results answer questions of Adams, Sullivan, and, later, A. Deleanu.
On the model equations which describe nonlinear wave motions in a rotating fluid
Jong Uhn
Kim
403-417
Abstract: This paper concerns mathematical aspects of the two model equations describing nonlinear wave motions in a rotating fluid. We establish local existence of solutions and show that singularities occur in a finite time under certain hypotheses. We also show that these equations admit nonconstant travelling wave solutions.
Compact group actions and maps into $K(\pi,1)$-spaces
Daniel H.
Gottlieb;
Kyung B.
Lee;
Murad
Özaydin
419-429
Abstract: Let $G$ act on an aspherical manifold $ M$. If $G$ is a compact Lie group acting effectively and homotopically trivially then $ G$ must be abelian. We prove a much more general form of this result, thus extending results of Donnelly and Schultz. Our method gives us a splitting result for torus actions complementing a result of Conner and Raymond. We also generalize a theorem of Schoen and Yau on homotopy equivariance.
Stability of the travelling wave solution of the FitzHugh-Nagumo system
Christopher K. R. T.
Jones
431-469
Abstract: Travelling wave solutions for the FitzHugh-Nagumo equations have been proved to exist, by various authors, close to a certain singular limit of the equations. In this paper it is proved that these waves are stable relative to the full system of partial differential equations; that is, initial values near (in the sup norm) to the travelling wave lead to solutions that decay to some translate of the wave in time. The technique used is the linearised stability criterion; the framework for its use in this context has been given by Evans [6-9]. The search for the spectrum leads to systems of linear ordinary differential equations. The proof uses dynamical systems arguments to analyse these close to the singular limit.
On subalgebras of simple Lie algebras of characteristic $p>0$
B.
Weisfeiler
471-503
Abstract: The main results of the paper are Theorems I.5.1, II.1.3 and III.2.1. Theorem I.5.1 states that if a maximal subalgebra $ M$ of a simple finite-dimensional Lie algebra $G$ has solvable quotients of dimension $\geqslant 2$, then every nilpotent element of $ H$ acts nilpotently on $ G$. Theorem II.1.3 states that if such a simple Lie algebra $G$ contains a maximal subalgebra which is solvable, then $G$ is Zassenbaus-Witt algebra. Theorem III.2.1 states that certain $ {\mathbf{Z}}$-graded finite-dimensional simple Lie algebras are either classical or the difference between the number of nonzero positive and negative homogeneous components is large.
Multi-invariant sets on compact abelian groups
Daniel
Berend
505-535
Abstract: Let $G$ be a finite-dimensional connected compact abelian group. Generalizing previous results, dealing with the case of finite-dimensional tori, a full characterization is given herewith of those commutative semigroups $\Sigma$ of continuous endomorphisms of $ G$ which satisfy the following property: The only infinite closed $ \Sigma$-invariant subset of $G$ is $G$ itself.
Tempered ultradistributions as boundary values of analytic functions
R. S.
Pathak
537-556
Abstract: The spaces ${S_{{a_k}}}$, $ {S^{{b_q}}}$ and $S_{{a_{k}}}^{{b_q}}$ were introduced by I. M. Gel'fand as a generalization of the test function spaces of type $ S$; the elements of the corresponding dual spaces are called tempered ultradistributions. It is shown that a function which is analytic in a tubular radial domain and satisfies a certain nonpolynomial growth condition has a distributional boundary value in the weak topology of the tempered ultradistribution space $ (S_{{b_{k}}}^{{a_{q}}})\prime$, which is the space of Fourier transforms of elements in $ (S_{{a_{k}}}^{{b_{q}}})\prime$. This gives rise to a representation of the Fourier transform of an element $U \in (S_{{a_{k}}}^{{b_{q}}})\prime$ having support in a certain convex set as a weak limit of the analytic function. Converse results are also obtained. These generalized Paley-Wiener-Schwartz theorems are established by means of a number of new lemmas concerning $S_{{a_{k}}}^{{b_{q}}}$ and its dual. Finally, in the appendix the equality $ S_{{a_k}}^{{b_q}} = {S_{{a_k}}} \cap {S^{{b_q}}}$ is proved.
Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in ${\bf R}\sp{4}$
Steven R.
Dunbar
557-594
Abstract: We establish the existence of traveling wave solutions for a reaction-diffusion system based on the Lotka-Volterra differential equation model of a predator and prey interaction. The waves are of transition front type, analogous to the solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction-diffusion equation. The waves discussed here are not necessarily monotone. There is a speed ${c^\ast} > 0$ such that for $c > {c^\ast}$ there is a traveling wave moving with speed $c$. The proof uses a shooting argument based on the nonequivalence of a simply connected region and a nonsimply connected region together with a Liapunov function to guarantee the existence of the traveling wave solution. The traveling wave solution is equivalent to a heteroclinic orbit in $4$-dimensional phase space.
Minimal periodic orbits for continuous maps of the interval
Lluís
Alsedà;
Jaume
Llibre;
Rafel
Serra
595-627
Abstract: For continuous maps of the interval into itself, Sarkovskii's Theorem gives the notion of minimal periodic orbit. We complete the characterization of the behavior of minimal periodic orbits. Also, we show for unimodal maps that the min-max essentially describes the behavior of minimal periodic orbits.
The radiance obstruction and parallel forms on affine manifolds
William
Goldman;
Morris W.
Hirsch
629-649
Abstract: A manifold $ M$ is affine if it is endowed with a distinguished atlas whose coordinate changes are locally affine. When they are locally linear $ M$ is called radiant. The obstruction to radiance is a one-dimensional class $ {c_M}$ with coefficients in the flat tangent bundle of $M$. Exterior powers of ${c_M}$ give information on the existence of parallel forms on $M$, especially parallel volume forms. As applications, various kinds of restrictions are found on the holonomy and topology of compact affine manifolds.
The Godbillon-Vey invariant of a transversely homogeneous foliation
Robert
Brooks;
William
Goldman
651-664
Abstract: A real projective foliation is a foliation $ \mathfrak{F}$ with a system of local coordinates transverse to $\mathfrak{F}$ modelled on ${\mathbf{R}}{P^1}$ (so that the coordinate changes are real linear fractional transformations). Given a closed manifold $M$, there is but a finite set of values in ${H^3}(M;{\mathbf{R}})$ which the Godbillon-Vey invariant of such foliations may assume. A bound on the possible values, in terms of the fundamental group, is computed. For $M$ an oriented circle bundle over a surface, this finite set is explicitly computed.
Radial functions and invariant convolution operators
Christopher
Meaney
665-674
Abstract: For $1 < p < 2$ and $n > 1$, let ${A_p}({{\mathbf{R}}^n})$ denote the Figà-Talamanca-Herz algebra, consisting of functions of the form $ ( \ast)$ $\displaystyle \sum\limits_{k = 0}^\infty {{f_k} \ast {g_k}}$ with $\sum\nolimits_k {\vert\vert{f_k}\vert{\vert _p}\cdot\vert\vert{g_k}\vert{\vert _{p\prime}} < \infty }$. We show that if $2n/(n + 1) < p < 2$, then the subalgebra of radial functions in ${A_p}({{\mathbf{R}}^n})$ is strictly larger than the subspace of functions with expansions $( \ast )$ subject to the additional condition that ${f_k}$ and ${g_k}$ are radial for all $k$. This is a partial answer to a question of Eymard and is a consequence of results of Herz and Fefferman. We arrive at the statement above after examining a more abstract situation. Namely, we fix $G \in [FIA]_{B}^{ - }$ and consider $^B{A_p}(G)$ the subalgebra of $ B$-invariant elements of $ {A_p}(G)$. In particular, we show that the dual of $ ^B{A_p}(G)$ is equal to the space of bounded, right-translation invariant operators on $ {L^{p}}(G)$ which commute with the action of $B$.
Hermitian forms in function theory
Christine R.
Leverenz
675-688
Abstract: Let $f$ and $g$ be analytic in the unit disk $\vert z\vert\; < 1$. We give a new derivation of the positive semidefinite Hermitian form equivalent to $ \vert g(z)\vert \leq \vert f(z)\vert$, for $\vert z \vert < 1$, and use it to derive Hermitian forms for various classes of univalent functions. Sharp coefficient bounds for these classes are obtained from the Hermitian forms. We find the specific functions required to make the Hermitian forms equal to zero.
The transformation of vector-functions, scaling and bifurcation
R. J.
Magnus
689-713
Abstract: Various known methods for studying the bifurcation of zeros of a Banach-space-valued mapping are unified under a single idea, akin to using a coordinate transformation to obtain a simple form of the function under consideration. The general nature of the hypotheses permits the dropping of the pervasive "Fredholm condition" of bifurcation theory.
Inverse producing extension of a Banach algebra which eliminates the residual spectrum of one element
C. J.
Read
715-725
Abstract: If $A$ is a commutative unital Banach algebra and $G \subset A$ is a collection of nontopological zero divisors, the question arises whether we can find an extension $A\prime$ of $A$ in which every element of $G$ has an inverse. Shilov [1] proved that this was the case if $G$ consisted of a single element, and Arens [2] conjectures that it might be true for any set $ G$. In [3], Bollobás proved that this is not the case, and gave an example of an uncountable set $G$ for which no extension $A\prime$ can contain inverses for more than countably many elements of $G$. Bollobás proved that it was possible to find inverses for any countable $G$, and gave best possible bounds for the norms of the inverses in [4]. In this paper, it is proved that inverses can always be found if the elements of $G$ differ only by multiples of the unit; that is, we can eliminate the residual spectrum of one element of $A$. This answers the question posed by Bollobás in [5].
Triangulations of subanalytic sets and locally subanalytic manifolds
M.
Shiota;
M.
Yokoi
727-750
Abstract: If two polyhedrons are locally subanalytically homeomorphic (that is, the graph is locally subanalytic), they are ${\text{PL}}$ homeomorphic. A locally subanalytic manifold is one whose coordinate transformations are locally subanalytic. It is proved that a locally subanalytic manifold has a unique ${\text{PL}}$ manifold structure. A semialgebraic manifold also is considered.
Asymptotic periodicity of the iterates of Markov operators
A.
Lasota;
T.-Y.
Li;
J. A.
Yorke
751-764
Abstract: We say $P:{L^1} \to {L^1}$ is a Markov operator if (i) $ Pf \geq 0$ for $f \geq 0$ and (ii) $\Vert Pf\Vert = \Vert f\Vert$ if $f \geq 0$. It is shown that any Markov operator $P$ has certain spectral decomposition if, for any $f \in {L^1}$ with $f \geq 0$ and $ \Vert f\Vert = 1$, ${P^n}f \to \mathcal{F}$ when $n \to \infty$, where $ \mathcal{F}$ is a strongly compact subset of ${L^1}$. It follows from this decomposition that $ {P^n}f$ is asymptotically periodic for any $ f \in {L^1}$.
Finite subgroups of formal $A$-modules over ${\germ p}$-adic integer rings
Tetsuo
Nakamura
765-769
Abstract: Let $B \supset A$ be $ \mathfrak{p}$-adic integer rings such that $A/{Z_p}$ is finite and $B/A$ is unramified. Generalizing a result of Fontaine on finite commutative $p$-group schemes, we show that galois homomorphisms of finite subgroups of one-dimensional formal $ A$-modules over $ B$ are given by power series.
Galois theory for cylindric algebras and its applications
Stephen D.
Comer
771-785
Abstract: A Galois correspondence between cylindric set algebras and permutation groups is presented in this paper. Moreover, the Galois connection is used to help establish two important algebraic properties for certain classes of finite-dimensional cylindric algebras, namely the amalgamation property and the property that epimorphisms are surjective.
Behaviour of the velocity of one-dimensional flows in porous media
Juan Luis
Vázquez
787-802
Abstract: The one-dimensional flow of gas of density $u$ through a porous medium obeys the equation $ {u_t} = {({u^m})_{xx}}$, where $ m > 1,x \in {\mathbf{R}}$ and $t > 0$. We prove that the local velocity of the gas, given by $\upsilon = - m{u^{m - 2}}{u_x}$, not only is bounded for $ t \geqslant \tau > 0$ but approaches an $N$-wave profile as $ t \to \infty$. $ N$-waves are the typical asymptotic profiles for some first-order conservation laws, a class of nonlinear hyperbolic equations. The case $ m \leqslant 1$ is also studied: there are solutions with unbounded velocity while others have bounded velocity.
Exact sequences in stable homotopy pair theory
K. A.
Hardie;
A. V.
Jansen
803-816
Abstract: A cylinder-web diagram with associated diagonal sequences is described in stable homotopy pair theory. The diagram may be used to compute stable homotopy pair groups and also stable track groups of two-cell complexes. For the stable Hopf class $\eta$ the stable homotopy pair groups $ {G_k}(\eta ,\eta )(k \leqslant 8)$ are computed together with some of the additive structure of the stable homotopy ring of the complex projective plane.
Graphs of tangles
J. C.
Gómez-Larrañaga
817-830
Abstract: We prove that under necessary conditions a graph of tangles is a prime link. For this we generalize the result that the sum of $ 2$-string prime $ L$-tangles is a prime link. Some applications are found. We explore Property ${\text{L}}$ for tangles in order to prove primeness of knots.
Nonvanishing local cohomology classes
Ira
Moskowitz
831-837
Abstract: We discuss the nonvanishing of a top-dimensional canonical cohomology class of the space $\bar B\mathcal{D}if{f_\omega }\;M$. We treat parallelizable and odd-dimensional stably parallelizable manifolds.
Oriented manifolds that fiber over $S\sp{4}$
Steven M.
Kahn
839-850
Abstract: Necessary and sufficient conditions are given for an oriented manifold $ M$ to fiber up to cobordism over the $4$-sphere ${S^4}$ (i.e. for $M$ to be oriented cobordant to a fiber bundle over ${S^4}$). The result here extends those previously obtained for fiberings over ${S^1}$ and ${S^2}$. In addition, fiberings over products of surfaces are studied with complete solutions (in the sense above) being given in most cases including those of ${S^2} \times {S^2}$ and ${({S^1})^4}$.
On the structure of real transitive Lie algebras
Jack F.
Conn
1-71
Abstract: In this paper, we examine some of the ways in which abstract algebraic objects in a transitive Lie algebra $L$ are expressed geometrically in the action of each transitive Lie pseudogroup $ \Gamma$ associated to $ L$. We relate those chain decompositions of $\Gamma$ which result from considering $ \Gamma$-invariant foliations to Jordan-Hölder sequences (in the sense of Cartan and Guillemin) for $L$. Local coordinates are constructed which display the nature of the partial differential equations defining $\Gamma$; in particular, locally homogeneous pseudocomplex structures (also called ${\text{CR}}$-structures) are associated to the nonabelian quotients of complex type in a Jordan-Hölder sequence for $L$.
A method of lines for a nonlinear abstract functional evolution equation
A. G.
Kartsatos;
M. E.
Parrott
73-89
Abstract: Let $X$ be a real Banach space with $ {X^\ast}$ uniformly convex. A method of lines is introduced and developed for the abstract functional problem (E) $\displaystyle u\prime(t) + A(t)u(t) = G(t,{u_t}), \quad {u_0} = \phi , \quad t \in [0,T].$ The operators $A(t):D \subset X \to X$ are $m$-accretive and $G(t,\phi )$ is a global Lipschitzian-like function in its two variables. Further conditions are given for the convergence of the method to a strong solution of (E). Recent results for perturbed abstract ordinary equations are substantially improved. The method applies also to large classes of functional parabolic problems as well as problems of integral perturbations. The method is straightforward because it avoids the introduction of the operators $\hat A(t)$ and the corresponding use of nonlinear evolution operator theory.
Codimension $1$ orbits and semi-invariants for the representations of an equioriented graph of type $D\sb{n}$
S.
Abeasis
91-123
Abstract: We consider the Dynkin diagram ${D_n}$ equioriented and the variety $ \operatorname{Hom}({V_1},{V_3}) \times \Pi_{1 = 2}^n \operatorname{Hom} ({V_i},{V_{i + 1}})$, ${V_j}$ a vector space over $K$, on which the group $ G = \prod\nolimits_{i = 1}^n {{\text{GL}}} ({V_i})$ acts. We determine the maximal orbit and the codim. $1$ orbits of this action, giving their decomposition in terms of the irreducible representations of ${D_n}$. We also deduce a set of algebraically independent semi-invariant polynomials which generate the ring of semi-invariants.
$R$-sets and category
Rana
Barua
125-158
Abstract: We prove some category theoretic results for $R$-sets much in the spirit of Vaught and Burgess. Since the proofs entail many results on $ R$-sets and the $ R$-operator, we have studied them in some detail and have formulated many results appropriate for our purpose in, perhaps, a more unified manner than is available in the literature. Our main theorem is the following: Any $R$-set in the product of two Polish spaces can be approximated, in category, uniformly over all sections, by sets generated by rectangles with one side an $ R$-set and the other a Borel set. In fact, we prove a levelwise version of this result. For $C$-sets, this has been proved by V. V. Srivatsa.
Adapted probability distributions
Douglas N.
Hoover;
H. Jerome
Keisler
159-201
Abstract: We introduce a family of notions of equivalence for stochastic processes on spaces with an underlying filtration. These refine the notion of having the same distribution by taking account of the relation of the processes to their underlying filtrations. The weakest of these notions is the same as the notion of synonymity introduced by Aldous. Analysis of the strongest equivalence property leads to spaces with a strong universality property for adapted stochastic processes, which we call saturation. Spaces having this property contain 'strong' solutions to a large class of stochastic integral equations.
Interpolation and Gleason parts in $L$-domains
Michael Frederick
Behrens
203-225
Abstract: We describe the closure of $[ - 1/2,0)$ in the maximal ideal space $\mathcal{M}(\mathcal{D})$ of ${H^\infty }(\mathcal{D})$) for an arbitrary $L$-domain $ \mathcal{D}$. For $ L$-domains satisfying $ \sup ({c_{n + 1}}/{c_n}) < 1$ and $\Sigma {({r_n}/{c_n})^p} < \infty$, some $p \geqslant 1$, we describe all interpolation sequences for ${H^\infty }(\mathcal{D})$, we show that a homomorphism (except the distinguished homomorphism, when it exists) lies in a nontrivial Gleason part if and only if it is contained in the closure of an interpolating sequence, and we describe all the analytic structure occurring in $ \mathcal{M}(\mathcal{D})$.
Sous-espaces bien dispos\'es de $L\sp{1}$-applications
Gilles
Godefroy
227-249
Abstract: RÉsumÉ. On montre que le quotient d'un espace ${L^1}$ par un sous-espace fermé dont la boule unité est fermée dans $ {L^0}$ est faiblement séquentiellement complet; cette situation se présente dans de nombreux cas concrets, tels que le quotient ${L^1}/{H^1}$. On applique le résultat général dans diverses situations: duaux de certaines algères uniformes, analyse harmonique, fonctions de plusieurs variables complexes. On montre ensuite comment peuvent s'appliquer les métheodes de $M$-structure; on considère aussi de nouvelles classes d'uniques préduaux. A titre d'exemples, on montre: (1) Le caractère f.s.c. d'espaces $ {\mathcal{C}_E}{(G)^\ast}$, pour de "gros" sous-ensembles $ E$ du groupe dual $\Gamma = \hat G$. (2) Le caractère f.s.c. d'espaces $ {L^1}/{H^1}$ mutli-dimensionnels, tels que $ {L^1}/{H^1}({D^n})$ et $ {L^1}/{H^1}({B^n})$. (3) L'unicité du prédual pour certaines sous-algèbres ultrafaiblement fermées non-autoadjointes de $ \mathcal{L}(H)$. One shows that the quotient of an ${L^1}$-space by a closed subspace, whose unit ball is closed in ${L^0}$, is weakly sequentially complete. This situation occurs in many natural cases, like ${L^1}/{H^1}$. This result is applied in several situations: uniform algebras, harmonic analysis, functions of several complex variables. One shows how to apply $M$-structure theory; several new classes of unique preduals are also obtained. As an example, one shows: (1) If $E$ is a "big" subset of the dual group $\Gamma = \hat G$, then $ {\mathcal{C}_E}{(G)^\ast}$ is w.s.c. (2) The spaces ${L^1}/{H^1}({D^n})$ and ${L^1}/{H^1}({B^n})$ are w.s.c. (3) Several classes of ${\omega ^\ast}$-closed non-self-adjoint subalgebras of $ \mathcal{L}(H)$ have unique preduals.
The amalgamation property for varieties of lattices
Alan
Day;
Jaroslav
Ježek
251-256
Abstract: There are precisely three varieties of lattices that satisfy the amalgamation property: trivial lattices, distributive lattices, and all lattices.
The evolution of random graphs
Béla
Bollobás
257-274
Abstract: According to a fundamental result of Erdös and Rényi, the structure of a random graph ${G_M}$ changes suddenly when $M \sim n/2$: if $M = \left\lfloor {cn} \right\rfloor$ and $c < \frac{1}{2}$ then a.e. random graph of order $ n$ and since $ M$ is such that its largest component has $O(\log n)$ vertices, but for $c > \frac{1}{2}$ a.e. ${G_M}$ has a giant component: a component of order $ (1-{\alpha _c}+o(1))n$ where ${\alpha _c} < 1$. The aim of this paper is to examine in detail the structure of a random graph ${G_M}$ when $M$ is close to $n/2$. Among others it is proved that if $M = n/2 + s$, $s = o(n)$ and $s \geq {(\log n)^{1/2}}{n^{2/3}}$ then the giant component has $ (4 + o(1))s$ vertices. Furthermore, rather precise estimates are given for the order of the $r$th largest component for every fixed $ r$.
The formation of the dead core in parabolic reaction-diffusion problems
Catherine
Bandle;
Ivar
Stakgold
275-293
Abstract: For some nonlinear parabolic problems of reaction-diffusion, a region of zero reactant concentration may be formed in finite time. Conditions are formulated for the existence of such a dead core and estimates for its time of onset are also given. These results complement previous ones that dealt with the stationary (elliptic) problem.
Stochastic representation and singularities of solutions of second order equations with semidefinite characteristic form
Kazuo
Amano
295-312
Abstract: In the theory of partial differential equations, there is no explicit representation of solutions for general degenerate elliptic-parabolic equations. However, Stroock and Varadhan [15] have obtained a stochastic representation for such a wider class of equations in $ {L^\infty }$ space. In this paper we establish, by using Stroock and Varadhan's stochastic representation, a method which enables us to construct solutions with singularities of second order equations with semidefinite characteristic form. Our theorems are not probabilistic paraphrases of the results obtained in the theory of partial differential equations. In fact, each assumption of the theorems is much weaker than any assumption of corresponding known results.
Bilinear forms on $H\sp{\infty }$ and bounded bianalytic functions
J.
Bourgain
313-337
Abstract: Given an arbitrary Radon probability measure on the circle $\pi$, a generlization of the classical Cauchy transform is obtained. These projections are used to prove that each bounded linear operator from a reflexive subspace of ${L^1}$ or $ {L^1}(\pi )/{H^1}$ into ${H^\infty }(D)$ admits a bounded extension. These facts lead to different variants of the cotype-$ 2$ inequality for ${L^1}(\pi )/{H^1}$. Applications are given to absolutely summing operators and the existence of certain bounded bianalytic functions. For instance, we derive the Hilbert space factorization of arbitrary bounded linear operators from ${H^\infty }(D)$ into its dual without an a priori approximation hypothesis, thus completing some of the work in [1]. Our methods give new information about the Fourier coefficients of ${H^\infty }(D \times D)$-functions, thus improving a theorem in [6].
On Skolem's exponential functions below $2\sp{2\sp{X}}$
Lou
van den Dries;
Hilbert
Levitz
339-349
Abstract: A result of Ehrenfeucht implies that the smallest class of number-theoretic functions $f:{\mathbf{N}} \to {\mathbf{N}}$ containing the constants $ 0,1,2, \ldots$, the identity function $X$, and closed under addition, multiplication and $f \to {f^X}$, is well-ordered by the relation of eventual dominance. We show that its order type is $ {\omega ^{{\omega ^\omega }}}$, and that for any two nonzero functions $ f,g$ in the class the quotient $f(n)/g(n)$ tends to a limit in ${E^ + } \cup \{ 0,\infty \}$ as $n \to \infty$, where ${E^ + }$ is the smallest set of positive real numbers containing $1$ and closed under addition, multiplication and under the operations $x \to {x^{ - 1}},x \to {e^x}$.
The Selberg trace formula. V. Questions of trace class
M. Scott
Osborne;
Garth
Warner
351-376
Abstract: The purpose of this paper is to develop criteria which will ensure that the $K$-finite elements of $C_c^\infty (G)$ are represented on $ L_{{\text{dis}}}^2(G/\Gamma )$ by trace class operators.
A general approach to the optimality of minimum distance estimators
P. W.
Millar
377-418
Abstract: Let $\Theta$ be an open subset of a separable Hilbert space, and $ {\xi _n}(\theta )$, $\theta \in \Theta$, a sequence of stochastic processes with values in a (different) Hilbert space $ B$. This paper develops an asymptotic expansion and an asymptotic minimax result for "estimates" $ {\hat \theta _n}$ defined by ${\inf _\theta }\vert{\xi _n}(\theta )\vert = \vert{\xi _n}({\hat \theta _n})\vert$, where $ \vert \cdot \vert$ is the norm of $B$. The abstract results are applied to study optimality and asymptotic normality of procedures in a number of important practical problems, including simple regression, spectral function estimation, quantile function methods, min-chi-square methods, min-Hellinger methods, minimum distance methods based on $ M$-functionals, and so forth. The results unify several studies in the literature, but most of the $ {\text{LAM}}$ results are new. From the point of view of applications, the entire paper is a sustained essay concerning the problem of fitting data with a reasonable, but relatively simple, model that everyone knows cannot be exact.
On hypersingular integrals and anisotropic Bessel potential spaces
H.
Dappa;
W.
Trebels
419-429
Abstract: In this paper we characterize anisotropic potential spaces in terms of hypersingular integrals of mixed homogeneity with respect to a general dilation matrix.
Infinitesimally rigid polyhedra. I. Statics of frameworks
Walter
Whiteley
431-465
Abstract: From the time of Cauchy, mathematicians have studied the motions of convex polyhedra, with the faces held rigid while changes are allowed in the dihedral angles. In the 1940s Alexandrov proved that, even with additional vertices along the natural edges, and with an arbitrary triangulation of the natural faces on these vertices, such polyhedra are infinitesimally rigid. In this paper the dual (and equivalent) concept of static rigidity for frameworks is used to describe the behavior of bar and joint frameworks built around convex (and other) polyhedra. The static techniques introduced provide a new simplified proof of Alexandrov's theorem, as well as an essential extension which characterizes the static properties of frameworks built with more general patterns on the faces, including frameworks with vertices interior to the faces. The static techniques are presented and employed in a pattern appropriate to the extension of an arbitrary statically rigid framework built around any polyhedron (nonconvex, toroidal, etc.). The techniques are also applied to derive the static rigidity of tensegrity frameworks (with cables and struts in place of bars), and the static rigidity of frameworks projectively equivalent to known polyhedral frameworks. Finally, as an exercise to give an additional perspective to the results in $3$-space, detailed analogues of Alexandrov's theorem are presented for convex $4$-polytopes built as bar and joint frameworks in $4$-space.
On the realization of invariant subgroups of $\pi \sb\ast (X)$
A.
Zabrodsky
467-496
Abstract: Let $p$ be a prime and $T:X \to X$ a self map. Let $A$ be a multiplicatively closed subset of the algebraic closure of ${F_p}$. Denote by ${V_{T,A}}$ the set of characteristic values of $ {\pi_{\ast} }(T) \otimes {F_p}$ lying in $A$. It is proved that under certain conditions $ {V_{T,A}}$ is realizable by a pair $ \tilde X,\tilde T$: There exist a space $\tilde X$, maps $\tilde T:\tilde X \to \tilde X$ and $f:\tilde X\: \to \:X$ so that $ f\,\circ \,\tilde T\sim T\,\circ \,f,{\pi _ * }(F)$ is $\bmod\, p$ injective and $ {\rm {im}}({\pi_{\ast} }(f) \otimes {F_p}) = {V_{T,A}}$. This theorem yields, among others, examples of spaces whose $ \bmod\, p$ cohomology rings are polynomial algebras.
On Kummer's twenty-four solutions of the hypergeometric differential equation
B.
Dwork
497-521
Abstract: The $p$-adic analyticity of the Boyarsky matrix associated with the hypergeometric function $F(a,b;c;x)$ has been investigated in an earlier article. The transformation of this matrix under translation of $(a, b, c)$ by $ {{\bf {Z}}^3}$ was determined at that time. This article gives the transformation of this matrix under the extended Kummer group. The $p$-adic implications of quadratic and higher-degree transformations remain open.
Normal structure and weakly normal structure of Orlicz sequence spaces
Thomas
Landes
523-534
Abstract: For a convex Orlicz function $\varphi :{{\bf {R}}_ + } \to {{\bf {R}}_ + } \cup \{ \infty \}$ and the associated Orlicz sequence space $ {l_\varphi }$, we consider the following five properties: (1) ${l_\varphi }$ has a subspace isometric to $ {l_1}$. (2) ${l_\varphi }$ is Schur. (3) ${l_\varphi }$ has normal structure. (4) Every weakly compact subset of $ {l_\varphi }$ has normal structure. (5) Every bounded sequence in ${l_\varphi }$ has a subsequence $({x_n})$ which is pointwise and almost convergent to $ x \in {l_\varphi }$, i.e., $ \lim \,{\sup_{n \to \infty }}\parallel {x_n} - x{\parallel_{\varphi }} < \lim \inf _{n \to \infty }\parallel {x_n} - y{\parallel_\varphi }$ for all $y \ne x$. Our results are: (1) $ \Leftrightarrow \;\varphi$ is either linear at $0\;(\varphi (s)/s = c > 0,0 < s \leqslant t)$ or does not satisfy the $ {\Delta_2}$-condition at 0. (2) $\Leftrightarrow \;{l_\varphi }$ is isomorphic to ${l_1}\; \Leftrightarrow \;\varphi^{\prime}(0) = {\lim_{t \to 0}}\,\varphi \,(t)/t > 0$. (3) $\Leftrightarrow \varphi$ satisfies the ${\Delta_2}$-condition at $ 0, \varphi$ is not linear at 0 and $ C(\varphi ) = \sup \,\{ \varphi \,(t) < 1\} > \frac{1}{2}$. (4) $ \Leftrightarrow \,\varphi$ satisfies the $ {\Delta_2}$-condition at 0 and $C\,(\varphi ) > \frac{1}{2}\;{\rm {or}}\;\varphi^{\prime}(0) > 0$. (5) $\Leftrightarrow \;\varphi $ satisfies the ${\Delta_2}$-condition at 0 and $C(\varphi ) = 1$. The last equivalence contains a result of Lami-Dozo [10].
On the representation of order continuous operators by random measures
L.
Weis
535-563
Abstract: Using the representation $ Tf(y) = \smallint f\;d{v_y}$, where $({v_y})$ is a random measure, we characterize some interesting bands in the lattice of all order-continuous operators on a space of measurable functions. For example, an operator $T$ is (lattice-)orthogonal to all integral operators (i.e. all ${v_y}$ are singular) or belongs to the band generated by all Riesz homomorphisms (i.e. all ${v_y}$ are atomic) if and only if $ T$ satisfies certain properties which are modeled after the Riesz homomorphism property [31] and continuity with respect to convergence in measure. On the other hand, for operators orthogonal to all Riesz homomorphisms (i.e. all $ {v_y}$ are diffuse) we give characterizations analogous to the characterizations of Dunford and Pettis, and Buhvalov for integral operators. The latter results are related to Enflo operators, to a result of J. Bourgain on Dunford-Pettis operators and martingale representations of operators.
Embeddings of Harish-Chandra modules, ${\germ n}$-homology and the composition series problem: the case of real rank one
David H.
Collingwood
565-579
Abstract: Let $G$ be a connected semisimple matrix group of real rank one. Fix a minimal parabolic subgroup $P = MAN$ and form the (normalized) principal series representations $I_P^G(U)$. In the case of regular infinitesimal character, we explicitly determine (in terms of Langlands' classification) all irreducible submodules and quotients of $ I_P^G(U)$. As a corollary, all embeddings of an irreducible Harish-Chandra module into principal series are computed. The number of such embeddings is always less than or equal to three. These computations are equivalent to the determination of zero $ {\mathfrak{n}}$-homology.
Asymptotic expansions of traces for certain convolution operators
Raymond
Roccaforte
581-602
Abstract: A version of Szegö's theorem in Euclidean space gives the first two terms of the asymptotics as $\alpha \to \infty$ of the determinant of convolution operators on ${L_2}(\alpha \,\Omega )$, where $\Omega$ is a bounded subset of ${{\mathbf{R}}^n}$ with smooth boundary. In this paper the more general problem of the asymptotics of traces of certain analytic functions of the operators is considered and the next term in the expansion is obtained.
Structure sets vanish for certain bundles over Seifert manifolds
Christopher W.
Stark
603-615
Abstract: Let ${M^{n + 3}}$ be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold ${K^3}$ with an effective circle action of hyperbolic type. Assume that the fiber ${N^n}$ in this bundle is a closed orientable manifold with Noetherian integral group ring, with vanishing projective class and Whitehead groups, and such that the structure set $ {S_{\text{TOP}}}\,({N^n} \times {D^k},\partial )$ of topological surgery vanishes for sufficiently large $k$. Then the projective class and Whitehead groups of $M$ vanish and $ {S_{\text{TOP}}}\,({M^{n + 3}}\, \times \, {D^k},\partial ) = 0$ if $n + k \geqslant 3$ or if ${K^3}$ is closed and $n = 2$. The UNil groups of Cappell are the main obstacle here, and these results give new examples of generalized free products of groups such that ${\text{UNil}}_j$ vanishes in spite of the failure of Cappell's sufficient condition.
$\omega $-morasses, and a weak form of Martin's axiom provable in ${\rm ZFC}$
Dan
Velleman
617-627
Abstract: We prove, in ZFC, that simplified gap-$ 1$ morasses of height $ \omega$ exist. By earlier work on the relationship between morasses and forcing it immediately follows that a certain Martin's axiom-type forcing axiom is provable in ZFC. We show that this forcing axiom can be thought of as a weak form of ${\text{MA}}_{\omega_1}$ and give some applications.
Boundedness of fractional operators on $L\sp{p}$ spaces with different weights
Eleonor
Harboure;
Roberto A.
Macías;
Carlos
Segovia
629-647
Abstract: Let ${T_\alpha }$ be either the fractional integral operator $\smallint f(y)\vert x - y{\vert^{\alpha - n}}\; dy$, or the fractional maximal operator $\sup \left\{ {{r^{\alpha - n}}{\smallint_{\vert x - y\vert < r}}\vert f(y)\vert dy:\,r > 0} \right\}$. Given a weight $w$ (resp. $\upsilon$), necessary and sufficient conditions are given for the existence of a nontrivial weight $ \upsilon$ (resp. $ w$) such that ${(\smallint \vert{T_\alpha }f{\vert^q}\upsilon \;dx)^{1/q}} \leqslant {(\smallint\vert f{\vert^p}w\;dx)^{1/p}}$ holds. Weak type substitutes in limiting cases are considered.
The Dieudonn\'e property on $C(K,\,E)$
Fernando
Bombal;
Pilar
Cembranos
649-656
Abstract: In this paper we prove that if $E$ is a Banach space with separable dual, then the space $C(K,E)$ of all continuous $E$-valued functions on a compact Hausdorff topological space $K$ has the Dieudonné property.
Infinite-to-one codes and Markov measures
Mike
Boyle;
Selim
Tuncel
657-684
Abstract: We study the structure of infinite-to-one continuous codes between subshifts of finite type and the behaviour of Markov measures under such codes. We show that if an infinite-to-one code lifts one Markov measure to a Markov measure, then it lifts each Markov measure to uncountably many Markov measures and the fibre over each Markov measure is isomorphic to any other fibre. Calling such a code Markovian, we characterize Markovian codes through pressure. We show that a simple condition on periodic points, necessary for the existence of a code between two subshifts of finite type, is sufficient to construct a Markovian code. Several classes of Markovian codes are studied in the process of proving, illustrating and providing contrast to the main results. A number of examples and counterexamples are given; in particular, we give a continuous code between two Bernoulli shifts such that the defining vector of the image is not a clustering of the defining vector of the domain.
Surgery on Poincar\'e complexes
J. P. E.
Hodgson
685-701
Abstract: A geometric approach to surgery on Poincaré complexes is described. The procedure mimics the original techniques for manifolds. It is shown that the obstructions to surgering a degree-one normal map of Poincaré complexes to a homotopy equivalence lie in the Wall groups, and that all elements in these groups can arise as obstructions.
Approximating groups of bundle automorphisms by loop spaces
Roberto
Bencivenga
703-715
Abstract: D. H. Gottlieb proved in 1972 that the group of automorphisms of a numerable $ G$-bundle $p:X \to B$ is weakly homotopy equivalent to $\Omega \;\operatorname{Map}(B,{B_G};k)$, where $ k:B \to {B_G}$ is a classifying map for $p$. We refine this classical result by constructing a genuine homotopy equivalence between these two spaces which is natural with respect to numerable bundle morphisms, can be generalized to fibre bundles, and can be interpreted as a natural isomorphism between two suitably defined functors.
The interfaces of one-dimensional flows in porous media
Juan L.
Vázquez
717-737
Abstract: The solutions of the equation $ {u_t} = {({u^m})_{x\,x}}$ for $x \in {\mathbf{R}},0 < t < T,m > 1$, where $ u(x,0)$ is a nonnegative Borel measure that vanishes for $x > 0$ (and satisfies a growth condition at $- \infty$), exhibit a finite, monotone, continuous interface $ x = \zeta (t)$ that bounds to the right the region where $u > 0$. We perform a detailed study of $ \zeta$: initial behaviour, waiting time, behaviour as $ t \to \infty$. For certain initial data the solutions blow up in a finite time ${T^{\ast}}$: we calculate ${T^{\ast}}$ in terms of $u(x,0)$ and describe the behaviour of $\zeta$ as $t\, \uparrow \,{T^{\ast}}$.
On Fourier multiplier transformations of Banach-valued functions
Terry R.
McConnell
739-757
Abstract: We obtain analogues of the Mihlin multiplier theorem and Littlewood-Paley inequalities for functions with values in a suitable Banach space $B$. The requirement on $B$ is that it have the unconditionality property for martingale difference sequences.
Real zeros of derivatives of meromorphic functions and solutions of second order differential equations
Simon
Hellerstein;
Li-Chien
Shen;
Jack
Williamson
759-776
Abstract: We classify all functions $F$ meromorphic in the plane with only real zeros and real poles which satisfy the additional conditions that $F^{\prime}$ has no zeros and $ F''$ only real zeros. We apply this classification, in combination with some earlier results, to the study of the reality of zeros of solutions of the equation $w'' + H(z)w = 0,H$ entire.
Central limit theorem for products of random matrices
Marc A.
Berger
777-803
Abstract: Using the semigroup product formula of P. Chernoff, a central limit theorem is derived for products of random matrices. Applications are presented for representations of solutions to linear systems of stochastic differential equations, and to the corresponding partial differential evolution equations. Included is a discussion of stochastic semigroups, and a stochastic version of the Lie-Trotter product formula.
A quasilinear hyperbolic integro-differential equation related to a nonlinear string
Melvin L.
Heard
805-823
Abstract: We discuss global existence, boundedness and regularity of solutions to the integrodifferential equation \begin{displaymath}\begin{array}{*{20}{c}} {{u_{t\,t}}\,(t,x) + c\,(t)\,{u_t}(t,... ..._t}(0,x) = {u_1}(x), \qquad x \in {\mathbf{R}}.} \end{array}\end{displaymath} This type of equation occurs in the study of the nonlinear behavior of elastic strings. We show that if the initial data ${u_0}\,(x),{u_1}\,(x)$ is small in a suitable sense, and if the damping coefficient $c\,(t)$ grows sufficiently fast, then the above equation possesses a globally defined classical solution for forcing terms $ h\,(t,x,u)$ which are sublinear in $u$. In the nonlinearity we require that $M \in {C^1}\,[0,\infty )$ and, in addition, satisfies $ M( \lambda ) \geq {m_0} > 0$ for all $ \lambda \geq 0$.
Generic reducing fields of Jordan pairs
Holger P.
Petersson
825-843
Abstract: Generic reducing fields of Jordan pairs, generalizing at the same time generic splitting fields of associative algebras and generic zero fields of quadratic forms, are intrinsically defined and constructed. The most elementary properties are derived, and the relationship with other generic constructions, particularly those linked to Brauer-Severi varieties, are investigated. As an application it is shown that there exist nonisomorphic exceptional Jordan division algebras having the same splitting fields.
Generic representations are induced from square-integrable representations
Ronald L.
Lipsman
845-854
Abstract: It is proven for arbitrary real algebraic groups that the generic irreducible unitary representation is induced from a square-integrable representation (modulo the projective kernel). This generalizes the well-known result for reductive groups that the generic representations are either discrete series, or induced from discrete series (modulo the nilradical) representations of cuspidal parabolic subgroups.
The behavior under projection of dilating sets in a covering space
Burton
Randol
855-859
Abstract: Let $M$ be a compact Riemannian manifold with covering space $S$, and suppose $d{\mu_r}\;(0 < r < \infty )$ is a family of Borel probability measures on $S$, all of which arise from some fixed measure by $ r$-homotheties of $ S$ about some point, followed by renormalization of the resulting measure. In this paper we study the ergodic properties, as a function of $r$, of the corresponding family of projected measures on $M$ in the Euclidean and hyperbolic cases. A typical example arises by considering the behavior of a dilating family of spheres under projection.
Representations of compact groups on Banach algebras
David
Gurarie
1-55
Abstract: Let a compact group $ U$ act by automorphisms of a commutative regular and Wiener Banach algebra $\mathcal{A}$. We study representations ${R^\omega }$ of $U$ on quotient spaces $\mathcal{A}/I(\omega )$, where $\omega$ is an orbit of $U$ in the Gelfand space $X$ of $ \mathcal{A}$ and $I(\omega )$ is the minimal closed ideal with hull $\omega \subset X$. The main result of the paper is: if $\mathcal{A} = \,{\mathcal{A}_\rho }(X)$ is a weighted Fourier algebra on a LCA group $X = \hat A$ with a subpolynomial weight $ \rho$ on $A$, and $U$ acts by affine transformations on $ X$, then for any orbit $\omega \subset X$ the representation ${R^\omega }$ has finite multiplicity. Precisely, the multiplicity of $ \pi \in \hat U$ in ${R^\omega }$ is estimated as $k(\pi ;{R^\omega }) \leq c \cdot \deg (\pi )\;\forall \pi \in \hat U$ with a constant $c$ depending on $A$ and $\rho$. Applications of this result are given to topologically irreducible representations of motion groups and primary ideals of invariant subalgebras.
Homomorphisms and endomorphisms in varieties of pseudocomplemented distributive lattices (with applications to Heyting algebras)
M. E.
Adams;
V.
Koubek;
J.
Sichler
57-79
Abstract: According to a result by K. B. Lee, the lattice of varieties of pseudocomplemented distributive lattices is the $\omega + 1$ chain ${B_{ - 1}} \subset {B_0} \subset {B_1} \subset \cdots \subset {B_n} \subset \cdots \subset {B_\omega }$ in which the first three varieties are formed by trivial, Boolean, and Stone algebras respectively. In the present paper it is shown that any Stone algebra is determined within ${B_1}$ by its endomorphism monoid, and that there are at most two nonisomorphic algebras in $ {B_2}$ with isomorphic monoids of endomorphisms; the pairs of such algebras are fully characterized both structurally and in terms of their common endomorphism monoid. All varieties containing ${B_3}$ are shown to be almost universal. In particular, for any infinite cardinal $\kappa$ there are ${2^\kappa }$ nonisomorphic algebras of cardinality $\kappa$ in ${B_3}$ with isomorphic endomorphism monoids. The variety of Heyting algebras is also almost universal, and the maximal possible number of nonisomorphic Heyting algebras of any infinite cardinality with isomorphic endomorphism monoids is obtained.
Necessary and sufficient conditions for oscillations of higher order delay differential equations
G.
Ladas;
Y. G.
Sficas;
I. P.
Stavroulakis
81-90
Abstract: Consider the $n{\text{th}}$ order delay differential equation (1) $\displaystyle {x^{(n)}}(t) + {( - 1)^{n + 1}}\sum\limits_{i = 0}^k {{p_i}x(t - {\tau_i}) = 0, \qquad t \geq {t_0}},$ where the coefficients and the delays are constants such that $0 = {\tau_0} < {\tau_{1}}\, < \cdots < {\tau_k};{p_0}\, \geq 0,{p_i} > 0,i = 1,2,\ldots,k;k \geq 1$ and $n \geq 1$. The characteristic equation of (1) is (2) $\displaystyle {\lambda ^n} + {( - 1)^{n + 1}}\;\sum\limits_{i = 0}^k {{p_i}{e^{ - \lambda {\tau_i}}} = 0}.$ We prove the following theorem. Theorem. (i) For $n$ odd every solution of (1) oscillates if and only if (2) has no real roots. (ii) For $n$ even every bounded solution of (1) oscillates if and only if (2) has no real roots in $( - \infty ,0]$. The above results have straightforward extensions for advanced differential equations.
Factorizing the polynomial of a code
G.
Hansel;
D.
Perrin;
C.
Reutenauer
91-105
Abstract: We give an extension and a simplified presentation of a theorem of Schützenberger. This theorem describes the factorization of the commutative polynomial associated with a finite maximal code. It is the deepest result known so far in the theory of (variable-length) codes.
Riesz decompositions in Markov process theory
R. K.
Getoor;
J.
Glover
107-132
Abstract: Riesz decompositions of excessive measures and excessive functions are obtained by probabilistic methods without regularity assumptions. The decomposition of excessive measures is given for Borel right processes. The results for excessive functions are formulated within the framework of weak duality. These results extend and generalize the pioneering work of Hunt in this area.
On bases in the disc algebra
J.
Bourgain
133-139
Abstract: It is shown that the disc algebra has no Besselian basis. In fact, concrete minorations on certain Lebesgue functions are obtained. A consequence is the nonisomorphism of the disc algebra and the space of uniformly convergent Fourier series on the circle.
Convergence of multivariate polynomials interpolating on a triangular array
T. N. T.
Goodman;
A.
Sharma
141-157
Abstract: Given a triangular array of complex numbers, it is well known that for any function $f$ smooth enough, there is a unique polynomial $ {G_n}f$ of degree $ \leq n$ such that on each of the first $n + 1$ rows of the array the divided difference of $ {G_n}f$ coincides with that of $f$. This result has recently been generalized to give a unique polynomial ${\mathcal{G}_n}f$ in $k$ variables $(k > 1)$ of total degree $\leq n$ which interpolates a given function $f$ on a triangular array in ${C^k}$. In this paper we extend some results of A. O. Gelfond and derive formulas for ${\mathcal{G}_n}f$ and $f - {\mathcal{G}_n}f$ to prove some results on convergence of $ {\mathcal{G}_n}f$ to $ f$ as $n \to \infty$ under various conditions on $ f$ and on the triangular array.
Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems
Michael
Beals;
Michael
Reed
159-184
Abstract: The authors develop a calculus of pseudodifferential operators with nonsmooth coefficients in order to study the regularity of solutions to linear equations $P\,(x,D)\,u = f$. The regularity theorems are similar to those of Bony, but the calculus and the methods of proof are quite different. We apply the linear results to study the regularity properties of solutions to quasilinear partial differential equations.
On some nonextendable derivations of the gauge-invariant CAR algebra
Geoffrey L.
Price
185-201
Abstract: We provide examples of some approximately inner, commutative $ \ast$-derivations which are generators on the gauge-invariant CAR algebra but which have no closed densely-defined extensions to the CAR. Necessary conditions are given for a class of generators on the GICAR algebra to extend to closed $\ast$-derivations on the CAR.
Extremal problems for polynomials with exponential weights
H. N.
Mhaskar;
E. B.
Saff
203-234
Abstract: For the extremal problem: $\displaystyle {E_{n,r}}(\alpha ): = \min \parallel \exp ( - \vert x{\vert^\alpha })\,({x^n} + \cdots ){\parallel_{{L^r}}}, \qquad \alpha > 0,$ where $ {L^r}\,(0 < r \leqslant \infty )$ denotes the usual integral norm over ${\mathbf{R}}$, and the minimum is taken over all monic polynomials of degree $n$, we describe the asymptotic form of the error ${E_{n,r}}(\alpha )\;({\text{as}}\;n \to \infty )$ as well as the limiting distribution of the zeros of the corresponding extremal polynomials. The case $r = 2$ yields new information regarding the polynomials $\{ {p_n}(\alpha ;x) = {\gamma_n}(\alpha )\,{x^n} + \cdots \}$ which are orthonormal on ${\mathbf{R}}$ with respect to $\exp ( - 2\vert x{\vert^\alpha })$. In particular, it is shown that a conjecture of Freud concerning the leading coefficients ${\gamma_n}(\alpha )$ is true in a Cesàro sense. Furthermore a contracted zero distribution theorem is proved which, unlike a previous result of Ullman, does not require the truth of the Freud's conjecture. For $ r = \infty ,\alpha > 0$ we also prove that, if $\deg {P_n}(x) \leqslant n$, the norm $\parallel \exp ( - \vert x\vert^{\alpha })\,{P_n}(x)\parallel_{{L^\infty }}$ is attained on the finite interval $\displaystyle \left[ { - {{(n/{\lambda_\alpha })}^{1/\alpha }},{{(n/{\lambda_\a... ...ambda_\alpha } = \Gamma (\alpha )/{2^{\alpha - 2}}{\{ \Gamma (\alpha /2)\} ^2}.$ Extensions of Nikolskii-type inequalities are also given.
The strong conclusion of the F. and M. Riesz theorem on groups
I.
Glicksberg
235-240
Abstract: Let $S$ be a closed proper generating subsemigroup of the dual $\Gamma$ of a locally compact abelian group $ G$. Then there are Haar singular measures on $G$ orthogonal to $S$ unless $G = {\mathbf{R}} \times \Delta$ or ${\mathbf{T}} \times \Delta $ with $\Delta$ discrete, and then all $ \mu$ orthogonal to $ S$ are Haar absolutely continuous.
The theory of $G\sp{\infty }$-supermanifolds
Charles P.
Boyer;
Samuel
Gitler
241-267
Abstract: A theory of supermanifolds is developed in which a supermanifold is an ordinary manifold associated with a certain integrable second order $G$-structure. A structure theorem is proved showing that every $ {G^\infty }$-supermanifold has a complete distributive lattice of foliations with flat affine leaves. Furthermore, an existence and uniqueness theorem for local flows of ${G^\infty }$ vector fields is proved.
Cyclic vectors in the Dirichlet space
Leon
Brown;
Allen L.
Shields
269-303
Abstract: We study the Hilbert space of analytic functions with finite Dirichlet integral in the open unit disc. We try to identify the functions whose polynomial multiples are dense in this space. Theorems 1 and 2 confirm a special case of the following conjecture: if $\vert f(z)\vert \geqslant \vert g\,(z)\vert$ at all points and if $g$ is cyclic, then $f$ is cyclic. Theorems 3-5 give a sufficient condition ($f$ is an outer function with some smoothness and the boundary zero set is at most countable) and a necessary condition (the radial limit can vanish only for a set of logarithmic capacity zero) for a function $ f$ to be cyclic.
Involutions with isolated fixed points on orientable $3$-dimensional flat space forms
E.
Luft;
D.
Sjerve
305-336
Abstract: In this paper we completely classify (up to conjugacy) all involutions $\iota: M \to M$, where $M$ is an orientable connected flat $ 3$-dimensional space form, such that $\iota$ has fixed points but only finitely many. If $ M_1,\ldots,M_6$ are the $ 6$ space forms then only $M_1, M_2, M_6$ admit such involutions. Moreover, they are unique up to conjugacy. The main idea behind the proof is to find incompressible tori $T \subseteq M$ so that either $\iota(T) = T$ or $\iota(T) \cap T = \varnothing$ and then cut $ M$ into simpler pieces. These results lead to a complete classification of $ 3$-manifolds containing $\mathbf{Z} \oplus \mathbf{Z} \oplus \mathbf{Z}$ in their fundamental groups.
Compressed algebras: Artin algebras having given socle degrees and maximal length
Anthony
Iarrobino
337-378
Abstract: J. Emsalem and the author showed in [18] that a general polynomial $ f$ of degree $ j$ in the ring $ \mathcal{R} = k[ {{y_1},\ldots,{y_r}} ]$ has $\left( {\begin{array}{*{20}{c}} {j + r - 1} {r - 1} \end{array} } \right)$ linearly independent partial derivates of order $ i$, for $i = 0,1,\ldots,t = [ {j/2} ]$. Here we generalize the proof to show that the various partial derivates of $ s$ polynomials of specified degrees are as independent as possible, given the room available. Using this result, we construct and describe the varieties $G(E)$ and $Z(E)$ parametrizing the graded and nongraded compressed algebra quotients $A = R/I$ of the power series ring $R = k[[{x_1},\ldots,{x_r}]]$, having given socle type $E$. These algebras are Artin algebras having maximal length $\dim {_{k}}A$ possible, given the embedding degree $r$ and given the socle-type sequence $ E = ({e_1},\ldots,{e_s})$, where ${e_i}$ is the number of generators of the dual module $\hat A$ of $A$, having degree $i$. The variety $Z(E)$ is locally closed, irreducible, and is a bundle over $G(E)$, fibred by affine spaces ${{\mathbf{A}}^N}$ whose dimension is known. We consider the compressed algebras a new class of interesting algebras and a source of examples. Many of them are nonsmoothable--have no deformation to $(k + \cdots + k)$--for dimension reasons. For some choices of the sequence $ E,{\text{D}}$. Buchsbaum, ${\text{D}}$. Eisenbud and the author have shown that the graded compressed algebras of socle-type $ E$ have almost linear minimal resolutions over $R$, with ranks and degrees determined by $ E$. Other examples have given type $e = {\dim_k}\;({\text{socle}}\;A)$ and are defined by an ideal $I$ with certain given numbers of generators in $R = k[[{x_1},\ldots\;,{x_r}]]$. An analogous construction of thin algebras $ A = R/({f_1},\ldots,{f_s})$ of minimal length given the initial degrees of $ {f_1},\ldots,{f_s}$ is compared to the compressed algebras. When $ r = 2$, the thin algebras are characterized and parametrized, but in general when $r > 3$, even their length is unknown. Although $k = {\mathbf{C}}$ through most of the paper, the results extend to characteristic $ p$.
Closed timelike geodesics
Gregory J.
Galloway
379-388
Abstract: It is shown that every stable free $t$-homotopy class of closed timelike curves in a compact Lorentzian manifold contains a longest curve which must be a closed timelike geodesic. This result enables one to obtain a Lorentzian analogue of a classical theorem of Synge. A criterion for stability is presented, and a theorem of Tipler is derived as a special case of the result stated above.
Slice links in $S\sp{4}$
Tim
Cochran
389-401
Abstract: We produce necessary and sufficient conditions of a homotopy-theoretic nature for a link of $2$-spheres in ${S^4}$ to be slice (i.e., cobordant to the unlink). We give algebraic conditions on the link group sufficient to guarantee sliceness, generalizing the known results for boundary links. The notion of a "stable link" is introduced and shown to be useful in constructing cobordisms in dimension $ 4$.
Contributions from conjugacy classes of regular elliptic elements in ${\rm Sp}(n,\,{\bf Z})$ to the dimension formula
Min King
Eie
403-410
Abstract: The dimension of the space of cusp forms on the degree $n$ Siegel upper half-space can be obtained from the Selberg trace formula; in this paper we compute the contribution from the conjugacy classes of regular elliptic elements in $ \operatorname{Sp}(n,{\mathbf{Z}})$ using Weyl's character formula for representations of $ {\text{GL}}(n,{\mathbf{C}})$.
On certain Boolean algebras $\mathcal{P}(\omega)/I$
Winfried
Just;
Adam
Krawczyk
411-429
Abstract: We consider possible isomorphisms between algebras of the form $ \mathcal{P}(\omega)/I$, assuming ${\rm {CH}}$. In particular, the solution of a problem of Erdös and Ulam is given. We include some remarks on the completeness of such algebras.
Countable homogeneous tournaments
A. H.
Lachlan
431-461
Abstract: A tournament $ T$ is called homogeneous just in case every isomorphism of subtournaments of smaller cardinality can be lifted to an automorphism of $T$. It is shown that there are precisely three homogeneous tournaments of power ${\aleph_0}$. Some analogous results for $ 2$-tournaments are obtained.
An interface tracking algorithm for the porous medium equation
E.
DiBenedetto;
David
Hoff
463-500
Abstract: We study the convergence of a finite difference scheme for the Cauchy problem for the porous medium equation ${u_t} = {({u^m})_{x\,x}},m > 1$. The scheme exhibits the following two features. The first is that it employs a discretization of the known interface condition for the propagation of the support of the solution. We thus generate approximate interfaces as well as an approximate solution. The second feature is that it contains a vanishing viscosity term. This term permits an estimate of the form $\parallel {({u^{m - 1}})_{x\,x}}\;\parallel _{1,{\mathbf{R}}} \leqslant c/t$. We prove that both the approximate solution and the approximate interfaces converge to the correct ones. Finally error bounds for both solution and free boundaries are proved in terms of the mesh parameters.
Deformation and linkage of Gorenstein algebras
Andrew R.
Kustin;
Matthew
Miller
501-534
Abstract: General double linkage of Gorenstein algebras is defined. Rigidity, genericity, and regularity up to codimension six all pass across general double linkage. Rigid strongly unobstructed codimension four Gorenstein algebras which lie in different Herzog classes are produced.
Nondegenerate symmetric bilinear forms on finite abelian $2$-groups
Rick
Miranda
535-542
Abstract: Let ${\mathcal{B}_2}$ be the semigroup of isomorphism classes of finite abelian $2$-groups with a nondegenerate symmetric bilinear form having values in $Q/{\mathbf{Z}}$. Generators for ${\mathcal{B}_2}$ were given by C. T. C. Wall and the known relations among these generators were proved to be complete by A. Kawauchi and S. Kojima. In this article we describe a normal form for such bilinear forms, expressed in terms of Wall's generators, and as a by-product we obtain a simpler proof of the completeness of the known relations.
$L\sp{2}$-cohomology of noncompact surfaces
David R.
DeBaun
543-565
Abstract: This paper is motivated by the question of whether nonzero $ {L^2}$-harmonic differentials exist on coverings of a Riemann surface of genus $\geqslant 2$. Our approach will be via an analogue of the de Rham theorem. Some results concerning the invariance of ${L^2}$-homology and the intersection number of $ {L^2}$-cycles are demonstrated. A growth estimate for triangulations of planar coverings of the two-holed torus is derived. Finally, the equivalence between the existence of $ {L^2}$-harmonic one-cycles and the transience of random walks on a planar surface is shown.
Rate of approach to minima and sinks---the Morse-Smale case
Helena S.
Wisniewski
567-581
Abstract: The dynamical systems herein are Morse-Smale diffeomorphisms and flows on ${C^\infty }$ compact manifolds. We show the asymptotic rate of approach of orbits to the sinks of the systems to be bounded by an expression of the form $K\;\exp ( - DN)$, where $D$ may be any number smaller than $C = {\min_p}\{ 1/m\;\log \;\operatorname{Jac}\;{D_P}\,{f^m}\vert{W^u}(P)\}$. Here the minimum is taken over all nonsink $P$ in the nonwandering set of $f$, and $m$ is the period of $P$. We extend our theorems to the entire manifold, so that there is no restriction on the location of the initial points of trajectories.
Fine and parabolic limits for solutions of second-order linear parabolic equations on an infinite slab
B. A.
Mair
583-599
Abstract: This paper investigates the boundary behaviour of positive solutions of the equation $Lu = 0$, where $L$ is a uniformly parabolic second-order differential operator in divergence form having Hölder-continuous coefficients on $X = {{\mathbf{R}}^n} \times (0,T)$, where $0 < T < \infty$. In particular, the notion of semithinness for the potential theory on $X$ associated with $L$ is introduced, and the relationships between fine, semifine and parabolic convergence at points of $ {{\mathbf{R}}^n} \times \{ 0 \}$ are studied. The abstract Fatou-Naim-Doob theorem is used to deduce that every positive solution of $ Lu = 0$ on $X$ has parabolic limits Lebesgue-almost-everywhere on ${{\mathbf{R}}^n} \times \{ 0 \}$. Furthermore, a Carleson-type result is obtained for solutions defined on a union of parabolic regions.
Twistor CR manifolds and three-dimensional conformal geometry
Claude R.
LeBrun
601-616
Abstract: A CR (i.e. partially complex) $5$-manifold is contructed as a sphere bundle over an arbitrary $3$-manifold with conformal metric. This so-called twistor $CR$ manifold is show to capture completely the original geometry, and necessary and sufficient conditions are given for an abstract CR manifold to arise via the construction. The above correspondence is then used to prove that a twistor CR manifold is locally imbeddable as a real hypersurface in ${{\mathbf{C}}^3}$ only if it is real-analytic with respect to a suitable atlas.
Diffusion approximation and computation of the critical size
C.
Bardos;
R.
Santos;
R.
Sentis
617-649
Abstract: This paper is devoted to the mathematical definition of the extrapolation length which appears in the diffusion approximation. To obtain this result, we describe the spectral properties of the transport equation and we show how the diffusion approximation is related to the computation of the critical size. The paper also contains some simple numerical examples and some new results for the Milne problem.
Characteristic, maximum modulus and value distribution
W. K.
Hayman;
J. F.
Rossi
651-664
Abstract: Let $f$ be an entire function such that $\log M(r,f)\sim T(r,f)$ on a set $ E$ of positive upper density. Then $f$ has no finite deficient values. In fact, if we assume that $E$ has density one and $f$ has nonzero order, then the roots of all equations $f(z) = a$ are equidistributed in angles. In view of a recent result of Murai [6] the conclusions hold in particular for entire functions with Fejér gaps.
Minimal cyclic-$4$-connected graphs
Neil
Robertson
665-687
Abstract: A theory of cyclic-connectivity is developed, matroid dual to the standard vertex-connectivity. The cyclic-$4$-connected graphs minimal under the elementary operations of single-edge deletion or contraction and removal of a trivalent vertex are classified. These turn out to belong to three simple infinite families of indecomposable graphs, or to be decomposable into constituent subgraphs which themselves belong to three simple infinite families. This is modeled after W. T. Tutte's theorem classifying the minimal $3$-connected graphs under single-edge deletion or contraction as forming the single infinite family of "wheels." Such theorems serve two main purposes: (1) illustrating the structure of graphs in the class by isolating a type of extremal graph, and (2) by providing a set-up so that induction on $\vert E(G)\vert$ can be carried out effectively within the class.
Variational invariants of Riemannian manifolds
Jerrold
Siegel;
Frank
Williams
689-705
Abstract: This paper treats higher-dimensional analogues to the minimum geodesic distance in a compact Riemannian manifold $M$ with finite fundamental group. These invariants are based on the concept of homotopy distance in $M$. This defines a parametrized variational problem which is approached by globalizing the Morse theory of the spaces of paths between two points of $ M$ to the space of all paths in $M$. We develop machinery that we apply to calculate the invariants for numerous examples. In particular, we shall observe that knowledge of the invariants for the standard spheres determines the question of the existence of elements of Hopf invariant one.
Infinite crossed products and group-graded rings
D. S.
Passman
707-727
Abstract: In this paper, we precisely determine when a crossed product $R\;\ast\;G$ is semiprime or prime. Indeed we show that these conditions ultimately depend upon the analogous conditions for the crossed products $R\;\ast\;N$ of the finite subgroups $N$ of $G$ and upon the interrelationship between the normalizers of these subgroups and the ideal structure of $R$. The proof offered here is combinatorial in nature, using the $\Delta$-methods, and is entirely self-contained. Furthermore, since the argument applies equally well to strongly $G$-graded rings, we have opted to work in this more general context.
On positive solutions of some pairs of differential equations
E. N.
Dancer
729-743
Abstract: In this paper, we discuss the existence of solutions, with both components positive, of a Dirichlet problem for a coupled pair of partial differential equations. The main result is proved by using degree theory in cones. We also discuss the asymptotic behaviour of solutions as a parameter tends to zero.
Function theoretic results for complex interpolation families of Banach spaces
Richard
Rochberg
745-758
Abstract: The theory of complex interpolation of Banach spaces is viewed as a branch of the theory of vector valued holomorphic functions. Versions of the Schwarz lemma, Liouville's theorem, the identity theorem and the reflection principle are proved and are interpreted from the point of view of interpolation theory.
On the arithmetic of projective coordinate systems
Christian
Herrmann
759-785
Abstract: A complete list of subdirectly irreducible modular (Arguesian) lattices generated by a frame of order $n \geq 4\;(n \geq 3)$ is given. Also, it is shown that a modular lattice variety containing the rational projective geometries cannot be both finitely based and generated by its finite dimensional members.
Difference equations, isoperimetric inequality and transience of certain random walks
Jozef
Dodziuk
787-794
Abstract: The difference Laplacian on a square lattice in ${{\mathbf{R}}^n}$ has been studied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheeger's bound for the lowest eigenvalue) hold for this difference operator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient.
Tensor products for the de Sitter group
Robert P.
Martin
795-814
Abstract: The decomposition of the tensor product of a principal series representation with any other irreducible unitary representation of $G$ is determined for the simply connected double covering, $G = \operatorname{Spin}\,(4,1)$, of the DeSitter group.
Ramsey games
A.
Hajnal;
Zs.
Nagy
815-827
Abstract: The paper deals with game-theoretic versions of the partition relations $ \alpha \to (\beta )_2^{ < \tau }$ and $\alpha \to (\beta )_2^\tau $ introduced in [2]. The main results are summarized in the Introduction.
Rank change on adjoining real powers to Hardy fields
Maxwell
Rosenlicht
829-836
Abstract: This paper concerns asymptotic approximations and expansions in cases where the usual Poincaré power series in $ 1/x$ do not suffice because there may be more than one comparability class of functions that are very large or very small. The attempt to find asymptotic approximations in terms of real powers of given representatives of the comparability classes fails in general, but the situation can be saved by the adjunction of suitable real power products of the original functions, at the possible cost of an increase in the number of comparability classes.
Some applications of the topological characterizations of the sigma-compact spaces $l\sp{2}\sb{f}$ and $\Sigma $
Doug
Curtis;
Tadeusz
Dobrowolski;
Jerzy
Mogilski
837-846
Abstract: We use a technique involving skeletoids in $\sigma$-compact metric ARs to obtain some new examples of spaces homeomorphic to the $\sigma$-compact linear spaces $l_f^2$ and $\Sigma$. For example, we show that (1) every ${\aleph_0}$-dimensional metric linear space is homeomorphic to $l_f^2$; (2) every $\sigma$-compact metric linear space which is an AR and which contains an infinite-dimensional compact convex subset is homeomorphic to $ \Sigma$; and (3) every weak product of a sequence of $\sigma$-compact metric ARs which contain Hilbert cubes is homeomorphic to $ \Sigma$.
On the spectrum of $C\sb{0}$-semigroups
Jan
Prüss
847-857
Abstract: In this paper we give characterizations of the spectrum of a $ {C_0}$-semigroup $ {e^{At}}$ in terms of certain solution properties of the differential equation $ (\ast)\;u^{\prime} = Au + f$ and, in case $X$ is a Hilbert space, also in terms of properties of $ {(\lambda - A)^{ - 1}}$. We give several applications of these results including a study of the existence of dichotomic projections for $ (\ast)$.
Filters and the weak almost periodic compactification of a discrete semigroup
John F.
Berglund;
Neil
Hindman
1-38
Abstract: The weak almost periodic compactification of a semigroup is a compact semitopological semigroup with certain universal properties relative to the original semigroup. It is not, in general, a topological compactification. In this paper an internal construction of the weak almost periodic compactification of a discrete semigroup is constructed as a space of filters, and it is shown that for discrete semigroups, the compactification is usually topological. Other results obtained on the way to the main one include descriptions of weak almost periodic functions on closed subsemigroups of topological groups, conditions for functions on the additive natural numbers or on the integers to be weak almost periodic, and an example to show that the weak almost periodic compactification of the natural numbers is not the closure of the natural numbers in the weak almost periodic compactification of the integers.
Sieved ultraspherical polynomials
Waleed
Al-Salam;
W. R.
Allaway;
Richard
Askey
39-55
Abstract: The continuous $ q$-ultraspherical polynomials contain a number of important examples as limiting or special cases. One of these arose in Allaway's Ph.D. thesis. In a previous paper we solved a characterization problem essentially equivalent to Allaway's and showed that these polynomials arose from the $ q$-ultraspherical polynomials when $q$ approached a root of unity. A second class of such polynomials is found, and the recurrence relation and orthogonality relation are found for each of these polynomials. The orthogonality is interesting because the weight function has a finite number of zeros in $ (-1, 1)$. Generating functions and other formulas are also found.
Stable rank $2$ reflexive sheaves on ${\bf P}\sp{3}$ with small $c\sb{2}$ and applications
Mei-Chu
Chang
57-89
Abstract: We investigate the moduli spaces of stable rank two reflexive sheaves on $ {{\mathbf{P}}^3}$ with small Chern classes. As an application to curves of low degree in $ {{\mathbf{P}}^3}$, we prove the curve has maximal rank and that the corresponding Hilbert scheme is irreducible and unirational.
Localization of equivariant cohomology rings
J.
Duflot
91-105
Abstract: The main result of this paper is the "calculation" of the Borel equivariant cohomology ring ${H^{\ast} }(EG \times_G\,X,{\mathbf{Z}}/p{\mathbf{Z}})$ localized at one of its minimal prime ideals. In case $X$ is a point, the work of Quillen shows that the minimal primes $ {\mathfrak{P}_A}$ are parameterized by the maximal elementary abelian $ p$-subgroups $ A$ of $G$ and the result is $\displaystyle {H^{\ast} }{(BG,{\mathbf{Z}}/p{\mathbf{Z}})_{{\mathfrak{P}_A}}} \... ...{H^{\ast} }(B{C_G}(A),{\mathbf{Z}}/p{\mathbf{Z}})_{{\mathfrak{P}_A}}^{{W_G}(A)}$ . Here, $ {C_G}(A)$ is the centralizer of $A$ in $G$, and ${W_G}(A) = {N_G}(A)/{C_G}(A)$, where $ {N_G}(A)$ is the normalizer of $A$ in $G$. An example is included.
The law of exponential decay for expanding transformations of the unit interval into itself
M.
Jabłoński
107-119
Abstract: Let $T:[0,1] \to [0,1]$ be an expanding map of the unit interval and let $ {\xi _\varepsilon }(x)$ be the smallest integer $n$ for which ${T^n}(x) \in [0,\varepsilon ]$; that is, it is the random variable given by the formula $\displaystyle {\xi _\varepsilon }(x) = \min \{ n:{T^n}\;(x) \leqslant \varepsilon \}.$ It is shown that for any $z \geqslant 0$ and for any integrable function $f:[0,1] \to {R^ + }$ the measure $ {\mu _f}$ (where $ \mu$ is Lebesgue measure and ${\mu _f}$ is defined by $d{\mu _f} = fd\mu$) of the set of points $ x$ for which ${\xi _\varepsilon }(x) \leqslant z/\varepsilon$ tends to an exponential function of $z$ as $ \varepsilon$ tends to zero.
The heat equation with a singular potential
Pierre
Baras;
Jerome A.
Goldstein
121-139
Abstract: Of concern is the singular problem $\partial u/\partial t = \Delta u + (c/\vert x{\vert^2})\,u + f(t,x), u(x,0) = u_{0}(x)$, and its generalizations. Here $c \geqslant 0,x \in {{\mathbf{R}}^N},t > 0$, and $f$ and ${u_0}$ are nonnegative and not both identically zero. There is a dimension dependent constant ${C_{\ast} }(N)$ such that the problem has no solution for $c > {C_{\ast} }(N)$. For $c \leqslant {C_{\ast} }(N)$ necessary and sufficient conditions are found for $f$ and ${u_0}$ so that a nonnegative solution exists.
Actions of finite groups on homotopy $3$-spheres
M. E.
Feighn
141-151
Abstract: It is conjectured that the action of a finite group of diffeomorphisms of the $3$-sphere is equivariantly diffeomorphic to a linear action. This conjecture is verified if both of the following conditions hold: (i) Each isotropy subgroup is dihedral or cyclic. (ii) There is a point with cyclic isotropy subgroup of order not $ 1,2,3$ or $5$.
Operators with $C\sp{\ast} $-algebra generated by a unilateral shift
John B.
Conway;
Paul
McGuire
153-161
Abstract: If $T$ is an operator on a Hilbert space $\mathcal{H}$, this paper gives necessary and sufficient conditions on $T$ such that $ {C^{\ast} }(T)$, the ${C^{\ast} }$-algebra generated by $T$, is generated by a unilateral shift of some multiplicity. This result is then specialized to the cases in which $T$ is a hyponormal or subnormal operator. In particular, it is shown how to prove a recent conjecture of C. R. Putnam as a consequence of our result.
On the universal theory of classes of finite models
S.
Tulipani
163-170
Abstract: First order theories for which the truth of a universal sentence on their finite models implies the truth on all models are investigated. It is proved that an equational theory has such a property if and only if every finitely presented model is residually finite. The most common classes of algebraic structures are discussed.
The theory of ordered abelian groups does not have the independence property
Y.
Gurevich;
P. H.
Schmitt
171-182
Abstract: We prove that no complete theory of ordered abelian groups has the independence property, thus answering a question by B. Poizat. The main tool is a result contained in the doctoral dissertation of Yuri Gurevich and also in P. H. Schmitt's Elementary properties of ordered abelian groups, which basically transforms statements on ordered abelian groups into statements on coloured chains. We also prove that every $n$-type in the theory of coloured chains has at most ${2^n}$ coheirs, thereby strengthening a result by B. Poizat.
Points of continuity for semigroup actions
Jimmie D.
Lawson
183-202
Abstract: The purpose of this paper is to provide a more unified approach to questions involving the existence of points of joint continuity in separately continuous semigroup actions by deriving a small number of general principles which suffice to deduce previously derived results and generalizations thereof. The first major result gives sufficient conditions for a point to be a point of joint continuity in a general setting of "migrants", a useful symmetric generalization of semigroup actions. Results concerning actions of semigroups with group-like properties follow. In the latter part of the paper the notion of a subordinate point is introduced and joint continuity at subordinate points for various settings is proved. Finally, these results are applied to linear actions on locally convex spaces.
Complexity of subcases of Presburger arithmetic
Bruno
Scarpellini
203-218
Abstract: We consider formula subclasses of Presburger arithmetic which have a simple structure in one sense or the other and investigate their computational complexity. We also prove some results on the lower bounds of lengths of formulas which are related to questions on quantifier elimination.
Mean value properties of the Laplacian via spectral theory
Robert S.
Strichartz
219-228
Abstract: Let $\phi ({z^2})$ be an even entire function of temperate exponential type, $L$ a selfadjoint realization of $- \Delta + c\,(x)$, where $\Delta$ is the Laplace-Beltrami operator on a Riemannian manifold, and $ \phi \,(L)$ the operator given by spectral theory. A Paley-Wiener theorem on the support of $\phi \,(L)$ is proved, and is used to show that $Lu = \lambda u$ on a suitable domain implies $\phi \,(L)\,u = \phi \,(\lambda)\,u$, as well as a generalization of Àsgeirsson's theorem. A concrete realization of the operators $\phi \,(L)$ is given in the case of a compact Lie group or a noncompact symmetric space with complex isometry group.
Unicellular operators
José
Barría;
Kenneth R.
Davidson
229-246
Abstract: An operator is unicellular if its lattice of invariant subspaces is totally ordered by inclusion. The list of nests which are known to be the set of invariant subspaces of a unicellular operator is surprisingly short. We construct unicellular operators on ${l^p},1 \leqslant p < \infty$, and on $ {c_0}$ with lattices isomorphic to $ \alpha + X + {\beta ^{\ast}}$ where $\alpha$ and $\beta$ are countable (finite or zero) ordinals, and $X$ is in this short list. Certain other nests are attained as well.
Positive-definiteness and its applications to interpolation problems for holomorphic functions
Frank
Beatrous;
Jacob
Burbea
247-270
Abstract: Holomorphic interpolation problems of the Pick-Nevanlinna and Loewner types as well as abstract interpolation theorems on functional Hilbert spaces are considered. Various characterizations are presented for restrictions of bounded holomorphic functions. In addition, certain norm estimates for restrictions and extensions of holomorphic functions are obtained.
Moments of balanced measures on Julia sets
M. F.
Barnsley;
A. N.
Harrington
271-280
Abstract: By a theorem of S. Demko there exists a balanced measure on the Julia set of an arbitrary nonlinear rational transformation on the Riemann sphere. It is proved here that if the transformation admits an attractive or indifferent cycle, then there is a point with respect to which all the moments of a balanced measure exist; moreover, these moments can be calculated exactly. An explicit balanced measure is exhibited in an example where the Julia set is the whole sphere and for which the moments, with respect to any point, do not all exist.
Singular Vietoris-Begle theorems for relations
D. G.
Bourgin;
Robert M.
Nehs
281-318
Abstract: The Vietoris-Begle theorem with singularities, for three spaces $ X$, $Y$, $T$, is extended to the case that a closed relation replaces a continuous map and more generally to set valued maps. The developments are carried out based on modification of the topology of $T$ so that in general it is no longer even Hausdorff. This entails interpretation of dimension of singulars sets in terms of considertions in $Y$ rather than $T$. The techniques are those of sheaf and spectral sequence theory.
On the Weil-Petersson metric on Teichm\"uller space
A. E.
Fischer;
A. J.
Tromba
319-335
Abstract: Teichmüller space for a compact oriented surface $M$ without boundary is described as the quotient $ \mathcal{A}/{\mathcal{D}_0}$, where $ \mathcal{A}$ is the space of almost complex structures on $M$ (compatible with a given orientation) and ${\mathcal{D}_0}$ are those ${C^\infty }$ diffeomorphisms homotopic to the identity. There is a natural ${\mathcal{D}_0}$ invariant ${L_2}$ Riemannian structure on $\mathcal{A}$ which induces a Riemannian structure on $ \mathcal{A}/{\mathcal{D}_0}$. Infinitesimally this is the bilinear pairing suggested by Andre Weil--the Weil-Petersson Riemannian structure. The structure is shown to be Kähler with respect to a naturally induced complex structure on $ \mathcal{A}/{\mathcal{D}_0}$.
Fixed point sets of metric and nonmetric spaces
John R.
Martin;
William
Weiss
337-353
Abstract: A space $ X$ is said to have the complete invariance property $ ($CIP$)$ if every nonempty closed subset of $ X$ is the fixed point set of some self-mapping of $X$. It is shown that connected subgroups of the plane and compact groups need not have CIP, and CIP need not be preserved by self-products of Peano continua, nonmetric manifolds or 0-dimensional spaces. Sufficient conditions are given for an infinite product of spaces to have CIP. In particular, an uncountable product of real lines, circles or two-point spaces has CIP. Examples are given which contrast the behavior of CIP in the nonmetric and metric cases, and examples of spaces are given where the existence of CIP is neither provable nor refutable with the usual axioms of set theory.
The family approach to total cocompleteness and toposes
Ross
Street
355-369
Abstract: A category with small homsets is called total when its Yoneda embedding has a left adjoint; when the left adjoint preserves pullbacks, the category is called lex total. Total categories are characterized in this paper in terms of special limits and colimits which exist therein, and lex-total categories are distinguished as those which satisfy further exactness conditions. The limits involved are finite limits and intersections of all families of subobjects. The colimits are quotients of certain relations (called congruences) on families of objects (not just single objects). Just as an arrow leads to an equivalence relation on its source, a family of arrows into a given object leads to a congruence on the family of sources; in the lex-total case all congruences arise in this way and their quotients are stable under pullback. The connection with toposes is examined.
Liouville theorems, partial regularity and H\"older continuity of weak solutions to quasilinear elliptic systems
Michael
Meier
371-387
Abstract: This paper describes the connections between Liouville type theorems and interior regularity results for bounded weak solutions of quasilinear elliptic systems with quadratic growth. It is shown that equivalence does in general hold only in some restricted sense. A complete correspondence can be established in certain cases, e.g. for small solutions and for minima of quadratic variational integrals.
Universal families for conull FK spaces
A. K.
Snyder
389-399
Abstract: This paper considers the problem of determining a useful family of sequence spaces which is universal for conull FK spaces in the following sense: An FK space is conull if and only if it contains a member of the family. In the equivalent context of weak wedge spaces, an appropriate family of subspaces of boundedness domains ${m_A}$ of matrices is shown to be universal. Most useful is the fact that the members of this family exhibit unconditional sectional convergence. The latter phenomenon is known for wedge spaces. Another family of spaces which is universal for conull spaces among semiconservative spaces is provided. The spaces are designed to simplify gliding humps arguments. Improvements are thereby obtained for some pseudoconull type theorems of Bennett and Kalton. Finally, it is shown that conull spaces must contain pseudoconull BK algebras.
Milnor's invariants and the completions of link modules
Lorenzo
Traldi
401-424
Abstract: Let $L$ be a tame link of $\mu \geqslant 2$ components in ${S^3}$, $H$ the abelianization of its group ${\pi _1}({S^3} - L)$, and $IH$ the augmentation ideal of the integral group ring $ {\mathbf{Z}}H$. The $ IH$-adic completions of the Alexander module and Alexander invariant of $ L$ are shown to possess presentation matrices whose entries are given in terms of certain integers $\mu ({i_1}, \ldots ,{i_q})$ introduced by J. Milnor. Various applications to the theory of the elementary ideals of these modules are given, including a condition on the Alexander polynomial necessary for the linking numbers of the components of $L$ with each other to all be zero. In the special case $\mu = 2$, it is shown that the various Milnor invariants $ \bar \mu ([r + 1,s + 1])$ are determined (up to sign) by the Alexander polynomial of $L$, and that this Alexander polynomial is 0 iff $ \bar \mu ([r + 1,s + 1]) = 0$ for all $ r,s \geqslant 0$ with $ r + s$ even; also, the Chen groups of $L$ are determined (up to isomorphism) by those nonzero $ \bar \mu ([r + 1,s + 1])$ with $r + s$ minimal. In contrast, it is shown by example that for $ \mu \geqslant 3$ the Alexander polynomials of a link and its sublinks do not determine its Chen groups.
Proper holomorphic mappings that must be rational
Steven
Bell
425-429
Abstract: Suppose $f:{D_1} \to {D_2}$ is a proper holomorphic mapping between bounded domains in ${{\mathbf{C}}^n}$. We shall prove that under certain circumstances $f$ must be a rational mapping, i.e., that the $ n$ component functions $ {f_i}$ of $f$ are rational functions.
An integral version of the Brown-Gitler spectrum
Don H.
Shimamoto
383-421
Abstract: In this paper, certain spectra ${B_1}(k)$ are studied whose behavior qualifies them as being integral versions of the Brown-Gitler spectra $ B(k)$. The bulk of our work emphasizes the similarities between $ {B_1}(k)$ and $ B(k)$, shown mainly using the techniques of Brown and Gitler. In analyzing the homotopy type of ${B_1}(k)$, we provide a free resolution over the Steenrod algebra for its cohomology and study its Adams spectral sequence. We also list conditions which characterize it at the prime $2$. The paper begins, however, on a somewhat different topic, namely, the construction of a configuration space model for ${\Omega ^2}({S^3}\left\langle 3 \right\rangle )$ and other related spaces.
On cyclic trigonal Riemann surfaces. I
Robert D. M.
Accola
423-449
Abstract: Definition. Call the Riemann surfaces for the equation ${y^3} = P(x)$ cyclic trigonal. For one case of genus $4$ ($2$ distinct $g_3^1$'s) and all genera greater than $4$, cyclic trigonal Riemann surfaces are characterized by the vanishing properties of the theta function at certain $(1/6)$-periods of the Jacobian. Also for trigonal Riemann surfaces of genera $5$, $6$, and $7$, homogeneous theta relations are derived using the fact that Prym varieties for trigonal Riemann surfaces are Jacobians.
Hecke modular forms and the Kac-Peterson identities
George E.
Andrews
451-458
Abstract: The identity of certain Hecke modular forms with well-known infinite products is derived in an elementary manner. New identities and applications are discussed.
The equivariant Dolbeault lemma
Wilfried
Schmid;
Joseph A.
Wolf
459-463
Abstract: A form of the Dolbeault Lemma is obtained for circular domains $D \subset {{\mathbf{C}}^n}$, which is equivariant for the subgroup of $ \operatorname{GL} (n;{\mathbf{C}})$ that stabilizes $D$.
Dichotomies and asymptotic behaviour for linear differential systems
James S.
Muldowney
465-484
Abstract: Sufficient conditions that a system of differential equations
On stable blocks of Auslander-algebras
Christine
Riedtmann
485-505
Abstract: The Auslander-algebra ${E_\Lambda }$ of an algebra $\Lambda$ of finite representation type is the endomorphism algebra of the direct sum $M = \oplus {M_i}$ of one copy of each indecomposable $ \Lambda$-module. A stable block of $ {E_\Lambda }$ is a connected direct factor of the residue algebra of ${E_\Lambda }$ modulo the two-sided ideal generated by the projections of $M$ to the ${M_i}$'s that are not stable under $DTr$. This paper describes the stable blocks whose quiver is a stable translation-quiver of class ${A_n}$ or ${D_n}$.
On pairs of recursively enumerable degrees
Klaus
Ambos-Spies
507-531
Abstract: Lachlan and Yates proved that some, but not all, pairs of incomparable recursively enumerable (r.e.) degrees have an infimum. We answer some questions which arose from this situation. We show that not every nonzero incomplete r.e. degree is half of a pair of incomparable r.e. degrees which have an infimum, whereas every such degree is half of a pair without infimum. Further, we prove that every nonzero r.e. degree can be split into a pair of r.e. degrees which have no infimum, and every interval of r.e. degrees contains such a pair of degrees.
Pure states on some group-invariant $C\sp{\ast} $-algebras
Geoffrey L.
Price
533-562
Abstract: Let $\mathfrak{A}$ be a UHF algebra of Glimm type ${n^\infty }$, i.e., $ \mathfrak{A} = \otimes _{k \geqslant 1}^{\ast}{N_k}$, where $N = {N_1} = {N_2} = \cdots $ are $n \times n$ matrix algebras. We define an AF-subalgebra $ {\mathfrak{A}^G}$ of $\mathfrak{A}$, consisting of those elements of $\mathfrak{A}$ invariant under a group of automorphisms $\{ {\alpha _g}:g \in G = \operatorname{SU} (n)\}$ of product type. $ {\mathfrak{A}^G}$ is shown to be generated by an embedding of $S(\infty )$, the discrete group of finite permutations on countably many symbols. Let $\omega$ be a pure product state on $\mathfrak{A}$, $ {\omega ^G}$ its restriction to $ {\mathfrak{A}^G}$. Let $ e \in N$ be a one-dimensional projection with corresponding projections ${e^k} \in {N_k}$. Then if both (i) ${\Sigma _{k \geqslant 1}}\omega ({e^k}) = \infty$, and (ii) $ 0 < {\Sigma _{k \geqslant 1}}\omega ({e^k})[1 - \omega ({e^k})] < \infty$ hold, ${\omega ^G}$ is not pure. ${\omega ^G}$ is shown to be pure if there exist orthogonal one-dimensional projections $\{ {p_i}:1 \leqslant i \leqslant n\}$ of $N$ with corresponding projections $p_i^k \in {N_k}$ such that $\omega (p_i^k) = 0$ or $1$, $1 \leqslant i \leqslant n,\,k \geqslant 1$, and $0 < {\Sigma _{k \geqslant 1}}\omega (p_i^k) < \infty$ for at most one $i$.
Tangential equivalence of group actions
Sławomir
Kwasik
563-573
Abstract: We consider the problem of tangential equivalence of group actions on manifolds. In particular we discuss a conjecture of B. Mazur and its modifications. The negative answer to this conjecture is presented. On the other hand we prove that the "isovariant" version of this conjecture, as well as the modified one, remains true. As an application some results on the tangential equivalence of ${Z_p}$-actions on homotopy spheres are obtained.
An infinite graph of girth $12$
Asia Ivić
Weiss
575-588
Abstract: From the regular hyperbolic honeycomb $ \{ 3,6,3\}$ we derive regular honeycombs with finite numbers of toroidal cells. Joining centers of faces of these honeycombs to the midpoints of its edges we obtain trivalent symmetrical graphs. We investigate the relation between these honeycombs, their groups and the graphs embedded in them.
Eisenstein series of weight ${3\over 2}$. II
Ting Yi
Pei
589-603
Abstract: In a previous paper we proved that for some special levels, in the space of elliptic modular forms with weight $3/2$ the orthogonal complement of the subspace of cusp forms with respect to the Petersson inner product is generated by the Eisenstein series. In this paper we prove that this fact is true for any level.
Subordination-preserving integral operators
Sanford S.
Miller;
Petru T.
Mocanu;
Maxwell O.
Reade
605-615
Abstract: Let $\beta$ and $\gamma$ be complex numbers and let $ H$ be the space of functions regular in the unit disc. Subordination of functions $ f$, $g \in H$ is denoted by $f \prec g$. Let $K \subset H$ and let the operator $A:K \to H$ be defined by $F = A(f)$, where $\displaystyle F(z) = {\left[ {\frac{1} {{{z^\gamma }}}\int_0^z {{f^\beta }(t){t^{\gamma - 1}}dt} } \right]^{1/\beta }}.$ The authors determine conditions under which $\displaystyle f \prec g \Rightarrow A(f) \prec A(g),$ and then use this result to obtain new distortion theorems for some classes of regular functions.
A partial order on the regions of ${\bf R}\sp{n}$ dissected by hyperplanes
Paul H.
Edelman
617-631
Abstract: We study a partial order on the regions of $ {{\mathbf{R}}^n}$ dissected by hyperplanes. This includes a computation of the Möbius function and, in some cases, of the homotopy type. Applications are presented to zonotopes, the weak Bruhat order on Weyl groups and acyclic orientations of graphs.
Structural stability of equivariant vector fields on two-manifolds
G. L.
dos Reis
633-643
Abstract: A class of vector fields on two-dimensional manifolds equivariant under the action of a compact Lie group is defined. Properties of openness, structural ability, and density are proved.
Tameness of pairs of nuclear power series spaces and related topics
Kaisa
Nyberg
645-660
Abstract: The equivalence of the following six assertions is proved: (i) The set of the finite limit points of the ratios $ {\alpha _m}/{\beta _n},n,m \in {\mathbf{N}}$, is bounded, (ii) Every operator from $ {\Lambda _\infty }(\beta )$ to $ {\Lambda _1}(\alpha )$ is compact, (iii) The pair $({\Lambda _\infty }(\beta ),\,{\Lambda _1}(\alpha ))$ is tame, i.e., for every operator $ T$ from ${\Lambda _\infty }(\beta )$ to ${\Lambda _1}(\alpha )$ there is a positive integer $ a$ such that for every $k \in {\mathbf{N}}$ there is a constant ${C_k}$ such that $\vert\vert Tx\vert{\vert _k} \leqslant {C_k}\vert x{\vert _{ak}}$ for every $x \in {\Lambda _\infty }(\beta )$. (iv) Every short exact sequence $ 0 \to {\Lambda _\tau }(\beta ) \to X \to {\Lambda _1}(\alpha ) \to 0$, where $ \tau = 1$ or $ \infty$, splits. (v) ${\Lambda _1}(\alpha ) \times {\Lambda _\infty }(\beta )$ has a regular basis, (vi) ${\Lambda _1}(\alpha ) \otimes {\Lambda _\infty }(\beta )$ has a regular basis. Also the finite type power series spaces that contain subspaces isomorphic to an infinite type power series space are characterized as well as the infinite type power series spaces that have finite type quotient spaces.
On the arithmetic and homology of algebras of linear type
J.
Herzog;
A.
Simis;
W. V.
Vasconcelos
661-683
Abstract: Three modifications of the symmetric algebra of a module are introduced and their arithmetical and homological properties studied. Emphasis is placed on converting syzygetic properties of the modules into ideal theoretic properties of the algebras, e.g. Cohen-Macaulayness, factoriality. The main tools are certain Fitting ideals of the module and an extension to modules of a complex of not necessarily free modules that we have used in studying blowing-up rings.
Commuting analytic functions
Carl C.
Cowen
685-695
Abstract: Let $f$ and $g$ (not conformal automorphisms of the unit disk) be analytic mappings of the unit disk into itself. We say $f$ and $g$ commute if $f \circ g = g \circ f$. This paper characterizes those functions $g$ that commute with a given function $f$. Several corollaries of this characterization give qualitative information about $g$ given similar information about $ f$, and examples are given in each case to show the limitations of the conclusions. Some of the qualitative properties considered are univalence, fixed point sets, and whether two such $ g$ must commute with each other.
Contraction operators quasisimilar to a unilateral shift
V. T.
Alexander
697-703
Abstract: Let ${U_n}$ denote the unilaterial shift of finite multiplicity $n$. It is shown that a contraction operator $ T$ is quasisimilar to $ {U_n}$ if and only if $ T$ is of Class $ {C_1}$., the canonical isometry $V$ associated with $T$ is pure and $T$ is $n$-cyclic with analytically independent vectors. For this, the notions of operators of analytic type and analytic independence of vectors are introduced. A characterization of the cyclic vectors of the Backward Shift is also presented.
On certain elementary extensions of models of set theory
Ali
Enayat
705-715
Abstract: In $\S1$ we study two canonical methods of producing models of $ \operatorname{ZFC}$ with no elementary end extensions. $\S2$ is devoted to certain "completeness" theorems dealing with elementary extensions, e.g., using ${\diamondsuit _{{\omega _1}}}$ we show that for a consistent $T \supseteq \operatorname{ZFC}$ the property "Every model $ \mathfrak{A}$ of $ T$ has an elementary extension fixing $ {\omega ^\mathfrak{A}}$" is equivalent to $T\vdash$ "There exists an uncountable measurable cardinal". We also give characterizations of $ T\vdash$ "$\kappa$ is weakly compact" and $ T\vdash$ "$\kappa$ is measurable" in terms of elementary extensions.
Contraction semigroups for diffusion with drift
R.
Seeley
717-728
Abstract: Recently Dodziuk, Karp and Li, and Strichartz have given results on existence and uniqueness of contraction semigroups generated by the Laplacian $\Delta$ on a manifold $M$; earlier, Yau gave related results for $L = \Delta + V$ for a vector field $V$. The present paper considers $L = \Delta - V - c$, with $c$ a real function, and gives conditions for (a) uniqueness of semigroups on the bounded continuous functions, (b) preservation of $ {C_0}$ (functions vanishing at $\infty$) by the minimal semigroup, and (c) existence and uniqueness of contraction semigroups on $ {L^p}(\mu ),\;1 \leqslant p < \infty$, for an arbitrary smooth density $ \mu$ on $M$. The conditions concern $L\rho /\rho$, where $\rho$ is a smooth function, $\rho \to \infty$ as $ x \to \infty$. They variously extend, strengthen, and complement the previous results mentioned above.
Some sequence spaces and absolute almost convergence
G.
Das;
B.
Kuttner;
S.
Nanda
729-739
Abstract: The object of this paper is to introduce a new concept of absolute almost convergence which emerges naturally as an absolute analogue of almost convergence, in the same way as convergence leads to absolute convergence.
Weak solutions of the Gellerstedt and the Gellerstedt-Neumann problems
A. K.
Aziz;
M.
Schneider
741-752
Abstract: We consider the question of existence of weak and semistrong solutions of the Gellerstedt problem $\displaystyle u{\vert _{{\Gamma _0} \cup {\Gamma _1} \cup {\Gamma _2}}} = 0$ and the Gellerstedt-Neumann problem $\displaystyle ({d_n}u = k(y){u_x}dy - {u_y}dx{\vert _{{\Gamma _0}}} = 0,\qquad u{\vert _{{\Gamma _1} \cup {\Gamma _2}}} = 0)$ for the equation of mixed type $\displaystyle L[u] \equiv k(y){u_{xx}} + {u_{yy}} + \lambda u = f(x,y),\qquad \lambda = \operatorname{const} < 0$ in a region $G$ bounded by a piecewise smooth curve ${\Gamma _0}$ lying in the half-plane $y > 0$ and intersecting the line $y = 0$ at the points $ A( - 1,0)$ and $ B(1,0)$. For $ y < 0$, $G$ is bounded by the characteristic curves ${\gamma _1}(x < 0)$ and ${\gamma _2}(x > 0)$ of (1) through the origin and the characteristics $ {\Gamma _1}$ and ${\Gamma _2}$ through $A$ and $B$ which intersect $ {\gamma _1}$ and ${\gamma _2}$ at the points $P$ and $Q$, respectively. Using a variation of the energy integral method, we give sufficient conditions for the existence of weak and semistrong solutions of the boundary value problems (Theorems 4.1, 4.2, 5.1).
Concatenations applied to analytic hypoellipticity of operators with double characteristics
Kil Hyun
Kwon
753-763
Abstract: We use the method of concatenations to get a sufficient condition for a class of analytic pseudodifferential operators with double characteristics to be analytic hypoelliptic.
The structure of Galois groups of ${\rm CM}$-fields
B.
Dodson
1-32
Abstract: A $CM$-field $K$ defines a triple $ (G,H,\rho )$, where $ G$ is the Galois group of the Galois closure of $K$, $H$ is the subgroup of $G$ fixing $K$, and $\rho \in G$ is induced by complex conjugation. A "$\rho$-structure" identifies $CM$-fields when their triples are identified under the action of the group of automorphisms of $G$. A classification of the $\rho $-structures is given, and a general formula for the degree of the reflex field is obtained. Complete lists of $\rho$-structues and reflex fields are provided for $[K:\mathbb{Q}] = 2n$, with $n = 3,4,5$ and $7$. In addition, simple degenerate Abelian varieties of $CM$-type are constructed in every composite dimension. The collection of reflex fields is also determined for the dihedral group $ G = {D_{2n}}$, with $ n$ odd and $H$ of order $2$, and a relative class number formula is found.
Projective normal flatness and Hilbert functions
U.
Orbanz;
L.
Robbiano
33-47
Abstract: Projective normal flatness of a local ring $R$ along an ideal $I$ is defined to be the flatness of the morphism on the exceptional divisor induced by blowing up $ R$ with center $ I$. It is shown that most of the important properties of normal flatness have an analogue for projective normal flatness. In particular, we study the local Hilbert function in connection with projective normal flatness. If $R/I$ is regular and $R$ projectively normally flat along $I$, then we obtain the same inequality for the local Hilbert functions under blowing up as in the permissible case.
Aposyndetic continua as bundle spaces
James T.
Rogers
49-55
Abstract: Let $\mathcal{S}$ be the $P$-adic solenoid bundle, and let $\eta :X \to {S^1}$ be a map of the continuum $ X$ onto ${S^1}$. The bundle space $B$ of the induced bundle ${\eta ^{ - 1}}\mathcal{S}$ is investigated. Sufficient conditions are obtained for $ B$ to be connected, to be aposyndetic, and to be homogeneous. Uncountably many aposyndetic, homogeneous, one-dimensional, nonlocally connected continua are constructed. Other classes of continua are placed into this framework.
Global analysis of two-parameter elliptic eigenvalue problems
H.-O.
Peitgen;
K.
Schmitt
57-95
Abstract: We consider the nonlinear boundary value problem $({\ast})Lu + \lambda f(u) = 0$, $x \in \Omega ,\,u = \sigma \phi ,\,x \in \partial \Omega $, where $L$ is a second order elliptic operator and $ \lambda$ and $ \sigma$ are parameters. We analyze global properties of solution continua of these problems as $\lambda$ and $\sigma$ vary. This is done by investigating particular sections, and special interest is devoted to questions of how solutions of the $ \sigma = 0$ problem are embedded in the two-parameter family of solutions of $ ({\ast})$. As a natural biproduct of these results we obtain (a) a new abstract method to analyze bifurcation from infinity, (b) an unfolding of the bifurcations from zero and from infinity, and (c) a new framework for the numerical computations, via numerical continuation techniques, of solutions by computing particular one-dimensional sections.
Path derivatives: a unified view of certain generalized derivatives
A. M.
Bruckner;
R. J.
O’Malley;
B. S.
Thomson
97-125
Abstract: A collection $E = \{ {E_x}:x \in R\}$ is a system of paths if each set ${E_x}$ has $x$ as a point of accumulation. For such a system $E$ the derivative $F_E'(x)$ of a function $F$ at a point $x$ is just the usual derivative at $x$ relative to the set ${E_x}$. The goal of this paper is the investigation of properties that $F$ and its derivative $F_E'$ must have under certain natural assumptions about the collection $E$. In particular, it is shown that most of the familiar properties of approximate derivatives and approximately differentiable functions follow in this setting from three conditions on the collection $E$ relating to the "thickness" of the sets $ {E_x}$ and the way in which the sets intersect.
Quartic surfaces of elliptic ruled type
Yumiko
Umezu
127-143
Abstract: Let $X$ be a normal quartic surface whose resolutions are birationally equivalent to elliptic ruled surfaces. We classify the singularities on $ X$ and then describe the global structure of $X$.
A nonlinear integral equation occurring in a singular free boundary problem
Klaus
Höllig;
John A.
Nohel
145-155
Abstract: We study the Cauchy problem $\displaystyle \left\{ \begin{gathered}{u_t} = \phi {({u_x})_x},\qquad (x,t) \in... ...{{\mathbf{R}}_ + }, u( \cdot ,0) = f \end{gathered} \right.$ with the piecewise linear constitutive function $\phi (\xi ) = {\xi _ + } = \max (0,\xi )$ and with smooth initial data $f$ which satisfy $x \in {\mathbf{R}}$, and $ f''(0) > 0$. We prove that free boundary $s$, given by ${u_x}(s{(t)^ + },t) = 0$, is of the form $\displaystyle s(t) = - \kappa \sqrt t + o\left( {\sqrt t } \right),\qquad t \to {0^ + },$ where the constant $\kappa = 0.9034 \ldots$ is the (numerical) solution of a particular nonlinear equation. Moreover, we show that for any $ \alpha \in (0,1/2)$, $\displaystyle \left\vert {\frac{{{d^2}}} {{d{t^2}}}f(s(t))} \right\vert = O({t^{\alpha - 1}}),\qquad t \to {0^ + }.$ The proof involves the analysis of a nonlinear singular integral equation.
An integral inequality with applications
M. A.
Leckband
157-168
Abstract: Using a technical integral inequality, J. Moser proved a sharp result on exponential integrability of a certain space of Sobolev functions. In this paper, we show that the integral inequality holds in a general setting using nonincreasing functions and a certain class of convex functions. We then apply the integral inequality to extend the above result by J. Moser to other spaces of Sobolev functions. A second application is given generalizing some different results by M. Jodeit.
Where the continuous functions without unilateral derivatives are typical
Jan
Malý
169-175
Abstract: An alternative proof of the existence of a Besicovitch function (i.e. a continuous function which has nowhere a unilateral derivative) is presented. The method consists in showing the residuality of Besicovitch functions in special subspaces of the Banach space of all continuous functions on $[0,1]$ and yields Besicovitch functions with additional properties of Morse or Hölder type. A way how to obtain functions with a similar behavior on normed linear spaces is briefly mentioned.
Conjugacy classes of hyperbolic matrices in ${\rm Sl}(n,\,{\bf Z})$ and ideal classes in an order
D. I.
Wallace
177-184
Abstract: A bijection is proved between $\operatorname{Sl} (n,{\mathbf{Z}})$-conjugacy classes of hyperbolic matrices with eigenvalues $ \{ {\lambda _1}, \ldots ,{\lambda _n}\}$ which are units in an $n$-degree number field, and narrow ideal classes of the ring ${R_k} = {\mathbf{Z}}[{\lambda _i}]$. A bijection between $\operatorname{Gl} (n,{\mathbf{Z}})$-conjugacy classes and the wide ideal classes, which had been known, is repeated with a different proof.
A Cayley-Dickson process for a class of structurable algebras
B. N.
Allison;
J. R.
Faulkner
185-210
Abstract: In this paper, we study the class of all simple structurable algebras with the property that the space of skew-hermitian elements has dimension $1$. These algebras with involution have arisen in the study of Lie algebra constructions. The reduced algebras are isotopic to $2 \times 2$ matrix algebras. We study a Cayley-Dickson process for rationally constructing some algebras in the class including division algebras and nonreduced nondivision algebras. An important special case of the process endows the direct sum of two copies of a $28$-dimensional degree $4$ central simple Jordan algebra $\mathcal{B}$ with the structure of an algebra with involution. In preparatory work, we obtain a procedure for giving the space $ {\mathcal{B}_0}$ of trace zero elements of any such Jordan algebra $\mathcal{B}$ the structure of a $ 27$-dimensional exceptional Jordan algebra.
Abelian subgroups of topological groups
Siegfried K.
Grosser;
Wolfgang N.
Herfort
211-223
Abstract: In [1] Šmidt's conjecture on the existence of an infinite abelian subgroup in any infinite group is settled by counterexample. The well-known Hall-Kulatilaka Theorem asserts the existence of an infinite abelian subgroup in any infinite locally finite group. This paper discusses a topological analogue of the problem. The simultaneous consideration of a stronger condition--that centralizers of nontrivial elements be compact--turns out to be useful and, in essence, inevitable. Thus two compactness conditions that give rise to a profinite arithmetization of topological groups are added to the classical list (see, e.g., [13 or 4]).
Examples of unoriented area-minimizing surfaces
Frank
Morgan
225-237
Abstract: A comprehensive study is made of constructions of area-minimizing flat chains modulo two. Many have singularities. For instance, any bounded, area-minimizing submanifold of $ {{\mathbf{R}}^n}$ occurs as the singular set of some area-minimizing flat chain modulo two in some $ {{\mathbf{R}}^N}$.
On the diagonal of an operator
Peng
Fan
239-251
Abstract: Characterizations of zero-diagonal operators (i.e., operators that have a diagonal whose entries consist entirely of zeros) and the norm-closure of these operators are obtained. Also included are new characterizations of trace class operators, self-commutators of bounded operators, and others.
Banach spaces which are $M$-ideals in their biduals
Peter
Harmand;
Åsvald
Lima
253-264
Abstract: We investigate Banach spaces $X$ such that $X$ is an $M$-ideal in $ {X^{{\ast}{\ast}}}$. Subspaces, quotients and ${c_0}$-sums of spaces which are $M$-ideals in their biduals are again of this type. A nonreflexive space $X$ which is an $M$-ideal in $ {X^{{\ast}{\ast}}}$ contains a copy of ${c_0}$. Recently Lima has shown that if $ K(X)$ is an $M$-ideal in $L(X)$ then $X$ is an $M$-ideal in $ {X^{{\ast}{\ast}}}$. Here we show that if $X$ is reflexive and $K(X)$ is an $M$-ideal in $L(X)$, then $ K{(X)^{{\ast}{\ast}}}$ is isometric to $L(X)$, i.e. $K(X)$ is an $M$-ideal in its bidual. Moreover, for real such spaces, we show that $K(X)$ contains a proper $M$-ideal if and only if $X$ or $ {X^{\ast}}$ contains a proper $M$-ideal.
Applications of uniform convexity of noncommutative $L\sp{p}$-spaces
Hideki
Kosaki
265-282
Abstract: We consider noncommutative ${L^p}$-spaces, $ 1 < p < \infty$, associated with a von Neumann algebra, which is not necessarily semifinite, and obtain some consequences of their uniform convexity. Among other results, we obtain (i) the norm continuity of the "absolute value part" map from each ${L^p}$-space onto its positive part; (ii) a certain continuity result on Radon-Nikodym derivatives in the context of positive cones introduced by H. Araki; and (iii) the necessary and sufficient condition for certain ${L^p}$-norm inequalities to become equalities. Some dominated convergence theorems for a probability gage are also considered.
Decomposability of Radon measures
R. J.
Gardner;
W. F.
Pfeffer
283-293
Abstract: A topological space is called metacompact or metalindelöf if each open cover has a point-finite or point-countable refinement, respectively. It is well known that each Radon measure is expressible as a sum of Radon measures supported on a disjoint family of compact sets, called a concassage. If the unions of measurable subsets of the members of a concassage are also measurable, the Radon measure is called decomposable. We show that Radon measures in a metacompact space are always saturated, and therefore decomposable whenever they are complete. The previous statement is undecidable in ZFC if "metacompact" is replaced by "metalindelöf". The proofs are based on structure theorems for a concassage of a Radon measure. These theorems also show that the union of a concassage of a Radon measure in a metacompact space is a Borel set, which is paracompact in the subspace topology whenever the main space is regular.
On bordism groups of immersions
Guillermo
Pastor
295-301
Abstract: The bordism group of immersions of oriented $n$-manifolds into ${{\mathbf{R}}^{n + k}}$ is identified with the stable homotopy group $\Pi _{n + k}^s(\operatorname{MSO} (k))$. We study these groups for $n - 2 \leqslant k \leqslant n$, and discuss the behaviour of double points and their relation with the corresponding bordism groups of embeddings.
Suspension spectra and homology equivalences
Nicholas J.
Kuhn
303-313
Abstract: Let $ f:{\Sigma ^\infty }X \to {\Sigma ^\infty }Y$ be a stable map between two connected spaces, and let $ {E_{\ast}}$ be a generalized homology theory. We show that if ${E_{\ast}}(f)$ is an isomorphism then ${E_{\ast}}({\Omega ^\infty }f):{E_{\ast}}(QX) \to {E_{\ast}}(QY)$ is a monomorphism, but possibly not an epimorphism. Applications of this theorem include results of Miller and Snaith concerning the $ K$-theory of the Kahn-Priddy map.
The quotient semilattice of the recursively enumerable degrees modulo the cappable degrees
Steven
Schwarz
315-328
Abstract: In this paper, we investigate the quotient semilattice $\underline R /\underline M $ of the r.e. degrees modulo the cappable degrees. We first prove the $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{R} /\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{M} $ counterpart of the Friedberg-Muchnik theorem. We then show that minimal elements and minimal pairs are not present in $\underline R /\underline M$. We end with a proof of the $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{R} /\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{M} $ counterpart to Sack's splitting theorem.
Compact spaces and spaces of maximal complete subgraphs
Murray
Bell;
John
Ginsburg
329-338
Abstract: We consider the space $M(G)$ of all maximal complete subgraphs of a graph $G$ and, in particular, the space $M(P)$ of all maximal chains of an ordered set $P$. The main question considered is the following: Which compact spaces can be represented as $ M(G)$ for some graph $ G$ or as $M(P)$ for some ordered set $P$? The former are characterized as spaces which have a binary subbase for the closed sets which consists of clopen sets. We give an example to show that this does not include all zero-dimensional supercompact spaces. The following negative result is obtained concerning ordered sets: Let $D$ be an uncountable discrete space and let $ \alpha D$ denote the one-point compactification of $D$. Then there is no ordered set $P$ such that $M(P) \simeq \alpha D$.
Quadratic forms of height two
Robert W.
Fitzgerald
339-351
Abstract: Quadratic forms of height two and leading form defined over the base field are determined over several fields. Also forms of height and degree two over an arbitrary field are classified.
Extensions of tight set functions with applications in topological measure theory
Wolfgang
Adamski
353-368
Abstract: Let $ {\mathcal{K}_1},\,{\mathcal{K}_2}$ be lattices of subsets of a set $ X$ with $ {\mathcal{K}_1} \subset {\mathcal{K}_2}$. The main result of this paper states that every semifinite tight set function on ${\mathcal{K}_1}$ can be extended to a semifinite tight set function on $ {\mathcal{K}_2}$. Furthermore, conditions assuring that such an extension is uniquely determined or $\sigma$-smooth at $\phi$ are given. Since a semifinite tight set function defined on a lattice $ \mathcal{K}$ [and being $ \sigma$-smooth at $ \phi$] can be identified with a semifinite $ \mathcal{K}$-regular content [measure] on the algebra generated by $\mathcal{K}$, the general results are applied to various extension problems in abstract and topological measure theory.
Factoring compact and weakly compact operators through reflexive Banach lattices
C. D.
Aliprantis;
O.
Burkinshaw
369-381
Abstract: When does a weakly compact operator between two Banach spaces factor through a reflexive Banach lattice? This paper provides some answers to this question. One of the main results: If an operator between two Banach spaces factors through a Banach lattice with weakly compact factors, then it also factors through a reflexive Banach lattice. In particular, the square of a weakly compact operator on a Banach lattice factors through a reflexive Banach lattice. Similar results hold for compact operators. For instance, the square of a compact operator on a Banach lattice factors with compact factors through a reflexive Banach lattice.
Variational problems with two phases and their free boundaries
Hans Wilhelm
Alt;
Luis A.
Caffarelli;
Avner
Friedman
431-461
Abstract: The problem of minimizing $\int {[\nabla \upsilon {\vert^2}} + {q^2}(x){\lambda ^2}(\upsilon )]dx$ in an appropriate class of functions $\upsilon$ is considered. Here $q(x) \ne 0$ and ${\lambda ^2}(\upsilon ) = \lambda _1^2$if $\upsilon < 0, = \lambda _2^2$ if $\upsilon > 0$. Any minimizer $u$ is harmonic in $\{ u \ne 0\}$ and $ \vert\nabla u{\vert^2}$ has a jump $\displaystyle {q^2}(x)\left( {\lambda _1^2 - \lambda _2^2} \right)$ across the free boundary $\{ u \ne 0\} $. Regularity and various properties are established for the minimizer $ u$ and for the free boundary.
Codimension $1$ orbits and semi-invariants for the representations of an oriented graph of type $\mathcal{A}_n$
S.
Abeasis
463-485
Abstract: We consider the Dynkin diagram $ \mathcal{A}_n$ with an arbitrary orientation $\Omega$. For a given dimension $d = ({d_1}, \ldots ,{d_n})$ we consider the corresponding variety ${L_d}$ of all the representations of $ (\mathcal{A}_n,\Omega )$ on which a group ${G_d}$ acts naturally. In this paper we determine the maximal orbit and the codim. $1$ orbits of this action, giving explicitly their decomposition in terms of the irreducible representations of $ \mathcal{A}_n$. We also deduce a set of algebraically independent semi-invariant polynomials which generate the ring of semi-invariants.
Some properties of viscosity solutions of Hamilton-Jacobi equations
M. G.
Crandall;
L. C.
Evans;
P.-L.
Lions
487-502
Abstract: Recently M. G. Crandall and P. L. Lions introduced the notion of "viscosity solutions" of scalar nonlinear first order partial differential equations. Viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics. The value of this concept is established by the fact that very general existence, uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application. The notion of a " viscosity solution" admits several equivalent formulations. Here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner. Moreover, by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper.
Matrix localizations of $n$-firs. I
Peter
Malcolmson
503-518
Abstract: An $n$-fir is an associative ring in which every $n$-generator right ideal is free of unique rank. Matrix localization of a ring involves the adjunction of universal inverses to certain matrices over the ring, so that a new ring results over which the matrices have inverses, but so that the minimum of additional relations is imposed. A full matrix is a square matrix which, when considered as an endomorphism of a free module, cannot be factored through a free module of smaller rank. The main result of this paper is that if the original ring is an $n$-fir with $n > 2k$ and if we form a matrix localization by adjoining universal inverses to all full matrices of size $ k$, then the resulting ring is an $(n - 2k)$-fir. This generalizes an announced result of V. N. Gerasimov. There are related results on the structure of the universal skew field of fractions of a semifir.
Matrix localizations of $n$-firs. II
Peter
Malcolmson
519--527
Abstract: In a previous paper by this author and with a similar title, it was shown that adjoining universal inverses for all $k \times k$ full matrices over an $n$-fir results in the localized ring being an $(n - 2k)$-fir. In this note a counterexample is used to show that the result is best possible in general. Techniques of the previous paper are strengthened and a result on a kind of finite inertia of certain rings within their localizations is obtained.
Haefliger structures and linear homotopy
Javier
Bracho
529-538
Abstract: The notion of linear-homotopy into a classifying space is introduced and used to give a precise classification of Haefliger structures. Appendix on the product theorem for simplicial spaces and realizations of bisimplicial spaces.
Degrees of recursively saturated models
Angus
Macintyre;
David
Marker
539-554
Abstract: Using relativizations of results of Goncharov and Peretyat'kin on decidable homogeneous models, we prove that if $M$ is $S$-saturated for some Scott set $S$, and $F$ is an enumeration of $S$, then $M$ has a presentation recursive in $F$. Applying this result we are able to classify degrees coding (i) the reducts of models of PA to addition or multiplication, (ii) internally finite initial segments and (iii) nonstandard residue fields. We also use our results to simplify Solovay's characterization of degrees coding nonstandard models of Th(N).
Orthogonal polynomials on the sphere with octahedral symmetry
Charles F.
Dunkl
555-575
Abstract: For any finite reflection group $G$ acting on $ {{\mathbf{R}}^N}$ there is a family of $G$-invariant measures ( $({h^2}d\omega$, where $h$ is a certain product of linear functions whose zero-sets are the reflecting hyperplanes for $ G$) on the unit sphere and an associated partial differential operator ( $ {L_h}f: = \Delta (fh) - f\Delta h$; $\Delta$ is the Laplacian). Analogously to spherical harmonics, there is an orthogonal (with respect to ${h^2}d\omega $) decomposition of homogeneous polynomials, that is, if $p$ is of degree $n$ then $\displaystyle p(x) = \sum\limits_{j = 0}^{[n/2]} {\vert x{\vert^{2j}}{p_{n - 2j}}(x),}$ where ${L_h}{p_i} = 0$ and ${\operatorname{deg}}{p_i} = i$ for each $ i$, but with the restriction that $p$ and ${p_i}$ must all be $G$-invariant. The main topic is the hyperoctahedral group with $\displaystyle h(x) = {({x_1}{x_2} \cdots {x_N})^\alpha }{\left( {\prod\limits_{i < j} {(x_i^2 - x_j^2)} } \right)^\beta }.$ The special case $N = 2$ leads to Jacobi polynomials. A detailed study of the case $N = 3$ is made; an important result is the construction of a third-order differential operator that maps polynomials associated to $h$ with indices $ (\alpha ,\beta )$ to those associated with $(\alpha + 2,\beta + 1)$.
Asymptotic behavior of solutions of second order differential equations with integrable coefficients
Manabu
Naito
577-588
Abstract: The differential equation $ x'' + a(t)f(x) = 0$, $t > 0$, is considered under the condition that ${\lim_{t \to \infty }}{\int ^t}a(s)ds$ exists and is finite, and necessary and/or sufficient conditions are given for this equation to have solutions which behave asymptotically like nontrivial linear functions ${c_1} + {c_2}t$.
Entropy via random perturbations
Yuri
Kifer
589-601
Abstract: The entropy of a dynamical system ${S^t}$ on a hyperbolic attractor with respect to the Bowen-Ruelle-Sinai measure is obtained as a limit of entropy characteristics of small random perturbations $x_t^\varepsilon $ of ${S^t}$. Both the case of perturbations only in some neighborhood of an attractor and global perturbations of a flow with hyperbolic attracting sets are considered.
Jordan domains and the universal Teichm\"uller space
Barbara Brown
Flinn
603-610
Abstract: Let $L$ denote the lower half plane and let $ B(L)$ denote the Banach space of analytic functions $f$ in $L$ with ${\left\Vert f \right\Vert _L} < \infty$, where $ {\left\Vert f \right\Vert _L}$ is the suprenum over $z \in L$ of the values $\left\vert {f(z)} \right\vert{(text{Im} z)^2}$. The universal Teichmüller space, $ T$, is the subset of $ B(L)$ consisting of the Schwarzian derivatives of conformal mappings of $ L$ which have quasiconformal extensions to the extended plane. We denote by $ J$ the set $\displaystyle \left\{ {{S_f}:f{\text{is conformal in }}L{\text{and }}f(L){\text{is a Jordan domain}}} \right\},$ which is a subset of $B(L)$ contained in the Schwarzian space $ S$. In showing $S - \bar T \ne \emptyset$, Gehring actually proves $ S - \bar J \ne \emptyset$. We give an example which demonstrates that $J - \bar T \ne \emptyset$.
Moduli of continuity in ${\bf R}\sp{n}$ and $D\subset {\bf R}\sp{n}$
Z.
Ditzian
611-623
Abstract: The $r$ modulus of continuity for $f \in C({R^n})$ is expressed in terms of $ r$ moduli of continuity in $ n$ independent directions. Generalizations to other spaces of functions on $ {R^n}$ or $D \subset {R^n}$ are also given.
On maximal rearrangement inequalities for the Fourier transform
W. B.
Jurkat;
G.
Sampson
625-643
Abstract: Suppose that $ w$ is a measurable function on $ {{\mathbf{R}}^n}$ and denote by $W = {w^ \ast }$ the decreasing rearrangement of $\left\vert w \right\vert$ (provided that it exists). We show that the $n$-dimensional Fourier transform $ \hat f$ satisfies (1) $\displaystyle {\left\Vert {w\hat f} \right\Vert _q} \leqslant {\left\Vert {W{{(... ...t)\int_0^{1/t} {{f^ \ast }} } \right\Vert\quad (C {\text{absolute constant}}),$ if $ 1 < q < \infty$ and $ {t^{2/q - 1}}W(t) \searrow$ for $t > 0$. We also show that (2) $\displaystyle {\left\Vert {w\hat f} \right\Vert _q} \geqslant {c_{n,q}}{\left\V... ...\vert x \right\vert} {f(y)} dy} \right\Vert _q}\quad (f {\text{nonnegative),}}$ if $ 1 < q < \infty$ and $ w$ is nonnegative and symmetrically decreasing. Inequality (2) implies that (1) is maximal in the sense that the left side reaches the right side if $f$ is nonnegative and symmetrically decreasing. Hence, (1) implies all other possible estimates in terms of $W$ and ${f^ \ast }$. The cases $q \ne 2$ of (1) can be derived from the case $q = 2$ (and same $f$) by a convexity principle which does not involve interpolation. The analogue of (1) for Fourier series is due to H. L. Montgomery if $q \geqslant 2$ (then the extra condition on $ W$ is automatically satisfied).
Intermediate normalizing extensions
A. G.
Heinicke;
J. C.
Robson
645-667
Abstract: Relationships between the prime ideals of a ring $R$ and of a normalizing extension $ S$ have been studied by several authors recently. In this work, most of these known results are extended to give relationships between the prime ideals of $R$ and of $T$ where $T$ is a ring with $R \subset T \subset S$, and $S$ is a normalizing extension of $ R$: such rings $ T$ are called intermediate normalizing extensions of $ R$. One result ("Cutting Down") asserts that for any prime ideal $ J$ of $T$, $J \cap R$ is the intersection of a finite set of prime ideals ${P_i}$ of $R$, uniquely defined by $J$, whose corresponding factor rings $R/{P_i}$ are mutually isomorphic. The minimal members of the family of ${P_i}$'s are the primes of $R$ minimal over $J \cap R$, and an "incomparability" theorem is proved which shows that no two comparable primes of $ T$ can have their intersections with $R$ share a common minimal prime. Other results include versions of the "lying over" and "going up" theorems, proofs that chain conditions such as right Goldie or right Noetherian pass between $T/J$ and each of the rings $R/{P_i}$, and a demonstration that the "additivity principle" holds.
Mean convergence of Lagrange interpolation. III
Paul
Nevai
669-698
Abstract: Necessary and sufficient conditions are found for weighted mean convergence of Lagrange and quasi-Lagrange interpolation based at the zeros of generalized Jacobi polynomials.
Monotone decompositions of IUC continua
W. Dwayne
Collins
699-709
Abstract: For the class of hereditarily unicoherent metric continua a spectrum of monotone decompositions has been developed by several authors which "improves" the quotient spaces. This spectrum is developed for a broader class of continua, namely continua with property IUC. A metric continuum $ M$ has property IUC provided each proper subcontinuum of $M$ with interior is unicoherent. One important result which develops is that semiaposyndetic IUC continua are hereditarily arcwise connected. Also the notion of smoothness is studied for IUC continua.
Projectively equivalent metrics subject to constraints
William
Taber
711-737
Abstract: This work examines the relationship between pairs of projectively equivalent Riemannian or Lorentz metrics $g$ and $ {g^ \ast }$ on a manifold $ M$ that induce the same Riemannian metric on a hypersurface $H$. In general such metrics must be equal. In the case of distinct metrics, the structure of the metrics and the manifold are strongly determined by the set, $C$, of points at which $g$ and ${g^ \ast }$ are conformally related. The space $(M - C,g)$ is locally a warped product manifold over the hypersurface $H$. In the Lorentz setting, $C$ is empty. In the Riemannian case, $ C$ contains at most two points. If $C$ is nonempty, then $H$ is isometric to a standard sphere. Furthermore, in the case that $C$ contains one point, natural hypotheses imply $ M$ is diffeomorphic to $ {R^n}$. If $C$ contains two points $M$ is diffeomorphic to $ {S^n}$.
On specializations of curves. I
A.
Nobile
739-748
Abstract: The following is proved: Given a family of projective reduced curves $X \to T$ ($T$ irreducible), if $ {X_t}$ (the general curve) is integral and ${X_0}$ is a special curve (having irreducible components ${X_1}, \ldots ,{X_r}$), then $\sum\nolimits_{i = 1}^r {{g_i}({X_i}) \leqslant g({X_t})}$, where $g(Z) =$ geometric genus of $Z$. Conversely, if $A$ is a reduced plane projective curve, of degree $ n$ with irreducible components $ {X_1}, \ldots ,{X_r}$, and $ g$ satisfies $ \sum\nolimits_{i = 1}^r {{g_i}({X_i}) \leqslant g \leqslant \frac{1} {2}(n - 1)(n - 2)}$, then a family of plane curves $ X \to T$ (with $ T$ integral) exists, where for some $ {t_0} \in T,{X_{{t_0}}} = Z$ and for $t$ generic, ${X_t}$ is integral and has only nodes as singularities. Results of this type appear in an old paper by G. Albanese, but the exposition is rather obscure.
Stable viscosities and shock profiles for systems of conservation laws
Robert L.
Pego
749-763
Abstract: Wide classes of high order "viscosity" terms are determined, for which small amplitude shock wave solutions of a nonlinear hyperbolic system of conservation laws ${u_t} + f{(u)_x} = 0$ are realized as limits of traveling wave solutions of a dissipative system ${u_t} + f{(u)_x} = \nu {({D_1}{u_x})_x} + \cdots + {\nu ^n}{({D_n}{u^{(n)}})_x}$. The set of such "admissible" viscosities includes those for which the dissipative system satisfies a linearized stability condition previously investigated in the case $n = 1$ by A. Majda and the author. When $n = 1$ we also establish admissibility criteria for singular viscosity matrices $ {D_1}(u)$, and apply our results to the compressible Navier-Stokes equations with viscosity and heat conduction, determining minimal conditions on the equation of state which ensure the existence of the "shock layer" for weak shocks.
Brauer factor sets and simple algebras
Louis H.
Rowen
765-772
Abstract: It is shown that the Brauer factor set $ ({c_{ijk}})$ of a finite-dimensional division algebra of odd degree $n$ can be chosen such that $ {c_{iji}} = {c_{iij}} = {c_{jii}} = 1$ for all $i,j$ and ${c_{ijk}} = c_{kji}^{ - 1}$. This implies at once the existence of an element $a \ne 0$ with ${\text{tr}}(a) = {\text{tr}}({a^2}) = 0$; the coefficients of $ {x^{n - 1}}$ and ${x^{n - 2}}$ in the characteristic polynomial of $ a$ are thus 0. Also one gets a generic division algebra of degree $ n$ whose center has transcendence degree $ n + (n - 1)(n - 2)/2$, as well as a new (simpler) algebra of generic matrices. Equations are given to determine the cyclicity of these algebras, but they may not be tractable.
Realizability of branched coverings of surfaces
Allan L.
Edmonds;
Ravi S.
Kulkarni;
Robert E.
Stong
773-790
Abstract: A branched covering $M \to N$ of degree $ d$ between closed surfaces determines a collection $ \mathfrak{D}$ of partitions of $d$--its "branch data"--corresponding to the set of branch points. The collection of partitions must satisfy certain obvious conditions implied by the Riemann-Hurwitz formula. This paper investigates the extent to which any such finite collection $\mathfrak{D}$ of partitions of $d$ can be realized as the branch data of a suitable branched covering. If $N$ is not the $2$-sphere, such data can always be realized. If $\mathfrak{D}$ contains sufficiently many elements compared to $d$, then it can be realized. And whenever $ d$ is nonprime, examples are constructed of nonrealizable data.
Strong martingale convergence of generalized conditional expectations on von Neumann algebras
Fumio
Hiai;
Makoto
Tsukada
791-798
Abstract: Accardi and Cecchini generalized the concept of conditional expectations on von Neumann algebras. In this paper we give some conditions for strong convergence of increasing or decreasing martingales of Accardi and Cecchini's conditional expectations.
A sharp form of the Ahlfors' distortion theorem, with applications
D. H.
Hamilton
799-806
Abstract: The constant appearing in the asymptotic version of the Ahlfors' distortion theorem is $1$. Also it is shown that for mean $1$-valent functions $f = z + {a_2}{z^2} \cdots \left\Vert {{a_{n + 1}}\vert - \vert{a_n}} \right\Vert \leqslant 1$ for $n \geqslant N(f)$.
On the proper holomorphic equivalence for a class of pseudoconvex domains
M.
Landucci
807-811
Abstract: A complete and explicit description of the holomorphic proper mappings between weakly pseudoconvex domains of the class ${\Delta _p}$ (see ( *) below) is given.
Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions
C. Ward
Henson;
Lee A.
Rubel
1-32
Abstract: In this paper we study identities between certain functions of many variables that are constructed by using the elementary functions of addition $x+y$, multiplication $x \cdot y$, and two-place exponentiation $ x^y$. For a restricted class of such functions, we show that every true identity follows from the natural set of eleven axioms. The rates of growth of such functions, in the case of a single independent variable $x$, as $x \to \infty $, are also studied, and we give an algorithm for the Hardy relation of eventual domination, again for a restricted class of functions. Value distribution of analytic functions of one and of several complex variables, especially the Nevanlinna characteristic, plays a major role in our proofs.
Decompositions of the maximal ideal space of $L\sp{\infty }$
Pamela
Gorkin
33-44
Abstract: In this paper we show the existence of one point maximal antisymmetric sets for $ {H^\infty } + C$.
The generalized Zahorski class structure of symmetric derivatives
Lee
Larson
45-58
Abstract: A generalized Zahorski class structure is demonstrated for symmetric derivatives. A monotonicity theorem is proved and a condition sufficient to ensure that a symmetric derivative has the Darboux property is presented.
Estimates for eigenfunctions and eigenvalues of nonlinear elliptic problems
Chris
Cosner
59-75
Abstract: We consider solutions to the nonlinear eigenvalue problem $\displaystyle (*)\quad A(x,\vec u)\vec u + \lambda f(x,\vec u) = 0\:\quad {\tex... ...,\quad \vec u{\text{ = }}0,\quad {\text{on}}\partial \Omega ,\quad \vec{u} = 0,$ where (*) is a quasilinear strongly coupled second order elliptic system of partial differential equations and $ \Omega \subseteq \mathbf{R}^{n}$ is a smooth bounded domain. We obtain lower bounds for $\lambda$ in the case where $f(x,\vec u)$ has linear growth, and relations between $\lambda ,\Omega $, and ess sup$ \vert\vec u\vert$ when $f(x,\vec u)$ has sub- or superlinear growth. The estimates are based on integration by parts and application of certain Sobolev inequalities. We briefly discuss extensions to higher order systems.
Inducible periodic homeomorphisms of tree-like continua
Juan A.
Toledo
77-108
Abstract: In this paper we prove that every periodic homeomorphism on a tree-like continuum can be strongly induced on an inverse sequence composed of a certain kind of graph that we call ``bellows''. We introduce the concepts of ``#-graph'' of a periodic homeomorphism and of ``perfect'' homeomorphism. A theorem concerning the parallel inducing of two periodic homeomorphisms having orbit spaces with the same multiplicity structure is also proved. The results are related to conjugacy and to the pseudo-arc.
Noncommutative topological dynamics. I
Daniel
Avitzour
109-119
Abstract: Ergodicity and minimality are defined for $C^{\ast}$-flows $(G,A)$ where $G$ is a group acting on a $C^{\ast}$-algebra $A$ by *-automorphisms. Elementary properties are proved and several examples are given. In particular, an example shows that there are arbitrarily large $ C^{\ast}$-algebras admitting a minimal action of the integers.
Noncommutative topological dynamics. II
Daniel
Avitzour
121-135
Abstract: This part deals with almost periodic and weakly mixing ${C^ \ast }$-flows, and with disjointness and weak disjointness of $ {C^ \ast }$-flows (flows on ${C^ \ast }$-algebras). The main result is a generalization to ${C^ \ast }$-flows of Keynes and Robertson's characterization of minimal weakly mixing flows. Examples are discussed exhibiting anomalous behaviour of disjointness in the $ {C^ \ast }$-flow case.
Brauer's height conjecture for $p$-solvable groups
David
Gluck;
Thomas R.
Wolf
137-152
Abstract: We complete the proof of the height conjecture for $p$-solvable groups, using the classification of finite simple groups.
The free boundary of a semilinear elliptic equation
Avner
Friedman;
Daniel
Phillips
153-182
Abstract: The Dirichlet problem $ \Delta u = \lambda \,f(u)$ in a domain $ \Omega ,\,u = 1$ on $\partial\Omega$ is considered with $ f(t) = 0$ if $t \leq 0,\,f(t) > 0$ if $t > 0,\,f(t) \sim {t^p}$ if $t \downarrow 0,0 < p < 1;f(t)$ is not monotone in general. The set $\{ u = 0\}$ and the ``free boundary'' $\partial \{ u = 0\}$ are studied. Sharp asymptotic estimates are established as $\lambda \to \infty$. For suitable $f$, under the assumption that $ \Omega$ is a two-dimensional convex domain, it is shown that $\{ u = 0\}$ is a convex set. Analogous results are established also in the case where $\partial u/\partial v + \mu (u - 1) = 0$ on $\partial \Omega $.
The ill-posed Hele-Shaw model and the Stefan problem for supercooled water
Emmanuele
DiBenedetto;
Avner
Friedman
183-204
Abstract: The Hele-Shaw flow of a slow viscous fluid between slightly separated plates is analyzed in the ill-posed case when the fluid recedes due to absorption through a core $ G$. Necessary and sufficient conditions are given on the initial domain occupied by the fluid to ensure the existence of a solution. Regularity of the free boundary is established in certain rather general cases. Similar results are obtained for the analogous parabolic version, which models the one-phase Stefan problem for supercooled water.
Classifying torsion-free subgroups of the Picard group
Andrew M.
Brunner;
Michael L.
Frame;
Youn W.
Lee;
Norbert J.
Wielenberg
205-235
Abstract: Torsion-free subgroups of finite index in the Picard group are the fundamental groups of hyperbolic $3$-manifolds. The Picard group is a polygonal product of finite groups. Recent work by Karrass, Pietrowski and Solitar on the subgroups of a polygonal product make it feasible to calculate all the torsion-free subgroups of any finite index. This computation is carried out here for index 12 and 24, where there are, respectively, 2 and 17 nonisomorphic subgroups. The manifolds are identified by using surgery.
Group-graded rings, smash products, and group actions
M.
Cohen;
S.
Montgomery
237-258
Abstract: Let $A$ be a $k$-algebra graded by a finite group $G$, with ${A_1}$ the component for the identity element of $ G$. We consider such a grading as a ``coaction'' by $G$, in that $A$ is a $k{[G]^ \ast }$-module algebra. We then study the smash product $A\char93 k{[G]^ \ast }$; it plays a role similar to that played by the skew group ring $R\, \ast \,G$ in the case of group actions, and enables us to obtain results relating the modules over $ A,\,{A_1}$, and $A\char93 k{[G]^ \ast }$. After giving algebraic versions of the Duality Theorems for Actions and Coactions (results coming from von Neumann algebras), we apply them to study the prime ideals of $A$ and ${A_1}$. In particular we generalize Lorenz and Passman's theorem on incomparability of primes in crossed products. We also answer a question of Bergman on graded Jacobson radicals.
The number of factorizations of numbers less than $x$ into divisors greater than $y$
Douglas
Hensley
259-274
Abstract: Let $A(x,\,y)$ be the number in the title. There is a function $h:[0,\,\infty ) \to [0,\,2]$, decreasing and convex, with $h(0) = 2$ and $\operatorname{lim}_{r \to \infty }h(r) = 0$, such that if $r = \operatorname{log} y/\sqrt {\operatorname{log} \,x}$ then as $x \to \infty$ with $ r$ fixed, $\displaystyle A(x,y) = \frac{{C(r)x\,\operatorname{exp} \left( {h(r)\sqrt {\ope... ...{log} \,x)}^{3/4}}}}\left( {1 + O{{(\operatorname{log} \,x)}^{ - 1/4}}} \right)$ . The estimate is uniform on intervals $0 < r \leq {R_0}$. As corollaries we have for $\operatorname{log} \,y = \theta {(\operatorname{log} \,x)^{1/4}}$, $\displaystyle \lim \limits_{x \to \infty } \,\frac{{A(x,\,y)}} {{A(x,\,1)/y}} = {e^{{\theta ^2}/2}}$ ,and if $ \operatorname{log} \,y = o {(\operatorname{log} \,x)^{1/4}}$ then $ A(x,\,y) \approx A(x,\,1)/y$.
The radius ratio and convexity properties in normed linear spaces
D.
Amir;
C.
Franchetti
275-291
Abstract: The supremum of the ratios of the self-radius ${r_A}(A)$ of a convex bounded set in a normed linear space $X$ to its absolute radius ${r_X}(A)$ is related to the supremum of the relative projection constants of the maximal subspaces of $ X$. Necessary conditions and sufficient conditions for these suprema to be smaller than 2 are given. These conditions are selfadjoint superproperties similar to $B$-convexity, superreflexivity and $ P$-convexity.
Free products of inverse semigroups
Peter R.
Jones
293-317
Abstract: A structure theorem is provided for the free product $ S\,{\operatorname{inv}}\,T$ of inverse semigroups $S$ and $T$. Each element of $ S\,{\operatorname{inv}}\,T$ is uniquely expressible in the form $\varepsilon (A)a$, where $A$ is a certain finite set of ``left reduced'' words and either $a = 1$ or $ a = {a_1} \cdots {a_m}$ is a ``reduced'' word with $aa_m^{ - 1} \in A$. (The word ${a_1} \cdots {a_m}$ in $ S\,{\operatorname{sgp}}\,T$ is called reduced if no letter is idempotent, and left reduced if exactly ${a_m}$ is idempotent; the notation $\varepsilon (A)$ stands for $\Pi \{ a{a^{ - 1}}:\,a \in A\}$.) Under a product remarkably similar to Scheiblich's product for free inverse semigroups, the corresponding pairs $ (A,\,a)$ form an inverse semigroup isomorphic with $ S\,{\operatorname{inv}}\,T$. This description enables various properties of $ S\,{\operatorname{inv}}\,T$ to be determined. For example $ (S\:{\operatorname{inv}}\:T)\backslash (S \cup T)$ is always completely semisimple and each of its subgroups is isomorphic with a finite subgroup of $S$ or $T$. If neither $S$ nor $T$ has a zero then $ (S\:{\operatorname{inv}}\:T)$ is fundamental, but in general fundamentality itself is not preserved.
Equivalence problems in projective differential geometry
Kichoon
Yang
319-334
Abstract: Equivalence problems for abstract, and induced, projective structures are investigated. (i) The notion of induced projective structures on submanifolds of a projective space is rigorously defined. (ii) Equivalence problems for such structures are discussed; in particular, it is shown that nonplanar surfaces in $ \mathbf{R}{P^3}$ are all projectively equivalent to each other. (iii) The imbedding problem of abstract projective structures is studied; in particular, we show that every abstract projective structure on a $2$-manifold arises as an induced structure on an arbitrary nonplanar surface in $\mathbf{R}{P^3}$; this result should be contrasted to that of Chern (see [6]).
Positive solutions of nonlinear elliptic equations---existence and nonexistence of solutions with radial symmetry in $L\sb{p}({\bf R}\sp{N})$
J. F.
Toland
335-354
Abstract: It is shown that when $r$ is nonincreasing, radially symmetric, continuous and bounded below by a positive constant, the solution set of the nonlinear elliptic eigenvalue problem $\displaystyle - \Delta u = \lambda u + r{u^{1 + \sigma }},\qquad u > 0\qquad {\... ...athbf{R}^N},\qquad u \to 0\qquad {\text{as}}\,{\text{\vert x\vert}} \to \infty$ , contains a continuum $ \mathcal{C}$ of nontrivial solutions which is unbounded in $\mathbf{R}\, \times \,{L_p}({\mathbf{R}^N})$ for all $p \geq 1$. Various estimates of the $ {L_p}$ norm of $ u$ are obtained which depend on the relative values of $\sigma$ and $p$, and the Pohozaev and Sobolev embedding constants.
Ford and Dirichlet regions for discrete groups of hyperbolic motions
P. J.
Nicholls
355-365
Abstract: It is shown that for a discrete group of hyperbolic motions of the unit ball of $ {\mathbf{R}^n}$, there is a single construction of fundamental regions which gives the Ford and Dirichlet regions as special cases and which also yields fundamental regions based at limit points. It is shown how the region varies continuously with the construction. The construction is connected with a class of limit points called Garnett points. The size of the set of such points is investigated.
Approximation of infinite-dimensional Teichm\"uller spaces
Frederick P.
Gardiner
367-383
Abstract: By means of an exhaustion process it is shown that Teichmüller's metric and Kobayashi's metric are equal for infinite dimensional Teichmüller spaces. By the same approximation method important estimates coming from the Reich-Strebel inequality are extended to the infinite dimensional cases. These estimates are used to show that Teichmüller's metric is the integral of its infinitesimal form. They are also used to give a sufficient condition for a sequence to be an absolute maximal sequence for the Hamilton functional. Finally, they are used to give a new sufficient condition for a sequence of Beltrami coefficients to converge in the Teichmüller metric.
Knots prime on many strings
Steven A.
Bleiler
385-401
Abstract: A study is made of the factorization of prime knots into tangles. Several infinite families of knots which do not factor into prime tangles are examined, and a new characterization of knot primality is developed.
Discontinuous translation invariant functionals
Sadahiro
Saeki
403-414
Abstract: Let $G$ be an infinite $ \sigma$-compact locally compact group. We shall study the existence of many discontinuous translation invariant linear functionals on a variety of translation invariant Fréchet spaces of Radon measures on $G$. These spaces include the convolution measure algebra $M(G)$, the Lebesgue spaces ${L^p}(G)$, where $1 \leq p \leq \infty$, and certain combinations thereof. Among other things, it will be shown that $ C(G)$ has many discontinuous translation invariant functionals, provided that $ G$ is amenable. This solves a problem of G. H. Meisters.
Linearization and mappings onto pseudocircle domains
Andrew
Haas
415-429
Abstract: We demonstrate the existence of linearizations for groups of conformal and anticonformal homeomorphisms of Riemann surfaces. The finitely generated groups acting on plane domains are classified in terms of specific linearizations. This extends Maskit's work in the directly conformal case. As an application we prove that there exist conformal representations of finite genus open Riemann surfaces for which accessible boundary points are either isolated or lie on circular arcs of pseudocircular boundary components. In many cases these are actually circle domains. Along the way we extend the applicability of Carathéodory's boundary correspondence theorem for prime ends.
Binomial coefficients and Jacobi sums
Richard H.
Hudson;
Kenneth S.
Williams
431-505
Abstract: Throughout this paper $e$ denotes an integer $ \geqslant 3$ and $ p$ a prime $\equiv \;1 \pmod e$. With $f$ defined by $ p = ef + 1$ and for integers $r$ and $s$ satisfying $1 \leqslant s < r \leqslant e - 1$ , certain binomial coefficients $ \left( {\begin{array}{*{20}{c}} {rf} {sf} \end{array} } \right)$ have been determined in terms of the parameters in various binary and quaternary quadratic forms by, for example, Gauss [13], Jacobi [19, 20], Stern [37-40], Lehmer [23] and Whiteman [42, 45, 46]. In $\S2$ we determine for each $e$ the exact number of binomial coefficients $\left( {\begin{array}{*{20}{c}} {rf} {sf} \end{array} } \right)$ not trivially congruent to one another by elementary properties of number theory and call these representative binomial coefficients. A representative binomial coefficient is said to be of order $e$ if and only if $(r,s) = 1$. In $\S\S3-4$, we show how the Davenport-Hasse relation [7], in a form given by Yamamoto [50], leads to determinations of $ {n^{(p - 1)/m}}$ in terms of binomial coefficients modulo $p = ef + 1 = mnf + 1$. These results are of some interest in themselves and are used extensively in later sections of the paper. Making use of Theorem 5.1 relating Jacobi sums and binomial coefficients, which was first obtained in a slightly different form by Whiteman [45], we systematically investigate in $\S\S6-21$ all representative binomial coefficients of orders $e = 3,4,6,7,8,9,11,12,14,15,16,20$ and $24$, which we are able to determine explicitly in terms of the parameters in well-known binary quadratic forms, and all representative binomial coefficients of orders $ e = 5,10,13,15,16$ and $ 20$, which we are able to explicitly determine in terms of quaternary quadratic decompositions of $16p$ given by Dickson [9], Zee [51] and Guidici, Muskat and Robinson [14]. Some of these results have been obtained by previous authors and many new ones are included. For $ e = 7$ and $14$ we are unable to explicitly determine representative binomial coefficients in terms of the six variable quadratic decomposition of $ 72p$ given by Dickson [9] for reasons given in $\S10$, but we are able to express these binomial coefficients in terms of the parameter ${x_1}$ in this system in analogy to a recent result of Rajwade [34]. Finally, although a relatively rare occurrence for small $ e$, it is possible for representative binomial coefficients of order $ e$ to be congruent to one another $\pmod p$. Representative binomial coefficients which are congruent to $\pm 1$ times at least one other representative for all $p = ef + 1$ are called Cauchy-Whiteman type binomial coefficients for reasons given in [17] and $\S21$. All congruences between such binomial coefficients are carefully examined and proved (with the sign ambiguity removed in each case) for all values of $ e$ considered. When $ e = 24$ there are $ 48$ representative binomial coefficients, including those of lower order, and it is shown in $\S21$ that an astonishing $43$ of these are Cauchy-Whiteman type binomial coefficients. It is of particular interest that the sign ambiguity in many of these congruences does not arise from any expression of the form ${n^{(p - 1)/m}}$ in contrast to the case for all $ e < 24$.
An algebraic classification of some even-dimensional spherical knots. I
M. Š.
Farber
507-527
Abstract: The main result of the paper is the classification of simple even-dimensional spherical knots in terms of their algebraic invariants.
An algebraic classification of some even-dimensional spherical knots. II
M. Š.
Farber
529-570
Abstract: The paper reduces the problem of classification of simple even-dimensional spherical knots of codimension two to an algebraic problem.
Homomorphisms of cocompact Fuchsian groups on ${\rm PSL}\sb{2}(Z\sb{p\sp{n}}[x]/(f(x)))$
Jeffrey
Cohen
571-585
Abstract: We obtain conditions under which $ {\text{PSL}}_2({Z_{{p^n}}}[x]/(f(x)))$ is a factor of $(l,m,n)$. Using this, certain results about factors of cocompact Fuchsian groups are obtained. For example, it is shown that: (i) $\Gamma$ has infinitely many simple nonabelian factors. (ii) $\Gamma$ has factors with nontrivial center. (iii) For each $n$, there exists $m$ such that $\Gamma$ has at least $n$ factors of order $m$. Further, all factored normal subgroups can be taken torsion-free. Also, new Hurwitz groups and noncongruence subgroups of the modular group are obtained.
The $\bar \partial $-Neumann solution to the inhomogeneous Cauchy-Riemann equation in the ball in ${\bf C}\sp{n}$
F. Reese
Harvey;
John C.
Polking
587-613
Abstract: Let $\vartheta$ denote the formal adjoint of the Cauchy-Riemann operator $\overline \partial$ on ${{\mathbf{C}}^n}$, and let $N$ denote the Kohn-Neumann operator on the unit ball in $ {{\mathbf{C}}^n}$. The operator $ \vartheta \; \circ \;N$ provides a natural fundamental solution for $\overline \partial f = g$ on the ball. It is our purpose to present the kernel $P$ for this operator $\vartheta \; \circ \;N$ explicitly--the coefficients are exhibited as rational functions.
Constructions arising from N\'eron's high rank curves
M.
Fried
615-631
Abstract: Many papers quote Néron's geometric construction of elliptic curves of rank $11$ over $ \mathbb{Q}\;[{\mathbf{N}}]$--still, at the writing of this paper, the elliptic curves of highest demonstrated rank. The purported reason for the ordered display of "creeping rank" in [ $ {\mathbf{PP}},{\mathbf{GZ}},{\mathbf{Na}}$ and $ {\mathbf{BK}}$] is to make $ [{\mathbf{N}}]$ explicit. Excluding $ [{\mathbf{BK}}]$, however, these papers derive little from Néron's constructions. All show some lack of confidence in the details of $ [{\mathbf{N}}]$. The core of this paper ($\S3$), meets objections to $[{\mathbf{N}}]$ raised by correspondents. Our method adds a novelty as it magnifies the constructions of $ [{\mathbf{N}}]$--"generation of pencils of cubics from their singular fibers". This has two advantages: it displays (Remark 4.2) the free parameters whose specializations give high rank curves; and it demonstrates the existence of rank $11$ curves through one appeal only to Hilbert's irreducibility theorem. That is, we have eliminated the unusual analogue of Hilbert's result that takes up most of $ [{\mathbf{N}}]$. In particular $(\S4(c))$, the explicit form of the irreducibility theorem in $ [{\mathbf{Fr}}]$ applies to give explicit rank $11$ curves over $ \mathbb{Q}$: with Selmer's conjecture, rank $12$.
Nonstable reflexive sheaves on ${\bf P}\sp{3}$
Timothy
Sauer
633-655
Abstract: The spectrum is defined for nonstable rank two reflexive sheaves on ${{\mathbf{P}}^3}$ and is used to establish vanishing theorems for intermediate cohomology in terms of the Chern classes and the order of nonstability. These results are shown to be best possible and the extremal cases are classified. Some applications to Cohen-Macaulay generically local complete intersection curves in $ {{\mathbf{P}}^3}$ are given.
A minimal model for $\neg{\rm CH}$: iteration of Jensen's reals
Uri
Abraham
657-674
Abstract: A model of ${\text{ZFC}} + {2^{\aleph_0}} = {\aleph_2}$ is constructed which is minimal with respect to being a model of $ \neg {\text{CH}}$. Any strictly included submodel of $ {\text{ZF}}$ (which contains all the ordinals) satisfies ${\text{CH}}$. In this model the degrees of constructibility have order type $ {\omega_2}$. A novel method of using the diamond is applied here to construct a countable-support iteration of Jensen's reals: In defining the $ \alpha {\text{th}}$ stage of the iteration the diamond "guesses" possible $\beta > \alpha$ stages of the iteration.
Weak restricted and very restricted operators on $L\sp{2}$
J. Marshall
Ash
675-689
Abstract: A battlement is a real function with values in $\{ 0,1\}$ that looks like a castle battlement. A commuting with translation linear operator $ T$ mapping step functions on ${\mathbf{R}}$ into the set of all measurable functions on $ {\mathbf{R}}$ and satisfying $\parallel Tb{\parallel_2} \leqslant C\parallel b{\parallel_2}$ for all battlements $b$ is bounded on ${L^2}({\mathbf{R}})$. This remains true if the underlying space is the circle but is demonstrably false if the underlying space is the integers. Michael Cowling's theorem that linear commuting with translation operators are bounded on ${L^2}$ if they are weak restricted $(2,2)$ is reproved and an application of this result to sums of exponentials is given.
Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds
Hyeong In
Choi
691-716
Abstract: We define the asymptotic Dirichlet problem and give a sufficient condition for solving it. This proves an existence of nontrivial bounded harmonic functions on certain classes of noncompact complete Riemannian manifolds.
Supercompactness of compactifications and hyperspaces
Murray G.
Bell
717-724
Abstract: We prove a theorem which implies that if $ \gamma \omega$ is a supercompact compactification of the countable discrete space $ \omega$ then $\gamma \omega - \omega$ is separable. This improves an earlier result of the author's that such a $\gamma \omega$ must have $\gamma \omega - \omega \;{\text{ccc}}$. We prove a theorem which implies that the hyperspace of closed subsets of $ {2^{\omega_2}}$ is not a continuous image of a supercompact space. This improves an earlier result of $ {\text{L}}$. Šapiro that the hyperspace of closed subsets of ${2^{\omega_2}}$ is not dyadic.
The level sets of the moduli of functions of bounded characteristic
Robert D.
Berman
725-744
Abstract: For $f$ a nonconstant meromorphic function on $ \Delta = \{ \vert z\vert < 1\}$ and $r \in (\inf \vert f\vert,\sup \vert f\vert)$, let $\mathcal{L}(f,r) = \{ z \in \Delta :\vert f(z)\vert = r\}$. In this paper, we study the components of $ \Delta \backslash \mathcal{L}(f,r)$ along with the level sets $\mathcal{L}(f,r)$. Our results include the following: If $f$ is an outer function and $\Omega$ a component of $ \Delta \backslash \mathcal{L}(f,r)$, then $\Omega$ is a simply-connected Jordan region for which $({\text{fr}}\;\Omega ) \cap \{ \vert z\vert = 1\}$ has positive measure. If $f$ and $g$ are inner functions with $ \mathcal{L}\,(f,r) = \mathcal{L}\,(g,s)$, then $g = \eta {f^\alpha }$, where $\vert\eta \vert = 1$ and $ \alpha > 0$. When $ g$ is an arbitrary meromorphic function, the equality of two pairs of level sets implies that $ g = c{f^\alpha }$, where $ c \ne 0$ and $\alpha \in ( - \infty ,\infty )$. In addition, an inner function can never share a level set of its modulus with an outer function. We also give examples to demonstrate the sharpness of the main results.
Endomorphisms of the cohomology of complex Grassmannians
Michael
Hoffman
745-760
Abstract: For any complex Grassmann manifold $G$, we classify all endomorphisms of the rational cohomology ring of $G$ which are nonzero on dimension $2$. Some applications of this result are given.
Approximation in the mean by solutions of elliptic equations
Thomas
Bagby
761-784
Abstract: A result analogous to the Vituškin approximation theorem is proved for mean approximation by solutions of certain elliptic equations.
On the paths of symmetric stable processes
Burgess
Davis
785-794
Abstract: It is shown that if $ X(t), t \geqslant 0$, is a symmetric stable process of index $\alpha, 0 < \alpha < 2$, then $\sup_t \lim \inf_{h \downarrow 0} (X(t + h) - X(t))h^{-1/\alpha} = \infty$ a.s. This settles a question of Fristedt about strictly stable subordinators.
Finitely generated extensions of partial difference fields
Peter
Evanovich
795-811
Abstract: A proof of the following theorem is given: If $ \mathcal{M}$ is a finitely generated extension of a partial difference field $\mathcal{K}$ then every subextension of $\mathcal{M}/\mathcal{K}$ is finitely generated. An integral measure of partial difference field extensions having properties similar to the dimension of field extensions and the limit degree of ordinary difference field extensions and a new method of computing transformal transcendence degree are developed.
Invariant densities for random maps of the interval
S.
Pelikan
813-825
Abstract: A random map is a discrete time process in which one of a number of functions is selected at random and applied. Here we study random maps of $[0,1]$ which represent dynamical systems on the square $[0,1] \times [0,1]$. Sufficient conditions for a random map to have an absolutely continuous invariant measure are given, and the number of ergodic components of a random map is discussed.
On Block's condition for simple periodic orbits of functions on an interval
Chung-Wu
Ho
827-832
Abstract: Recently, L. Block has shown that for any mapping $f$ of an interval, whether $ f$ has a periodic point whose period contains an odd factor greater than $ 1$ depends entirely on the periodic orbits of $f$ whose periods are powers of $2$. In this paper the author shows that Block's result is a special case of a more general phenomenon.
A property of complete minimal surfaces
Thomas
Hasanis;
Dimitri
Koutroufiotis
833-843
Abstract: If $M$ is a complete minimal surface in $ {R^n}$, we denote by $ W$ the set of points in $ {R^n}$ that do not lie on any tangent plane of $M$. By taking a point in $W$ as origin, the position vector of $ M$ determines a global unit normal vector field $e$ to $M$. We prove that if $e$ is a minimal section, then $M$ is a plane. In particular, the set of tangent planes of a nonflat complete minimal surface in ${R^3}$ covers all ${R^3}$. We also prove a similar result for a complete minimal surface $M$ in ${S^3}$, and deduce from it that if the spherical image of $M$ lies in a closed hemisphere, then $ M$ is a great $ {S^2}$.
Harish-Chandra modules with the unique embedding property
David H.
Collingwood
1-48
Abstract: Let $G$ be a connected semisimple real matrix group. In view of Casselman's subrepresentation theorem, every irreducible admissible representation of $ G$ may be realized as a submodule of some principal series representation. We give a classification of representations with a unique embedding into principal series, in the case of regular infinitesimal character. Our basic philosophy is to link the theory of asymptotic behavior of matrix coefficients with the theory of coherent continuation of characters. This is accomplished by using the "Jacquet functor" and the Kazhdan-Lusztig conjectures.
Zero distribution for pairs of holomorphic functions with applications to eigenvalue distribution
A. A.
Shkalikov
49-63
Abstract: Let $f$ and $g$ be holomorphic in an angle $\Lambda$. Theorem 1 shows that the zero-distributions of $f$ and $g$ are comparable if, near $\partial \Lambda$, $f$ and $g$ grow similarly. This result is applied to analyse the asymptotic behavior of eigenvalues of certain perturbed normal operators.
Sets of essentially unitary operators
Ridgley
Lange
65-75
Abstract: Let ${U_e}$ be the set of essentially unitary operators on a separable Hilbert space $H$; for $1 \leqslant p \leqslant \infty$, let $ {U_p}$ be the set of operators $T$ such that $I - T^{\ast}T$ lies in the Schatten $p$-ideal and the spectrum of $T$ does not fill the unit disc; and let $U_e^n$ be the set of operators in $ {U_e}$ of Fredholm index $ n$. The author proves that each $U_e^n$ is closed and path connected, that $ {U_p}$ is dense in $ {U_e}^0$ and $ {U_p}$ is path connected for each $p$, and that all these sets are invariant under Cayley transform. It is proved that the spectrum is continuous on ${U_\infty }$ but not on ${U_e}$, while the spectral radius is continuous on ${U_e}$. Sufficient conditions that an operator in ${U_e}$ have a nontrivial hyperinvariant subspace are given, and it is proved that the general hyperinvariant subspace problem can be reduced to that problem for perturbations of the bilateral shift. The product of commuting operators in ${U_p}$ is ${U_p}$, but this result is false in general. Quasisimilarity in ${U_e}$ is also studied; quasisimilar operators in $ {U_e}\backslash {U_\infty }$ are unitarily equivalent modulo the ideal of compacts, and this result also holds in ${U_\infty }$ if the spectrum is also preserved.
Weighted norm inequalities for strongly singular convolution operators
Sagun
Chanillo
77-107
Abstract: We derive sharp function estimates for convolution operators whose kernels are more singular than Calderon-Zygmund kernels. This leads to weighted norm inequalities. Weighted weak $(1,1)$ results are also proved. All the results obtained are in the context of ${A_p}$ weights.
An algebraic decomposition of the recursively enumerable degrees and the coincidence of several degree classes with the promptly simple degrees
Klaus
Ambos-Spies;
Carl G.
Jockusch;
Richard A.
Shore;
Robert I.
Soare
109-128
Abstract: We specify a definable decomposition of the upper semilattice of recursively enumerable (r.e.) degrees $\mathbf{R}$ as the disjoint union of an ideal $\mathbf{M}$ and a strong filter $\mathbf{NC}$. The ideal $ \mathbf{M}$ consists of $\mathbf{0}$ together with all degrees which are parts of r.e. minimal pairs, and thus the degrees in $\mathbf{NC}$ are called noncappable degrees. Furthermore, $ \mathbf{NC}$ coincides with five other apparently unrelated subclasses of $\mathbf{R: ENC}$, the effectively noncappable degrees; $\mathbf{PS}$, the degrees of promptly simple sets; $ \mathbf{LC}$, the r.e. degrees cuppable to $ {\mathbf{0}}^{\prime}$ by a low r.e. degree; $ {\mathbf{SP\bar H}}$, the degrees of non-$hh$-simple r.e. sets with the splitting property; and $\mathbf{G}$, the degrees in the orbit of an r.e. generic set under automorphisms of the lattice of r.e. sets.
Further results on convergence acceleration for continued fractions $K(a\sb{n}/1)$
Lisa
Jacobsen
129-146
Abstract: If $K(a_n^{\prime}/1)$ is a convergent continued fraction with known tails, it can be used to construct modified approximants $ f_n^{\ast}$ for other continued fractions $ K({a_n}/1)$ with unknown values. These modified approximants may converge faster to the value $f$ of $ K({a_n}/1)$ than the ordinary approximants ${f_n}$ do. In particular, if ${a_n} - a_n^{\prime} \to 0$ fast enough, we obtain $\vert f - f_n^{\ast}\vert/\vert f - {f_n}\vert \to 0$; i.e. convergence acceleration. the present paper also gives bounds for this ratio of the two truncation errors, in many cases.
Semidirect products and reduction in mechanics
Jerrold E.
Marsden;
Tudor
Raţiu;
Alan
Weinstein
147-177
Abstract: This paper shows how to reduce a Hamiltonian system on the cotangent bundle of a Lie group to a Hamiltonian system in the dual of the Lie algebra of a semidirect product. The procedure simplifies, unifies, and extends work of Greene, Guillemin, Holm, Holmes, Kupershmidt, Marsden, Morrison, Ratiu, Sternberg and others. The heavy top, compressible fluids, magnetohydrodynamics, elasticity, the Maxwell-Vlasov equations and multifluid plasmas are presented as examples. Starting with Lagrangian variables, our method explains in a direct way why semidirect products occur so frequently in examples. It also provides a framework for the systematic introduction of Clebsch, or canonical, variables.
A linear homogenization problem with time dependent coefficient
Maria Luisa
Mascarenhas
179-195
Abstract: We consider: the homogenization problem $\displaystyle \left\{ {\begin{array}{*{20}{c}} {(\partial u\varepsilon /\partia... ...0,} {{u_\varepsilon }(x,0) = \phi (x), } & {} \end{array} } \right.$ where $ \beta$ is a strictly positive bounded real function, periodic of period $ 1$, and $ {\beta_\varepsilon }(x) = \beta (x/\varepsilon )$; the equivalent integral equation $\displaystyle {u_\varepsilon }(x,t) + \int_0^t {{\beta_\varepsilon }(x)\,{u_\varepsilon }(x,s)\;ds = \phi (x)};$ and the homogenized equation $\displaystyle {u_0}(x,t) + \int_0^t {K(t - s)\,{u_0}(s)\,ds = \phi (x)},$ where $ K$ is a unique, well-defined function depending on $\beta$. We study this problem for a time dependent $ \beta$, and characterize a two-variable function $K(s,t)$ satisfying $\displaystyle {u_0}(x,t) + \int_0^t {K(s,t - s)\,{u_0}(x,s)\;ds = \phi (x)}$ and study its uniqueness.
Homologically homogeneous rings
K. A.
Brown;
C. R.
Hajarnavis
197-208
Abstract: In this paper we study the structure of a right Noetherian ring $ R$ of finite right global dimesion integral over a central subring $ C$ and satisfying the following condition: if $V,W$ are irreducible right $R$-modules with ${r_C}(V) = {r_C}(W)$ then $\operatorname{pr}\, \dim (V) = \operatorname{pr}\, \dim (W)$.
Additivity of measure implies additivity of category
Tomek
Bartoszyński
209-213
Abstract: In this paper it is proved that $ {2^\omega }$-additivity of category follows from $ {2^\omega }$-additivity of measure, and a combinatorial characterization of additivity of measure is found.
Nonlinear stability of asymptotic suction
Milan
Miklavčič
215-231
Abstract: The semigroup approach to the Navier-Stokes equation in halfspace is used to prove that the stability of the asymptotic suction velocity profile is determined by the eigenvalues of the classical Orr-Sommerfeld equation. The usual obstacle, namely, that the corresponding linear operator contains 0 in the spectrum is removed with the use of weighted spaces.
$M$-structure in the Banach algebra of operators on $C\sb{0}(\Omega )$
P. H.
Flinn;
R. R.
Smith
233-242
Abstract: The $M$-ideals in $B({C_0}(\Omega ))$, the space of continuous linear operators on $ {C_0}(\Omega )$, are determined where $\Omega$ is a locally compact Hausdorff countably paracompact space. A one-to-one correspondence between $M$-ideals in $ B({C_0}(\Omega ))$, open subsets of the Stone-Čech compactification of $\Omega$, and lower semicontinuous Hermitian projections in $B{({C_0}(\Omega ))^{\ast\ast}}$ is established.
James maps, Segal maps, and the Kahn-Priddy theorem
J.
Caruso;
F. R.
Cohen;
J. P.
May;
L. R.
Taylor
243-283
Abstract: The standard combinatorial approximation $ C({R^n},X)$ to ${\Omega ^n}{\Sigma ^n}X$ is a filtered space with easily understood filtration quotients ${D_q}({R^n},X)$. Stably, $ C({R^n},X)$ splits as the wedge of the $ {D_q}({R^n},X)$. We here analyze the multiplicative properties of the James maps which give rise to the splitting and of various related combinatorially derived maps between iterated loop spaces. The target of the total James map $\displaystyle j = ({j_q}):{\Omega ^n}{\Sigma ^n}X \to \mathop \times \limits_{q \geqslant 0} \;{\Omega ^{2nq}}{\Sigma ^{2nq}}{D_q}({R^n},X)$ is a ring space, and $j$ is an exponential $H$-map. There is a total Segal map $\displaystyle s = \mathop \times \limits_{q \geqslant 0} \;{s_{q}}:\mathop \tim... ... \mathop \times \limits_{q \geqslant 0} \;\Omega ^{3nq}\,\Sigma ^{3nq}{X^{[q]}}$ which is a ring map between ring spaces. There is a total partial power map $\displaystyle k = ({k_q}): {\Omega ^{n}}\,{\Sigma ^{n}}X \to \mathop \times \limits_{q \geqslant 0} \;{\Omega ^{n\,q}}\,{\Sigma ^{n\,q}}{X^{[q]}}$ which is an exponential $H$-map. There is a noncommutative binomial theorem for the computation of the smash power ${\Omega ^n}{\Sigma ^n}X \to {\Omega ^{nq}}{\Sigma ^{nq}}{X^{[q]}}$ in terms of the ${k_m}$ for $m \leqslant q$. The composite of $s$ and $j$ agrees with the composite of $k$ and the natural inclusion $\displaystyle \mathop \times \limits_{q \geqslant 0} \;{\Omega ^{n\,q}}\,{\Sigm... ...es \limits_{q \geqslant 0} \,{\Omega ^{3\,n\,q}}\,{\Sigma ^{3\,n\,q}}{X^{[q]}}.$ This analysis applies to essentially arbitrary spaces $ X$. When specialized to $ X = {S^0}$, it implies an unstable version of the Kahn-Priddy theorem. The exponential property of the James maps leads to an analysis of the behavior of loop addition with respect to the stable splitting of ${\Omega ^n}{\Sigma ^n}X$ when $X$ is connected, and there is an analogous analysis relating loop addition to the stable splitting of $ Q({X^ + })$.
James maps and $E\sb{n}$ ring spaces
F. R.
Cohen;
J. P.
May;
L. R.
Taylor
285-295
Abstract: We parametrize by operad actions the multiplicative analysis of the total James map given by Caruso and ourselves. The target of the total James map $\displaystyle j = \sum {{j_q}} :C({R^n},X) \to \prod\limits_{q \geqslant 0} {Q{D_q}({R^n},X)}$ is an ${E_n}$ ring space and $j$ is a $ {\mathcal{C}_n}$-map, where $ {\mathcal{C}_n}$ is the little $n$-cubes operad. This implies that $j$ has an $n$-fold delooping with domain ${\Sigma^n}X$. It also implies an algorithm for the calculation of $ {j_{\ast}}$ and thus of each $ {({j_q})_{\ast}}$ on $ \bmod\, p$ homology. When $n = \infty$ and $p = 2$, this algorithm is the essential starting point for Kuhn's proof of the Whitehead conjecture.
On some subalgebras of a von Neumann algebra crossed product
Baruch
Solel
297-308
Abstract: We study conditions for a nonselfadjoint subalgebra of a von Neumann crossed product $ \mathcal{L}$ to be an algebra of analytic operators with respect to a flow on $\mathcal{L}$. We restrict ourselves to the case where $\mathcal{L}$ is constructed from a finite von Neumann algebra $M$ with a trace preserving $ ^{\ast}$-automorphism $ \alpha$ that acts ergodically on the center of $M$.
Extensions of algebraic systems
Awad A.
Iskander
309-327
Abstract: There are several generalizations to universal algebras of the notion "The group $ \mathfrak{A}$ is an extension of the group $ \mathfrak{B}$ by the group $ \mathfrak{C}$". In this paper we study three such generalizations and the corresponding products of classes of algebraic systems. Various results are presented. One such theorem characterizes the weakly congruence regular varieties admitting extensions of a particular sort. Another result gives, under a weak congruence permutability condition, an equational basis for the variety obtained by applying one such product to two other varieties.
Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator
Eric
Sawyer
329-337
Abstract: Characterizations are obtained for those pairs of weight functions $w,\upsilon$ for which the Hardy operator $ Tf(x) = \int_0^x {f(s)\;ds}$ is bounded from the Lorentz space $ {L^{r,s}}((0,\infty ),\upsilon \,dx)$ to $ {L^{p,q}}((0,\infty ),w\,dx),0 < p,q,r,s \leqslant \infty$. The modified Hardy operators ${T_\eta }f(x) = {x^{ - \eta }}Tf(x)$ for $ \eta$ real are also treated.
A two weight weak type inequality for fractional integrals
Eric
Sawyer
339-345
Abstract: For $1 < p \leqslant q < \infty ,0 < \alpha < n$ and $w(x),\upsilon (x)$ nonnegative weight functions on ${R^n}$ we show that the weak type inequality $\displaystyle \int_{\{ {T_\alpha }f > \lambda \} }\,w(x)\;dx \leqslant A{\lambda ^{ - q}}{\left( \int \vert f(x){\vert^p}\;\upsilon (x)\;dx \right)^{q/p}}$ holds for all $f \geqslant 0$ if and only if $\displaystyle \int_Q\,[{T_\alpha }({\chi_Q}w)\,(x)]^{p'}\upsilon (x)^{1 - p'}\,dx \leqslant B\left( \int_Qw \right)^{p^{\prime}/q^{\prime}} < \infty$ for all cubes $ Q$ in ${R^n}$. Here $ {T_\alpha }$ denotes the fractional integral of order $\alpha ,{T_\alpha }f(x) = \int \vert x - y{\vert^{\alpha - n}}f(y)\,dy$. More generally we can replace ${T_\alpha }$ by any suitable convolution operator with radial kernel decreasing in $\vert x\vert$.
An improved stability result for resonances
Mark S.
Ashbaugh;
Carl
Sundberg
347-360
Abstract: We prove stability of shape resonances for the sequence of Schrödinger equations $( - {d^2}/d{x^2} + U(x) + {W_n}(x))\psi (x) = E\psi (x),0 \leqslant x\, < \infty$, in the limit $n \to \infty$ where the barrier potentials ${W_n}(x)$ are integrable, nonnegative, supported in the interval $[1,a]\;(1 < a < \infty )$, and approach infinity pointwise a.e. for $x \in [1,a]$ as $n \to \infty$. In the course of our investigation we prove that for suitable complex initial conditions the solution to the Riccati equation $ S^{\prime}(x) = 1 - ({W_n}(x) - E){[S(x)]^2}$ goes to 0 as $n \to \infty$ uniformly on compact subsets of $ [1,a]$. Our approach is via ordinary differential equations using outgoing wave boundary conditions to define resonances. Our stability result extends a similar result of Ashbaugh and Harrell, who use an argument based on asymptotics and the implicit function theorem to study the above problem with $ \lambda V(x)$ replacing $ {W_n}(x)$. Our approach is to use the Riccati equation analysis mentioned above and an application of Hurwitz's Theorem from complex variable theory.
The null space and the range of a convolution operator in a fading memory space
Olof J.
Staffans
361-388
Abstract: We study the convolution equation $(\ast)$ $\displaystyle \mu \; \ast \;x^{\prime}(t) + v\; \ast \;x(t) = f(t) \quad ( - \infty < t\, < \infty )$ , as well as a perturbed version of $(\ast)$, namely $ (\ast\ast)$ $\displaystyle \mu \;\ast\;x^{\prime}(t) + v\,\ast\;x(t) = \,F(x)\,(t)\quad ( - \infty < t < \infty ).$ Here $x$ is a $ {{\mathbf{R}}^n}$-valued function on $( - \infty ,\infty ),x^{\prime}(t) = dx(t)/dt$, and $\mu$ and $\nu$ are matrix-valued measures. If $\mu$ and $\nu$ are supported on $ [0,\infty )$, with $ \mu$ atomic at zero, then $ (\ast)$ can be regarded as a linear, autonomous, neutral functional differential equation with infinite delay. However, most of the time we do not consider the ordinary Cauchy problem for the neutral equation, i.e. we do not suppose that $ \mu$ and $\nu$ are supported on $[0,\infty )$, prescribe an initial condition of the type $ x(t) = \xi (t)\,(t \leqslant 0)$, and require $(\ast)$ and $ (\ast\ast)$ to hold only for $t \geqslant 0$. Instead we permit $ (\ast)$ and $(\ast\ast)$ to be of "Fredholm" type, i.e. $ \mu$ and $\nu$ need not vanish on $( - \infty ,0)$, we restrict the growth rate of $ x$ and $f$ at plus and minus infinity, and we look at the problem of the existence and uniqueness of solutions of $(\ast)$ and $ (\ast\ast)$ on the whole real line, satisfying conditions like $\vert x(t)\vert \leqslant C\eta (t)\;( - \infty < t < \infty )$, where $C$ is a constant, depending on $x$, and $\eta$ is a predefined function. Some authors use the word "admissible" when discussing problems of this type. In the case when the homogeneous version of $ (\ast)$ has nonzero solutions, we decompose the solutions into components with different exponential growth rates, and give a priori bounds on the growth rates of the solutions. As an application of the basic theory, we look at the Cauchy problem for a neutral functional differential equation, and prove the existence of stable and unstable manifolds.
Approximate subdifferentials and applications. I. The finite-dimensional theory
A. D.
Ioffe
389-416
Abstract: We introduce and study a new class of subdifferentials associated with arbitrary functions. Among the questions considered are: connection with other derivative-like objects (e.g. derivatives, convex subdifferentials, generalized gradients of Clarke and derivate containers of Warga), calculus of approximate subdifferentials and applications to analysis of set-valued maps and to optimization. It turns out that approximate subdifferentials are minimal (as sets) among other conceivable subdifferentials satisfying some natural requirements. This shows that certain results involving approximate subdifferentials are the best possible and, at the same time, marks certain limitations of nonsmooth analysis. Another important property of approximate subdifferentials is that, being essentially nonconvex, they admit a rich calculus that covers the calculus of convex subdifferentials and leads to more precise and sometimes new results for generalized gradients of Clarke.
Generalized Hua-operators and parabolic subgroups. The cases of ${\rm SL}(n,\,{\bf C})$ and ${\rm SL}(n,\,{\bf R})$
Kenneth D.
Johnson
417-429
Abstract: Suppose $G = {\text{SL}}(n,{\mathbf{C}})$ or $ {\text{SL}}(n,{\mathbf{R}})$ and $K$ is a maximal compact subgroup of $G$. If $P$ is any parabolic subgroup of $G$, we determine a system of differential equations on $G/K$ with the property that any function on $ G/K$ satisfies these differential equations if and only if it is the Poisson integral of a hyperfunction on $G/P$.
Linearized stability of extreme shock profiles in systems of conservation laws with viscosity
Robert L.
Pego
431-461
Abstract: For a genuinely nonlinear hyperbolic system of conservation laws with added artificial viscosity, ${u_t} + f{(u)_x} = \varepsilon {u_{xx}}$, we prove that traveling wave profiles for small amplitude extreme shocks (the slowest and fastest) are linearly stable to perturbations in initial data chosen from certain spaces with weighted norm; i.e., we show that the spectrum of the linearized equation lies strictly in the left-half plane, except for a simple eigenvalue at the origin (due to phase translations of the profile). The weight ${e^{cx}}$ is used in components transverse to the profile, where, for an extreme shock, the linearized equation is dominated by unidirectional convection.
Complex and integral laminated lattices
J. H.
Conway;
N. J. A.
Sloane
463-490
Abstract: In an earlier paper we studied real laminated lattices (or ${\mathbf{Z}}$-modules) $ {\Lambda_n}$, where ${\Lambda_1}$ is the lattice of even integers, and ${\Lambda_n}$ is obtained by stacking layers of a suitable $(n - 1)$-dimensional lattice ${\Lambda_{n - 1}}$ as densely as possible, keeping the same minimal norm. The present paper considers two generalizations: (a) complex and quaternionic lattices, obtained by replacing $ {\mathbf{Z}}$-module by $ J$-module, where $ J$ may be the Eisenstein, Gaussian or Hurwitzian integers, etc., and (b) integral laminated lattices, in which ${\Lambda_n}$ is required to be an integral lattice with the prescribed minimal norm. This enables us to give a partial answer to a question of J. G. Thompson on integral lattices, and to explain some of the computer-generated results of Plesken and Pohst on this problem. Also a number of familiar lattices now arise in a canonical way. For example the Coxeter-Todd lattice is the $6$-dimensional integral laminated lattice over ${\mathbf{Z}}[ \omega ]$ of minimal norm $ 2$. The paper includes tables of the best real integral lattices in up to $ 24$ dimensions.
A topological group having no homeomorphisms other than translations
Jan
van Mill
491-498
Abstract: We give an example of a (separable metric) connected and locally connected topological group, the only autohomeomorphisms of which are group translations.
Secant functions, the Reiss relation and its converse
Mark L.
Green
499-507
Abstract: Generalizing a classical Euclidean theorem for the circle, certain meromorphic functions on $ {{\mathbf{P}}_1}$ relating to the geometry of algebraic plane curves are shown to be constant. Differentiated twice, this gives a new proof of the Reiss relation and its converse. The relation of these functions to Abel's Theorem is discussed, and a generalization of secant functions to space curves is given, for which the Chow form arises in a natural way.
Multi-invariant sets on tori
Daniel
Berend
509-532
Abstract: Given a compact metric group $G$, we are interested in those semigroups $ \Sigma$ of continuous endomorphisms of $G$, possessing the following property: The only infinite, closed, $\Sigma$-invariant subset of $G$ is $G$ itself. Generalizing a one-dimensional result of Furstenberg, we give here a full characterization--for the case of finitedimensional tori--of those commutative semigroups with the aforementioned property.
Degenerate elliptic operators as regularizers
R. N.
Pederson
533-553
Abstract: The spaces ${\mathcal{K}_{mk}}$, introduced in the Nehari Volume of Journal d'Analyse Mathématique, for nonnegative integer values of $m$ and arbitrary real values of $k$ are extended to negative values of $m$. The extension is consistent with the equivalence $ \parallel {\zeta ^j}u{\parallel_{m,k}}\sim\parallel u{\parallel_{m,k - j}}$, the inequality $ \parallel {D^\alpha }u{\parallel_{m,k}} \leqslant {\text{const}}\parallel u{\parallel_{m + \vert\alpha \vert,k + \vert\alpha \vert}}$, and the generalized Cauchy-Schwarz inequality $\vert\langle {u,v} \rangle \vert \leqslant \parallel u\,{\parallel_{m,k}}\parallel v\parallel_{ - m, - k}$. (Here $\langle u, \upsilon \rangle$ is the $ {L_2}$ scalar product.) There exists a second order degenerate elliptic operator which maps ${\mathcal{K}_{m,k}}\,1 - 1$ onto ${\mathcal{K}_{m - 2,k}}$. These facts are used to simplify proof of regularity theorems for elliptic and hyperbolic problems and to give new results concerning growth rates at the boundary for the coefficients of the operator and the forcing function. (See Notices Amer. Math. Soc. 28 (1981), 226.)
Real vs. complex rational Chebyshev approximation on an interval
Lloyd N.
Trefethen;
Martin H.
Gutknecht
555-561
Abstract: If $f \in C[ - 1,1]$ is real-valued, let ${E^{r}(f)}$ and $ {E^{c}(f)}$ be the errors in best approximation to $f$ in the supremum norm by rational functions of type $(m,n)$ with real and complex coefficients, respectively. It has recently been observed that ${E^c}(f) < {E^r}(f)$ can occur for any $n \geqslant 1$, but for no $n \geqslant 1$ is it known whether ${\gamma_{mn}} = \inf_f\,{E^c}(f)/{E^{r}(f)}$ is zero or strictly positive. Here we show that both are possible: $ {\gamma_{01}} > 0$, but ${\gamma_{mn}} = 0$ for $n \geqslant m + 3$. Related results are obtained for approximation on regions in the plane.
Szeg\H o limit theorems for the harmonic oscillator
A. J. E. M.
Janssen;
Steven
Zelditch
563-587
Abstract: Let $H = - \frac{1}{2}{d^2}/d{x^2} + \frac{1}{2}{x^2}$ be the harmonic oscillator Hamiltonian on $ {L^2}( {\mathbf{R}})$, and let $A$ be a selfadjoint $DO$ of order $O$ in the Beals-Fefferman class with weights $\varphi = 1,\Phi (x,\xi ) = {(1 + \vert\xi \,{\vert^2} + \vert x\,{\vert^2})^{1/2}}$. Form the measure $\mu(f) = {\lim_{\lambda \to \infty }}(1/{\text{rank}}\;{\pi_\lambda })\,{\text{tr}}\,f({\pi_\lambda }\,A{\pi_\lambda })$ where ${\pi_\lambda }\,A{\pi_\lambda }$ is the compression of $A$ onto the span of the Hermite functions with eigenvalue less than or equal to $\lambda$. Then one has the following Szegö limit theorem: $\displaystyle \mu (f) = \mathop {\lim }\limits_{T \to \infty } \;\frac{1} {{2\,... ...qslant T} {f(a(x,\xi ))\;dx} \;d\xi \qquad {\text{for}} f \in C({\mathbf{R}}).$ For the special case where $f(x) = x$, this will be proved for a considerably wider class of operators by employing the Weyl correspondence. Moreover, by using estimates on Wigner functions of Hermite functions we are able to prove the full Szegö theorem for a fairly general class of multiplication operators.
Lower semicontinuity, existence and regularity theorems for elliptic variational integrals of multiple valued functions
Pertti
Mattila
589-610
Abstract: Let $A$ be an open set in ${{\mathbf{R}}^m}$ with compact smooth boundary, and let $ {\mathbf{Q}}$ be the space of unordered $Q$ tuples of points of ${{\mathbf{R}}^n}$. F. J. Almgren, Jr. has developed a theory for functions $f:A \to {\mathbf{Q}}$ and used them to prove regularity theorems for area minimizing integral currents. In particular, he has defined in a natural way the space $ {\mathcal{Y}_2}(A,{\mathbf{Q}})$ of functions $f:A \to {\mathbf{Q}}$ with square summable distributional partial derivatives and the Dirichlet integral $\operatorname{Dir}(f;A)$ of such functions. In this paper we study more general constant coefficient quadratic integrals $ {\mathbf{G}}(f;A)$ which are $Q$ elliptic in the sense that there is $c > 0$ such that ${\mathbf{G}}(f;A) \geqslant c\,\operatorname{Dir}(f;A)$ for $f \in {\mathcal{Y}_2}(A;{\mathbf{Q}})$ with zero boundary values. We prove a lower semicontinuity theorem which leads to the existence of a ${\mathbf{G}}$ minimizing function with given reasonable boundary values. In the case $m = 2$ we also show that such a function is Hölder continuous and regular on an open dense set. In the case $ m \geqslant 3$ the regularity problem remains open.
On certain sums of Fourier-Stieltjes coefficients
J. B.
Twomey
611-621
Abstract: We obtain estimates for certain sums of Fourier-Stieltjes (and hence also Fourier) coefficients of continuous functions $ f$ of bounded variation in terms of the modulus of continuity of $f$. As a consequence of one of our results we obtain an improvement on a theorem of Zygmund on the absolute convergence of Fourier series of functions of bounded variation. We also consider absolutely continuous functions and show by examples that a number of the results we obtain are "best possible".
All varieties of central completely simple semigroups
Mario
Petrich;
Norman R.
Reilly
623-636
Abstract: Completely simple semigroups may be considered as a variety of algebras with the binary operation of multiplication and the unary operation of inversion. A completely simple semigroup is central if the product of any two idempotents lies in the centre of the containing maximal subgroup. Central completely simple semigroups form a subvariety $\mathcal{C}$ of the variety of all completely simple semigroups. We find an isomorphic copy of $ \mathcal{L}(\mathcal{C})$ as a subdirect product of the lattices $ \mathcal{L}(\mathcal{R}\,\mathcal{B})$, $ \mathcal{L}(\mathcal{A}\,\mathcal{G})$, and $\mathcal{L}(\mathcal{G})$ of all varieties of rectangular bands, abelian groups, and groups, respectively. We consider also several homomorphisms and study congruences they induce.
Determination of bounds for the solutions to those binary Diophantine equations that satisfy the hypotheses of Runge's theorem
David Lee
Hilliker;
E. G.
Straus
637-657
Abstract: In 1887 Runge [13] proved that a binary Diophantine equation $F(x,y) = 0$, with $F$ irreducible, in a class including those in which the leading form of $F$ is not a constant multiple of a power of an irreducible polynomial, has only a finite number of solutions. It follows from Runge's method of proof that there exists a computable upper bound for the absolute value of each of the integer solutions $ x$ and $y$. Runge did not give such a computation. Here we first deduce Runge's Theorem from a more general theorem on Puiseux series that may be of interest in its own right. Second, we extend the Puiseux series theorem and deduce from the generalized version a generalized form of Runge's Theorem in which the solutions $x$ and $y$ of the polynomial equation $F(x,y) = 0$ are integers, satisfying certain conditions, of an arbitrary algebraic number field. Third, we compute bounds for the solutions $(x,y) \in {{\mathbf{Z}}^2}$ in terms of the height of $F$ and the degrees in $x$ and $y$ of $F$.
The rank of a Hardy field
Maxwell
Rosenlicht
659-671
Abstract: A Hardy field is a field of germs of real-valued functions on positive half-lines that is closed under differentiation. Its rank is the rank of the associated ordered abelian group, the value group of the canonical valuation of the field. The properties of this rank are worked out, its relevance to asymptotic expansions indicated, examples provided, and applications given to the order of growth of solutions of ordinary differential equations.
Invariant theory and the lambda algebra
William M.
Singer
673-693
Abstract: Let $A$ be the Steenrod algebra over the field ${F_2}$. In this paper we construct for any left $ A$-module $M$ a chain complex whose homology groups are isomorphic to the groups $ \operatorname{Tor}_s^A({F_2},M)$. This chain complex in homological degree $ s$ is built from a ring of invariants associated with the action of the linear group $ G{L_s}({F_2})$ on a certain algebra of Laurent series. Thus, the homology of the Steenrod algebra (and so the Adams spectral sequence for spheres) is seen to have a close relationship to invariant theory. A key observation in our work is that the Adem relations can be described in terms of the invariant theory of $ G{L_2}({F_2})$. Our chain complex is not new: it turns out to be isomorphic to the one constructed by Kan and his coworkers from the dual of the lambda algebra. Thus, one effect of our work is to give an invariant-theoretic interpretation of the lambda algebra. As a consequence we find that the dual of lambda supports an action of the Steenrod algebra that commutes with the differential. The differential itself appears as a kind of "residue map". We are also able to describe the coalgebra structure of the dual of lambda using our invariant-theoretic language.
Strong Fatou-$1$-points of Blaschke products
C. L.
Belna;
F. W.
Carroll;
G.
Piranian
695-702
Abstract: This paper shows that to every countable set $M$ on the unit circle there corresponds a Blaschke product whose set of strong Fatou-$1$-points contains $M$. It also shows that some Blaschke products have an uncountable set of strong Fatou-$ 1$-points.
Forcing positive partition relations
Stevo
Todorčević
703-720
Abstract: We show how to force two strong positive partition relations on ${\omega_1}$ and use them in considering several well-known open problems.
Inverses and parametrices for right-invariant pseudodifferential operators on two-step nilpotent Lie groups
Kenneth G.
Miller
721-736
Abstract: Let $P$ be a right-invariant pseudodifferential operator with principal part $ {P_0}$ on a simply connected two-step nilpotent Lie group $G$ of type $H$. It will be shown that if $\pi (P_0)$ is injective in ${\mathcal{S}_\pi }$ for every nontrivial irreducible unitary representation $\pi$ of $G$, then $P$ has a pseudodifferential left parametrix. For such groups this generalizes the Rockland-Helffer-Nourrigat criterion for the hypoellipticity of a homogeneous right-invariant partial differential operator on $G$. If, in addition, $\pi (P)$ is injective in ${\mathcal{S}_\pi }$ for every irreducible unitary representation of $G$, it will be shown that $P$ has a pseudodifferential left inverse. The constructions of the inverse and parametrix make use of the Kirillov theory, their symbols being obtained on the orbits individually and then pieced together.
Reye congruences
François R.
Cossec
737-751
Abstract: This paper studies the congruences of lines which are included in two distinct quadrics of a given generic three-dimensional projective space of quadrics in ${{\mathbf{P}}^3}$.
Representations of generic algebras and finite groups of Lie type
R. B.
Howlett;
G. I.
Lehrer
753-779
Abstract: The complex representation theory of a finite Lie group $G$ is related to that of certain "generic algebras". As a consequence, formulae are derived ("the Comparison Theorem"), relating multiplicities in $ G$ to multiplicities in the Weyl group $W$ of $G$. Applications include an explicit description of the dual (see below) of an arbitrary irreducible complex representation of $G$.
Systems of conservation laws with invariant submanifolds
Blake
Temple
781-795
Abstract: Systems of conservation laws with coinciding shock and rarefaction curves arise in the study of oil reservoir simulation, multicomponent chromatography, as well as in the study of nonlinear motion in elastic strings. Here we characterize this phenomenon by deriving necessary and sufficient conditions on the geometry of a wave curve in order that the shock wave curve coincide with its associated rarefaction wave curve for a system of conservation laws. This coincidence is the one dimensional case of a submanifold of the state variables being invariant for the system of equations, and the necessary and sufficient conditions are derived for invariant submanifolds of arbitrary dimension. In the case of $2 \times 2$ systems we derive explicit formulas for the class of flux functions that give rise to the coupled nonlinear conservation laws for which the shock and rarefaction wave curves coincide.
Uniqueness of torsion free connection on some invariant structures on Lie groups
Michel Nguiffo
Boyom;
Georges
Giraud
797-808
Abstract: Let $\mathcal{G}$ be a connected Lie group with Lie algebra $ \mathfrak{g}$. Let $ \operatorname{Int}(\mathfrak{g})$ be the group of inner automorphisms of $\mathfrak{g}$. The group $ \mathcal{G}$ is naturally equipped with $ \operatorname{Int}(\mathfrak{g})$-reductions of the bundle of linear frames on $\mathcal{G}$. We investigate for what kind of Lie group the 0-connection of E. Cartan is the unique torsion free connection adapted to any of those $ \operatorname{Int}(\mathfrak{g})$-reductions.
Length dependence of solutions of FitzHugh-Nagumo equations
Clyde
Collins
809-832
Abstract: We investigate the behavior of the solutions of the problem \begin{displaymath}\begin{array}{*{20}{c}} {{u_t} = {u_{xx}} - \alpha u - \upsil... ... h(t),} & {{u_x}(L,t) = {\upsilon _x}(L,t) = 0} \end{array} \end{displaymath} where $t \geqslant 0$ and $0 < x < L \leqslant \infty$. Solutions of the above equations are considered a qualitative model of conduction of nerve axon impulses. Using explicit constructions and semigroup methods, we obtain decay results on the norms of differences between the solution for $L$ infinite and the solutions when $ L$ is large but finite. We conclude that nerve impulses for long finite nerves become uniformly close to those of the semi-infinite nerves away from the right endpoint of the finite nerve.
Diffusion dependence of the FitzHugh-Nagumo equations
Clyde
Collins
833-839
Abstract: We investigate the behavior of the solutions of \begin{displaymath}\begin{array}{*{20}{c}} {{u_t} = {u_{x\,x}} - \alpha \,u - v ... ...} = \eta \,{v_{x\,x}} + \sigma \,u - \gamma v,} \end{array} \end{displaymath} as $\eta$ tends to zero from above.
Correction to: ``The stable geometric dimension of vector bundles over real projective spaces'' [Trans. Amer. Math. Soc. {\bf 268} (1981), no. 1, 39--61; MR0628445 (83c:55006)]
Donald M.
Davis;
Sam
Gitler;
Mark
Mahowald
841-843
Abstract: The theory of $ bo$-resolutions as utilized in The stable geometric dimension of vector bundles over real projective spaces did not give adequate care to the $ K{{\mathbf{Z}}_2}$'s occurring at each stage of the resolution. This restricts somewhat the set of integers $e$ for which we can prove that the geometric dimension of vector bundles of order ${2^e}$ on large real projective spaces is precisely $2\,e + \delta $.
A convergence theory for saddle functions
Hédy
Attouch;
Roger J.-B.
Wets
1-41
Abstract: We develop a convergence theory called epi/hypo-convergence, for bivariate functions that essentially implies the convergence of their saddle points. We study the properties of this limiting process in particular. We characterize the limit functions associated to any collection of bivariate functions and obtain a compactness theorem for the space of saddle functions. Even when restricted to the univariate case, the results generalize those known for epi-convergence. In particular, we show that the analysis of the convergence process via Yosida approximates must not be restricted to the convex case.
Some research problems about algebraic differential equations
Lee A.
Rubel
43-52
Abstract: Twenty-four new research problems are posed, and their background and partial solutions are sketched. Many of these problems are in the (somewhat unexpected) area of interaction between algebraic differential equations, topology, and mathematical logic.
Criteria for solvability of left invariant operators on nilpotent Lie groups
Lawrence
Corwin
53-72
Abstract: We define a special nilpotent Lie group $N$ to be one which has a $1$-dimensional center, dilations, square-integrable representations, and a maximal subordinate algebra common to almost all functionals on the Lie algebra $\mathfrak{N}$. Every nilpotent Lie group with dilations imbeds in such a special group so that the dilations extend. Let $L$ be a homogeneous left invariant differential operator on $N$. We give a representation-theoretic condition on $L$ which $L$ must satisfy if it has a tempered fundamental solution and which implies global solvability of $ L$. (The sufficiency is a corollary of a more general theorem, valid on all nilpotent $N$.) For the Heisenberg group, the condition is equivalent to having a tempered fundamental solution.
Controlled boundary and $h$-cobordism theorems
T. A.
Chapman
73-95
Abstract: In this paper two theorems are established which are consequences of some earlier approximation results of the author. The first theorem is a controlled boundary theorem for finite-dimensional manifolds. By this we mean an ordinary boundary theorem plus small $ \varepsilon$-control in a given parameter space. The second theorem is a controlled $h$-cobordism theorem for finite-dimensional manifolds with small $ \varepsilon$-control in a given parameter space. These results generalize the End Theorem and the Thin $h$-Corbordism of Quinn.
On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs
Curtis
Greene;
Thomas
Zaslavsky
97-126
Abstract: The doubly indexed Whitney numbers of a finite, ranked partially ordered set $ L$ are (the first kind) ${w_{ij}} = \sum {\{ \mu ({x^i},{x^j}):{x^i},{x^j} \in L}$ with ranks $i,j\}$ and (the second kind) ${W_{ij}} =$ the number of $({x^i},{x^j})$ with ${x^i} \leqslant {x^j}$. When $L$ has a 0 element, the ordinary (simply indexed) Whitney numbers are ${w_j} = {w_{0j}}$ and ${W_j} = {W_{0j}} = {W_{jj}}$ . Building on work of Stanley and Zaslavsky we show how to interpret the magnitudes of Whitney numbers of geometric lattices and semilattices arising in geometry and graph theory. For example: The number of regions, or of $ k$-dimensional faces for any $k$, of an arrangement of hyperplanes in real projective or affine space, that do not meet an arbitrary hyperplane in general position. The number of vertices of a zonotope $P$ inside the visible boundary as seen from a distant point on a generating line of $P$. The number of non-Radon partitions of a Euclidean point set not due to a separating hyperplane through a fixed point. The number of acyclic orientations of a graph (Stanley's theorem, with a new, geometrical proof); the number with a fixed unique source; the number whose set of increasing arcs (in a fixed ordering of the vertices) has exactly $q$ sources (generalizing Rényi's enumeration of permutations with $q$ "outstanding" elements). The number of totally cyclic orientations of a plane graph in which there is no clockwise directed cycle. The number of acyclic orientations of a signed graph satisfying conditions analogous to an unsigned graph's having a unique source.
The word problem for lattice-ordered groups
A. M. W.
Glass;
Yuri
Gurevich
127-138
Abstract: Theorem. There is a finitely generated one relator lattice-ordered group with insoluble (group) word problem.
Highly connected embeddings in codimension two
Susan
Szczepanski
139-159
Abstract: In this paper we study semilocal knots over $f$ into $\xi$, that is, embeddings of a manifold $ N$ into $E(\xi)$, the total space of a $ 2$-disk bundle over a manifold $M$, such that the restriction of the bundle projection $p:E(\xi) \to M$ to the submanifold $ N$ is homotopic to a normal map of degree one, $f:N \to N$. We develop a new homology surgery theory which does not require homology equivalences on boundaries and, in terms of these obstruction groups, we obtain a classification (up to cobordism) of semilocal knots over $f$ into $\xi$. In the simply connected case, the following geometric consequence follows from our classification. Every semilocal knot of a simply connected manifold $ M\char93 K$ in a bundle over $M$ is cobordant to the connected sum of the zero section of this bundle with a semilocal knot of the highly connected manifold $K$ into the trivial bundle over a sphere.
Harmonic maps and classical surface theory in Minkowski $3$-space
Tilla Klotz
Milnor
161-185
Abstract: Harmonic maps from a surface $S$ with nondegenerate prescribed and induced metrics are characterized, showing that holomorphic quadratic differentials play the same role for harmonic maps from a surface with indefinite prescribed metric as they do in the Riemannian case. Moreover, holomorphic quadratic differentials are shown to arise as naturally on surfaces of constant $ H$ or $K$ in ${M^3}$ as on their counterparts in $ {E^3}$. The connection between the sine-Gordon, $\sinh$-Gordon and $\cosh$-Gordon equations and harmonic maps is explained. Various local and global results are established for surfaces in ${M^3}$ with constant $H$, or constant $K \ne 0$. In particular, the Gauss map of a spacelike or timelike surface in ${M^3}$ is shown to be harmonic if and only if $ H$ is constant. Also, $ K$ is shown to assume values arbitrarily close to ${H^2}$ on any entire, spacelike surface in $ {M^3}$ with constant $ H$, except on a hyperbolic cylinder.
On the oscillation of differential transforms of eigenfunction expansions
C. L.
Prather;
J. K.
Shaw
187-206
Abstract: This paper continues the study of Pólya and Wiener, Hille and Szegö into the connections between the oscillation of derivatives of a real function and its analytic character. In the present paper, a Sturm-Liouville operator $L$ is applied successively to an infinitely differentiable function which admits a certain eigenfunction expansion. The eigenfunction expansion is assumed to be "conservative", in the sense of Hille. Several theorems are given which link the frequency of oscillation of $ ({L^k}f)(x)$ to the size of the coefficients of $f(x)$, and thus to its analytic character.
Convolution theorems with weights
R. A.
Kerman
207-219
Abstract: Analogues of Young's Inequality and the Convolution Theorem are shown to hold when the ${L_p}$ and $L(p,q)$ spaces have underlying measure defined in terms of power weights.
Toeplitz operators on bounded symmetric domains
Harald
Upmeier
221-237
Abstract: In this paper Jordan algebraic methods are applied to study Toeplitz operators on the Hardy space ${H^2}(S)$ associated with the Shilov boundary $ S$ of a bounded symmetric domain $D$ in $ {{\mathbf{C}}^n}$ of arbitrary rank. The Jordan triple system $Z \approx {{\mathbf{C}}^n}$ associated with $D$ is used to determine the relationship between Toeplitz operators and differential operators. Further, it is shown that each Jordan triple idempotent $ e \in Z$ induces an irreducible representation ("$e$-symbol") of the $ {C^{\ast} }$-algebra $\mathcal{T}$ generated by all Toeplitz operators $ {T_f}$ with continuous symbol function $f$.
A strong type $(2,\,2)$ estimate for a maximal operator associated to the Schr\"odinger equation
Carlos E.
Kenig;
Alberto
Ruiz
239-246
Abstract: Let $ {T^{\ast} }f(x) = \sup_{t > 0}\vert{T_t}f(x)\vert$, where $({T_t}f)\hat{\empty}(\xi) = {e^{it\vert\xi \vert^2}}\hat f(\xi)/\vert\xi {\vert^{1/4}}$. We show that, given any finite interval $I$, $\int_I {\vert{T^{\ast} }f{\vert^2}\;dx \leqslant {C_I}\int_{\mathbf{R}} {\vert f(x){\vert^2}\;dx} }$, and that the above inequality is false with $2$ replaced by any $p < 2$. This maximal operator is related to solutions of the Schrödinger equation.
Analyticity on rotation invariant families of curves
Josip
Globevnik
247-254
Abstract: Let $\mathfrak{G}$ be a rotation invariant family of smooth Jordan curves contained in $\Delta$, the open unit disc in ${\mathbf{C}}$. For each $\Gamma \in \mathfrak{G}$ let ${D_\Gamma }$ be the simply connected domain bounded by $\Gamma$. We present various conditions which imply that if $f$ is a continuous function on $\Delta$ such that for every $\Gamma \in \mathfrak{G}$ the function $f\vert\Gamma $ has a continuous extension to $ \overline {{D_\Gamma }}$ which is analytic in $ {D_\Gamma }$, then $ f$ is analytic in $ \Delta$.
Some prime elements in the lattice of interpretability types
Pavel
Pudlák
255-275
Abstract: A general theorem is proved which implies that the types of PA (Peano Arithmetic), ZF (Zermelo-Fraenkel Set Theory) and GB (Gödel-Bernays Set Theory) are prime in the lattice of interpretability types.
The $S\sp{1}$-transfer in surgery theory
H. J.
Munkholm;
E. K.
Pedersen
277-302
Abstract: Let ${S^1} \to X \to Y$ be an ${S^1}$-bundle of Poincaré spaces. If $ f:N \to Y$ is a surgery problem then so is the pullback $\hat f:M \to X$. We define algebraically a homomorphism ${\varphi ^!}:{L_n}({\mathbf{Z}}{\pi _1}(Y)) \to {L_{n + 1}}({\mathbf{Z}}{\pi _1}(X))$ and prove that it maps the surgery obstruction for $ f$ to the one for $ \hat f$.
Qualitative differentiation
Michael J.
Evans;
Lee
Larson
303-320
Abstract: Qualitative derivates and derivatives, as well as qualitative symmetric derivates and derivatives, are studied in the paper. Analogues of several results known for ordinary derivates and derivatives are obtained in the qualitative setting.
Maximal abelian subalgebras of von Neumann algebras and representations of equivalence relations
Colin E.
Sutherland
321-337
Abstract: We associate to each pair $ (\mathcal{M},\mathcal{A})$, (with $ \mathcal{M}$ a von Neumann algebra, and $ \mathcal{A}$ a maximal abelian subalgebra) a representation $ \alpha$ of the Takesaki equivalence relation $ \mathcal{R}\,(\mathcal{M},\mathcal{A})$ of $ (\mathcal{M},\mathcal{A})$ as automorphisms of a ${{\text{I}}_\infty }$ factor. Conversely each such representation $\alpha$ of $ \mathcal{R}$ on $ (X,\mu)$ as automorphisms of $ \mathcal{B}\,(\mathcal{H})$ determines a von Neumann algebra-abelian subalgebra pair $ S^{\prime}\,(\mathcal{R},\alpha) = (\mathcal{N},\mathcal{B})$ where $ \mathcal{N}$ is the commutant of the algebra of "self-intertwiners" for $ \alpha$ and $ \mathcal{B} = {L^\infty }(X,\mu) \otimes 1$ on ${L^2}(X,\mu) \otimes \mathcal{H}$. The main concern is the assignments $(\mathcal{M},\mathcal{A}) \to \mathcal{T}\;(\mathcal{M},\mathcal{A}) = (\mathcal{R}\,(\mathcal{M},\mathcal{A}),\alpha)$ and $(\mathcal{R},\alpha) \to S^{\prime}(\mathcal{R},\alpha)$, and in particular, the extent to which they are inverse to each other--this occurs if $\mathcal{R}$ is countable nonsingular and $ \alpha$ is (conjugation by) a projective square-integrable representation (cf. [8]), or if $ \mathcal{A}$ is a Cartan subalgebra (cf. [5]), among other cases. A partial dictionary between the representations $(\mathcal{R},\alpha)$ and pairs $ (\mathcal{M},\mathcal{A})$ is given--thus if $ \mathcal{R}$ is countable nonsingular and $\alpha$ is what we term replete, $ S^{\prime}(\mathcal{R},\alpha)$ is injective whenever $ \mathcal{R}$ is amenable, and a complete Galois theory generalizing that for crossed products by discrete groups is available. We also show how to construct various pathological examples such as a singular maximal abelian subalgebra $ \mathcal{A} \subseteq \mathcal{M}$ for which the Takesaki equivalence relation $ \mathcal{R}\,(\mathcal{M},\mathcal{A})$ is nontrivial.
Recursivity in quantum mechanics
John C.
Baez
339-350
Abstract: The techniques of effective descriptive set theory are applied to the mathematical formalism of quantum mechanics in order to see whether it actually provides effective algorithms for the computation of various physically significant quantities, e.g. matrix elements. Various Hamiltonians are proven to be recursive (effectively computable) and shown to generate unitary groups which act recursively on the Hilbert space of physical states. In particular, it is shown that the $n$-particle Coulombic Hamiltonian is recursive, and that the time evolution of $n$-particle quantum Coulombic systems is recursive.
The initial trace of a solution of the porous medium equation
D. G.
Aronson;
L. A.
Caffarelli
351-366
Abstract: Let $u = u(x,t)$ be a continuous weak solution of the porous medium equation in ${{\mathbf{R}}^d} \times (0,T)$ for some $T > 0$. We show that corresponding to $u$ there is a unique nonnegative Borel measure $ \rho$ on ${{\mathbf{R}}^d}$ which is the initial trace of $ u$. Moreover, we show that the initial trace $\rho$ must belong to a certain growth class. Roughly speaking, this growth restriction shows that there are no solutions of the porous medium equation whose pressure grows, on average, more rapidly then $ \vert x{\vert^2}$ as $\vert x\vert \to \infty $.
Global invariants for measured foliations
Steven
Hurder
367-391
Abstract: New exotic invariants for measured foliations are constructed, which we call the $\mu$-classes of a pair $(\mathcal{F},\mu)$. The dependence of the $ \mu$-classes on the geometry of the foliation $ \mathcal{F}$ is examined, and the dynamics of a foliation is shown to determine the $\mu$-classes in many cases. We use the $ \mu$-classes to study the classifying space $ B{\Gamma_{S{L_q}}}$ of foliations with a transverse invariant volume form, and we show the homotopy groups of $B{\Gamma _{S{L_q}}}$ are uncountably generated starting in degrees $q + 3$. New invariants for groups of volume preserving diffeomorphisms also arise from the $ \mu$-classes; these invariants are nontrivial and related to the geometric aspects of the group action. Relations between the $ \mu$-classes and the secondary classes of a foliation are exhibited.
Selfadjoint representations of polynomial algebras
Atsushi
Inoue;
Kunimichi
Takesue
393-400
Abstract: In this paper we will investigate the selfadjointness of unbounded $ ^{\ast}$-representations of the polynomial algebra. In particular, it is shown that the notion of selfadjoint representation is equivalent to that of standard representation in the case of the polynomial algebra generated by one hermitian element. Although the notion of standardness implies that of selfadjointness, the converse is not true in general. Therefore, we consider under what conditions a $ ^{\ast}$-representation is standard.
Geometric condition for universal interpolation in $\hat{\mathcal{E}}'$
William A.
Squires
401-413
Abstract: It is known that if $ h$ is an entire function of exponential type and $Z(h) = {\{ {z_k}\} _{k = 1}}$ with $\vert h^{\prime}({z_k})\vert \geqslant \varepsilon \exp (- c\vert{z_k}\vert)$ for constants $\epsilon$, $C$ independent of $k$, then $ \{ {z_k}\} _{k = 1}^\infty$ is a universal interpolation sequence. That is, given any sequence of complex numbers $\{ {a_k}\} _{k = 1}^\infty$ such that $ \vert{a_k}\vert \leqslant A\,\exp (B\vert{z_k}\vert)$ for constants $A,B$ independent of $ K$ then there exists $ g$ of exponential type such that $g({z_k}) = {a_k}$. This note is concerned with finding geometric conditions which make $ \{ {z_k}\} _{k = 1}^\infty$ a universal interpolation sequence for various spaces of entire functions. For the space of entire functions of exponential type a necessary and sufficient condition for $ \{ {z_k}\} _{k = 1}^\infty$ to be a universal interpolation sequence is that $ \int_0^{\vert{z_k}\vert} {n({z_k},t)\,dt/t \leqslant C\vert{z_k}\vert + D,k = 1} , 2,\ldots$, where $ n({z_k},t)$ is the number of points of $ \{ {z_k}\} _{k = 1}^\infty$ in the disc of radius $t$ about ${z_k}$, excluding ${z_k}$, and $C,D$ are constants independent of $ k$. Results for the space $ \hat{\mathcal{E}}^\prime= \{ f\;{\text{entire}}\vert\vert f(z)\vert \leqslant A\;\exp [B\vert\operatorname{Im} z\vert + B\log (1 + \vert z\vert^{2})]\}$ are given but the theory is not as complete as for the above example.
Local spectra of seminormal operators
Kevin F.
Clancey;
Bhushan L.
Wadhwa
415-428
Abstract: The local spectral theory of seminormal operators is studied by examining the connection between two naturally occurring contractive operator functions. These results are used to control the local spectra of cohyponormal operators. An invariant subspace result for seminormal operators whose real part has thin spectra is provided.
Erratum to: ``The derived functors of the primitives for ${\rm BP}\sb\ast (\Omega S\sp{2n+1})$''
Martin
Bendersky
429
Markov neighborhoods for zero-dimensional basic sets
Dennis
Pixton
431-462
Abstract: We extend the local stable and unstable laminations for a zero-dimensional basic set to semi-invariant laminations of a neighborhood, and use these extensions to construct the appropriate analog of a Markov partition, which we call a Markov neighborhood. The main applications we give are in the perturbation theory for stable and unstable manifolds; in particular, we prove a transversality theorem. For these applications we require not only that the basic sets be zero dimensional but that they satisfy certain tameness assumptions. This leads to global results on improving stability properties via small isotopies.
On first countable, countably compact spaces. I. $(\omega \sb{1},\,\omega \sp{\ast} \sb{1})$-gaps
Peter J.
Nyikos;
Jerry E.
Vaughan
463-469
Abstract: This paper is concerned with the $ ({\omega_1},\omega_1^{\ast})$-gaps of F. Hausdorff and the topological spaces defined from them by Eric van Douwen. We construct special gaps in order that the associated gap spaces will have interesting topological properties. For example, the gap spaces we construct show that in certain models of set theory, there exist countably compact, first countable, separable, nonnormal ${T_2}$-spaces.
On the location of zeros of oscillatory solution
H.
Gingold
471-496
Abstract: The location of zeros of solutions of second order singular differential equations is provided by a new asymptotic decomposition formula. The approximate location of zeros is provided with high accuracy error estimates in the neighbourhood of the point at infinity. The same asymptotic formula suggested is applicable to the neighbourhood of most types of singularities as well as to the neighbourhoods of regular points.
The structure of $\omega \sb{1}$-separable groups
Paul C.
Eklof
497-523
Abstract: A classification theorem is proved for $ {\omega_1}$-separable ${\omega_1}$-free abelian groups of cardinality ${\omega_1}$ assuming Martin's Axiom $($MA$)$ and ${2^{\aleph_0}} > {\aleph_1}$. As a consequence, several structural results about direct sum decompositions of $ {\omega_1}$-separable groups are proved. These results are proved independent of ZFC, and, in addition, another structural property is proved undecidable in ${\text{ZFC}} + {\text{MA}} + {2^{\aleph_0}} > {\aleph_1}$. The problem of classifying these groups in a model of ${2^{\aleph_0}} = {\aleph_1}$ is also investigated.
An inequality with applications in potential theory
Boris
Korenblum;
Edward
Thomas
525-536
Abstract: An analytic inequality (announced previously) is proved and a certain monotonicity condition is shown to be essential for its validity, contrary to an earlier conjecture. Then, a generalization of the inequality, which takes into account the extent of nonmonotonicity, is established.
Global solvability on two-step compact nilmanifolds
Jacek M.
Cygan;
Leonard F.
Richardson
537-554
Abstract: We apply the methods of representation theory of nilpotent Lie groups to study the convergence of Fourier series of smooth global solutions to first order invariant partial differential equations $Df = g$ in $ {C^\infty }$ of a two-step compact nilmanifold. We show that, under algebraically well-defined conditions on $D$ in the complexified Lie algebra, smooth infinite-dimensional irreducible solutions, when they exist, satisfy estimates strong enough to guarantee uniform convergence of the irreducible (or primary) Fourier series to a smooth global solution. Such strong estimates are not possible on multidimensional tori.
Mean values of subsolutions of elliptic and parabolic equations
William P.
Ziemer
555-568
Abstract: Integral averages of weak subsolutions (and supersolutions) in $ {R^n}$ of quasilinear elliptic and parabolic equations are investigated. The important feature is that these integral averages are defined in terms of measures that reflect interesting geometric phenomena. Harnack type inequalities are established in terms of these integral averages.
Doubly slice knots and the Casson-Gordon invariants
Daniel
Ruberman
569-588
Abstract: We find knots in all dimensions which are algebraically but not geometrically doubly slice. Our new obstructions involve the Casson-Gordon invariants of the finite cyclic covers in odd dimensions and of the infinite cyclic cover in even dimensions. These same invariants provide new criteria for amphicheirality and invertibility of even-dimensional knots.
Gradings of ${\bf B}\sb{n}$ and ${\bf C}\sb{n}$ of finite representation type
Ibrahim
Assem;
Oscar
Roldán
589-609
Abstract: It was shown by Bongartz and Gabriel that the classification of simplyconnected algebras (i.e. finite-dimensional, basic, of finite representation type and with a simply-connected Auslander-Reiten graph) can be reduced to the study of certain numerical functions, called gradings, operating on a tree. Here, we classify in terms of their bounden species the simply-connected algebras arising from gradings of the Dynkin trees $ {{\mathbf{B}}_n}$ and ${{\mathbf{C}}_n}$, and show that these are exactly the tilted algebras of types ${{\mathbf{B}}_n}$ and ${{\mathbf{C}}_n}$, respectively.
Espaces $l\sp{p}$ dans les sous-espaces de $L\sp{1}$
S.
Guerre;
M.
Levy
611-616
Abstract: It is shown that every subspace $E$ of ${L^1}$ contains a subspace isomorphic to ${l^{p(E)}}$, where $p(E)$ is the upper bound of the set of real $ p$'s such that $ E$ is of type $ p$-Rademacher. As $ p(E)$ is also the upper bound of the set of real $p$'s such that $E$ embeds into ${L^p}$, this result answers a question of H. P. Rosenthal. The proof uses the theory of stable Banach spaces developed by J. L. Krivine and B. Maurey.
Time-ordered operators. II
Tepper L.
Gill
617-634
Abstract: In this paper, we substantially improve on the work of [G1]. After constructing the general mathematical foundations for linear time-ordered evolution equations, we apply our results to show that both the perturbation expansion and the Feynman diagram method are mathematically sound. We provide a remainder term so that the expansion may be considered exact at all orders. We then show that time-ordered operators naturally induce an operator-valued path integral whenever a transition kernel is given.
Estimates of caloric measure and the initial-Dirichlet problem for the heat equation in Lipschitz cylinders
Eugene
Fabes;
Sandro
Salsa
635-650
Abstract: In this paper the authors prove unique solvability of the initial-Dirichlet problem for the heat equation in a cylindrical domain with Lipschitz base, lateral data in ${L^p},p \geqslant 2$, and zero initial values. A Poisson kernel for this problem is shown to exist with the property that its $ {L^2}$-averages over parabolic rectangles are equivalent to $ {L^1}$-averages over the same sets.
A restriction theorem for semisimple Lie groups of rank one
Juan A.
Tirao
651-660
Abstract: Let ${\mathfrak{g}_{\mathbf{R}}} = {\mathfrak{f}_{\mathbf{R}}} + {\mathfrak{p}_{\mathbf{R}}}$ be a Cartan decomposition of a real semisimple Lie algebra $ {\mathfrak{g}_{\mathbf{R}}}$ and let $\mathfrak{g} = \mathfrak{f} + \mathfrak{p}$ be the corresponding complexification. Also let $ {\mathfrak{a}_{\mathbf{R}}}$ be a maximal abelian subspace of $ {\mathfrak{p}_{\mathbf{R}}}$ and let $ \mathfrak{a}$ be the complex subspace of $ \mathfrak{p}$ generated by $ {\mathfrak{a}_{\mathbf{R}}}$. We assume $\dim {\mathfrak{a}_{\mathbf{R}}} = 1$. Now let $G$ be the adjoint group of $\mathfrak{g}$ and let $K$ be the analytic subgroup of $G$ with Lie algebra $ {\text{ad}}_\mathfrak{g}(\mathfrak{f})$. If $ S^\prime(\mathfrak{g})$ denotes the ring of all polynomial functions on $\mathfrak{g}$ then clearly $S^\prime(\mathfrak{g})$ is a $G$-module and a fortiori a $K$-module. In this paper, we determine the image of the subring $ S^\prime{(\mathfrak{g})^K}$ of $K$-invariants in $S^\prime(\mathfrak{g})$ under the restriction map $f \mapsto f{\vert _{\mathfrak{f} + \mathfrak{a}}}(f \in S^\prime{(\mathfrak{g})^K})$.
A generalization of $F$-spaces and some topological characterizations of GCH
Mary Anne
Swardson
661-675
Abstract: Several topological characterizations involving $F$-spaces of the continuum hypothesis are due to R. G Woods and E. K. van Douwen. We extend this work by defining a space $X$ to be an $ {F_\alpha }$-space if the union of $< \alpha$ cozero-sets is ${C^{\ast}}$-embedded in $X$ and by giving, for every infinite cardinal $ \alpha$, topological characterizations involving $ {F_\alpha }$-spaces of the cardinal equality ${2^\alpha } = {\alpha ^ + }$ .
Some applications of direct integral decompositions of $W\sp{\ast} $-algebras
Edward
Sarian
677-689
Abstract: Let $\mathcal{A}$ be a $ {W^{\ast}}$-algebra and let $A \in \mathcal{A}$. $ \mathcal{K}(\mathcal{A})$ and $C(A)$ represent certain convex subsets of $\mathcal{A}$. We prove the following via direct integral theory: (1) If $ \mathcal{A}$ is of type $ {{\text{I}}_\infty }$, $ {\text{II}}_\infty$, or III, then $C(A) = \{ 0\}$ iff ${\text{A}} \in \mathcal{K}(\mathcal{A})$. (2) If $ \mathcal{A}$ is of type I or II, then $ \mathcal{K}(\mathcal{A})$ is strongly dense in $ \mathcal{A}$. (3) If $\mathcal{A}$ is of type ${{\text{I}}_\infty }$, ${\text{II}}_\infty$, or III and $\mathcal{B}$ is a $ {W^{\ast}}$-subalgebra of $\mathcal{A}$, we give sufficient conditions for a Schwartz map $P$ of $ \mathcal{A}$ into $\mathcal{B}$ to annihilate $ \mathcal{K}(\mathcal{A})$. Several preliminary lemmas that are useful for direct integral theory are also proved.
Radial limits of $n$-subharmonic functions in the polydisc
W. C.
Nestlerode;
M.
Stoll
691-703
Abstract: We prove a relation between a certain weighted radial limit of an $ n$-subharmonic function in the polydisc ${U^n}$ and the representing measure of its least $n$-harmonic majorant. We apply this result to functions in $N({U^n})$, the Nevalinna class of ${U^n}$. In particular, we obtain a necessary condition for a function to belong to the component of the origin in $N({U^n})$. These results are extensions of the work of J. H. Shapiro and A. L. Shields to $n > 1$.
Linearization of second-order nonlinear oscillation theorems
Man Kam
Kwong;
James S. W.
Wong
705-722
Abstract: The problem of oscillation of super- and sublinear Emden-Fowler equations is studied. Established are a number of oscillation theorems involving comparison with related linear equations. Recent results on linear oscillation can thus be used to obtain interesting oscillation criteria for nonlinear equations.
Nonimmersions and nonembeddings of quaternionic spherical space forms
Teiichi
Kobayashi
723-728
Abstract: We determine the orders of the canonical elements in $KO$-rings of quaternionic spherical space forms $ {S^{4n + 3}}/{Q_k}$ and apply them to prove the nonexistence theorems of immersions and embeddings of ${S^{4n + 3}}/{Q_k}$ in Euclidean spaces.
All three-manifolds are pullbacks of a branched covering $S\sp{3}$ to $S\sp{3}$
Hugh M.
Hilden;
María Teresa
Lozano;
José María
Montesinos
729-735
Abstract: There are two main results in this paper. First, we show that every closed orientable $3$-manifold can be constructed by taking a pair of disjoint bounded orientable surfaces in $ {S^3}$, call them $ {F_1}$ and ${F_2}$; taking three copies of $ {S^3}$; splitting the first along ${F_1}$, the second along ${F_1}$ and ${F_2}$, and the third along ${F_2}$; and then pasting in the natural way. Second, we show that given any closed orientable $ 3$-manifold ${M^3}$ there is a $3$-fold irregular branched covering space, $ p:{M^3} \to {S^3}$, such that $p:{M^3} \to {S^3}$ is the pullback of the $3$-fold irregular branched covering space $q:{S^3} \to {S^3}$ branched over a pair of unknotted unlinked circles.
Invariant subspaces on Riemann surfaces of Parreau-Widom type
Mikihiro
Hayashi
737-757
Abstract: In this paper we generalize Beurling's invariant subspace theorem to the Hardy classes on a Riemann surface with infinite handles. The problem is to classify all closed ( weak$^{\ast}$ closed, if $p = \infty$) $ {H^\infty }(d\chi )$-submodules, say $ \mathfrak{m}$, of ${L^p}(d\chi )$, $1 \leqslant p \leqslant \infty$, where $ d\chi$ is the harmonic measure on the Martin boundary of a Riemann surface $ R$, and ${H^\infty }(d\chi )$ is the set of boundary functions of all bounded analytic functions on $R$. Our main result is stated roughly as follows. Let $R$ be of Parreau-Widom type, that is, the space ${H^\infty }(R,\gamma )$ of bounded analytic sections contains a nonzero element for every complex flat line bundle $\gamma \in \pi {(R)^{\ast}}$. We may assume, without loss of generality, that the Green's function of $R$ vanishes at the infinity. Set $ {m^\infty }(\gamma ) = \sup \{ \vert f({\mathbf{O}})\vert:f \in {H^\infty }(R,\gamma ),\vert f\vert \leqslant 1\}$ for a fixed point $ {\mathbf{O}}$ of $ R$. Then, a necessary and sufficient condition in order that every such an $\mathfrak{m}$ takes either the form $ \mathfrak{m} = {C_E}{L^p}(d\chi )$, where ${C_E}$ is the characteristic function of a set $E$, or the form $\mathfrak{m} = q{H^p}(d\chi ,\gamma )$, where $\vert q\vert = 1$ a.e. and $\gamma$ is some element of $\pi {(R)^{\ast}}$ is that ${m^\infty }(\gamma )$ is continuous for the variable $ \gamma \in \pi {(R)^{\ast}}$.
Spectral properties of a certain class of complex potentials
V.
Guillemin;
A.
Uribe
759-771
Abstract: In this paper we discuss spectral properties of the Schroedinger operator $- \Delta + q$ on compact homogeneous spaces for certain complex valued potentials $q$. We show, for instance, that for these potentials the spectrum of $- \Delta + q$ is identical with the spectrum of $- \Delta$.
Quotients by ${\bf C}\sp{\ast} $ and ${\rm SL}(2,{\bf C})$ actions
Andrzej
Białynicki-Birula;
Andrew John
Sommese
773-800
Abstract: Let $ \rho :{{\mathbf{C}}^{\ast}} \times X \to X$ be a meromorphic action of $ {{\mathbf{C}}^{\ast}}$ on a compact normal analytic space. We completely classify $ {{\mathbf{C}}^{\ast}}$-invariant open $ U \subseteq X$ with a compact analytic space $U/T$ as a geometric quotient for a wide variety of actions, including all algebraic actions. As one application, we settle affirmatively a conjecture of D. Mumford on compact geometric quotients by $ {\text{SL(2}},{\mathbf{C}})$ of Zariski open sets of $ {({\mathbf{P}}_{\mathbf{C}}^1)^n}$.
On a question of Quillen
S. M.
Bhatwadekar;
R. A.
Rao
801-810
Abstract: Let $R$ be a regular local ring, and $ f$ a regular parameter of $ R$. Quillen asked whether every projective ${R_f}$-module is free. We settle this question when $ R$ is a regular local ring of an affine algebra over a field $k$. Further, if $R$ has infinite residue field, we show that projective modules over Laurent polynomial extensions of ${R_f}$ are also free.
Preorderings compatible with probability measures
Rolando
Chuaqui;
Jerome
Malitz
811-824
Abstract: The main theorem proved in this paper is: Let $B$ be a $\sigma$-complete Boolean algebra and $\succcurlyeq a$ binary relation with field $ B$ such that: (i) Every finite subalgebra $ B^{\prime}$ admits a probability measure $ \mu^{\prime}$ such that for $p,q \in B^{\prime},p \succcurlyeq q\;iff\mu 'p \geqslant \mu 'q$. (ii) If for every $i,{p_i},q \in B$ and $ {p_i} \subseteq {p_{i + 1}} \preccurlyeq q$, then ${ \cup_{i < \infty }}{p_i} \preccurlyeq q$. Under these conditions there is a $\sigma $-additive probability measure $\mu$ on $B$ such that: (a) If there is $a\;p \in B$, such that for every $q \subseteq p$ there is a $q^{\prime} \subseteq q$ with $ q^{\prime} \preccurlyeq q,q^{\prime} \npreceq 0$, and $q \npreceq q^{\prime}$, then we have that for every $p,q \in B,\mu \,p \geqslant \mu \,q\,iff\,p \succcurlyeq q$. (b) If for every $p \in B$, there is $a\;q \subseteq p$ such that $q^{\prime} \subseteq q$ implies $q \preccurlyeq q^{\prime}\;or\;q^{\prime} \preccurlyeq 0$, then we have that for every $p,q \in B,p \succcurlyeq q$ implies $\mu p \geqslant \mu q$.
The invariant subspace structure of nonselfadjoint crossed products
Baruch
Solel
825-840
Abstract: Let $\mathcal{L}$ be the von Neumann algebra crossed product determined by a finite von Neumann algebra $ M$ and a trace preserving $ ^{\ast}$-automorphism $ \alpha$ of $M$. We study the invariant subspace structure of the subalgebra $ {\mathcal{L}_ + }$ of $\mathcal{L}$ consisting of those operators whose spectrum with respect to the dual automorphism group on $\mathcal{L}$ is nonnegative. We investigate the conditions for two invariant subspaces ${\mathcal{M}_1}$, and ${\mathcal{M}_2}$ (with ${Q_{1}},{Q_2}$ the corresponding orthogonal projections) to satisfy ${Q_1} = {R_\upsilon }\,{Q_2}\,R_\upsilon^{\ast}$ for some partial isometry ${R_{\upsilon }}$ in $\mathcal{L}^{\prime}$. We use this analysis to find the general form of a $\sigma$-weakly closed subalgebra of $\mathcal{L}$ that contains ${\mathcal{L}_ + }$.
The properties $\sp{\ast} $-regularity and uniqueness of $C\sp{\ast} $-norm in a general $\sp{\ast} $-algebra
Bruce A.
Barnes
841-859
Abstract: In this paper two properties of a $^{\ast}$-algebra $A$ are considered which are concerned with the relationship between the algebra and its ${C^{\ast}}$-enveloping algebra. These properties are that $A$ have a unique $ {C^{\ast}}$-norm, and that $ A$ be $ ^{\ast}$-regular. Both of these concepts are closely involved with the representation theory of the algebra.
On the deformation of algebra morphisms and diagrams
M.
Gerstenhaber;
S. D.
Schack
1-50
Abstract: A diagram here is a functor from a poset to the category of associative algebras. Important examples arise from manifolds and sheaves. A diagram $ {\mathbf{A}}$ has functorially associated to it a module theory, a (relative) Yoneda cohomology theory, a Hochschild cohomology theory, a deformation theory, and two associative algebras ${\mathbf{A}}!$ and ${\mathbf{(\char93 A)!}}$. We prove the Yoneda and Hochschild cohomologies of $ {\mathbf{A}}$ to be isomorphic. There are functors from ${\mathbf{A}}$-bimodules to both ${\mathbf{A}}!$-bimodules and ${\mathbf{(\char93 A)!}}$bimodules which, in the most important cases (e.g., when the poset is finite), induce isomorphisms of Yoneda cohomologies. When the poset is finite every deformation of ${\mathbf{(\char93 A)!}}$ is induced by one of ${\mathbf{A}}$; if $ {\mathbf{A}}$ also takes values in commutative algebras then the deformation theories of $ {\mathbf{(\char93 A)!}}$ and ${\mathbf{A}}$ are isomorphic. We conclude the paper with an example of a noncommutative projective variety. This is obtained by deforming a diagram representing projective $2$-space to a diagram of noncommutative algebras.
Weighted iterates and variants of the Hardy-Littlewood maximal operator
M. A.
Leckband;
C. J.
Neugebauer
51-61
Abstract: In a recent paper, M. A. Leckband and C. J. Neugebauer obtained a rearrangement inequality for a generalized maximal operator with respect to two measures. For an application they studied norm bounds for the iterated Hardy-Littlewood maximal operator with respect to two measures. In this paper this theory is further developed and other applications of the rearrangement inequality are obtained.
On the distribution of the principal series in $L\sp{2}(\Gamma \backslash G)$
Roberto J.
Miatello;
Jorge A.
Vargas
63-75
Abstract: Let $G$ be a semisimple Lie group of split rank one with finite center. If $\Gamma \subset G$ is a discrete cocompact subgroup, then ${L^2}(\Gamma \backslash G) = {\Sigma_{\omega \in \mathcal{E}(G)}}{n_\Gamma }(\omega ) \cdot \omega$. For fixed $ \sigma \in \mathcal{E}(M)$, let $P(\sigma )$ denote the classes of irreducible unitary principal series $ {\pi_{\sigma ,iv}}(v \in {\mathcal{U}^{\ast}})$. Let, for $s > 0,{\psi_\sigma }(s) = {\Sigma_{\omega \in P(\sigma )}}{n_\Gamma }(\omega ) \cdot {e^{s{\lambda_\omega }}}$, where $ {\lambda_\omega }$ is the eigenvalue of $\Omega$ (the Casimir element of $G$) on the class $\omega$. In this paper, we determine the singular part of the asymptotic expansion of ${\psi_\sigma }(s)$ as $s \to {0^ + }$ if $\Gamma$ is torsion free, and the first term of the expansion for arbitrary $\Gamma$. As a consequence, if ${N_\sigma }(r) = \Sigma_{\omega \in P(\sigma ),\vert{\lambda_{{\omega }}\vert < r}}{n_\Gamma }(\omega )$ and $G$ is without connected compact normal subgroups, then $\displaystyle {N_\sigma }(r)\;\sim {C_G}\; \cdot\; \vert Z(G) \cap \Gamma \vert... ...\; \cdot\; \dim(\sigma )\; \cdot\; {r^c} \qquad (c = \frac{1} {2}\,\dim \,G/K),$ as $r \to + \infty$. In the course of the proof, we determine the image and kernel of the restriction homomorphism ${i^{\ast}}:R(K) \to R(M)$ between representation rings.
Handle attaching on generic maps
Youn W.
Lee
77-94
Abstract: Using the handle attaching technique along the singular value set of generic maps in the stable range together with the handle subtraction of Haefliger, smooth immersions and embeddings are studied. We generalize Whitney's immersion theorem, and Haefliger and Hirsh's result on embedding and classification of embeddings of $k$-connected ($(k + 1)$-connected for the classification) smooth $ n$-manifolds into $ {{\mathbf{R}}^{2n - k}}$. For example, we obtain the following as a generalization of Whitney's immersion theorem. If $f: {V^n} \to {M^m}, {3n} < {2m}$, is a generic map such that each component of its double point set is either a closed manifold or diffeomorphic to the $(2n - m)$-disk, then $f$ is homotopic to an immersion.
Operators of $P$-variation and the evolution representation problem
M. A.
Freedman
95-112
Abstract: In contrast to a continuous linear semigroup, a continuous linear evolution $U( \cdot )$ may be nondifferentiable or of unbounded variation. In order to study these evolutions we introduce a class of operator-valued functions $A( \cdot )$ which satisfy a generalized bounded variation condition and represent $U$ as the product integral $U = \prod [I + dA]$.
Sign-embeddings of $l\sp{n}\sb{1}$
John
Elton
113-124
Abstract: If $({e_i})_{i = 1}^n$ are vectors in a real Banach space with $\parallel {e_i}\parallel \leqslant 1$ and Average$_{{\varepsilon_1} = \pm 1}\parallel \sum\nolimits_{i = 1}^n {{\varepsilon_i}{e_i}\parallel \geqslant \delta n}$, where $\delta > 0$, then there is a subset $ A \subseteq \{ 1,\ldots,n\}$ of cardinality $ m \geqslant cn$ such that ${({e_i})_{i \in A}}$ is $K$-equivalent to the standard $ l_1^m$ basis, where $ c > 0$ and $K < \infty$ depend only on $\delta$. As a corollary, if $1 < p < \infty$ and $l_1^n$ is $K$-isomorphic to a subspace of ${L_p}(X)$, then $l_1^m(m \geqslant cn)$ is $K^{\prime}$-isomorphic to a subspace of $ X$, where $c > 0$ and $K^{\prime} < \infty$ depend only on $ K$ and $p$.
Certain reflexive sheaves on ${\bf P}\sp{n}\sb{{\bf C}}$ and a problem in approximation theory
Peter F.
Stiller
125-142
Abstract: This paper establishes a link between certain local problems in the theory of splines and properties of vector bundles and reflexive sheaves on complex projective spaces.
Stability criteria for Volterra equations
T. A.
Burton;
W. E.
Mahfoud
143-174
Abstract: We consider a system of integro-differential equations of the form (1.1) $\displaystyle x^{\prime} = A(t)x + \int_0^t {C(t,s)x(s)\;ds}$ with $A$ and $C$ being $n \times n$ matrices. Various types of stability are defined and results are obtained showing when one type of stability is equivalent to another type. We also construct a number of Lyapunov functional from which we obtain necessary and sufficient conditions for stability of (1.1). Finally, we prove several results concerning qualitative behavior of solutions of (1.1).
An extension of the theory of completely decomposable torsion-free abelian groups
Fred
Richman
175-185
Abstract: We construct a class of strongly indecomposable finite rank torsion-free groups that includes the rank-one groups, and develop a complete set of invariants for them and their direct sums.
Spherical harmonics and integral geometry on projective spaces
Eric L.
Grinberg
187-203
Abstract: The Radon transform $ R$ on $ {\mathbf{C}}{P^{\text{n}}}$ associates to a point function $f(x)$ the hyperplane function $ Rf(H)$ by integration over the hyperplane $H$. If ${R^t}$ is the dual transform, we can invert $ {R^t}R$ by a polynomial in the Laplace-Beltrami operator, and verify the formula of Helgason [7] with very simple computations. We view the Radon transform as a $ G$-invariant map between representations of the group of isometries $G = U(n + 1)$ on function spaces attached to $ {\mathbf{C}}{P^n}$. Pulling back to a sphere via a suitable Hopf fibration and using the theory of spherical harmonics, we can decompose these representations into irreducibles. The scalar by which $R$ acts on each irreducible is given by a simple integral. Thus we obtain an explicit formula for $R$. The action of ${R^t}R$ is immediately related to the spectrum of $ {\mathbf{C}}{P^n}$. This shows that ${R^t}R$ can be inverted by a polynomial in the Laplace-Beltrami operator. Similar procedures give corresponding results for the other compact $2$-point homogeneous spaces: ${\mathbf{R}}{P^n}$, $ {\mathbf{H}}{P^n}$, ${\mathbf{O}}{P^n}$, as well as spheres.
Chaotic difference equations: generic aspects
Hans Willi
Siegberg
205-213
Abstract: It is shown that in the set of all continuous selfmaps of a compact acyclic polyhedron (i.e. the homology groups of the space vanish in all dimensions $> 0$) the chaotic maps form a dense set. The notion of chaos used here is that of Li and Yorke. If this notion is slightly weakened ("almost chaotic") the density result can be improved by the theorem that the set of almost chaotic (continuous) selfmaps of a compact acyclic polyhedron $P$ contains a residual subset of the space of all continuous selfmaps of $P$. Moreover, the topological entropy of such a generic almost chaotic map is shown to be infinite. The basic ingredients of the proofs are from fixed point index theory.
Elementary first integrals of differential equations
M. J.
Prelle;
M. F.
Singer
215-229
Abstract: We show that if a system of differential equations has an elementary first integral (i.e. a first integral expressible in terms of exponentials, logarithms and algebraic functions) then it must have a first integral of a very simple form. This unifies and extends results of Mordukhai-Boltovski, Ritt and others and leads to a partial algorithm for finding such integrals.
Symbolic dynamics in flows on three-manifolds
John
Franks
231-236
Abstract: This article deals with the problem of what suspended subshifts of finite type can be realized as a basic set of a nonsingular Smale flow on three-dimensional manifolds. It is shown that any suspended subshift can be realized in such a flow on some three-manifold. Also if signs reflecting orientation are included in the matrix of the subshift of finite type then there is an obstruction to the realization on $ {S^3}$ of basic sets corresponding to some matrices.
On compact cohomology theories and Pontrjagin duality
Keith
Johnson
237-247
Abstract: Cohomology theories taking values in the category of topological groups are examined and a representation theorem is established for those whose coefficient groups are compact and locally euclidean. A method for constructing unstable homology operations is developed using this theorem, and application is made to the case of complex $ K$-theory.
Nakayama algebras and graded trees
B.
Rohnes;
S. O.
Smalø
249-256
Abstract: Let $k$ be an algebraically closed field. We show that if $T$ is a finite tree, then there is a grading $ g$ on $T$ such that $(T,g)$ is a representation finite graded tree, and such that the corresponding simply connected $k$-algebra is a Nakayama algebra (i.e. generalized uniserial algebra).
Cohomology detects failures of the axiom of choice
Andreas
Blass
257-269
Abstract: We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show that cohomology, when defined so that its usual exactness properties hold even in the absence of the axiom of choice, is adequate for detecting failures of this axiom in the following sense. If a set $X$, viewed as a discrete space, has trivial first cohomology for all coefficient groups, then every $ X$-indexed family of nonempty sets has a choice function. We also obtain related results when the coefficient groups are required to be abelian or well-orderable. In particular, we show that, if all discrete spaces have trivial first cohomology for all abelian coefficient groups, then the axiom of choice holds.
Finitely additive $F$-processes
Thomas E.
Armstrong
271-295
Abstract: If one replaces random variables by finitely additive measures one obtains instead of an $F$-process a finitely additive $F$-process. Finitely additive $ F$-processes on a decreasing collection of Boolean algebras form a dual base norm ordered Banach space. When the collection is linearly ordered they form a dual Kakutani $L$-space. This $L$-space may be represented as the $ L$-space of all finitely additive bounded measures on the Boolean ring of predictable subsets of the extreme points of the positive face of the unit ball. Of independent interest is the fact that any bounded supermartingale is a decreasing process in contrast to the usual case where only the supermartingales of class $DL$ are decreasing processes.
Graphs with relations, coverings and group-graded algebras
Edward L.
Green
297-310
Abstract: The paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs. To include gradings of all basic finite-dimensional algebras over an algebraically closed field, a theory of coverings of graphs with relations is introduced.
Characterization of recursively enumerable sets with supersets effectively isomorphic to all recursively enumerable sets
Wolfgang
Maass
311-336
Abstract: We show that the lattice of supersets of a recursively enumerable (r.e.) set $A$ is effectively isomorphic to the lattice of all r.e. sets if and only if the complement $ \bar A$ of $A$ is infinite and $\{ e\vert{W_e} \cap \bar A\;{\text{finite}}\}\;{\leqslant_{1}}\emptyset''$ (i.e. $\bar A$ is semilow$_{1.5}$). It is obvious that the condition `` $ \bar{A}\;$ semilow$_{1.5}$'' is necessary. For the other direction a certain uniform splitting property (the "outer splitting property") is derived from semilow$_{1.5}$ and this property is used in an extension of Soare's automorphism machinery for the construction of the effective isomorphism. Since this automorphism machinery is quite complicated we give a simplified proof of Soare's Extension Theorem before we add new features to this argument.
Groups and simple languages
Robert H.
Haring-Smith
337-356
Abstract: With any finitely generated group presentation, one can associate a formal language (called the reduced word problem) consisting of those words on the generators and their inverses which are equal to the identity but which have no proper prefix equal to the identity. We show that the reduced word problem is a simple language if and only if each vertex of the presentation's Cayley diagram has only a finite number of simple closed paths passing through it. Furthermore, if the reduced word problem is simple, then the group is a free product of a free group of finite rank and a finite number of finite groups.
The slice map problem for $\sigma $-weakly closed subspaces of von Neumann algebras
Jon
Kraus
357-376
Abstract: A $\sigma $-weakly closed subspace $\mathcal{S}$ of $ B(\mathcal{H})$ is said to have Property $ {S_\sigma }$ if for any $ \sigma$-weakly closed subspace $ \mathcal{T}$ of a von Neumann algebra $\mathcal{N},\{ x \in \mathcal{S}\;\overline \otimes \mathcal{N}:{R_\varphi }(x... ...hi \in B{(\mathcal{H})_{\ast}}\} = \mathcal{S}\,\overline \otimes \,\mathcal{T}$, where $ {R_\varphi }$ is the right slice map associated with $\varphi$. It is shown that semidiscrete von Neumann algebras have Property $ {S_\sigma }$, and various stability properties of the class of $\sigma $-weakly closed subspaces with Property $ {S_\sigma }$ are established. It is also shown that if $(\mathcal{M},G,\alpha )$ is a ${W^{\ast}}$-dynamical system such that $\mathcal{M}$ has Property ${S_\sigma }$ and $G$ is compact abelian, then all of the spectral subspaces associated with $\alpha$ have Property $ {S_\sigma }$. Some applications of these results to the study of tensor products of spectral subspaces and tensor products of reflexive algebras are given. In particular, it is shown that if $ {\mathcal{L}_1}$ is a commutative subspace lattice with totally atomic core, and $ {\mathcal{L}_2}$ is an arbitrary subspace lattice, then $ {\text{alg}}({\mathcal{L}_{1}} \otimes {\mathcal{L}_2}) = {\text{alg}}\;{\mathcal{L}_{1}}\,\overline \otimes {\text{alg}}\;{\mathcal{L}_2}$.
Hamburger-Noether expansions over rings
Antonio
Campillo
377-388
Abstract: We study Hamburger-Noether expansions over rings, obtaining some applications to equisingular deformation theory and the moduli problem of plane curve singularities, and construct a universal equation for a given equisingularity class.
Group rings which are Azumaya algebras
F. R.
DeMeyer;
G. J.
Janusz
389-395
Abstract: The group ring $ RG$ of a group $ G$ over a ring $ R$ (with identity $ 1(R)$) is a separable algebra over its center if and only if the following conditions hold: (a) $R$ is a separable algebra over its center; (b) the center of $G$ has finite index in $G$: (c) the commutator subgroup $G^{\prime}$ of $G$ has finite order $m$ and $m1(R)$ is invertible in $R$.
Improved Sobolev inequalities
Robert S.
Strichartz
397-409
Abstract: For a function $ f$ defined on ${{\mathbf{R}}^n}$, Sobolev's inequality $\parallel f{\parallel_q} \leqslant c(\parallel \,f\;{\parallel_{p}} + \parallel \nabla f{\parallel_{p}})$, where $1 < p < q < \infty$ and $1/p - 1/q = 1/n$, can be improved if the Fourier transform $\hat f$ is assumed to have support in a set $ A$ which satisfies an estimate $\vert\{ \xi \in A:\vert\vert\xi \vert\vert \leqslant s\} \vert \leqslant c{s^d}$ for some $ d < n$ the improvement being that we can take $ 1/p - 1/q = 1/d$, provided we also assume $ p \leqslant 2 \leqslant q$. Analogous results are proved for other Sobolev inequalities, for embeddings into Lipschitz-Zygmund spaces, and for functions on symmetric spaces whose Fourier expansions are suitably limited. Improved Sobolev inequalities are established locally for solutions of the wave equation. An application to the Radon transform on spheres is given.
Borel games and the Baire property
Kenneth
Schilling;
Robert
Vaught
411-428
Abstract: The Borel game operations are a natural generalization of the operation $($A$)$. It is shown that these operations preserve the property of Baire in all topological spaces. Applications are given to invariant descriptive set theory and the model theory of infinitary logic.
Erratum to: ``Homotopy in functor categories''
Alex
Heller
429
Equivariant vector fields on spheres
Unni
Namboodiri
431-460
Abstract: We address the following question: If $G$ is a compact Lie group and $S(M)$ is the unit sphere of an $ R[G]$-module $ M$, then how many orthonormal $G$-invariant vector fields can be found on $ S(M)$? We call this number the $G$-field number of $M$. Under reasonable hypotheses on $M$, we reduce this question to determining when the difference of two $G$-vector bundles vanishes in a certain subquotient of the $K{O_G}$-theory of a real projective space. In this general setting, we solve the problem for $2$-groups, for odd-order groups, and for abelian groups. If $M$ also has "enough" orbit types (for example, all of them), then we solve the problem for arbitrary finite groups. We also show that under mild hypotheses on $M$, the $G$-field number depends only on the dimensions of the fixed point sets of $M$.
$F$-purity and rational singularity
Richard
Fedder
461-480
Abstract: We investigate singularities which are $F$-pure (respectively $F$-pure type). A ring $R$ of characteristic $p$ is $F$-pure if for every $R$-module $M,0 \to M \otimes R \to M \otimes\, ^1R$ is exact where $^1R$ denotes the $R$-algebra structure induced on $R$ via the Frobenius map (if $ r \in R$ and $s \in \, ^{1}R$, then $r \cdot s = {r^p}s$ in $^1R$). $F$-pure type is defined in characteristic 0 by reducing to characteristic $p$. It is proven that when $R = S/I$ is the quotient of a regular local ring $S$, $R$ is $F$-pure at the prime ideal $Q$ if and only if $({I^{[p]}}:I) \not\subset {Q^{[p]}}$. Here, ${J^{[p]}}$ denotes the ideal $\{ {a^p}\vert a \in J\} $. Several theorems result from this criterion. If $f$ is a quasihomogeneous hypersurface having weights $ ({r_1},\ldots,{r_n})$ and an isolated singularity at the origin: (1) $ \sum\nolimits_{i = 1}^n {{r_i} > 1}$ implies $ K[{X_1},\ldots,{X_n}]/(f)$ has $F$-pure type at $m = ({X_1},\ldots,{X_n})$. (2) $\sum\nolimits_{i = 1}^n {{r_i} < 1}$ implies $ K[{X_1},\ldots,{X_n}]/(f)$ does not have $F$-pure type at $m$. (3) $\sum\nolimits_{i = 1}^n {{r_i} = 1}$ remains unsolved, but does connect with a problem that number theorists have studied for many years. This theorem parallels known results about rational singularities. It is also proven that classifying $F$-pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces, and that the $ F$-pure locus in the maximal spectrum of $ K[{X_1},\ldots,{X_n}]/I$, where $K$ is a perfect field of characteristic $ P$, is Zariski open. An important conjecture is that $R/fR$ is $F$-pure (type) should imply $R$ is $F$-pure (type) whenever $R$ is a Cohen-Macauley, normal local ring. It is proven that $ \operatorname{Ext}^1{(^1}R,R) = 0$ is a sufficient, though not necessary, condition. A local ring $(R,m)$ of characteristic $p$ is $F$-injective if the Frobenius map induces an injection on the local cohomology modules $H_m^i(R) \to H_m^i{(^1}R)$. An example is constructed which is $F$-injective but not $F$-pure. From this a counterexample to the conjecture that $R/fR$ is $F$-pure implies $R$ is $F$-pure is constructed. However, it is not a domain, much less normal. Moreover, it does not lead to a counterexample to the characteristic 0 version of the conjecture.
Poles of a two-variable $P$-adic complex power
Leon
Strauss
481-493
Abstract: For almost all $ P$-adic completions of an algebraic number field, if $s \in {\mathbf{C}}$ is a pole of ${f^s} = \int_{}^{} {\int_{}^{} {\vert f(x,y){\vert^s}\vert dx{\vert _{{K_p}}}\vert dy{\vert _{{K_p}}}} }$ , where $f$ is a polynomial whose only singular point is the origin, $f(0,0) = 0$, and $f$ is irreducible in $\overline K [[x,y]]$, then $\operatorname{Re} (s)$ is $- 1$ or one of an explicitly given set of rational numbers, whose cardinality is the number of characteristic exponents of $f = 0$.
Measure and category approximations for $C$-sets
V. V.
Srivatsa
495-505
Abstract: The class of $ C$-sets in a Polish space is the smallest $\sigma$-field containing the Borel sets and closed under operation $ (\mathcal{A})$. In this article we show that any $C$-set in the product of two Polish spaces can be approximated (in measure and category), uniformly over all sections, by sets generated by rectangles with one side a $C$-set and the other a Borel set. Such a formulation unifies many results in the literature. In particular, our methods yield a simpler proof of a selection theorem for $C$-sets with $ {G_\delta }$-sections due to Burgess [4].
The coordinatization of Arguesian lattices
Alan
Day;
Douglas
Pickering
507-522
Abstract: We show that the auxiliary planar ternary ring of an $n$-frame in an Arguesian lattice, $n \geqslant 3$, is indeed an associative ring with unit. The addition of two weak necessary conditions allows us to coordinatize a hyperplane of this $ n$-frame. This generalizes the classical work of von Neumann, Baer-Inaba, Jónsson and Jónsson-Monk.
Support points of families of analytic functions described by subordination
D. J.
Hallenbeck;
T. H.
MacGregor
523-546
Abstract: We determine the set of support points for several families of functions analytic in the open unit disc and which are generally defined in terms of subordination. The families we study include the functions with a positive real part, the typically-real functions, and the functions which are subordinate to a given majorant. If the majorant $F$ is univalent then each support point has the form $F \circ \;\phi $, where $\phi$ is a finite Blaschke product and $\phi (0) = 0$. This completely characterizes the set of support points when $F$ is convex. The set of support points is found for some specific majorants, including $F(z) = {((1 + z)/(1 - z))^p}$ where $p > 1$. Let $ K$ and ${\text{St}}$ denote the set of normalized convex and starlike mappings, respectively. We find the support points of the families $ {K^{\ast} }$ and ${\text{St}}^{\ast}$ defined by the property of being subordinate to some member of $K$ or $ {\text{St}}$, respectively.
$I\sp{X}$, the hyperspace of fuzzy sets, a natural nontopological fuzzy topological space
R.
Lowen
547-564
Abstract: Let $X$ be a uniform topological space, then on the family ${I^X}$ (resp. $\Phi (X)$) of all nonzero functions (resp. nonzero uppersemicontinuous functions) from $X$ to the unit interval $I$, a fuzzy uniform topology is constructed such that ${2^X}$ (resp. $ \mathcal{F}(X)$), the family of all nonvoid (resp. nonvoid closed) subsets of $ X$ equipped with the Hausdorff-Bourbaki structure is isomorphically injected in $ {I^X}$ (resp. $ \Phi (X)$). The main result of this paper is a complete description of convergence in ${I^X}$, by means of a notion of degree of incidence of members of ${I^X}$. Immediate consequences are that first it can be shown that this notion of convergence refines some particular useful notions of convergence of fuzzy sets used in applications, and that second it follows from its construction and properties that for each ordinary uniform topological space $ X$ there exists a natural nontopological fuzzy uniform topology on ${I^X}$.
Cocycles and local product decomposition
Jun-ichi
Tanaka
565-572
Abstract: As an application of cocycles, we establish a relation between the classical Hardy spaces on the real line $R$ and simply invariant subspaces on a quotient of the Bohr group. When this result is specialized suitably, it yields the well-known results concerning the elements of invariant subspaces. We also study, by using Gamelin's representation theorem, unitary functions which are the values of cocycles.
Conformally flat manifolds with nilpotent holonomy and the uniformization problem for $3$-manifolds
William M.
Goldman
573-583
Abstract: A conformally flat manifold is a manifold with a conformal class of Riemannian metrics containing, for each point $x$, a metric which is flat in a neighborhood of $x$. In this paper we classify closed conformally flat manifolds whose fundamental group (more generally, holonomy group) is nilpotent or polycyclic of rank $3$. Specifically, we show that such conformally flat manifolds are covered by either the sphere, a flat torus, or a Hopf manifold--in particular, their fundamental groups contain abelian subgroups of finite index. These results are applied to show that certain $ {T^2}$-bundles over $ {S^1}$ (namely, those whose attaching map has infinite order) do not have conformally flat structures. Apparently these are the first examples of $3$-manifolds known not to admit conformally flat structures.
Separatrix and limit cycles of quadratic systems and Dulac's theorem
Carmen
Chicone;
Douglas S.
Shafer
585-612
Abstract: Separatrix cycles for a planar quadratic vector field are studied. The results obtained are used to show that in any bounded region of the plane a quadratic vector field has at most a finite number of limit cycles.
The divisor classes of the hypersurface $z\sp{p\sp{m}}=G(x\sb{1},\cdots ,x\sb{n})$ in characteristic $p>0$
Jeffrey
Lang
613-634
Abstract: In this article we use P. Samuel's purely inseparable descent techniques to study the divisor class groups of normal affine hypersurfaces of the form ${z^p} = G({x_1},\ldots,{x_n})$ and develop an inductive procedure for studying those of the form $ {z^{p^m}} = G$. We obtain results concerning the order and type of these groups and apply this theory to some specific examples.
Diophantine approximation properties of certain infinite sets
Wolfgang M.
Schmidt
635-645
Abstract: We exhibit various infinite sets of reals whose finite subsets do not have good simultaneous rational approximations. In particular there is an infinite set such that each finite subset is "badly approximable" in the sense that Dirichlet's theorem is best possible up to a multiplicative constant.
The sufficiency of the Matkowsky condition in the problem of resonance
Ching Her
Lin
647-670
Abstract: We consider the sufficiency of the Matkowsky condition concerning the differential equation $f(0,\varepsilon ) = 0$ identically in $\varepsilon ,{f_x}(0,\varepsilon ) \ne 0$ with $f > 0$ for $x < 0$ and $f < 0$ for $x > 0$. Y. Sibuya proved that the Matkowsky condition implies resonance in the sense of N. Kopell if $ f$ and $g$ are convergent power series for $\vert\varepsilon \vert < \rho \;(\rho > 0),f(x,0)=-2x$ and the interval $[ - a,b]$ is contained in a disc $D$ with center at 0. The main problem in this work is to remove from Sibuya's result the assumption that $D$ is a disc.
Stable complete constant mean curvature surfaces in ${\bf R}\sp{3}$ and $H\sp{3}$
Hiroshi
Mori
671-687
Abstract: We construct some $ 1$-parameter families of complete rotation surfaces with constant mean curvature in the hyperbolic $3$-space ${H^3}$ of constant sectional curvature $ -1$, and show that some of them are stable for the variational problem of area together with oriented volume, and that a complete connected, oriented surface with constant mean curvature in the Euclidean $3$-space ${R^3}$ which is stable for the variational problem is a plane.
Embedding $L\sp{1}$ in $L\sp{1}/H\sp{1}$
J.
Bourgain
689-702
Abstract: It is proved that $ {L^1}$ is isomorphic to a subspace of $ {L^1}/{H^1}$. More precisely, there exists a diffuse $\sigma$-algebra $ \mathfrak{S}$ on the circle such that the corresponding expectation $ {\mathbf{E}}:{H^\infty } \to {L^\infty }({\mathbf{C}})$ is onto. The method consists in studying certain martingales on the product $ {\prod ^{\mathbf{N}}}$.
Almost everywhere summability on nilmanifolds
Andrzej
Hulanicki;
Joe W.
Jenkins
703-715
Abstract: Let $G$ be a stratified, nilpotent Lie group and let $L$ be a homogeneous sublaplacian on $ G$. Let $E(\lambda )$ denote the spectral resolution of $ L$ on ${L^2}(G)$. Given a function $ K$ on $\mathbf{R}^+$, define the operator ${T_K}$ on ${L^2}(G)$ by ${T_k}f = \int_0^\infty \, {K(\lambda )\;dE(\lambda )\,f}$. Sufficient conditions on $ K$ to imply that $ {T_K}$ is bounded on $ {L^1}(G)$ and the maximal operator $K^{\ast} \varphi (x) = \sup_{t > 0}\vert{T_{K_t}}\varphi (x)\vert$ (where ${K_t}(\lambda ) = K(t\lambda )$) is of weak type $(1,1)$ are given. Picking a basis ${e_0},{e_1},\ldots$ of ${L^2}(G/\Gamma )$ ($\Gamma$ being a discrete cocompact subgroup of $G$) consisting of eigenfunctions of $ L$, we obtain almost everywhere and norm convergence of various summability methods of $ \Sigma (\varphi ,{e_j}){e_j},\varphi \in {L^p}(G/\Gamma ), 1 \leqslant p < \infty$.
Subcontinua with degenerate tranches in hereditarily decomposable continua
Lex G.
Oversteegen;
E. D.
Tymchatyn
717-724
Abstract: A hereditarily decomposable, irreducible, metric continuum $ M$ admits a mapping $ f$ onto $[0,1]$ such that each ${f^{ - 1}}(t)$ is a nowhere dense subcontinuum. The sets ${f^{ - 1}}(t)$ are the tranches of $ M$ and ${f^{ - 1}}(t)$ is a tranche of cohesion if $t \in \{ 0,1\}$ or ${f^{ - 1}}(t) = {\text{C1}}({f^{ - 1}}([0,t))) \cap {\text{C1}}\,({f^{ - 1}}((t,1]))$. The following answer a question of Mahavier and of E. S. Thomas, Jr. Theorem. Every hereditarily decomposable continuum contains a subcontinuum with a degenerate tranche. Corollary. If in an irreducible hereditarily decomposable continuum each tranche is nondegenerate then some tranche is not a tranche of cohesion. The theorem answers a question of Nadler concerning arcwise accessibility in hyperspaces.
Hereditarily additive families in descriptive set theory and Borel measurable multimaps
Roger W.
Hansell
725-749
Abstract: A family $\mathcal{B}$ of Borel subsets of a space $ X$ is (boundedly) Borel additive if, for some countable ordinal $ \alpha$, the union of every subfamily of $ \mathcal{B}$ is a Borel set of class $\alpha$ in $X$. A problem which arises frequently in nonseparable descriptive set theory is to find conditions under which this property is "hereditary" in the sense that any selection of a Borel subset from each member of $\mathcal{B}$ (of uniform bounded class) will again be a Borel additive family. Similar problems arise for other classes of projective sets; in particular, for Souslin sets and their complements. Positive solutions to the problem have previously been obtained by the author and others when $X$ is a complete metric space or under additional set-theoretic axioms. We give here a fairly general solution to the problem, without any additional axioms or completeness assumptions, for an abstract "descriptive class" in the setting of generalized metric spaces (e.g., spaces with a $\sigma$-point-finite open base). A typical corollary states that any point-finite (co-) Souslin additive family in (say) a metrizable space is hereditarily (co-) Souslin additive. (There exists a point-countable ${F_\sigma }$ additive family of subsets of the real line which has a point selection which is not even Souslin additive.) Two structure theorems for "hereditarily additive" families are proven, and these are used to obtain a nonseparable extension of the fundamental measurable selection theorem of Kuratowski and Ryll-Nardzewski, and a complete solution to the problem of Kuratowski on the Borel measurability of complex and product mappings for nonseparable metric spaces.
Fully nonlinear, uniformly elliptic equations under natural structure conditions
Neil S.
Trudinger
751-769
Abstract: We derive first and second derivative estimates for classical solutions of fully nonlinear, uniformly elliptic equations which are subject to natural structure conditions analogous to those proposed and treated by Ladyzhenskaya and Ural'tseva for quasilinear equations. As an application we extend recent work of Evans and Lions on the Bellman equation for families of linear operators to families of quasilinear operators.
On some cheap control problems for diffusion processes
José-Luis
Menaldi;
Maurice
Robin
771-802
Abstract: We consider several cases of control problems for diffusion processes when the payoff functional does not depend explicitly on the control. We prove the continuity of the optimal cost function and give a characterization of this cost with a quasi-variational inequality interpreting the problem as limit of an impulse control problem when the cost of impulse tends to zero. Moreover, we show the existence of an optimal control for some particular situations.
Uniformly exhaustive submeasures and nearly additive set functions
N. J.
Kalton;
James W.
Roberts
803-816
Abstract: Every uniformly exhaustive submeasure is equivalent to a measure. From this, we deduce that every vector measure with compact range in an $F$-space has a control measure. We also show that $ {c_0}$ (or any ${\mathcal{L}_\infty }$-space) is a $\mathcal{K}$-space, i.e. cannot be realized as the quotient of a nonlocally convex $F$-space by a one-dimensional subspace.
An application of asymptotic techniques to certain problems of spectral and scattering theory of Stark-like Hamiltonians
Matania
Ben-Artzi
817-839
Abstract: Let $ {L_0} = - \Delta + V({x_1}),L = {L_0} + {V_p}(x)$ be selfadjoint in ${L^2}({R^n})$. Here $V,{V_p}$ are real functions, $V({x_1})$ depends only on the first coordinate. Existence of the wave-operators $ {W_ \pm }\,(L,{L_0}) = s$ -${\lim_{t \to \pm \infty }}\,\exp (itL)\exp ( - it{L_0})$ is proved, using the stationary phase method. For this, an asymptotic technique is applied to the study of $ -{d^2}/d{t^2} + V(t)$ in $ {L^2}(R)$. Its absolute continuity is proved as well as a suitable eigenfunction expansion. $V$ is a "Stark-like" potential. In particular, the cases $ V({x_1}) = ( - \operatorname{sgn}{x_1})\vert{x_1}\,{\vert^\alpha },0 < \alpha \leqslant 2$, are included. ${V_p}$ may be taken as the sum of an $ {L^2}$-function and a function satisfying growth conditions in the $ + {x_1}$ direction. ${V_p}(x) = \vert x\vert^{ - 1}$ is included.
Group actions on aspherical $A\sb{k}(N)$-manifolds
Hsü Tung
Ku;
Mei Chin
Ku
841-859
Abstract: By an aspherical $ {A_k}(N)$-manifold, we mean a compact connected manifold $M$ together with a map $f$ from $M$ into an aspherical complex $N$ such that ${f^{\ast}}: H^k(N;Q)\to H^k(M;Q)$ is nontrivial. In this paper we shall show that if ${S^1}$ acts effectively and smoothly on a smooth aspherical ${A_k}(N)$-manifold, $k > 1$, $N$ a closed oriented Riemannian $k$-manifold, with strictly negative curvature, and the $K$-degree $ K(f) \ne 0$, then the fixed point set $F$ is not empty, and at least one component of $F = { \cup_{j}}{F_j}$ is an aspherical ${A_k}(N)$-manifold. Moreover, ${\operatorname{Sign}}(f) = {\Sigma_j}\,{\operatorname{Sign}}(f\vert{F_j})$. We also study the degree of symmetry and semisimple degree of symmetry of aspherical $ {A_k}(N)$-manifolds.
An extension of the Fuglede commutativity theorem modulo the Hilbert-Schmidt class to operators of the form $\sum M\sb{n}XN\sb{n}$
Gary
Weiss
1-20
Abstract: We study the operators $\Delta (X) = \sum\nolimits_1^n {{M_n}X{N_n}}$ and ${\Delta^{\ast}}(X) = \sum\nolimits_1^n {M_n^{\ast}XN_n^{\ast}}$ which map the algebra of all bounded linear operators on a separable Hubert space to itself, where $\langle {M_n}\rangle_1^m$ and $\langle {N_n}\rangle_1^m$ are separately commuting sequences of normal operators. We prove that (1) when $m \leqslant 2$, the Hilbert-Schmidt norms of $\Delta (X)$ and $ {\Delta^{\ast}}(X)$ are equal (finite or infinite); (2) for $m \geqslant 3$, if $ \Delta (X)$ and ${\Delta^{\ast}}(X)$ are Hilbert-Schmidt operators, then their Hilbert-Schmidt norms are equal; (3) if $ \Delta ,{\Delta^{\ast}}$ have the property that for each $X,\Delta (X) = 0$ implies ${\Delta^{\ast}}(X) = 0$, then for each $X$, if $ \Delta (X)$ is a Hilbert-Schmidt operator then ${\Delta^{\ast}}^2(X)$ is also and the latter has the same Hilbert-Schmidt norm as ${\Delta ^2}(X)$. Note that Fuglede's Theorem is immediate from $(1)$ in the case $m = 2,{M_1} = {N_2}$ and ${N_1} = I = - {M_2}$. The proofs employ the duality between the trace class and the class of all bounded linear operators and, unlike the early proofs of Fuglede's Theorem, they are free of complex function theory.
Semilinear parabolic problems define semiflows on $C\sp{k}$ spaces
Xavier
Mora
21-55
Abstract: Linear parabolic problems of a general class are proved to determine analytic semigroups on certain closed subspaces of ${C^k}(\overline \Omega )$ ($k$ integer); ${C^k}(\overline \Omega )$ denotes the space of functions whose derivatives or order $\leqslant k$ are bounded and uniformly continuous, with the usual supremum norm; the closed subspaces where the semigroups are obtained, denoted by ${\hat C^k}(\overline \Omega )$, are determined by the boundary conditions and a possible condition at infinity. One also obtains certain embedding relations concerning the fractional power spaces associated to these semigroups. Usually, results of this type are based upon the theory of solution of elliptic problems, while this work uses the corresponding theory for parabolic problems. The preceding results are applied to show that certain semilinear parabolic problems define semiflows on spaces of the type ${\hat C^k}(\overline \Omega )$.
Uniqueness of $\Gamma \sb{p}$ in the Gross-Koblitz formula for Gauss sums
Alan
Adolphson
57-63
Abstract: It is determined what continuous functions besides the $p$-adic gamma function make the Gross-Koblitz formula valid.
A Boolean algebra with few subalgebras, interval Boolean algebras and retractiveness
Matatyahu
Rubin
65-89
Abstract: Using $ {\diamondsuit_{{\aleph_1}}}$ we construct a Boolean algebra $B$ of power $ {\aleph_1}$, with the following properties: (a) $B$ has just $ {\aleph_1}$ subalgebras. (b) Every uncountable subset of $B$ contains a countable independent set, a chain of order type $\eta$, and three distinct elements $ a,b$ and $c$, such that $ a \cap b = c$. (a) refutes a conjecture of J. D. Monk, (b) answers a question of R. McKenzie. $B$ is embeddable in $ P(\omega )$. A variant of the construction yields an almost Jónson Boolean algebra. We prove that every subalgebra of an interval algebra is retractive. This answers affirmatively a conjecture of B. Rotman. Assuming MA or the existence of a Suslin tree we find a retractive BA not embeddable in an interval algebra. This refutes a conjecture of B. Rotman. We prove that an uncountable subalgebra of an interval algebra contains an uncountable chain or an uncountable antichain. Assuming CH we prove that the theory of Boolean algebras in Magidor's and Malitz's language is undecidable. This answers a question of M. Weese.
Persistently finite theories with hyperarithmetic models
Terrence
Millar
91-99
Abstract: Nerode asked if there could be a complete decidable theory with only finitely many countable models up to isomorphism, such that not all of the countable models were decidable. Morley, Lachlan, and Peretyatkin produced examples of such theories. However, all the countable models of those theories were decidable in $0^{\prime}$. The question then arose whether all countable models of such theories had to be, for example, arithmetic. In this paper we provide a negative answer to that question by showing that there are such examples with countable models of arbitrarily high hyperarithmetic degree. It is not difficult to show that any countable model of a hyperarithmetic theory which has only finitely many countable models must be decidable in some hyperarithmetic degree.
On quasiprojective covers
Theodore G.
Faticoni
101-113
Abstract: The main results determine the Goldie dimension of superfluous submodules of semiprime left Goldie rings and apply this to the study of quasiprojective covers of torsion free modules. Conditions are given to guarantee that a quasiprojective cover of a torsion free module is an isomorphism. A class of nonperfect rings is given such that finitely generated singular modules possess quasiprojective covers.
Analytic functionals with unbounded carriers and mean periodic functions
Alex
Meril
115-136
Abstract: We study certain ideals in some spaces of analytic functionals with unbounded carriers introduced by T. Kawaï, M. Morimoto and J. W. de Roever. Using Banach algebra methods, we show an example of space without spectral synthesis. Using Hörmander's ${L^2}$ estimates, we prove a spectral synthesis theorem for mean periodic functions.
A new bound for the Steiner ratio
D. Z.
Du;
F. K.
Hwang
137-148
Abstract: Let $V$ denote a given set of $n$ points in the euclidean plane. A Steiner minimal tree for $V$ is the shortest network (clearly, it has to be a tree) interconnecting $V$. Junctions of the network which are not in $ V$ are called Steiner points (those in $V$ will be called regular points). A shortest tree interconnecting $V$ without using any Steiner points is called a minimal tree. Let $ \sigma (V)$ and $ \mu (V)$ denote the lengths of a Steiner minimal tree and a minimal tree, respectively. Define $\rho$ to be the greatest lower bound for the ratio $ \sigma (V)/\mu (V)$ over all $V$. We prove $\rho > .8$.
Steiner minimal trees on zig-zag lines
D. Z.
Du;
F. K.
Hwang;
J. F.
Weng
149-156
Abstract: A Steiner minimal tree for a given set $P$ of points in the Euclidean plane is a shortest network interconnecting $P$ whose vertex set may include some additional points. The construction of Steiner minimal trees has been proved to be an $NP$-complete problem for general $P$. However, the $ NP$-completeness does not exclude the possibility that Steiner trees for sets of points with special structures can be efficiently determined. In this paper we determine the Steiner mimmal trees for zig-zag lines with certain regularity properties. We also give an explicit formula for the length of such a tree.
Earthquakes and tessellations of Teichm\"uller space
Peter
Waterman;
Scott
Wolpert
157-167
Abstract: A brief exposition of the Teichmüller and earthquake theory is given for the once punctured torus. Using these ideas, computer plots, of the tessellation of Teichmüller space by the mapping class group and of the earthquake paths leaving a point, were generated. The algorithms for generating the plots are discussed.
On the rate of convergence of moments in the central limit theorem for lattice distributions
Peter
Hall
169-181
Abstract: We derive sharp asymptotic expressions for moments of the type $ E\{ b(\vert{S_n}/{n^{1/2}}\vert)\}$, where ${S_n}$ is a sum of independent lattice-valued random variables with finite variance, and $b$ is a concave function. It is shown that the behaviour of $b$ at the origin has a profound effect on the behaviour of such moments, and that this influence accounts for the major difference between the properties of moments of lattice and nonlattice sums. Asymptotic expansions for moments of sums of lattice-valued variables are also derived.
Bounds for integral solutions of diagonal cubic equations
Ka Hin
Leung
183-195
Abstract: It was proved by Davenport [3] that for the nonzero integral ${\lambda_i}$ the equation ${\lambda_1}x_1^3 + \cdots + {\lambda_8}x_8^3 = 0$ always has a nontrivial integral solution. In this paper, we investigate the bounds of nontrivial integral solutions in terms of ${\lambda_1}, \ldots ,{\lambda_8}$.
Stop rule inequalities for uniformly bounded sequences of random variables
Theodore P.
Hill;
Robert P.
Kertz
197-207
Abstract: If ${X_{0}},{X_1},\ldots$ is an arbitrarily-dependent sequence of random variables taking values in $ [0,1]$ and if $V({X_0},{X_1},\ldots)$ is the supremum, over stop rules $t$, of $E{X_t}$, then the set of ordered pairs $\{ (x,y):x = V({X_0},{X_1},\ldots,{X_n})$ and $y = E({\max_{j\, \leqslant \,n}}{X_j})$ for some $ {X_0},\ldots,{X_n}\}$ is precisely the set $\displaystyle {C_n} = \{ (x,y):x \leqslant y \leqslant x\,( {1 + n\,(1 - {x^{1/n}})} );0 \leqslant x \leqslant 1\} ;$ and the set of ordered pairs $\{ (x,y):x = V({X_{0}},{X_1},\ldots)$ and $ y = E({\sup_n}\;{X_n})$ for some $ {X_0},{X_1},\ldots\}$ is precisely the set $\displaystyle C = \bigcup\limits_{n = 1}^\infty {{C_n}} .$ As a special case, if ${X_0},{X_1},\ldots$ is a martingale with $E{X_0} = x$, then $ E({\max_{j \leqslant n}} X) \leqslant x + nx(1 - {x^{1\,/\,n}})$ and $ E({\sup_n}\;{X_n}) \leqslant x - x\ln \;x$, and both inequalities are sharp.
Spaces of complex null geodesics in complex-Riemannian geometry
Claude
LeBrun
209-231
Abstract: The notion of a complex - Riemannian $n$-manifold, meaning a complex $ n$-manifold with a nondegenerate complex quadratic form on each tangent space which varies holomorphically from point to point, is briefly developed. It is shown that, provided $n \geqslant 4$, every such manifold locally arises canonically as the moduli space of all quadrics of a fixed normal-bundle type in an associated space of complex null geodesies. This relationship between local geometry and global complex analysis is stable under deformations.
Toral actions on $5$-manifolds
Hae Soo
Oh
233-252
Abstract: We are mainly concerned with closed orientable manifolds of dimension $ 5$ supporting effective three-dimensional torus actions. We obtain a complete classification of simply-connected manifolds of this type and a partial classification for the nonsimply-connected case.
Bi-interpretable groups and lattices
M.
Jambu-Giraudet
253-269
Abstract: A large class of 0-$2$ transitive lattice-ordered groups is finitely axiomatizable as a class of groups and as a class of lattices. In each model, the group structure and the lattice structure plus one parameter are bi-interpretable, sometimes up to duality only. A characterization of lattice-automorphisms of the structures is also given.
Homology cobordisms, link concordances, and hyperbolic $3$-manifolds
Robert
Myers
271-288
Abstract: Let $M_0^3$ and $M_1^3$ be compact, oriented $3$-manifolds. They are homology cobordant (respectively relative homology cobordant) if $ \partial M_1^3 = \emptyset \;({\text{resp.}}\;\partial M_1^3 \ne \emptyset )$ and there is a smooth, compact oriented $4$-manifold ${W^4}$ such that $\partial {W^4} = M_0^3 - M_1^3$ (resp. $\partial {W^4} = M_0^3 - M_1^3) \cup (M_i^3 \times [0,1])$ and $ {H_{\ast}}(M_i^3;{\mathbf{Z}}) \to {H_{\ast}}({W^4};{\mathbf{Z}})$ are isomorphisms, $i = 0,1$. Theorem. Every closed, oriented $3$-manifold is homology cobordant to a hyperbolic $3$-manifold. Theorem. Every compact, oriented $3$-manifold whose boundary is nonempty and contains no $2$-spheres is relative homology cobordant to a hyperbolic $3$-manifold. Two oriented links $ {L_0}$ and ${L_1}$ in a $3$-manifold ${M^3}$ are concordant if there is a set ${A^2}$ of smooth, disjoint, oriented annuli in $M \times [0,1]$ such that $\partial {A^2} = {L_0} - {L_1}$, where ${L_{i}} \subseteq \;{M^3} \times \{ i\} ,i = 0,1$. Theorem. Every link in a compact, oriented $ 3$-manifold $ {M^3}$ whose boundary contains no $2$-spheres is concordant to a link whose exterior is hyperbolic. Corollary. Every knot in ${S^3}$ is concordant to a knot whose exterior is hyperbolic.
Sign changes in harmonic analysis on reductive groups
Robert E.
Kottwitz
289-297
Abstract: Let $G$ be a connected reductive group over a field $F$. In this note the author constructs an element $ e(G)$ of the Brauer group of $F$. The square of this element is trivial. For a local field, $e(G)$ may be regarded as an element of $\{ \pm 1\}$ and is needed for harmonic analysis on reductive groups over that field. For a global field there is a product formula.
Existence of infinitely many solutions for a forward backward heat equation
Klaus
Höllig
299-316
Abstract: Let $\phi$ be a piecewise linear function which satisfies the condition $ s\phi (s) \geqslant c{s^2},c > 0,s \in {\mathbf{R}}$, and which is monotone decreasing on an interval $(a,b) \subset {{\mathbf{R}}_ + }$. It is shown that for $f \in {C^2}[0,1]$, with $\max f^\prime > a$, there exists a $T > 0$ such that the initial boundary value problem $\displaystyle {u_t} = \phi \,{({u_x})_x},\qquad {u_x}(0,t) = {u_x}(1,t) = 0,\qquad u( \cdot ,0) = f,$ has infinitely many solutions $u$ satisfying $\parallel \;u\;{\parallel_{\alpha }},\parallel \;{u_x}{\parallel_{\infty }},\parallel \;{u_t}{\parallel_{2}} \leqslant c(f,\phi )$ on $[0,1] \times [0,T]$.
Inverting the half-jump
S.
Homer;
G. E.
Sacks
317-331
Abstract: Assume $ \beta$ is weakly admissible over 0 and $ {0^{1\,/\,2}}$. It follows that the $\beta$-recursively enumerable degrees are dense. In addition each $\beta$-recursively enumerable degree above ${0^{1\,/\,2}}$ is the half-jump of some tamely $ \beta$-recursively enumerable degree below $ {0^{1\,/\,2}}$.
Convergence of linear and nonlinear Pad\'e approximants from series of orthogonal polynomials
D. S.
Lubinsky;
A.
Sidi
333-345
Abstract: Analogues of the Nuttall-Pommerenke theorem and Wallin-type theorems for classical Padé approximants, are proved for linear and nonlinear Padé approximants formed from series of orthogonal polynomials, corresponding to a distribution $ d\alpha (x)$ with at most finitely many sign changes.
The structure of rings with faithful nonsingular modules
J. M.
Zelmanowitz
347-359
Abstract: It is shown that the existence of a faithful nonsingular uniform module characterizes rings which have a full linear maximal quotient ring. New information about the structure of these rings is obtained and their maximal quotient rings are constructed in an explicit manner. More generally, rings whose maximal quotient rings are finite direct sums of full linear rings are characterized by the existence of a faithful nonsingular finite dimensional module.
On the generators of the first homology with compact supports of the Weierstrass family in characteristic zero
Goro C.
Kato
361-368
Abstract: Let ${{\mathbf{W}}_{\mathbf{Q}}} = \operatorname{Proj}({\mathbf{Q}}[{{\text{g}}_2},... ...eous ideal generated by }} - {Y^2}Z + 4\,{X^3} - {g_2}\,X{Z^2} - {g_3}\,{Z^3}))$. This is said to be the Weierstrass Family over the field $ {\mathbf{Q}}$. Then the first homology with compact supports of the Weierstrass Family is computed explicitly, i.e., it is generated by ${\{ {C^{ - k}}\,dX\, \wedge \;dY\}_{k \geqslant 1}}$ and ${\{ X{C^{ - k}}dX \wedge \,dY\}_{k\, \geqslant 1}}$ over the ring $ {\mathbf{Q}}[{g_{2}},{g_3}]$, where $C$ is a polynomial ${Y^2} - 4{X^3} + {g_2}X + {g_3}$. When one tensors the homology of the Weierstrass Family with ${\Delta ^{ - 1}}\,{\mathbf{Q}}[{g_2},{g_3}]$, being localized at the discriminant $\Delta = g_2^3 - 27g_3^2$, over $ {\mathbf{Q}}[{{\text{g}}_2},{g_3}]$, the first homology is generated by $ {C^{ - 1}}dX\; \wedge \;dY$ and $ X{C^{ - 1}}dX\, \wedge dY$. One also obtains the first homologies with compact supports of singular fibres over $\wp = ({g_2} = {g_{3}} = 0)$ and $\wp = \,({g_2} = 3,{g_{3}} = 1)$ as corollaries.
A note on Michael's problem concerning the Lindel\"of property in the Cartesian products
K.
Alster
369-375
Abstract: In this note we present a sketch of a negative solution of the Michael's conjecture which says that if the product $Y \times X$ is Lindelöf for every hereditarily Lindelöf space $Y$, then $ Y \times {X^\omega }$ is Lindelöf for every hereditarily Lindelöf space $Y$.
Equilibrium states of an elastic conductor in a magnetic field: a paradigm of bifurcation theory
Peter
Wolfe
377-387
Abstract: In this paper we study the equilibrium states of a nonlinearly elastic conducting wire in a magnetic field. The wire is perfectly flexible and is suspended between fixed supports. The wire carries an electric current and is subjected to a constant magnetic field whose direction is parallel to the line between the supports. We solve this problem exactly and show that the set of solutions gives rise to a paradigmatic bifurcation diagram. We then carry out a study of the equations obtained by linearization about the nontrivial solutions in order to gain some insight into the stability of the various solution branches.
The nest subgroups of Kleinian groups
Takehiko
Sasaki
389-399
Abstract: The residual limit points of finitely generated Kleinian groups are classified into two types: the first kind and the second kind. To each point of the second kind, Abikoff associated a web subgroup. We shall classify the points of the first kind into two types and associate to each point of one type a finitely generated subgroup, the nest subgroup. To the points of the other type we shall give a significance showing that they are important for the sets of generators.
Solvable groups acting on the line
J. F.
Plante
401-414
Abstract: Actions of discrete groups on the real line are considered. When the group of homeomorphisms is solvable several sufficient conditions are given which guarantee that the group is semiconjugate to a subgroup of the affine group of the line. In the process of obtaining these results sufficient conditions are also determined for the existence of invariant (quasi-invariant) measures for abelian (solvable) groups acting on the line. It turns out, for example, that any solvable group of real analytic diffeomorphisms or a polycyclic group of homeomorphisms has a quasi-invariant measure, and that any abelian group of $ {C^2}$ diffeomorphisms has an invariant measure. An example is given to show that some restrictions are necessary in order to obtain such conclusions. Some applications to the study of codimension one foliations are indicated.
The Gauss-Lucas theorem and Jensen polynomials
Thomas
Craven;
George
Csordas
415-429
Abstract: A characterization is given of the sequences $\{ {\gamma_k}\}_{k = 0}^\infty$ with the property that, for any complex polynomial $f(z) = \Sigma \,{a_k}{z^k}$ and convex region $K$ containing the origin and the zeros of $ f$, the zeros of $ \Sigma \,{\gamma_k}{a_k}{z^k}$ again lie in $K$. Many applications and related results are also given. This work leads to a study of the Taylor coefficients of entire functions of type I in the Laguerre-Pólya class. If the power series of such a function is given by $ \Sigma \,{\gamma_k}{z^k}/k!$ and the sequence $ \{ {\gamma_k}\}$ is positive and increasing, then the sequence satisfies an infinite collection of strong conditions on the differences, namely ${\Delta ^n}{\gamma_k} \geqslant 0$ for all $ n$, $k$.
On the detonation of a combustible gas
Robert A.
Gardner
431-468
Abstract: This paper is concerned with the existence of detonation waves for a combustible gas. The equations are those of a viscous, heat conducting, polytropic gas coupled with an additional equation which governs the evolution of the mass fraction of the unburned gas (see (1)). The reaction is assumed to be of the simplest form: $A \to B$, i.e., there is a single product and a single reactant. The main result (see Theorem 2.1) is a rigorous existence theorem for strong, and under certain conditions, weak detonation waves for explicit ranges of the viscosity, heat conduction, and species diffusion coefficients. In other words, a class of admissible "viscosity matrices" is determined. The problem reduces to finding an orbit of an associated system of four ordinary differential equations which connects two distinct critical points. The proof employs topological methods, including Conley's index of isolated invariant sets.
The Atiyah-Singer invariant, torsion invariants, and group actions on spheres
Donald E.
Smith
469-488
Abstract: This paper deals with the classification of cyclic group actions on spheres using the Atiyah-Singer invariant and Reidemeister-type torsion. Our main tool is the computation of the group of relative homotopy triangulations of the product of a disk and a lens space. These results are applied to obtain lower bounds on the image of an equivariant $J$-homomorphism.
Extension of Wiener's Tauberian identity and multipliers on the Marcinkiewicz space
Ka-Sing
Lau
489-506
Abstract: This is a continuation of the work of Bertrandias, Lee and Lau on Wiener's generalized harmonic analysis. Among the other results, we extend Wiener's Tauberian identity to cover a larger class of functions; we characterize the multipliers on the Marcinkiewicz space ${\mathcal{M}^2}$, and we obtain a Tauberian theorem on $ {\mathcal{M}^2}$ with full generality.
Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium
Juan Luis
Vázquez
507-527
Abstract: The one-dimensional porous media equation ${u_t} = {({u^m})_{xx}}$, $m > 1$, is considered for $x \in R$, $t > 0$ with initial conditions $u(x,0) = {u_0}(x)$ integrable, nonnegative and with compact support. We study the behaviour of the solutions as $t \to \infty $ proving that the expressions for the density, pressure, local velocity and interfaces converge to those of a model solution. In particular the first term in the asymptotic development of the free-boundary is obtained.
Linear superpositions with mappings which lower dimension
Y.
Sternfeld
529-543
Abstract: It is shown that for every $n$-dimensional compact metric space $X$, there exist $2n + 1$ functions $\{ {\varphi _j}\}_{j = 1}^{2n + 1}$ in $ C(X)$ and $n$ mappings $\{ {\psi _i}\}_{i = 1}^n$ on $X$ with $1$-dimensional range each, with the following property: for every $0 \leqslant k \leqslant n$, every $k$ tuple $\{ {\psi_{i_l}}\}_{l = 1}^k$ of the $ {\psi _i}$'s and every $2(n - k) + 1$ tuple $\{ {\varphi _{{j_m}}}\}_{m = 1}^{2(n - k) + 1}$ of the $ {\varphi_j}$'s, each $f \in C(X)$ can be represented as $f(x) = \Sigma _{l = 1}^k{g_l}({\psi _{{i_l}}}(x)) + \Sigma_{m = 1}^{2(n - k) + 1}{h_m}({\varphi_{{j_m}}}(x))$, with ${g_l} \in C({\psi _{{i_l}}}(X))$ and ${h_m} \in C(R)$. It is also shown that in many cases the number $ 2(n - k) + 1$ is the smallest possible.
Meromorphic functions that share four values
Gary G.
Gundersen
545-567
Abstract: An old theorem of ${\text{R}}$. Nevanlinna states that if two distinct nonconstant meromorphic functions share four values counting multiplicities, then the functions are Möbius transformations of each other, two of the shared values are Picard values for both functions, and the cross ratio of a particular permutation of the shared values equals -1. In this paper we show that if two nonconstant meromorphic functions share two values counting multiplicities and share two other values ignoring multiplicities, then the functions share all four values counting multiplicities.
Quadratic spaces over Laurent extensions of Dedekind domains
Raman
Parimala
569-578
Abstract: Let $R$ be a Dedekind domain in which $ 2$ is invertible. We show in this paper that any isotropic quadratic space over $R[T,{T^{ - 1}}]$ is isometric to ${q_1} \perp T{q_2}$ where ${q_1},{q_2}$ are quadratic spaces over $ R$. We give an example to show that this result does not hold for anisotropic spaces.
Independence results on the global structure of the Turing degrees
Marcia J.
Groszek;
Theodore A.
Slaman
579-588
Abstract: From CON(ZFC) we obtain: 1. CON$ ($ZFC$+ 2^\omega$ is arbitrarily large $ +$ there is a locally finite upper semilattice of size $ {\omega_2}$ which cannot be embedded into the Turing degrees as an upper semilattice). 2. CON$($ZFC$+ 2^\omega$ is arbitrarily large $ +$ there is a maximal independent set of Turing degrees of size ${\omega _1}$).
The symmetric derivative
Lee
Larson
589-599
Abstract: It is shown that all symmetric derivatives belong to Baire class one, and a condition characterizing all measurable symmetrically differentiable functions is presented. A method to find a well-behaved primitive for any finite symmetric derivative is introduced, and several of the standard theorems of differential calculus are extended to include the symmetric derivative.
Tangent cones and quasi-interiorly tangent cones to multifunctions
Lionel
Thibault
601-621
Abstract: R. T. Rockafellar has proved a number of rules of subdifferential calculus for nonlocally lipschitzian real-valued functions by investigating the Clarke tangent cones to the epigraphs of such functions. Following these lines we study in this paper the tangent cones to the sum and the composition of two multifunctions. This will be made possible thanks to the notion of quasi-interiorly tangent cone which has been introduced by the author for vector-valued functions in [29] and whose properties in the context of multifunctions are studied. The results are strong enough to cover the cases of real-valued or vector-valued functions.
Some examples of square integrable representations of semisimple $p$-adic groups
George
Lusztig
623-653
Abstract: We construct irreducible representations of the Hecke algebra of an affine Weyl group analogous to Kilmoyer's reflection representation corresponding to finite Weyl groups, and we show that in many cases they correspond to a square integrable representation of a simple $p$-adic group.
Ambiently universal sets in $E\sp{n}$
David G.
Wright
655-664
Abstract: For each closed set $ X$ in ${E^n}$ of dimension at most $ n - 3$, we show that $ X$ fails to be ambiently universal with respect to Cantor sets in $ {E^n}$; i.e., we find a Cantor set $Y$ in ${E^n}$ so that for any self-homeomorphism $ h$ of ${E^n}$, $h(Y)$ is not contained in $X$. This result answers a question posed by H. G. Bothe and completes the understanding of ambiently universal sets in ${E^n}$.
Stable orbits of differentiable group actions
Dennis
Stowe
665-684
Abstract: We prove that a compact orbit of a smooth Lie group action is stable provided the first cohomology space vanishes for the normal representation at some (equivalently, every) point of the orbit. When the orbit is a single point, the acting group need only be compactly generated and locally compact for this conclusion to hold. Applied to foliations, this provides a sufficient condition for the stability of a compact leaf and includes the stability theorems of Reeb and Thurston and of Hirsch as cases.
Rotation hypersurfaces in spaces of constant curvature
M.
do Carmo;
M.
Dajczer
685-709
Abstract: Rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed. A local characterization of such hypersurfaces, with dimensions greater than two, is given in terms of principal curvatures. Some special cases of rotation hypersurfaces, with constant mean curvature, in hyperbolic space are studied. In particular, it is shown that the well-known conjugation between the belicoid and the catenoid in euclidean three-space extends naturally to hyperbolic three-space $H^3$; in the latter case, catenoids are of three different types and the explicit correspondence is given. It is also shown that there exists a family of simply-connected, complete, embedded, nontotally geodesic stable minimal surfaces in $H^3$.
Gauss sums and Fourier analysis on multiplicative subgroups of $Z\sb{q}$
Harold G.
Diamond;
Frank
Gerth;
Jeffrey D.
Vaaler
711-726
Abstract: Let $G(q)$ denote the multiplicative group of invertible elements in $ {{\mathbf{Z}}_q}$, the ring of integers modulo $q$. Let $H \subseteq G(q)$ be a multiplicative subgroup with cosets $aH$ and $bH$. If $f: {\mathbf{Z}}_q \to {\mathbf{C}}$ is supported in $aH$ we show that $f$ can be recovered from the values of $\hat f$ restricted to $bH$ if and only if Gauss sums for $ H$ are nonvanishing. Here $ \hat f$ is the (finite) Fourier transform of $f$ with respect to the additive group ${{\mathbf{Z}}_q}$. The main result is a simple criterion for deciding when these Gauss sums are nonvanishing. If $H = G(q)$ then $f$ can be recovered from $\hat f$ restricted to $G(q)$ by a particularly elementary formula. This formula provides some inequalities and extremal functions.
Rees matrix covers for locally inverse semigroups
D. B.
McAlister
727-738
Abstract: A regular semigroup $ S$ is locally inverse if each local submonoid $eSe$, $e$ an idempotent, is an inverse semigroup. It is shown that every locally inverse semigroup is an image of a regular Rees matrix semigroup, over an inverse semigroup, by a homomorphism $\theta$ which is one-to-one on each local submonoid; such a homomorphism is called a local isomorphism. Regular semigroups which are locally isomorphic images of regular Rees matrix semigroups over semilattices are also characterized.
Strongly Cohen-Macaulay schemes and residual intersections
Craig
Huneke
739-763
Abstract: This paper studies the local properties of closed subschemes $ Y$ in Cohen-Macaulay schemes $X$ such that locally the defining ideal of $ Y$ in $X$ has the property that its Koszul homology is Cohen-Macaulay. Whenever this occurs $ Y$ is said to be strongly Cohen-Macaulay in $X$. This paper proves several facts about such embeddings, chiefly with reference to the residual intersections of $Y$ in $X$. The main result states that any residual intersection of $Y$ in $X$ is again Cohen-Macaulay.
Isomorphism types in wreath products and effective embeddings of periodic groups
Kenneth K.
Hickin;
Richard E.
Phillips
765-778
Abstract: For any finitely generated group $ Y,\omega (Y)$ denotes the Turing degree of the word problem of $Y$. Let $G$ be any non-Abelian $2$-generator group and $B$ an infinite group generated by $k \geqslant 1$ elements. We prove that if $ \tau$ is any Turing degree with $\tau \geqslant 1.{\text{u.b.}}\{ {\omega (G),\omega (B)} \}$ then the unrestricted wreath product $ W = G{\text{Wr}}\,B$ has a $( {k + 1} )$-generator subgroup $H$ with $ \omega (H) = \tau$. If $ B$ is also periodic, then $ W$ has a $k$-generator subgroup $H$ such that $\tau = 1.{\text{u.b.}}\{ {\omega (B),\omega (H)} \}$. Easy consequences include: $ G{\text{Wr}}\,{\mathbf{Z}}$ has ${2^{{\aleph _0}}}$ pairwise nonembeddable $2$-generator subgroups and if $B$ is periodic then $G{\text{Wr}}\,B$ has ${2^{{\aleph _0}}}$ pairwise nonembeddable $ k$-generator subgroups. Using similar methods, we prove an effective embedding theorem for embedding countable periodic groups in $ 2$-generator periodic groups.
The optimal accuracy of difference schemes
Arieh
Iserles;
Gilbert
Strang
779-803
Abstract: We consider difference approximations to the model hyperbolic equation ${u_{t}} = {u_x}$ which compute each new value $ U(x,t + \Delta t)$ as a combination of the known values $ U(x - r\Delta x,t),\ldots,U(x + s\Delta x,\Delta t)$. For such schemes we find the optimal order of accuracy: stability is possible for small $ \Delta t/\Delta x$ if and only if $p \leqslant \min \{ {r + s,2r + 2,2s} \}$. A similar bound is established for implicit methods. In this case the most accurate schemes are based on Padé approximations $P(z)/Q(z)$ to $ {z^\lambda }$ near $ z = 1$, and we find an expression for the difference $\vert Q{\vert^2} - \vert P{\vert^2}$; this allows us to test the von Neumann condition $\vert P/Q\vert \leqslant 1$. We also determine the number of zeros of $Q$ in the unit circle, which decides whether the implicit part is uniformly invertible.
Nonanalytic solutions of certain linear PDEs
E. C.
Zachmanoglou
805-814
Abstract: It is shown that if $ P$ is a linear partial differential operator with analytic coefficients, and if $M$ is an analytic submanifold of codimensions $ 3$ in ${{\mathbf{R}}^n}$, which is partially characteristic with respect to $P$ and satisfies certain additional conditions, then one can find, in a neighborhood of any point of $ M$, solutions of the equation $Pu = 0$ which are flat or singular precisely on $ M$. The additional condition requires that a nonhomogeneous Laplace equation in two variables possesses a solution with a strong extremum at the origin. The right side of this nonhomogeneous equation is a homogeneous polynomial in two variables with coefficients being repeated Poisson brackets of the real and imaginary parts of the principal symbol of $P$.
Projections onto translation-invariant subspaces of $L\sb{1}({\bf R})$
Dale E.
Alspach;
Alec
Matheson
815-823
Abstract: The complemented translation-invariant subspaces of ${L_1}({\mathbf{R}})$ are characterized. This completes an investigation begun by H. P. Rosenthal.
Small into-isomorphisms between spaces of continuous functions. II
Yoav
Benyamini
825-833
Abstract: We construct two compact Hausdorff spaces, $X$ and $Y$, so that $C(X)$ does not embed isometrically into $ C(Y)$, but for each $\varepsilon > 0$, there is an isomorphism ${T_\varepsilon }$ from $C(X)$ into $C(Y)$ satisfying $\parallel f\parallel \leqslant \parallel {T_\varepsilon }f\;\parallel \leqslant (1 + \varepsilon)\parallel f\parallel$ for all $f \in C(X)$.
Borel functions of bounded class
D. H.
Fremlin;
R. W.
Hansell;
H. J. K.
Junnila
835-849
Abstract: Let $X$ and $Y$ be metric spaces and $f:X \to Y$ a Borel measurable function. Does $ f$ have to be of bounded class, i.e. are the sets $ {f^{ - 1}}[ H ]$, for open $H \subseteq Y$, of bounded Baire class in $X?$ This is an old problem of A. H. Stone. Positive answers have been given under a variety of extra hypotheses and special axioms. Here we show that (i) unless something similar to a measurable cardinal exists, then $f$ is of bounded class and (ii) if $f$ is actually a Borel isomorphism, then $f\,({\text{and}} {f^{ - 1}})$ are of bounded class.
Spectral decomposition with monotonic spectral resolvents
I.
Erdélyi;
Sheng Wang
Wang
851-859
Abstract: The spectral decomposition problem of a Banach space over the complex field entails two kinds of constructive elements: (1) the open sets of the field and (2) the invariant subspaces (under a given linear operator) of the Banach space. The correlation between these two structures, in the framework of a spectral decomposition, is the spectral resolvent concept. Special properties of the spectral resolvent determine special types of spectral decompositions. In this paper, we obtain conditions for a spectral resolvent to have various monotonic properties.
Viscosity solutions of Hamilton-Jacobi equations
Michael G.
Crandall;
Pierre-Louis
Lions
1-42
Abstract: Problems involving Hamilton-Jacobi equations--which we take to be either of the stationary form $H(x,u,Du) = 0$ or of the evolution form ${u_{t}} + H(x,t,u,Du) = 0$, where $Du$ is the spatial gradient of $ u$--arise in many contexts. Classical analysis of associated problems under boundary and/or initial conditions by the method of characteristics is limited to local considerations owing to the crossing of characteristics. Global analysis of these problems has been hindered by the lack of an appropriate notion of solution for which one has the desired existence and uniqueness properties. In this work a notion of solution is proposed which allows, for example, solutions to be nowhere differentiable but for which strong uniqueness theorems, stability theorems and general existence theorems, as discussed herein, are all valid.
Conformally invariant variational integrals
S.
Granlund;
P.
Lindqvist;
O.
Martio
43-73
Abstract: Let $f:G \to {R^n}$ be quasiregular and $I = \int {F(x,\nabla \,u)\,dm}$ a conformally invariant variational integral. Hölder-continuity, Harnack's inequality and principle are proved for the extremals of $I$. Obstacle problems and their connection to subextremals are studied. If $u$ is an extremal or a subextremal of $ I$, then $u \circ f$ is again an extremal or a subextremal if an appropriate change in $F$ is made.
Trace class self-commutators
C. A.
Berger;
Marion Glazerman
Ben-Jacob
75-91
Abstract: This paper extends earlier results of Berger and Shaw to all $ {W^\ast}$ algebras. The multiplicity of an operator in a ${W^\ast}$ algebra is defined in terms of the trace on the ${W^\ast}$-algebra, and it is shown that if $ T$ is a hyponormal operator in such an algebra, the trace of its self-commutator is bounded by this multiplicity times the area of the spectrum of $T$, divided by $\pi$.
Completeness and basis properties of complex exponentials
Raymond M.
Redheffer;
Robert M.
Young
93-111
Abstract: This paper is concerned with what might be termed the "fine structure" of the completeness and basis properties of complex exponentials. We give new criteria for two sequences to have the same excess in the sense of Paley and Wiener, a result that illuminates and supplements a well-known completeness criterion of Levinson, and new examples and counterexamples pertaining to Riesz bases.
Finitely generic abelian lattice-ordered groups
Dan
Saracino;
Carol
Wood
113-123
Abstract: The authors characterize the finitely generic abelian lattice-ordered groups and make application of this characterization to specific examples.
Almost convergent and weakly almost periodic functions on a semigroup
Heneri A. M.
Dzinotyiweyi
125-132
Abstract: Let $S$ be a topological semigroup, ${\text{US}}(S)$ the set of all bounded uniformly continuous functions on $S,{\text{WAP(}}S)$ the set of all (bounded) weakly almost periodic functions on $S,{E_0}(S): = \{ f \in {\text{UC(}}S):m(\vert f\vert) = 0$ for each left and right invariant mean $m$ on $ {\text{UC(}}S)\}$ and ${W_0}(S): = \{ f \in {\text{WAP}}(S):\:m(\vert f\vert) = 0$ for each left and right invariant mean $ m$ on ${\text{WAP(}}S)\}$. Among other results, for a large class of noncompact locally compact topological semigroups $S$, we show that the quotient space ${E_0}(S)/{W_0}(S)$ contains a linear isometric copy of ${l^\infty }$ and so is nonseparable.
Highest weight modules over graded Lie algebras: resolutions, filtrations and character formulas
Alvany
Rocha-Caridi;
Nolan R.
Wallach
133-162
Abstract: In this paper the study of multiplicities in Verma modules for Kac-Moody algebras is initiated. Our analysis comprises the case when the integral root system is Euclidean of rank two. Complete results are given in the case of rank two, Kac-Moody algebras, affirming the Kazhdan-Lusztig conjectures for the case of infinite dihedral Coxeter groups. The main tools in this paper are the resolutions of standard modules given in [21] and a generalization to the case of Kac-Moody Lie algebras of Jantzen's character sum formula for a quotient of two Verma modules (one of the main results of this article). Finally, a precise analogy is drawn between the rank two, Kac-Moody algebras and the Witt algebra (the Lie algebra of vector fields on the circle).
Completion of Akahori's construction of the versal family of strongly pseudoconvex CR structures
Kimio
Miyajima
163-172
Abstract: Let $M$ be a compact smooth boundary of a strongly pseudo-convex domain of a complex manifold $ N$ with dim $N \geqslant 4$. We established a sharp a priori estimate for the Laplacian operator associated with Akahori's subcomplex of the ${\bar \partial _b}$-complex to construct the complex analytic versal family (in the sense of Kuranishi) of $CR$ structures of class $ {C^\infty }$ on $ M$.
Applications of $q$-Lagrange inversion to basic hypergeometric series
Ira
Gessel;
Dennis
Stanton
173-201
Abstract: A family of $ q$-Lagrange inversion formulas is given. Special cases include quadratic and cubic transformations for basic hypergeometric series. The $ q$-analogs of the so-called "strange evaluations" are also corollaries. Some new RogersRamanujan identities are given. A connection between the work of Rogers and Andrews, and $q$-Lagrange inversion is stated.
Nonfactorization theorems in weighted Bergman and Hardy spaces on the unit ball of ${\bf C}\sp{n}$ $(n>1)$
M. Seetharama
Gowda
203-212
Abstract: Let ${A^{p,\alpha }}(B),{A^{q,\alpha }}(B)$ and ${A^{l,\alpha }}(B)$ be weighted Bergman spaces on the unit ball of ${{\text{C}}^{n}}\,(n > 1)$. We prove: Theorem 1. If $ 1/l = 1/p + 1/q$ then ${A^{p,\alpha }}(B) \cdot {A^{q,\alpha }}(B)$ is of first category in ${A^{l,\alpha }}(B)$. Theorem 2. Theorem 1 holds for Hardy spaces in place of weighted Bergman spaces. We also show that Theorems 1 and 2 hold for the polydisc ${U^n}$ in place of $B$.
On neighbourly triangulations
K. S.
Sarkaria
213-239
Abstract: A simplicial complex is called $d$-neighbourly if any $d + 1$ vertices determine a $d$-simplex. We give methods for constructing $1$-neighbourly triangulations of $ 3$- and $4$-manifolds; further we discuss some relationships between $d$-neighbourly triangulations, chromatic numbers and the problem of finding upper and lower bounds on the number of simplices and locating the zeros of the characteristic polynomial of a triangulation. A triangulation of an orientable manifold is called order-orientable if there exists some ordering of the vertices which orients the manifold. We give necessary conditions for their existence; also we construct such triangulations on $3$-dimensional handlebodies and discuss the problem of recognising finite monotone subsets of an affine space by using these ideas.
Discrete series characters and Fourier inversion on semisimple real Lie groups
Rebecca A.
Herb
241-262
Abstract: Let $G$ be a semisimple real Lie group. Explicit formulas for discrete series characters on noncompact Cartan subgroups are given. These formulas are used to give a simple formula for the Fourier transform of orbital integrals of regular semisimple orbits.
On derivations of certain algebras related to irreducible triangular algebras
Baruch
Solel
263-273
Abstract: This paper deals with derivations on algebras that are generated by a maximal abelian selfadjoint algebra of operators $\mathcal{A}$ on a Hilbert space and a group of unitary operators acting on it. A necessary and sufficient condition for such a derivation to be implemented by an operator affiliated with $ \mathcal{A}$ is given. The results are related to the study of derivations on a certain class of irreducible triangular algebras.
Systems of fixed point sets
A. D.
Elmendorf
275-284
Abstract: Let $G$ be a compact Lie group. A canonical method is given for constructing a $ G$-space from homotopy theoretic information about its fixed point sets. The construction is a special case of the categorical bar construction. Applications include easy constructions of certain classifying spaces, as well as $ G$-Eilenberg-Mac Lane spaces and Postnikov towers.
Axioms for Stiefel-Whitney homology classes of some singular spaces
Darko
Veljan
285-305
Abstract: A system of axioms for the Stiefel-Whitney classes of certain type of singular spaces is established. The main examples of these singular spaces are Euler manifolds mod$\, 2$ and homology manifolds mod$\, 2$. As a consequence, it is shown that on homology manifolds mod$ \, 2$ the generalized Stiefel conjecture holds.
Semistability at the end of a group extension
Michael L.
Mihalik
307-321
Abstract: A $1$-ended $ {\text{CW}}$-complex, $ Q$, is semistable at $ \infty$ if all proper maps $r: [0,\infty) \to Q$ are properly homotopic. If ${X_1}$ and ${X_2}$ are finite $ {\text{CW}}$-complexes with isomorphic fundamental groups, then the universal cover of ${X_1}$ is semistable at $\infty$ if and only if the universal cover of $ {X_2}$ is semistable at $ \infty$. Hence, the notion of a finitely presented group being semistable at $ \infty$ is well defined. We prove Main Theorem. Let $1 \to H \to G \to K \to 1$ be a short exact sequence of finitely generated infinite groups. If $ G$ is finitely presented, then $G$ is semistable at $\infty$. Theorem. If $ A$ and $ B$ are locally compact, connected noncompact $CW$-complexes, then $A \times B$ is semistable at $ \infty$. Theorem. $\langle\;x,y:x{y^b}{x^{ - 1}} = {y^c};b\; and \; c \; nonzero\; integers\; \rangle $ is semistable at $ \infty$. The proofs are geometrical in nature and the main tool is covering space theory.
On lexicographically shellable posets
Anders
Björner;
Michelle
Wachs
323-341
Abstract: Lexicographically shellable partially ordered sets are studied. A new recursive formulation of $ {\text{CL}}$-shellability is introduced and exploited. It is shown that face lattices of convex polytopes, totally semimodular posets, posets of injective and normal words and lattices of bilinear forms are $ {\text{CL}}$-shellable. Finally, it is shown that several common operations on graded posets preserve shellability and $ {\text{CL}}$-shellability.
Real-analytic submanifolds which are local uniqueness sets for holomorphic functions of ${\bf C}\sp{3}$
Gary A.
Harris
343-351
Abstract: The following problem is considered. Given a real-analytic two-dimensional submanifold, $M$, of complex Euclidean three-space, are ambient holomorphic functions determined by their values on $M?$ For a large class of submanifolds a necessary and sufficient condition is found for $M$ to be a local uniqueness set for holomorphic functions on complex three-space. Finally, the general problem is shown to be related to two-dimensional Nevanlinna theory.
Conservation laws of free boundary problems and the classification of conservation laws for water waves
Peter J.
Olver
353-380
Abstract: The two-dimensional free boundary problem for incompressible irrotational water waves without surface tension is proved to have exactly eight nontrivial conservation laws. Included is a discussion of what constitutes a conservation law for a general free boundary problem, and a characterization of conservation laws for two-dimensional free boundary problems involving a harmonic potential proved using elementary methods from complex analysis.
Spectra of invariant uniform and transform algebras
I.
Glicksberg
381-396
Abstract: For $G$ a locally compact abelian group, any closed invariant proper subalgebra of ${C_0}(G)$ has analytic discs in its spectrum. Related results are given for $A(G)$ and $B(G)$.
Twisting cochains and duality between minimal algebras and minimal Lie algebras
Richard M.
Hain
397-411
Abstract: An algebraic duality theory is developed between $1$-connected minimal cochain algebras of finite type and connected minimal chain Lie algebras of finite type by means of twisting cochains. The duality theory gives a concrete relationship between Sullivan's minimal models, Chen's power series connections and the various Lie algebra models of a $1$-connected topological space defined by Quillen, Allday, Baues-Lemaire and Neisendorfer. It can be used to compute the Lie algebra model of a space from the algebra model of the space and vice versa.
Decay of Walsh series and dyadic differentiation
William R.
Wade
413-420
Abstract: Let ${W_2}\,n\,[f]$ denote the ${2^n}{\text{th}}$ partial sums of the Walsh-Fourier series of an integrable function $f$. Let $ {\rho _n}(x)$ represent the ratio $ {W_2}n[f,x]/{2^n}$, for $x \in [0,1]$, and let $T(f)$ represent the function ${(\Sigma \rho _n^2)^{1/2}}$. We prove that $T(f)$ belongs to $ {L^p}[0,1]$ for all $0 < p < \infty$. We observe, using inequalities of Paley and Sunouchi, that the operator $f \to T(f)$ arises naturally in connection with dyadic differentiation. Namely, if $f$ is strongly dyadically differentiable (with derivative $\dot Df$) and has average zero on the interval [0, 1], then the ${L^p}$ norms of $f$ and $T(\dot Df)$ are equivalent when $1 < p < \infty$. We improve inequalities implicit in Sunouchi's work for the case $p = 1$ and indicate how they can be used to estimate the ${L^1}$ norm of $ T(\dot Df)$ and the dyadic $ {H^1}$ norm of $ f$ by means of mixed norms of certain random Walsh series. An application of these estimates establishes that if $f$ is strongly dyadically differentiable in dyadic ${H^1}$, then $\int_0^1 {\Sigma _{N = 1}^\infty \vert{W_N}[f,x] - {\sigma _N}[f,x]/N\,dx < \infty}$.
Relative genus theory and the class group of $l$-extensions
Gary
Cornell
421-429
Abstract: The structure of the relative genus field is used to study the class group of relative $l$-extensions. Application to class field towers of cyclic $l$-extensions of the rationals are given.
Primarit\'e de $L\sp{p}(X)$
Michèle
Capon
431-487
Abstract: Soit $X$ un espace de Banach à base symétrique. Nous étudions les opérateurs de ${L^p}(X)$ dans lui-même, en leur associant une "matrice" d'opérateurs de $ {L^p}$. Cette technique nous permet de démontrer que pour tout $p$ réel dans
Topological semiconjugacy of piecewise monotone maps of the interval
Bill
Byers
489-495
Abstract: This paper establishes a topological semiconjugacy between two piecewise monotone maps of the interval which have the same kneading sequences and do not map one turning point into another, whenever itineraries under the second map are given uniquely by their invariant coordinate. Various examples are given and consequences obtained.
Steenrod and Dyer-Lashof operations on $B{\bf U}$
Timothy
Lance
497-510
Abstract: This paper describes a simple, fast algorithm for the computation of Steenrod and Dyer-Lashof operations on $BU$. The calculations are carried out in $ H^{\ast}(BU,{{\mathbf{Z}}_{(p)}})$ and $ {H_{\ast}}(BU,{{\mathbf{Z}}_{(p)}})$ where $p$ local lifts are determined by the values on primitives and Cartan formulas. This algorithm also provides a description of Steenrod and Dyer-Lashof operations on the fiber of any $H$ map (or infinite loop map) $BU \to BU$, and applications to the classifying spaces of surgery which arise in this fashion will appear shortly.
Some conjectures on elliptic curves over cyclotomic fields
D.
Goldfeld;
C.
Viola
511-515
Abstract: We give conjectures for the mean values of Hasse-Weil type $ L$-functions over cyclotomic fields. In view of the Birch-Swinnerton-Dyer conjectures, this translates to interesting arithmetic information.
Neighborhoods of algebraic sets
Alan H.
Durfee
517-530
Abstract: In differential topology, a smooth submanifold in a manifold has a tubular neighborhood, and in piecewise-linear topology, a subcomplex of a simplicial complex has a regular neighborhood. The purpose of this paper is to develop a similar theory for algebraic and semialgebraic sets. The neighborhoods will be defined as level sets of polynomial or semialgebraic functions.
A characterization of bounded symmetric domains by curvature
J. E.
D’Atri;
I.
Dotti Miatello
531-540
Abstract: This paper will prove that a bounded homogeneous domain is symmetric if and only if, in the Bergman metric, all sectional curvatures are nonpositive.
Weakening the topology of a Lie group
T. Christine
Stevens
541-549
Abstract: With any topological group $ (G, \mathcal{U})$ one can associate a locally arcwise-connected group $ (G, {\mathcal{U}}^{\ast})$, where $ {\mathcal{U}}^{\ast}$ is stronger than $ \mathcal{U}$. $(G, \mathcal{U})$ is a weakened Lie $ (WL)$ group if $(G, {\mathcal{U}}^{\ast})$ is a Lie group. In this paper the author shows that the WL groups with which a given connected Lie group $(L,\mathcal{J})$ is associated are completely determined by a certain abelian subgroup $ H$ of $L$ which is called decisive. If $ L$ has closed adjoint image, then $H$ is the center $Z(L)$ of $L$; otherwise, $H$ is the product of a vector group $V$ and a group $J$ that contains $Z(L)$. $J/Z(L)$ is finite (trivial if $L$ is solvable). We also discuss the connection between these theorems and recent results of Goto.
Interpolating sequences for $QA\sb{B}$
Carl
Sundberg;
Thomas H.
Wolff
551-581
Abstract: Let $B$ be a closed algebra lying between ${H^\infty}$ and $ {L^\infty}$ of the unit circle. We define $QA_B = H^\infty \cap \bar{B}$, the analytic functions in $ Q_B = B \cap \bar{B}$. By work of Chang, ${Q_B}$ is characterized by a vanishing mean oscillation condition. We characterize the sequences of points $\left\{{{z_n}} \right\}$ in the open unit disc for which the interpolation problem $ f({z_n}) = {\lambda _n}, n = 1, 2,\ldots$, is solvable with $f \in {Q_B}$ for any bounded sequence of numbers $ \left\{{{\lambda _n}} \right\}$. Included as a necessary part of our proof is a study of the algebras $Q{A_B}$ and ${Q_B}$.
On genus $2$ Heegaard diagrams for the $3$-sphere
Takeshi
Kaneto
583-597
Abstract: Let $D$ be any genus $2$ Heegaard diagram for the $ 3$-sphere and $\left\langle {{a_1}, {a_2}; {{\tilde r}_1}, {{\tilde r}_2}} \right\rangle$ be the cyclically reduced presentation associated with $D$. We shall show that ${{\tilde{r}}_1}$ contains ${{\tilde{r}}_2}$ or ${\tilde{r}}_2^{-1}$ as a subword in cyclic sense if $ \left\{{\tilde r}_1, {\tilde r}_2 \right\} \ne \left\{{a_1}^{\pm 1}, {a_2}^{\pm 1} \right\}$ holds, and that, using this property, $\left\langle {a_1}, {a_2};{r_1}, {r_2} \right\rangle$ can be transformed to the trivial one $\left\langle {{a_1}, {a_2};{a_1}^{\pm 1}, a_2^{\pm 1}} \right\rangle$. By the recent positive solution of genus $2$ Poincaré conjecture, our result implies the purely algebraic, algorithmic solution to the decision problem; whether a given $3$-manifold with a genus $2$ Heegaard splitting is simply connected or not, equivalently, is homeomorphic to the $ 3$-sphere or not.
The derived functors of the primitives for ${\rm BP}\sb\ast (\Omega S\sp{2n+1})$
Martin
Bendersky
599-619
Abstract: Formulas for the Hopf invariant, and the $P$ map in the Novikov double suspension sequence are derived. The formulas allow an effective inductive computation of the ${E_2}$-term of the unstable Adams-Novikov spectral sequence. The $3$ primary ${E_2}$-term through the $54$ stem is displayed.
A correction and some additions to: ``Fundamental solutions for differential equations associated with the number operator''
Yuh Jia
Lee
621-624
Abstract: Let $(H,B)$ be an abstract Wiener pair and $\mathfrak{N}$ the operator defined by $\mathfrak{N}u(x) = - {\text{trace}}_H{D^2}u(x) + (x,Du(x))$, where $x \in B$ and $ (\cdot, \cdot )$ denotes the $ B$-$B^{\ast}$ pairing. In this paper, we point out a mistake in the previous paper concerning the existence of fundamental solutions of ${\mathfrak{N}^k}$ and intend to make a correction. For this purpose, we study the fundamental solution of the operator ${(\mathfrak{N} + \lambda I)^k}\,(\lambda > 0)$ and investigate its behavior as $\lambda \to 0$. We show that there exists a family $ \{{Q_\lambda}(x,dy)\}$ of measures which serves as the fundamental solution of $ {(\mathfrak{N} + \lambda I)^k}$ and, for a suitable function $f$, we prove that the solution of $ {\mathfrak{N}^k}u = f$ can be represented by $u(x) = {\lim _{\lambda \to 0}}\int_B f(y){Q_\lambda}(x,dy) + C$, where $C$ is a constant.
Smooth type ${\rm III}$ diffeomorphisms of manifolds
Jane
Hawkins
625-643
Abstract: In this paper we prove that every smooth paracompact connected manifold of dimension $ \geqslant 3$ admits a smooth type $ {\text{III}}_\lambda$ diffeomorphism for every $0 \leqslant \lambda \leqslant 1$. (Herman proved the result for $ \lambda = 1$ in [7].) The result follows from a theorem which gives sufficient conditions for the existence of smooth ergodic real line extensions of diffeomorphisms of manifolds.
Domain Bloch constants
C. David
Minda
645-655
Abstract: The classical Bloch constant $ \mathcal{B}$ is defined for holomorphic functions $f$ defined on ${\mathbf{B}} = \{z:\vert z\vert < 1\}$ and normalized by $ \vert f^{\prime}(0)\vert = 1$. Let ${R_f}$ denote the Riemann surface of $f$ and ${B_f}$ the set of branch points. Then $\mathcal{B}$ can be regarded as a lower bound for the radius of the largest disk contained in ${R_f}\backslash {B_f}$. The metric on ${R_f}$ used to measure the size of disks on ${R_f}$ is obtained by lifting the euclidean metric from $ {\mathbf{C}}$ to $ {R_f}$. The surface $ {R_f}$ can also be regarded as spread over $ {\mathbf{B}}$ and the hyperbolic metric lifted to ${R_f}$. One may then ask for the radius of the largest hyperbolic disk on ${R_f}\backslash {B_f}$. A lower bound for this radius is called a domain Bloch constant. The determination of domain Bloch constants is nontrivial for nonconstant analytic functions $ f:{\mathbf{B}} \to X$, where $X$ is a hyperbolic Riemann surface. Upper and lower bounds for domain Bloch constants are given. Also, domain Bloch constants are given an interpretation as a radius of local schlichtness.
Generalized intersection multiplicities of modules
Sankar P.
Dutta
657-669
Abstract: In this paper we study intersection multiplicities of modules as defined by Serre and prove that over regular local rings of $ \dim \leqslant 5$, given two modules $M,N$ with $l(M\otimes_{R}N) < \infty$ and $\dim\;M + \dim \;N < \dim \;R,\chi (M,N) = \sum\nolimits_{i = 0}^{\dim\; R}( - 1)^i l(\operatorname{Tor}_i^R(M,N)) = 0$. We also study multiplicity in a more general set up. Finally we extend Serre's result from pairs of modules to pairs of finite free complexes whose homologies are killed by $ {I^n},{J^n}$, respectively, for some $n > 0$, with $\dim \,R/I + \dim \,R/J < \dim \,R$.
The splitting of $B{\rm O}\langle 8\rangle \wedge b{\rm o}$ and $M{\rm O}\langle 8\rangle \wedge b{\rm o}$
Donald M.
Davis
671-683
Abstract: Let $BO\left\langle 8 \right\rangle$ denote the classifying space for vector bundles trivial on the $7$-skeleton, and $MO\left\langle 8 \right\rangle$ the associated Thom spectrum. It is proved that, localized at $ 2$, $BO\left\langle 8 \right\rangle \wedge \,bo$ and $MO\left\langle 8 \right\rangle \wedge \,bo$ split as a wedge of familiar spectra closely related to $bo$, where $bo$ is the spectrum for connective $ KO$-theory.
Biholomorphic invariants of a hyperbolic manifold and some applications
B. L.
Fridman
685-698
Abstract: A biholomorphically invariant real function ${h_x}$ is defined for a hyperbolic manifold $ X$. Properties of such functions are studied. These properties are applied to prove the following theorem. If a hyperbolic manifold $ X$ can be exhausted by biholomorphic images of a strictly pseudoconvex domain $D \subset {{\mathbf{C}}^n}$ with $ \partial D\; \in \;{C^3}$, then $X$ is biholomorphically equivalent either to $ D$ or to the unit ball in $ {{\mathbf{C}}^n}$. The properties of ${h_D}$ are also applied to some questions concerning the group of analytical automorphisms of a strictly pseudoconvex domain and to similar questions concerning polyhedra.
Quotients of $L\sp{\infty }$ by Douglas algebras and best approximation
Daniel H.
Luecking;
Rahman M.
Younis
699-706
Abstract: We show that ${L^\infty}/A$ is not the dual space of any Banach space when $A$ is a Douglas algebra of a certain type. We do this by showing its unit ball has no extreme points. The method used requires that any function in ${L^\infty}$ has a nonunique best approximation in $A$. We therefore also show that the Douglas algebra $ {H^\infty} + L_F^\infty$, when $F$ is an open subset of the unit circle, permits best approximation. We use a method originating in Hayashi [6] and independently obtained by Marshall and Zame.
The sharp form of Ole\u\i nik's entropy condition in several space variables
David
Hoff
707-714
Abstract: We investigate the conditions under which the Volpert-Kruzkov solution of a single conservation law in several space variables with flux $F$ will satisfy the simplified entropy condition $ \operatorname{div}\,F^{\prime}(u) \leqslant 1/t$, and when this condition guarantees uniqueness for given $ {L^\infty}$ Cauchy data. We show that, when $F$ is ${C^1}$, our condition guarantees uniqueness iff $ F$ is isotropic, and that, for such $F$, the Volpert-Kruzkov solution always satisfies our condition.
Efficient computation in groups and simplicial complexes
John C.
Stillwell
715-727
Abstract: Using HNN extensions of the Boone-Britton group, a group $ E$ is obtained which simulates Turing machine computation in linear space and cubic time. Space in $E$ is measured by the length of words, and time by the number of substitutions of defining relators and conjugations by generators required to convert one word to another. The space bound is used to derive a PSPACE-complete problem for a topological model of computation previously used to characterize NP-completeness and RE-completeness.
Weighted norm inequalities for the Fourier transform
Benjamin
Muckenhoupt
729-742
Abstract: Given $ p$ and $q$ satisfying $1 < p \leqslant q < \infty$, sufficient conditions on nonnegative pairs of functions $U,V$ are given to imply $\displaystyle {\left[ {\int_{{R^n}}^{} {\vert\hat f(x){\vert^q}U(x)\,dx}} \righ... ...qslant c{\left[ {\int_{{R^n}}^{} {\vert f(x){\vert^p}V(x)\,dx}} \right]^{1/p}},$ where $\hat f$ denotes the Fourier transform of $f$, and $c$ is independent of $f$. For the case $ q = p^{\prime}$ the sufficient condition is that for all positive $r$, $\displaystyle \left[ {\int_{U(x) > Br} {U(x)\;dx}} \right]\left[ {\int_{V(x) < {r^{p - 1}}} {V{{(x)}^{- 1/(p - 1)}}\;dx}} \right] \leqslant A,$ where $A$ and $B$ are positive and independent of $ r$. For $q \ne p^{\prime}$ the condition is more complicated but also is invariant under rearrangements of $ U$ and $V$. In both cases the sufficient condition is shown to be necessary if the norm inequality holds for all rearrangements of $U$ and $V$. Examples are given to show that the sufficient condition is not necessary for a pair $U,V$ if the norm inequality is assumed only for that pair.
Shrinking countable decompositions of $S\sp{3}$
Richard
Denman;
Michael
Starbird
743-756
Abstract: Conditions are given which imply that a countable, cellular use decomposition $G$ is shrinkable. If the embedding of each element in $ G$ has the bounded nesting property, defined in this paper, then ${S^3}/G$ is homeomorphic to $ {S^3}$. The bounded nesting property is a condition on the defining sequence of cells for an element of $G$ which implies that $G$ satisfies the Disjoint Disk criterion for shrinkability [ $ {\mathbf{S1}}$, Theorem 3.1]. From this result, one deduces that countable, star-like equivalent use decompositions of $ {S^3}$ are shrinkable--a result proved independently by E. Woodruff [ ${\mathbf{W}}$]. Also, one deduces the shrinkability of countable bird-like equivalent use decompositions (a generalization of the star-like result), and the recently proved theorem that if each element of a countable use decomposition $G$ of ${S^3}$ has a mapping cylinder neighborhood, then $ G$ is shrinkable [ ${\mathbf{E}}$; $ {\mathbf{S1}}$, Theorem 4.1; ${\mathbf{S}}$-${\mathbf{W}}$, Theorem 1].
Constructing approximate fibrations
T. A.
Chapman;
Steve
Ferry
757-774
Abstract: In this paper two results concerning the construction of approximate fibrations are established. The first shows that there are approximate fibrations $ p:M \to S^2$ which are homotopic to bundle maps but which cannot be approximated by bundle maps. Here $M$ can be a compact $Q$-manifold or some topological $n$-manifold, $n \geqslant 5$. The second shows how to construct approximate fibrations $p:M \to B$ whose fibers do not have finite homotopy type, for any $B$ of Euler characteristic zero. Here $ M$ can be a compact $ Q$-manifold and $ B$ only has to be an ANR, or $M$ can be an $n$-manifold, $ n \geqslant 6$, and $ B$ must then also be a topological manifold.
Approximation by smooth multivariate splines
C.
de Boor;
R.
DeVore
775-788
Abstract: The degree of approximation achievable by piecewise polynomial functions of given total order on certain regular grids in the plane is shown to be adversely affected by smoothness requirements--in stark contrast to the univariate situation. For a rectangular grid, and for the triangular grid derived from it by adding all northeast diagonals, the maximum degree of approximation (as the grid size $1/n$ goes to zero) to a suitably smooth function is shown to be $ O({n^{- \rho - 2}})$ in case we insist that the approximating functions are in ${C^\rho}$. This only holds as long as $\rho \leqslant (r - 3)/2$ and $\rho \leqslant (2r - 4)/3$, respectively, with $r$ the total order of the polynomial pieces. In the contrary case, some smooth functions are not approximable at all. In the discussion of the second mesh, a new and promising kind of multivariate ${\text{B}}$-spline is introduced.
Scattering theory and the geometry of multitwistor spaces
Matthew L.
Ginsberg
789-815
Abstract: Existing results which show the zero rest mass field equations to be encoded in the geometry of projective twistor space are extended, and it is shown that the geometries of spaces of more than one twistor contain information concerning the scattering of such fields. Some general constructions which describe spacetime interactions in terms of cohomology groups on subvarieties in twistor space are obtained and are used to construct a purely twistorial description of spacetime propagators and of first order ${\phi ^4}$ scattering. Spacetime expressions concerning these processes are derived from their twistor counterparts, and a physical interpretation is given for the twistor constructions.
Conjugate Fourier series on certain solenoids
Edwin
Hewitt;
Gunter
Ritter
817-840
Abstract: We consider an arbitrary noncyclic subgroup of the additive group ${\mathbf{Q}}$ of rational numbers, denoted by $ {{\mathbf{Q}}_{\mathbf{a}}}$, and its compact character group ${\Sigma _{\mathbf{a}}}$. For $1 < p < \infty$, an abstract form of Marcel Riesz's theorem on conjugate series is known. For $ f$ in $ {\mathfrak{L}_p}({\Sigma _{\mathbf{a}}})$, there is a function $\tilde{f}$ in ${\mathfrak{L}_p}({\Sigma _{\mathbf{a}}})$ whose Fourier transform $ (\tilde{f})\hat{\empty}(\alpha )$ at $\alpha$ in $ {{\mathbf{Q}}_{\mathbf{a}}}$ is $- i\,\operatorname{sgn}\,\alpha \hat{f}(\alpha )$. We show in this paper how to construct $\tilde{f}$ explicitly as a pointwise limit almost everywhere on $ {\Sigma_{\mathbf{a}}}$ of certain harmonic functions, as was done by Riesz for the circle group. Some extensions of this result are also presented.
Are primitive words universal for infinite symmetric groups?
D. M.
Silberger
841-852
Abstract: Let $W = W({x_1}, \ldots ,{x_j})$ be any word in the $j$ free generators ${x_1}, \ldots ,{x_j}$, and suppose that $ W$ cannot be expressed in the form $W = {V^k}$ for $V$ a word and for $k$ an integer with $\left\vert k \right\vert \ne 1$. We ask whether the equation $f = W$ has a solution $ ({x_1}, \ldots ,{x_j}) = (a_{1}, \ldots, a_{j}) \in G^{j}$ whenever $ G$ is an infinite symmetric group and $f$ is an element in $G$. We establish an affirmative answer in the case that $ W(x,y) = {x^m}{y^n}$ for $ m$ and $n$ nonzero integers.
Bundle-like foliations with K\"ahlerian leaves
Richard H.
Escobales
853-859
Abstract: For bundle-like foliations with Kählerian leaves a certain function $ f$ is studied and its Laplacian along a leaf is computed. From this computation one obtains geometric conditions which guarantee the integrability of the distribution orthogonal to that determined by the leaves. When the leaves are compact, the key condition needed to guarantee the integrability of this orthogonal distribution can be interpreted as a condition on the first Chern class of each of the leaves.
An algebraic classification of certain simple even-dimensional knots
C.
Kearton
1-53
Abstract: The simple $ 2q$-knots, $q \geqslant 4$, for which ${H_q}(\tilde{K})$ contains no ${\mathbf{Z}}$-torsion, are classified by means of Hermitian duality pairings on their homology and homotopy modules.
On the $\psi $-mixing condition for stationary random sequences
Richard C.
Bradley
55-66
Abstract: For strictly stationary sequences of random variables two mixing conditions are studied which together form the $ \psi$-mixing condition. For the dependence coefficients associated with these two mixing conditions this article gives results on the possible limiting values and possible rates of convergence to these limits.
Syntax and semantics in higher-type recursion theory
David P.
Kierstead
67-105
Abstract: Recursion in higher types was introduced by S. C. Kleene in 1959. Since that time, it has come to be recognized as a natural and important generalization of ordinary recursion theory. Unfortunately, the theory contains certain apparent anomalies, which stem from the fact that higher type computations deal with the intensions of their arguments, rather than the extensions. This causes the failure of the substitution principle (that if $ \varphi ({\alpha ^{j + 1}},\mathfrak{A})$ and $\theta ({\beta ^j},\mathfrak{A})$ are recursive, then there should be a recursive $\psi (\mathfrak{A})$ such that $\psi (\mathfrak{A}) \simeq \varphi (\lambda {\beta ^j}\theta ({\beta ^j},\mathfrak{A}),\mathfrak{A})$ at least whenever $ \lambda {\beta ^j}\theta ({\beta ^j},\mathfrak{A})$ is total), and of the first recursion principle (that if ${\mathbf{F}}(\zeta ;\mathfrak{A})$ is a recursive functional, then the minimal solution $ \zeta$ of the equation ${\mathbf{F}}(\zeta ;\mathfrak{A}) \simeq \zeta (\mathfrak{A})$ should be recursive as well). In an effort to remove--or at least explain--these anomalies, Kleene, in 1978, developed a system for computation in higher types which was based entirely on the syntactic manipulation of formal expressions, called $ j$-expressions. As Kleene pointed out, no adequate semantics for these expressions can be based on the classical (total) type structure $Tp$ over $ {\mathbf{N}}$. In a paper to appear in The Kleene Symposium (North-Holland), we showed that an appropriate semantics could be based on the type structure $\hat Tp$, which is obtained by adding a new object $ \mathfrak{u}$ at level 0 and, at level $(j + 1)$, allowing all monotone, partial functions from type $\hat j$ into $ {\mathbf{N}}$. Over $ \hat Tp$, both of the principles mentioned above do hold. There is a natural embedding to $Tp$ into $\hat Tp$. In this paper, we complement the syntactic structure with a syntax-free definition of recursion over $\hat Tp$, and show that the two notions are equivalent. This system admits an enumeration theorem, in spite of the fact that the presence of partial objects complicates the coding of finite sequences. Indeed, it is not possible to code all finite sequences from type $ \hat j$ as type-$ \hat j$ objects. We use the combination of the syntactic and semantic systems to prove that, for any $ \varphi: Tp^{(\sigma)}\mathop \to \limits_p {\mathbf{N}} $, the following are equivalent: A. $\varphi$ is recursive in the sense of Kleene [1959], B. $\varphi$ is recursive in the sense of Kleene [1978], and C. $\varphi$ is the pull-back in $Tp$ of some recursive $\psi :\hat T{p^{(\sigma )}}\mathop \to \limits_p {\mathbf{N}}$. Using these equivalences, we give a necessary and sufficient condition on $\theta :T{p^{(\sigma )}}\mathop \to \limits_p {\mathbf{N}}$, under which the substitution principle mentioned above will hold for any recursive $\varphi :T{p^{(\tau )}}\mathop \to \limits_p {\mathbf{N}}$. With one trivial exception, the condition is that if $ j \geqslant 1$, then $\mathfrak{A}$ must contain a variable of type greater than $j$. We feel that this result is particularly natural in the current setting.
On homogeneous polynomials on a complex ball
J.
Ryll;
P.
Wojtaszczyk
107-116
Abstract: We prove that there exist $n$-homogeneous polynomials ${p_n}$ on a complex $d$-dimensional ball such that $ {\left\Vert {{p_n}} \right\Vert _\infty} = 1$ and $ {\left\Vert {{p_n}} \right\Vert _2} \geqslant \sqrt \pi {2^{- d}}$. This enables us to answer some questions about ${H_p}$ and Bloch spaces on a complex ball. We also investigate interpolation by $ n$-homogeneous polynomials on a $2$-dimensional complex ball.
CR-hypersurfaces in a space with a pseudoconformal connection
Michael J.
Markowitz
117-132
Abstract: In this paper we study a submanifold in a space with a pseudoconformal connection. We assume that the submanifold $M$ is so situated that it inherits the structure of a $ {\text{CR}}$-hypersurface from the ambient space. $M$ then supports two natural Cartan connections, the normal pseudoconformal connection of Cartan-Chern-Tanaka and an induced pseudoconformal connection. Analogues of the Gauss-Codazzi equations are derived and applied to determine necessary and sufficient conditions for the equivalence of these connections.
Rigidity of pseudoconformal connections
Michael
Markowitz;
Roger
Schlafly
133-135
Abstract: Let ${M^{2n - 1}}(n \geqslant 3)$ be a strictly pseudoconvex abstract $ {\text{CR}}$-hypersurface ${\text{CR}}$-immersed in the unit sphere in ${{\mathbf{C}}^N}$. We show that the pseudoconformal connection induced on $M$ by the standard flat connection agrees with the intrinsic normal connection of Cartan-Chern-Tanaka if and only if $M$ is pseudoconformally flat. In this case $ M$ is a piece of the transverse intersection of $ {S^{2N - 1}}$ with a complex $n$-plane in $ {{\mathbf{C}}^N}$.
On the singular structure of three-dimensional, area-minimizing surfaces
Frank
Morgan
137-143
Abstract: A sufficient condition is given for the union of two three-dimensional planes through the origin in ${{\mathbf{R}}^n}$ to be area-minimizing. The condition is in terms of the three angles $0 \leqslant {\gamma _1} \leqslant {\gamma _2} \leqslant {\gamma _3}$ which characterize the geometric relationship between the planes. If $ {\gamma _3} \leqslant {\gamma _1} + {\gamma _2}$, the union of the planes is area-minimizing.
Unital $l$-prime lattice-ordered rings with polynomial constraints are domains
Stuart A.
Steinberg
145-164
Abstract: It is shown that a unital lattice-ordered ring in which the square of every element is positive must be a domain provided the product of two nonzero $l$-ideals is nonzero. More generally, the same conclusion follows if the condition ${a^2} \geqslant 0$ is replaced by $p(a) \geqslant 0$ for suitable polynomials $ p(x)$; and if it is replaced by $ f(a,b) \geqslant 0$ for suitable polynomials $f(x,y)$ one gets an $l$-domain. It is also shown that if $a \wedge b = 0$ in a unital lattice-ordered algebra which satisfies these constraints, then the $ l$-ideals generated by $ ab$ and $ba$ are identical.
${\rm SL}(2,\,{\bf C})$ actions on compact Kaehler manifolds
James B.
Carrell;
Andrew John
Sommese
165-179
Abstract: Whenever $G = SL(2,C)$ acts holomorphically on a compact Kaehler manifold $X$, the maximal torus $T$ of $G$ has fixed points. Consequently, $X$ has associated Bialynicki-Birula plus and minus decompositions. In this paper we study the interplay between the Bialynicki-Birula decompositions and the $G$-action. A representative result is that the Borel subgroup of upper (resp. lower) triangular matrices in $G$ preserves the plus (resp. minus) decomposition and that each cell in the plus (resp. minus) decomposition fibres $G$-equivariantly over a component of ${X^T}$. We give some applications; e.g. we classify all compact Kaehler manifolds $ X$ admitting a $ G$-action with no three dimensional orbits. In particular we show that if $ X$ is projective and has no three dimensional orbit, and if Pic$(X) \cong {\mathbf{Z}}$, then $X = C{{\mathbf{P}}^n}$. We also show that if $ X$ admits a holomorphic vector field with unirational zero set, and if $\operatorname{Aut}_0(X)$ is reductive, then $ X$ is unirational.
On wave fronts propagation in multicomponent media
M. I.
Freĭdlin
181-191
Abstract: The behavior as $t \to \infty$ of solutions of some parabolic systems of differential equations of the Kolmogorov-Petrovskii-Piskunov type is investigated. The present approach uses the Kac-Feynman formula and estimates on large deviations.
A geometric interpretation of the Chern classes
R.
Sivera Villanueva
193-200
Abstract: Let ${f_\xi}: M \to BU$ be a classifying map of the stable complex bundle $\xi$ over the weakly complex manifold $ M$. If $\tau$ is the stable right homotopical inverse of the infinite loop spaces map $\eta :QBU(1) \to BU$, we define $f_\xi ^{\prime} = \tau \cdot {f_\xi}$ and we prove that the Chern classes ${c_k}(\xi )$ are $ f_\xi^{\prime\ast}(h_k^{\ast}(t_k))$, where ${h_k}$ is given by the stable splitting of $ QBU(1)$ and ${t_k}$ is the Thom class of the bundle ${\gamma ^{(k)}} = E{\Sigma _k}{X_{{\Sigma _k}}}{\gamma ^k}$. Also, we associate to $f^{\prime}$ an immersion $g:N \to M$ and we prove that ${c_k}(\xi )$ is the dual of the image of the fundamental class of the $k$-tuple points manifold of the immersion $ g,g_k^{\ast}([{N_k}])$.
On the dimension of the $l\sp{n}\sb{p}$-subspaces of Banach spaces, for $1\leq p<2$
Gilles
Pisier
201-211
Abstract: We give an estimate relating the stable type $p$ constant of a Banach space $X$ with the dimension of the $ l_p^n$-subspaces of $ X$. Precisely, let $ C$ be this constant and assume $1 < p < 2$. We show that, for each $\varepsilon > 0,X$ must contain a subspace $(1 + \varepsilon )$-isomorphic to $l_p^k$, for every $k$ less than $\delta (\varepsilon ){C^{p^{\prime}}}$ where $ \delta (\varepsilon ) > 0$ is a number depending only on $p$ and $ \varepsilon$.
On the gr\"ossencharacter of an abelian variety in a parametrized family
Robert S.
Rumely
213-233
Abstract: We consider families of abelian varieties parametrized by classical theta-functions, and show that specifying the family and a CM point in Siegel space determines the grössencharacter of the corresponding CM abelian variety. We associate an adelic group to the family, and describe the kernel of the grössencharacter as the pull-back of the group under the map in Shimura's Reciprocity Law.
A general sufficiency theorem for nonsmooth nonlinear programming
R. W.
Chaney
235-245
Abstract: Second-order conditions are given which are sufficient to guarantee that a given point be a local minimizer for a real-valued locally Lipschitzian function over a closed set in $n$-dimensional real Euclidean space. These conditions are expressed in terms of the generalized gradients of Clarke. The conditions provide a very general and unified framework into which many previous first- and second-order theorems fit.
Inequalities for holomorphic functions of several complex variables
Jacob
Burbea
247-266
Abstract: Sharp norm-inequalities, valid for functional Hilbert spaces of holomorphic functions on the polydisk, unit ball and ${{\mathbf{C}}^n}$ are established by using the notion of reproducing kernels. These inequalities extend earlier results of Saitoh and ours.
Applications of variational inequalities to the existence theorem on quadrature domains
Makoto
Sakai
267-279
Abstract: In this paper we shall study quadrature domains for the class of subharmonic functions. By using the theory of variational inequalities, we shall give a new proof of the existence and uniqueness theorem. As an application, we deal with Hele-Shaw flows with a free boundary and show that their two weak solutions, one of which was defined by the author using quadrature domains and the other was defined by Gustafsson [3] using variational inequalities, are identical with each other.
A necessary and sufficient condition for the asymptotic version of Ahlfors' distortion property
Burton
Rodin;
S. E.
Warschawski
281-288
Abstract: Let $f$ be a conformal map of $R = \{w = u + iv \in {\mathbf{C}}\vert{\varphi _0}(u) < v < {\varphi _1}(u)\}$ onto $S = \{z = x + iy \in {\mathbf{C}}\vert < y < 1\}$ where the ${\varphi _j} \in {C^0}( - \infty ,\infty )$ and $ \operatorname{Re} f(w) \to \pm \infty$ as $\operatorname{Re} w \to \pm \infty$. There are well-known results giving conditions on $R$ sufficient for the distortion property $\operatorname{Re} f(u + iv) = \int_0^u ({\varphi _1} - {\varphi _0})^{- 1}du + {\text{const}}. + o(1)$, where $o(1) \to 0$ as $u \to + \infty$. In this paper the authors give a condition on $R$ which is both necessary and sufficient for $ f$ to have this property.
Cartesian-closed coreflective subcategories of uniform spaces
M. D.
Rice;
G. J.
Tashjian
289-300
Abstract: This paper characterizes the coreflective subcategories $\mathcal{C}$ of uniform spaces for which a natural function space structure generates the exponential law $ {X^{Y \otimes Z}} = {({X^Y})^Z}$ on $ \mathcal{C}$. Such categories are cartesian-closed. Specifically, we show that $\mathcal{C}$ is cartesian-closed in this way if and only if $ \mathcal{C}$ is inductively generated by a finitely productive family of locally fine spaces. The results divide naturally into two cases: those subcategories containing the unit interval are generated by precompact spaces, while the subcategories not containing the unit interval are generated by spaces which admit an infinite cardinal. These results may be used to derive the characterizations of cartesian-closed coreflective subcategories of Tychonoff spaces found in [10].
The approximation property for some $5$-dimensional Henselian rings
Joseph
Becker;
J.
Denef;
L.
Lipshitz
301-309
Abstract: Let $k$ be a field of characteristic 0, $k[[{X_1},{X_2}]]$ the ring of formal power series and $R = k[[{X_1},{X_2}]]{[{X_3},{X_4},{X_5}]^ \sim}$ the algebraic closure of $ k[[{X_1},{X_2}]][{X_3},{X_4},{X_5}]$ in $k[[{X_1},\ldots,{X_5}]]$. It is shown that $ R$ has the Approximation Property.
Sufficient conditions for smoothing codimension one foliations
Christopher
Ennis
311-322
Abstract: Let $M$ be a compact ${C^\infty}$ manifold. Let $X$ be a ${C^0}$ nonsingular vector field on $M$, having unique integral curves $ (p,t)$ through $p \in M$. For $f: M \to {\mathbf{R}}$ continuous, call $\left. Xf(p) = df(p,t)/dt\right\vert _{t = 0}$ whenever defined. Similarly, call $ {X^k}f(p)=X(X^{k-1}f)(p)$. For $ 0 \leqslant r < k$, a $ {C^r}$ foliation $\mathcal{F}$ of $M$ is said to be ${C^k}$ smoothable if there exist a ${C^k}$ foliation $\mathcal{G}$, which ${C^r}$ approximates $ \mathcal{F}$, and a homeomorphism $h:M \to M$ such that $h$ takes leaves of $ \mathcal{F}$ onto leaves of $\mathcal{G}$. Definition. A transversely oriented Lyapunov foliation is a pair $(\mathcal{F},X)$ consisting of a $ {C^0}$ codimension one foliation $ \mathcal{F}$ of $ M$ and a ${C^0}$ nonsingular, uniquely integrable vector field $X$ on $M$, such that there is a covering of $M$ by neighborhoods $\{{W_i}\}$, $0 \leqslant i \leqslant N$, on which $\mathcal{F}$ is described as level sets of continuous functions ${f_i}:{W_i} \to {\mathbf{R}}$ for which $ X{f_i}(p)$ is continuous and strictly positive. We prove the following theorems. Theorem 1. Every ${C^0}$ transversely oriented Lyapunov foliation $ (\mathcal{F},X)$ is $ {C^1}$ smoothable to a ${C^1}$ transversely oriented Lyapunov foliation $ (\mathcal{G},X)$. Theorem 2. If $ (\mathcal{F},X)$ is a ${C^0}$ transversely oriented Lyapunov foliation, with $ X \in {C^{k - 1}}$ and $ {X^j}{f_i}(p)$ continuous for $ 1 \leqslant j \leqslant k$ and $ 0 \leqslant i \leqslant N$, then $ (\mathcal{F},X)$ is $ {C^k}$ smoothable to a ${C^k}$ transversely oriented Lyapunov foliation $ (\mathcal{G},X)$. The proofs of the above theorems depend on a fairly deep result in analysis due to F. Wesley Wilson, Jr. With only elementary arguments we obtain the ${C^k}$ version of Theorem 1. Theorem 3. If $ (\mathcal{F},X)$ is a $ {C^{k - 1}}\;(k \geqslant 2)$ transversely oriented Lyapunov foliation, with $X \in {C^{k - 1}}$ and ${X^k}{f_i}(p)$ is continuous, then $(\mathcal{F},X)$ is ${C^k}$ smoothable to a $ {C^k}$ transversely oriented Lyapunov foliation $(\mathcal{G},X)$.
The dispersion of the coefficients of univalent functions
D. H.
Hamilton
323-333
Abstract: The Hayman $ {T_a}$ function for the asymptotic distribution of the coefficients of univalent functions has a continuous derivative which is closely related to the asymptotic behavior of coefficient differences.
On the generalized Seidel class $U$
Jun Shung
Hwang
335-346
Abstract: As usual, we say that a function $f \in U$ if $f$ is meromorphic in $\vert z \vert < 1$ and has radial limits of modulus $ 1$ a.e. (almost everywhere) on an arc $A$ of $ \left\vert z \right\vert = 1$. This paper contains three main results: First, we extend our solution of A. J. Lohwater's problem (1953) by showing that if $f \in U$ and $f$ has a singular point $P$ on $A$, and if $\upsilon$ and $ 1/\bar{\upsilon}$ are a pair of values which are not in the range of $ f$ at $P$, then one of them is an asymptotic value of $f$ at some point of $A$ near $P$. Second, we extend our solution of J. L. Doob's problem (1935) from analytic functions to meromorphic functions, namely, if $f \in U$ and $f(0) = 0$, then the range of $f$ over $\left\vert z \right\vert < 1$ covers the interior of some circle of a precise radius depending only on the length of $A$. Finally, we introduce another class of functions. Each function in this class has radial limits lying on a finite number of rays a.e. on $\left\vert z \right\vert = 1$, and preserves a sector between domain and range. We study the boundary behaviour and the representation of functions in this class.
A generalization of minimal cones
Norio
Ejiri
347-360
Abstract: Let ${R_ +}$ be a positive real line, $ {S^n}$ an $n$-dimensional unit sphere. We denote by ${R_+} \times {S^n}$ the polar coordinate of an $(n + 1)$-dimensional Euclidean space ${R^{n + 1}}$. It is well known that if $M$ is a minimal submanifold in $ {S^n}$, then ${R_ +} \times M$ is minimal in ${R^{n + 1}}$. $ {R_+} \times M$ is called a minimal cone. We generalize this fact and give many minimal submanifolds in real and complex space forms.
Products of powers of nonnegative derivatives
Jan
Mařík;
Clifford E.
Weil
361-373
Abstract: This paper contains some results concerning functions that can be written as $f_1^{{\beta _1}} \cdots f_n^{{\beta _n}}$, where $ n$ is an integer greater than $1$, ${f_j}$ are nonnegative derivatives and ${\beta _j}$ are positive numbers. If we choose $ {\beta _1} = \cdots = {\beta _n} = 1$, we obtain theorems about products of nonnegative derivatives.
Branched coverings. I
R. E.
Stong
375-402
Abstract: This paper analyzes the possible cobordism classes $[M] - (\deg \;\phi )[N]$ for $\phi : M \to N$ a smooth branched covering of closed smooth manifolds. It is assumed that the branch set is a codimension $2$ submanifold. The results are a fairly complete description in the unoriented case, a partial description in the oriented case, and a detailed analysis of the case in which $N$ is a sphere.
Branched coverings. II
R. E.
Stong
403-407
Abstract: This paper improves the analysis of the possible cobordism classes $[M] - (\deg \;\phi )[N]$ for $\phi : M \to N$ a smooth branched covering of closed oriented smooth manifolds. It is assumed that the branch set is a codimension $2$ submanifold.
Convergence of functions: equi-semicontinuity
Szymon
Dolecki;
Gabriella
Salinetti;
Roger J.-B.
Wets
409-430
Abstract: We study the relationship between various types of convergence for extended real-valued functionals engendered by the associated convergence of their epigraphs; pointwise convergence being treated as a special case. A condition of equi-semicontinuity is introduced and shown to be necessary and sufficient to allow the passage from one type of convergence to another. A number of compactness criteria are obtained for families of semicontinuous functions; in the process we give a new derivation of the Arzelá-Ascoli Theorem.
Spectral families of projections, semigroups, and differential operators
Harold
Benzinger;
Earl
Berkson;
T. A.
Gillespie
431-475
Abstract: This paper presents new developments in abstract spectral theory suitable for treating classical differential and translation operators. The methods are specifically geared to conditional convergence such as arises in Fourier expansions and in Fourier inversion in general. The underlying notions are spectral family of projections and well-bounded operator, due to D. R. Smart and J. R. Ringrose. The theory of well-bounded operators is considerably expanded by the introduction of a class of operators with a suitable polar decomposition. These operators, called polar operators, have a canonical polar decomposition, are free from restrictions on their spectra (in contrast to well-bounded operators), and lend themselves to semigroup considerations. In particular, a generalization to arbitrary Banach spaces of Stone's theorem for unitary groups is obtained. The functional calculus for well-bounded operators with spectra in a nonclosed arc is used to study closed, densely defined operators with a well-bounded resolvent. Such an operator $L$ is represented as an integral with respect to the spectral family of its resolvent, and a sufficient condition is given for $(- L)$ to generate a strongly continuous semigroup. This approach is applied to a large class of ordinary differential operators. It is shown that this class contains significant subclasses of operators which have a polar resolvent or generate strongly continuous semigroups. Some of the latter consist of polar operators up to perturbation by a semigroup continuous in the uniform operator topology.
The number of factorizations of numbers less than $x$ into factors less than $y$
Douglas
Hensley
477-496
Abstract: Let $K(x,y)$ be the number in the title. There is a function $f(r)$, concave and decreasing with $ f(0) = 2$ and $f^{\prime}(0) = 0$ such that if $r = \sqrt {\log x} /\log y$ then as $x \to \infty$ with $r$ fixed, $\displaystyle K(x,y) = x \exp\,\left({f(r)\,\sqrt {\log x} + O\,{{(\log \log x)}^2}} \right)$ . The proof uses a uniform version of Chernoff's theorem on large deviations from the sample mean of a sum of $ N$ independent random variables.
Generic algebras
John
Isbell
497-510
Abstract: The familiar (merely) generic algebras in a variety $\mathcal{V}$ are those which separate all the different operations of $ \mathcal{V}$, or equivalently lie in no proper Birkhoff subcategory. Stronger notions are considered, the strongest being canonicalness of a (small) subcategory $\mathcal{A}$ of $ \mathcal{V}$, defined: the structure functor takes inclusion $\mathcal{A} \subset \mathcal{V}$ to an isomorphism of varietal theories. Intermediate are dominance and exemplariness: lying in no proper varietal subcategory, respectively full subcategory. It is shown that, modulo measurable cardinals, every finitary variety has a canonical set (subcategory) of one or two algebras, the possible second one being the empty algebra. Without reservation, every variety with rank has a dominant set of one or two algebras (the second as before). Finally, in modules over a ring $R$, the generic module $R$ is shown to be (a) dominant if exemplary, and (b) dominant if $R$ is countable or right artinian. However, power series rings $R$ and some others are not dominant $ R$-modules.
An algebraic approach to Grothendieck's residue symbol
Glenn
Hopkins
511-537
Abstract: A certain map--the "residue map"--is defined and its properties are investigated. The impetus for the definition and study of this map is a definition by A. Grothendieck of a homomorphism, the "residue symbol", which has been found applicable in several areas, including the duality theory of algebraic varieties.
Computability and noncomputability in classical analysis
Marian Boykan
Pour-El;
Ian
Richards
539-560
Abstract: This paper treats in a systematic way the following question: which basic constructions in real and complex analysis lead from the computable to the noncomputable, and which do not? The topics treated include: computability for $ {C^n}$, ${C^\infty }$, real analytic functions, Fourier series, and Fourier transforms. A final section presents a more general approach via "translation invariant operators". Particular attention is paid to those processes which occur in physical applications. The approach to computability is via the standard notion of recursive function from mathematical logic.
Sufficient conditions for the generalized problem of Bolza
Vera
Zeidan
561-586
Abstract: This paper presents sufficient conditions for strong local optimality in the generalized problem of Bolza. These conditions represent a unification, in the sense that they can be applied to both the calculus of variations and to optimal control problems, as well as problems with nonsmooth data. Also, this paper brings to light a new point of view concerning the Jacobi condition in the classical calculus of variations, showing that it can be considered as a condition which guarantees the existence of a canonical transformation which transforms the original Hamiltonian to a locally concave-convex Hamiltonian.
Skewness in Banach spaces
Simon
Fitzpatrick;
Bruce
Reznick
587-597
Abstract: Let $E$ be a Banach space. One often wants to measure how far $E$ is from being a Hilbert space. In this paper we define the skewness $s(E)$ of a Banach space $E$, $0 \leqslant s(E) \leqslant 2$, which describes the asymmetry of the norm. We show that $s(E) = s({E^{\ast}})$ for all Banach spaces $ E$. Further, $ s(E) = 0$ if and only if $ E$ is a (real) Hilbert space and $s(E) = 2$ if and only if $E$ is quadrate, so $s(E) < 2$ implies $E$ is reflexive. We discuss the computation of $ s({L^p})$ and describe its asymptotic behavior near $p = 1,2$ and $\infty$. Finally, we discuss a higher-dimensional generalization of skewness which gives a characterization of smooth Banach spaces.
Pseudojump operators. I. The r.e. case
Carl G.
Jockusch;
Richard A.
Shore
599-609
Abstract: Call an operator $ J$ on the power set of $ \omega$ a pseudo jump operator if $J(A)$ is uniformly recursively enumerable in $ A$ and $A$ is recursive in $J(A)$ for all subsets $A$ of $\omega$. Thus the (Turing) jump operator is a pseudo jump operator, and any existence proof in the theory of r.e. degrees yields, when relativized, one or more pseudo jump operators. Extending well-known results about the jump, we show that for any pseudo jump operator $J$, every degree $\geqslant {\mathbf{0}}^{\prime}$ has a representative in the range of $J$, and that there is a nonrecursive r.e. set $A$ with $J(A)$ of degree ${\mathbf{0}}^{\prime}$. The latter result yields a finite injury proof in two steps that there is an incomplete high r.e. degree, and by iteration analogous results for other levels of the ${H_n}$, ${L_n}$ hierarchy of r.e. degrees. We also establish a result on pairs of pseudo jump operators. This is combined with Lachlan's result on the impossibility of combining splitting and density for r.e. degrees to yield a new proof of Harrington's result that ${\mathbf{0}}^{\prime}$ does not split over all lower r.e. degrees.
An embedding characterization of almost compact spaces
Sergio
Salbany
611-621
Abstract: We characterize almost compact and almost realcompact spaces in terms of their situation in the product ${(J,u)^C}$. In the characterization of almost compactness $J$ is the two point set or the unit interval; in the characterization of almost realcompactness $ J$ is the set of nonnegative integers or the nonnegative reals. $ u$ is the upper topology on the real line restricted to $J$.
$L\sp{p}$ multipliers with weight $x\sp{kp-1}$
Benjamin
Muckenhoupt;
Wo Sang
Young
623-639
Abstract: Let $k$ be a positive integer and $1 < p < \infty$. It is shown that if $ T$ is a multiplier operator on ${L^p}$ of the line with weight $\vert x{\vert^{kp-1}}$, then $Tf$ equals a constant times $ f$ almost everywhere. This does not extend to the periodic case since $m(j) = 1/j, j \ne 0$, is a multiplier sequence for $ {L^p}$ of the circle with weight $ \vert x{\vert^{kp-1}}$. A necessary and sufficient condition is derived for a sequence $m(j)$ to be a multiplier on ${L^2}$ of the circle with weight $\vert x{\vert^{2k - 1}}$.
Schr\"odinger operators with rapidly oscillating central potentials
Denis A. W.
White
641-677
Abstract: Spectral and scattering theory is discussed for the Schrödinger operators $H = - \Delta + V$ and ${H_0} = - \Delta$ when the potential $V$ is central and may be rapidly oscillating and unbounded. A spectral representation for $ H$ is obtained along with the spectral properties of $H$. The existence and completeness of the modified wave operators is also demonstrated. Then a condition on $V$ is derived which is both necessary and sufficient for the Møller wave operators to exist and be complete. This last result disproves a recent conjecture of Mochizuki and Uchiyama.
Regular functions of restricted growth and their zeros in tangential regions
C. N.
Linden
679-686
Abstract: For a given function $k$, positive, continuous, nondecreasing and unbounded on $[0,1)$, let ${A^{(k)}}$ denote the class of functions regular in the unit disc for which log $\vert f(z)\vert < k(\vert z\vert)$ when $ \vert z\vert < 1$. Hayman and Korenblum have shown that a necessary and sufficient condition for the sets of positive zeros of all functions in ${A^{(k)}}$ to be Blaschke is that $\displaystyle \int_0^1 {\sqrt {(k(t)/(1 - t))\,dt} }$ is finite. It is shown that the imposition of a further regularity condition on the growth of $k$ ensures that in some tangential region the zero set of each function in ${A^{(k)}}$ is also Blaschke.
Dehn surgery and satellite knots
C. McA.
Gordon
687-708
Abstract: For certain kinds of $3$-manifolds, the question whether such a manifold can be obtained by nontrivial Dehn surgery on a knot in ${S^3}$ is reduced to the corresponding question for hyperbolic knots. Examples are, whether one can obtain $ {S^3}$, a fake $ {S^3}$, a fake $ {S^3}$ with nonzero Rohlin invariant, $ {S^1} \times {S^2}$, a fake ${S^1} \times {S^2}, {S^1} \times {S^2} \char93 M$ with $M$ nonsimply-connected, or a fake lens space.
AF algebras with directed sets of finite-dimensional $\sp{\ast} $-subalgebras
Aldo J.
Lazar
709-721
Abstract: We characterize the unital $AF$ algebras whose families of finite dimensional $^{\ast}$-subalgebras are directed by inclusion. A representation theorem for the algebras of this class allows us to classify them up to $^{\ast} $-isomorphisms.
Characterizations of simply connected rotationally symmetric manifolds
Hyeong In
Choi
723-727
Abstract: We prove that a simply connected, complete Riemannian manifold $ M$ is rotationally symmetric at $p$ if and only if the exponential image of every linear subspace of ${M_p}$ is a smooth, closed, totally geodesic submanifold of $M$. This result is in essence Schur's theorem at one point $p$, as it becomes apparent in the proof.
Differential group actions on homotopy spheres. III. Invariant subspheres and smooth suspensions
Reinhard
Schultz
729-750
Abstract: A linear action of an abelian group on a sphere generally contains a large family of invariant linear subspheres. In this paper the problem of finding invariant subspheres for more general smooth actions on homotopy spheres is considered. Classification schemes for actions with invariant subspheres are obtained; these are formally parallel to the classifications discussed in the preceding paper of this series. The realizability of a given smooth action as an invariant codimension two subsphere is shown to depend only on the ambient differential structure and an isotopy invariant. Applications of these results to specific cases are given; for example, it is shown that every exotic $10$-sphere admits a smooth circle action.
Brownian motion and a generalised little Picard's theorem
Wilfrid S.
Kendall
751-760
Abstract: Goldberg, Ishihara, and Petridis have proved a generalised little Picard's theorem for harmonic maps; if a harmonic map of bounded dilatation maps euclidean space, for example, into a space of negative sectional curvatures bounded away from zero then that map is constant. In this paper a probabilistic proof is given of a variation on this result, requiring in addition that the image space has curvatures bounded below. The method involves comparing asymptotic properties of Brownian motion with the asymptotic behaviour of its image under such a map.
Measurable representations of preference orders
R. Daniel
Mauldin
761-769
Abstract: A continuous preference order on a topological space $Y$ is a binary relation $\preccurlyeq$ which is reflexive, transitive and complete and such that for each $x,\{y:x \preccurlyeq y\} $ and $\{y:y \preccurlyeq x\}$ are closed. Let $ T$ and $X$ be complete separable metric spaces. For each $t$ in $T$, let ${B_t}$ be a nonempty subset of $X$, let ${ \preccurlyeq _t}$ be a continuous preference order on ${B_t}$ and suppose $E = \{(t,x,y): x{ \preccurlyeq _t}y\}$ is a Borel set. Let $B = \{(t,x):x \in {B_t}\} $. Theorem 1. There is an $\mathcal{S}(T) \otimes \mathcal{B}(X)$-measurable map $g$ from $B$ into $R$ so that for each $t,g(t,\cdot)$ is a continuous map of $ {B_t}$ into $ R$ and $g(t,x) \leqslant g(t,y)$ if and only if $x{ \preccurlyeq _t}y$. (Here $\mathcal{S}(T)$ forms the $C$-sets of Selivanovskii and $\mathcal{B}(X)$ is a Borel field on $ X$.) Theorem 2. If for each $t,{B_t}$ is a $\sigma$-compact subset of $Y$, then the map $g$ of the preceding theorem may be chosen to be Borel measurable. The following improvement of a theorem of Wesley is proved using classical methods. Theorem 3. Let $ g$ be the map constructed in Theorem 1. If $\mu$ is a probability measure defined on the Borel subsets of $T$, then there is a Borel set $N$ such that $\mu (N) = 0$ and such that the restriction of $ g$ to $B \cap ((T - N) \times X)$ is Borel measurable.
Stochastic waves
E. B.
Dynkin;
R. J.
Vanderbei
771-779
Abstract: Let $\phi$ be a real valued function defined on the state space of a Markov process $ {x_t}$. Let ${\tau _t}$ be the first time $ {x_t}$ gets to a level set of $\phi$ which is $t$ units higher than the one on which it started. We call the time changed process $\tilde{x}_{t} = x_{{\tau_t}}$ a stochastic wave. We give conditions under which this process is Markovian and we evaluate its infinitesimal operator.
Weighted norm inequalities for homogeneous families of operators
José L.
Rubio de Francia
781-790
Abstract: If a family of operators in ${R^n}$ is invariant under rotations and dilations and satisfy a certain inequality in ${L^p}({l^r})$, then it is uniformly bounded in the weighted space ${L^r}(\vert x\vert{^{n(r/p - 1)}}\,dx)$. This is the main consequence of a more general result for operators in homogeneous spaces. Applications are given to certain maximal operators, the Fourier transform and Bochner-Riesz multipliers.
Structural stability and group cohomology
Philip J.
Fleming
791-809
Abstract: We prove a version of the theorem of Stowe concerning the stability of stationary points of a differentiable group action which is valid on Hilbert manifolds. This result is then used to show that the vanishing of certain cohomology groups is sufficient to guarantee structural semistability for a differentiable action of a group of finite type on a closed smooth manifold. We then apply this to groups of diffeomorphisms of the circle.
A sphere theorem for manifolds of positive Ricci curvature
Katsuhiro
Shiohama
811-819
Abstract: Instead of injectivity radius, the contractibility radius is estimated for a class of complete manifolds such that $ {\text{Ri}}{{\text{c}}_M} \geqslant 1,{K_M} \geqslant - {\kappa ^2}$ and the volume of $M \geqslant$ the volume of the $(\pi - \varepsilon )$-ball on the unit $ m$-sphere, $m = {\text{dim }}M$. Then for a suitable choice of $ \varepsilon = \varepsilon (m,k)$ every $M$ belonging to this class is homeomorphic to $ {S^m}$.
A general maximal operator and the $A\sb{p}$-condition
M. A.
Leckband;
C. J.
Neugebauer
821-831
Abstract: A rearrangement inequality for a general maximal operator $Mf(x) = {\sup _{x \in Q}}\int {f\phi_{Q}\,d\nu }$ is established. This is then applied to the Hardy-Littlewood maximal operator with weights.
Krull dimension of differential operator rings. III. Noncommutative coefficients
K. R.
Goodearl;
T. H.
Lenagan
833-859
Abstract: This paper is concerned with the Krull dimension (in the sense of Gabriel and Rentschler) of a differential operator ring $S[\theta ;\delta ]$, where $S$ is a right noetherian ring with finite Krull dimension $n$ and $\delta$ is a derivation on $S$. The main theorem states that $S[\theta ;\delta ]$ has Krull dimension $ n$ unless there exists a simple right $S$-module $A$ such that $A{ \otimes _S}S[\theta ;\delta ]$ is not simple (as an $ S[\theta ;\delta ]$-module) and $A$ has height $n$ in the sense that there exist critical right $ S$-modules $A = {A_0},{A_1},\ldots,{A_n}$ such that each ${A_i} \otimes_s S[\theta ;\delta ]$ is a critical $S[\theta ;\delta ]$-module, each $ {A_i}$ is a minor subfactor of ${A_{i + 1}}$ and ${A_n}$ is a subfactor of $S$. If such an $A$ does exist, then $S[\theta ;\delta ]$ has Krull dimension $ n + 1$. This criterion is simplified when $S$ is fully bounded, in which case it is shown that $S[\theta ;\delta ]$ has Krull dimension $n$ unless $S$ has a maximal ideal $M$ of height $n$ such that either ${\text{char(}}S/M) > 0$ or $\delta (M) \subseteq M$, and in these cases $S[\theta ;\delta ]$ has Krull dimension $ n + 1$.
Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I
Douglas N.
Hoover;
Edwin
Perkins
1-36
Abstract: R. M. Anderson has developed a nonstandard approach to Itô integration in which the Itô integral is interpreted as an internal Riemann-Stieltjes sum. In this paper we extend this approach to integration with respect to semimartingales. Lifting and pushing down theorems are proved for local martingales, semimartingales and other right-continuous processes on a Loeb space.
Nonstandard construction of the stochastic integral and applications to stochastic differential equations. II
Douglas N.
Hoover;
Edwin
Perkins
37-58
Abstract: H. J. Keisler has recently used a nonstandard theory of Itô integration (due to R. M. Anderson) to construct solutions of Itô integral equations by solving an associated internal difference equation. In this paper we use the same general approach to find solutions $y(t)$ of semimartingale integral equations of the form $\displaystyle y(t,\omega) = h(t,\omega) + \int_0^t {f(s,\omega ,y(\cdot ,\omega))\,dz(s)}$ , where $z$ is a given semimartingale, $ h$ is a right-continuous process and $ f(s,\omega , \cdot)$ is continuous on the space of right-continuous functions with left limits, with the topology of uniform convergence on compacts. In addition, we generalize Keisler's continuity theorem and give necessary and sufficient conditions for an internal martingale to be $ S$-continuous.
Tauberian $L\sp{1}$-convergence classes of Fourier series. I
William O.
Bray;
Časlav V.
Stanojević
59-69
Abstract: It is shown that the Stanojević [2] necessary and sufficient conditions for ${L^1}$-convergence of Fourier series of $f \in {L^1}(T)$ can be reduced to the classical form. A number of corollaries of a recent Tauberian theorem are obtained for the subclasses of the class of Fourier coefficients satisfying $ {n^\alpha }\vert\Delta \hat{f}(n)\vert = o(l)\,(n \to \infty)$ for some $0 < \alpha \leqslant \frac{1}{2}$. For Fourier series with coefficients asymptotically even with respect to a sequence $\{{l_n}\} ,{l_n} = o(n)\,(n \to \infty)$, and satisfying $\displaystyle l_n^{ - 1/q}{\left({\sum\limits_{k = n}^{n + [n/{l_n}]} {{k^{p - ... ...} {\vert^p}} \right)^{1/p}} = o(1)\, \quad (n \to \infty), \quad 1/p + 1/q = 1,$ necessary and sufficient conditions for ${L^1}$-convergence are obtained. In particular for ${l_n} = [\parallel {\sigma _n}(f) - f{\parallel ^{ - 1}}]$, an important corollary is obtained which connects smoothness of $f$ with smoothness of $\{\hat f(n)\}$.
Mixed Hodge structures
Fouad
El Zein
71-106
Abstract: The theory of Mixed Hodge Structures (M.H.S.) on the cohomology of an algebraic variety $X$ over complex numbers was found by Deligne in 1970. The case when $X$ is a Normal Crossing Divisor is fundamental. When the variety $X$ is embedded in a smooth ambient space we get the Mixed Hodge Structure using standard exact sequences in topology. This technique uses resolution of singularities one time for a complete variety and $ 2$ times for a quasi-projective one. As applications to the study of local cohomology we give the spectral sequence to the Mixed Hodge Structure on cohomology with support on a subspace $Y$.
Quadratic forms permitting triple composition
Kevin
McCrimmon
107-130
Abstract: In an algebraic investigation of isoparametric hypersurfaces, J. Dorfmeister and E. Neher encountered a nondegerate quadratic form which permitted composition with a trilinear product, $ Q\left(\{{xyz} \} \right) = Q(x)Q(y)Q(z)$. In this paper we give a complete description of such composition triples: they are all obtained as isotopes of permutations of standard triples $\{xyz \} = (xy)z$ or $x(yz)$ determined by a composition algebra, with the quadratic form $Q$ the usual norm form. For any fixed $ Q$ this leads to $ 1$ isotopy class in dimensions $1$ and $2$, $3$ classes in the dimension $4$ quaternion case, and $6$ classes in the dimension $8$ octonion case.
Markov processes with identical hitting probabilities
Joseph
Glover
131-142
Abstract: Let $(X(t),{P^x})$ and $ (Y(t),{Q^x})$ be transient Hunt processes on a state space $E$ satisfying the hypothesis of absolute continuity (Meyer's hypothesis (L)). Let $ T(K)$ be the first entrance time into a set $K$, and assume ${P^x}(T(K) < \infty) = {Q^x}(T(K) < \infty)$ for all compact sets $K \subseteq E$. There exists a strictly increasing continuous additive functional of $X(t),A(t)$, so that if $T(t) = {\text{inf}}\{s:A(s) > t\}$, then $(X(T(t)),{P^x})$ and $ (Y(t),{Q^x})$ have the same joint distributions. An analogous result is stated if $X$ and $Y$ are right processes (with an additional hypothesis). These theorems generalize the Blumenthal-Getoor-McKean Theorem and have interpretations in terms of potential theory.
Dynamical systems and extensions of states on $C\sp{\ast} $-algebras
Nghiem
Dang-Ngoc
143-152
Abstract: Let $(A,G,\tau)$ be a noncommutative dynamical system, i.e. $A$ is a $ {C^{\ast} }$-algebra, $ G$ a topological group and $ \tau$ an action of $ G$ on $A$ by $^{\ast}$-automorphisms, and let $({M_\alpha })$ be an $M$-net on $G$. We characterize the set of $a$ in $A$ such that $ {M_\alpha }a$ converges in norm. We show that this set is intimately related to the problem of extensions of pure states of R. V. Kadison and I. M. Singer: if $B$ is a maximal abelian subalgebra of $A$, we can associate a dynamical system $(A,G,\tau)$ such that $ {M_\alpha }a$ converges in norm if and only if all extensions to $A$, of a homomorphism of $ B$, coincide on $ a$. This result allows us to construct different examples of a ${C^{\ast} }$-algebra $A$ with maximal abelian subalgebra $B$ (isomorphic to $ C({\mathbf{R}}/{\mathbf{Z}})$ or ${L^\infty }[0,1])$ for which the property of unique pure state extension of homomorphisms is or is not verified.
Locally strange hyperbolic sets
Lowell
Jones
153-162
Abstract: The purpose of this paper is to present a very general method of constructing basic sets having complicated local homeomorphism types.
Normal subgroups of ${\rm Diff}\sp{\Omega }({\bf R}\sp{n})$
Francisca
Mascaró
163-173
Abstract: Let $\Omega$ be a volume element on ${{\mathbf{R}}^n}$. $ {\text{Dif}}{{\text{f}}^\Omega }({{\mathbf{R}}^n})$ is the group of $\Omega $-preserving diffeomorphisms of $ {{\mathbf{R}}^n}$. $ {\text{Diff}}_W^\Omega ({{\mathbf{R}}^n})$ is the subgroup of all elements whose set of nonfixed points has finite $\Omega $-volume. $ {\text{Diff}}_f^\Omega ({{\mathbf{R}}^n})$ is the subgroup of all elements whose support has finite $\Omega$-volume. ${\text{Diff}}_c^\Omega ({{\mathbf{R}}^n})$ is the subgroup of all elements with compact support. $ {\text{Diff}}_{{\text{co}}}^\Omega ({{\mathbf{R}}^n})$ is the subgroup of all elements compactly $\Omega$-isotopic to the identity. We prove, in the case $ {\text{vo}}{{\text{l}}_{\Omega }}{{\mathbf{R}}^n} < \infty$ and for $ {\text{n}} \geqslant {\text{3}}$ that any subgroup of $ {\text{Dif}}{{\text{f}}^\Omega }({{\mathbf{R}}^n})$, $N$, is normal if and only if $ {\text{Diff}}_{{\text{co}}}^\Omega ({{\mathbf{R}}^n}) \subset N \subset {\text{Diff}}_c^\Omega ({{\mathbf{R}}^n})$. If $ {\text{vo}}{{\text{l}}_{\Omega }}{{\mathbf{R}}^n} = \infty$, any subgroup of $ {\text{Dif}}{{\text{f}}^\Omega }({{\mathbf{R}}^n})$, $N$, satisfying $ {\text{Diff}}_{{\text{co}}}^\Omega ({{\mathbf{R}}^n}) \subset N \subset {\text{Diff}}_c^\Omega ({{\mathbf{R}}^n})$ is normal, for $ n \geqslant {\text{3}}$, there are no normal subgroups between $ {\text{Diff}}_W^\Omega ({{\mathbf{R}}^n})$ and $ {\text{Dif}}{{\text{f}}^\Omega }({{\mathbf{R}}^n})$ and for $n \geqslant 4$ there are no normal subgroups between ${\text{Diff}}_c^\Omega ({{\mathbf{R}}^n})$ and ${\text{Diff}}_f^\Omega ({{\mathbf{R}}^n})$.
Arithmetic equivalent of essential simplicity of zeta zeros
Julia
Mueller
175-183
Abstract: Let $R(x)$ and $S(t)$ be the remainder terms in the prime number theorem and the Riemann-von Mangoldt formula respectively, that is $ \psi (x) = x + R(x)$ and $ N(t) = (1/2\pi)\int_0^t {\log (\tau /2\pi)\,d\tau + S(t) + 7/8 + O(1/t)}$. We are interested in the following integrals: $J(T,\beta) = \int_1^{{T^\beta }} {{{(R(x + x/T) - R(x))}^2}dx/{x^2}} $ and $I(T,\alpha) = \int_1^T {{{(S(t + \alpha /L) - S(t))}^2}dt}$, where $L = {(2\pi)^{ - 1}}\log \,T$. Furthermore, denote by $N(T,\alpha)(N^{\ast}(T))$ the number of pairs of zeros $\frac{1} {2} + i\Upsilon ,\frac{1} {2} + i\Upsilon^{\prime}$ with $ 0 < \Upsilon \leqslant T$ and $0 < (\Upsilon^{\prime} - \Upsilon)L \leqslant \alpha \,((\Upsilon^{\prime} - \Upsilon)L = 0)$--i.e., off-diagonal and diagonal pairs. Theorem. Assume the Riemann hypothesis. The following three hypotheses (A), (B) and $ ({{\text{C}}_1},{{\text{C}}_2})$ are equivalent: for $\beta \to \infty$ and $\alpha \to 0$ as $ T \to \infty$ we have (A) $J(T,\beta) \sim \beta {T^{ - 1}}{\log ^2}T$, (B) $ I(T,\alpha) \sim \alpha T$ and $ ({{\text{C}}_1})\;N^{\ast}(T) \sim TL,({{\text{C}}_2})N(T,\alpha) = o(TL)$. Hypothesis $ ({{\text{C}}_1},{{\text{C}}_2})$ is called the essential simplicity hypothesis.
On strongly indefinite functionals with applications
Helmut
Hofer
185-214
Abstract: Recently, in their remarkable paper Critical point theory for indefinite functionals, V. Benci and P. Rabinowitz gave a direct approach--avoiding finite-dimensional approximations--to the existence theory for critical points of indefinite functionals. In this paper we develop under weaker hypotheses a simpler but more general theory for such problems. In the second part of the paper the abstract results are applied to a class of resonance problems of the Landesman and Lazer type, and moreover they are illustrated by an application to a wave equation problem.
On the absence of positive eigenvalues of Schr\"odinger operators with long range potentials
Hubert
Kalf;
V.
Krishna Kumar
215-229
Abstract: In this paper we consider the problem of obtaining upper bounds for the positive bound states associated with the Schrödinger operators with long range potentials. We have extended the size of the class of long range potentials for which one can establish the nonexistence of positive eigenvalues, improving upon the recent results of G. B. Khosrovshahi, H. A. Levine and L. E. Payne (Trans. Amer. Math. Soc. 253 (1979), 211-228).
Dimension of stratifiable spaces
Shinpei
Oka
231-243
Abstract: We define a subclass, denoted by $E{M_3}$, of the class of stratifiable spaces, and obtain several dimension theoretical results for $ E{M_3}$ including the coincidence theorem for dim and Ind. The class $ E{M_3}$ is countably productive, hereditary, preserved under closed maps and, furthermore, the largest subclass of stratifiable spaces in which a harmonious dimension theory can be established.
Classical solutions of the Hamilton-Jacobi-Bellman equation for uniformly elliptic operators
Lawrence C.
Evans
245-255
Abstract: We prove under appropriate hypotheses that the Hamilton-JacobiBellman dynamic programming equation with uniformly elliptic operators, ${\max _{1 \leqslant k \leqslant m}}\{{L^k}u - {f^k}\} = 0$, has a classical solution $u \in {C^{2,\beta }}$, for some (small) Hölder exponent $\beta > 0$.
The index of harmonic foliations on spheres
Franz W.
Kamber;
Philippe
Tondeur
257-263
Abstract: For foliations on a compact oriented manifold there is a natural energy functional, defined with respect to a Riemannian metric. Harmonic Riemannian foliations are then the critical foliations for this functional under an appropriate class of special variations. The index of the title is the index of the Hessian of the energy functional at a critical, i.e., harmonic foliation. It is a finite number. In this note it is shown that for a harmonic Riemannian foliation $ \mathcal{F}$ of codimension $q$ on the $n$-sphere ($n > 2$) this index is greater or equal to $q + 1$. Thus $\mathcal{F}$ is unstable. Moreover the given bound is best possible.
Convergence acceleration for continued fractions $K(a\sb{n}/1)$
Lisa
Jacobsen
265-285
Abstract: A known method for convergence acceleration of limit periodic continued fractions $K({a_n}/1),{a_n} \to a$, is to replace the approximants ${S_n}(0)$ by "modified approximants" ${S_n}({f^{\ast}})$, where $f^{\ast} = K(a/1)$. The present paper extends this idea to a larger class of converging continued fractions. The "modified approximants" will then be $ {S_n}({f^{(n)^{\prime}}})$, where $ K({a^{\prime}_n}/1)$ is a converging continued fraction whose tails ${f^{(n)\prime}}$ are all known, and where ${a_n} - a_n^\prime \to 0$. As a measure for the improvement obtained by this method, upper bounds for the ratio of the two truncation errors are found.
Monotone decompositions of $\theta $-continua
E. E.
Grace
287-295
Abstract: A $\theta $-continuum ( ${\theta _n}$-continuum) is a compact, connected, metric space that is not separated into infinitely many (more than $n$) components by any subcontinuum. The following results are among those proved. The first generalizes earlier joint work with E. J. Vought for ${\theta _n}$-continua, and the second generalizes earlier work by Vought for ${\theta _1}$-continua. A $\theta $-continuum $X$ admits a monotone, upper semicontinuous decomposition $ \mathcal{D}$ such that the elements of $ \mathcal{D}$ have void interiors and the quotient space $X/\mathcal{D}$ is a finite graph, if and only if, for each nowhere dense subcontinuum $H$ of $X$, the continuum $T(H) = \{x \in X\vert$ if $K$ is a subcontinuum of $X$ and $x$ is in the interior of $K$, then $K \cap H \ne \emptyset \} $ is nowhere dense. Also, if $X$ satisfies this condition, then $ X$ is in fact a ${\theta _n}$-continuum, for some natural number $ n$, and, for each natural number $m$, $X$ is a $ {\theta _m}$-continuum, if and only if $ X/\mathcal{D}$ is a ${\theta _m}$-continuum.
Evolution generated by semilinear dissipative plus compact operators
Eric
Schechter
297-308
Abstract: Existence results and sharp continuous dependence results are given for an evolution equation in an arbitrary Banach space. The right-hand side of the equation consists of a linear dissipative term plus a continuous dissipative term plus a compact term.
Linking numbers and the elementary ideals of links
Lorenzo
Traldi
309-318
Abstract: Let $L = {K_1} \cup \, \cdots \cup {K_\mu } \subseteq {S^3}$ be a tame link of $ \mu \geqslant 2$ components, and $H$ the abelianization of $G = {\pi _1}({S^3} - L)$. Let $\mathcal{L} = ({\mathcal{L}_{ij}})$ be the $\mu \times \mu $ matrix with entries in $\mathbf{Z}H$ given by $\mathcal{L}{_{ii}} = \sum\nolimits_{k \ne i} {l({K_i},{K_k}) \cdot ({t_k} - 1)}$ and for $i \ne j\,{\mathcal{L}_{ij}} = l({K_i},{K_j}) \cdot (1 - {t_i})$. Then if $0 < k < \mu$ $\displaystyle \sum\limits_{i = 0}^{k - 1} {{E_{\mu - k + i}}(L) \cdot {{(IH)}^{... ...{k - 1} {{E_{\mu - k + i}}(\mathcal{L}) \cdot {{(IH)}^{2i}} + {{(IH)}^{2k}}} }$ Various consequences of this equality are derived, including its application to the reduced elementary ideals. These results are used to give several different characterizations of links in which all the linking numbers are zero.
Reality of the zeros of an entire function and its derivatives
Simon
Hellerstein;
Li Chien
Shen;
Jack
Williamson
319-331
Abstract: In 1914 Pólya raised the problem of classifying the entire functions which together with all their derivatives have only real zeros. In earlier work Hellerstein and Williamson settled this problem for entire functions which are real on the real axis. We complete the classification in all cases and show that it is sufficient to consider the function and its first two derivatives.
On the triangulation of stratified sets and singular varieties
F. E. A.
Johnson
333-343
Abstract: We show that every compact stratified set in the sense of Thom can be triangulated as a simplicial complex. The proof uses that author's description of a stratified set as the geometric realisation of a certain type of diagram of smooth fibre bundles and smooth imbeddings, and the triangulability of smooth fibre bundles. As a consequence, we obtain proofs of the classical triangulation theorems for analytic and subanalytic sets, and a correct proof of Yang's theorem that the orbit space of a smooth compact transformation group is triangulable.
Some canonical cohomology classes on groups of volume preserving diffeomorphisms
Dusa
McDuff
345-356
Abstract: We discuss some canonical cohomology classes on the space $\bar B\mathcal{D}iff_{\omega 0}^cM$, where $\mathcal{D}iff_{\omega 0}^cM$ is the identity component of the group of compactly supported diffeomorphisms of the manifold $M$ which preserve the volume form $\omega$. We first look at some classes ${c_k}(M),1 \leqslant k \leqslant n = {\text{dim}}\,M$, which are defined for all $M$, and show that the top class ${c_n}(M) \in \,{H^n}(\bar B\mathcal{D}iff_{\omega 0}^cM;{\mathbf{R}})$ is nonzero for $M = {S^n},n$ odd, and is zero for $M = {S^n},n$ even. When $ H_c^i(M;{\mathbf{R}}{\text{) = 0}}$ for $ 0 \leqslant i < n$, the classes ${c_k}(M)$ all vanish and a secondary class $s(M) \in \,{H^{n - 1}}(\bar B\mathcal{D}iff_{\omega 0}^cM; {\mathbf{R}})$ may be defined. This is trivially zero when $n$ is odd, and is twice the Calabi invariant for symplectic manifolds when $n = 2$. We prove that $s({{\mathbf{R}}^n}) \ne 0$ when $n$ is even by showing that it is one of a set of nonzero classes which were defined by Hurder in [7].
Weak-star convergence in the dual of the continuous functions on the $n$-cube, $1\leq n\leq \infty $
Richard B.
Darst;
Zorabi
Honargohar
357-372
Abstract: Let $n$ be a positive integer and let $J = \times _{j = 1}^n{[0,1]_j}$ denote the $n$-cube. Let $\mathbf{C} = \mathbf{C}(J)$ denote the (sup norm) space of continuous (real-valued) functions defined on $J$, and let $ \mathfrak{M}$ denote the (variation norm) space of (real-valued) signed Borel measures defined on the Borel subsets of $J$. Let $\left\langle {{\mu _l}} \right\rangle$ be a sequence of elements of $ \mathfrak{M}$. Necessary and sufficient conditions are given in order that $ {\text{li}}{{\text{m}}_l}\int f \,d{\mu _l}$ exists for every $f \in \mathbf{C}$. After considering a finite dimensional case, the infinite dimensional case is entertained.
Nonlinear mappings that are globally equivalent to a projection
Roy
Plastock
373-380
Abstract: The Rank theorem gives conditions for a nonlinear Fredholm map of positive index to be locally equivalent to a projection. In this paper we wish to find conditions which guarantee that such a map is globally equivalent to a projection. The problem is approached through the method of line lifting. This requires the existence of a locally Lipschitz right inverse, ${F^ \downarrow }(x)$, to the derivative map ${F^\prime }(x)$ and a global solution to the differential equation ${P^\prime }(t) = {F^ \downarrow }(P(t))(y - {y_0})$. Both these problems are solved and the generalized Hadamard-Levy criterion $\displaystyle \int_0^\infty {\mathop {\inf }\limits_{\vert x\vert < s} \left({1/\vert{F^ \downarrow }(x)\vert} \right)\,ds = \infty }$ is shown to be sufficient for $F$ to be globally equivalent to a projection map (Theorem 3.2). The relation to fiber bundle mappings is explored in §4.
On generalized Peano derivatives
Cheng Ming
Lee
381-396
Abstract: A function $ F$ is said to have a generalized $n$th Peano derivative at $x$ if $F$ is continuous in a neighborhood of $ x$ and if there exists a positive integer $k$ such that a $k$th primitive of $F$ in the neighborhood has the $(k + n)$th Peano derivative at $ x$; and in this case this $ (k + n)$th Peano derivative at $x$ is proved to be independent of the integer $ k$ and the $k$th primitives, and is called the generalized $n$th Peano derivative of $F$ at $x$ which is denoted as $ {F_{[n]}}(x)$. If ${F_{[n]}}(x)$ exists and is finite for all $ x$ in an interval, then it is shown that ${F_{[n]}}$ shares many interesting properties that are known for the ordinary Peano derivatives. Using the generalized Peano derivatives, a notion called absolute generalized Peano derivative is studied. It is proved that on a compact interval, the absolute generalized Peano derivatives are just the generalized Peano derivatives. In particular, Laczkovich's absolute (ordinary) Peano derivatives are generalized Peano derivatives.
Orthogonal geodesic and minimal distributions
Irl
Bivens
397-408
Abstract: Let $\mathfrak{F}$ be a smooth distribution on a Riemannian manifold $M$ with $ \mathfrak{H}$ the orthogonal distribution. We say that $ \mathfrak{F}$ is geodesic provided $ \mathfrak{F}$ is integrable with leaves which are totally geodesic submanifolds of $M$. The notion of minimality of a submanifold of $M$ may be defined in terms of a criterion involving any orthonormal frame field tangent to the given submanifold. If this criterion is satisfied by any orthonormal frame field tangent to $\mathfrak{H}$ then we say $ \mathfrak{H}$ is minimal. Suppose that $ \mathfrak{F}$ and $\mathfrak{H}$ are orthogonal geodesic and minimal distributions on a submanifold of Euclidean space. Then each leaf of $ \mathfrak{F}$ is also a submanifold of Euclidean space with mean curvature normal vector field $\eta$. We show that the integral of $\vert\eta {\vert^2}$ over $M$ is bounded below by an intrinsic constant and give necessary and sufficient conditions for equality to hold. We study the relationships between the geometry of $M$ and the integrability of $\mathfrak{H}$. For example, if $\mathfrak{F}$ and $ \mathfrak{H}$ are orthogonal geodesic and minimal distributions on a space of nonnegative sectional curvature then $\mathfrak{H}$ is integrable iff $\mathfrak{F}$ and $ \mathfrak{H}$ are parallel distributions. Similarly if ${M^n}$ has constant negative sectional curvature and dim $ \mathfrak{H} = 2 < n$ then $\mathfrak{H}$ is not integrable. If $\mathfrak{F}$ is geodesic and $\mathfrak{H}$ is integrable then we characterize the local structure of the Riemannian metric in the case that the leaves of $ \mathfrak{H}$ are flat submanifolds of $M$ with parallel second fundamental form.
The spectrum of a Riemannian manifold with a unit Killing vector field
David D.
Bleecker
409-416
Abstract: Let $(P,g)$ be a compact, connected, ${C^\infty }$ Riemannian $(n + 1)$-manifold $ (n \geqslant 1)$ with a unit Killing vector field with dual $1$-form $\eta$. For $t > 0$, let ${g_{t}} = {t^{ - 1}}g + (t^{n}-t^{-1})\eta \otimes \eta$, a family of metrics of fixed volume element on $P$. Let $ {\lambda _1}(t)$ be the first nonzero eigenvalue of the Laplace operator on ${C^\infty }(P)$ of the metric ${g_t}$. We prove that if $d\eta$ is nowhere zero, then ${\lambda _1}(t) \to \infty$ as $t \to \infty$. Using this construction, we find that, for every dimension greater than two, there are infinitely many topologically distinct compact manifolds for which $ {\lambda _1}$ is unbounded on the space of fixed-volume metrics.
Convexity and tightness for restrictions of Hamiltonian functions to fixed point sets of an antisymplectic involution
J. J.
Duistermaat
417-429
Abstract: The Kostant convexity theorem for real flag manifolds is generalized to a Hamiltonian framework. More precisely, it is proved that if $f$ is the momentum mapping for a Hamiltonian torus action on a symplectic manifold $M$ and $Q$ is the fixed point set of an antisymplectic involution of $M$ leaving $f$ invariant, then $ f(Q) = f(M) =$ a convex polytope. Also it is proved that the coordinate functions of $f$ are tight, using "half-turn" involutions of $ Q$.
Tilted algebras
Dieter
Happel;
Claus Michael
Ringel
399-443
Abstract: Let $A$ be a finite dimensional hereditary algebra over a field, with $n$ simple $A$-modules. An $A$-module $T_A$ with $n$ pairwise nonisomorphic indecomposable direct summands and satisfying $ {\text{Ex}}{{\text{t}}^1}({T_A},\,{T_A}) = 0$ is called a tilting module, and its endomorphism ring $B$ is a tilted algebra. A tilting module defines a (usually nonhereditary) torsion theory, and the indecomposable $B$-modules are in one-to-one correspondence to the indecomposable $A$-modules which are either torsion or torsionfree. One of the main results of the paper asserts that an algebra of finite representation type with an indecomposable sincere representation is a tilted algebra provided its Auslander-Reiten quiver has no oriented cycles. In fact, tilting modules are introduced and studied for any finite dimensional algebra, generalizing recent results of Brenner and Butler.
Recognition of linear actions on spheres
Sören
Illman
445-478
Abstract: Let $G$ be a finite group acting smoothly on a homotopy sphere $\Sigma^m$. We wish to establish necessary and sufficient conditions for the given $G$-action on $\Sigma$ to be topologically equivalent to a linear action. That is, we want to be able to decide whether or not there exists a $G$-homeomorphism $\gamma:\Sigma\to S^m(\rho)$, where ${S^m}(\rho ) \subset {\mathbf{R}^{m + 1}}(\rho )$ denotes the unit sphere in an orthogonal representation space $\mathbf{R}^{m + 1}(\rho )$ for $G$. In order for a $G$-action on $\Sigma$ to be topologically equivalent to a linear action it is clearly necessary that: (i) For each subgroup $H$ of $G$ the fixed-point set $\Sigma^H$ is homeomorphic to a sphere, or empty. (ii) For any subgroups $H$ and $H \subsetneq {H_i},\,1 \leq i \leq k$, of $ G$ the pair $ (\Sigma^{H},\,\cup_{i=1}^{k}\Sigma^{H_{i}})$ is homeomorphic to a standard pair $ (S^{n},\,\cup_{i=1}^{k}S_{i}^{n_{i}})$, where each $S_i^{{n_i}},\,1 \le i \le k$, is a standard $ n_i$-subsphere of $ S^n$. In this paper we consider the case where the fixed-point set $ \Sigma^G$ is nonempty and all other fixed-point sets have dimension at least 5. In giving efficient sufficient conditions we do not need the full strength of condition (ii). We only need: (ii)$^{\ast}$ For any subgroups $H$ and $H \subsetneq {H_i},\,1 \leq i \leq p$, of $ G$ such that $ {\operatorname{dim}}\,{\Sigma ^{{H_i}}} = {\operatorname{dim}}\,{\Sigma ^H} - 2$, the pair $ \Sigma^{H},\,\cup_{i=1}^{p}\Sigma^{H_{i}})$ is homeomorphic to a standard pair $({S^n},\, \cup _{i = 1}^pS_i^{n - 2})$, where each $ S_i^{n - 2},\,1 \le i \le p$, is a standard $(n-2)$-subsphere of $S^n$. Our main results are then that, in the case when $G$ is abelian, conditions (i) and (ii)$ ^{\ast}$ are necessary and sufficient for a given $G$-action on $\Sigma$ to be topologically equivalent to a linear action, and in the case of an action of an arbitrary finite group the same holds under the additional assumption that any simultaneous codimension 1 and 2 fixed-point situation is simple. Our results generalize, for actions of finite groups, a well-known theorem of Connell, Montgomery and Yang, and are the first to also cover the case where codimension 2 fixed-point situations occur.
Multiplicative properties of power maps. II
C. A.
McGibbon
479-508
Abstract: The notions of $ A_n$-maps and $ C_n$-forms can be regarded as crude approximations to the concepts of homomorphisms and commutativity, respectively. These approximations are used to study power maps on connected Lie groups and their localizations. For such groups the power map $ x \mapsto {x^n}$ is known to be an $A_2$-map if and only if $n$ is a solution to a certain quadratic congruence. In this paper, $A_3$-power maps are studied. For the Lie group Sp(l) it is shown that the $A_3$-powers coincide with solutions which are common to the quadratic congruence, mentioned earlier, and another cubic congruence. Other Lie groups, when localized so as to become homotopy commutative, are also shown to have infinitely many $ A_3$-powers. The proofs reflect the combinatorial nature of the obstructions involved.
Equivariant minimal models
Georgia V.
Triantafillou
509-532
Abstract: Let $G$ be a finite group. We give an algebraicization of rational $G$-homotopy theory analogous to Sullivan's theory of minimal models in ordinary homotopy theory.
On critical point theory for indefinite functionals in the presence of symmetries
Vieri
Benci
533-572
Abstract: We consider functionals which are not bounded from above or from below even modulo compact perturbations, and which exhibit certain symmetries with respect to the action of a compact Lie group. We develop a method which permits us to prove the existence of multiple critical points for such functionals. The proofs are carried out directly in an infinite dimensional Hilbert space, and they are based on minimax arguments. The applications given here are to Hamiltonian systems of ordinary differential equations where the existence of multiple time-periodic solutions is established for several classes of Hamiltonians. Symmetry properties of these Hamiltonians such as time translation invariancy or evenness are exploited.
Eisenstein series of weight $3/2$. I
Ting Yi
Pei
573-606
Abstract: We prove that in the space of elliptic modular forms with weight 3/2, the orthogonal complement of the subspace of cusp forms with respect to the Petersson inner product is generated by Eisenstein series in some special cases.
Independent families in complete Boolean algebras
B.
Balcar;
F.
Franěk
607-618
Abstract: We present a proof (without any set-theoretical assumptions) that every infinite complete Boolean algebra includes a free subalgebra of the same cardinality. It follows that the set of all ultrafilters on an infinite complete Boolean algebra $B$ has power $2^{\vert B\vert}$.
Partition algebras for almost-disjoint families
James E.
Baumgartner;
Martin
Weese
619-630
Abstract: A set $a \subseteq \omega$ is a partitioner of a maximal almost-disjoint faculty $F$ of subsets of $\omega$ if every element of $F$ is almost contained in or almost-disjoint from $ a$. The partition algebra of $F$ is the quotient of the Boolean algebra of partitioners modulo the ideal generated by $F$ and the finite sets. We show that every countable algebra is a partition algebra, and that CH implies every algebra of cardinality $\leq {2^{{\aleph _0}}}$ is a partition algebra. We also obtain consistency and independence results about the representability of Boolean algebras as partition algebras.
On extending free group actions on spheres and a conjecture of Iwasawa
Frank
Connolly;
Robert
Geist
631-640
Abstract: A transfer map for Reidemeister torsion is defined and used to determine whether free actions of $ \mathbf{Z}/k$ on $ S^{2n+1}$, $n > 1$, extend to free actions of $\mathbf{Z}/hk$. It is shown that for $ k$ odd, every free $\mathbf{Z}/k$ action on $S^{2n+1}$, $n > 1$, extends to a free $\mathbf{Z}/2k$ action. For prime $p$, extension of an arbitrary free $\mathbf{Z}/p$ action to a free $\mathbf{Z}/p^{2}$ action is reduced to a long-standing conjecture of Iwasawa.
$2$-sided embeddings of projective planes into $3$-manifolds
Mitsuyuki
Ochiai
641-650
Abstract: Let $M$ be a nonorientable closed $ 3$-manifold which admits a $ 2$-sided embedding of a projective plane. Then we first prove the following theorem: If $M$ has a Heegaard splitting of genus two, then $ M$ is homeomorphic to ${P^{2}}\times {S^{1}}$. Next, let $ M$ be a nonorientable $ 3$-manifold whose fundamental group is abelian. We verify that if $ M$ has a Heegaard splitting of genus two, then $M$ is either the nonorientable $2$-sphere bundle over the circle or ${P^{2}}\times {S^{1}}$.
On the group of real analytic diffeomorphisms of a compact real analytic manifold
J.
Leslie
651-669
Abstract: In this paper we exhibit a Lie group structure on the group of real analytic diffeomorphisms of a compact real analytic manifold. Further, we show that a variant of the Kupka-Smale theorem holds for the underlying topology of the above mentioned Lie group.
Whitney levels in hyperspaces of certain Peano continua
Jack T.
Goodykoontz;
Sam B.
Nadler
671-694
Abstract: Let $X$ be a Peano continuum. Let $ {2^x}$ (resp., $ C(X)$) be the space of all nonempty compacta (resp., subcontinua) of $ X$ with the Hausdorff matric. Let $\omega$ be a Whitney map defined on $\mathcal{H}={2^{X}}$ or $C(X)$ such that $\omega$ is admissible (this requires the existence of a certain type of deformation of $ \mathcal{H}$). If $\mathcal{H}=C(X)$, assume $X$ contains no free arc. Then, for any $ {t_0} \in (0,\omega (X))$, it is proved that $ {\omega ^{ - 1}}({t_0}),\,{\omega ^{ - 1}}([0,\,{t_0}])$, and ${\omega ^{ - 1}}([{t_0},\,\omega (X)])$ are Hilbert cubes. This is an analogue of the Curtis-Schori theorem for $ \mathcal{H}$. A general result for the existance of admissible Whitney maps is proved which implies that these maps exist when $ X$ is starshaped in a Banach space or when $X$ is a dendrite. Using these results it is shown, for example that being an AR, an ANR, an LC space, or an $ {\text{L}}{{\text{C}}^n}$ space is not strongly Whitney-reversible.
Unstable bordism groups and isolated singularities
David
Ellis
695-708
Abstract: An isolated singularity of an embedded submanifold can be topologically smoothed if and only if a certain obstruction element in ${\pi _ \ast }(MG)$ vanishes, where $ G$ is the group of the normal bundle. In fact this obstruction lies in a certain subgroup which is referred to here as the unstable $ G$-bordism group. In this paper some of the unstable $SO$-bordism groups are computed; the obstruction to smoothing the complex cone on an oriented submanifold $X \subset \mathbf{C}{P^n}$ at $\infty$ is computed in terms of the characteristic numbers of $X$. Examples of nonsmoothable complex cone singularities are given using these computations.
Polynomials on affine manifolds
David
Fried
709-719
Abstract: For a closed affine manifold $M$ of dimension $m$ the developing map defines an open subset $ D(\tilde M) \subset {{\mathbf{R}}^m}$. We show that $ D(\tilde M)$ cannot lie between parallel hyperplanes. When $m \le 3$ we show that any nonconstant polynomial $p:{{\mathbf{R}}^m} \to {\mathbf{R}}$ is unbounded on $D(\tilde M)$. If $D(\tilde M)$ lies in a half-space we show $ M$ has zero Euler characteristic. Under various special conditions on $ M$ we show that $ M$ has no nonconstant functions given by polynomials in affine coordinates.
On the variety of invariant subspaces of a finite-dimensional linear operator
Mark A.
Shayman
721-747
Abstract: If $V$ is a finite-dimensional vector space over $ \mathbf{R}$ or $\mathbf{C}$ and $A \in {\operatorname {Hom}}(V)$, the set $ {S_A}(k)$ of $ k$-dimensional $ A$-invariant subspaces is a compact subvariety of the Grassmann manifold $ {G^k}(V)$, but it need not be a Schubert variety. We study the topology of $ {S_A}(k)$. We reduce to the case where $A$ is nilpotent. In this case we prove that $ {S_A}(k)$ is connected but need not be a manifold. However, the subset of $ {S_A}(k)$ consisting of those subspaces with a fixed cyclic structure is a regular submanifold of ${G^k}(V)$.
Local analyticity in weighted $L\sp{1}$-spaces and applications to stability problems for Volterra equations
G. S.
Jordan;
Olof J.
Staffans;
Robert L.
Wheeler
749-782
Abstract: We study the qualitative properties of the solutions of linear convolution equations such as $x \ast \mu = f$. We are especially concerned with finding conditions which ensure that these equations have resolvents which belong to, or are determined up to a term belonging to, certain weighted ${L^1}$-spaces. Our results are obtained as consequences of more general Banach algebra results on functions that are locally analytic with respect to the elements of a weighted ${L^1}$-space. In particular, we derive a proposition of Wiener-Lévy type for weighted $ {L^1}$-spaces which underlies all subsequent results. Our method applies equally well to equations more general than those mentioned above. We unify and sharpen the results of several recent papers on the asymptotic behavior of Volterra convolution equations of the types mentioned above, and indicate how many of them can be extended to the Fredholm case. In addition, we give necessary and sufficient conditions on the perturbation term $f$ for the existence of bounded or integrable solutions $x$ in some critical cases when the corresponding limit equations have nontrivial solutions.
Disproof of a coefficient conjecture for meromorphic univalent functions
Anna
Tsao
783-796
Abstract: Let $\Sigma$ denote the class of functions $g(z) = z + {b_0} + {b_1}{z^{ - 1}} + \cdots$ analytic and univalent in $\vert z\vert> 1$ except for a simple pole at $ \infty$. A well-known conjecture asserts that $ \vert{b_n}\vert\, \leq 2/(n + 1)\qquad (n = 1,2, \ldots )$ with equality for $g(z) = {(1 + {z^{n + 1}})^{2/(n + 1)}}/z = z + 2{z^{ - n}}/(n + 1) + \cdots$. Although the conjecture is true for $n=1,2$ and certain subclasses of the class $ \Sigma$, the general conjecture is known to be false for all odd $n\ge 3$ and $n=4$. In $\S 2$, we generalize a variational method of Goluzin and develop second-variational techniques. This enables us in $\S 3$ to construct explicit counterexamples to the conjecture for all $n > 4$. In fact, the conjectured extremal function does not even provide a local maximum for ${\text{Re}}\{ {b_n}\}$, $n > 4$.
On some open problems of P. Tur\'an concerning Birkhoff interpolation
A. K.
Varma
797-808
Abstract: In 1974 P. Turán (see [6]) raised many interesting open problems in Approximation Theory some of which are on Birkhoff Interpolation. The object of this paper is to answer some of these problems (XXXVI-XXXIX). We obtain some new quadrature formulas where function values and second derivatives are only prescribed on the zeros of $\displaystyle {\pi _n}(x) = {c_n}\int_{ - 1}^x {{P_{n - 1}}(t)\,dt}$ , ${P_n}(x)$ being Legendre polynomial of degree $ n$.
An elementary integral solution operator for the Cauchy-Riemann equations on pseudoconvex domains in ${\bf C}\sp{n}$
R. Michael
Range
809-816
Abstract: An integral representation formula for $(0,q)$ forms is constructed on a strictly pseudoconvex domain $D$ in $ \mathbf{C}^n$ by using only the local geometry of the boundary of $D$. By combining this representation with elementary results about compact operators in Banach spaces, one obtains the solution of the Levi problem and, more importantly, an integral solution operator for $ \bar{\partial}$ on $ D$. The construction does not need any a priori knowledge of the solvability of $ \bar{\partial}$ and thus allows us to establish fundamental global results by a direct and elementary method.
$C\sp{\ast} $-algebras of multivariable Wiener-Hopf operators
Paul S.
Muhly;
Jean N.
Renault
1-44
Abstract: The ${C^ \ast }$-algebra $ \mathfrak{W}$ generated by the Wiener-Hopf operators defined over a subsemigroup of a locally compact group is shown to be the image of a groupoid ${C^ \ast }$-algebra under a suitable representation. When the subsemigroup is either a polyhedral cone or a homogeneous, self-dual cone in an Euclidean space, this representation may be used to show that $ \mathfrak{W}$ is postliminal and to find a composition series with very explicit subquotients. This yields a concrete parameterization of the spectrum of $ \mathfrak{W}$ and exhibits the topology on it.
The inverse of a totally positive bi-infinite band matrix
Carl
de Boor
45-58
Abstract: It is shown that a bounded bi-infinite banded totally positive matrix $ A$ is boundedly invertible iff there is one and only one bounded sequence mapped by $A$ to the sequence $ ({( - )^i})$. The argument shows that such a matrix has a main diagonal, i.e., the inverse of $A$ is the bounded pointwise limit of inverses of finite sections of $A$ principal with respect to a particular diagonal; hence $ ({( - )^{i + j}}{A^{ - 1}}(i,j))$ or its inverse is again totally positive.
Inverses of infinite sign regular matrices
C.
de Boor;
S.
Friedland;
A.
Pinkus
59-68
Abstract: Let $A$ be an infinite sign regular (sr) matrix which can be viewed as a bounded linear operator from ${l_\infty }$ to itself. It is proved here that if the range of $A$ contains the sequence $( \ldots ,1, - 1,1, - 1, \ldots )$, then $ A$ is onto. If ${A^{ - 1}}$ exists, then $ D{A^{ - 1}}D$ is also sr, where $D$ is the diagonal matrix with diagonal entries alternately $1$ and $- 1$. In case $A$ is totally positive (tp), then $D{A^{ - 1}}D$ is also tp under additional assumptions on $A$.
Conditioning by $\langle $equal, linear$\rangle $
Chii-Ruey
Hwang
69-83
Abstract: We deal with a limit problem of regularity controlled probabilities in metric pattern theory. The probability on the generator space is given by a density function $f(x,y)$ on which some integrability conditions are imposed. Let $T$ denote the integral operator with kernel $ f$. When $n$ i.i.d. generators $({X_k},{Y_k})$ are connected together to form the configuration space $ {\mathcal{C}_n}$ via the regularity $ \left\langle {{\text{EQUAL}},{\text{LINEAR}}} \right\rangle$, i.e. "conditioning" on $ {X_{k + 1}} = {Y_k}$ for $1 \leqslant k < n$, an approximate identity is used to define the regularity controlled probability on $ {\mathcal{C}_n}$. The probabilistic effect induced by the regularity conditions on some fixed subconfiguration of a larger configuration $ {\mathcal{C}_n}$ is described by its corresponding marginal probability within $ {\mathcal{C}_n}$. When $ n$ goes to infinity in a suitable way, the above mentioned marginal probability converges weakly to a limit whose density can be expressed in terms of the largest eigenvalues and the corresponding eigenspaces of $T$ and ${T^ \ast }$. When $f$ is bivariate normal, the eigenvalue problem is solved explicitly. The process determined by the limiting marginal probabilities is strictly stationary and Markovian.
A trace formula for compact manifolds
K. S.
Sarkaria
85-88
Abstract: An integral formula for the Euler characteristic is given, in which the data consists of a finite dimensional transitive vector space $V$ of vector fields and a volume form $\Omega$ supported in a small neighborhood of the origin of $V$.
Faisceaux amples sur les espaces analytiques
Vincenzo
Ancona
89-100
Abstract: The following result is established. Let $f:X \to Y$ be a morphism between two compact complex spaces and $ \mathfrak{L}$ a weakly positive invertible sheaf on $X$; then for suitable $nf{}_ \ast {\mathfrak{L}^n}$ is weakly positive on $Y$. It follows that Moišezon spaces can be characterized via weakly positive coherent sheaves. Moreover, a problem posed by Grauert on the exceptional subspaces of complex spaces can be solved.
Decisive subgroups of analytic groups
T. Christine
Stevens
101-108
Abstract: It is known that every analytic group $(L,\tau )$ contains a closed abelian subgroup $ H$ which is "decisive" in the sense that the Hausdorff topologies for $ L$ which are weaker than $ \tau$ are completely determined by their restrictions to $H$. We show here that $H$ must ordinarily contain the entire center of $ L$ but that the rest of $ H$ can in general be reduced. The proof involves constructing "unusual" topologies for abelian Lie groups.
The Albanese mapping for a punctual Hilbert scheme. II. Symmetrized differentials and singularities
Mark E.
Huibregtse
109-140
Abstract: Let $f:X \to A$ be the canonical mapping from the irreducible and nonsingular surface $ X$ to its Albanese variety $ A$, $X(n)$ the $n$-fold symmetric product of $X$, and $H_X^n$ the punctual Hilbert scheme parameterizing 0-dimensional closed subschemes of length $ n$ on $X$. The latter is an irreducible and nonsingular variety of dimension $2n$, and the "Hilbert-Chow" morphism ${\sigma _n}:H_X^n \to X(n)$ is a birational map which desingularizes $ X(n)$. Let ${f_n}:X(n) \to A$ denote the map induced by $ f$ by addition on $ A$. This paper studies the singularities of the composite morphism $\displaystyle {\varphi _n}:H_X^n\mathop \to \limits^{{\sigma _n}} X(n)\mathop \to \limits^{{f_n}} A,$ which is a natural generalization of the mapping $C(n) \to J$, where $C$ is an irreducible and nonsingular curve and $J$ is its Jacobian. Unlike the latter, however, ${\varphi _n}$ need not be smooth for $n \gg 0$. We prove that ${\varphi _n}$ is smooth for $n \gg 0$ only if $f:X \to A$ is smooth (Theorem 3), and over ${\mathbf{C}}$ we prove the converse (Theorem 4). In case $X = A$ is an abelian surface, we show ${\varphi _n}$ is smooth for $n$ prime to the characteristic (Theorem 5), and give a counterexample to smoothness for all $n$ (Theorem 6). We exhibit a connection (over ${\mathbf{C}}$) between singularities of ${\varphi _n}$ and generalized Weierstrass points of $X$ (Theorem 9). Our method is as follows: We first show that the singularities of ${\varphi _n}$ are the zeros of certain holomorphic $ 1$-forms on $H_X^n$ which are the "symmetrizations" of holomorphic $1$-forms on $X$. We then study "symmetrized differentials" and their zeros on $H_X^n$ (Theorems 1,2). Our method works for curves $ C$ as well; we give an alternative proof of a result of Mattuck and Mayer [10, p. 226] which shows that the zeros of symmetrized differentials on $C(n)$ represent (for $C$ complete nonsingular) the special divisors of degree $n$ on $C$.
Weakly almost periodic functions and Fourier-Stieltjes algebras of locally compact groups
Ching
Chou
141-157
Abstract: A noncompact locally compact group $G$ is called an Eberlein group if $W(G) = B{(G)^ - }$ where $W(G)$ is the algebra of continuous weakly almost periodic functions on $G$ and $B{(G)^ - }$ is the uniform closure of the Fourier-Stieltjes algebra of $G$. We show that if $G$ is a noncompact $[IN]$-group or a noncompact nilpotent group then $W(G)/B{(G)^ - }$ contains a linear isometric copy of $ {l^\infty }$. In particular, $G$ is not an Eberlein group. On the other hand, finite direct products of Euclidean motion groups and, by a result of W. Veech, noncompact semisimple analytic groups with finite centers are Eberlein groups.
Regularizing effects for $u\sb{t}=\Delta \varphi (u)$
Michael G.
Crandall;
Michel
Pierre
159-168
Abstract: One expression of the fact that a nonnegative solution of the initial-value problem $\displaystyle ({\text{IVP}})\quad \left\{ {\begin{array}{*{20}{c}} {{u_t} - \De... ...\right.\quad \begin{array}{*{20}{c}} {t > 0,x \in {R^N},} {} \end{array}$ where $m > 0$, is more regular for $t > 0$ than a rough initial datum ${u_0}$ is the remarkable pointwise inequality ${u_t} = \Delta {u^m} \geqslant - (N/(N(m - 1) + 2)t)u$ obtained by Aronson and Bénilan for $ t > 0$ and $m > \max ((N - 2)/N,0)$. This inequality was used by Friedman and Caffarelli in proving that solutions of (IVP) are continuous for $ t > 0$. The main results of this paper generalize the Aronson-Bénilan inequality and show the extended inequality is valid for a much broader class of equations of the form ${u_t} = \Delta \varphi (u)$. In particular, the results apply to the Stefan problem which is modeled by $ \varphi (r) = {(r - 1)^ + }$ and imply ${({(u - 1)^ + })_t} \geqslant - ({(u - 1)^ + } + N/2)/t$ in this case.
Equilibrium states of grid functions
Nelson G.
Markley;
Michael E.
Paul
169-191
Abstract: It is well known that locally constant functions on symbolic spaces have unique equilibrium states. In this paper we investigate the nature of equilibrium states for a type of continuous function which need not have a finite range. Although most of these functions have a unique equilibrium state, phase transitions or multiple equilibrium states do occur and can be analyzed.
Clifford module invariants of spin bundles
Jacques
Allard;
Anthony
Bahri
193-202
Abstract: In this paper, we study $KO$-theory invariants of Spin bundles obtained by the $ \alpha$-construction from Clifford module representations of the Spinor group. We begin by describing their elementary properties including various Whitney sum formulae and their relation with the $d$-invariant for vector bundles over spheres. We next observe an important difference between the two half-Spin representations and then proceed to investigate the fiber homotopy properties of the invariants. We conclude with some applications.
Eilenberg-Moore models for fibrations
J.-C.
Thomas
203-225
Abstract: E. M. model is a new invariant in rational homotopy theory which gives us both a Künneth object and a Tate-Josefiak resolution. With the E. M. model, we study relations between formality of base, total space and fibre of a Serre fibration, obstructions to $ {\mathbf{k}}$-realizability of a cohomology equivalence between two continuous maps and formalizable maps.
Dunford-Pettis operators on Banach lattices
C. D.
Aliprantis;
O.
Burkinshaw
227-238
Abstract: Consider a Banach lattice $E$ and two positive operators $S,T:E \to E$ that satisfy $0 \leqslant S \leqslant T$. In $[{\mathbf{2,3}}]$ we examined the case when $ T$ is a compact (or weakly compact) operator and studied what effect this had on an operator (such as $S$) dominated by $T$. In this paper, we extend these techniques and study similar questions regarding Dunford-Pettis operators. In particular, conditions will be given on the operator $T$, to ensure that $S$ (or some power of $S$) is a Dunford-Pettis operator. As a sample, the following is one of the major results dealing with these matters. Theorem. Let $E$ be a Banach lattice, and let $S,T:E \to E$ be two positive operators such that $ 0 \leqslant S \leqslant T$. If $T$ is compact then (1) $ {S^3}$ is a compact operator (although ${S^2}$ need not be compact); (2) $ {S^2}$ is a Dunford-Pettis and weakly compact operator ( although $ S$ need not be ); (3) $S$ is a weak Dunford-Pettis operator. In another direction, our techniques and results will be related to the lattice stracture of the Dunford-Pettis operators. For instance, it will be shown that under certain conditions the Dunford-Pettis operators form a band.
The space of positive definite matrices and Gromov's invariant
Richard P.
Savage
239-263
Abstract: The space $ X_d^n{\text{of}}n \times n$ positive definite matrices with ${\text{determinant}} = 1$ is considered as a subset of ${{\mathbf{R}}^{n(n + 1)/2}}$ with isometries given by $X \to AX{A^t}$ where the determinant of $A = 1$ and $X_d^n$ is given its invariant Riemannian metric. This space has a collection of simplices which are preserved by the isometries and formed by projecting geometric simplices in ${{\mathbf{R}}^{n(n + 1)/2}}$ to the hypersurface $X_d^n$. The main result of this paper is that for each $n$ the volume of all top dimensional simplices of this type has a uniform upper bound. This result has applications to Gromov's Invariant as defined in William P. Thurston's notes, The geometry and topology of $3$-manifolds. The result implies that the Gromov Invariant of the fundamental class of compact manifolds which are formed as quotients of $ X_d^n$ by discrete subgroups of the isometries is nonzero. This gives the first nontrivial examples of manifolds that have a nontrivial Gromov Invariant but do not have strictly negative curvature or nonvanishing characteristic numbers.
Bordered Klein surfaces with maximal symmetry
Newcomb
Greenleaf;
Coy L.
May
265-283
Abstract: A compact bordered Klein surface of (algebraic) genus $g \geqslant 2$ is said to have maximal symmetry if its automorphism group is of order $12(g - 1)$, the largest possible. In this paper we study the bordered surfaces with maximal symmetry and their automorphism groups, the $ {M^\ast}$-groups. We are concerned with the topological type, rather than just the genus, of these surfaces and its relation to the structure of the associated $ {M^\ast}$-group. We begin by classifying the bordered surfaces with maximal symmetry of low topological genus. We next show that a bordered surface with maximal symmetry is a full covering of another surface with primitive maximal symmetry. A surface has primitive maximal symmetry if its automorphism group is ${M^\ast}$-simple, that is, if its automorphism group has no proper ${M^\ast}$-quotient group. Our results yield an approach to the problem of classifying the bordered Klein surfaces with maximal symmetry. Next we obtain several constructions of full covers of a bordered surface. These constructions give numerous infinite families of surfaces with maximal symmetry. We also prove that only two of the ${M^\ast}$-simple groups are solvable, and we exhibit infinitely many nonsolvable ones. Finally we show that there is a correspondence between bordered Klein surfaces with maximal symmetry and regular triangulations of surfaces.
Krull and global dimensions of semiprime Noetherian PI-rings
Richard
Resco;
Lance W.
Small;
J. T.
Stafford
285-295
Abstract: In this paper it is shown that if $R$ is a semiprime Noetherian PI-ring of finite global dimension, then the Krull dimension of $ R$ is less than or equal to its global dimension. The proof depends upon two preliminary results on arbitrary Noetherian PI-rings, which are of independent interest: (i) any height two prime ideal of $R$ contains infinitely many height one prime ideals; (ii) the localization of the polynomial ring $ R[x]$ at its set of monic elements is a Jacobson ring.
Free boundary convergence in the homogenization of the one-phase Stefan problem
José-Francisco
Rodrigues
297-305
Abstract: We consider the one phase Stefan problem in a "granular" medium, i.e., with nonconstant thermal diffusity, and we study the asymptotic behaviour of the free boundary with respect to homogenization. We prove the convergence of the coincidence set in measure and in the Hausdorff metric. We apply this result to the free boundary and we obtain the convergence in mean for the star-shaped case and the uniform convergence for the one-dimensional case, respectively. This gives an answer to a problem posed by J. L. Lions in [L].
Injectivity in Banach spaces and the Mazur-Ulam theorem on isometries
Julian
Gevirtz
307-318
Abstract: A mapping $ f$ of an open subset $ U$ of a Banach space $ X$ into another Banach space $Y$ is said to be $(m,M)$-isometric if it is a local homeomorphism for which $M \geqslant {D^ + }f(x)$ and $m \leqslant {D^ - }f(x)$ for all $x \in U$, where ${D^ + }f(x)$ and $ {D^ - }f(x)$ are, respectively, the upper and lower limits of $\vert f(y) - f(x)\vert/\vert y - x\vert\;{\text{as}}\;y \to x$. For $0 < \rho \leqslant 1$ we find a number $\mu (\rho ) > 1$ which has the following property: Let $X$ and $Y$ be Banach spaces and let $U$ be an open convex subset of $ X$ containing a ball of radius $r$ and contained in the concentric ball of radius $ R$. Then all $ (m,M)$-isometric mappings of $U$ into $Y$ are injective if $ M/m < \mu (r/R)$. We also derive similar injectivity criteria for a more general class of connected open sets $ U$. The basic tool used is an approximate version of the Mazur-Ulam theorem on the linearity of distance preserving transformations between normed linear spaces.
On the division of distributions by analytic functions in locally convex spaces
Denis
Chansolme
319-325
Abstract: Although the division of distributions by real polynomials and real analytic functions (which are nonzero) is always possible in finite dimensional spaces (from classical results of Hörmander and Lojasiewicz respectively), we show that this is not always possible in infinite dimensional locally convex spaces. In particular, we characterize those locally convex spaces where division is always possible.
The preprojective partition for hereditary Artin algebras
D.
Zacharia
327-343
Abstract: The purpose of this paper is to study the preprojective partition of a hereditary artin algebra. For a hereditary algebra of finite representation type, we give some numerical invariants in terms of the length of chains of irreducible maps, also in terms of the length of the maximal indecomposable module, and the orientation of the quiver of the algebra. Similar results are given for algebras stably equivalent to hereditary artin algebras.
Borsuk-Ulam theorems for arbitrary $S\sp{1}$ actions and applications
E. R.
Fadell;
S. Y.
Husseini;
P. H.
Rabinowitz
345-360
Abstract: An ${S^1}$ version of the Borsuk-Ulam Theorem is proved for a situation where Fix ${S^1}$ may be nontrivial. The proof is accomplished with the aid of a new relative index theory. Applications are given to intersection theorems and the existence of multiple critical points is established for a class of functional invariant under an $ {S^1}$ symmetry.
Ampleness and connectedness in complex $G/P$
Norman
Goldstein
361-373
Abstract: This paper determines the "ampleness" of the tangent bundle of the complex homogeneous space, $G/P$, by calculating the maximal fibre dimension of the desingularization of a nilpotent subvariety of the Lie algebra of $G$.
Hardy spaces and Jensen measures
Takahiko
Nakazi
375-378
Abstract: Suppose $ A$ is a subalgebra of ${L^\infty }(m)$ on which $m$ is multiplicative. In this paper, we show that if $ m$ is a Jensen measure and $A + \overline A $ is dense in $ {L^2}(m)$, then $A + \overline A$ is weak-* dense in ${L^\infty }(m)$, that is, $A$ is a weak-* Dirichlet algebra. As a consequence of it, it follows that if $ A + \overline A$ is dense in ${L^4}(m)$, then $A$ is a weak-* Dirichlet algebra. (It is known that even if $ A + \overline A$ is dense in ${L^3}(m)$, $A$ is not a weak-* Dirichlet algebra.) As another consequence, it follows that if $B$ is a subalgebra of the classical Hardy space $ {H^\infty }$ containing the constants and dense in ${H^2}$, then $B$ is weak-* dense in $ {H^\infty }$.
Distal compactifications of semigroups
H. D.
Junghenn
379-397
Abstract: Properties of distal functions on semitopological semigroups are studied via the device of right topological semigroup compactification. Algebras of distal functions are used to construct the universal right simple, left simple, and group compactifications of a semigroup, and these compactifications are in turn systematically employed to obtain information about distal functions. Applications are made to semidirect products and flows.
Free products of topological groups with central amalgamation. I
M. S.
Khan;
Sidney A.
Morris
405-416
Abstract: It is proved that the amalgamated free product of any two Hausdorff topological groups exists and is Hausdorff, providing the subgroup which is being amalgamated is closed and central.
Free products of topological groups with central amalgamation. II
M. S.
Khan;
Sidney A.
Morris
417-432
Abstract: In Free products of topological groups with central amalgamation. I, we introduced the notion of amalgamated free product of topological groups and showed that if $ A$ is a common central closed subgroup of Hausdorff topological groups $ G$ and $H$, then the amalgamated free product $G{\coprod _A}H$ exists and is Hausdorff. In this paper, we give an alternative much shorter (but less informative) proof of this result. We then proceed to describe the properties of $G{\coprod _A}H$. In particular, we find necessary and sufficient conditions for $G{\coprod _A}H$ to be a locally compact Hausdorff group, a complete metric group, and a maximally almost periodic group. Properties such as being a Baire space and connectedness are also investigated. In the case that $G$, $H$ and $A$ are $ {k_\omega }$-groups, the topology of $ G{\coprod _A}H$ is fully described. A consequence of this description is that for ${k_\omega }$-groups $ G{\coprod _A}H$ is homeomorphic to $(G{ \times _A}H) \times F(G/A\Lambda H/A)$, where $G{ \times _A}H$ is the direct product of $G$ and $H$ with $A$ amalgamated, and $F(G/A\Lambda H/A)$ is the free topological group on the smash product of $G/A$ and $H/A$.
Lattices over orders: finitely presented functors and preprojective partitions
M.
Auslander;
S. O.
Smalø
433-446
Abstract: Suppose $ R$ is a commutative noetherian equidimensional Gorenstein ring and $ \Lambda$ an $ R$-algebra which is finitely generated as an $R$-module. A $\Lambda$-module $M$ is a lattice if ${M_{\underline{\underline p} }}$ is ${\Lambda _{\underline{\underline p} }}$-projective and $ {\text{Ho}}{{\text{m}}_R}{(M,R)_{\underline{\underline p} }}$ is $ \Lambda _{\underline{\underline p} }^{{\text{op}}}$-projective for all nonmaximal prime ideals $\underline{\underline p} $ in $R$. We assume that $\Lambda$ is an $R$-order in the sense that $\Lambda$ is a lattice when viewed as a $ \Lambda$-module. The first main result is to show that simple contravariant functors from lattices to abelian groups are finitely presented. This is then applied to showing that if $ R$ is also local and complete, then the category of lattices has a preprojective partition. This generalizes previous results of the authors in the cases $R$ is artinian or a discrete valuation ring.
Asymptotic analysis of Gaussian integrals. I. Isolated minimum points
Richard S.
Ellis;
Jay S.
Rosen
447-481
Abstract: This paper derives the asymptotic expansions of a wide class of Gaussian function space integrals under the assumption that the minimum points of the action are isolated. Degenerate as well as nondegenerate minimum points are allowed. This paper also derives limit theorems for related probability measures which correspond roughly to the law of large numbers and the central limit theorem. In the degenerate case, the limits are non-Gaussian.
Reproducing kernels and bilinear sums for $q$-Racah and $q$-Wilson polynomials
Mizan
Rahman
483-508
Abstract: A five-parameter family of reproducing kernels is constructed for $ q$-Racah polynomials. Special cases for $q$-Hahn and little $q$-Jacobi polynomials are considered by selecting the parameters appropriately. Corresponding bilinear sums are also obtained for a whole range of $q$-orthogonal polynomials. As a special case, some product formulas are obtained for $ q$-Racah and $ q$-Wilson polynomials.
Irreducible representations of $A\sb{n}$ with a $1$-dimensional weight space
D. J.
Britten;
F. W.
Lemire
509-540
Abstract: In this paper we classify all irreducible linear representations of the simple Lie algebra ${A_n}$ which admit a one-dimensional weight space with respect to some Cartan subalgebra $H$ of ${A_n}$. We first show that the problem is equivalent to determining all algebra homomorphisms from the centralizer of the Cartan subalgebra $H$ in the universal enveloping algebra of ${A_n}$ to the base field. We construct all such algebra homomorphisms and provide conditions under which two such algebra homomorphisms provide inequivalent irreducible representations of $ {A_n}$.
A general principle for limit theorems in finitely additive probability
Rajeeva L.
Karandikar
541-550
Abstract: In this paper we formulate and prove a general principle which enables us to deduce limit theorems for sequences of independent random variables in a finitely additive setting from their analogues in the conventional countably additive setting.
Transfinite duals of quasireflexive Banach spaces
Steven F.
Bellenot
551-577
Abstract: The transfinite duals of a space with a neighborly basis are constructed until they become nonseparable. Let $ s(X)$ be the first ordinal $ \alpha$ so that ${X^\alpha }$ is nonseparable. It is shown that if $X$ is nonreflexive, $s(X) \leqslant {\omega ^2} + 1$ (this is best possible) and that $ \{ s(X):X{\text{separable quasireflexive of order one}}\} = \{ \omega + 1,\omega + 2,2\omega + 1,2\omega + 2,{\omega ^2} + 1\}$. A quasireflexive space $X$ is constructed so that ${X^\omega }$ is isomorphic to $X \oplus {c_0}$ and no basic sequence in $ X$ is equivalent to a neighborly basis. It is shown that the ${\omega ^2}$th dual of James space and James function space are isomorphic to subspaces of one another. Also, perhaps of interest on its own, a reflexive space with a subsymmetric basis is constructed whose inversion spans a nonreflexive space.
Fr\'echet spaces with nuclear K\"othe quotients
Steven F.
Bellenot;
Ed
Dubinsky
579-594
Abstract: Each separable Fréchet non-Banach space $X$ with a continuous norm is shown to have a quotient $ Y$ with a continuous norm and a basis. If, in addition, $Y$ can be chosen to be nuclear, we say that $X$ has a nuclear Köthe quotient. We obtain a (slightly technical) characterization of those separable Fréchet spaces with nuclear Köthe quotients. In particular, separable reflexive Fréchet spaces which are not Banach (and thus Fréchet Montel spaces) have nuclear Köthe quotients.
A characterization of Fourier and Radon transforms on Euclidean space
Alexander
Hertle
595-607
Abstract: We show that a continuous operator behaving under rotations, positive dilations, and translations like the Fourier or the Radon transform on $ {{\mathbf{R}}^n}$ must be a constant multiple of one of these transforms. We prove this characterization for various function spaces, e.g. we characterize the Fourier transform as an operator acting on spaces between $ \mathfrak{D}({{\mathbf{R}}^n})$ and $ \mathfrak{D}({{\mathbf{R}}^n})$ to $ \mathfrak{D}({{\mathbf{R}}^n})$ and ${S^{n - 1}} \times {\mathbf{R}}$. In the special case $n = 1$, our methods sharpen results of J. L. B. Cooper and H. Kober, who characterize the Fourier transform as an operator from ${L^p}({\mathbf{R}})$ into $ {L^p}^\prime ({\mathbf{R}}),1 \leqslant p \leqslant 2$.
Rational homotopy of the space of sections of a nilpotent bundle
André
Haefliger
609-620
Abstract: We show that an algebraic construction proposed by Sullivan is indeed a model for the rational homotopy type of the space of sections of a nilpotent bundle.
Asymptotic Toeplitz operators
José
Barría;
P. R.
Halmos
621-630
Abstract: An asymptotic Toeplitz is an operator $T$ such the sequence $\{ {U^{ \ast n}}T{U^n}\} $ is strongly convergent, where $U$ is the unilateral shift. Every element of the norm-closed algebra generated by all Toeplitz and Hankel opertors together is an asymptotic Toeplitz operator. The authors study the relations among this Hankel algebra, the classical Toeplitz algebra, the set of all asymptotic Toeplitz operators, and the essential commutant of the unilateral shift. They offer several examples of operators in some of these classes but not in others, and they raise several open questions.
The structure of pseudo-inverse semigroups
F.
Pastijn
631-655
Abstract: A regular semigroup $ S$ is called a pseudo-inverse semigroup if $eSe$ is an inverse semigroup for each $e = {e^2} \in S$. We show that every pseudo-inverse semigroup divides a semidirect product of a completely simple semigroup and a semilattice. We thereby give a structure theorem for pseudo-inverse semigroups in terms of groups, semilattices and morphisms. The structure theorem which is presented here generalizes several structure theorems which have been given for particular classes of pseudo-inverse semigroups by several authors, and thus contributes to a unification of the theory.
Weak $P$-points in \v Cech-Stone compactifications
Jan
van Mill
657-678
Abstract: Let $X$ be a nonpseudocompact space which is either nowhere ccc or nowhere of weight $\leqslant {2^\omega }$. Then $\beta X - X$ contains a point $x$ which is a weak $P$-point of $\beta X$, i.e. if $F \subset \beta X - \{ x\} $ is countable, then $x \notin \bar F$. In addition, under MA, if $ X$ is any nonpseudocompact space, then $ \beta X - X$ contains a point $x$ such that whenever $F \subset \beta X - \{ x\} $ is countable and nowhere dense, then $ x \notin \bar F$.
The Fourier expansion of Eisenstein series for ${\rm GL}(3,\,{\bf Z})$
K.
Imai;
A.
Terras
679-694
Abstract: The Fourier expansions of Eisenstein series for $ {\text{GL}}(3,{\mathbf{Z}})$ are obtained by two methods--one analogous to the classical method used by many number theorists, including Weber, in his derivation of the Kronecker limit formula. The other method is analogous to that used by Siegel to obtain Fourier expansions of Eisenstein series for the Siegel modular group. The expansions involve matrix argument $K$-Bessel functions recently studied by Tom Bengtson. These $K$-Bessel functions are natural generalizations of the ordinary $K$-Bessel function which arise when considering harmonic analysis on the symmetric space of the general linear group using a certain system of coordinates.
Extending free cyclic actions on spheres
John
Ewing
695-703
Abstract: Connolly and Geist have reduced the problem of determining which free cyclic actions on spheres extend to free actions of specified supergroups to a problem involving a certain transfer map. In this note we develop some algebraic tools for calculating the transfer and show that some cyclic actions do not extend to certain supergroups.
$C(\alpha )$ preserving operators on $C(K)$ spaces
John
Wolfe
705-719
Abstract: Let $A:C(K) \to X$ be a bounded linear operator where $K$ is a compact Hausdorff space and $ X$ is a separable Banach space. Sufficient conditions are given for $A$ to be an isomorphism (into) when restricted to a subspace $Y$ of $C(K)$, such that $Y$ is isometrically isomorphic to a space $C(\alpha )$ of continuous functions on the space of ordinal numbers less than or equal to the countable ordinal $\alpha$.
The asymptotic number of convex polyhedra
L. B.
Richmond;
N. C.
Wormald
721-735
Abstract: We obtain an asymptotic formula for the number of combinatorially distinct convex polyhedra with $n$ edges.
Odd primary Steenrod operations in first-quadrant spectral sequences
John
Sawka
737-752
Abstract: This paper defines two kinds of Steenrod operations in the spectral sequence of a bisimplical $\operatorname{mod} p$ coalgebra and shows them to be a complete list of all such possible Steenrod operations. These operations are compatible with the differentials and with Steenrod operations on the total complex. A general rule is given for computing the operations on ${E_2}$. A generalization of the Kudo transgression theorem is also proved, placing it in a larger and more natural setting.
Solvability of quasilinear elliptic equations with nonlinear boundary conditions
Gary M.
Lieberman
753-765
Abstract: On an $ n$-dimensional domain $ \Omega$, we consider the boundary value problem $\displaystyle (\ast)\quad Qu = 0\;{\text{in}}\Omega {\text{,}}\quad Nu = 0\;{\text{on}}\;\partial \Omega$ where $Q$ is a quasilinear elliptic second-order differential operator and $N$ is a nonlinear first order differential operator satisfying an Agmon-Douglis-Nirenberg consistency condition. If the coefficients of $ Q$ and $N$ satisfy additional hypotheses (such as sufficient smoothness), Fiorenza was able to reduce the solvability of $(\ast)$ to the establishment of a priori bounds for solutions of a related family of boundary value problems. We simplify Fiorenza's argument, obtaining the reduction under more general hypotheses and requiring a priori bounds only for solutions of $ Qu = f$, $Nu = g$ where $f$ and $g$ range over suitable function spaces. As an example, classical solutions of the capillary problem are shown to exist without using the method of elliptic regularization.
A generalization of a theorem of Maximoff and applications
S. J.
Agronsky
767-779
Abstract: Many classes of functions can be characterized in terms of their associated sets. Maximoff gave another type of characterization for the approximately continuous functions. In this paper, we give the conditions under which the two types of characterizations are equivalent. We then show that many classes of functions defined or characterized in terms of their associated sets also admit Maximoff-type characterizations.
The logarithm of the Poisson kernel of a $C\sp{1}$ domain has vanishing mean oscillation
David S.
Jerison;
Carlos E.
Kenig
781-794
Abstract: Let $D$ be a ${C^1}$ domain in $ {{\mathbf{R}}^n}$, and $ \omega$ the harmonic measure of $\partial D$, with respect to a fixed pole in $D$. Then, $ d\omega = kd\sigma$, where $k$ is the Poisson kernel of $D$. We show that log $k$ has vanishing mean oscillation of $\partial D$.
$C\sp{\ast} $-algebra fibre bundles
Maw Ding
Jean
795-801
Abstract: It will be shown in this paper that for any ${C^\ast}$-algebra fibre bundle with base space $ X$ and fibre $ A$, a $ {C^\ast}$-algebra, the Jacobson spectrum of the ${C^\ast}$-algebra of sections of the fibre bundle can be identified as a topological fibre bundle with the same base space $X$ and fibre the Jacobson spectrum of $A$.
Rational LS category and its applications
Yves
Félix;
Stephen
Halperin
1-37
Abstract: Let $S$ be a $1$-connected CW-complex of finite type and put $ {\text{ca}}{{\text{t}}_0}(S) =$ Lusternik-Schnirelmann category of the localization $ {S_{\mathbf{Q}}}$. This invariant is characterized in terms of the minimal model of $S$. It is shown that if $\phi :S \to T$ is injective on $ {\pi _ \ast } \otimes {\mathbf{Q}}$ then $ {\text{ca}}{{\text{t}}_0}(S) \leqslant {\text{ca}}{{\text{t}}_0}(T)$, and this result is strengthened when $ \phi$ is the fibre inclusion of a fibration. It is also shown that if $ {H^ \ast }(S;{\mathbf{Q}}) < \infty$ then either ${\pi _ \ast }(S) \otimes {\mathbf{Q}} < \infty$ or the groups ${\pi _k}(S) \otimes {\mathbf{Q}}$ grow exponentially with $k$.
Cohomology of nilmanifolds and torsion-free, nilpotent groups
Larry A.
Lambe;
Stewart B.
Priddy
39-55
Abstract: Let $M$ be a nilmanifold, i.e. $M = G/D$ where $ G$ is a simply connected, nilpotent Lie group and $D$ is a discrete uniform, nilpotent subgroup. Then $M \simeq K(D,1)$. Now $D$ has the structure of an algebraic group and so has an associated algebraic group Lie algebra $ L(D)$. The integral cohomology of $M$ is shown to be isomorphic to the Lie algebra cohomology of $L(D)$ except for some small primes depending on $ D$. This gives an effective procedure for computing the cohomology of $ M$ and therefore the group cohomology of $D$. The proof uses a version of form cohomology defined for subrings of $ {\mathbf{Q}}$ and a type of Hirsch Lemma. Examples, including the important unipotent case, are also discussed.
Contractive projections on $C\sb{0}(K)$
Yaakov
Friedman;
Bernard
Russo
57-73
Abstract: We show that the range of a norm one projection on a commutative $ {C^\ast}$-algebra has a ternary product structure (Theorem 2). We describe and characterize all such projections in terms of extreme points in the unit ball of the image of the dual (Theorem 1). We give necessary and sufficient conditions for the range to be isometric to a $ {C^\ast}$-algebra (Theorem 4) and we show that the range is a ${C_\sigma }$-space (Theorem 5).
Simple knots in compact, orientable $3$-manifolds
Robert
Myers
75-91
Abstract: A simple closed curve $J$ in the interior of a compact, orientable $ 3$-manifold $M$ is called a simple knot if the closure of the complement of a regular neighborhood of $ J$ in $M$ is irreducible and boundary-irreducible and contains no non-boundary-parallel, properly embedded, incompressible annuli or tori. In this paper it is shown that every compact, orientable $ 3$-manifold $M$ such that $ \partial M$ contains no $ 2$-spheres contains a simple knot (and thus, from work of Thurston, a knot whose complement is hyperbolic). This result is used to prove that such a $3$-manifold is completely determined by its set $ \mathcal{K}(M)$ of knot groups, i.e, the set of groups ${\pi _1}(M - J)$ as $J$ ranges over all the simple closed curves in $ M$. In addition, it is proven that a closed $3$-manifold $M$ is homeomorphic to ${S^3}$ if and only if every simple closed curve in $ M$ shrinks to a point inside a connected sum of graph manifolds and $3$-cells.
The asymptotic expansion for the trace of the heat kernel on a generalized surface of revolution
Ping Charng
Lue
93-110
Abstract: Let $M$ be a smooth compact Riemannian manifold without boundary. Let $I$ be an open interval. Let $h(r)$ be a smooth positive function. Let $g$ be the metric on $M$. Consider the fundamental solution $ E(x,y,{r_1},{r_2};t)$ of the heat equation on $M \times I$ with metric $ {h^2}(r)g + dr \otimes dr$ (when $E$ exists globally we call it the heat kernel on $M \times I$). The coefficients of the asymptotic expansion of the trace $E$ are studied and expressed in terms of corresponding coefficients on the basis $M$. It is fulfilled by means of constructing a parametrix for $E$ which is different from a parametrix in the standard form. One important result is that each of the former coefficients is a linear combination of the latter coefficients.
$q$-extension of the $p$-adic gamma function. II
Neal
Koblitz
111-129
Abstract: Taylor series and asymptotic expansions are developed for $q$-extensions of the $p$-adic psi (derivative of log-gamma) function "twisted" by roots of unity. Connections with $ p$-adic $L$-functions and $q$-expansions of Eisenstein series are discussed. The $p$-adic series are compared with the analogous classical expansions.
Trace-like functions on rings with no nilpotent elements
M.
Cohen;
Susan
Montgomery
131-145
Abstract: Let $R$ be a ring with no nilpotent elements, with extended center $C$, and let $E$ be the set of idempotents in $ C$. Our first main result is that for any finite group $G$ acting as automorphisms of $R $, there exist a finite set $L \subseteq E$ and an $ {R^G}$-bimodule homomorphism $ \tau :R \to {(RL)^G}$ such that $\tau (R)$ is an essential ideal of $ {(RE)^G}$. This theorem is applied to show the following: if $R$ is a Noetherian, affine $ PI$-algebra (with no nilpotent elements) over the commutative Noetherian ring $ A$, and $G$ is a finite group of $ A$-automorphisms of $ R$ such that $ {R^G}$ is Noetherian, then $ {R^G}$ is affine over $ A$.
When is the natural map $X\rightarrow \Omega \Sigma X$ a cofibration?
L. Gaunce
Lewis
147-155
Abstract: It is shown that a map $f:X \to F(A,W)$ is a cofibration if its adjoint $ f:X \wedge A \to W$ is a cofibration and $X$ and $A$ are locally equiconnected (LEC) based spaces with $A$ compact and nontrivial. Thus, the suspension map $ \eta :X \to \Omega \sum X$ is a cofibration if $X$ is LEC. Also included is a new, simpler proof that C.W. complexes are LEC. Equivariant generalizations of these results are described.
Branching degrees above low degrees
Peter A.
Fejer
157-180
Abstract: In this paper, we investigate the location of the branching degrees within the recursively enumerable (r.e.) degrees. We show that there is a branching degree below any given nonzero r.e. degree and, using a new branching degree construction and a technique of Robinson, that there is a branching degree above any given low r.e. degree. Our results extend work of Shoenfield and Soare and Lachlan on the generalized nondiamond question and show that the branching degrees form an automorphism base for the r.e. degrees.
Algebraic and geometric models for $H\sb{0}$-spaces
J.
Aguadé;
A.
Zabrodsky
181-190
Abstract: For every $ {H_0}$-space (i.e. a space whose rationalization is an $H$-space) we construct a space $J$ depending only on ${H^\ast}(X;{\mathbf{Z}})$ and a rational homotopy equivalence $J \to X$ (i.e. $J$ is a universal space to the left of all $ {H_0}$-spaces having the same integral cohomology ring as $X$ is constructed generalizing the James reduced product. We study also the integral cohomology of $ {H_0}$-spaces and we prove that under certain conditions it contains an algebra with divided powers.
No division implies chaos
Tien Yien
Li;
Michał
Misiurewicz;
Giulio
Pianigiani;
James A.
Yorke
191-199
Abstract: Let $I$ be a closed interval in ${R^1}$ and $f:I \to I$ be continuous. Let ${x_0} \in I$ and $\displaystyle {x_{i + 1}} = f({x_i})\quad {\text{for}}\;i > 0.$ We say there is no division for $ ({x_1},{x_2}, \ldots ,{x_n})$ if there is no $a \in I$ such that ${x_j} < a$ for all $j$ even and ${x_j} < a$ for all $j$ odd. The main result of this paper proves the simple statement: no division implies chaos. Also given here are some converse theorems, detailed estimates of the existing periods, and examples which show that, under our conditions, one cannot do any better.
The Catalan equation over function fields
Joseph H.
Silverman
201-205
Abstract: Let $K$ be the function field of a projective variety. Fix $ a,b,c \in {K^ \ast }$. We show that if $ \max \{ m,n\}$ is sufficiently large, then the Catalan equation $a{x^m} + b{y^n} = c$ has no nonconstant solutions $ x,y \in K$.
Topological invariant means on the von Neumann algebra ${\rm VN}(G)$
Ching
Chou
207-229
Abstract: Let $VN(G)$ be the von Neumann algebra generated by the left regular representation of a locally compact group $G$, $A(G)$ the Fourier algebra of $G$ and $ TIM(\hat G)$ the set of topological invariant means on $VN(G)$. Let ${\mathcal{F}_1} = \{ \mathcal{O} \in {({l^\infty })^ \ast }\} :\mathcal{O} \geqslant 0,\;\vert\vert\mathcal{O}\vert\vert = 1$ and $\mathcal{O}(f) = 0\;{\text{if}}\;f \in {l^\infty }$ and $f(n) \to 0\} $. We show that if $ G$ is nondiscrete then there exists a linear isometry $\Lambda$ of $ {({l^\infty })^ \ast }$ into $VN{(G)^ \ast }$ such that $ \Lambda ({\mathcal{F}_1}) \subset TIM(\hat G)$. When $G$ is further assumed to be second countable then ${\mathcal{F}_1}$ can be embedded into some predescribed subsets of $ TIM(\hat G)$. To prove these embedding theorems for second countable groups we need the existence of a sequence of means $\{ {u_n}\}$ in $A(G)$ such that their supports in $VN(G)$ are mutually orthogonal and $ \vert\vert u{u_n} - {u_n}\vert\vert \to 0\;{\text{if}}\;u$ is a mean in $A(G)$. Let $F(\hat G)$ be the space of all $T \in VN(G)$ such that $m(T)$ is a constant as $m$ runs through $ TIM(\hat G)$ and let $ W(\hat G)$ be the space of weakly almost periodic elements in $VN(G)$. We show that the following conditions are equivalent: (i) $G$ is discrete, (ii) $F(\hat G)$ is an algebra and (iii) $ (A(G) \cdot VN(G)) \cap F(\hat G) \subset W(\hat G)$.
An almost sure invariance principle for Hilbert space valued martingales
Gregory
Morrow;
Walter
Philipp
231-251
Abstract: We obtain an almost sure approximation of a martingale with values in a real separable Hilbert space $H$ by a suitable $H$-valued Brownian motion.
Local blow-up of stratified sets up to bordism
S.
Buoncristiano;
M.
Dedò
253-280
Abstract: Homological obstructions are given, whose vanishing is a necessary and sufficient condition for the existence of a blow-up of an abstract prestratification $ V$ along a 'locally top-dimensional' substratification $V'$.
Operator-self-similar processes in a finite-dimensional space
William N.
Hudson;
J. David
Mason
281-297
Abstract: A general representation for an operator-self-similar process is obtained and its class of exponents is characterized. It is shown that such a process is the limit in a certain sense of an operator-normed process and any limit of an operator-normed process is operator-self-similar.
Products of $k$-spaces and spaces of countable tightness
G.
Gruenhage;
Y.
Tanaka
299-308
Abstract: In this paper, we obtain results of the following type: if $ f:X \to Y$ is a closed map and $X$ is some "nice" space, and ${Y^2}$ is a $k$-space or has countable tightness, then the boundary of the inverse image of each point of $ Y$ is "small" in some sense, e.g., Lindelöf or $ {\omega _1}$-compact. We then apply these results to more special cases. Most of these applications combine the "smallness" of the boundaries of the point-inverses obtained from the earlier results with "nice" properties of the domain to yield "nice" properties on the range.
Global solvability on compact Heisenberg manifolds
Leonard F.
Richardson
309-317
Abstract: We apply the methods of primary and irreducible Fourier series on compact nilmanifolds to determine the ranges of all first order invariant operators on the compact Heisenberg manifolds. We show that the sums of primary solutions behave better on these manifolds than on any multidimensional torus.
On the double suspension homomorphism at odd primes
J. R.
Harper;
H. R.
Miller
319-331
Abstract: We work with the $ {E_1}$-term for spheres and the stable Moore space, given by the $\Lambda $-algebra at odd primes. Writing $W(n) = \Lambda (2n + 1)/\Lambda (2n - 1)$ and $M(0) = {H_ \ast }({S^0}{ \cup _p}{e^1})$, we construct compatible maps ${f_n} \cdot W(n) \to M(0)\tilde \otimes \Lambda$ and prove the Metastability Theorem: in homology ${f_n}$ induces an isomorphism for $ \sigma < 2({p^2} - 1)(s - 2) + pqn - 2p - 2$ where $\sigma =$ stem degree$ $, $s =$ homological degree resulting from the bigrading of $\Lambda$ and $ q = 2p - 2$. There is an operator ${\upsilon _1}$ corresponding to the Adams stable self-map of the Moore space and ${\upsilon _1}$ extends to $W(n)$. A corollary of the Metastability Theorem and the Localization Theorem of the second author is that the map ${f_n}$ induces an isomorphism on homology after inverting $ {\upsilon _1}$.
Resolvent operators for integral equations in a Banach space
R. C.
Grimmer
333-349
Abstract: Conditions are given which ensure the existence of a resolvent operator for an integrodifferential equation in a Banach space. The resolvent operator is similar to an evolution operator for nonautonomous differential equations in a Banach space. As in the finite dimensional case, this operator is used to obtain a variation of parameters formula which can be used to obtain results concerning the asymptotic behaviour of solutions and weak solutions.
On the oscillation theory of $f\sp{\prime\prime}+Af=0$ where $A$ is entire
Steven B.
Bank;
Ilpo
Laine
351-363
Abstract: In this paper, we investigate the distribution of zeros of solutions of $f'' + A(z)f = 0$. More specifically, results are obtained concerning the exponent of convergence of the zero-sequence of a solution in both the case where $A(z)$ is a polynomial, and the case where $ A(z)$ is transcendental.
If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal
William G.
Fleissner
365-373
Abstract: We formulate an axiom, HYP, and from it construct a normal, metacompact, nonmetrizable Moore space. HYP unifies two better known axioms. The Continuum Hypothesis implies HYP; the nonexistence of an inner model with a measurable cardinal implies HYP. As a consequence, it is impossible to replace Nyikos' "provisional" solution to the normal Moore space problem with a solution not involving large cardinals. Finally, we discuss how this space continues a process of lowering the character for normal, not collectionwise normal spaces.
Exact dynamical systems and the Frobenius-Perron operator
A.
Lasota;
James A.
Yorke
375-384
Abstract: Conditions are investigated which guarantee exactness for measurable maps on measure spaces. The main application is to certain piecewise continuous maps $T$ on $[0,1]$ for which $T'(0) > 1$. We assume $[0,1]$ can be broken into intervals on which $ T$ is continuous and convex and at the left end of these intervals $T = 0$ and $dt/dx > 0$. Such maps have an invariant absolutely continuous density which is exact.
A symplectic Banach space with no Lagrangian subspaces
N. J.
Kalton;
R. C.
Swanson
385-392
Abstract: In this paper we construct a symplectic Banach space $(X,\Omega )$ which does not split as a direct sum of closed isotropic subspaces. Thus, the question of whether every symplectic Banach space is isomorphic to one of the canonical form $Y \times {Y^ \ast }$ is settled in the negative. The proof also shows that $ \mathfrak{L}(X)$ admits a nontrivial continuous homomorphism into $\mathfrak{L}(H)$ where $H$ is a Hilbert space.
Circle actions and fundamental groups for homology $4$-spheres
Steven
Plotnick
393-404
Abstract: We generalize work of Fintushel and Pao to give a topological classification of smooth circle actions on oriented $4$-manifolds $\Sigma$ satisfying ${H_1}(\Sigma ) = 0$. We then use these ideas to construct infinite families of homology $4$-spheres that do not admit effective circle actions, and whose fundamental groups cannot be $ 3$-manifold groups.
Counting interval graphs
Phil
Hanlon
383-426
Abstract: In this paper we enumerate interval graphs (up to isomorphism) along with labelled interval graphs, identity interval graphs, transitive interval graphs and various sorts of unit interval graphs. The enumeration makes use of a structural decomposition of interval graphs which leads to a characterization of those interval graphs having a unique interval representation. Several tables are included.
Large cardinals and strong model theoretic transfer properties
Matthew
Foreman
427-463
Abstract: In this paper we prove the following theorem: $[{\rm {Con}}({\rm {ZFC}}\,{\rm { + }}\,there\,is\,a\,{\rm {2 - }}huge\,cardinal) \Rightarrow for\,all\,n$ $\displaystyle {\rm {Con}}({\rm {ZFC + }}({\aleph _{n + 3}},{\aleph _{n + 2}},{\... ...{n + 1}}) \twoheadrightarrow ({\aleph _{n + 2}},{\aleph _{n + 1}},{\aleph _n}))$ . We do this by using iterated forcing to collapse the $2$-huge cardinal to $ {\aleph _{n + 1}}$ and extending the elementary embedding generically.
The metabelian $p$-groups of maximal class. II
R. J.
Miech
465-474
Abstract: This paper gives a classification, up to isomorphism, of the metabelian $p$-groups of maximal class. A recent idea is used to produce a considerable simplification of an earlier classification scheme for these groups.
Left definite multiparameter eigenvalue problems
Paul
Binding
475-486
Abstract: We study the problem $\displaystyle (\ast)\qquad{T_m}{x_m} = \sum\limits_{n = 1}^k {{\lambda _n}{V_{mn}}{x_m},\qquad 0 \ne } {x_m} \in {H_m},\,m = 1, \ldots ,k,$ where $ {T_m}$ and ${V_{mn}}$ are selfadjoint linear operators on separable Hilbert spaces ${H_m}$, with ${T_m}$ positive, $ T_m^{ - 1}$ compact and $ {V_{mn}}$ bounded. We assume ``left definiteness'' which involves positivity of certain linear combinations of cofactors in the determinant with $(m,\,n)$th entry $({x_m},\,{V_{mn}}{x_m})$. We establish a spectral theory for $(\ast)$ that is in some way simpler and more complete than those hitherto available for this case. In particular, we make use of operators ${B_n} = \Delta _n^{ - 1}{\Delta _0}$, where the ${\Delta _n}$ are determinantal operators on $ \otimes _{m = 1}^k{H_m}$. This complements an established approach to the alternative ``right definite'' problem (where ${\Delta _0}$ is positive) via the operators ${\Gamma_n} = \Delta _0^{ - 1}{\Delta _n}$.
On weighted norm inequalities for the Hilbert transform of functions with moments zero
Ernst
Adams
487-500
Abstract: Let $\tilde f$ denote the Hilbert transform of $f$, i.e. $\displaystyle \tilde f(x) = {\rm {p}}{\rm {.v}}{\rm {.}}\int {\frac{{f(t)}}{{x - t}}dt}$ and let $1 < p < \infty$. A weight function $ w$ is shown to satisfy $\displaystyle \int {\vert\tilde f(x)} {\vert^p}w(x)dx \le C{\int {\vert f(x)\vert} ^p}w(x)dx$ for all $f$ with the first $N$ moments zero, if and only if it is of the form $ w(x) = \vert q(x){\vert^p}U(x)$, where $q$ is a polynomial of degree at most $ N$ and $U \in {A_p}$.
Some varieties containing relation algebras
Roger
Maddux
501-526
Abstract: Three varieties of algebras are introduced which extend the variety $ RA$ of relation algebras. They are obtained from $RA$ by weakening the associative law for relative product, and are consequently called nonassociative, weakly-associative and semiassociative relation algebras, or $NA$, $WA$, and $SA$, respectively. Each of these varieties arises naturally in solving various problems concerning relation algebras. We show, for example, that $WA$ is the only one of these varieties which is closed under the formation of complex algebras of atom structures of algebras, and that $ WA$ is the closure of the variety of representable $RA$'s under relativization. The paper also contains a study of the elementary theories of these varieties, various representation theorems, and numerous examples.
Cellular maps between polyhedra
James P.
Henderson
527-537
Abstract: A compact subset $ X$ of a polyhedron $ P$ is cellular in $ P$ if there is a pseudoisotopy of $P$ shrinking precisely $X$ to a point. A proper surjection $f:P\rightarrow Q$ is cellular if each point inverse of $f$ is cellular in $P$. We give certain conditions under which cellular maps between polyhedra are approximable by homeomorphisms. An example of a cellular map which is not approximable by homeomorphisms is also given.
Smoothness of the boundary values of functions bounded and holomorphic in the disk
Shinji
Yamashita
539-544
Abstract: The non-Euclidean counterparts of Hardy-Littlewood's theorems on Lipschitz and mean Lipschitz functions are considered. Let $ 1\le p < \infty$ and $0 < \alpha \le 1$. For $f$ holomorphic and bounded, $\vert f\vert< 1$, in $\vert z\vert< 1$, the condition that is necessary and sufficient for $f$ to be continuous on $\vert z\vert\le 1$ with the boundary function $f({e^{it}}) \in \sigma {\Lambda _\alpha }$, the hyperbolic Lipschitz class. Furthermore, the condition that the $p$th mean of $f^{\ast}$ on the circle $\vert z\vert=r < 1$ is $O({(1 - r)^{\alpha - 1}})$ is necessary and sufficient for $f$ to be of the hyperbolic Hardy class $H_\sigma^{p}$ and for the radial limits to be of the hyperbolic mean Lipschitz class $ \sigma\Lambda_\alpha^{p}$.
Best simultaneous Diophantine approximations. I. Growth rates of best approximation denominators
J. C.
Lagarias
545-554
Abstract: This paper defines the notion of a best simultaneous Diophantine approximation to a vector $\alpha$ in $R^n$ with respect to a norm $\left\Vert \,\cdot\, \right\Vert$ on $ R^n$. Suppose $ \alpha$ is not rational and order the best approximations to $ \alpha$ with respect to $\left\Vert\, \cdot\, \right\Vert$ by increasing denominators $1=q_1 < q_2 < \cdots$. It is shown that these denominators grow at least at the rate of a geometric series, in the sense that $\displaystyle g\left( {\alpha ,\,\left\Vert {\,\cdot\,} \right\Vert} \right) = ... ...liminf\limits_{k \to \infty }} {({q_k})^{1/k}} \geq 1 + \frac{1}{{{2^{n + 1}}}}$ . Let $g\left( {\left\Vert\, \cdot\, \right\Vert} \right)$ denote the infimum of $g\left( {\alpha ,\,\left\Vert {\,\cdot\,} \right\Vert} \right)$ over all $\alpha$ in $R^n$ with an irrational coordinate. For the sup norm $ \left\Vert\, \cdot \,\right\Vert _s$ on $R^2$ it is shown that $g\left( {\left\Vert \, \cdot \, \right\Vert}_s \right)\ge\theta=1.270^{+}$ where $ \theta^4=\theta^{2}+1$.
Coextensions of regular semigroups by rectangular bands. II
John
Meakin;
K. S. S.
Nambooripad
555-568
Abstract: A construction of all coextensions of a regular semigroup $S$ by rectangular bands is obtained. The construction is analogous to Hall's construction of orthodox semigroups as spined products of bands and inverse semigroups and reduces to that construction when $S$ is inverse. The results are specialized to provide a construction of the category of all normal coextensions of a regular semigroup.
Spaces which look like quaternionic projective $n$-space
C. A.
McGibbon
569-587
Abstract: The projective $ n$-spaces which correspond to the various multiplicative structures on the three sphere are studied. Necessary and sufficient conditions for a projective $n$-space to extend to a projective $ n+1$-space are described. At each odd prime, an infinite family of exotic projective spaces is constructed. These exotic spaces are not homotopy equivalent, at the prime in question, to the classical quaternionic projective $ n$-space. It is also shown that these exotic projective $n$-spaces do not occur as the finite skeleton of a classifying space for a group with the homotopy type of the three sphere.
On actions of regular type on complex Stiefel manifolds
McKenzie Y.
Wang
589-610
Abstract: The usual unitary representations of the special unitary, symplectic, or special orthogonal groups define a sequence of smooth actions on the complex Stiefel manifolds called the regular linear models. If one of the above groups acts smoothly on the complex Stiefel manifold of orthonormal $2$-frames in $ \mathbf C^n$ for odd $ n$, and if the identity component of the principal isotropy type is of regular type, then it is shown under mild dimension restrictions that the orbit structure and the cohomology structure of the fixed point varieties (over the $\mod 2$ Steenrod algebra) resemble those of the regular linear models. The resemblance is complete in the cases of the special unitary and symplectic groups. There is an obstruction to complete resemblance in the case of the special orthogonal groups. An application of the above regularity theorems is given.
On actions of adjoint type on complex Stiefel manifolds
McKenzie Y.
Wang
611-628
Abstract: Let $G(m)$ denote ${\rm {SU}}(m)$ or $ {\rm {Sp}}(m)$. It is shown that when $m \geq 5\,G(m)$ cannot act smoothly on $W_{n,2}$, the complex Stiefel manifold of orthonormal $2$-frames in $ \mathbf C^n$, for $ n$ odd, with connected principal isotropy type equal to the class of maximal tori in $G(m)$. This demonstrates an important difference between $W_{n,2}$, $n$ odd, and $S^{2n-3}\times S^{2n-1}$ in the behavior of differentiable transformation groups. Exactly the same holds for ${\rm {SO}}(m)$ or Spin$(m)$ if it is further assumed that a maximal $2$-torus of $ {\rm {SO}}(m)$ has fixed points.$^{2}$
On group $C\sp{\ast} $-algebras of bounded representation dimension
Iain
Raeburn
629-644
Abstract: We consider the structure of group $C^{\ast}$-algebras whose irreducible representations have bounded dimension. We give some general results, including a description of the topology on the spectrum, and then calculate explicitly the $ C^{\ast}$-algebras of two specific groups, one of them a nonsymmorphic space group.
Generalization of continuous posets
Dan
Novak
645-667
Abstract: In this paper we develop a general theory of continuity in partially ordered sets. Among the interesting special cases of this theory is the theory of continuous lattices developed by D. Scott, J. Lawson and others.
Determining Thurston classes using Nielsen types
Jane
Gilman
669-675
Abstract: In previous work [3] we showed how the Thurston or Bers classifications of diffeomorphisms of surfaces could be obtained using the Nielsen types of the lifts of the diffeomorphism to the unit disc. In this paper we find improved conditions on the Nielsen types for the Thurston and Bers classes. We use them to verify that an example studied by Nielsen is a pseudo-Anosov diffeomorphism with stretching factor of degree 4. This example is of interest in its own right, but it also serves to illustrate exactly how the Nielsen types are used for verifying examples. We discuss the general usefulness of this method.
Quaternionic Kaehler manifolds
Lee
Whitt
677-692
Abstract: The topological classification of $4$- and $8$- (real) dimensional compact quaternionic Kaehler manifolds is given. There is only the torus in dimension 4. In dimension 8, there are 12 homeomorphism classes; representatives are given explicitly.
Connected algebraic monoids
Mohan S.
Putcha
693-709
Abstract: Let $S$ be a connected algebraic monoid with group of units $G$ and lattice of regular $ \mathcal{J}$-classes $\mathcal{U}(S)$. The connection between the solvability of $G$ and the semilattice decomposition of $ S$ into archimedean semigroups is further elaborated. If $S$ has a zero and if $\mathcal{U}(S)\le 7$, then it is shown that $ G$ is solvable if and only if $ \mathcal{U}(S)$ is relatively complemented. If $J\in \mathcal{U}(S)$, then we introduce two basic numbers $\theta(J)$ and $\delta(J)$ and study their properties. Crucial to this process is the theorem that for any indempotent $ e$ of $S$, the centralizer of $ e$ in $G$ is connected. Connected monoids with central idempotents are also studied. A conjecture about their structure is forwarded. It is pointed out that the maximal connected submonoids of $S$ with central idempotents need not be conjugate. However special maximal connected submonoids with central idempotents are conjugate. If $S$ is regular, then $S$ is a Clifford semigroup if and only if for all $f\in E(S)$, the set $\{ e\vert e \in E(S),\,e \geq f\}$ is finite. Finally the maximal semilattice image of any connected monoid is determined.
On the Schwarz reflection principle
J. S.
Hwang
711-719
Abstract: Recently, we have solved a long outstanding problem of A. J. Lohwater (1953) by showing that if $f(z)$ is meromorphic in $\vert z\vert< 1$ whose radial limits have modulus 1 for almost all points on an arc $A$ of $\vert z\vert< 1$, and if $P$ is a singular point of $f(z)$ on $A$, then every value of modulus 1 which is not in the range of $f(z)$ at $P$ is an asymptotic value of $f(z)$ at some point of each subarc of $ A$ containing the point $ P$. Lohwater proved this theorem for functions of bounded characteristic and he made a comment that his method is not, in general, applicable to functions of unbounded characteristic. In this paper, we shall present an alternative proof of the above theorem based on the very method of Lohwater.
Invariant operators and univalent functions
Reuven
Harmelin
721-731
Abstract: Necessary and sufficient conditions for univalence of meromorphic functions in certain domains in the complex plane are established in terms of some differential operators of degrees $\ge 3$, possessing the same invariance property as the Schwarzian derivative. Those operators include the derivatives of the Schwarzian derivative and Aharonov's invariants. Conditions for the existence of quasiconformal extensions are also achieved.
Unipotent characters of the even orthogonal groups over a finite field
George
Lusztig
733-751
Abstract: The characters of unipotent representations of a simple algebraic group over $F_q$ of type $\ne D_n$ on any regular semisimple element are explicitly known for large $q$. This paper deals with the remaining case: type $ D_n$.
Multiple critical points of perturbed symmetric functionals
Paul H.
Rabinowitz
753-769
Abstract: Variational problems which are invariant under a group of symmetries often possess multiple solutions. This paper studies the effect of perturbations which are not small and which destroy the symmetry for two classes of such problems and shows how multiple solutions persist despite the perturbation.
A nonshrinkable decomposition of $S\sp{n}$ involving a null sequence of cellular arcs
R. J.
Daverman;
J. J.
Walsh
771-784
Abstract: This paper presents a decomposition $G$ of $ S^n(n\ge 3)$ into points and a null sequence of cellular arcs such that $ S^n/G$ is not a manifold; furthermore, the union of the nondegenerate elements from $G$ lies in a $2$-cell in $S^n$ and the image in $S^n/G$ of this union has 0-dimensional closure. Examples of nonshrinkable decompositions with a null sequence of cellular arcs have been constructed in the case $n=3$ by D. S. Gillman and J. M. Martin and by R. H. Bing and M. Starbird. We construct another example in this dimension, for which all the arcs lie in the boundary of a crumpled cube $C$, and then produce higher dimensional examples by spinning $C$.
Asymptotic expansions of some integral transforms by using generalized functions
Ahmed I.
Zayed
785-802
Abstract: The technique devised by Wong to derive the asymptotic expansions of multiple Fourier transforms by using the theory of Schwartz distributions is extended to a large class of integral transforms. The extension requires establishing a general procedure to extend these integral transforms to generalized functions. Wong's technique is then applied to some of these integral transforms to obtain their asymptotic expansions. This class of integral transforms encompasses, among others, the Laplace, the Airy, the $K$ and the Hankel transforms.
Correction to: ``Closed $3$-manifolds with no periodic maps'' [Trans. Amer. Math. Soc. {\bf 221} (1976), no. 2, 403--418; MR {\bf 54} \#3703]
Frank
Raymond;
Jeffrey L.
Tollefson
803-807
Erratum to: ``Liapounoff's theorem for nonatomic, finitely-additive, bounded, finite-dimensional, vector-valued measures'' [Trans. Amer. Math. Soc. {\bf 266} (1981), no. 2, 499--514; MR 82f:28008]
Thomas E.
Armstrong;
Karel
Prikry
809
Erratum to: ``Embedding processes in Brownian motion in ${\bf R}\sp{n}$''
Neil
Falkner
811
Finite groups containing an intrinsic $2$-component of Chevalley type over a field of odd order
Morton E.
Harris
1-65
Abstract: This paper extends the celebrated theorem of Aschbacher that classifies all finite simple groups $G$ containing a subgroup $L \cong {\text{SL}}(2,q)$, $ q$ odd, such that $ L$ is subnormal in the centralizer in $G$ of its unique involution. Under the same embedding assumptions, the main result of this work allows $L$ to be almost any Chevalley group over a field of odd order and determines the resulting simple groups $ G$. The results of this paper are an essential ingredient in the current classification of all finite simple groups. Major sections are devoted to deriving various properties of Chevalley groups that are required in the proofs of the three theorems of this paper. These sections are of some independent interest.
Two consequences of determinacy consistent with choice
John R.
Steel;
Robert
Van Wesep
67-85
Abstract: We begin with a ground model satisfying ${\text{ZF}} + {\text{AD}} + {\text{A}}{{\text{C}}_{\mathbf{R}}}$, and from it construct a generic extension satisfying ${\text{ZFC}} + {\mathbf{\delta }}_2^1 = {\omega _2} +$ "the nonstationary ideal on ${\omega _1}$ is $ {\omega _2}$-saturated".
Asymptotic estimates of sums involving the Moebius function. II
Krishnaswami
Alladi
87-105
Abstract: Let $n$ be a positive integer and $ \mu (n)$ the Moebius function. If $n > 1$, let $P(n)$ denote its largest prime factor and put $ P(1) = 1$. We study the asymptotic behavior of the sum ${M^ \ast }(x,y) = \sum\nolimits_{1 \leqslant n \leqslant x,P(n) < y} {\mu (n)}$ as $x,y \to \infty$ and discuss a few applications.
A projective description of weighted inductive limits
Klaus-D.
Bierstedt;
Reinhold
Meise;
William H.
Summers
107-160
Abstract: Considering countable locally convex inductive limits of weighted spaces of continuous functions, if $\mathcal{V} = {\{ {V_n}\} _n}$ is a decreasing sequence of systems of weights on a locally compact Hausdorff space $X$, we prove that the topology of ${\mathcal{V}_0}C(X) = {\text{in}}{{\text{d}}_{n \to }}C{({V_n})_0}(X)$ can always be described by an associated system $\overline V = {\overline V _\mathcal{V}}$ of weights on $X$; the continuous seminorms on ${\mathcal{V}_0}C(X)$ are characterized as weighted supremum norms. If $\mathcal{V} = {\{ {\upsilon _n}\} _n}$ is a sequence of continuous weights on $X$, a condition is derived in terms of $\mathcal{V}$ which is both necessary and sufficient for the completeness (respectively, regularity) of the $(LB)$-space $ {\mathcal{V}_0}C(X)$, and which is also equivalent to ${\mathcal{V}_0}C(X)$ agreeing algebraically and topologically with the associated weighted space $C{\overline V _0}(X)$; for sequence spaces, this condition is the same as requiring that the corresponding echelon space be quasi-normable. A number of consequences follow. As our main application, in the case of weighted inductive limits of holomorphic functions, we obtain, using purely functional analytic methods, a considerable extension of a theorem due to B. A. Taylor [37] which is useful in connection with analytically uniform spaces and convolution equations. The projective description of weighted inductive limits also serves to improve upon existing tensor and slice product representations. Most of our work is in the context of spaces of scalar or Banach space valued functions, but, additionally, some results for spaces of functions with range in certain $ (LB)$-spaces are mentioned.
Level sets of derivatives
David
Preiss
161-184
Abstract: The main result of the paper is the characterization of those triples $S$, $G$ and $E$ of subsets of the reals for which there exists an everywhere differentiable real-valued function $f$ of one real variable such that $({\text{Z))}}$ introduced in the paper. The main result leads to a complete description of the structure of the sets
Homotopy in functor categories
Alex
Heller
185-202
Abstract: If ${\mathbf{C}}$ is a small category enriched over topological spaces the category $ {\mathcal{J}^{\mathbf{C}}}$ of continuous functors from ${\mathbf{C}}$ into topological spaces admits a family of homotopy theories associated with closed subcategories of $ {\mathbf{C}}$. The categories $ {\mathcal{J}^{\mathbf{C}}}$, for various $ {\mathbf{C}}$, are connected to one another by a functor calculus analogous to the $\otimes$, Hom calculus for modules over rings. The functor calculus and the several homotopy theories may be articulated in such a way as to define an analogous functor calculus on the homotopy categories. Among the functors so described are homotopy limits and colimits and, more generally, homotopy Kan extensions. A by-product of the method is a generalization to functor categories of E. H. Brown's representability theorem.
Holomorphic curves in Lorentzian CR-manifolds
Robert L.
Bryant
203-221
Abstract: A CR-manifold is said to be Lorentzian if its Levi form has one negative eigenvalue and the rest positive. In this case, it is possible that the CR-manifold contains holomorphic curves. In this paper, necessary and sufficient conditions are derived (in terms of the "derivatives" of the CR-structure) in order that holomorphic curves exist. A "flatness" theorem is proven characterizing the real Lorentzian hyperquadric ${Q_5} \subseteq {\mathbf{C}}{P^3}$, and examples are given showing that nonflat Lorentzian hyperquadrics can have a richer family of holomorphic curves than the flat ones.
Nonseparability of quotient spaces of function algebras on topological semigroups
Heneri A. M.
Dzinotyiweyi
223-235
Abstract: Let $S$ be a topological semigroup, $ C(S)$ the space of all bounded real-valued continuous functions on $S$. We define $WUC(S)$ the subspace of $C(S)$ consisting of all weakly uniformly continuous functions and $WAP(S)$ the space of all weakly almost periodic functions in $C(S)$. Among other results, for a large class of topological semigroups $S$, for which noncompact locally compact topological groups are a very special case, we prove that the quotient spaces $ WUC(S)/WAP(S)$ and, for nondiscrete $S$, $ C(S)/WUC(S)$ are nonseparable. (The actual setting of these results is more general.) For locally compact topological groups, parts of our results answer affirmatively certain questions raised earlier by Ching Chou and E. E. Granirer.
Simplexes of extensions of states of $C\sp{\ast} $-algebras
C. J. K.
Batty
237-246
Abstract: Let $B$ be a ${C^\ast}$-subalgebra of a ${C^\ast}$-algebra $A$, $F$ a compact face of the state space $S(B)$ of $B$, and ${S_F}(A)$ the set of all states of $A$ whose restrictions to $ B$ lie in $F$. It is shown that ${S_F}(A)$ is a Choquet simplex if and only if (a) $F$ is a simplex, (b) pure states in $ {S_F}(A)$ restrict to pure states in $F$, and (c) the states of $A$ which restrict to any given pure state in $ F$ form a simplex. The properties (b) and (c) are also considered in isolation. Sets of the form ${S_F}(A)$ have been considered by various authors in special cases including those where $B$ is a maximal abelian subalgebra of $A$, or $A$ is a ${C^\ast}$-crossed product, or the Cuntz algebra $ {\mathcal{O}_n}$.
Generalized Lefschetz numbers
S. Y.
Husseini
247-274
Abstract: Given $ [C;f]$, where $ C$ is a finitely-generated $ \pi$-projective chain complex, and $f:C \to C{\text{a(}}\pi {\text{,}}\varphi {\text{)}}$-chain map, with $\varphi :\pi \to \pi$ being a homomorphism, then the generalized Lefschetz number ${L_{(\pi ,\varphi )}}[C;f]$ of $ [C;f]$ is defined as the alternating sum of the $ (\pi ,\varphi )$-Reidemeister trace of $f$. In analogy with the ordinary Lefschetz number, $ {L_{(\pi ,\varphi )}}[C;f]$ is shown to satisfy the commutative property and to be invariant under $ (\pi ,\varphi )$-chain homotopy. Also, when ${H_\ast}C$ is $\pi$-projective, $\displaystyle {L_{(\pi ,\varphi )}}[C;f] = {L_{(\pi ,\varphi )}}[{H_\ast}C;{H_\ast}f]$ If ${[\alpha ;\pi ]_\varphi }$ is essential. If $(\pi ,\varphi )$-classes of $ f:C \to C$. This is expressed as a decomposition of ${L_{(\pi ,\varphi )}}[C;f]$ in terms of $ {L_{(\pi ',{\varphi _\xi })}}[C';{f_\xi }]$ where $f( \cdot ){\xi ^{ - 1}} = {f_\xi }( \cdot )$ and ${\varphi _\xi }( \cdot ) = \xi \varphi ( \cdot ){\xi ^{ - 1}}$. The algebraic theory is applied to the Nielsen theory of a map $f:X \to X$, where $X$ is a finite CW-complex relative to a regular cover $\tilde X \to X$. One can define a generalized Lefschetz number ${L_{(\pi ,\varphi )}}$ using any cellular approximation to $f$, where $\pi$ is the group of covering transformations of $\tilde X \to X$. The quantity ${L_{(\pi ,\varphi )}}$ can be expressed naturally as a formal sum in the $\pi$-Nielsen classes of $f$ with their indices appearing as coefficients. From this expression, one is able to deduce from the properties of the generalized Lefschetz number the usual results of the relative Nielsen theory.
Metrically complete regular rings
K. R.
Goodearl
275-310
Abstract: This paper is concerned with the structure of those (von Neumann) regular rings $R$ which are complete with respect to the weakest metric derived from the pseudo-rank functions on $ R$, known as the ${N^ \ast }$-metric. It is proved that this class of regular rings includes all regular rings with bounded index of nilpotence, and all $ {\aleph _0}$-continuous regular rings. The major tool of the investigation is the partially ordered Grothendieck group $ {K_0}(R)$, which is proved to be an archimedean normcomplete interpolation group. Such a group has a precise representation as affine continuous functions on a Choquet simplex, from earlier work of the author and D. E. Handelman, and additional aspects of its structure are derived here. These results are then translated into ring-theoretic results about the structure of $R$. For instance, it is proved that the simple homomorphic images of $R$ are right and left self-injective rings, and $ R$ is a subdirect product of these simple self-injective rings. Also, the isomorphism classes of the finitely generated projective $R$-modules are determined by the isomorphism classes modulo the maximal two-sided ideals of $ R$. As another example of the results derived, it is proved that if all simple artinian homomorphic images of $R$ are $n \times n$ matrix rings (for some fixed positive integer $n$), then $R$ is an $n \times n$ matrix ring.
A fake topological Hilbert space
R. D.
Anderson;
D. W.
Curtis;
J.
van Mill
311-321
Abstract: We give an example of a topologically complete separable metric AR space $ X$ which is not homeomorphic to the Hilbert space ${l^2}$, but which has the following properties: (i) $X$ imbeds as a convex subset of ${l^2}$ (ii) every compact subset of $ X$ is a $Z$-set; (iii) $X \times X \approx {l^2};$ (iv) $ X$ is homogeneous; (v) $ X \approx X\backslash G$ for every countable subset $G$.
Analysis of spectral variation and some inequalities
Rajendra
Bhatia
323-331
Abstract: A geometric method, based on a decomposition of the space of complex matrices, is employed to study the variation of the spectrum of a matrix. When adapted to special cases, this leads to some classical inequalities as well as some new ones. As an example of the latter, we show that if $ U$, $V$ are unitary matrices and $K$ is a skew-Hermitian matrix such that $U{V^{ - 1}} = \exp K$, then for every unitary-invariant norm the distance between the eigenvalues of $U$ and those of $V$ is bounded by $\vert\vert K\vert\vert$. This generalises two earlier results which used particular unitary-invariant norms.
The restriction of admissible modules to parabolic subalgebras
J. T.
Stafford;
N. R.
Wallach
333-350
Abstract: This paper studies algebraic versions of Casselman's subrepresentation theorem. Let $ \mathfrak{g}$ be a semisimple Lie algebra over an algebraically closed field $ F$ of characteristic zero and $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$ be an Iwasawa decomposition for $\mathfrak{g}$. Then $ (\mathfrak{g},\mathfrak{k})$ is said to satisfy property $(\mathfrak{n})$ if $M \ne M$ for every admissible $ (\mathfrak{g},\mathfrak{k})$-module $M$. We prove that, if $ (\mathfrak{g},\mathfrak{k})$ satisfies property $ (\mathfrak{n})$, then $\mathfrak{n}N \ne N$ whenever $ N$ is a $ (\mathfrak{g},\mathfrak{k})$-module with $\dim N < \operatorname{card} F$. This is then used to show (purely algebraically) that $ (\mathfrak{s}l(n,F),\mathfrak{s}o(n,F))$ satisfies property $(\mathfrak{n})$. The subrepresentation theorem for $ \mathfrak{s}l(n)$ is an easy consequence of this.
Levi geometry and the tangential Cauchy-Riemann equations on a real analytic submanifold of ${\bf C}\sp{n}$
Al
Boggess
351-374
Abstract: The relationship between the Levi geometry of a submanifold of ${{\mathbf{C}}^n}$ and the tangential Cauchy-Riemann equations is studied. On a real analytic codimension two submanifold of $ {{\mathbf{C}}^n}$, we find conditions on the Levi algebra which allow us to locally solve the tangential Cauchy-Riemann equations (in most bidegrees) with kernels. Under the same conditions, we show that, locally, any CR-function is the boundary value jump of a holomorphic function defined on some suitable open set in ${{\mathbf{C}}^n}$. This boundary value jump result is the best possible result because we also show that there is no one-sided extension theory for such submanifolds of $ {{\mathbf{C}}^n}$. In fact, we show that if $S$ is a real analytic, generic, submanifold of ${{\mathbf{C}}^n}$ (any codimension) where the excess dimension of the Levi algebra is less than the real codimension, then $S$ is not extendible to any open set in ${{\mathbf{C}}^n}$.
A new solution to the word problem in the fundamental groups of alternating knots and links
Mark J.
Dugopolski
375-382
Abstract: A new solution to the word problem for alternating knots and links is given. The solution is based on Waldhausen's algorithm, but is greatly simplified.
Generation of finite groups of Lie type
Gary M.
Seitz
351-407
Abstract: Let $p$ be an odd prime and $G$ a finite group of Lie type in characteristic other than $p$. Fix an elementary abelian $p$-subgroup of $\operatorname{Aut} (G)$. It is shown that in most cases $G$ is generated by the centralizers of the maximal subgroups of $E$. Results are established concerning the notions of layer generation and balance, and the strongly $p$-embedded subgroups of $\operatorname{Aut} (G)$ are determined.
Symmetric skew balanced starters and complete balanced Howell rotations
Ding Zhu
Du;
F. K.
Hwang
409-413
Abstract: Symmetric skew balanced starters on $n$ elements have been previously constructed for $n = 4k + 3$ a prime power and $8k + 5$ a prime power. In this paper we give an approach for the general case $n = {2^m}k + 1$ a prime power with $k$ odd. In particular we show how this approach works for $m = 2$ and $3$. Furthermore, we prove that for $n$ of the general form and $k > 9 \cdot {2^{3m}}$, then a symmetric skew balanced starter always exists. It is known that a symmetric skew balanced starter on $ n$ elements, $ n$ odd, can be used to construct complete balanced Howell rotations (balanced Room squares) for $n$ players and $2(n + 1)$ players, and in the case that $ n$ is congruent to $ 3$ modulo $4$, also for $n + 1$ players.
Balanced Howell rotations of the twin prime power type
Ding Zhu
Du;
F. K.
Hwang
415-421
Abstract: We prove by construction that a balanced Howell rotation for $n$ players always exists if $n = {p^r}{q^s}$ where $p$ and $q \ne 3$ are primes and ${q^s} = {p^r} + 2$. This generalizes a much weaker previous result. The construction uses properties of a Galois domain which is a direct sum of two Galois fields.
Free products of $C\sp{\ast} $-algebras
Daniel
Avitzour
423-435
Abstract: Small ("spatial") free products of $ {C^{\ast}}$-algebras are constructed. Under certain conditions they have properties similar to those proved by Paschke and Salinas for the algebras $ C_r^{\ast}({G_1}{\ast}{G_2})$ where ${G_1}$, ${G_2}$ are discrete groups. The free-product analogs of noncommutative Bernoulli shifts are discussed.
Nilpotent inverse semigroups with central idempotents
G.
Kowol;
H.
Mitsch
437-449
Abstract: An inverse semigroup $S$ with central idempotents, i.e. a strong semilattice of groups, will be called nilpotent, if it is finite and if for each prime divisor ${p_i}$ of the orders of the structure groups of $S$ the sets ${P_i} = \{ s \in S\vert o(s) = p_i^{{k_s}},\,{k_s} \geqslant 0\}$ are subsemigroups of $ S$. If $S$ is a group, then ${P_i}$ are exactly the Sylow $ {p_i}$-subgroups of the group. A theory similar to that given by W. Burnside for finite nilpotent groups is developed introducing the concepts of ascending resp. descending central series in an inverse semigroup, and it is shown that almost all of the well-known properties of finite nilpotent groups do hold also for the class of finite inverse semigroups with central idempotents.
The Budan-Fourier theorem and Hermite-Birkhoff spline interpolation
T. N. T.
Goodman;
S. L.
Lee
451-467
Abstract: We extend the classical Budan-Fourier theorem to Hermite-Birkhoff splines, that is splines whose knots are determined by a finite incidence matrix. This is then applied to problems of interpolation by Hermite-Birkhoff splines, where the nodes of interpolation are also determined by a finite incidence matrix. For specified knots and nodes in a finite interval, conditions are examined under which there is a unique interpolating spline for any interpolation data. For knots and nodes spaced periodically on the real line, conditions are examined under which there is a unique interpolating spline of power growth for data of power growth.
A remainder formula and limits of cardinal spline interpolants
T. N. T.
Goodman;
S. L.
Lee
469-483
Abstract: A Peano-type remainder formula $\displaystyle f(x) - {S_n}(f;\,x) = \int_{ - \infty }^\infty {{K_n}(x,\,t){f^{(n + 1)}}(t)\,dt}$ for a class of symmetric cardinal interpolation problems C.I.P. $ (E,\,F,\,{\mathbf{x}})$ is obtained, from which we deduce the estimate $ \vert\vert f - {S_{n,r}}(f;\,)\vert{\vert _\infty } \leqslant K\vert\vert{f^{(n + 1)}}\vert{\vert _\infty }$. It is found that the best constant $K$ is obtained when $ {\mathbf{x}}$ comprises the zeros of the Euler-Chebyshev spline function. The remainder formula is also used to study the convergence of spline interpolants for a class of entire functions of exponential type and a class of almost periodic functions.
Varieties of combinatorial geometries
J.
Kahn;
J. P. S.
Kung
485-499
Abstract: A hereditary class of (finite combinatorial) geometries is a collection of geometries which is closed under taking minors and direct sums. A sequence of universal models for a hereditary class $\mathcal{J}$ of geometries is a sequence $ ({T_n})$ of geometries in $\mathcal{J}$ with rank ${T_n} = n$, and satisfying the universal property: if $G$ is a geometry in $ \mathcal{J}$ of rank $ n$, then $G$ is a subgeometry of $ {T_n}$. A variety of geometries is a hereditary class with a sequence of universal models. We prove that, apart from two degenerate cases, the only varieties of combinatorial geometries are (1) the variety of free geometries, (2) the variety of geometries coordinatizable over a fixed finite field, and (3) the variety of voltage-graphic geometries with voltages in a fixed finite group.
On the construction of relative genus fields
Gary
Cornell
501-511
Abstract: We show how to construct the relative genus field in many cases. This is then applied to constructing fields with interesting class groups.
Existence of Chebyshev centers, best $n$-nets and best compact approximants
Dan
Amir;
Jaroslav
Mach;
Klaus
Saatkamp
513-524
Abstract: In this paper we investigate the existence and continuity of Chebyshev centers, best $n$-nets and best compact sets. Some of our positive results were obtained using the concept of quasi-uniform convexity. Furthermore, several examples of nonexistence are given, e.g., a sublattice $M$ of $C[0,\,1]$, and a bounded subset $B \subset M$ is constructed which has no Chebyshev center, no best $n$-net and not best compact set approximant.
Transitivity of families of invariant vector fields on the semidirect products of Lie groups
B.
Bonnard;
V.
Jurdjevic;
I.
Kupka;
G.
Sallet
525-535
Abstract: In this paper we give necessary and sufficient conditions for a family of right (or left) invariant vector fields on a Lie group $ G$ to be transitive. The concept of transitivity is essentially that of controllability in the literature on control systems. We consider families of right (resp. left) invariant vector fields on a Lie group $G$ which is a semidirect product of a compact group $ K$ and a vector space $ V$ on which $K$ acts linearly. If $\mathcal{F}$ is a family of right-invariant vector fields, then the values of the elements of $\mathcal{F}$ at the identity define a subset $ \Gamma$ of $L(G)$ the Lie algebra of $G$. We say that $\mathcal{F}$ is transitive on $G$ if the semigroup generated by ${ \cup _{X \in \Gamma }}\{ \exp (tX):t \geqslant 0\}$ is equal to $G$. Our main result is that $\mathcal{F}$ is transitive if and only if $ \operatorname{Lie} (\Gamma )$, the Lie algebra generated by $\Gamma$, is equal to $L(G)$.
Extensions for AF $C\sp{\ast}$ algebras and dimension groups
David
Handelman
537-573
Abstract: Let $A$, $C$ be approximately finite dimensional $ ({\text{AF)}}\,{C^{\ast}}$ algebras, with $A$ nonunital and $C$ unital; suppose that either (i) $A$ is the algebra of compact operators, or (ii) both $A$, $C$ are simple. The classification of extensions of $A$ by $C$ is studied here, by means of Elliott's dimension groups. In case (i), the weak Ext group of $ C$ is shown to be ${\operatorname{Ext} _{\mathbf{Z}}}({K_0}(C),\,{\mathbf{Z}})$, and the strong Ext group is an extension of a cyclic group by the weak Ext group; conditions under which either Ext group is trivial are determined. In case (ii), there is an unnatural and complicated group structure on the classes of extensions when $ A$ has only finitely many pure finite traces (and somewhat more generally).
Generalized skew polynomial rings
John
Dauns
575-586
Abstract: For a totally ordered cancellative semigroup $\Gamma$, a skew field $K$, let $ K[\Gamma ;\phi ]$ be a skew semigroup ring. If $x \in \Gamma ,\,k \in K$, then $kx = x{k^x}$, where $k \to {k^x}$ is an endomorphism of $K$ depending on $x$. Ideals of $ K[\Gamma ;\phi ]$ are investigated for various semigroups or groups $ \Gamma$.
Saturation properties of ideals in generic extensions. II
James E.
Baumgartner;
Alan D.
Taylor
587-609
Abstract: The general type of problem considered here is the following. Suppose $ I$ is a countably complete ideal on ${\omega _1}$ satisfying some fairly strong saturation requirement (e.g. $I$ is precipitous or ${\omega _2}$-saturated), and suppose that $ P$ is a partial ordering satisfying some kind of chain condition requirement (e.g. $ P$ has the c.c.c. or forcing with $P$ preserves $ {\omega _1}$). Does it follow that forcing with $P$ preserves the saturation property of $ I$? In this context we consider not only precipitous and ${\omega _2}$-saturated ideals, but we also introduce and study a class of ideals that are characterized by a property lying strictly between these two notions. Some generalized versions of Chang's conjecture and Kurepa's hypothesis also arise naturally from these considerations.
Theta-characteristics on algebraic curves
Joe
Harris
611-638
Abstract: The theory of theta-characteristics is developed algebraically, so that it may be applied to possibly singular and/or reducible algebraic curves. The configuration of theta-characteristics on a curve is described in terms of its singularities, with applications to the geometry of plane quartic curves and the problem of Appolonius. Some results on Gorenstein local rings are appended.
Minimal immersions of closed Riemann surfaces
J.
Sacks;
K.
Uhlenbeck
639-652
Abstract: Let $M$ be a closed orientable surface of genus larger than zero and $N$ a compact Riemannian manifold. If $u:M \to N$ is a continuous map, such that the map induced by it between the fundamental groups of $M$ and $N$ contains no nontrivial element represented by a simple closed curve in its kernel, then there exists a conformal branched minimal immersion $ s:M \to N$ having least area among all branched immersions with the same action on ${\pi _1}(M)$ as $u$. Uniqueness within the homotopy class of $ u$ fails in general: It is shown that for certain $3$-manifolds which fiber over the circle there are at least two geometrically distinct conformal branched minimal immersions within the homotopy class of any inclusion map of the fiber. There is also a topological discussion of those $3$-manifolds for which uniqueness fails.
Singular elliptic operators of second order with purely discrete spectra
Roger T.
Lewis
653-666
Abstract: The Friedrichs extension of a second order singular elliptic operator is considered on a weighted $\mathcal{L}_w^2(\Omega )$ space. The region $ \Omega$ is not necessarily bounded. Necessary conditions and sufficient conditions on the coefficients that will insure a discrete spectrum are given with a certain degree of sharpness achieved. The boundary conditions include the Dirichlet, Neumann, and mixed Dirichlet-Neumann boundary value problems.
Star-finite representations of measure spaces
Robert M.
Anderson
667-687
Abstract: In nonstandard analysis, $^{\ast}$-finite sets are infinite sets which nonetheless possess the formal properties of finite sets. They permit a synthesis of continuous and discrete theories in many areas of mathematics, including probability theory, functional analysis, and mathematical economics. $^{\ast}$-finite models are particularly useful in building new models of economic or probabilistic processes. It is natural to ask what standard models can be obtained from these $^{\ast}$-finite models. In this paper, we show that a rich class of measure spaces, including the Radon spaces, are measure-preserving images of $ ^{\ast}$-finite measure spaces, using a construction introduced by Peter A. Loeb [15]. Moreover, we show that a number of measure-theoretic constructs, including integrals and conditional expectations, are naturally expressed in these models. It follows that standard models which can be expressed in terms of these measure spaces and constructs can be obtained from $^{\ast}$-finite models.
Minimum simplicial complexes with given abelian automorphism group
Zevi
Miller
689-718
Abstract: Let $K$ be a pure $n$-dimensional simplicial complex. Let ${\Gamma _0}(K)$ be the automorphism group of $ K$, and let ${\Gamma _n}(K)$ be the group of permutations on $ n$-cells of $K$ induced by the elements of ${\Gamma _0}(K)$. Given an abelian group $A$ we consider the problem of finding the minimum number of points $ M_0^{(n)}(A)$ in $ K$ such that ${\Gamma _0}(K) \cong A$, and the minimum number of $ n$-cells $M_1^{(n)}(A)$ in $K$ such that ${\Gamma _n}(K) \cong A$. Write $A = {\prod _{{p^\alpha }}}{\mathbf{Z}}_{{p^\alpha }}^{e({p^\alpha })}$, where each factor ${{\mathbf{Z}}_{{p^\alpha }}}$ appears $e({p^\alpha })$ times in the canonical factorization of $A$. For $A$ containing no factors ${{\mathbf{Z}}_{{p^\alpha }}}$ satisfying ${p^\alpha } < 17$ we find that $M_1^{(n)}(A) = M_0^{(2)}(A) = {\sum _{{p^\alpha }}}{p^\alpha }e({p^\alpha })$ when $n \geqslant 4$, and we derive upper bounds for $M_1^{(n)}(A)$ and $ M_0^{(n)}(A)$ in the remaining possibilities for $A$ and $n$.
Brownian motion with partial information
Terry R.
McConnell
719-731
Abstract: We study the following problem concerning stopped $N$-dimensional Brownian motion: Compute the maximal function of the process, ignoring those times when it is in some fixed region $R$. Suppose this modified maximal function belongs to ${L^q}$. For what regions $R$ can we conclude that the unrestricted maximal function belongs to ${L^q}$? A sufficient condition on $R$ is that there exist $p > q$ and a function $u$, harmonic in $R$, such that $\displaystyle \vert x{\vert^p} \leqslant u(x) \leqslant C\vert x{\vert^p} + C,\qquad x \in R,$ for some constant $C$. We give applications to analytic and harmonic functions, and to weak inequalities for exit times.
Interior and boundary continuity of weak solutions of degenerate parabolic equations
William P.
Ziemer
733-748
Abstract: In this paper we consider degenerate parabolic equations of the form $ ({\ast})$ $\displaystyle \beta {(u)_t} - \operatorname{div} A(x,\,t,\,u,\,{u_x}) + B(x,\,t,\,u,\,{u_x}) \ni 0$ where $A$ and $B$ are, respectively, vector and scalar valued Baire functions defined on $U \times {R^1} \times {R^n}$, where $ U$ is an open subset of ${R^{n + 1}}(x,\,t)$. The functions $ A$ and $B$ are subject to natural structural inequalities. Sufficiently general conditions are allowed on the relation $\beta \subset {R^1} \times {R^1}$ so that the porus medium equation and the model for the two-phase Stefan problem can be considered. The main result of the paper is that weak solutions of $({\ast})$ are continuous throughout $U$. In the event that $U = \Omega \times (0,\,T)$ where $ \Omega$ is an open set of $ {R^n}$, it is also shown that a weak solution is continuous at $ ({x_0},{t_0}) \in \partial \Omega \times (0,\,T)$ provided ${x_0}$ is a regular point for the Laplacian on $\Omega$.
The rank $3$ permutation representations of the finite classical groups
William M.
Kantor;
Robert A.
Liebler
1-71
Abstract: The permutation representations in the title are all determined, and no surprises are found to occur.
Type structure complexity and decidability
T. S.
Millar
73-81
Abstract: We prove that for every countable homogeneous model $\mathcal{A}$ such that the set of recursive types of $\operatorname{Th} (\mathcal{A})$ is $ \sum _2^0$, $\mathcal{A}$ is decidable iff the set of types realized in $\mathcal{A}$ is a $\sum _2^0$ set of recursive types. As a corollary to a lemma, we show that if a complete theory $ T$ has a recursively saturated model that is decidable in the degree of $ T$, then $T$ has a prime model.
Trees, Gleason spaces, and coabsolutes of $\beta {\bf N}\sim {\bf N}$
Scott W.
Williams
83-100
Abstract: For a regular Hausdorff space $X$, let $ \mathcal{E}(X)$ denote its absolute, and call two spaces $X$ and $Y$ coabsolute ( $ \mathcal{G}$-absolute) when $ \mathcal{E}(X)$ and $\mathcal{E}(Y)$ ( $\beta \mathcal{E}(X)$ and $\beta \mathcal{E}(Y)$) are homeomorphic. We prove $ X$ is $\mathcal{G}$-absolute with a linearly ordered space iff the POSET of proper regular-open sets of $ X$ has a cofinal tree; a Moore space is $ \mathcal{G}$-absolute with a linearly ordered space iff it has a dense metrizable subspace; a dyadic space is $\mathcal{G}$-absolute with a linearly ordered space iff it is separable and metrizable; if $ X$ is a locally compact noncompact metric space, then $\beta X \sim X$ is coabsolute with a compact linearly ordered space having a dense set of $ P$-points; CH implies but is not implied by "if $X$ is a locally compact noncompact space of $ \pi$-weight at most ${2^\omega }$ and with a compatible complete uniformity, then $\beta X \sim X$ and $\beta N \sim N$ are coabsolute."
Universal bundles for deformations of asymmetric structures
Paul R.
Dippolito
101-115
Abstract: Riemannian parallel transport of frames and Cartan connections can be considered as special cases of a type of structure in which the connection form on the principal bundle can transform by an arbitrary extension of the adjoint representation of the fiber group; there always exist universal deformations for asymmetric objects in any of these geometries.
The Hodge theory of flat vector bundles on a complex torus
Jerome William
Hoffman
117-131
Abstract: We study the Hodge spectral sequence of a local system on a compact, complex torus by means of the theory of harmonic integrals. It is shown that, in some cases, Baker's theorems concerning linear forms in the logarithms of algebraic numbers may be applied to obtain vanishing theorems in cohomology. This is applied to the study of Betti and Hodge numbers of compact analytic threefolds which are analogues of hyperelliptic surfaces. Among other things, it is shown that, in contrast to the two-dimensional case, some of these varieties are nonalgebraic.
Transfer and the spectral sequence of a fibration
Carlos
Prieto
133-142
Abstract: The purpose of this paper is to show that transfers for fibrations behave well with respect to spectral sequences which are induced by filtrations of the base space. In particular, for the spectral sequence of a fibration (induced by the skeletal filtration of the base space), one obtains the expected effect on the ${E_2}$-terms: We prove that the transfer in the $ {E_2}$-terms is determined by the transfer of the fiber (considered trivially as a fibration over a point). As an application, results of Atiyah on the $K$-theory of classifying spaces are transcribed to generalized cohomology theories.
Torsion-groups of abelian coverings of links
John P.
Mayberry;
Kunio
Murasugi
143-173
Abstract: If $M$ is an abelian branched covering of $ {S^3}$ along a link $ L$, the order of $ {H_1}(M)$ can be expressed in terms of (i) the Alexander polynomials of $ L$ and of its sublinks, and (ii) a "redundancy" function characteristic of the monodromy-group. In 1954, the first author thus generalized a result of Fox (for $L$ a knot, in which case the monodromy-group is cyclic and the redundancy trivial); we now prove earlier conjectures and give a simple interpretation of the redundancy. Cyclic coverings of links are discussed as simple special cases. We also prove that the Poincaré conjecture is valid for the above-specified family of $3$-manifolds $M$. We state related results for unbranched coverings.
Permutation-partition pairs. II. Bounds on the genus of the amalgamation of graphs
Saul
Stahl
175-182
Abstract: Bounds are derived on the extent to which the parameter $\mu (P,\,\prod )$ can fail to be additive over disjoint permutations. This is done by associating an Eulerian digraph to each such pair and relating the maximum orbiticity $ \mu (P,\,\prod )$ to the decompositions of this digraph's arc set into arc disjoint cycles. These bounds are then applied to obtain information about the genus of the amalgamation of graphs.
The minimum modulus of certain small entire functions
P. C.
Fenton
183-195
Abstract: Suppose that $ f(z)$ is an entire function satisfying $\displaystyle \mathop {\min }\limits_{\vert z\vert = r} \vert f(z)\vert \leqslant C(\sigma )\mathop {\max }\limits_{\vert z\vert = r} \vert f(z)\vert,$ for $r \geqslant {\rho _0} > 0$, where $\sigma > 0$ and $\displaystyle C(\sigma ) = \prod\limits_{k = 1}^\infty {{{\left\{ {\frac{{1 - \... ...- (2k - 1) / 4\sigma )}} {{1 + \exp ( - (2k - 1) / 4\sigma )}}} \right\}}^2}.}$ It is shown that $\displaystyle \mathop {\underline {\lim } }\limits_{r \to \infty } \frac{{{{\ma... ...)}^2}}} {{\log r}} \geqslant - 2\sigma \log (\max ({\rho _0},\vert{a_1}\vert)),$ where $ {a_1}$ is the first nonzero zero of $f$.
Siegel domains and representations of Jordan algebras
O. S.
Rothaus
197-213
Abstract: In the analysis of infinitesimal automorphisms of arbitrary Siegel domains, a certain class of nonsemisimple Jordan algebras occurs. The description of all the infinitesimal automorphisms of the domain may be based on a study of representations of the associated Jordan algebra satisfying a certain "strange identity". In this paper, all the possibilities for the Jordan algebra and representations satisfying the identity are given.
Tangent $2$-fields on even-dimensional nonorientable manifolds
Benedict J.
Pollina
215-224
Abstract: This paper uses the Postnikov decomposition of a nonsimple fibration to describe the obstructions to a tangent $2$-field on an even-dimensional nonorientable manifold.
The sums of powers theorem for commuting block maps
Frank
Rhodes
225-236
Abstract: A block map is a map $ f:{\{ 0,\,1\} ^n} \to \{ 0,\,1\}$ for some $n \geqslant 1$. A block map $ f$ induces an endomorphism ${f_\infty }$ of the full $2$-shift $ (X,\,\sigma )$. Composition of block maps is defined in such a way that ${(f \circ g)_\infty } = {f_\infty } \circ {g_\infty }$. In this paper some recent results concerning the set $ \{ g\vert g \circ f = f \circ g\}$ for certain types of block maps $f$ are extended.
Tauberian conditions for $L\sp{1}$-convergence of Fourier series
Časlav V.
Stanojević
237-244
Abstract: It is proved that Fourier series with asymptotically even coefficients and satisfying $ {\lim _{\lambda \to 1}}\lim {\sup _{n \to \infty }}\sum _{j = n}^{[\lambda n]}{j^{p - 1}}\vert\Delta \hat f(j){\vert^p} = 0$, for some $1 < p \leqslant 2$, converge in ${L^1}$-norm if and only if $ \vert\vert\hat f(n){E_n} + \hat f( - n){E_{ - n}}\vert\vert = o(1)$, where ${E_n}(t) = \sum _{k = 0}^n{e^{ikt}}$. Recent results of Stanojević [1], Bojanic and Stanojević [2], and Goldberg and Stanojević [3] are special cases of some corollaries to the main theorem.
Growth of solutions of linear differential equations at a logarithmic singularity
A.
Adolphson;
B.
Dwork;
S.
Sperber
245-252
Abstract: We consider differential equations $Y' = AY$ with a regular singular point at the origin, where $A$ is an $n \times n$ matrix whose entries are $ p$-adic meromorphic functions. If the solution matrix at the origin is of the form $Y = P\exp (\theta \log x)$, where $P$ is an $n \times n$ matrix of meromorphic functions and $\theta$ is an $ n \times n$ constant matrix whose Jordan normal form consists of a single block, then we prove that the entries of $P$ have logarithmic growth of order $ n - 1$.
Analogues of the Denjoy-Young-Saks theorem
C. L.
Belna;
G. T.
Cargo;
M. J.
Evans;
P. D.
Humke
253-260
Abstract: In this paper, an analogue of the Denjoy-Young-Saks theorem concerning the almost everywhere classification of the Dini derivates of an arbitrary real function is established in both the case where the exceptional set is of first category and the case where it is $ \sigma$-porous. Examples are given to indicate the sharpness of these results.
On the construction and distribution of a local martingale with a given absolute value
Edwin
Perkins
261-281
Abstract: A local martingale is constructed on an appropriate Loeb space whose absolute value equals a given nonnegative local submartingale. Nonstandard analysis is used to reduce the problem to the discrete time setting where the original construction of D. Gilat is fairly simple. This approach has the advantage of allowing explicit computations. In particular, the distribution of the local martingale is described in terms of the Doob-Meyer decomposition of the original local submartingale.
Derivations into the integral closure
Richard
Draper;
Klaus
Fischer
283-298
Abstract: Let $A$ be a reduced analytical $ k$-algebra of dimension $ r$ and $A'$ its integral closure in the full ring of quotients of $A$. We investigate the condition on $A$ that there exist $r$ elements ${x_1}, \ldots ,{x_r}$ in $A$ and $k$-derivations $ {d_1}, \ldots ,{d_r}$ from $ A$ into $A'$ so that $ {d_i}({x_j})$ is the $r \times r$ identity matrix and so that ${d_1}, \ldots ,{d_r}$ freely generate
Algebras generated by a subnormal operator
Robert F.
Olin;
James E.
Thomson
299-311
Abstract: We use the notion of generalized Toeplitz operators to obtain some basic results concerning the $ {C^{\ast}}$-algebra generated by a subnormal operator. We apply these results to problems concerning the intersection of ${C^{\ast}}(S)$ with rationally closed algebras generated by $S$. In particular, we prove that ${C^{\ast}}(S) \cap \mathcal{W}(S) = \{ f(S):f \in R({\sigma _{\mathcal{W}(S)}}(S))\}$. The spectral inclusion property for generalized Toeplitz operators with symbols in ${P^\infty }(\mu ) + C(\sigma (N))$ is also considered.
Classification of oriented equivariant spherical fibrations
Stefan
Waner
313-324
Abstract: Classifying spaces for oriented equivariant spherical fibrations are constructed, and the notion of an equivariant $ SF$-fibration is introduced. It is shown that equivariant $SF$-fibrations are naturally oriented in $ RO(G)$-graded equivariant singular cohomology.
Self-maps of projective spaces
C. A.
McGibbon
325-346
Abstract: The classical projective $n$-spaces (real, complex, and quaternionic) are studied in terms of their self maps, from a homotopy point of view. Self maps of iterated suspensions of these spaces are also considered. The goal in both cases is to classify, up to homology, all such maps. This goal is achieved in the stable case. Some partial results are obtained in the unstable case. The results from both cases are used to compute the genus groups and the stable genus groups of the classical projective spaces. Applications to other spaces are also given.
Corrections and additions to: ``Some properties of measure and category''
Arnold W.
Miller
347-348
Szeg\H o limit theorems for Toeplitz operators on compact homogeneous spaces
I. I.
Hirschman;
D. S.
Liang;
E. N.
Wilson
351-376
Abstract: Let $f$ be a real valued integrable function on a compact homogeneous space $M = K\backslash G$ and ${M_f}$ the operator of pointwise multiplication by $ f$. The authors consider families of Toeplitz operators ${T_{f,P}} = P{M_f}P$ as $P$ ranges over a net of orthogonal projections from $ {L^2}(M)$ to finite dimensional $G$-invariant subspaces. Necessary and sufficient conditions are given on the net in order that the distribution of eigenvalues of these Toeplitz operators is asymptotic to the distribution of values of $f$ in the sense of Szegö's classical theorem for the circle. Explicit sequences satisfying these conditions are constructed for all compact Lie groups and for all Riemannian symmetric compact spaces.
Maximal positive boundary value problems as limits of singular perturbation problems
Claude
Bardos;
Jeffrey
Rauch
377-408
Abstract: We study three types of singular perturbations of a symmetric positive system of partial differential equations on a domain $ \Omega \subset {{\mathbf{R}}^n}$. In all cases the limiting behavior is given by the solution of a maximal positive boundary value problem in the sense of Friedrichs. The perturbation is either a second order elliptic term or a term large on the complement of $\Omega$. The first corresponds to a sort of viscosity and the second to physical systems with vastly different properties in $\Omega$ and outside $\Omega$. The results show that in the limit of zero viscosity or infinitely large difference the behavior is described by a maximal positive boundary value problem in $\Omega$. The boundary condition is determined in a simple way from the system and the singular terms.
Mathematical theory of single channel systems. Analyticity of scattering matrix
I. M.
Sigal
409-437
Abstract: We show that the $ S$-matrix of a quantum many-body, short-range, single-channel system has a meromorphic continuation whose poles occur at most at the dilation-analytic resonances [28], [24] and at the eigenvalues of the Hamiltonian. In passing, we prove the main spectral theorem (on location of the essential spectrum) and asymptotic completeness for the mentioned class of systems.
Relations between $H\sp{p}\sb{u}$ and $L\sp{p}\sb{u}$ with polynomial weights
Jan-Olov
Strömberg;
Richard L.
Wheeden
439-467
Abstract: Relations between $ L_u^p$ and $H_u^p$ of the real line are studied in the case when $p > 1$ and $ u(x) = \vert q(x){\vert^p}w(x)$, where $q(x)$ is a polynomial and $w(x)$ satisfies the ${A_p}$ condition. It turns out that $ H_u^p$ and $L_u^p$ can be identified when all the zeros of $q$ are real, and that otherwise $H_u^p$ can be identified with a certain proper subspace of $L_u^p$. In either case, a complete description of the distributions in $H_u^p$ is given. The questions of boundary values and of the existence of dense subsets of smooth functions are also considered.
The group of automorphisms of a class of finite $p$-groups
Arye
Juhász
469-481
Abstract: Let $G$ be a finite $p$-group and denote by ${K_i}(G)$ the members of the lower central series of $G$. We call $G$ of type $(m,\,n)$ if (a) $G$ has nilpotency class $m - 1$, (b) $ G/{K_2}(G) \cong {{\mathbf{Z}}_{{p^n}}} \times {{\mathbf{Z}}_{{p^n}}}$ and ${K_i}(G)/{K_{i + 1}}(G) \cong {{\mathbf{Z}}_{{p^n}}}$ for every $i$, $2 \leqslant i \leqslant n - 1$. In this work we describe the structure of $\operatorname{Aut} (G)$ and certain relations between $ \operatorname{Out} (G)$ and $G$.
Differentiability of the metric projection in Hilbert space
Simon
Fitzpatrick;
R. R.
Phelps
483-501
Abstract: A study is made of differentiability of the metric projection $ P$ onto a closed convex subset $K$ of a Hilbert space $H$. When $K$ has nonempty interior, the Gateaux or Fréchet smoothness of its boundary can be related with some precision to Gateaux or Fréchet differentiability properties of $P$. For instance, combining results in $ \S3$ with earlier work of R. D. Holmes shows that $K$ has a ${C^2}$ boundary if and only if $P$ is ${C^1}$ in $ H\backslash K$ and its derivative $P'$ has a certain invertibility property at each point. An example in $\S5$ shows that if the ${C^2}$ condition is relaxed even slightly then $ P$ can be nondifferentiable (Fréchet) in $ H\backslash K$.
Nonorientable surfaces in some non-Haken $3$-manifolds
J. H.
Rubinstein
503-524
Abstract: If a closed, irreducible, orientable $3$-manifold $M$ does not possess any $2$-sided incompressible surfaces, then it can be very useful to investigate embedded one-sided surfaces in $M$ of minimal genus. In this paper such $ 3$-manifolds $ M$ are studied which admit embeddings of the nonorientable surface $ K$ of genus $3$. We prove that a $3$-manifold $M$ of the above type has at most $3$ different isotopy classes of embeddings of $K$ representing a fixed element of ${H_2}(M,\,{Z_2})$. If $M$ is either a binary octahedral space, an appropriate lens space or Seifert manifold, or if $ M$ has a particular type of fibered knot, then it is shown that the embedding of $ K$ in $M$ realizing a specific homology class is unique up to isotopy.
Fundamental groups of topological $R$-modules
Ann
Bateson
525-536
Abstract: The main result of this paper is that if $R$ is a countable, Noetherian ring, then the underlying abelian group of every $R$-module is isomorphic to the fundamental group of some topological $R$-module. As a corollary, it is shown that for certain varieties $V$(e.g., varieties of finite type) every abelian group in $V$ is isomorphic to the fundamental group of some arcwise connected topological algebra in $V$.
Measurable parametrizations of sets in product spaces
V. V.
Srivatsa
537-556
Abstract: Various parametrization theorems are proved. In particular the following is shown: Let $B$ be a Borel subset of $I \times I$ (where $I = [0,\,1]$) with uncountable vertical sections. Let $\sum \dot \cup N$ be the discrete (topological) union of $\sum$, the space of irrationals, and $ N$, the set of natural numbers with discrete topology. Then there is a map $ f:I \times (\sum \dot \cup N) \to I$ measurable with respect to the product of the analytic $\sigma$-field on $I$ (that is, the smallest $\sigma $-field on $I$ containing the analytic sets) and the Borel $\sigma$-field on $ \sum \dot \cup N$ such that $f(t,\,\, \cdot ):\,\sum \dot \cup N \to I$ is a one-one continuous map of $ \sum \dot \cup N$ onto $\{ x:(t,\,x) \in B\}$ for each $ t \in T$. This answers a question of Cenzer and Mauldin.
Saturation properties of ideals in generic extensions. I
James E.
Baumgartner;
Alan D.
Taylor
557-574
Abstract: We consider saturation properties of ideals in models obtained by forcing with countable chain condition partial orderings. As sample results, we mention the following. If $ M[G]$ is obtained from a model $M$ of GCH via any $\sigma$-finite chain condition notion of forcing (e.g. add Cohen reals or random reals) then in $ M[G]$ every countably complete ideal on $ {\omega _1}$ is ${\omega _3}$-saturated. If "$\sigma $-finite chain condition" is weakened to "countable chain condition," then the conclusion no longer holds, but in this case one can conclude that every $ {\omega _2}$-generated countably complete ideal on $ {\omega _1}$ (e.g. the nonstationary ideal) is $ {\omega _3}$-saturated. Some applications to ${\mathcal{P}_{{\omega _1}}}({\omega _2})$ are included and the role played by Martin's Axiom is discussed. It is also shown that if these weak saturation requirements are combined with some cardinality constraints (e.g. ${2^{{\aleph _1}}} > {({2^{{\aleph _0}}})^ + })$), then the consistency of some rather large cardinals becomes both necessary and sufficient.
Formal spaces with finite-dimensional rational homotopy
Yves
Félix;
Stephen
Halperin
575-588
Abstract: Let $S$ be a simply connected space. There is a certain principal fibration ${K_1} \to E\mathop \to \limits^\pi {K_0}$ in which ${K_1}$ and ${K_0}$ are products of rational Eilenberg-Mac Lane spaces and a continuous map $\phi :S \to E$ such that in particular ${\phi _0} = \pi \circ \phi $ maps the primitive rational homology of $S$ isomorphically to that of ${K_0}$. A main result of this paper is the Theorem. If $\dim \pi {}_{\ast}(S) \otimes {\mathbf{Q}} < \infty$ then $\phi$ is a rational homotopy equivalence if and only if all the primitive homology in $ H{}_{\ast}(S;\,{\mathbf{Q}})$ and $ H{}_{\ast}({K_0},\,S;\,{\mathbf{Q}})$ can (up to integral multiples) be represented by spheres and disk-sphere pairs. Corollary. If $S$ is formal, $\phi$ is a rational homotopy equivalence.
Free Lie subalgebras of the cohomology of local rings
Luchezar L.
Avramov
589-608
Abstract: A criterion is established, in terms of the Massey products structure carried by the homology of partial resolutions, for the Yoneda cohomology algebra ${\operatorname{Ext} _A}(k,\,k)$ to be a free module over the universal envelope of a free graded Lie subalgebra. It is shown that several conjectures on the (co)homology of local rings, in particular on the asymptotic behaviour of the Betti numbers, follow from such a structure. For all rings with $\operatorname{edim} A - \operatorname{depth} A \leqslant 3$, and for Gorenstein rings with $\operatorname{edim} A - \operatorname{depth} A = 4$, the following dichotomy is proved: Either $ A$ is a complete intersection, or ${\operatorname{Ext} _A}(k,\,k)$ contains a nonabelian free graded Lie subalgebra.
Hopf manifolds and spectral geometry
Kazumi
Tsukada
609-621
Abstract: We characterize Hopf manifolds in the class of Hermitian manifolds by the spectra of the real Laplacians and the complex Laplacians.
Spectral properties of orthogonal polynomials on unbounded sets
T. S.
Chihara
623-639
Abstract: We consider orthogonal polynomials when the three term recurrence formula for the monic polynomials has unbounded coefficients. We obtain information relative to three questions: Under what conditions on the coefficients will the derived set of the spectrum have a finite infimum $ \sigma$? If $ \sigma$ is finite, when will there be at most finitely many spectral points smaller than $\sigma$; and when will the distribution function be continuous at $\sigma$?
The structure of quasinormal operators and the double commutant property
John B.
Conway;
Pei Yuan
Wu
641-657
Abstract: In this paper a characterization of those quasinormal operators having the double commutant property is obtained. That is, a necessary and sufficient condition is given that a quasinormal operator $T$ satisfy the equation $\{ T\} '' = \mathcal{A}(T)$, the weakly closed algebra generated by $T$ and $1$. In particular, it is shown that every pure quasinormal operator has the double commutant property. In addition two new representation theorems for certain quasinormal operators are established. The first of these represents a pure quasinormal operator $ T$ as multiplication by $ z$ on a subspace of an $ {L^2}$ space whenever there is a vector $f$ such that $\{ \vert T{\vert^k}{T^j}f:\,k,\,j \geqslant 0\}$ has dense linear span. The second representation theorem applies to those pure quasinormal operators $T$ such that $ {T^{\ast}}T$ is invertible. The second of these representation theorems will be used to determine which quasinormal operators have the double commutant property.
Spectral permanence for joint spectra
Raul E.
Curto
659-665
Abstract: For a ${C^{\ast}}$-subalgebra $A$ of a $ {C^{\ast}}$-algebra $ B$ and a commuting $ n$-tuple $a = ({a_1}, \ldots ,{a_n})$ of elements of $ A$, we prove that $\operatorname{Sp} (a,\,A) = \operatorname{Sp} (a,\,B)$, where $ \operatorname{Sp}$ denotes Taylor spectrum. As a consequence we prove that $0 \notin \operatorname{Sp} (a,\,A)$ if and only if $\displaystyle \hat a = \left( {\begin{array}{*{20}{c}} {{d_1}} & {} & {} {d_... ...d{array} } \right) \in L\left( {A \otimes {{\mathbf{C}}^{{2^{n - 1}}}}} \right)$ is invertible, where ${d_i}$ is the $i$th boundary map in the Koszul complex for $ A$. More generally, we show that $ {\sigma _{\delta ,k}}(a,\,A) = {\sigma _{\delta ,k}}\left( {a,\,B} \right)$ and ${\sigma _{\pi ,k}}(a,\,A) = {\sigma _{\pi ,k}}(a,\,B)$ (all $k$), where ${\sigma _{\delta ,\cdot}}$ and ${\sigma _{\pi ,\cdot}}$ are the joint spectra considered by Z. Słodkowski.
The reciprocal of an entire function of infinite order and the distribution of the zeros of its second derivative
John
Rossi
667-683
Abstract: Let $f$ be a real entire function of infinite order whose zeros together with those of $f'$ are all real. It is proved that $ (1/f)''$ has an infinity of nonreal zeros. The location of the zeros of $ f''$ and $(1/f)''$ is also investigated. The result complements a finite order result of Hellerstein and Williamson.
The Bergman kernel function and proper holomorphic mappings
Steven R.
Bell
685-691
Abstract: It is proved that a proper holomorphic mapping $f$ between bounded complete Reinhardt domains extends holomorphically past the boundary and that if, in addition, $ {f^{ - 1}}(0) = \{ 0\}$, then $f$ is a polynomial mapping. The proof is accomplished via a transformation rule for the Bergman kernel function under proper holomorphic mappings.
The nonfinite generation of ${\rm Aut}(G)$, $G$ free metabelian of rank $3$
S.
Bachmuth;
H. Y.
Mochizuki
693-700
Abstract: The group of automorphisms of the free metabelian group of rank $ 3$ is not finitely generated.
On $(K\sb\ast ({\bf Z}/p\sp{2}{\bf Z})$ and related homology groups
Leonard
Evens;
Eric M.
Friedlander
1-46
Abstract: It is shown that, for $p \geqslant 5$, $\displaystyle R = {\mathbf{Z}} / {p^2}{\mathbf{Z}},\,{K_3}(R) = {\mathbf{Z}} / {p^2}{\mathbf{Z}} + {\mathbf{Z}} / ({p^2} - 1){\mathbf{Z}}$ and $ {K_4}(R) = 0$. Similar calculations are made for $R$ the ring of dual numbers over ${\mathbf{Z}} / p{\mathbf{Z}}$. The calculation reduces to finding homology groups of $\operatorname{Sl} (R)$. A key tool is the spectral sequence of the group extension of $ \operatorname{Sl} (n,\,{p^2})$ over $\operatorname{Sl} (n,\,p)$. The terms of this spectral sequence depend in turn on the homology of $ \operatorname{Gl} (n,\,p)$ with coefficients various multilinear modules. Calculation of the differentials uses the Charlap-Vasquez description of ${d^2}$.
A hypersurface defect relation for a class of meromorphic maps
Aldo
Biancofiore
47-60
Abstract: Let ${D_1}, \ldots ,{D_q}$ be hypersurfaces of degree $ p$ in ${{\mathbf{P}}_n}$ with normal crossings. We prove for a certain class of meromorphic maps $ f:{{\mathbf{C}}^m} \to {{\mathbf{P}}_n}$ a defect relation ${\delta _f}\left( {{D_1}} \right) + \cdots + {\delta _f}({D_q}) \leqslant (n + 1)/p$ conjectured by Ph. Griffiths and B. Shiffman.
Uncountable admissibles. I. Forcing
Sy D.
Friedman
61-73
Abstract: Assume $V = L$. Let $\kappa$ be a regular cardinal and for $X \subseteq \kappa$ let $\alpha (X)$ denote the least ordinal $\alpha$ such that ${L_\alpha }[X]$ is admissible. In this paper we characterize those ordinals of the form $\alpha (X)$ using forcing and fine structure of $ L$ techniques. This generalizes a theorem of Sacks which deals with the case $\kappa = \omega $.
A family of links and the Conway calculus
Cole A.
Giller
75-109
Abstract: In 1969, J. H. Conway gave efficient methods of calculating abelian invariants of classical knots and links. The present paper includes a detailed exposition (with new proofs) of these methods and extensions in several directions. The main application given here is as follows. A link $ L$ of two unknotted components in ${S^3}$ has the distinct lifting property for $p$ if the lifts of each component to the $ p$-fold cover of $ {S^3}$ branched along the other are distinct. The $p$-fold covers of these lifts are homeomorphic, and so $L$ gives an example of two distinct knots with the same $p$-fold cover. The above machinery is then used to construct an infinite family of links, each with the distinct lifting property for all $p \geqslant 2$.
Duality between logics and equivalence relations
Daniele
Mundici
111-129
Abstract: Assuming $ \omega$ is the only measurable cardinal, we prove: (i) Let $\sim$ be an equivalence relation such that $ \sim \, = \,{ \equiv _L}$ for some logic $L \leqslant {L^{\ast}}$ satisfying Robinson's consistency theorem (with ${L^{\ast}}$ arbitrary); then there exists a strongest logic $ {L^ + } \leqslant {L^{\ast}}$ such that $\sim \, = \,{ \equiv _{{L^ + }}}$; in addition, $ {L^ + }$ is countably compact if $ \sim \, \ne \, \cong$. (ii) Let $\dot \sim$ be an equivalence relation such that $ \sim \, = \,{ \equiv _{{L^0}}}$ for some logic ${L^0}$ satisfying Robinson's consistency theorem and whose sentences of any type $\tau$ are (up to equivalence) equinumerous with some cardinal $ {\kappa _\tau }$; then $ {L^0}$ is the unique logic $ L$ such that $\sim \, = \,{ \equiv _L}$; furthermore, $ {L^0}$ is compact and obeys Craig's interpolation theorem. We finally give an algebraic characterization of those equivalence relations $\sim$ which are equal to ${ \equiv _L}$ for some compact logic $L$ obeying Craig's interpolation theorem and whose sentences are equinumerous with some cardinal.
Boolean algebras with no rigid or homogeneous factors
Petr
Štěpánek
131-147
Abstract: A simple construction of Boolean algebras with no rigid or homogeneous factors is described. It is shown that for every uncountable cardinal $\kappa$ there are $ {2^\kappa }$ isomorphism types of Boolean algebras of power $\kappa$ with no rigid or homogeneous factors. A similar result is obtained for complete Boolean algebras for certain regular cardinals. It is shown that every Boolean algebra can be completely embedded in a complete Boolean algebra with no rigid or homogeneous factors in such a way that the automorphism group of the smaller algebra is a subgroup of the automorphism group of the larger algebra. It turns out that the cardinalities of antichains in both algebras are the same. It is also shown that every $ \kappa$-distributive complete Boolean algebra can be completely embedded in a $ \kappa$-distributive complete Boolean algebra with no rigid or homogeneous factors.
Calibers of compact spaces
S.
Argyros;
A.
Tsarpalias
149-162
Abstract: Let $X$ be a compact Hausdorff space and $ \kappa$ its Souslin number.$^{2}$ We prove that if $\alpha$ is a cardinal such that either $ \alpha$ and $ \operatorname{cf} (\alpha )$ are greater than $\kappa$ and strongly $\kappa$-inaccessible or else $\alpha$ is regular and greater than $ \kappa$, then $ X$ has $ (\alpha ,\,\sqrt[{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\kappa } }]{\alpha })$ caliber. Restricting our interest to the category of compact spaces $X$ with $ S(X) = {\omega ^ + }$ (i.e. $X$ satisfy the countable chain condition), the above statement takes, under G.C.H., the following form. For any compact space $X$ with $ S(X) = {\omega ^ + }$, we have that (a) if $\alpha$ is a cardinal and $\operatorname{cf} (\alpha )$ does not have the form ${\beta ^ + }$ with $\operatorname{cf} (\beta ) = \omega$, then $ \alpha$ is caliber for the space $X$. (b) If $\varepsilon = {\beta ^ + }$ and $ \operatorname{cf} (\beta ) = \omega$ then $ (\alpha ,\,\beta )$ is caliber for $X$. A related example shows that the result of (b) is in a sense the best possible.
A Poisson summation formula for integrals over quadratic surfaces
Robert S.
Strichartz
163-173
Abstract: Let $S(t)$ denote Lebesgue measure on the sphere of radius $t > 0$ in $ {{\mathbf{R}}^n}$, and $\displaystyle {S_k}(t) = {\left( {\frac{\partial } {{\partial t}}\quad \frac{1} {t}} \right)^k}S(t).$ Let $P{\sum _k} = {S_k}(0) + 2\sum _{m = 1}^\infty {S_k}(m)$. Theorem. If $n$ is odd and $j$ and $k$ are nonnegative integers with $j + k = (n - 1) / 2$, then the Fourier transform of $P{\sum _j}$ is ${(2\pi )^{j - k}}P{\sum _k}$. There is an analogous, although slightly different, identity involving integrals over hyperboloids in odd dimensions. These results were inspired by recent work of M. Vergne.
A note on complete intersections
S. M.
Bhatwadekar
175-181
Abstract: Let $R$ be a regular local ring and let $ R[T]$ be a polynomial algebra in one variable over $R$. In this paper the author proves that every maximal ideal of $R[T]$ is complete intersection in each of the following cases: (1) $R$ is a local ring of an affine algebra over an infinite perfect field, (2) $R$ is a power series ring over a field.
Degeneracy theorems for holomorphic mappings between algebraic varieties
Robert
Molzon
183-192
Abstract: Degeneracy theorems are proved for holomorphic mappings from affine algebraic manifolds to projective algebraic manifolds of equal dimensions. A mapping is degenerate if it satisfies a growth estimate and omits a set of $\kappa$-plane sections of positive capacity; the capacity being defined in terms of a singular integral. The capacity is a more delicate method of measuring the size of a set of $\kappa$-plane sections than Hausdorff measure and arises naturally by considering the singular integrals in the First Main Theorem of Nevanlinna.
On nonseparable Banach spaces
Spiros A.
Argyros
193-216
Abstract: Combining combinatorial methods from set theory with the functional structure of certain Banach spaces we get some results on the isomorphic structure of nonseparable Banach spaces. The conclusions of the paper, in conjunction with already known results, give complete answers to problems of the theory of Banach spaces. An interesting point here is that some questions of Banach spaces theory are independent of Z.F.C. So, for example, the answer to a conjecture of Pełczynski that states that the isomorphic embeddability of ${L^1}{\{ - 1,\,1\} ^\alpha }$ into ${X^{\ast}}$ implies, for any infinite cardinal $ \alpha$, the isomorphic embedding of $ l_\alpha ^1$ into $ X$, gets the following form: if $\alpha = \omega $, has been proved from Pełczynski; if $\alpha > {\omega ^ + }$, the proof is given in this paper; if $ \alpha = {\omega ^ + }$, in $ {\text{Z}}{\text{.F}}{\text{.C}}{\text{.}} + {\text{C}}{\text{.H}}{\text{.}}$, an example discovered by Haydon gives a negative answer; if $\alpha = {\omega ^ + }$, in $ {\text{Z}}{\text{.F}}{\text{.C}}{\text{.}} + \urcorner {\text{C}}{\text{.H}}{\text{.}} + {\text{M}}{\text{.A}}{\text{.}}$, is also proved in this paper.
Closures of conjugacy classes in classical real linear Lie groups. II
Dragomir Ž.
Djoković
217-252
Abstract: By a classical group we mean one of the groups $G{L_n}(R)$, $G{L_n}(C)$, $G{L_n}(H)$, $U(p,\,q)$, ${O_n}(C)$, $O(p,\,q)$, $ S{O^{\ast}}(2n)$, $S{p_{2n}}(C)$, $ S{p_{2n}}(R)$, or $ Sp(p,\,q)$. Let $ G$ be a classical group and $L$ its Lie algebra. For each $x \in L$ we determine the closure of the orbit $G \cdot x$ (for the adjoint action of $ G$ on $L$). The problem is first reduced to the case when $x$ is nilpotent. By using the exponential map we also determine the closures of conjugacy classes of $ G$.
A study of the local components of the Hecke algebra mod $l$
Naomi
Jochnowitz
253-267
Abstract: We use information about modular forms $\bmod l$ to study the local structure of the Hecke ring. In particular, we find nontrivial lower bounds for the dimensions of the Zariski tangent spaces of the local components of the Hecke ring $\bmod l$. These results suggest that the local components of the Hecke ring $\bmod l$ are more complex than originally expected. We also investigate the inverse limits of the Hecke rings of weight $k\bmod l$ as $k$ varies within a fixed congruence class $\bmod l - 1$. As an immediate corollary to some of the above results, we show that when $ k$ is sufficiently large, an arbitrary prime $l$ must divide the index of the classical Hecke ring $ {{\mathbf{T}}_k}$ in the ring of integers of ${{\mathbf{T}}_k} \otimes {\mathbf{Q}}$.
Congruences between systems of eigenvalues of modular forms
Naomi
Jochnowitz
269-285
Abstract: We modify and generalize proofs of Tate and Serre in order to show that there are only a finite number of systems of eigenvalues for the Hecke operators with respect to ${\Gamma _0}(N)\bmod l$. We also summarize results for ${\Gamma _1}(N)$. Using these results, we show that an arbitrary prime divides the discriminant of the classical Hecke ring to a power which grows linearly with $k$. In this way, we find a lower bound for the discriminant of the Hecke ring. After limiting ourselves to cusp forms, we also find an upper bound. Lastly we use the constructive nature of Tate and Serre's result to describe the structure and dimensions of the generalized eigenspaces for the Hecke operators $\bmod l$.
Structure theory for a class of grade four Gorenstein ideals
Andrew
Kustin;
Matthew
Miller
287-307
Abstract: An ideal $ I$ in a commutative noetherian ring $R$ is a Gorenstein ideal of $\operatorname{grade} g$ if ${\operatorname{pd} _R}(R / I) = \operatorname{grade} \,I = g$ and the canonical module $ \operatorname{Ext} _R^g(R / I,\,R)$ is cyclic. Serre showed that if $g = 2$ then $I$ is a complete intersection, and Buchsbaum and Eisenbud proved a structure theorem for the case $g = 3$. We present generic resolutions for a class of Gorenstein ideals of $\operatorname{grade} 4$, and we illustrate the structure of the resolution with various specializations. Among these examples there are Gorenstein ideals of $ \operatorname{grade} \,4$ in $ k[[x,\,y,\,z,\,v]]$ that are $n$-generated for any odd integer $n \geqslant 7$. We construct other examples from almost complete intersections of $ \operatorname{grade} \,3$ and their canonical modules. In the generic case the ideals are shown to be normal primes. Finally, we conclude by giving an explicit associative algebra structure for the resolutions. It is this algebra structure that we use to classify the different Gorenstein ideals of $\operatorname{grade} \,4$, and which may be the key to a complete structure theorem.
A network of congruences on an inverse semigroup
Mario
Petrich;
Norman R.
Reilly
309-325
Abstract: A congruence $ \rho$ on an inverse semigroup $S$ is determined uniquely by its kernel and its trace. Denoting by $ {\rho ^{\min }}$ and ${\rho _{\min }}$ the least congruence on $ S$ having the same kernel and the same trace as $\rho$, respectively, and denoting by $\omega$ the universal congruence on $ S$, we consider the sequence $\omega$, $ {\omega ^{\min }}$, ${\omega _{\min }}$, ${({\omega ^{\min }})_{\min }}$, ${({\omega _{\min }})^{\min }} \ldots$. These congruences, together with the intersections of corresponding pairs, form a sublattice of the lattice of all congruences on $S$. We study the properties of these congruences and establish several properties of the quasivarieties of inverse semigroups induced by them.
Banach bundles of continuous functions and an integral representation theorem
Anthony Karel
Seda
327-332
Abstract: A construction is given of a Banach bundle $p:A \to X$ whose fibres are spaces of continuous functions which vanish at infinity. A Riesz type integral representation theorem is established which describes all functional on $A$.
The integrability tensor for bundle-like foliations
Richard H.
Escobales
333-339
Abstract: A certain function is introduced which is useful in the study of a bundle-like foliation on a Riemannian manifold. Under the assumption that the leaves are totally geodesic, the Laplacian of this function is computed along a leaf. From this computation a sufficient condition is provided for the ambient manifold to be locally isometric to a product.
On the radial limits of analytic and meromorphic functions
J. S.
Hwang
341-348
Abstract: Early in the fifties, A. J. Lohwater proved that if $f(z)$ is analytic in $\vert z\vert < 1$ and has the radial limit 0 almost everywhere on $\vert z\vert = 1$, then every complex number $ \zeta$ is an asymptotic value of $f(z)$ provided the $\zeta$-points satisfy the following Blaschke condition: $\sum _{k = 1}^\infty (1 - \vert{z_k}\vert) < \infty$, where $ f({z_k}) = \zeta$, $k = 1\,,2, \ldots$. We may, therefore, ask under the hypothesis on $f(z)$ how many complex numbers $\zeta$ are there whose $\zeta $-points can satisfy the Blaschke condition. We show that there is at most one such number and this one number phenomenon can actually occur if the number is zero.
Errata to: ``Linear spaces with an $H\sp{\ast} $-algebra-valued inner product'' [Trans. Amer. Math. Soc. {\bf 262} (1980), no. 2, 543--549; MR 81m:46080]
Parfeny P.
Saworotnow
349
On the homotopy index for infinite-dimensional semiflows
Krzysztof P.
Rybakowski
351-382
Abstract: In this paper we consider semiflows whose solution operator is eventually a conditional $\alpha$-contraction. Such semiflows include solutions of retarded and neutral functional differential equations, of parabolic and certain other classes of partial differential equations. We prove existence of (nonsmooth) isolating blocks and index pairs for such semiflows, via the construction of special Lyapunov functionals. We show that index pairs enjoy all the properties needed to define the notion of a homotopy index, thus generalizing earlier results of Conley [2]. Finally, using a result of Mañé [9], we prove that, under additional smoothness assumptions on the semiflow, the homotopy index is essentially a finite-dimensional concept. This gives a formal justification of the applicability of Ważewski's Principle to infinite-dimensional problems. Several examples illustrate the theory.
Finiteness theorems for approximate fibrations
D. S.
Coram;
P. F.
Duvall
383-394
Abstract: This paper concerns conditions on the point inverses of a mapping between manifolds which insure that it is an approximate fibration almost everywhere. The primary condition is $ {\pi _i}$-movability, which says roughly that nearby point inverses include isomorphically on the $i$th shape group into a mutual neighborhood. Suppose $f:{M^m} \to {N^n}$ is a $U{V^1}$ mapping which is $ {\pi _i}$-movable for $i \leqslant k - 1$, and $n \geqslant k + 1$. An earlier paper proved that $ f$ is an approximate fibration when $ m \leqslant 2k - 1$. If instead $m = 2k$, this paper proves that there is a locally finite set $ S \subset N$ such that $f\vert{f^{ - 1}}(N - S)$ is an approximate fibration. Also if $ m = 2k + 1$ and all of the point inverses are FANR's with the same shape, then there is a locally finite set $ E \subset N$ such that $f\vert{f^{ - 1}}(N - E)$ is an approximate fibration.
On an extension of localization theorem and generalized Conner conjecture
Satya
Deo;
Tej Bahadur
Singh;
Ram Anugrah
Shukla
395-402
Abstract: Let $G$ be a compact Lie group. Then Borel-Segal-Quillen-Hsiang localization theorems are known for any $G$-space $X$ where $X$ is any compact Hausdorff space or a paracompact Hausdorff space of finite cohomology dimension. The Conner conjecture proved by Oliver and its various generalizations by Skjelbred are also known for only these two classes of spaces. In this paper we extend all of these results for the equivariant category of all finitistic $G$-spaces. For the case when $G = {Z_p}$ or $G = T$ (torus) some of these results were already proved by Bredon.
The automorphism group of a composition of quadratic forms
C.
Riehm
403-414
Abstract: Let $U \times X \to X$ be a (bilinear) composition $(u,\,x) \mapsto ux$ of two quadratic spaces $U$ and $X$ over a field $F$ of characteristic $\ne 2$ and assume there is a vector in $ U$ which induces the identity map on $X$ via this composition. Define $G$ to be the subgroup of $O(U) \times O(X)$ consisting of those pairs $(\phi ,\,\psi )$ satisfying $\phi (u)\psi (x) = \psi (ux)$ identically and define ${G_X}$ to be the projection of $G$ on $O(X)$. The group $G$ is investigated and in particular it is shown that its connected component, as an algebraic group, is isogenous to a product of two or three classical groups and so is reductive. Necessary and sufficient conditions are given for ${G_X}$ to be transitive on the unit sphere of $ X$ when $U$ and $X$ are Euclidean spaces.
Weakly Ramsey $P$ points
Ned I.
Rosen
415-427
Abstract: If the continuum hypothesis (CH) holds, then for any $n$ Ramsey $P$ point $D$ and any $k \geqslant 1$ there exist many $n + k$ Ramsey $P$ points which are immediate Rudin-Keisler successors of $D$. There exist (CH) many 5 Ramsey $P$ points whose constellations are not linearly ordered.
Structure of complex linear differential equations
Margalit
Ronen
429-444
Abstract: In this paper homogeneous linear differential equations in the complex domain are considered. Relations between (a) properties of the zeros of solutions, (b) factorization of the equation into linear factors, and (c) nonvanishing of corresponding Wronskians are proved.
On the fullness of surjective maps of an interval
Harold
Proppe;
Abraham
Boyarsky
445-452
Abstract: Let $I = [0,\,1]$, $ \mathcal{B}$ = Lebesgue measurable subsets of $[0,\,1]$, and let $\lambda$ denote the Lebesgue measure on $(I,\,\mathcal{B})$. Let $\tau :I \to I$ be measurable and surjective. We say $\tau$ is full, if for all $A \in \mathcal{B}$, $ \lambda (A) > 0$, $ \tau (A),\,{\tau ^2}(A), \ldots$, measurable, the condition (1) $\displaystyle \mathop {\lim }\limits_{n \to \infty } \lambda ({\tau ^n}(A)) = 1$ holds. We say $\tau$ is interval full if (1) holds for any interval $ A \subset I$. In this note, we give an example of $ \tau :I \to I$ which is continuous and interval full, but not full. We also show that for a class of transformations $ \tau$ satisfying Renyi's condition, interval fullness implies fullness. Finally, we show that fullness is not preserved under limits on the surjections.
Probabilistic and deterministic averaging
N. H.
Bingham;
Charles M.
Goldie
453-480
Abstract: Let $\{ {S_n}\}$ be a random walk whose step distribution has positive mean $\mu$ and an absolutely continuous component. For any bounded measurable function $f$, a Marcinkiewicz-Zygmund strong law in an $r$-quick version (a 'Lai strong law') is proved for $ f({S_n})$, assuming existence of a suitable higher moment of the step distribution. This is extended to show ${n^{ - \alpha }}\{ \sum\nolimits_1^n {f({S_k})} - \int_0^n {f(\mu t)dt\} \to 0}$ ($r$-quickly). These results remain true when the step distribution is lattice, provided $ f$ is constant between lattice points. Certain intermediate results on renewal theory, mixing, local limit theory, ladder height, and a strong law of Lai for mixing random variables are of independent interest.
Focal sets and real hypersurfaces in complex projective space
Thomas E.
Cecil;
Patrick J.
Ryan
481-499
Abstract: Let $M$ be a real submanifold of $ C{P^m}$, and let $ J$ denote the complex structure. We begin by finding a formula for the location of the focal points of $M$ in terms of its second fundamental form. This takes a particularly tractable form when $M$ is a complex submanifold or a real hypersurface on which $J\xi$ is a principal vector for each unit normal $ \xi$ to $M$. The rank of the focal map onto a sheet of the focal set of $M$ is also computed in terms of the second fundamental form. In the case of a real hypersurface on which $ J\xi$ is principal with corresponding principal curvature $\mu$, if the map onto a sheet of the focal set corresponding to $\mu$ has constant rank, then that sheet is a complex submanifold over which $M$ is a tube of constant radius (Theorem 1). The other sheets of the focal set of such a hypersurface are given a real manifold structure in Theorem 2. These results are then employed as major tools in obtaining two classifications of real hypersurfaces in $ C{P^m}$. First, there are no totally umbilic real hypersurfaces in $ C{P^m}$, but we show: Theorem 3. Let $M$ be a connected real hypersurface in $ C{P^m}$, $m \geqslant 3$, with at most two distinct principal curvatures at each point. Then $M$ is an open subset of a geodesic hypersphere. Secondly, we show that there are no Einstein real hypersurfaces in $C{P^m}$ and characterize the geodesic hyperspheres and two other classes of hypersurfaces in terms of a slightly less stringent requirement on the Ricci tensor in Theorem 4.
Ramsey numbers for the pair sparse graph-path or cycle
S. A.
Burr;
P.
Erdős;
R. J.
Faudree;
C. C.
Rousseau;
R. H.
Schelp
501-512
Abstract: Let $G$ be a connected graph on $ n$ vertices with no more than $ n(1 + \varepsilon )$ edges, and ${P_k}$ or ${C_k}$ a path or cycle with $k$ vertices. In this paper we will show that if $n$ is sufficiently large and $\varepsilon$ is sufficiently small then for $ k$ odd $\displaystyle r(G,\,{C_k}) = 2n - 1.$ Also, for $k \geqslant 2$, $\displaystyle r(G,\,{P_k}) = \max \{ n + [k/2] - 1,\,n + k - 2 - \alpha \prime - \delta \} ,$ where $\alpha \prime$ is the independence number of an appropriate subgraph of $G$ and $\delta$ is 0 or $1$ depending upon $n$, $k$ and $\alpha \prime $.
Picard's theorem
Douglas
Bridges;
Allan
Calder;
William
Julian;
Ray
Mines;
Fred
Richman
513-520
Abstract: This paper deals with the numerical content of Picard's Theorem. Two classically equivalent versions of this theorem are proved which are distinct from a computational point of view. The proofs are elementary, and constructive in the sense of Bishop. A Brouwerian counterexample is given to the original version of the theorem.
A Loeb-measure approach to theorems by Prohorov, Sazonov and Gross
Tom L.
Lindstrøm
521-534
Abstract: We use the Loeb-measure of nonstandard analysis to prove three classical results on limit measures: Let ${\{ {\mu _i}\} _{i \in I}}$ be a projective system of Radon measures, we use the Loeb-measure $L({\tilde \mu _E})$ for an infinite $E \in {}^{\ast}I$ and a standard part map to construct a Radon limit measure on the projective limit (Prohorov's Theorem). Using the Loeb-measures on hyperfinite dimensional linear spaces, we characterize the Fourier-transforms of measures on Hilbert spaces (Sazonov's Theorem), and extend cylindrical measures on Hilbert spaces to $\sigma$-additive measures on Banach spaces (Gross' Theorem).
Geometry and the Pettis integral
Robert F.
Geitz
535-548
Abstract: Convex sets involving the range of a vector-valued function are constructed. These constructions provide a complete characterization of the bounded Pettis integrable functions.
Continuous measures and lacunarity on hypergroups
Richard C.
Vrem
549-556
Abstract: The relationship between measures on a compact hypergroup $K$ whose Fourier-Stieltjes transforms vanish at infinity and the space ${M_c}(K)$ of continuous measures is studied. Examples are provided of measures $\mu$ with $\hat \mu$ vanishing at infinity and $\mu \in {M_c}(K)$. Sufficient conditions are given for $ \hat \mu \in {c_0}(\hat K)$ to imply $ \mu \in {M_c}(K)$. An investigation of Helson sets on compact abelian hypergroups is initiated and the study of Sidon sets on compact abelian hypergroups is continued. A class of compact abelian hypergroups is shown to have no infinite Helson sets and no infinite Sidon sets. This result generalizes results of D. L. Ragozin and D. Rider on central Sidon sets for compact connected Lie groups.
Weak $P$-points in compact CCC $F$-spaces
Alan
Dow
557-565
Abstract: Using a technique due to van Mill we show that each compact ccc $ F$-space of weight greater than ${2^\omega }$ contains a weak $ P$-point, i.e. a point $ x \in X$ such that $x \notin \overline F$ for each countable $ F \subset X - \{ x\}$. We show that, assuming $BF(c)$, each nowhere separable compact $ F$-space has a weak $ P$-point. We show the existence of points which are not limit points of any countable nowhere dense set in compact $F$-spaces of weight ${\aleph _1}$. We also discuss remote points and points not the limit point of any countable discrete set.
An alternating sum formula for multiplicities in $L\sp{2}(\Gamma \backslash G)$
Roberto J.
Miatello
567-574
Abstract: We prove an alternating sum formula for multiplicities in ${L^2}(\Gamma \backslash G)$, where $ G$ is a semisimple Lie group of split rank one with finite center and $ \Gamma$ is a discrete cocompact torsion free subgroup.
Collections of subsets with the Sperner property
Jerrold R.
Griggs
575-591
Abstract: Let $X = \{ 1, \ldots ,n\}$ and $Y = \{ 1, \ldots ,k\}$, $k \leqslant n$. Let $C(n,\,k)$ be the subsets of $X$ which intersect $Y$, ordered by inclusion. Lih showed that $ C(n\,,k)$ has the Sperner property. Here it is shown that $C(n,\,k)$ has several stronger properties. A nested chain decomposition is constructed for $C(n,\,k)$ by bracketing. $C(n,\,k)$ is shown to have the LYM property. A more general class of collections of subsets is studied: Let $X$ be partitioned into parts ${X_1}, \ldots ,{X_m}$, let ${I_1}, \ldots ,{I_m}$ be subsets of $\{ 0,\,1, \ldots ,\,n\} $, and let $P = \{ Z \subset X\vert\vert Z \cap {X_i}\vert\, \in {I_i},\,1 \leqslant i \leqslant m\}$. Sufficient conditions on the ${I_i}$ are given for $P$ to be LYM, or at least Sperner, and examples are provided in which $P$ is not Sperner. Other results related to Sperner's theorem, the Kruskal-Katona theorem, and the LYM inequality are presented.
A generalization of Torres' second relation
Lorenzo
Traldi
593-610
Abstract: Let $L = {K_1} \cup \cdots \cup {K_\mu }$ be a tame link in ${S^3}$ of $ \mu \geqslant 2$ components, and let ${L_\mu }$ be its sublink ${L_\mu } = L - {K_\mu }$. Let $H$ and ${H_\mu }$ be the abelianizations of ${\pi _1}({S^3} - L)$ and ${\pi _1}({S^3} - {L_\mu })$, respectively, and let $ {t_1}, \ldots ,{t_\mu }$ (resp., $ {t_1}, \ldots ,{t_{\mu - 1}}$) be the usual generators of $H$ (resp., ${H_\mu }$). If $\phi :{\mathbf{Z}}H \to {\mathbf{Z}}{H_\mu }$ is the (unique) ring homomorphism with $\phi ({t_i}) = {t_i}$ for $1 \leqslant i < \mu $, and $\phi ({t_\mu }) = 1$, then Torres' second relation is equivalent to the statement that $\phi {E_1}(L) = (({\prod _{i < \mu }}t_i^{{l_i}}) - 1) \cdot {E_1}({L_\mu })$, where for $1 \leqslant i < \mu $, ${l_i}$ is the linking number ${l_i} = l({K_i},\,{K_\mu })$. We prove that if $I{H_\mu }$ is the augmentation ideal of $ {\mathbf{Z}}{H_\mu }$, then for any $ k \geqslant 2$, $\displaystyle {E_{k - 1}}({L_\mu }) + \left( {\left( {\prod\limits_{i < \mu } {... ...q \phi {E_k}(L) \subseteq {E_{k - 1}}({L_\mu }) + I{H_\mu }\cdot{E_k}({L_\mu })$ and examples are given to indicate that either of these inclusions may be an equality. This theorem is used to generalize certain known properties of ${E_1}$ to the higher ideals.
Products of two Borel measures
Roy A.
Johnson
611-625
Abstract: Let $\mu$ and $\nu$ be finite Borel measures on Hausdorff spaces $ X$ and $Y$, respectively, and suppose product measures $ \mu \times {}_1\nu$ and $\mu \times {}_2\nu$ are defined on the Borel sets of $X \times Y$ by integrating vertical and horizontal cross-section measure, respectively. Sufficient conditions are given so that $\mu \times {}_1\nu = \mu \times {}_2\nu$ and so that the usual product measure $\mu \times \nu$ can be extended to a Borel measure on $X \times Y$ by means of completion. Examples are given to illustrate these ideas.
Classes of Baire functions
Gregory V.
Cox;
Paul D.
Humke
627-635
Abstract: Let $\mathcal{A}$ and $ \mathcal{P}$ denote the sets of approximately continuous and almost everywhere continuous functions, and ${B_1}(F)$ denote Baire's first class generated by $F$. The classes ${B_1}(\mathcal{A})$, ${B_1}(\mathcal{P})$, ${B_1}(\mathcal{A} \cap \mathcal{P})$, and Grande's class $ \mathcal{A}{\mathcal{P}_1}$ are investigated in some detail. Although Grande's question of whether ${B_1}(\mathcal{A} \cap \mathcal{P}) = {B_1}(\mathcal{A}) \cap {B_1}(\mathcal{A}) \cap \mathcal{A}{\mathcal{P}_1}$ is not settled, we do show, among other results, that $ \mathcal{A}{\mathcal{P}_1} \subset {B_1}(\mathcal{P})$.
Nonexponential leaves at finite level
John
Cantwell;
Lawrence
Conlon
637-661
Abstract: Previous examples of leaves with nonexponential and nonpolynomial growth (due to G. Hector) have occurred at infinite level. Here the same growth types are produced at finite level in open, saturated sets of leaves without holonomy. Such sets consist of leaves with only one or two locally dense ends, and it is shown that the exotic growth types only occur in the case of one locally dense end. Finally, ${C^1}$-foliations are produced with open, saturated sets as above in which the leaves have strictly fractional growth.
The semicellularity theorem
Gene G.
Garza
663-676
Abstract: In this paper are proved several theorems concerning semicellularity of subsets of $2$-spheres in ${E^3}$. In particular, it is shown that a cellular arc or disk on a $2$-sphere which has no nonpiercing points is semicellular in both complementary domains of the $ 2$-sphere. The proof is entirely geometrical and involves the idea of piercing points.
A class of $L\sp{1}$-convergence
R.
Bojanić;
Č. V.
Stanojević
677-683
Abstract: It is proved that if the Fourier coefficients $ \{ {a_n}\}$ of $f \in {L^1}(0,\,\pi )$ satisfy $({\ast}){n^{ - 1}}\sum\nolimits_{k = n}^{2n} {{k^p}\vert\Delta {a_n}\vert p = o(1)}$, for some $1 < p \leqslant 2$, then $\vert\vert{s_n} - f\vert\vert = o(1)$, if and only if $ {a_n}\lg n = o(1)$. For cosine trigonometric series with coefficients of bounded variation and satisfying $({\ast})$ it is proved that a necessary and sufficient condition for the series to be a Fourier series is $ \{ {a_n}\} \in \mathcal{C}$, where $ \mathcal{C}$ is the Garrett-Stanojević [4] class.
Geometric transfer and the homotopy type of the automorphism groups of a manifold
D.
Burghelea;
R.
Lashof
1-38
Abstract: Lifting concordances (pseudo-isotopies) in a smooth fibre bundle gives a transfer of stable concordance groups. Properties of the transfer are proved and exploited to obtain the homotopy structure of the group of diffeomorphisms or homeomorphisms of a manifold in a stable range.
Lipschitz spaces on stratified groups
Steven G.
Krantz
39-66
Abstract: Let $G$ be a connected, simply connected nilpotent Lie group. Call $G$ stratified if its Lie algebra $\mathfrak{g}$ has a direct sum decomposition $\mathfrak{g} = {V_1} \oplus \cdots \oplus {V_m}$ with $[{V_i},{V_j}] = {V_{i + j}}$ for $i + j \leqslant m$, $[{V_{i,}}{V_j}] = 0$ for $i + j > m$. Let $\{ {X_1}, \ldots ,{X_n}\} $ be a vector space basis for ${V_1}$. Let $f \in C(G)$ satisfy $\vert\vert f(g\exp {X_i} \cdot )\vert\vert \in {\Lambda _\alpha }({\mathbf{R}})$, uniformly in $g \in G$, where ${\Lambda _\alpha }$ is the usual Lipschitz space and $0 < \alpha < \infty$. It is proved that, under these circumstances, it holds that $ f \in {\Gamma _\alpha }(G)$ where $ {\Gamma _\alpha }$ is the nonisotropic Lipschitz space of Folland. Applications of this result to interpolation theory, hypoelliptic partial differential equations, and function theory are provided.
Almost sure invariance principles for sums of $B$-valued random variables with applications to random Fourier series and the empirical characteristic process
Michael B.
Marcus;
Walter
Philipp
67-90
Abstract: We establish an almost sure approximation of the partial sums of independent, identically distributed random variables with values in a separable Banach space $B$ by a suitable $B$-valued Brownian motion under the hypothesis that the partial sums can be ${L^1}$-closely approximated by finite-dimensional random variables. We show that this hypothesis is satisfied if the given random variables are random Fourier series or related stochastic processes. As an application we obtain an almost sure approximation of the empirical characteristic process by a suitable $ {\mathbf{C}}(K)$-valued Brownian motion whenever the empirical characteristic process satisfies the central limit theorem.
Weighted Sobolev spaces and pseudodifferential operators with smooth symbols
Nicholas
Miller
91-109
Abstract: Let ${u^\char93 }$ be the Fefferman-Stein sharp function of $u$, and for $1 < r < \infty$, let $ {M_r}u$ be an appropriate version of the Hardy-Littlewood maximal function of $u$. If $A$ is a (not necessarily homogeneous) pseudodifferential operator of order 0, then there is a constant $ c > 0$ such that the pointwise estimate ${(Au)^\char93 }(x) \leqslant c{M_r}u(x)$ holds for all $x \in {R^n}$ and all Schwartz functions $ u$. This estimate implies the boundedness of 0-order pseudodifferential operators on weighted ${L^p}$ spaces whenever the weight function belongs to Muckenhoupt's class ${A_p}$. Having established this, we construct weighted Sobolev spaces of fractional order in $ {R^n}$ and on a compact manifold, prove a version of Sobolev's theorem, and exhibit coercive weighted estimates for elliptic pseudodifferential operators.
Canonical semi-invariants and the Plancherel formula for parabolic groups
Ronald L.
Lipsman;
Joseph A.
Wolf
111-131
Abstract: A parabolic subgroup of a reductive Lie group is called "good" if the center of the universal enveloping algebra of its nilradical contains an element that is semi-invariant of weight proportional to the modular function. The "good" case is characterized here by invariance of the set of simple roots defining the parabolic, under the negative of the opposition element of the Weyl group. In the "good" case, the unbounded Dixmier-Pukanszky operator of the parabolic subgroup is described, the conditions under which it is a differential operator rather than just a pseudodifferential operator are specified, and an explicit Plancherel formula is derived for that parabolic.
Dirac quantum fields on a manifold
J.
Dimock
133-147
Abstract: On globally hyperbolic Lorentzian manifolds we construct field operators which satisfy the Dirac equation and have a causal anticommutator. Ambiguities in the construction are removed by formulating the theory in terms of ${C^{\ast}}$ algebras of local observables. A generalized form of the Haag-Kastler axioms is verified.
On the monodromy at isolated singularities of weighted homogeneous polynomials
Benjamin G.
Cooper
149-166
Abstract: Assume $ f:{{\mathbf{C}}^m} \to {\mathbf{C}}$ is a weighted homogeneous polynomial with isolated singularity, and define $\phi :{S^{2m - 1}} - {f^{ - 1}}(0) \to {S^1}$ by $\phi (\overrightarrow z ) = f(\overrightarrow z ) / \vert f(\overrightarrow z )\vert$. If the monomials of $f$ are algebraically independent, then the closure ${\overline F _0}$ of ${\phi ^{ - 1}}(1)$ in $ {S^{2m - 1}}$ admits a deformation into the subset $G$ where each monomial of $f$ has nonnegative real values. For the polynomial $f({z_1}, \ldots ,{z_m}) = z_1^{{a_1}}{z_2} + \cdots + z_{m - 1}^{{a_{m - 1}}}{z_m} + z_m^{{a_m}}{z_1}$, $ G$ is a cell complex of dimension $m - 1$, invariant under a characteristic map $ h$ of the fibration $ \phi$, and the inclusion $G \to {F_0}$ induces isomorphisms in homology. To compute the homology of the link $K = {f^{ - 1}}(0) \cap {S^{2m - 1}}$ it thus suffices to calculate the action of ${h_{\ast}}$ on $ {H_{m - 1}}(G)$. Let $d = {a_1}{a_2} \cdots {a_m} + {( - 1)^{m - 1}}$. Let ${w_1},\,{w_2}, \ldots ,{w_m}$ be the weights associated with $f$, satisfying ${a_j} / {w_j} + 1 / {w_{j + 1}} = 1$ for $j = 1,\,2, \ldots ,\,m - 1$ and ${a_m}/{w_m} + 1/{w_1} = 1$. Let $n = d/{w_1}$, $ q = \gcd (n,\,d)$, $r = q + {( - 1)^m}$. Then ${H_{m - 2}}(K) = {Z^r} \oplus {z_{d/q}}$ and $ {H_{m - 1}}(K) = {Z^r}$.
Full continuous embeddings of toposes
M.
Makkai
167-196
Abstract: Some years ago, G. Reyes and the author described a theory relating first order logic and (Grothendieck) toposes. This theory, together with standard results and methods of model theory, is applied in the present paper to give positive and negative results concerning the existence of certain kinds of embeddings of toposes. A new class, that of prime-generated toposes is introduced; this class includes M. Barr's regular epimorphism sheaf toposes as well as the so-called atomic toposes introduced by M. Barr and R. Diaconescu. The main result of the paper says that every coherent prime-generated topos can be fully and continuously embedded in a functor category. This result generalizes M. Barr's full exact embedding theorem. The proof, even when specialized to Barr's context, is essentially different from Barr's original proof. A simplified and sharpened form of Barr's proof of his theorem is also described. An example due to J. Malitz is adapted to show that a connected atomic topos may have no points at all; this shows that some coherence assumption in our main result is essential.
Coextensions of regular semigroups by rectangular bands. I
John
Meakin;
K. S. S.
Nambooripad
197-224
Abstract: This paper initiates a general study of the structure of a regular semigroup $S$ via the maximum congruence $\rho$ on $S$ with the property that each $\rho$-class $e\rho$, for $e = {e^2} \in S$, is a rectangular subband of $S$. Congruences of this type are studied and the maximum such congruence is characterized. A construction of all biordered sets which are coextensions of an arbitrary biordered set by rectangular biordered sets is provided and this is specialized to provide a construction of all solid biordered sets. These results are used to construct all regular idempotent-generated semigroups which are coextensions of a regular idempotent-generated semigroup by rectangular bands: a construction of normal coextensions of biordered sets is also provided.
Ideal theory in $f$-algebras
C. B.
Huijsmans;
B.
de Pagter
225-245
Abstract: The paper deals mainly with the theory of algebra ideals and order ideals in $f$-algebras. Necessary and sufficient conditions are established for an algebra ideal to be prime, semiprime or idempotent. In a uniformly complete $ f$-algebra with unit element every algebra ideal is an order ideal iff the $ f$-algebra is normal. This result is based on the fact that the range of every orthomorphism in a uniformly complete normal Riesz space is an order ideal.
Attractors: persistence, and density of their basins
Mike
Hurley
247-271
Abstract: An investigation of qualitative features of flows on manifolds, in terms of their attractors and quasi-attractors. A quasi-attractor is any nonempty intersection of attractors. It is shown that quasi-attractors other than attractors occur for a large set of flows. It is also shown that for a generic flow (for each flow in a residual subset of the set of all flows), each attractor "persists" as an attractor of all nearby flows. Similar statements are shown to hold with "quasi-attractor", "chain transitive attractor", and "chain transitive quasi-attractor" in place of "attractor". Finally, the set of flows under which almost all points tend asymptotically to a chain transitive quasi-attractor is characterized in terms of stable sets of invariant sets.
A two-cardinal theorem for homogeneous sets and the elimination of Malitz quantifiers
Philipp
Rothmaler;
Peter
Tuschik
273-283
Abstract: Sufficient conditions for the eliminability of Malitz quantifiers in a complete first order theory are given. Proving that certain superstable and not $\omega$-stable theories satisfy these conditions, a question of Baldwin and Kueker is answered negatively.
An improvement of the Poincar\'e-Birkhoff fixed point theorem
Patricia H.
Carter
285-299
Abstract: If $g$ is a twist homeomorphism of an annulus $ A$ in the plane which leaves at most one point in the interior of $A$ fixed, then there is an essential simple closed curve in the interior of $A$ which meets its image in at most one point; hence the annular region bounded by this simple closed curve and the inside component of the boundary of $ A$ is mapped onto either a proper subset or a proper superset of itself.
Curvature operators and characteristic classes
Irl
Bivens
301-310
Abstract: Given tensors $ A$ and $B$ of type $(k,\,k)$ on a Riemannian manifold $M$ we construct in a natural way a $ 2k$ form ${F_k}(A,\,B)$. If $A$ and $B$ satisfy the generalized Codazzi equations then this $2k$ form is closed. In particular if ${R_{2k}}$ denotes the $2k$th curvature operator then $ {F_{2k}}({R_{2k,\,}}{R_{2k}})$ is (up to a constant multiple) the $k$th Pontrjagin class of $ M$. By means of a theorem of Gilkey we give conditions sufficient to guarantee that a form constructed from more complicated expressions involving the curvature operators does in fact belong to the Pontrjagin algebra. As a corollary we obtain Thorpe's vanishing theorem for manifolds with constant $ 2p$th sectional curvature. If at each point in $M$ the tangent space contains a subspace of a particular type (similar to curvature nullity) we show that certain Pontrjagin classes must vanish. We generalize the result that submanifolds of Euclidean space with flat normal bundle have a trivial Pontrjagin algebra. The curvature operator, ${R_2}$, is interesting in that the components of ${R_2}$ with respect to any orthonormal frame are given by certain universal (independent of frame) homogeneous linear polynomials in the components of the curvature tensor. We characterize all such operators and using this characterization derive in a natural way the Weyl component of ${R_2}$.
Finite sublattices of a free lattice
J. B.
Nation
311-337
Abstract: Every finite semidistributive lattice satisfying Whitman's condition is isomorphic to a sublattice of a free lattice.
On the critical degree of differentiability of a complex planar curve
Joseph
Becker
339-350
Abstract: An example of a pair of complex analytic curves in ${{\mathbf{C}}^2}$ is given which have the same characteristic pairs but which do not have the same critical degree of differentiability.
Differentiable group actions on homotopy spheres. II. Ultrasemifree actions
Reinhard
Schultz
255-297
Abstract: A conceptually simple but very useful class of topological or differentiable transformation groups is given by semifree actions, for which the group acts freely off the fixed point set. In this paper, the slightly more general notion of an ultrasemifree action is introduced, and it is shown that the existing machinery for studying semifree actions on spheres may be adapted to study ultrasemifree actions equally well. Some examples and applications are given to illustrate how ultrasemifree actions (i) may be used to study questions not answerable using semifree actions alone, and (ii) provide examples of unusual smooth group actions on spheres with no semifree counterparts.
A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves
Walter D.
Neumann
299-344
Abstract: Any graph-manifold can be obtained by plumbing according to some plumbing graph $\Gamma$. A calculus for plumbing which includes normal forms for such graphs is developed. This is applied to answer several questions about the topology of normal complex surface singularities and analytic families of complex curves. For instance it is shown that the topology of the minimal resolution of a normal complex surface singularity is determined by the link of the singularity and even by its fundamental group if the singularity is not a cyclic quotient singularity or a cusp singularity.
Les mod\`eles d\'enombrables d'une th\'eorie ayant des fonctions de Skolem
Daniel
Lascar
345-366
Abstract: Let $T$ be a countable complete theory having Skolem functions. We prove that if all the types over finitely generated models are definable (this is the case for example if $T$ is stable), then either $T$ has $ {2^{{\aleph _0}}}$ countable models or all its models are homogeneous. The proof makes heavy use of stability techniques.
On the weak behaviour of partial sums of Legendre series
S.
Chanillo
367-376
Abstract: We show that the partial sum operator associated with the Legendre series is restricted weak type, but not weak type, on the ${L^p}$ spaces when $p = 4$.
Involutions on Klein spaces $M(p,\,q)$
Paik Kee
Kim
377-409
Abstract: The Klein spaces $ M(p,\,q)$ are defined (up to homeomorphisms) to be the class of closed, orientable, irreducible $3$-manifolds with finite fundamental groups, in which a Klein bottle can be embedded. Their fundamental groups act freely on the $3$-sphere ${S^3}$ in the natural way. We obtain a complete classification of the PL involutions on Klein spaces $ M(p,\,q)$. It can be applied to the study of some transformation group actions on ${S^3}$ and double branched coverings of $ {S^3}$.
Geodesic rigidity in compact nonpositively curved manifolds
Patrick
Eberlein
411-443
Abstract: Our goal is to find geometric properties that are shared by homotopically equivalent compact Riemannian manifolds of sectional curvature $K \leqslant 0$. In this paper we consider mainly properties of free homotopy classes of closed curves. Each free homotopy class is represented by at least one smooth periodic geodesic, and the nonpositive curvature condition implies that any two periodic geodesic representatives are connected by a flat totally geodesic homotopy of periodic geodesic representatives. By imposing certain geometric conditions on these periodic geodesic representatives we define and study three types of free homotopy classes: Clifford, bounded and rank $1$. Let $M$, $M\prime$ denote compact Riemannian manifolds with $K \leqslant 0$, and let $ \theta :{\pi _1}(M,\,m) \to {\pi _1}(M\prime ,\,m\prime )$ be an isomorphism. Let $ \theta$ also denote the induced bijection on free homotopy classes. Theorem A. The free homotopy class $[\alpha ]$ in $M$ is, respectively, Clifford, bounded or rank $1$ if and only if the class $\theta [\alpha ]$ in $M\prime$ is of the same type. Theorem B. If $M$, $M\prime$ have dimension $3$ and do not have a rank $ 1$ free homotopy class then they have diffeomorphic finite covers of the form ${S^1} \times {M^2}$. The proofs of Theorems A and B use the fact that $\theta$ is induced by a homotopy equivalence $f:(M,\,m) \to (M\prime ,\,m\prime )$. Theorem C. The manifold $M$ satisfies the Visibility axiom if and only if $M\prime$ satisfies the Visibility axiom.
Quasicoherent sheaves over affine Hensel schemes
Silvio
Greco;
Rosario
Strano
445-465
Abstract: The following two theorems concerning affine Hensel schemes are proved. Theorem A. Every quasi-coherent sheaf over an affine Hensel scheme is generated by its global sections. Theorem B. ${H^p}(X,\,F) = 0$ for all positive $ p$ and all quasi-coherent sheaves $F$ over an affine Hensel scheme $ X$.
Fundamental solutions for differential equations associated with the number operator
Yuh Jia
Lee
467-476
Abstract: Let $(H,\,B)$ be an abstract Wiener space. If $ u$ is a twice $ H$-differentiable function on $B$ such that $ Du(x) \in {B^{\ast}}$ and $ {D^2}u(x)$ is of trace class, then we define $\mathfrak{N}u(x) = - \Delta u(x) + (x,\,Du(x))$, where $\Delta u(x) = {\operatorname{trace} _H}\,{D^2}u(x)$ is the Laplacian and $( \cdot ,\, \cdot )$ denotes the $B$- $ {B^{\ast}}$ pairing. The closure $\overline{\mathfrak{N}} $ of $\mathfrak{N}$ is known as the number operator. In this paper, we investigate the existence, uniqueness and regularity of solutions for the following two types of equations: (1) ${u_t} = - \mathfrak{N}u$ (initial value problem) and (2) ${\mathfrak{N}^k}u = f(k \geqslant 1)$. We show that the fundamental solutions of (1) and (2) exist in the sense of measures and we represent their solutions by integrals with respect to these measures.
Probability and interpolation
G. G.
Lorentz;
R. A.
Lorentz
477-486
Abstract: An $m \times n$ matrix $E$ with $n$ ones and $(m - 1)n$ zeros, which satisfies the Pólya condition, may be regular and singular for Birkhoff interpolation. We prove that for random distributed ones, $ E$ is singular with probability that converges to one if $m$, $n \to \infty $. Previously, this was known only if $m \geqslant (1 + \delta )n/\log n$. For constant $ m$ and $n \to \infty$, the probability is asymptotically at least $ \tfrac{1} {2}$.
The Dror-Whitehead theorem in prohomotopy and shape theories
S.
Singh
487-496
Abstract: Many analogues of the classical Whitehead theorem from homotopy theory are now available in pro-homotopy and shape theories. E. Dror has significantly extended the homology version of the Whitehead theorem from the well-known simply connected case to the more general, for instance, nilpotent case. We prove a full analogue of Dror's theorems in pro-homotopy and shape theories. More specifically, suppose $ \underline f :\underline X \to \underline Y$ is a morphism in the pro-homotopy category of pointed and connected topological spaces which induces isomorphisms of the integral homology pro-groups. Then $ \underline f$ induces isomorphisms of the homotopy pro-groups, for instance, when $\underline X$ and $\underline Y$ are simple, nilpotent, complete, or $\underline H $-objects; these notions are well known in homotopy theory and we have naturally extended them to pro-homotopy and shape theories.
Vortex rings: existence and asymptotic estimates
Avner
Friedman;
Bruce
Turkington
1-37
Abstract: The existence of a family of steady vortex rings is established by a variational principle. Further, the asymptotic behavior of the solutions is obtained for limiting values of an appropriate parameter $\lambda$; as $ \lambda \to \infty$ the vortex ring tends to a torus whose cross-section is an infinitesimal disc.
The stable geometric dimension of vector bundles over real projective spaces
Donald M.
Davis;
Sam
Gitler;
Mark
Mahowald
39-61
Abstract: An elementary argument shows that the geometric dimension of any vector bundle of order ${2^e}$ over $R{P^n}$ depends only on $e$ and the residue of $ n\,\bmod \,8$ for $ n$ sufficiently large. In this paper we calculate this geometric dimension, which is approximately $2e$. The nonlifting results are easily obtained using the spectrum $bJ$. The lifting results require $ bo$-resolutions. Half of the paper is devoted to proving Mahowald's theorem that beginning with the second stage $ bo$-resolutions act almost like $ K({Z_2})$-resolutions.
An effective version of Dilworth's theorem
Henry A.
Kierstead
63-77
Abstract: We prove that if $(P,\,{ < ^P})$ is a recursive partial order with finite width $w$, then $P$ can be covered by $ ({5^w} - 1)/4$ recursive chains. For each $w$ we show that there is a recursive partial ordering of width $w$ that cannot be covered by $4(w - 1)$ recursive chains.
Geometric properties of homogeneous vector fields of degree two in ${\bf R}\sp{3}$
M. Izabel T.
Camacho
79-101
Abstract: In the space of homogeneous polynomial vector fields of degree two, those that project on Morse-Smale vector fields on $ {S^2}$ by the Poincaré central projection form a generic subset. The classification of those vector fields on ${S^2}$ without periodic orbits is given and applications to the study of local actions of the affine group of the line are derived.
Brouwerian semilattices
Peter
Köhler
103-126
Abstract: Let ${\mathbf{P}}$ be the category whose objects are posets and whose morphisms are partial mappings $\alpha :P \to Q$ satisfying (i) $ \forall p,\,q \in \operatorname{dom} \alpha [p < q \Rightarrow \alpha (p) < \alpha (q)]$ and (ii) $\forall p \in \operatorname{dom} \alpha \forall q \in Q[q < \alpha (p) \Rightarrow \exists r \in \operatorname{dom} \alpha [r < p\& \alpha (r) = q]]$. The full subcategory ${{\mathbf{P}}_f}$ of $ {\mathbf{P}}$ consisting of all finite posets is shown to be dually equivalent to the category of finite Brouwerian semilattices and homomorphisms. Under this duality a finite Brouwerian semilattice $ \underline A$ corresponds with $ M(\underline A )$, the poset of all meet-irreducible elements of $\underline A$. The product (in ${{\mathbf{P}}_f}$) of $n$ copies $ (n \in \mathbb{N})$ of a one-element poset is constructed; in view of the duality this product is isomorphic to the poset of meet-irreducible elements of the free Brouwerian semilattice on $n$ generators. If $ {\mathbf{V}}$ is a variety of Brouwerian semilattices and if $\underline A$ is a Brouwerian semilattice, then $\underline A $ is ${\mathbf{V}}$-critical if all proper subalgebras of $\underline A $ belong to ${\mathbf{V}}$ but not $ \underline A$. It is shown that a variety $ {\mathbf{V}}$ of Brouwerian semilattices has a finite equational base if and only if there are up to isomorphism only finitely many $ {\mathbf{V}}$-critical Brouwerian semilattices. This is used to show that a variety generated by a finite Brouwerian semilattice as well as the join of two finitely based varieties is finitely based. A new example of a variety without a finite equational base is exhibited.
Preservation of convergence of convex sets and functions in finite dimensions
L.
McLinden;
Roy C.
Bergstrom
127-142
Abstract: We study a convergence notion which has particular relevance for convex analysis and lends itself quite naturally to successive approximation schemes in a variety of areas. Motivated particularly by problems in optimization subject to constraints, we develop technical tools necessary for systematic use of this convergence in finite-dimensional settings. Simple conditions are established under which this convergence for sequences of sets, functions and subdifferentials is preserved under various basic operations, including, for example, those of addition and infimal convolution in the case of functions.
A partition relation for Souslin trees
Attila
Máté
143-149
Abstract: The aim of these notes is to give a direct proof of the partition relation Souslin tree $ \to (\alpha )_k^2$, valid for any integer $k$ and any ordinal $\alpha < {\omega _1}$. This relation was established by J. E. Baumgartner, who noticed that it follows by a simple forcing and absoluteness argument from the relation ${\omega _1} \to (\alpha )_k^2$, which is a special case of a theorem of Baumgartner and A. Hajnal.
Uniform partitions of an interval
Vladimir
Drobot
151-160
Abstract: Let $\{ {x_n}\}$ be a sequence of numbers in $ [0,\,1]$; for each $ n$ let ${u_0}(n), \ldots ,\,{u_n}(n)$ be the lengths of the intervals resulting from partitioning of $ [0,\,1]$ by $ \{ {x_1},\,{x_2}, \ldots ,\,{x_n}\}$. For $p > 1$ put $ {A^{(p)}}(n) = {(n + 1)^{p - 1}}\sum\nolimits_0^n {{{[{u_j}(n)]}^p}}$; the paper investigates the behavior of ${A^{(p)}}(n)$ as $ n \to \infty$ for various sequences $\{ {x_n}\}$. Theorem 1. If ${x_n} = n\theta \,(\bmod \,1)$ for an irrational $ \theta > 0$, then $\lim \,\inf \,{A^{(p)}}(n) < \infty$. However $ \lim \,\sup \,{A^{(p)}} < \infty$ if and only if the partial quotients of $\theta$ are bounded (in the continued fraction expansion of $\theta$). Theorem 2 gives the exact values for $\lim \,\inf$ and $\lim \,\sup$ when $\theta = \tfrac{1} {2}(1 + \sqrt 5 )$. Theorem 3. If $ \lim \,{A^{(p)}}(n) = \Gamma (p + 1)$ almost surely.
Singular sets and remainders
George L.
Cain;
Richard E.
Chandler;
Gary D.
Faulkner
161-171
Abstract: This paper characterizes the singular sets of several traditional classes of continuous mappings associated with compactifications. By relating remainders of compactifications to singular sets of mappings with compact range, new results are obtained about each.
Facial characterizations of complex Lindenstrauss spaces
A. J.
Ellis;
T. S. S. R. K.
Rao;
A. K.
Roy;
U.
Uttersrud
173-186
Abstract: We characterize complex Banach spaces $A$ whose Banach dual spaces are ${L^1}(\mu )$ spaces in terms of $L$-ideals generated by certain extremal subsets of the closed unit ball $K$ of $ {A^{\ast}}$. Our treatment covers the case of spaces $A$ containing constant functions and also spaces not containing constants. Separable spaces are characterized in terms of $ {w^{\ast}}$-compact sets of extreme points of $K$, whereas the nonseparable spaces necessitate usage of the $ {w^{\ast}}$-closed faces of $K$. Our results represent natural extensions of known characterizations of Choquet simplexes. We obtain also a characterization of complex Lindenstrauss spaces in terms of boundary annihilating measures, and this leads to a characterization of the closed subalgebras of $ {C_{\mathbf{C}}}(X)$ which are complex Lindenstrauss spaces.
Measures with bounded powers on locally compact abelian groups
G. V.
Wood
187-210
Abstract: If $\mu$ is a measure on a locally compact abelian group with its positive and negative convolution powers bounded in norm by $K < \tfrac{1} {3}(4\cos (\pi /9) + 1) \sim 1.58626$ , then $\mu$ has the form $ \mu = \lambda (\cos \theta {\delta _x} + i\,\sin \theta {\delta _{xu}})$ where $\vert\lambda \vert = 1$ and ${u^2} = e$. Applications to isomorphism theorems are given. In particular, if $ {G_1}$ and ${G_2}$ are l.c.a. groups and $ T$ is an isomorphism of ${L^1}({G_1})$ onto $ {L^1}({G_2})$ with $\left\Vert T \right\Vert < \tfrac{1} {3}(4\,\cos (\pi /9) + 1)$, then either ${G_1}$ and ${G_2}$ are isomorphic, or they both have subgroups of order $2$ with isomorphic quotients.
On the $q$-analogues of some transformations of nearly-poised hypergeometric series
B.
Nassrallah;
Mizan
Rahman
211-229
Abstract: A number of transformation formulas for very well-poised basic hypergeometric series have been obtained which, in the limit $q \to 1 -$, approach the known transformation formulas for nearly-poised ordinary hypergeometric series.
Codazzi tensors and reducible submanifolds
Irl
Bivens
231-246
Abstract: An integral formula is derived for Codazzi tensors of type $ (k,\,k)$. Many of the classical Minkowski type integral formulas then become special cases of this one. If $M$ is a submanifold of Euclidean space and $ \pi$ is a parallel distribution on $M$ then each leaf of $\pi$ is a submanifold of Euclidean space with mean curvature normal vector field $\eta$. Using the above integral formula we show that the integral of ${\left\vert \eta \right\vert^2}$ over $ M$ is bounded below by an intrinsic constant and we give necessary and sufficient conditions for equality to hold. The reducible surfaces for which equality holds are characterized and related results concerned with Riemannian product manifolds are proved. Parallel tensors of type $ (1,\,1)$ are characterized in terms of the de Rham decomposition. It is shown that if $M$ is irreducible and $A$ is a parallel tensor of type $(1,\,1)$ on $M$ which is not multiplication by a constant then $M$ is a Kaehler manifold. Some further results are derived for manifolds whose simply connected cover is Kaehler.
Generic cohomology for twisted groups
George S.
Avrunin
247-253
Abstract: Let $G$ be a simple algebraic group defined and split over $ {k_0} = {{\mathbf{F}}_p}$, and let $\sigma$ be a surjective endomorphism of $ G$ with finite fixed-point set ${G_\sigma }$. We give conditions under which cohomology groups of $G$ are isomorphic to cohomology groups of ${G_\sigma }$.
Embedding processes in Brownian motion in ${\bf R}\sp{n}$
Neil
Falkner
335-363
Abstract: We give a potential-theoretic characterization of the right-continuous processes which can be embedded in Brownian motion in ${{\mathbf{R}}^n}$ by means of an increasing family of standard stopping times. In general it is necessary to use a Brownian motion process whose filtration is richer than the natural one.
Curvature tensors on almost Hermitian manifolds
Franco
Tricerri;
Lieven
Vanhecke
365-398
Abstract: A complete decomposition of the space of curvature tensors over a Hermitian vector space into irreducible factors under the action of the unitary group is given. The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined. Several applications for almost Hermitian manifolds are given. Conformal invariants are considered and a general Bochner curvature tensor is introduced and shown to be a conformal invariant. Finally curvature tensors on four-dimensional manifolds are studied in detail.
Counting divisors with prescribed singularities
Israel
Vainsencher
399-422
Abstract: Given a family of divisors $\{ {D_s}\}$ in a family of smooth varieties $\{ {Y_s}\}$ and a sequence of integers ${m_1}, \ldots ,{m_t}$, we study the scheme parametrizing the points $(s,{y_1}, \ldots ,{y_t})$ such that ${y_i}$ is a (possibly infinitely near) $ {m_i}$-fold point of $ {D_s}$. We obtain a general formula which yields, as special cases, the formula of de Jonquières and other classical results of Enumerative Geometry. We also study the questions of finiteness and the multiplicities of the solutions.
Self-maps of flag manifolds
Henry H.
Glover;
William D.
Homer
423-434
Abstract: Rationally, a map between flag manifolds is seen to be determined up to homotopy by the homomorphism it induces on cohomology. Two algebraic results for cohomology endomorphisms then serve (a) to determine those flag manifolds which have (nontrivial) self-maps that factor through a complex projective space, and (b) for a special class of flag manifolds, to classify the self-maps of their rationalizations up to homotopy.
Singular integrals and maximal functions associated with highly monotone curves
W. C.
Nestlerode
435-444
Abstract: Let $ \gamma :[ - 1,1] \to {{\mathbf{R}}^n}$ be an odd curve. Set $\displaystyle {H_\gamma }f(x) = {\text{PV}}\int {f(x - \gamma (t))\,(dt/t)}$ and $\displaystyle {M_\gamma }f(x) = \sup {h^{ - 1}}\int_0^h {\vert f(x - \gamma (t))\vert\,dt}$ . We introduce a class of highly monotone curves in $ {{\mathbf{R}}^n}$, $n \geqslant 2$, for which we prove that ${H_\gamma }$ and $ {M_\gamma }$ are bounded operators on ${L^2}({{\mathbf{R}}^n})$. These results are known if $ \gamma$ has nonzero curvature at the origin, but there are highly monotone curves which have no curvature at the origin. Related to this problem, we prove a generalization of van der Corput's estimate of trigonometric integrals.
Subspaces of $L\sp{1}$, via random measures
David J.
Aldous
445-463
Abstract: It is shown that every subspace of ${L^1}$ contains a subspace isomorphic to some $ {l_q}$. The proof depends on a fixed point theorem for random measures.
A relative Nash theorem
Selman
Akbulut;
Henry C.
King
465-481
Abstract: We prove that if $ M$ is a closed smooth manifold and ${M_i}$, $i = 1, \ldots ,k$, are transversally intersecting closed smooth submanifolds of $ M$, then there exist a nonsingular algebraic set $Z$ and nonsingular algebraic subsets $ {Z_i}$, $i = 1, \ldots ,k$, of $Z$ such that $(M;{M_1}, \ldots ,{M_k})$ is diffeomorphic to $ (Z;{Z_1}, \ldots ,{Z_k})$. We discuss a generalization and the consequences of this result.
Spherical means and geodesic chains on a Riemannian manifold
Toshikazu
Sunada
483-501
Abstract: Some spectral properties of spherical mean operators defined on a Riemannian manifold are given. As an application we deduce a statistic property of geodesic chains which is interesting from the view point of geometric probability.
The structure of tensor products of semilattices with zero
G.
Grätzer;
H.
Lakser;
R.
Quackenbush
503-515
Abstract: If $A$ and $B$ are finite lattices, then the tensor product $ C$ of $A$ and $B$ in the category of join semilattices with zero is a lattice again. The main result of this paper is the description of the congruence lattice of $ C$ as the free product (in the category of bounded distributive lattices) of the congruence lattice of $A$ and the congruence lattice of $B$. This provides us with a method of constructing finite subdirectly irreducible (resp., simple) lattices: if $A$ and $B$ are finite subdirectly irreducible (resp., simple) lattices then so is their tensor product. Another application is a result of E. T. Schmidt describing the congruence lattice of a bounded distributive extension of ${M_3}$.
Smooth perturbations of a function with a smooth local time
D.
Geman;
J.
Horowitz
517-530
Abstract: A real Borel function on $[0,\,1]$ has a local time if its occupation measure up to each time $t$ (equivalently: its increasing, equimeasurable rearrangement on $[0,\,t]$) is absolutely continuous; the local time ${\alpha _t}(x)$ is then the density. An inverse relationship exists between the smoothness of the local time in $(t,\,x)$ and that of the original function. The sum of a function with a smooth local time and a well-behaved (e.g. absolutely continuous) function is shown to have a local time, which inherits certain significant properties from the old local time, and for which an explicit formula is given. Finally, using a probabilistic approach, examples are given of functions having local times of prescribed smoothness.
Hypersingular integrals and parabolic potentials
Sagun
Chanillo
531-547
Abstract: In this paper we characterize the potential spaces associated with the heat equation in terms of singular integrals of mixed homogeneity.
Partitions of products
David
Pincus;
J. D.
Halpern
549-568
Abstract: This paper extends some applications of a theorem of Halpern and Lauchli on partitions of products of finitary trees. The extensions are to weak infinite products of dense linear orderings, and ultrafilter preservation for finite product Sacks forcing.
Morse theory by perturbation methods with applications to harmonic maps
K.
Uhlenbeck
569-583
Abstract: There are many interesting variational problems for which the Palais-Smale condition cannot be verified. In cases where the Palais-Smale condition can be verified for an approximating integral, and the critical points converge, a Morse theory is valid. This theory applies to a class of variational problems consisting of the energy integral for harmonic maps with a lower order potential.
Twist maps, coverings and Brouwer's translation theorem
H. E.
Winkelnkemper
585-593
Abstract: We apply the Brouwer Translation Theorem to a class of twist maps of the annulus (which contains ${C^1}$ area preserving maps) to show that, if $ h$ belongs to this class, then a certain set $ {\mathcal{P}_0}$ of periodic points of $h$ cannot be dense. The definition of ${\mathcal{P}_0}$ does not impose any a priori restrictions on the periods of the points of ${\mathcal{P}_0}$.
Some universal sets of terms
Walter
Taylor
595-607
Abstract: For every $ \Pi _2^1$ class of cardinals containing 0 and $1$, there exists a finite set $T$ of terms, such that $X$ is precisely the class of cardinals in which $T$ is universal.
Compact groups of homeomorphisms on tree-like continua
J. B.
Fugate;
T. B.
McLean
609-620
Abstract: This paper is concerned with the fixed point sets of certain collections of homeomorphisms on a tree-like continuum. Extending a theorem of P. A. Smith, the authors prove that a periodic homeomorphism has a (nonvoid) continuum as its fixed point set. They then deduce possible periods for homeomorphisms on tree-like continua which satisfy certain decomposability or irreducibility conditions. The main result of the paper is that a compact group of homeomorphisms has a continuum as its fixed point set. This is applied to isometries. The paper concludes with sufficient conditions that a pointwise periodic homeomorphism have a fixed point.
KV-theory of categories
Charles A.
Weibel
621-635
Abstract: Quillen has constructed a $K$-theory $ {K_{\ast}}C$ for nice categories, one of which is the category of projective $ R$-modules. We construct a theory $ K{V_{\ast}}C$ for the nice categories parametrized by rings. When applied to projective modules we recover the Karoubi-Villamayor $ K$-theory $K{V_{\ast}}(R)$. As an application, we show that the Cartan map from $ {K_{\ast}}(R)$ to ${G_{\ast}}(R)$ factors through the groups $K{V_{\ast}}(R)$. We also compute $K{V_{\ast}}$ for the categories of faithful projectives and Azumaya algebras, generalizing results of Bass.
On the topological structure of even-dimensional complete intersections
A. S.
Libgober;
J. W.
Wood
637-660
Abstract: A topological connected sum decomposition into indecomposable pieces is given for complete intersections, and these pieces are described by plumbing constructions. The principal technical results are structure theorems for the intersection form on the middle dimensional homology and the submodule of spherical classes.
Conditional expectations in $C\sp{\ast} $-crossed products
Shigeru
Itoh
661-667
Abstract: Let $(A,\,G,\,\alpha )$ be a $ {C^{\ast}}$-dynamical system. Let $B$ be a $ {C^{\ast}}$-subalgebra of $ A$ and $P$ be a conditional expectation of $ A$ onto $B$ such that ${\alpha _t}P = P{\alpha _t}$ for each $ t \in G$. Then it is proved that there exists a conditional expectation of ${C^{\ast}}(G,\,A,\,\alpha )$ onto ${C^{\ast}}(G,\,B,\,\alpha )$. In particular, if $ G$ is amenable and $ A$ is unital, then there always exists a conditional expectation of $ {C^{\ast}}(G,\,A,\,\alpha )$ onto $ {C^{\ast}}(G)$. Some related results are also obtained.
A perturbation method in critical point theory and applications
Abbas
Bahri;
Henri
Berestycki
1-32
Abstract: This paper is concerned with existence and multiplicity results for nonlinear elliptic equations of the type $- \Delta u = {\left\vert u \right\vert^{p - 1}}u + h(x)$ in $\Omega ,\,u = 0$ on $ \partial \Omega$. Here, $ \Omega \subset {{\mathbf{R}}^N}$ is smooth and bounded, and $h \in {L^2}(\Omega )$ is given. We show that there exists ${p_N} > 1$ such that for any $p \in (1,\,{p_N})$ and any $h \in {L^2}(\Omega )$, the preceding equation possesses infinitely many distinct solutions. The method rests on a characterization of the existence of critical values by means of noncontractibility properties of certain level sets. A perturbation argument enables one to use the properties of some associated even functional. Several other applications of this method are also presented.
On the zeros of Dirichlet $L$-functions. II
Akio
Fujii
33-40
Abstract: Some consequences of the main theorem of On the zeros of Dirichlet $ L$-functions. I, Trans. Amer. Math. Soc. 196 (1974), 225-235 are proved.
The diameter of random graphs
Béla
Bollobás
41-52
Abstract: Extending some recent theorems of Klee and Larman, we prove rather sharp results about the diameter of a random graph. Among others we show that if $d = d(n) \geqslant 3$ and $m = m(n)$ satisfy $(\log n)/d - 3\,\log \log n \to \infty$, $ {2^{d - 1}}{m^d}/{n^{d + 1}} - \log n \to \infty$ and ${d^{d - 2}}{m^{d - 1}}/{n^d} - \log n \to - \infty$ then almost every graph with $n$ labelled vertices and $m$ edges has diameter $d$.
The second conjugate algebra of the Fourier algebra of a locally compact group
Anthony To Ming
Lau
53-63
Abstract: Let $G$ be a locally compact group and let $VN(G)$ denote the von Neumann algebra generated by the left translations of $G$ on ${L_2}(G)$. Then $ VN{(G)^{\ast}}$, when regarded as the second conjugate space of the Fourier algebra of $G$, is a Banach algebra with the Arens product. We prove among other things that when $G$ is amenable, $VN{(G)^{\ast}}$ is neither commutative nor semisimple unless $G$ is finite. We study in detail the class of maximal regular left ideals in $VN{(G)^{\ast}}$. We also show that if ${G_1}$ and ${G_2}$ are discrete groups, then ${G_1}$ and ${G_2}$ are isomorphic if and only if $VN{({G_1})^{\ast}}$ and $VN{({G_2})^{\ast}}$ are isometric order isomorphic.
Anneaux de valuation discr\`ete complets non commutatifs
Robert
Vidal
65-81
Abstract: On construit, grace au concept de bimodule au sens de M. Artin, une bonne généralisation non commutative de la notion d'anneau de valuation discrète complet; puis on étudie la validité du théorème de structure de I. S. Cohen en égale caractéristique. Lorsque l'anneau de valuation possède un corps de représentants qui est un corps local (commutatif) à corps résiduel de caractéristique nulle et algébriquement clos, on donne une classification des déviations de commutativité engendrées par une dérivation continue de ce corps local. Enfin, on propose une méthode générale de construction d'anneaux sans corps de Cohen en égale caractéristique et l'article se termine par des problèmes ouverts dans cette théorie.
Obstructions to deforming a space curve
Daniel J.
Curtin
83-94
Abstract: Mumford described a curve, $\gamma$, in $ {{\mathbf{P}}^3}$ that has obstructed infinitesimal deformations (in fact the Hilbert scheme of the curve is generically nonreduced). This paper studies ${{\mathbf{P}}^3}$ over parameter spaces of the form $ \operatorname{Spec} (k[t]/({t^n})),\,n = 2,\,3,\, \ldots $. Given a deformation of $ \gamma$ over $ \operatorname{Spec} (k[t]/({t^n}))$ one attempts to extend it to a deformation of $\gamma$ over $\operatorname{Spec} (k[t]/({t^{n + 1}}))$. If it will not extend, this deformation is said to be obstructed at the nth order. I show that on a generic version of Mumford's curve, an infinitesimal deformation (i.e., a deformation over $ \operatorname{Spec} (k[t]/({t^2}))$) is either obstructed at the second order, or at no order, in which case we say it is unobstructed.
Algebraic extensions of power series rings
Jimmy T.
Arnold
95-110
Abstract: Let $D$ and $J$ be integral domains such that $D \subset J$ and $J[[X]]$ is not algebraic over $D[[X]]$. Is it necessarily the case that there exists an integral domain $R$ such that $D[[X]] \subset R \subseteq J[[X]]$ and $ R \cong D[[X]][[\{ {Y_i}\} _{i = 1}^\infty ]]$? While the general question remains open, the question is answered affirmatively in a number of cases. For example, if $D$ satisfies any one of the conditions (1) $D$ is Noetherian, (2) $D$ is integrally closed, (3) the quotient field $ K$ of $D$ is countably generated as a ring over $D$, or (4) $D$ has Krull dimension one, then an affirmative answer is given. Further, in the Noetherian case it is shown that $J[[X]]$ is algebraic over $D[[X]]$ if and only if it is integral over $D[[X]]$ and necessary and sufficient conditions are given on $D$ and $J$ in order that this occur. Finally if, for every positive integer $n$, $ D[[{X_1}, \ldots ,{X_n}]] \subset R \subseteq J[[{X_1}, \ldots ,{X_n}]]$ implies that $R \ncong D[[{X_1}, \ldots ,{X_n}]][[\{ {Y_i}\} _{i = 1}^\infty ]]$, then it is shown that $ J[[{X_1}, \ldots ,{X_n}]]$ is algebraic over $D[[{X_1}, \ldots ,{X_n}]]$ for every $ n$.
When is a linear functional multiplicative?
M.
Roitman;
Y.
Sternfeld
111-124
Abstract: We prove here by elementary arguments a generalization of a theorem by Gleason, Kahane and Żelazko: If $ \varphi$ is a linear functional on an algebra with unit $A$ such that $\varphi (1) = 1$ and $ \varphi (u) \ne 0$ for any invertible $u$ in $A$, then $\varphi$ is multiplicative, provided the spectrum of each element in $A$ is bounded. We present also other conditions which may replace the assumptions on $A$ in the theorem above.
Algebraic determination of fiberwise PL involutions
Hayon
Kim;
Jehpill
Kim;
Kyung Whan
Kwun
125-131
Abstract: Some fiberwise PL involutions on fibered $3$-manifolds induce the obvious automorphism of the fundamental group. It is shown that this expected behavior of the fundamental group in turn characterizes such fiberwise involutions.
The local Kronecker-Weber theorem
Jonathan
Lubin
133-138
Abstract: The extension of a local field generated by adjoining the torsion points on a suitable formal group is essentially the maximal abelian extension of the field. This fact is proven by appealing to the functorial properties of the Herbrand transition function of higher ramification theory.
Equivariant cofibrations and nilpotency
Robert H.
Lewis
139-155
Abstract: Let $f:B \to Y$ be a cofibration whose cofiber is a Moore space. We give necessary and sufficient conditions for $f$ to be induced by a map of the desuspension of the cofiber into $B$. These conditions are especially simple if $ B$ and $Y$ are nilpotent. We obtain some results on the existence of equivariant Moore spaces, and use them to construct examples of noninduced cofibrations between nilpotent spaces. Our machinery also leads to a cell structure proof of the characterization of pre-nilpotent spaces due to Dror and Dwyer [7], and to a simple proof, for finite fundamental group, of the result of Brown and Kahn [4] that homotopy dimension equals simple cohomological dimension in nilpotent spaces.
Essential spectra of elementary operators
L. A.
Fialkow
157-174
Abstract: This paper describes the essential spectrum and index function of the operator $X \to AXB$, where $A$, $B$, and $X$ are Hilbert space operators. Analogous results are given for the restriction of this operator to a norm ideal and partial analogues are given for sums of such operators and for the case when the operators act on a Banach space.
Whitney stratified chains and cochains
R. Mark
Goresky
175-196
Abstract: This paper contains the technical constructions necessary for a "geometric cycle" definition of cohomology and homology in the context of Whitney stratifications. Cup and cap products are interpreted as the transverse intersection of geometric cocycles and cycles.
Derivatives and Lebesgue points via homeomorphic changes of scale
Don L.
Hancock
197-218
Abstract: Let $I$ be a closed interval, and suppose $\mathcal{K}$, $ \mathcal{H}$, and $ \Lambda$ denote, respectively, the class of homeomorphisms of $ I$ onto itself, the class of homeomorphisms of the line onto itself, and the class of real functions on $I$ for which each point is a Lebesgue point. Maximoff proved that $\Lambda \circ \mathcal{K}$ is exactly the class of Darboux Baire $1$ functions, where $\Lambda \circ \mathcal{K} = \{ f \circ k:f \in \Lambda ,k \in \mathcal{K}\}$. The present paper is devoted primarily to a study of $\mathcal{H} \circ \Lambda = \{ h \circ f:f \in \Lambda ,h \in \mathcal{H}\}$. The characterizations of this class which are obtained show that a function is a member of $\mathcal{H} \circ \Lambda $ if and only if, in addition to the obvious requirement of approximate continuity, it satisfies certain growth and density-like conditions. In particular, any approximately continuous function with countably many non-Lebesgue points belongs to $\mathcal{H} \circ \Lambda $. It is also established that $ \mathcal{H} \circ \Lambda$ is a uniformly closed algebra properly containing the smallest algebra generated from $ \Lambda$, and a characterization of the latter algebra is provided.
On analytic diameters and analytic centers of compact sets
Shōji
Kobayashi;
Nobuyuki
Suita
219-228
Abstract: In this paper several results on analytic diameters and analytic centers are obtained. We show that the extremal function for analytic diameter is unique and that there exist compact sets with many analytic centers. We answer negatively several problems posed by F. Miinsker.
Invariant connections and Yang-Mills solutions
Mitsuhiro
Itoh
229-236
Abstract: A condition on the self-duality and the stability of Yang-Mills solutions are discussed. The canonical invariant $ G$-connections on $ {S^4}$ and ${P_2}({\mathbf{C}})$ are considered as Yang-Mills solutions. The non-self-duality of the connections requires the injectivity of the isotropy homomorphisms. We construct examples of non-self-dual connections on $ G$-vector bundles ($ G$ is a compact simple group). Under a certain property of the isotropy homomorphism, these canonical connections are not weakly stable.
The complexification and differential structure of a locally compact group
Kelly
McKennon
237-258
Abstract: The concept of a complexification of a locally compact group is defined and its connections with the differential structure developed. To provide an interpretation in terms of irreducible representations of separable, Type I groups, a duality theorem and Bochner theorem are presented.
A characterization of best $\Phi $-approximants
D.
Landers;
L.
Rogge
259-264
Abstract: Let $T$ be an operator from an Orlicz space ${L_\Phi }$ into itself. It is shown in this paper that four algebraic conditions and one integration condition assure that $T$ is the best $\Phi$-approximator, given a suitable $ \sigma$-lattice.
Refinement properties and extensions of filters in Boolean algebras
Bohuslav
Balcar;
Petr
Simon;
Peter
Vojtáš
265-283
Abstract: We consider the question, under what conditions a given family $ A$ in a Boolean algebra $\mathcal{B}$ has a disjoint refinement. Of course, $A$ cannot have a disjoint refinement if $ A$ is a dense subset of an atomless $ \mathcal{B}$, or if $\mathcal{B}$ is complete and $A$ generates an ultrafilter on $\mathcal{B}$. We show in the first two sections that these two counterexamples can be the only possible ones. The third section is concerned with the question, how many sets must necessarily be added to a given filter in order to obtain an ultrafilter base.
A Phragm\'en-Lindel\"of theorem conjectured by D. J. Newman
W. H. J.
Fuchs
285-293
Abstract: Let $D$ be a region of the complex plane, $\infty \in \partial D$. If $ f(z)$ is holomorphic in $ D$, write $ M(r) = {\sup _{\vert z\vert = r,\,z \in D}}\vert f(z)\vert$. Theorem 1. If $f(z)$ is holomorphic in $D$ and $ \lim {\sup _{z \to \zeta ,\,z \in D}}\vert f(z)\vert \leqslant 1$ for $\zeta \in \partial D$, $\zeta \ne \infty$, then one of the following holds (a) $ \vert f(z)\vert < 1(z \in D)$, (b)$f(z)$ has a pole at $\infty$, (c) $\log \,M(r)/\log r \to \infty$ as $r \to \infty$. If $M(r)/r \to 0(r \to \infty )$, then (a) must hold.
The Hilbert transform and maximal function for approximately homogeneous curves
David A.
Weinberg
295-306
Abstract: Let $ {\mathcal{H}_\gamma }f(x) = {\text{p}}{\text{.v}}{\text{.}}\int_{ - 1}^1 {f(x - \gamma (t))dt/t}$ and ${\mathfrak{M}_\gamma }f(x) = {\sup _{1 \geqslant h > 0}}{h^{ - 1}}\int_0^h {\vert f(x - \gamma (t))\vert dt}$. It is proved that for $f \in \mathcal{S}({{\mathbf{R}}^n})$, the Schwartz class, and for an approximately homogeneous curve $\gamma (t) \in {{\mathbf{R}}^n}$, $ {\left\Vert {{\mathcal{H}_\gamma }f} \right\Vert _2} \leqslant C{\left\Vert f \right\Vert _2}$, ${\left\Vert {{\mathfrak{M}_\gamma }f} \right\Vert _2} \leqslant C{\left\Vert f \right\Vert _2}$. A homogeneous curve is one which satisfies a differential equation $0 < t < \infty$, where $A$ is a nonsingular matrix all of whose eigenvalues have positive real part. An approximately homogeneous curve $ \gamma (t)$ has the form $ {\gamma _1}(t) + {\gamma _2}(t)$, where $ {\gamma _2}(t)$ is a carefully specified "error", such that $\gamma _2^{(j)}$ is also restricted for $j = 2, \ldots ,n + 1$. The approximately homogeneous curves generalize the curves of standard type treated by Stein and Wainger.
Markov processes with Lipschitz semigroups
Richard
Bass
307-320
Abstract: For $f$ a function on a metric space, let $\displaystyle \operatorname{Lip} f = \mathop {\sup }\limits_{x \ne y} \vert f(x) - f(y)\vert/d(x,\,y),$ and say that a semigroup $ {P_t}$ is Lipschitz if $\operatorname{Lip} ({P_t}f) \leqslant {e^{Kt}}\operatorname{Lip} f$ for all $f$, $t$, where $K$ is a constant. If one has two Lipschitz semigroups, then, with some additional assumptions, the sum of their infinitesimal generators will also generate a Lipschitz semigroup. Furthermore a sequence of uniformly Lipschitz semigroups has a subsequence which converges in the strong operator topology. Examples of Markov processes with Lipschitz semigroups include all diffusions on the real line which are on natural scale whose speed measures satisfy mild conditions, as well as some jump processes. One thus gets Markov processes whose generators are certain integro-differential operators. One can also interpret the results as giving some smoothness conditions for the solutions of certain parabolic partial differential equations.
Prime knots and tangles
W. B. Raymond
Lickorish
321-332
Abstract: A study is made of a method of proving that a classical knot or link is prime. The method consists of identifying together the boundaries of two prime tangles. Examples and ways of constructing prime tangles are explored.
Erratum to: ``$\sp{\ast} $-valuations and ordered $\sp{\ast} $-fields''
S. S.
Holland
333
The topology on the primitive ideal space of transformation group $C\sp{\ast} $-algebras and C.C.R. transformation group $C\sp{\ast} $-algebras
Dana P.
Williams
335-359
Abstract: If $(G,\Omega )$ is a second countable transformation group and the stability groups are amenable then ${C^ \ast }(G,\Omega )$ is C.C.R. if and only if the orbits are closed and the stability groups are C.C.R. In addition, partial results relating closed orbits to C.C.R. algebras are obtained in the nonseparable case. In several cases, the topology of the primitive ideal space is calculated explicitly. In particular, if the stability groups are all contained in a fixed abelian subgroup $H$, then the topology is computed in terms of $ H$ and the orbit structure, provided $ {C^ \ast }(G,\Omega )$ and $ {C^ \ast }(H,\Omega )$ are $ EH$-regular. These conditions are automatically met if $G$ is abelian and $ (G,\Omega )$ is second countable.
The discontinuous initial value problem of a reacting gas flow
Lung An
Ying;
Ching Hua
Wang
361-387
Abstract: We show that the local solvability of the solution of a reacting gas flow system $(1.1) - (1.4)$ with initial values $ (1.5)$, which has a large jump at the point $x' = 0$ and the structure of the solution near the origin $(0,0)$ are identical to those of the Riemann problem of the homogeneous system corresponding to the Cauchy problem $(1.1) - (1.5)$.
The cyclic connectivity of homogeneous arcwise connected continua
David P.
Bellamy;
Lewis
Lum
389-396
Abstract: A continuum is cyclicly connected provided each pair of its points lie together on some simple closed curve. In 1927, G. T. Whyburn proved that a locally connected plane continuum is cyclicly connected if and only if it contains no separating points. This theorem was fundamental in his original treatment of cyclic element theory. Since then numerous authors have obtained extensions of Whyburn's theorem. In this paper we characterize cyclic connectedness in the class of all Hausdorff continua. Theorem. The Hausdorff continuum $ X$ is cyclicly connected if and only if for each point $x \in X$, $x$ lies in the relative interior of some arc in $X$ and $X - \{ x\}$ is arcwise connected. We then prove that arcwise connected homogeneous metric continua are cyclicly connected.
Nonweakly compact operators from order-Cauchy complete $C(S)$ lattices, with application to Baire classes
Frederick K.
Dashiell
397-413
Abstract: This paper is concerned with the connection between weak compactness properties in the duals of certain Banach spaces of type $C(S)$ and order properties in the vector lattice $C(S)$. The weak compactness property of principal interest here is the condition that every nonweakly compact operator from $C(S)$ into a Banach space must restrict to an isomorphism on some copy of $ {l^\infty }$ in $ C(S)$. (This implies Grothendieck's property that every ${w^ \ast }$-convergent sequence in $C{(S)^ \ast }$ is weakly convergent.) The related vector lattice property studied here is order-Cauchy completeness, a weak type of completeness property weaker than $\sigma$-completeness and weaker than the interposition property of Seever. An apphcation of our results is a proof that all Baire classes (of fixed order) of bounded functions generated by a vector lattice of functions are Banach spaces satisfying Grothendieck's property. Another application extends previous results on weak convergence of sequences of finitely additive measures defined on certain fields of sets.
A counterexample to the bounded orbit conjecture
Stephanie M.
Boyles
415-422
Abstract: A long outstanding problem in the topology of Euclidean spaces is the Bounded Orbit Conjecture, which states that every homeomorphism of ${E^2}$ onto itself, with the property that the orbit of every point is bounded, must have a fixed point. It is well known that the conjecture is true for orientation preserving homeomorphisms. We provide a counterexample to the conjecture by constructing a fixed point free orientation reversing homeomorphism which satisfies the hypothesis of the conjecture.
Isomorphism theorems for octonion planes over local rings
Robert
Bix
423-439
Abstract: It is proved that there is a collineation between two octonion planes over local rings if and only if the underlying octonion algebras are isomorphic as rings. It is shown that every isomorphism between the little or middle projective groups of two octonion planes over local rings is induced by conjugation with a collineation or a correlation of the planes when the local rings contain $\frac{1} {2}$.
Symmetry properties of the zero sets of nil-theta functions
Sharon
Goodman
441-460
Abstract: Let $N$ denote the three dimensional Heisenberg group, and let $\Gamma$ be a discrete two-generator subgroup of $ N$ such that $ N/\Gamma$ is compact. Then we may decompose $ {L^2}(N/\Gamma )$ into primary summands with respect to the right regular representation $R$ of $N$ on $ {L^2}(N/\Gamma )$ as follows: ${L^2}(N/\Gamma ) = \oplus \sum\nolimits_{m \in {\mathbf{Z}}} {{H_m}(\Gamma )}$. It can be shown that for $m \ne 0,{H_m}(\Gamma )$ is a multiplicity space for the representation $ R$ of multiplicity $\left\vert m \right\vert$. The distinguished subspace theory of $ {\text{L}}$. Auslander and ${\text{J}}$. Brezin singles out a finite number of the decompositions of ${H_m}(\Gamma ),m \ne 0$, which are in some ways nicer than the others. They define algebraically an integer valued function, called the index, on the set ${\Omega _m}$ of irreducible closed $ R$-invariant subspaces of ${H_m}(\Gamma )$ such that the distinguished subspaces have index one. In this paper, we give an analytic-geometric interpretation of the index. Every space in $ {\Omega _m}$ contains a unique (up to constant multiple) special function, called a nil-theta function, that arises as a solution of a certain differential operator on $N/\Gamma$. These nil-theta functions have been shown to be closely related to the classical theta functions. Since the classical theta functions are determined (up to constant multiple) by their zero sets, it is natural to attempt to classify the spaces in ${\Omega _m}$ using various properties of the zero sets of the nil-theta functions lying in these spaces. We define the index of a nil-theta function in ${H_m}(\Gamma )$ using the symmetry properties of its zero set. Our main theorem asserts that the algebraic index of a space in $ {\Omega _m}$ equals the index of the unique nil-theta function lying in that space. We have thus an analytic-geometric characterization of the index. We then use these results to give a complete description of the zero sets of those nil-theta functions of a fixed index. We also investigate the behavior of the index under the multiplication of nil-theta functions; i.e. we discuss how the index of the nil-theta function $FG$ relates to the indices of the nil-theta functions $F$ and $G$.
Generalizations of Ces\`aro continuous functions and integrals of Perron type
Cheng Ming
Lee
461-481
Abstract: The linear space of all the Cesàro continuous functions of any order is extended by introducing pointwisely Cesàro continuous functions and exact generalized Peano derivatives. Then six generalized integrals of Perron type are defined and studied. They are based on three recent monotonicity theorems and each depends on an abstract upper semilinear space of certain functions. Some of the integrals are more general than all the integrals in the Cesàro-Perron scale provided that the abstract semilinear space is taken to be the linear space of all the pointwisely Cesàro continuous functions or all the exact generalized Peano derivatives. That such a concrete general integral is possible follows from the fact proved here that each exact generalized Peano derivative is in Baire class one and has the Darboux property. Relations between the pointwisely Cesàro continuous functions or the exact generalized Peano derivatives and functions defined by means of the values of certain Schwartz's distributions at "points" are also established.
On purely inseparable algebras and P.H.D. rings
Shizuka
Satô
483-498
Abstract: M. E. Sweedler has considered purely inseparable algebras over rings. We define a stronger notion for purely inseparable algebras over rings and we study the fundamental properties of purely inseparable algebras. Moreover, we consider the relations between purely inseparable algebras and P.H.D. rings.
Liapounoff's theorem for nonatomic, finitely-additive, bounded, finite-dimensional, vector-valued measures
Thomas E.
Armstrong;
Karel
Prikry
499-514
Abstract: Liapounoff's theorem states that if $ (X,\Sigma )$ is a measurable space and $\mu :\Sigma \to {{\mathbf{R}}^d}$ is nonatomic, bounded, and countably additive, then $ \mathcal{R}(\mu ) = \{ \mu (A):A \in \Sigma \}$ is compact and convex. When $ \Sigma$ is replaced by a $ \sigma$-complete Boolean algebra or an $F$-algebra (to be defined) and $\mu$ is allowed to be only finitely additive, $ \mathcal{R}(\mu )$ is still convex. If $\Sigma$ is any Boolean algebra supporting nontrivial, nonatomic, finitely-additive measures and $Z$ is a zonoid, there exists a nonatomic measure on $\Sigma$ with range dense in $Z$. A wide variety of pathology is examined which indicates that ranges of finitely-additive, nonatomic, finite-dimensional, vector-valued measures are fairly arbitrary.
Weak and pointwise compactness in the space of bounded continuous functions
Robert F.
Wheeler
515-530
Abstract: Let $T$ be a completely regular Hausdorff space, ${C_b}(T)$ the space of bounded continuous real-valued functions on $T$, $M(T)$ the Banach space dual of ${C_b}(T)$. Let $\mathcal{H}$ denote the family of subsets of $ {C_b}(T)$ which are uniformly bounded and relatively compact for the topology $ {\mathfrak{J}_p}$ of pointwise convergence. The basic question considered here is: what is the largest subspace $Z$ of $M(T)$ such that every member of $\mathcal{H}$ is relatively $\sigma ({C_b},Z)$-compact? Classical results of Grothendieck and Ptak show that $ Z = M(T)$ if $ T$ is pseudocompact. In general, ${M_t} \subset Z \subset {M_s};$ assuming Martin's Axiom, a deep result of Talagrand improves the lower bound to ${M_\tau }$. It is frequently, but not always, true that $Z = {M_s};$ counterexamples are given which use Banach spaces in their weak topologies to construct the underlying $T$'s.
On the genus of symmetric groups
Viera Krňanová
Proulx
531-538
Abstract: A new method for determining genus of a group is described. It involves first getting a bound on the sizes of the generating set for which the corresponding Cayley graph could have smaller genus. The allowable generating sets are then examined by methods of computing average face sizes and by voltage graph techniques to find the best embeddings. This method is used to show that genus of the symmetric group ${S_5}$ is equal to four. The voltage graph method is used to exhibit two new embeddings for symmetric groups on even number of elements. These embeddings give us a better upper bound than that previously given by A. T. White.
Integral geometric properties of capacities
Pertti
Mattila
539-554
Abstract: Let $m$ and $n$ be positive integers, $0 < m < n$, and ${C_K}$ and ${C_H}$ the usual potential-theoretic capacities on ${R^n}$ corresponding to lower semicontinuous kernels $ K$ and $H$ on ${R^n} \times {R^n}$ with $ H(x,y) = K(x,y){\left\vert {x - y} \right\vert^{n - m}} \geqslant 1$ for $ \left\vert {x - y} \right\vert \leqslant 1$. We consider relations between the capacities ${C_K}(E)$ and $ {C_H}(E \cap A)$ when $E \subset {R^n}$ and $A$ varies over the $m$-dimensional affine subspaces of ${R^n}$. For example, we prove that if $ E$ is compact, ${C_K}(E) \leqslant c\smallint {C_H}(E \cap A)d{\lambda _{n,m}}A$ where ${\lambda _{n,m}}$ is a rigidly invariant measure and $c$ is a positive constant depending only on $ n$ and $m$.
Global Warfield groups
Roger
Hunter;
Fred
Richman
555-572
Abstract: A global Warfield group is a summand of a simply presented abelian group. The theory of global Warfield groups encompasses both the theory of totally projective $p$-groups, which includes the classical Ulm-Zippin theory of countable $p$-groups, and the theory of completely decomposable torsion-free groups. This paper develops the central results of the theory including existence and uniqueness theorems. In addition it is shown that every decomposition basis of a global Warfield group has a nice subordinate with simply presented torsion cokernel, and that every global Warfield group is a direct sum of a group of countable torsion-free rank and a simply presented group.
Manifolds of nonanalyticity of solutions of certain linear PDEs
E. C.
Zachmanoglou
573-582
Abstract: It is shown that if $ P$ is a linear partial differential operator with analytic coefficients, and if $M$ is an analytic manifold of codimension $ 3$ which is partially characteristic with respect to $P$ and satisfies certain additional conditions, then one can find, in a neighborhood of any point of $ M$, solutions of the equation $Pu = 0$ which are flat or singular precisely on $ M$.
Cartan structures on contact manifolds
G.
Burdet;
M.
Perrin
583-602
Abstract: Owing to the existence of a dilatation generator of eigenvalues $\pm 2, \pm 1,0$ the symplectic Lie algebra is considered as a $\vert 2\vert$-graded Lie algebra. The corresponding decomposition of the symplectic group ${\text{Sp(2(}}n + 1{\text{),}}{\mathbf{R}}{\text{)}}$ makes the semidirect product (denoted $ {L^0}$) of the $ (2n + 1)$-dimensional Weyl group by the conformal symplectic group $ {\text{CSp(}}2n,{\mathbf{R}}{\text{)}}$ appear as a privileged subgroup and permits one to construct a $2n + 1$-dimensional homogeneous space possessing a natural contact form. Then ${\text{Sp}}(2(n + 1),{\mathbf{R}})$-valued Cartan connections on a ${L^0}$principal fibre bundle over a $2n + 1$-dimensional manifold ${B_{2n + 1}}$ are constructed and called symplectic Cartan connections. The conditions for obtaining a unique symplectic Cartan connection are given. The existence of this unique Cartan connection is used to define the notion of contact structure over ${B_{2n + 1}}$ and it is shown that any $ {L^0}$-structure of degree $ 2$ over ${B_{2n + 1}}$ can be considered as a contact structure on it. Moreover it is shown that a contact structure can be associated in a canonical way to any contact manifold.
On asymptotically almost periodic solutions of a convolution equation
Olof J.
Staffans
603-616
Abstract: We study questions related to asymptotic almost periodicity of solutions of the linear convolution equation $( \ast )\mu \ast x = f$. Here $\mu$ is a complex measure, and $ x$ and $f$ are bounded functions. Basically we are interested in conditions which imply that bounded solutions of $( \ast )$ are asymptotically almost periodic. In particular, we show that a certain necessary condition on $f$ for this to happen is also sufficient, thereby strengthening earlier results. We also include a result on existence of bounded solutions, and indicate a generalization to a distribution equation.
$L\sp{p}$ norms of certain kernels of the $N$-dimensional torus
L.
Colzani;
P. M.
Soardi
617-627
Abstract: In this paper we study a class of kernels ${F_R}$ which generalize the Bochner-Riesz kernels on the $N$-dimensional torus. Our main result consists in upper estimates for the ${L^p}$ norms of ${F_R}$ as $R$ tends to infinity. As a consequence we prove a convergence theorem for means of functions belonging to suitable Besov spaces.
Arborescent structures. II. Interpretability in the theory of trees
James H.
Schmerl
629-643
Abstract: The first-order theory of arborescent structures is shown to be completely faithfully interpretable in the first-order theory of trees. It follows from this interpretation that Vaught's conjecture is true for arborescent structures, the theory of arborescent structures is decidable, and every $ {\aleph _0}$-categorical arborescent structure has a decidable theory.
A rigid subspace of $L\sb{0}$
N. J.
Kalton;
James W.
Roberts
645-654
Abstract: We construct a closed infinite-dimensional subspace of ${L_0}(0,1)$ (or ${L_p}$ for $0 < p < 1$) which is rigid, i.e. such that every endomorphism in the space is a multiple of the identity.
Baire category principle and uniqueness theorem
J. S.
Hwang
655-665
Abstract: Applying a theorem of Bagemihl and Seidel (1953), we prove that if $ {S_2}$ is a set of second category in $ (\alpha ,\beta )$, where $0 \leqslant \alpha < \beta \leqslant 2\pi$, and if $f(z)$ is a function meromorphic in the sector $ \Delta (\alpha ,\beta ) = \{ z:0 < \left\vert z \right\vert < \infty ,\alpha < \arg z < \beta \}$ for which ${\underline {{\operatorname{lim}}} _{r \to \infty }}\left\vert {f(r{e^{i\theta }})} \right\vert > 0$, for all $ \theta \in {S_2}$, then there exists a sector $ \Delta (\alpha ',\beta ') \subseteq \Delta (\alpha ,\beta )$ such that $(\alpha ',\beta ') \subseteq {\bar S_2},{S_2}$ is second category in $(\alpha ',\beta ')$, and $f(z)$ has no zero in $\Delta (\alpha ',\beta ')$. Based on this property, we prove several uniqueness theorems.
Erratum to: ``The kinematic formula in complex integral geometry''
Theodore
Shifrin
667
Nonsmooth analysis: differential calculus of nondifferentiable mappings
A. D.
Ioffe
1-56
Abstract: A new approach to local analysis of nonsmooth mappings from one Banach space into another is suggested. The approach is essentially based on the use of set-valued mappings of a special kind, called fans, for local approximation. Convex sets of linear operators provide an example of fans. Generally, fans can be considered a natural set-valued extension of linear operators. The first part of the paper presents a study of fans; the second is devoted to calculus and includes extensions of the main theorems of classical calculus.
On two-dimensional normal singularities of type ${}\sb\ast A\sb{n}$, ${}\sb\ast D\sb{n}$ and ${}\sb\ast E\sb{n}$
Shigeki
Ohyanagi
57-69
Abstract: Let $G$ be the weighted dual graph associated with a contractible curve $A = \cup {A_i}$. There are many combinations of the weights $ {A_i} \cdot {A_i}$ which make the graph contractible. If $G$ is a graph which is the weighted dual graph for a rational singularity with any combination of the weights, then $G$ is either $ {}_ \ast {A_n}$, ${}_ \ast {D_n}$ or $ _\ast{E_n}$.
Standard $3$-components of type ${\rm Sp}(6,\,2)$
Larry
Finkelstein;
Daniel
Frohardt
71-92
Abstract: It is shown that if $ G$ is a finite simple group with a standard $3$-component of type ${\text{Sp}}(6,2)$ and $G$ satisfies certain $2$-local and $3$-local conditions then either $G$ is isomorphic to ${\text{Sp}}(8,2)$ or $G$ is isomorphic to ${F_4}(2)$.
Some properties of measure and category
Arnold W.
Miller
93-114
Abstract: Several elementary cardinal properties of measure and category on the real line are studied. For example, one property is that every set of real numbers of cardinality less than the continuum has measure zero. All of the properties are true if the continuum hypothesis is assumed. Several of the properties are shown to be connected with the properties of the set of functions from integers to integers partially ordered by eventual dominance. Several, but not all, combinations of these properties are shown to be consistent with the usual axioms of set theory. The main technique used is iterated forcing.
Neighborhood fixed pendant vertices
S. E.
Anacker;
G. N.
Robertson
115-128
Abstract: If $x$ is pendant in $G$, then ${x^ \ast }$ denotes the unique vertex of $ G$ adjacent to $ x$. Such an $x$ is said to be neighborhood-fixed whenever ${x^ \ast }$ is fixed by $A(G - x)$. It is shown that if $ G$ is not a tree and has a pendant vertex, but no *-fixed pendant vertex, then there is a subgraph ${G^\char93 }$ of $G$ such that for some $y \in V({G^\char93 })$, $O(A{({G^\char93 })_y}) \geqslant t!$ where $ t$ is the maximum number of edges in a tree rooted in ${G^\char93 }$.
Fredholm and invertible $n$-tuples of operators. The deformation problem
Raul E.
Curto
129-159
Abstract: Using J. L. Taylor's definition of joint spectrum, we study Fredholm and invertible $n$-tuples of operators on a Hilbert space. We give the foundations for a "several variables" theory, including a natural generalization of Atkinson's theorem and an index which well behaves. We obtain a characterization of joint invertibility in terms of a single operator and study the main examples at length. We then consider the deformation problem and solve it for the class of almost doubly commuting Fredholm pairs with a semi-Fredholm coordinate.
Time-ordered operators. I. Foundations for an alternative view of reality
Tepper L.
Gill
161-181
Abstract: The purpose of this paper is to present the proper framework for the mathematical foundations of time-ordered operators. We introduce a new mathematical process which we call the chronological process. This process generalizes the notion of a limit and allows us to recapture many properties lost in time-ordering. We then construct time-ordered integrals and evolution operators. We show that under reasonable assumptions, the time-ordered sum of two generators of contraction semigroups is a generator. This result resolves a question that has been debated in physics for forty years.
Lifting cycles to deformations of two-dimensional pseudoconvex manifolds
Henry B.
Laufer
183-202
Abstract: Let $M$ be a strictly pseudoconvex manifold with exceptional set $A$. Let $D \geqslant 0$ be a cycle on $ A$. Let $\omega :\mathfrak{M} \to Q$ be a deformation of $ M$. Kodaira's theory for deforming submanifolds of $ \mathfrak{M}$ is extended to the subspace $D$. Let $ \mathfrak{J}$ be the sheaf of germs of infinitesimal deformations of $ D$. Suppose that $ {H^1}(D,\mathfrak{J}) = 0$. If $\omega$ is the versal deformation, then $ D$ lifts to above a submanifold of $Q$. This lifting is a complete deformation of $ D$ with a smooth generic fiber. If all of the fibers of $\mathfrak{M}$ are isomorphic, then $ \omega$ is the trivial deformation. If $M$ has no exceptional curves of the first kind, then there exists $\omega$ such that only any given irreducible component of $A$ disappears as part of the exceptional set.
Nonseparable approximate equivalence
Donald W.
Hadwin
203-231
Abstract: This paper extends Voiculescu's theorem on approximate equivalence to the case of nonseparable representations of nonseparable ${C^ \ast }$-algebras. The main result states that two representations $ f$ and $g$ are approximately equivalent if and only if ${\text{rank}}f(x) = {\text{rank}}g(x)$ for every $x$. For representations of separable ${C^ \ast }$-algebras a multiplicity theory is developed that characterizes approximate equivalence. Thus for a separable $ {C^ \ast }$-algebra, the space of representations modulo approximate equivalence can be identified with a class of cardinal-valued functions on the primitive ideal space of the algebra. Nonseparable extensions of Voiculescu's reflexivity theorem for subalgebras of the Calkin algebra are also obtained.
Absolute continuity and uniqueness of measures on metric spaces
Pertti
Mattila
233-242
Abstract: We show that if two Borel regular measures on a separable metric space are in a suitable sense homogeneous, then they are mutually absolutely continuous. We use such absolute continuity theorems together with some density theorems to prove the uniqueness of measures more general than Haar measures.
Hulls of deformations in ${\bf C}\sp{n}$
H.
Alexander
243-257
Abstract: A problem of ${\text{E}}$. Bishop on the polynomially convex hulls of deformations of the torus is considered. Let the torus ${T^2}$ be the distinguished boundary of the unit polydisc in $ {{\mathbf{C}}^2}$. If $t \mapsto T_t^2$ is a smooth deformation of $ {T^2}$ in ${{\mathbf{C}}^2}$ and ${g_0}$ is an analytic disc in ${{\mathbf{C}}^2}$ with boundary in ${T^2}$, a smooth family of analytic discs $t \mapsto {g_t}$, is constructed with the property that the boundary of ${g_t}$ lies in $T_t^2$. This construction has implications for the polynomially convex hulls of the tori $ T_t^2$. An analogous problem for a $2$-sphere in $ {{\mathbf{C}}^2}$ is also considered.
Algebraic fiber bundles
Steven E.
Landsburg
259-273
Abstract: When $X$ is a finite simplicial complex and $ G$ is any of a certain class of groups, a classification of $ G$-principal bundles over $ X$ in terms of projective modules over a ring $R(G,X)$ is given. This generalizes Swan's classification of vector bundles and uses the results of Mulvey. Often, $R$ can be taken to be noetherian; in this case $ {\text{Spec}}(R)$ is usually reducible with "cohomologically trivial" irreducible components. Information is derived concerning the nature of projective modules over such rings, and some results are obtained indicating how such information reflects information about $ X$.
On the convergence of closed-valued measurable multifunctions
Gabriella
Salinetti;
Roger J.-B.
Wets
275-289
Abstract: In this paper we study the convergence almost everywhere and in measure of sequences of closed-valued multifunctions. We first give a number of criteria for the convergence of sequences of closed subsets. These results are used to obtain various characterizations for the convergence of measurable multifunctions. In particular we are interested in the convergence properties of (measurable) selections.
$C\sp{\ast} $-extreme points
Alan
Hopenwasser;
Robert L.
Moore;
V. I.
Paulsen
291-307
Abstract: Let $\mathcal{A}$ be a $ {C^ \ast }$-algebra and let $\mathcal{S}$ be a subset of $\mathcal{A}$. $ \mathcal{S}$ is ${C^ \ast }$-convex if whenever ${T_1},{T_2}, \ldots ,{T_n}$ are in $\mathcal{S}$ and ${A_1}, \ldots ,{A_n}$ are in $\mathcal{A}$ with $\sum\nolimits_{i = 1}^n {A_i^ \ast {A_i} = I}$, then $\sum\nolimits_{i = 1}^n {A_i^ \ast {T_i}{A_i}}$ is in $\mathcal{S}$. An element $T$ in $ \mathcal{S}$ is called ${C^ \ast }$-extreme in $ \mathcal{S}$ if whenever $T = \sum\nolimits_{i = 1}^n {A_i^ \ast {T_i}{A_i}}$ with ${T_i}$ and ${A_i}$ as above and with ${A_i}$ invertible, then ${T_i}$ is unitarily equivalent to $T$ for each $i$. We investigate the linear extreme points and ${C^ \ast }$-extreme points for three sets: first, the unit ball of operators in Hilbert space; next, the set of $2 \times 2$ matrices with numerical radius bounded by $1$; and last, the unit interval of positive operators on Hilbert space. In particular we find that for the second set, the linear and ${C^ \ast }$-extreme points are different.
Power series methods of summability: positivity and gap perfectness
A.
Jakimovski;
W.
Meyer-König;
K.
Zeller
309-317
Abstract: A class of power series methods of summability is defined. By means of a positivity argument (Bohman-Korovkin) it is shown that each method of the class is gap perfect. This fact facilitates the proof of Tauberian gap theorems. Several examples are given.
Projective geometry on partially ordered sets
Ulrich
Faigle;
Christian
Herrmann
319-332
Abstract: A set of axioms is presented for a projective geometry as an incidence structure on partially ordered sets of "points" and "lines". The axioms reduce to the axioms of classical projective geometry in the case where the points and lines are unordered. It is shown that the lattice of linear subsets of a projective geometry is modular and that every modular lattice of finite length is isomorphic to the lattice of linear subsets of some finite-dimensional projective geometry. Primary geometries are introduced as the incidence-geometric counterpart of primary lattices. Thus the theory of finite-dimensional projective geometries includes the theory of finite-dimensional projective Hjelmslev-spaces of level $n$ as a special case. Finally, projective geometries are characterized by incidence properties of points and hyperplanes.
Erratum to: ``Permutation-partition pairs: a combinatorial generalization of graph embeddings''
Saul
Stahl
333
Characterizations of the Fischer groups. I, II, III
David
Parrott
303-347
Abstract: B. Fischer, in his work on finite groups which contain a conjugacy class of $ 3$-transpositions, discovered three new sporadic finite simple groups, usually denoted $M(22)$, $M(23)$ and $M(24)'$. In Part I two of these groups, $ M(22)$ and $M(23)$, are characterized by the structure of the centralizer of a central involution. In addition, the simple groups ${U_6}(2)$ (often denoted by $M(21))$ and $ P\Omega (7,3)$, both of which are closely connected with Fischer's groups, are characterized by the same method. The largest of the three Fischer groups $M(24)$ is not simple but contains a simple subgroup $ M(24)'$ of index two. In Part II we give a similar characterization by the centralizer of a central involution of $M(24)$ and also a partial characterization of the simple group $M(24)'$. The purpose of Part III is to complete the characterization of $M(24)'$ by showing that our abstract group $ G$ is isomorphic to $ M(24)'$. We first prove that $G$ contains a subgroup $X \cong M(23)$ and then we construct a graph (on the cosets of $X$) which is shown to be isomorphic to the graph for $ M(24)$.
A unicity theorem for meromorphic mappings between algebraic varieties
S. J.
Drouilhet
349-358
Abstract: Using the techniques of value distribution theory in several complex variables, we obtain a theorem which can be used to determine whether two nondegenerate meromorphic mappings from an affine algebraic variety to a projective algebraic variety of the same or lower dimension are identical. The theorem generalizes a result of $R$. Nevanlinna in one complex variable.
Algebraic invariants of boundary links
Nobuyuki
Sato
359-374
Abstract: In this paper we study the homology of the universal abelian cover of the complement of a boundary link of $n$-spheres in $ {S^{n + 2}}$, as modules over the (free abelian) group of covering transformations. A consequence of our results is a characterization of the polynomial invariants ${p_{i,q}}$ of boundary links for $1 \leqslant q \leqslant [n/2]$. Along the way we address the following algebraic problem: given a homomorphism of commutative rings $ f:R \to S$ and a chain complex ${C_ \ast }$ over $R$, determine when the complex $S{ \otimes _R}{C_ \ast }$ is acyclic. The present work is a step toward the characterization of link modules in general.
Random evolution processes with feedback
Kyle
Siegrist
375-392
Abstract: A general random evolution Markov process is constructed which switches back and forth at random among a given collection of Markov processes ("modes of evolution") defined on a common evolution state space and indexed by an evolution rule space. Feedback is incorporated by allowing the path of the evolution component to influence the changes in evolution rule. The semigroup of the random evolution process is studied and is used to compare the process with the operator random evolutions of Griego and Hersh. Using deterministic modes of evolution, we generalize the Markov processes constructed by Erickson and by Heath. We also study new random evolution processes constructed from Brownian motions and from regular step processes.
Semigroup compactifications of semidirect products
H. D.
Junghenn;
B. T.
Lerner
393-404
Abstract: Let $S$ and $T$ be semigroups, $S\circlebin{\tau} T$ a semidirect product, and $ F$ a ${C^ \ast }$-algebra of bounded, complex-valued functions on $ S\circlebin{\tau} T$. Necessary and sufficient conditions are given for the $ F$-compactification of $ S\circlebin{\tau} T$ to be expressible as a semidirect product of compactifications of $S$ and $T$. This result is used to show that the strongly almost periodic compactification of $S\circlebin{\tau} T$ is a semidirect product and that, in certain general cases, the analogous statement holds for the almost periodic compactification and the left uniformly continuous compactification of $S\circlebin{\tau} T$. Applications are made to wreath products.
A tangential convergence for bounded harmonic functions on a rank one symmetric space
Jacek
Cygan
405-418
Abstract: Let $u$ be a bounded harmonic function on a noncompact rank one symmetric space $M = G/K \approx {N^ - }A,{N^ - }AK$ being a fixed Iwasawa decomposition of $ G$. We prove that if for an ${a_0} \in A$ there exists a limit $u(n{a_0}) \equiv {c_0}$, as $n \in {N^ - }$ goes to infinity, then for any $a \in A$, $u(na) = {c_0}$. For $M = SU(n,1)/S(U(n) \times U(1)) = {B^n}$, the unit ball in $ {{\mathbf{C}}^n}$ with the Bergman metric, this is a result of Hulanicki and Ricci, and in this case it reads (via the Cayley transformation) as a theorem on convergence of a bounded harmonic function to a boundary value at a fixed boundary point, along appropriate, tangent to $\partial {B^n}$, surfaces.
Tensegrity frameworks
B.
Roth;
W.
Whiteley
419-446
Abstract: A tensegrity framework consists of bars which preserve the distance between certain pairs of vertices, cables which provide an upper bound for the distance between some other pairs of vertices and struts which give a lower bound for the distance between still other pairs of vertices. The present paper establishes some basic results concerning the rigidity, flexibility, infinitesimal rigidity and infinitesimal flexibility of tensegrity frameworks. These results are then applied to a number of questions, problems and conjectures regarding tensegrity frameworks in the plane and in space.
R\'ealisation de morphismes donn\'es en cohomologie et suite spectrale d'Eilenberg-Moore
Micheline
Vigué-Poirrier
447-484
Abstract: On construit une suite d'obstructions à la réalisation par une application entre types d'homotopie rationnelle, d'un morphisme donné en cohomologie. On donne, sous des hypothèses de finitude, des conditions simples d'existence de réalisation. On montre aussi que, pour des algèbres différentielles commutatives graduées sur un corps de caractéristique 0, la réalisation d'un morphisme donné en cohomologie dépend, en général, du corps de base. La technique utilisée est la construction du modèle minimal bigradué d'un homomorphisme d'algèbres commutatives graduées, puis du modèle filtré d'une application continue, par déformation des différentielles du modèle bigradué de l'application induite en cohomologie. Cette construction est utilisée pour donner une méthode explicite de calcul de la suite spectrale d'Eilenberg-Moore $({E_i},{d_i})$ d'un carré fibré. On en déduit des critères pour que ${d_i} = 0,i \geqslant 2$.
Unflat connections in $3$-sphere bundles over $S\sp{4}$
Andrzej
Derdziński;
A.
Rigas
485-493
Abstract: The paper concerns connections in $3$-sphere bundles over $4$-manifolds having the property of unflatness, which is a necessary condition in order that a natural construction give a Riemannian metric of positive sectional curvature in the total space. It is shown that, as conjectured by A. Weinstein, the only $3$-sphere bundle over ${S^4}$ with an unflat connection is the Hopf bundle.
The order convergence of martingales indexed by directed sets
Kenneth A.
Astbury
495-510
Abstract: We obtain a condition on the underlying family of $\sigma$-algebras which is properly weaker than the Vitali property but which is also a sufficient condition for the order convergence of martingales of semibounded variation. We also obtain a sufficient condition for the order convergence of martingales of semibounded variation in terms of the finiteness of the extreme order limits of martingales of bounded variation.
Weighted estimates for fractional powers of partial differential operators
Raymond
Johnson
511-525
Abstract: It is shown that fractional powers defined by the wave polynomial $P(\xi ) = \xi _{^1}^2 + \cdots + \xi _p^2 - \xi _{p + 1}^2 - \cdots - \xi _n^2$, defined in terms of Fourier transforms by $\widehat{{T^\lambda }f} = {\left\vert {P(\xi )} \right\vert^\lambda }\hat f$, are in the Bernstein subalgebra of functions with integrable Fourier transforms for $ \lambda > (n - 1)/2$, provided $f \in C_c^m$ with $m$ large enough. The proof uses embedding theorems for Besov spaces and Stein's theorem on interpolation of analytic families of operators.
Recognizing the real line
Yuri
Gurevich;
W. Charles
Holland
527-534
Abstract: A certain elementary statement about the group of automorphisms of the real line $\mathbf{R}$ is sufficient to characterize $\mathbf{R}$ among homogeneous chains. A similar result holds for the chain of rational numbers.
Multiplicatively invariant subspaces of Besov spaces
Per
Nilsson
535-543
Abstract: We study subspaces of Besov spaces $B_p^{s,q}$ which are invariant under pointwise multiplication by characters. The case $s > 0$ is completely described, and for the case $s \leqslant 0$ we extend known results.
Arc-smooth continua
J. B.
Fugate;
G. R.
Gordh;
Lewis
Lum
545-561
Abstract: Continua admitting arc-structures and arc-smooth continua are introduced as higher dimensional analogues of dendroids and smooth dendroids, respectively. These continua include such spaces as: cones over compacta, convex continua in ${l_2}$, strongly convex metric continua, injectively metrizable continua, as well as various topological semigroups, partially ordered spaces, and hyperspaces. The arc-smooth continua are shown to coincide with the freely contractible continua and with the metric $ K$-spaces of Stadtlander. Known characterizations of smoothness in dendroids involving closed partial orders, the set function $ T$, radially convex metrics, continuous selections, and order preserving mappings are extended to the setting of continua with arc-structures. Various consequences of the special contractibility properties of arc-smooth continua are also obtained.
A faithful Hille-Yosida theorem for finite-dimensional evolutions
M. A.
Freedman
563-573
Abstract: As a natural generalization of the classical Hille-Yosida theorem to evolution operators, necessary and sufficient conditions are found for an evolution $U$ acting in ${R^N}$ so that for each $s \geqslant t$, $U(s,t)$ can be uniquely represented as a product integral $ \prod _t^s{[I + V]^{ - 1}}$ for some additive, accretive generator $ V$. Under these conditions, we further show that $ U(\xi ,\zeta )$ is differentiable a.e.
On supercuspidal representations of the metaplectic group
James
Meister
575-598
Abstract: The Weil representations associated to anisotropic quadratic forms in one and three variables are used to study supercuspidal representations of the two-fold metaplectic covering group ${\overline {{\text{GL}}} _2}(k)$, where $ k$ is a local nonarchimedean field of odd residual characteristic. The principal result is the explicit calculation of certain Whittaker functionals for any square-integrable irreducible admissible genuine representation of ${\overline {{\text{GL}}} _2}(k)$. In particular, a recent conjecture of Gelbart and Piatetski-Shapiro is answered by obtaining a bijection between the set of quasicharacters of ${k^ \ast }$ and the set of irreducible admissible genuine distinguished representations of ${\overline {{\text{GL}}} _2}(k)$, i.e. those representations possessing only one Whittaker functional, or, equivalently, those having a unique Whittaker model. The distinguished representations are precisely the representations attached to the Weil representation associated to a one dimensional form. The local piece of the generalized Shimura correspondence between automorphic forms of $ {\overline {{\text{GL}}} _2}({\mathbf{A}})$ and $ {\text{G}}{{\text{L}}_2}({\mathbf{A}})$ is also treated. Based upon a conjecture of the equivalences among the constituents of the Weil representations associated to two nonequivalent ternary forms, evidence for the explicit form of the local piece of this global correspondence, restricted to supercuspidal representations of ${\overline {{\text{GL}}} _2}(k)$, is presented. In this form, the map is shown to be injective and its image is described.
An elementary proof of the local Kronecker-Weber theorem
Michael
Rosen
599-605
Abstract: Let $K$ be a local field. Lubin and Tate have shown how to explicitly construct an abelian extension of $K$ which they prove to be the maximal abelian extension. Their proof of this result uses local class field theory. When $K$ is a $p$-adic field we give an elementary proof which even avoids the use of higher ramification groups. Instead we rely on facts about the principal units in a finite abelian extension of $K$ as a module for the Galois group.
Boolean powers: direct decomposition and isomorphism types
Kenneth
Hickin;
J. M.
Plotkin
607-621
Abstract: We determine properties of Boolean powers of groups and other algebraic structures, and we generalize Jónsson's theorem on Boolean powers of centerless, directly indecomposable groups. We show that every nonabelian, finitely generated group has $ {2^{{\aleph _0}}}$ nonisomorphic countable Boolean, and hence subcartesian, powers. We show that nonabelian groups $G$ such that either (i) $G$ is not the central product of two nonabelian groups or (ii) every pair of nontrivial normal subgroups of $G$ intersect nontrivially yield nonisomorphic Boolean powers with respect to nonisomorphic Boolean algebras.
Uniqueness of invariant means for measure-preserving transformations
Joseph
Rosenblatt
623-636
Abstract: For some compact abelian groups $X$ (e.g. $T^n$, $ n \geqslant 2$, and $ \prod\nolimits_{n = 1}^\infty {{Z_2}}$), the group $G$ of topological automorphisms of $ X$ has the Haar integral as the unique $G$-invariant mean on ${L_\infty }(X,{\lambda _X})$. This gives a new characterization of Lebesgue measure on the bounded Lebesgue measurable subsets $\beta$ of ${R^n}$, $ n \geqslant 3$; it is the unique normalized positive finitely-additive measure on $ \beta$ which is invariant under isometries and the transformation of ${R^n}:({x_1}, \ldots ,{x_n}) \mapsto ({x_1} + {x_2},{x_2}, \ldots ,{x_n})$. Other examples of, as well as necessary and sufficient conditions for, the uniqueness of a mean on ${L_\infty }(X,\beta ,p)$, which is invariant by some group of measure-preserving transformations of the probability space $(X,\beta ,p)$, are described.
Indecomposable representations of semisimple Lie groups
Birgit
Speh
1-34
Abstract: Let $G$ be a semisimple connected linear Lie group, ${\pi _1}$ a finite-dimensional irreducible representation of $G$, ${\pi _2}$ an infinite-dimensional irreducible representation of $G$ which has a nontrivial extension with $ {\pi _1}$. We study the category of all Harish-Chandra modules whose composition factors are equivalent to ${\pi _1}$ and ${\pi _2}$
Approximating topological surfaces in $4$-manifolds
Gerard A.
Venema
35-45
Abstract: Let ${M^2}$ be a compact, connected $ 2$-manifold with $ \partial {M^2} \ne \emptyset$ and let $ h:{M^2} \to {W^4}$ be a topological embedding of ${M^2}$ into a $4$-manifold. The main theorem of this paper asserts that if ${W^4}$ is a piecewise linear $4$-manifold, then $h$ can be arbitrarily closely approximated by locally flat PL embeddings. It is also shown that if the $4$-dimensional annulus conjecture is correct and if $ W$ is a topological $ 4$-manifold, then $ h$ can be arbitrarily closely approximated by locally flat embeddings. These results generalize the author's previous theorems about approximating disks in $4$-space.
On connectivity in matroids and graphs
James G.
Oxley
47-58
Abstract: In this paper we derive several results for connected matroids and use these to obtain new results for -connected graphs. In particular, we generalize work of Murty and Seymour on the number of two-element cocircuits in a minimally connected matroid, and work of Dirac, Plummer and Mader on the number of vertices of degree two in a minimally $ 2$-connected graph. We also solve a problem of Murty by giving a straightforward but useful characterization of minimally connected matroids. The final part of the paper gives a matroid generalization of Dirac and Plummer's result that every minimally $2$-connected graph is $3$-colourable.
Random ergodic sequences on LCA groups
Jakob I.
Reich
59-68
Abstract: Let $ {\{ X(t,\omega )\} _{t \in {{\mathbf{R}}^ + }}}$ be a stochastic process on a locally compact abelian group $G$, which has independent stationary increments. We show that under mild restrictions on $ G$ and $\{ X(t,\omega )\}$ the random families of probability measures $\displaystyle {\mu _T}( \cdot ,\omega ) = B_T^{ - 1}\int\limits_0^T {f(t){x_{( \cdot )}}} (X(t,\omega ))dt\quad {\text{for}}\;T > 0{\text{,}}$ where $ f(t)$ is a function from ${{\mathbf{R}}^ + }$ to ${{\mathbf{R}}^ + }$ of polynomial growth and ${B_T} = \int_0^T {f(t)} \;dt$, converge weakly to Haar measure of the Bohr compactification of $ G$. As a consequence we obtain mean and individual ergodic theorems and asymptotic occupancy times for these processes.
Iteration and the solution of functional equations for functions analytic in the unit disk
Carl C.
Cowen
69-95
Abstract: This paper considers the classical functional equations of Schroeder $f \circ \varphi = \lambda f$, and Abel $f \circ \varphi = f + 1$, and related problems of fractional iteration where $ \varphi$ is an analytic mapping of the open unit disk into itself. The main theorem states that under very general conditions there is a linear fractional transformation $ \Phi$ and a function $ \sigma$ analytic in the disk such that $\Phi \circ \sigma = \sigma \circ \varphi$ and that, with suitable normalization, $\Phi$ and $\sigma$ are unique. In particular, the hypotheses are satisfied if $\varphi$ is a probability generating function that does not have a double zero at 0. This intertwining relates solutions of functional equations for $ \varphi$ to solutions of the corresponding equations for $\Phi$. For example, it follows that if $ \varphi$ has no fixed points in the open disk, then the solution space of $ f \circ \varphi = \lambda f$ is infinite dimensional for every nonzero $ \lambda$. Although the discrete semigroup of iterates of $\varphi$ usually cannot be embedded in a continuous semigroup of analytic functions mapping the disk into itself, we find that for each $z$ in the disk, all sufficiently large fractional iterates of $\varphi$ can be defined at $z$. This enables us to find a function meromorphic in the disk that deserves to be called the infinitesimal generator of the semigroup of iterates of $ \varphi$. If the iterates of $\varphi$ can be embedded in a continuous semigroup, we show that the semigroup must come from the corresponding semigroup for $\Phi$, and thus be real analytic in $t$. The proof of the main theorem is not based on the well known limit technique introduced by Koenigs (1884) but rather on the construction of a Riemann surface on which an extension of $\varphi$ is a bijection. Much work is devoted to relating characteristics of $ \varphi$ to the particular linear fractional transformation constructed in the theorem.
Lewy's curves and chains on real hypersurfaces
James J.
Faran
97-109
Abstract: Lewy's curves on an analytic real hypersurface $M = \{ r(z,z) = 0\}$ in ${{\mathbf{C}}^2}$ are the intersections of $ M$ with any of the Segre hypersurfaces ${Q_w} = \{ z:r(z,w) = 0\} $. If $M$ is the standard unit sphere, these curves are chains in the sense of Chern and Moser. This paper shows the converse in the strictly pseudoconvex case: If all of Lewy's curves are chains, $ M$ is locally biholomorphically equivalent to the sphere. This is proven by analyzing the holomorphic structure of the space of chains. A similar statement is true about real hypersurfaces in $ {{\mathbf{C}}^n}$, $n > 2$, in which case the proof relies on a pseudoconformal analogue to the theorem in Riemannian geometry which states that a manifold having "sufficiently many" totally geodesic submanifolds is projectively flat.
Compactness properties of an operator which imply that it is an integral operator
A. R.
Schep
111-119
Abstract: In this paper we study necessary and (or) sufficient conditions on a given operator to be an integral operator. In particular we give another proof of a characterization of integral operators due to W. Schachermayer.
Tensor products of principal series for the De Sitter group
Robert P.
Martin
121-135
Abstract: The decomposition of the tensor product of two principal series representations is determined for the simply connected double covering, $G = {\text{Spin}}(4,1)$, of the DeSitter group. The main result is that this decomposition consists of two pieces, ${T_c}$ and ${T_d}$, where ${T_c}$ is a continuous direct sum with respect to Plancherel measure on $\hat G$ of representations from the principal series only and ${T_d}$ is a discrete sum of representations from the discrete series of $G$. The multiplicities of representations occurring in ${T_c}$ and ${T_d}$ are all finite.
Genealogy of periodic points of maps of the interval
Robert L.
Devaney
137-146
Abstract: We describe the behavior of families of periodic points in one parameter families of maps of the interval which feature a transition from simple dynamics with finitely many periodic points to chaotic mappings. In particular, we give topological criteria for the appearance and disappearance of these families. Our results apply specifically to quadratic maps of the form ${F_\mu }(x) = \mu x(1 - x)$.
An approximation to $\Omega \sp{n}\Sigma \sp{n}X$
J.
Caruso;
S.
Waner
147-162
Abstract: For an arbitrary (nonconnected) based space $X$, a geometrical construction ${\tilde C_n}X$ is given, such that ${\tilde C_n}X$ is weakly homotopy-equivalent to ${\Omega ^n}{\Sigma ^n}X$ as a $ {\mathcal{C}_n}$-space.
A simpler approximation to $QX$
Jeffrey L.
Caruso
163-167
Abstract: McDuff's construction ${C^ \pm }(M)$ of a space of positive and negative particles is modified to a space ${C^ \pm }({R^\infty },X)$, which is weakly homotopy equivalent to ${\Omega ^\infty }{\Sigma ^\infty }X$, for a locally equi-connected, nondegenerately based space $ X$.
Higher derivation Galois theory of fields
Nickolas
Heerema
169-179
Abstract: A Galois correspondence for finitely generated field extensions $ k/h$ is presented in the case characteristic $h = p \ne 0$. A field extension $k/h$ is Galois if it is modular and $h$ is separably algebraically closed in $k$. Galois groups are the direct limit of groups of higher derivations having rank a power of $p$. Galois groups are characterized in terms of abelian iterative generating sets in a manner which reflects the similarity between the finite rank and infinite rank theories of Heerema and Deveney [9] and gives rise to a theory which encompasses both. Certain intermediate field theorems obtained by Deveney in the finite rank case are extended to the general theory.
Poincar\'e-Bendixson theory for leaves of codimension one
John
Cantwell;
Lawrence
Conlon
181-209
Abstract: The level of a local minimal set of a ${C^2}$ codimension-one foliation of a compact manifold is a nonnegative integer defined inductively, level zero corresponding to the minimal sets in the usual sense. Each leaf of a local minimal set at level $ k$ is at level $ k$. The authors develop a theory of local minimal sets, level, and how leaves at level $k$ asymptotically approach leaves at lower level. This last generalizes the classical Poincaré-Bendixson theorem and provides information relating growth, topological type, and level, e.g. if $ L$ is a totally proper leaf at level $k$ then $L$ has exactly polynomial growth of degree $ k$ and topological type $ k - 1$.
The arithmetic perfection of Buchsbaum-Eisenbud varieties and generic modules of projective dimension two
Craig
Huneke
211-233
Abstract: We prove the ideals associated with the construction of generic complexes are prime and arithmetically perfect. This is used to construct the generic resolution for modules of projective dimension two.
Unique minimality of Fourier projections
S. D.
Fisher;
P. D.
Morris;
D. E.
Wulbert
235-246
Abstract: The question of when the Fourier projection is the only one of least norm from a space of continuous functions on the circle onto spaces spanned by trigonometric polynomials is studied in two settings. In the first the domain space is the disc algebra and the range is finite-dimensional. In the second the domain space consists of all real continuous functions and the range has finite codimension.
Finitely additive Markov chains
S.
Ramakrishnan
247-272
Abstract: In this paper we develop the theory of Markov chains with stationary transition probabilities, where the transition probabilities and the initial distribution are assumed only to be finitely additive. We prove a strong law of large numbers for recurrent chains. The problem of existence and uniqueness of finitely additive stationary initial distributions is studied and the ergodicity of recurrent chains under a stationary initial distribution is proved.
On spaces of maps of $n$-manifolds into the $n$-sphere
Vagn Lundsgaard
Hansen
273-281
Abstract: The space of (continuous) maps of a closed, oriented manifold $ {C^n}$ into the $ n$-sphere ${S^n}$ has a countable number of (path-) components. In this paper we make a general study of the homotopy classification problem for such a set of components. For $ {C^n} = {S^n}$, the problem was solved in [4], and for an arbitrary closed, oriented surface ${C^2}$, it was solved in [5]. We get a complete solution for manifolds ${C^n}$ of even dimension $n \geqslant 4$ with vanishing first Betti number. For odd dimensional manifolds ${C^n}$ we show that there are at most two different homotopy types among the components. Finally, for a class of manifolds introduced by Puppe [8] under the name spherelike manifolds, we get a complete analogue to the main theorem in [4] concerning the class of spheres.
A class of extremal functions for the Fourier transform
S. W.
Graham;
Jeffrey D.
Vaaler
283-302
Abstract: We determine a class of real valued, integrable functions $f(x)$ and corresponding functions $ {M_f}(x)$ such that $ f(x) \leqslant {M_f}(x)$ for all $x$, the Fourier transform ${\hat M_f}(t)$ is zero when $\left\vert t \right\vert \geqslant 1$, and the value of ${\hat M_f}(0)$ is minimized. Several applications of these functions to number theory and analysis are given.
The kinematic formula in complex integral geometry
Theodore
Shifrin
255-293
Abstract: Given two nonsingular projective algebraic varieties $X,Y \subset {{\mathbf{P}}^n}$, $Y \subset {{\mathbf{P}}^n}$ meeting transversely, it is classical that one may express the Chern classes of their intersection $X \cap Y$ in terms of the Chern classes of $X$ and $Y$ and the Kähler form (hyperplane class) of $ {{\mathbf{P}}^n}$. This depends on global considerations. However, by putting a hermitian connection on the tangent bundle of $X$, we may interpret the Chern classes of $ X$ as invariant polynomials in the curvature form of the connection. Armed with this local formulation of Chern classes, we now consider two complex submanifolds (not necessarily compact) $ X$, $Y \subset {{\mathbf{P}}^n}$, and investigate the geometry of their intersection. The pointwise relation between the Chern forms of $ X \cap Y$ and those of $ X$ and $Y$ is rather complicated. However, when we average integrals of Chern forms of $X \cap gY$ over all elements $ g$ of the group of motions of $ {{\mathbf{P}}^n}$, these can be expressed in a universal fashion in terms of integrals of Chern forms of $X$ and $Y$. This is, then, the kinematic formula for the unitary group.
Periods of iterated integrals of holomorphic forms on a compact Riemann surface
Shu Yin
Hwang Ma
295-300
Abstract: Holomorphic forms are integrated iteratedly along paths in a compact Riemann surface $M$ of genus $g$, thus inducing a homomorphism from the fundamental group $\Gamma = {\pi _1}(M,{P_0})$ to a proper multiplicative subgroup $G$ of the group of units in $\widehat{T({\Omega ^{1 \ast }})}$, where ${\Omega ^1}$ denotes the space of holomorphic forms on $T$ is the complex dual of $ {\Omega ^1}$, $ T$ means the associated tensor algebra and 11$\hat{ }$'' means completion with respect to the natural grading. The associated homomorphisms from $ \Gamma /{\Gamma ^{(n + 1)}}$ to $G/{G^{(n + 1)}}$ reduces to the classical case ${H_1}(M) \to {\Omega ^{1 \ast }}$ when $ n = 1$. We show that the images of $ \Gamma /{\Gamma ^{(n + 1)}}$ are always cocompact in $G/{G^{(n + 1)}}$ and are discrete for all $n \geqslant 2$ if and only if the Jacobian variety $ J(M)$ of $M$ is isogenous to ${E^g}$ for some elliptic curve $ E$ with complex multiplication.
On the structure of equationally complete varieties. II
Don
Pigozzi
301-319
Abstract: Each member $\mathcal{V}$ of a large family of nonassociative or, when applicable, nondistributive varieties has the following universal property: Every variety $\mathcal{K}$ that satisfies certain very weak versions of the amalgamation and joint embedding properties is isomorphic, as a category, to a coreflective subcategory of some equationally complete subvariety of $ \mathcal{V}$. Moreover, the functor which serves to establish the isomorphism preserves injections. As a corollary one obtains the existence of equationally complete subvarieties of $\mathcal{V}$ that fail to have the amalgamation property and fail to be residually small. The family of varieties universal in the above sense includes commutative groupoids, bisemigroups (i.e., algebras with two independent associative operations), and quasi-groups.
The C. Neumann problem as a completely integrable system on an adjoint orbit
Tudor
Raţiu
321-329
Abstract: It is shown by purely Lie algebraic methods that the ${\text{C}}$. Neumann problem--the motion of a material point on a sphere under the influence of a quadratic potential--is a completely integrable system of Euler-Poisson equations on a minimal-dimensional orbit of a semidirect product of Lie algebras.
Real submanifolds of codimension two in complex manifolds
Hon Fei
Lai
331-352
Abstract: The equivalence problem for a real submanifold $M$ of dimension at least eight and codimension two in a complex manifold is solved under a certain nondegeneracy condition on the Levi form. If the Levi forms at all points of $M$ are equivalent, a normalized Cartan connection can be defined on a certain principal bundle over $ M$. The group of this bundle can be described by means of the osculating quartic of $ M$ or the prolongation of the graded Lie algebra of type $ {\mathfrak{g}_2} \oplus {\mathfrak{g}_1}$ defined by the Levi form.
Configurations of surfaces in $4$-manifolds
Patrick M.
Gilmer
353-380
Abstract: We consider collections of surfaces $ \{ {F_i}\}$ smoothly embedded, except for a finite number of isolated singularities, self-intersections, and mutual intersections, in a $4$-manifold $M$. A small $3$-sphere about each exceptional point will intersect these surfaces in a link. If $[{F_i}] \in {H_2}(M)$ are linearly dependent modulo a prime power, we find lower bounds for $ \Sigma$ genus $ ({F_i})$ in terms of the $ [{F_i}]$, and invariants of the links that describe the exceptional points.
Division by holomorphic functions and convolution equations in infinite dimension
J.-F.
Colombeau;
R.
Gay;
B.
Perrot
381-391
Abstract: Let $E$ be a complex complete dual nuclear locally convex space (i.e. its strong dual is nuclear), $ \Omega$ a connected open set in $E$ and $ \mathcal{E}(\Omega )$ the space of the $ {C^\infty }$ functions on $ \Omega$ (in the real sense). Then we show that any element of $\operatorname{Exp} (E')$ in terms of the zero set of their characteristic functions.
Boundary interpolation sets for holomorphic functions smooth to the boundary and BMO
Joaquim
Bruna
393-409
Abstract: Let ${A^p}$ denote the class of holomorphic functions on the unit disc whose first $p$-derivatives belong to the disc algebra. We characterize the boundary interpolation sets for ${A^p}$, that is, those closed sets $E \subset T$ such that every function in $ {C^p}(E)$ extends to a function in ${A^p}$. We also give a constructive proof of the corresponding result for $ {A^\infty }$ (see [1]). We show that the structure of these sets is in some sense related to BMO and that this fact can be used to obtain precise estimates of outer functions vanishing on $E$.
The $\aleph \sb{2}$-Souslin hypothesis
Richard
Laver;
Saharon
Shelah
411-417
Abstract: We prove the consistency with $CH$ that there are no $ {\aleph _2}$-Souslin trees.
Residually small varieties with modular congruence lattices
Ralph
Freese;
Ralph
McKenzie
419-430
Abstract: We focus on varieties $\mathcal{V}$ of universal algebras whose congruence lattices are all modular. No further conditions are assumed. We prove that if the variety $\mathcal{V}$ is residually small, then the following law holds identically for congruences over algebras in $ \mathcal{V}:\beta \cdot [\delta ,\delta ] \leqslant [\beta ,\delta ]$. (The symbols in this formula refer to lattice operations and the commutator operation defined over any modular variety, by Hagemann and Herrmann.) We prove that a finitely generated modular variety $\mathcal{V}$ is residually small if and only if it satisfies this commutator identity, and in that case $ \mathcal{V}$ is actually residually $< n$ for some finite integer $n$. It is further proved that in a modular variety generated by a finite algebra $ A$ the chief factors of any finite algebra are bounded in cardinality by the size of $A$, and every simple algebra in the variety has a cardinality at most that of $A$.
Stability theorems for the continuous spectrum of a negatively curved manifold
Harold
Donnelly
431-448
Abstract: The spectrum of the Laplacian $\Delta$ for a simply connected complete negatively curved Riemannian manifold is studied. The Laplacian ${\Delta _0}$ of a simply connected constant curvature space ${M_0}$ is known up to unitary equivalence. Decay conditions are given, on the metric $g$ and curvature $K$ of $M$, which imply that the continuous part of ${\Delta _0}$ is unitarily equivalent to ${\Delta _0}$.
Some restrictions on finite groups acting freely on $(S\sp{n})\sp{k}$
Gunnar
Carlsson
449-457
Abstract: Restrictions other than rank conditions on elementary abelian subgroups are found for finite groups acting freely on ${({S^n})^k}$, with trivial action on homology. It is shown that elements $x$ of order $p$, $p$ an odd prime, with $x$ in the normalizer of an elementary abelian $ 2$-subgroup $E$ of $G$, must act trivially on $E$ unless $p\vert(n + 1)$. It is also shown that if $ p = 3$ or $7$, $x$ must act trivially, independent of $ n$. This produces a large family of groups which do not act freely on ${({S^n})^k}$ for any values of $n$ and $k$. For certain primes $p$, using the mod two Steenrod algebra, one may show that $x$ acts trivially unless ${2^{\mu (p)}}\vert(n + 1)$, where $\mu (p)$ is an integer depending on $ p$.
Adjoint operators in Lie algebras and the classification of simple flexible Lie-admissible algebras
Susumu
Okubo;
Hyo Chul
Myung
459-472
Abstract: Let $\mathfrak{A}$ be a finite-dimensional flexible Lie-admissible algebra over an algebraically closed field $F$ of characteristic 0. It is shown that if ${\mathfrak{A}^ - }$ is a simple Lie algebra which is not of type $ {A_n}(n \geqslant 2)$ then $\mathfrak{A}$ is a Lie algebra isomorphic to ${\mathfrak{A}^ - }$, and if ${\mathfrak{A}^ - }$ is a simple Lie algebra of type ${A_n}(n \geqslant 2)$ then $\mathfrak{A}$ is either a Lie algebra or isomorphic to an algebra with multiplication $x \ast y = \mu xy + (1 - \mu )yx - (1/(n + 1))\operatorname{Tr} (xy)I$ which is defined on the space of $(n + 1) \times (n + 1)$ traceless matrices over $F$, where $xy$ is the matrix product and $\mu \ne \frac{1} {2}$ is a fixed scalar in $ F$. This result for the complex field has been previously obtained by employing an analytic method. The present classification is applied to determine all flexible Lie-admissible algebras $ \mathfrak{A}$ such that ${\mathfrak{A}^ - }$ is reductive and the Levi-factor of $ {\mathfrak{A}^ - }$ is simple. The central idea is the notion of adjoint operators in Lie algebras which has been studied in physical literature in conjunction with representation theory.
BP torsion in finite $H$-spaces
Richard
Kane
473-497
Abstract: Let $p$ be odd and $(X,\mu )$ a $1$-connected $ \operatorname{mod} p$ finite $H$-space. It is shown that for $n \geqslant 1$ the Morava $K$-theories, $k{(n)_ \ast }(X)$ and $k{(n)^ \ast }(X)$, have no higher ${\upsilon _n}$ torsion. Also examples are constructed to show that $ {\upsilon _1}$ torsion in $ BP{\langle 1\rangle ^ \ast }(X)$ can be of arbitrarily high order.
Free coverings and modules of boundary links
Nobuyuki
Sato
499-505
Abstract: Let $L = \{ {K_1}, \ldots ,{K_m}\}$ be a boundary link of $n$-spheres in $ {S^{n + 2}}$, where $n \geqslant 3$, and let $X$ be the complement of $L$. Although most of the classical link invariants come from the homology of the universal abelian cover $ \tilde X$ of $ X$, with increasing $ m$ these groups become difficult to manage. For boundary links, there is a canonical free covering $ {X_\omega }$, which is simultaneously a cover of $\tilde X$. Thus, knowledge of ${H_ \ast }{X_\omega }$ yields knowledge of ${H_ \ast }\tilde X$. We study general properties of such covers and obtain, for $ 1 < q < n/2$, a characterization of the groups ${H_q}{X_\omega }$ as modules over the group of covering transformations. Some applications follow.
The Radon-Nikod\'ym property in conjugate Banach spaces. II
Charles
Stegall
507-519
Abstract: In the first part of this article the following result was proved. Theorem. The dual of a Banach space $X$ has the Radon-Nikodym property if and only if for every closed, linear separable subspace $Y$ of $X$, ${Y^ \ast }$ is separable. We find other, more detailed descriptions of Banach spaces whose duals have the Radon-Nikodym property.
On perfect measures
G.
Koumoullis
521-537
Abstract: Let $\mu$ be a nonzero positive perfect measure on a $\sigma$-algebra of subsets of a set $ X$. It is proved that if $\{ {A_i}:i \in I\}$ is a partition of $ X$ with ${\mu ^ \ast }({A_i}) = 0$ for all $i \in I$ and the cardinal of $ I$ non-(Ulam-) measurable, then there is $ J \subset I$ such that $ { \cup _{_{i \in J}}}{A_i}$ is not $\mu$-measurable, generalizing a theorem of Solovay about the Lebesgue measure. This result is used for the study of perfect measures on topological spaces. It is proved that every perfect Borel measure on a metric space is tight if and only if the cardinal of the space is nonmeasurable. The same result is extended to some nonmetric spaces and the relation between perfectness and other smoothness properties of measures on topological spaces is investigated.
Generalized $3$-manifolds whose nonmanifold set has neighborhoods bounded by tori
Matthew G.
Brin
539-555
Abstract: We show that all compact, ANR, generalized $3$-manifolds whose nonmanifold set is 0-dimensional and has a neighborhood system bounded by tori are cell-like images of compact $3$-manifolds if and only if the Poincaré conjecture is true. We also discuss to what extent the assumption of the Poincaré conjecture can be replaced by other hypotheses.
The strong convergence of Schr\"odinger propagators
Alan D.
Sloan
557-570
Abstract: Time dependent versions of the Trotter-Kato theorem are discussed using nonstandard analysis. Both standard and nonstandard results are obtained. In particular, it is shown that if a sequence of generators converges in the strong resolvent topology at each time to a limiting generator and if the sequence of generators and limiting generator uniformly satisfy Kisynski type hypotheses then the corresponding Schrodinger propagators converge strongly. The results are used to analyze time dependent, form bounded perturbations of the Laplacian.
Non-quasi-well behaved closed $\ast $-derivations
Frederick M.
Goodman
571-578
Abstract: Examples are given of a non-quasi-well behaved closed * derivation in $C([0,1] \times [0,1])$ extending the partial derivative, and of a compact subset $ \Omega$ of the plane such that $C(\Omega )$ has no nonzero quasi-well behaved * derivations but $ C(\Omega )$ does admit nonzero closed * derivations.
A short proof of Castelnuovo's criterion of rationality
William E.
Lang
579-582
Abstract: We give a new proof in positive characteristic of Castelnuovo's criterion of rationality of algebraic surfaces. We use crystalline cohomology and the de Rham-Witt complex as a substitute for the transcendeal methods of Kodaira.
A correction and some additions to: ``Reparametrization of $n$-flows of zero entropy'' [Trans. Amer. Math. Soc. {\bf 256} (1979), 289--304; MR 81h:28012]
J.
Feldman;
D.
Nadler
583-585
Abstract: In addition to correcting an error in the previously mentioned paper, we show that if $\upsilon \mapsto {\varphi _w}$ and $w \mapsto {\Psi _\sigma }$ on $X$ and $Y$ are $n$- and $m$-flows, respectively, then the $(n + m)$-flow $ (\upsilon ,w) \mapsto {\varphi _\upsilon } \times {\Psi _w}$ on $X \times Y$ is "loosely Kronecker" if and only if $ \varphi$ and $ \Psi$ are.
The genus of a map
Sara
Hurvitz
1-28
Abstract: The elements $[f'](f':X' \to Y')$ of the genus $- G(f)$ of a map $f:X \to Y$ are equivalence classes of homotopy classes of maps $f'$ which satisfy: For every prime $p$ there exist homotopy equivalences $X - {G^X}(f)$ and the genus of $f$ over $ Y - {G_Y}(f)$ are defined similarly. In this paper we prove that under certain conditions on $f$, the sets $G(f)$, ${G^X}(f)$ and ${G_Y}(f)$ are finite and admit an abelian group structure. We also compare the genus of $f$ to those of $X$ and $Y$, calculate it for some principal fibrations of the form $ K(G,n - 1) \to X \to Y$, and deal with the noncancellation phenomenon.
Orientation-reversing Morse-Smale diffeomorphisms on the torus
Steve
Batterson
29-37
Abstract: For orientation-reversing diffeomorphisms on the torus necessary and sufficient conditions are given for an isotopy class to admit a Morse-Smale diffeomorphism with a specified periodic behavior.
Schur products of operators and the essential numerical range
Quentin F.
Stout
39-47
Abstract: Let $ \mathcal{E} = \{ {e_n}\} _{n = 1}^\infty$ be an orthonormal basis for a Hilbert space $ \mathcal{H}$. For operators $A$ and $B$ having matrices $({a_{ij}})_{i,\;j = 1}^\infty$ and $ ({b_{ij}})_{i,\;j}^\infty = 1$, their Schur product is defined to be $ ({a_{ij}}{b_{ij}})_{i,\:j}^\infty = 1$. This gives $ \mathcal{B}(\mathcal{H})$ a new Banach algebra structure, denoted $ {\mathcal{P}_\mathcal{E}}$. For any operator $T$ it is shown that $T$ is in the kernel (hull(compact operators)) in some $ {\mathcal{B}_\mathcal{E}}$ iff 0 is in the essential numerical range of $ T$. These conditions are also equivalent to the property that there is a basis such that Schur multiplication by $ T$ is a compact operator mapping Schatten classes into smaller Schatten classes. Thus we provide new results linking $ \mathcal{B}(\mathcal{H})$, $ {\mathcal{B}_\mathcal{E}}$ and $ \mathcal{B}(\mathcal{B}(\mathcal{H}))$.
Some general theorems on the cohomology of classifying spaces of compact Lie groups
Mark
Feshbach
49-58
Abstract: This paper is divided into two parts. The first part proves a number of general theorems on the cohomology of the classifying spaces of compact Lie groups. These theorems are proved by transfer methods, relying heavily on the double coset theorem [F$_{1}$]. Several of these results are well known while others are quite new. For the most part the proofs of the theorems are independent of each other and are quite short. Nevertheless they are true in great generality. Several are proven for arbitrary compact Lie groups and arbitrary cohomology theories. Perhaps the most interesting of the new results relates the cohomology of the classifying space of an arbitrary compact Lie group with that of the normalizer of a maximal torus. The second part of the paper generalizes many theorems to certain equivariant cohomology theories. Some of these theorems appear in [F$ _{2}$].
On spectral theory and convexity
C. K.
Fong;
Louisa
Lam
59-75
Abstract: A compact convex set $K$ in a locally convex algebra is said to be a spectral carrier if, for all $x$, $y \in K$, we have $ xy = yx \in K$ and $x + y - xy \in K$. We show that if a compact convex set $K$ is a spectral carrier, then the idempotents in $ K$ are exactly the extreme points of $K$ and form a complete lattice. Conversely, if a compact set $K$ is a closed convex hull of a lattice of commuting idempotents, then $K$ is a spectral carrier. Furthermore, a metrizable spectral carrier is a Choquet simplex if and only if its extreme points form a chain of idempotents.
Submonotone subdifferentials of Lipschitz functions
Jonathan E.
Spingarn
77-89
Abstract: The class of "lowwer-${C^1}$" functions, that is functions which arise by taking the maximum of a compactly indexed family of $ {C^1}$ functions, is characterized in terms of properties of the generalized subdifferential. A locally Lipschitz function is shown to be lower-${C^1}$ if and only if its subdifferential is "strictly submonotone". Other properties of functions with "submonotone" subdifferentials are investigated.
Invariance of solutions to invariant nonparametric variational problems
John E.
Brothers
91-111
Abstract: Let $f$ be a weak solution to the Euler-Lagrange equation of a convex nonparametric variational integral in a bounded open subset $D$ of $ {{\mathbf{R}}^n}$. Assume the boundary $B$ of $D$ to be rectifiable. Let $D$ be a compact connected Lie group of diffeomorphisms of a neighborhood of $D \cup B$ which leave $D$ invariant and assume the variational integral to be $G$-invariant. Conditions are formulated which imply that if $f$ is continuous on $D \cup B$ and $ f \circ g\vert B = f\vert B$ for $g \in G$ then $ f \circ g = f$ for every $ g \in G$. If the integrand $ L$ is strictly convex then $ f$ can be shown to have a local uniqueness property which implies invariance. In case $L$ is not strictly convex the graph $ {T_f}$ of $f$ in ${{\mathbf{R}}^n} \times {\mathbf{R}}$ is interpreted as the solution to an invariant parametric variational problem, and invariance of ${T_f}$, hence of $f$, follows from previous results of the author. For this purpose a characterization is obtained of those nonparametric integrands on ${{\mathbf{R}}^n}$ which correspond to a convex positive even parametric integrand on $ {{\mathbf{R}}^n} \times {\mathbf{R}}$ in the same way that the nonparametric area integrand corresponds to the parametric area integrand.
A representation-theoretic criterion for local solvability of left invariant differential operators on nilpotent Lie groups
Lawrence
Corwin
113-120
Abstract: Let $L$ be a left invariant differential operator on the nilpotent Lie group $N$. It is shown that if $\pi (L)$ is invertible for all irreducible representations $\pi$ in general position (and if the inverses satisfy some mild technical conditions), then $ L$ is locally solvable. This result generalizes a theorem of ${\text{L}}$. Rothschild.
Arithmetic of elliptic curves upon quadratic extension
Kenneth
Kramer
121-135
Abstract: This paper is a study of variations in the rank of the Mordell-Weil group of an elliptic curve $E$ defined over a number field $F$ as one passes to quadratic extensions $K$ of $F$. Let $S(K)$ be the Selmer group for multiplication by $ 2$ on $E(K)$. In analogy with genus theory, we describe $S(K)$ in terms of various objects defined over $ F$ and the local norm indices $ {i_\upsilon } = {\dim _{{{\mathbf{F}}_2}}}E({F_\upsilon })/$Norm$\{ E({K_w})\}$ for each completion ${F_\upsilon }$ of $F$. In particular we show that $ \dim S(K) + \dim E{(K)_2}$ has the same parity as $\Sigma {i_\upsilon }$. We compute ${i_\upsilon }$ when $E$ has good or multiplicative reduction modulo $\upsilon$. Assuming that the $2$-primary component of the Tate-Shafarevitch group $ \mathcyr{SH}(K)$ is finite, as conjectured, we obtain the parity of rank $ E(K)$. For semistable elliptic curves defined over $ {\mathbf{Q}}$ and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of Birch and Swinnerton-Dyer.
The cohomology algebras of finite-dimensional Hopf algebras
Clarence
Wilkerson
137-150
Abstract: The cohomology algebra of a finite dimensional graded connected cocommutative biassociative Hopf algebra over a field $ K$ is shown to be a finitely generated $K$-algebra. Counterexamples to the analogue of a result of Quillen (that nonnilpotent cohomology classes should have nonzero restriction to some abelian sub-Hopf algebra) are constructed, but an elementary proof of the validity of this "detection principle" for the special case of finite sub-Hopf algebras of the $ \operatorname{mod} 2$ Steenrod algebra is given. As an application, an explicit formula for the Krull dimension of the cohomology algebras of the finite skeletons of the $\operatorname{mod} 2$ Steenrod algebra is given.
Homotopy groups of the space of self-homotopy-equivalences
Darryl
McCullough
151-163
Abstract: Let $M$ be a connected sum of $ r$ closed aspherical manifolds of dimension $n \geqslant 3$, and let $ EM$ denote the space of self-homotopy-equivalences of $M$, with basepoint the identity map of $ M$. Using obstruction theory, we calculate $ {\pi _q}(EM)$ for $1 \leqslant q \leqslant n - 3$ and show that ${\pi _{n - 1}}(EM)$ is not finitely-generated. As an application, for the case $n = 3$ and $r \geqslant 3$ we show that infinitely many generators of ${\pi _1}(E{M^3},{\text{i}}{{\text{d}}_M})$ can be realized by isotopies, to conclude that ${\pi _1}({\text{Homeo}}({M^3}),{\text{i}}{{\text{d}}_M})$ is not finitely-generated.
Uniqueness of product and coproduct decompositions in rational homotopy theory
Roy
Douglas;
Lex
Renner
165-180
Abstract: Let $X$ be a nilpotent rational homotopy type such that (1) $S(X)$, the image of the Hurewicz map has finite total rank, and (2) the basepoint map of $ M$, a minimal algebra for $ X$, is an element of the Zariski closure of $ {\text{Aut}}(M)$ in ${\text{End}}(M)$ (i.e. $X$ has "positive weight"). Then (A) any retract of $X$ satisfies the two properties above, (B) any two irreducible product decompositions of $ X$ are equivalent, and (C) any two irreducible coproduct decompositions of $ X$ are equivalent.
Which curves over ${\bf Z}$ have points with coordinates in a discrete ordered ring?
Lou
van den Dries
181-189
Abstract: A criterion is given for curves defined over $ {\mathbf{Z}}$ to have an infinite point in a discrete ordered ring. Using this, one can decide effectively whether a given polynomial in $ {\mathbf{Z}}[X,Y]$ has a zero in a model for the axioms of open induction. Riemann-Roch for curves over ${\mathbf{Q}}$ is the main tool used.
Quasisymmetric embeddings in Euclidean spaces
Jussi
Väisälä
191-204
Abstract: We consider quasi-symmetric embeddings $ f:G \to {R^n}$, $ G$ open in ${R^p}$, $ p \leqslant n$. If $ p = n$, quasi-symmetry implies quasi-conformality. The converse is true if $ G$ has a sufficiently smooth boundary. If $p < n$, the Hausdorff dimension of $fG$ is less than $n$. If $fG$ has a finite $p$-measure, $f$ preserves the property of being of $ p$-measure zero. If $ p < n$ and $n \geqslant 3$, ${R^n}$ contains a quasi-symmetric $ p$-cell which is topologically wild. We also prove auxiliary results on the relations between Hausdorff measure and Čech cohomology.
Characterization of some zero-dimensional separable metric spaces
Jan
van Mill
205-215
Abstract: Let $X$ be a separable metric zero-dimensional space for which all nonempty clopen subsets are homeomorphic. We show that, up to homeomorphism, there is at most one space $Y$ which can be written as an increasing union $\cup _{n = 1}^\infty {F_n}$ of closed sets so that for all $n \in {\mathbf{N}}$, ${F_n}$ is a copy of $X$ which is nowhere dense in ${F_{n + 1}}$. If moreover $X$ contains a closed nowhere dense copy of itself, then $Y$ is homeomorphic to ${\mathbf{Q}} \times X$ where ${\mathbf{Q}}$ denotes the space of rational numbers. This gives us topological characterizations of spaces such as ${\mathbf{Q}} \times {\mathbf{C}}$ and $ {\mathbf{Q}} \times {\mathbf{P}}$.
Some countability conditions on commutative ring extensions
Robert
Gilmer;
William
Heinzer
217-234
Abstract: If $S$ is a finitely generated unitary extension ring of the commutative ring $ R$, then $S$ cannot be expressed as the union of a strictly ascending sequence $\{ {R_n}\} _{n = 1}^\infty $ of intermediate subrings. A primary concern of this paper is that of determining the class of commutative rings $T$ for which the converse holds--that is, each unitary extension of $T$ not expressible as $\cup _1^\infty {T_i}$ is finitely generated over $ T$.
Localizable analytically uniform spaces and the fundamental principle
Sönke
Hansen
235-250
Abstract: The Fundamental Principle of Ehrenpreis states that the solutions of homogeneous linear partial differential equations with constant coefficients have natural integral representations. Using the Oka-Cartan procedure Ehrenpreis derived this theorem for spaces of functions and distributions which he called localizable analytically uniform (LAU-spaces). With a new definition of LAU-spaces we explain how Hörmander's results on cohomology with bounds fit into Ehrenpreis' method of proof of the Fundamental Principle. Furthermore, we show that many of the common Fréchet-Montel spaces of functions are LAU-spaces.
Class groups of cyclic groups of square-free order
Andrew
Matchett
251-254
Abstract: Let $G$ be a finite cyclic group of square free order. Let ${\text{Cl(}}ZG{\text{)}}$ denote the projective class group of the integral group ring $ZG$. Our main theorem describes explicitly the quotients of a certain filtration of $ {\text{Cl(}}ZG{\text{)}}$. The description is in terms of class groups and unit groups of the rings of cyclotomic integers involved in $ZG$. The proof is based on a Mayer-Vietoris sequence.
Degenerations of $K3$ surfaces of degree $4$
Jayant
Shah
271-308
Abstract: A generic $ K3$ surface of degree $ 4$ may be embedded as a nonsingular quartic surface in ${{\mathbf{P}}_3}$. Let $ f:X \to \operatorname{Spec} \;{\mathbf{C}}[[t]]$ be a family of quartic surfaces such that the generic fiber is regular. Let ${\Sigma _0}$, $ {\Sigma _2^0}$, ${\Sigma _4}$ be respectively a nonsingular quadric in $ {{\mathbf{P}}_3}$, a cone in $ {{\mathbf{P}}_3}$ over a nonsingular conic and a rational, ruled surface in $ {{\mathbf{P}}_9}$ which has a section with self intersection $- 4$. We show that there exists a flat, projective morphism $f':X' \to {\text{Spec}}\;{\mathbf{C}}[[t]]$ and a map $\rho :{\text{Spec}}\:{\mathbf{C}}[[t]] \to {\text{Spec}}\:{\mathbf{C}}[[t]]$ such that (i) the generic fiber of $ f'$ and the generic fiber of the pull-back of $f$ via $\rho$ are isomorphic, (ii) the fiber $ {X'_0}$ of $f'$ over the closed point of $ {\text{Spec}}\;{\mathbf{C}}[[t]]$ has only insignificant limit singularities and (iii) ${X'_0}$ is either a quadric surface or a double cover of ${\Sigma _0}$, ${\Sigma _2^0}$ or $ {\Sigma _4}$. The theorem is proved using the geometric invariant theory.
Product-convolution operators and mixed-norm spaces
Robert C.
Busby;
Harvey A.
Smith
309-341
Abstract: Conditions for boundedness and compactness of product-convolution operators $g \to {P_h}{C_f}g = h \cdot (f\ast g)$ on spaces $ {L_p}(G)$ are studied. It is necessary for boundedness to define a class of "mixed-norm" spaces $ {L_{(p,q)}}(G)$ interpolating the ${L_p}(G)$ spaces in a natural way $({L_{(p,p)}} = {L_p})$. It is then natural to study the operators acting between ${L_{(p,q)}}(G)$ spaces, where $G$ has a compact invariant neighborhood. The theory of $ {L_{(p,q)}}(G)$ is developed and boundedness and compactness conditions of a nonclassical type are obtained. It is demonstrated that the results extend easily to a somewhat broader class of integral operators. Several known results are strengthened or extended as incidental consequences of the investigation.
Capacity densities and angular limits of quasiregular mappings
Matti
Vuorinen
343-354
Abstract: It is shown that if a bounded quasiregular mapping of the unit ball ${B^n} \subset {R^n}$, $n \geqslant 2$, has a limit at $b \in \partial {B^n}$ through a set $E \subset {B^n}$ with $ b \in \bar E$, then it has an angular limit at $b$ provided that $E$ is contained in an open cone $C \subset {B^n}$ with vertex $b$ and that $E$ is thick enough at $b$. The thickness condition is expressed in terms of the $n$-capacity density.
Spherical bundles adapted to a $G$-fibration
J. P. E.
Hodgson
355-361
Abstract: A spherical fibration $p:E \to B$ is said to be adapted to a $G$-fibration $\pi :E \to E/G$ if there is a fibration $q:E/G \to B$ with fibre the quotient of a sphere by a free $G$-action and such that the composition $q \circ \pi = p$. In this paper it is shown that for spherical bundles in the PL, TOP or Homotopy categories that are adapted to ${Z_2}$- or ${S^1}$-fibrations there is a procedure analogous to the splitting principle for vector bundles that enables one to define characteristic classes for these fibrations and to relate them to the usual characteristic classes. The methods are applied to show that a spherical fibration over a $4$-connected base which is adapted to an $ {S^1}$-fibration admits a PL structure.
The Hewitt realcompactification of products
Haruto
Ohta
363-375
Abstract: For a completely regular Hausdorff space $X$, $\upsilon X$ denotes the Hewitt realcompactification of $X$. Given a topological property $\mathcal{P}$ of spaces, our interest is in characterizing the class $ \mathcal{R}(\mathcal{P})$ of all spaces $X$ such that $\upsilon (X \times Y) = \upsilon X \times \upsilon Y$ holds for each $\mathcal{P}$-space $Y$. In the present paper, we obtain such characterizations in the case that $ \mathcal{P}$ is locally compact and in the case that $ \mathcal{P}$ is metrizable.
Maximum principles for degenerate elliptic-parabolic equations with Venttsel's boundary condition
Kazuo
Amano
377-396
Abstract: In this paper, we first establish interior and boundary maximum principles for degenerate elliptic-parabolic equations; we state both principles in one single theorem in terms of the propagation set (cf. Theorem 1). We next generalize the boundary condition to Venttsel's one and obtain the similar result (cf. Theorem 2). Venttsel''s boundary condition contains Dirichlet, Neumann, oblique derivative and mixed boundary conditions as special cases and, from a probabilistic point of view (cf. Venttsel' [9]), it is the most general admissible boundary condition. We give several examples in the last section.
On computation of eigenvalues for differential equations by Newton's method
Zeev
Nehari
397-409
Abstract: Professor Zeev Nehari, in the last months before his untimely death, had been analyzing and developing a new algorithm for computing eigenvalues of selfadjoint boundary value problems of arbitrary order. Apparently, his main goals were Theorems 3.1 and 3.2 below (which incidentally yield the eigenvalues of the given problem, as well as those of a related problem). Unfortunately, as far as we know, Professor Nehari has not left a proof of the basic Lemma 3.1 which presumably was to be based on $\S 2$ or related results. It seems, however, worthwhile to publish this paper since the results of $ \S \S 1$ and $ 2$ are complete and of independent interest and since the question of the validity of Lemma 3.1 is also of interest. While the presentation in $\S 1$ and $2$ is in most respects complete, there are a few places where some minor clarifications by selected footnotes were felt to be desirable. Also, Professor Nehari's original incomplete $\S 3$ has been replaced by a revised and abbreviated version.
Derivations and automorphisms of nonassociative matrix algebras
G. M.
Benkart;
J. M.
Osborn
411-430
Abstract: This paper studies the derivation algebra and the automorphism group of $ {M_n}(A)$, $n \times n$ matrices over an arbitrary nonassociative algebra $A$ with multiplicative identity $1$. The investigation also includes results on derivations and automorphisms of the algebras obtained from ${M_n}(A)$ using the Lie product $[xy] = xy - yx$, and the Jordan product $x \circ y = \tfrac{1} {2}(xy + yx)$.
Basic sequences and subspaces in Lorentz sequence spaces without local convexity
Nicolae
Popa
431-456
Abstract: After some preliminary results $(\S1)$, we give in $\S2$ another proof of the result of N. J. Kalton [5] concerning the unicity of the unconditional bases of $ {l_p}$, $0 < p < 1$. Using this result we prove in §3 the unicity of certain bounded symmetric block bases of the subspaces of the Lorentz sequence spaces $d(w,p)$, $0 < p < 1$. In $\S4$ we show that every infinite dimensional subspace of $d(w,p)$ contains a subspace linearly homeomorphic to ${l_p}$, $0 < p < 1$. Unlike the case $p \geqslant 1$ there are subspaces of $d(w,p)$, $0 < p < 1$, which contain no complemented subspaces of $d(w,p)$ linearly homeomorphic to $ {l_p}$. In fact there are spaces $d(w,p)$, $0 < p < 1$, which contain no complemented subspaces linearly homeomorphic to ${l_p}$. We conjecture that this is true for every $ d(w,p)$, $0 < p < 1$. The answer to the previous question seems to be important: for example we can prove that a positive complemented sublattice $ E$ of $d(w,p)$, $0 < p < 1$, with a symmetric basis is linearly homeomorphic either to ${l_p}$ or to $d(w,p)$; consequently, a positive answer to this question implies that $E$ is linearly homeomorphic to $ d(w,p)$. In $\S5$ we are able to characterise the sublattices of $d(w,p)$, $ p = {k^{ - 1}}$ (however under a supplementary restriction concerning the sequence $({w_n})_{n = 1}^\infty )$, which are positive and contractive complemented, as being the order ideals of $ d(w,p)$. Finally, in $ \S6$, we characterise the Mackey completion of $d(w,p)$ also in the case $p = {k^{ - 1}}$, $k \in {\mathbf{N}}$.
The lattice of closed congruences on a topological lattice
Dennis J.
Clinkenbeard
457-467
Abstract: Our primary objectives are: (1) if $L$ is a lattice endowed with a topology making both the meet and join continuous then (i) the natural map which associates a congruence with the smallest topologically closed congruence containing it preserves finite meets and arbitrary joins; (ii) the lattice of such closed congruences is a complete Brouwerian lattice; (2) if $L$ is a topological (semi) lattice with the unit interval as a (semi) lattice homomorphic image then the lattice of closed (semi) lattice congruences has no compatible Hausdorff topology.
Boundary crossing probabilities for stationary Gaussian processes and Brownian motion
Jack
Cuzick
469-492
Abstract: Let $X(t)$ be a stationary Gaussian process, $ f(t)$ a continuous function, and $T$ a finite or infinite interval. This paper develops asymptotic estimates for $P(X(t) \geqslant f(t)$, some $t \in T$ when this probability is small. After transformation to an Ornstein Uhlenbeck process the results are also applicable to Brownian motion. In that special case, if $W(t)$ is Brownian motion, $f$ is continuously differentiable, and $T = [0,T]$ our estimate for $P(W(t) \geqslant f(t)$, some $t \in T)$ is $ {\Phi ^ \ast }$ is its upper tail distribution. Our approach is to find an approximate first passage density and then compute crossing probabilities as a one-dimensional integral. In the case of boundaries without cusps, our results unify and extend separate results for crossings of constant levels developed by Pickands, and Qualls-Watanabe, and crossings of rapidly increasing barriers studied by Berman. Applications are also briefly explored.
Representations of functions by derivatives
S. J.
Agronsky;
R.
Biskner;
A. M.
Bruckner;
J.
Mařík
493-500
Abstract: Let $\Delta '$ be the class of all derivatives. The main goal of this paper is the investigation of the vector space generated by $\Delta '$ and O'Malley's class $B_1^ \ast$; this space is identical with our system
The zeros of the second derivative of the reciprocal of an entire function
Simon
Hellerstein;
Jack
Williamson
501-513
Abstract: Let $f$ be a real entire function of finite order with only real zeros. Assuming that $f'$ has only real zeros, we show that the number of nonreal zeros of $f''$ equals the number of real zeros of $F''$, where $F = 1/f$. From this, we show that $F''$ has only real zeros if and only if $ f(z) = \exp(a{z^2} + bz + c)$, $a \geqslant 0$, or $f(z) = {(Az + B)^n}$, $A \ne 0$, $n$ a positive integer.
On the contact between complex manifolds and real hypersurfaces in ${\bf C}\sp{3}$
Thomas
Bloom
515-529
Abstract: Let $m$ be a real ${\mathcal{C}^\infty }$ hypersurface of an open subset of $ {{\mathbf{C}}^3}$ and let $ p \in M$. Let ${a^1}(M,p)$ denote the maximal order of contact of a one-dimensional complex submanifold of a neighborhood of $p$ in $ {{\mathbf{C}}^3}$ with $ M$ at $p$. Let $ {c^1}(M,p)$ denote the $ \sup \{ m \in {\mathbf{Z}}\vert$ for all tangential holomorphic vector fields $ L$ with $L(p) \ne 0$ then $ {L^{{i_0}}}{\bar L^{{j_0}}} \ldots {L^{{i_n}}}{\bar L^{{j_n}}}({\mathfrak{L}_M}(L))(p) = 0\}$ where ${i_0}, \ldots ,{i_n};{j_0}, \ldots ,{j_n}$ are positive integers such that $ \sum\nolimits_{t = 0}^n {{i_t} + {j_t} = m - 3}$ and ${\mathfrak{L}_M}(L)$ denotes the Levi form of $ M$ evaluated on the vector field $L$. Theorem. If $M$ is pseudoconvex near $ p \in M$ then $ {a^1}(M,p) = {c^1}(M,p)$.
A lattice renorming theorem and applications to vector-valued processes
William J.
Davis;
Nassif
Ghoussoub;
Joram
Lindenstrauss
531-540
Abstract: A norm, $ \vert\vert\;\vert\vert$, on a Banach space $E$ is said to be locally uniformly convex if $ \left\Vert {{x_n}} \right\Vert \to \left\Vert x \right\Vert$ and $\left\Vert {{x_n} + x} \right\Vert \to 2\left\Vert x \right\Vert$ implies that ${x_n} \to x$ in norm. It is shown that a Banach lattice has an (order) equivalent locally uniformly convex norm if and only if the lattice is order continuous. This result is used to reduce convergence theorems for (lattice-valued) positive martingales and submartingales to the scalar case.
On a relation between $\widetilde {\rm SL}\sb{2}$ cusp forms and cusp forms on tube domains associated to orthogonal groups
S.
Rallis;
G.
Schiffmann
1-58
Abstract: We use the decomposition of the discrete spectrum of the Weil representation of the dual reductive pair $({\tilde{SL}_2},\;O(Q))$ to construct a generalized Shimura correspondence between automorphic forms on $ O(Q)$ and $\widetilde{S{L_2}}$. We prove a generalized Zagier identity which gives the relation between Fourier coefficients of modular forms on $\widetilde{S{L_2}}$ and $O(Q)$. We give an explicit form of the lifting between $ \widetilde{S{L_2}}$ and $ O(n,2)$ in terms of Dirichlet series associated to modular forms. For the special case $n = 3$, we construct certain Euler products associated to the lifting between $S{L_2}$ and ${\text{S}}{{\text{p}}_2} \cong O(3,2)$ (locally).
Families of real and symmetric analytic functions
Yusuf
Abu-Muhanna;
Thomas H.
MacGregor
59-74
Abstract: We introduce families of functions analytic in the unit disk and having rotational symmetries. The families include the $ k$-fold symmetric univalent functions which have real coefficients. We relate the families to special classes of functions with a positive real part and then determine their extreme points. The case $k = 2$ corresponds to the odd functions which "preserve quadrants" and the extreme points of this set are characterized by having a radial limit which is real or imaginary almost everywhere. We also find estimates on the initial coefficients of functions in the families.
On the prevalence of horseshoes
Lai Sang
Young
75-88
Abstract: In this paper the symbolic dynamics of several differentiable systems are investigated. It is shown that many well-known dynamical systems, including Axiom $ {\text{A}}$ systems, piecewise monotonic maps of the interval, the Lorenz attractor and Abraham-Smale examples, have inside them subsystems conjugate to subshifts of finite type. These subsystems have hyperbolic structures and hence are stable. They can also be chosen to have entropy arbitrarily close to that of the ambient system.
Nonstandard analysis and lattice statistical mechanics: a variational principle
A. E.
Hurd
89-110
Abstract: Using nonstandard methods we construct a configuration space appropriate for the statistical mechanics of lattice systems with infinitely many particles and infinite volumes. Nonstandard representations of generalized equilibrium measures are obtained, yielding as a consequence a simple proof of the existence of standard equilibrium measures. As another application we establish an extension for generalized equilibrium measures of the basic variational principle of Landord and Ruelle. The same methods are applicable to continuous systems, and will be presented in a later paper.
Stationary logic and ordinals
D. G.
Seese
111-124
Abstract: The $L({\mathbf{aa}})$-theory of ordinals is investigated. It is proved that this theory is decidable and that each ordinal is finitely determinate.
Subnormal operators, Toeplitz operators and spectral inclusion
Gerard E.
Keough
125-135
Abstract: Let $S$ be a subnormal operator on the Hilbert space $H$, and let $N = \int z \;dE(z)$ be its minimal normal extension on $K$. Let $\mu$ be a scalar spectral measure for $ N$. If $f \in {L^\infty }(\mu )$, define ${T_f} = Pf(N){\vert _H}:\;H \to H$, where $ P:K \to H$ denotes orthogonal projection. $S$ has the $ {C^ \ast }$-Spectral Inclusion Property ( $ {C^ \ast }$-SIP) if $ \sigma (f(N)) \subseteq \sigma ({T_f})$, for all $f \in C(\sigma (N))$, and $S$ has the ${W^\ast}$-Spectral Inclusion Property ($ {W^\ast}$-SIP) if $ \sigma (f(N)) \subseteq \sigma ({T_f})$, for all $f \in {L^\infty }(\mu )$. It is shown that $ S$ has the ${C^\ast}$-SIP if and only if $\sigma (N) = \Pi (S)$, the approximate point spectrum of $S$. This is equivalent to requiring that $E(\Delta )K$ have angle 0 with $H$, for all nonempty, relatively open $ \Delta \subseteq \sigma (N)$. $S$ has the ${W^\ast}$-SIP if this angle condition holds for all proper Borel subsets of $ \sigma (N)$. If $ S$ is pure and has the $ {C^\ast}$ or $ {W^\ast}$-SIP, then it is shown that $\sigma (f(N)) \subseteq {\sigma _e}({T_f})$, for all appropriate $f$.
A partition theorem for the infinite subtrees of a tree
Keith R.
Milliken
137-148
Abstract: We prove a generalization for infinite trees of Silver's partition theorem. This theorem implies a version for trees of the Nash-Williams partition theorem.
Multivariate rearrangements and Banach function spaces with mixed norms
A. P.
Blozinski
149-167
Abstract: Multivariate nonincreasing rearrangement and averaging functions are defined for functions defined over product spaces. An investigation is made of Banach function spaces with mixed norms and using multivariate rearrangements. Particular emphasis is given to the $ L(P,Q;\ast)$ spaces. These are Banach function spaces which are in terms of mixed norms, multivariate rearrangements and the Lorentz $L(p,g)$ spaces. Embedding theorems are given for the various function spaces. Several well-known theorems are extended to the $ L(P,Q;\ast)$ spaces. Principal among these are the Strong Type (Riesz-Thorin) Interpolation Theorem and the Convolution (Young's inequality) Theorem.
Fine convergence and admissible convergence for symmetric spaces of rank one
Adam
Korányi;
J. C.
Taylor
169-181
Abstract: The connections between fine convergence in the sense of potential theory and admissible convergence to the boundary for quotients of eigenfunctions of the Laplace-Beltrami operator are investigated. This leads to a version of the local Fatou theorem on symmetric spaces of rank one which is considerably stronger than previously known results. The appendix establishes the relationship between harmonic measures on the intersection of the Martin boundaries of a domain and a subdomain.
On the Picard group of a continuous trace $C\sp{\ast} $-algebra
Iain
Raeburn
183-205
Abstract: Let $A$ be a continuous trace $ {C^\ast}$-algebra with paracompact spectrum $T$, and let $C(T)$ be the algebra of bounded continuous functions on $T$, so that $C(T)$ acts on $A$ in a natural way. An $A - A$ bimodule $X$ is an $ A{ - _{C(T)}}A$ imprimitivity bimodule if it is an $A - A$ imprimitivity bimodule in the sense of Rieffel and the induced actions of $C(T)$ on the left and right of $X$ agree. We denote by $ {\text{Pi}}{{\text{c}}_{C(T)}}A$ the group of isomorphism classes of $A{ - _{C(T)}}A$ imprimitivity bimodules under ${ \otimes _A}$. Our main theorem asserts that $ {\text{Pi}}{{\text{c}}_{C(T)}}A \cong {\text{Pi}}{{\text{c}}_{C(T)}}{C_0}(T)$. This result is well known to algebraists if $A$ is an $n$-homogeneous ${C^\ast}$-algebra with identity, and if $ A$ is separable it can be deduced from two recent descriptions of the automorphism group $ {\text{Au}}{{\text{t}}_{C(T)}}A$ due to Brown, Green and Rieffel on the one hand and Phillips and Raeburn on the other. Our main motivation was to provide a direct link between these two characterisations of $ {\text{Au}}{{\text{t}}_{C(T)}}A$.
Holomorphic actions of ${\rm Sp}(n,\,{\bf R})$ with noncompact isotropy groups
Hugo
Rossi
207-230
Abstract: $U(p,q)$ is a subgroup of ${S_p}(n,R)$, for $p + q = n$. ${B_q} = {S_p}(n,r)/U(p,q)$ is realized as an open subset of the manifold of Lagrangian subspaces of ${{\mathbf{C}}^n} \times {{\mathbf{C}}^n}$. It is shown that ${B_q}$ carries a $(pq)$-pseudoconvex exhaustion function. ${B_{pq}} = {S_p}(n,r)/U(p) \times U(q)$ carries two distinct holomorphic structures making the projection to ${B_q}$, ${B_0}$ holomorphic respectively. The geometry of the correspondence between ${B_q}$ and ${B_0}$ via ${B_{pq}}$ is investigated.
Robinson's consistency theorem in soft model theory
Daniele
Mundici
231-241
Abstract: In a soft model-theoretical context, we investigate the properties of logics satisfying the Robinson consistency theorem; the latter is for many purposes the same as the Craig interpolation theorem together with compactness. Applications are given to H. Friedman's third and fourth problem.
Plane models for Riemann surfaces admitting certain half-canonical linear series. II
Robert D. M.
Accola
243-259
Abstract: For $r \geqslant 2$, closed Riemann surfaces of genus $ 3r + 2$ admitting two simple half-canonical linear series $g_{3r + 1}^r,h_{3r + 1}^r$ are characterized by the existence of certain plane models of degree $ 2r + 3$ where the linear series are apparent. The plane curves have $r - 2$ $3$-fold singularities, one $ (2r - 1)$-fold singularity $ Q$, and two other double points (typically tacnodes) whose tangents pass through $ Q$. The lines through $ Q$ cut out a $ g_4^1$ which is unique. The case where the $g_4^1$ is the set of orbits of a noncyclic group of automorphisms of order four is characterized by the existence of $3r + 3$ pairs of half-canonical linear series of dimension $r - 1$, where the sum of the two linear series in any pair is linearly equivalent to $g_{3r + 1}^r + h_{3r + 1}^r$.
Monotone decompositions of $\theta \sb{n}$-continua
E. E.
Grace;
Eldon J.
Vought
261-270
Abstract: We prove the following theorem for a compact, metric ${\theta _n}$-continuum (i.e., a compact, connected, metric space that is not separated into more than $ n$ components by any subcontinuum). The continuum $X$ admits a monotone, upper semicontinuous decomposition $ \mathfrak{D}$ such that the elements of $ \mathfrak{D}$ have void interiors and the quotient space $X/\mathfrak{D}$ is a finite graph, if and only if, for each nowhere dense subcontinuum $H$ of $X$, the continuum $ T(H) = \{ x\vert$ if $ K$ is a subcontinuum of $ X$ and $x \in {K^ \circ }$, then $K \cap H \ne \emptyset \} $ is nowhere dense. The elements of the decomposition are characterized in terms of the set function $T$. An example is given showing that the condition that requires $T(x)$ to have void interior for all $ x \in X$ is not strong enough to guarantee the decomposition.
Approximation results in Hilbert cube manifolds
T. A.
Chapman
303-334
Abstract: The purpose of this paper is to answer questions of the following type for maps $ f:\,M \to \,N$ between Q-manifolds: When is f close to an approximate fibration? When is f close to a fiber bundle map? When is f close to a block bundle map? For the second and third questions a series of obstructions are encountered in the lower algebraic K-theoretic functors $ {K_{ - \,i}}$ of Bass.
Splitting criteria for $\mathfrak{g}$-modules induced from a parabolic and the Ber\v{n}ste{\u\i}n-Gel'fand-Gel'fand resolution of a finite-dimensional, irreducible $\mathfrak{g}$-module
Alvany
Rocha-Caridi
335-366
Abstract: Let $\mathcal{g}$ be a finite dimensional, complex, semisimple Lie algebra and let V be a finite dimensional, irreducible $ \mathcal{g}$-module. By computing a certain Lie algebra cohomology we show that the generalized versions of the weak and the strong Bernstein-Gelfand-Gelfand resolutions of V obtained by H. Garland and J. Lepowsky are identical. Let G be a real, connected, semisimple Lie group with finite center. As an application of the equivalence of the generalized Bernstein-Gelfand-Gelfand resolutions we obtain a complex in terms of the degenerate principal series of G, which has the same cohomology as the de Rham complex.
Algebras of automorphic forms with few generators
Philip
Wagreich
367-389
Abstract: Those finitely-generated Fuchsian groups G for which the graded algebra of automorphic forms A is generated by 2 or 3 elements are classified. In these cases the structure of A is described.
Preservation of closure in a locally convex space. I
I.
Brodsky
391-397
Abstract: This paper is concerned with the lifting of the closures of sets. If H is a topological vector space, G a subspace and A closed in G for the induced topology, under what conditions on A in G is it true that the closure of A is preserved in H, i.e., A is closed in H? In this paper a fundamental lifting proposition is proved. 'Preservation of closure' will prove to be a fruitful technique in obtaining some interesting results in the theory of locally convex spaces. Using this technique, we will first show when closure is equivalent to completeness. Then we will prove a generalization to locally convex spaces of the classical Heine-Borel Theorem for Euclidean n-space. Generalizing a result of Petunin, we will also give some necessary and sufficient conditions on semireflexivity. Finally, we will give a necessary and sufficient condition for the sum of two closed subspaces to be closed.
Deformation theory and the tame fundamental group
David
Harbater
399-415
Abstract: Let U be a curve of genus g with $n\, + \,1$ points deleted, defined over an algebraically closed field of characteristic $p\, \geqslant \,0$. Then there exists a bijection between the Galois finite étale covers of U having degree prime to p, and the finite $ p'$-groups on $n\, + \,2g$ generators. This fact has been proven using analytic considerations; here we construct such a bijection algebraically. We do this by algebraizing an analytic construction of covers which uses Hurwitz families. The process of algebraization relies on a deformation theorem, which we prove using Artin's Algebraization Theorem, and which allows the patching of local families into global families. That our construction provides the desired bijection is afterwards verified analytically.
The product of two countably compact topological groups
Eric K.
van Douwen
417-427
Abstract: We use MA ( = Martin's Axiom) to construct two countably compact topological groups whose product is not countably compact. To this end we first use MA to construct an infinite countably compact topological group which has no nontrivial convergent sequences.
Nonexistence of nontrivial $cm''$-harmonic 1-forms on a complete foliated Riemannian manifold
Haruo
Kitahara
429-435
Abstract: We study the nonexistence of nontrivial $\square''$-harmonic 1-forms on a complete foliated riemannian manifold with positive definite Ricci curvature. It is well known that the harmonic 1-form on a compact and orientable riemannian manifold with positive definite Ricci curvature is trivial. Our main theorem is an extension of this fact in the complete foliated riemannian case.
The category of $D$-completely regular spaces is simple
N. C.
Heldermann
437-446
Abstract: In a recent paper H. Brandenburg characterized the objects of the epireflective hull of all developable spaces-that are those spaces which are homeomorphic to a subspace of a product of developable spaces-by intrinsic properties. It is shown here that these spaces, called D-completely regular, can be generated from a single second countable developable space D which has the same cardinality as the reals. As an application of this result we obtain a new characterization of D-normal spaces analogous to Urysohn's lemma and a new (external) characterization of perfect spaces (meaning every closed set is a ${G_\delta }$).
Composition factors of the principal series representations of the group ${\rm Sp}(n,\,1)$
M. W.
Baldoni Silva;
H.
Kraljević
447-471
Abstract: Using Vogan's algorithm the composition factors of any principal series representation of the group $Sp(n,\,1)$ are determined.
Regularity of certain small subharmonic functions
P. C.
Fenton
473-486
Abstract: Suppose that u is subharmonic in the plane and that $ {\underline {\lim } _{r \to \infty }}B(r)/{(\log \,r)^{2\,}}\, = \,\sigma \, < \,\infty$. It is known that, given $\varepsilon > 0$, there are arbitrarily large values of r such that $ A(r)\, > \,B(r)\, - (\sigma \, + \,\varepsilon){\pi ^2}$. The following result is proved. Let u be subharmonic and let $ \sigma$ be any positive number. Then either $A(r)\, > \,B(r)\, - {\pi ^2}\sigma$ for certain arbitrarily large values of r or, if this is false, then $\displaystyle \mathop {\lim }\limits_{r \to \infty } \left( {B\left( r \right)\, - \,\sigma {{\left( {\log \,r} \right)}^2}} \right)/\log \,r$ exists and is either $+ \,\infty$ or finite.
On the singularities of Gegenbauer (ultraspherical) expansions
Ahmed I.
Zayed
487-503
Abstract: The results of Gilbert on the location of the singular points of an analytic function $f(z)$ given by Gegenbauer (ultraspherical) series expansion $f(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}\,C_n^\mu (z)$ are extended to the case where the series converges to a distribution. On the other hand, this generalizes Walter's results on distributions given by Legendre series: $ f(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}\,C_n^{1/2}(z)$. The singularities of the analytic representation of $ f(z)$ are compared to those of the associated power series $ g(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}{z^n}$. The notion of value of a distribution at a point is used to study the boundary behavior of the associated power series. A sufficient condition for Abel summability of Gegenbauer series is also obtained in terms of the distribution to which the series converges.
A new result on the convergence of nonhomogeneous stochastic chains
Arunava
Mukherjea
505-520
Abstract: Nonhomogeneous stochastic chains with a finite number of states are considered in this paper. Convergence of such chains is established here in terms of strong ergodicity of certain related chains of smaller size. These results are shown to be best possible and extend earlier results of Maksimov. Nonnegative idempotent matrices are also considered.
$L\sp{p}$ behavior of certain second order partial differential operators
Carlos E.
Kenig;
Peter A.
Tomas
521-531
Abstract: We give examples of bounded inverses of polynomials in ${{\textbf{R}}^n}$, $n\, > \,1$, which are not Fourier multipliers of $ {L^p}\,({{\textbf{R}}^n})$ for any $p\, \ne \,2$. Our main tool is the Kakeya set construction of C. Fefferman. Using these results, we relate the invertibility on $ {L^p}$ of a linear second order constant coefficient differential operator D on $ {{\textbf{R}}^n}$ to the geometric structure of quadratic surfaces associated to its symbol d. This work was motivated by multiplier conjectures of N. Rivière and R. Strichartz.
On locally and globally conformal K\"ahler manifolds
Izu
Vaisman
533-542
Abstract: Some relations between the locally conformal Kähler (l.c.K.) and the globally conformal Kähler (g.c.K.) properties are established. Compact l.c.K. manifolds which are not g.c.K. do not have Kähler metrics. l.c.K. manifolds which are not g.c.K. are analytically irreducible. Various curvature restrictions on l.c.K. manifolds imply the g.c.K. property. Total spaces of induced Hopf fibrations are l.c.K. and not g.c.K. manifolds. Conjecture. A compact l.c.K. manifold which is not g.c.K. has at least one odd odd-dimensional Betti number.
Linear spaces with an $H\sp{\ast} $-algebra-valued inner product
Parfeny P.
Saworotnow
543-549
Abstract: The paper deals with a particular class of VH-spaces of Loynes [5] whose inner product assumes its values in a trace-algebra associated with an $ H^{\ast}$-algebra. It is shown that these spaces admit a structure of a ``nonassociative module", and this structure could be used to characterize such spaces. Also we characterize other related spaces.
The asymptotic behavior of gas in an $n$-dimensional porous medium
Avner
Friedman;
Shoshana
Kamin
551-563
Abstract: Consider the flow of gas in an n-dimensional porous medium with initial density ${u_0}(x)\, \geqslant \,0$. The density $ u(x,\,t)$ then satisfies the nonlinear degenerate parabolic equation ${u_t}\, = \,\Delta {u^m}$ where $ m\, > \,1$ is a physical constant. Assuming that $ I\, \equiv \,\int {\,{u_0}(x)} dx\, < \,\infty$ it is proved that $ u(x,\,t)$ behaves asymptotically, as $ t\, \to \,\infty$, like the special (explicitly given) solution $V(\vert x\vert,\,t)$ which is invariant by similarity transformations and which takes the initial values $\delta (x)I\,(\delta (x)\, = \,$ the Dirac measure) in the distribution sense.
Weak subordination and stable classes of meromorphic functions
Kenneth
Stephenson
565-577
Abstract: This paper introduces the notion of weak subordination: If F and G are meromorphic in the unit disc $\mathcal{u}$, then F is weakly subordinate to G, written $F\, < \,G$, provided there exist analytic functions $\phi$ and $\omega :\,\mathcal{u}\, \to \,\mathcal{u}$, with $ \phi$ an inner function, so that $F\, \circ \,\phi \, = \,G\, \circ \,\omega$. A class $\mathcal{X}$ of meromorphic functions is termed stable if $F\,\mathop w\limits_ < \,G$ and $G\, \in \,\mathcal{X}\, \Rightarrow \,F\, \in \,\mathcal{X}$. The motivation is recent work of Burkholder which relates the growth of a function with its range and boundary values. Assume F and G are meromorphic and G has nontangential limits, a.e. Assume further that $ F(\mathcal{u})\, \cap \,G(\mathcal{u})\, \ne \,\emptyset $ and $ G({e^{i\theta }})\, \notin \,F(\mathcal{u})$, a.e. This is denoted by $ F\, < \,G$. Burkholder proved for several classes $ \mathcal{X}$ that $\displaystyle F\, < \,G \qquad{\text{and}}\quad G\, \in \,\mathcal{X}\, \Rightarrow \,F\, \in \,\mathcal{X}.$ ($ (\ast)$) The main result of this paper is the Theorem: $ F\, < \,G\, \Rightarrow \,F\,{ \prec ^w}\,G$. In particular, implication (*) holds for all stable classes $ \mathcal{X}$. The paper goes on to study various stable classes, which include BMOA, ${H^p}$, $0\, < \,p\, \leqslant \,\infty$, ${N_{\ast}}$, the space of functions of bounded characteristic, and the ${M^\Phi }$ spaces introduced by Burkholder. VMOA and the Bloch functions are examples of classes which are not stable.
A maximal function characterization of $H\sp{p}$ on the space of homogeneous type
Akihito
Uchiyama
579-592
Abstract: Let ${\psi _0}(x)\, \in \,{\mathcal{S(}}{R^n}{\text{)}}$ and let $\int_{{R^n}} {{\psi _0}(y)\,dy\, \ne \,0}$.For $x\, \in \,{R^n}$ and $M\, \geqslant \,0$, let $\displaystyle {f^ + }(x)\, = \,\mathop {\sup }\limits_{t\, > \,0} \,\left\vert {f\,{\ast}\,{\psi _{0t}}(x)} \right\vert$ and let $ {f^{{\ast}M}}(x)\, = \,\sup \{ \left\vert {f\,{\ast}\,{\psi _t}(x)} \right\vert:\,t\, > \,0$, $\psi (y)\, \in \,{\mathcal{S(}}{R^n})$, $\operatorname{supp} \,\psi \, \subset \,\{ y\, \in \,{R^n}:\,\left\vert y \right\vert\, < \,1\}$, ${\left\Vert {{D^\alpha }\psi } \right\Vert _{{L^\infty }}}\, \leqslant \,1$ for any multi-index $\alpha \, = \,({\alpha _1},\, \ldots ,\,{\alpha _n})$ such that $\Sigma _{i = 1}^n\,{\alpha _i}\, \leqslant \,M\}$ where ${\psi _t}(y)\, = \,{t^{ - n}}\psi (y/t)$. Fefferman-Stein [11] showed Theorem A. Let $ p\, > \,0$. Then there exists $M(p,\,n)$, depending only on p and n, such that if $ M\, \geqslant \,M(p,\,n)$, then $\displaystyle c\left\Vert {{f^ + }} \right\Vert{L^p}\, \leqslant \,\left\Vert {... ...right\Vert{L^p}\, \leqslant \,{\textbf{C}}\left\Vert {{f^ + }} \right\Vert{L^p}$ for any
Kernels for the tangential Cauchy-Riemann equations
Al
Boggess
1-49
Abstract: On certain codimension one and codimension two submanifolds in ${{\textbf{C}}^n}$, we can solve the tangential Cauchy-Riemann equations ${\bar \partial _b}u\, = \,f$ with an explicit integral formula for the solution. Let $M\, = \,\partial D$, where D is a strictly pseudoconvex domain in $ {{\textbf{C}}^n}$. Let $ \omega \, \subset \, \subset \,M$ be defined by $ \omega \, = \,\{ z\, \in \,M;\,\operatorname{Re} \,h(z)\, < \,0\}$, where h is holomorphic near D. Points on the boundary of $\omega$, $ \partial \omega$, where the tangent space of $ \partial \omega$ becomes complex linear, are called characteristic points. Theorem 1. Suppose $\partial \omega$ is admissible (in particular if $\partial \omega $ has two characteristic points). Suppose $f\, \in \,{\mathcal{E}}_M^{p,q}(\bar \omega )$, $1\, \leqslant \,q\, \leqslant \,n\, - \,3$, is smooth on $\omega$ and satisfies ${\bar \partial _M}f\, = \,0$ on $ \omega$; then there exists $u\, \in \,{\mathcal{E}}_M^{p,q - 1}(\omega )$ which is smooth on $\omega$ except possibly at the characteristic points on $\partial \omega$ and which solves the equation $ {\bar \partial _M}u\, = \,f$ on $\omega$. Theorem 2. Suppose $f\, \in \,{\mathcal{E}}_M^{p,q}(\omega )$, $2\, \leqslant \,q\, \leqslant \,n\, - \,3$, is smooth on $\omega$; vanishes near each characteristic point; and $ {\bar \partial _M}f\, = \,0$ on $\omega$. Then there exists $ u\, \in \,{\mathcal{E}}_M^{p,q - 1}(\omega )$ satisfying ${\bar \partial _M}u\, = \,f$ on $ \omega$. Theorem 3. Suppose $f\, \in \,{\mathcal{D}}_M^{p,q}(\omega )$, $2\, \leqslant \,q\, \leqslant \,n - \,3$, is smooth with compact support in $ \omega$, and ${\bar \partial _M}f\, = \,0$. Then there exists $u\, \in \,{\mathcal{D}}_M^{p,q - 1}(\omega )$ with compact support in $ \omega$ and which solves $ {\bar \partial _M}u\, = \,f$. In all three theorems we have an explicit integral formula for the solution. Now suppose $S\, = \,\partial \omega$. Let $ {C_s}$ be the set of characteristic points on S. We construct an explicit operator $ E:\,{\mathcal{D}}_S^{p,q}(S\, - \,{C_S})\, \to \,{\mathcal{E}}_S^{p,q - 1}(S\, - \,{C_S})$ with the following properties. Theorem 4. The operator E maps $ L_{p,\operatorname{comp} }^{\ast}(S\, - \,{C_S})\, \to \,L_{p,\operatorname{loc} }^{\ast}(S\, - {C_S})$ and if $f\, \in \,{\mathcal{D}}_S^{p,q}(S\, - \,{C_S})$, $1\, \leqslant \,q\, \leqslant \,n\, - \,3$, then $f\, = \,{\bar \partial _S}\{ E(f)\} \, + \,E({\bar \partial _S}f)$.
Countable ultrahomogeneous undirected graphs
A. H.
Lachlan;
Robert E.
Woodrow
51-94
Abstract: Let $G\, = \,\left\langle {{V_G},\,{E_G}} \right\rangle$ be an undirected graph. The complementary graph $\tilde G$ is $\left\langle {{V_G},\,{E_{\tilde G}}} \right\rangle$ where $({V_1},\,{V_2})\, \in \,{E_{\tilde G}}$ iff ${V_1}\, \ne \,{V_2}$ and $ ({V_1},\,{V_2})\, \notin \,{E_G}$. Let $K(n)$ be the complete undirected graph on n vertices and let E be the graph [ill] i.e. $\left\langle {\{ a,\,b,\,c\} ,\,\{ (b,\,c),\,(c,\,b)\} } \right\rangle$. G is ultrahomogeneous just in case every isomorphism of subgraph of smaller cardinality can be lifted to an automorphism of G. Let $\mathcal{D}\, = \,\{ K(n):\,n\, \in \,\omega \} \, \cup \,\{ E,\,\tilde E\} \, \cup \,\{ \tilde K(n):\,n\, \in \,\omega \}$. Theorem: Let ${G_1}$, ${G_2}$ be two countable (infinite) ultrahomogeneous graphs such that for each $H\, \in \,\mathcal{D}\,H$ can be embedded in ${G_1}$, just in case it can be embedded in $ {G_2}$. Then ${G_1}\, \cong \,{G_2}$. Corollary: There are a countable number of countable ultrahomogeneous (undirected) graphs.
On a $4$-manifold homology equivalent to a bouquet of surfaces
Akio
Kawauchi
95-112
Abstract: This paper gives some algebraic invariants for a piecewise linear imbedding of a surface into some 4-manifold inducing a Z or Q-homology isomorphism. Several examples are obtained by using these invariants.
Approximation of $G$-maps by maps in equivariant general positions and imbeddings of $G$-complexes
Sören
Illman
113-157
Abstract: Let G be a finite group. In this paper we consider maps $ f:P \to M$ from equivariant polyhedra into equivariant p.l. manifolds. We prove an equivariant general position result which shows how to approximate a given continuous proper equivariant (or isovariant) map $f:P \to M$ by a G-map which is in equivariant general position. We also apply this equivariant general position result to get a general G-imbedding theorem. Applied to the case of G-imbeddings of simplicial G-complexes into euclidean representation space this general G-imbedding theorem gives a result which provides a good hold on the required dimension of the euclidean representation space. For example in the case when $G = {Z_m}$ we prove that there exists a representation space $ {{\textbf{R}}^{r(k,m)}}(\rho)$ with the property that any k-dimensional simplicial ${Z_m}$-complex X admits a proper p.l. $ {Z_m}$-imbedding into $ {{\textbf{R}}^{r(k,m)}}(\rho)$ and we also show that the dimension $r(k,m)$ is best possible, i.e., one cannot find a euclidean representation space of lower dimension than $r(k,m)$ with the same property as $ {{\textbf{R}}^{r(k,m)}}(\rho)$. Simple explicit expressions for the dimension $r(k,m)$ are given. We also consider the case of semi-free actions with a given imbedding of the fixed point set into some euclidean space. Furthermore we show that the p.l. G-imbeddings of equivariant p.l. manifolds into euclidean representation space obtained by our G-imbedding results are in general equivariantly locally knotted although they are locally flat in the ordinary sense. This phenomenon can occur in arbitrarily high codimensions.
Invariance of solutions to invariant parametric variational problems
John E.
Brothers
159-179
Abstract: Let G be a compact Lie group of diffeomorphisms of a connected orientable manifold M of dimension $n + 1$. Assume the orbits of highest dimension to be connected. Let $ \Psi$ be a convex positive even parametric integrand of degree n on M which is invariant under the action of G. Let T be a homologically $\Psi$-minimizing rectifiable current of dimension n on M, and assume there exists a G-invariant rectifiable current $T'$ which is homologous to T. It is shown that T is G-invariant provided $ \Psi$ satisfies a symmetry condition which makes it no less efficient for the tangent planes of T to lie along the orbits. This condition is satisfied by the area integrand in case G is a group of isometries of a Riemannian metric on M. Consequently, one obtains the corollary that if a rectifiable current T is a solution to the n-dimensional Plateau problem in M with G-invariant boundary $\partial T$, and if $ \partial T$ bounds a G-invariant rectifiable current $T'$ such that $T - T'$ is a boundary, then T is G-invariant. An application to the Plateau problem in ${{\textbf{S}}^3}$ is given.
Hyperbolic Birkhoff centers
I. P.
Malta
181-193
Abstract: The purpose of this paper is to show that if f is a diffeomorphism of a compact manifold whose Birkhoff center, $ c(f)$, is hyperbolic and has no cycles, then f satisfies Axiom A and is $ \Omega$-stable. To obtain a filtration for $c(f)$, the concept of an isolated set for a homeomorphism of a compact metric space is introduced. As a partial converse it is proved that if $c(f)$ is hyperbolic and f is $ \Omega$-stable, then $ c(f)$ has the no cycle property. A characterization of $\Omega$-stability when $c(f)$ is finite is also given.
Undecidability and definability for the theory of global fields
R. S.
Rumely
195-217
Abstract: We prove that the theory of global fields is essentially undecidable, using predicates based on Hasse's Norm Theorem to define valuations. Polynomial rings or the natural numbers are uniformly defined in all global fields, as well as Gödel functions encoding finite sequences of elements.
$\sp{\ast} $-valuations and ordered $\sp{\ast} $-fields
Samuel S.
Holland
219-243
Abstract: We generalize elementary valuation theory to *-fields (division rings with involution), apply the generalized theory to the task of ordering *-fields, and give some applications to Hermitian forms.
Noninvariance of an approximation property for closed subsets of Riemann surfaces
Stephen
Scheinberg
245-258
Abstract: A closed subset E of an open Riemann surface M is said to have the approximation property $\mathcal{a}$ if each continuous function on E which is analytic at all interior points of E can be approximaed uniformly on E by functions which are everywhere analytic on M. It is known that $\mathcal{a}$ is a topological invariant (i.e., preserved by homeomorphisms of the pair $ (M,E)$) when M is of finite genus but not in general, not even for ${C^\infty }$ quasi-conformal automorphisms of M. The principal result of this paper is that $\mathcal{a}$ is not invariant even under a real-analytic isotopy of quasi-conformal automorphisms (of a certain M). M is constructed as the two-sheeted unbranched cover of the plane minus a certain discrete subset of the real axis, and the isotopy is induced by $(x + \,iy,\,t) \mapsto x + \,ity$, for $t > 0$; E can be taken to be that portion of M which lies over a horizontal strip.
Applications of the Fourier-Wiener transform to differential equations on infinite-dimensional spaces. I
Yuh Jia
Lee
259-283
Abstract: Let $(H,i,B)$ be an abstract Wiener space and $ {p_t}$ be the Wiener measure on B with variance t. Let [B] be the complexification of B and ${\mathcal{E}_a}$ be the class of exponential type analytic functions defined on [B]. We define the Fourier-Wiener c-transform for any f in $ {\mathcal{E}_a}$ by $\displaystyle {F_c}f(y)\, = \,\int_\textbf{B} {f(x\, + \,iy){p_c}(dx)}$ and the inverse transform by $\mathcal{F}_c^{ - 1}f(y)\, = \,{\mathcal{F}_c}f( - y)$. Then the inversion formula holds and ${\mathcal{F}_2}$ extends to ${L^2}(B,{p_1})$ as a unitary operator. Next, we apply the above transform to investigate the existence, uniqueness and regularity of solutions for Cauchy problems associated with the following two equations: (1) $ {u_t}\, = \, - {\mathcal{N}^k}u$, (2) ${u_{tt}}\, = \, - {\mathcal{N}^k}u$; and the elliptic type equation (3) $- {N^k}u\, = \,f(k\, \geqslant \,1)$, where $ \Delta$ is the Laplacian and $ \mathcal{N}u(x)\, = \, - \Delta u(x)\, + \,(x,Du(x))$.
Stability of the null solution of parabolic functional inequalities
Ray
Redheffer;
Wolfgang
Walter
285-302
Abstract: Uniqueness and stability theorems are established for coupled systems of parabolic differential equations which may involve a Volterra-type dependence on the past history of the process. We allow retarded or deviating arguments, convolution-type memory terms, and strong coupling. (This means that all the space derivatives up to a given order can occur in all the equations.) Our results for strong coupling depend on the concept of ``admissible monomial'' which is here introduced for the first time and has no counterpart in the linear case. It is possible for uniqueness to fail in general, but to be restored (relative to a tolerably large class of functions of $ (x,t)$) if a single solution independent of x exists. Another curious feature of these theorems, depending again on the concept of admissible monomial, is that conditions for uniqueness can involve derivatives of order much higher than those occurring in the equation. Examples given elsewhere show that the results are, in various respects, sharp. Thus, the seemingly peculiar hypotheses do not arise from deficient technique, but from the actual behavior of strongly coupled systems. The paper concludes with a new method of dealing with unbounded regions for the difficult case in which the functional occurs in the boundary operator as well as in the differential equation.
The embeddings of the discrete series in the principal series for semisimple Lie groups of real rank one
M. Welleda
Baldoni Silva
303-368
Abstract: We consider the problem of finding all the ``embeddings'' of a discrete series representation in the principal series in the case of a simple real Lie group G of real rank one. More precisely, we solve the problem when G is $ \operatorname{Spin} (2n,\,1),{\text{SU}}(n,\,1),\,{\text{SP}}(n,\,1)\,{\text{or}}\,{F_4}\,(n\, \geqslant \,2)$. The problem is reduced to considering only discrete series representations with trivial infinitesimal character, by means of tensoring with finite dimensional representations. Various other techniques are employed.
A stable converse to the Vietoris-Smale theorem with applications to shape theory
Steve
Ferry
369-386
Abstract: Our main result says that if $f:\,X\, \to \,Y$ is a map between finite polyhedra which has k-connected homotopy fiber, then there is an n such that $f\, \times \,{\text{id:}}\,X\, \times \,{I^n} \to Y$ is homotopic to a map with k-connected point-inverses. This result is applied to give an algebraic characterization of compacta shape equivalent to locally n-connected compacta. We also show that a $U{V^1}$ compactum can be ``improved'' within its shape class until its homotopy theory and strong shape theory are the same with respect to finite dimensional polyhedra.
Second-order equations of fixed type in regions with corners. I
Leonard
Sarason
387-416
Abstract: A class of well-posed boundary value problems for second order equations in regions with corners and edges is studied. The boundary condition may involve oblique derivatives, and edge values may enter the graph of the associated Hilbert space operator. Uniqueness of weak solutions and existence of strong solutions is shown.
Octonion planes over local rings
Robert
Bix
417-438
Abstract: Let $\mathcal{D}$ be an octonion algebra which is a free module over a local ring R and let $J = H({\mathcal{D}_3},\gamma )$ be the quadratic Jordan algebra of Hermitian 3-by-3 matrices over R. We define the octonion plane determined by J and prove that every collineation is induced by a norm semisimilarity of J. We classify the subgroups of the collineation group normalized by the little projective group.
Invariant solutions to the oriented Plateau problem of maximal codimension
David
Bindschadler
439-462
Abstract: The principal result gives conditions which imply that a solution to the Plateau problem inherits the symmetries of its boundary. Specifically, let G be a compact connected Lie subgroup of $ {\text{SO}}(n)$. Assume the principal orbits have dimension m, there are no exceptional orbits and the distribution of $(n\, - \,m)$-planes orthogonal to the principal orbits is involutive. We show that if B is a finite sum of oriented principal orbits, then every absolutely area minimizing current with boundary B is invariant. As a consequence of the methods used, the above Plateau problems are shown to be equivalent to 1-dimensional variational problems in the orbit space. Some results concerning invariant area minimizing currents in Riemannian manifolds are also obtained.
Topological spaces with prescribed nonconstant continuous mappings
Věra
Trnková
463-482
Abstract: Given a $ {T_1}$-space Y and a ${T_3}$-space V, consider ${T_3}$-spaces X such that X has a closed covering by spaces homeomorphic to V and any continuous mapping $f:\,X \to Y$ is constant. All such spaces and all their continuous mappings are shown to form a very comprehensive category, containing, e.g., a proper class of spaces without nonconstant, nonidentical mappings or containing a space X, for every monoid M, such that all the nonconstant continuous mappings of X into itself are closed under composition and form a monoid isomorphic to M. The category of paracompact connected spaces, having a closed covering by a given totally disconnected paracompact space, has, e.g., analogous properties. Categories of metrizable spaces are also investigated.
In-between theorems in uniform spaces
D.
Preiss;
J.
Vilímovský
483-501
Abstract: Necessary and sufficient conditions for the existence of a uniformly continuous function in-between given functions $f\, \geqslant \,g$ on a uniform space are studied. It appears that the investigation of this problem is closely related to some combinatorial properties of covers and leads to the concept of perfect refinability, the latter being used, e.g., to obtain an intrinsic description of uniform real extensors. Several interesting classes of uniform spaces are characterized by special types of in-between theorems. As examples of applications we show that the usual in-between theorems in topology and their generalizations, as well as some important methods of construction of derivatives of real functions, follow easily from the general results.
A strong Stieltjes moment problem
William B.
Jones;
W. J.
Thron;
Haakon
Waadeland
503-528
Abstract: This paper is concerned with double sequences of complex numbers $ C\, = \,\{ {c_n}\} _{ - \infty }^\infty$ and with formal Laurent series ${L_0}(C)\, = \,\Sigma _1^\infty \, - \,{c_{ - m}}{z^m}$ and ${L_\infty }(C)\, = \,\Sigma _0^\infty \,{c_m}{z^{ - m}}$ generated by them. We investigate the following related problems: (1) Does there exist a holomorphic function having ${L_0}(C)$ and $ {L_\infty }(C)$ as asymptotic expansions at $z\, = \,0$ and $ z\, = \,\infty$, respectively? (2) Does there exist a real-valued bounded, monotonically increasing function $\psi (t)$ with infinitely many points of increase on $[0,\,\infty )$ such that, for every integer n, ${c_n}\, = \,\int_0^\infty {{{( - t)}^n}\,d\psi (t)}$? The latter problem is called the strong Stieltjes moment problem. We also consider a modified moment problem in which the function $\psi (t)$ has at most a finite number of points of increase. Our approach is made through the study of a special class of continued fractions (called positive T-fractions) which correspond to ${L_0}(C)$ at $z\, = \,0$ and $ {L_\infty }(C)$ at $z\, = \,\infty$. Necessary and sufficient conditions are given for the existence of these corresponding continued fractions. It is further shown that the even and odd parts of these continued fractions always converge to holomorphic functions which have $ {L_0}(C)$ and ${L_\infty }(C)$ as asymptotic expansions. Moreover, these holomorphic functions are shown to be represented by Stieltjes integral transforms whose distributions $ {\psi ^{(0)}}(t)$ and ${\psi ^{(1)}}(t)$ solve the strong Stieltjes moment problem. Necessary and sufficient conditions are given for the existence of a solution to the strong Stieltjes moment problem. This moment problem is shown to have a unique solution if and only if the related continued fraction is convergent. Finally it is shown that the modified moment problem has a unique solution if and only if there exists a terminating positive T-fraction that corresponds to both ${L_0}(C)$ and ${L_\infty }(C)$. References are given to other moment problems and to investigations in which negative, as well as positive, moments have been used.
On the Wall finiteness obstruction for the total space of certain fibrations
Hans J.
Munkholm;
Erik Kjaer
Pedersen
529-545
Abstract: The problem of computing the Wall finiteness obstruction for the total space of a fibration $ p:\,E\, \to \,B$ in terms of that for the base and homological data of the fiber has been considered by D. R. Anderson and by E. K. Pedersen and L. R. Taylor. We generalize their results and show how the problem is related to the algebraically defined transfer map ${\varphi ^{\ast}}:\,{K_0}({\textbf{Z}}{\pi _1}(B))\, \to \,{K_0}({\textbf{Z}}{\pi _1}(E))$, $\varphi \, = \,{p_{\ast}}:\,{\pi _1}(E)\, \to \,{\pi _1}(B)$, whenever the latter is defined.
Canonical embeddings
J.
Morrow;
H.
Rossi
547-565
Abstract: In this paper the authors compare the embedding of a compact Riemann surface in its tangent bundle to the embedding as the diagonal in the product. These embeddings are proved to be first, but not second, order equivalent. The embedding of a hyperelliptic curve in its tangent bundle is described in an explicit way. Although it is not possible to be so explicit in the other cases, it is shown that in all cases, if the Riemann surface R has genus greater than two, then the blowdown of the zero section of the tangent bundle and the blowdown of the diagonal in the product have the same Hilbert polynomial.
Derivations on algebras of unbounded operators
Atsushi
Inoue;
Shôichi
Ota
567-577
Abstract: This paper is a study of derivations on unbounded operator algebras in connection with those in operator algebras. In particular we study spatiality of derivations in several situations. We give the characterization of derivations on general *-algebras by using positive linear functionals. We also show that a derivation with some range-property on a left $E{W^\char93 }$-algebra induced by an unbounded Hilbert algebra is strongly implemented by an operator which belongs to an algebra of measurable operators.
The Rayleigh-Schr\"odinger expansion of the Gibbs state of a classical Heisenberg ferromagnet
William G.
Faris
579-587
Abstract: The equilibrium Gibbs state of a classical Heisenberg ferromagnet is a probability measure on an infinite product of spheres. The Kirkwood-Salsburg equations may be iterated to produce a convergent high temperature expansion of this measure about a product measure. Here we show that this expansion may also be obtained as the Rayleigh-Schrödinger expansion of the ground state eigenvector of a differential operator. The operator describes a Markovian time evolution of the ferromagnet.
Chaotic behavior in piecewise continuous difference equations
James P.
Keener
589-604
Abstract: A class of piecewise continuous mappings with positive slope, mapping the unit interval into itself is studied. Families of 1-1 mappings depending on some parameter have periodic orbits for most parameter values, but have an infinite invariant set which is a Cantor set for a Cantor set of parameter values. Mappings which are not 1-1 exhibit chaotic behavior in that the asymptotic behavior as measured by the rotation number covers an interval of values. The asymptotic behavior depends sensitively on initial data in that the rotation number is either a nowhere continuous function of initial data, or else it is a constant on all but a Cantor set of the unit interval.
Realization of square-integrable representations of unimodular Lie groups on $L\sp{2}$-cohomology spaces
Jonathan
Rosenberg
1-32
Abstract: An analogue of the ``Langlands conjecture'' is proved for a large class of connected unimodular Lie groups having square-integrable representations (modulo their centers). For nilpotent groups, it is shown (without restrictions on the group or the polarization) that the $ {L^2}$-cohomology spaces of a homogeneous holomorphic line bundle, associated with a totally complex polarization for a flat orbit, vanish except in one degree given by the ``deviation from positivity'' of the polarization. In this degree the group acts irreducibly by a square-integrable representation, confirming a conjecture of Moscovici and Verona. Analogous results which improve on theorems of Satake are proved for extensions of a nilpotent group having square-integrable representations by a reductive group, by combining the theorem for the nilpotent case with Schmid's proof of the Langlands conjecture. Some related results on Lie algebra cohomology and the ``Harish-Chandra homomorphism'' for Lie algebras with a triangular decomposition are also given.
Lie cohomology of representations of nilpotent Lie groups and holomorphically induced representations
Richard
Penney
33-51
Abstract: Let U be a locally injective, Moore-Wolf square integrable representation of a nilpotent Lie group N. Let $ (\mathcal{H},\,\lambda )$ be a complex, maximal subordinate pair corresponding to U and let $ {\mathcal{H}_0}\, = \,\ker \,\lambda \, \cap \,\mathcal{H}$. The space ${C^\infty }(U)$ of differentiable vectors for U is an ${\mathcal{H}_0}$ module. In this work we compute the Lie algebra cohomology $ {H^p}({\mathcal{H}_0},\,{C^\infty }(U))$ of this Lie module. We show that the cohomology is zero for all but one value of p and that for this specific value the cohomology is one dimensional. These results, when combined with earlier results of ours, yield the existence and irreducibility of holomorphically induced representations for arbitrary (nonpositive), totally complex polarizations.
On random Fourier series
Jack
Cuzick;
Tze Leung
Lai
53-80
Abstract: Motivated by Riemann's ${R_1}$ summation method for i.i.d. random variables ${X_1},\,{X_2},\, \ldots$, this paper studies random Fourier series of the form $\sum\nolimits_1^\infty {{a_n}{X_n}\,\sin (nt\, + \,{\Phi _n})}$, where $ \{ {a_n}\}$ is a sequence of constants and $ \{ {\Phi _n}\}$ is a sequence of independent random variables which are independent of $\{ {X_n}\}$. Questions of continuity and of unboundedness are analyzed through the interplay between the asymptotic properties of $\{ {a_n}\}$ and the tail distribution of $ {X_1}$. A law of the iterated logarithm for the local behavior of the series is also obtained and extends the classical result for Brownian motion to a general class of random Fourier series.
Free modular lattices
Ralph
Freese
81-91
Abstract: It is shown that the word problem for the free modular lattice on five generators is recursively unsolvable.
On a simplicial complex associated to the monodromy
Gerald Leonard
Gordon
93-101
Abstract: Suppose we have a complex analytic family, ${V_t}$, $\left\vert t \right\vert\, \leqslant \,1$, such that the generic fibre is a nonsingular complex manifold of complex dimension n. Let T denote the monodromy induced from going once around the singular fibre and let I denote the identity map. We shall associate to the singular fibre a simplicial complex $\Gamma$, which is at most n-dimensional. Then under certain conditions on the family ${V_t}$ (which are satisfied for the Milnor fibration of an isolated singularity or if the $ {V_t}$ are compact Kähler), there is an integer $N\, > \,0$ such that ${({T^N}\, - \,I)^k}{H_k}({V_t})\, = \,0$ if and only if ${H_k}(\Gamma )\, = \,0$.
On the group of volume-preserving diffeomorphisms of ${\bf R}\sp{n}$
Dusa
McDuff
103-113
Abstract: The group of all diffeomorphisms of $ {\textbf{R}^n}$ which preserve a given volume form is shown to be perfect when $n\, \geqslant \,3$. Some useful factorizations of such diffeomorphisms are also obtained.
Binary sequences which contain no $BBb$
Earl D.
Fife
115-136
Abstract: A (one-sided) sequence or (two-sided) bisequence is irreducible provided it contains no block of the form BBb, where b is the initial symbol of the block B. Gottschalk and Hedlund [Proc. Amer. Math. Soc. 15 (1964), 70-74] proved that the set of irreducible binary bisequences is the Morse minimal set M. Let ${M^ + }$ denote the one-sided Morse minimal set, i.e. ${M^ + }\, = \,\{ {x_0}{x_1}{x_2}\, \ldots : \ldots \,{x_{ - 1}}{x_0}{x_1}\, \ldots \, \in \,M\}$. Let ${P^ + }$ denote the set of all irreducible binary sequences. We establish a method for generating all $x\, \in \,{P^ + }$. We also determine ${P^ + }\, - \,{M^ + }$. Considering $ {P^ + }$ as a one-sided symbolic flow, ${P^ + }$ is not the countable union of transitive flows, thus ${P^ + }$ is considerably larger than ${M^ + }$. However ${M^ + }$ is the $\omega $-limit set of each $x\, \in \,{P^ + }$, and in particular ${M^ + }$ is the nonwandering set of $ {P^ + }$.
Critical points of harmonic functions on domains in ${\bf R}\sp{3}$
Robert
Shelton
137-158
Abstract: It is shown that the critical point relations of Morse theory, together with the maximum principle, comprise a complete set of critical point relations for harmonic functions of three variables. The proof proceeds by first constructing a simplified example and then developing techniques to modify this example to realize all admissible possibilities. Techniques used differ substantially from those used by Morse in his solution of the analogous two-variable problem.
The essential norm of an operator and its adjoint
Sheldon
Axler;
Nicholas
Jewell;
Allen
Shields
159-167
Abstract: We consider the relationship between the essential norm of an operator T on a Banach space X and the essential norm of its adjoint $T^{\ast}$. We show that these two quantities are not necessarily equal but that they are equivalent if $ X^{\ast}$ has the bounded approximation property. For an operator into the sequence space ${c_0}$, we give a formula for the distance to the compact operators and show that this distance is attained. We introduce a property of a Banach space which is useful in showing that operators have closest compact approximants and investigate which Banach spaces have this property.
Uniform approximation on unbounded sets by harmonic functions with logarithmic singularities
P. M.
Gauthier;
M.
Goldstein;
W. H.
Ow
169-183
Abstract: This paper deals with the qualitative theory of uniform approximation by harmonic functions. The theorems of Brelot and Deny on Runge- and Walsh-type approximation on compact sets are extended to unbounded closed sets.
Isotopy types of knots of codimension two
M. Š.
Farber
185-209
Abstract: In this paper the classification of n-dimensional knots in ${S^{n + 2}}$, bounding r-connected manifolds, where $3r\, \geqslant \,n\, + \,1\, \geqslant \,6$, in terms of stable homotopy theory is suggested.
Hardy spaces and rearrangements
Burgess
Davis
211-233
Abstract: Let f be an integrable valued function on the unit circle in the complex plane, and let g be the rearrangement of f satisfying $ g({e^{i\theta }})\, \geqslant \,g({e^{i\varphi }})$ if $0\, \leqslant \,\theta \, < \,\varphi \, < \,2\pi$. Define $\displaystyle G(\theta )\, = \,\int_{ - \theta }^\theta {g({e^{i\varphi }})} \,d\varphi$ . It is shown that some rearrangement of f is in $\operatorname{Re} \,{H^1}$, that is, the distribution of f is the distribution of a function in $\operatorname{Re} \,{H^1}$, if and only if $\int_0^\pi {\vert G(\theta )/\theta \vert} \,d\theta \, < \,\infty$, and that, if any rearrangement of f is in $\operatorname{Re} \,{H^1}$, then g is. The existence and form of rearrangements minimizing the ${H^1}$ norm are investigated. It is proved that f is in $\operatorname{Re} \,{H^1}$ if and only if some rotation of f is in the space dyadic $ {H^1}$ of martingales. These results are extended to other ${H^p}$ spaces.
Quadratic forms and the Birman-Craggs homomorphisms
Dennis
Johnson
235-254
Abstract: Let ${\mathcal{M}_g}$ be the mapping class group of a genus g orientable surface M, and ${\mathcal{J}_g}$ the subgroup of those maps acting trivially on the homology group ${H_1}(M,\,Z)$. Birman and Craggs produced homomorphisms from $ {\mathcal{J}_g}$ to $ {Z_2}$ via the Rochlin invariant and raised the question of enumerating them; in this paper we answer their question. It is shown that the homomorphisms are closely related to the quadratic forms on $ {H_1}(M,\,{Z_2})$ which induce the intersection form; in fact, they are in 1-1 correspondence with those quadratic forms of Arf invariant zero. Furthermore, the methods give a description of the quotient of $ {\mathcal{J}_g}$ by the intersection of the kernels of all these homomorphisms. It is a ${Z_2}$-vector space isomorphic to a certain space of cubic polynomials over ${H_1}(M,\,{Z_2})$. The dimension is then computed and found to be $\left( {\begin{array}{*{20}{c}} {2g} 3 \end{array} } \right)\, + \,\left( {\begin{array}{*{20}{c}} {2g} 2 \end{array} } \right)$ . These results are also extended to the case of a surface with one boundary component, and in this situation the linear relations among the various homomorphisms are also determined.
Existentially complete abelian lattice-ordered groups
A. M. W.
Glass;
Keith R.
Pierce
255-270
Abstract: The theory of abelian totally ordered groups has a model completion. We show that the theory of abelian lattice-ordered groups has no model companion. Indeed, the Archimedean property can be captured by a first order $\forall\exists\forall$ sentence for existentially complete abelian lattice-ordered groups, and distinguishes between finitely generic abelian lattice-ordered groups and infinitely generic ones. We then construct (by sheaf techniques) the model companions of certain classes of discrete abelian lattice-ordered groups.
A spectral sequence for group presentations with applications to links
Selma
Wanna
271-285
Abstract: A spectral sequence is associated with any presentation of a group G. It turns out that this spectral sequence is independent of the chosen presentation. In particular if G is the fundamental group of a link L in ${R^3}$ the spectral sequence leads to invariants that compare to the Milnor invariants of L.
Transforms of measures on quotients and spline functions
Alan
MacLean
287-296
Abstract: Let G be a LCA group with closed subgroup H and let $v\, \in \,M(G/H)$. A general procedure is established for constructing a large family of measures in $M(G)$ whose Fourier transforms interpolate $ \hat v$. This method is used to extend a theorem of Shepp and Goldberg by showing that if $ v\, \in \,M([0,\,2\pi ))$, then each even order cardinal spline function which interpolates the sequence $(\hat v(n))_{n\, = \, - \,\infty }^\infty$ Fourier transform of a bounded Borel measure on R.
Vanishing theorems and K\"ahlerity for strongly pseudoconvex manifolds
Vo Van
Tan
297-302
Abstract: A precise vanishing theorem of Kodaira-Nakano type for strongly pseudoconvex manifolds and Nakano semipositive vector bundles is established. This result answers affirmatively a question posed by Grauert and Riemenschneider. However an analogous version of vanishing theorem of Akizuki-Nakano type for strongly pseudoconvex manifolds and Nakano semipositive line bundles does not hold in general. A counterexample for this fact is explicitly constructed. Furthermore we prove that any strongly pseudoconvex manifold with 1-dimensional exceptional subvariety is Kählerian; in particular any strongly pseudoconvex surface is Kählerian.
Median algebra
John R.
Isbell
319-362
Abstract: A study of algebras with a ternary operation $ (x,\,y,\,z)$ satisfying some identities, equivalent to embeddability in a lattice with $(x,\,y,\,z)$ realized as, simultaneously, $(x\, \wedge \,(y\, \vee \,z))\, \vee \,(y\, \wedge \,z)$ and $(x\, \vee \,(y\, \wedge \,z))\, \wedge \,(y\, \vee \,z)$. This is weaker than embeddability in a modular lattice, where those expressions coincide for all x, y, and z, but much of the theory survives the extension. For actual embedding in a modular lattice, some necessary conditions are found, and the investigation is carried much further in a special, geometrically described class of examples ("2-cells"). In distributive lattices $(x,\,y,\,z)$ reduces to the median $(x\, \wedge \,y)\, \vee \,(x\, \wedge \,z)\, \vee \,(y\, \wedge \,z)$, previously studied by G. Birkhoff and S. Kiss. It is shown that Birkhoff and Kiss found a basis for the laws; indeed, their algebras are embeddable in distributive lattices, i.e. in powers of the 2-element lattice. Their theory is much further developed and is connected into an explicit Pontrjagin-type duality.
A separation theorem for $\Sigma \sp{1}\sb{1}$ sets
Alain
Louveau
363-378
Abstract: In this paper, we show that the notion of Borel class is, roughly speaking, an effective notion. We prove that if a set A is both $\prod _\xi ^0$ and $\Delta _1^1$, it possesses a $\Pi _\xi ^0$-code which is also $\Delta _1^1$. As a by-product of the induction used to prove this result, we also obtain a separation result for $\Sigma _1^1$ sets: If two $\Sigma _1^1$ sets can be separated by a $\Pi _\xi ^0$ set, they can also be separated by a set which is both $ \Delta _1^1$ and $\Pi _\xi ^0$. Applications of these results include a study of the effective theory of Borel classes, containing separation and reduction principles, and an effective analog of the Lebesgue-Hausdorff theorem on analytically representable functions. We also give applications to the study of Borel sets and functions with sections of fixed Borel class in product spaces, including a result on the conservation of the Borel class under integration.
Every contractible fan is locally connected at its vertex
Lex G.
Oversteegen
379-402
Abstract: We prove that each contractible fan is locally connected at its vertex. It follows that every contractible fan is embeddable in the plane. This gives a solution to a problem raised by J. J. Charatonik and C. A. Eberhart.
Nonexistence of continuous selections of the metric projection for a class of weak Chebyshev spaces
Manfred
Sommer
403-409
Abstract: The class $\mathfrak{B}$ of all those n-dimensional weak Chebyshev subspaces of $ C\,[a,\,b]$ whose elements have no zero intervals is considered. It is shown that there does not exist any continuous selection of the metric projection for G if there is a nonzero g in G having at least $ n\, + \,1$ distinct zeros. Together with a recent result of Nürnberger-Sommer, the following characterization of continuous selections for $ \mathfrak{B}$ is valid: There exists a continuous selection of the metric projection for G in $ \mathfrak{B}$ if and only if each nonzero g in G has at most n distinct zeros.
Inner product spaces associated with Poincar\'e complexes
Seiya
Sasao;
Hideo
Takahashi
411-419
Abstract: We consider the homotopy type classification of a certain kind of Poincaré complex. First we define an inner product space associated with such a Poincaré complex and we investigate the relation between the inner product space and the homotopy type of the Poincaré complex. As an application, some results for manifolds are proved.
Homotopy operations for simplicial commutative algebras
W. G.
Dwyer
421-435
Abstract: The indicated operation algebra is studied by methods dual to the usual ones for studying the Steenrod algebra. In particular, the operations are constructed using higher symmetries of the shuffle map and their ``Adem relations'' are computed using the transfer map in the cohomology of symmetric groups.
Higher divided squares in second-quadrant spectral sequences
W. G.
Dwyer
437-447
Abstract: The geometric action of the Steenrod algebra on many mod 2 cohomology spectral sequences is complemented by the action of a completely different algebra.
$q$-extension of the $p$-adic gamma function
Neal
Koblitz
449-457
Abstract: p-adic functions depending on a parameter q, $0\, < \,\vert q\, - \,1{\vert _p}\, < \,1$, are defined which extend Y. Morita's p-adic gamma function and the derivative of J. Diamond's p-adic log-gamma function in the same way as the classical q-gamma function ${\Gamma _q}(x)$ extends $ \Gamma (x)$. Properties of these functions which are analogous to the basic identities satisfied by $ {\Gamma _q}(x)$ are developed.
Univalence criteria in multiply-connected domains
Brad G.
Osgood
459-473
Abstract: Theorems due to Nehari and Ahlfors give sufficient conditions for the univalence of an analytic function in relation to the growth of its Schwarzian derivative. Nehari's theorem is for the unit disc and was generalized by Ahlfors to any simply-connected domain bounded by a quasiconformal circle. In both cases the growth is measured in terms of the hyperbolic metric of the domain. In this paper we prove a corresponding theorem for finitely-connected domains bounded by points and quasiconformal circles. Metrics other than the hyperbolic metric are also considered and similar results are obtained.
Products in the Atiyah-Hirzebruch spectral sequence and the calculation of $M{\rm SO}\sb\ast $
Brayton
Gray
475-483
Abstract: It is possible to put a multiplicative structure in the Atiyah-Hirzebruch spectral sequence in certain cases even though the spectra involved are not both ring spectra. As a special case, an easy calculation of the homotopy of MSO is obtained.
On the computational complexity of determining the solvability or unsolvability of the equation $X\sp{2}-DY\sp{2}=-1$
J. C.
Lagarias
485-508
Abstract: The problem of characterizing those D for which the Diophantine equation $ {X^2}\, - \,D{Y^2}\, = \, - \,1$ is solvable has been studied for two hundred years. This paper considers this problem from the viewpoint of determining the computational complexity of recognizing such D. For a given D, one can decide the solvability or unsolvability of ${X^2}\, - \,D{Y^2}\, = \, - \,1$ using the ordinary continued fraction expansion of $ \sqrt D$, but for certain D this requires more than $ \tfrac{1}{3}\,\sqrt D \,{(\log \,D)^{ - \,1}}$ computational operations. This paper presents a new algorithm for answering this question and proves that this algorithm always runs to completion in $O({D^{1/4\, + \,\varepsilon }})$ bit operations. If the input to this algorithm includes a complete prime factorization of D and a quadratic nonresidue $ {n_i}$ for each prime $ {p_i}$ dividing D, then this algorithm is guaranteed to run to completion in $O({(\log \,D)^5}\,(\log \,\log \,D)(\log \,\log \,\log \,D))$ bit operations. This algorithm is based on an algorithm that finds a basis of forms for the 2-Sylow subgroup of the class group of binary quadratic forms of determinant D.
Minimal skew products
S.
Glasner
509-514
Abstract: Let $(\sigma ,\,Z)$ be a metric minimal flow. Let Y be a compact metric space and let $\mathcal{g}$ be a pathwise connected group of homeomorphisms of Y. We consider a family of skew product flows on $Z\, \times \,Y\, = \,X$ and show that when $(\mathcal{g},\,Y)$ is minimal most members of this family have the property of being disjoint from every minimal flow which is disjoint from $(\sigma ,\,Z)$. From this and some further results about skew product flows we deduce the existence of a minimal metric flow which is disjoint from every weakly mixing minimal flow but is not PI.
A ${\bf Z}\times {\bf Z}$ structurally stable action
P. R.
Grossi Sad
515-525
Abstract: We consider in the product of spheres $ {S^m}\, \times \,{S^n}$ the $Z\, \times \,Z$-action generated by two simple Morse-Smale diffeomorphisms; if they have some kind of general position, the action is shown to be stable. An application is made to foliations.
An asymptotic theory for a class of nonlinear Robin problems. II
F. A.
Howes
527-552
Abstract: Various asymptotic phenomena exhibited by solutions of singularly perturbed Robin boundary value problems are studied in the case when the right-hand side grows faster than the square of the derivative.
Periodic orbits of continuous mappings of the circle
Louis
Block
553-562
Abstract: Let f be a continuous map of the circle into itself and let $ P(f)$ denote the set of positive integers n such that f has a periodic point of period n. It is shown that if $1\, \in \,P(f)$ and $n\, \in \,P(f)$ for some odd positive integer n then for every integer $m\, > \,n$, $ m\, \in \,P(f)$. Furthermore, if $P(f)$ is finite then there are integers m and n (with $ m\, \geqslant \,1$ and $n\, \geqslant \,0$) such that $P(f)\, = \,\{ m,\,2\,m,\,4\,m,\,8\,m,\,\ldots,\,{2^n}\,m\}$.
Composition series for analytic continuations of holomorphic discrete series representations of ${\rm SU}(n,\,n)$
Bent
Ørsted
563-573
Abstract: We study a certain family of holomorphic discrete series representations of the semisimple Lie group $G\, = \,SU(n,\,n)$ and the corresponding analytic continuation in the inducing parameter $\lambda$. At the values of $\lambda$ where the representations become reducible, we compute the composition series in terms of a Peter-Weyl basis on the Shilov boundary of the Hermitian symmetric space for G.
A Radon transform on spheres through the origin in ${\bf R}\sp{n}$ and applications to the Darboux equation
A. M.
Cormack;
E. T.
Quinto
575-581
Abstract: On domain ${C^\infty }\,({R^n})$ we invert the Radon transform that maps a function to its mean values on spheres containing the origin. Our inversion formula implies that if $ f\, \in \,{C^\infty }\,({R^n})$ and its transform is zero on spheres inside a disc centered at 0, then f is zero inside that disc. We give functions $f\, \notin \,{C^\infty }\,({R^n})$ whose transforms are identically zero and we give a necessary condition for a function to be the transform of a rapidly decreasing function. We show that every entire function is the transform of a real analytic function. These results imply that smooth solutions to the classical Darboux equation are determined by the data on any characteristic cone with vertex on the initial surface; if the data is zero near the vertex then so is the solution. If the data is entire then a real analytic solution with that data exists.
The gluing of maximal ideals---spectrum of a Noetherian ring---going up and going down in polynomial rings
Ada Maria
de Souza Doering;
Yves
Lequain
583-593
Abstract: If ${M_1},\,...\,,\,{M_s}$ are maximal ideals of a ring R that have isomorphic residue fields, then they can be ``glued'' in the sense that a subring D of R with R is integral over D and ${M_1}\, \cap \,D\, = \,...\, = \,{M_s}\, \cap \,D$ can be constructed. We use this gluing process to prove the following result: Given any finite ordered set $ \mathcal{B}$, there exists a reduced Noetherian ring B and an embedding $\psi :\,\mathcal{B}\, \to \,Spec\,B$ such that $ \psi$ establishes a bijection between the maximal (respectively minimal) elements of $ \mathcal{B}$ and the maximal (respectively minimal) prime ideals of B and such that given any elements $\beta '$, $\beta ''$ of $ \mathcal{B}$, there exists a saturated chain of prime ideals of length r between $\psi (\beta '')$ if and only if there exists a saturated chain of length r between $ \beta '$ and $ \beta ''$. We also use the gluing process to construct a Noetherian domain A with quotient field L and a Noetherian domain B between A and L such that: $ A\,\hookrightarrow \,B$ possesses the Going Up and the Going Down properties, $ A[X]\,\hookrightarrow \,B[X]$ is unibranched and $A[X]\,\hookrightarrow \,B[X]$ possesses neither the Going Up nor the Going Down properties.
Factorization of curvature operators
Jaak
Vilms
595-605
Abstract: Let V be a real finite-dimensional vector space with inner product and let R be a curvature operator, i.e., a symmetric linear map of the bivector space $\Lambda {\,^2}V$ into itself. Necessary and sufficient conditions are given for R to admit factorization as $ R\, = \,\Lambda {\,^2}L$, with L a symmetric linear map of V into itself. This yields a new characterization of Riemannian manifolds that admit local isometric embedding as hypersurfaces of Euclidean space.
Fourier inversion on Borel subgroups of Chevalley groups: the symplectic group case
Ronald L.
Lipsman
607-622
Abstract: In recent papers, the author and J. A. Wolf have developed the Plancherel theory of parabolic subgroups of real reductive Lie groups. This includes describing the irreducible unitary representations, computing the Plancherel measure, and-since parabolic groups are nonunimodular-explicating the (unbounded) Dixmier-Pukanszky operator that appears in the Plancherel formula. The latter has been discovered to be a special kind of pseudodifferential operator. In this paper, the author considers the problem of extending this analysis to parabolic subgroups of semisimple algebraic groups over an arbitrary local field. Thus far he has restricted his attention to Borel subgroups (i.e. minimal parabolics) in Chevalley groups (i.e. split semisimple groups). The results he has obtained are described in this paper for the case of the symplectic group. The final result is (perhaps surprisingly), to a large extent, independent of the local field over which the group is defined. Another interesting feature of the work is the description of the ``pseudodifferential'' Dixmier-Pukanszky operator in the nonarchimedean situation.
On the ranges of analytic functions
J. S.
Hwang
623-629
Abstract: Following Doob, we say that a function $f(z)$ analytic in the unit disk U has the property $K(\rho )\,$ if $ f(0)\, = \,0$ and for some $\operatorname{arc} \,A\,$ on the unit circle whose measure $\left\vert A \right\vert\, \geqslant \,2\rho \, > \,0$, $\displaystyle \mathop {\lim \,\inf }\limits_{i \to \infty } \,\left\vert {f({P_... ... \,1\,{\text{where}}\,{P_i}\, \to \,P\, \in \,A\,{\text{and}}\,{P_i}\, \in \,U.$ We recently have solved a problem of Doob by showing that there is an integer $ N(\rho )$ such that no function with the property $K(\rho )$ can satisfy $\displaystyle {f_n}(z)\, = \,1\, + \,(1\, - \,{z^n})/{n^2},$ shows that the condition ${f_n}(0)\, = \,0$ is necessary and cannot be replaced by ${f_n}(0)\, = \,r{e^{i\alpha }}$, for $ r\, > \,1$. Naturally, we may ask whether this can be replaced by ${f_n}(0)\, = \,r{e^{i\alpha }}$, for $ r\, < \,1$? The answer turns out to be yes, when $n\, > \,N\,(r,\,\rho )$, where $\displaystyle N(r,\,\rho )\,\doteqdot\,(1/(1\, - \,r))\log (1/(1\, - \,\cos \rho )).$ .
Algebras of Fourier transforms with closed restrictions
Benjamin B.
Wells
631-636
Abstract: Let G denote a compact abelian group and let B denote a Banach subalgebra of A, the algebra of complex-valued functions on G whose Fourier series is absolutely convergent. If B contains the constant functions, separates the points of G, and if the restriction algebra, $B(E)$, is closed in $A(E)$ for every closed subset E of G, then $B = A$.
The Minakshisundaram-Pleijel coefficients for the vector-valued heat kernel on compact locally symmetric spaces of negative curvature
Roberto J.
Miatello
1-33
Abstract: We use harmonic analysis on semisimple Lie groups to determine the Minakshisundaram-Pleijel asymptotic expansion for the trace of the heat kernel on natural vector bundles over compact, locally symmetric spaces of strictly negative curvature.
Homological algebra on a complete intersection, with an application to group representations
David
Eisenbud
35-64
Abstract: Let R be a regular local ring, and let $A\, = \,R/(x)$, where x is any nonunit of R. We prove that every minimal free resolution of a finitely generated A-module becomes periodic of period 1 or 2 after at most $\operatorname{dim} \, A$ steps, and we examine generalizations and extensions of this for complete intersections. Our theorems follow from the properties of certain universally defined endomorphisms of complexes over such rings.
A relation between the coefficients in the recurrence formula and the spectral function for orthogonal polynomials
Jeffrey S.
Geronimo
65-82
Abstract: A relation is found between the rate of convergence of the coefficients in the recurrence formula for polynomials orthogonal on a segment of the real line and certain properties of the spectral function. The techniques of Banach algebras and scattering theory are used. The close connection between polynomials orthogonal on the unit circle and polynomials orthogonal on the real line is exploited.
Principal $2$-blocks of the simple groups of Ree type
Peter
Landrock;
Gerhard O.
Michler
83-111
Abstract: The decomposition numbers in characteristic 2 of the groups of Ree type are determined, as well as the Loewy and socle series of the indecomposable projective modules. Moreover, we describe the Green correspondents of the simple modules. As an application of this and similar works on other simple groups with an abelian Sylow 2-subgroup, all of which have been classified apart from those considered in the present paper, we show that the Loewy length of an indecomposable projective module in the principal block of any finite group with an abelian Sylow 2-subgroup of order ${2^n}$ is bounded by $\max \{ 2n\, + \,1,\,{2^n}\}$. This bound is the best possible.
On Harish-Chandra's $\mu $-function for $p$-adic groups
Allan J.
Silberger
113-121
Abstract: The Harish-Chandra $ \mu$-function is, up to known constant factors, the Plancherel's measure associated to an induced series of representations. In this paper we show that, when the series is induced from special representations lifted to a parabolic subgroup, the $ \mu$-function is a quotient of translated $\mu$-functions associated to series induced from supercuspidal representations. It is now known, in both the real and p-adic cases, that the $\mu$-function is always an Euler factor.
Harmonically induced representations on nilpotent Lie groups and automorphic forms on nilmanifolds
Richard C.
Penney
123-145
Abstract: It is shown that the irreducible ``discrete series'' representations of certain nilpotent Lie groups may be realized in square integrable $\bar \partial $ cohomology spaces. This theory is applied to obtain a concept of automorphic forms on nilmanifolds which generalizes the niltheta functions of Cartier and Auslander-Tolimieri. We also use the automorphic cohomology to solve certain holomorphic difference equations on ${{\textbf{C}}^n}$.
Nonstandard extensions of transformations between Banach spaces
D. G.
Tacon
147-158
Abstract: Let X and Y be (infinite-dimensional) Banach spaces and denote their nonstandard hulls with respect to an ${\aleph _1}$-saturated enlargement by $\hat X$ and $\hat Y$ respectively. If ${\mathcal{B}}\,(X,\,Y)$ denotes the space of bounded linear transformations then a subset S of elements of $ {\mathcal{B}}\,(X,\,Y)$ extends naturally to a subset $\hat S$ of ${\mathcal{B}}\,(\hat X,\,\hat Y)$. This paper studies the behaviour of various kinds of transformations under this extension and introduces, in this context, the concepts of super weakly compact, super strictly singular and socially compact operators. It shows that $({\mathcal{B}}\,(X,\,Y)\hat )\,\mathop \subset \limits_ \ne \,{\mathcal{B}}\,(\hat X,\,\hat Y)$ provided X and Y are infinite dimensional and contrasts this with the inclusion ${\mathcal{K}}(\hat H)\,\mathop \subset \limits_ \ne \,({\mathcal{K}}(H)\hat )$ where ${\mathcal{K}}(H)$ denotes the space of compact operators on a Hilbert space.
Shellable and Cohen-Macaulay partially ordered sets
Anders
Björner
159-183
Abstract: In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to R. Stanley, is generalized to posets and shown to imply shellability, while Stanley's main theorem on the Jordan-HÖlder sequences of such labelings remains valid. Further, we show a number of ways in which shellable posets can be constructed from other shellable posets and complexes. These results give rise to several new examples of Cohen-Macaulay posets. For instance, the lattice of subgroups of a finite group G is Cohen-Macaulay (in fact shellable) if and only if G is supersolvable. Finally, it is shown that all the higher order complexes of a finite planar distributive lattice are shellable.
Distinguished subfields
James K.
Deveney;
John N.
Mordeson
185-193
Abstract: Let L be a finitely generated nonalgebraic extension of a field K of characteristic $p\, \ne \,0$. A maximal separable extension D of K in L is distinguished if $ L\, \subseteq \,{K^{{p^{ - \,n}}}}(D)$ for some n. Let d be the transcendence degree of L over K. If every maximal separable extension of K in L is distinguished, then every set of d relatively p-independent elements is a separating transcendence basis for a distinguished subfield. Conversely, if $K({L^p})$ is separable over K, this condition is also sufficient. A number of properties of such fields are determined and examples are presented illustrating the results.
On the topology of simply connected algebraic surfaces
Richard
Mandelbaum;
Boris
Moishezon
195-222
Abstract: Suppose x is a smooth simply-connected compact 4-manifold. Let $ p\, = \,{\textbf{C}}{P^2}$ and $ Q\, = \, - {\textbf{C}}{P^2}$ be the complex projective plane with orientation opposite to the usual. We shall say that X is completely decomposable if there exist integers a, b such that X is diffeomorphic to $ aP\,{\text{\char93 }}\,bQ$. By a result of Wall [W1] there always exists an integer k such that $X\,\char93 \,(k\, + \,1)P\,\char93 kQ$ is completely decomposable. If $X\,\char93 \,P$ is completely decomposable we shall say that X is almost completely decomposable. In [MM] we demonstrated that any nonsingular hypersurface of $ {\textbf{C}}{P^3}$ is almost completely decomposable. In this paper we generalize this result in two directions as follows: Theorem 3.5. Suppose W is a simply-connected nonsingular complex projective 3-fold. Then there exists an integer $ {m_0}\, \geqslant \,1$ such that any hypersurface section ${V_m}$ of W of degree $m\, \geqslant \,{m_0}$ which is nonsingular will be almost completely decomposable. Theorem 5.3. Let V be a nonsingular complex algebraic surface which is a complete intersection. Then V is almost completely decomposable.
Extending combinatorial piecewise linear structures on stratified spaces. II
Douglas R.
Anderson;
Wu Chung
Hsiang
223-253
Abstract: Let X be a stratified space and suppose that both the complement of the n-skeleton and the n-stratum have been endowed with combinatorial piecewise linear (PL) structures. In this paper we investigate the problem of ``fitting together'' these separately given PL structures to obtain a single combinatorial PL structure on the complement of the $ (n\, - \,1)$-skeleton. The first main result of this paper reduces the geometrically given ``fitting together'' problem to a standard kind of obstruction theory problem. This is accomplished by introducing a tangent bundle for the n-stratum and using immersion theory to show that the ``fitting together'' problem is equivalent to reducing the structure group of the tangent bundle of the n-stratum to an appropriate group of PL homeomorphisms. The second main theorem describes a method for computing the homotopy groups arising in the obstruction theory problem via spectral sequence methods. In some cases, the spectral sequences involved are fairly small and the first few differentials are described. This paper is an outgrowth of earlier work by the authors on this problem.
Stability theorems for holomorphic foliations
T.
Duchamp;
M.
Kalka
255-266
Abstract: Here we investigate topological stability in the space of holomorphic foliations on a compact manifold. We show that under certain conditions nearby holomorphic foliations are topologically equivalent. We then present examples of foliations which are stable as holomorphic foliations but unstable as smooth foliations.
A representation theorem and applications to topological groups
J.-M.
Belley
267-279
Abstract: We show that, given a set S dense in a compact Hausdorff space X and a complex-valued bounded linear functional $ \Lambda$ on the space $ C(X)$ of continuous complex-valued functions on X with uniform norm, there exist an algebra $ {\mathcal{A}}$ of sets in S and a unique bounded finitely additive set function $\mu :\,{\mathcal{A}}\, \to \,{\textbf{C}}$ which is inner and outer regular with respect to the zero and cozero sets respectively and such that $\int_s {f\left\vert S \right.\,d\mu }$ exists and is equal to $\Lambda (f)$ for all $ f\, \in \,C(X)$. In the context of topological groups, this theorem permits us to obtain (1) a concrete representation theorem for bounded complex-valued linear functionals on the space of almost periodic functions with uniform norm, (2) a representation theorem for (not necessarily continuous) positive definite functions, (3) a characterization of the space B of finite linear combinations of positive definite functions, and (4) a necessary and sufficient condition to have a linear transformation from B to B.
Moduli for analytic left algebraic groups
Andy R.
Magid
281-291
Abstract: This paper classifies left algebraic group structures on faithfully representable complex analytic groups by establishing the existence of an algebraic variety whose complex points correspond to such structures on a given analytic group.
Disconjugacy and integral inequalities
Achim
Clausing
293-307
Abstract: The basic data in this paper are a disconjugate differential operator and an associated two-point boundary value problem. These define in a natural way a cone of functions satisfying a differential inequality with respect to the operator. By using a result of P. W. Bates and G. B. Gustafson on the monotonicity properties of Green's kernels it is shown that such a cone has a compact convex base which is a Bauer simplex. This result is used to derive a variety of integral inequalities which include known inequalities of Frank and Pick, Levin and Steckin, Karlin and Ziegler, as well as several new ones.
A characterization of periodic automorphisms of a free group
James
McCool
309-318
Abstract: Let $\theta$ be an automorphism of finite order of a free group X. We characterise the action of $\theta$ on X by showing that X has a free basis which is the disjoint union of finite subsets ${S_j}$, where if ${S_j}\, = \,\{ {u_0},\,{u_1},\, \ldots ,\,{u_k}\}$ then $ {u_i}\theta \, = \,{u_{i\, + \,1}}\,(0\, \leqslant \,1\, < \,k)$ and $ {u_k}\theta \, = \,{A_j}u_0^e{B_j}$ for some ${A_j}$, ${B_j}$ in X and $\varepsilon \, = \, \pm \,1$. As an application of this result, we obtain a list of the conjugacy classes of periodic automorphisms of the free group of rank three.
Linear operators on $L\sb{p}$ for $0<p<1$
N. J.
Kalton
319-355
Abstract: If $0\, < \,p\, < \,1$ we classify completely the linear operators $ T:\,{L_p}\, \to \,X$ where X is a p-convex symmetric quasi-Banach function space. We also show that if $T:\,{L_p}\, \to \,{L_0}$ is a nonzero linear operator, then for $p\, < \,q\, \leqslant \,2$ there is a subspace Z of ${L_p}$, isomorphic to ${L_q}$, such that the restriction of T to Z is an isomorphism. On the other hand, we show that if $ p\, < \,q\, < \,\infty$, the Lorentz space $L(p,\,q)$ is a quotient of ${L_p}$ which contains no copy of ${l_p}$.
Convergence and Cauchy structures on lattice ordered groups
Richard N.
Ball
357-392
Abstract: This paper employs the machinery of convergence and Cauchy structures in the task of obtaining completion results for lattice ordered groups. §§1 and 2 concern l-convergence and l-Cauchy structures in general. §4 takes up the order convergence structure; the resulting completion is shown to be the Dedekind-MacNeille completion. §5 concerns the polar convergence structure; the corresponding completion has the property of lateral completeness, among others. A simple theory of subset types routinizes the adjoining of suprema in §3. This procedure, nevertheless, is shown to be sufficiently general to prove the existence and uniqueness of both the Dedekind-MacNeille completion in §4 and the lateral completion in §5. A proof of the existence and uniqueness of a proper class of similar completions comes free. The principal new hull obtained by the techniques of adjoining suprema is the type $\mathcal{Y}$ hull, strictly larger than the lateral completion in general.
Riemann surfaces and bounded holomorphic functions
Walter
Pranger
393-400
Abstract: The principal result of this article asserts the equivalence of the following four conditions on a hyperbolic Riemann surface X: (a) the following set $ z\vert\,\vert f(z)\vert\, \leqslant \,{\text{sup}}\,\vert f\vert$ on K for every bounded holomorphic section f of $\xi$ is compact for every unitary vector bundle $ \xi$ and every compact set K; (b) every unitary line bundle has nontrivial bounded holomorphic sections and the condition in (a) holds for $ \xi \, = \,{i_d}$; (c) every unitary line bundle has nontrivial bounded holomorphic sections and X is regular for potential theory; (d) every unitary line bundle has nontrivial bounded holomorphic sections and X is its own B-envelope of holomorphy. If X is a subset of C, these are also equivalent to the following: (e) for every unitary line bundle $\xi$ the bounded holomorphic sections are dense in the holomorphic sections.
The von Neumann kernel and minimally almost periodic groups
Sheldon
Rothman
401-421
Abstract: We calculate the von Neumann kernel $n(G)$ of an arbitrary connected Lie group. As a consequence we see that the closed characteristic subgroup $n(G)$ is also connected. It is shown that any Levi factor of a connected Lie group is closed. Then, various characterizations of minimal almost periodicity for a connected Lie group are given. Among them is the following. A connected Lie group G with radical R is minimally almost periodic (m.a.p.) if and only if $G/R$ is semisimple without compact factors and $G\, = \,{[G,\,G]^ - }$. In the special case where R is also simply connected it is proven that $ G\, = \,[G,\,G]$. This has the corollary that if the radical of a connected m.a.p. Lie group is simply connected then it is nilpotent. Next we prove that a connected m.a.p. Lie group has no nontrivial automorphisms of bounded displacement. As a consequence, if G is a m.a.p. connected Lie group, H is a closed subgroup of G such that $G/H$ has finite volume, and $\alpha$ is an automorphism of G with ${\text{disp}}(\alpha ,\,H)$ bounded, then $ \alpha$ is trivial. Using projective limits of Lie groups we extend most of our results on the characterization of m.a.p. connected Lie groups to arbitrary locally compact connected topological groups, and finally get a new and relatively simple proof of the Freudenthal-Weil theorem.
Central Fourier-Stieltjes transforms with an isolated value
Alan
Armstrong
423-437
Abstract: Let $\mu$ be a central Borel measure on a compact, connected group G. If 0 is isolated in the range of $ {\hat \mu }$, then there exists a closed, normal subgroup H of G such that ${\pi _H}\mu$, the restriction of $ \mu$ to the cosets of H, is the convolution of an invertible measure with a nonzero idempotent measure. This result extends I. Glicksberg's result for LCA groups. An example is given which shows that this result is false in general for disconnected groups.
Some curvature properties of locally conformal K\"ahler manifolds
Izu
Vaisman
439-447
Abstract: Curvature identities and holomorphic sectional curvature of locally conformal Kähler manifolds are investigated. Particularly, sufficient conditions for such manifolds to be globally conformal Kähler are derived.
Locally free affine group actions
J. F.
Plante
449-456
Abstract: Differentiable actions by the nonabelian 2-dimensional Lie group on compact manifolds are considered. When the action is locally free and the orbits have codimension one it is shown that there are at most finitely many minimal sets each containing a countably infinite number of cylindrical orbits. Examples are given to show that various codimension, differentiability, and minimality restrictions are necessary.
On linear algebraic semigroups
Mohan S.
Putcha
457-469
Abstract: Let K be an algebraically closed field. By an algebraic semigroup we mean a Zariski closed subset of ${K^n}$ along with a polynomially defined associative operation. Let S be an algebraic semigroup. We show that S has ideals $ {I_0},\, \ldots \,,\,{I_t}$ such that $S\, = \,{I_t}\, \supseteq \, \cdots \, \supseteq \,{I_0}$, ${I_0}$ is the completely simple kernel of S and each Rees factor semigroup ${I_k}/{I_{k - 1}}$ is either nil or completely 0-simple $ (k\, = \,1,\, \ldots \,,\,t)$. We say that S is connected if the underlying set is irreducible. We prove the following theorems (among others) for a connected algebraic semigroup S with idempotent set $E(S)$. (1) If $E(S)$ is a subsemigroup, then S is a semilattice of nil extensions of rectangular groups. (2) If all the subgroups of S are abelian and if for all $a\, \in \,S$, there exists $e\, \in \,E(S)$ such that $ea\, = \,ae\, = a$, then S is a semilattice of nil extensions of completely simple semigroups. (3) If all subgroups of S are abelian and if S is regular, then S is a subdirect product of completely simple and completely 0-simple semigroups. (4) S has only trivial subgroups if and only if S is a nil extension of a rectangular band.
On linear algebraic semigroups. II
Mohan S.
Putcha
471-491
Abstract: We continue from [11] the study of linear algebraic semigroups. Let S be a connected algebraic semigroup defined over an algebraically closed field K. Let $\mathcal{U}(S)$ be the partially ordered set of regular $ \mathcal{J}$-classes of S and let $E(S)$ be the set of idempotents of S. The following theorems (among others) are proved. (1) $\mathcal{U}(S)$ is a finite lattice. (2) If S is regular and the kernel of S is a group, then the maximal semilattice image of S is isomorphic to the center of $E(S)$. (3) If S is a Clifford semigroup and $f\, \in \,E(S)$, then the set $\{ \,e\,\vert\,e\, \in \,E(S),\,e\, \geqslant \,f\}$ is finite. (4) If S is a Clifford semigroup, then there is a commutative connected closed Clifford subsemigroup T of S with zero such that T intersects each $ \mathcal{J}$-class of S. (5) If S is a Clifford semigroup with zero, then S is commutative and is in fact embeddable in $({K^n},\, \cdot )$ for some $n\, \in \,{\textbf{Z}^ + }$. (6) If ${\text{ch}}\, \cdot \,K\, = \,0$ and S is a commutative Clifford semigroup, then S is isomorphic to a direct product of an abelian connected unipotent group and a closed connected subsemigroup of $({K^n},\, \cdot )$ for some $n\, \in \,{\textbf{Z}^ + }$. (7) If S is a regular semigroup and $ {\text{dim}}\, \cdot \,S\, \leqslant \,2$, then $\left\vert {\mathcal{U}(S)} \right\vert\, \leqslant \,4$. (8) If S is a Clifford semigroup with zero and ${\text{dim}}\, \cdot \,S\, = \,3$, then $\left\vert {E(S)} \right\vert\, = \,\left\vert {\mathcal{U}(S)} \right\vert$ can be any even number $ \geqslant \,8$. (9) If S is a Clifford semigroup then $\mathcal{U}(S)$ is a relatively complemented lattice and all maximal chains in $\mathcal{U}(S)$ have the same number of elements.
Some categorical equivalences for $E$-unitary inverse semigroups
Mario
Petrich
493-503
Abstract: The structure of E-unitary inverse semigroups has been described by McAlister and by Reilly and the author. The parameters in the first structure theorem may be made into a category, and the same holds for the parameters in the second structure theorem. We prove that each of these categories is equivalent to the category of E-unitary inverse semigroups and their homomorphisms. We also provide functors between the two first-mentioned categories which are naturally equivalent to the composition of the functors figuring in the categorical equivalence referred to above.
An application of homological algebra to the homotopy classification of two-dimensional CW-complexes
Micheal N.
Dyer
505-514
Abstract: Let $\pi$ be ${Z_m}\, \times \,{Z_n}$. In this paper the homotopy types of finite connected two dimensional CW-complexes with fundamental group $\pi$ are shown to depend only on the Euler characteristic. The basic method is to study the structure of the group ${\text{Ext}}_{Z\pi }^1(I{\pi ^2},\,Z)$ as a principal ${\text{End(}}I{\pi ^2}{\text{)}}$-module.
Branched extensions of curves in orientable surfaces
Cloyd L.
Ezell;
Morris L.
Marx
515-532
Abstract: Given a set of regular curves ${f_1}\,,\,\ldots,\,{f_\rho }$ in an orientable surface N, we are concerned with the existence and structure of all sense-preserving maps $F:\,M\, \to \,N$ where (a) M is a bordered orientable surface with $\rho$ boundary components ${K_1},\ldots,\,{K_\rho }$, (b) $ F\vert{K_i}\, = \,{f_i},\,i\, = \,1,\,\ldots,\,\rho$, (c) at each interior point of M, there is an integer n such that F is locally topologically equivalent to the complex map $ w\, = \,{z^n}$.
Branched extensions of curves in compact surfaces
Cloyd L.
Ezell
533-546
Abstract: A polymersion is a map $ F:\,M\, \to \,N$ where M and N are compact surfaces, orientable or nonorientable, M a surface with boundary, where (a) At each interior point of M, there is an integer $ n\, \geqslant \,1$ such that F is topologically equivalent to the complex map ${z^n}$ in a neighborhood about the point. (b) At each point x in the boundary of M, $ \delta M$, there is a neighborhood U containing x such that U is homeomorphic to F(U). A normal polymersion is one where $F(\delta M)$ is a normal set of curves in N. We are concerned with establishing a combinatorial representation for normal polymersions which map to arbitrary compact surfaces.
Quasilinear evolution equations in Banach spaces
Michael G.
Murphy
547-557
Abstract: This paper is concerned with the quasi-linear evolution equation $[0,\,T],\,u(0)\, = \,{x_0}$ in a Banach space setting. The spirit of this inquiry follows that of T. Kato and his fundamental results concerning linear evolution equations. We assume that we have a family of semigroup generators that satisfies continuity and stability conditions. A family of approximate solutions to the quasi-linear problem is constructed that converges to a ``limit solution.'' The limit solution must be the strong solution if one exists. It is enough that a related linear problem has a solution in order that the limit solution be the unique solution of the quasi-linear problem. We show that the limit solution depends on the initial value in a strong way. An application and the existence aspect are also addressed.
Effective $p$-adic bounds for solutions of homogeneous linear differential equations
B.
Dwork;
P.
Robba
559-577
Abstract: We consider a finite set of power series in one variable with coefficients in a field of characteristic zero having a chosen nonarchimedean valuation. We study the growth of these series near the boundary of their common ``open'' disk of convergence. Our results are definitive when the wronskian is bounded. The main application involves local solutions of ordinary linear differential equations with analytic coefficients. The effective determination of the common radius of convergence remains open (and is not treated here).
Affine connections and defining functions of real hypersurfaces in ${\bf C}\sp{n}$
Hing Sun
Luk
579-588
Abstract: The affine connection and curvature introduced by Tanaka on a strongly pseudoconvex real hypersurface are computed explicitly in terms of its defining function. If Fefferman's defining function is used, then the Ricci form is shown to be a function multiple of the Levi form. The factor is computable by Fefferman's algorithm and its positivity implies the vanishing of certain cohomology groups (of the $ {\bar \partial _b}$ complex) in the compact case.
Complementary series for $p$-adic groups. I
Allan J.
Silberger
589-598
Abstract: Let $\Omega$ be a nonarchimedean local field, G the group of $\Omega$-points of a connected reductive algebraic group defined over $\Omega$. This paper establishes that to each zero of the Plancherel measure of G one can associate complementary series. Our result is the analogue for p-adic groups of a similar statement, announced separately by Knapp-Stein and Harish-Chandra, for real groups.
Approximately finite-dimensional $C\sp{\ast} $-algebras and Bratteli diagrams
A. J.
Lazar;
D. C.
Taylor
599-619
Abstract: We determine properties of an AF algebra by observing the characteristics of its diagram. In particular, we characterize AF algebras that are liminal, postliminal, antiliminal and with continuous trace; moreover, we characterize liminal AF algebras with Hausdorff spectrum. Some elementary examples of AF algebras with certain desired properties are constructed by using these characterizations.
Hermite-Birkhoff interpolation in the $n$th roots of unity
A. S.
Cavaretta;
A.
Sharma;
R. S.
Varga
621-628
Abstract: Consider, as nodes for polynomial interpolation, the nth roots of unity. For a sufficiently smooth function $ f(z)$, we require a polynomial $p(z)$ to interpolate f and certain of its derivatives at each node. It is shown that the so-called Pólya conditions, which are necessary for unique interpolation, are in this setting also sufficient.
Flows on fibre bundles
J. L.
Noakes
629-635
Abstract: Conditions are given under which a fibrewise flow on a fibre bundle must have a nonempty catastrophe space.
Erratum to: ``On parabolic measures and subparabolic functions''
Jang Mei G.
Wu
636-636
Companionship of knots and the Smith conjecture
Robert
Myers
1-32
Abstract: This paper studies the Smith Conjecture in terms of H. Schubert's theory of companionship of knots. Suppose J is a counterexample to the Smith Conjecture, i.e. is the fixed point set of an action of ${{\textbf{Z}}_p}$ on ${S^3}$. Theorem. Every essential torus in an invariant knot space $C(J)$ of J is either invariant or disjoint from its translates. Since the companions of J correspond to the essential tori in $C(J)$, this often allows one to split the action among the companions and satellites of J. In particular: Theorem. If J is composite, then each prime factor of J is a counterexample, and conversely. Theorem. The Smith Conjecture is true for all cabled knots. Theorem. The Smith Conjecture is true for all doubled knots. Theorem. The Smith Conjecture is true for all cable braids. Theorem. The Smith Conjecture is true for all nonsimple knots with bridge number less than five. In addition we show: Theorem. If the Smith Conjecture is true for all simple fibered knots, then it is true for all fibered knots. Theorem. The Smith Conjecture is true for all nonfibered knots having a unique isotopy type of incompressible spanning surface.
The group of rational solutions of $y\sp{2}=x(x-1)(x-t\sp{2}-c)$
Charles F.
Schwartz
33-46
Abstract: In this paper, we show that the Mordell-Weil group of the Weierstrass equation ${y^2}\, = \,x(x\, - \,1)(x\, - \,{t^2}\, - \,c),\,c \ne \,0,\,1$ (i.e., the group of solutions (x,y), with $x,\,y\, \in \,{\textbf{C}}(t)$) is generated by its elements of order 2, together with one element of infinite order, which is exhibited.
Twisted Lubin-Tate formal group laws, ramified Witt vectors and (ramified) Artin-Hasse exponentials
Michiel
Hazewinkel
47-63
Abstract: For any ring R let $\Lambda (R)$ denote the multiplicative group of power series of the form $1\, + \,{a_1}t\, + \, \cdots$ with coefficients in R. The Artin-Hasse exponential mappings are homomorphisms $ {W_{p,\,\infty }}(k)\, \to \,\Lambda ({W_{p,\,\infty }}(k))$, which satisfy certain additional properties. Somewhat reformulated, the Artin-Hasse exponentials turn out to be special cases of a functorial ring homomorphism $E:\,{W_{p,\,\infty }}( - )\, \to \,{W_{p,\,\infty }}({W_{p,\,\infty }}( - ))$, where ${W_{p,\,\infty }}$ is the functor of infinite-length Witt vectors associated to the prime p. In this paper we present ramified versions of both ${W_{p,\,\infty }}( - )$ and E, with ${W_{p,\,\infty }}( - )$ replaced by a functor $ W_{q,\,\infty }^F( - )$, which is essentially the functor of q-typical curves in a (twisted) Lubin-Tate formal group law over A, where A is a discrete valuation ring that admits a Frobenius-like endomorphism $ \sigma$ (we require $\sigma (a)\, \equiv \,{a^q}\,\bmod \,{\mathcal{m}}$ for all $a\, \in \,A$, where ${\mathcal{m}}$ is the maximal idea of A). These ramified-Witt-vector functors $W_{q,\,\infty }^F( - )$ do indeed have the property that, if $ k\, = \,A/{\mathcal{m}}$ is perfect, A is complete, and $l/k$ is a finite extension of k, then $ W_{q,\,\infty }^F(l)$ is the ring of integers of the unique unramified extension $ L/K$ covering $ l/k$.
Some undecidability results concerning Radon measures
R. J.
Gardner;
W. F.
Pfeffer
65-74
Abstract: We show that in metalindelöf spaces certain questions about Radon measures cannot be decided within the Zermelo-Fraenkel set theory, including the axiom of choice.
On generalized harmonic analysis
Ka Sing
Lau;
Jonathan K.
Lee
75-97
Abstract: Motivated by Wiener's work on generalized harmonic analysis, we consider the Marcinkiewicz space $ {{\mathcal{M}}^p}({\textbf{R}})$ of functions of bounded upper average p power and the space $ {{\mathcal{V}}^p}({\textbf{R}})$ of functions of bounded upper p variation. By identifying functions whose difference has norm zero, we show that $ {{\mathcal{V}}^p}({\textbf{R}})$, $ 1\, < \,p\, < \infty$, is a Banach space. The proof depends on the result that each equivalence class in $ {{\mathcal{V}}^p}({\textbf{R}})$ contains a representative in ${L^p}({\textbf{R}})$. This result, in turn, is based on Masani's work on helixes in Banach spaces. Wiener defined an integrated Fourier transformation and proved that this transformation is an isometry from the nonlinear subspace $ {{\mathcal{W}}^2}({\textbf{R}})$ of $ {{\mathcal{M}}^2}({\textbf{R}})$ consisting of functions of bounded average quadratic power, into the nonlinear subspace $ {{\mathcal{u}}^2}({\textbf{R}})$ of $ {{\mathcal{V}}^2}({\textbf{R}})$ consisting of functions of bounded quadratic variation. By using two generalized Tauberian theorems, we prove that Wiener's transformation W is actually an isomorphism from $ {{\mathcal{M}}^2}({\textbf{R}})$ onto $ {{\mathcal{V}}^2}({\textbf{R}})$. We also show by counterexamples that W is not an isometry on the closed subspace generated by $ {{\mathcal{W}}^2}({\textbf{R}})$.
On the Hardy-Littlewood maximal function and some applications
C. J.
Neugebauer
99-105
Abstract: With a monotone family $ F\, = \,\{ {S_\alpha }\} ,\,{S_\alpha }\, \subset \,{{\textbf{R}}^n}$, we associate the Hardy-Littlewood maximal function ${M_F}f(x)\, = \,{\sup _\alpha }(1/\left\vert {{S_\alpha }} \right\vert)\int_{{S_\alpha }\, + \,x} {\left\vert f \right\vert}$. In general, ${M_F}$ is not weak type (1.1). However, if we replace in the denominator $ {S_\alpha }$ by $S_F^ {\ast} \, = \,\{ x\, - \,y:\,x,\,y\, \in \,{S_\alpha }\}$, and denote the resulting maximal function by $M_F^ {\ast}$, then $M_F^ {\ast}$ is weak type (1, 1) with weak type constant 1.
Generic deformations of varieties
Yieh Hei
Wan
107-119
Abstract: Typical families of varieties, which are defined by families of smooth maps, are studied through the method from the singularity theory of differentiable maps. It is proved that generic families of varieties (of certain types) are stable in an appropriate sense.
Subspaces of basically disconnected spaces or quotients of countably complete Boolean algebras
Eric K.
van Douwen;
Jan
van Mill
121-127
Abstract: Under ${\text{MA}}\,{\text{ + }}\,{{\text{2}}^\omega }\, = \,{\omega _2}$ there is a (compact) strongly zero-dimensional F-space of weight ${2^\omega }$ which cannot be embedded in any basically disconnected space. Dually, under $ {\text{MA}}\, + \,{2^\omega }\, = \,{\omega _2}$ there is a weakly countably complete (or almost $\sigma$-complete, or countable separation property) Boolean algebra of cardinality ${2^\omega }$ which is not a homomorphic image of any countably complete Boolean algebra. The key to our construction is the observation that if X is a subspace of a basically disconnected space and $ \beta \omega \, \subseteq \,X$ then $\beta \omega $ is a retract of X. Dually, if B is a homomorphic image of a countably complete Boolean algebra, and if h is a homomorphism from B onto $\mathcal{P}(\omega )$, the field of subsets of w, then there is an embedding $ e:\,\mathcal{P}(\omega ) \to \,B$ such that $h\, \circ \,e\, = \,{\text{i}}{{\text{d}}_{\mathcal{P}(\omega )}}$.
Permutation-partition pairs: a combinatorial generalization of graph embeddings
Saul
Stahl
129-145
Abstract: Permutation-partition pairs are a purely combinatorial generalization of graph embeddings. Some parameters are defined here for these pairs and several theorems are proved. These results are strong enough to prove virtually all the known theoretical informaton about the genus parameter as well as a new theorem regarding the genus of the amalgamation of two graphs over three points.
Application of the extremum principle to investigating certain extremal problems
L.
Mikołajczyk;
S.
Walczak
147-155
Abstract: Denote by C, K, X, respectively, a complex plane, the disc $\{ z\, \in \,{\textbf{C:}}\,\left\vert z \right\vert\, < \,1\} $ and any compact Hausdorff space. Denote by P a set of probabilistic measures defined on Borel subsets of the space X. For $\mu \, \in \,P$, let $ f(z)\, = \,\int_X {q(z,\,t)\,d\mu } ,\,z\, \in \,K$, and ${\mathcal{F}}\, = \,\{ f:\,\mu \, \in \,P\}$. Consider a finite sequence of real functions $ {F_0},\,{F_{1,}}\, \ldots ,\,{F_m}$ defined in the space ${R^{2n}}$. Let ${\zeta _1},\, \ldots ,\,{\zeta _k}$ be fixed points of the disc K and $\eta (f)\, = \,[\operatorname{re} \,{f^{(0)}}(\zeta ),\,\operatorname{im} \,{f... ...e} \,{f^{({n_k})}}({\zeta _k}),\,\operatorname{im} \,{f^{({n_k})}}({\zeta _k})]$, where $ f\, \in \,{\mathcal{F}},\,n\, = \,{n_1}\, + \, \cdots \, + \,{n_k}\, + \,k$. Let ${F_j}(f)\, = \,{F_j}(\eta (f)),\,j\, = \,0,\,1,\, \ldots ,\,m$. We consider the following extremal problem. Determine a minimum of the functional ${F_0}(f)$ under the conditions $ {F_j}(f)\, \leqslant \,0,\,j\, = \,1,\,2,\, \ldots ,\,m,\,f\, \in \,{\mathcal{F}}$. We apply the extremum principle to solve this problem. In the linear case this problem was investigated in [11].
Lifting surgeries to branched covering spaces
Hugh M.
Hilden;
José María
Montesinos
157-165
Abstract: It is proved that if ${M^n}$ is a branched covering of a sphere, branched over a manifold, so is ${M^n}\, \times \,{S^m}$, but the number of sheets is one more. In particular, the n-dimensional torus is an n-fold simple covering of ${S^n}$ branched over an orientable manifold. The proof involves the development of a new technique to perform equivariant handle addition. Other consequences of this technique are given.
Some examples of sequence entropy as an isomorphism invariant
F. M.
Dekking
167-183
Abstract: With certain geometrically diverging sequences A and the shift T on dynamical systems arising from substitutions we associate a Markov shift S such that the A-entropy of T equals the usual entropy of S. We present examples to demonstrate the following results. Sequence entropy can distinguish between an invertible ergodic transformation and its inverse. A-entropy does not depend monotonically on A. The variational principle for topological sequence entropy need not hold.
Equivariant dynamical systems
M. J.
Field
185-205
Abstract: The basic properties of vector fields and diffeomorphisms invariant under the action of a compact Lie group are presented. A Kupka-Smale density theorem for equivariant dynamical systems and an existence theorem for equivariant Morse-Smale systems on an arbitrary compact G-manifold are proved.
Zeros of successive derivatives of entire functions of the form $h(z){\rm exp}(-e\sp{z})$
Albert
Edrei
207-226
Abstract: Consider $f(z)\, = \,h(z)\exp ( - {e^z})\,(z\, = \,x\, + \,iy)$ where $h(z)$ is a real entire function of finite order having no zeros in some strip $\{ x\, + \,iy:\,\left\vert {y - \pi } \right\vert\, < \,{\eta _1},\,x\, > \,{x_0}\} \,(0\, < \,{\eta _1})$. The author studies the power series (1) $f(\tau \, + \,z)\, = \,\Sigma _{n = 0}^\infty \,{a_n}{z^n}$ ($\tau$ real) and the number $N({\tau _1},\,{\tau _2};\,n)$ of real zeros of ${f^{(n)}}(z)$ which lie in the interval $[{\tau _1},\,{\tau _2}]$. He proves (2) $N({\tau _1},{\tau _2};\,n)\, \sim \,({\tau _2} - {\tau _1})n/{(\log \,n)^2}\,(n \to \infty )$. With regard to the expansion (1) he determines a positive, strictly increasing, unbounded sequence $ \{ {v_k}\} _{k = 1}^\infty$ such that $({v_{k + 1}} - {v_k})/\log \,{v_k} \to 1\,(k \to \infty )$ and having the following properties: (i) if $ {v_k}\, < \,n\, < \,{v_{k + 1}}$, then $ {a_n}\, \ne \,0$ and all the ${a_n}$ have the same sign; (ii) if in addition $ {v_{k + 1}}\, < \,m\, < \,{v_{k + 2}}$, tnen ${a_m}{a_n}\, < 0$. It is possible to deduce from (2) the complete characterization of the final set (in the sense of Pólya) of $\exp ( - {e^z})$.
$p$-subgroups of compact Lie groups and torsion of infinite height in $H\sp{\ast} (BG)$
Mark
Feshbach
227-233
Abstract: The relation between elementary abelian p-subgroups of a connected compact Lie group G and the existence of p-torsion in $ {H^ {\ast} }(G)$ has been known for some time [B-S]. In this paper we prove that if G is any compact Lie group then ${H^ {\ast} }(BG)$ contains p-torsion of infinite height iff G contains an elementary abelian p-group not contained in a maximal torus. The hard direction is proven using the double coset theorem for the transfer. A third equivalent condition is also given.
Littlewood-Paley and multiplier theorems on weighted $L\sp{p}$ spaces
Douglas S.
Kurtz
235-254
Abstract: The Littlewood-Paley operator $\gamma (f)$, for functions f defined on $ {{\textbf{R}}^n}$, is shown to be a bounded operator on certain weighted $ {L^p}$ spaces. The weights satisfy an ${A_p}$ condition over the class of all n-dimensional rectangles with sides parallel to the coordinate axes. The necessity of this class of weights demonstrates the 1-dimensional nature of the operator. Results for multipliers are derived, including weighted versions of the Marcinkiewicz Multiplier Theorem and Hörmander's Multiplier Theorem.
Statically tame periodic homeomorphisms of compact connected $3$-manifolds. II. Statically tame implies tame
Edwin E.
Moise
255-280
Abstract: Let f be a periodic homeomorphism $M\, \leftrightarrow \,M$, where M is a compact connected 3-manifold (without boundary). Suppose that for each i, the fixed-point set of $ {f^i}$ is a tame set. Then f is simplicial, relative to some triangulation of M.
On the boundary values of Riemann's mapping function
R. J. V.
Jackson
281-297
Abstract: Classically, the calculus of variations is required to prove the existence of a biholomorphism from the unit disk to a given simply-connected, smooth domain in the complex plane. Here, the problem is reduced to the solution of an ordinary differential equation along the boundary of the domain. The sole coefficient in this equation is identified with the bounded term in the asymptotic expansion of the Bergman kernel function. It is shown that this coefficient can not depend upon any differential expression involving only the curvature function of the boundary.
Generation of analytic semigroups by strongly elliptic operators under general boundary conditions
H. Bruce
Stewart
299-310
Abstract: Strongly elliptic operators are shown to generate analytic semigroups of evolution operators in the topology of uniform convergence, when realized under general boundary conditions on (possibly) unbounded domains. An application to the existence and regularity of solutions to parabolic initial-boundary value problems is indicated.
The fundamental theorem on torsion classes of lattice-ordered groups
Jorge
Martinez
311-317
Abstract: This paper generalizes the earlier notion of a torsion class to a setting where its significance can be fully realized. The dual notion of a torsion-free class is herein defined and the fundamental Connection Theorem is proved. In addition, a few restrictions are considered, in particular, how to view the application of the main theorem to the hereditary classes.
Erratum to: ``A construction of uncountably many weak von Neumann transformations'' [Trans. Amer. Math. Soc. {\bf 257} (1980), no. 2, 397--410]
Karl
David
318-318
Cosmoi of internal categories
Ross
Street
271-318
Abstract: An internal full subcategory of a cartesian closed category $\mathcal{A}$, is shown to give rise to a structure on the 2-category $ Cat(\mathcal{A})$ of categories in $ \mathcal{A}$ which introduces the notion of size into the analysis of categories in $\mathcal{A}$ and allows proofs by transcendental arguments. The relationship to the currently popular study of locally internal categories is examined. Internal full subcategories of locally presentable categories (in the sense of Gabriel-Ulmer) are studied in detail. An algorithm is developed for their construction and this is applied to the categories of double categories, triple categories, and so on.
Matrix-valued special functions and representation theory of the conformal group. I. The generalized gamma function
Kenneth I.
Gross;
Wayne J.
Holman
319-350
Abstract: This article examines in detail the matrix-valued gamma function $\displaystyle {\Gamma ^{{\lambda ^0}}}\,(\alpha )\, = \,\int_P {{e^{ - {\text{tr}}(r)}}{\lambda ^0}(r,\,\bar r){{(\det \,r)}^{\alpha - 2}}\,} dr$ associated to the conformal group $ G\, = \,U(2,\,2)$. Here, $ \alpha$ is a continuous complex parameter, $ {\lambda ^0}$ runs through a family of ``weights'' of $K\, = \,U(2)\, \times \,U(2)$, P is the cone of $2\, \times \,2$ positive-definite Hermitian matrices, and the integral is well known to converge absolutely for ${\text{Re}}(\alpha )\, > \,1$. However, until now very little has been known about the analytic continuation for the general weight $ {\lambda ^0}$. The results of this paper include the following: The complete analytic continuation of ${\Gamma ^{{\lambda ^0}}}$ is determined for all weights $ {\lambda ^0}$. In analogy to the case of the classical gamma function it is proved that for any weight $ {\lambda ^0}$ the mapping $\alpha \, \to \,{\Gamma ^{{\lambda ^0}}}\,{(\alpha )^{ - 1}}$ is entire. A new integral formula is given for the inverse of the gamma function. An explicit calculation is given for the normalized variant of the gamma matrix that arises in the reproducing kernel for the spaces in which the holomorphic discrete series of G is realized, and one observes that the behavior of the analytic continuation for weights ``in general position'' is markedly different from the special cases in which the gamma function ``is scalar". The full analytic continuation of the holomorphic discrete series for G is determined. The gamma function for the forward light cone (the boundary orbit) is found, and the associated Hardy space of vector-valued holomorphic functions is described. Analogs are given for some of the well-known formulas for the classical gamma function. As an epilogue, applications of the matrix-valued gamma function, such as generalizations to $2\, \times \,2$ matrix space of the classical binomial theorem, are announced. These applications require a detailed understanding of the (generalized) Bessel functions associated to the conformal group that will be treated in the sequel to this paper.
Equivariant homotopy theory and Milnor's theorem
Stefan
Waner
351-368
Abstract: The foundations of equivariant homotopy and cellular theory are examined; an equivariant Whitehead theorem is proved, and the classical results by Milnor about spaces with the homotopy-type of a CW complex are generalized to the equivariant case. The ambient group G is assumed compact Lie. Further results include equivariant cellular approximation and the procedure for replacement of an arbitrary G-space by a G-CW complex.
Equivariant fibrations and transfer
Stefan
Waner
369-384
Abstract: The basic properties of equivariant fibrations are described, including an equivariant version of the Ďold Theorem. The foundations of equivariant stable homotopy theory are described, and the theory of equivariant transfer is developed.
Equivariant classifying spaces and fibrations
Stefan
Waner
385-405
Abstract: Explicit classifying spaces for equivariant fibrations are constructed using the geometric two-sided bar construction. The constructions are then extended to classify stable equivariant spherical fibrations and equivariant K-theory. The ambient groups is assumed compact Lie.
On the existence of eigenvalues of differential operators dependent on a parameter
Sh.
Strelitz;
S.
Abramovich
407-429
Abstract: In this paper we obtain results about the existence of eigenvalues for a system which depends polynomially on $ \lambda$, $ k\, = \,1,...,\,N$. In order to get these results we prove that this system can be reduced to a standard system of the form $ k\, = \,1,...,\,n$.
On meromorphic solutions of algebraic differential equations
Sh.
Strelitz
431-440
Abstract: The Malmquist Theorem is generalized for equations of the type
Cyclic extensions of parafree groups
Peng Choon
Wong
441-456
Abstract: Let $ 1\, \to \,F\, \to \,G\, \to \,T\, \to \,1$ be a short exact sequence where F is parafree and T is infinite cyclic. We examine some properties of G when $F/F'$ is a free ZT-module. Here $ F'$ is the commutator subgroup of F and ZT is the integral group ring of T. In particular, we show G is parafree and ${\gamma _n}F/{\gamma _{n + 1}}F$ is a free ZT-module for every $n \geqslant 1$ (where ${\gamma _n}F$ is the nth term of the lower central series of F).
Parametrizations of $G\sb{\delta }$-valued multifunctions
H.
Sarbadhikari;
S. M.
Srivastava
457-466
Abstract: Let T, X be Polish spaces, $ \mathcal{J}$ a countably generated sub-$\sigma$-field of $ {\mathcal{B}_T}$, the Borel $\sigma$-field of T, and $F:\,T\, \to \,X$ a multifunction such that $ F(t)$ is a ${G_\delta }$ in X for each $t\, \in \,T$. F is $\mathcal{J}$-measurable and $ {\text{Gr}}(F)\, \in \,J\, \otimes \,{\mathcal{B}_X}$, where ${\text{Gr}}(F)$ denotes the graph of F. We prove the following three results on F. (I) There is a map $f:\,T\, \times \,\Sigma \, \to \,X$ such that for each $ t\, \in \,T,\,f(t,\, \cdot )$ is a continuous, open map from $\Sigma$ onto $F(t)$ and for each $\sigma \, \in \,\Sigma ,\,f( \cdot ,\,\sigma )$ is $ \mathcal{J}$-measurable, where $\Sigma$ is the space of irrationals. (II) The multifunction F is of Souslin type. (III) If X is uncountable and $F(t),\,t\, \in \,T$, are all dense-in-itself then there is a $\mathcal{J}\, \otimes \,{\mathcal{B} _X}$-measurable map $ f:\,T\, \times \,X\, \to \,X$ such that for each $t\, \in \,T,\,f(t,\, \cdot )$ is a Borel isomorphism of X onto $F(t)$.
Scattering theory and polynomials orthogonal on the real line
J. S.
Geronimo;
K. M.
Case
467-494
Abstract: The techniques of scattering theory are used to study polynomials orthogonal on a segment of the real line. Instead of applying these techniques to the usual three-term recurrence formula, we derive a set of two two-term recurrence formulas satisfied by these polynomials. One of the advantages of these new recurrence formulas is that the Jost function is related, in the limit as $n\, \to \,\infty $, to the solution of one of the recurrence formulas with the boundary conditions given at $n\, = \,0$. In this paper we investigate the properties of the Jost function and the spectral function assuming the coefficients in the recurrence formulas converge at a particular rate.
On oscillatory elliptic equations on manifolds
A.
Baider;
E. A.
Feldman
495-504
Abstract: In this note we investigate the possibility of oscillatory behavior for a second-order selfadjoint elliptic operators on noncompact Riemannian manifolds (M, g). Let A be such an operator which is semibounded below and symmetric on $C_0^\infty (M)\, \subseteq \,{L^2}(M,\,d\mu )$ where $ d\mu$ is a volume element on M. If $\varphi$ is a $ {C^\infty }$ function such that $ A\varphi \, = \,\lambda \varphi$, we would naively say that $\varphi$ is oscillatory (and by extension $\lambda$ is oscillatory if it possesses such an eigenfunction $\varphi$) if $M\, - \,{\varphi ^{ - 1}}(0)$ has an infinite number of bounded connected components. For technical reasons this is not quite adequate for a definition. However, in §1 we give the usual definition of oscillatory which is a slight generalization of the one above. Let ${\Lambda _0}$ be the number below which this phenomenon cannot occur; $ {\Lambda _0}$ is the oscillatory constant for the operator A. In that A is semibounded and symmetric on $ C_0^\infty (M)\, \subseteq \,{L^2}(M,\,d\mu )$, A has a Friedrichs extension. Let $ {\Lambda _c}$ be the bottom of the continuous spectrum of the Friedrichs extension of A. Our main result is ${\Lambda _0}\, = \,{\Lambda _c}$.
The Witt ring of a space of orderings
Murray
Marshall
505-521
Abstract: The theory of ``space of orderings'' generalizes the reduced theory of quadratic forms over fields (or, more generally, over semilocal rings). The category of spaces of orderings is equivalent to a certain category of ``abstract Witt rings". In the particular case of the space of orderings of a formally real field K, the corresponding abstract Witt ring is just the reduced Witt ring of K. In this paper it is proved that if $X\, = \,(X,\,G)$ is any space of orderings with Witt ring W(X), and $X\, \to \,Z$ is any continuous function, then g is represented by an element of W(X) if and only if ${\Sigma _{\sigma \, \in \,V}}g(\sigma )\, \equiv \,0\,\bmod \,\left\vert V \right\vert$ holds for all finite fans $V \subseteq X$. This generalizes a recent field theoretic result of Becker and Bröcker. Following the proof of this, applications are given to the computation of the stability index of X, and to the representation of continuous functions $ g:\,X\, \to \, \pm 1$ by elements of G.
A nonlinear Volterra equation with rapidly decaying solutions
Olof J.
Staffans
523-530
Abstract: We study the asymptotic behavior of the solutions of a nonlinear integrodifferential Volterra equation with a convolution kernel. More specifically, we give conditions which imply that a solution x satisfies $ x(t)\, = \,O({t^{ - p}})\,(t \to \infty )$, where p is an arbitrary, positive real number.
The fixed point property and unbounded sets in Hilbert space
William O.
Ray
531-537
Abstract: It is shown that a closed convex subset K of a real Hilbert space H has the fixed point property for nonexpansive mappings if and only if K is bounded.
Erratum to: ``The behavior of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimension''
Barry F.
Knerr
539-539
Asymptotic completeness for classes of two, three, and four particle Schr\"odinger operators
George A.
Hagedorn
1-75
Abstract: Formulas for the resolvent $ {(z\, - \,H)^{ - 1}}$ are derived, where $ H\, = \,{H_0}\, + \,{\Sigma _{i < j}}{\lambda _{ij}}{V_{ij}}$ is an N particle Schrödinger operator with the center of mass motion removed. For a large class of two-body potentials and generic couplings $\{ {\lambda _{ij}}\}$, these formulas are used to prove asymptotic completeness in the $N\, \leqslant \,4$ body problem when the space dimension is $ m\, \geqslant \,3$. The allowed potentials belong to a space of dilation analytic multiplication operators which fall off more rapidly than $ {r^{ - 2 - \varepsilon }}$ at $\infty$. In particular, Yukawa potentials, generalized Yukawa potentials, and potentials of the form $ {(1\, + \,r)^{ - 2 - \varepsilon }}$ are allowed.
On the excursion process of Brownian motion
Frank B.
Knight
77-86
Abstract: Let $W_0^ + \,(t)$ denote the scaled excursion process of Brownian motion, and let $l_0^ + \,(a),\,0\, \leqslant \,a,$ be its local time at a. The joint distribution of $l_0^ + \,(a),\,\beta (a),$ and $\gamma (a)$ is obtained, where $\beta (a)$ and $ \gamma (a)$ are the last exit time and the first passage time of a by $W_0^{ + }\,(t)$.
Approximate torus fibrations of high dimensional manifolds can be approximated by torus bundle projections
R. E.
Goad
87-97
Abstract: In this paper, we prove that approximate torus fibrations of high dimensional manifolds can be approximated by torus bundle projections. The principal tools are the torus trick developed by Kirby and Siebenmann, a surgery theorem concerning homotopy structures on torii due to Hsiang and Wall, a theorem on the space of homeomorphisms of the torus due to Hamstrom and a generalization of hereditary homotopy equivalence developed by the author.
The interior operator logic and product topologies
Joseph
Sgro
99-112
Abstract: In this paper we present a model theory of the interior operator on product topologies with continuous functions. The main results are a completeness theorem, an axiomatization of topological groups, and a proof of an interpolation and definability theorem.
Topological equivalence of gradient vectorfields
Douglas S.
Shafer
113-126
Abstract: This paper is a study of the behavior of the topological equivalence class of the planar gradient vectorfield $X\, = \,{\operatorname{grad} _g}\,V$, in a neighborhood of a degenerate singularity of V, as g varies over all Riemannian metrics. It is shown that under simple restrictions on V the topological equivalence class of X is determined by its first nonvanishing jet, and that only finitely many equivalence classes occur (for fixed V). In this case, when the degree of the first nonvanishing jet of V is less than five, necessary and sufficient conditions for change in equivalence class are given, both in terms of the coefficients of the homogeneous part of V and geometrically in terms of its level curves. A catalogue of possible phase portraits, up to topological equivalence, is included. Necessary conditions are given for change in higher degree.
Fibrewise localization and completion
J. P.
May
127-146
Abstract: The behavior of fibrewise localization and completion on the classifying space level is analyzed. The relationship of these constructions to fibrewise joins and smash products and to orientations of spherical fibrations is also analyzed. This theory is essential to validate Sullivan's proof of the Adams conjecture.
Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms
Aroldo
Kaplan
147-153
Abstract: We introduce a class of nilpotent Lie groups which arise naturally from the notion of composition of quadratic forms, and show that their standard sublaplacians admit fundamental solutions analogous to that known for the Heisenberg group.
Fractional differentiation and Lipschitz spaces on local fields
C. W.
Onneweer
155-165
Abstract: In this paper we continue our study of differentiation on a local field K. We define strong derivatives of fractional order $\alpha \, > \,0$ for functions in $ {L_r}(\textbf{K})$, $ 1\, \leqslant \,r\, < \,\infty$. After establishing a number of basic properties for such derivatives we prove that the spaces of Bessel potentials on K are equal to the spaces of strongly $ {L_r}(\textbf{K})$-differentiable functions of order $\alpha \, > \,0$ when $1\, \leqslant \,r\, \leqslant \,2$. We then focus our attention on the relationship between these spaces and the generalized Lipschitz spaces over K. Among others, we prove an inclusion theorem similar to a wellknown result of Taibleson for such spaces over $ {\textbf{R}^n}$.
The number of groups of a given genus
T. W.
Tucker
167-179
Abstract: It is shown that the number of groups with a given genus greater than one is finite. The proof depends heavily on V. K. Proulx's classification of groups of genus one. The key observation is that as the number of vertices of a graph imbedded on a given surface increases, the average face size of the imbedding approaches the average face size of a toroidal imbedding. The result appears to be related to Hurwitz's theorem bounding the order of a group of conformal automorphisms on a Riemann surface of genus g.
On special classes of entire functions whose zeros and growth are restricted
Carl L.
Prather
181-189
Abstract: The present paper is an investigation of the uniform limits on bounded sets of entire functions of genus $\leqslant \,2p$ whose zeros are real, or lie in an even number of sectors (or correspondingly on rays) of a certain size, both the number and size depending on p. A characterization of the uniform limits of entire functions of genus $\leqslant \,2p$ whose zeros lie in these sectors is given.
Convolution equations in spaces of infinite-dimensional entire functions of exponential and related types
J.-F.
Colombeau;
B.
Perrot
191-198
Abstract: We prove results of existence and approximation of the solutions of the convolution equations in spaces of entire functions of exponential type on infinite dimensional spaces. In particular we obtain: let E be a complex, quasi-complete and dual nuclear locally convex space and $ \Omega$ a convex balanced open subset of E; let $\mathcal{H} (\Omega )$ be the space of the holomorphic functions on $\Omega$, equipped with the compact open topology and $\mathcal{F}$; equip this space $\mathcal{F}$. Then, ``every nonzero convolution operator on
Basic sequences in non-Schwartz Fr\'echet spaces
Steven F.
Bellenot
199-216
Abstract: Obliquely normalized basic sequences are defined and used to characterize non-Schwartz-Fréchet spaces. It follows that each non-Schwartz-Fréchet space E has a non-Schwartz subspace with a basis and a quotient which is not Montel (which has a normalized basis if E is separable). Stronger results are given when more is known about E, for example, if E is a subspace of a Fréchet $ {l_p}$-Köthe sequence space, then E has the Banach space $ {l_p}$ as a quotient and E has a subspace isomorphic to a non-Schwartz $ {l_p}$-Köthe sequence space. Examples of Fréchet-Montel spaces which are not subspaces of any Fréchet space with an unconditional basis are given. The question of the existence of conditional basic sequences in non-Schwartz-Fréchet spaces is reduced to questions about Banach spaces with symmetric bases. Nonstandard analysis is used in some of the proofs and a new nonstandard characterization of Schwartz spaces is given.
Minimal excessive measures and functions
E. B.
Dynkin
217-244
Abstract: Let H be a class of measures or functions. An element h of H is minimal if the relation $h\, = \,{h_1}\, + \,{h_2}$, ${h_1}$, $ {h_2} \in H$ implies that $ {h_1}$, ${h_2}$ are proportional to h. We give a limit procedure for computing minimal excessive measures for an arbitrary Markov semigroup $ {T_t}$ in a standard Borel space E. Analogous results for excessive functions are obtained assuming that an excessive measure $ \gamma$ on E exists such that $ {T_t}f\, = 0$ if $ f\, = \,0$ $\gamma$-a.e. In the Appendix, we prove that each excessive element can be decomposed into minimal elements and that such a decomposition is unique.
Derivation alternator rings with idempotent
Irvin R.
Hentzel;
Harry F.
Smith
245-256
Abstract: A nonassociative ring is called a derivation alternator ring if it satisfies the identities $ (yz,\,x,\,x)\, = \,y(z,\,x,\,x)\, + \,(y,\,x,\,x)z,\,(x,\,x,\,yz)\, = \,y(x,\,x,\,z)\, + \,(x,\,x,\,y)z$ and $(x,\,x,\,x)\, = 0$. Let R be a prime derivation alternator ring with idempotent $e \ne 1$ and characteristic $ \ne 2$. If R is without nonzero nil ideals of index 2, then R is alternative.
Two approaches to supermanifolds
Marjorie
Batchelor
257-270
Abstract: The problem of supplying an analogue of a manifold whose sheaf of functions contains anticommuting elements has been approached in two ways. Either one extends the sheaf of functions formally, as in the category of graded manifolds [3], [8], or one mimicks the usual definition of a manifold, having replaced Euclidean space with a suitable product of the odd and even parts of an exterior algebra as in the category of supermanifolds [6]. This paper establishes the equivalence of the category of supermanifolds with the category of graded manifolds.
On maximally elliptic singularities
Stephen Shing Toung
Yau
269-329
Abstract: Let p be the unique singularity of a normal two-dimensional Stein space V. Let m be the maximal ideal in $ _V{\mathcal{O}_p}$, the local ring of germs of holomorphic functions at p. We first define the maximal ideal cycle which serves to identify the maximal ideal. We give an ``upper'' estimate for maximal ideal cycle in terms of the canonical divisor which is computable via the topological information, i.e., the weighted dual graph of the singularity. Let $M \to V$ be a resolution of V. It is known that $h\, = \,\dim \,{H^1}(M,\,\mathcal{O})$ is independent of resolution. Rational singularities in the sense of M. Artin are equivalent to $ h\, = \,0$. Minimally elliptic singularity in the sense of Laufer is equivalent to saying that $h\, = \,1$ and $ _V{\mathcal{O}_p}$ is Gorenstein. In this paper we develop a theory for a general class of weakly elliptic singularities which satisfy a maximality condition. Maximally elliptic singularities may have h arbitrarily large. Also minimally elliptic singlarities are maximally elliptic singularities. We prove that maximally elliptic singularities are Gorenstein singularities. We are able to identify the maximal ideal. Therefore, the important invariants of the singularities (such as multiplicity) are extracted from the topological information. For weakly elliptic singularities we introduce a new concept called ``elliptic sequence". This elliptic sequence is defined purely topologically, i.e., it can be computed explicitly via the intersection matrix. We prove that --K, where K is the canonical divisor, is equal to the summation of the elliptic sequence. Moreover, the analytic data $\dim \,{H^1}(M,\,\mathcal{O})$ is bounded by the topological data, the length of elliptic sequence. We also prove that $ h\, = \,2$ and $_V{\mathcal{O}_p}$ Gorenstein implies that the singularity is weakly elliptic.
The dependence of the generalized Radon transform on defining measures
Eric Todd
Quinto
331-346
Abstract: Guillemin proved that the generalized Radon transform R and its dual ${R^t}$ are Fourier integral operators and that $ {R^t}R$ is an elliptic pseudodifferential operator. In this paper we investigate the dependence of the Radon transform on the defining measures. In the general case we calculate the symbol of $ {R^t}R$ as a pseudodifferential operator in terms of the measures and give a necessary condition on the defining measures for $ {R^t}R$ to be invertible by a differential operator. Then we examine the Radon transform on points and hyperplanes in ${\textbf{R}^n}$ with general measures and we calculate the symbol of ${R^t}R$ in terms of the defining measures. Finally, if ${R^t}R$ is a translation invariant operator on ${\textbf{R}^n}$ then we prove that ${R^t}R$ is invertible and that our condition is equivalent to $ {({R^t}R)^{ - 1}}$ being a differential operator.
The lattice of $l$-group varieties
J. E.
Smith
347-357
Abstract: For any type of abstract algebra, a variety is an equationally defined class of such algebras. Recently, attempts have been made to study varieties of lattice-ordered groups (l-groups). Martinez has shown that the set L of all l-group varieties forms a lattice under set inclusion with a compatible associative multiplication. Certain varieties ${\mathcal{S}_p}$ (p prime) have been proved by Scrimger to be minimal nonabelian varieties in L. In the present paper, it is shown that these varieties can be used to produce varieties minimal with respect to properly containing various other varieties in L. Also discussed are the relations among the ${\mathcal{S}_n}\,(n\, \in \,N)$, and it is established that all infinite collections of the ${\mathcal{S}_n}$ have the same least upper bound in L. Martinez has also classified l-groups using torsion classes, a generalization of the idea of varieties. It is proved here that L is not a sublattice of T, the lattice of torsion classes.
$p$-adic gamma functions and Dwork cohomology
Maurizio
Boyarsky
359-369
Abstract: The relations of Gross and Koblitz between gauss sums and the p-adic gamma function is reexamined from the point of view of Dwork's formulation of p-adic cohomology. Some higher dimensional generalizations are proposed.
Decomposition of nonnegative group-monotone matrices
S. K.
Jain;
Edward K.
Kwak;
V. K.
Goel
371-385
Abstract: A decomposition of nonnegative matrices with nonnegative group inverses has been obtained. This decomposition provides a new approach to the solution of problems relating to nonnegative matrices with nonnegative group inverses. As a consequence, a number of results are derived. Our results, among other things, answer a question of Berman, extend the theorems of Berman and Plemmons, DeMarr and Flor.
A spectral theorem for $J$-nonnegative operators
Bernard N.
Harvey
387-396
Abstract: A J-space is a Hilbert space with the usual inner product denoted $ [x,y]$ and an indefinite inner product defined by $(x,y)\, =\, [Jx,y]$ where J is a bounded selfadjoint operator whose square is the identity. We define a J-adjoint ${T^ + }$ of an operator T with respect to the indefinite inner product in the same way as the regular adjoint $T^{\ast}$ is defined with respect to $[x,y]$. We say T is J-selfadjoint if $T = {T^ + }$. An operator-valued function is called a J-spectral function with critical point zero if it is defined for all $t \ne 0$, is bounded, J-selfadjoint and has the properties of a resolution of the identity on its domain. It has been proved by M. G. Krein and Ju. P. Smul'jan that bounded Jselfadjoint operators A with $(Ax,x) \geqslant 0$ for all x can be represented as a strongly convergent improper integral of t with respect to a J-spectral function with critical point zero plus a nilpotent of index 2. Further, the product of the nilpotent with the J-spectral function on intervals not containing zero is zero. The present paper extends this theory to the unbounded case. We show that unbounded J-selfadjoint operators with $(Ax,x) \geqslant 0$ are a direct sum of an operator of the above mentioned type and the inverse of a bounded operator of the same type whose nilpotent part is zero.
A construction of uncountably many weak von Neumann transformations
Karl
David
397-410
Abstract: We define weak von Neumann transformations and discuss some of their properties, using several examples of countable classes of these transformations. Then we construct an uncountable class by the cutting-and-stacking method. We show that each member of this class is ergodic and has zero entropy.
The free boundary for elastic-plastic torsion problems
Avner
Friedman;
Gianni A.
Pozzi
411-425
Abstract: Consider the variational inequality: Find $ u\, \in\, K$ such that $\int_Q {\nabla u \cdot\, \nabla\, (\upsilon\, -\, u)\, \geqslant\, \mu\, \int_Q\, {(\upsilon\, -\, u)\,(\mu\, >\, 0)} }$ for any $\upsilon\, \in\, K$, where $K\, =\, \{ w\, \in\, H_0^1(Q);\,\left\vert {\nabla\, w} \right\vert\, \leqslant\, 1\}$ and Q is a 2-dimensional simply connected domain in $ {R^2}$ with piecewise $ {C^3}$ boundary. The solution u represents the stress function in a torsion problem of an elastic-plastic bar with cross section Q. The sets $E\, =\, \{ x\, \in\, Q;\,\left\vert {\nabla\, u(x)} \right\vert\, <\, 1\} $, $P\, =\, \{ x\, \in\, Q;\,\left\vert {\nabla\, u(x)} \right\vert\, =\, 1\}$ are the elastic and plastic sets respectively. The purpose of this paper is to study the free boundary $\partial E\, \cap\, Q$; more specifically, an estimate is derived on the number of points of local maximum of the free boundary.
A lower central series for split Hopf algebras with involution
Bruce W.
Jordan
427-454
Abstract: A lower central series is defined for split Hopf algebras with involution over a field k. Various structure theorems for coalgebras and Hopf algebras are established.
A noncommutative generalization and $q$-analog of the Lagrange inversion formula
Ira
Gessel
455-482
Abstract: The Lagrange inversion formula is generalized to formal power series in noncommutative variables. A q-analog is obtained by applying a linear operator to the noncommutative formula before substituting commuting variables.
On the oscillatory behavior of singular Sturm-Liouville expansions
J. K.
Shaw
483-505
Abstract: A singular Sturm-Liouville operator $Ly\, =\, - (Py')'\, +\, Qy$, defined on an interval $[0,b^{\ast})$ of regular points, but singular at $b^{\ast}$, is considered. Examples are the Airy equation on $[0,\infty )$ and the Legendre equation on $ [0,1)$. A mode of oscillation of the successive iterates $f(t)$, $(Lf)(t)$, $ ({L^2}f)(t),\, \ldots$ of a smooth function f is assumed, and the resulting influence on f is studied. The nature of the mode is that for a fixed integer $N\, \geqslant\, 0$, each iterate $({L^k}f)(t)$ shall have on $(0,b^{\ast})$ exactly N sign changes which are stable, in a certain sense, as k varies. There is quoted from the literature the main characterization of such functions f which additionally satisfy strong homogeneous endpoint conditions at 0 and $ b^{\ast}$. An extended characterization is obtained by weakening the conditions of f at 0 and $b^{\ast}$. The homogeneous endpoint conditions are replaced by a summability condition on the values, or limits of values, of f at 0 and $ b^{\ast}$.
Continuously translating vector-valued measures
U. B.
Tewari;
M.
Dutta
507-519
Abstract: Let G be a locally compact group and A an arbitrary Banach space. $ {L^p}(G,A)$ will denote the space of p-integrable A-valued functions on G. $M(G,A)$ will denote the space of regular A-valued Borel measures of bounded variation on G. In this paper, we characterise the relatively compact subsets of $ {L^p}(G,A)$. Using this result, we prove that if $\mu\, \in\, M(G,A)$, such that either $x\, \to\, {\mu _x}$ or $x{ \to _x}\mu$ is continuous, then $\mu\, \in\, {L^1}(G,A)$.
On the cohomology of real Grassmanians
Howard L.
Hiller
521-533
Abstract: Let ${G_k}({\textbf{R}^{n + k}})$ denote the grassman manifold of k-planes in real $(n\, +\, k)$-space and $ {w_1}\, \in\, {H^1}({G_k}({\textbf{R}^{n + k}});\,{\textbf{Z}_2})$ the first Stiefel-Whitney class of the universal bundle. Using Schubert calculus techniques and the cohomology of flag manifolds we estimate the height of $ {w_1}$ in the cohomology ring. We then apply this to improve earlier lower bounds on the Lusternik-Schnirelmann category of real grassmanians.
Correction to: ``On the free boundary of a quasivariational inequality arising in a problem of quality control'' [Trans. Amer. Math. Soc. {\bf 246} (1978), 95--110; MR 80f:93086c]
Avner
Friedman
535-537
A decidable fragment of the elementary theory of the lattice of recursively enumerable sets
M.
Lerman;
R. I.
Soare
1-37
Abstract: A natural class of sentences about the lattice of recursively enumerable sets modulo finite sets is shown to be decidable. This class properly contains the class of sentences previously shown to be decidable by Lachlan. New structure results about the lattice of recursively enumerable sets are proved which play an important role in the decision procedure.
Milnor's $\bar \mu $-invariants and Massey products
Richard
Porter
39-71
Abstract: The main result of this paper gives an interpretation of Milnor's $\overline \mu $-invariants of a link in terms of Massey products in the complement of the link. The approach presented here can be used to give topological proofs of results about the $\overline \mu $-invariants obtained by Milnor using different methods.
Crossed extensions
John G.
Ratcliffe
73-89
Abstract: We develop a natural five term exact sequence relating the second and third cohomology of groups. We show that this sequence is the proper framework for the problem of realizing an abstract kernel. As an application, we give an interpretation of the third cohomology of a group in terms of crossed sequences.
Twisted free tensor products
Elyahu
Katz
91-103
Abstract: A twisted free tensor product of a differential algebra and a free differential algebra is introduced. This complex is proved to be chain homotopy equivalent to the complex associated with a twisted free product of a simplicial group and a free simplicial group. In this way we turn a geometric situation into an algebraic one, i.e. for the cofibration $Y \to Y\,{ \cup _g}\,CX \to \Sigma X$ we obtain a spectral sequence converging into $ H(\Omega (Y\,{ \cup _g}\,CX))$. The spectral sequence obtained in the above situation is similar to the one obtained by L. Smith for a cofibration. However, the one we obtain has more information in the sense that differentials can be traced, requires more lax connectivity conditions and does not need the ring of coefficients to be a field.
Rotundity in Lebesgue-Bochner function spaces
Mark A.
Smith;
Barry
Turett
105-118
Abstract: This paper concerns the isometric theory of the Lebesgue-Bochner function space ${L^p}(\mu ,\,X)$ where $1 < p < \infty$. Specifically, the question of whether a geometrical property lifts from X to ${L^p}\,(\mu ,\,X)$ is examined. Positive results are obtained for the properties local uniform rotundity, weak uniform rotundity, uniform rotundity in each direction, midpoint local uniform rotundity, and B-convexity. However, it is shown that the Radon-Riesz property does not lift from X to ${L^p}\,(\mu ,\,X)$. Consequently, Lebesgue-Bochner function spaces with the Radon-Riesz property are examined more closely.
$K$-theory of hyperplanes
Barry H.
Dayton;
Charles A.
Weibel
119-141
Abstract: Let R be the coordinate ring of a union of N hyperplanes in general position in $ {\textbf{A}}_k^{n + 1}$. Then $\displaystyle {K_i}(R)\, = \,{K_i}(k)\, \oplus \,\left( {\begin{array}{*{20}{c}} {N\, - \,1} {n\, + \,1} \end{array} } \right)\,{K_{n + i}}(k).$ This formula holds for ${K_0},\,{K_1},\,{K_i}\,(i < 0)$, and for the Karoubi-Villamayor groups $K{V_i}\,(i \in \,{\mathbf{Z}})$. For $ {K_2}$ there is an extra summand $\bar R/R$, where $\bar R$ is the normalization of R. For ${K_3}$ the above is a quotient of ${K_3}(R)$. In §4 we show that $ {K_1}$-regularity implies $ {K_0}$-regularity, answering a question of Bass. We also show that $ {K_i}$-regularity is equivalent to Laurent ${K_i}$-regularity for $i \leqslant 1$. The results of this section are independent of the rest of the paper.
$\sigma $-id\'eaux engendr\'es par des ensembles ferm\'es et th\'eor\`emes d'approximation
Alain
Louveau
143-169
Abstract: This paper is motivated by the study of *-games on $\omega$, and by a question of D. A. Martin on the strength of the hypothesis ${\text{AD}}_\omega ^{\ast}$ that every *-game on $ {\omega ^\omega }$ is determined. A general study of the $\sigma $-ideals of subsets of ${\omega ^\omega }$ generated by closed sets encompasses the *-games and the ``perfect set property". Using associated games, we extend for these ideals many properties known for countable sets, under various hypotheses of determinacy. Our methods thus apply also to other examples of regularity properties, such as those introduced by A. S. Kechris. Finally, a general theorem of approximation by analytic sets in Solovay's model is proved which, together with the preceding results, gives the solution of Martin's problem: ${\text{AD}}_\omega ^{\ast}$ is true in Solovay's model.
Affine extensions of a Bernoulli shift
J.
Feldman;
D. J.
Rudolph;
C. C.
Moore
171-191
Abstract: (a) For any automorphism $\phi$ of a compact metric group G, and any $a > 0$, we show the existence of a free finite measure-preserving (m.p.) action of the twisted product $Z{ \times ^\phi }\,G$ whose restriction to Z is Bernoulli with entropy $a\, + \,h(\phi )$, $h(\phi )$ being the entropy of $\phi$ on G with Haar measure. (b) A classification is given of all free finite m.p. actions of $ Z\, \times {\,^\phi }\,G$ such that the action of Z on the $ \sigma$-algebra of invariant sets of G is a Bernoulli action. (c) The classification of (b) is extended to ``quasifree'' actions: those for which the isotropy subgroups are in a single conjugacy class within G. An existence result like that of (a) holds in this case, provided certain necessary and sufficient algebraic conditions are satisfied; similarly, an isomorphism theorem for such actions holds, under certain necessary and sufficient conditions. (d) If G is a Lie group, then all actions of $Z\, \times {\,^\phi }\,G$ are quasifree; if G is also connected, then the second set of additional algebraic conditions alluded to in (c) is always satisfied, while the first will be satisfied only in an obvious case. (e) Examples are given where the isomorphism theorem fails: by violation of the algebraic conditions in the quasifree case, for other reasons in the non-quasifree case.
Homeomorphisms of $S\sp{3}$ leaving a Heegaard surface invariant
Jerome
Powell
193-216
Abstract: We find a finite set of generators for the group ${\mathcal{H} _g}$ of isotopy classes of orientation-preserving homeomorphisms of the 3-sphere $ {S^3}$ which leave a Heegaard surface T of genus g in $ {S^3}$ invariant. We also show that every element of the group ${\mathcal{H} _g}$ can be represented by a deformation of the surface T in ${S^3}$ of a very special type: during the deformation the surface T is the boundary of the regular neighborhood of a graph embedded in a fixed 2-sphere. The only exception occurs when a subset of the graph contained in a disc on the 2-sphere is ``flipped over."
Equivariant $G$-structure on versal deformations
Dock S.
Rim
217-226
Abstract: Let ${X_0}$ be an algebraic variety, and $(\chi ,\,\Sigma )$ its versal deformation. Now let G be an affine algebraic group acting algebraically on ${X_0}$. It gives rise to a definite linear G-action on the tangent space of $\Sigma$. In this paper we establish that if G is linearly reductive then there is an equivariant G-action on $ (\chi ,\Sigma )$ which induces given G-action on the special fibre $ {X_0}$ and its linear G-action on the tangent space of the formal moduli $ \Sigma$. Furthermore, such equivariant G-structure is shown to be unique up to noncanonical isomorphism.
Diophantine sets over algebraic integer rings. II
J.
Denef
227-236
Abstract: We prove that Z is diophantine over the ring of algebraic integers in any totally real number field or quadratic extension of a totally real number field.
Constructing Smale diffeomorphisms on compact surfaces
Steve
Batterson
237-245
Abstract: A necessary condition for an isotopy class on a compact surface to admit a Smale diffeomorphism whose dynamics are a specified set of subshifts of finite type is that the Euler characteristic of the manifold be equal to a sum and difference of certain numbers obtained from the matrices representing the subshifts. In this paper it is shown that this condition is sufficient up to a finite power of the subshifts.
Destructible and indestructible Blaschke products
H. Stephen
Morse
247-253
Abstract: A special case of destructibility for Blaschke products is introduced and studied. An example is given of a destructible Blaschke product which becomes indestructible when a single point is deleted from its zero-set.
A theorem on free envelopes
Chester C.
John
255-259
Abstract: The free envelope of a finite commutative semigroup was defined by Grillet [Trans. Amer. Math. Soc. 149 (1970), 665-682] to be a finitely generated free commutative semigroup $F(S)$ with identity and a homomorphism $\alpha :\,S\, \to \,F(S)$ endowed with certain properties. Grillet raised the following question: does $\alpha (S)$ always generate a pure subgroup of the free Abelian group with the same basis as $ F(S)$? We prove this is indeed the case. It follows as a result of two lemmas. Lemma 1: Given a full rank proper subgroup H of a finitely generated free Abelian group F and a basis X of F there exists a surjective homomorphism $f:\,F\, \to \,{\textbf{Z}}$ such that f is positive on X and ${f_{\left\vert H \right.}}$ is not surjective. Lemma 2: A finitely generated totally cancellative reduced subsemigroup of a finitely generated free Abelian group F is contained in the positive cone of some basis of F. The following duality theorem is also proved. Let ${S^{\ast}}\, \cong \,\operatorname{Hom} (S,\,N)$ where N is the nonnegative integers under addition. Then $S\, \cong \,{S^{{\ast}{\ast}}}$ if and only if S is isomorphic to a unitary subsemigroup of a finitely generated free commutative semigroup with identity.
On positive contractions in $L\sp{p}$-spaces
H. H.
Schaefer
261-268
Abstract: Let T denote a positive contraction $ (T\, \geqslant \,0,\,\left\Vert T \right\Vert\, \leqslant \,1)$ on a space ${L^p}(\mu )\,(1\, < \,p\, < \, + \,\infty )$. A primitive nth root of unity $\varepsilon$ is in the point spectrum $P\sigma (T)$ iff it is in $\varepsilon$ is in both $ P\sigma (T)$ and ${L^q}({p^{ - \,1}}\, + \,{q^{ - \,1}}\, = \,1)$ which are in canonical duality and on which T (resp., $T'$) acts as an isometry. If, in addition, T is quasi-compact then the spectral projection associated with the unimodular spectrum of T (resp., $T'$) is a positive contraction onto a Riesz subspace of ${L^p}$ (resp., ${L^q}$) on which T (resp., $ T'$) acts as an isometry.
Oscillation and asymptotic behavior of systems of ordinary linear differential equations
Carl H.
Rasmussen
1-47
Abstract: Conditions are established for oscillatory and asymptotic behavior for first-order matrix systems of ordinary differential equations, including Hamiltonian systems in the selfadjoint case. Asymptotic results of Hille, Shreve, and Hartman are generalized. Disconjugacy criteria of Ahlbrandt, Tomastik, and Reid are extended.
Global analysis on PL-manifolds
Nicolae
Teleman
49-88
Abstract: The paper deals mainly with combinatorial structures; in some cases we need refinements of combinatorial structures. Riemannian metrics are defined on any combinatorial manifold M. The existence of distance functions and of Riemannian metrics with ``constant volume density'' implies smoothing. A geometric realization of $ {\text{PL}}\left( m \right){\text{/O}}\left( m \right)$ is given in terms of Riemannian metrics. A graded differential complex $ {\Omega ^ {\ast} }( M )$ is constructed: it appears as a subcomplex of Sullivan's complex of piecewise differentiable forms. In the complex $ {\Omega ^{\ast}}( M )$ the operators $d$, $\ast$, $\delta$, $\Delta$ are defined. A Rellich chain of Sobolev spaces is presented. We obtain a Hodge-type decomposition theorem, and the Hodge homomorphism is defined and studied. We study also the combinatorial analogue of the signature operator.
${\bf Z}\sb{(2)}$-knot cobordism in codimension two, and involutions on homotopy spheres
Chao Chu
Liang
89-97
Abstract: Let ${Z_{(2)}}$ denote the ring of rational 2-adic integers. In this paper, we consider the group $ {\Psi _k}$ of $ {Z_{(2)}}$-cobordism classes of ${Z_{(2)}} - \operatorname{knot} (\Sigma ^{k + 2}, \,{K^k})$, where $\Sigma$ is a 1-connected ${Z_{(2)}}$-sphere ${Z_{(2)}}$-cobordant to $ {S^{k + 2}}$, and K is a 1-connected ${Z_{(2)}}$-sphere embedded in $\Sigma$ with trivial normal bundle. For $n \geqslant 3$, we will prove that ${\Psi _{2n}} = 0$ and $ {\Psi _{2n - 1}} = {C_\varepsilon }({Z_{(2)}})$, $\varepsilon = {( - 1)^n}$. Also, we will show that the group $\Theta _{4m - 1}^{4m + 1}$ of L-equivalence classes of differentiable involutions on $ (4m + 1)$-homotopy spheres with codimension two fixed point sets defined by Bredon contains infinitely many copies of Z.
Continuous functions on countable compact ordered sets as sums of their increments
Gadi
Moran
99-112
Abstract: Every continuous function from a countable compact linearly ordered set A into a Banach space V (vanishing at the least element of A ) admits a representation as a sum of a series of its increments (in the topology of uniform convergence). This series converges to no other sum under rearrangements of its terms. A uniqueness result to the problem of representation of a regulated real function on the unit interval as a sum of a continuous and a steplike function is derived.
On the Littlewood-Paley theory for mixed norm spaces
John A.
Gosselin
113-124
Abstract: An inequality of Littlewood-Paley type is proved for the mixed norm spaces $ {L_P}({l_r})$, $1 < p$, $r < \infty$, on the interval $[0,1]$. This result makes use of recent work by C. Fefferman and A. Cordoba on the boundedness of singular integrals on these spaces. As an application of this inequality, boundedness of the lacunary maximal partial sum operator for Walsh-Fourier series on ${L_p}({l_r})$ is established. This result can be viewed as an extension of a similar result for the Hardy-Littlewood maximal function due to C. Fefferman and E. M. Stein.
Generic differentiability of Lipschitzian functions
G.
Lebourg
125-144
Abstract: It is shown that, in separable topological vector spaces which are Baire spaces, the usual properties that have been introduced to study the local ``first order'' behaviour of real-valued functions which satisfy a Lipschitz type condition are ``generically'' equivalent and thus lead to a unique class of ``generically smooth'' functions. These functions are characterized in terms of tangent cones and directional derivatives and their ``generic'' differentiability properties are studied. The results extend some of the well-known differentiability properties of continuous convex functions.
Subnormal operators quasisimilar to an isometry
William W.
Hastings
145-161
Abstract: Let $V = {V_0} \oplus {V_1}$ be an isometry, where $ {V_0}$ is unitary and $ {V_1}$ is a unilateral shift of finite multiplicity n. Let $S = {S_0} \oplus {S_1}$ be a subnormal operator where ${S_0} \oplus {S_1}$ is the normal decomposition of S into a normal operator $ {S_0}$ and a completely nonnormal operator ${S_1}$. It is shown that S is quasisimilar to V if and only if ${S_0}$ is unitarily equivalent to ${V_0}$ and ${S_1}$ is quasisimilar to ${V_1}$. To prove this, a standard representation is developed for n-cyclic subnormal operators. Using this representation, the class of subnormal operators which are quasisimilar to ${V_1}$ is completely characterized.
Formal and convergent power series solutions of singular partial differential equations
Stanley
Kaplan
163-183
Abstract: A class of singular first-order partial differential equations is described for which an analogue of a theorem of M. Artin on the solutions of analytic equations holds: given any formal power series solution and any nonnegative integer v, a convergent power series solution may be found which agrees with the given formal solution up to all terms of order $\leqslant v$.
On the embedding problem for $1$-convex spaces
Vo Van
Tan
185-197
Abstract: In this paper we provide a necessary and sufficient condition for 1-convex spaces (i.e., strongly pseudoconvex spaces) which can be realized as closed analytic subvarieties in some ${C^N} \times {P_M}$. A construction of some normal 3-dimensional 1-convex space which cannot be embedded in any $ {C^N} \times {P_M}$ is given. Furthermore, we construct explicitly a non-kählerian 3-dimensional 1-convex manifold which answers a question posed by Grauert.
On natural radii of $p$-adic convergence
B.
Dwork;
P.
Robba
199-213
Abstract: We study the radius of p-adic convergence of power series which represent algebraic functions. We apply the p-adic theory of ordinary linear differential equations to show that the radius of convergence is the natural one, provided the degree of the function is less than p. The study of similar questions for solutions of linear differential equations is indicated.
The theorems of Beth and Craig in abstract model theory. I. The abstract setting
J. A.
Makowsky;
S.
Shelah
215-239
Abstract: In the context of abstract model theory various definability properties, their interrelations and their relation to compactness are investigated.
Global ideal theory of meromorphic function fields
Norman L.
Alling
241-266
Abstract: It is shown that the ideal theories of the fields of all meromorphic functions on any two noncompact Riemann surfaces are isomorphic. Further, various new representation and factorization theorems are proved.
On $2$-blocks with semidihedral defect groups
Karin
Erdmann
267-287
Abstract: Let G be one of the following groups: ${L_3}(q)$ and $GL(2,\,q)$ with $q\, \equiv \,3\,(\bmod \,4)$, ${U_3}(q)$ and $GU(2,\,q)$ with $q\, \equiv \,1\,(\bmod \,4)$. This paper is concerned with 2-blocks B of G having semidihedral defect groups. In particular, vertices, sources and Green correspondents of the simple modules in B are determined and used to obtain the submodule structure of the indecomposable projective modules.
Reparametrization of $n$-flows of zero entropy
J.
Feldman;
D.
Nadler
289-304
Abstract: Let $\phi$, $\psi$ be two ergodic n-parameter flows which preserve finite probability measures on their spaces X, Y. Let T be a nullset-preserving map: $X\, \to \,Y$ sending each $\phi$-orbit homeomorphically to a $ \phi$-orbit. Then $ \phi$, $\psi$ are called homeomorphically orbit-equivalent. For $n\, = \,1$, there has been developed a theory of such equivalence: ``Loosely Bernoulli'' theory. A completely parallel theory exists for higher dimensions, except that it is necessary to impose a certain natural ``growth'' restriction on T, a restriction which is vacuous in the case $n\, = \,1$. In this paper we carry out this program, but only for the case of zero entropy.
An equivariant Wall obstruction theory
Jenny A.
Baglivo
305-324
Abstract: Let G be a finite group. For a certain class of CW-complexes with a G-action which are equivariantly dominated by a finite complex we define algebraic invariants to decide when the space is equivariantly homotopy or homology equivalent to a finite complex.
Hypoconvexity and essentially $n$-normal operators
Norberto
Salinas
325-351
Abstract: In this paper a classifying structure for the class of essentially n-normal operators on a separable Hilbert space is introduced, and various invariance properties of this classifying structure are studied. The notion of a hypoconvex subset of the algebra ${\mathcal{M}_n}$ of all complex $n\, \times \,n$ matrices is defined, and it is shown that the set of all equivalence classes of essentially n-normal operators (under a natural equivalence relation), whose reducing essential $n\, \times \,n$ matricial spectrum is a given hypoconvex set, forms an abelian group. It is also shown that this correspondence between hypoconvex subsets of $ {\mathcal{M}_n}$ and abelian groups is a homotopy invariant, covariant functor. This result is then used to prove that Toeplitz operators (on strongly pseudoconvex domains) which have homotopic continuous matricial symbols, are unitarily equivalent up to compact perturbation.
Some Ramsey-type numbers and the independence ratio
William
Staton
353-370
Abstract: If each of k, m, and n is a positive integer, there is a smallest positive integer $r\, = \,{r_k}\,(m,\,n)$ with the property that each graph G with at least r vertices, and with maximum degree not exceeding k, has either a complete subgraph with m vertices, or an independent subgraph with n vertices. In this paper we determine ${r_3}(3,\,n)\, = \,r(n)$, for all n. As a corollary we obtain the largest possible lower bound for the independence ratio of graphs with maximum degree three containing no triangles.
Geometric properties of a class of support points of univalent functions
Johnny E.
Brown
371-382
Abstract: Let S denote the set of functions $f(z)$ analytic and univalent in $\vert z\vert\, < \,1$, normalized by $f(0)\, = \,0$ and $ \operatorname{Re} \,L(g)$, $g \in S$. The support points corresponding to the point-evaluation functionals are determined explicitly and are shown to also be extreme points of S. New geometric properties of their omitte $\operatorname {arcs}\,\Gamma$ are found. In particular, it is shown that for each such support point $\Gamma$ lies entirely in a certain half-strip, $ \Gamma$ has monotonic argument, and the angle between radius and tangent vectors increases from zero at infinity to a finite maximum value at the tip of the $\operatorname{arc}\,\Gamma $. Numerical calculations appear to indicate that the known bound $\pi /4$ for the angle between radius and tangent vectors is actually best possible.
Approximation theory in the space of sections of a vector bundle
David
Handel
383-394
Abstract: Let $p:\,E\, \to \,B$ be a real m-plane bundle and S an n-dimensional subspace of the space of sections $ \Gamma (E)$ of E. S is said to be k-regular if whenever ${x_1},\, \ldots ,\,{x_k}$ are distinct points of B and ${\upsilon _i}\, \in \,{p^{ - 1}}({x_i})$, $ 1\, \leqslant \,i\, \leqslant \,k$, there exists a $\sigma \, \in \,S$ such that $\sigma ({x_i})\, = \,{\upsilon _i}$ for $ 1\, \leqslant \,i\, \leqslant k$. It is proved that if E has a Riemannian metric and B is compact Hausdorff with at least $ k\, + \,1$ points, then S is k-regular if and only if for each $ \varphi \, \in \,\Gamma (E)$, the set of best approximations to $ \varphi$ by elements of S has dimension at most n - km. This extends a classical theorem of Haar, Kolmogorov, and Rubinstein (the case of the product line bundle). Complex and quaternionic analogues of the above are obtained simultaneously. Existence and nonexistence of k-regular subspaces of a given dimension are obtained in special cases via cohomological methods involving configuration spaces. For example, if E is the product real $ (2m\, - \,1)$-plane bundle over a 2-dimensional disk, then $\Gamma (E)$ contains a k-regular subspace of dimension $2km\, - \,1$, but not one of dimension $2km\, - \,1\,\alpha (k)$, where $\alpha (k)$ denotes the number of ones in the dyadic expansion of k.
The dynamics of Morse-Smale diffeomorphisms on the torus
Steve
Batterson
395-403
Abstract: For orientation preserving diffeomorphisms on the torus necessary and sufficient conditions are given for an isotopy class to admit a Morse-Smale diffeomorphism with a specified periodic behavior.
Holomorphic sectional curvatures of bounded homogeneous domains and related questions
J. E.
D’Atri
405-413
Abstract: This paper considers a class of homogeneous Kähler metrics which include the Bergman metrics on homogeneous bounded domains. We obtain various necessary conditions for (a) nonpositive holomorphic sectional curvature, (b) nonpositive sectional curvature, and (c) covariant constant curvature (symmetric metric). In particular, we give examples showing that there exist homogeneous bounded domains which in the Bergman metric have some positive holomorphic sectional curvature.
Twisted sums of sequence spaces and the three space problem
N. J.
Kalton;
N. T.
Peck
1-30
Abstract: In this paper we study the following problem: given a complete locally bounded sequence space Y, construct a locally bounded space Z with a subspace X such that both X and $Z/X$ are isomorphic to Y, and such that X is uncomplemented in Z. We give a method for constructing Z under quite general conditions on Y, and we investigate some of the properties of Z. In particular, when Y is $ {l_p}\,(1\, < \,p\, < \,\infty )$, we identify the dual space of Z, we study the structure of basic sequences in Z, and we study the endomorphisms of Z and the projections of Z on infinite-dimensional subspaces.
Injectivity, projectivity, and the axiom of choice
Andreas
Blass
31-59
Abstract: We study the connection between the axiom of choice and the principles of existence of enough projective and injective abelian groups. We also introduce a weak choice principle that says, roughly, that the axiom of choice is violated in only a set of different ways. This principle holds in all ordinary Fraenkel-Mostowski-Specker and Cohen models where choice fails, and it implies, among other things, that there are enough injective abelian groups. However, we construct an inner model of an Easton extension with no nontrivial injective abelian groups. In the presence of our weak choice principle, the existence of enough projective sets is as strong as the full axiom of choice, and the existence of enough free projective abelian groups is nearly as strong. We also prove that the axiom of choice is equivalent to ``all free abelian groups are projective'' and to ``all divisible abelian groups are injective."
Algebraic description of homogeneous cones
Josef
Dorfmeister
61-89
Abstract: This paper finishes the author's investigations on homogeneous cones. As a result a classification of homogeneous cones is derived. The most important tool to get insight into the structure of homogeneous cones are J-morphisms. Therefore, in this paper we mainly deal with morphisms of homogeneous cones. The main result gives an algebraic description of J-morphisms. It includes a description of ``Linear imbeddings of self-dual homogeneous cones'' and the above mentioned classification of homogeneous cones. In a subsequent paper it will be used to describe homogeneous Siegel domains.
On the asymptotic behavior of a fundamental set of solutions
Charles
Powder
91-110
Abstract: We consider nth order homogeneous linear ordinary differential equations whose coefficients have an asymptotic expansion as $x \to \infty$ in terms of real powers of x and are analytic in sectors of the complex plane. In earlier work Bank (Funkcial. Ekvac. 11 (1968), 87-100) developed a method for reading off the asymptotic behavior of solutions directly from the equation, except in certain cases where roots asymptotically coalesce. For our results, we consider coefficients in a field of the type developed by Strodt (Trans. Amer. Math. Soc. 105 (1962), 229-250). By successive algebraic transforms, we show that an equation in the exceptional case can be reduced to the nonexceptional case and so the asymptotic behavior of the solutions can be read from the equation. This generalizes the classical results when $\infty$ is a singular point and the coefficients are analytic in neighborhoods of $\infty$. The strength of our results is that the coefficients need not be defined in a full neighborhood of $ \infty$, and that the asymptotic behavior can be read directly from the equation.
Structure mappings, coextensions and regular four-spiral semigroups
John
Meakin
111-134
Abstract: The structure mapping approach to regular semigroups developed by K. S. S. Nambooripad and J. Meakin is used to describe the $ \mathcal{K}$-coextensions of the fundamental four-sprial semigroup and hence to describe the structure of all regular semigroups whose idempotents form a four-spiral biordered set. Isomorphisms between regular four-spiral semigroups are studied. The notion of structural uniformity of a regular semigroup is defined and exploited.
Ergodic behaviour of nonstationary regenerative processes
David
McDonald
135-152
Abstract: Let ${V_t}$ be a regenerative process whose successive generations are not necessarily identically distributed and let A be a measurable set in the range of ${V_t}$. Let ${\mu _n}$ be the mean length of the nth generation and $ {\alpha _n}$ be the mean time ${V_t}$ is in A during the nth generation. We give conditions ensuring $ {\lim _{t \to \infty }}\,\operatorname{prob} \{ \,{V_t}\, \in \,A\,\} \, = \,\alpha /\mu$ where $\mathop {\lim }\limits_{n \to \infty } (1/n)\Sigma _{j = 1}^n\,{\mu _j}\, = \mu $ and $\mathop {\lim }\limits_{n \to \infty } (1/n)\Sigma _{j = 1}^n\,{\alpha _j}\, = \,\alpha$.
CR functions and tube manifolds
M.
Kazlow
153-171
Abstract: Various generalizations of Bochner's theorem on the extension of holomorphic functions over tube domains are considered. It is shown that CR functions on tubes over connected, locally closed, locally starlike subsets of ${\textbf{R}^n}$ uniquely extend to CR functions on almost all of the convex hull of the tube set. A CR extension theorem on maximally stratified real submanifolds of $ {\textbf{C}^n}$ is proven. The above two theorems are used to show that the CR functions (resp. CR distributions) on tubes over a fairly general class of submanifolds of ${\textbf{R}^n}$ uniquely extend to CR functions (CR distributions) on almost all of the convex hull.
$\beta $-recursion theory
Sy D.
Friedman
173-200
Abstract: We define recursion theory on arbitrary limit ordinals using the J-hierarchy for L. This generalizes $\alpha $-recursion theory, where the ordinal is assumed to be $ {\Sigma _1}$-admissible. The notion of tameness for a recursively enumerable set is defined and the degrees of tame r.e. sets are studied. Post's Problem is solved when $ {\Sigma _1}\operatorname{cf} \beta \,\beta {\ast}$. Lastly, simple sets are constructed for all $\beta$ with the aid of a $\beta$-recursive version of Fodor's Theorem.
Moduli of punctured tori and the accessory parameter of Lam\'e's equation
L.
Keen;
H. E.
Rauch;
A. T.
Vasquez
201-230
Abstract: To solve the problems of uniformization and moduli for Riemann surfaces, covering spaces and covering mappings must be constructed, and the parameters on which they depend must be determined. When the Riemann surface is a punctured torus this can be done quite explicitly in several ways. The covering mappings are related by an ordinary differential equation, the Lamé equation. There is a constant in this equation which is called the ``accessory parameter". In this paper we study the behavior of this accessory parameter in two ways. First, we use Hill's method to obtain implicit relationships among the moduli of the different uniformizations and the accessory parameter. We prove that the accessory parameter is not suitable as a modulus-even locally. Then we use a computer and numerical techniques to determine more explicitly the character of the singularities of the accessory parameter.
Necessary conditions for the convergence of cardinal Hermite splines as their degree tends to infinity
T. N. T.
Goodman
231-241
Abstract: Let ${\mathcal{S}_{n,s}}$ denote the class of cardinal Hermite splines of degree n having knots of multiplicity S at the integers. In this paper we show that if $ {f_n}\, \to \,f$ uniformly on R, where $ {f_n}\, \in \,{\mathcal{S}_{{i_{n,s}}}}\,{i_n}\, \to \,\infty$ as $n\, \to \,\infty$, and f is bounded, then f is the restriction to R of an entire function of exponential type $ \leqslant \,S$. In proving this result, we need to derive some extremal properties of certain splines $ {\mathcal{E}_{n,s}}\, \in \,{\mathcal{S}_{n,s}}$, in particular that $ \vert\vert{\mathcal{E}_{n,s}}\vert{\vert _\infty }$ minimises $\vert\vert S\vert{\vert _\infty }$ over $ S\, \in \,{\mathcal{S}_{n,s}}$ with $ \vert\vert{S^{(n)}}\vert{\vert _\infty }\, = \,\vert\vert\mathcal{E}_{n,s}^{(n)}\vert{\vert _\infty }$.
On a class of transformations which have unique absolutely continuous invariant measures
Abraham
Boyarsky;
Manny
Scarowsky
243-262
Abstract: A class of piecewise ${C^2}$ transformations from an interval into itself with slopes greater than 1 in absolute value, and having the property that it takes partition points into partition points is shown to have unique absolutely continuous invariant measures. For this class of functions, a central limit theorem holds for all real measurable functions. For the subclass of piecewise linear transformations having a fixed point, it is shown that the unique absolutely continuous invariant measures are piecewise constant.
Stability theory for functional-differential equations
T. A.
Burton
263-275
Abstract: We consider a system of functional differential equations $\mathcal{V}\,(t,\,x( \cdot ))$ with $ \mathcal{F}$ be bounded for $x( \cdot )$ bounded and that $\mathcal{F}$ depend on $x(s)$ only for $t\, - \,\alpha (t)\, \leqslant \,s\, \leqslant \,t$ where $\alpha$ is a bounded function in order to obtain stability properties. We show that if there is a function $H(t,\,x)$ whose derivative along $ \mathcal{F}$, and it improves the standard results on the location of limit sets for ordinary differential equations.
The variety of modular lattices is not generated by its finite members
Ralph
Freese
277-300
Abstract: This paper proves the result of the title. It shows that there is a five-variable lattice identity which holds in all finite modular lattices but not in all modular lattices. It is also shown that every free distributive lattice can be embedded into a free modular lattice. An example showing that modular lattice epimorphisms need not be onto is given.
Spectral theory for subnormal operators
R. G.
Lautzenheiser
301-314
Abstract: We give an example of a subnormal operator T such that $ {\text{C}}\,\backslash \,\sigma (T)$ has an infinite number of components, $ \operatorname{int} (\sigma (T))$ has two components U and V, and T cannot be decomposed with respect to U and V. That is, it is impossible to write $T\, = \,{T_1}\, \oplus \,{T_2}$ with $ \sigma ({T_1})\, = \,\overline U$ and $\sigma ({T_2})\, = \,\overline V$. This example shows that Sarason's decomposition theorem cannot be extended to the infinitely-connected case. We also use Mlak's generalization of Sarason's theorem to prove theorems on the existence of reducing subspaces. For example, if X is a spectral set for T and $ K\, \subset \,X$, conditions are given which imply that T has a nontrivial reducing subspace $ \mathcal{M}$ such that $\sigma (T\vert\mathcal{M})\, \subset \,K$. In particular, we show that if T is a subnormal operator and if $\Gamma$ is a piecewise ${C^2}$ Jordan closed curve which intersects $\sigma (T)$ in a set of measure zero on $ \Gamma$, then $T\, = \,{T_1}\, \oplus \,{T_2}$ with $\sigma ({T_1})\, \subset \,\sigma (T)\, \cap \,\overline {\operatorname{ext} (\Gamma )}$ and $ \sigma ({T_2})\, \subset \,\sigma (T)\, \cap \,\overline {\operatorname{int} (\Gamma )}$.
Markov cell structures for expanding maps in dimension two
F. T.
Farrell;
L. E.
Jones
315-327
Abstract: Let $f:\,{M^2}\, \to \,{M^2}$ be an expanding self-immersion of a closed 2-manifold, then for some positive integer n, ${f^n}$ leaves invariant a cell structure on $ {M^2}$. A similar result is true when M is a branched 2-manifold.
Notes on square-integrable cohomology spaces on certain foliated manifolds
Shinsuke
Yorozu
329-341
Abstract: We discuss some square-integrable cohomology spaces on a foliated manifold with one-dimensional foliation whose leaves are compact and with a complete bundle-like metric. Applications to a contact manifold are given.
Results on weighted norm inequalities for multipliers
Douglas S.
Kurtz;
Richard L.
Wheeden
343-362
Abstract: Weighted $ {L^p}$-norm inequalities are derived for multiplier operators on Euclidean space. The multipliers are assumed to satisfy conditions of the Hörmander-Mikhlin type, and the weight functions are generally required to satisfy conditions more restrictive than $ {A_p}$ which depend on the degree of differentiability of the multiplier. For weights which are powers of $ \left\vert x \right\vert$, sharp results are obtained which indicate such restrictions are necessary. The method of proof is based on the function ${f^\char93 }$ of C. Fefferman and E. Stein rather than on Littlewood-Paley theory. The method also yields results for singular integral operators.
Adding and subtracting jumps from Markov processes
Richard F.
Bass
363-376
Abstract: If ${X_t}$ is a continuous Markov process with infinitesimal generator A, if n is a kernel satisfying certain conditions, and if B is an operator given by $\displaystyle Bg(x)\, = \,\int {[ {g( y)\, - \,g(x)}]} \,n({x,\,dy}),$ then $ A\, + \,B$ will be the generator of a Markov process that has Lévy system $ (n,\,dt)$. Conversely, if $ {X_t}$ has Lévy system $(n,\,dt)$, n satisfies certain conditions, and B is defined as above, then $A\, - \,B$ will be the generator of a continuous Markov process.
The classification of two-dimensional manifolds
Edward M.
Brown;
Robert
Messer
377-402
Abstract: Invariants are constructed to classify all noncompact 2-manifolds including those with boundary. The invariants of a 2-manifold M are the space of ends of M and the subspaces of nonplanar ends, of nonorientable ends, and of ends that are limits of compact boundary components. Also the space of ends of the boundary components together with its natural map into the ends of M and the orientation of these ends induced by orientations of neighborhoods of the orientable ends of M are used in addition to the usual compact invariants. Special properties are established for the invariants of a 2-manifold, and a 2-manifold is constructed for each set of invariants with the special properties.
Diffeomorphisms and volume-preserving embeddings of noncompact manifolds
R. E.
Greene;
K.
Shiohama
403-414
Abstract: The theorem of J. Moser that any two volume elements of equal total volume on a compact manifold are diffeomorphism-equivalent is extended to noncompact manifolds: A necessary and sufficient condition (equal total and same end behavior) is given for diffeomorphism equivalence of two volume forms on a noncompact manifold. Results on the existence of embeddings and immersions with the property of inducing a given volume form are also given. Generalizations to nonorientable manifolds and manifolds with boundary are discussed.
Equivariant concordance of invariant knots
Neal W.
Stoltzfus
1-45
Abstract: The classification of equivariant concordance classes of high-dimensional codimension two knots invariant under a cyclic action, T, of order m has previously been reported on by Cappell and Shaneson [CS2]. They give an algebraic solution in terms of their algebraic k-theoretic $\Gamma$-groups. This work gives an alternative description by generalizing the well-known Seifert linking forms of knot theory to the equivariant case. This allows explicit algorithmic computations by means of the procedures and invariants of algebraic number theory (see the subsequent work [St], particularly Theorem 6.13). Following Levine [L3], we define bilinear forms on the middle-dimensional homology of an equivariant Seifert surface $ {B_i}(x,y) = L(x,{i_ + }(T_{\ast}^iy))$, for $i = 1, \cdots ,m$. Our first result (2.5) is that an invariant knot is equivariantly concordant to an invariant trivial knot if and only if there is a subspace of half the rank on which the ${B_i}$ vanish simultaneously. We then introduce the concepts of equivariant isometric structure and algebraic concordance which mirror the preceding geometric ideas. The resulting equivalence classes form a group under direct sum which has infinitely many elements of each of the possible orders (two, four and infinite), at least for odd periods. The central computation (3.4) gives an isomorphism of the equivariant concordance group with the subgroup of the algebraic knot concordance group whose Alexander polynomial, $ \Delta$, satisfies the classical relation $\left\vert {\prod\nolimits_{i = \,1}^m {\Delta \left( {{\lambda ^i}} \right)} } \right\vert\, = \,1\,$, where $\lambda$ is a primitive mth root of unity. This condition assures that the m-fold cover of the knot complement is also a homology circle, permitting the geometric realization of each equivariant isometric structure. Finally, we make an explicit computation of the Browder-Livesay desuspension invariant for knots invariant under an involution and also elucidate the connection of our methods with the results of [CS2] by explicitly describing a homomorphism from the group of equivariant isometric structures to the appropriate $\Gamma$-group.
Partially conservative extensions of arithmetic
D.
Guaspari
47-68
Abstract: Let T be a consistent r.e. extension of Peano arithmetic; $\Sigma _n^0$, $\Pi _n^0$ the usual quantifier-block classification of formulas of the language of arithmetic (bounded quantifiers counting ``for free"); and $ \Gamma$, $\Gamma '$ variables through the set of all classes $ \Sigma _n^0$, $ \Pi _n^0$. The principal concern of this paper is the question: When can we find an independent sentence $\phi \, \in \,\Gamma$ which is $\Gamma '$-conservative in the following sense: Any sentence $\chi$ in $\Gamma '$ which is provable from $T + \phi$ is already provable from T? (Additional embellishments: Ensure that $ \phi$ is not provably equivalent to a sentence in any class ``simpler'' than $ \Gamma$; that $ \phi$ is not conservative for classes ``more complicated'' than $ \Gamma '$.) The answer, roughly, is that one can find such a $\phi$, embellishments and all, unless $\Gamma$ and $\Gamma '$ are so related that such a $\phi$ obviously cannot exist. This theorem has applications to the theory of interpretations, since ``$\phi$ is $\Gamma$-conservative'' is closely related to the property ``$T + \phi$ is interpretable in T"-or to variants of it, depending on $\Gamma$. Finally, we provide simple model theoretic characterizations of $ \Gamma$-conservativeness. Most results extend straightforwardly if extra symbols are added to the language of arithmetic, and most have analogs in the Levy hierarchy of set theoretic formulas (T then being an extension of ZF).
Antiholomorphic involutions of analytic families of abelian varieties
Allan
Adler
69-94
Abstract: In this paper, we investigate antiholomorphic involutions of Kuga-Satake analytic families of polarized abelian varieties V. A complete set of invariants of the Aut(V)-conjugacy classes of antiholomorphic involutions of V is obtained. These invariants are expressed as cohomological invariants of the arithmetic data defining V. In the last section, the fibre varieties of Kuga-Satake type belonging to totally indefinite quaternion division algebras over totally real fields are investigated in more detail, and the cohomological invariants are related to results of Steve Kudla. The group of holomorphic sections of V is computed for this case. It is also shown that in general the fibre structure of V is intrinsic.
Some constructions of infinite M\"obius planes
Nicholas
Krier
95-115
Abstract: New infinite Möbius planes are constructed using transfinite induction. Any infinite affine plane A can be embedded in a Möbius plane M and the construction allows some groups of perspectivities of A to be extended to automorphism groups of M. Given $\left\{ {{A_\alpha },\,\alpha \, \in \,J} \right\}$, an infinite collection of s affine planes each of order s, there exist a Möbius plane M and a bijection b from J to the point set of M so that for each $\alpha \, \in \,J,\,{M_{b\left( \alpha \right)}}$ is isomorphic to $ {A_\alpha }$.
Normal two-dimensional elliptic singularities
Stephen Shing Toung
Yau
117-134
Abstract: Given a weighted dual graph such that the canonical cycle $ K'$ exists, is there a singularity corresponding to the given weighted dual graph and which has Gorenstein structure? This is one of the important problems in normal surface singularities. In this paper, we give a necessary and sufficient condition for the existence of Gorenstein structures for weakly elliptic singularities. A necessary and sufficient condition for the existence of maximally elliptic structure is also given. Hence, the above question is answered affirmatively for a special kind of singularities. We also develop a theory for those elliptic Gorenstein singularities with geometric genus equal to three.
Free states of the gauge invariant canonical anticommutation relations. II
B. M.
Baker
135-155
Abstract: A class of representations of the gauge invariant subalgebra of the canonical anticommutation relations (henceforth GICAR) is studied. These representations are induced by restricting the well-known pure, nongauge invariant generalized free states of the canonical anticommutation relations (henceforth CAR). Denoting a state of the CAR by $\omega$, and the unique generalized free state of the CAR such that $\omega \left( {a{{\left( f \right)}^{\ast}}a\left( g \right)} \right)\, = \,\left( {f,Tg} \right)$ and $\omega \left( {a\left( f \right)a\left( g \right)} \right)\, = \,\left( {Sf,g} \right)$ by ${\omega _{S,T}}$, it is shown that a pure, nongauge invariant state ${\omega _{S,T}}$ induces a factor representation of the GICAR if and only if $Tr\,T\left( {I - T} \right)\, = \,\infty$.
Generic sets and minimal $\alpha $-degrees
C. T.
Chong
157-169
Abstract: A non-$ \alpha$-recursive subset G of an admissible ordinal $\alpha$ is of minimal $\alpha $-degree if every set of strictly lower $\alpha$-degree than that of G is $ \alpha$-recursive. We give a characterization of regular sets of minimal $ \alpha$-degree below $ 0'$ via the notion of genericity. We then apply this to outline some 'minimum requirements' to be satisfied by any construction of a set of minimal $ \aleph _\omega ^L$-degree below $0'$.
Analytic extensions and selections
J.
Globevnik
171-177
Abstract: Let G be a closed subset of the closed unit disc in C, let F be a closed subset of the unit circle of measure 0 and let $\Phi$ map G into the class of all open subsets of a complex Banach space X. Under suitable additional assumptions on $\Phi$ we prove that given any continuous function $f:\,F \to X$ satisfying $f(z)\, \in \,{\text{closure(}}\Phi (z))\,(z\, \in \,F\, \cap \,G)$ there exists a continuous function f from the closed unit disc into X, analytic in the open unit disc, which extends f and satisfies $\tilde f(z)\, \in \,\Phi (z)\,(z\, \in \,G\, - \,F)$. This enables us to generalize and sharpen known dominated extension theorems for the disc algebra.
Rational inner functions on bounded symmetric domains
Adam
Korányi;
Stephen
Vági
179-193
Abstract: It is shown that the rational inner functions on any bounded symmetric domain are given by a generalized version of a formula found by Rudin and Stout in the case of the polydisc. In particular, it is shown that all rational inner functions are constant on symmetric domains which have no irreducible factor of tube type.
Local $H$-maps of classifying spaces
Timothy
Lance
195-215
Abstract: Let BU denote the localization at an odd prime p of the classifying space for stable complex bundles, and let $f:BU \to BU$ be an H-map with fiber F. In this paper the Hopf algebra $ {H^{\ast}}(F,\textbf{Z}/P)$ is computed for any such f. For certain H-maps f of geometric interest the p-local cohomology of F is given by means of the Bockstein spectral sequence. A direct description of ${H^ {\ast} }(F,{{\textbf{Z}}_{(P)}})$ is also given for an important special case. Applications to the classifying spaces of surgery will appear later.
Cell-like $0$-dimensional decompositions of $S\sp{3}$ are $4$-manifold factors
R. J.
Daverman;
W. H.
Row
217-236
Abstract: The main result is the title theorem asserting that if G is any upper semicontinuous decomposition of $ {S^3}$ into cell-like sets which is 0-dimensional, in the sense that the image of the nondegenerate elements in ${S^3}/G$ is 0-dimensional, then $G\, \times \,{S^1}$ is shrinkable, and $ \left( {{S^3}/G} \right)\, \times \,{S^1}$ is homeomorphic to ${S^3}\, \times \,{S^1}$.
Constructing framed $4$-manifolds with given almost framed boundaries
Steve J.
Kaplan
237-263
Abstract: Two methods are presented for constructing framed 4-manifolds with given almost framed boundaries. The main tools are the ``moves'' of Kirby's calculus of framed links. A new description is given for the $\mu$-in-variant of a knot and this description is used to study almost framed 3-manifolds.
Morse and generic contact between foliations
Russell B.
Walker
265-281
Abstract: Motivated by the recent work of J. Franks and C. Robinson, the study of the contact between two foliations of equal codimension is begun. Two foliations generically contact each other in certain dimensional sub-manifold complexes. All but a finite number of these contact points are ``Morse". In a recent paper by the author, a complete large isotopy ``index of contact'' is specified for two foliations of ${T^2}$. If contact is restricted to index 0 ("domed contact"), some sharp conclusions are made as to the topology of the manifold and isotopy classes of the two foliations. It is hoped that this work will lead to the construction of new quasi-Anosov diffeomorphisms and possibly to a new Anosov diffeomorphism.
Selection theorems for $G\sb{\delta }$-valued multifunctions
S. M.
Srivastava
283-293
Abstract: In this paper we establish under suitable conditions the existence of measurable selectors for $ {G_\delta }$-valued multifunctions. In particular we prove that a measurable partition of a Polish space into $ {G_\delta }$ sets admits a Borel selector.
Analogs of Clifford's theorem for polycyclic-by-finite groups
Martin
Lorenz
295-317
Abstract: Let P be a primitive ideal in the group algebra $K[G]$ of the polycyclic group G and let N be a normal subgroup of G. We show that among the irreducible right $K[G]$-modules with annihilator P there exists at least one, V, such that the restricted $K[N]$-module ${V_N}$ is completely reducible, a sum of G-conjugate simple $K[N]$-submodules. Various stronger versions of this result are obtained. We also consider the action of G on the factor $ K[N]/P \cap K[N]$ and show that, in case K is uncountable, any ideal I of $K[N]$ satisfying ${ \cap _{g \in G}}{I^g}\, = \,P\, \cap \,K[N]$ is contained in a primitive ideal Q of $K[N]$ with ${ \cap _{g \in G}}{I^g}\, = \,P\, \cap \,K[N]$.
Critical mappings of Riemannian manifolds
David D.
Bleecker
319-338
Abstract: We consider maps, from one Riemannian manifold to another, which are critical for all invariantly defined functionals on the space of maps. There are many such critical mappings, perhaps too numerous to suitably classify, although a characterization of sorts is provided. They are proven to have constant rank, with the image being a homogeneous minimal submanifold of the target manifold. Critical maps need not be Riemannian submersions onto their images. Also, there are homogeneous spaces for which the identity map is not critical. Many open problems remain.
Isometries of $L\sb{p}$-spaces associated with semifinite von Neumann algebras
P. K.
Tam
339-354
Abstract: The paper determines the structure of (classes of) linear isometries between ${L_p}$-spaces associated with semifinite normal faithful traces on von Neumann algebras, generalizing works of M. Broise and B. Russo. Also established are some auxiliary results on ${L_p}$ norm inequalities which are of interest by themselves.
The Dirichlet norm and the norm of Szeg\H o type
Saburou
Saitoh
355-364
Abstract: Let S be a smoothly bounded region in the complex plane. Let $ g(z,t)$ denote the Green's function of S with pole at t. We show that $\displaystyle \iint_S {\vert f'(z){\vert^2}\,dx\,dy\, \leqslant \,\frac{1}{2}\i... ... {\frac{{\partial g(z,t)}} {{\partial {n_z}}}} \right)}^{ - 1}}\vert dz\vert} }$ holds for any analytic function $ f(z)$ on $S\, \cup \,\partial S$. This curious inequality is obtained as a special case of a much more general result.
A new characterization of amenable groups
Jon
Sherman
365-389
Abstract: Paradoxical sets, which are a natural generalization of the type of sets made famous as Hausdorff-Banach-Tarski paradoxes, are defined in terms of piecewise translations. Piecewise translations are the generalization to arbitrary discrete groups of the maps used in the Banach-Tarski paradoxes as congruences by finite decomposition. A subset of a group is defined to be large if finitely many translates of it can cover the group. The main result of this paper is that a group is amenable if and only if it does not contain a large paradoxical set.
Simple periodic orbits of mappings of the initial
Louis
Block
391-398
Abstract: Let f be a continuous map of a closed, bounded interval into itself. A criterion is given to determine whether or not f has a periodic point whose period is not a power of 2, which just depends on the periodic orbits of f whose period is a power of 2. Also, a lower bound for the topological entropy of f is obtained.
Compactifications of the generalized Jacobian variety
Tadao
Oda;
C. S.
Seshadri
1-90
Abstract: The generalized Jacobian variety of an algebraic curve with at most ordinary double points is an extension of an abelian variety by an algebraic torus. Using the geometric invariant theory, we systematically compactify it in finitely many different ways and describe their structure in terms of torus embeddings. Our compactifications include all known good ones.
Multiplications on cohomology theories with coefficients
Alvin Frank
Martin
91-120
Abstract: Araki and Toda have considered the existence and classification of multiplications on generalized cohomology theories with coefficients in the category of finite CW-complexes. We consider the same matters for representable cohomology theories in a category of stable CW-spectra, such as that constructed by Adams. We obtain similar, and in certain instances stronger, results than Araki and Toda, with methods of proof that are often simpler and more straightforward.
Natural endomorphisms of Burnside rings
Andreas
Blass
121-137
Abstract: The Burnside ring $ \mathcal{B}(G)$ of a finite group G consists of formal differences of finite G-sets. $ \mathcal{B}$ is a contravariant functor from finite groups to commutative rings. We study the natural endomorphisms of this functor, of its extension $\textbf{Q} \otimes \mathcal{B}$ to rational scalars, and of its restriction $ \mathcal{B} \upharpoonright {\text{Ab}}$ to abelian groups. Such endomorphisms are canonically associated to certain operators that assign to each group one of its conjugacy classes of subgroups. Using these operators along with a carefully constructed system of linear congruences defining the image of $ \mathcal{B}(G)$ under its canonical embedding in a power of Z, we exhibit a multitude of natural endomorphisms of $\mathcal{B}$, we show that only two of them map G-sets to G-sets, and we completely describe all natural endomorphisms of $ \mathcal{B} \upharpoonright {\text{Ab}}$.
On sufficient conditions for harmonicity
P. C.
Fenton
139-147
Abstract: Suppose that u is continuous in the plane and that given any complex number z there is a number $\rho = \rho (z) > 0$ such that $\displaystyle u(z) = \frac{1} {{2\pi }}\int_0^{2\pi } {u(z + \rho {e^{i\theta }})} d\theta$ (1) The main result is: if u possesses a harmonic majorant and $\rho (z)$ is continuous and satisfies a further condition (which may not be omitted) then u is harmonic. Another result in the same vein is proved.
An algebraic determination of closed orientable $3$-manifolds
William
Jaco;
Robert
Myers
149-170
Abstract: Associated with each polyhedral simple closed curve j in a closed, orientable 3-manifold M is the fundamental group of the complement of j in M, ${\pi _1}(M - j)$. The set, $\mathcal{K}(M)$, of knot groups of M is the set of groups $ {\pi _1}(M - j)$ as j ranges over all polyhedral simple closed curves in M. We prove that two closed, orientable 3-manifolds M and N are homeomorphic if and only if $\mathcal{K}(M) = \mathcal{K}(N)$. We refine the set of knot groups to a subset $\mathcal{F}(M)$ of fibered knot groups of M and modify the above proof to show that two closed, orientable 3-manifolds M and N are homeomorphic if and only if $\mathcal{F}(M) = \mathcal{F}(N)$.
The spaces of functions of finite upper $p$-variation
Robert R.
Nelson
171-190
Abstract: Let y be a Banach space, $1\, \leqslant \,p\, < \,\infty$, and $ {U_p}$ be the semi-normed space of Y-valued Bochner measurable functions of a real variable which have finite upper p-variation. Let $ {\tilde U_p}$ be the space of ${U_p}$-equivalence classes. An averaging operator is defined with the aid of the theory of helixes in Banach spaces, which enables us to show that the spaces ${\tilde U_p}$ are Banach spaces, to characterize their members, and to show that they are isometrically isomorphic to Banach spaces of Y-valued measures with bounded p-variation.
Differentiability of measures associated with parabolic equations on infinite-dimensional spaces
M. Ann
Piech
191-209
Abstract: The transition measures of the Brownian motion on manifolds modelled on abstract Wiener spaces locally correspond to fundamental solutions of certain infinite dimensional parabolic equations. We establish the existence of such fundamental solutions under a broad new set of hypotheses on the differential coefficients. The fundamental solutions can be approximated in total variation by fundamental solutions of ``almost'' finite dimensional parabolic equations. By the finite dimensional theory, the approximations are seen to be differentiable. We prove that the property of differentiability is closed under a particular type of sequential convergence, and conclude the differentiability of the fundamental solutions of the infinite dimensional parabolic equations. This result provides strong evidence in support of the conjecture that the transition measures of the Brownian motion are differentiable, and hence is of importance in the construction of infinite dimensional Laplace-Beltrami operators.
On the positive spectrum of Schr\"odinger operators with long range potentials
G. B.
Khosrovshahi;
H. A.
Levine;
L. E.
Payne
211-228
Abstract: In this paper we are concerned with solutions of the equation $\Delta u\, + \,p(x)u\, = \,0$ in an unbounded domain $\Omega$ in ${R^n}$, $ \Omega \, \supset \,\{ x\vert\,\,\left\Vert x \right\Vert\, \geqslant \,{R_0}\}$. The main result is a determination of conditions on the asymptotic behavior of $p(x)$ sufficient to guarantee that no nontrivial ${L_2}$ solution exists. Our results contain those of previous authors as special cases. The principal application is to the determination of upper bounds for positive eigenvalues of Schrödinger operators.
Dixmier's representation theorem of central double centralizers on Banach algebras
Sin-ei
Takahasi
229-236
Abstract: The present paper is devoted to a representation theorem of central double centralizers on a complex Banach algebra with a bounded approximate identity. In particular, our result implies the representation theorem of the ideal center of an arbitrary $ {C^\ast}$-algebra established by J. Dixmier.
Dirichlet forms associated with hypercontractive semigroups
James G.
Hooton
237-256
Abstract: We exhibit a class of probability measures on ${\textbf{R}^n}$ such that the associated Dirichlet form is represented by a selfadjoint operator A and such that $ {e^{ - tA}}$ is a hypercontractive semigroup of operators. The measures are of the form $ d\mu \, = \,{\Omega ^2}\,dx$ where $\Omega$ has classical first derivatives and $ {L^p}$ second derivatives, p determined by n.
Dispersion points for linear sets and approximate moduli for some stochastic processes
Donald
Geman
257-272
Abstract: Let $\Gamma \, \in \,[0,\,1]$ be Lebesgue measurable; then $ \Gamma$ has Lebesgue density 0 at the origin if and only if $\displaystyle \int_\Gamma {{t^{ - 1}}\Psi ({t^{ - 1}}\,{\text{meas}}} \{ \Gamma \, \cap \,(0,\,t)\} )\,dt\, < \,\infty$ for some continuous, strictly increasing function $\Psi (t)\,(0\, \leqslant \,t\, \leqslant \,1)$ with $\Psi (0)\, = \,0$. This result is applied to the local growth of certain Gaussian (and other) proceses $\{ {X_t},\,t\, \geqslant \,0\}$ as follows: we find continuous, increasing functions $\phi (t)$ and $\eta (t)\,(t\, \geqslant \,0)$ such that, with probability one, the set $\{ t:\eta (t)\, \leqslant \,\left\vert {{X_t}\, - \,{X_0}} \right\vert\, \leqslant \,\phi (t)\}$ has density 1 at the origin.
Difference equations over locally compact abelian groups
G. A.
Edgar;
J. M.
Rosenblatt
273-289
Abstract: A homogeneous linear difference equation with constant coefficients over a locally compact abelian group G is an equation of the form $\Sigma_{j\, = \,1}^n {{c_j}f({t_j}x)\, = \,0}$ which holds for all $ x\, \in \,G$ where ${c_1}, \ldots ,\,{c_n}$ are nonzero complex scalars, ${t_1},\, \ldots \,,\,{t_n}$ are distinct elements of G, and f is a complex-valued function on G. A function f has linearly independent translates precisely when it does not satisfy any nontrivial linear difference equation. The locally compact abelian groups without nontrivial compact subgroups are exactly the locally compact abelian groups such that all nonzero $f\, \in \,{L_p}(G)$ with $1\, \leqslant \,p\, \leqslant \,2$ have linearly independent translates. Moreover, if G is the real line or, more generally, if G is $ {R^n}$ and the difference equation has a characteristic trigonometric polynomial with a locally linear zero set, then the difference equation has no nonzero solutions in $ {C_0}(G)$ and no nonzero solutions in ${L_p}(G)$ for $1\, \leqslant \,p\, < \,\infty$. But if G is some ${R^n}$ for $ n\, \geqslant \,2$ and the difference equation has a characteristic trigonometric polynomial with a curvilinear portion of its zero set, then there will be nonzero ${C_0}({R^n})$ solutions and even nonzero ${L_p}({R^n})$ solutions for $p\, > \,2n/(n - 1)$. These examples are the best possible because if $1\, \leqslant p\, < \,2n/(n - 1)$, then any nonzero function in $ {L_p}({R^n})$ has linearly independent translates. Also, the solutions to linear difference equations over the circle group can be simply described in a fashion which an example shows cannot be extended to all compact abelian groups.
Smooth orbit equivalence of ergodic ${\bf R}\sp{d}$ actions, $d\geq 2$
Daniel
Rudolph
291-302
Abstract: We show here that any two free ergodic finite measure preserving actions of $ {\textbf{R}^d}$, $d\, \geqslant \,2$, are orbit equivalent by a measure preserving map which on orbits is ${C^\infty }$.
$\sigma $-connectedness in hereditarily locally connected spaces
J.
Grispolakis;
E. D.
Tymchatyn
303-315
Abstract: B. Knaster, A. Lelek and J. Mycielski [Colloq. Math. 6 (1958), 227-246] had asked whether there exists a hereditarily locally connected planar set, which is the union of countably many disjoint arcs. They gave an example of a locally connected, connected planar set, which is the union of a countable sequence of disjoint arcs. Lelek proved in a paper in Fund. Math. in 1959, that connected subsets of planar hereditarily locally connected continua are weakly $\sigma$-connected (i.e., they cannot be written as unions of countably many disjoint, closed connected subsets). In this paper we generalize the notion of finitely Suslinian to noncompact spaces. We prove that there is a class of spaces, which includes the class of planar hereditarily locally connected spaces and the finitely Suslinian spaces, and which are weakly $ \sigma$-connected, thus, answering the above question in the negative. We also prove that arcwise connected, hereditarily locally connected, planar spaces are locally arcwise connected. This answers in the affirmative a question of Lelek [Colloq. Math. 36 (1976), 87-96].
On the global asymptotic behavior of Brownian local time on the circle
E.
Bolthausen
317-328
Abstract: The asymptotic behavior of the local time of Brownian motion on the circle is investigated. For fixed time point t this is a (random) continuous function on ${S^1}$. It is shown that after appropriate norming the distribution of this random element in $ C({S^1})$ converges weakly as $t\, \to \,\infty $. The limit is identified as $ 2(B(x)\, - \,\int {B(y)\,dy)}$ where B is the Brownian bridge. The result is applied to obtain the asymptotic distribution of a Cramer-von Mises type statistic for the global deviation of the local time from the constant t on ${S^1}$.
The structure of supermanifolds
Marjorie
Batchelor
329-338
Abstract: The increasing recognition of Lie superalgebras and their importance in physics inspired a search to find an object, a ``supermanifold", which would realize the geometry implicit in Lie superalgebras. This paper analyzes the structure of supermanifolds as defined by B. Kostant. The result is the following structure theorem. The Main Theorem. If E is a real vector bundle over the smooth manifold X, let $\Lambda E$ be the associated exterior bundle and let $ \Gamma (\Lambda E)$ be the sheaf of sections of $\Lambda E$. Then every supermanifold over X is isomorphic to $ \Gamma (\Lambda E)$ for some vector bundle E over X. Although the vector bundle E is not unique but is determined only up to isomorphism, and the isomorphism guaranteed is not canonical, the existence of the isomorphism provides a base for a better understanding of geometry in the graded setting.
Pseudo-integral operators
A. R.
Sourour
339-363
Abstract: Let $(X,\,\mathcal{a},\,m)$ be a standard finite measure space. A bounded operator T on ${L^2}(X)$ is called a pseudo-integral operator if $(Tf)(x)\, = \,\int {f(y)\,\mu (x,\,dy)}$, where, for every x, $\mu (x,\, \cdot \,)$ is a bounded Borel measure on X. Main results: 1. A bounded operator T on $ {L^2}$ is a pseudo-integral operator with a positive kernel if and only if T maps positive functions to positive functions. 2. On nonatomic measure spaces every operator unitarily equivalent to T is a pseudo-integral operator if and only if T is the sum of a scalar and a Hilbert-Schmidt operator. 3. The class of pseudo-integral operators with absolutely bounded kernels form a selfadjoint (nonclosed) algebra, and the class of integral operators with absolutely bounded kernels is a two-sided ideal. 4. An operator T satisfies $ (Tf)(x)\, = \,\int {f(y)\,\mu (x,\,dy)}$ for $f\, \in \,{L^\infty }$ if and only if there exists a positive measurable (almost-everywhere finite) function $\Omega$ such that $\left\vert {(Tf)(x)} \right\vert\, \leqslant \,{\left\Vert f \right\Vert _\infty }\Omega (x)$.
Optimal stochastic switching and the Dirichlet problem for the Bellman equation
Lawrence C.
Evans;
Avner
Friedman
365-389
Abstract: Let ${L^i}$ be a sequence of second order elliptic operators in a bounded n-dimensional domain $ \Omega$, and let $ {f^i}$ be given functions. Consider the problem of finding a solution u to the Bellman equation ${\sup _i}({L^i}u\, - \,{f^i})\, = \,0$ a.e. in $ \Omega$, subject to the Dirichlet boundary condition $u\, = \,0$ on $ \partial \Omega$. It is proved that, provided the leading coefficients of the $ {L^i}$ are constants, there exists a unique solution u of this problem, belonging to ${W^{1,\infty }}(\Omega )\, \cap \,W_{{\text{loc}}}^{2,\infty }(\Omega )$. The solution is obtained as a limit of solutions of certain weakly coupled systems of nonlinear elliptic equations; each component of the vector solution converges to u. Although the proof is entirely analytic, it is partially motivated by models of stochastic control. We solve also certain systems of variational inequalities corresponding to switching with cost.
The atomic decomposition for parabolic $H\sp{p}$ spaces
Robert H.
Latter;
Akihito
Uchiyama
391-398
Abstract: The theorem of A. P. Calderón giving the atomic decomposition for certain parabolic ${H^p}$ spaces is extended to all such spaces. The proof given also applies to Hardy spaces defined on the Heisenberg group.
Statically tame periodic homeomorphisms of compact connected $3$-manifolds. I. Homeomorphisms conjugate to rotations of the $3$-sphere
Edwin E.
Moise
1-47
Abstract: Let f be a homeomorphism of the 3-sphere onto itself, of finite period n, and preserving orientation. Suppose that the fixed-point set F of f is a tame 1-sphere. It is shown that (1) the 3-sphere has a triangulation $ K({{\textbf{S}}^3})$ such that F forms a subcomplex of $K({{\textbf{S}}^3})$ and f is simplicial relative to $ K({{\textbf{S}}^3})$. Suppose also that F is unknotted. It then follows that (2) f is conjugate to a rotation.
Maximal inequalities related to generalized a.e. continuity
W. B.
Jurkat;
J. L.
Troutman
49-64
Abstract: An integral inequality of the classical Hardy-Littlewood type is obtained for the maximal function of positive convolution operators associated with approximations of the identity in ${R^n}$. It is shown that the (formally) rearranged maximal function can in general be estimated by an elementary integral involving the decreasing rearrangements of the kernel of the approximation and the function being approximated. (The estimate always holds when the kernel has compact support or a decreasing radial majorant integrable in a neighborhood of infinity; a one-dimensional counterexample shows that integrability alone may not suffice.) The finiteness of the integral determines a Lorentz space of functions which are a.e. continuous in the generalized sense of the approximation. Conversely, in dimension one it is established that this space is the largest strongly rearrangement invariant Banach space of such functions. In particular, the new inequality provides access to the study of Cesàro continuity of order less than one.
The free boundary for elastic-plastic torsion problems
Luis A.
Caffarelli;
Avner
Friedman
65-97
Abstract: Consider the variational inequality: find $ u\, \in \,K$ such that $ \int_Q {\nabla u\, \cdot \,\nabla \left( {v\, - \,u} \right)} \, \geqslant \,\mu \,\int_Q {\left( {v\, - \,u} \right)} \,\left( {\mu \, > \,0} \right)$ for any $ v\, \in \,K$, where $K\, = \,\left\{ {w\, \in \,H_0^1\left( Q \right),\,\left\vert {\nabla w} \right\vert\, \leqslant \,1\,} \right\}$ and Q is a simply connected domain whose boundary is piecewise ${C^3}$. The solution u represents the stress function in a torsion problem of an elastic bar with cross section Q; the sets $E\, = \,\left\{ {x\, \in \,Q;\,\left\vert {\nabla u\left( x \right)} \right\ve... ...x\, \in \,Q;\,\left\vert {\nabla u\left( x \right)} \right\vert = \,1} \right\}$ are the elastic and plastic subsets of Q. The ridge R of Q is, by definition, the set of points in Q where dist $\left( {x,\,\partial Q} \right)$ is not ${C^{1,1}}$. The paper studies the location and shape of E, P and the free boundary $ \Gamma \, = \,\partial E\, \cap \,Q$. It is proved that the ridge is elastic and that E is contained in a $\left( {c/\mu } \right)$-neighborhood of R, as $\mu \, \to \,\infty \,\left( {c\, > \,0} \right)$. The behavior of E and P near the vertices of $ \partial Q$ is studied in detail, as well as the nature of $\Gamma$ away from the vertices. Applications are given to special domains. The case where Q is multiply connected is also studied; in this case the definition of K is somewhat different. Some results on the ``upper plasticity'' and ``lower plasticity'' and on the behavior as $\mu \, \to \,\infty$ are obtained.
Continuity of the density of a gas flow in a porous medium
Luis A.
Caffarelli;
Avner
Friedman
99-113
Abstract: The equation of gas in a porous medium is a degenerate nonlinear parabolic equation. It is known that a unique generalized solution exists. In this paper it is proved that the generalized solution is continuous.
Sur les germes d'applications differentiables \`a singularit\'es isol\'ees
Jacek
Bochnak;
Wojciech
Kucharz
115-131
Abstract: Le but de cet article est d'étudier les germes d'applications différentiables $({{\textbf{R}}^n},0)\, \to \,({{\textbf{R}}^p},0)$, ou plus généralement les familles de telles applications, ayant en 0 une singularité isolée. Nous formulerons certains critères de ${C^0}$-suffisance de jets et nous démontrerons quelques théorèmes sur le nombre de types topologiques de germes qui apparaissent dans des familles de germes à singularité isolée.
Single-valued representation of set-valued mappings
A. D.
Ioffe
133-145
Abstract: It is shown that the graph of a set-valued mapping satisfying typical conditions which guarantee the existence of measurable selections can be represented as the union of graphs of measurable single-valued mappings depending continuously on a parameter running through some Polish space.
On the topology of the set of completely unstable flows
Zbigniew
Nitecki
147-162
Abstract: We show that: (1) on any open manifold other than the line or plane, there exist nonsingular flows with $\Omega \, \ne \,\emptyset $ which can be perturbed, in the strong ${C^r}$ topology (any r), to flows with $ \Omega \, \ne \,\emptyset$, and that (2) on certain open 3-manifolds there exist flows with $\Omega \, \ne \,\emptyset $ which cannot be approximated, in the strong $ {{\mathcal{C}}^1}$ topology, by flows satisfying both $\Omega \, \ne \,\emptyset $ and no ${{\mathcal{C}}^1}$ $\Omega$-explosions. These examples give partial negative answers to the conjecture of Takens and White, that the completely unstable flows with the strong $ {{\mathcal{C}}^r}$ topology equal the closure of their interior.
Complex-foliated structures. I. Cohomology of the Dolbeault-Kostant complexes
Hans R.
Fischer;
Floyd L.
Williams
163-195
Abstract: We study the cohomology of differential complexes, which we shall call Dolbeault-Kostant complexes, defined by certain integrable sub-bundles F of the complex tangent bundle of a manifold M. When M has a complex or symplectic structure and F is chosen to be the bundle of anti-holomorphic tangent vectors or, respectively, a ``polarization'' then the corresponding complexes are, respectively, the Dolbeault complex and (under further conditions) a complex introduced by Kostant in the context of geometric quantization. A simple condition on F insures that our complexes are elliptic. Assuming ellipticity and compactness of M, for example, one of our key results is a Hirzebruch-Riemann-Roch Theorem.
Zeros of Stieltjes and Van Vleck polynomials
Mahfooz
Alam
197-204
Abstract: The study of the polynomial solutions of the generalized Lamé differential equation gives rise to Stieltjes and Van Vleck polynomials. Marden has, under quite general conditions, established varied generalizations of the results proved earlier by Stieltjes, Van Vleck, Bocher, Klein, and, Pólya, concerning the location of the zeros of such polynomials. We study the corresponding problem for yet another form of the generalized Lamé differential equation and generalize some recent results due to Zaheer and to Alam. Furthermore, applications of our results to the standard form of this differential equation immediately furnish the corresponding theorems of Marden. Consequently, our main theorem of this paper may be considered as the most general result obtained thus far in this direction.
Structural stability and hyperbolic attractors
Artur Oscar
Lopes
205-219
Abstract: A necessary condition for structural stability is presented that in the two dimensional case means that the system has a finite number of topological attractors.
Wiman-Valiron theory for entire functions of finite lower growth
P. C.
Fenton
221-232
Abstract: A general method of Wiman-Valiron type for dealing with entire functions of finite lower growth is presented and used to obtain the lower-order version of a result of W. K. Hayman on the real part of entire functions of small lower growth.
Globally hypoelliptic and globally solvable first-order evolution equations
Jorge
Hounie
233-248
Abstract: We consider global hypoellipticity and global solvability of abstract first order evolution equations defined either on an interval or in the unit circle, and prove that it is equivalent to certain conditions bearing on the total symbol. We relate this to known results about hypoelliptic vector fields on the 2-torus.
Theorems of Fubini type for iterated stochastic integrals
Marc A.
Berger;
Victor J.
Mizel
249-274
Abstract: An extension of the Itô calculus which treats iterated Itô integration, as applied to a class of two-parameter processes, is introduced. This theory includes the integration of certain anticipative integrands and introduces a notion of stochastic differential for such integrands. Among the key results is a version of Fubini's theorem for iterated stochastic integrals, in which a ``correction'' term appears. Applications to stochastic integral equations and to the Itô calculus are given, and the relation of the present development to recent work of Ogawa is described.
Singular perturbations and nonstandard analysis
S.
Albeverio;
J. E.
Fenstad;
R.
Høegh-Krohn
275-295
Abstract: We study by methods of nonstandard analysis second order differential operators with zero order coefficients which are too singular to be defined by standard functions. In particular we study perturbations of the Laplacian in $ {R^3}$ given by potentials of the form $\lambda {\Sigma _j}\delta \left( {x\, - \,{x_j}} \right)$. We also study Sturm-Liouville problems with zero order coefficients given by measures and prove that they satisfy the same oscillation theorems as the regular Sturm-Liouville problems.
Toeplitz operators and related function algebras on certain pseudoconvex domains
Nicholas P.
Jewell;
Steven G.
Krantz
297-312
Abstract: Toeplitz operators are defined on pseudoconvex domains in ${{\textbf{C}}^n}$ and their spectral properties are studied. In addition, the linear space ${H^\infty }\, + \,C$ is discussed and is seen to be a closed algebra on a variety of domains.
Expansive homeomorphisms and topological dimension
Ricardo
Mañé
313-319
Abstract: Let K be a compact metric space. A homeomorphism $f:\,K\mid$ is expansive if there exists $\varepsilon \, > \,0$ such that if $x, y\, \in \,K$ satisfy $d\left( {{f^n}\left( x \right),\,{f^n}\left( y \right)} \right)\, < \,\varepsilon$ for all $ n\, \in \,{\textbf{Z}}$ (where $d\left( { \cdot ,\, \cdot } \right)$ denotes the metric on K) then $x\, = \,y$. We prove that a compact metric space that admits an expansive homeomorphism is finite dimensional and that every minimal set of an expansive homeomorphism is 0-dimensional.
Inductive construction of homogeneous cones
Josef
Dorfmeister
321-349
Abstract: A method is explained how to construct all homogeneous cones in a unique way out of lower dimensional ones. The infinitesimal automorphisms of such a cone and its associated left-symmetric algebras are described in terms of the lower dimensional constituents of the cone. It is characterized when a homogeneous cone is self-dual or a sum of homogeneous cones.
Expanding maps on sets which are almost invariant. Decay and chaos
Giulio
Pianigiani;
James A.
Yorke
351-366
Abstract: Let $A\, \subset \,{R^n}$ be a bounded open set with finitely many connected components ${A_j}$ and let $T:\,\overline A \to \,{R^n}$ be a smooth map with $ A\,\, \subset \,\,T\left( A \right)$. Assume that for each ${A_j}$, $A\,\, \subset \,\,{T^m}\left( {{A_j}} \right)$ for all m sufficiently large. We assume that T is ``expansive", but we do not assume that $ T\left( A \right) = A$. Hence for $x\, \in \,A,\,{T^i}\,\left( x \right)$ may escape A as i increases. Let ${\mu _0}$ be a smooth measure on A (with ${\operatorname{inf} _A}\,{{d{\mu _0}} \mathord{\left/ {\vphantom {{d{\mu _0}} {dx}}} \right. \kern-\nulldelimiterspace} {dx}}\, > \,0$) and let $x\, \in \,A$ be chosen at random (using $ {\mu _0}$). Since T is ``expansive'' we may expect ${T^i}\left( x \right)\,$ to oscillate chaotically on A for a certain time and eventually escape A. For each measurable set $E\, \subset \,A$ define ${\mu _m}\left( E \right)$ to be the conditional probability that ${T^m}\left( x \right) \in \,E$ given that $x,T\left( x \right),\ldots,{T^m}\left( x \right)$ are in A. We show that ${\mu _m}$ converges to a smooth measure $\mu$ which is independent of the choice of $ {\mu _0}$. One dimensional examples are stressed.
Codimension one isometric immersions between Lorentz spaces
L. K.
Graves
367-392
Abstract: The theorem of Hartman and Nirenberg classifies codimension one isometric immersions between Euclidean spaces as cylinders over plane curves. Corresponding results are given here for Lorentz spaces, which are Euclidean spaces with one negative-definite direction (also known as Minkowski spaces). The pivotal result involves the completeness of the relative nullity foliation of such an immersion. When this foliation carries a nondegenerate metric, results analogous to the Hartman-Nirenberg theorem obtain. Otherwise, a new description, based on particular surfaces in the three-dimensional Lorentz space, is required.
On an extremal property of Doob's class
J. S.
Hwang
393-398
Abstract: Recently, we have solved a long open problem of Doob (1935). To introduce the result proved here, we say that a function $ f(z)$ belongs to Doob's class D, if $f(z)$ is analytic in the unit disk U and has radial limit zero at an endpoint of some arc R on the unit circle such that $\operatorname{lim} \,{\operatorname{inf} _{n \to \infty }}\,\left\vert {f({P_n})} \right\vert$, where $\{ {P_n}\}$ is an arbitrary sequence of points in U tending to an arbitrary interior point of R. With this definition, our main result is the following extremal property of Doob's class. Theorem. $ {\operatorname{inf} _{f \in D}}\left\Vert f \right\Vert\, = \,{2 /e}$, where
The analytic continuation of the discrete series. I
Nolan R.
Wallach
1-17
Abstract: In this paper the analytic continuation of the holomorphic discrete series is defined. The most elementary properties of these representations are developed. The study of when these representations are unitary is begun.
The analytic continuation of the discrete series. II
Nolan R.
Wallach
19-37
Abstract: This is the second in a series of papers on the analytic continuation of the holomorphic discrete series. In this paper necessary and sufficient conditions for unitarizability are given in the case of line bundles. The foundations for the vector valued case are begun.
Uniformly continuous functionals on the Fourier algebra of any locally compact group
Anthony To Ming
Lau
39-59
Abstract: Let G be any locally compact group. Let $VN\,(G)$ be the von Neumann algebra generated by the left regular representation of G. We study in this paper the closed subspace $UBC\mathop {(G)}\limits^ \wedge$ of $ VN\, (G)$ consisting of the uniformly continuous functionals as defined by E. Granirer. When G is abelian, $UBC\mathop {(G)}\limits^ \wedge$ is precisely the bounded uniformly continuous functions on the dual group Ĝ. We prove among other things that if G is amenable, then the Banach algebra $ UBC\mathop {(G)}\limits^ \wedge {\ast}$ (with the Arens product) contains a copy of the Fourier-Stieltjes algebra in its centre. Furthermore, $UBC\mathop {(G)}\limits^ \wedge {\ast}$ is commutative if and only if G is discrete. We characterize $ W\mathop {(G)}\limits^ \wedge$, the weakly almost periodic functionals, as the largest subspace X of $VN\, (G)$ for which the Arens product makes sense on ${X^ {\ast} }$ and ${X^ {\ast} }$ is commutative. We also show that if G is amenable, then for certain subspaces Y of $VN\, (G)$ which are invariant under the action of the Fourier algebra $A\, (G)$, the algebra of bounded linear operators on Y commuting with the action of $A\, (G)$ is isometric and algebra isomorphic to ${X^ {\ast} }$ for some $X \subseteq UBC(\mathop {G)}\limits^ \wedge$.
Regular points of Lipschitz functions
Alexander D.
Ioffe
61-69
Abstract: Let f be a locally Lipschitz function on a Banach space X, and S a subset of X. We define regular (i.e. noncritical) points for f relative to S, and give a sufficient condition for a point $z \, \in \, S$ to be regular. This condition is then expressed in the particular case when f is ${C^1}$, and is used to obtain a new proof of Hoffman's inequality in linear programming.
Stable measures and central limit theorems in spaces of stable type
Michael B.
Marcus;
Wojbor A.
Woyczyński
71-102
Abstract: Let X be a symmetric random variable with values in a quasinormed linear space E. X satisfies the central limit theorem on E with index p, $0 \, < \, p \, \leqslant \, 2$, if $\mathcal{L}{n^{ - 1/p}}({X_1} + \cdots + {{\text{X}}_n}))$ converges weakly to some probability measure on E. Hoffman-Jorgensen and Pisier have shown that Banach spaces of stable type 2 provide a natural environment for the central limit theorem with index $p = 2$. In this paper we show that, for $0 < p < 2$, quasi-normed linear spaces of stable type p provide a natural environment for the central limit theorem with index p. A similar result holds also for the weak law of large numbers with index p.
Canonical subgroups of formal groups
Jonathan
Lubin
103-127
Abstract: Let R be a complete local domain of mixed characteristic. This paper gives a complete answer to the question: ``If F is a one-dimensional formal group over R of finite height, when is there a canonical morphism
On $3$-manifolds that have finite fundamental group and contain Klein bottles
J. H.
Rubinstein
129-137
Abstract: The closed irreducible 3-manifolds with finite fundamental group and containing an embedded Klein bottle can be identified with certain Seifert fibre spaces. We calculate the isotopy classes of homeomorphisms of such 3-manifolds. Also we prove that a free involution acting on a manifold of this type, gives as quotient either a lens space or a manifold in this class. As a corollary it follows that a free action of ${Z_8}$ or a generalized quaternionic group on $ {S^3}$ is equivalent to an orthogonal action.
The transfer and compact Lie groups
Mark
Feshbach
139-169
Abstract: Let G be a compact Lie group with H and K arbitrary closed subgroups. Let BG, BH, BK be l-universal classifying spaces, with $\rho (H,G):BH \to BG$ the natural projection. Then transfer homomorphisms $T(H,G):h(BH) \to h(BG)$ are defined for h an arbitrary cohomology theory. One of the basic properties of the transfer for finite coverings is a double coset formula. This paper proves a double coset theorem in the above more general context, expressing ${\rho ^{\ast}}(K,G) \circ T(H,G)$ as a sum of other compositions. The main theorems were announced in the Bulletin of the American Mathematical Society in May 1977.
On parabolic measures and subparabolic functions
Jang Mei G.
Wu
171-185
Abstract: Let D be a domain in $ R_x^n \, \times \, R_t^1$ and ${\partial _p}D$ be the parabolic boundary of D. Suppose ${\partial _p}D$ is composed of two parts B and S: B is given locally by $t = \tau$ and S is given locally by the graph of ${x_n} = f({x_1},{x_2}, \cdots ,{x_{n - 1}},t)$ where f is Lip 1 with respect to the local space variables and Lip $ \tfrac{1} {2}$ with respect to the universal time variable. Let $\sigma$ be the n-dimensional Hausdorff measure in $ {R^{n + 1}}$ and $ \sigma '$ be the $ (n - 1)$-dimensional Hausdorff measure in $ {\textbf{R}^n}$. And let $E \subseteq {\partial _p}D$. We study (i) the relation between the parabolic measure on ${\partial _p}D$ and the measure dm on ${\partial _p}D$ and (ii) the boundary behavior of subparabolic functions on D.
Complete characterization of functions which act, via superposition, on Sobolev spaces
Moshe
Marcus;
Victor J.
Mizel
187-218
Abstract: Given a domain $\Omega \subset {R_N}$ and a Borel function $h:\,{R_m} \to R$, conditions on h are sought ensuring that for every m-tuple of functions ${u_i}$ belonging to the first order Sobolev space ${W^{1,p}}(\Omega )$, the function $ h({u_1}( \cdot ), \ldots ,{u_m}( \cdot ))$ will belong to a first order Sobolev space $ {W^{1,r}}(\Omega )$, $1 \leqslant r \leqslant p < \infty$.In this paper conditions are found which are both necessary and sufficient in order that h have the above property. This result is based on a characterization obtained here for those Borel functions $g:\,{R_m} \times {({R_N})_m} \to R$ satisfying the requirement that for every m-tuple of functions $ {u_i} \in {W^{1,p}}(\Omega )$ the function $g({u_1}( \cdot ), \ldots ,{u_m}( \cdot ),\nabla {u_1}( \cdot ), \ldots ,\nabla {u_m}( \cdot ))$ belongs to ${L^r}(\Omega )$. A needed result on the measurability of the set of ${R_k}$-Lebesgue points of a function on $ {R_N}$ is presented in an appendix.
The Littlewood-Paley theory for Jacobi expansions
William C.
Connett;
Alan L.
Schwartz
219-234
Abstract: The machinery for harmonic analysis utilizing Jacobi polynomial expansions is developed using the explicit form of the convolution kernel discovered by Gasper. Various maximal functions, and the standard Littlewood-Paley functionals are studied and an application is given to multiplier theorems.
Adjacent connected sums and torus actions
Dennis
McGavran
235-254
Abstract: Let M and N be closed, compact manifolds of dimension m and let X be a closed manifold of dimension $ n < m$ with embeddings of $ X\, \times \,{D^{m - n}}$ into M and N. Suppose the interior of $X\, \times \,{D^{m - n}}$ is removed from M and N and the resulting manifolds are attached via a homeomorphism $f:\,X \times \,{S^{m - n - 1}}\, \to \,X\, \times \,{S^{m - n - 1}}$. Let this homeomorphism be of the form $ f(x,\,t)\, = \,(x,\,F(x)(t))$ where $ F:\,X \to \,SO(m - n)$. The resulting manifold, written as $M\,{\char93 _X}\,N$, is called the adjacent connected sum of M and N along X. In this paper definitions and examples are given and the examples are then used to classify actions of the torus ${T^n}$ on closed, compact, connected, simply connected $(n\, + \,2)$-manifolds, $n \geqslant \,4$.
Sweedler's two-cocycles and generalizations of theorems on Amitsur cohomology
Dave
Riffelmacher
255-265
Abstract: For any (not necessarily commutative) algebra C over a commutative ring k Sweedler defined a cohomology set, denoted here by $ {\mathcal{H}^2}(C/k)$, which generalizes Amitsur's second cohomology group ${H^2}(C/k)$. In this paper, if I is a nilpotent ideal of C and $\bar C\, \equiv \,C/I$ is K-projective, a natural bijection $ {\mathcal{H}^2}(C/k)\tilde \to {\mathcal{H}^2}(\bar C{\text{/}}k)$ is established. Also, when $k \subset B$ are fields and C is a commutative B-algebra, the sequence $\{ 1\} \to {H^2}(B{\text{/}}k)\xrightarrow{{{l^{\ast}}}}{H^2}(C/k)\xrightarrow{r}{H^2}(C/B)$ is shown to be exact if the natural map $C{ \otimes _k}C \to C{ \otimes _B}C$ induces a surjection on units, $ {l^ {\ast} }$ is induced by the inclusion, and r is the ``restriction'' map.
The Albanese mapping for a punctual Hilbert scheme. I. Irreducibility of the fibers
Mark E.
Huibregtse
267-285
Abstract: Let $f:\,X \to A$ be the canonical mapping from an algebraic surface X to its Albanese variety A, $X(n)$ the n-fold symmetric product of X, and $H_X^n$ the punctual Hilbert scheme parameterizing 0-dimensional closed subschemes of length n on X. The latter is a nonsingular and irreducible variety of dimension $2n$, and the ``Hilbert-Chow'' morphism $ {\sigma _n}:\,H_X^n \to X(n)$ is a birational map which desingularizes $ X(n)$. This paper studies the composite morphism $\displaystyle {\varphi _n}:\,H_X^n\xrightarrow{{{\sigma _n}}}X(n)\xrightarrow{{{f_n}}}A ,$ where ${f_n}$ is obtained from f by addition on A. The main result (Part 1 of the paper) is that for $n \gg 0$, all the fibers of ${\varphi _n}$ are irreducible and of dimension $ 2n - q$, where $q = \dim A$. An interesting special case (Part 2 of the paper) arises when $X = A$ is an abelian surface; in this case we show (for example) that the fibers of ${\varphi _n}$ are nonsingular, provided n is prime to the characteristic.
Wall manifolds
R. E.
Stong
287-298
Abstract: In the calculation of the oriented cobordism ring, it is standard to consider so-called Wall manifolds, for which the first Stiefel-Whitney class is the reduction of an integral class. This paper studies the Wall-type structures in the equivariant case.
A pointwise ergodic theorem for the group of rational rotations
Lester E.
Dubins;
Jim
Pitman
299-308
Abstract: Let f be a bounded, measurable function defined on the multiplicative group $\Omega$ of complex numbers of absolute value 1, and define $\displaystyle {{f_n}(\omega ) = \frac{1} {n}\sum\limits_{i = 1}^n {f(z_n^i\omega )} ,} \qquad \omega \in \Omega ,$ ($(1)$) where ${z_n}$ is a primitive nth root of unity. The present paper generalizes this result of Jessen [1934]: if $n(k)$ is an increasing sequence of positive integers with $n(k)$ dividing $n(k')$ whenever $k < k'$, then $ {f_{n(k)}}$ converges almost surely as $ k \to \infty$.
An axiom for nonseparable Borel theory
William G.
Fleissner
309-328
Abstract: Kuratowski asked whether the Lebesgue-Hausdorff theorem held for metrizable spaces. A. Stone asked whether a Borel isomorphism between metrizable spaces must be a generalized homeomorphism. The existence of a Q set refutes the generalized Lebesgue-Hausdorff theorem. In this paper we discuss the consequences of the axiom of the title, among which are ``yes'' answers to both Kuratowski's and Stone's questions. The axiom states that a point finite analytic additive family is $\sigma$ discretely decomposable. We show that this axiom is valid in the model constructed by collapsing a supercompact cardinal to ${\omega _2}$ using Lévy forcing. Our proof displays relationships between $\sigma$ discretely decomposable families, analytic additive families and d families.
Projective geometries as projective modular lattices
Ralph
Freese
329-342
Abstract: It is shown that the lattice of subspaces of a finite dimensional vector space over a finite prime field is projective in the class of modular lattices provided the dimension is at least 4.
On a sufficient condition for proximity
Ka Sing
Lau
343-356
Abstract: A closed subspace M in a Banach space X is called U-proximinal if it satisfies: $(1 + \rho )S \cap (S + M) \subseteq S + \varepsilon (\rho )(S \cap M)$, for some positive valued function $\varepsilon (\rho )$, $\rho > 0$, and $\varepsilon (\rho ) \to 0$ as $\rho\, \to\, 0$, where S is the closed unit ball of X. One of the important properties of this class of subspaces is that the metric projections are continuous. We show that many interesting subspaces are U-proximinal, for example, the subspaces with the 2-ball property (semi M-ideals) and certain subspaces of compact operators in the spaces of bounded linear operators.
On Castelnuovo's inequality for algebraic curves. I
Robert D. M.
Accola
357-373
Abstract: Let ${W_p}$ be a Riemann surface of genus p admitting a simple linear series $g_n^r$ where $ n\, =\, m(r - 1)\, +\, q,\,\, q\, =\, 2,\,3,...,\,r - 1$, or r. Castelnuovo's inequality states that (1) $2p\, \leqslant\, 2f(r,n,1)\, =\, m(m - 1)(r - 1)\, +\, 2m(q - 1)$. By further work of Castelnuovo, equality in (1) and $q\, <\, r$ implies that ${W_p}$ admits a plane model of degree $n\, -\, r\, +\, 2$ with $r\, -\, 2$ m-fold singularities and one $ (n\, -\, r\, +\, 1\, -\, m)$-fold singularity. Formula (1) generalizes as follows. Suppose ${W_p}$ admits s simple linear series $ g_n^r$ where $n\, =\, m(rs - 1)\, +\, q$ and $ q = - (s - 1)r + 2,\, - (s - 1)r + 3,\ldots,r - 1$, or r. For q consider the cases $ v = 0,1,\ldots,s - 1$ as follows: case $v\, =\, 0:2 \leqslant q \leqslant r$, case $ v > 0:2 \leqslant q + vr \leqslant r + 1$. Then (2) $2p \,\leqslant\, 2f(r,\,n,\,s)\, =\, {m^2}(r{s^2}\, -\, s) \,+\, ms(2q \,-\, 1\, -\, r)\, -\, v (v \, -\, 1)r\, -\, 2v (q \,-\, 1)$. Examples show that (2) is sharp. Finally, if $n\, = \,m'r\, + \,q'$,
Jordan rings with nonzero socle
J. Marshall
Osborn;
M. L.
Racine
375-387
Abstract: Let $\mathcal{J}$ be a nondegenerate Jordan algebra over a commutative associative ring $ \Phi$ containing $\tfrac{1}{2}$. Defining the socle $\mathcal{G}$ of $ \mathcal{J}$ to be the sum of all minimal inner ideals of $\mathcal{J}$, we prove that $\mathcal{G}$ is the direct sum of simple ideals of $\mathcal{J}$. Our main result is that if $\mathcal{J}$ is prime with nonzero socle, then either (i) $ \mathcal{J}$ is simple unital and satisfies DCC on principal inner ideals, (ii) $\mathcal{J}$ is isomorphic to a Jordan subalgebra $ \mathcal{J}$ is isomorphic to a Jordan subalgebra $ \mathcal{J}''$ of the Jordan algebra of all symmetric elements H of a. primitive associative algebra A with nonzero socle S, and $ \mathcal{J}''$ contains $H\, \cap \,S$. Conversely, any algebra of type (i), (ii), or (iii) is a prime Jordan algebra with nonzero socle. We also prove that if $\mathcal{J}$ is simple then $\mathcal{J}$ contains a completely primitive idempotent if and only if either $ \mathcal{J}$ is unital and satisfies DCC on principal inner ideals or $\mathcal{J}$ is isomorphic to the Jordan algebra of symmetric elements of a $*$-simple associative algebra A with involution $*$ containing a minimal one-sided ideal.
On invariant operator ranges
E.
Nordgren;
M.
Radjabalipour;
H.
Radjavi;
P.
Rosenthal
389-398
Abstract: A matricial representation is given for the algebra of operators leaving a given dense operator range invariant. It is shown that every operator on an infinite-dimensional Hilbert space has an uncountable family of invariant operator ranges, any two of which intersect only in 0.
Coxeter functors without diagrams
Maurice
Auslander;
María Inés
Platzeck;
Idun
Reiten
1-46
Abstract: The first part of this paper is devoted to a generalization of the notion of partial Coexeter functor from diagrams to certain types of artin rings and artin algebras. The rest of the paper is devoted to a discussion of the connection between the various Coxeter functors which exist for diagrams as well as for artin rings and artin algebras.
Modeloids. I
Miroslav
Benda
47-90
Abstract: If A is a set and  is the collection of finite nonrepeating sequences of its elements then a modeloid E on A is an equivalence relation on  which preserves length, is hereditary, and is invariant under the action of permutations. The pivotal operation on modeloids is the derivative. The theory of this operation turns out to be very rich with connections leading to diverse branches of mathematics. For example, in §3 we associate an action space with a modeloid and in §5 we characterize the action spaces which are associated with the basic modeloids, i.e., those which are derivatives of themselves. What emerges is a kind of stability for the action space. We then show that action spaces with this stability can be approximated by finite actions and, subject to certain requirements, this approximation is unique (see Proposition 5.7). Algebraically, the countable basic modeloids correspond to closed subgroups of the symmetric groups. This and the study of automorphisms of modeloids let us show, without any algebra, that the only nontrivial normal subgroups of the finite ( $ \left( { \geqslant \,\,5} \right)$) symmetric groups are the alternating groups. The last section gives, hopefully, credence to the thesis that the essence of model theory is the study of modeloids.
Ultrafiltres \`a la fa\c con de Ramsey
Maryvonne
Daguenet-Teissier
91-120
Abstract: Let $\beta {\text{N}}$ be the set of ultrafilters on N; $ {\mathcal{u}}\, \in \,\beta {\text{N}}$ is ``absolu'' [6] (Ramsey [4]) if all its free images by continuous maps $ \beta {\text{N}}\, \to \,\beta {\text{N}}$ are isomorphic. We study here a weaker Ramsey-like property, which implies the existence of fiber products $\mathcal{D}\,\, \otimes {\,_E}\,\mathcal{D}\,\left( { \otimes _\textbf{E}^k\,\mathcal{D}} \right)$ extending the usual product ${\mathcal{D}}\, \otimes \,{\mathcal{D}}\,\left( {{ \otimes ^{k\,}}{\mathcal{D}}} \right)$. This can be translated in the language of model-theory on the one hand as the existence of repeated almagamated sums and on the other hand by some properties of sets of indiscernibles associated with ultrafilters having this property (§5). We show that the class of ultrafilters we study strictly contains the class of Ramsey ultrafilters (§1) and is (§2) strictly (§3) contained in the class of p-point ultrafilters [9] ("$\delta$-stables'' [6]) and contains the free images of its elements (§4). In §2 we also give a characterization of p-point ultrafilters in terms of the product ${ \otimes ^k}\,{\mathcal{D}}$. In §3 we show the link with weakly Ramsey ultrafilters of Blass [3] and more generally we study ultrafilters $ {\mathcal{D}}$ on N having only a finite number $i\left( {\mathcal{D}} \right)$ of free images up to isomorphism and such that $\char93 \,\tau {\,^{ - \,1}}\left( {{\mathcal{D}},\,{\mathcal{D}}} \right)\, = \,2i\left( {\mathcal{D}} \right)\, + \,1$, where $\char93 {\tau ^{ - 1}}\left( {{\mathcal{D}},{\mathcal{D}}} \right)$ is the number of ultrafilters on $ {{\text{N}}^2}$ finer than the filter generated by $\left( {D\, \times \,D} \right)$ with $D\, \in \,{\mathcal{D}}$.
Isotoping mappings to open mappings
John J.
Walsh
121-145
Abstract: Let f be a quasi-monotone mapping from a compact, connected manifold $ {M^m}\,(m\, \geqslant \,3)$ onto a space Y; then there is an open mapping g from M onto Y such that, for each $ y\, \in \,Y,\,{g^{ - 1}}(y)$ is not a point and $ {g^{ - 1}}(y)$ and ${f^{ - 1}}(y)$ are equivalently embedded in M (in particular, $ {g^{ - 1}}(y)$ and ${f^{ - 1}}(y)$ have the same shape). Applying the result with f equal to the identity mapping on M yields a continuous decomposition of M into cellular sets each of which is not a point.
Zeeman's filtration of homology
Clint
McCrory
147-166
Abstract: Geometric interpretations of Zeeman's filtrations of the homology and cohomology of a triangulable space are given, using an analysis of his spectral sequence for Poincaré duality.
A nonunitary pairing of polarizations for the Kepler problem
J. H.
Rawnsley
167-180
Abstract: The half-form pairing of two polarizations of the Kepler manifold is found and shown to define a bounded linear isomorphism of the two Hilbert spaces, but is not unitary.
A $3$-local characterization of $L\sb{7}(2)$
Larry
Finkelstein;
Daniel
Frohardt
181-194
Abstract: Recent work of Gorenstein and Lyons on finite simple groups has led to standard form problems for odd primes. The present paper classifies certain simple groups which have a standard 3-component of type ${L_5}\left( 2 \right)$.
Starlike, convex, close-to-convex, spiral-like, and $\Phi $-like maps in a commutative Banach algebra with identity
L. F.
Heath;
T. J.
Suffridge
195-212
Abstract: Let C(X) be the space of continuous functions on a compact $ {T_2}$-space X where each point of X is a ${G_\delta }$. If $F:\,B\, \to \,C\,(X)$ is a biholomorphic (in the sense that F and $ {F^{ - 1}}$ are Fréchet differentiable) map of $ B\, = \,\{ \,f\left\vert {\,\left\Vert f \right\Vert} \right.\, < \,1\}$ onto a convex domain with $ DF(0)\, = \,I$, then F is Lorch analytic (i.e., $DF\,(f)(g)\, = \,{a_f}g $ for some ${a_{f}} \, \in \,C\,(X))$). Let R be a commutative Banach algebra with identity such that the Gelfand homomorphism of R into $C(\mathcal{m})$ is an isometry. Starlike, convex, close-to-convex, spirallike and $\Phi$-like functions are defined in $B\, = \,\{ x\, \in \,R\,\left\vert {\,\left\Vert x \right\Vert} \right.\, < \,1\}$ for L-analytic functions in B and they are related to associated complex-valued holomorphic functions in $\Delta \, = \,\{ z\, \in \,\left. {\textbf{C}} \right\vert\,\,\left\vert z \right\vert\, < \,1\}$.
Defining Lagrangian immersions by phase functions
J. Alexander
Lees
213-222
Abstract: In order to analyze the singularities of the solutions of certain partial differential equations, Hörmander, in his paper on Fourier integral operators, extends the method of stationary phase by introducing the class of nondegenerate phase functions. Each phase function, in turn, defines a lagrangian submanifold of the cotangent bundle of the manifold which is the domain of the corresponding differential operator. Given a lagrangian submanifold of a cotangent bundle, when is it globally defined by a nondegenerate phase function? A necessary and sufficient condition is here found to be the vanishing of two topological obstructions; one in the cohomology and the other in the k-theory of the given lagrangian submanifold.
Borel parametrizations
R. Daniel
Mauldin
223-234
Abstract: Let X and Y be uncountable Polish spaces and B a Borel subset oi $ X\, \times \,Y$ such that for each x, ${B_x}$ is uncountable. A Borel parametrization of B is a Borel isomorphism, g, of $X\, \times \,E$ onto B where E is a Borel subset of Y such that for each x, $ g\left( {x,\, \cdot } \right)$ maps E onto $ {B_x}\, = \,\left\{ {y:\,\left( {x,\,y} \right)\, \in \,B} \right\}$. It is shown that B has a Borel parametrization if and only if B contains a Borel set M such that for each x, ${M_x}$ is a nonempty compact perfect set, or, equivalently, there is an atomless conditional probability distribution, $\mu$, so that for each x, $\mu \left( {x,\,{B_x}} \right)\, > \, 0$. It is also shown that if Y is dense-in-itself and ${B_x}$ is not meager, for each x, then B has a Borel parametrization.
Control problems governed by a pseudo-parabolic partial differential equation
Luther W.
White
235-246
Abstract: Let G be a bounded domain in ${R^n}$ and $Q\, = \,G\, \times \,\left( {0,\,T} \right)$. We consider the solution $ y\left( u \right)$ of the pseudo-parabolic initial-value problem \begin{multline}\left( {1\, + \,M\left( x \right)} \right)\,{y_t}\,\left( u \rig... ... \right)\, = \,0\,{\text{in}}\,{L^2}\,\left( G \right), \end{multline} , to be the state corresponding to the control u. Here $M\left( x \right)$ and $L\left( x \right)$ are symmetric uniformly strongly elliptic second-order partial differential operators. The control problem is to find a control $ {u_0}$ in a fixed ball in ${L^2}\left( Q \right)$ such that (i) the endpoint of the corresponding state $ y\left( { \cdot ,\,T;\,{u_0}} \right)$ lies in a given neighborhood of a target Z in $ {L^2}\left( G \right)$ and (ii) ${u_0}$ minimizes a certain energy functional. In this paper we establish results concerning the controllability of the states and the compatibility of the constraints, existence and uniqueness of the optimal control, existence and properties of Lagrange multipliers associated with the constraints, and regularity properties of the optimal control.
Weak cuts of combinatorial geometries
Hien Q.
Nguyen
247-262
Abstract: A weak cut of a Combinatorial Geometry G is a generalization of a modular cut, corresponding to the family of the new dependent sets in a weak map image of G. The use of weak cuts allows the construction of all weak images of G, an important result being that, to any family $ {\mathcal{M}}$ of independent sets of G, is associated a unique weak cut ${\mathcal{C}}$ containing ${\mathcal{M}}$. In practice, the flats of the weak image defined by $ {\mathcal{C}}$ can be constructed directly. The weak cuts corresponding to known weak maps, such as truncation, projection, elementary quotient, are determined. The notion of weak cut is particularly useful in the study of erections. Given a geometry F and a weak image G, an F-erection of G is an erection of G which is a weak image of F. The main results are that the set of all F-erections of G is a lattice with the weak map order, and that the free F-erection can be constructed explicitly. Finally, a problem involving higher order erection is solved.
Space curves that point almost everywhere
J. B.
Wilker
263-274
Abstract: We construct a simple, closed, continuously differentiable curve $r:\,[0,\,1]\, \to \,{E^d}\,(d\, \geqslant \,3)$ whose tangent vector never points twice in the same direction of $ {S^{d\, - \,1}}$ yet sweeps out a set of directions equal to almost all of ${S^{d\, - \,1}}$.
A converse of the Borel formula
Ronald M.
Dotzel
275-287
Abstract: When an elementary Abelian p-group acts on a $ {Z_p}$-homology sphere (p a prime), it is known that the Borel formula must hold. Here we ask that the Borel formula hold and determine how this restricts, homologically, the type of space which can occur, assuming spherical fixed sets and connectedness. This is done by constructing a linear model of the action and an equivariant map to the model, the mapping cone of which yields certain homological information.
The first order theory of $N$-colorable graphs
William H.
Wheeler
289-310
Abstract: Every N-colorable graph without loops or multiple edges is a substructure of a direct power of a particular, finite, N-coloarable graph. Consequently, the class of N-colorable graphs without loops or endpoints can be recursively axiomatized by a first order, universal Horn theory. This theory has a model-companion which has a primitive recursive elimination of quantifiers and is decidable, complete, ${\aleph _0}$-categorical, and independent. The N-colorable graphs without loops or multiple edges which have a proper, prime model extension for the model-companion are precisely the finite, amalgamation bases.
Deforming twist-spun knots
R. A.
Litherland
311-331
Abstract: In [15] Zeeman introduced the process of twist-spinning an n-knot to obtain an (n + l)-knot, and proved the remarkable theorem that a twist-spun knot is fibred. In [2] Fox described another deformation which can be applied during the spinning process, and which he called rolling. We show that, provided one combines the rolling with a twist, the resulting knot is again fibred. In fact, this result holds for a larger class of deformations, defined below.
CR submanifolds of a Kaehler manifold. II
Aurel
Bejancu
333-345
Abstract: The differential geometry of CR submanifolds of a Kaehler manifold is studied. Theorems on parallel normal sections and on a special type of flatness of the normal connection on a CR submanifold are obtained. Also, the nonexistence of totally umbilical proper CR submanifolds in an elliptic or hyperbolic complex space is proven.
An algebraic characterization of connected sum factors of closed $3$-manifolds
W. H.
Row
347-356
Abstract: Let M and N be closed connected 3-manifolds. A knot group of M is the fundamental group of the complement of a tame simple closed curve in M. Denote the set of knot groups of M by K(M). A knot group G of M is realized in N if G is the fundamental group of a compact submanifold of N with connected boundary. Theorem. Every knot group of N is realized in M iff N is a connected sum factor of M. Corollary 1. $ K\,(M)\, = \,K\,(N)$ iff M is homeomorphic to N. Given M, there exists a knot group ${G_M}$ of M that serves to characterize M in the following sense. Corollary 2. $ {G_M}$ is realized in N and ${G_N}$, is realized in M iff M is homeomorphic to N. Our proof depends heavily on the work of Bing, Feustal, Haken, and Waldhausen in the 1960s and early 1970s. A. C. Conner announced Corollary 1 for orientable 3-manifolds in 1969 which Jaco and Myers have recently obtained using different techniques.
Nonstandard measure theory: avoiding pathological sets
Frank
Wattenberg
357-368
Abstract: The main results in this paper concern representing Lebesgue measure by nonstandard measures which avoid certain pathological sets. An (external) set E is S-thin if InfmA|A standard,* $A\, \supseteq \,E$ = 0 and Q-thin if Inf*mA|A internal, $A\, \supseteq \,E$ = 0. It is shown that any *finite sample which represents Lebesgue measure avoids every S-thin set and that given any Q-thin set E there is a *finite sample avoiding E which represents Lebesgue measure. In the last part of the paper a particular pathological set ${\mathcal{H}}\,\, \subseteq \, * \left[ {0,\,1} \right]$ is constructed which is important for the study of approximate limits, derivatives etc. It is shown that every *finite sample which represents Lebesgue measure assigns inner measure zero and outer measure one to this set and that Loeb measure does the same. Finally, it is shown that Loeb measure can be extended to a $ \sigma$-algebra including ${\mathcal{H}}$ in such a way that ${\mathcal{H}}$ is assigned zero measure.
Orthogonal polynomials defined by a recurrence relation
Paul G.
Nevai
369-384
Abstract: R. Askey has conjectured that if a system of orthogonal polynomials is defined by the three term recurrence relation $\displaystyle x{p_{n\, - \,1}}\left( x \right)\, = \,\frac{{{\gamma _{n\, - \,1... ...amma _{n\, - \,2}}}} {{{\gamma _{n\, - \,1}}}}\,{p_{n\, - \,2}}\left( x \right)$ and $\displaystyle {\alpha _n}\, = \,\frac{{{{( - 1)}^n}}} {n}\,{\text{const}}\,{\text{ + }}O\left( {\frac{1} {{{n^2}}}} \right),$ $\displaystyle \frac{{{\gamma _n}}} {{{\gamma _{n + 1}}}}\, = \,\frac{1} {2}\, +... ...1)}^n}}} {n}\,{\text{const}}\,{\text{ + }}O\left( {\frac{1} {{{n^2}}}} \right),$ then the logarithm of the absolutely continuous portion of the corresponding weight function is integrable. The purpose of this paper is to prove R. Askey's conjecture and solve related problems.
A simultaneous lifting theorem for block diagonal operators
G. D.
Allen;
J. D.
Ward
385-397
Abstract: Stampfli has shown that for a given $T\, \in \,B\left( H \right)$ there exists a $K\, \in \,C\left( H \right)$ so that $\sigma \left( {T\, + \, K} \right)\,= \,{\sigma _w}\left( T \right)$. An analogous result holds for the essential numerical range ${W_e}\left( T \right)$. A compact operator K is said to preserve the Weyl spectrum and essential numerical range of an operator $T\, \in \,B\left( H \right)$ if $\sigma \left( {T\, + \, K} \right)\, = \,{\sigma _w}\left( T \right)$ and $\overline {W\left( {T \, + \, K} \right)} \, = \,{W_e}\left( T \right)$. Theorem. For each block diagonal operator T, there exists a compact operator K which preserves the Weyl spectrum and essential numerical range of T. The perturbed operator $T \, + \, K$ is not, in general, block diagonal. An example is given of a block diagonal operator T for which there can be no block diagonal perturbation which preserves the Weyl spectrum and essential numerical range of T.
Erratum to: ``Pullback de Rham cohomology of the free path fibrations'' [Trans. Amer. Math. Soc. {\bf 242} (1978), 307--318; MR 57:17678]
Kuo Tsai
Chen
398-398
Noncollision singularities in the four-body problem
Robert Orrin
Shelton
225-259
Abstract: It is shown that if there is a singularity in a solution of the four-body problem which is not a collision then the motion of the bodies near the singularity is nearly one-dimensional. This is established by grouping the bodies into natural clusters and showing the angular momentum of each cluster with respect to its center of mass tends to zero near the singularity. This is related to Sperling's proof of von Zeipel's theorem.
The Bergman norm and the Szeg\H{o} norm
Saburou
Saitoh
261-279
Abstract: Let G denote an arbitrary bounded regular region in the plane and ${H_2}\left( G \right)$ the analytic Hardy class on G with index 2. We show that the generalized isoperimetric inequality \begin{multline}\frac{1}{\pi }\,\iint\limits_G {{{\left\vert {\varphi \left( z \... ...t}^{2}}\,\left\vert dz \right\vert}\,\,\,\,\,\,\,(z\,=\,x\,+\,iy) \end{multline} holds for any $\varphi$ and $\psi \, \in \,{H_2}\left( G \right)$. We also determine necessary and sufficient conditions for equality.
Fourier inversion of invariant integrals on semisimple real Lie groups
Rebecca A.
Herb
281-302
Abstract: Let G be a connected, semisimple real Lie group with finite center. Associated with every regular semisimple element g of G is a tempered invariant distribution ${ \Lambda _g}$ given by an orbital integral. This paper gives an inductive formula for computing the Fourier transform of ${ \Lambda _g}$ in terms of the space of tempered invariant eigendistributions of G.
Extensions, restrictions, and representations of states on $C\sp{\ast} $-algebras
Joel
Anderson
303-329
Abstract: In the first three sections the question of when a pure state g on a $ {C^{\ast}}$-subalgebra B of a $ {C^{\ast}}$-algebra A has a unique state extension is studied. It is shown that an extension f is unique if and only if inf $ \left\Vert {b\left( {a\, - \,f\left( a \right)1} \right)b} \right\Vert\, = \,0$ for each a in A, where the inf is taken over those b in B such that $ 0\, \leqslant \,b\, \leqslant \,1$ and $g(b) = 1$. The special cases where B is maximal abelian and/or $A\, = \,B\left( H \right)$ are treated in more detail. In the remaining sections states of the form $T \mapsto \mathop {\lim }\limits_{\mathcal{u}} \left( {T{x_\alpha },\,{x_\alpha }} \right)$, where $\left\{ {{x_\alpha }} \right\}{\,_{\alpha \, \in \,\kappa }}$ is a set of unit vectors in H and $ {\mathcal{u}}$ is an ultrafilter are studied.
The equivalence of $\times \sp{t}C\approx \times \sp{t}D$ and $J\times C\approx J\times D$
Ronald
Hirshon
331-340
Abstract: Let C satisfy the maximal condition for normal subgroups and let $\times {\,^t}C\, \approx \, \times {\,^t}D$ for some positive integer t. Then $C\, \times \,J\, \approx \,D\, \times \,J$ where J is the infinite cyclic group. If $\times {\,^s}C\, \approx \, \times {\,^t}D$ and $s \geqslant \,t$, there exists a finitely generated free abelian group S such that C is a direct factor of $ D\, \times \,S$.
Distribution of zeros of orthogonal polynomials
Paul G.
Nevai
341-361
Abstract: The purpose of the paper is to investigate distribution of zeros of orthogonal polynomials given by a three term recurrence relation.
Ramsey's theorem for spaces
Joel H.
Spencer
363-371
Abstract: A short proof is given of the following known result. For all k, r, t there exists n so that if the t-spaces of an n-space are r-colored there exists a k-space all of whose t-spaces are the same color. Here t-space refers initially to a t-dimensional affine space over a fixed finite field. The result is also shown for a more general notion of t-space.
Metacompactness, paracompactness, and interior-preserving open covers
Heikki J. K.
Junnila
373-385
Abstract: In this paper metacompactness and paracompactness are characterized in terms of the existence of closure-preserving closed refinements and interior-preserving open star-refinements of interior-preserving directed open covers of a topological space. Several earlier results on metacompact spaces and paracompact spaces are obtained as corollaries to these characterizations. For a Tychonoff-space X, metacompactness of X is characterized in terms of orthocompactness of $ X\, \times \,\beta X$.
Maximal subspaces of Besov spaces invariant under multiplication by characters
R.
Johnson
387-407
Abstract: Unlike the familiar $ {L^p}$ spaces, neither the homogeneous Besov spaces nor the ${H^p}$ spaces, $0\, < \,p\, < \,\,1$, are closed under multiplication by the functions $ x\, \to \,{e^{i\left\langle {x,h} \right\rangle }}$. We determine the maximal subspace of these spaces which are closed under multiplication by these functions, which are the characters of $ {R^n}$.
The behavior of the support of solutions of the equation of nonlinear heat conduction with absorption in one dimension
Barry F.
Knerr
409-424
Abstract: We consider the Cauchy problem in one space dimension for a nonlinear degenerate parabolic partial differential equation. The connectedness of the support of the solution and estimates of the growth of the support as $t \,\to \,\infty$ are established.
The module of indecomposables for mod $2$ finite $H$-spaces
Richard
Kane
425-433
Abstract: The module of indecomposables obtained from the mod 2 cohomology of a finite H-space is studied. It is shown that this module is trivial in dimensions $ \equiv \,0$ (mod 4).
Nash rings on planar domains
Gustave A.
Efroymson
435-445
Abstract: Let D be a semialgebraic domain in ${R^2}$. Let ${N_D}$ denote the Nash ring of algebraic analytic functions on D. Let ${A_D}$ denote the ring of analytic functions on D. The main theorem of this paper implies that if $\mathcal{B}$ is a prime ideal of ${N_D}$, then $\mathcal{B}{A_D}$ is also prime. This result is proved by considering $p\left( {x,\,y} \right)$ in $\textbf{R}[{x,\,y}]$ and showing that $p({x,\,y})$ can be put into a form so that its factorization in ${N_D}$ is given by looking at its local factorization as a polynomial in y with coefficients which are analytic functions of x. Then for more general domains, a construction using the ``complex square root'' enables one to reduce to the case already considered.
Erratum to: ``The ninety-one types of isogonal tilings in the plane'' (Trans. Amer. Math. Soc. {\bf 242} (1978), 335--353)
Branko
Grünbaum;
G. C.
Shephard
446-446
The Riemann problem in gas dynamics
Randolph G.
Smith
1-50
Abstract: We consider the Riemann problem (R.P.) for the $3\, \times \, 3$ system of gas dynamics equations in a single space variable. We assume that the specific internal energy $e = e(v,\,s)$ (s = specific entropy, v = specific volume) satisfies the usual hypotheses, $ {p_v}\, < \,0,\,{p_{vv}}\, > \,0,\,{p_s}\, > \,0\,(p\, = \, - \,{e_v}\, =$ pressure); we also assume some reasonable hypotheses about the asymptotic behavior of e. We call functions e satisfying these hypotheses energy functions Theorem 1. For any initial data $ ({U_l},\,{U_r})\,({U_l}\, = \,({v_l},\,{p_l},\,{u_l})$, ${U_r}\, = \,({v_r},\,{p_r},\,{u_r})$, u = flow velocity), the R. P. has a solution. We introduce two conditions: \begin{displaymath}\begin{array}{*{20}{c}}\tag{$\text{(I)}$} {\frac{\partial } {... ...ant \frac{{{p^2}}} {{2e}}} & {(v,\,e\, > \,0),} \end{array} \end{displaymath} \begin{displaymath}\begin{array}{*{20}{c}}\tag{$\text{(II)}$} {\frac{\partial }{... ... \geqslant - \frac{p} {2}} & {(v,\,p\, > \,0).} \end{array} \end{displaymath} Theorem 2. (I) is necessary and sufficient for uniqueness of solutions of the R. P. Nonuniqueness persists under small perturbations of the initial data. (I) is implied by the known condition $\displaystyle {\frac{\partial } {{\partial v}}e(v,p) > 0} \qquad (v,p > 0),$ ($(\ast)$) which holds for all usual gases. (I) implies (II). We construct energy functions e that violate (II), that satisfy (II) but violate (I), and that satisfy (I) but violate (*). In all solutions considered, the shocks satisfy the entropy condition and the Lax shock conditions.
Fourier inversion for unipotent invariant integrals
Dan
Barbasch
51-83
Abstract: Consider G a semisimple Lie group and $\Gamma\, \subseteq \,G$ a discrete subgroup such that ${\text{vol(}}G/\Gamma )\, < \,\infty$. An important problem for number theory and representation theory is to find the decomposition of ${L^2}(G/\Gamma )$ into irreducible representations. Some progress in this direction has been made by J. Arthur and G. Warner by using the Selberg trace formula, which expresses the trace of a subrepresentation of $ {L^2}(G/\Gamma )$ in terms of certain invariant distributions. In particular, measures supported on orbits of unipotent elements of G occur. In order to obtain information about representations it is necessary to expand these distributions into Fourier components using characters of irreducible unitary representations of G. This problem is solved in this paper for real rank $ G\, =\, 1$. In particular, a relationship between the semisimple orbits and the nilpotent ones is made explicit generalizing an earlier result of R. Rao.
Degrees of exceptional characters of certain finite groups
Harvey I.
Blau
85-96
Abstract: Let G be a finite group whose order is divisible by a prime p to the first power only. Restrictions beyond the known congruences modulo p are shown to hold for the degrees of the exceptional characters of G, under the assumptions that either all $ p'$-elements centralizing a Sylow p-subgroup are in fact central in G and there are at least three conjugacy classes of elements of order p, or that the characters in question lie in the principal p-block. Results of Feit and the author are thereby generalized.
On the existence of nonregular ultrafilters and the cardinality of ultrapowers
Menachem
Magidor
97-111
Abstract: Assuming the consistency of huge cardinals, we prove that ${\omega _3}$ can carry an ultrafilter D such that $ {\omega _1}^{{\omega _3}}/D$ has cardinality $ {\omega _3}$. (Hence D is not $({\omega _3},\,{\omega _1})$ regular.) Similarly ${\omega _2}$ can carry an ultrafilter D such that $ {\omega ^{{\omega _2}}}/D$ has cardinality $ {\omega _2}$. (Hence D is not $({\omega _2},\,\omega )$ regular.)
The commuting block maps problem
Ethan M.
Coven;
G. A.
Hedlund;
Frank
Rhodes
113-138
Abstract: A block map is a map $f:\,{\{ {\text{0}},\,{\text{1}}\} ^n}\, \to \,\{ 0,\,1\}$ for some $n\, \geqslant \,1$. A block map f induces an endomorphism $ {f_\infty }$ of the full 2-shift $ (X,\,\sigma )$. We define composition of block maps so that $ {(f \circ g)_\infty }\, = \,{f_\infty } \circ {g_\infty }$. The commuting block maps problem (for f) is to determine $\mathcal{C}(f)\, = \,\{ g\vert f \circ g\, = \,g \circ f\}$. We solve the commuting block maps problem for a number of classes of block maps.
Balanced Cohen-Macaulay complexes
Richard P.
Stanley
139-157
Abstract: A balanced complex of type $ ({a_1},\ldots,{a_m})$ is a finite pure simplicial complex $\Delta$ together with an ordered partition $ ({V_1},\ldots,{V_m})$ of the vertices of $\Delta$ such that card $({V_i}\, \cap \,F)\, = \,{a_i}$, for every maximal face F of $\Delta$. If ${\mathbf{b}}\, = \,({b_1},\ldots,{b_m})$, then define ${f_\textbf{b}}(\Delta )$ to be the number of $F\, \in \,\Delta$ satisfying card $({V_i}\, \cap \,F)\, = \,{b_i}$. The formal properties of the numbers ${f_\textbf{b}}(\Delta )$ are investigated in analogy to the f-vector of an arbitrary simplicial complex. For a special class of balanced complexes known as balanced Cohen-Macaulay complexes, simple techniques from commutative algebra lead to very strong conditions on the numbers ${f_\textbf{b}}(\Delta )$. For a certain complex $\Delta (P)$ coming from a poset P, our results are intimately related to properties of the Möbius function of P.
On the zeros of Jacobi polynomials $P\sb{n}\sp{(\alpha \sb{n},\beta \sb{n})}(x)$
D. S.
Moak;
E. B.
Saff;
R. S.
Varga
159-162
Abstract: If ${r_n}$ and ${s_n}$ denote, respectively, the smallest and largest zeros of the Jacobi polynomial $P_n^{({\alpha _n},{\beta _n})}$, where ${\alpha _n}\, > \,1$, ${\beta _n}\, - \,1$, and if ${\lim _{n \to \infty }}\,{\alpha _n}/(2n\, + \,{\alpha _n}\, + \,{\beta _n}\, + \,1)\, = \,a$ and if $ {\lim _{n \to \infty }}{\beta _n}/(2n\, + \,{\alpha _n}\, + \,{\beta _n}\, + \,1)\, = \,b$, then the numbers ${r_{a,b}}$ and ${s_{a,b}}$ are determined where $\displaystyle \mathop {\lim }\limits_{n \to \infty } \,{r_{n\,}}\, = \,{r_{a,b}},\mathop {\lim }\limits_{n \to \infty } \,{s_{n\,}}\, = \,{s_{a,b}}$ . Furthermore, the zeros of $\{ P_n^{({\alpha _n},{\beta _n})}(x)\} _{n = 0}^\infty$ are dense in $ [{r_{a,b}},{s_{a,b}}]$.
The sharpness of Lorentz's theorem on incomplete polynomials
E. B.
Saff;
R. S.
Varga
163-186
Abstract: For any fixed $ \theta$ with $0 < \theta < 1$, G. G. Lorentz recently showed that bounded sequences $\{\Sigma_{\theta {n_i} \leqslant k \leqslant {n_i}} {{a_k}(i){{(1 + t)}^k}\} _{i = 1}^\infty }$ of incomplete polynomials on $[ - 1, + 1]$ tend uniformly to zero on closed intervals of $[ - 1,\Delta (\theta ))$, where $2{\theta ^2} - 1 \leqslant \Delta (\theta ) < 2\theta - 1$. In this paper, we show that $\Delta (\theta ) = 2{\theta ^2} - 1$ is best possible, and that the geometric convergence to zero of such sequences on closed intervals $[{t_0},{t_1}]$ can be precisely bounded above as a function of ${t_j}$ and $\theta$. Extensions of these results to the complex plane are also included.
Focal points for a linear differential equation whose coefficients are of constant signs
Uri
Elias
187-202
Abstract: The differential equation considered is ${y^{(n)}} + \Sigma {{p_i}(x){y^{(i)}}} = 0$, where ${\sigma _i}{p_i}(x) \geqslant 0,i = 0,\ldots,n - 1,{\sigma _i} = \pm 1$. The focal point $ \zeta (a)$ is defined as the least value of s, $s > a$, such that there exists a nontrivial solution y which satisfies $ {y^{(i)}}(a) = 0,{\sigma _i}{\sigma _{i + 1}} > 0$ and ${y^{(i)}}(s) = 0$, ${\sigma _i}{\sigma _{i + 1}} < 0$. Our method is based on a characterization of $\zeta (a)$ by solutions which satisfy $ {\sigma _i}{y^{(i)}} > 0,i = 0,\ldots,n - 1$, on $[a,b]$, $ b < \zeta (a)$. We study the behavior of the function $\zeta$ and the dependence of $ \zeta (a)$ on ${p_0},\ldots,{p_{n - 1}}$ when at least a certain ${p_i}(x)$ does not vanish identically near a or near $\zeta (a)$. As an application we prove the existence of an eigenvalue of a related boundary value problem.
Cell-like $0$-dimensional decompositions of $E\sp{3}$
Michael
Starbird
203-215
Abstract: Let G be a cell-like, 0-dimensional upper semicontinuous decomposition of ${E^3}$. It is shown that if $\Gamma$ is a tame 1-complex which is a relatively closed subset of a saturated open set U whose boundary misses the nondegenerate elements of G, then there is a homeomorphism $h:{E^3} \to {E^3}$ so that $h\vert{E^3} - U = {\text{id}}$ and $h(\Gamma )$ misses the nondegenerate elements of G. This theorem implies a disjoint disk type criterion for shrinkability of G. This criterion in turn provides a direct proof of the recent result of Starbird and Woodruff that if G is an u.s.c. decomposition of ${E^3}$ into points and countably many cellular, tamely embedded polyhedra, then ${E^3}/G$ is homeomorphic to ${E^3}$.
Center-by-metabelian groups of prime exponent
Jay I.
Miller
217-224
Abstract: We show that a center-by-metabelian group of prime exponent p is nilpotent of class at most p, and this result is best possible. The proof is based on techniques dealing with varieties of groups.
Duality theory for covariant systems
Magnus B.
Landstad
223-267
Abstract: If $(A,\rho ,G)$ is a covariant system over a locally compact group G, i.e. $\rho$ is a homomorphism from G into the group of $^{\ast}$-automorphisms of an operator algebra A, there is a new operator algebra $\mathfrak{A}$ called the covariance algebra associated with $(A,\rho ,G)$. If A is a von Neumann algebra and $\rho$ is $\sigma$-weakly continuous, $\mathfrak{A}$ is defined such that it is a von Neumann algebra. If A is a $ {C^{\ast}}$-algebra and $ \rho$ is norm-continuous $\mathfrak{A}$ will be a $ {C^{\ast}}$-algebra. The following problems are studied in these two different settings: 1. If $ \mathfrak{A}$ is a covariance algebra, how do we recover A and $ \rho$? 2. When is an operator algebra $ \mathfrak{A}$ the covariance algebra for some covariant system over a given locally compact group G?
Sheaf constructions and their elementary properties
Stanley
Burris;
Heinrich
Werner
269-309
Abstract: We are interested in sheaf constructions in model-theory, so an attempt is made to unify and generalize the results to date, namely various forms of the Feferman-Vaught Theorem, positive decidability results, and constructions of model companions. The task is considerably simplified by introducing a new definition of sheaf constructions over Boolean spaces.
Some new constructions and estimates in the problem of least area
Harold
Parks
311-346
Abstract: Surfaces of least k dimensional area in ${\textbf{R}^n}$ are constructed by minimization of the n dimensional volume of suitably thickened sets subject to a homological constraint. Specifically, let $1 \,\, \leqslant \,\,k\,\, \leqslant \,n$ be integers and $ B\, \subset \,{\textbf{R}^n}$ be compact and $k\, - \,1$ rectifiable. Let G be a compact abelian group and L be a subgroup of the Čech homology group ${H_{k - 1}}\left( {B;\,\,G} \right)$ (in case $k = \,1$, suppose, additionally, L is contained in the kernel of the usual augmentation map). J. F. Adams has defined what it means for a compact set ${\rm X}\, \subset \,{\textbf{R}^n}$ to span L. Using also a natural notion of what it means for a compact set to be $ \varepsilon$-thick, we show that, for each $ \varepsilon \, > \,0$, there exists an $ \varepsilon$-thick set which minimizes n dimensional volume subject to the requirement that it span L. Our main result is that as $ \varepsilon$ approaches 0 a subsequence of the above volume minimizing sets converges in the Hausdorff distance topology to a set, X, which minimizes k dimensional area subject to the requirement that it span L. It follows, of course, from the regularity results of Reifenberg or Almgren that, except for a compact singular set of zero k dimensional measure, X is a real analytic minimal submanifold of ${\textbf{R}^n}$.
Subgroups of classical groups generated by long root elements
William M.
Kantor
347-379
Abstract: All conjugacy classes of subgroups G of classical groups of characteristic p are determined, which are generated by a conjugacy class of long root elements and satisfy
Nonselfadjoint crossed products (invariant subspaces and maximality)
Michael
McAsey;
Paul S.
Muhly;
Kichi-Suke
Saito
381-409
Abstract: Let $\mathcal{L}$ be the von Neumann algebra crossed product determined by a finite von Neumann algebra M and a trace preserving automorphism. In this paper we investigate the invariant subspace structure of the subalgebra $ {\mathcal{L}_ + }$ of $\mathcal{L}$ consisting of those operators whose spectrum with respect to the dual automorphism group on $\mathcal{L}$ is nonnegative, and we determine conditions under which ${\mathcal{L}_ + }$ is maximal among the $ \sigma$-weakly closed subalgebras of $ \mathcal{L}$. Our main result asserts that the following statements are equivalent: (1) M is a factor; (2) ${\mathcal{L}_ + }$ is a maximal $\sigma $-weakly closed subalgebra of $\mathcal{L}$; and (3) a version of the Beurling, Lax, Halmos theorem is valid for ${\mathcal{L}_ + }$. In addition, we prove that if $\mathfrak{A}$ is a subdiagonal algebra in a von Neumann algebra $ \mathcal{B}$ and if a form of the Beurling, Lax, Halmos theorem holds for $\mathfrak{A}$, then $ \mathcal{B}$ is isomorphic to a crossed product of the form $\mathcal{L}$ and $ \mathfrak{A}$ is isomorphic to ${\mathcal{L}_ + }$.
Commutativity in series of ordinals: a study of invariants
J. L.
Hickman
411-434
Abstract: It is well known that two ordinals are additively commutative if and only if they are finite multiples of some given ordinal, and it is very easy to extend this result to any finite sequence of ordinals. However, no necessary and sufficient conditions for the commutativity of a series of ordinals seem to be known when the length of that series is infinite, although sufficient conditions for certain cases have been given by Sierpiński and Ginsburg. In this paper we present such necessary and sufficient conditions. The general problem is split into five distinct cases: those in which the length of the series is a regular initial ordinal, a singular initial ordinal, an infinite, noninitial prime component, an infinite successor ordinal, and an infinite limit ordinal that is not a prime component. These are dealt with respectively in the second through to the sixth sections of the paper, and it turns out that in every case our criteria can be expressed in terms of an ordinal parameter, which is in fact an invariant of the series in question. This concept of invariance is introduced in the first section, which also contains several lemmas and a slight strengthening of the original Sierpiński-Ginsburg result. The final section of this paper differs from the preceding four sections in two aspects. Firstly, the proofs of its two main results are merely sketched, since they contain no arguments that have not previously appeared in some form or other. Secondly, we have not given any explicit determination of the ordinal parameter introduced in this section, since we felt that such a determination would prolong the paper intolerably and encroach upon work done by J. A. H. Anderson: we have therefore simply referred to Anderson's interesting paper.
A class of Schur algebras
M.
Brender
435-444
Abstract: This paper delineates a class of Schur algebras over a finite group G, parametrized by two subgroups $K\, \triangleleft \,H\, \subset \,G$. The constructed Schur algebra ${\text{C}}\left[ G \right]_K^H$ is maximal for the two properties (a) centralizing the elements of H, and (b) containing the elements of K in the identity. Most commonly considered examples of Schur algebras fall into this class. A complete set of characters of ${\text{C}}\left[ G \right]_K^H$ is given in terms of the spherical functions on the group G with respect to the subgroup H. Necessary and sufficient conditions are given for this Schur algebra to be commutative, in terms of a condition on restriction multiplicities of characters. This leads to a second-orthogonality-type relation among a subset of the spherical functions. Finally, as an application, a particular Schur algebra of this class is analyzed, and shown to be a direct sum of centralizer rings.
A general Stone-Gel'fand duality
J.
Lambek;
B. A.
Rattray
1-35
Abstract: We give a simple characterization of full subcategories of equational categories. If $ \mathcal{a}$ is one such and $\mathcal{B}$ is the category of topological spaces, we consider a pair of adjoint functors $ {\mathcal{a}^{op}}\underset{F}{\overset{U}{\longleftrightarrow}}\mathcal{B}$ which are represented by objects I and J in the sense that the underlying sets of $U(A)$ and $F(B)$ are $ \mathcal{a}(A,I)$ and $\mathcal{B}(B,J)$. (One may take I and J to have the same underlying set.) Such functors always establish a duality between Fix FU and Fix UF. We study conditions under which one can conclude that FU and UF are reflectors into Fix FU and Fix UF, that Fix FU = Image F = the limit closure of I in $ \mathcal{a}$ and that Fix UF = Image U = the limit closure of J in $\mathcal{B}$. For example, this happens if (1) $\mathcal{a}$ is a limit closed subcategory of an equational category, (2) J is compact Hausdorff and has a basis of open sets of the form $\{ x \in J\vert\alpha (I)(x) \ne \beta (I)(x)\}$, where $\alpha$ and $\beta$ are unary $ \mathcal{a}$-operations, and (3) there are quaternary operations $\xi$ and $\eta$ such that, for all $x \in {J^4},\xi (I)(x) = \eta (I)(x)$ if and only if ${x_1} = {x_2}$ or ${x_3} = {x_4}$. (The compactness of J may be dropped, but then one loses the conclusion that Fix FU is the limit closure of I.) We also obtain a quite different set of conditions, a crucial one being that J is compact and that every f in $ \mathcal{B}({J^n},J)$, n finite, can be uniformly approximated arbitrarily closely by $ \mathcal{a}$-operations on I. This generalizes the notion of functional completeness in universal algebra. The well-known dualities of Stone and Gelfand are special cases of both situations and the generalization of Stone duality by Hu is also subsumed.
Higher order Massey products and links
Edward J.
O’Neill
37-66
Abstract: In this paper we generalize Steenrod's functional cup products, calling the generalizations functional triple products, and relate them with Massey 4-products. We then study certain links using this machinery and a new relation that is satisfied by 4-products under conditions on X which permit applications to links. Finally, many examples illustrating the connection between Massey higher products, both ordinary and matrix, and links are presented. This material constituted a portion of the author's doctoral dissertation. The author would like to thank his thesis advisor Professor W. S. Massey for his encouragement and guidance.
Circle-preserving functions of spheres
Joel
Gibbons;
Cary
Webb
67-83
Abstract: Suppose a function of the standard sphere ${S^2}$ into the standard sphere ${S^{2 + m}}$, $m \geqslant 0$, sends every circle into a circle but is not a circlepreserving bijection of ${S^2}$. Then the image of the function must lie in a five-point set or, if it contains more than five points, it must lie in a circle together with at most one other point. We prove the local version of this theorem together with a generalization to n dimensions. In the generalization, the significance of 5 is replaced by $2n + 1$. There is also proved a 3-dimensional result in which, compared to the n-dimensional theorem, we are allowed to weaken the structure assumed on the image set of the function.
Uniquely arcwise connected plane continua have the fixed-point property
Charles L.
Hagopian
85-104
Abstract: This paper contains a solution to a fixed-point problem of G. S. Young [17, p. 884] and R. H. Bing [4, Question 4, p. 124]. Let M be an arcwise connected plane continuum that does not contain a simple closed curve. We prove that every continuous function of M into M has a fixed point.
On tails and domains of attraction of stable measures in Banach spaces
Aloisio
Araujo;
Evarist
Giné
105-119
Abstract: The exact tail behavior of stable measures in Banach spaces and measures in their domains of attraction is given. Conditions for a p.m. to be in the domain of attraction of a stable p.m. of order $\alpha$ are derived which are sufficient in type p spaces, $p > \alpha $, and necessary in general. This paper also contains a short proof of the Lévy-Khinchin formula in Banach spaces.
Small zeros of additive forms in many variables
Wolfgang M.
Schmidt
121-133
Abstract: It is shown that if s is large as a function of k and of $\varepsilon > 0$, then the diophantine equation $ {a_1}{x_1}^k + \cdots + {a_s}x_s^k = {b_1}y_1^k + \cdots + {b_s}y_s^k$ with positive coefficients $ {a_1}, \ldots ,{a_s}$, ${b_1}, \ldots ,{b_s}$ has a nontrivial solution in nonnegative integers ${x_1}, \ldots ,{x_s}$, ${y_1}, \ldots ,{y_s}$ not exceeding $ {m^{\left( {1/k} \right) + \varepsilon }}$, where m is the maximum of the coefficients.
Some infinite free boundary problems
David E.
Tepper;
Gerald
Wildenberg
135-144
Abstract: Let $\Gamma$ be the boundary of an unbounded simply connected region $ \mathcal{D}$, and let $ \mathcal{C}(\Gamma )$ denote the family of all simply connected regions $\Delta \subset \mathcal{D}$ such that $ \partial \Delta = \Gamma \cup \gamma$ where $ \gamma \cap \Gamma$ contains only the infinite point. For $\Delta \in \mathcal{C}(\Gamma )$ we call $\gamma$ the free boundary of $\Delta$. Given a positive constant $ \lambda$, we seek to find a region ${\Delta _\lambda } \in \mathcal{C}(\Gamma )$ with free boundary $ {\gamma _\lambda }$ such that there is a bounded harmonic function V in $ {\Delta _\lambda }$ with the properties that (i) $V = 0$ on $\Gamma$, (ii) $V = 1$ on $\gamma$, (iii) $\left\vert {{\text{grad }}V(z)} \right\vert = \lambda$ for $ z \in {\gamma _\lambda }$. We give sufficient conditions for existence and uniqueness of $ {\Delta _\lambda }$. We also give quantitative properties of ${\gamma _\lambda }$.
The periodic behavior of Morse-Smale diffeomorphisms on compact surfaces
Carolyn C.
Narasimhan
145-169
Abstract: Necessary and sufficient conditions are given for the existence of Morse-Smale diffeomorphisms homotopic to the identity with prescribed periodic characteristics on any compact 2-manifold.
On a case of extensions of group schemes
B.
Weisfeiler
171-189
Abstract: The extensions of a smooth connected commutative group scheme whose generic fiber is ${G_m}$ by the additive group scheme are studied. The results are most explicit in the case when the basic scheme is the spectrum of an integral domain containing a field.
The PL Grassmannian and PL curvature
Norman
Levitt
191-205
Abstract: A space ${\mathcal{G}_{n,k}}$ is constructed, together with a block bundle over it, which is analogous to the Grassmannian ${G_{n,k}}$ in that, given a PL manifold ${M^n}$ as a subcomplex of an affine triangulation of $ {R^{n + k}}$, there is a natural ``Gauss map'' ${M^n} \to {\mathcal{G}_{n,k}}$ covered by a block-bundle map of the PL tubular neighborhood of ${M^n}$ to the block bundle over ${G_{n,k}}$. Certain subcomplexes of $ {G_{n,k}}$ are then studied in connection with immersion problems, the chief result being that a connected manifold $ {M^n}$ (nonclosed) PL immerses in ${R^{n + k}}$ satisfying certain ``local'' conditions if and only if its stable normal bundle is represented by a map to the subcomplex of ${G_{n,k}}$ corresponding to the condition. An important example of such a condition is a restriction on PL curvature, e.g., nonnegative or nonpositive, PL curvature having been defined by D. Stone.
Automorphisms of the deformation space of a Kleinian group
James A.
Gentilesco
207-220
Abstract: In the following paper, we determine the biholomorphic automorphisms of a cross-product of Teichmüller spaces. This in turn helps us to determine the biholomorphic automorphisms of the deformation space of a Kleinian group using the fact that its holomorphic universal covering space is a cross-product of Teichmüller spaces. Among other interesting results, we show that in general the deformation space is not a homogeneous space.
Erratum to: ``Torsion in the bordism of oriented involutions'' (Trans. Amer. Math. {\bf 231} (1977), no. 2, 541--548)
Russell J.
Rowlett
221-221
Symmetrized separable convex programming
L.
McLinden
1-44
Abstract: The duality model for convex programming studied recently by E. L. Peterson is analyzed from the viewpoint of perturbational duality theory. Relationships with the traditional Lagrangian model for ordinary programming are explored in detail, with particular emphasis placed on the respective dual problems, Kuhn-Tucker vectors, and extremality conditions. The case of homogeneous constraints is discussed by way of illustration. The Slater existence criterion for optimal Lagrange multipliers in ordinary programming is sharpened for the case in which some of the functions are polyhedral. The analysis generally covers nonclosed functions on general spaces and includes refinements to exploit polyhedrality in the finite-dimensional case. Underlying the whole development are basic technical facts which are developed concerning the Fenchel conjugate and preconjugate of the indicator function of an epigraph set.
Distribution of eigenvalues of a two-parameter system of differential equations
M.
Faierman
45-86
Abstract: In this paper two simultaneous Sturm-Liouville systems are considered, the first defined for the interval $0\, \leqslant \,{x_1}\, \leqslant \,1$, the second for the interval $0\, \leqslant \,{x_{2\,}}\, \leqslant \,1$, and each containing the parameters $\lambda$ and $\mu$. Denoting the eigenvalues and eigenfunctions of the simultaneous systems by $({\lambda _{j,k}},{\mu _{j,k}})$ and $ {\psi _{j,k}}({x_{1,}}{x_2})$, respectively, $j,\,k\, = \,0,\,1,\, \ldots \,$, asymptotic methods are employed to derive asymptotic formulae for these expressions, as $j + k \to \infty$ when $ (j,\,k)$ is restricted to lie in a certain sector of the $(x,\,y)$ -plane. These results constitute a further stage in the development of the theory related to the behaviour of the eigenvalues and eigenfunctions of multiparameter Sturm-Liouville systems and answer an open question concerning the uniform boundedness of the ${\psi _{j,k}}\,({x_1},\,{x_2})$.
On the construction of branched coverings of low-dimensional manifolds
Israel
Berstein;
Allan L.
Edmonds
87-124
Abstract: Several general results are proved concerning the existence and uniqueness of various branched coverings of manifolds in dimensions 2 and 3. The results are applied to give a rather complete account as to which 3-manifolds are branched coverings of ${S^3}$, $ {S^2}\, \times \,{S^1}$, ${P^2}\, \times \,{S^1}$, or the nontrivial ${S^3}$-bundle over ${S^1}$, and which degrees can be achieved in each case. In particular, it is shown that any closed nonorientable 3-manifold is a branched covering of ${P^2}\, \times \,{S^1}$ of degree which can be chosen to be at most 6 and with branch set a simple closed curve. This result is applied to show that a closed nonorientable 3-manifold admits an open book decomposition which is induced from such a decomposition of ${P^2}\, \times \,{S^1}$.
Examples of noncatenary rings
Raymond C.
Heitmann
125-136
Abstract: A technique is developed for constructing a new family of noetherian integral domains. To each domain, there naturally corresponds its poset (partially ordered set) of prime ideals. The resulting family of posets has the following property: every finite poset is isomorphic to a saturated subset of some poset in the family. In the process, it is determined when certain power series may be adjoined to noetherian rings without destroying the noetherian property.
Irrational connected sums and the topology of algebraic surfaces
Richard
Mandelbaum
137-156
Abstract: Suppose W is an irreducible nonsingular projective algebraic 3-fold and V a nonsingular hypersurface section of W. Denote by ${V_m}$ a nonsingular element of $\left\vert {mV} \right\vert$. Let $ {V_1}$, ${V_m}$, $ {V_{m\, + \,1}}$ be generic elements of $ \left\vert V \right\vert$, $\left\vert {mV} \right\vert$, $\left\vert {(m\, + \,1)V} \right\vert$ respectively such that they have normal crossing in W. Let $ {S_{1m}}\, = \,{V_1}\, \cap \,{V_m}$ and $C\, = \,{V_1}\, \cap \,{V_m}\, \cap \,{V_{m + 1}}$. Then ${S_{1m}}$ is a nonsingular curve of genus $ {g_m}$ and C is a collection of $N\, = \,m\left( {m + 1} \right)V_1^3$ points on $ {S_{1m}}$. By [MM2] we find that $ ( \ast )\,{V_{m\, + \,1}}$ is diffeomorphic to $\overline {{V_m}\, - \,T({S_{1m}})} \,{ \cup _\eta }\,\overline {{V_1}'\, - \,T({S_{1m}}')}$, where $T\left( {{S_{1m}}} \right)$ is a tubular neighborhood of ${S_{1m}}$ in ${V_m}$, ${V_1}'$ is ${V_1}$ blown up along C, ${S_{1m}}'$ is the strict image of $ {S_{1m}}$ in $ {V_1}'$,
On the existence of good Markov strategies
Theodore Preston
Hill
157-176
Abstract: In contrast to the known fact that there are gambling problems based on a finite state space for which no stationary family of strategies is at all good, in every such problem there always exist $ \varepsilon$-optimal Markov families (in which the strategy depends only on the current state and time) and also $\varepsilon$-optimal tracking families (in which the strategy depends only on the current state and the number of times that state has been previously visited). More generally, this result holds for all finite state gambling problems with a payoff which is shift and permutation invariant.
Thickenings of CW complexes of the form $S\sp{m}\cup \sb{\alpha }e\sp{n}$
George
Cooke
177-209
Abstract: Necessary conditions are given for the existence of a thickening of $ {S^m}\,{ \cup _\alpha }\,{e^n}$ in codimension k. I give examples of such complexes requiring arbitrarily large codimension in order to thicken. Sufficient conditions are given for the existence of a tractable thickening in codimension $k\, + \,1$. The methods used include the study of the reduced product space of a pair of CW complexes.
A consistent consequence of AD
E. L.
Bull;
E. M.
Kleinberg
211-226
Abstract: We are concerned with the existence of a normal measure concentrating on the $ \omega$-closed unbounded sets. We strengthen the known result that the axiom of choice implies no such measure exists. It is shown that the existence of these measures is consistent (relative to a large cardinal). In particular, ${\aleph _2}$ may admit exactly two normal measures: one which contains the $\omega$-closed unbounded sets and the other, the ${\aleph _1}$-closed unbounded sets. This property of ${\aleph _2}$ is a well-known consequence of AD.
Partitions and sums and products of integers
Neil
Hindman
227-245
Abstract: The principal result of the paper is that, if $r\, < \,\omega$ and ${\{ {A_i}\} _{i < r}}$ is a partition of $ \omega$, then there exist $ i\, < \,r$ and infinite subsets B and C of $\omega$ such that $\sum F\, \in \,{A_i}$ and $\prod {G\, \in \,{A_i}}$ whenever F and G are finite nonempty subsets of B and C respectively. Conditions on the partition are obtained which are sufficient to guarantee that B and C can be chosen equal in the above statement, and some related finite questions are investigated.
Differential algebraic Lie algebras
Phyllis Joan
Cassidy
247-273
Abstract: A class of infinite-dimensional Lie algebras over the field $\mathcal{K}$ of constants of a universal differential field $ \mathcal{U}$ is studied. The simplest case, defined by homogeneous linear differential equations, is analyzed in detail, and those with underlying set $\mathcal{U}\, \times \,\mathcal{U}$ are classified.
Frattini subgroups of $3$-manifold groups
R. B. J. T.
Allenby;
J.
Boler;
B.
Evans;
L. E.
Moser;
C. Y.
Tang
275-300
Abstract: In this paper it is shown that if the Frattini subgroup of the fundamental group of a compact, orientable, irreducible, sufficiently large 3-manifold is nontrivial then the 3-manifold is a Seifert fibered space. We show further that the Frattini subgroup of the group of a Seifert fibered space is trivial or cyclic. As a corollary to our work we prove that every knot group has trivial Frattini subgroup.
Semi-algebraic groups and the local closure of an orbit in a homogeneous space
Morikuni
Goto
301-315
Abstract: Let L be a topological group acting on a locally compact Hausdorff space M as a transformation group. Let m be in M. A subset Q of M is called the local closure of the orbit Lm if Q is the smallest locally compact invariant subset of M with $m\, \in \,Q$. A partition $\displaystyle M = \,\bigcup\limits_{\lambda \in \wedge } \,{Q_\lambda },\,\,\,\... ...,}}\, \cap \,\,{Q_\mu } = \,\emptyset \,\,\,\,\left( {\lambda \ne \mu } \right)$ is called an LC-partition of M with respect to the L action if each ${Q_\lambda }$ is the local closure of Lm for any m in $ {Q_\lambda }$. Theorem. Let G be a connected Lie group, and let A and B be subgroups of G with only finitely many connected components. Suppose that B is closed. Then the factor space $G/B$ has an LC-partition with respect to the A action.
Erratum to: ``Generalized super-solutions of parabolic equations'' (Trans. Amer. Math. Soc. {\bf 220} (1976), 235--242)
Neil
Eklund
317-318
Difference equations: disconjugacy, principal solutions, Green's functions, complete monotonicity
Philip
Hartman
1-30
Abstract: We find analogues of known results on nth order linear differential equations for nth order linear difference equations. These include the concept of disconjugacy, Pólya's criterion for disconjugacy, Frobenius factorizations, generalized Sturm theorems, existence and properties of principal solutions, signs of Green's functions, and completely monotone families of solutions of equations depending on a parameter.
Multidimensional quality control problems and quasivariational inequalities
Robert F.
Anderson;
Avner
Friedman
31-76
Abstract: A machine can manufacture any one of n m-dimensional Brownian motions with drift $ {\lambda _j}$, $P_x^{{\lambda _j}}$, defined on the space of all paths $x\left( t \right)\, \in \,C\left( {\left[ {0,\,\infty } \right);\,{R^m}} \right)$. It is given that the product is a random evolution dictated by a Markov process $ \theta \left( t \right)$ with n states, and that the product is $P_x^{{\lambda _j}}$ when $\theta \left( t \right)\, = \,j,\,1\, \leqslant \,j\, \leqslant \,n$. One observes the $\sigma $-fields of $x\left( t \right)$, but not of $\theta \left( t \right)$. With each product $P_x^{{\lambda _j}}$ there is associated a cost $ {c_j}$. One inspects $ \theta$ at a sequence of times (each inspection entails a certain cost) and stops production when the state $\theta \, = \,n$ is reached. The problem is to find an optimal sequence of inspections. This problem is reduced to solving a certain elliptic quasi variational inequality. The latter problem is actually solved in a rather general case.
Quality control for Markov chains and free boundary problems
Robert F.
Anderson;
Avner
Friedman
77-94
Abstract: A machine can manufacture any one of n Markov chains $P_x^{{\lambda _j}}\,\left( {1\, \leq \,j\, \leq \,n} \right)$; the $ P_x^{{\lambda _j}}$ are defined on the space of all sequences $x\, = \,\left\{ {x\left( m \right)} \right\}\,\left( {1\, \leq \,m\, \leq \,\infty } \right)$ and are absolutely continuous (in finite times) with respect to one another. It is assumed that chains $P_x^{{\lambda _j}}$ evolve in a random way, dictated by a Markov chain $\theta \left( m \right)$ with n states, so that when $\theta \left( m \right)\, = \,j$ the machine is producing $P_x^{{\lambda _j}}$. One observes the $ \sigma$-fields of $x\left( m \right)$ in order to determine when to inspect $ \theta \left( m \right)$. With each product $ P_x^{{\lambda _j}}$ there is associated a cost ${c_j}$. One inspects $\theta$ at a sequence of times (each inspection entails a certain cost) and stops production when the state $\theta \, = \,n$ is reached. The problem is to find an optimal sequence of inspections. This problem is reduced, in this paper, to solving a certain free boundary problem. In case $n\, = \,2$ the latter problem is solved.
On the free boundary of a quasivariational inequality arising in a problem of quality control
Avner
Friedman
95-110
Abstract: In some recent work in stochastic optimization with partial observation occurring in quality control problems, Anderson and Friedman [1], [2] have shown that the optimal cost can be determined as a solution of the quasi variational inequality \begin{displaymath}\begin{gathered}Mw\left( p \right)\, + \,f\left( p \right)\, ... ... \,\psi \left( {p;\,w} \right)} \right)\, = \,0 \end{gathered} \end{displaymath} in the simplex $ {p_i}\, > \,0$, $ \sum\nolimits_{i\, = \,1}^n {{p_i}\, = \,1}$. Here f, $\psi$ are given functions of p, $ \psi$ is a functional of w, and M is a given elliptic operator degenerating on the boundary. This system has a unique solution when M does not degenerate in the interior of the simplex. The aim of this paper is to study the free boundary, that is, the boundary of the set where $w\left( p \right)\, < \,\psi \left( {p;\,w} \right)$.
On regular semigroups and their multiplication
Pierre Antoine
Grillet
111-138
Abstract: A method is given for the construction of regular semigroups in terms of groups and partially ordered sets. This describes any regular semigroup S and its multiplication by means of triples $\left( {i,\,g,\,\lambda } \right)$ with $i\, \in \,S/{\mathcal{R}}$, $ \lambda \, \in \,S/{\mathcal{L}}$ and g in the Schützenberger group of the corresponding $ {\mathcal{D}}$-class. It is shown that the multiplication on S is determined by certain simple products. Furthermore the associativity of these simple products implies associativity of the entire multiplication.
Compactifications of ${\bf C}\sp{n}$
L.
Brenton;
J.
Morrow
139-153
Abstract: Let X be a compactification of $ {{\text{C}}^n}$. We assume that X is a compact complex manifold and that $ A\, = \,X\, - \,{{\text{C}}^n}$ is a proper subvariety of X. If we suppose that A is a Kähler manifold, then we prove that X is projective algebraic, ${H^{\ast}}\left( {A,\,{\textbf{Z}}} \right)\, \cong \,{H^{\ast}}\left( {{{\textbf{P}}^{n\, - \,1}},\,{\textbf{Z}}} \right)$, and ${H^{\ast}}\left( {X,\,{\textbf{Z}}} \right)\, \cong \,{H^{\ast}}\left( {{{\textbf{P}}^n},\,{\textbf{Z}}} \right)$. Various additional conditions are shown to imply that $X\, = \,{{\textbf{P}}^n}$. It is known that no additional conditions are needed to imply $ X\, = \,{{\textbf{P}}^n}$ in the cases $ n\, = \,1,\,2$. In this paper we prove that if $n\, = \,3$, $X\, = \,{{\textbf{P}}^3}$.
Automorphisms of ${\rm GL}\sb{n}(R)$
B. R.
McDonald
155-171
Abstract: Let R denote a commutative ring having 2 a unit. Let $ {\text{G}}{{\text{L}}_n}\left( R \right)$ denote the general linear group of all $n\, \times \,n$ invertible matrices over R. Let $\wedge$ be an automorphism of $ {\text{G}}{{\text{L}}_n}\left( R \right)$. An automorphism $\wedge$ is ``stable'' if it behaves properly relative to families of commuting involutions (see §IV). We show that if R is connected, i.e., 0 and 1 are only idempotents, then all automorphisms $\wedge$ are stable. Further, if $n\, \geqslant \,3$, R is an arbitrary commutative ring with 2 a unit, and $\wedge$ is a stable automorphism, then we obtain a description of $\wedge$ as a composition of standard automorphisms.
Quantization and projective representations of solvable Lie groups
Henri
Moscovici;
Andrei
Verona
173-192
Abstract: Kostant's quantization procedure is applied for constructing irreducible projective representations of a solvable Lie group from symplectic homogeneous spaces on which the group acts. When specialized to a certain class of such groups, including the exponential ones, the technique exposed in the present paper provides a complete parametrization of all irreducible projective representations.
The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. I
Gary
Weiss
193-209
Abstract: We prove the following statements about bounded linear operators on a separable, complex Hilbert space: (1) Every normal operator N that is similar to a Hilbert-Schmidt perturbation of a diagonal operator D is unitarily equivalent to a Hilbert-Schmidt perturbation of D; (2) For every normal operator N, diagonal operator D and bounded operator X, the Hilbert-Schmidt norms (finite or infinite) of $NX\, - \,XD$ and ${N^{\ast}}X\, - \,X{D^{\ast}}$ are equal; (3) If $NX\, - \,XN$ and ${N^{\ast}}X\, - \,X{N^{\ast}}$ are Hilbert-Schmidt operators, then their Hilbert-Schmidt norms are equal; (4) If X is a Hilbert-Schmidt operator and N is a normal operator so that $NX\, - \,XN$ is a trace class operator, then Trace $ \left( {NX\, - \,XN} \right)\, = \,0$; (5) For every normal operator N that is a Hilbert-Schmidt perturbation of a diagonal operator, and every bounded operator X, the Hilbert-Schmidt norms (finite or infinite) of $NX\, - \,XN$ and ${N^{\ast}}X\, - \,X{N^{\ast}}$ are equal. The main technique employs the use of a new concept which we call 'generating functions for matrices'.
Geometric convexity. III. Embedding
John
Cantwell;
David C.
Kay
211-230
Abstract: The straight line spaces of dimension three or higher which were considered by the first author in previous papers are shown to be isomorphic with a strongly open convex subset of a real vector space. To achieve this result we consider the classical descriptive geometry studied in various papers and textbooks by Pasch, Hilbert, Veblen, Whitehead, Coxeter, Robinson, and others, with the significant difference that the geometry considered here is not restricted to 3 dimensions. Our main theorem (which is well known in dimension 3) is that any such geometry is isomorphic to a strongly open convex subset of a real vector space whose ``chords'' play the role of lines.
On modular functions in characteristic $p$
Wen Ch’ing Winnie
Li
231-259
Abstract: Let $k\, = \,{{\textbf{F}}_q}\left( T \right)$ be a function field of one variable over a finite field $ {{\textbf{F}}_q}$. For a nonzero polynomial $A\, \in \,{{\textbf{F}}_q}\left[ T \right]$ one can define the modular group $\Gamma \left( A \right)$. In this paper, we continue a theme introduced by Weil, and study the $\lambda $-harmonic modular functions for $ \Gamma \left( A \right)$. The main purpose of this paper is to give a natural definition of $\lambda$-harmonic Eisenstein series for $\Gamma \left( A \right)$ so that we obtain a decomposition theory of $\lambda$-harmonic modular functions, analogous to the classical results of Hecke. That is, we prove $\displaystyle {\text{Modular}}\,{\text{Function}}\,{\text{ = }}\,{\text{Eisenstein}}\,{\text{series}}\, \oplus \,{\text{Cusp Functions}}{\text{.}}$ Moreover, the dimension of the space generated by $\lambda$-harmonic Eisenstein series for $\Gamma \left( A \right)$ is equal to the number of cusps of $\Gamma \left( A \right)$, and so is independent of $\lambda$. For the definition of $\lambda $-harmonic Eisenstein series and the proof of decomposition theory, we consider two cases: (i) $\lambda \, \ne \, \pm 2\sqrt q$ and (ii) $ \lambda \, = \, \pm 2\sqrt q$, separately. Case (i) is treated in the usual way. Case (ii), being a ``degenerate'' case, is more interesting and requires more complicated analysis.
Persistent manifolds are normally hyperbolic
Ricardo
Mañé
261-283
Abstract: Let M be a smooth manifold, $f:\,M\,\mid$ a$\,{{C}^{1}}$ diffeomorphism and $V \subset M\,{\text{a}}\,{{\text{C}}^1}$ compact submanifold without boundary invariant under f (i.e. $f\left( V \right)\, = \,V$). We say that V is a persistent manifold for f if there exists a compact neighborhood U of V such that ${ \cap _{n\, \in \,{\textbf{z}}}}\,{f^n}\left( U \right)\, = \,V$, and for all diffeomorphisms $g:\,M\,\mid$ near to f in the $ {C^1}$ topology the set ${V_g}\, = \,{ \cap _{n\, \in \,{\textbf{z}}}}{g^n}\left( U \right)$ is a ${C^1}$ submanifold without boundary $ {C^1}$ near to V. Several authors studied sufficient conditions for persistence of invariant manifolds. Hirsch, Pugh and Shub proved that normally hyperbolic manifolds are persistent, where normally hyperbolic means that there exist a Tf-invariant splitting $ TM/V\, = \,{N^s}V\, \oplus \,{N^u}V\, \oplus \,TV$ and constants $K\, > \,0$, $0\, < \,\lambda \, < \,1$ such that: \begin{displaymath}\begin{gathered}\left\Vert {{{\left( {Tf} \right)}^n}/N_x^sV}... ... \right)}}V} \right\Vert\, \leq \,K{\lambda ^n} \end{gathered} \end{displaymath} for all $n\, > \,0$, $ x\, \in \,V$. In this paper we prove the converse result, namely that persistent manifolds are normally hyperbolic.
The spectral theory of distributive continuous lattices
Karl H.
Hofmann;
Jimmie D.
Lawson
285-310
Abstract: In this paper various properties of the spectrum (i.e. the set of prime elements endowed with the hull-kernel topology) of a distributive continuous lattice are developed. It is shown that the spectrum is always a locally quasicompact sober space and conversely that the lattice of open sets of a locally quasicompact sober space is a continuous lattice. Algebraic lattices are a special subclass of continuous lattices and the special properties of their spectra are treated. The concept of the patch topology is extended from algebraic lattices to continuous lattices, and necessary and sufficient conditions for its compactness are given.
Intertwining differential operators for ${\rm Mp}(n,\,{\bf R})$ and ${\rm SU}(n,\,n)$
Hans Plesner
Jakobsen
311-337
Abstract: For each of the two series of groups, three series of representations $ {U_n}$, ${D_n}$, and ${H_n}(n \in Z)$ are considered. For each series of representations there is a differential operator with the property, that raised to the nth power $ (n > 0)$, it intertwines the representations indexed by $- n$ and n. The operators are generalizations of the d'Alembertian, the Diracoperator and a combination of the two. Unitarity of subquotients of representations indexed by negative integers is derived from the intertwining relations.
The heat equation on a compact Lie group
H. D.
Fegan
339-357
Abstract: Recently there has been much work related to Macdonald's $\eta $-function identities. In the present paper the aim is to give another proof of these identities using analytical methods. This is done by using the heat equation to obtain Kostant's form of the identities. The basic idea of the proof is to look at subgroups of the Lie group which are isomorphic to the group $SU(2)$. When this has been done the problem has essentially been reduced to that for the group $ SU(2)$, which is a classical result.
Hardy spaces of close-to-convex functions and their derivatives
Finbarr
Holland;
John B.
Twomey
359-372
Abstract: Let $ f(z) = \sum\nolimits_1^\infty {{a_n}} {z^n}$ be close-to-convex on the unit disc. It is shown that (a) if $\lambda > 0$, then f belongs to the Hardy space $ {H^\lambda }$ if and only if ${\sum {{n^{\lambda - 2}}\left\vert {{a_n}} \right\vert} ^\lambda }$ is finite and that (b) if $0 < \lambda < 1$, then $ \sum {{n^{2\lambda - 2}}} {\left\vert {{a_n}} \right\vert^\lambda }$ or, equivalently,
Invariance of the $L$-regularity of compact sets in ${\bf C}\sp{N}$ under holomorphic mappings
W.
Pleśniak
373-383
Abstract: The property for a polynomially convex compact set E in $ {C^N}$ that the Siciak extremal function ${\Phi _E}$ be continuous or, equivalently, that E satisfy some Bernstein type inequality, is proved to be invariant under a large class of holomorphic mappings with values in $ {C^M}(M \leqslant N)$ including all open holomorphic mappings. Local specifications of this result are also given.
On the first occurrence of values of a character
G.
Kolesnik;
E. G.
Straus
385-394
Abstract: Let $\chi$ be a character of order $k\,(\bmod\, n)$, and let $ {g_m}(\chi )$ be the smallest positive integer at which $\chi$ attains its $(m + 1)$st nonzero value. We consider fixed k and large n and combine elementary group-theoretic considerations with the known results on character sums and sets of integers without large prime factors to obtain estimates for ${g_m}(\chi )$.
Amalgamation and inverse and regular semigroups
T. E.
Hall
395-406
Abstract: A method for proving the embeddability of semigroup amalgams is introduced. After providing necessary and sufficient conditions in terms of representations for the weak embeddability of a semigroup amalgam, it successfully deals with the embedding of inverse semigroup amalgams into inverse semigroups and the embedding of an amalgam of regular semigroups whose core is full in each member.
A characterization and sum decomposition for operator ideals
Andreas
Blass;
Gary
Weiss
407-417
Abstract: Let $L(H)$ be the ring of bounded operators on a separable Hubert space. Assuming the continuum hypothesis, we prove that in $L(H)$ every two-sided ideal that contains an operator of infinite rank is the sum of two smaller two-sided ideals. The proof involves a new combinatorial description of ideals of $L(H)$. This description is also used to deduce some related results about decompositions of ideals. Finally, we discuss the possibility of proving our main theorem under weaker assumptions than the continuum hypothesis and the impossibility of proving it without the axiom of choice.
Dense subgroups of Lie groups. II
David
Zerling
419-428
Abstract: Let G be a dense analytic subgroup of an analytic group L. Then G contains a maximal (CA) closed normal analytic suhgroup M and a closed abelian subgroup $A = Z(G) \times E$, where E is a closed vector subgroup of G, such that $G = M \cdot A$, $M \cap A = Z(G)$, $\overline M = M \cdot \overline {Z(G)}$, and $L = M \cdot \overline A$. We also indicate the extent to which a (CA) analytic group is uniquely determined by its center and a dense analytic subgroup.
Liftings and the construction of stochastic processes
Donald L.
Cohn
429-438
Abstract: It is shown that if the continuum hypothesis holds, then the use of liftings to construct modifications of stochastic processes can replace measurable processes with nonmeasurable ones. The use of liftings to choose the paths, rather than the random variables, of a stochastic process is investigated.
Rational subspaces of induced representations and nilmanifolds
R.
Penney
439-450
Abstract: Recently, Auslander and Brezin developed a technique of distinguishing between certain unitarily equivalent irreducible subspaces of ${L^2}$ of the Heisenberg nilmanifold. In this paper we extend the Auslander-Brezin technique to arbitrary induced representations of arbitrary locally compact groups. We then return to nilmanifolds, showing that the existence of a ``nice'' theory of distinguished subspaces is equivalent to the existence of square integrable representations for the group.
Parabolic function spaces with mixed norm
V. R.
Gopala Rao
451-461
Abstract: The spaces $ \mathcal{H}_\alpha ^p$ of parabolic Bessel potentials were introduced by B. F. Jones and R. J. Bagby. We prove a Sobolev-type imbedding theorem for $\mathcal{H}_\alpha ^{{p_1},{p_2}}$ (multinormed versions of $ \mathcal{H}_\alpha ^p$) when $\alpha$ is a positive integer k, $1 < {p_1}$, ${p_2} < \infty$. In particular this theorem holds for $W_{2l,l}^p$, since $ \mathcal{H}_{2l}^p \equiv W_{2l,l}^p$. We use the concepts of parabolic Riesz transforms and half-time derivatives introduced by us elsewhere.
On the existence of uniformly distributed sequences in compact topological spaces. I
V.
Losert
463-471
Abstract: We prove the existence of uniformly distributed sequences for an arbitrary probability measure on a separable dyadic space, e.g. on a separable compact topological group. Some counterexamples for the nonexistence of u.d. sequences in certain dense subsets are given.
The fixed-point construction in equivariant bordism
Russell J.
Rowlett
473-481
Abstract: Consider the bordism $ {\Omega _ {\ast}}(G)$ of smooth G-actions. If K is a subgroup of G, with normalizer NK, there is a standard $NK/K$-action on $ {\Omega _ {\ast}}(K)$(All, Proper). If M has a smooth G-action, a tubular neighborhood of the fixed set of K in M represents an element of $ {\Omega _ {\ast}}(K){({\text{All, Proper}})^{NK/K}}$. One thus obtains the ``fixed point homomorphism'' $\phi$ carrying $ {\Omega _ {\ast}}(G)$ to the sum of the ${\Omega _ {\ast}}(K){({\text{All, Proper}})^{NK/K}}$, summed over conjugacy classes of subgroups K. Let P be the collection of primes not dividing the order of G. We show that the P-localization of $ \phi$ is an isomorphism, and give several applications.
On a theorem of Steinitz and Levy
Gadi
Moran
483-491
Abstract: Let $ \sum\nolimits_{n\,\, \in \,\omega } {h(n)}$ be a conditionally convergent series in a real Banach space B. Let $S(h)$ denote the set of sums of the convergent rearrangements of this series. A well-known theorem of Riemann states that $ S(h)\, = \,B$ if $ B\, = \,R$, the reals. A generalization of Riemann's Theorem, due independently to Levy [L] and Steinitz [S], states that if B is finite dimensional, then $ S(h)$ is a linear manifold in B of dimension $> \,0$. Another generalization of Riemann's Theorem [M] can be stated as an instance of the Levy-Steinitz Theorem in the Banach space of regulated real functions on the unit interval I. This instance generalizes to the Banach space of regulated B-valued functions on I, where B is finite dimensional, implying a generalization of the Levy-Steinitz Theorem.
Some metric properties of piecewise monotonic mappings of the unit interval
Sherman
Wong
493-500
Abstract: In this note, the result of Lasota and Yorke on the existence of invariant measures for piecewise ${C^2}$ functions is extended to a larger class of piecewise continuous functions. Also the result of Li and Yorke on the existence of ergodic measures for piecewise ${C^2}$ functions is extended for the above class of functions.
Even triangulations of $S\sp{3}$ and the coloring of graphs
Jacob Eli
Goodman;
Hironori
Onishi
501-510
Abstract: A simple necessary and sufficient condition is given for the vertices of a graph, planar or not, to be properly four-colorable. This criterion involves the notion of an ``even'' triangulation of ${S^3}$ and generalizes, in a natural way, a corresponding criterion for the three-colorability of planar graphs.
A noncommutative probability theory
S. P.
Gudder;
R. L.
Hudson
1-41
Abstract: A noncommutative probability theory is developed in which no boundedness, finiteness, or ``tracial'' conditions are imposed. The underlying structure of the theory is a ``probability algebra'' $(\mathcal{a},\omega )$ where $\mathcal{a}$ is a *-algebra and $\omega$ is a faithful state on $\mathcal{a}$. Conditional expectations and coarse-graining are discussed. The bounded and unbounded commutants are considered and commutation theorems are proved. Two classes of probability algebras, which we call closable and symmetric probability algebras are shown to have important regularity properties. The canonical algebra of quantum mechanics is considered in some detail and a strong commutation theorem is proven for this case. Moreover, in this case, isotropic normal states, KMS states, and stable states are defined and characterized.
Explosions in completely unstable flows. I. Preventing explosions
Zbigniew
Nitecki
43-61
Abstract: Several conditions are equivalent to the property that a flow (on an open manifold) and its ${C^0}$ perturbations have only wandering points. These conditions are: (i) there exists a strong Liapunov function; (ii) there are no generalized recurrent points in the sense of Auslander; (iii) there are no chain recurrent points, in the sense of Conley; (iv) there exists a fine sequence of filtrations; (v) relative to some metric; the flow is the gradient flow of a function without critical points. We establish these equivalences, and consider a few questions related to structural stability when all orbits wander.
Explosions in completely unstable flows. II. Some examples
Zbigniew
Nitecki
63-88
Abstract: A dynamical system with all points wandering is ``explosive'' if some $ {C^0}$ perturbation has nonwandering points. It is known that the plane admits no explosive cascades or flows; in this paper, examples are constructed to show that all open manifolds except ${R^1}$ and ${R^2}$ admit explosive flows (and hence cascades).
A $q$-analog of restricted growth functions, Dobinski's equality, and Charlier polynomials
Stephen C.
Milne
89-118
Abstract: We apply finite operator techniques due to G. C. Rota to a combinatorial identity, which counts a collection of generalized restricted growth functions in two ways, and obtain a q-analog of Charlier polynomials and Dobinski's equality for the number of partitions of an n-set. Our methods afford a unified proof of certain identities in the combinatorics of finite dimensional vector spaces over $ {\text{GF}}(q)$.
Conformality and semiconformality of a function holomorphic in the disk
Shinji
Yamashita
119-138
Abstract: Conformality and semiconformality at a boundary point, of a function f nonconstant and holomorphic in $\left\vert z \right\vert < 1$ are local properties. Therefore one would suspect the requirement of such global conditions on f as f is univalent in $ \left\vert z \right\vert < 1$, or f is a member of a larger class which contains all univalent functions in $\left\vert z \right\vert < 1$. We shall prove some extensions and new results without any assumption on f, or with a local assumption on f at most. Our methods are, for the most part, different from the ones in the classical cases. One of the main tools is Theorem 8 on the angular limits of the real part of a holomorphic function and its derivative.
Unitary invariance in algebraic algebras
Charles
Lanski
139-146
Abstract: A structure theorem is obtained for subspaces invariant under conjugation by the unitary group of a prime algebraic algebra over an infinite field. For an invariant subalgebra W, it is shown that either W is central, W contains an ideal, or the ring satisfies the standard identity of degree eight. Also, for prime algebras not satisfying such an identity, the unitary group is not solvable.
Symmetric duality for structured convex programs
L.
McLinden
147-181
Abstract: A fully symmetric duality model is presented which subsumes the classical treatments given by Duffin (1956), Eisenberg (1961) and Cottle (1963) for linear, homogeneous and quadratic convex programming. Moreover, a wide variety of other special objective functional structures, including homogeneity of any nonzero degree, is handled with equal ease. The model is valid in spaces of arbitrary dimension and treats explicitly systems of both nonnegativity and linear inequality constraints, where the partial orderings may correspond to nonpolyhedral convex cones. The approach is based on augmenting the Fenchel-Rockafellar duality model (1951, 1967) with cone structure to handle constraint systems of the type mentioned. The many results and insights from Rockafellar's general perturbational duality theory can thus be brought to bear, particularly on sensitivity analysis and the interpretation of dual variables. Considerable attention is devoted to analysis of suboptimizations occurring in the model, and the model is shown to be the projection of another model.
Dolbeault homotopy theory
Joseph
Neisendorfer;
Laurence
Taylor
183-210
Abstract: For complex manifolds, we define ``complex homotopy groups'' in terms of the Dolbeault complex. Many theorems of classical homotopy theory are reflected in the properties of complex homotopy groups. Analytic fibre bundles yield long exact sequences of complex homotopy groups and various Hurewicz theorems relate complex homotopy groups to the Dolbeault cohomology. In a more analytic vein, the classical Fröhlicher spectral sequence has a complex homotopy analogue. We compute these complex homotopy invariants for such examples as Calabi-Eckmann manifolds, Stein manifolds, and complete intersections.
The real and rational cohomology of differential fibre bundles
Joel
Wolf
211-220
Abstract: Consider a differential fibre bundle (E, $\pi$, X, ${G \mathord{\left/ {\vphantom {G H}} \right. \kern-\nulldelimiterspace} H}$ , G). Under certain reasonable hypotheses, the cohomology of the total space E is computed in terms of the cohomology of the base space X and algebraic invariants of the imbedding of H into G.
Projective modules for finite Chevalley groups
John W.
Ballard
221-249
Abstract: The purpose of this paper is to obtain character formulas for certain indecomposable projective modules for a finite Chevalley group. It is shown that these modules are also modules for the corresponding semisimple algebraic group.
Hyperbolicity and cycles
J. E.
Franke;
J. F.
Selgrade
251-262
Abstract: In this paper cycle points are defined without the assumption of Axiom A. The closure of the set of cycle points $\mathcal{C}$ being quasi-hyperbolic is shown to be equivalent to Axiom A plus no cycles. Also we give a sufficient condition for $ \mathcal{C}$ to equal the chain recurrent set. In proving these theorems, a spectral decomposition for quasi-hyperbolic invariant sets is used.
Stability of isometries
Peter M.
Gruber
263-277
Abstract: A map $ T:E \to F$ (E, F Banach spaces) is called an $\varepsilon$-isometry if $ \left\vert\,{\left\Vert {T(x)-T(y)} \right\Vert-\left\Vert{x -y}\right\Vert}\,\right\vert\, \leqslant \varepsilon$ whenever $x,\,y \in E$. Hyers and Ulam raised the problem whether there exists a constant $\kappa$, depending only on E, F, such that for every surjective $\varepsilon$-isometry $T:E \to F$ there exists an isometry $I:E \to F$ with ${\left\Vert {T(x) - I(x)} \right\Vert}\leqslant \kappa \varepsilon$ for every $x \in E$. It is shown that, whenever this problem has a solution for E, F, one can assume $\kappa \leqslant 5$. In particular this holds true in the finite dimensional case.
The rigidity of graphs
L.
Asimow;
B.
Roth
279-289
Abstract: We regard a graph G as a set $ \{ 1, \ldots , v \}$ together with a nonempty set E of two-element subsets of $ \{ 1, \ldots , v \}$. Let $p = ({p_1},\ldots,{p_v})$ be an element of $ {\textbf{R}^{nv}}$ representing v points in ${\textbf{R}^n}$. Consider the figure $G(p)$ in $ {\textbf{R}^n}$ consisting of the line segments $ [{p_i},{p_j}]$ in ${\textbf{R}^n}$ for $ \{ i,j\} \in E$. The figure $G(p)$ is said to be rigid in ${\textbf{R}^n}$ if every continuous path in ${\textbf{R}^{nv}}$, beginning at p and preserving the edge lengths of $G(p)$, terminates at a point $q \in {\textbf{R}^{nv}}$ which is the image $(T{p_1}, \ldots ,T{p_v})$ of p under an isometry T of ${\textbf{R}^n}$. Otherwise, $G(p)$ is flexible in ${\textbf{R}^n}$. Our main result establishes a formula for determining whether $G(p)$ is rigid in ${\textbf{R}^n}$ for almost all locations p of the vertices. Applications of the formula are made to complete graphs, planar graphs, convex polyhedra in $ {\textbf{R}^3}$, and other related matters.
Hamiltonian systems in a neighborhood of a saddle point
Viorel
Barbu
291-307
Abstract: The behavior of Hamiltonian differential systems associated with a concave convex function H in a Hilbert space is studied by variational methods. It is shown that under quite general conditions on the function H the system behaves in a neighborhood of a minimax saddle point of H much like as in the classical theory of ordinary differential systems. The results extend previous work of R. T. Rockafellar.
Chern classes of certain representations of symmetric groups
Leonard
Evens;
Daniel S.
Kahn
309-330
Abstract: A formula is derived for the Chern classes of the representation id $ \int {\xi :P\int {H \to {U_{pn}}} }$ where P is cyclic of order P and $\xi :H \to {U_n}$ is a fintie dimensional unitary representation of the group H. The formula is applied to the problem of calculating the Chern classes of the ``natural'' representations ${\pi _j}:{\mathcal{S}_j} \to {U_j}$ of symmetric groups by permutation matrices.
An integral Riemann-Roch formula for induced representations of finite groups
Leonard
Evens;
Daniel S.
Kahn
331-347
Abstract: Let H be a subgroup of the finite group G, $\xi$ a finite dimensional complex representation of H and $\rho$ the induced representation of G. If ${s_k}(\rho ) \in {H^{2k}}(G,\textbf{Z})$, $k \geqslant 1$ denote the characteristic classes bearing the same relation to power sums that Chern classes bear to elementary symmetric functions, then we prove the following, $\displaystyle \bar N (k)( {{s_k}(\rho ) - {\text{T}}{{\text{r}}_{H \to G}}({s_k}(\xi ))}) = 0,$ (1) where $\displaystyle \bar N(k) = {\prod _{\begin{array}{*{20}{c}} {p\vert N(k)} {p{\text{prime}}} \end{array}}}p$ (2) and $\displaystyle N(k) = \left( {\begin{array}{*{20}{c}} {\prod\limits_{p{\text{prime}}} {{p^{[k/p - 1]}}}} \end{array} } \right)/k!.$ (3) (Tr denotes transfer.) Moreover, $\bar N (k)$ is the least integer with this property. This settles a question originally raised in a paper of Knopfmacher in which it was conjectured that the required bound was N(k).
Quantitative Korovkin theorems for positive linear operators on $L\sb{p}$-spaces
H.
Berens;
R.
DeVore
349-361
Abstract: Let $({L_n})$ be a sequence of positive linear operators on $ {L_p}(\Omega )$, $1 \leqslant p < \infty$ or $C(\Omega )$ with $\Omega \subseteq {R^m}$. For suitable $\Omega$, the functions $({\varphi _i})_{i = 0}^{m + 1}$ given by ${\varphi _0}(x) \equiv 1$, ${\varphi _i}(x) \equiv {x_i}$, $1 \leqslant i \leqslant m$,and $ {\varphi _{m + 1}}(x) \equiv {\left\vert x \right\vert^2}$ form a test set for $ {L_p}(\Omega )$. That is, if ${L_n}({\varphi _i})$ converges to ${\varphi _i}$ in ${\left\Vert \cdot \right\Vert _p}$ for each $i = 0,1, \ldots ,m + 1$, then $ {L_n}(f)$ converges to f in $ {\left\Vert \cdot \right\Vert _p}$ for each $ f \in {L_p}(\Omega )$. We give here quantitative versions of this result. Namely, we estimate ${\left\Vert {f - {L_n}f} \right\Vert _p}$ in terms of the error $ {\left\Vert {{\varphi _i} - {L_n}{\varphi _i}} \right\Vert _p}$, $0 \leqslant i \leqslant m + 1$,and the smoothness of the function f.
On complete hypersurfaces of nonnegative sectional curvatures and constant $m$th mean curvature
Philip
Hartman
363-374
Abstract: The main result is that if $M = {M^n}$ is a complete Riemann manifold of nonnegative sectional curvature and $X:\,M \to {R^{n + 1}}$ is an isometric immersion such that $X(M)$ has a positive constant mth mean curvature, then $X(M)$ is the product of a Euclidean space ${R^{n - d}}$ and a d-dimensional sphere, $m \leqslant d \leqslant n$.
Harmonic functions and mass cancellation
J. R.
Baxter
375-384
Abstract: If a function on an open set in $ {\textbf{R}^n}$ has the mean value property for one ball at each point of the domain, the function will be said to possess the restricted mean value property. (The ordinary or unrestricted mean value property requires that the mean value property hold for every ball in the domain.) We specify the single ball at each point x by its radius $\delta (x)$, a function of x. Under appropriate conditions on $\delta$ and the function, the restricted mean value property implies that the function is harmonic, giving a converse to the mean value theorem (see references). In the present paper a converse to the mean value theorem is proved, in which the function $ \delta$ is well behaved, but the function is only required to be nonnegative. A converse theorem for more general means than averages over balls is also obtained. These results extend theorems of D. Heath, W. Veech, and the author (see references). Some connections are also pointed out between converse mean value theorems and mass cancellation.
On the algebraic criteria for local Pareto optima. II
Yieh Hei
Wan
385-397
Abstract: Constrained vector optimization problems in the large are studied in this paper. Fixing any constraints so that the feasible set is a manifold with corners, we prove that the set of local Pareto optima for typical vector-valued functions admit Whitney prestratifications. Furthermore, these prestratifications persist under small perturbation of vector-valued functions. The main tools used here are a variant of Thom's transversality theorem and the stratification theory of semialgebraic sets.
Measurable parametrizations and selections
Douglas
Cenzer;
R. Daniel
Mauldin
399-408
Abstract: Let W be a Borel subset of $I \times I$ (where $I = [0,1]$) such that, for each x, $ {W_x} = \{ y:\,(x,y) \in W\}$ is uncountable. It is shown that there is a map, g, of $I \times I$ onto W such that (1) for each x, $ g(x, \cdot )$ is a Borel isomorphism of I onto ${W_x}$ and (2) both g and ${g^{ - 1}}$ are $ S(I \times I)$-measurable maps. Here, if X is a topological space, $ S(X)$ is the smallest family containing the open subsets of X which is closed under operation (A) and complementation. Notice that $S(X)$ is a subfamily of the universally or absolutely measurable subsets of X. This result answers a problem of A. H. Stone. This result improves a theorem of Wesley and as a corollary a selection theorem is obtained which extends the measurable selection theorem of von Neumann. We also show an analogous result holds if W is only assumed to be analytic.
Analytic equations and singularities of plane curves
John J.
Wavrik
409-417
Abstract: Theorems (Artin, Wavrik) exist which show that sufficiently good approximate (power series) solutions to a system of analytic equations may be approximated by convergent solutions. This paper considers the problem of explicity determining the order, $\beta$, to which an approximate solution must solve the system of equations. The paper deals with the case of one equation, $ f(x,y) = 0$, in two variables. It is shown how $\beta$ depends on the singularities of the curve $f(x,y) = 0$. A method for obtaining the minimal $ \beta$ is given. A rapid way of finding $\beta$ using the Newton Polygon for f applies in special cases.
A remark on zeta functions
Jun-ichi
Igusa
419-429
Abstract: In the adelic definition of the zeta function by Tate and Iwasawa, especially in the form given by Weil, one uses all Schwartz-Bruhat functions as ``test functions"; we have found that such an adelic zeta function relative to Q contains the Dedekind zeta function of any finite normal extension of Q and that the normality assumption can be removed if Artin's conjecture is true.
Semifree actions on finite groups on homotopy spheres
John
Ewing
431-442
Abstract: We show that for any finite group the group of semifree actions on homotopy spheres of some fixed even dimension is finite, provided that the dimension of the fixed point set is greater than 2. The argument shows that for such an action the normal bundle to the fixed point set is equivariantly, stably trivial.
A unified approach to measurable and continuous selections
G.
Mägerl
443-452
Abstract: An abstract selection theorem is presented which contains as special cases-among others-the measurable selection theorem of Kuratowski and Ryll-Nardzewski, as well as the continuous selection theorem of Michael.
$4$-manifolds, $3$-fold covering spaces and ribbons
José María
Montesinos
453-467
Abstract: It is proved that a PL, orientable 4-manifold with a handle presentation composed by 0-, 1-, and 2-handles is an irregular 3-fold covering space of the 4-ball, branched over a 2-manifold of ribbon type. A representation of closed, orientable 4-manifolds, in terms of these 2-manifolds, is given. The structure of 2-fold cyclic, and 3-fold irregular covering spaces branched over ribbon discs is studied and new exotic involutions on $ {S^4}$ are obtained. Closed, orientable 4-manifolds with the 2-handles attached along a strongly invertible link are shown to be 2-fold cyclic branched covering spaces of ${S^4}$. The conjecture that each closed, orientable 4-manifold is a 4-fold irregular covering space of ${S^4}$ branched over a 2-manifold is reduced to studying $ \gamma \char93 {S^1} \times {S^2}$ as a nonstandard 4-fold irregular branched covering of ${S^3}$.
Group actions on $A\sb{k}$-manifolds
Hsü Tung
Ku;
Mei Chin
Ku
469-492
Abstract: By an $ {A_k}$-manifold we mean a connected manifold with elements $ {w_i} \in {H^1}(M),\,1 \leqslant i \leqslant k$, such that ${w_1} \cup \, \cdots \cup \,{w_k} \ne 0$. In this paper we study the fixed point set, degree of symmetry, semisimple degree of symmetry and gaps of transformation groups on ${A_k}$-manifolds.
A new approach to the limit theory of recurrent Markov chains
K. B.
Athreya;
P.
Ney
493-501
Abstract: Let $\{ {X_n};\,n \geqslant 0\} $ be a Harris-recurrent Markov chain on a general state space. It is shown that there is a sequence of random times $\{ {N_i};\,i \geqslant 1\} $ such that $ \{ {X_{{N_i}}};{\text{ }}i \geqslant 1\}$ are independent and identically distributed. This idea is used to show that $\{ {X_n}\}$ is equivalent to a process having a recurrence point, and to develop a regenerative scheme which leads to simple proofs of the ergodic theorem, existence and uniqueness of stationary measures.
Contractible $3$-manifolds of finite genus at infinity
E. M.
Brown
503-514
Abstract: A class of contractible open 3-manifolds is defined. It is shown that all contractible open 3-manifolds which can be written as a union of cubes with a bounded number of handles are in this class. It is shown that a proper map between manifolds of this class which induces an isomorphism of proper fundamental groups (e.g. a proper homotopy equivalence) is proper homotopic to a homeomorphism. A naturality condition for homomorphisms of proper fundamental groups is developed. It is shown that a natural homomorphism between the proper fundamental groups of these manifolds is induced by a proper map.
Generalized conjugate function theorems for solutions of first-order elliptic systems on the plane
Chung Ling
Yu
1-35
Abstract: Our essential aim is to generalize Privoloff's theorem, Schwarz reflection principle, Kolmogorov's theorem and the theorem of M. Riesz for conjugate functions to the solutions of differential equations in the $z = x + iy$ plane of the following elliptic type: $\displaystyle \frac{{\partial u}} {{\partial x}} - \frac{{\partial v}} {{\parti... ...\partial u}} {{\partial y}} + \frac{{\partial v}} {{\partial x}} = cu + dv + g.$ ( (M)) Theorem 1. Let the coefficients of (M) be Hölder continuous on $ \left\vert z \right\vert \leqslant 1$. Let $(u, v)$ be a solution of (M) in $\left\vert z \right\vert < 1$. If u is continuous on $\left\vert z \right\vert \leqslant 1$ and Hölder continuous with index $\alpha$ on $\left\vert z \right\vert = 1$, then $ (u, v)$ is Höolder continuous with index $\alpha$ on $\left\vert z \right\vert \leqslant 1$. Theorem 2. Let the coefficients of (M) be continuous on $ \left\vert z \right\vert \leqslant 1$ and satisfy the condition $\displaystyle \int_0^y {b(x,t)dt + \int_0^x {d(t,y)dt = \int_0^y {b(0,t)dt + \int_0^x {d(t,0)dt}}}}$ ( (N)) for $\left\vert z \right\vert \leqslant 1$. And let ${\left\Vert f \right\Vert _p} = {\sup _{0 \leqslant r < 1}}\{ (1/2\pi )\int_{ ... ...i }^\pi {{{\left\vert {f(r{e^{i\theta }})} \right\vert}^p}d\theta {\} ^{1/p}}}$. Then to each p, $0 < p < \infty$, there correspond two constants ${A_p}$ and ${B_p}$ such that \begin{displaymath}\begin{array}{*{20}{c}} {{{\left\Vert v \right\Vert}_p} \leqs... ...Vert u \right\Vert}_1} + {B_p},} & {0 < p < 1,} \end{array} \end{displaymath} hold for every solution $ (u, v)$ of (M) in $ \left\vert z \right\vert < 1$ with $v (0) = 0$. If $ f \equiv g \equiv 0$, the theorem holds for ${B_p} = 0$. Furthermore, if b and d do not satisfy the condition (N) in $ \left\vert z \right\vert \leqslant 1$, then we can relax the condition $ v (0) = 0$, and still have the above inequalities. Theorem 3. Let the coefficients of (M) be analytic for x, y in $ \left\vert z \right\vert < 1$. Let $(u, v)$ be a solution of (M) in $ \{ \left\vert z \right\vert < 1\} \cap \{ y > 0\} $. If u is continuous in $\{ \left\vert z \right\vert < 1\} \cap \{ y \geqslant 0\}$ and analytic on $\{ - 1 < x < 1\} $, then $ (u,v)$ can be continued analytically across the boundary $\{ - 1 < x < 1\} $. Furthermore, if the coefficients and u satisfy some further boundary conditions, then $(u, v)$ can be continued analytically into the whole of $\{ \left\vert z \right\vert < 1\}$.
Existence theorems for Pareto optimization; multivalued and Banach space valued functionals
L.
Cesari;
M. B.
Suryanarayana
37-65
Abstract: Existence theorems are obtained for optimization problems where the cost functional takes values in an ordered Banach space. The order is defined in terms of a closed convex cone in the Banach space; and in this connection, several relevant properties of cones are studied and they are shown to coincide in the finite dimensional case. The notion of a weak (Pareto) extremum of a subset of an ordered Banach space is then introduced. Existence theorems are proved for extrema for Mayer type as well as Lagrange type problems-in a manner analogous to and including those with scalar valued cost. The side conditions are in the form of general operator equations on a class of measurable functions defined on a finite measure space. Needed closure and lower closure theorems are proved. Also, several analytic criteria for lower closure are provided. Before the appendix, several illustrative examples are given. In the appendix, a criterion (different from the one used in main text) is given and proved, for the Pareto optimality of an element.
On the boundary behaviour of functions orthogonal to invariant subspaces
Kar Koi
Leung
67-84
Abstract: In this paper we study the relationship between the boundary behaviour of functions in the orthogonal subspace of the invariant subspace $\varphi {H^2}$ and the boundary behaviour of the inner function $\varphi$ itself.
Two-dimensional $\varepsilon $-isometries
D. G.
Bourgin
85-102
Abstract: An affirmative answer to the antipodal $ \varepsilon$-isometry conjecture is established for 2-dimensional Banach spaces.
Application of the sector condition to the classification of sub-Markovian semigroups
Martin L.
Silverstein
103-146
Abstract: Let ${p_{t}}$, $t > 0$, be a strongly continuous submarkovian semigroup on a real Hilbert space ${L^2}(X, m)$. The measure m is assumed to be excessive and the ${L^2}$ generator A is assumed to satisfy an estimate (the sector condition) which permits the application of Dirichlet spaces (not necessarily symmetric). Other submarkovian semigroups $P_t^ \sim$ with the same local generator and cogenerator and relative to which m is again excessive are classified in terms of generators for processes which live on a suitable boundary.
The Hanf number of the first order theory of Banach spaces
Saharon
Shelah;
Jacques
Stern
147-171
Abstract: In this paper, we discuss the possibility of developing a nice i.e. first order theory for Banach spaces: the restrictions on the set of sentences for recent compactness arguments applied to Banach spaces as well as for other model-theoretic results are both natural and necessary; without them we essentially get a second order logic with quantification over countable sets. Especially, the Hanf number for sets of sentences of the first order theory of Banach spaces is exactly the Hanf number for the second order logic of binary relations (with the second order quantifiers ranging over countable sets).
On the exceptional central simple non-Lie Mal'cev algebras
Renate
Carlsson
173-184
Abstract: Malcev algebras belong to the class of binary Lie algebras. Any Lie algebra is a Malcev algebra. In this paper we show that for each seven-dimensional central simple non-Lie Malcev algebra any finite dimensional Malcev module is completely reducible also for positive characteristics. This contrasts with each modular semisimple Lie algebra. As a consequence we get that the classical structure theory for characteristic zero is valid also in the modular case if semisimplicity is replaced by $ {G_1}$-separability. The Wedderburn principal theorem is proved for Malcev algebras.
The isotopy problem for Jordan matrix algebras
Holger P.
Petersson
185-197
Abstract: Conditions are given which are necessary and sufficient for two members of a certain class of Jordan matrix algebras to be isotopic. The main ingredient of these conditions is McCrimmon's notion of isotopy for alternative algebras.
Rational fibrations, minimal models, and fibrings of homogeneous spaces
Stephen
Halperin
199-224
Abstract: Sullivan's theory of minimal models is used to study a class of maps called rational fibrations, which contains most Serre fibrations. It is shown that if the total space has finite rank and the fibre has finite dimensional cohomology, then both fibre and base have finite rank. This is applied to prove that certain homogeneous spaces cannot be the total space of locally trivial bundles. In addition two main theorems are proved which exhibit a close relation between the connecting homomorphism of the long exact homotopy sequence, and certain properties of the cohomology of fibre and base.
Analytically decomposable operators
Ridgley
Lange
225-240
Abstract: The author introduces the notion of an analytically decomposable operator which generalizes the decomposable operator due to C. Foias in that the spectral decompositions of the underlying Banach space (1) admit a wider class of invariant subspaces called ``analytically invariant'' and (2) span the space only densely. It is shown that analytic decomposability is stable under the functional calculus, direct sums and restrictions to certain kinds of invariant subspaces, as well as perturbation by commuting scalar operators. It is fundamental for many of these results that every analytically decomposable operator has the single-valued extension property. An extensive investigation of analytically invariant subspaces is given. The author shows by example that this class is distinct from those of spectral maximal and hyperinvariant subspaces, but he further shows that analytically invariant subspaces have many useful spectral properties. Some applications of the general theory are made. For example, it is shown that under certain restrictions an analytically decomposable operator is decomposable.
$R$-separation of variables for the four-dimensional flat space Laplace and Hamilton-Jacobi equations
E. G.
Kalnins;
Willard
Miller
241-261
Abstract: All A-separable orthogonal coordinate systems for the complex equations $\Sigma_{i = 1}^4 {{\partial _{ii}}\Psi = 0}$ and $\Sigma_{i = 1}^4 {{{({\partial _i}W)}^2} = 0}$ are classified and it is shown that these equations separate in exactly the same systems.
Algebraic structures for $\bigoplus \sum \sb{n\geq 1}L\sp{2}(Z/n)$ compatible with the finite Fourier transform
L.
Auslander;
R.
Tolimieri
263-272
Abstract: Let ${Z / n}$ denote the integers $ \bmod\,n$ and let ${\mathcal{F}_n}$ denote the finite Fourier transform on ${L^2}({Z / n})$. We let $ \oplus \Sigma {{\mathcal{F}_n}} = F$ operate on $\oplus \Sigma {L^2}({Z / n})$ and show that $\oplus \Sigma {L^2}({Z / n})$ can be given a graded algebra structure (with no zero divisors) such that $\mathcal{F}(fg) = \mathcal{F}(f)\mathcal{F}(g)$. We do this by establishing a natural isomorphism with the algebra of theta functions with period i. In addition, we find all algebra structures on $ \oplus \Sigma {L^2}({Z / n})$ satisfying the above condition.
An asymptotic double commutant theorem for $C\sp{\ast} $-algebras
Donald W.
Hadwin
273-297
Abstract: An asymptotic version of von Neumann's double commutant theorem is proved in which $ {C^{\ast}}$-algebras play the role of von Neumann algebras. This theorem is used to investigate asymptotic versions of similarity, reflexivity, and reductivity. It is shown that every nonseparable, norm closed, commutative, strongly reductive algebra is selfadjoint. Applications are made to the study of operators that are similar to normal (subnormal) operators. In particular, if T is similar to a normal (subnormal) operator and $\pi$ is a representation of the ${C^{\ast}}$-algebra generated by t, then $ \pi (T)$ is similar to a normal (subnormal) operator.
Some $3$-manifolds which admit Klein bottles
Paik Kee
Kim
299-312
Abstract: Consider a closed, orientable, irreducible 3-manifold M with $ \left\vert {{\pi _1}(M)} \right\vert < \infty$, in which a Klein bottle can be embedded. We present a classification of the spaces M and show that, if $ {\pi _1}(M)$ is cyclic, then M is homeomorphic to a lens space. Note that all surfaces of even genus can be embedded in each space M. We also classify all free involutions on lens spaces whose orbit spaces contain Klein bottles.
The eigenvalue spectrum as moduli for flat tori
Scott
Wolpert
313-321
Abstract: A flat torus T carries a natural Laplace Beltrami operator. It is a conjecture that the spectrum of the Laplace Beltrami operator determines T modulo isometries. We prove that, with the exception of a subvariety in the moduli space of flat tori, this conjecture is true. A description of the subvariety is given.
Induced representations of $C\sp{\ast} $-algebras and complete positivity
James G.
Bennett
1-36
Abstract: It is shown that $ ^{*}$representations may be induced from one $ {C^{\ast}}$-algebra B to another $ {C^{\ast}}$-algebra A via a vector space equipped with a completely positive B-valued inner product and a $ ^{*}$representation of A. Theorems are proved on induction in stages, on continuity of the inducing process and on completely positive linear maps of finite dimensional ${C^{\ast}}$-algebras and of group algebras.
Lexicographic partial order
Henry
Crapo
37-51
Abstract: Given a (partially) ordered set P with the descending chain condition, and an ordered set Q, the set $ {Q^P}$ of functions from P to Q has a natural lexicographic order, given by $f \leqslant g$ if and only if $f(y) < g(y)$ for all minimal elements of the set $ \{ x;f(x) \ne g(x)\}$ where the functions differ. We show that if Q is a complete lattice, so also is the set ${Q^P}$, in the lexicographic order. The same holds for the set $ {\operatorname{Hom}}(P,Q)$ of order-preserving functions, and for the set $ {\text{Op}}(P)$ of increasing order-preserving functions on the set P. However, the set $ {\text{Cl}}(P)$ of closure operators on P is not necessarily a lattice even if P is a complete lattice.
Vector valued eigenfunctions of ergodic transformations
E.
Flytzanis
53-60
Abstract: We study the solutions X, T, of the eigenoperator equation $\displaystyle X(h( \cdot ))\,= \,TX( \cdot )\,{\text{a}}{\text{.e}}{\text{.}}$ , where h is a measurable transformation in a $ \sigma$-finite measure space $(S,\Sigma ,m)$, T is a bounded linear operator in a separable Hilbert space H and $X:S \to H$ is Borel measurable. We solve the equation for some classes of measure preserving transformations. For the general case we obtain necessary conditions concerning the eigenoperators, in terms of operators induced by h in the scalar function spaces over the measure space. Finally we investigate integrability properties of the eigenfunctions.
Square integrable representations and a Plancherel theorem for parabolic subgroups
Frederick W.
Keene
61-73
Abstract: Let G be a semisimple Lie group with Iwasawa decomposition $G\, = \,KAN$. Let ${\mathfrak{g}_0} \,=\, {\mathcal{f}_0} \,+\, \mathfrak{a} \,+\, \mathfrak{n}$ be the corresponding decomposition of the Lie algebra of G. Then the nilpotent subgroup N has square integrable representations if and only if the reduced restricted root system is of type ${A_1}$ or ${A_2}$. The Plancherel measure for N can be found explicitly in these cases. We then prove the Plancherel theorem in the ${A_1}$ case for the solvable subgroup NA by combining Mackey's ``Little Group'' method with an idea due to C. C. Moore: we find an operator D, defined on the $ {C^\infty }$ functions on NA with compact support, such that $\displaystyle \phi (e) = \int_{{{(NA)}^\wedge}} {{\text{tr}}} (D\pi (\phi ))d\mu (\pi )$ where ${(NA)^ \wedge }$ is the unitary dual, e is the identity, and $\mu$ is the Plancherel measure for NA, and D is an unbounded selfadjoint operator. In the $ {A_1}$ case, D involves fractional powers of the Laplace operator and hence is not a differential operator.
Nullity and generalized characteristic classes of differential manifolds
Sin Leng
Tan
75-88
Abstract: Using the Kamber-Tondeur construction of characteristic classes for foliated bundes, the author has given a method for constructing generalized characteristic classes for a differentiable manifold M without imposing conditions on M. In particular a vanishing theorem on the manifold M is obtained. The construction is particularly useful if the ordinary characteristic ring Pont*(M) of the manifold M vanishes much below the dimension of M.
Functions operating on positive definite matrices and a theorem of Schoenberg
Jens Peter Reus
Christensen;
Paul
Ressel
89-95
Abstract: We prove that the set of all functions $f:\,[ - 1,\,1] \to [ - 1,\,1]$ operating on real positive definite matrices and normalized such that $f(1)\, = \,1$, is a Bauer simplex, and we identify its extreme points. As an application we obtain Schoenberg's theorem characterising positive definite kernels on the infinite dimensional Hilbert sphere.
Invariant differential equations on certain semisimple Lie groups
F.
Rouvière
97-114
Abstract: If G is a semisimple Lie group with one conjugacy class of Cartan subalgebras (e.g. a complex semisimple Lie group), a bi-invariant differential equation on G can be reduced by means of the Radon transform to one on the subgroup MA. In particular, all polynomials of the Casimir operator have a central fundamental solution, and are solvable in $ {C^\infty }(G)$; but, for G complex, the ``imaginary'' Casimir operator is not.
The minimum norm projection on $C\sp{2}$-manifolds in ${\bf R}\sp{n}$
Theagenis J.
Abatzoglou
115-122
Abstract: We study the notion of best approximation from a point $x \in {R^n}$ to a ${C^2}$-manifold. Using the concept of radius of curvature, introduced by J. R. Rice, we obtain a formula for the Fréchet derivative of the minimum norm projection (best approximation) of $x \in {R^n}$ into the manifold. We also compute the norm of this derivative in terms of the radius of curvature.
Branch point structure of covering maps onto nonorientable surfaces
Cloyd L.
Ezell
123-133
Abstract: Let $f:M\, \to \,N$ be a degree n branched cover onto a compact, connected nonorientable surface with branch points ${y_1},\,{y_2},\, \ldots ,\,{y_m}$ in N, and let the multiplicities at points in ${f^{ - 1}}({y_i})$ be $ {\mu _{i1}},\,{\mu _{i2}},\, \ldots ,\,{\mu _{i{k_i}}}$. The branching array of f, designated by B, is the following array of numbers: \begin{displaymath}\begin{gathered}{\mu _{11}},\,{\mu _{12}},\, \ldots ,\,{\mu _... ...m1}},\,{\mu _{m2}},\, \ldots ,\,{\mu _{m{k_m}}} \end{gathered} \end{displaymath} We show that the numbers in the branching array must always satisfy the following conditions: (1) $\displaystyle \sum {\{ {\mu _{ij}} \,+ \,1\vert j \,=\, 1,\,2,\, \ldots ,\,{k_i}\} \,=\, n}$ , (2) $ \sum {\{ {\mu _{ij}}\vert i\,=\, 1,\,2, \ldots ,m;j\,=\, 1,\,2, \ldots ,{k_i}\} }$ is even. Furthermore, if B is any array of numbers satisfying these conditions, and if N is not the projective plane, then there is a branched cover onto N with B as its branching array.
Pure weak mixing
R.
Ellis;
S.
Glasner
135-146
Abstract: We show that for the group of integers Z, the maximal common factor of the maximal weakly mixing minimal flows coincides with the universal purely weakly mixing flow. We show that the maximal weakly mixing minimal flows are not all isomorphic to each other. None of the maximal w.m. flows are regular.
A note on the operator $X\rightarrow AX-XB$
L.
Fialkow
147-168
Abstract: If A and B are bounded linear operators on an infinite dimensional complex Hilbert space $\mathcal{H}$, let $\tau (X)\, = \,AX\, - \,XB$ (X in $ \mathcal{L}(\mathcal{H})$). It is proved that $\sigma (\tau )\, = \,\sigma (\tau \vert{C_p})\,(1\, \leqslant \,p\, \leqslant \infty )$, where, for $ 1\, \leqslant p\, < \,\infty$, ${C_p}$ is the Schatten p-ideal, and ${C_\infty }$ is the ideal of all compact operators in $ \mathcal{L}(\mathcal{H})$. Analogues of this result for the parts of the spectrum are obtained and sufficient conditions are given for $\tau$ to be injective. It is also proved that if A and B are quasisimilar, then the right essential spectrum of A intersects the left essential spectrum of B.
Absolute continuity in the dual of a Banach algebra
Stephen Jay
Berman
169-194
Abstract: If A is a Banach algebra, G is in the dual space ${A^{\ast}}$, and I is a closed ideal in A, then let ${\left\Vert G \right\Vert _{{I^{\ast}}}}$ denote the norm of the restriction of G to I. We define a relation $\ll$ in $ {A^{\ast}}$ as follows: $G \ll L$ if for every $\varepsilon > 0$ there exists a $\delta > 0$ such that if I is a closed ideal in A and ${\left\Vert L \right\Vert _{{I^{\ast}}}} < \delta$ then ${\left\Vert G \right\Vert _{{I^{\ast}}}} < \varepsilon$. We explore this relation (which coincides with absolute continuity of measures when A is the algebra of continuous functions on a compact space) and related concepts in the context of several Banach algebras, particularly the algebra ${C^1}[0,1]$ of differentiable functions and the algebra of continuous functions on the disc with holomorphic extensions to the interior. We also consider generalizations to noncommutative algebras and Banach modules.
Maxima of random algebraic curves
M.
Das;
S. S.
Bhatt
195-212
Abstract: Let ${X_1},{X_2}, \ldots ,{X_n}$ be a sequence of independent and identically distributed random variables with common characteristic function ${\exp}( - {\left\vert Z \right\vert^\alpha })$ where $0 < \alpha \leqslant 2$, and $P(x) = \sum\nolimits_1^n {{X_k}{x^k}}$. Then we show that the numbers ${M_n}$ of maxima of the curves $y = P(x)$ have expectation $E{M_n} \sim c \log n$, as $n \to \infty$, where $c = c(\alpha ) = {c_1}(\alpha ) + {c_2}(\alpha )$ and \begin{displaymath}\begin{array}{*{20}{c}} {{c_1}(\alpha )\, = \,\frac{1} {{{\pi... ...,\alpha )}}\,\exp \,( - z)\,dz} \right\}} \,dv.} } \end{array} \end{displaymath}
Interpolation by complex splines
J.
Tzimbalario
213-222
Abstract: In this paper we solve the problem of interpolation by certain class of cardinal complex splines. This solution is used to complete the study of cardinal trigonometric splines started in [10] and also to give shorter proofs and to complete the results found for the interpolation problem by complex splines over the unit circle by I. J. Schoenberg [9], J. H. Ahlberg, E. N. Nilson and J. L. Walsh [1].
Absolutely area minimizing singular cones of arbitrary codimension
David
Bindschadler
223-233
Abstract: The examples of area minimizing singular cones of codimension one discovered by Bombieri, DeGiorgi and Guisti are generalized to arbitrary codimension, thus filling a dimensional gap. Previously the only nontrivial examples of singular area minimizing integral currents of codimension other than one were obtained from holomorphic varieties and hence of even codimension. Specifically, let S be the N-fold Cartesian product of p-dimensional spheres and C be the cone over S. We prove that for p sufficiently large, C is absolutely area minimizing. It follows from the technique used that C restricted to the ball of radius ${N^{1/2}}$ is the unique solution to the oriented Plateau problem with boundary S.
Degrees of irreducible characters of $(B,\,N)$-pairs of types $E\sb{6}$ and $E\sb{7}$
David B.
Surowski
235-249
Abstract: Let G be a finite (B, N)-pair whose Coxeter system is of type ${E_6}$ or ${E_7}$. Let $1_B^G$ be the permutation character of the action of G on the left cosets of the Borel subgroup B in G. In this paper we give the character degrees of the irreducible constituents of $ 1_B^G$.
Centers of hypergroups
Kenneth A.
Ross
251-269
Abstract: This paper initiates a study of Z-hypergroups, that is, commutative topological hypergroups K such that $K/Z$ is compact where Z denotes the maximum subgroup (equivalently, the center) of K. The character hypergroup $ {K^\wedge}$ is studied and its connection with the locally compact abelian group ${Z^\wedge}$ is given. Each Z-group is shown to correspond in a natural way to a Z-hypergroup. It is observed that the dual of a Z-group is itself a hypergroup. The basic orthogonality relations on Z-groups due to S. Grosser and M. Moskowitz are shown to hold for most Z-hypergroups. Some results on measure algebras of compact hypergroups due to C. F. Dunkl are extended to a class of noncompact hypergroups.
Amenable pairs of groups and ergodic actions and the associated von Neumann algebras
Robert J.
Zimmer
271-286
Abstract: If X and Y are ergodic G-spaces, where G is a locally compact group, and X is an extension of Y, we study a notion of amenability for the pair $(X,Y)$. This simultaneously generalizes and expands upon previous work of the author concerning the notion of amenability in ergodic theory based upon fixed point properties of affine cocycles, and the work of Eymard on the conditional fixed point property for groups. We study the relations between this concept of amenability, properties of the von Neumann algebras associated to the actions by the Murray-von Neumann construction, and the existence of relatively invariant measures and conditional invariant means.
On the stable decomposition of $\Omega \sp{2}S\sp{r+2}$
E. H.
Brown;
F. P.
Peterson
287-298
Abstract: In this paper we show that $ {\Omega ^2}{S^{r + 2}}$ is stably homotopy equivalent to a wedge of suspensions of other spaces $C_k^1$, and that $C_k^1$ is homotopy 2-equivalent to the Brown-Gitler spectrum.
On approximation by shifts and a theorem of Wiener
R. A.
Zalik
299-308
Abstract: We study the completeness in ${L_2}(R)$ of sequences of the form $\{ f({c_n} - t)\}$, where $\{ {c_n}\}$ is a sequence of distinct real numbers. A Müntztype theorem is proved, valid for a large class of functions and, in particular, for $f(t) = \exp ( - {t^2})$.
Erratum to: ``Periodic homeomorphisms of $3$-manifolds fibered over $S\sp{1}$'' (Trans. Amer. Math. Soc. {\bf 223} (1976), 223--234)
Jeffrey L.
Tollefson
309-310
Haar measure for measure groupoids
Peter
Hahn
1-33
Abstract: It is proved that Mackey's measure groupoids possess an analogue of Haar measure for locally compact groups; and many properties of the group Haar measure generalize. Existence of Haar measure for groupoids permits solution of a question raised by Ramsay. Ergodic groupoids with finite Haar measure are characterized.
The regular representations of measure groupoids
Peter
Hahn
35-72
Abstract: Techniques are developed to study the regular representation and $ \sigma$-regular representations of measure groupoids. Convolution, involution, a modular Hilbert algebra, and local and global versions of the regular representation are defined. The associated von Neumann algebras, each uniquely determined by the groupoid and the cocycle $\sigma$, provide a generalization of the group-measure space construction. When the groupoid is principal and ergodic, these algebras are factors. Necessary and sufficient conditions for the $ \sigma$-regular representations of a principal ergodic groupoid to be of type I, II, or III are given, as well as a description of the flow of weights; these are independent of $ \sigma$. To treat nonergodic groupoids, an ergodic decomposition theorem is provided.
Processes with independent increments on a Lie group
Philip
Feinsilver
73-121
Abstract: The Lévy-Khinchin representation for processes with independent increments is extended to processes taking values in a Lie group. The basis of the proof is to approximate continuous time processes by Markov chains. The processes involved are handled by the technique, developed by Stroock and Varadhan, of characterizing Markov processes by associated martingales.
The asymptotic behaviour of certain integral functions
P. C.
Fenton
123-140
Abstract: Let$f(z)$ be an integral function satisfying $\displaystyle {\int_{}^\infty \{\log \,m(r,f)\, - \,\cos \,\pi \rho \,\log \,M(r,f)\} ^ + }\frac{{dr}}{{{r^{\rho + 1}}}}\, < \,\infty$ and $\displaystyle 0\, < \,\mathop {\lim }\limits_{\overline {r\, \to \infty } } \,\frac{{\log \,M(r,f)}}{{{r^\rho }}}\, < \,\infty$ for some $ \rho :\,0\, < \,\rho \, < \,1$. It is shown that such functions have regular asymptotic behaviour outside a set of circles with centres ${\zeta _i}$ and radii ${t_i}$ for which $\displaystyle \sum\limits_{i = 1}^\infty {\frac{{{t_i}}}{{\left\vert {{\zeta _i}} \right\vert}}} < \infty$ .
Alternators of a right alternative algebra
Irvin Roy
Hentzel
141-156
Abstract: We show that in any right alternative algebra, the additive span of the alternators is nearly an ideal. We give an easy test to use to determine if a given set of additional identities will imply that the span of the alternators is an ideal. We apply our technique to the class of right alternative algebras satisfying the condition $(a,a,b) = \lambda [a,[a,b]]$. We show that any semiprime algebra over a field of characteristic $ \ne 2$, $\ne 3$ which satisfies the right alternative law and the above identity with $\lambda \ne 0$ is a subdirect sum of (associative and commutative) integral domains.
The cohomology of the symmetric groups
Benjamin Michael
Mann
157-184
Abstract: Let ${{\mathcal{S}}_n}$ be the symmetric group on n letters and SG the limit of the sets of degree +1 homotopy equivalences of the $n - 1$ sphere. Let p be an odd prime. The main results of this paper are the calculations of $ {H^{\ast}}({\mathcal{S}_n},\,Z/p)$ and $ {H^{\ast}}(SG,Z/p)$ as algebras, determination of the action of the Steenrod algebra, $ \mathcal{a}(p)$, on $ {H^{\ast}}({\mathcal{S}_n},\,Z/p)$ and $ {H^{\ast}}(SG,Z/p)$ and integral analysis of $ {H^{\ast}}({\mathcal{S}_n},\,Z,\,p)$ and $ {H^{\ast}}(SG,\,Z,\,p)$.
The invariant $\Pi \sp{0}\sb{\alpha }$ separation principle
Douglas E.
Miller
185-204
Abstract: We ``invariantize'' the classical theory of alternated unions to obtain new separation results in both invariant descriptive set theory and in infinitary logic. Application is made to the theory of definitions of countable models.
The traces of holomorphic functions on real submanifolds
Gary Alvin
Harris
205-223
Abstract: Suppose M is a real-analytic submanifold of complex Euclidean n = space and consider the following question: Given a real-analytic function f defined on M, is f the restriction to M of an ambient holomorphic function? If M is a C.R. submanifold the question has been answered completely. Namely, f is the trace of a holomorphic function if and only if f is a C.R. function. The more general situation in which M need not be a C.R. submanifold is discussed in this paper. A complete answer is obtained in case the dimension of M is larger than or equal to n and M is generic in some neighborhood of each point off its C.R. singularities. The solution is of infinite order and follows from a consideration of the following problem: Given a holomorphic function f and a holomorphic mapping $\Phi$, when does there exist a holomorphic mapping F such that $f = F \circ \Phi$?
Sampling theorems for nonstationary random processes
Alan J.
Lee
225-241
Abstract: Consider a second order stochastic process $\{ X(t),t \in \textbf{R}\} $, and let $H(X)$ be the Hilbert space generated by the random variables of the process. The process is said to be linearly determined by its samples $\{ X(nh),n \in \textbf{Z}\}$ if the random variables $X(nh)$ generate $H(X)$. In this paper we give a sufficient condition for a wide class of nonstationary processes to be determined by their samples, and present sampling theorems for such processes. We also consider similar problems for harmonizable processes indexed by LCA groups having suitable subgroups.
Isomorphic factorisations. I. Complete graphs
Frank
Harary;
Robert W.
Robinson;
Nicholas C.
Wormald
243-260
Abstract: An isomorphic factorisation of the complete graph ${K_p}$ is a partition of the lines of $ {K_p}$ into t isomorphic spanning subgraphs G; we then write $ G\vert{K_p}$, and $G \in {K_p}/t$. If the set of graphs ${K_p}/t$ is not empty, then of course $t\vert p(p - 1)/2$. Our principal purpose is to prove the converse. It was found by Laura Guidotti that the converse does hold whenever $(t,p) = 1$ or $ (t,p - 1) = 1$. We give a new and shorter proof of her result which involves permuting the points and lines of ${K_p}$. The construction developed in our proof happens to give all the graphs in ${K_6}/3$ and ${K_7}/3$. The Divisibility Theorem asserts that there is a factorisation of $ {K_p}$ into t isomorphic parts whenever t divides $p(p - 1)/2$. The proof to be given is based on our proof of Guidotti's Theorem, with embellishments to handle the additional difficulties presented by the cases when t is not relatively prime to p or $p - 1$.
Interpretation of the $p$-adic log gamma function and Euler constants using the Bernoulli measure
Neal
Koblitz
261-269
Abstract: A regularized version of J. Diamond's p-adic log gamma function and his p-adic Euler constants are represented as integrals using B. Mazur's p-adic Bernoulli measure.
The homotopy continuation method: numerically implementable topological procedures
J. C.
Alexander;
James A.
Yorke
271-284
Abstract: The homotopy continuation method involves numerically finding the solution of a problem by starting from the solution of a known problem and continuing the solution as the known problem is homotoped to the given problem. The process is axiomatized and an algebraic topological condition is given that guarantees the method will work. A number of examples are presented that involve fixed points, zeroes of maps, singularities of vector fields, and bifurcation. As an adjunct, proofs using differential rather than algebraic techniques are given for the Borsuk-Ulam Theorem and the Rabinowitz Bifurcation Theorem.
Curves with large tangent space
Joseph
Becker;
Rajendra
Gurjar
285-296
Abstract: Theorem. Let V be a complex analytic variety irreducible at a point $p \in V$. Givén any integer l, there exists an analytic curve $ {C_l}$ on V passing through p and irreducible at p such that the germs of $ {C_l}$ and V at p are isomorphic up to order l.
A probable Hasse principle for pencils of quadrics
William C.
Waterhouse
297-306
Abstract: Let k be a global field, ${\text{char}}(k) \ne 2$. Although pencils of quadrics over k may fail to satisfy a local-to-global equivalence principle, the failures are exceptional in the precise sense of having limiting probability zero. The proof uses the classification of pairs of quadratic forms. It also requires knowing that a square class in a finite extension usually comes from k when it does so locally; the Galois-theoretic criterion for this is determined.
Pullback de Rham cohomology of the free path fibration
Kuo-Tsai
Chen
307-318
Abstract: Let M and N be smooth manifolds and let $\bar B (A)$ be the reduced bar construction on the de Rham complex $ \Lambda (M)$ or a suitable subcomplex A of M. For every smooth map $ f:N \to M \times M$, the tensor product $\Lambda (N) \otimes \bar B(A)$, equipped with a suitable differential, will yield the correct cohomology for the pullback of the free path fibration $P(M) \to M \times M$ via the smooth map F. Moreover, $ \Lambda (N) \otimes \bar B(A)$ can be taken as a de Rham subcomplex of the pullback space.
The product of nonplanar complexes does not imbed in $4$-space
Brian R.
Ummel
319-328
Abstract: We prove that if $ {K_1}$ and ${K_2}$ are nonplanar simplicial complexes, then $ {K_1}\, \times\, {K_2}$ does not imbed in $ {{\textbf{R}}^4}$.
Systems of $n$ partial differential equations in $n$ unknown functions: the conjecture of M. Janet
Joseph
Johnson
329-334
Abstract: It was conjectured by Janet that an analytic solution to a system of n ``independent'' analytic differential equations in n unknown functions if not isolated must depend on at least one unknown function of $m - 1$ variables plus possibly other functions of fewer than m variables. Here m is the dimension of the complex domain on which the equations and the solution are given. An algebraic generalization of the linear form of the conjecture is proven. Also the result is extended to give a nonlinear version.
The ninety-one types of isogonal tilings in the plane
Branko
Grünbaum;
G. C.
Shephard
335-353
Abstract: A tiling of the plane by closed topological disks of isogonal if its symmetries act transitively on the vertices of the tiling. Two isogonal tilings are of the same type provided the symmetries of the tiling relate in the same way every vertex in each to its set of neighbors. Isogonal tilings were considered in 1916 by A. V. Šubnikov and by others since then, without obtaining a complete classification. The isogonal tilings are vaguely dual to the isohedral (tile transitive) tilings, but the duality is not strict. In contrast to the existence of 81 isohedral types of planar tilings we prove the following result: There exist 91 types of isogonal tilings of the plane in which each tile has at least three neighbors.
$R$-separable coordinates for three-dimensional complex Riemannian spaces
C. P.
Boyer;
E. G.
Kalnins;
Willard
Miller
355-376
Abstract: We classify all R-separable coordinate systems for the equations $\Sigma _{i,j = 1}^3\,{g^{ - 1/2}}{\partial _j}({g^{1/2}}{g^{ij}}{\partial _i}\psi ) = 0$ and $\Sigma_{i,j\, = \,1}^3 {{g^{ij}}{\partial _i}W{\partial _j}W\, = \,0}$ with special emphasis on nonorthogonal coordinates, and give a group-theoretic interpretation of the results. We show that for flat space the two equations separate in exactly the same coordinate systems.
Classification of circle actions on $4$-manifolds
Ronald
Fintushel
377-390
Abstract: This article studies locally smooth ${S^1}$-actions on closed oriented 4-manifolds in terms of the orbit space, orbit type data, and the characteristic class of the action which lies in $ {H_1}({M^{\ast}},{S^{\ast}})$ where $ {M^{\ast}}$ is the orbit space and $ {S^{\ast}}$ is the image of a certain collection of singular orbits. It is proved that such actions are determined by their weighted orbit spaces and are in 1-1 correspondence with ``legally-weighted'' 3-manifolds. The information contained in the weighted orbit space is used to give a presentation of the fundamental group of the 4-manifold, and in certain cases the quadratic form is computed.
The Diophantine problem for polynomial rings and fields of rational functions
J.
Denef
391-399
Abstract: We prove that the diophantine problem for a ring of polynomials over an integral domain of characteristic zero or for a field of rational functions over a formally real field is unsolvable.
A theorem of Ahlfors for hyperbolic spaces
Su Shing
Chen
401-406
Abstract: L. Ahlfors has proved that if the Dirichlet fundamental polyhedron of a Kleinian group G in the unit ball ${B^3}$ has finitely many sides, then the normalized Lebesgue measure of $L(G)$ is either zero or one. We generalize this theorem and a theorem of Beardon and Maskit to the n-dimensional case.
The order and symbol of a distribution
Alan
Weinstein
1-54
Abstract: A definition is given, for an arbitrary distribution g on a manifold X, of the order and symbol of g at a point $({\chi ,\xi })$ of the cotangent bundle $ T^{\ast}X$. If $X = \textbf{R}^n$, the order of g at $ ({0,\xi})$ is the growth order as $\tau \to \infty $ of the distributions $ {g^\tau }(x) = {e^{ - i\sqrt \tau \langle x,\xi \rangle }}g\left( {x /\sqrt \tau } \right)$ ; if the order is less than or equal to N, the N-symbol of g is the family $ {g^\tau }$ modulo $O({{\tau ^{N - 1/2}}})$. It is shown that the order and symbol behave in a simple way when g is acted upon by a pseudo-differential operator. If g is a Fourier integral distribution, suitable identifications can be made so that the symbol defined here agrees with the bundle-valued symbol defined by Hörmander.
Groups of PL $\Lambda $-homology spheres
Gerald A.
Anderson
55-67
Abstract: Let $ \wedge \, = \,{{\textbf{Z}}_k}\,$ be a subring of Q. The group of $ {H_K}$-cobordism classes of closed PL n-manifolds with the $ \wedge$-homology of $ {S^n}$ is computed for $ n \ge 4$ (modulo K-torsion). The simply connected version is also computed.
Solution of the nonlinear problem $Au=N(u)$ in a Banach space
Martin
Schechter;
Jack
Shapiro;
Morris
Snow
69-78
Abstract: We solve a nonlinear problem $Au\, = \,N(u)$ where A is semi-Fredholm and N is a nonlinear compact operator.
Ultrafilter invariants in topological spaces
Victor
Saks
79-97
Abstract: Let $ \mathfrak{m}\, \geqslant \,{\aleph _0}$ and $X\, = \,\prod\nolimits_{i\, \in \,I} {{X_i}}$. Then X is $[{\aleph _0},\,\mathfrak{m}]$-compact if and only if $\prod\nolimits_{i\, \in \,J} {{X_i}}$ is $ [{\aleph _0},\,\mathfrak{m}]$-compact for all $ J\, \subset \,I$ with $\vert J\vert\, \leqslant \,{2^{{2^\mathfrak{m}}}}$. Let $\mathfrak{m}\, \geq \,{\aleph _0}$, $ ({x_\xi }:\,\xi \, < \,\mathfrak{m})$ a net in X, $p\, \in \,X$, and $\mathcal{D}\, \in \,\beta (\mathfrak{m})$. Then $p\, = \,\mathcal{D}\, - \,{\lim _{\xi < \mathfrak{m}}}\,{x_\xi }$ if $ \{ \xi \, < \,\mathfrak{m}:\,{x_\xi }\, \in \,U \} \, \in \, \mathcal{D}$ for every neighborhood U of p. Every topological space is characterized by its $\mathcal{D}$-limits. Several topological properties are described using ultrafilter invariants, including compactness and perfect maps. If X is a Hausdorff space and D is a discrete space equipotent with a dense subset of X, then X is a continuous perfect image of a subspace of $\beta D$ which contains D if and only if X is regular.
The BP homology of $H$-spaces
Richard
Kane
99-119
Abstract: The BP homology of 1-connected H-spaces whose loop space is torsion free is studied. It is shown that localizing in a suitable manner kills all torsion in $ B{P_{\ast}}(X)$. Similar results are then obtained for the bordism of X. Finally the K-theory of X is shown to have no p torsion.
Undecidable existential problems for addition and divisibility in algebraic number rings
L.
Lipshitz
121-128
Abstract: Existential formulas involving addition and divisibility are shown to be undecidable in the ring of integers of a real quadratic extension of the rationals. A weaker result is proved for extensions of higher degree.
Distances between two-state Markov processes attainable by Markov joinings
Martin H.
Ellis
129-153
Abstract: The function which assigns to each pair of two-state Markov processes the set of partition distances between them attainable by a Markov process on their joint atoms is computed. It is found that the infimum of these distances, the ``Markov distance'' between the pair, fails to satisfy the Triangle Inequality, hence fails to be a metric; thus in some cases the $\overline d$-distance between two two-state Markov processes cannot be attained by a Markov process on their joint atoms.
Singularly perturbed boundary value problems with angular limiting solutions
F. A.
Howes
155-182
Abstract: A basic result of Haber and Levinson which describes the behavior of solutions of $y(a,\varepsilon )$, $ y(b,\varepsilon )$, prescribed, in the presence of a reduced solution with corners is modified to treat related classes of problems. Under various stability assumptions, solutions are shown to remain, for small $\varepsilon \, > \,0$, in a o(l)-neighborhood of an angular reduced solution with the possible exception of narrow layers near the boundaries in some cases. Each aspect of the theory developed here is illustrated by several examples.
Characterizations of amenable groups
William R.
Emerson
183-194
Abstract: Generalizing a construction of Banach from 1923 we obtain new criteria for the amenability of a locally compact group G. The relationship of these new criteria to known characterizations is then investigated, and in particular a formally strenghthened version of the von Neumann/Dixmier condition for amenability is established.
Topologies on spaces of vector-valued continuous functions
Surjit Singh
Khurana
195-211
Abstract: Topologies ${\beta _0},{\beta _1},\beta ,{\beta _\infty },{\beta _{\infty c}}$ are defined on $ {C_b}(X,E)$, the space of all bounded, continuous functions from a completely regular Hausdorff space X, into E, a normed space, and their duals are determined. Also many properties of these topologies are proved.
$3$-pseudomanifolds with preassigned links
Amos
Altshuler
213-237
Abstract: A 3-pseudomanifold is a finite connected simplicial 3-complex $\mathcal{K}$ such that every triangle in $\mathcal{K}$ belongs to precisely two 3-simplices of $\mathcal{K}$, the link of every edge in $\mathcal{K}$ is a circuit, and the link of every vertex in $ \mathcal{K}$ is a closed 2-manifold. It is proved that for every finite set $ \sum$ of closed 2-manifolds, there exists a 3-pseudomanifold $\mathcal{K}$ such that the link of every vertex in $\mathcal{K}$ is homeomorphic to some $S\, \in \,\sum$, and every $S\, \in \,\sum$ is homeomorphic to the link of some vertex in $ \mathcal{K}$.
On the growth of the integral means of subharmonic functions of order less than one
Faruk F.
Abi-Khuzam
239-252
Abstract: Let u be a subharmonic function of order $\lambda (0 < \lambda < 1)$, and let ${m_s}(r,u)\, = \,{\left\{ {(1/2\pi )\int_{ - \pi }^\pi {{{\left\vert {u(r{e^{i\theta }})} \right\vert}^s}} d\theta } \right\}^{1/s}}$. We compare the growth of $ {m_s}(r,u)$ with that of the Riesz mass of u as measured by $N\,(r,u)\, = (1/2\pi )\int_{ - \pi }^\pi {u(r{e^{i\theta }})d\theta }$. A typical result of this paper states that the following inequality is sharp: $\displaystyle \underset{x\to \infty }{\mathop{\lim \,\inf }}\,\,\frac{{{m}_{s}}... ...N\left( r,\,u \right)}\,\leqslant\,{{m}_{s}}\left( {{\psi }_{\lambda }} \right)$ ($ \ast$) where $ \psi_\lambda (\theta )\, = \,(\pi \lambda /\sin \,\lambda )\cos \,\lambda \theta$. The case $s\, = \,1$ is due to Edrei and Fuchs, the case $ s\, = \,2$ is due to Miles and Shea and the case $ s\, = \,\infty$ is due to Valiron.
Standard and normal reductions
R.
Hindley
253-271
Abstract: Curry and Feys' original standardization proof for $\lambda \beta$-reduction is analyzed and generalized to $\lambda \beta \eta$-reductions with extra operators. There seem to be two slightly different definitions of 'standard reduction' in current use, without any awareness that they are different; it is proved that although these definitions turn out to be equivalent for $ \lambda \beta$-reduction, they become different for $\lambda \beta \eta$ and for reductions involving extra operators, for example the recursion operator. Normal reductions are also studied, and it is shown that the basic normal-reduction theorem stays true when fairly simple operators like Church's $\delta$ and Curry's iterator Z are added, but fails for more complicated ones like the recursion operator R. Finally, a table is given summarizing the results, and showing how far the main theorems on $ \lambda \beta$-reductions extend to reductions with various extra operators.
Weak uniqueness sets for discrete groups
Marek
Bożejko;
Tadeusz
Pytlik
273-282
Abstract: For discrete groups we introduce a new class of sets, called weak uniqueness sets, which for abelian groups contains the class of sets of uniqueness. Considered is the problem of determining groups for which every finite set is a weak uniqueness set. Some examples are given.
Partition theorems and ultrafilters
James E.
Baumgartner;
Alan D.
Taylor
283-309
Abstract: We introduce a class of ultrafilters on $\omega$ called k-arrow ultrafilters and characterized by the partition relation $U \to {(U,k)^2}$. These are studied in conjunction with P-points, Q-points, weakly Ramsey and Ramsey ultrafilters.
Structural stability for flows on the torus with a cross-cap
Carlos
Gutiérrez
311-320
Abstract: Let $ {{\mathcal{X}}^r}({\tilde M}), r \geq 1$, denote the space of ${C^r}$-vector fields on the torus with a cross-cap $\tilde M$. We show that the Morse-Smale vector fields of ${{\mathcal{X}}^r}({\tilde M})$ are dense on it. We also give a simple proof that a ${C^0}$-flow on the Klein bottle cannot support a nontrivial $\omega$-recurrent trajectory.
Regular Hjelmslev planes. II
Dieter
Jungnickel
321-330
Abstract: In this paper we introduce the notion of an x-partition for a Hjelmslev-matrix (H-matrix). This allows us to prove a new composition theorem for H-matrices. We obtain the existence of $ ({t,\,r})$-H-matrices and hence of regular $ ({t,\,r})$-H-planes for infinitely many series of invariants which were not yet known. In fact, many of these invariants were not even known to occur as the invariants of any H-plane at all (whether regular or not).
The semigroup of varieties of Brouwerian semilattices
Peter
Köhler
331-342
Abstract: It is shown that the semigroup of varieties of Brouwerian semilattices is free.
Desarguesian Klingenberg planes
P. Y.
Bacon
343-355
Abstract: Klingenberg planes are generalizations of Hjelmslev planes. If R is a local ring, one can construct a projective Klingenberg plane $ {\textbf{V}}(R)$ and a derived affine Klingenberg plane ${\textbf{A}}(R)$ from R. If V is a projective Klingenberg plane, if ${R_1},\,{R_2}$ and ${R_3}$ are local rings, if ${s_1},\,{s_2}$ and ${s_3}$ are the sides of a nondegenerate triangle in V, and if each of the derived affine Klingenberg planes $\mathcal{a}\left( {V,\,{s_i}} \right)$ is isomorphic to $ {\textbf{A}}({R_i}),\,$, $i\, = \,1,\,2,\,3$, then the rings $ {R_1},\,{R_2}$ and $ {R_3}$ are isomorphic, and V is isomorphic to ${\textbf{V}}({R_1});$; also, if g is a line of V, then the derived affine Klingenberg plane $ \mathcal{a}({V,\,g})$ is isomorphic to $ \textbf{A}({R_1})$. Examples are given of projective Klingenberg planes V, each of which has the following two properties: (1) V is not isomorphic to ${\textbf{V}}(R)$ for any local ring R; and (2) there is a flag $(B,\,b)$ of V, and a local ring S such that each derived affine Klingenberg plane $\mathcal{a}({V,\,m})$ is isomorphic to ${\textbf{A}}(S)$ whenever $m\, = \,b$, or m is a line through B which is not neighbor to b.
Isosingular loci and the Cartesian product structure of complex analytic singularities
Robert
Ephraim
357-371
Abstract: Let X be a (not necessarily reduced) complex analytic space, and let V be a germ of an analytic space. The locus of points q in X at which the germ $ {X_q}$ is complex analytically isomorphic to V is studied. If it is nonempty it is shown to be a locally closed submanifold of X, and X is locally a Cartesian product along this submanifold. This is used to define what amounts to a coarse partial ordering of singularities. This partial ordering is used to show that there is an essentially unique way to completely decompose an arbitrary reduced singularity as a cartesian product of lower dimensional singularities. This generalizes a result previously known only for irreducible singularities.
Uniqueness in the Cauchy problem for parabolic equations
Roger M.
Hayne
373-399
Abstract: In a classical paper S. Täcklind (Nova Acta Soc. Sci. Upsaliensis (4) 10 (1936), 1-57) closed the uniqueness question for the Cauchy problem for the heat equation with a general growth hypothesis which was both necessary and sufficient. Täcklind's proof of the sufficiency involved an ingenious bootstrapping comparison technique employing the maximum principle and a comparison function constructed from the Green's function for a half cylinder. G. N. Zolotarev (Izv. Vysš. Učebn. Zaved. Matematika 2 (1958), 118-135) has extended this result using essentially the same technique to show that Täcklind's uniqueness condition remains sufficient for a general second order parabolic equation provided the coefficients are regular enough to permit the existence and estimation of a Green's function. We have now shown, using a new approach which replaces the construction based upon a Green's function by an appropriate comparison solution of the maximizing equation (C. Pucci, Ann. Mat. Pura Appl. 72 (1966), 141-170), that Täcklind's condition is sufficient without any regularity conditions on the coefficients.
A characterization of upper-embeddable graphs
Mark
Jungerman
401-406
Abstract: It is proved that a pseudograph G is upper-embeddable if and only if it has a spanning tree T such that G - T has at most one component with an odd number of edges. This result is then used to show that all 4-edge connected graphs are upper-embeddable.
Hypoellipticity on the Heisenberg group-representation-theoretic criteria
Charles
Rockland
1-52
Abstract: A representation-theoretic characterization is given for hypoellipticity of homogeneous (with respect to dilations) left-invariant differential operators P on the Heisenberg group ${H_n}$; it is the precise analogue for ${H_n}$ of the statement for ${{\mathbf{R}}^n}$ that a homogeneous constant-coefficient differential operator is hypoelliptic if and only if it is elliptic. Under these representation-theoretic conditions a parametrix is constructed for P by means of the Plancherel formula. However, these conditions involve all the irreducible representations of ${H_n}$, whereas only the generic, infinite-dimensional representations occur in the Plancherel formula. A simple class of examples is discussed, namely $P = \Sigma _{i = 1}^nX_i^{2m} + Y_i^{2m}$, where ${X_i},{Y_i},i = 1, \ldots ,n$, and Z generate the Lie algebra of ${H_n}$ via the commutation relations $[{X_i},{Y_j}] = {\delta _{ij}}Z$, and where m is a positive integer. In the course of the proof a connection is made between homogeneous left-invariant operators on ${H_n}$ and a class of degenerate-elliptic operators on $ {{\mathbf{R}}^{n + 1}}$ studied by Grušin. This connection is examined in the context of localization in enveloping algebras.
The conjugacy problem for boundary loops in $3$-manifolds
Benny D.
Evans
53-64
Abstract: A geometric solution of the word problem for fundamental groups of compact, orientable, irreducible, sufficiently large 3-manifolds has been given by F. Waldhausen. We present here a solution of a restricted version of the conjugacy problem for this same class of 3-manifolds; however, the conjugacy problem for 3-manifolds remains in general unsolved. The main results is that there is an algorithm that will determine for any two loops ${L_1},{L_2}$ in the boundary of a compact, orientable, irreducible sufficiently large 3-manifold M if ${L_1}$, is freely homotopic in M to $ {L_2}$.
Erickson's conjecture on the rate of escape of $d$-dimensional random walk
Harry
Kesten
65-113
Abstract: We prove a strengthened form of a conjecture of Erickson to the effect that any genuinely d-dimensional random walk $ {S_n},d \geqslant 3$, goes to infinity at least as fast as a simple random walk or Brownian motion in dimension d. More precisely, if $S_n^\ast$ is a simple random walk and $ {B_t}$, a Brownian motion in dimension d, and $\psi :[1,\infty ) \to (0,\infty )$ a function for which ${t^{ - 1/2}}\psi (t) \downarrow 0$, then $\psi {(n)^{ - 1}}\vert S_n^\ast\vert \to \infty$ w.p.l, or equivalently, $\psi {(t)^{ - 1}}\vert{B_t}\vert \to \infty$ w.p.l, iff $\smallint _1^\infty \psi {(t)^{d - 2}}{t^{ - d/2}} < \infty $; if this is the case, then also $\psi {(n)^{ - 1}}\vert{S_n}\vert \to \infty$ w.p.l for any random walk Sn of dimension d.
The cohomology of semisimple Lie algebras with coefficients in a Verma module
Floyd L.
Williams
115-127
Abstract: The structure of the cohomology of a complex semisimple Lie algebra with coefficients in an arbitrary Verma module is completely determined. Because the Verma modules are infinite-dimensional, the cohomology need not vanish (as it does for nontrivial finite-dimensional modules). The methods presented exploit the homological machinery of Cartan-Eilenberg [3]. The results of [3], when applied to the universal enveloping algebra of a semisimple Lie algebra and when coupled with key results of Kostant [12], Hochschild-Serre [9], yield the basic structure theorem-Theorem 4.19. Our results show, incidently, that an assertion of H. Kimura, Theorem 2 of [13] is false. A counterexample is presented in §6.
Existence, stability, and compactness in the $\alpha $-norm for partial functional differential equations
C. C.
Travis;
G. F.
Webb
129-143
Abstract: The abstract ordinary functional differential equation $ (a/dt)u(t) = - Au(t) + F({u_t}),{u_0} = \phi$, is studied, where $- A$ is the infinitesimal generator of an analytic semigroup of linear operators and F is continuous with respect to a fractional power of A.
Some exponential moments of sums of independent random variables
J.
Kuelbs
145-162
Abstract: If $\{ {X_n}\}$ is a sequence of vector valued random variables, $ \{ {a_n}\}$ a sequence of positive constants, and $M = {\sup _{n \geqslant 1}}\left\Vert {({X_1} + \cdots + {X_n})/{a_n}} \right\Vert$, we examine when $E(\Phi (M)) < \infty$ under various conditions on $ \Phi ,\{ {X_n}\}$, and $\{ {a_n}\}$. These integrability results easily apply to empirical distribution functions.
Variations, characteristic classes, and the obstruction to mapping smooth to continuous cohomology
Mark A.
Mostow
163-182
Abstract: In a recent paper, the author gave an example of a singular foliation on $ {{\mathbf{R}}^2}$ for which it is impossible to map the de Rham cohomology ${T_{{\text{DR}}}}$ to the continuous singular cohomology $ {T_{\text{c}}}$ (in the sense of Bott and Haefliger's continuous cohomology of spaces with two topologies) compatibly with evaluation of cohomology classes on homology classes. In this paper the obstruction to mapping ${T_{{\text{DR}}}}$ to ${T_{\text{c}}}$ is pinpointed by defining a whole family of cohomology theories ${T_{k,m,n}}$, based on cochains which vary in a $ {C^k}$ manner, which mediate between the two. It is shown that the obstruction vanishes on nonsingularly foliated manifolds. The cohomology theories are extended to Haefliger's classifying space $ (B{\Gamma _q} \to B{J_q})$, with its germ and jet topologies, by using a notion of differentiable space similar to those of J. W. Smith and K. T. Chen. The author proposes that certain of the ${T_{kmn}}$ be used instead of ${T_{\text{c}}}$ to study Bott and Haefliger's conjecture that the continuous cohomology of $(B{\Gamma _q} \to B{J_q})$ equals the relative Gel'fand-Fuks cohomology $ {H^\ast}({\mathfrak{a}_q},{O_q})$. It is shown that ${T_{kmn}}(B{\Gamma _q} \to B{J_q})$ may contain new characteristic classes for foliations which vary only in a ${C^k}$ manner when a foliation is varied smoothly.
Operators with small self-commutators
J. W.
Del Valle
183-194
Abstract: Let A be a bounded operator on a Hilbert space H. The self-commutator of A, denoted [A], is ${A^\ast}A - A{A^\ast}$. An operator is of commutator rank n if the rank of [A] is n. In this paper operators of commutator rank one are studied. Two particular subclasses are investigated in detail. First, completely nonnormal operators of commutator rank one for which ${A^\ast}A$ and $A{A^\ast}$ commute are completely characterized. They are shown to be special types of simple weighted shifts. Next, operators of commutator rank one for which $ \{ {A^n}e\} _{n = 0}^\infty$ is an orthogonal sequence (where e is a generator of the range of [A]) are characterized as a type of weighted operator shift.
On the growth of solutions of algebraic differential equations
Steven B.
Bank
195-212
Abstract: In this paper we determine estimates for the growth of both real-valued and complex-valued solutions of algebraic differential equations on an interval $({x_0}, + \infty )$. One of the main results of the paper (Theorem 3) confirms E. Borel's conjecture on the growth of real-valued solutions for a broad class of solutions of second-order algebraic differential equations. The conjecture had previously been shown to be false for third-order equations.
Classifying open principal fibrations
David A.
Edwards;
Harold M.
Hastings
213-220
Abstract: Let G be a compact metric group. We shall construct classifying spaces for open principal G-fibrations over compact metric spaces.
Continuous maps of the interval with finite nonwandering set
Louis
Block
221-230
Abstract: Let f be a continuous map of a closed interval into itself, and let $\Omega (f)$ denote the nonwandering set of f. It is shown that if $ \Omega (f)$ is finite, then $\Omega (f)$ is the set of periodic points of f. Also, an example is given of a continuous map g, of a compact, connected, metrizable, one-dimensional space, for which $ \Omega (g)$ consists of exactly two points, one of which is not periodic.
Ultrapowers and local properties of Banach spaces
Jacques
Stern
231-252
Abstract: The present paper is an approach to the local theory of Banach spaces via the ultrapower construction. It includes a detailed study of ultrapowers and their dual spaces as well as a definition of a new notion, the notion of a u-extension of a Banach space. All these tools are used to give a unified definition of many classes of Banach spaces characterized by local properties (such as the ${\mathcal{L}_p}$-spaces). Many examples are given; also, as an application, it is proved that any ${\mathcal{L}_p}$-space, $1 < p < \infty$, has an ultrapower which is isomorphic to an ${L_p}$-space.
The signature of symplectic manifolds
Leslie P.
Jones
253-262
Abstract: The motivation for this work was a calculation by Oshanin of the image of the signature homomorphism from the special unitary cobordism ring into the integers. Here we compute this image for symplectic cobordism. This is accomplished by proving two divisibility theorems and then giving examples to show the theorems are the best possible.
Conditionally compact semitopological one-parameter inverse semigroups of partial isometries
M. O.
Bertman
263-275
Abstract: The algebraic structure of one-parameter inverse semigroups has been completely described. Furthermore, if B is the bicyclic semigroup and if B is contained in any semitopological semigroup, the relative topology on B is discrete. We show that if F is an inverse semigroup generated by an element and its inverse, and F is contained in a compact semitopological semigroup, then the relative topology is discrete; in fact, if F is any one-parameter inverse semigroup contained in a compact semitopological semigroup, then the multiplication on F is jointly continuous if and only if the inversion is continuous on F, and we describe $\bar F$ in that case. We also show that if $\{ {J_t}\}$ is a one-parameter semigroup of bounded linear operators on a (separable) Hilbert space, then $\{ {J_t}\} \cup \{ J_t^\ast\}$ generates a one-parameter inverse semigroup T with $J_t^{ - 1} = J_t^\ast$ if and only if $\{ {J_t}\}$ is a one-parameter semigroup of partial isometries, and we describe the weak operator closure of T in that case.
$(E\sp{3}/X)\times E\sp{1}\approx E\sp{4}$ ($X$, a cell-like set): an alternative proof
J. W.
Cannon
277-285
Abstract: The author gives an alternative proof that a cell-like closed-0dimensional decomposition of ${E^3}$ is an ${E^4}$ factor. The argument is essentially 2-dimensional. The 3- and 4-dimensional topology employed is truly minimal.
On the Seifert manifold of a $2$-knot
M. A.
Gutierrez
287-294
Abstract: From geometric facts about embeddings $ {S^2} \to {S^4}$ we study the relationship between the smallest number of normal generators (weight) of a group and its preabelian presentations.
Strong differentiability of Lipschitz functions
C. J.
Neugebauer
295-306
Abstract: Let F be a differentiation basis in ${R^n}$, i.e., a family of measurable sets S contracting to 0 such that ${\left\Vert {{M_F}f} \right\Vert _p} \leqslant {A_p}{\left\Vert f \right\Vert _p}$, where ${M_F}$ is the Hardy-Littlewood maximal operator. For $f \in \Lambda _\alpha ^{pq}$, we let ${E_F}(f)$ be the complement of the Lebesgue set of f relative to F, and we show that $ {E_F}$ has $L_\alpha ^{pq}$-capacity 0, where $L_\alpha ^{pq}$ is a capacity associated with $\Lambda _\alpha ^{pq}$ in much the same way as the Bessel capacity ${B_{\alpha p}}$ is associated with $L_\alpha ^p$.
Paracompactness of box products of compact spaces
Kenneth
Kunen
307-316
Abstract: We consider box products of countably many compact Hausdorff spaces. Under the continuum hypothesis, the product is paracompact iff its Lindelöf degree is no more than the continuum; in particular, the product is paracompact if each space has weight continuum or less, or if each space is dispersed. Some partial results are proved under Martin's axiom.
Projective modules over subrings of $k[X, Y]$
David F.
Anderson
317-328
Abstract: In this paper we study projective modules over subrings of $k[X,Y]$. Conditions are given for projective modules to decompose into free $\oplus$ rank 1 modules. Our main result is that if k is an algebraically closed field and A a subring of $ B = k[X,Y]$ with $A \subset B$ integral and ${\text{sing}}(A)$ finite, then all f.g. projective A-modules have the form free $\oplus$ rank 1. We also give several examples of subrings of $k[X,Y]$ which have indecomposable projective modules of rank 2.
Projective varieties of low codimension in characteristic $p>0$
Robert
Speiser
329-343
Abstract: Let X be an s-dimensional closed Cohen-Macaulay subvariety of projective n-space, over an algebraically closed field of characteristic $p > 0$. Assume $s \geqslant \tfrac{1}{2}(n + 1)$. Then (1) every stratified vector bundle on X is trivial; (2) X is simply connected. Assertion (1) generalizes Gieseker's result for projective space, while (2) is a strengthened analogue of results of Barth and Ogus in characteristic zero.
Reductions of residuals are finite
R.
Hindley
345-361
Abstract: An important theorem of the $\lambda \beta K$-calculus which has not been fully appreciated up to now is D. E. Schroer's finiteness theorem (1963), which states that all reductions of residuals are finite. The present paper gives a new proof of this theorem and extends it from $\lambda \beta$-reduction to $\lambda \beta \eta $-reduction and reductions with certain extra operators added, for example the pairing, iteration and recursion operators. Combinatory weak reduction, with or without extra operators, is also included.
Infinitesimal calculus on locally convex spaces. I. Fundamentals
K. D.
Stroyan
363-383
Abstract: Differential calculus on nonnormed locally convex spaces suffers from technical difficulties (and the subsequent plethora of different definitions) partly because the families of multilinear maps over the spaces do not inherit a suitable topology. In this note we give the elementary ingredients of a strong differentiation based on Abraham Robinson's theory of infinitesimals. Though nontopologizable, our theory is a natural generalization of standard infinitesimal calculus (finite dimensional or Banach space), see Robinson [1966], Keisler [1976], or Stroyan and Luxemburg [1976]. It is simpler than many recent developments, e.g., Yamamuro [1974] and Keller [1974]. The technical improvement of our approach should lead to advances in a variety of subjects.
Sets of divergence on the group $2\sp{\omega }$
David C.
Harris;
William R.
Wade
385-392
Abstract: We show that there exist uncountable sets of divergence for $C({2^\omega })$. We also show that a necessary and sufficient condition that a set E be a set of divergence for ${L^p}({2^\omega }),1 < p < \infty$, is that E be of Haar measure zero.
Erratum: ``A partial surface variation for extremal schlicht functions'' (Trans. Amer. Math. Soc. {\bf 234} (1977), no. 1, 119--138)
T. L.
McCoy
393
The commutant of an analytic Toeplitz operator
Carl C.
Cowen
1-31
Abstract: For a function f in ${H^\infty }$ of the unit disk, the operator on ${H^2}$ of multiplication by f will be denoted by ${T_f}$ and its commutant by ${B^{ - 1}} \circ B$ motivates work on more general functions. A theorem is proved which gives conditions on a family $ \mathcal{F}$ of ${H^\infty }$ functions which imply that there is a function h such that $ \{ {T_h}\} ' = { \cap _{f \in \mathcal{F}}}\{ {T_f}\} '$. As a special case of this theorem, we find that if the inner factor of $ f - f(c)$ is a finite Blaschke product for some c in the disk, then there is a finite Blaschke product B with $ \{ {T_f}\} ' = \{ {T_B}\} '$. Necessary and sufficient conditions are given for an operator to commute with ${T_f}$ when f is a covering map (in the sense of Riemann surfaces). If f and g are in ${H^\infty }$ and $ f = h \circ g$, then $ \{ {T_f}\} ' \supset \{ {T_g}\} '$. This paper introduces a class of functions, the ${H^2}$-ancestral functions, for which the converse is true. If f and g are $ {H^2}$-ancestral functions, then $ \{ {T_f}\} ' \ne \{ {T_g}\} '$ unless $f = h \circ g$ where h is univalent. It is shown that inner functions and covering maps are ${H^2}$-ancestral functions, although these do not exhaust the class. Two theorems are proved, each giving conditions on a function f which imply that ${T_f}$ does not commute with nonzero compact operators. It follows from one of these results that if f is an ${H^2}$-ancestral function, then $ {T_f}$ does not commute with any nonzero compact operators.
Asymptotic formulas for Toeplitz determinants
Estelle
Basor
33-65
Abstract: The object of this paper is to find an asymptotic formula for determinants of finite dimensional Toeplitz operators generated by a class of functions with singularities. The formula is a generalization of the Strong Szegö Limit Theorem.
Isotopy groups
Lawrence L.
Larmore
67-97
Abstract: For any mapping $ f:V \to M$ (not necessarily an embedding), where V and M are differentiable manifolds without boundary of dimensions k and n, respectively, V compact, let ${[V \subset M]_f} = {\pi _1}({M^V},E,f)$, i.e., the set of isotopy classes of embeddings with a specific homotopy to f (E = space of embeddings). The purpose of this paper is to enumerate ${[V \subset M]_f}$. For example, if $k \geqslant 3,n = 2k$, and M is simply connected, $ {[{S^k} \subset M]_f}$ corresponds to ${\pi _2}M$ or ${\pi _2}M \otimes {Z_2}$, depending on whether k is odd or even. In the metastable range, i.e., $3(k + 1) > 2n$, a natural Abelian affine structure on $ {[V \subset M]_f}$ is defined: if, further, f is an embedding ${[V \subset M]_f}$ is then an Abelian group. The set of isotopy classes of embeddings homotopic to f is the set of orbits of the obvious left action of ${\pi _1}({M^V},f)$ on ${[V \subset M]_f}$. A spectral sequence is constructed converging to a theory ${H^\ast}(f)$. If $3(k + 1) < 2n, {H^0}(f) \cong {[V \subset M]_f}$ provided the latter is nonempty. A single obstruction $\Gamma (f) \in {H^1}(f)$ is also defined, which must be zero if f is homotopic to an embedding; this condition is also sufficient if $3(k + 1) \leqslant 2n$. The ${E_2}$ terms are cohomology groups of the reduced deleted product of V with coefficients in sheaves which are not even locally trivial. ${[{S^k} \subset M]_f}$ is specifically computed in terms of generators and relations if $n = 2k, k \geqslant 3$ (Theorem 6.0.2).
Hyperarithmetically encodable sets
Robert M.
Solovay
99-122
Abstract: We say that a set of integers, A, is hyperarithmetically (recursively) encodable, if every infinite set of integers X contains an infinite subset Y in which A is hyperarithmetical (recursive). We show that the recursively encodable sets are precisely the hyperarithmetic sets. Let $\sigma$ be the closure ordinal of a universal $\Sigma _1^1$ inductive definition. Then A is hyperarithmetically encodable iff it is constructible before stage $\sigma$. We also prove an effective version of the Galvin-Prikry results that open sets, and more generally Borel sets, are Ramsey, and in the case of open sets prove that our improvement is optimal.
The theorem of Torelli for singular curves
Thomas
Jambois
123-146
Abstract: Let C be a compact (singular) curve embedded in a surface. Then C carries a canonical sheaf $\Omega$ which is locally free of rank 1. Moreover, C has a generalized Jacobian J which fits in an exact sequence $\displaystyle 0 \to F \to J \to A \to 0$ ($ \ast$) of algebraic groups such that A is an abelian variety and $F = {({{\mathbf{C}}^\ast})^r} \times {{\mathbf{C}}^s}$. Let $\underline{C}$ be the set of nonsingular points of C and let $\theta$ = Zariski-closure of the image of $(\underline{C})^{(g - 1)}$ in J. Then: Theorem. If C is irreducible and sections of $\Omega$ map C onto X in ${P^{g - 1}}$ then the isomorphism class of J together with the translation class of the divisor $ \theta$ on J determine the isomorphism class of X. As a corollary, if $\psi :C \to X$ is an isomorphism (in which case we call C nonhyperelliptic) the above data determine the isomorphism class of C. I do not know if this remains true when C is hyperelliptic. It should be noted that the linear equivalence class of $\theta$ is not enough to determine X. The principal idea of the proof is that of Andreotti, that is, to recover the curve as the dual of the branch locus of the Gauss map from $\theta$ to $ {P^{g - 1}}$; however our arguments are usually analytic. The organization of this paper is as follows: In §1 we prove a stronger than usual version of Abel's theorem for Riemann surfaces and in §2 we extend this theorem to apply to singular curves. In succeeding sections we construct the generalized Jacobian as a complex Lie group J and embed J in an analytic fibre bundle over A with projective spaces as fibre. This we use to endow J with the structure of an algebraic group. §7 contains a miscellany of facts about branch loci and dual varieties, and in §8 the main theorems are stated and proved. We should mention here that the variations on Abel's theorem proved in this paper (1.2.4 and 3.0.1) were proved by Severi, at least in the special case corresponding to ordinary double points [12].
Mesures invariantes sur les hypergroupes
R.
Spector
147-165
Abstract: The notion of an invariant or relatively invariant measure on a hypergroup is studied. The main result is that every commutative hypergroup carries an invariant measure.
A chain functor for bordism
Stanley O.
Kochman
167-196
Abstract: Chains of differential graded abelian monoids are defined for bordism and cobordism theories. These chains are used to define matric Massey products and can be filtered so as to define the Adams spectral sequence. From this point of view, we prove three basic theorems which show how Massey products behave in the Adams spectral sequence.
Tensor products for ${\rm SL}(2,\,k)$
Robert P.
Martin
197-211
Abstract: Let G be $ {\text{SL}}(2,k)$ where k is a locally compact, nondiscrete, totally disconnected topological field whose residual characteristic is not 2, $ {\pi _\sigma }$, be a principal series representation of G, and $\pi \in \hat G$ be arbitrary. We determine the decomposition of ${\pi _\sigma } \otimes \pi $ into irreducibles by reducing this problem to decomposing the restriction of each $T \in \hat G$ to a minimal parabolic subgroup B of G and decomposing certain tensor products of irreducibles of B.
Complete universal locally finite groups
Ken
Hickin
213-227
Abstract: This paper will partly strengthen a recent application of model theory to the construction of sets of pairwise nonembeddable universal locally finite groups [8]. Our result is Theorem. There is a set $\mathcal{U}$ of ${2^{{\aleph _1}}}$ universal locally finite groups of order $ {\aleph _1}$ with the following properties: 0.1. If $U \ne V \in \mathcal{U}$ and A and B are uncountable sugroups of U and V, then A and B are not isomorphic. Let A be an uncountable subgroup of $U \in \mathcal{U}$. 0.2. A does not belong to any proper variety of groups, and 0.3. A is not isomorphic to any of its proper subgroups. 0.4. Every $U \in \mathcal{U}$ is a complete group (every automorphism of U is inner).
Twist spinning revisited
Deborah L.
Goldsmith;
Louis H.
Kauffman
229-251
Abstract: This paper contains several applications of the following theorem: The 1-twist spin ${L_1}(k)$ of any knot $k \subset {S^{n - 1}}$ is interchangeable with the standard unknotted $(n - 2)$-sphere K in ${S^n}$ by means of a homeomorphism of triples $ h:({S^n},K,{L_1}(k)) \to ({S^n},{L_1}(k),K)$ which reverses the orientation of $ {S^n}$, and preserves the orientations of K and ${L_1}(k)$. One of these applications is Zeeman's Theorem about twist spun knots; another is a proof of a conjecture of R. H. Fox about certain manifolds which have the same fundamental group. We also prove that the iterated twist spun knot ${L_{a,b}}(k) \subset {S^{n + 1}}$ is fiber equivalent to one of $ {L_{0,g}}(k)$ or ${L_{g,g}}(k)$ where $g = {\text{g.c.d.}}(a,b)$.
Completeness theorems, incompleteness theorems and models of arithmetic
Kenneth
McAloon
253-277
Abstract: Let $\mathcal{A}$ be a consistent extension of Peano arithmetic and let $ \mathcal{A}_n^0$ denote the set of $\Pi _n^0$ consequences of $ \mathcal{A}$. Employing incompleteness theorems to generate independent formulas and completeness theorems to construct models, we build nonstandard models of $\mathcal{A}_{n + 2}^0$ in which the standard integers are $\Delta _{n + 1}^0$-definable. We thus pinpoint induction axioms which are not provable in $\mathcal{A}_{n + 2}^0$; in particular, we show that (parameter free) $ \Delta _1^0$-induction is not provable in Primitive Recursive Arithmetic. Also, we give a solution of a problem of Gaifman on the existence of roots of diophantine equations in end extensions and answer questions about existentially complete models of $ \mathcal{A}_2^0$. Furthermore, it is shown that the proof of the Gödel Completeness Theorem cannot be formalized in $\mathcal{A}_2^0$ and that the MacDowell-Specker Theorem fails for all truncated theories $\mathcal{A}_n^0$.
Diffeomorphisms almost regularly homotopic to the identity
Robert
Wells
279-292
Abstract: Let $f:M \to M$ be a self-map of a closed smooth n-manifold. Does there exist a diffeomorphism $\varphi :M \to M$ homotopic to f? Define $\varphi$ to be almost regularly homotopic to the identity if $\varphi \vert M - {\text{pt}}$. is regularly homotopic to the inclusion $M - {\text{pt}}. \subset M$. Let $\psi :M \to M \vee M$ be the result of collapsing the boundary of a smooth n-cell in M, and let $ \xi \in {\pi _n}(M)$ define $\tau (\xi )$ to be the composition $ n = 2l > 5$ with $ l\;{\nequiv}\;0 \bmod (4)$, then $ \tau (\xi )$ contains a diffeomorphism almost regularly homotopic to the identity iff $\xi$ is in the kernel of the stabilization map $ {\pi _n}(M) \to \pi _n^s(M)$.
Heisenberg manifolds and theta functions
R.
Tolimieri
293-319
Abstract: The algebraic structure of the $(2n + 1)$-dimensional Heisenberg group naturally induces a special class of differential operators whose solutions $(Df = 0)$ are related to classical theta function theory.
The classification of flat solvmanifolds
Alexander
Morgan
321-351
Abstract: This paper contains a complete algebraic characterization of the fundamental groups of flat solvmanifolds. This characterization is in terms of finite integral representations of free abelian groups and the associated cohomology. A classification of compact flat solvmanifolds follows, and a list of all compact flat solvmanifolds of dimensions 3, 4, and 5 (except the 5-dimensional with first betti number 1) is given. Some theorems on the classification of noncompact flat solvmanifolds have also been obtained. These give full results in some cases, partial results in others. For example, the odd order holonomy group case is completely settled.
Limiting distributions for branching random fields
Joseph
Fleischman
353-389
Abstract: In this paper we derive limiting distributions for branching Brownian motion. The cases considered are where the state space is (1) the line and (2) the plane where (a) initially there's but one particle and (b) where there's initially a random number of independent particles. In all cases, the branching process is critical and we obtain results for the growth of selectively neutral mutant types. We use moment generating functions to derive these results.
The existence of combinatorial formulae for characteristic classes
Norman
Levitt;
Colin
Rourke
391-397
Abstract: Given a characteristic class on a locally ordered combinatorial manifold M there exists a cocycle which represents the class on M and is locally defined, i.e. its value on $ \sigma \in M$ depends only on the ordered star ${\text{st}}(\sigma ,M)$. For rational classes the dependence on order disappears. There is also a locally defined cycle which carries the dual homology class.
On the rationality of divisors and meromorphic functions
Chia Chi
Tung
399-406
Abstract: Let E be a holomorphic vector bundle over a connected complex manifold X and D a divisor on E. Let $ A(D)$ be the set of all $x \in X$ for which $ ({\text{supp}}\;D) \cap {E_x}$ is a proper algebraic set in ${E_x}$. The purpose of this paper is to prove that the following conditions are equivalent: (i) $A(D)$ has positive measure in X; (ii) D extends to a unique divisor on the projective completion Ē of E; (iii) D is locally given by the divisors of rational meromorphic functions defined over open sets in X. Similar results for meromorphic functions are derived. The proof requires an extension theorem for analytic set: Assume E is a holomorphic vector bundle over a pure p-dimensional complex space X and S an analytic set in E of pure codimension 1. Then the closure S of S in E is analytic if and only if $S \cap {E_x}$ is a proper algebraic set for all x in a set of positive 2p-measure in every branch of X.
Hermitian Lie algebras and metaplectic representations. I
Shlomo
Sternberg;
Joseph A.
Wolf
1-43
Abstract: A notion of ``hermitian Lie algebra'' is introduced which relates ordinary and graded Lie algebra structures. In the case of real-symplectic and arbitrary-signature-unitary Lie algebras, it leads to an analysis of the minimal dimensional coadjoint orbits, and then to the metaplectic representations and their restrictions to unitary groups of arbitrary signature and parabolic subgroups of these unitary groups.
Examples of nonintegrable analytic Hamiltonian vector fields with no small divisors
R.
Cushman
45-55
Abstract: Any analytic symplectic diffeomorphism $\Phi$ of a symplectic manifold M is the Poincaré map of a real analytic Hamiltonian vector field ${X_H}$. If $\Phi$ does not have an analytic integral, then $ {X_H}$ has no analytic integral which is not a power series in H. Let $M = {{\mathbf{R}}^2}$. If $ \Phi$ has a finite contact homoclinic point, then $\Phi$ is nonintegrable. Also Moser's polynomial mapping is nonintegrable.
On the number of real zeros of a random trigonometric polynomial
M.
Sambandham
57-70
Abstract: For the random trigonometric polynomial $\displaystyle \sum\limits_{n = 1}^N {{g_n}(t)\cos n\theta ,}$ where ${g_n}(t),0 \leqslant t \leqslant 1$, are dependent normal random variables with mean zero, variance one and joint density function ${M^{ - 1}}$ is the moment matrix with ${\rho _{ij}} = \rho ,0 < \rho < 1,i \ne j,i,j = 1,2, \ldots ,N$ and $\bar a$ is the column vector, we estimate the probable number of zeros.
Bounded point evaluations and smoothness properties of functions in $R\sp{p}(X)$
Edwin
Wolf
71-88
Abstract: Let X be a compact subset of the complex plane C. We denote by $ {R_0}(X)$ the algebra consisting of the (restrictions to X of) rational functions with poles off X. Let m denote 2-dimensional Lebesgue measure. For $p \geqslant 1$, let ${L^p}(X) = {L^p}(X,dm)$. The closure of $ {R_0}(X)$ in $ {L^p}(X)$ will be denoted by ${R^p}(X)$. Whenever p and q both appear, we assume that $ 1/p + 1/q = 1$. If x is a point in X which admits a bounded point evaluation on ${R^p}(X)$, then the map which sends f to $ f(x)$ for all $f \in {R_0}(X)$ extends to a continuous linear functional on ${R^p}(X)$. The value of this linear functional at any $f \in {R^p}(X)$ is denoted by $ f(x)$. We examine the smoothness properties of functions in ${R^p}(X)$ at those points which admit bounded point evaluations. For $p > 2$ we prove in Part I a theorem that generalizes the ``approximate Taylor theorem'' that James Wang proved for $R(X)$. In Part II we generalize a theorem of Hedberg about the convergence of a certain capacity series at a point which admits a bounded point evaluation. Using this result, we study the density of the set X at such a point.
Representation theory of algebras stably equivalent to an hereditary Artin algebra
María Inés
Platzeck
89-128
Abstract: Two artin algebras are stably equivalent if their categories of finitely generated modules modulo projectives are equivalent. The author studies the representation theory of algebras stably equivalent to hereditary algebras, using the notions of almost split sequences and irreducible morphisms. This gives a new unified approach to the theories developed for hereditary and radical square zero algebras by Gabriel, Gelfand, Bernstein, Ponomarev, Dlab, Ringel and Müller, as well as other algebras not covered previously. The techniques are purely module theoretical and do not depend on representations of diagrams. They are similar to those used by M. Auslander and the author to study hereditary algebras.
Weak Chebyshev subspaces and continuous selections for the metric projection
Günther
Nürnberger;
Manfred
Sommer
129-138
Abstract: Let G be an n-dimensional subspace of $C[a,b]$. It is shown that there exists a continuous selection for the metric projection if for each f in $C[a,b]$ there exists exactly one alternation element ${g_f}$, i.e., a best approximation for f such that for some $a \leqslant {x_0} < \cdots < {x_n} \leqslant b$, $\displaystyle \varepsilon {( - 1)^i}(f - {g_f})({x_i}) = \left\Vert {f - {g_f}} \right\Vert,\quad i = 0, \ldots ,n,\varepsilon = \pm 1.$ Further it is shown that this condition is fulfilled if and only if G is a weak Chebyshev subspace with the property that each g in G, $g \ne 0$, has at most n distinct zeros. These results generalize in a certain sense results of Lazar, Morris and Wulbert for $n = 1$ and Brown for $n = 5$.
A Hopf global bifurcation theorem for retarded functional differential equations
Roger D.
Nussbaum
139-164
Abstract: We prove a result concerning the global nature of the set of periodic solutions of certain retarded functional differential equations. Our main theorem is an analogue, for retarded F.D.E.'s, of a result by J. Alexander and J. Yorke for ordinary differential equations.
Analytic left algebraic groups. II
Andy R.
Magid
165-177
Abstract: An analytic left algebraic group is a complex analytic group carrying a structure of affine algebraic variety such that left translations by fixed elements are morphisms. The core of such a group is the (algebraic) subgroup of all elements such that right translation by them is a morphism. It is shown that the core determines the left algebraic structure, and this is used to determine when left algebraic structures are conjugate by inner automorphisms.
Bender groups as standard subgroups
Robert L.
Griess;
David R.
Mason;
Gary M.
Seitz
179-211
Abstract: A subgroup X of a finite group G is called $^ \ast $-standard if $\tilde X = X/O(X)$ is quasisimple, $Y = {C_G}(X)$ is tightly embedded in G and ${N_G}(X) = {N_G}(Y)$. This generalizes the notion of standard subgroups. Theorem. Let G be a finite group with $O(G) = 1$. Suppose X is $^ \ast$-standard in G and $\tilde X/Z(\tilde X) \cong {L_2}({2^n}),{U_3}({2^n})$ or $ {\text{Sz}}({2^n})$. Assume $ X \ntriangleleft G$. Then $O(X) = 1$ and one of the following holds: $({\text{i}})\;E(G) \cong X \times X$. $ ({\text{ii}})\;X \cong {L_2}({2^n})$ and $E(G) \cong {L_2}({2^{2n}}),{U_3}({2^n})\;or\;{L_3}({2^n})$. $({\text{iii}})\;X \cong {U_3}({2^n})$ and $ E(G) \cong {L_3}({2^{2n}})$. $({\text{iv}})\;X \cong {\text{Sz}}({2^n})$ and $ E(G) \cong {\text{Sp}}(4,{2^n})$. $({\text{v}})\;X \cong {L_2}(4)$ and $E(G) \cong {M_{12}},{A_9},{J_1},{J_2},{A_7},{L_2}(25),{L_3}(5)\;or\;{U_3}(5)$. $ ({\text{vi}})\;X \cong {\text{Sz}}(8)$ and $E(G) \cong {\text{Ru}}$ (the Rudvalis group). $({\text{vii}})\;X \cong {L_2}(8)$ and $E(G) \cong {G_2}(3)$. $ ({\text{viii}})\;X \cong {\text{SL}}(2,5)$ and G has sectional 2-rank at most 4. In particular, if G is simple, $ G \cong {M_{12}},{A_9},{J_1},{J_2},{\text{Ru}},{U_3}(5),{L_3}(5),{G_2}(5), or\;{^3}{D_4}(5)$.
Maximum principles, gradient estimates, and weak solutions for second-order partial differential equations
William
Bertiger
213-227
Abstract: Weak solutions to second order elliptic equations and the first derivatives of these solutions are shown to satisfy $ {L^p}$ bounds. Classical second order equations with nonnegative characteristic form are also considered. It is proved that auxiliary functions of the gradient of a solution must satisfy a maximum principle. This result is extended to higher order derivatives and systems.
On a degenerate principal series of representations of ${\rm U}(2, 2)$
Yang
Hua
229-252
Abstract: A degenerate principal series of representations $T(\rho ,m; \cdot ),(\rho ,m) \in {\mathbf{R}} \times {\mathbf{Z}}$, of $U(2,2)$, is realized on the Hilbert space of all square integrable functions on the space X of $2 \times 2$ Hermitian matrices. Using Fourier analysis, gamma functions, and Mellin analysis, we spectrally analyze the operator equation $AT(\rho ,m;g) = T(\rho ,m;g)A$ for all $g \in \mathfrak{G} = U(2,2)$ on an invariant subspace of ${L^2}(X)$, and obtain the first main result: For $\rho \ne 0$ or m odd, $T(\rho ,m; \cdot )$ is irreducible. Then we define certain integral transforms on ${L^2}(X)$ the analytic continuation of which leads to the second main result: $T(0,2n; \cdot )$ is reducible.
On the structure of principal ideals of operators
G. D.
Allen;
L. C.
Shen
253-270
Abstract: This paper considers various types of principal ideals generated by single compact operators on a separable Hilbert space. In particular, necessary and sufficient condtions that a principal ideal be normable are given. Relations between principal ideals and duals of Lorentz and Orlicz spaces are also given. All conditions are expressed using the singular numbers of the operator.
Growth hyperspaces of Peano continua
D. W.
Curtis
271-283
Abstract: For X a nondegenerate Peano continuum, let ${2^X}$ be the hyperspace of all nonempty closed subsets of X, topologized with the Hausdorff metric. It is known that ${2^X}$ is homeomorphic to the Hilbert cube. A nonempty closed subspace $ \mathcal{G}$ of $ {2^X}$ is called a growth hyperspace provided it satisfies the following condition: if $A \in \mathcal{G}$, and $B \in {2^X}$ such that $B \supset A$ and each component of B meets A, then also $B \in \mathcal{G}$. The class of growth hyperspaces includes many previously considered subspaces of $ {2^X}$. It is shown that if X contains no free arcs, and $\mathcal{G}$ is a nontrivial growth hyperspace, then $\mathcal{G}\backslash \{ X\}$ is a Hilbert cube manifold. A corollary characterizes those growth hyperspaces which are homeomorphic to the Hilbert cube. Analogous results are obtained for growth hyperspaces with respect to the hyperspace ${\text{cc}}(X)$ of closed convex subsets of a convex n-cell X.
Periodic solutions for a differential equation in Banach space
James H.
Lightbourne
285-299
Abstract: Suppose X is a Banach space, $ \Omega \subset X$ is closed and convex, and $A:[0,\infty ) \times \Omega \to X$ is continuous. Then if $\displaystyle \mathop {\lim }\limits_{h \to 0} \vert x + hA(t,x);\Omega \vert/h = 0\quad {\text{for}}\;{\text{all}}\;(t,x) \in [0,\infty ) \times \Omega ,$ there exist approximate solutions to the initial value problem $A(t,x) = B(t,x) + C(t,x)$, where B satisfies a dissipative condition and C is compact, we obtain a growth estimate on the measure of noncompactness of trajectories for a class of approximate solutions. This estimate is employed to obtain existence of periodic solutions to (IVP).
Translation planes of order $q\sp{2}$: asymptotic estimates
Gary L.
Ebert
301-308
Abstract: R. H. Bruck has pointed out the one-to-one correspondence between the isomorphism classes of certain translation planes, called subregular, and the equivalence classes of disjoint circles in a finite miquelian inversive plane $ IP(q)$. The problem of determining the number of isomorphism classes of translation planes is old and difficult. Let q be an odd prime-power. In this paper, a study of sets of disjoint circles in $IP(q)$ enables the author to find an asymptotic estimate of the number of isomorphism classes of translation planes of order ${q^2}$ which are subregular of index 3 or 4. It is conjectured (and proved for $n \leqslant 3$) that, given a set of n disjoint circles in $IP(q)$, the numbers of circles disjoint from each of the given n circles is asymptotic to ${q^3}/{2^n}$. This conjecture, if true, would allow one to estimate the number of subregular translation planes of order ${q^2}$ with any positive index.
Central twisted group algebras
Harvey A.
Smith
309-320
Abstract: A twisted group algebra $ {L^1}(A,G;T,\alpha )$ is central iff T is trivial and A commutative. (Group algebras of central extension of G are such.) We show that if ${H^2}(G)$ is discrete any central ${L^1}(A,G;\alpha )$ is a direct sum of closed ideals ${L^1}({A_i},G;{\alpha _i})$ having as duals fibre bundles over the duals of closed ideals ${A_i}$ in A, with fibres projective duals of G, and principal ${G^\wedge}$ bundles (where ${G^\wedge}$ denotes the group of characters of G) satisfying the conditions which define characteristic bundles for G abelian. (If G is compact ${H^2}(G)$ is always discrete, the direct sum is countable, and the bundles are locally trivial.) Applications are made to the duals of central extensions of groups and in particular to duals of ``central'' groups. For G commutative, ${H^2}(G)$ discrete, and A a $ {C^\ast}$-algebra with identity, all central twisted group algebras ${L^1}(A,G;\alpha )$ (and their duals) are classified in purely algebraic terms involving ${H^2}(G)$, the group G, and the first Čech cohomology group of the dual of A. This result allows us, in principle, to construct all the central $ {L^1}(A,G;\alpha )$ and their duals where A is a ${C^\ast}$-algebra with identity and G a compact commutative group.
Galerkin methods in the constructive solvability of nonlinear Hammerstein equations with applications to differential equations
P. M.
Fitzpatrick;
W. V.
Petryshyn
321-340
Abstract: We consider the solution of abstract Hammerstein equations by means of a Galerkin approximating scheme. The convergence of the scheme is proven by first establishing an equivalent scheme in a Hilbert space and then proving a convergence result for firmly monotone operators in a Hilbert space. The general results are applied to the case when the involved linear mapping is angle-bounded, and also to the treatment of certain differential equations.
Generic properties of eigenfunctions of elliptic partial differential operators
Jeffrey H.
Albert
341-354
Abstract: The problem considered here is that of describing generically the zeros, critical points and critical values of eigenfunctions of elliptic partial differential operators. We consider operators of the form $L + \rho$, where L is a fixed, second-order, selfadjoint, $ {C^\infty }$ linear elliptic partial differential operator on a compact manifold (without boundary) and $\rho$ is a $ {C^\infty }$ function. It is shown that, for almost all $\rho$, i.e. for a residual set, the eigenvalues of $L + \rho$ are simple and the eigenfunctions have the following properties: (1) they are Morse functions; (2) distinct critical points have distinct critical values; (3) 0 is not a critical value.
Semimodular functions and combinatorial geometries
Hien Quang
Nguyen
355-383
Abstract: A point-lattice $\mathfrak{L}$ being given, to any normalized, nondecreasing, integer-valued, semimodular function f defined on $ \mathfrak{L}$, we can associate a class of combinatorial geometries called expansions of f. The family of expansions of f is shown to have a largest element for the weak map order, $E(f)$, the free expansion of f. Expansions generalize and clarify the relationship between two known constructions, one defined by R. P. Dilworth, the other by J. Edmonds and G.-C. Rota. Further applications are developed for solving two extremal problems of semimodular functions: characterizing (1) extremal rays of the convex cone of real-valued, nondecreasing, semimodular functions defined on a finite set; (2) combinatorial geometries which are extremal for the decomposition into a sum.
Taming and the Poincar\'e conjecture
T. L.
Thickstun
385-396
Abstract: L. Glaser and L. Siebenmann have shown that the double suspension of a homotopy 3-sphere is homeomorphic to the 5-sphere. This result, together with a well-known characterization of $ {S^3}$ due to R. H. Bing, is used to establish a relationship between the Poincaré conjecture and two conjectures concerned with taming embeddings in higher dimensions. One of the two conjectures, each of which implies the Poincaré conjecture, states, in effect, that a codimension two sphere is tame if it is tame ``modulo'' a tame disk contained in it.
A Gross measure property
Lawrence R.
Ernst
397-406
Abstract: We prove that there exists a subset E of $[0,1] \times {{\mathbf{R}}^2}$ such that the 2-dimensional Gross measure of E is 0, while the 1-dimensional Gross measure of $\{ z:(y,z) \in E\}$ is positive for all $y \in [0,1]$. It is known that for Hausdorff measures no set exists satisfying these conditions.
Square-integrable factor representations of locally compact groups
Jonathan
Rosenberg
1-33
Abstract: The well-known theory of square-integrable representations is generalized to the case of primary representations (not necessarily type I) quasi-contained in either the regular representation or the representation induced from a character of the center of a (not necessarily unimodular) locally compact group, and relations with the topology of the primitive ideal space of the group $ {C^\ast}$-algebra are obtained. The cases of discrete and almost connected groups are examined in more detail, and it is shown that for such groups, square-integrable factor representations must be traceable. For connected Lie groups, these representations can (in principle) be determined up to quasi-equivalence using a complicated construction of L. Pukanszky-for type I simply connected solvable Lie groups, the characterization reduces to that conjectured by C. C. Moore and J. Wolf. In the case of unimodular exponential groups, essentially everything is as in the nilpotent case (including a result on multiplicities in the decomposition of $ {L^2}(G/\Gamma )$, $ \Gamma$ a discrete uniform subgroup of G). Finally, it is shown that the same criterion as for type I solvable Lie groups characterizes the squareintegrable representations of certain solvable $ \mathfrak{p}$-adic groups studied by R. Howe.
Free states of the gauge invariant canonical anticommutation relations
B. M.
Baker
35-61
Abstract: The gauge invariant subalgebra of the canonical anticommutation relations (henceforth GICAR) is viewed as an inductive limit of finitedimensional ${C^\ast}$-algebras, and a study is made of a simple class of its representations. In particular, representations induced by restricting the wellknown gauge invariant generalized free states from the entire canonical anticommutation relations (henceforth CAR) are considered. Denoting (a) a state of the CAR by $\omega$ and its restriction to the GICAR by ${\omega ^ \circ }$, (b) the unique gauge invariant generalized free state of the CAR such that $\omega (a{(f)^\ast}a(g)) = (f,Ag)$ by ${\omega _A}$, it is shown that $(1)\;\omega _A^ \circ$ induces (an impure) factor representation of the GICAR if and only if ${\text{Tr}}\;A(I - A) = \infty$, (2) two (impure) GICAR factor representations $\omega _A^ \circ$ and $\omega_B^\circ$ are quasi-equivalent if and only if $ {A^{1/2}} - {B^{1/2}}$ and $ {(I - A)^{1/2}} - {(I - B)^{1/2}}$ are Hilbert-Schmidt class operators.
The $\alpha $-union theorem and generalized primitive recursion
Barry E.
Jacobs
63-81
Abstract: A generalization to $ \alpha$-recursion theory of the McCreight-Meyer Union Theorem is proved. Theorem. Let $\Phi$ be an $\alpha $-computational complexity measure and $\{ {f_\varepsilon }\vert\varepsilon < \alpha \}$ an $\alpha$-r.e. strictly increasing sequence of $\alpha$-recursive functions. Then there exists an $\alpha$-recursive function k such that $C_k^\Phi = { \cup _{\varepsilon < \alpha }}C_{{f_\varepsilon }}^\Phi $. The proof entails a no-injury cancellation atop a finite-injury priority construction and necessitates a blocking strategy to insure proper convergence. Two infinite analogues to ($ \omega$-) primitive recursive functions are studied. Although these generalizations coincide at $\omega$, they diverge on all admissible $\alpha > \omega$. Several well-known complexity properties of primitive recursive functions hold for one class but fail for the other. It is seen that the Jensen-Karp ordinally primitive recursive functions restricted to admissible $\alpha > \omega$ cannot possess natural analogues to Grzegorczyk's hierarchy.
Necessary and sufficient conditions for the ${\rm GHS}$ inequality with applications to analysis and probability
Richard S.
Ellis;
Charles M.
Newman
83-99
Abstract: The GHS inequality is an important tool in the study of the Ising model of ferromagnetism (a model in equilibrium statistical mechanics) and in Euclidean quantum field theory. This paper derives necessary and sufficient conditions on an Ising spin system for the GHS inequality to be valid. Applications to convexity-preserving properties of certain differential equations and diffusion processes are given.
Application of the dual-process method to the study of a certain singular diffusion
David
Williams
101-110
Abstract: This paper should be regarded as a sequel to a paper by Holley, Stroock and the author. Its primary purpose is to provide further illustration of the application of the dual-process method. The main result is that if $d \geqslant 2$ and $\varphi$ is the characteristic function of an aperiodic random walk on ${{\mathbf{Z}}^d}$, then there is precisely one Feller semigroup on the d-dimensional torus with generator extending $A = \{ 1 - \varphi (\theta )\} \Delta$. A necessary and sufficient condition for the associated Feller process to leave the singular point 0 is determined. This condition provides a criterion for uniqueness in law of a stochastic differential equation which is naturally associated with the process.
Projective modules over Laurent polynomial rings
Richard G.
Swan
111-120
Abstract: Quillen's solution of Serre's problem is extended to Laurent polynomial rings. An example is given of a $A[T,{T^{ - 1}}]$-module P which is not extended even though A is regular and ${P_\mathfrak{m}}$ is extended for all maximal ideals $\mathfrak{m}$ of A.
Cyclic actions on lens spaces
Paik Kee
Kim
121-144
Abstract: A 3-dimensional lens space $L = L(p,q)$ is called symmetric if ${q^2} \equiv \pm 1 \bmod p$. Let h be an orientation-preserving PL homeomorphism of even period $n( > 2)$ on L with nonempty fixed-point set. We show: (1) If n and p are relatively prime, up to weak equivalence (PL), there exists exactly one such h if L is symmetric, and there exist exactly two such h if L is nonsymmetric. (2) $ {\text{Fix}}(h)$ is disconnected only if $ p \equiv 0 \bmod n$, and there exists exactly one such h up to weak equivalence (PL). A ${Z_n}$-action is called nonfree if $ {\text{Fix}}(\phi ) \ne \emptyset$ for some $\phi ( \ne 1) \in {Z_n}$. We also classify all orientation-preserving nonfree $ {Z_4}$-actions (PL) on all lens spaces $L(p,q)$. It follows that each of ${S^3}$ and ${P^3}$ admits exactly three orientation-preserving $ {Z_4}$-actions (PL), up to conjugation.
On the group of automorphisms of affine algebraic groups
Dong Hoon
Lee
145-152
Abstract: We study the conservativeness property of affine algebraic groups over an algebraically closed field of characteristic 0 and of their group of automorphisms. We obtain a certain decomposition of affine algebraic groups, and this, together with the result of Hochschild and Mostow, becomes a major tool in our study of the conservativeness property of the group of automorphisms.
Homotopy operations under a fixed space
D. E.
Kruse;
J. F.
McClendon
153-174
Abstract: The problem of classifying extensions of a function up to relative homotopy leads in a natural way to the homotopy operations of the title. The operations, stable and unstable, primary and higher order, are defined and studied. Some specific applications are worked out.
Relations among characteristic classes
Stavros
Papastavridis
175-187
Abstract: Let M be an n-dimensional, compact, closed, ${C^\infty }$ manifold, and $v:M \to BO$ be the map classifying its stable normal bundle. Let $S \subseteq {H^\ast}(BO;{Z_2})$ be a set of characteristic classes and let q, k, be fixed nonnegative integers. We define $I_n^q(S,k) = \{ x \in {H^q}(B):{v^\ast}(x) \cdot y = 0$ for all $y \in {H^k}(M;{Z_2})$ and for all n-dimensional, $ {C^\infty }$ closed compact manifolds M, which have the propery that ${v^\ast}(S) = \{ 0\} \}$. In this paper we compute $ I_n^q(S,k)$, where all classes of S have dimension greater than $ n/2$. We examine also the case of BSO and BU manifolds.
Filtrations and canonical coordinates on nilpotent Lie groups
Roe
Goodman
189-204
Abstract: Let $\mathfrak{g}$ be a finite-dimensional nilpotent Lie algebra over a field of characteristic zero. Introducing the notion of a positive, decreasing filtration $\mathcal{F}$ on $\mathfrak{g}$, the paper studies the multiplicative structure of the universal enveloping algebra $U(\mathfrak{g})$, and also transformation laws between $ \mathcal{F}$-canonical coordinates of the first and second kind associated with the Campbell-Hausdorff group structure on $\mathfrak{g}$. The basic technique is to exploit the duality between $ U(\mathfrak{g})$ and $ S({\mathfrak{g}^\ast})$, the symmetric algebra of ${\mathfrak{g}^\ast}$, making use of the filtration $\mathcal{F}$. When the field is the complex numbers, the preceding results, together with the Cauchy estimates, are used to obtain estimates for the structure constants for $ U(\mathfrak{g})$. These estimates are applied to construct a family of completions $ U{(\mathfrak{g})_\mathfrak{M}}$ of $ U(\mathfrak{g})$, on which the corresponding simplyconnected Lie group G acts by an extension of the adjoint representation.
Linear isotopies in $E\sp{2}$
R. H.
Bing;
Michael
Starbird
205-222
Abstract: This paper deals with continuous families of linear embeddings (called linear isotopies) of finite complexes in the Euclidean plane ${E^2}$. Suppose f and g are two linear embeddings of a finite complex P with triangulation T into a simply connected open subset U of ${E^2}$ so that there is an orientation preserving homeomorphism H of ${E^2}$ to itself with $H \circ f = g$. It is shown that there is a continuous family of embeddings ${h_t}:P \to U(t \in [0,1])$ so that ${h_0} = f,{h_1} = g$, and for each t, $ {h_t}$ is linear with respect to T. It is also shown that if P is a PL star-like disk in ${E^2}$ with a triangulation T which has no spanning edges and f is a homeomorphism of P which is the identity on Bd P and is linear with respect to T, then there is a continuous family of homeomorphisms ${h_t}:P \to P(t \in [0,1])$ such that $ {h_0} = {\text{id}},{h_1} = f$, and for each t, ${h_t}$ is linear with respect to T. An example shows the necessity of the ``star-like'' requirement. A consequence of this last theorem is a linear isotopy version of the Alexander isotopy theorem-namely, if f and g are two PL embeddings of a disk P into ${E^2}$ so that $f\vert{\text{Bd}}\;P = g\vert{\text{Bd}}\;P$, then there is a linear isotopy with respect to some triangulation of P which starts at f, ends at g, and leaves the boundary fixed throughout.
Spectral properties of tensor products of linear operators. II. The approximate point spectrum and Kato essential spectrum
Takashi
Ichinose
223-254
Abstract: For tensor products of linear operators, their approximate point spectrum, approximate deficiency spectrum and essential spectra in the sense of T. Kato and Gustafson-Weidmann are determined together with explicit formulae for their nullity and deficiency. The theory applies to $A \otimes I + I \otimes B$ and $A \otimes B$.
Logarithmic Sobolev inequalities for the heat-diffusion semigroup
Fred B.
Weissler
255-269
Abstract: An explicit formula relating the Hermite semigroup ${e^{ - tH}}$ on R with Gauss measure and the heat-diffusion semigroup $ {e^{t\Delta }}$ on R with Lebesgue measure is proved. From this formula it follows that Nelson's hypercontractive estimates for ${e^{ - tH}}$ are equivalent to the best norm estimates for $ {e^{t\Delta }}$ as a map $ {L^q}(R)$ into ${L^p}(R),1 < q < p < \infty$. Furthermore,the inequality $\displaystyle \frac{d}{{dq}}\log \left\Vert \phi \right\Vert _q^q \leqslant \fr... ...\Vert \phi \right\Vert _q^q}}} \right] + \log {\left\Vert \phi \right\Vert _q},$ where $1 < q < \infty ,{J^q}\phi = (\operatorname{sgn} \phi )\vert\phi {\vert^{q - 1}}$, and the norms and sesquilinear form $\langle ,\rangle$ are taken with respect to Lebesgue measure on ${R^n}$, is shown to be equivalent to the best norm estimates for $ {e^{t\Delta }}$ as a map from ${L^q}({R^n})$ into ${L^p}({R^n})$. This inequality is analogous to Gross' logarithmic Sobolev inequality. Also, the above inequality is compared with a classical Sobolev inequality.
Segal algebras on non-abelian groups
Ernst
Kotzmann;
Harald
Rindler
271-281
Abstract: Let ${S^1}(G)$ be a Segal algebra on a locally compact group. The central functions of ${S^1}(G)$ are dense in the center of $ {L^1}(G)$. ${S^1}(G)$ has central approximate units iff G $G \in [SIN]$. This is a generalization of a result of Reiter on the one hand and of Mosak on the other hand. The proofs depend on the structure theorems of [SIN]- and [IN]-groups. In the second part some new examples of Segal algebras are constructed. A locally compact group is discrete or Abelian iff every Segal algebra is right-invariant. As opposed to the results, the proofs are not quite obvious.
The $\mu $-invariant of $3$-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented $2$-manifold
Joan S.
Birman;
R.
Craggs
283-309
Abstract: Let $\mathcal{H}(n)$ be the group of orientation-preserving selfhomeomorphisms of a closed oriented surface Bd U of genus n, and let $\mathcal{K}(n)$ be the subgroup of those elements which induce the identity on $ {H_1}({\text{Bd}}\;U;{\mathbf{Z}})$. To each element $h \in \mathcal{H}(n)$ we associate a 3-manifold $ M(h)$ which is defined by a Heegaard splitting. It is shown that for each $h \in \mathcal{H}(n)$ there is a representation $ \rho$ of $\mathcal{K}(n)$ into $ {\mathbf{Z}}/2{\mathbf{Z}}$ such that if $ k \in \mathcal{K}(n)$, then the $\mu$-invariant $ \mu (M(h))$ is equal to the $\mu$-invariant $ \mu (M(kh))$ if and only if k $\in$ kernel $\rho$. Thus, properties of the 4-manifolds which a given 3-manifold bounds are related to group-theoretical structure in the group of homeomorphisms of a 2-manifold. The kernels of the homomorphisms from $\mathcal{K}(n)$ onto $ {\mathbf{Z}}/2{\mathbf{Z}}$ are studied and are shown to constitute a complete conjugacy class of subgroups of $\mathcal{H}(n)$. The class has nontrivial finite order.
Cylindricity of isometric immersions between hyperbolic spaces
S.
Alexander;
E.
Portnoy
311-329
Abstract: The motivation for this paper was to prove the following analogue of the Euclidean cylinder theorem: any umbilic-free isometric immersion $ \eta :{H^{n - 1}} \to {H^n}$ between hyperbolic spaces takes the form of a hyperbolic $(n - 2)$-cylinder over a uniquely determined parallelizing curve in $ {\bar H^n}$. Our approach is through the more general study of isometric immersions generated by one-parameter families of hyperbolic k-planes without focal points. A by-product of this study is a natural extension to curves in ${\bar H^n}$ of the notion of a parallel family of k-planes along a curve in ${H^n}$; the extension is based on spherical symmetry of variation fields. Existence and uniqueness properties of this extended notion of parallelism are considered.
Positive cones and focal points for a class of $n$th-order differential equations
M. S.
Keener;
C. C.
Travis
331-351
Abstract: Necessary and sufficient conditions are obtained for both the existence and absence of focal points for a class of nth order linear differential equations. The techniques utilized the theory of ${\mu _0}$-positive operators with respect to a cone in a Banach space.
The $(\varphi , 1)$ rectifiable subsets of Euclidean space
Samir
Kar
353-371
Abstract: In this paper the structure of a subset $E \subset {{\mathbf{R}}^n}$ with $ {{\mathbf{H}}^1}(E) < \infty$ has been studied by examining its intersection with various translated positions of a smooth hypersurface B. The following result has been established: Let B be a proper $(n - 1)$ dimensional smooth submanifold of $ {{\mathbf{R}}^n}$ with nonzero Gaussian curvature at every point. If $E \subset {{\mathbf{R}}^n}$ with $ {{\mathbf{H}}^1}(E) < \infty$, then there exists a countably 1-rectifiable Borel subset R of $ {{\mathbf{R}}^n}$ such that $(E \sim R)$ is purely $({{\mathbf{H}}^1},1)$ unrectifiable and $ (E \sim R) \cap (g + B) = \emptyset$ for almost all $g \in {{\mathbf{R}}^n}$. Furthermore, if in addition E is $ {{\mathbf{H}}^1}$ measurable and $ E \cap (g + B) = \emptyset$ for $ {{\mathbf{H}}^n}$ almost all $ g \in {{\mathbf{R}}^n}$ then ${{\mathbf{H}}^1}(E \cap R) = 0$. Consequently, E is purely $ ({{\mathbf{H}}^1},1)$ unrectifiable.
Nonlinear operations and the solution of integral equations
Jon C.
Helton
373-390
Abstract: The letters S, G and H denote a linearly ordered set, a normed complete Abelian group with zero element 0, and the set of functions from G to G that map 0 into 0, respectively. In addition, if $V \in H$ and there exists an additive function $\alpha$ from $S \times S$ to the nonnegative numbers such that $\left\Vert {V(x,y)P - V(x,y)Q} \right\Vert \leqslant \alpha (x,y)\left\Vert {P - Q} \right\Vert$ for each $\{ x,y,P,Q\}$ in $S \times S \times G \times G$, then $V \in \mathcal{O}\mathcal{S}$ only if $\smallint _x^yVP$ exists for each $\{ x,y,P\}$ in $ S \times S \times G$, and $ V \in \mathcal{O}\mathcal{P}$ only if $ _x{\Pi ^y}(1 + V)P$ exists for each $\{ x,y,P\}$ in $S \times S \times G$. It is established that $ V \in \mathcal{O}\mathcal{S}$ if, and only if, $V \in \mathcal{O}\mathcal{P}$. Then, this relationship is used in the solution of integral equations of the form $ f(x) = h(x) + \smallint _c^x[U(u,v)f(u) + V(u,v)f(v)]$, where U and V are in $\mathcal{O}\mathcal{S}$. This research extends known results in that requirements pertaining to the additivity of U and V are weakened.
Replacing homotopy actions by topological actions
George
Cooke
391-406
Abstract: A homotopy action of a group G on a space X is a homomorphism from G to the group of homotopy classes of homotopy equivalences of X. The question studied in this paper is: When is a homotopy action equivalent, in an appropriate sense, to a topological action of G on X?
Unitary representations of Lie groups with cocompact radical and applications
L.
Pukanszky
1-49
Abstract: The paper gives a necessary and sufficient condition in order that a connected and simply connnected Lie group with cocompact radical be of type I. This result is then applied to a characterization of Lie groups, all irreducible unitary representations of which are completely continuous.
$L\sb{\infty }{}\sb{\lambda }$-equivalence, isomorphism and potential isomorphism
Mark
Nadel;
Jonathan
Stavi
51-74
Abstract: It is well known that two structures are ${L_{\infty \omega }}$-equivalent iff they are potentially isomorphic [that is, isomorphic in some (Cohen) extension of the universe]. We prove that no characterization of ${L_{\infty \lambda }}$-equivalence along these lines is possible (at least for successor cardinals $\lambda$) and the potential-isomorphism relation that naturally comes to mind in connection with ${L_{\infty \lambda }}$ is often not even transitive and never characterizes ${ \equiv _{\infty \lambda }}$ for $\lambda > \omega$. A major part of the work is the construction of ${\kappa ^ + }$-like linear orderings (also Boolean algebras) A, B such that ${N_{{\kappa ^ + }}}({\mathbf{A}},{\mathbf{B}})$, where ${N_\lambda }({\mathbf{A}},{\mathbf{B}})$ means: A and B are nonisomorphic ${L_{\infty \lambda }}$-equivalent structures of cardinality $\lambda$.
Fr\'echet differentiable functionals and support points for families of analytic functions
Paul
Cochrane;
Thomas H.
MacGregor
75-92
Abstract: Given a closed subset of the family $ {S^\ast}(\alpha )$ of functions starlike of order $\alpha$ of a particular form, a continuous Fréchet differentiable functional, J, is constructed with this collection as the solution set to the extremal problem $\max \operatorname{Re} J(f)$ over ${S^\ast}(\alpha )$. Similar results are proved for families which can be put into one-to-one correspondence with $ {S^\ast}(\alpha )$. The support points of $ {S^\ast}(\alpha )$ and $K(\alpha )$, the functions convex of order $ \alpha$, are completely characterized and shown to coincide with the extreme points of their respective convex hulls. Given any finite collection of support points of ${S^\ast}(\alpha )$ (or $ K(\alpha )$), a continuous linear functional, J, is constructed with this collection as the solution set to the extremal problem $ \max \operatorname{Re} J(f)$ over $ {S^\ast}(\alpha )$ (or $K(\alpha )$).
The metabelian $p$-groups of maximal class
R. J.
Miech
93-119
Abstract: This paper presents a solution to the isomorphism problem for the set of metabelian p-groups of maximal class.
Invariant measures and equilibrium states for some mappings which expand distances
Peter
Walters
121-153
Abstract: For a certain collection of transformations T we define a Perron-Frobenius operator and prove a convergence theorem for the powers of the operator along the lines of the theorem D. Ruelle proved in his investigation of the equilibrium states of one-dimensional lattice systems. We use the convergence theorem to study the existence and ergodic properties of equilibrium states for T and also to study the problem of invariant measures for T. Examples of the transformations T considered are expanding maps, transformations arising from f-expansions and shift systems.
Global structural stability of a saddle node bifurcation
Clark
Robinson
155-171
Abstract: S. Newhouse, J. Palis, and F. Takens have recently proved the global structural stability of a one parameter unfolding of a saddle node when the nonwandering set is finite and transversality conditions are satisfied. (The diffeomorphism is Morse-Smale except for the saddle node.) Using their local unfolding of a saddle node and our method of compatible families of unstable disks (instead of the more restrictive method of compatible systems of unstable tubular families), we are able to extend one of their results to the case where the nonwandering set is infinite. We assume that a saddle node is introduced away from the rest of the nonwandering set which is hyperbolic (Axiom A), and that a (strong) transversality condition is satisfied.
A nonlinear semigroup for a functional differential equation
Dennis W.
Brewer
173-191
Abstract: A representation theorem is obtained for solutions of the nonlinear functional differential equation
Nearnesses, proximities, and $T\sb{1}$-compactifications
Ellen E.
Reed
193-207
Abstract: Gagrat, Naimpally, and Thron together have shown that separated Lodato proximities yield ${T_1}$-compactifications, and conversely. This correspondence is not $1 - 1$, since nonequivalent compactifications can induce the same proximity. Herrlich has shown that if the concept of proximity is replaced by that of nearness then all principal (or strict) ${T_1}$-extensions can be accounted for. (In general there are many nearnesses compatible with a given proximity.) In this paper we obtain a 1-1 correspondence between principal ${T_1}$-extensions and cluster-generated nearnesses. This specializes to a 1-1 match between principal $ {T_1}$-compactifications and contigual nearnesses. These results are utilized to obtain a 1-1 correspondence between Lodato proximities and a subclass of $ {T_1}$-compactifications. Each proximity has a largest compatible nearness, which is contigual. The extension induced by this nearness is the construction of Gagrat and Naimpally and is characterized by the property that the dual of each clan converges. Hence we obtain a 1-1 match between Lodato proximities and clan-complete principal ${T_1}$-compactifications. When restricted to EF-proximities, this correspondence yields the usual map between ${T_2}$-compactifications and EF-proximities.
The Riemann hypothesis for Selberg's zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator
Burton
Randol
209-223
Abstract: Much of that part of the theory of the Riemann zeta-function based on the Riemann hypothesis carries over to zeta-functions of Selberg's type, and in this way one can get asymptotic information about various eigenvalue problems. The methods are illustrated in the case of a compact Riemann surface.
Relativized weak disjointness and relatively invariant measures
Douglas C.
McMahon
225-237
Abstract: In this paper we study the relativized weak disjointness and the relativized regionally proximal relation for homomorphisms of point-transitive transformation groups, under the assumption of a relativized invariant measure. We also include a proof of a Folner-type result for syndetic subsets of an amenable group.
Leaf prescriptions for closed $3$-manifolds
John
Cantwell;
Lawrence
Conlon
239-261
Abstract: Our basic question is: What open, orientable surfaces of finite type occur as leaves with polynomial growth in what closed 3-manifolds? This question is motivated by other work of the authors. It is proven that every such surface so occurs for suitable $ {C^\infty }$ foliations of suitable closed 3-manifolds and for suitable $ {C^1}$ foliations of all closed 3-manifolds. If the surface has no isolated nonplanar ends it also occurs for suitable ${C^\infty }$ foliations of all closed 3-manifolds. Finally, a large class of surfaces with isolated nonplanar ends occurs in suitable $ {C^\infty }$ foliations of all closed, orientable 3-manifolds that are not rational homology spheres.
Approximation theorems for uniformly continuous functions
Anthony W.
Hager
263-273
Abstract: Let X be a set, A a family of real-valued functions on X which contains the constants, ${\mu _A}$ the weak uniformity generated by A, and $ U({\mu _A}X)$ the collection of uniformly continuous functions to the real line R. The problem is how to construct $U({\mu _A}X)$ from A. The main result here is: For A a vector lattice, the collection of suprema of countable, finitely A-equiuniform, order-one subsets of ${A^ + }$ is uniformly dense in $U({\mu _A}X)$. Two less technical corollaries: If A is a vector lattice (resp., vector space), then the collection of functions which are finitely A-uniform and uniformly locally-A (resp., uniformly locally piecewise-A) is uniformly dense in $ U({\mu _A}X)$. Further, for any A, a finitely A-uniform function is just a composition $F \circ ({a_1}, \ldots ,{a_p})$ for some $ {a_1}, \ldots ,{a_p} \in A$ and F uniformly continuous on the range of $({a_1}, \ldots ,{a_p})$ in $ {R^p}$. Thus, such compositions are dense in $ U({\mu _A}X)$. For $BU({\mu _A}X)$, the compositions with $F \in BU({R^p})$ are dense (B denoting bounded functions). So, in a sense, to know $U({\mu _A}X)$ it suffices to know A and subspaces of the spaces ${R^p}$, and to know $ BU({\mu _A}X)$, A and the spaces ${R^p}$ suffice.
The homological dimensions of symmetric algebras
James E.
Carrig
275-285
Abstract: Let D be a Dedekind domain and M a rank-one torsion-free D-module. An analysis of $A = {S_D}(M)$, the symmetric algebra of M, yields the following information: Theorem. (1) Tor-dim $A \leqslant 2\;and\; = 1\;iff\;M = K$, the quotient field of D; (2) A is coherent; (3) Global $\dim A = 2$. For higher rank modules coherence is not assured and only rough estimates of the dimensions are found. On the other hand, if $ {S_D}(M)$ is a domain of global dimension two, then M has rank one but the dimension of D may be two. If D is local of dimension two then $M = K$.
Simple Lie algebras of toral rank one
Robert Lee
Wilson
287-295
Abstract: Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic $p > 7$. Let L have Cartan decomposition $L = H + {\sum _{\gamma \in \Gamma }}{L_\gamma }$. If $\Gamma$ generates a cyclic group then L is isomorphic to $ {\text{sl}}(2,F)$ or to one of the simple Lie algebras of generalized Cartan type $ W(1:{\mathbf{n}})$ or $H{(2:{\mathbf{n}}:\Phi )^{(2)}}$.
Frattini subalgebras of finitely generated soluble Lie algebras
Ralph K.
Amayo
297-306
Abstract: This paper is motivated by a recent one of Stewart and Towers [8] investigating Lie algebras with ``good Frattini structure'' (definition below). One consequence of our investigations is to prove that any finitely generated metanilpotent Lie algebra has good Frattini structure, thereby answering a question of Stewart and Towers and providing a complete Lie theoretic analogue of the corresponding group theoretic result of Phillip Hall. It will also be shown that in prime characteristic, finitely generated nilpotent-by-finite-dimensional Lie algebras have good Frattini structure.
A global theorem for singularities of maps between oriented $2$-manifolds
J. R.
Quine
307-314
Abstract: Let M and N be smooth compact oriented connected 2-mani-folds. Suppose $f:M \to N$ is smooth and every point $p \in M$ is either a fold point, cusp point, or regular point of f i.e., f is excellent in the sense of Whitney. Let ${M^ + }$ be the closure of the set of regular points at which f preserves orientation and M the closure of the set of regular points at which f reverses orientation. Let ${p_1}, \ldots ,{p_n}$ be the cusp points and $\mu ({p_k})$ the local degree at the cusp point $ {p_k}$. We prove the following: $\displaystyle \chi (M) - 2\chi ({M^ - }) + \sum \mu ({p_k}) = (\deg f)\chi (N)$ where $ \chi$ is the Euler characteristic and deg is the topological degree. We show that it is a generalization of the Riemann-Hurwitz formula of complex analysis and give some examples.
Invariant means on the continuous bounded functions
Joseph
Rosenblatt
315-324
Abstract: Let G be a noncompact nondiscrete $\sigma$-compact locally compact metric group. A Baire category argument gives measurable sets $ \{ {A_\gamma }:\gamma \in \Gamma \}$ of finite measure with card $(\Gamma ) = c$ which are independent on the open sets. One approximates $\{ {A_\gamma }:\gamma \in \Gamma \}$ by arrays of continuous bounded functions with compact support and then scatters these arrays to construct functions $ \{ {f_\gamma }:\gamma \in \Gamma \}$ in $ {\text{CB}}(G)$ with a certain independence property. If G is also amenable as a discrete group, the existence of these independent functions shows that on ${\text{CB}}(G)$ there are ${2^c}$ mutually singular elements of LIM each of which is singular to TLIM.
Perturbation of translation invariant positivity preserving semigroups on $L\sp{2}({\bf R}\sp{N})$
Ira W.
Herbst;
Alan D.
Sloan
325-360
Abstract: The theory of singular local perturbations of translation invariant positivity preserving semigroups on $ {L^2}({{\mathbf{R}}^N},{d^N}x)$ is developed. A powerful approximation theorem is proved which allows the treatment of a very general class of singular perturbations. Estimates on the local singularities of the kernels of the semigroups, ${e^{ - tH}}$, are given. Eigenfunction expansions are derived. The local singularities of the eigenfunction and generalized eigenfunctions are discussed. Results are illustrated with examples involving singular perturbations of --$\Delta$.
The immersion conjecture for $RP\sp{8l+7}$ is false
Donald M.
Davis;
Mark
Mahowald
361-383
Abstract: Let $\alpha (n)$ denote the number of l's in the binary expansion of n. It is proved that if $n \equiv 7$ (8), $ \alpha (n) = 6$, and $n \ne 63$, then ${\mathbf{R}}{P^n}$ can be immersed in ${{\mathbf{R}}^{2n - 14}}$. This provides the first counterexample to the well-known conjecture that the best immersion is in ${{\mathbf{R}}^{2n - 2\alpha (n) + 1}}$ (when $\alpha (n) \equiv 1$ or $ 2 \bmod 4$). The method of proof is obstruction theory.
Spectral theory for contraction semigroups on Hilbert space
Larry
Gearhart
385-394
Abstract: In this paper we determine the relationship between the spectra of a continuous contraction semigroup on Hilbert space and properties of the resolvent of its infinitesimal generator. The methods rely heavily on dilation theory. In particular, we reduce the general problem to the case that the cogenerator of the semigroup has a characteristic function with unitary boundary values. We then complete the analysis by generalizing the scalar result of J. W. Moeller on compressions of the translation semigroup to the case of infinite multiplicity.
Coordinatization applied to finite Baer * rings
David
Handelman
1-34
Abstract: We clarify and algebraicize the construction of the 'regular rings' of finite Baer $^\ast$ rings. We first determine necessary and sufficient conditions of a finite Baer $^\ast$ ring so that its maximal ring of right quotients is the 'regular ring', coordinating the projection lattice. This is applied to yield significant improvements on previously known results: If R is a finite Baer $^\ast$ ring with right projections $ ^\ast$-equivalent to left projections $({\text{LP}} \sim {\text{RP}})$, and is either of type II or has 4 or more equivalent orthogonal projections adding to 1, then all matrix rings over R are finite Baer $^\ast$ rings, and they also satisfy ${\text{LP}} \sim {\text{RP}}$; if R is a real $A{W^\ast}$ algebra without central abelian projections, then all matrix rings over R are also $ A{W^\ast}$. An alternate approach to the construction of the 'regular ring' is via the Coordinatization Theorem of von Neumann. This is discussed, and it is shown that if a Baer $^\ast$ ring without central abelian projections has a 'regular ring', the 'regular ring' must be the maximal ring of quotients. The following result comes out of this approach: A finite Baer $ ^\ast$ ring satisfying the 'square root' (SR) axiom, and either of type II or possessing 4 or more equivalent projections as above, satisfies $ {\text{LP}} \sim {\text{RP}}$, and so the results above apply. We employ some recent results of J. Lambek on epimorphisms of rings. Some incidental theorems about the existence of faithful epimorphic regular extensions of semihereditary rings also come out.
On the degree of approximation of a function by the partial sums of its Fourier series
Elaine
Cohen
35-74
Abstract: When f is a $ 2\pi$ periodic function with rth order fractional derivative, $r \geqslant 0$, of p-bounded variation, Golubov has obtained estimates of the degree of approximation of f, in the ${L^q}$ norm, $q > p$, by the partial sums of its Fourier series. Here we consider the analogous problem for functions whose fractional derivatives are of $ \Phi$-bounded variation and obtain estimates of the degree of approximation in an Orlicz space norm. In a similar manner we shall extend various results that he obtained on degree of approximation in the sup norm.
Spectral properties of tensor products of linear operators. I
Takashi
Ichinose
75-113
Abstract: The aim of the present paper is to obtain, for tensor products of linear operators, their essential spectra in the sense of F. E. Browder, F. Wolf and M. Schechter and explicit formulae of their nullity, deficiency and index. The theory applies to $A \otimes I + I \otimes B$ and $A \otimes B$.
Injective Banach spaces of continuous functions
John
Wolfe
115-139
Abstract: A description is given of the compact Hausdorff spaces S such that the Banach space $C(S)$ of continuous functions on S is a ${P_\lambda }$-space for $ \lambda < 3$ (under the assumption that S satisfies the countable chain condition). The existence of extension operators from $C({X^\ast}\backslash X)$ to $C({X^\ast})$ is examined under the assumption that $C({X^\ast})$ is injective where $ {X^\ast}$ is some compactification of a locally compact extremally disconnected Hausdorff space X (if $C(S)$ is injective, S is of this form). Some new examples of injective spaces $ C(S)$ are given.
Galois groups and complex multiplication
Michael
Fried
141-163
Abstract: The Schur problem for rational functions is linked to the theory of complex multiplication and thereby solved. These considerations are viewed as a special case of a general problem, prosaically labeled the extension of constants problem. The relation between this paper and a letter of J. Herbrand to E. Noether (published posthumously) is speculatively summarized in a conjecture that may be regarded as an arithmetic version of Riemann's existence theorem.
Real structure in complex $L\sb{1}$-preduals
Daniel E.
Wulbert
165-181
Abstract: Call a complex Banach space selfadjoint if it is isometrically isomorphic to a selfadjoint subspace of a $ C(X,{\mathbf{C}})$-space. B. Hirsberg and A. Lazar proved that if the unit ball of a complex Lindenstrauss space, E, has an extreme point, then E is selfadjoint. Here we will give a characterization of selfadjoint Lindenstrauss spaces, and construct a nonselfadjoint complex Lindenstrauss space.
Ergodic transformations from an interval into itself
Tien Yien
Li;
James A.
Yorke
183-192
Abstract: A class of piecewise continuous, piecewise ${C^1}$ transformations on the interval $J \subset R$ with finitely many discontinuities n are shown to have at most n invariant measures.
A vector lattice topology and function space representation
W. A.
Feldman;
J. F.
Porter
193-204
Abstract: A locally convex topology is defined for a vector lattice having a weak order unit and a certain partition of the weak order unit, analogous to the order unit topology. For such spaces, called ``order partition spaces,'' an extension of the classical Kakutani theorem is obtained: Each order partition space is lattice isomorphic and homeomorphic to a dense subspace of ${C_c}(X)$ containing the constant functions for some locally compact X, and conversely each such $ {C_c}(X)$ is an order partition space. $({C_c}(X)$ denotes all continuous real-valued functions on X with the topology of compact convergence.) One consequence is a lattice-theoretic characterization of ${C_c}(X)$ for X locally compact and realcompact. Conditions for an M-space to be an order partition space are provided.
Lattice-valued Borel measures. II
Surjit Singh
Khurana
205-211
Abstract: Let T be a completely regular Hausdorff space, ${C_b}(T)$ the set of all bounded real-valued continuous functions on T, E a boundedly monotone complete ordered vector space, and $\varphi :{C_b}(T) \to E$ a positive linear map. It is proved that under certain conditions there exist $ \sigma$-additive, $ \tau$-smooth or tight E-valued measures on T which represent $ \varphi$.
Compact manifolds in hyperbolicity
Robert
Brody
213-219
Abstract: In this paper we establish the strongest possible criterion for the hyperbolicity of a compact complex manifold: such a manifold is hyperbolic if and only if it contains no (nontrivial) complex lines. In addition, we study the behavior of such manifolds under deformation and, in particular, answer the two most natural questions about such deformations: Is the space of hyperbolic complex structures on a given $ {C^\infty }$ manifold open in the space of all its complex structures? (Yes.) Is it closed? (Not in general.) These results answer questions first posed by Kobayashi in [4] and [5].
Cardinal Hermite spline interpolation: convergence as the degree tends to infinity
M. J.
Marsden;
S. D.
Riemenschneider
221-244
Abstract: Let ${\mathcal{S}_{2m,r}}$, denote the class of cardinal Hermite splines of degree $2m - 1$ having knots of multiplicity r at the integers. For $f(x) \in {C^{r - 1}}(R)$, the cardinal Hermite spline interpolant to $f(x)$ is the unique element of ${\mathcal{S}_{2m,r}}$ which interpolates $ f(x)$ and its first $ r - 1$ derivatives at the integers. For $y = ({y^0}, \ldots ,{y^{r - 1}})$ an r-tuple of doubly-infinite sequences, the cardinal Hermite spline interpolant to y is the unique $ S(x) \in {\mathcal{S}_{2m,r}}$ satisfying ${S^{(s)}}(\nu) = {y^s},s = 0,1, \ldots ,r - 1$, and $ \nu$ an integer. The following results are proved: If $f(x)$ is a function of exponential type less than $ r\pi$, then the derivatives of the cardinal Hermite spline interpolants to $ f(x)$ converge uniformly to the respective derivatives of $f(x)$ as $ m \to \infty$. For functions from more general, but related, classes, weaker results hold. If y is an r-tuple of $ {l^p}$ sequences, then the cardinal Hermite spline interpolants to y converge to ${W_r}(y)$, a certain generalization of the Whittaker cardinal series which lies in the Sobolev space ${W^{p,r - 1}}(R)$. This convergence is in the Sobolev norm. The class of all such $ {W_r}(y)$ is characterized. For small values of r, the explicit forms of ${W_r}(y)$ are described.
Homotopy and uniform homotopy
Allan
Calder;
Jerrold
Siegel
245-270
Abstract: It is shown that the sets, homotopy and uniform homotopy classes of maps from a finite dimensional normal space to a space of finite type with finite fundamental group, coincide. Applications of this result to the study of remainders of Stone-Cech compactifications, Kan extensions, and other areas are given.
The Diophantine problem for addition and divisibility
L.
Lipshitz
271-283
Abstract: An algorithm is given for deciding existential formulas involving addition and the divisibility relation over the natural numbers.
An existence result on a Volterra equation in a Banach space
Stig-Olof
Londen
285-304
Abstract: Let W be a real reflexive Banach space, dense in a Hilbert space H and with dual $W'$. Let the injection $W \to H$ be continuous and compact. We consider the nonlinear integral equation $u''(t) + Au(t) = f'(t),t \geqslant 0$.
Bases for the positive cone of a partially ordered module
W. Russell
Belding
305-313
Abstract: $(R,{R^ + })$ is a partially ordered ring and $(M,{M^ + })$ is a strict $ (R,{R^ + })$-module. So M is a left R-module and $({R^ + }\backslash \{ 0\} )({M^ + }\backslash \{ 0\} ) \subseteq {M^ + }\backslash \{ 0\}$. Let ${M^ + }.B \subseteq {M^ + }$ is an ${R^ + }$-basis for ${M^ + }$ means ${R^ + }B = {M^ + }$ (spanning) and if r is in R, b in B with $0 < 'rb \leqslant 'b$ then $rb \notin {R^ + }(B\backslash \{ b\} )$ (independence). Result: If B and D are $ {R^ + }$-bases for $ {M^ + }$ then card $ B =$ card D and to within a permutation ${b_i} = {u_i}{d_i}$, for units ${u_i}$ of ${R^ + }$.
The centralizer of a Cartan subalgebra of a Jordan algebra
Edgar G.
Goodaire
314-322
Abstract: If L is a diagonable subspace of an associative algebra A over a field $\Phi \;(L$ is spanned by commuting elements and the linear transformations ad $x:a \mapsto x - xa,x \in L$, are simultaneously diagonalizable), then a map $\lambda :L \to \Phi$ is said to be a weight of L on an A-module V if the space ${V_\lambda } = \{ v \in V:vx = \lambda (x)v\;{\text{for}}\;{\text{all}}\;x \in L\}$ is nonzero. It is shown that if A is finite dimensional semisimple and the characteristic of $\Phi$ is zero then the centralizer of L in A is the centralizer of an element $x \in A$ if and only if x distinguishes the weights of L on every irreducible A-module. This theorem can be used to show that for each representative V of an isomorphism class of irreducible A-modules and for each weight $\lambda$ of L on V, the centralizer of L contains the matrix ring ${D_{{n_\lambda }}},D = {\text{End}_A}V,{n_\lambda } = {\dim _D}{V_\lambda }$ and in fact is the direct sum of all such algebras. If J is a finite dimensional simple reduced Jordan algebra, one can determine precisely those x in J whose centralizer in the universal enveloping algebra of J coincides with the centralizer of a Cartan subalgebra. The simple components of such a centralizer can also be found and in fact are listed for the degree $J \geqslant 3$ case.
Topological entropy at an $u$-explosion
Louis
Block
323-330
Abstract: In this paper an example is given of a ${C^2}$ map g from the circle onto itself, which permits an $\Omega$-explosion. It is shown that topological entropy (considered as a map from ${C^2}({S^1},{S^1})$ to the nonnegative real numbers) is continuous at g.
Characteristic numbers of $G$-manifolds and multiplicative induction
Michael
Bix;
Tammo
tom Dieck
331-343
Abstract: We determine those finite groups G for which characteristic numbers determine G-equivariant bordism in the unoriented and unitary cases.
Existence theorems for Warfield groups
Roger
Hunter;
Fred
Richman;
Elbert
Walker
345-362
Abstract: Warfield studied p-local groups that are summands of simply presented groups, introducing invariants that, together with Ulm invariants, determine these groups up to isomorphism. In this paper, necessary and sufficient conditions are given for the existence of a Warfield group with prescribed Ulm and Warfield invariants. It is shown that every Warfield group is the direct sum of a simply presented group and a group of countable torsion-free rank. Necessary and sufficient conditions are given for when a valuated tree can be embedded in a tree with prescribed relative Ulm invariants, and for when a valuated group in a certain class, including the simply presented valuated groups, admits a nice embedding in a countable group with prescribed relative Ulm invariants. These conditions, which are intimately connected with the existence of Warfield groups, are given in terms of new invariants for valuated groups, the derived Ulm invariants, which vanish on groups and fit into a six term exact sequence with the Ulm invariants.
Some one-relator Hopfian groups
Donald J.
Collins
363-374
Abstract: The group presented by $\displaystyle (a,t;{t^{ - 1}}{a^l}t = {a^m})$ is non-Hopfian if $l,m \ne \pm 1$ and $\pi (l) \ne \pi (m)$, where $\pi (l)$ and $\pi (m)$ denote the sets of prime divisors of l and m. By contrast, we prove that if w is a word of the free group $F({a_1},{a_2})$ which is not primitive and not a proper power, then the group $({a_1},{a_2},t;{t^{ - 1}}{w^l}t = {w^m})$ is Hopfian.
A probabilistic approach to a boundary layer problem
Walter
Vasilsky
375-385
Abstract: An elliptic second order linear operator is approximated by the transition operator of a Markov chain, and the solution to the corresponding approximate boundary value problem is expanded in terms of a small parameter, up to the first order term. In characterizing the boundary values of the first order term in the expansion, a problem of a boundary layer arises, which is treated by probabilistic methods.
An extension of Carlson's theorem for entire functions of exponential type
R.
Gervais;
Q. I.
Rahman
387-394
Abstract: The paper contains some new extensions of the well-known theorem of F. Carlson for entire functions of exponential type.
Rational approximation to $e\sp{-x}$. II
Q. I.
Rahman;
G.
Schmeisser
395-402
Abstract: It is shown that as compared to reciprocals of polynomials of degree n, rational functions of degree n provide an effectively better uniform approximation to the function ${e^{ - x}}$ on $ [0,\infty )$.
Norbert Wiener's ergodic theorem for convex regions
Norberto A.
Fava;
Jorge H.
Nanclares
403-406
Abstract: It is proved that the geometric hypothesis of a theorem which generalizes Norbert Wiener's multiparameter ergodic theorem are satsified in the case of arbitrary convex regions, provided only that they form a substantial family as defined in the introduction.
Ergodic equivalence relations, cohomology, and von Neumann algebras. I
Jacob
Feldman;
Calvin C.
Moore
289-324
Abstract: Let $(X,\mathcal{B})$ be a standard Borel space, $R \subset X \times X$ an equivalence relation $\in \mathcal{B} \times \mathcal{B}$. Assume each equivalence class is countable. Theorem 1: $ \exists$ a countable group G of Borel isomorphisms of $(X,\mathcal{B})$ so that $R = \{ (x,gx):g \in G\} $. G is far from unique. However, notions like invariance and quasi-invariance and R-N derivatives of measures depend only on R, not the choice of G. We develop some of the ideas of Dye [1], [2] and Krieger [1]-[5] in a fashion explicitly avoiding any choice of G; we also show the connection with virtual groups. A notion of ``module over R'' is defined, and we axiomatize and develop a cohomology theory for R with coefficients in such a module. Surprising application (contained in Theorem 7): let $\alpha ,\beta $ be rationally independent irrationals on the circle $ \mathbb{T}$, and f Borel: $ \mathbb{T} \to \mathbb{T}$. Then $\exists$ Borel $g,h:\mathbb{T} \to \mathbb{T}$ with $f(x) = (g(ax)/g(x))(h(\beta x)/h(x))$ a.e. The notion of ``skew product action'' is generalized to our context, and provides a setting for a generalization of the Krieger invariant for the R-N derivative of an ergodic transformation: we define, for a cocycle c on R with values in the group A, a subgroup of A depending only on the cohomology class of c, and in Theorem 8 identify this with another subgroup, the ``normalized proper range'' of c, defined in terms of the skew action. See also Schmidt [1].
Ergodic equivalence relations, cohomology, and von Neumann algebras. II
Jacob
Feldman;
Calvin C.
Moore
325-359
Abstract: Let R be a Borel equivalence relation with countable equivalence classes, on the standard Borel space $(X,\mathcal{A},\mu )$. Let $\sigma$ be a 2-cohomology class on R with values in the torus $ \mathbb{T}$. We construct a factor von Neumann algebra ${\mathbf{M}}(R,\sigma )$, generalizing the group-measure space construction of Murray and von Neumann [1] and previous generalizations by W. Krieger [1] and G. Zeller-Meier [1]. Very roughly, ${\mathbf{M}}(R,\sigma )$ is a sort of twisted matrix algebra whose elements are matrices $({a_{x,y}})$, where $ (x,y) \in R$. The main result, Theorem 1, is the axiomatization of such factors; any factor M with a regular MASA subalgebra A, and possessing a conditional expectation from M onto A, and isomorphic to $ {\mathbf{M}}(R,\sigma )$ in such a manner that A becomes the ``diagonal matrices"; $ (R,\sigma )$ is uniquely determined by M and A. A number of results are proved, linking invariants and automorphisms of the system (M, A) with those of $(R,\sigma )$. These generalize results of Singer [1] and of Connes [1]. Finally, several results are given which contain fragmentary information about what happens with a single M but two different subalgebras $ {{\mathbf{A}}_1},{{\mathbf{A}}_2}$.
Convergence of random processes without discontinuities of the second kind and limit theorems for sums of independent random variables
L. Š.
Grinblat
361-379
Abstract: Let $ {\xi _1}(t), \ldots ,{\xi _n}(t), \ldots$ and $\xi (t)$ be random processes on the interval [0, 1], without discontinuities of the second kind. A. V. Skorohod has given necessary and sufficient conditions under which the distribution of $f({\xi _n}(t))$ converges to the distribution of $f(\xi (t))$ as $n \to \infty$ for any functional f continuous in the Skorohod metric. In the following we shall consider only stochastically right-continuous processes without discontinuities of the second kind, i.e., processes such that the space X of their sample functions is the space of all right-continuous functions $x(t)(0 \leqslant t \leqslant 1)$ without discontinuities of the second kind. For a set $T = \{ {t_1}, \ldots {t_n}, \ldots \} \subset [0,1]$ the metric ${\rho _T}$ is defined on X as in 2.3. The metric ${\rho _T}$ defines on the X the minimal topology in which all functional continuous in Skorohod's metric and also the functional $ x({t_1} - 0),x({t_1}), \ldots ,x({t_n} - 0),x({t_n}), \ldots$ are continuous. We will give necessary and sufficient conditions under which the distribution of $f({\xi _n}(t))$ converges to the distribution of $f(\xi (t))$ as $n \to \infty$ for any completely continuous functional f, i.e. for any functional f which is continuous in any of the metrics ${\rho _T}$ defined in 2.3.
The porous medium equation in one dimension
Barry F.
Knerr
381-415
Abstract: We consider a second order nonlinear degenerate parabolic partial differential equation known as the porous medium equation, restricting our attention to the case of one space variable and to the Cauchy problem where the initial data are nonnegative and have compact support consisting of a bounded interval. Solutions are known to have compact support for each fixed time. In this paper we study the lateral boundary, called the interface, of the support $ P[u]$ of the solution in ${R^1} \times (0,T)$. It is shown that the interface consists of two monotone Lipschitz curves which satisfy a specified differential equation. We then prove results concerning the behavior of the interface curves as t approaches zero and as t approaches infinity, and prove that the interface curves are strictly monotone except possibly near $t = 0$. We conclude by proving some facts about the behavior of the solution in $ P[u]$.
The broken-circuit complex
Tom
Brylawski
417-433
Abstract: The broken-circuit complex introduced by H. Wilf (Which polynomials are chromatic?, Proc. Colloq. Combinational Theory (Rome, 1973)) of a matroid G is shown to be a cone over a related complex, the reduced broken-circuit complex
Cartan subalgebras of simple Lie algebras
Robert Lee
Wilson
435-446
Abstract: Let L be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic $p > 7$. Let H be a Cartan subalgebra of L, let $L = H + {\Sigma _{\gamma \in \Gamma }}{L_\gamma }$ be the Cartan decomposition of L with respect to H, and let $ \bar H$ be the restricted subalgebra of Der L generated by ad H. Let T denote the maximal torus of $ \bar H$ and I denote the nil radical of $\bar H$. Then $ \bar H = T + I$. Consequently, each $ \gamma \in \Gamma$ is a linear function on H.
General position of equivariant maps
Edward
Bierstone
447-466
Abstract: A natural generic notion of general position for smooth maps which are equivariant with respect to the action of a compact Lie group is introduced. If G is a compact Lie group, and M, N are smooth G-manifolds, then the set of smooth equivariant maps $ F:M \to N$ which are in general position with respect to a closed invariant submanifold P of N, is open and dense in the Whitney topology. The inverse image of P, by an equivariant map in general position, is Whitney stratified. The inverse images, by nearby equivariant maps in general position, are topologically ambient isotopic. In the local context, let V, W be linear G-spaces, and $F:V \to W$ a smooth equivariant map. Let ${F_1}, \ldots ,{F_k}$ be a finite set of homogeneous polynomial generators for the module of smooth equivariant maps, over the ring of smooth invariant functions on V. There are invariant functions ${h_1}, \ldots ,{h_k}$ such that $F = U \circ$ graph h, where graph h is the graph of $h(x) = ({h_1}(x), \ldots ,{h_k}(x))$, and $ U(x,h) = \Sigma _{i = 1}^k{h_i}{F_i}(x)$. The isomorphism class of the real affine algebraic subvariety $(U = 0)$ of $V \times {{\mathbf{R}}^k}$ is uniquely determined (up to product with an affine space) by V, W. F is said to be in general position with respect to $0 \in W$ at $0 \in V$ if graph $h:V \to V \times {{\mathbf{R}}^k}$ is transverse to the minimum Whitney stratification of $ (U = 0)$, at $ x \in V$.
The oscillatory behavior of certain derivatives
Richard J.
O’Malley;
Clifford E.
Weil
467-481
Abstract: The derivatives considered are the approximate derivative and the kth Peano derivative. The main results strengthen the Darboux property, which both of these derivatives possess. Theorem. If the approximate derivative ${f'_{{\text{ap}}}} = f'$ and on which $ f'$ attains both M and -- M. The other main theorem is obtained from this one by replacing the approximate derivative by the kth Peano derivative.
The two-generator subgroups of one-relator groups with torsion
Stephen J.
Pride
483-496
Abstract: The main aim of this paper is to show that every two-generator subgroup of any one-relator group with torsion is either a free product of cyclic groups or is a one-relator group with torsion. This result is proved by using techniques for reducing pairs of elements in certain HNN groups. These techniques not only apply to one-relator groups with torsion but also to a large number of other groups, and some additional applications of the techniques are included in the paper. In particular, examples are given to show that the following result of K. Honda is no longer true for infinite groups: if g is a commutator in a finite group G then every generator of ${\text{sgp}}\{ g\}$ is a commutator in G. This confirms a conjecture of B. H. Neumann.
Derivatives of entire functions and a question of P\'olya. II
Simon
Hellerstein;
Jack
Williamson
497-503
Abstract: It is shown that if f is an entire function of infinite order, which is real on the real axis and has, along with $ f'$, only real zeros, then $ f''$ has nonreal zeros (in fact, infinitely many). The finite order case was treated by the authors in a preceding paper. The combined results show that the only real entire functions f for which $f,f'$, and $f''$ have only real zeros are those in the Laguerre-Pólya class, i.e. $\displaystyle f(z) = {z^m}\exp \{ - a{z^2} + bz + c\} \prod\limits_n {\left( {1 - \frac{z}{{{z_n}}}} \right)} {e^{z/{z_n}}},$ $a \geqslant 0,b,c$ and the ${z_n}$ real, and $\Sigma z_n^{ - 2} < \infty$. This gives a strong affirmative version of an old conjecture of Pólya.
The asymptotic behavior of the first eigenvalue of differential operators degenerating on the boundary
Allen
Devinatz;
Avner
Friedman
505-529
Abstract: When L is a second order ordinary or elliptic differential operator, the principal eigenvalue for the Dirichlet problem and the corresponding principal (positive) eigenfunction u are known to exist and u is unique up to normalization. If further L has the form $\varepsilon \Sigma {a_{ij}}{\partial ^2}/\partial {x_i}\partial {x_i} + \Sigma {b_i}\partial /\partial {x_i}$ then results are known regarding the behavior of the principal eigenvalue $\lambda = {\lambda _\varepsilon }$ as $\varepsilon \downarrow 0$. These results are very sharp in case the vector $({b_i})$ has a unique asymptotically stable point in the domain $\omega$ where the eigenvalue problem is considered. In this paper the case where L is an ordinary differential operator degenerating on the boundary of $\omega$ is considered. Existence and uniqueness of a principal eigenvalue and eigenfunction are proved and results on the behavior of ${\lambda _\varepsilon }$ as $\varepsilon \downarrow 0$ are established.
On the degree of convergence of piecewise polynomial approximation on optimal meshes
H. G.
Burchard
531-559
Abstract: The degree of convergence of best approximation by piecewise polynomial and spline functions of fixed degree is analyzed via certain F-spaces $ {\mathbf{N}}_0^{p,n}$ (introduced for this purpose in [2]). We obtain two o-results and use pairs of inequalities of Bernstein- and Jackson-type to prove several direct and converse theorems. For f in ${\mathbf{N}}_0^{p,n}$ we define a derivative ${D^{n,\sigma }}f$ in ${L^\sigma },\sigma = {(n + {p^{ - 1}})^{ - 1}}$, which agrees with ${D^n}f$ for smooth f, and prove several properties of $ {D^{n,\sigma }}$.
Compact perturbations of certain von Neumann algebras
Joan K.
Plastiras
561-577
Abstract: Let $\mathcal{E}$ be a sequence of mutually orthogonal, finite dimensional projections whose sum is the identity on a Hilbert space $ \mathcal{H}$. If we denote the commutant of $ \mathcal{E}$ by $ \mathcal{D}(\mathcal{E})$ and the ideal of compact operators on $\mathcal{H}$ by $ \mathcal{C}(\mathcal{H})$, then it is easily verified that $\mathcal{D}(\mathcal{E}) + \mathcal{C}(\mathcal{H}) = \{ T + K:T \in \mathcal{D}(\mathcal{E}),K \in \mathcal{C}(\mathcal{H})\}$ is a ${C^\ast}$-algebra. In this paper we classify all such algebras up to $^\ast$-isomorphism and characterize them by examining their relationship to certain quasidiagonal and quasitriangular algebras.
The weakly coupled Yukawa$\sb{2}$ field theory: cluster expansion and Wightman axioms
Alan
Cooper;
Lon
Rosen
1-88
Abstract: We prove convergence of the Glimm-Jaffe-Spencer cluster expansion for the weakly coupled Yukawa model in two dimensions, thereby verifying the Wightman axioms including a positive mass gap.
P.R.-regulated systems of notation and the subrecursive hierarchy equivalence property
Fred
Zemke
89-118
Abstract: We can attempt to extend the Grzegorczyk Hierarchy transfinitely by defining a sequence of functions indexed by the elements of a system of notation $\mathcal{S}$, using either iteration (majorization) or enumeration techniques to define the functions. (The hierarchy is then the sequence of classes of functions elementary in the functions of the sequence of functions.) In this paper we consider two sequences $ {\{ {F_s}\} _{s \in \mathcal{S}}}$ and ${\{ {G_s}\} _{s \in \mathcal{S}}}$ defined by iteration and a sequence ${\{ {E_s}\} _{s \in \mathcal{S}}}$ defined by enumeration; the corresponding hierarchies are $ \{ {\mathcal{F}_s}\} ,\{ {\mathcal{G}_s}\} ,\{ \mathcal{E}{_s}\}$. We say that $ \mathcal{S}$ has the subrecursive hierarchy equivalence property if these two conditions hold: (I) $ {\mathcal{E}_s} = {\mathcal{F}_s} = {\mathcal{G}_s}$ for all $s \in \mathcal{S}$; (II) ${\mathcal{E}_s} = {\mathcal{E}_t}$ for all $ s,t \in \mathcal{S}$ such that $\vert s\vert = \vert t\vert(\vert s\vert$ is the ordinal denoted by s). We show that a certain type of system of notation, called p.r.-regulated, has the subrecursive hierarchy equivalence property. We present a nontrivial example of such a system of notation, based on Schütte's Klammersymbols. The resulting hierarchy extends those previously in print, which used the so-called standard fundamental sequences for limits $< {\varepsilon _0}$.
A partial surface variation for extremal schlicht functions
T. L.
McCoy
119-138
Abstract: Let a topological sphere be formed from $ \vert z\vert \leqslant 1$ by dissecting the circumference into finitely many pairs $ {I_j},{I'_j}$. In a natural way, Q-polygons become Riemann surfaces, thus can be mapped conformally onto the number sphere. When Q is of the form $Q(z) = \Sigma _{j = - N}^N{B_j}{z^j}$, then the corresponding mapping functions, suitably normalized, become the extremal schlicht functions for the coefficient body $ {V_{N + 1}}$ [3, p. 120]. Suppose that for a given dissection of $\vert z\vert = 1$ there is a family $ Q(z,t)$ of consistent meromorphic functions. For Q sufficiently smooth as a function of $ \varepsilon$, we study the variation of the corresponding normalized mapping functions $ f(p,\varepsilon )$, using results of [2], and show smoothness of f as a function of $ \varepsilon$. Specializing Q to the form above, we deduce the existence of smooth submanifolds of $\partial {V_{N + 1}}$ and obtain a variational formula for the extremal schlicht functions corresponding to motion along these submanifolds.
On $n$-widths in $L\sp{\infty }$
Charles A.
Micchelli;
Allan
Pinkus
139-174
Abstract: The n-width in ${L^\infty }$ of certain sets determined by matrices and integral operators is determined. The notion of total positivity is essential in the analysis.
Weak convergence of the area of nonparametric $L\sb{1}$ surfaces
Kim E.
Michener
175-184
Abstract: The main purpose of this work is to obtain an analogue to a theorem of L. C. Young on the behavior of the nonparametric surface area of continuous functions. The analogue is for $ {L^1}$ functions of generalized bounded variation. By considering arbitrary Borel vector measures and kernels other than the area kernel, results concerning the weak behavior of measures induced by a class of sublinear functionals are obtained.
Inclusion relations between power methods and matrix methods of limitation
Abraham
Ziv
185-211
Abstract: A matrix method of limitation is a generalization of both ordinary Toeplitz methods and semicontinuous methods. A power method is a generalization of both Abel's method and Borel's exponential method. The main concern of this paper is to find necessary and sufficient conditions for the field of a given power method to be included in the field of a given matrix method. The problem is solved for a wide family of power methods which includes all Abel type methods, the logarithmic method, all Borel type methods and others (also nonregular power methods). Preliminary results, which serve as tools in the solution of the main problem, clarify some aspects of the nature of the field of a power method as an FK space.
Group presentations corresponding to spines of $3$-manifolds. II
R. P.
Osborne;
R. S.
Stevens
213-243
Abstract: Let $\phi = \langle {a_1}, \ldots ,{a_n}\vert{R_1}, \ldots ,{R_m}\rangle$ denote a group presentation. Let $ {K_\phi }$ denote the corresponding 2-complex. It is well known that every compact 3-manifold has a spine of the form ${K_\phi }$ for some $\phi$, but that not every $ {K_\phi }$ is a spine of a compact 3-manifold. Neuwirth's algorithm (Proc. Cambridge Philos. Soc. 64 (1968), 603-613) decides whether ${K_\phi }$ can be a spine of a compact 3-manifold. However, it is impractical for presentations of moderate length. In this paper a simple planar graph-like object, called a RR-system (railroad system), is defined. To each RR-system corresponds a whole family of compact orientable 3-manifolds with spines of the form ${K_\phi }$, where $\phi$ has a particular form (e.g., $\langle a,b{a^m}{b^n}{a^p}{b^n},{a^m}{b^n}{a^m}{b^q}\rangle $), subject only to certain requirements of relative primeness of certain pairs of exponents. Conversely, every ${K_\phi }$ which is a spine of some compact orientable 3-manifold can be obtained in this way. An equivalence relation on RR-systems is defined so that equivalent RR-systems determine the same family of manifolds. Results of Zieschang are applied to show that the simplest spine of 3-manifolds arises from a particularly simple kind of RR-system called a reduced RR-system. Following Neuwirth, it is shown how to determine when a RR-system gives rise to a collection of closed 3-manifolds.
Group presentations corresponding to spines of $3$-manifolds. III
R. P.
Osborne;
R. S.
Stevens
245-251
Abstract: Continuing after the previous papers of this series, attention is devoted to RR-systems having two towns (i.e., to compact 3-manifolds with spines corresponding to group presentations having two generators). An interesting kind of symmetry is noted and then used to derive some useful results. Specifically, the following theorems are proved: Theorem 1. Let $ \phi$ be a group presentation corresponding to a spine of a compact orientable 3-manifold, and let w be a relator of $ \phi$ involving just two generators a and b. If w is cyclically reduced, then either (a) w can be ``written backwards" (i.e., if $w = {a^{{m_1}}}{b^{{m_1}}}{a^{{m_2}}}{b^{{n_2}}} \ldots {a^{{m_k}}}{b^{{n_k}}}$, then w is a cyclic conjugate of ${b^{{n_k}}}{a^{{m_k}}} \ldots {b^{{n_2}}}{a^{{m_2}}}{b^{{n_1}}}{a^{{m_1}}}$), or (b) w lies in the commutator subgroup of the free group on a and b. Theorem 2. (Loose translation). If $\phi$ is a group presentation with two generators and if the corresponding 2-complex ${K_\phi }$ is a spine of a closed orientable 3-manifold then, ${K_\phi }$ is a spine of a closed orientable 3-manifold if and (except for two minor cases) only if $\phi$ has two relators and among the six allowable types of syllables (3 in each generator), exactly four occur an odd number of times. Further, each of the two relators can be ``written backwards."
Structure of symmetric tensors of type $(0, 2)$ and tensors of type $(1, 1)$ on the tangent bundle
Kam Ping
Mok;
E. M.
Patterson;
Yung Chow
Wong
253-278
Abstract: The concepts of M-tensor and M-connection on the tangent bundle TM of a smooth manifold M are used in a study of symmetric tensors of type (0, 2) and tensors of type (1, 1) on TM. The constructions make use of certain local frames adapted to an M-connection. They involve extending known results on TM using tensors on M to cases in which these tensors are replaced by M-tensors. Particular attention is devoted to (pseudo-) Riemannian metrics on TM, notably those for which the vertical distribution on TM is null or nonnull, and to the construction of almost product and almost complex structures on TM.
Lattices of convex sets
Mary Katherine
Bennett
279-288
Abstract: If V is a vector space over an ordered division ring, C a convex subset of V and L the lattice of convex subsets of C, then we call L a convexity lattice. We give necessary and sufficient conditions for an abstract lattice to be a convexity lattice in the finite dimensional case.
Horocycle flows on certain surfaces without conjugate points
Patrick
Eberlein
1-36
Abstract: We study the topological but not ergodic properties of the horocycle flow $\{ {h_t}\}$ in the unit tangent bundle SM of a complete two dimensional Riemannian manifold M without conjugate points that satisfies the ``uniform Visibility'' axiom. This axiom is implied by the curvature condition $K \leqslant c < 0$ but is weaker so that regions of positive curvature may occur. Compactness is not assumed. The method is to relate the horocycle flow to the geodesic flow for which there exist useful techniques of study. The nonwandering set ${\Omega _h} \subseteq SM$ for $\{ {h_t}\}$ is classified into four types depending upon the fundamental group of M. The extremes that $ {\Omega _h}$ be a minimal set for $\{ {h_t}\}$ and that ${\Omega _h}$ admit periodic orbits are related to the existence or nonexistence of compact ``totally convex'' sets in M. Periodic points are dense in $ {\Omega _h}$ if they exist at all. The only compact minimal sets in ${\Omega _h}$ are periodic orbits if M is noncompact The flow $\{ {h_t}\}$ is minimal in SM if and only if M is compact. In general $\{ {h_t}\}$ is topologically transitive in ${\Omega _h}$ and the vectors in ${\Omega _h}$ with dense orbits are classified. If the fundamental group of M is finitely generated and ${\Omega _h} = SM$ then $\{ {h_t}\}$ is topologically mixing in SM.
On analytically invariant subspaces and spectra
Domingo A.
Herrero
37-44
Abstract: Let T be a bounded linear operator from a complex Banach space $\mathfrak{X}$ into itself. Let ${\mathcal{A}_T}$ and $ \mathcal{A}_T^a$ denote the weak closure of the polynomials and the rational functions (with poles outside the spectrum $\sigma (T)$ of T) in T, respectively. The lattice $ {\operatorname{Lat}}\;\mathcal{A}_T^a$ of (closed) invariant subspaces of $\mathcal{A}_T^a$ is a very particular subset of the invariant subspace lattice $ {\operatorname{Lat}}\;{\mathcal{A}_T} = {\operatorname{Lat}}\;T$ of T. It is shown that: (1) If the resolvent set of T has finitely many components, then $ {\operatorname{Lat}}\;\mathcal{A}_T^a$ is a clopen (i.e., closed and open) sublattice of $ {\operatorname{Lat}}\;T$, with respect to the ``gap topology'' between subspaces. (2) If $ {\mathfrak{M}_1},{\mathfrak{M}_2} \in {\operatorname{Lat}}\;T,{\mathfrak{M}_1} \cap {\mathfrak{M}_2} \in {\operatorname{Lat}}\;\mathcal{A}_T^a$ and ${\mathfrak{M}_1} + {\mathfrak{M}_2}$ is closed in $ \mathfrak{X}$ and belongs to $ {\operatorname{Lat}}\;\mathcal{A}_T^a$, then $ {\mathfrak{M}_1}$ and ${\mathfrak{M}_2}$ also belong to $ {\operatorname{Lat}}\;\mathcal{A}_T^a$. (3) If $\mathfrak{M} \in {\operatorname{Lat}}\;T,R$ is the restriction of T to $\mathfrak{M}$ and $\bar T$ is the operator induced by T on the quotient space $ \mathfrak{X}/\mathfrak{M}$, then $\sigma (T) \subset \sigma (R) \cup \sigma (\bar T)$. Moreover, $\sigma (T) = \sigma (R) \cup \sigma (\bar T)$ if and only if $\mathfrak{M} \in {\operatorname{Lat}}\;\mathcal{A}_T^a$. The results also include an analysis of the semi-Fredholm index of R and $ \bar T$ at a point $\lambda \in \sigma (R) \cup \sigma (\bar T)\backslash \sigma (T)$ and extensions of the results to algebras between $ {\mathcal{A}_T}$ and $\mathcal{A}_T^a$.
On the spectra of the restrictions of an operator
Domingo A.
Herrero
45-58
Abstract: Let T be a bounded linear operator from a complex Banach space $\mathfrak{X}$ into itself and let $\mathfrak{M}$ be a closed invariant subspace of T. Let $ T\vert\mathfrak{M}$ denote the restriction of T to $\mathfrak{M}$ and let $\sigma$ denote the spectrum of an operator. The main results say that: (1) If $\mathfrak{X}$ is the closed linear span of a family $ \{ {\mathfrak{M}_v}\}$ of invariant subspaces, then every component of $\sigma (T)$ intersects the closure of the set $ { \cup _v}\sigma (T\vert{\mathfrak{M}_v})$ and every point of $\sigma (T)\backslash { \cup _v}\sigma (T\vert{\mathfrak{M}_v})$ is an approximate eigenvalue of T. (2) If $ \mathfrak{X}$ is the closed linear span of a finite family $ \{ {\mathfrak{M}_1}, \ldots ,{\mathfrak{M}_n}\}$ of invariant subspaces, and the spectra $\sigma (T\vert{\mathfrak{M}_j}),j = 1,2, \ldots ,n$, are pairwise disjoint, then $\mathfrak{X}$ is actually equal to the algebraic direct sum of the $ {\mathfrak{M}_j}$'s, the $ {\mathfrak{M}_j}$'s are hyperinvariant subspaces of T and $\sigma (T) = \cup _{j = 1}^n\sigma (T\vert{\mathfrak{M}_j})$. This last result is sharp in a certain specified sense. The results of (1) have a ``dual version'' $ (1')$; (1) and $ (1')$ are applied to analyze the spectrum of an operator having a chain of invariant subspaces which is ``piecewise well-ordered by inclusion", extending in several ways recent results of J. D. Stafney on the spectra of lower triangular matrices.
The Mackey Borel structure on the spectrum of an approximately finite-dimensional separable $C\sp*$-algebra
George A.
Elliott
59-68
Abstract: It is shown that the Mackey Borel structures on the spectra of any two approximately finite-dimensional separable $ {C^\ast}$-algebras not of type I are isomorphic.
Topological irreducibility of nonunitary representations of group extensions
Floyd L.
Williams
69-84
Abstract: A functional-analytic approach to the study of the topological irreducibility of certain nonunitary induced representations is set forth. The methods contrast, and in some sense, encompass those first initiated by E. Thieleker in [4], and are amenable to complete irreducibility questions as well. Several sufficient conditions for topological irreducibility are established. A sufficient condition for reducibility is also presented--the latter serving to explain an interesting counterexample due to J. M. G. Fell.
Embedding of closed categories into monoidal closed categories
Miguel L.
Laplaza
85-91
Abstract: S. Eilenberg and G. M. Kelly have defined a closed category as a category with internal homomorphism functor, left Yoneda natural arrows, unity object and suitable coherence axioms. A monoidal closed category is a closed category with an associative tensor product which is adjoint to the int-hom. This paper proves that a closed category can be embedded in a monoidal closed category: the embedding preserves any associative tensor product which may exist. Besides the usual tools of the theory of closed categories the proof uses the results of B. Day on promonoidal structures.
Plugging flows
Peter B.
Percell;
F. Wesley
Wilson
93-103
Abstract: A plug construction is a local modification of a nonsingular flow which severs certain kinds of recurrence properties. In this paper we investigate the effect of plug constructions on minimal sets, the nonwandering set, and the chain recurrent set and the explosions of these sets when a plug construction is perturbed.
Quasi-multipliers
Kelly
McKennon
105-123
Abstract: A quasi-multiplier m on an algebra A is a bilinear mapping from $A \times A$ into itself such that $m(ax,yb) = am(x,y)b$ for all $a,x,y,b \in A$. An introduction to the theory of quasi-multipliers on Banach algebras with minimal approximate identities is given and applications to $ {C^\ast}$-algebras and group algebras are developed.
Almost split sequences for group algebras of finite representation type
Idun
Reiten
125-136
Abstract: Let k be an algebraically closed field of characteristic p and G a finite group such that p divides the order of G. We compute all almost split sequences over kG when kG is of finite representation type, or more generally, for a finite dimensional k-algebra $\Lambda$ given by a Brauer tree. We apply this to show that if $\Lambda$ and $\Lambda '$ are stably equivalent k-algebras given by Brauer trees, then they have the same number of simple modules.
Knotting a $k$-connected closed ${\rm PL}$ $m$-manifold in $E\sp{2m-k}$
Jože
Vrabec
137-165
Abstract: Embeddings of a k-connected closed PL m-manifold $(0 \leqslant k \leqslant m - 3)$ in $ (2m - k)$-dimensional euclidean space are classified up to isotopy. Thus this paper completes the results stated, and partly proved, in J. F. P. Hudson's Piecewis linear topology.
Packing and covering constants for certain families of trees. II
A.
Meir;
J. W.
Moon
167-178
Abstract: In an earlier paper we considered the problem of determining the packing and covering constants for families of trees whose generating function y satisfied a relation $y = x\phi (y)$ for some power series $\phi$ in y. In the present paper we consider the problem for some families of trees whose generating functions satisfy a more complicated relation.
Weak $L\sb{1}$ characterizations of Poisson integrals, Green potentials and $H\sp{p}$ spaces
Peter
Sjögren
179-196
Abstract: Our main result can be described as follows. A subharmonic function u in a suitable domain $\Omega$ in $ {{\mathbf{R}}^n}$ is the difference of a Poisson integral and a Green potential if and only if u divided by the distance to $\partial \Omega $ is in weak $ {L_1}$ in $\Omega$. Similar conditions are given for a harmonic function to be the Poisson integral of an ${L_p}$ function on $ \partial \Omega$. Iterated Poisson integrals in a polydisc are also considered. As corollaries, we get weak ${L_1}$ characterizations of $ {H^p}$ spaces of different kinds.
A superposition theorem for unbounded continuous functions
Raouf
Doss
197-203
Abstract: Let ${R^n}$ be the n-dimensional Euclidean space. We prove that there are 4n real functions ${\varphi _q}$ continuous on ${R^n}$ with the following property: Every real function f, not necessarily bounded, continuous on ${R^n}$, can be written $ f(x) = \Sigma _{q = 1}^{2n + 1}g({\varphi _q}(x)) + \Sigma _{q = 2n + 2}^{4n}h({\varphi _q}(x)),x \in {R^n}$, where g, h are 2 real continuous functions of one variable, depending on f.
The closed leaf index of foliated manifolds
Lawrence
Conlon;
Sue
Goodman
205-221
Abstract: For M a closed, connected, oriented 3-manifold, a topological invariant is computed from the cohomology ring ${H^\ast}(M;{\mathbf{Z}})$ that provides an upper bound to the number of topologically distinct types of closed leaves any smooth transversely oriented foliation of M can contain. In general, this upper bound is best possible.
The diameter of orbits of compact groups of isometries; Newman's theorem for noncompact manifolds
David
Hoffman
223-233
Abstract: The diameter of orbits of a compact isometry group G of a Riemannian manifold M cannot be uniformly small. If the sectional curvature of M is bounded above by $ {b^2}$ (b real or pure imaginary), then explicit bounds are found for $D(M)$, where $D(M)$ is defined to be the largest number such that: If every orbit G has diameter less than $ D(M)$, then G acts trivially on M. These bounds depend only on b and the injectivity radius of M. The proofs involve an investigation of various types of convex sets and an estimate for distance contraction of the exponential map on a manifold with bounded curvature.
Some properties of families of convex cones
Meir
Katchalski
235-240
Abstract: The purpose of this paper is to study properties of finite families of convex cones in n-dimensional Euclidean space ${R^n}$, whose members all have the origin as a common apex. Of special interest are such families of convex cones in ${R^n}$ which have the following property: Each member of the family is of dimension n, the intersection of any two members is at least $(n - 1)$-dimensional, ..., the intersection of any n members is at least 1-dimensional and the intersection of all the members is the origin.
On the asymptotic distribution of closed geodesics on compact Riemann surfaces
Burton
Randol
241-247
Abstract: The set of lengths of closed geodesics on a compact Riemann surface is related to the Selberg zeta function in a manner which is evocative of the relationship between the rational primes and the Riemann zeta function. In this paper, this connection is developed to derive results about the asymptotic distribution of these lengths.
Disintegration of measures on compact transformation groups
Russell A.
Johnson
249-264
Abstract: Let G be a compact metrizable group which acts freely on a locally compact Hausdorff space X. Let X, $ \mu$ be a measure on $ X,\pi :X \to X/G \equiv Y$ the projection, $ \nu = \pi (\mu )$. We show that there is a $\nu$-Lusin-measurable disintegration of $ \mu$ with respect to it. We use this result to prove a structure theorem concerning T-ergodic measures on bitransformation groups (G, X, T) with G metric and X compact. We finish with some remarks concerning the case when G is not metric.
On characterizing the standard quantum logics
W. John
Wilbur
265-282
Abstract: Let $\mathcal{L}$ be a complete projective logic. Then $\mathcal{L}$ has a natural representation as the lattice of $\langle { \cdot , \cdot } \rangle$-closed subspaces of a left vector space V over a division ring D, where $\langle {\cdot,\cdot} \rangle$ is a definite $ \theta$-bilinear symmetric form on V, $\theta$ being some involutive antiautomorphism of D. Now a well-known theorem of Piron states that if D is isomorphic to the real field, the complex field or the sfield of quaternions, if $ \theta$ is continuous, and if the dimension of $ \mathcal{L}$ is properly restricted, then $ \mathcal{L}$ is just one of the standard Hilbert space logics. Here we also assume $\mathcal{L}$ is a complete projective logic. Then if every $\theta$-fixed element of D is in the center of D and can be written as $\pm \,d\theta (d)$, some $d \in D$, and if the dimension of $\mathcal{L}$ is properly restricted, we show that $\mathcal{L}$ is just one of the standard Hilbert space logics over the reals, the complexes, or the quaternions. One consequence is the extension of Piron's theorem to discontinuous $\theta$. Another is a purely lattice theoretic characterization of the lattice of closed subspaces of separable complex Hilbert space.
Restrictions of convex subsets of $C(X)$
Per
Hag
283-294
Abstract: The main result of this paper is a theorem giving a measure-theoretic condition which is necessary and sufficient for a closed convex subset S of $C(X)$ to have the so-called bounded extension property with respect to a closed subset F of X. This theorem generalizes well-known results on closed subspaces by Bishop, Gamelin and Semadeni.
Ideals of coadjoint orbits of nilpotent Lie algebras
Colin
Godfrey
295-307
Abstract: For f a linear functional on a nilpotent Lie algebra g over a field of characteristic 0, let $J(f)$ be the ideal of all polynomials in $ S(g)$ vanishing on the coadjoint orbit through f in ${g^\ast}$, and let $I(f)$ be the primitive ideal of Dixmier in the universal enveloping algebra $U(g)$, corresponding to the orbit. An inductive method is given for computing generators ${P_1}, \ldots ,{P_r}$ of $J(f)$ such that $\varphi {P_1}, \ldots ,\varphi {P_r}$ generate $ I(f),\varphi$ being the symmetrization map from $S(g)$ to $U(g)$. Upper bounds are given for the number of variables in the polynomials ${P_i}$ and a counterexample is produced for upper bounds proposed by Kirillov.
On curves on formal groups
Robert A.
Morris;
Bodo
Pareigis
309-319
Abstract: The structure of the group of curves on a formal group is determined when the formal group is on a truncated power series algebra over a commutative ring. The resulting curve functor is faithful but not full. Applications to the Lie algebra of the formal group are given.
The $p$-adic log gamma function and $p$-adic Euler constants
Jack
Diamond
321-337
Abstract: We define $ {G_p}$, a p-adic analog of the classical log gamma function and show it satisfies relations similar to the standard formulas for log gamma. We also define p-adic Euler constants and use them to obtain results on $G{'_p}$ and on the logarithmic derivative of Morita's $ {\Gamma _p}$.
Matrix representation of simple halfrings
H. E.
Stone
339-353
Abstract: The structure of halfrings which are strong direct sums of minimal subtractive right ideals is studied. A class of right simple hemirings which contains both division hemirings and differential subsemirings of division rings is introduced and studied extensively as a tool in this investigation. A matrix representation is obtained for a class of halfrings which properly includes differential subsemirings of simple Artinian rings.
Deformations of Lie subgroups
Don
Coppersmith
355-366
Abstract: We give rigidity and universality theorems for embedded deformations of Lie subgroups. If $K \subset H \subset G$ are Lie groups, with $ {H^1}(K,g/h) = 0$, then for every ${C^\infty }$ deformation of H, a conjugate of K lies in each nearby fiber $ {H_s}$. If $H \subset G$ with $ {H^2}(H,g/h) = 0$, then there is a universal ``weak'' analytic deformation of H, whose base space is a manifold with tangent plane canonically identified with $\operatorname{Ker} {\delta ^1}$.
Addendum to: ``Knots with infinitely many spanning surfaces'' (Trans. Amer. Math. Soc. {\bf 229} (1977), 329--349)
Julian R.
Eisner
367-369
Vector fields on polyhedra
Michael A.
Penna
1-31
Abstract: This paper presents a bundle theory for studying vector fields and their integral flows on polyhedra $_ \ast$ and applications. Every polyhedron has a tangent object in the category of simplicial bundles in much the same way as every smooth manifold has a tangent object in the category of smooth vector bundles. One can show that there is a correspondence between piecewise smooth flows on a polyhedron P and sections of the tangent object of P (i.e., vector fields on P); using this result one can prove existence results for piecewise smooth flows on polyhedra. Finally an integral formula for the Euler characteristic of a closed, oriented, even-dimensional combinatorial manifold is given; as a consequence of this result one obtains a representation of Euler classes of such combinatorial manifolds in terms of piecewise smooth forms.
A law of the iterated logarithm for stable summands
R. P.
Pakshirajan;
R.
Vasudeva
33-42
Abstract: Let ${X_1},{X_2}, \ldots$ be a sequence of independent indentically distributed stable random variables with parameters $\alpha \;(0 < \alpha < 2)$ and $\beta (\vert\beta \vert \leqslant 1)$. Let ${S_n} = \sum\nolimits_{i = 1}^n {{X_i}}$. Suppose that $({S_{1,n}})$ and $({S_{2,n}})$ are independent copies of the sequence $({S_n})$. In this paper we obtain the set of all limit points in the plane of the sequence $\displaystyle \left\{ {\vert{n^{ - 1/\alpha }}({S_{1,n}} - {a_n}){\vert^{1/(\lo... ...},\vert{n^{ - 1/\alpha }}({S_{2,n}} - {a_n}){\vert^{1/(\log \log n)}}} \right\}$ where $({a_n})$ is zero if $ \alpha \ne 1$ and is $(2\beta n\log n)/\pi$ if $\alpha = 1$.
The lattice of patterns induced by a positive cone of functions
D. J.
Hartfiel;
C. J.
Maxson
43-59
Abstract: This paper contains a study of a special type of lattice which arises by considering the supports of functions in a positive cone of functions. It is shown that many known combinatorial structures provide examples of such a lattice. The basic problem addressed in the paper is that of determining the structure of this lattice.
On inner ideals and ad-nilpotent elements of Lie algebras
Georgia
Benkart
61-81
Abstract: An inner ideal of a Lie algebra L over a commutative ring k is a k-submodule B of L such that $ [B[BL]] \subseteq B$. This paper investigates properties of inner ideals and obtains results relating ad-nilpotent elements and inner ideals. For example, let L be a simple Lie algebra in which $D_y^2 = 0$ implies $y = 0$, where ${D_y}$ denotes the adjoint mapping determined by y. If L satisfies the descending chain condition on inner ideals and has proper inner ideals, then L contains a subalgebra $S = \langle e,f,h\rangle $, isomorphic to the split 3-dimensional simple Lie algebra, such that $D_e^3 = D_f^3 = 0$. Lie algebras having such 3-dimensional subalgebras decompose into the direct sum of two copies of a Jordan algebra, two copies of a special Jordan module, and a Lie subalgebra of transformations of the Jordan algebra and module. The main feature of this decomposition is the correspondence between the Lie and the Jordan structures. In the special case when L is a finite dimensional, simple Lie algebra over an algebraically closed field of characteristic $p > 5$ this decomposition yields: Theorem. L is classical if and only if there is an $x \ne 0$ in L such that $D_x^{p - 1} = 0$ and if $ D_y^2 = 0$ implies $ y = 0$. The proof involves actually constructing a Cartan subalgebra which has 1-dimensional root spaces for nonzero roots and then using the Block axioms.
Disjoint circles: a classification
Gary L.
Ebert
83-109
Abstract: For q a prime-power, let $ {\text{IP}}(q)$ denote the miquelian inversive plane of order q. The classification of certain translation planes of order $ {q^2}$, called subregular, has been reduced to the classification of sets of disjoint circles in $ {\text{IP}}(q)$. While R. H. Bruck has extensively studied triples of disjoint circles, this paper is concerned with sets of four or more circles in $ {\text{IP}}(q)$. In a previous paper, the author has shown (for odd q) that the number of quadruples of disjoint circles in ${\text{IP}}(q)$ is asymptotic to ${q^{12}}/1536$. Hence a judicious approach to the classification problem is to study ``interesting'' quadruples. In general, let ${C_1}, \ldots ,{C_n}$ be a nonlinear set of n disjoint circles in $ {\text{IP}}(q)$. Let H be the subgroup of the collineation group of ${\text{IP}}(q)$ composed of collineations that permute the ${C_i}$'s among themselves, and let K be that subgroup composed of collineations fixing each of the ${C_i}'s$. An interesting set of n disjoint circles would be one for which $K = 1$. It is shown that $K = 1$ if and only if $\displaystyle \left\{ {\begin{array}{*{20}{c}} {{\text{(i)}}\;{\text{there}}\;{... ...}}\;{\text{other}}\;n - 1\;{\text{circles}}{\text{.}}} \end{array} } \right.$ ($ *$) When $n = 4$ and under mild restrictions on q, an algorithm is developed that finds all nonlinear quadruples of disjoint circles satisfying the orthogonality conditions $( \ast )$ and having nontrivial group H. Given such a quadruple, the algorithm determines exactly what group H is acting. It is also shown that most quadruples in $ {\text{IP}}(q)$, for large q, do indeed satisfy the conditions $ ( \ast )$. In addition, the cases when $n = 5,6,$ or 7 are explored to a lesser degree.
Generalized Hankel conjugate transformations on rearrangement invariant spaces
R. A.
Kerman
111-130
Abstract: The boundedness properties of the generalized Hankel conjugate transformations ${H_\lambda }$ on certain weighted Lebesgue spaces are studied. These are used to establish a boundedness criterion for the $ {H_\lambda }$ on the more general class of rearrangement invariant spaces. The positive operators in terms of which the criterion is given are used to construct pairs of spaces between which the $ {H_\lambda }$ are continuous; in particular, a natural analogue of a well-known result of Zygmund concerning the classical conjugate function operator is obtained for the ${H_\lambda }$.
Closed convex invariant subsets of $L\sb{p}(G)$
Anthony To Ming
Lau
131-142
Abstract: Let G be a locally compact group. We characterize in this paper closed convex subsets K of ${L_p}(G),1 \leqslant p < \infty$, that are invariant under all left or all right translations. We prove, among other things, that $K = \{ 0\}$ is the only nonempty compact (weakly compact) convex invariant subset of ${L_p}(G)\;({L_1}(G))$. We also characterize affine continuous mappings from ${P_1}(G)$ into a bounded closed invariant subset of $ {L_p}(G)$ which commute with translations, where ${P_1}(G)$ denotes the set of nonnegative functions in $ {L_1}(G)$ of norm one. Our results have a number of applications to multipliers from ${L_q}(G)$ into ${L_p}(G)$.
Study of the permanent conjecture and some of its generalizations. II
O. S.
Rothaus
143-154
Abstract: In this paper we investigate in a more systematic manner some of the topics initiated in part I of the paper with same title [5]. More specifically, we study in greater detail the properties of the function $E(y)$ defined in [5] attached to convex polytopes, whose properties in the special case of the space of doubly stochastic matrices are connected with the permanent conjecture. Some close links with Perron-Frobenius theory are developed, and we obtain as a by-product of our study what is, I believe, a new expression for the maximum eigenvalue of a nonnegative matrix, which leads to some new estimates of the same. A final section of the paper investigates some purely algebraic properties of $E(y)$, and we obtain some very interesting information connecting a doubly stochastic matrix and its transversals. In order to keep this paper as self-contained as possible, facts used here drawn from part I are stated with as much explicit detail as possible.
Local and global factorizations of matrix-valued functions
K. F.
Clancey;
I.
Gohberg
155-167
Abstract: Let C be a simple closed Liapounov contour in the complex plane and A an invertible $n \times n$ matrix-valued function on C with bounded measurable entries. There is a well-known concept of factorization of the matrix function A relative to the Lebesgue space ${L_p}(C)$. The notion of local factorization of A relative to ${L_p}$ at a point ${t_0}$ in C is introduced. It is shown that A admits a factorization relative to $ {L_p}(C)$ if and only if A admits a local factorization relative to $ {L_p}$ at each point $ {t_0}$ in C. Several problems connected with local factorizations relative to ${L_p}$ are raised.
A bound on the rank of purely simple systems
Frank
Okoh
169-186
Abstract: A pair of complex vector spaces (V, W) is called a system if and only if there is a C-bilinear map from ${{\mathbf{C}}^2} \times V$ to W. The category of systems contains subcategories equivalent to the category of modules over the ring of complex polynomials. Many concepts in the latter generalize to the category of systems. In this paper the pure projective systems are characterized and a bound on the rank of purely simple systems is obtained.
On everywhere-defined integrals
Lester E.
Dubins
187-194
Abstract: Hardly any finite integrals can be defined for all real-valued functions. In contrast, if infinity is admitted as a possible value for the integral, then every finite integral can be extended to all real-valued functions.
Decomposition spaces having arbitrarily small neighborhoods with $2$-sphere boundaries
Edythe P.
Woodruff
195-204
Abstract: Let G be an u.s.c. decomposition of ${S^3}$. Let H denote the set of nondegenerate elements and P be the natural projection of $ {S^3}$ onto ${S^3}/G$. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with 2-sphere boundaries which miss $P(H)$. We prove in this paper that this condition implies that ${S^3}/G$ is homeomorphic to ${S^3}$. This answers a question asked by Armentrout [1, p. 15]. Actually, the hypothesis concerning neighborhoods with 2-sphere boundaries is necessary only for the points of $P(H)$.
A note on limits of unitarily equivalent operators
Lawrence A.
Fialkow
205-220
Abstract: Let $\mathcal{U}(\mathcal{H})$ denote the set of all unitary operators on a separable complex Hilbert space $\mathcal{H}$. If T is a bounded linear operator on $\mathcal{H}$, let ${\pi _T}$ denote the mapping of $\mathcal{U}(\mathcal{H})$ onto $\mathcal{U}(T)$ given by conjugation. It is proved that if T is normal or isometric, then there exists a locally defined continuous cross-section for $ {\pi _T}$ if and only if the spectrum of T is finite. Examples of nonnormal operators with local cross-sections are given.
PL involutions of fibered $3$-manifolds
Paik Kee
Kim;
Jeffrey L.
Tollefson
221-237
Abstract: Let h be a PL involution of $F \times [0,1]$ such that $ h(F \times \{ 0,1\} ) = F \times \{ 0,1\}$, where F is a compact 2-manifold. It is shown that h is equivalent to an involution $h'$ of the form
The Huygens property for the heat equation
D. V.
Widder
239-244
Abstract: This note summarizes several criteria which guarantee that a solution of the heat equation should also have the semigroup property described in equation (1.2) below. In particular, it corrects a mistake in an earlier proof of one of these.
Products of sequentially compact spaces and the $V$-process
M.
Rajagopalan;
R. Grant
Woods
245-253
Abstract: In this paper we produce a family of sequentially compact, locally compact, ${T_2}$ first countable, scattered and separable spaces whose product is not countably compact and thus answer a problem of C. T. Scarborough and A. H. Stone [11] in the negative. We do this using the continuum hypothesis. We also produce a completely regular, $ {T_2}$, sequentially compact space K which is not p-compact for any $p \in \beta N - N$.
Anisotropic $H\sp{p}$ real interpolation, and fractional Riesz potentials
W. R.
Madych
255-263
Abstract: We observe that the anisotropic variants of ${H^p}$ interpolate by the real method in the usual manner. Using this fact we show that the corresponding fractional Riesz potentials and related operators perform an embedding in $ {H^p},p > 0$, analogous to the one for $ {L^p},p > 1$. We also state a theorem concerning the mapping properties of $f \to h \ast f$, where h is in $B_\alpha ^{1,\infty }$, which hold only for a restricted range of p.
Symplectic Stiefel harmonics and holomorphic representations of symplectic groups
Tuong
Ton-That
265-277
Abstract: Let ${I_k}$ denote the identity matrix of order k and set $\displaystyle {s_k} = \left[ {\begin{array}{*{20}{c}} 0 & { - {I_k}} {{I_k}} & 0 \end{array} } \right].$ Let $ {\text{Sp}}(k,{\mathbf{C}})$ denote the group of all complex $2k \times k$ matrices which satisfy the equation $g{s_k}{g^t} = {s_k}$. Let E be the linear space of all $n \times 2k$ complex matrices with $k \geqslant n$, and let $S({E^\ast})$ denote the symmetric algebra of all complex-valued polynomial functions on E. The study of the action of $ {\text{Sp}}(k,{\mathbf{C}})$, which is obtained by right translation on $S({E^\ast})$, leads to a concrete and simple realization of all irreducible holomorphic representations of $ {\text{Sp}}(k,{\mathbf{C}})$. In connection with this realization, a theory of symplectic Stiefel harmonics is also established. This notion may be thought of as a generalization of the spherical harmonics for the symplectic Stiefel manifold.
Two-descent for elliptic curves in characteristic two
Kenneth
Kramer
279-295
Abstract: This paper is a study of two-descent to find an upper bound for the rank of the Mordell-Weil group $A(F)$ of an elliptic curve A defined over a field F of characteristic two. It includes local and global duality theorems which are the analogs of known results for descent by an isogeny whose degree is relatively prime to the characteristic of the field of definition.
Strange billiard tables
Benjamin
Halpern
297-305
Abstract: A billiard table is any compact convex body T in the plane bounded by a continuously differentiable curve $\partial T$. An idealized billiard ball is a point which moves at unit speed in a straight line except when it hits the boundary $ \partial T$ where it rebounds making the angle of incidence equal to the angle of reflection. A rather surprising phenomenon can happen on such a table.
The level structure of a residual set of continuous functions
A. M.
Bruckner;
K. M.
Garg
307-321
Abstract: Let C denote the Banach space of continuous real-valued functions on $[0,1]$ with the uniform norm. The present article is devoted to the structure of the sets in which the graphs of a residual set of functions in C intersect with different straight lines. It is proved that there exists a residual set A in C such that, for every function $f \in A$, the top and the bottom (horizontal) levels of f are singletons, in between these two levels there are countably many levels of f that consist of a nonempty perfect set together with a single isolated point, and the remaining levels of f are all perfect. Moreover, the levels containing an isolated point correspond to a dense set of heights between the minimum and the maximum values assumed by the function. As for the levels in different directions, there exists a residual set B in C such that, for every function $f \in B$, the structure of the levels of f is the same as above in all but a countable dense set of directions, and in each of the exceptional nonvertical directions the level structure of f is the same but for the fact that one (and only one) of the levels has two isolated points in place of one. For a general function $f \in C$ a theorem is proved establishing the existence of singleton levels of f, and of the levels of f that contain isolated points.
Holomorphic continuation of smooth functions over Levi-flat hypersurfaces
Eric
Bedford
323-341
Abstract: Here we consider the singularities of a Levi-flat real hypersurface S in C that lie in an analytic variety of codimension 2. It is shown that, from the geometric point of view, there are two kinds of singularities, and the type of singularity determines whether S bounds a domain of holomorphy of type ${A^\infty }$.
The structure of generalized Morse minimal sets on $n$ symbols
John C.
Martin
343-355
Abstract: A class of bisequences on n symbols is constructed which includes the generalized Morse sequences introduced by Keane. Those which give rise to strictly ergodic sets are characterized, and the spectrum of the shift operator on these systems is investigated. It is shown that in certain cases the shift operator has partly discrete and partly continuous spectrum. The theorems generalize results of Keane on generalized Morse sequences and a theorem of Kakutani regarding a particular strictly transitive sequence on four symbols. Another special case yields information on the spectrum of certain substitution minimal sets.
The Kobayashi pseudometric on algebraic manifolds of general type and in deformations of complex manifolds
Marcus
Wright
357-370
Abstract: This paper deals with regularity properties of the infinitesimal form of the Kobayashi pseudo-distance. This form is shown to be upper semicontinuous in the parameters of a deformation of a complex manifold. The method of proof involves the use of a parametrized version of the Newlander-Nirenberg Theorem together with a theorem of Royden on extending regular mappings from polydiscs into complex manifolds. Various consequences and improvements of this result are discussed; for example, if the manifold is compact hyperbolic the infinitesimal Kobayashi metric is continuous on the union of the holomorphic tangent bundles of the fibers of the deformation. This result leads to the fact that the coarse moduli space of a compact hyperbolic manifold is Hausdorff. Finally, the infinitesimal form is studied for a class of algebraic manifolds which contains algebraic manifolds of general type. It is shown that the form is continuous on the tangent bundle of a manifold in this class. Many members of this class are not hyperbolic.
Nonzero-sum stochastic differential games with stopping times and free boundary problems
Alain
Bensoussan;
Avner
Friedman
275-327
Abstract: One is given a diffusion process and two payoffs which depend on the process and on two stopping times ${\tau _1},{\tau _2}$. Two players are to choose their respective stopping times ${\tau _1},{\tau _2}$ so as to achieve a Nash equilibrium point. The problem whether such times exist is reduced to finding a ``regular'' solution $({u_1},{u_2})$ of a quasi-variational inequality. Existence of a solution is established in the stationary case and, for one space dimension, in the nonstationary case; for the latter situation, the solution is shown to be regular if the game is of zero sum.
Uniqueness properties of CR-functions
L. R.
Hunt
329-338
Abstract: Let M be a real infinitely differentiable closed hypersurface in X, a complex manifold of complex dimension $n \geqslant 2$. The uniqueness properties of solutions to the system ${\bar \partial _M}u = f$, where ${\bar \partial _M}$ is the induced Cauchy-Riemann operator on M, are of interest in the fields of several complex variables and partial differential equations. Since dM is linear, the study of the solution to the equation ${\bar \partial _M}u = 0$ is sufficient for uniqueness. A ${C^\infty }$ solution to this homogeneous equation is called a CR-function on M. The main result of this article is that a CR-function is uniquely determined, at least locally, by its values on a real k-dimensional ${C^\infty }$ generic submanifold ${S^k}$ of M with $k \geqslant n$. The facts that ${S^k}$ is generic and $k \geqslant n$ together form the lower dimensional analogue of the concept of noncharacteristic.
Kernel functions on domains with hyperelliptic double
William H.
Barker
339-347
Abstract: In this paper we show that the structure of the Bergman and Szegö kernel functions is especially simple on domains with hyperelliptic double. Each such domain is conformally equivalent to the exterior of a system of slits taken from the real axis, and on such domains the Bergman kernel function and its adjoint are essentially the same, while the Szegö kernel function and its adjoint are elementary and can be written in a closed form involving nothing worse than fourth roots of polynomials. Additionally, a number of applications of these results are obtained.
Extremal arcs and extended Hamiltonian systems
Frank H.
Clarke
349-367
Abstract: A general variational problem is considered; it involves the minimization of an integrand L of a very general nature. The Lagrangian L is allowed to assume the value $+ \infty$, and need satisfy no differentiability or convexity conditions. A Hamiltonian corresponding to the problem is defined via the conjugate function of convex analysis, and it is shown how one obtains necessary conditions in the form of an extended Hamiltonian system. This system is expressed in terms of certain ``generalized gradients'' previously developed by the author. A further result is given which has the feature that the principal hypotheses required, as well as the ensuing conclusions, are entirely in terms of H. This allows the treatment of classes of problems in which H is more amenable to direct analysis than L. The approach also sheds light on the relation between existence theory and the theory of necessary conditions, since the results may easily be compared with R. T. Rockafellar's recent work on existence theory, in which H also plays a central role. As an example of its application the main result is specialized to a differential inclusion problem. A specific example of its use is also given, an unorthodox optimal control problem with a discontinuous cost functional.
Boundary behavior of harmonic forms on a rank one symmetric space
Aroldo
Kaplan;
Robert
Putz
369-384
Abstract: We study the boundary behavior of 1-forms on a rank-one symmetric space M satisfying the equations $d\omega = 0 = \delta \omega $; the role of boundary is played by a nilpotent (Iwasawa) group $ \bar N$ of isometries of M. For forms satisfying certain $ {H^p}$ integrability conditions, we obtain the existence of boundary values in an appropriate sense, characterize these boundary values by means of fractional and singular integral operators on the group $\bar N$, and exhibit explicit isomorphisms between ${H^p}$ spaces of forms on M and the ordinary $ {L^p}$ spaces of functions on the group $\bar N$.
Structurally stable Grassmann transformations
Steve
Batterson
385-404
Abstract: A Grassmann transformation is a diffeomorphism on a Grassmann manifold which is induced by an $n \times n$ nonsingular matrix. In this paper the structurally stable Grassmann transformations are characterized to be the maps which are induced by matrices whose eigenvalues have distinct moduli. There is exactly one topological conjugacy class of complex structurally stable Grassmann transformations. For the real case the topological classification is determined by the ordering (relative to modulus) of the signs of the eigenvalues of the inducing matrix.
The decay of solutions of the two dimensional wave equation in the exterior of a straight strip
Peter
Wolfe
405-428
Abstract: We study an initial boundary value problem for the wave equation in the exterior of a straight strip. We assume the initial data has compact support and that the solution vanishes on the strip. We then show that at any point in space the solution is $O(1/t)$ as $t \to \infty$. This is the same rate of decay as obtains for the solution of the initial boundary value problem posed in the exterior of a smooth star shaped region. Our method is to use a Laplace transform. This reduces the problem to a consideration of a boundary value problem for the Helmholtz equation. We derive estimates for the solution of the Helmholtz equation for both high and low frequencies which enable us to obtain our results by estimating the Laplace inversion integral asymptotically.
Transversality in $G$-manifolds
M. J.
Field
429-450
Abstract: A definition of transversality is given for the category of G-manifolds (G, a compact Lie group). Transversality density and isotopy theorems are shown to hold for this definition. An example is given to show that we cannot require differential stability of intersections.
Singularities of spaces of flat bundles over complex manifolds
B.
Wong
451-461
Abstract: We use Fox differential calculus on free group to study the singularities of complex analytic varieties arising from flat bundles over complex manifolds. Criteria of regularity in terms of cohomology of the fundamental group of the underlying manifold are established.
Cyclic purity versus purity in excellent Noetherian rings
Melvin
Hochster
463-488
Abstract: A characterization is given of those Noetherian rings R such that whenever R is ideally closed ($\equiv$ cyclically pure) in an extension algebra S, then R is pure in S. In fact, R has this property if and only if the completion $(A,m)$ of each local ring of R at a maximal ideal has the following two equivalent properties: (i) For each integer $N > 0$ there is an m-primary irreducible ideal $ {I_N} \subset {m^N}$. (ii) Either $\dim \;A = 0$ and A is Gorenstein or else depth $A \geqslant 1$ and there is no $P \in {\text{Ass}}(A)$ such that $\dim (A/P) = 1$ and $(A/P) \oplus (A/P)$ is embeddable in A. It is then shown that if R is a locally excellent Noetherian ring such that either R is reduced (or, more generally, such that R is generically Gorenstein), or such that Ass(R) contains no primes of coheight $ \leqslant 1$ in a maximal ideal, and R is ideally closed in S, then R is pure in S. Matlis duality and the theory of canonical modules are utilized. Module-theoretic analogues of condition (i) above are, of necessity, also analyzed. Numerous related questions are studied. In the non-Noetherian case, an example is given of a ring extension $R \to S$ such that R is pure in S but $R[[T]]$ is not even cyclically pure in $ S[[T]]$.
Modular forms for $G\sb{0}(N)$ and Dirichlet series
Michael J.
Razar
489-495
Abstract: A criterion is given for a function to be a modular form for ${\Gamma _0}(N)$. It is similar to the criterion given by Weil in his 1967 Math. Ann. paper Über die Bestimmung Dirichletscher Reihen durch Funktional-gleichungen in that it involves checking that certain twists of the associated Dirichlet series satisfy functional equations. It differs in the number and type of such equations which need to be satisfied.
Efficient generation of maximal ideals in polynomial rings
E. D.
Davis;
A. V.
Geramita
497-505
Abstract: The cardinality of a minimal basis of an ideal I is denoted $ \nu (I)$. Let A be a polynomial ring in $n > 0$ variables with coefficients in a noetherian (commutative with $1 \ne 0$) ring R, and let M be a maximal ideal of A. In general $ \nu (M{A_M}) + 1 \geqslant \nu (M) \geqslant \nu (M{A_M})$. This paper is concerned with the attaining of equality with the lower bound. It is shown that equality is attained in each of the following cases: (1) ${A_M}$ is not regular (valid even if A is not a polynomial ring), (2) $M \cap R$ is maximal in R and (3) $n > 1$. Equality may fail for $n = 1$, even for R of dimension 1 (but not regular), and it is an open question whether equality holds for R regular of dimension $ > 1$. In case $ n = 1$ and $\dim (R) = 2$ the attaining of equality is related to questions in the K-theory of projective modules. Corollary to (1) and (2) is the confirmation, for the case of maximal ideals, of one of the Eisenbud-Evans conjectures; namely, $ \nu (M) \leqslant \max \{ \nu (M{A_M}),\dim (A)\}$. Corollary to (3) is that for R regular and $n > 1$, every maximal ideal of A is generated by a regular sequence--a result well known (for all $n \geqslant 1$) if R is a field (and somewhat less well known for R a Dedekind domain).
On the dimension of left invariant means and left thick subsets
Maria
Klawe
507-518
Abstract: If S is a left amenable semigroup, let $\dim \langle Ml(S)\rangle $ denote the dimension of the set of left invariant means on S. Theorem. If S is left amenable, then $\dim \langle Ml(S)\rangle = n < \infty$ if and only if S contains exactly n disjoint finite left ideal groups. This result was proved by Granirer for S countable or left cancellative. Moreover, when S is infinite, left amenable, and either left or right cancellative, we show that $ \dim \langle Ml(S)\rangle$ is at least the cardinality of S. An application of these results shows that the radical of the second conjugate algebra of ${l_1}(S)$ is infinite dimensional when S is a left amenable semigroup which does not contain a finite ideal.
Spectra and measure inequalities
C. R.
Putnam
519-529
Abstract: Let T be a bounded operator on a Hilbert space $\mathfrak{H}$ and let ${T_z} = T - zI$. Then the operators $ {T_z}T_z^\ast,{T_z}{T_t}{({T_z}{T_t})^\ast}$, and $ {T_z}{T_t}{T_s}{({T_z}{T_t}{T_s})^\ast}$ are nonnegative for all complex numbers z, t, and s. We shall obtain some norm estimates for nonnegative lower bounds of these operators, when z, t, and s are restricted to certain sets, in terms of certain capacities or area measures involving the spectrum and point spectrum of T. A typical such estimate is the following special case of Theorem 4 below: Let $\mathfrak{H}$ be separable and suppose that $ {T_z}{T_t}{({T_z}{T_t})^\ast} \geqslant D \geqslant 0$ for all z and t not belonging to the closure of the interior of the point spectrum of T. In addition, suppose that the boundary of the interior of the point spectrum of T has Lebesgue planar measure 0. Then ${\left\Vert D \right\Vert^{1/2}} \leqslant {\pi ^{ - 1}}\;{\text{meas}_2}\;({\sigma _p}(T))$. If T is the adjoint of the simple unilateral shift, then equality holds with $D = I - {T^\ast}T$.
On one-dimensional coupled Dirac equations
R. T.
Glassey
531-539
Abstract: The Cauchy Problem for Dirac equations coupled through scalar and Fermi interactions is considered in one space dimension. Global solutions of finite energy are shown to exist, provided that either the magnitude of the coupling constant or the $ {L_2}({R^1})$-norm of the initial data is suitably restricted.
Torsion in the bordism of oriented involutions
Russell J.
Rowlett
541-548
Abstract: In the bordism theory ${\Omega _ \ast }(Z_2^k)$ of smooth, orientation-preserving $Z_2^k$-actions all torsion has order two. Furthermore, the torsion classes inject in the unoriented theory ${N_ \ast }(Z_2^k)$, and any class represented by a stationary-point free action has infinite order. In addition, a procedure is given for producing Smith constructions in some generality.
On ordinary linear $p$-adic differential equations
B.
Dwork;
P.
Robba
1-46
Abstract: We study the solutions of ordinary linear differential equations whose coefficients are analytic elements. As one application we show nonexistence of index for certain linear differential operators with rational function coefficients.
Associated and skew-orthologic simplexes
Leon
Gerber
47-63
Abstract: A set of $n + 1$ lines in n-space is said to be associated if every $ (n - 2)$-flat which meets n of the lines also meets the remaining line. Two Simplexes are associated if the joins of their corresponding vertices are associated. Two Simplexes are (skew-)orthologic if the perpendiculars from the vertices of one on the faces of the other are concurrent (associated); it follows that the reciprocal relation holds. In an earlier paper, Associated and Perspective Simplexes, we gave an affine necessary and sufficient condition for two simplexes to be associated that was so easy to apply that extensions to n-dimensions of nearly all known theorems, and a few new ones, were proved in a few lines of calculations. In this sequel we take a closer look at some of the results of the earlier paper and prove some new results. Then we give simple Euclidean necessary and sufficient conditions for two simplexes to be orthologic or skew-orthologic which yield as corollaries known results on altitudes, the Monge point and orthocentric simplexes. We conclude by discussing some of the qualitative differences between the geometries of three and higher dimensions.
Variation of conformal spheres by simultaneous sewing along several arcs
T. L.
McCoy
65-82
Abstract: Let M be a closed Riemann surface of genus zero, $\Gamma$ a tree on M with branches ${\Gamma _j}$, and ${p_0}$ a point of $M - \Gamma$. A family of neighboring topological surfaces $ M(\varepsilon )$ is formed by regarding each $ {\Gamma _j}$ as a slit with edges $\Gamma _j^ -$ and $\Gamma _j^ +$, and re-identifying p on ${\Gamma ^{{ - _j}}}$ with $ p + \varepsilon {\chi _j}(p,\varepsilon )$ on $ \Gamma _j^ +$, with $ {\chi _j}$ vanishing at the endpoints of $ {\Gamma _j}$. We assume the ${\Gamma _j}$ and ${\chi _j}$ are such that, under a certain natural choice of uniformizers, the $M(\varepsilon )$ are closed Riemann surfaces of genus zero. Then there exists a unique function $f(p,\varepsilon ;{p_0})$ mapping $M(\varepsilon )$ conformally onto the complex number sphere, with normalization $ \varepsilon$. Further, we obtain smoothness results for f as a function of $\varepsilon $. The problem is connected with the study of the extremal schlicht functions; that is, the schlicht mappings of the unit disc corresponding to boundary points of the coefficient bodies.
Asymptotic behavior of solutions of nonlinear functional differential equations in Banach space
John R.
Haddock
83-92
Abstract: Let X be a Banach space and let $C = C([ - r,0],X)$ denote the space of continuous functions from $[ - r,0]$ to X. In this paper the problem of convergence in norm of solutions of the nonlinear functional differential equation $\dot x = F(t,{x_t})$ is considered where $F:[0,\infty ) \times C \to X$. As a special case of the main theorem, stability results are given for the equation $\dot x(t) = f(t,x(t)) + g(t,{x_t})$, where $- f(t, \cdot ) - \alpha (t)I$ satisfies certain accretive type conditions and $g(t, \cdot )$ is Lipschitzian with Lipschitz constant $\beta (t)$ closely related to $\alpha (t)$.
Inequalities for polynomials on the unit interval
Q. I.
Rahman;
G.
Schmeisser
93-100
Abstract: Let $ {p_n}(z) = \sum\nolimits_{k = 0}^n {{a_k}{z^k}}$ be a polynomial of degree at most n with real coefficients. Generalizing certain results of I. Schur related to the well-known inequalities of Chebyshev and Markov we prove that if $ {p_n}(z)$ has at most $n - 1$ distinct zeros in $( - 1,1)$, then
A class of infinitely connected domains and the corona
W. M.
Deeb
101-106
Abstract: Let D be a bounded domain in the complex plane. Let ${H^\infty }(D)$ be the Banach algebra of bounded analytic functions on D. The corona problem asks whether D is weak$^\ast$ dense in the space $\mathfrak{M}(D)$ of maximal ideals of ${H^\infty }(D)$. Carleson [3] proved that the open unit disc ${\Delta _0}$ is dense in $\mathfrak{M}({\Delta _0})$. Stout [9] extended Carleson's result to finitely connected domains. Behrens [2] found a class of infinitely connected domains for which the corona problem has an affirmative answer. In this paper we will use Behrens' idea to extend the results to more general domains. See [11] for further extensions and applications of these techniques.
$D$-domains and the corona
W. M.
Deeb;
D. R.
Wilken
107-115
Abstract: Let D be a bounded domain in the complex plane C. Let ${H^\infty }(D)$ denote the usual Banach algebra of bounded analytic functions on D. The Corona Conjecture asserts that D is weak$^\ast$ dense in the space $\mathfrak{M}(D)$ of maximal ideals of ${H^\infty }(D)$. In [2] Carleson proved that the unit disk ${\Delta _0}$ is dense in $\mathfrak{M}({\Delta _0})$. In [7] Stout extended Carleson's result to finitely connected domains. In [4] Gamelin showed that the problem is local. In [1] Behrens reduced the problem to very special types of infinitely connected domains and established the conjecture for a large class of such domains. In this paper we extract some of the crucial ingredients of Behrens' methods and extend his results to a broader class of infinitely connected domains.
Multiplier criteria of Marcinkiewicz type for Jacobi expansions
George
Gasper;
Walter
Trebels
117-132
Abstract: It is shown how an integral representation for the product of Jacobi polynomials can be used to derive a certain integral Lipschitz type condition for the Cesàro kernel for Jacobi expansions. This result is then used to give criteria of Marcinkiewicz type for a sequence to be multiplier of type (p, p), $1 < p < \infty$, for Jacobi expansions.
Subgroups of finitely presented solvable linear groups
Michael W.
Thomson
133-142
Abstract: Let G be a finitely generated solvable linear group. It is shown that there exists a finitely presented solvable linear group H with G embedded in H.
Continuous dependence of solutions of operator equations. I
Zvi
Artstein
143-166
Abstract: Continuous dependence of the solutions of the operator equation $x = Tx + z$ in a topological vector space is the main subject of the paper. We find sufficient and necessary conditions for the continuous dependence on the data (T, z) or on a parameter. We do it for the space of all closed operators. Equivalent conditions for particular subfamilies are discussed. Among other families we deal with compact operators, compact perturbations of the identity, condensing operators and demicompact operators.
Conjugate points of vector-matrix differential equations
Roger T.
Lewis
167-178
Abstract: The system of equations $\displaystyle \sum\limits_{k = 0}^n {{{( - 1)}^{n - k}}{{\left( {{P_k}(x){y^{(n - k)}}(x)} \right)}^{(n - k)}}} = 0\quad (0 \leqslant x < \infty )$ is considered where the coefficients are real, continuous, symmetric matrices, y is a vector, and ${P_0}(x)$ is positive definite. It is shown that the well-known quadratic functional criterion for existence of conjugate points for this system can be further utilized to extend results of the associated scalar equation to the vector-matrix case, and in some cases the scalar results are also improved. The existence and nonexistence criteria for conjugate points of this system are stated in terms of integral conditions on the eigenvalues or norms of the coefficient matrices.
On the transformation group of a real hypersurface
S. M.
Webster
179-190
Abstract: The group of biholomorphic transformations leaving fixed a strongly pseudoconvex real hypersurface in a complex manifold is a Lie group. In this paper it is shown that the Chern-Moser invariants must vanish if this group is noncompact and the hypersurface is compact. Also considered are transformation groups of flat hypersurfaces and intransitive groups.
Minimal invariant functions of the space-time Wiener process
Kai Yuen
Woo
191-200
Abstract: Minimal invariant functions of the space-time Wiener process are obtained.
An example where topological entropy is continuous
Louis
Block
201-213
Abstract: Let ent denote topological entropy, and let ${C^r}({S^1},{S^1})$ denote the space of continuous functions of the circle to itself having r continuous derivatives with the ${C^r}$ (uniform) topology. Let $ {f_0}$ denote a particular $ {C^2}$ map of the circle ($ {f_0}$ is the first bifurcation point one comes to in a bifurcation from a full three shift to a map with finite nonwandering set). The main results of this paper are the following: Theorem A. The map ent: $ {C^0}({S^1},{S^1}) \to R \cup \{ \infty \}$ is lower-semicontinuous at $ {f_0}$. Theorem B. The map ent: ${C^2}({S^1},{S^1}) \to R$ is continuous at $ {f_0}$. In proving these two theorems several general results on entropy of mappings of the circle are proved.
Functional calculus and positive-definite functions
Colin C.
Graham
215-231
Abstract: For a LCA group G with dual group Ĝ, let $D(G) = D(\hat G)$ denote the convex (not closed) hull of $\{ \langle x,\gamma \rangle :x \in G,\gamma \in \hat G\}$. The set $D(G)$ is the natural domain for functions that operate by composition from the class, $P{D_1}(\hat G)$, of Fourier-Stieltjes transforms of probability measures on G to $B(\hat G)$, the class of all Fourier-Stieltjes transforms on Ĝ. Little is known about the behavior of F on the boundary of $ D(G)$. In §1, we show (1) if F operators from $P{D_1}(G)$ to $B(G)$ and G is compact, then $ K(z) = {\lim _{t \to {1^ - }}}F(tz)$ exists for all $ z \in D(G)$ and K operates from $ P{D_1}(\hat G)$ to $ B(\hat G)$; (2) if F operates from $ P{D_1}(\hat G)$ to $ PD(\hat G) = { \cup _{r > 0}}rP{D_1}(\hat G)$ and G is compact, then K operates from $ P{D_1}(\hat G)$ to $PD(\hat G)$, and so also does $F - K$; (3) if $G = {{\mathbf{D}}_q},q \geqslant 2$, and F operates from $ P{D_1}(\hat G)$ to $ B(\hat G)$, then $ F = K$ on $D(G) \cap \{ z:\vert z\vert < 1\}$. This third result is shown to be sharp for compact groups of bounded order. In §2, an example is given that fills a gap in the theory of functions operating from $P{D_1}(\hat G)$ to $B(\hat G)$. In §3 we show that most Riesz products and all continuous measures on K-sets have a property that is very useful in proving symbolic calculus theorems. Applications of this are indicated. Some open questions are given in §4.
Almost sure behavior of linear functionals of supercritical branching processes
Søren
Asmussen
233-248
Abstract: The exact a.s. behavior of any linear functional ${Z_n} \cdot a$ of a supercritical positively regular p-type $ (1 < p < \infty )$ Galton-Watson process $ \{ {Z_n}\}$ is found under a second moment hypothesis. The main new results are of iterated logarithm type, with normalizing constants depending on the decomposition of a according to the Jordan canonical form of the offspring mean matrix.
Of regulated and steplike functions
Gadi
Moran
249-257
Abstract: Let C denote the class of regulated real-valued functions on the unit interval vanishing at the origin, whose positive and negative jumps sum to infinity in every nontrivial subinterval of I. Goffman [2] showed that every f in C is (essentially) a sum $g + s$ where g is continuous and s is steplike. In this sense, a function in C is like a function of bounded variation, that has a unique such g and s. The import of this paper is that for f in C the representation $f = g + s$ is not only not unique, but by far the opposite holds: g can be chosen to be any continuous function on I vanishing at 0, at the expense of a rearrangement of s.
Pointwise and norm convergence of a class of biorthogonal expansions
Harold E.
Benzinger
259-271
Abstract: Let $\{ {u_k}(x)\} ,\{ {v_k}(x)\} ,k = 0, \pm 1, \ldots ,0 \leqslant x \leqslant 1$, be sequences of functions in $ {L^\infty }(0,1)$, such that $ ({u_k},{v_j}) = {\delta _{kj}}$. Let ${\phi _k}(x) = \exp \;2k\pi ix$. It is shown that if for a given p, $1 < p < \infty$, the sequence $\{ {u_k}\}$ is complete in ${L^p}(0,1)$, and $ \{ {v_k}\}$ is complete in ${L^q}(0,1),pq = p + q$, and if the $ {u_k}$'s, ${v_j}$'s are asymptotically related to the ${\phi _k}$'s, in a sense to be made precise, then $\{ {u_k}\}$ is a basis for ${L^p}(0,1)$, equivalent to the basis $\{ {\phi _k}\}$, and for every f in ${L^p}(0,1)$ a.e. This result is then applied to the eigenfunction expansions of a large class of ordinary differential operators.
Remarks on ``Embedding theorems and generalized discrete ordered abelian groups'' (Trans. Amer. Math. Soc. {\bf 175} (1973), 283--297) by P. Hill and J. L. Mott
Isidore
Fleischer
273-274
On the integrable and square-integrable representations of ${\rm Spin}(1, 2m)$
Ernest
Thieleker
1-40
Abstract: All the unitary equivalence classes of irreducible integrable and square-integrable representations of the groups ${\text{Spin}}(1,2m),m \geqslant 2$, are determined. The method makes use of some elementary results on differential equations and the classification of irreducible unitary representations of these groups. In the latter classification, certain ambiguities resulting from possible equivalences not taken into account in a previous paper, are cleared up here.
Inverting a cylinder through isometric immersions and isometric embeddings
B.
Halpern;
C.
Weaver
41-70
Abstract: It is shown that a right circular cylinder can be turned inside out through immersions which preserve its flat Riemannian metric if and only if its diameter is greater than its height. Such a cylinder can be turned inside out through embeddings which preserve its flat Riemannian metric provided its diameter is greater than $(\pi + 2)/\pi$ times its height. A flat Möbius strip has an immersion into Euclidean three dimensional space which preserves its Riemannian metric if and only if its length is greater than $\pi /2$ times its height.
The Blumberg problem
William A. R.
Weiss
71-85
Abstract: A compact Hausdorff space and a real-valued function on this space are constructed such that the function is not continuous on any dense subspace. This solves the Blumberg problem. Some related results are established.
Decompositions of linear maps
Sze Kai J.
Tsui
87-112
Abstract: In the first part we show that the decomposition of a bounded selfadjoint linear map from a ${C^\ast}$-algebra into a given von Neumann algebra as a difference of two bounded positive linear maps is always possible if and only if that range algebra is a ``strictly finite'' von Neumann algebra of type I. In the second part we define a ``polar decomposition'' for some bounded linear maps and show that polar decomposition is possible if and only if the map satisfies a certain ``norm condition". We combine the concepts of polar and positive decompositions to show that polar decomposition for a selfadjoint map is equivalent to a strict Hahn-Jordan decomposition (see Theorems 2.2.4 and 2.2.8).
Semigroups of unbounded linear operators in Banach space
Rhonda Jo
Hughes
113-145
Abstract: One-parameter families of unbounded linear operators acting in a Banach space X, and satisfying the semigroup and strong continuity properties on a suitable subspace of X, are discussed; the notion of infinitesimal generator is generalized to this unbounded setting, and a Hille-Yosida-type theorem is proved. The theory is illustrated by several examples, which include fractional integrals and derivatives acting in $ {L^p}(0,\infty )$.
Circle actions on simply connected $4$-manifolds
Ronald
Fintushel
147-171
Abstract: Locally smooth $ {S^1}$-actions on simply connected 4-manifolds are studied in terms of their weighted orbit spaces. An equivariant classification theorem is proved, and the weighted orbit space is used to compute the quadratic form of a given simply connected 4-manifold with ${S^1}$-action. This is used to show that a simply connected 4-manifold which admits a locally smooth $ {S^1}$-action must be homotopy equivalent to a connected sum of copies of ${S^4},C{P^2}, - C{P^2}$, and ${S^2} \times {S^2}$.
Finiteness in the minimal models of Sullivan
Stephen
Halperin
173-199
Abstract: Let X be a 1-connected topological space such that the vector spaces $ {\Pi _ \ast }(X) \otimes {\mathbf{Q}}$ and ${H^\ast}(X;{\mathbf{Q}})$ are finite dimensional. Then ${H^\ast}(X;{\mathbf{Q}})$ satisfies Poincaré duality. Set $ {\chi _\Pi } = \sum {( - 1)^p}\dim {\Pi _p}(X) \otimes {\mathbf{Q}}$ and ${\chi _c} =$ $\sum {( - 1)^p}\dim {H^p}(X;{\mathbf{Q}})$. Then $ {\chi _\Pi } \leqslant 0$ and $ {\chi _c} \geqslant 0$. Moreover the conditions: (1) ${\chi _\Pi } = 0$, (2) ${\chi _c} > 0,{H^\ast}(X;{\mathbf{Q}})$ evenly graded, are equivalent. In this case $ {H^\ast}(X;{\mathbf{Q}})$ is a polynomial algebra truncated by a Borel ideal. Finally, if X is a finite 1-connected C.W. complex, and an r-torus acts continuously on X with only finite isotropy, then ${\chi _\Pi } \leqslant - r$.
Topological examples of projective modules
Richard G.
Swan
201-234
Abstract: A new and more elementary proof is given for LØnsted's theorem that vector bundles over a finite complex can be represented by projective modules over a noetherian ring. The rings obtained are considerably smaller than those of LØnsted. In certain cases, methods associated with Hilbert's 17th problem can be used to give a purely algebraic description of the rings. In particular, one obtains a purely algebraic characterization of the homotopy groups of the classical Lie groups. Several examples are given of rings such that all projective modules of low rank are free. If $m \equiv 2 \bmod 4$, there is a noetherian ring of dimension m with nontrivial projective modules of rank m such that all projective modules of ${\text{rank}} \ne m$ are free.
On the cohomology groups of a polarisation and diagonal quantisation
J. H.
Rawnsley
235-255
Abstract: The sheaf ${\mathcal{S}_F}(L)$ of germs of sections of a line bundle L on a manifold X covariant constant with respect to a flat connection defined for vectors in a complex subbundle F of the tangent bundle has a resolution by differential forms defined on F with values in L provided F satisfies the integrability conditions of the complex Frobenius theorem. This includes as special cases the de Rham and Dolbeault resolutions.
The reduced Witt ring of a formally real field
Ron
Brown
257-292
Abstract: The reduced Witt rings of certain formally real fields are computed here in terms of some basic arithmetic invariants of the fields. For some fields, including the rational function field in one variable over the rational numbers and the rational function field in two variables over the real numbers, this is done by computing the image of the total signature map on the Witt ring. For a wider class of fields, including all those with only finitely many square classes, it is done by computing the Witt rings of certain ultracompletions of the field and representing the reduced Witt ring as an appropriate subdirect product of the Witt rings of the ultracompletions. The reduced Witt rings of a still wider class of fields, including for example the fields of transcendence degree one and the rational function field in three variables over the real numbers, are computed similarly, except that the description of the subdirect product no longer involves only local conditions.
Coherence in nonmonoidal closed categories
Miguel L.
Laplaza
293-311
Abstract: A Trionmonoidal closed category is a category with an internal homomorphism functor, left Yoneda natural arrows, unity object and some natural transformations and coherence axioms. The object of this paper is to give a complete solution of the coherence problem in this structure: we use a cut-elimination theorem as basic tool to prove that the elementary natural transformations are characterized by their graph (roughly speaking the graph is the type of identification imposed by a natural transformation on the arguments of its domain and codomain).
Invariant-free representations of augmented rings
Peter M.
Curran
313-319
Abstract: Let $\Gamma$ be an augmented ring in the sense of Cartan-Eilenberg, and let there be given a representation of $\Gamma$ in $ {\text{End}_k}\;A$, where A is a finite dimensional vector space over the field k. We show that all cohomology of $\Gamma$ in A is trivial if there are no invariants in A under the action of a suitable commutative subring of $\Gamma$. This generalizes a previous result of the author for group cohomology, and is applied to obtain sufficient conditions for the vanishing of the cohomology of Lie algebras and associative algebras.
Continua whose cone and hyperspace are homeomorphic
Sam B.
Nadler
321-345
Abstract: Let X be a (nonempty) metric continuum. By the hyperspace of X we mean $C(X) = \{ A$: A is a nonempty subcontinuum of $X\}$ with the Hausdorff metric H. It is determined that there are exactly eight hereditarily decomposable continua X such that the cone over X is homeomorphic to $C(X)$. Information about cone-to-hyperspace homeomorphisms, and about arc components for general classes of continua whose cone and hyperspace are homeomorphic is obtained. It is determined that indecomposable continua whose cone and hyperspace are homeomorphic have arcwise connected composants and, if finite-dimensional, have a strong form of the cone = hyperspace property.
Growth problems for subharmonic functions of finite order in space
N. V.
Rao;
Daniel F.
Shea
347-370
Abstract: For a function $ u(x)$ subharmonic (or $ {C^2}$) in ${{\mathbf{R}}^m}$, we compare the ``harmonics'' (defined in §1) of u with those of a related subharmonic function whose total Riesz mass in $\vert x\vert \leqslant r$ is the same as that of u, but whose ${L^2}$ norm on $\vert x\vert = r$ is maximal, for all $0 < r < \infty$. We deduce estimates on the growth of the Riesz mass of u in $\vert x\vert \leqslant r$, as $r \to \infty$.
Knot modules. I
Jerome
Levine
1-50
Abstract: For a differentiable knot, i.e. an imbedding ${S^n} \subset {S^{n + 2}}$, one can associate a sequence of modules $ \{ {A_q}\}$ over the ring $Z[t,{t^{ - 1}}]$, which are the source of many classical knot invariants. If X is the complement of the knot, and $\tilde X \to X$ the canonical infinite cyclic covering, then ${A_q} = {H_q}(\tilde X)$. In this work a complete algebraic characterization of these modules is given, except for the Z-torsion submodule of ${A_1}$.
Parametrizations of Titchmarsh's $m(\lambda )$-functions in the limit circle case
Charles T.
Fulton
51-63
Abstract: For limit-circle eigenvalue problems the so-called $'m(\lambda )'$-functions of Titchmarsh [15] are introduced in such a fashion that their parametrization is built into the definition.
Unimodality and dominance for symmetric random vectors
Marek
Kanter
65-85
Abstract: In this paper a notion of unimodality for symmetric random vectors in $ {R^N}$ is developed which is closed under convolution as well as weak convergence. A related notion of stochastic dominance for symmetric random vectors is also studied.
Alternator and associator ideal algebras
Irvin Roy
Hentzel;
Giulia Maria
Piacentini Cattaneo;
Denis
Floyd
87-109
Abstract: If I is the ideal generated by all associators, $(a,b,c) = (ab)c - a(bc)$, it is well known that in any nonassociative algebra $R,I \subseteq (R,R,R) + R(R,R,R)$. We examine nonassociative algebras where $I \subseteq (R,R,R)$. Such algebras include $( - 1,1)$ algebras, Lie algebras, and, as we show, a large number of associator dependent algebras. An alternator is an associator of the type (a, a, b), (a, b, a), (b, a, a). We next study algebras where the additive span of all alternators is an ideal. These include all algebras where $I = (R,R,R)$ as well as alternative algebras. The last section deals with prime, right alternative, alternator ideal algebras satisfying an identity of the form $ [x,(x,x,a)] = \gamma (x,x,[x,a])$ for fixed $\gamma$. With two exceptions, if this algebra has an idempotent e such that $(e,e,R) = 0$, then the algebra is alternative. All our work deals with algebras with an identity element over a field of characteristic prime to 6. All our containment relations are given by identities.
Distribution of eigenvalues in the presence of higher order turning points
Anthony
Leung
111-135
Abstract: This article is concerned with the eigenvalue problem $u''(x) - {\lambda ^2}p(x)u(x) = 0,u(x) \in {L_2}( - \infty ,\infty )$, where $p(x)$ is real, analytic and possesses zeroes of arbitrary orders. Under certain conditions on $ p(x)$, approximate formulas for the eigenvalues are found. The problem considered is of interest in the study of particle scattering and wave mechanics. The formula is analogous to the quantum rule of Bohr-Sommerfeld.
Composition series and intertwining operators for the spherical principal series. I
Kenneth D.
Johnson;
Nolan R.
Wallach
137-173
Abstract: Let G be a connected semisimple Lie group with finite center and let K be a maximal compact subgroup. Let $ \pi$ be a not necessarily unitary principal series representation of G on the Hilbert space ${H^\pi }$. If ${X^\pi }$ denotes the space of K-finite vectors of ${H^\pi },\pi$ induces a representation $ {\pi _0}$ of $ U(g)$, the enveloping algebra of the Lie algebra of G, on ${X^\pi }$. In this paper, we determine when ${\pi _0}$ is irreducible, and if $ {\pi _0}$ is not irreducible we determine the composition series of $ {X^\pi }$ and the structure of the induced representations on the subquotients. Explicit computation of the intertwining operators for the different principal series representations are obtained and their relationship to polynomials defined by B. Kostant are obtained.
Moduli of continuity for exponential Lipschitz classes
Paul
De Land
175-189
Abstract: Let $\Psi$ be a convex function, and let f be a real-valued function on [0, 1]. Let a modulus of continuity associated to $\Psi$ be given as $\displaystyle {Q_\Psi }(\delta ,f) = \inf \left\{ {\lambda :\frac{1}{\delta }\i... ... f(x) - f(y)\vert}}{\lambda }} \right)}\;dx\;dy\; \leqslant \Psi (1)} \right\}.$ It is shown that $ \smallint _0^1{Q_\Psi }(\delta ,f)\;d\;( - {\Psi ^{ - 1}}(c/\delta )) < \infty$ guarantees the essential continuity of f, and, in fact, a uniform Lipschitz estimate is given. In the case that $\Psi (u) = \exp \;{u^2}$ the above condition reduces to $\displaystyle \int_0^1 {{Q_{\exp \;{u^2}}}\;(\delta ,f)\frac{{d\delta }}{{\delta \sqrt {\log (c/\delta )} }}\; < \infty .} $ This exponential square condition is satisfied almost surely by the random Fourier series $ {f_t}(x) = \Sigma _{n = 1}^\infty {a_n}{R_n}(t){e^{inx}}$, where $\{ {R_n}\}$ is the Rademacher system, as long as $\displaystyle \int_0^1 {\sqrt {a_n^2{{\sin }^2}(n\delta /2)} \frac{{d\delta }}{{\delta \sqrt {\log (1/\delta )} }}\; < \infty .} $ Hence, the random essential continuity of ${f_t}(x)$ is guaranteed by each of the above conditions.
On a notion of smallness for subsets of the Baire space
Alexander S.
Kechris
191-207
Abstract: Let us call a set $ A \subseteq {\omega ^\omega }$ of functions from $\omega$ into $ \omega \;\sigma$-bounded if there is a countable sequence of functions $\{ {\alpha _n}:n \in \omega \} \subseteq {\omega ^\omega }$ such that every member of A is pointwise dominated by an element of that sequence. We study in this paper definability questions concerning this notion of smallness for subsets of ${\omega ^\omega }$. We show that most of the usual definability results about the structure of countable subsets of $ {\omega ^\omega }$ have corresponding versions which hold about $\sigma $-bounded subsets of ${\omega ^\omega }$. For example, we show that every $ \Sigma _{2n + 1}^1\;\sigma$-bounded subset of $ {\omega ^\omega }$ has a $\Delta _{2n + 1}^1$ ``bound'' $\{ {\alpha _m}:m \in \omega \}$ and also that for any $ n \geqslant 0$ there are largest $\sigma$-bounded $ \Pi _{2n + 1}^1$ and $\Sigma _{2n + 2}^1$ sets. We need here the axiom of projective determinacy if $n \geqslant 1$. In order to study the notion of $ \sigma$-boundedness a simple game is devised which plays here a role similar to that of the standard $^\ast$-games (see [My]) in the theory of countable sets. In the last part of the paper a class of games is defined which generalizes the $^\ast$- and $ ^{ \ast \ast }$- (or Banach-Mazur) games (see [My]) as well as the game mentioned above. Each of these games defines naturally a notion of smallness for subsets of ${\omega ^\omega }$ whose special cases include countability, being of the first category and $\sigma $-boundedness and for which one can generalize all the main results of the present paper.
Integral representations of invariant measures
Ashok
Maitra
209-225
Abstract: In this paper we prove, under suitable conditions, several representation theorems for invariant measures arising out of the action of a family of measurable transformations $\mathcal{J}$ on a measurable space $(X,\mathcal{A})$. Our results unify and extend results of Farrell and Varadarajan on the representation of invariant measures.
The equivalence of complete reductions
R.
Hindley
227-248
Abstract: This paper is about two properties of the $ \lambda \beta$-calculus and combinatory reduction, namely (E): all complete reductions $\rho$ and $\sigma$ of the residuals of a set of redexes in a term X have the same end; and $({{\text{E}}^ + }):\rho$ and $\sigma$ leave the same residuals of any other redex in X. Property (E) is deduced from abstract assumptions which do not imply $({{\text{E}}^ + })$. Also $({{\text{E}}^ + })$ is proved for the usual extensions of combinatory and $ \lambda \beta$-reduction, and a weak but natural form of $({{\text{E}}^ + })$ is proved for $\lambda \beta \eta $-reduction.
Fields generated by linear combinations of roots of unity
R. J.
Evans;
I. M.
Isaacs
249-258
Abstract: It is shown that a linear combination of roots of unity with rational coefficients generates a large subfield of the field generated by the set of roots of unity involved, except when certain partial sums vanish. Some related results about polygons with all sides and angles rational are also proved.
Nonsmoothable, unstable group actions
Dennis
Pixton
259-268
Abstract: For $k > 1$ there is a nonempty open set of ${C^1}$ actions of $ {{\mathbf{Z}}^k}$ on $ {S^1}$, no element of which is either topologically conjugate to a $ {C^2}$ action or structurally stable. The ${C^1}$ closure of this set contains all $ {C^2}$ actions which have compact orbits, so no such action is structurally stable in the space of ${C^1}$ actions.
Lifting idempotents and exchange rings
W. K.
Nicholson
269-278
Abstract: Idempotents can be lifted modulo a one-sided ideal L of a ring R if, given $x \in R$ with $x - {x^2} \in L$, there exists an idempotent $e \in R$ such that $e - x \in L$. Rings in which idempotents can be lifted modulo every left (equivalently right) ideal are studied and are shown to coincide with the exchange rings of Warfield. Some results of Warfield are deduced and it is shown that a projective module P has the finite exchange property if and only if, whenever $P = N + M$ where N and M are submodules, there is a decomposition $P = A \oplus B$ with $A \subseteq N$ and $B \subseteq M$.
On the zeros of Stieltjes and Van Vleck polynomials
Neyamat
Zaheer;
Mahfooz
Alam
279-288
Abstract: Stieltjes and Van Vleck polynomials arise in the study of the polynomial solutions of the generalized Lamé differential equation. Our object is to generalize a theorem due to Marden on the location of the zeros of Stieltjes and Van Vleck polynomials. In fact, our generalization is two-fold: Firstly, we employ sets which are more general than the ones used by Marden for prescribing the location of the complex constants occurring in the Lamé differential equation; secondly, Marden deals only with the standard form of the said differential equation, whereas our result is equally valid for yet another form of the same differential equation. The part of our main theorem concerning Stieltjes polynomials may also be regarded as a generalization of Lucas' theorem to systems of partial fraction sums.
Zeroes of holomorphic vector fields and Grothendieck duality theory
N. R.
O’Brian
289-306
Abstract: The holomorphic fixed point formula of Atiyah and Bott is discussed in terms of Grothendieck's theory of duality. The treatment is valid for an endomorphism of a compact complex-analytic manifold with arbitrary isolated fixed points. An expression for the fixed point indices is then derived for the case where the endomorphism belongs to the additive group generated by a holomorphic vector field with isolated zeroes. An application and some examples are given. Two generalisations of these results are also proved. The first deals with holomorphic vector bundles having sufficient homogeneity properties with respect to the action of the additive group on the base manifold, and the second with additive group actions on algebraic varieties.
Existence of integrals and the solution of integral equations
Jon C.
Helton
307-327
Abstract: Functions are from R to N or $R \times R$ to N, where R denotes the real numbers and N denotes a normed complete ring. If S, T and G are functions from $R \times R$ to N, each of $ S({p^ - },p),S({p^ - },{p^ - }),T({p^ - },p)$ and $T({p^ - },{p^ - })$ exists for $a < p \leqslant b$, each of $ S(p,{p^ + }),S({p^ + },{p^ + }),T(p,{p^ + })$ and $T({p^ + },{p^ + })$ exists for $a \leqslant p < b$, G has bounded variation on [a, b] and $ \smallint _a^bG$ exists, then each of $\displaystyle \int_a^b S \left[ {G - \int G } \right]T\quad {\text{and}}\quad \int_a^b {S\left[ {1 + G - \prod {(1 + G)} } \right]} \;T$ exists and is zero. These results can be used to solve integral equations without the existence of integrals of the form $\displaystyle \int_a^b {\left\vert {G - \int G } \right\vert = 0} \quad {\text{and}}\quad \int_a^b {\left\vert {1 + G - \prod {(1 + G)} } \right\vert} = 0.$ This is demonstrated by solving the linear integral equation $\displaystyle f(x) = h(x) + (LR)\int_a^x {(fG + fH)}$ and the Riccati integral equations $\displaystyle f(x) = w(x) + (LRLR)\int_a^x {(fH + Gf + fKf)}$ without the existence of the previously mentioned integrals.
Knots with infinitely many minimal spanning surfaces
Julian R.
Eisner
329-349
Abstract: We show that if $ {k_1}$ and ${k_2}$ are nonfibered knots, then the composite knot $K = {k_1}\char93 {k_2}$ has an infinite collection of minimal spanning surfaces, no two of which are isotopic by an isotopy which leaves the knot K fixed. This result is then applied to show that whether or not a knot has a unique minimal spanning surface can depend on what definition of spanning surface equivalence is used.
Quasi-Anosov diffeomorphisms and hyperbolic manifolds
Ricardo
Mañé
351-370
Abstract: Let f be a diffeomorphism of a smooth manifold N and $M \subset N$ a compact boundaryless submanifold such that it is a hyperbolic set for f. The diffeomorphism f/M is characterized and it is proved that it is Anosov if and only if M is an invariant isolated set of f (i.e. the maximal invariant subset of some compact neighborhood). Isomorphisms of vector bundles with the property that the zero section is an isolated subset are studied proving that they can be embedded in hyperbolic vector bundle isomorphisms.
Representation theorems for nonlinear disjointly additive functionals and operators on Sobolev spaces
Moshe
Marcus;
Victor J.
Mizel
1-45
Abstract: An abstract characterization is obtained for a class of nonlinear differential operators defined on the subspace $S = {\ring{W}}_k^p[a,b]$ of the kth order Sobolev space $W_k^p[a,b], 1 \leqslant k, 1 \leqslant p \leqslant \infty$. It is shown that every mapping $T:S \to {L^1}[a,b]$ which is local, continuous and ${D^k}$-disjointly additive has the form $ (Tu)(t) = H(t,{D^k}u(t))$, where $ H:[a,b] \times R \to R$ is a function obeying Carathéodory conditions as well as $ (\ast )H( \cdot ,0) = 0$. Here ${D^k}$-disjoint additivity means $T(u + v) = Tu + Tv$ whenever $({D^k}u)({D^k}v) = 0$. Likewise, every real functional N on S which is continuous and ${D^k}$-disjointly additive has the form $N(u) = \smallint Tu$, with T as above. Liapunov's theorem on vector measures plays a crucial role, and the analysis suggests new questions about such measures. Likewise, a new type of Radon-Nikodým theorem is employed in an essential way.
Hall-Higman type theorems. III
T. R.
Berger
47-83
Abstract: This paper continues the investigations of this series. Suppose that $G = ANS$ where S and NS are normal subgroups of G. Suppose that $ (\vert A\vert,\vert NS\vert) = 1$, S is extraspecial, and $ S/Z(S)$ is a faithful minimal module for the subgroup AN of G. Assume that k is a field of characteristic prime to $ \vert G\vert$ and V is a faithful irreducible ${\mathbf{k}}[G]$-module. The structure of G is discussed in the minimal situation where N is cyclic, A is nilpotent, and $ V{\vert _A}$ does not have a regular A-direct summand.
Hopf invariants and Browder's work on the Kervaire invariant problem
Warren M.
Krueger
85-97
Abstract: In this paper we calculate certain functional differentials in the Adams spectral sequence converging to Wu cobordism whose values may be thought of as Hopf invariants. These results are applied to reobtain Browder's characterization: if $q + 1 = {2^k}$, there is a 2q dimensional manifold of Kervaire invariant one if and only if $h_k^2$ survives to ${E_\infty }({S^0})$.
Systems of nonlinear Volterra equations with positive definite kernels
Olof J.
Staffans
99-116
Abstract: We study the boundedness and the asymptotic behavior of the solutions of a nonlinear, $ {{\mathbf{R}}^n}$-valued Volterra equation with a positive definite kernel, generalizing earlier scalar results.
Essential central range and selfadjoint commutators in properly infinite von Neumann algebras
Herbert
Halpern
117-146
Abstract: The essential central range of an element A of a von Neumann algebra with respect to a central ideal is characterized as those elements arbitrarily close to the compression of A to a subspace large with respect to the ideal. The selfadjoint commutators in a properly infinite algebra are shown to be the elements whose essential central ranges with respect to the strong radical contain 0.
The asymptotic behavior of a Volterra-renewal equation
Peter
Ney
147-155
Abstract: Theorem. Assume that the functions $x( \cdot ),h( \cdot ),G( \cdot )$ satisfy: (i) $0 \leqslant x(t),t \in [0,\infty );x(t) \to 0$ as $t \to \infty ;x$ bounded, measurable; (ii) $0 \leqslant h(s);h(s)$ Lipschitz continuous for $s \in I$, where I is a closed interval containing the range of $(0,\infty )$ having nontrivial absolutely continuous component and finite second moment. Let $Hx(t) = \smallint _0^th[x(t - y)]dG(y)$. If $0 \leqslant (x - Hx)(t) = o({t^{ - 2}})$, with strict inequality on the left on a set of positive measure, then $x(t) \sim \gamma /t,t \to \infty$, where $\gamma$ is a constant depending only on h and G. The condition $o({t^{ - 2}})$ is close to best possible, and cannot, e.g., be replaced by $ O({t^{ - 2}})$.
Cone bundles
Clint
McCrory
157-163
Abstract: A theory of normal bundles for locally knotted codimension two embeddings of PL manifolds is developed. The classifying space for this theory is Cappell and Shaneson's space $ BR{N_2}$.
The structure of local integral orthogonal groups
D. G.
James
165-186
Abstract: Let M be a lattice on a regular quadratic space over a nondyadic local field. The normal subgroups of the integral orthogonal group $O(M)$ are determined.
Hausdorff content and rational approximation in fractional Lipschitz norms
Anthony G.
O’Farrell
187-206
Abstract: For $0 < \alpha < 1$, we characterise those compact sets X in the plane with the property that each function in the class ${\text{lip}}(\alpha ,X)$ that is analytic at all interior points of X is the limit in ${\text{Lip}}(\alpha ,X)$ norm of a sequence of rational functions. The characterisation is in terms of Hausdorff content.
Resolvents and bounds for linear and nonlinear Volterra equations
J. J.
Levin
207-222
Abstract: The asymptotic behavior of the resolvent of a linear Volterra equation is investigated without the assumption that the kernel of the equation is in $ {L^1}(0,\infty )$. A lower bound is obtained on the solutions of a related nonlinear Volterra equation. A special case of the latter result is employed in the proof of the former result.
Examples for the nonuniqueness of the equilibrium state
Franz
Hofbauer
223-241
Abstract: In this paper equilibrium states on shift spaces are considered. A uniqueness theorem for equilibrium states is proved. Then we study a particular class of continuous functions. We characterize the functions of this class which satisfy Ruelle's Perron-Frobenius condition, those which admit a measure determined by a homogeneity condition, and those which have unique equilibrium state. In particular, we get examples for the nonuniqueness of the equilibrium state.
The generalized Green's function for an $n$th order linear differential operator
John
Locker
243-268
Abstract: The generalized Green's function $K(t,s)$ for an nth order linear differential operator L is characterized in terms of the 2nth order differential operators $ L{L^\ast}$ and $ {L^\ast}L$. The development is operator oriented and takes place in the Hilbert space ${L^2}[a,b]$. Two features of the characterization are a determination of the jumps occurring in the derivatives of orders n, $n + 1, \ldots ,2n - 1$ at $t = s$ and a determination of the boundary conditions satisfied by the functions $K(a, \cdot )$ and $ K(b,\cdot)$. Several examples are given to illustrate the properties of the generalized Green's function.
Module structure of certain induced representations of compact Lie groups
E. James
Funderburk
269-285
Abstract: Let G be a compact connected Lie group and assume a choice of maximal torus and positive roots has been made. Given a dominant weight $\lambda$, the Borel-Weil Theorem shows how to construct a holomorphic line bundle on whose sections G acts so that the holomorphic sections provide a realization of the irreducible representation of G with highest weight $\lambda$. This paper studies the G-module structure of the space $\Gamma$ of square integrable sections of the Borel-Weil line bundle. It is found that $ \Gamma = {\lim _{n \to \infty }}\Gamma (n)$, where $\Gamma (n) \subset \Gamma (n + 1) \subset \Gamma$ and $ \Gamma (n)$ is isomorphic, as G-module, to $\displaystyle V(\lambda + n\lambda ) \otimes V(n{\lambda ^\ast}),$ where $V(\mu )$ denotes the irreducible representation of highest weight $\mu$, '+' is the Cartan semigroup operation, and '$ ^\ast$' is the contragredient operation. Similar formulas hold for powers of the Borel-Weil line bundle.
A new characterization of Ces\`aro-Perron integrals using Peano derivates
J. A.
Bergin
287-305
Abstract: The $ {Z_n}$-integrals are defined according to the method of Perron using Peano derivates. The properties of the integrals are given including the essential integration by parts theorem. The integrals are then shown to be equivalent to the Cesàro-Perron integrals of Burkill.
Bochner identities for Fourier transforms
Robert S.
Strichartz
307-327
Abstract: Let G be a compact Lie group and R an orthogonal representation of G acting on ${{\mathbf{R}}^n}$. For any irreducible unitary representation $\pi$ of G and vector v in the representation space of $\pi$ define $ \mathcal{S}(\pi ,v)$ to be those functions in $ \mathcal{S}({{\mathbf{R}}^n})$ which transform (under the action R) according to the vector v. The Fourier transform $\mathcal{F}$ preserves the class $\mathcal{S}(\pi ,v)$. A Bochner identity asserts that for different choices of G, R, $\pi ,v$ the Fourier transform is the same (up to a constant multiple). It is proved here that for G, R, $\pi ,v$ and $ G',R',\pi ',v'$ and a map $T:\mathcal{S}(\pi ,v) \to \mathcal{S}(\pi ',v')$ which has the form: restriction to a subspace followed by multiplication by a fixed function, a Bochner identity $f \in \mathcal{S}(\pi ,v)$ holds if and only if $ f \in \mathcal{S}(\pi ,v)$. From this result all known Bochner identities follow (due to Harish-Chandra, Herz and Gelbart), as well as some new ones.
Analysis on the Heisenberg manifold
Richard
Tolimieri
329-343
Abstract: A study of the function theory on the Heisenberg manifold in terms of theta functions. Subject to an explicit error, a ${C^\infty }$-function is written as an infinite sum, with theta functions of different degrees and characteristics playing the same role as exponentials do in the abelian theory.
On composite abstract homogeneous polynomials
Neyamat
Zaheer
345-358
Abstract: We study the null-sets of composite abstract homogeneous polynomials obtained from a pair of abstract homogeneous polynomials defined on a vector space over an algebraically closed field of characteristic zero. First such study for ordinary polynomials in the complex plane was made by Szegö, Cohn, and Egerváry and Szegö's theorem was later generalized to fields and vector spaces, respectively, by Zervos and Marden. Our main theorem in this paper further generalizes their results and, in the complex plane, improves upon Szegö's theorem and some other classical results. The method of proof is purely algebraic and utilizes the author's vector space analogue [Trans. Amer. Math. Soc. 218 (1976), 115-131] of Grace's theorem on apolar polynomials. We also show that our results cannot be further generalized in certain directions.
Extensions of Haar measure to relatively large nonmeasurable subgroups
H. Leroy
Peterson
359-370
Abstract: Let G be a locally compact group, with $\lambda$ a left Haar measure on G. A subgroup is large if it has finite index; a relatively large subgroup of G is a large subgroup of an open subgroup. In §1 we have an existence theorem for relatively large nonopen subgroups, and we observe that such subgroups are not $ \lambda$-measurable. This motivates the development, in §2, of a left-invariant countably additive extension ${\lambda ^ + }$ of $\lambda$, which includes in its domain all unions of left translates of a given relatively large subgroup K. For an arbitrarily chosen family ${\mathcal{K}_I}$ of relatively large subgroups of G, we define (in §3) a finitely additive measure $\lambda _I^ + $ such that, for any $K \in {\mathcal{K}_I},\lambda _I^ +$ is an extension of the corresponding ${\lambda ^ + }$ defined in §2. An example shows that $\lambda _I^ + $ need not be countably additive. Finally, in §4, we observe some aspects of the relationship between $ {\lambda ^ + }$-measurable and $\lambda$-measurable functions, in the context of existing literature on extensions of Haar measure. In particular, we generalize the well-known proposition that $\lambda$-measurable characters are continuous.
Intersection properties of balls and subspaces in Banach spaces
Ȧsvald
Lima
1-62
Abstract: We study intersection properties of balls in Banach spaces using a new technique. With this technique we give new and simple proofs of some results of Lindenstrauss and others, characterizing Banach spaces with ${L_1}(\mu )$ dual spaces by intersection properties of balls, and we solve some open problems in the isometric theory of Banach spaces. We also give new proofs of some results of Alfsen and Effros characterizing M-ideals by intersection properties of balls, and we improve some of their results. In the last section we apply these results on function algebras, G-spaces and order unit spaces and we give new and simple proofs for some representation theorems for those Banach spaces with $ {L_1}(\mu )$ dual spaces whose unit ball contains extreme points.
$K$-theory and Steenrod homology: applications to the Brown-Douglas-Fillmore theory of operator algebras
Jerome
Kaminker;
Claude
Schochet
63-107
Abstract: The remarkable work of L. G. Brown, R. Douglas and P. Fillmore on operators with compact self-commutators once again ties together algebraic topology and operator theory. This paper gives a comprehensive treatment of certain aspects of that connection and some adjacent topics. In anticipation that both operator theorists and topologists may be interested in this work, additional background material is included to facilitate access.
The isomorphism problem for two-generator one-relator groups with torsion is solvable
Stephen J.
Pride
109-139
Abstract: The theorem stated in the title is obtained by determining (in a sense to be made precise) all the generating pairs of an arbitrary two-generator one-relator group with torsion. As a consequence of this determination it is also deduced that every two-generator one-relator group G with torsion is Hopfian, and that the automorphism group of G is finitely generated.
Chern-Simons invariants of reductive homogeneous spaces
Harold
Donnelly
141-164
Abstract: The geometric characteristic classes of Chern-Simons are computed for certain connections on the canonical bundle and tangent bundle over a reductive homogeneous space. This includes globally symmetric spaces with the Levi-Civita connection of any bi-invariant metric.
The fundamental form of an inseparable extension
Murray
Gerstenhaber
165-184
Abstract: If K is a finite purely inseparable extension of a field k, then the symmetric multiderivations of K (symmetric maps $f:K \times \cdots \times K\;(n\;{\text{times}}) \to K$ which are derivations as functions of each single variable) form a ring under the symmetrized cup product. This ring contains an element $\Gamma (K/k)$ called the fundamental form of K over k, which is defined up to multiplication by a nonzero element of K and has the property that if B is any intermediate field between K and k, then $ \Gamma (K/B)$ divides $\Gamma (K/k)$.
Functions satisfying elementary relations
Michael F.
Singer
185-206
Abstract: In this paper we deal with the following problems: When do the solutions of a collection of differential equations satisfy an elementary relation, that is, when is there an equation of the form $R = 0$ where R is some algebraic combination of logarithmic, exponential and algebraic functions involving solutions of our differential equations? If such relations exist, what can they look like? These problems are given an algebraic setting and general forms for such relations are exhibited. With these, we are able to show that certain classes of functions satisfy no elementary relations.
Homological algebra and set theory
Paul C.
Eklof
207-225
Abstract: Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of $ {\operatorname{Ext}}_\Lambda ^1$ for rings $\Lambda$ of global dimension 1. Using Martin's Axiom, the necessity of these conditions is shown not to be a theorem of ZFC. Applications are given to abelian group theory, including a partial solution (assuming ${\text{V}} = {\text{L}}$) to a problem of Baer on the splitting of abelian groups. Some independence results in abelian group theory are also proved.
Derivatives of entire functions and a question of P\'olya
Simon
Hellerstein;
Jack
Williamson
227-249
Abstract: An old question of Pólya asks whether an entire function f which has, along with each of its derivatives, only real zeros must be of the form $\displaystyle f(z) = {z^m}{e^{ - a{z^2} + bz + c}}\prod\limits_n {\left( {1 - \frac{z}{{{z_n}}}} \right)} {e^{z/{z_n}}}$ where $a \geqslant 0,b$ and the ${z_n}$ are real, and ${\Sigma _n}z_n^{ - 2} < \infty$. This note answers this question (essentially in the affirmative) if f is of finite order; indeed, it is established that if $f,f'$, and $f''$ have only real zeros (f of finite order), then either f has the above form or f has one of the forms $\displaystyle f(z) = a{e^{bz}},\quad f(z) = a({e^{icz}} - {e^{id}})$ where a, b, c, and d are constants, b complex, c and d real.
A relation between two biharmonic Green's functions on Riemannian manifolds
Dennis
Hada
251-261
Abstract: The biharmonic Green's function $\gamma$ whose values and Laplacian are identically zero on the boundary of a region and the biharmonic Green's function $\Gamma$ whose values and normal derivative vanish on the boundary originated in the investigation of thin plates whose edges are simply supported or clamped, respectively. A relation between these two biharmonic Green's functions known for planar regions is extended to Riemannian manifolds thereby establishing that any Riemannian manifold for which $\gamma$ exists must also carry $\Gamma$.
Embedding Stieltjes-Volterra integral equations in Stieltjes integral equations
William L.
Gibson
263-277
Abstract: J. A. Reneke has shown that the linear Stieltjes-Volterra integral equations studied by D. B. Hinton can be transformed into Stieltjes integral equations of the type studied by J. S. Mac Nerney. By taking advantage of the nonlinear nature of Mac Nerney's results, Reneke was able to extend Hinton's existence theorem to a nonlinear setting. In this paper, we use Reneke's embedding technique to generalize several other of Hinton's results, and we characterize completely, in the linear case, the range of Reneke's embedding transformation.
The topological structure of $4$-manifolds with effective torus actions. I
Peter Sie
Pao
279-317
Abstract: Torus actions on orientable 4-manifolds have been studied by F. Raymond and P. Orlik [8] and [9]. The equivariant classification problem has been completely answered there. The problem then arose as to what can be said about the topological classification of these manifolds. Specifically, when are two manifolds homeomorphic if they are not equivariantly homeomorphic? In some cases this problem was answered in the above mentioned papers. For example, if the only nontrivial stability groups are finite cyclic, then the manifolds are essentially classified by their fundamental groups. In the presence of fixed points, a connected sum decomposition in terms of ${S^4},{S^2} \times {S^2},C{P^2}, - C{P^{ - 2}},{S^1} \times {S^3}$, and three families of elementary 4-manifolds, $ R(m,n),T(m,n;m',n'),L(n;p,q;m)$ has been obtained (where m, n, $ m',n'$, p, and q are integers). In addition, a stable homeomorphic relation for the manifolds $R(m,n)$ and $T(m,n;m',n')$ can also be found in [9]. But the topological classification of R's, T's, and especially L's were still unsolved problems. Furthermore, the connected sum decomposition of a manifold with fixed points, even in the simply connected case, is not unique. In this paper, we completely classify the manifolds with fixed points. For the manifolds R's and T's, the above mentioned stable homeomorphic relation is proved to be the topological classification. The manifolds $ L(n;p,q;m)$ form a very interesting family of 4-manifolds. They behave similarly to lens spaces. For example, the fundamental group of $L(n;p,q;m)$ is finite cyclic of order n. And it is proved that ${\pi _1}(L(n;p,q;m))$ and $({S^2} \times {S^2})\char93 \cdots \char93 \;({S^2} \times {S^2})$ ($n - 1$ copies), even though $L(n;p,q;m)$ and
Structure of subalgebras between $L\sp{\infty }$ and $H\sp{\infty }$
Sun Yung A.
Chang
319-332
Abstract: Let B be a closed subalgebra of $ {L^\infty }$ of the unit circle which contains $ {H^\infty }$ properly. Let $ {C_B}$ be the $ {C^\ast}$-algebra generated by the inner functions that are invertible in B. It is shown that the linear span ${H^\infty } + {C_B}$ is equal to B. Also, a closed subspace (called $VM{O_B}$) of BMO (space of functions of bounded mean oscillation) is identified to which B bears the same relation as $ {L^\infty }$ does to BMO.
Ulm's theorem for partially ordered structures related to simply presented abelian $p$-groups
Laurel A.
Rogers
333-343
Abstract: If we have an abelian p-group G, a multiplication by p for each element of G is defined by setting $ px = x + x + \cdots + x$, where p is the number of terms in the sum. If we forget about the addition on G, and just keep the multiplication by p, we have the algebraic structure called a p-basic tree. A natural partial order can be defined, the graph of which is a tree with 0 as root. A p-basic tree generates a simply presented abelian p-group, and provides a natural direct sum decomposition for it. Ulm invariants may be defined directly for a p-basic tree so that they are equal to the Ulm invariants of the corresponding group. A central notion is that of a stripping function between two p-basic trees. Given a stripping function from X onto Y we can construct an isomorphism between the groups corresponding to X and Y; in particular, X and Y have the same Ulm invariants. Conversely, if X and Y have the same Ulm invariants, then there is a map from X onto Y that is the composition of two stripping functions and two inverses of stripping functions. These results constitute Ulm's theorem for p-basic trees, and provide a new proof of Ulm's theorem for simply presented groups.
The classifying space of a permutation representation
James V.
Blowers
345-355
Abstract: In this article the concept of classifying space of a group is generalized to a classifying space of an arbitrary permutation representation. An example of this classifying space is given by a generalization of the infinite join construction that defines the standard example of a classifying space of a group. In a previous paper of the author, the join of two permutation representations was defined, and it was shown that the cohomology ring of the join was trivial. In this paper the classifying space of the join of two permutation representations is shown to be the topological join of the two classifying spaces and from this the triviality of the cup-product is derived topologically.
One-parameter groups of isometries on Hardy spaces of the torus: spectral theory
Earl
Berkson;
Horacio
Porta
357-370
Abstract: The spectral theory of the infinitesimal generator of an arbitrary one-parameter group of isometries on ${H^p}$ of the torus, $1 \leqslant p < \infty ,p \ne 2$, is considered. In particular, the spectrum of the generator is determined.
An $L\sp{1}$-space for Boolean algebras and semireflexivity of spaces $L\sp{\infty }(X,\Sigma ,\mu )$
Dennis
Sentilles
1-37
Abstract: In this paper we suggest a measure free construction of $ {L^1}$-spaces using Boolean algebras and strict topologies and initiate a duality theory of $ ({L^\infty },{L^1})$ like that of the duality of continuous functions and Baire measures, showing that the Boolean context yields a formal link between uniform tightness, uniform $ \sigma$-additivity and uniform integrability.
Potential processes
R. V.
Chacon
39-58
Abstract: The prototype of a potential process is a stochastic process which visits the same points in the same order as a Markov process, but at a rate obtained from a nonanticipating time change. The definition of a potential process may be given intrinsically and most generally without mention of a Markov process, in terms of potential theory. The definition may be given more directly and less generally in terms of potentials which arise from Markov processes, or more directly than this, as suitably time-changed Markov processes. The principal purpose of studying the class of potential processes, which may be shown to include martingales as well as Markov processes themselves, is to give a unified treatment to a wide class of processes which has potential theory at its core. That it is possible to do so suggests that potential rather than martingale results are central to the study of Markov processes. Furthermore, this also suggests that it is not the Markov property itself which makes Markov processes tractable, but rather the potential structure which can be constructed with the assistance of the Markov property. The general theory of potential processes is developed in a forthcoming paper. It will be shown there that a Markov process subject to an ordinary continuous nonanticipating time change is a local potential process. It may be seen, by examining examples, that it is necessary to consider randomized stopping times and randomized nonanticipating time changes in the general case. In the forthcoming paper a more general notion than randomized nonanticipating time changes is used to obtain a characterization of potential processes. It is an open problem whether randomization itself is sufficient in the general case, and whether ordinary nonanticipating time changes are sufficient for continuous parameter martingales and Brownian motion on the line. The emphasis in the present paper will be on developing the theory of discrete parameter martingales as a special case of the general theory.
The orderability and suborderability of metrizable spaces
S.
Purisch
59-76
Abstract: A space is defined to be suborderable if it is embeddable in a (totally) orderable space. It is shown that a metrizable space X is suborderable iff (1) each component of X is orderable, (2) the set of cut points of each component of X is open, and (3) each closed subset of X which is a union of components has a base of clopen neighborhoods. Note that condition (1) and hence this result is topological since there are many good topological characterizations of connected orderable spaces. In a space X let Q denote the union of all nondegenerate components each of whose noncut points has no compact neighborhood. It is also shown that a metrizable space X is orderable iff (1) X is suborderable, (2) $X - Q$ is not a proper compact open subset of X, and (3) if W is a neighborhood of $p \in X$ and K is the component in X containing p such that $ (W - K) - Q$ has compact closure and $\{ p\}$ is the intersection of the closures of $(W - K) - Q$ and $(W - K) \cap Q$, then K is a singleton. Corollaries are given; every condition in each of these corollaries is concisely stated and sufficient for a space to be orderable when it is metrizable and suborderable. Both of these results are extended to a class properly containing the metrizable spaces.
The periodic points of Morse-Smale endomorphisms of the circle
Louis
Block
77-88
Abstract: Let $MS({S^1})$ denote the set of continuously differentiable maps of the circle with finite nonwandering set, which satisfy certain generic properties. For $f \in MS({S^1})$ let $ P(f)$ denote the set of positive integers which occur as the period of some periodic point of f. It is shown that for $f \in MS({S^1})$ there are integers $m \geqslant 1$ and $n \geqslant 0$ such that $ P(f) = \{ m,2m,4m, \ldots ,{2^n}m\}$. Conversely, if m and n are integers, $m \geqslant 1,n \geqslant 0$, there is a map $f \in MS({S^1})$ with $P(f) = \{ m,2m,4m, \ldots ,{2^n}m\}$.
On Lie algebras of vector fields
Akira
Koriyama;
Yoshiaki
Maeda;
Hideki
Omori
89-117
Abstract: This paper has two purposes. The first is a generalization of the theorem of Pursell-Shanks [10]. Our generalization goes by assuming the existence of a nontrivial core of a Lie algebra. However, it seems to be a necessary condition for the theorems of Pursell-Shanks type. The second is the classification of cores under the assumption that the core itself is infinitesimally transitive at every point. As naturally expected, we have the nonelliptic, primitive infinite-dimensional Lie algebras.
Three-dimensional manifolds with finitely generated fundamental groups
Robert
Messer
119-145
Abstract: Recent results of G. P. Scott and T. W. Tucker indicate that a three-dimensional manifold with a finitely generated fundamental group is, in various senses, close to being compact. In this paper the structure of such a manifold M is described in terms of a certain compact, incompressible submanifold of M. This result is used to show that the product of M with the real line is essentially the interior of a compact 4-manifold. Finally, when M is $ {P^2}$-irreducible, a necessary and sufficient condition is given for M to be homeomorphic to the complement of a closed subset of the boundary of a compact 3-manifold.
A development of contraction mapping principles on Hausdorff uniform spaces
Cheng Ming
Lee
147-159
Abstract: Certain generalized Banach's contraction mapping principles on metric spaces are unified and/or extended to Hausdorff uniform spaces. Also given are some relationships between the set of all cluster points of the Picard iterates and the set of all fixed points for the mapping. These are obtained by assuming that the latter set is nonempty and by considering certain ``quasi"-contractive conditions. The ("quasi") contractive conditions are defined by using a suitable family of pseudometrics on the uniform space.
The independence ratio and genus of a graph
Michael O.
Albertson;
Joan P.
Hutchinson
161-173
Abstract: In this paper we study the relationship between the genus of a graph and the ratio of the independence number to the number of vertices.
Some smooth maps with infinitely many hyperbolic periodic points
John M.
Franks
175-179
Abstract: If a smooth map of the two-disk to itself has only hyperbolic periodic points and has no source or sink whose period is a power of two then it has infinitely many periodic points. This and similar results are proved.
Discrete analytic functions of exponential growth
Doron
Zeilberger
181-189
Abstract: Analogues of classical representation formulas for entire functions of exponential type are proved in the class of discrete analytic functions.
Cohomological dimension of a group with respect to finite modules
Juan José
Martínez
191-201
Abstract: The purpose of this paper is to compare the cohomological dimension of a group, relative to finite modules, with the cohomological dimension, in the usual sense, of its profinite completion. The basic tool used to perform this comparison is certain stable cohomology of the group. The reason is that there exists a spectral sequence which relates the continuous cohomology of the profinite completion, with coefficients in this stable cohomology, to the ordinary cohomology of the group. Moreover, the direct method of connecting the cohomology of the group with the profinite cohomology of its completion arises from the edge effects on the base of this spectral sequence.
Hausdorff measure functions in the space of compact subsets of the unit interval
P. R.
Goodey
203-208
Abstract: The work done in this paper is the result of an attempt to classify those functions h for which the corresponding Hausdorff measure of $ \mathcal{F}[0,1]$ is zero. A partial characterization is achieved and in doing this some problems of E. Boardman are solved.
Higher algebraic $K$-theories
D.
Anderson;
M.
Karoubi;
J.
Wagoner
209-225
Abstract: A homotopy fibration is established relating the Volodin or BN-pair definition of algebraic K-theory to the theory defined by Quillen. In [2] we outlined the construction of natural homomorphisms $\displaystyle K_ \ast ^Q \to K_ \ast ^{BN} \to K_ \ast ^V \to K_ \ast ^{KV}$ between higher algebraic K-theories $K_ \ast ^Q$ of [10] and [11], $K_ \ast ^{BN}$ of [17], $ K_ \ast ^V$ of [16], and $K_ \ast ^{KV}$ of [7] and [8]. This was one of the steps in proving the various definitions of higher K-theory are equivalent. It turns out they all agree-including the theory $ K_ \ast ^S$ of [14], [5], and [8]-provided one restricts to the category of regular rings when using $ K_\ast^{KV}$. See [1], [2], [5], [8] and [18]. The purpose of this paper is to prove the following theorem, announced in [2], which yields the construction of $K_ \ast ^Q \to K_ \ast ^{BN}$. Theorem. For any associative ring with identity A $\displaystyle G{L^{BN}}(A) \to B{\{ {U_F}\} ^ + } \to BGL{(A)^ + }$ is a homotopy fibration. For the reader's convenience and because the presentation of the BN-pair K-theory $K_ \ast ^{BN}$ used here is slightly different from that of [17], we shall briefly recall the definition of $G{L^{BN}}$ and $ B\{ {U_F}\}$ in the first section.
Second-order differential equations with fractional transition points
F. W. J.
Olver
227-241
Abstract: An investigation is made of the differential equation $\displaystyle {d^2}w/d{x^2} = \{ {u^2}{(x - {x_0})^\lambda }f(u,x) + g(u,x)/{(x - {x_0})^2}\} w,$ in which u is a large real (or complex) parameter, $\lambda$ is a real constant such that $\lambda > -2$, x is a real (or complex) variable, and $f(u,x)$ and $g(u,x)$ are continuous (or analytic) functions of x in a real interval (or complex domain) containing ${x_0}$. The interval (or domain) need not be bounded. Previous results of Langer and Riekstins giving approximate solutions in terms of Bessel functions of order $1/(\lambda + 2)$ are extended and error bounds supplied.
Best possible approximation constants
A. M.
Fink
243-255
Abstract: We study inequalities between the norm of the best approximating polynomial and the nth derivative of the function. These inequalities are then related to inequalities that have been considered elsewhere in different contexts.
A comparison of various definitions of contractive mappings
B. E.
Rhoades
257-290
Abstract: A number of authors have defined contractive type mappings on a complete metric space X which are generalizations of the well-known Banach contraction, and which have the property that each such mapping has a unique fixed point. The fixed point can always be found by using Picard iteration, beginning with some initial choice ${x_0} \in X$. In this paper we compare this multitude of definitions. X denotes a complete metric space with distance function d, and f a function mapping X into itself.
Function fields with isomorphic Galois groups
Robert J.
Bond
291-303
Abstract: Let K be a local field or a global field of characteristic p. Let ${G_K}$ be the Galois group of the separable closure of K over K. In the local case we show that ${G_K}$, considered as an abstract profinite group, determines the characteristic of K and the number of elements in the residue class field. In the global case we show that ${G_K}$ determines the number of elements in the constant field of K as well as the zeta function, genus and class number of K. Let $K'$ be another global field of characteristic p and assume we have
Linear operators for which $T\sp*T$ and $T+T\sp*$ commute. II
Stephen L.
Campbell;
Ralph
Gellar
305-319
Abstract: Let $\theta$ denote the set of bounded linear operators T, acting on a separable Hilbert space $ \mathcal{K}$ such that $ {T^\ast}T$ and $T + {T^\ast}$ commute. It is shown that such operators are ${G_1}$. A complete structure theory is developed for the case when $ \sigma (T)$ does not intersect the real axis. Using this structure theory, several nonhyponormal operators in $\theta$ with special properties are constructed.
A resolvent for an iteration method for nonlinear partial differential equations
J. W.
Neuberger
321-343
Abstract: For each of m and n a positive integer denote by $ S(m,i)$ the space of all real-valued symmetric i-linear functions on $ {E_m},i = 1,2, \ldots ,n$. Denote by L a nonzero linear functional on $ S(m,n)$, denote by f a real-valued analytic function on ${E_m} \times R \times S(m,1) \times \cdots \times S(m,[n/2])$ and denote by $\alpha$ a member of $D(f)$. Denote by H the space of all real-valued functions U, analytic at the origin of $ {E_m}$, so that $ K(\lambda )U$ converges, as $\lambda \to \infty$, to a solution Y to the partial differential equation $L{Y^{(n)}} = {f_Y}$. A resolvent j for this semigroup is determined so that $J(\lambda )U$ also converges to y as $\lambda \to \infty$ and so that $J{(\lambda /n)^n}U$ converges to $K(\lambda )U$ as $ n \to \infty$. The solutions $Y \in H$ of $ L{Y^{(n)}} = {f_Y}$ are precisely the fixed points of the semigroup K.
Oscillation and a class of linear delay differential equations
David Lowell
Lovelady
345-364
Abstract: The differential equation ${u^{(m)}}(t) + p(t)u(g(t)) = 0$. where P is one-signed, is broken into four cases, according to the parity of m and the sign of p. In each case, an analysis is given of the effect g can have on oscillation properties, and oscillation and nonoscillation criteria are given.
Zero-one laws and the minimum of a Markov process
P. W.
Millar
365-391
Abstract: If $\{ {X_t},t > 0\}$ is a real strong Markov process whose paths assume a (last) minimum at some time M strictly before the lifetime, then conditional on I, the value of this minimum, the process $\{ X(M + t),t > 0\}$ is shown to be Markov with stationary transitions which depend on I. For a wide class of Markov processes, including those obtained from Lévy processes via time change and multiplicative functional, a zero-one law is shown to hold at M in the sense that ${ \cap _{t > 0}}\sigma \{ X(M + s),s \leqslant t\} = \sigma \{ X(M)\}$, modulo null sets. When such a law holds, the evolution of $\{ X(M + t),t \geqslant 0\} $ depends on events before M only through $X(M)$ and I.
Addendum to: ``Some polynomials defined by generating relations'' (Trans. Amer. Math. Soc. {\bf 205} (1975), 360--370)
H. M.
Srivastava;
R. G.
Buschman
393-394
Abstract: Certain constraints are explicitly specified for the validity of a recent result involving a multivariate generating function, due to the present authors [1, p. 369, Theorem 6]. It is also indicated how this result can be further generalized. See Theorem ${6^\ast}$ below.
Normally flat deformations
Bruce
Bennett
1-57
Abstract: We study flat families $Z/T$, together with a section $\sigma :T \to Z$ such that the normal cone to the image of $\sigma$ in Z is flat over T. Such a family is called a ``normally flat deformation (along $ \sigma$)"; it corresponds intuitively to a deformation of a singularity which preserves the Hilbert-Samuel function. We construct the versal normally flat deformation of an isolated singularity (X,x) in terms of the flat strata of the relative jets of the ``usual'' versal deformation of X. We give explicit criteria, in terms of equations, for a flat family to be normally flat along a given section. These criteria are applied to demonstrate the smoothness of normally flat deformation theoryand of the canonical map from it to the cone deformation theory of the tangent cone-in the case of strict complete intersections. Finally we study the tangent space to the normally flat deformation theory, expressing it as the sum of two spaces: The first is a piece of a certain filtration of the tangent space to the usual deformation theory of X; the second is the tangent space to the special fibre of the canonical map $N \to S$, where N (resp. S) is the parameter space for the versal normally flat deformation of (X, x) (resp. for the versal deformation of X). We discuss the relation of this second space to infinitesimal properties of sections.
The classification of stunted projective spaces by stable homotopy type
S.
Feder;
S.
Gitler
59-81
Abstract: A complete classification of stable homotopy types of complex and quaternionic stunted projective spaces, denoted by ${\mathbf{C}}P_n^k$ and ${\mathbf{Q}}P_n^k$ respectively, is obtained. The necessary conditions for such equivalences are found using K-theory and various characteristic classes introduced originally by J. F. Adams. As a by-product one finds the J-orders of the Hopf bundles over $ {\mathbf{C}}{P^n}$ and $ {\mathbf{Q}}{P^n}$ respectively. The algebraic part is rather involved. Finally a homotopy theoretical argument yields the constructions of such homotopy equivalences as are allowed by the fulfillment of the necessary conditions.
On the bordism of almost free $Z\sb{2k}$ actions
R. Paul
Beem
83-105
Abstract: An ``almost free'' ${Z_{{2^k}}}$ action on a manifold is one in which only the included ${Z_2}$ may possibly fix points of the manifold. For k = 2, these are the stationary-point free actions. It is shown that almost free ${Z_{{2^k}}}$ bordism is generated by three subalgebras: the extension from ${Z_2}$ actions, a coset of ${Z_2}$ extensions being the restrictions of circle actions and a certain ideal of elements which annihilate the whole ring. The additive structure is determined. Free $ {Z_{{2^k}}}$ bordism is shown to split as an algebra. It is shown that the kernel of the extension homomorphism from $ {Z_2}$ to ${Z_{{2^k}}}$ bordism is equal to the image of the corresponding restriction homomorphism.
Free $S\sp{3}$-actions on $2$-connected nine-manifolds
Richard I.
Resch
107-112
Abstract: In this paper a classification of free ${S^3}$-actions on 2-connected 9-manifolds is obtained by examining the corresponding principal $ {S^3}$-bundles. The orbit spaces that may occur are determined and it is proved that there are exactly two homotopy classes of maps from each of these spaces into the classifying space for principal ${S^3}$-bundles. It is shown that the total spaces of the corresponding bundles are distinct, yielding the main result that for each nonnegative integer k, there exist exactly two 2-connected 9-manifolds which admit free ${S^3}$-actions and, furthermore, the actions on each of these manifolds are unique.
Strong differentiability properties of Bessel potentials
Daniel J.
Deignan;
William P.
Ziemer
113-122
Abstract: This paper is concerned with the ``strong'' ${L_p}$ differentiability properties of Bessel potentials of order $ \alpha > 0$ of $ {L_p}$ functions. Thus, for such a function f, we investigate the size (in the sense of an appropriate capacity) of the set of points x for which there is a polynomial $ {P_x}(y)$ of degree $k \leqslant \alpha$ such that $\displaystyle \mathop {\lim \sup }\limits_{{\text{diam}}(S) \to 0} \;{({\text{d... ...rt S{\vert^{ - 1}}\int {\vert f(y) - {P_x}(y){\vert^p}dy} } \right\}^{1/p}} = 0$ where, for example, S is allowed to run through the family of all oriented rectangles containing the origin.
On bounded univalent functions whose ranges contain a fixed disk
Roger W.
Barnard
123-144
Abstract: Let $\mathcal{S}$ denote the standard normalized class of regular, univalent functions in $K = {K_1} = \{ z:\vert z\vert < 1\}$. Let $\mathcal{F}$ be a given compact subclass of $\mathcal{S}$. We consider the following two problems. Problem 1. Find $ \max \vert{a_2}\vert$ for $f \in \mathcal{F}$. Problem 2. For $\vert z\vert = r < 1$, find the $\max \;(\min )\vert f(z)\vert$ for $f \in \mathcal{F}$. In this paper we are concerned with the subclass $\mathcal{S}_d^\ast(M) = \{ f \in \mathcal{S}:{K_d} \subset f(K) \subset {K_M}\} $. Through the use of the Julia variational formula and the Loewner theory we determine the extremal functions for Problems 1 and 2 for the class $ \mathcal{S}_d^\ast(M)$, for all d, M such that $\tfrac{1}{4} \leqslant d \leqslant 1 \leqslant M \leqslant \infty$.
A singular semilinear equation in $L\sp{1}({\bf R})$
Michael G.
Crandall;
Lawrence C.
Evans
145-153
Abstract: Let $\beta$ be a positive and nondecreasing function on R. The boundary-value problem $f \in {L^1}({\mathbf{R}})$. It is shown that this problem can have a solution only if $ \beta$ is integrable near $- \infty$, and that if this is the case, then the problem has a solution exactly when $\smallint _{ - \infty }^\infty f(x)dx > 0$.
Critical groups having central monoliths of a nilpotent by abelian product variety of groups
James J.
Woeppel
155-161
Abstract: Let $\mathfrak{N}$ be a variety of groups which has nilpotency class two and finite odd exponent. Let $\mathfrak{A}$ be an abelian variety of groups with finite exponent relatively prime to the exponent of $ \mathfrak{N}$. The existence in the product variety $\mathfrak{N}\mathfrak{A}$ of nonnilpotent critical groups having central monoliths is established. The structure of these critical groups is studied. This structure is shown to depend on an invariant, k. The join-irreducible subvariety of $\mathfrak{N}\mathfrak{A}$ generated by the nonnilpotent critical groups of $\mathfrak{N}\mathfrak{A}$ having central monoliths is determined, in particular, for k odd.
Homomorphisms of integral domains of characteristic zero
E.
Fried;
J.
Sichler
163-182
Abstract: Every category of universal algebras is isomorphic to a full subcategory of the category of all integral domains of characteristic zero and all their 1-preserving homomorphisms. Consequently, every monoid is isomorphic to the monoid of all 1-preserving endomorphisms of an integral domain of characteristic zero.
Balayage in Fourier transforms: general results, perturbation, and balayage with sparse frequencies
George S.
Shapiro
183-198
Abstract: Let $\Lambda$ be a discrete subset of an LCA group and E a compact subset of the dual group. Balayage is said to be possible for $(\Lambda ,E)$ if the Fourier transform of each measure on G is equal on E to the Fourier transform of some measure supported by $\Lambda$. Following Beurling, we show that this condition is equivalent to the possibility of bounding certain functions with spectra in E by their bounds on $\Lambda$. We derive consequences of this equivalence, among them a necessary condition on $\Lambda$ for balayage when E is compact and open (a condition analogous to a density condition Beurling and Landau gave for balayage in Euclidean spaces). We show that if balayage is possible for $(\Lambda ,E)$ and if $\Lambda '$ is close to $\Lambda$, then balayage is possible for $E \subset R$ with the property that there are ``arbitrarily sparse'' sets $\Lambda$ with balayage possible for $(\Lambda ,E)$.
Lebesgue summability of double trigonometric series
M. J.
Kohn
199-209
Abstract: We formulate a definition of symmetric derivatives of odd order for functions of two variables. Our definition is based on expanding in a Taylor's series a weighted average of the function about circles. The definition is applied to derive results on Lebesgue summability for spherically convergent double trigonometric series.
Completely unstable flows on $2$-manifolds
Dean A.
Neumann
211-226
Abstract: Completely unstable flows on 2-manifolds are classified under both topological and ${C^r}$-equivalence $(1 \leqslant r \leqslant \infty )$, in terms of the corresponding orbit spaces.
Stability in Witt rings
Thomas C.
Craven
227-242
Abstract: An abstract Witt ring R is defined to be a certain quotient of an integral group ring for a group of exponent 2. The ring R has a unique maximal ideal M containing 2. A variety of results are obtained concerning n-stability, the condition that ${M^{n + 1}} = 2{M^n}$, especially its relationship to the ring of continuous functions from the space of minimal prime ideals of R to the integers. For finite groups, a characterization of integral group rings is obtained in terms of n-stability. For Witt rings of formally real fields, conditions equivalent to n-stability are given in terms of the real places defined on the field.
On the sequence space $l\sb{(p\sb{n})}$ and $\lambda \sb{(p\sb{n})},$ $0<p\sb{n}\leq 1$
S. A.
Schonefeld;
W. J.
Stiles
243-257
Abstract: Let $({p_n})$ and $({q_n})$ be sequences in the interval $(0,1]$, let ${l_{({p_n})}}$ be the set of all real sequences $ ({x_n})$ such that $ \sum {\vert{x_n}{\vert^{{p_n}}} < \infty }$, and let ${\lambda _{({q_n})}}$ be the set of all real sequences $({y_n})$ such that $ {\sup _\pi }\sum {\vert{y_n}{\vert^{{q_{\pi (n)}}}} < \infty }$ where the sup is taken over all permutations $\pi$ of the positive integers. The purpose of this paper is to investigate some of the properties of these spaces. Our results are primarily concerned with (1) conditions which are necessary and/or sufficient for $ {l_{({p_n})}}$ (resp., $ {\lambda _{({p_n})}}$) to equal $ {l_{({q_n})}}$ (resp., $ {\lambda _{({q_n})}}$), and (2) isomorphic and topological properties of the subspaces of these spaces. In connection with (1), we show that the following four conditions are equivalent for any sequence $({\varepsilon _n})$ which decreases to zero and has ${\varepsilon _1} < 1$. (a) There exists a number $K > 1$ such that the series $\sum {1/{K^{1/{\varepsilon _n}}}}$ converges; (b) the elements $ {\varepsilon _n}$ of the sequence satisfy the condition ${\varepsilon _n} = O(1/\ln n)$; (c) the sequence $((\ln n)((1/n)\sum\nolimits_1^n {{\varepsilon _j}} ))$ is bounded; and (d) ${l_{(1 - {\varepsilon _n})}}$ equals ${l_1}$. In connection with (2), we show that the following are true when $({p_n})$ increases to one. (a) ${\lambda _{({p_n})}}$ contains an infinite-dimensional closed subspace where the ${l_{({p_n})}}$-topology and the ${\lambda _{({p_n})}}$-topology agree; (b) $ {l_{({p_n})}}$ and ${\lambda _{({p_n})}}$ contain closed subspaces isomorphic to ${l_1}$; and (c) ${\lambda _{({p_n})}}$ contains no infinite-dimensional subspace where the ${\lambda _{({p_n})}}$-topology agrees with the ${l_1}$-topology if and only if $\displaystyle \lim ({(1/n)^{{p_1}}} + {(1/n)^{{p_2}}} + \cdots + {(1/n)^{{p_n}}}) = \infty .$
Weierstrass normal forms and invariants of elliptic surfaces
Arnold
Kas
259-266
Abstract: Let $\pi :S \to B$ be an elliptic surface with a section $\sigma :B \to S$. Let ${L^{ - 1}} \to B$ be the normal bundle of $\sigma (B)$ in S, and let $W = P({L^{ \otimes 2}} \oplus {L^{ \otimes 3}} \oplus 1)$ be a ${{\mathbf{P}}^2}$-bundle over B. Let $ {S^\ast}$ be the surface obtained from S by contracting those components of fibres of S which do not intersect $\sigma (B)$. Then ${S^\ast}$ may be imbedded in W and defined by a ``Weierstrass equation": $\displaystyle {y^2}z = {x^3} - {g_2}x{z^2} - {g_3}{z^3}$ where ${g_2} \in {H^0}(B,\mathcal{O}({L^{ \otimes 4}}))$ and ${g_3} \in {H^0}(B,\mathcal{O}({L^{ \otimes 6}}))$. The only singularities (if any) of $ {S^\ast}$ are rational double points. The triples $ (L,{g_2},{g_3})$ form a set of invariants for elliptic surfaces with sections, and a complete set of invariants is given by $\{ (L,{g_2},{g_3})\} /G$ where $G \cong {{\mathbf{C}}^\ast} \times {\operatorname{Aut}}\;(B)$.
On subcategories of TOP
S. P.
Franklin;
D. J.
Lutzer;
B. V. S.
Thomas
267-278
Abstract: A categorical characterization of a subcategory S of TOP (or $ {T_2}$) is one which enables the identification of S in TOP (or $ {T_2}$) without requiring the reconstruction of the topological structure of its objects. In this paper we so characterize various familiar subcategories of TOP (Hausdorff spaces, normal spaces, compact Hausdorff spaces, paracompact Hausdorff spaces, metrizable spaces, first countable spaces) in terms of the global behavior of the (objects and) morphisms of the subcategory.
Sums of solid $n$-spheres
Lois M.
Broussard
279-294
Abstract: We prove that the sum of two solid Antoine n-spheres $(n \geqslant 3)$ by the identity on the boundary is homeomorphic to the n-sphere $ {S^n}$.
Cluster values of bounded analytic functions
T. W.
Gamelin
295-306
Abstract: Let D be a bounded domain in the complex plane, and let $ \zeta$ belong to the topological boundary $ \partial D$ of D. We prove two theorems concerning the cluster set $ {\text{Cl}}(f,\zeta )$ of a bounded analytic function f on D. The first theorem asserts that values in ${\text{Cl}}(f,\zeta )\backslash f({\mathcyr{SH}_\zeta })$ are assumed infinitely often in every neighborhood of $\zeta$, with the exception of those lying in a set of zero analytic capacity. The second asserts that all values in $ {\text{Cl}}(f,\zeta )\backslash f({\mathfrak{M}_\zeta } \cap {\text{supp}}\;\lambda )$ are assumed infinitely often in every neighborhood of $\zeta$, with the exception of those lying in a set of zero logarithmic capacity. Here ${\mathfrak{M}_\zeta }$ is the fiber of the maximal ideal space $ \mathfrak{M}(D)$ of ${H^\infty }(D)$ lying over $\zeta$, ${\mathcyr{SH}_\zeta }$ is the Shilov boundary of the fiber algebra, and $\lambda$ is the harmonic measure on $\mathfrak{M}(D)$.
A general extremal problem for the class of close-to-convex functions
John G.
Milcetich
307-323
Abstract: For $\beta \geqslant 0,{K_\beta }$ denotes the set of functions $f(z) = z + {a_2}{z^2} + \cdots$ defined on the unit disc U with the representation $0 \leqslant \beta \leqslant 1$, and $\zeta \in U$, let $F(u,v)$ be analytic in a neighborhood of $ \{ (f(\zeta ),\zeta ):f \in {K_\beta }\}$. Then $ \max \{ \operatorname{Re} F(f(\zeta ),\zeta ):f \in {K_\beta }\}$ occurs for a function of the form $\displaystyle f(z) = {(\beta + 1)^{ - 1}}{(x - y)^{ - 1}}[{(1 + xz)^{\beta + 1}}{(1 + yz)^{ - \beta - 1}} - 1],$ where $\vert x\vert = \vert y\vert = 1$ and $ x \ne y$. If $0 < \beta < 1$ these are the only extremal functions. A consequence of this result is the determination of the value region $\{ f(\zeta )/\zeta :f \in {K_\beta }\}$ as $\{ {(\beta + 1)^{ - 1}}{(s - t)^{ - 1}}[{(1 + s)^{\beta + 1}}{(1 + t)^{ - \beta - 1}} - 1]:\vert s\vert,\vert t\vert \leqslant \vert\zeta \vert\}$.
End extensions, conservative extensions, and the Rudin-Frol\'\i k ordering
Andreas
Blass
325-340
Abstract: The ordering of ultrafilters on the natural numbers defined by ``E-prod N is an end extension of D-prod N,'' the ordering defined by ``E-prod N is a conservative extension of D-prod N,'' and the Rudin-Frolik ordering are proved to be distinct if the continuum hypothesis holds. These three orderings are also characterized in terms of (not necessarily internal) ultrafilters in the Boolean algebra of internal sets of natural numbers in a nonstandard universe.
Spectral geometry of symmetric spaces
Peter B.
Gilkey
341-353
Abstract: Let M be a compact Riemannian manifold without boundary. Let D be a differential operator on M. Let spec (D, M) denote the eigenvalues of D repeated according to multiplicity. Several authors have studied the extent to which the geometry of M is reflected by spec (D, M) for certain natural operators D. We consider operators D which are convex combinations of the ordinary Laplacian and the Bochner or reduced Laplacian acting on the space of smooth functions and the space of smooth one forms. We prove that is is possible to determine if M is a local symmetric space from its spectrum. If the Ricci tensor is parallel transported, the eigenvalues of the Ricci tensor are spectral invariants of M.
Limit theorems for convolution iterates of a probability measure on completely simple or compact semigroups
A.
Mukherjea
355-370
Abstract: This paper extends the study (initiated by M. Rosenblatt) of the asymptotic behavior of the convolution sequence of a probability measure on compact or completely simple semigroups. Let S be a locally compact second countable Hausdorff topological semigroup. Let $ \mu$ be a regular probability measure on the Borel subsets of S such that S does not have a proper closed subsemigroup containing the support F of $\mu$. It is shown in this paper that when S is completely simple with its usual product representation $X \times G \times Y$, then the convolution sequence ${\mu ^n}$ converges to zero vaguely if and only if the group factor G is noncompact. When the group factor G is compact, ${\mu ^n}$ converges weakly if and only if $ {\underline {\lim } _{n \to \infty }}{F^n}$ is nonempty. This last result remains true for an arbitrary compact semigroup S generated by F. Furthermore, we show that in this case there exist elements ${a_n} \in S$ such that ${\mu ^n} \ast {\delta _{{a_n}}}$ converges weakly, where $ {\delta _{{a_n}}}$ is the point mass at ${a_n}$. This result cannot be extended to the locally compact case, even when S is a group.
Generic Fr\'echet-differentiability and perturbed optimization problems in Banach spaces
Ivar
Ekeland;
Gérard
Lebourg
193-216
Abstract: We define a function F on a Banach space V to be locally $\varepsilon$-supported by ${u^\ast} \in {V^\ast}$ at $u \in V$ if there exists an $\eta > 0$ such that $\left\Vert {v - u} \right\Vert \leqslant \eta \Rightarrow F(v) \geqslant F(u) + \langle {u^\ast},v - u\rangle - \varepsilon \left\Vert {v - u} \right\Vert$. We prove that if the Banach space V admits a nonnegative Fréchet-differentiable function with bounded nonempty support, then, for any $> 0$ and every lower semicontinuous function F, there is a dense set of points $u \in V$ at which F is locally $\varepsilon$-supported. The applications are twofold. First, to the study of functions defined as pointwise infima; we prove for instance that every concave continuous function defined on a Banach space with Fréchet-differentiable norm is Fréchet-differentiable generically (i.e. on a countable intersection of open dense subsets). Then, to the study of optimization problems depending on a parameter $u \in V$; we give general conditions, mainly in the framework of uniformly convex Banach spaces with uniformly convex dual, under which such problems generically have a single optimal solution, depending continuously on the parameter and satisfying a first-order necessary condition.
A decomposition of weighted translation operators
Joseph J.
Bastian
217-230
Abstract: Let (X, A, m) be a measure space and T an invertible measure-preserving transformation on X. Given $\phi$ in $ {L^\infty }(X)$, define operators ${M_\phi }$ and ${U_T}$ on ${L^2}(X)$ by $({M_\phi }f)(x) = \phi (x)f(x)$ and $({U_T}f) = f(Tx)$. Operators of the form ${M_\phi }{U_T}$ are called weighted translation operators. In this paper it is shown that every weighted translation operator on a sufficiently regular measure space an be decomposed into a direct integral of weighted translation operators where almost all of the transformations in the integrand are ergodic. It is also shown that every hyponormal weighted translation operator defined by an ergodic transformation is either normal or unitarily equivalent to a bilateral weighted shift. These two results along with some results concerning direct integrals of hyponormal and subnormal operators are used to show that every hyponormal (resp. subnormal) weighted translation operator is unitarily equivalent to a direct integral of normal operators and hyponormal (resp. subnormal) bilateral weighted shifts. The paper concludes with an example.
Absolute Tauberian constants for Ces\`aro means of a function
Soraya
Sherif
231-242
Abstract: This paper is concerned with introducing two estimates of the forms $F \leqslant C{A_k}(\alpha ),F \leqslant D{B_k}(\alpha ),(\alpha > 0)$, where $F = \smallint_0^\infty {\vert d\{ f(\alpha x) - {\sigma _k}(x)\} \vert,{\sigma _k}(x)}$ denote the Cesàro transform of order k of the function $ f(x) = \smallint_0^x {g(t)\;dt,g(t)}$ is a function of bounded variation in every finite interval of $t \geqslant 0,{A_k}(\alpha ),{B_k}(\alpha )$ are absolute Tauberian constants, $C = \smallint_0^\infty {\vert d\{ tg(t)\} \vert < \infty ,D = \smallint_0^\infty {\vert d\{ \phi (t)\} \vert < \infty } }$ and $\phi (t) = {t^{ - 1}}\smallint_0^t {ug(u)du}$. The constants ${A_k}(\alpha ),{B_k}(\alpha )$ will be determined.
Fringe families in stable homotopy
Raphael S.
Zahler
243-253
Abstract: It is shown how to detect and construct elements in the stable homotopy groups of spheres corresponding to the $ {_i}$ family of Toda. The only tools used are Brown-Peterson cohomology and the Adams spectral sequence.
Coefficient multipliers of Bloch functions
J. M.
Anderson;
A. L.
Shields
255-265
Abstract: The class $\mathcal{B}$ of Bloch functions is the class of all those analytic functions in the open unit disc for which the maximum modulus is bounded by $c/(1 - r)$ on $\vert z\vert \leqslant r$. We study the absolute values of the Taylor coefficients of such functions. In particular, we find all coefficient multipliers from ${l^p}$ into $ \mathcal{B}$ and from $\mathcal{B}$ into ${l^p}$. We find the second Köthe dual of $\mathcal{B}$ and show its relevance to the multiplier problem. We identify all power series $\sum {a_n}{z^n}$ such that $\sum {w_n}{a_n}{z^n}$ is a Bloch function for every choice of the bounded sequence $\{ {w_n}\}$. Analogous problems for ${H^p}$ spaces are discussed briefly.
Unique factorization in modules and symmetric algebras
Douglas L.
Costa
267-280
Abstract: Necessary and sufficient conditions are given for a torsion-free module M over a UFD D to admit a smallest factorial module containing it. This factorial hull is $\cap {M_P}$, the intersection taken over all height one primes of D. In case M is finitely generated, the hull is ${M^{ \ast \ast }}$, the bidual of M. It is shown that if the symmetric algebra $ {S_D}(M)$ admits a hull, then the hull is the smallest graded UFD containing $ {S_D}(M)$. ${S_D}(M)$ is a UFD if and only if it is a factorial D-module. If M is finitely generated over D, but not necessarily torsion-free, then ${ \oplus _{i \geqslant 0}}{({S^i}(M))^{ \ast \ast }}$ is a graded UFD. Examples are given to show that any finite number of symmetric powers of M may be factorial without ${S_D}(M)$ being factorial.
A minimax formula for dual $B\sp*$-algebras
Pak Ken
Wong
281-298
Abstract: Let A be a dual ${B^\ast}$-algebra. We give a minimax formula for the positive elements in A. By using this formula and some of its consequent results, we introduce and study the symmetric norms and symmetrically-normed ideals in A.
The Cartesian product structure and $C\sp{\infty }$ equivalances of singularities
Robert
Ephraim
299-311
Abstract: In this paper the cartesian product structure of complex analytic singularities is studied. A singularity is called indecomposable if it cannot be written as the cartesian product of two singularities of lower dimension. It is shown that there is an essentially unique way to write any reduced irreducible singularity as a cartesian product of indecomposable singularities. This result is applied to give an explicit description of the set of reduced irreducible complex singularities having a given underlying real analytic structure.
Duality theory for locally compact groups with precompact conjugacy classes. II. The dual space
Terje
Sund
313-321
Abstract: The present paper is concerned with the dual space Ĝ consisting of all unitary equivalence classes of continuous irreducible unitary representations of separable ${[FC]^ - }$ groups (i.e. groups with precompact conjugacy classes). The main purpose of the paper is to extend certain results from the duality theory of abelian groups and [Z] groups to the larger class of ${[FC]^ - }$ groups. In addition, we deal briefly with square-integrability for representations of ${[FC]^ - }$ groups. Most of our results are proved for type I groups. Our key result is that Ĝ may be written as a disjoint union of abelian topological $ {T_4}$ groups, which are open in Ĝ.
Continuously perfectly normal spaces and some generalizations
Gary
Gruenhage
323-338
Abstract: In this work we continue the study of continuously perfectly normal, continuously normal, and continuously completely regular spaces which was begun by Phillip Zenor. Among other results, we prove that separable continuously completely regular spaces are metrizable, and provide an example of a nonmetrizable continuously perfectly normal space.
Indecomposable homogeneous plane continua are hereditarily indecomposable
Charles L.
Hagopian
339-350
Abstract: F. Burton Jones [7] proved that every decomposable homogeneous plane continuum is either a simple closed curve or a circle of homogeneous nonseparating plane continua. Recently the author [5] showed that no subcontinuum of an indecomposable homogeneous plane continuum is hereditarily decomposable. It follows from these results that every homogeneous plane continuum that has a hereditarily decomposable subcontinuum is a simple closed curve. In this paper we prove that no subcontinuum of an indecomposable homogeneous plane continuum is decomposable. Consequently every homogeneous nonseparating plane continuum is hereditarily indecomposable. Parts of our proof follow one of R. H. Bing's arguments [2]. At the Auburn Topology Conference in 1969, Professor Jones [8] outlined an argument for this theorem and stated that the details would be supplied later. However, those details have not appeared.
A reduction theory for non-self-adjoint operator algebras
E. A.
Azoff;
C. K.
Fong;
F.
Gilfeather
351-366
Abstract: It is shown that every strongly closed algebra of operators acting on a separable Hilbert space can be expressed as a direct integral of irreducible algebras. In particular, every reductive algebra is the direct integral of transitive algebras. This decomposition is used to study the relationship between the transitive and reductive algebra problems. The final section of the paper shows how to view direct integrals of algebras as measurable algebra-valued functions.
Analysis with weak trace ideals and the number of bound states of Schr\"odinger operators
Barry
Simon
367-380
Abstract: We discuss interpolation theory for the operator ideals $ I_p^w$ defined on a separable Hilbert space as those operators A whose singular values ${\mu _n}(A)$ obey ${\mu _n} \leqslant c{n^{ - 1/p}}$ for some c. As an application we consider the functional $N(V) = \dim$ (spectral projection on $( - \infty ,0)$ for $ - \Delta + V$) on functions V on ${{\mathbf{R}}^n},n \geqslant 3$. We prove that for any $ \epsilon > 0:N(V) \leqslant C_\epsilon (\left\Vert V \right\Vert _{n/2 + \epsilon } + \left\Vert V \right\Vert _{n/2 - \epsilon})^{n/2}$ where ${\left\Vert \cdot \right\Vert _p}$ is an $ {L^p}$ norm and that $ {\lim\nolimits _{\lambda \to \infty }}N(\lambda V)/{\lambda ^{n/2}} = {(2\pi )^{ - n}}{\tau _n}\smallint \vert{V_ - }(x){\vert^{n/2}}{d^n}x$ for any $V \in {L^{n/2 - }} \cap {L^{n/2 + }}$. Here $ {V_ - }$ is the negative part of V and ${\tau _n}$ is the volume of the unit ball in ${{\mathbf{R}}^n}$.
Natural limits for harmonic and superharmonic functions
J. R.
Diederich
381-397
Abstract: In this paper it is shown that Fatou's theorem holds for superharmonic functions in certain Liapunov domains if mean continuous limits are used in place of nontangential limits for which Fatou's theorem fails. Also, existence of mean continuous limits is established for certain semi-linear elliptic equations in Liapunov domains.
Nonimmersion of lens spaces with 2-torsion
A. J.
Berrick
399-405
Abstract: From a study of the equivariant unitary K-theory of the Stiefel manifold $ {V_{k + 1,2}}({\mathbf{C}})$, it is shown that the lens space ${L^k}(n)$, with n a multiple of $ {2^{2k - 1 - \alpha (k - 1)}}$, does not immerse in Euclidean space of dimension $4k - 2\alpha (k) - 2$.
Quasi-similar models for nilpotent operators
C.
Apostol;
R. G.
Douglas;
C.
Foiaş
407-415
Abstract: Every nilpotent operator on a complex Hilbert space is shown to be quasi-similar to a canonical Jordan model. Further, the para-reflexive operators are characterized generalizing a result of Deddens and Fillmore.
The degree of approximation for generalized polynomials with integral coefficients
M.
von Golitschek
417-425
Abstract: The classcal Müntz theorem and the so-called Jackson-Müntz theorems concern uniform approximation on [0, 1] by polynomials whose exponents are taken from an increasing sequence of positive real numbers $\Lambda$. Under mild restrictions on the exponents, the degree of approximation for $ \Lambda$-polynomials with real coefficients is compared with the corresponding degree of approximation when the coefficients are taken from the integers.
Deformation of open embeddings of $Q$-manifolds
A.
Fathi;
Y. M.
Visetti
427-435
Abstract: We prove here for Hilbert cube manifolds the full analogue of the Černavski-Edwards-Kirby Theorem concerning the deformation Principle for open embeddings of topological manifolds.
Some consequences of the algebraic nature of $p(e\sp{i\theta })$
J. R.
Quine
437-442
Abstract: For polynomial p of degree n, the curve $p({e^{i\theta }})$ is a closed curve in the complex plane. We show that the image of this curve is a subset of an algebraic curve of degree 2n. Using Bézout's theorem and taking into account imaginary intersections at infinity, we show that if p and q are polynomials of degree m and n respectively, then the curves $p({e^{i\theta }})$ and $q({e^{i\theta }})$ intersect at most 2mn times. Finally, let ${U_k}$ be the set of points w, not on $p({e^{i\theta }})$, such that $ p(z) - w$ has exactly k roots in $\vert z\vert < 1$. We prove that if L is a line then $L \cap {U_k}$ has at most $n - k + 1$ components in L and in particular ${U_n}$ is convex.
A Plancherel formula for idyllic nilpotent Lie groups
Eloise
Carlton
1-42
Abstract: A procedure is developed which can be used to compute the Plancherel measure for a certain class of nilpotent Lie groups, including the Heisenberg groups, free groups, two-and three-step groups, the nilpotent part of an Iwasawa decomposition of the R-split form of the classical simple groups $ {A_l},{C_l},{G_2}$. Let G be a connected, simply connected nilpotent Lie group. The Plancherel formula for G can be expressed in terms of Plancherel measure of a normal subgroup N and projective Plancherel measures of certain subgroups of $G/N$. To get an explicit measure for G, we need an explicit formula for (1) the disintegration of Plancherel measure of N under the action of G on N, and (2) projective Plancherel measures of ${G_\gamma }/N$, where ${G_\gamma }$ is the stability subgroup at $ \gamma$ in N. When both N and $ {G_\gamma }/N$ are abelian, the measures (1) and (2) are obtained as special cases of more general problems. These measures combine into Plancherel measure for G.
Parametric perturbation problems in ordinary differential equations
Thomas G.
Hallam
43-59
Abstract: The asymptotic behavior of solutions of a nonlinear differential equation that arises through a nonlinear parametric perturbation of a linear system of differential equations is discussed. Fundamental hypotheses include the admissibility of a pair of Banach spaces for the linear system. Conclusions about the behavior of the perturbed system evolve through the behavior of certain manifolds of solutions of the unperturbed linear system. Asymptotic representations are found on a semi-infinite axis ${R_ + }$ and on the real line R. The bifurcation condition, which is shown to be trivial on $ {R_ + }$, plays an essential role for the perturbation problem on R. Illustrations and examples, primarily on the space $ {{\text{L}}^\infty }$, of the theoretical results are presented.
Nonregular ultrafilters and large cardinals
Jussi
Ketonen
61-73
Abstract: The relationship between the existence of nonregular ultrafilters and large cardinals in the constructible universe is studied.
Lie algebras of type $BC\sb{1}$
B. N.
Allison
75-86
Abstract: Let L be a central simple Lie algebra of type $B{C_1}$ with highest root space of dimension greater than one over a field of characteristic zero. It is shown that either L is isomorphic to the simple Lie algebra associated with a skew hermitian form of index one or L can be constructed from the tensor product of two composition algebras. This result is obtained by completing the description (begun in [3]) of the corresponding class of ternary algebras.
Inner product modules arising from compact automorphism groups of von Neumann algebras
William L.
Paschke
87-102
Abstract: Let M be a von Neumann algebra of operators on a separable Hilbert space H, and G a compact, strong-operator continuous group of $^\ast$-automorphisms of M. The action of G on M gives rise to a faithful, ultraweakly continuous conditional expectation of M on the subalgebra $N = \{ A \in M:g(A) = A\forall g \in G\}$, which in turn makes M into an inner product module over N. The inner product module M may be ``completed'' to yield a self-dual inner product module $\bar M$ over N; our most general result states that the ${W^\ast}$-algebra $A(\bar M)$ of bounded N-module maps of $ \bar M$ into itself is isomorphic to a restriction of the crossed product $M \times G$ of M by G. When G is compact abelian, we give conditions for $ A(\bar M)$ and $M \times G$ to be isomorphic and show, among other things, that if G acts faithfully on M, then $M \times G$ is a factor if and only if N is a factor. As an example, we discuss certain compact abelian automorphism groups of group von Neumann algebras.
Maximal chains of prime ideals in integral extension domains. I
L. J.
Ratliff;
S.
McAdam
103-116
Abstract: Let (R, M) be a local domain, let k be a positive integer, and let Q be a prime ideal in ${R_k} = R[{X_1}, \ldots ,{X_k}]$ such that $M{R_k} \subset Q$. Then the following statements are equivalent: (1) There exists an integral extension domain of R which has a maximal chain of prime ideals of length n. (2) There exists a minimal prime ideal z in the completion of R such that depth $z = n$. (3) There exists a minimal prime ideal w in the completion of $ {({R_k})_Q}$ such that depth $w = n + k - {\text{depth}}\;Q$. (4) There exists an integral extension domain of ${({R_k})_Q}$ which has a maximal chain of prime ideals of length $n + k - {\text{depth}}\;Q$. (5) There exists a maximal chain of prime ideals of length $n + k - {\text{depth}}\;Q$ in ${({R_k})_Q}$. (6) There exists a maximal chain of prime ideals of length $n + 1$ in $ R{[{X_1}]_{(M,{X_1})}}$.
Maximal chains of prime ideals in integral extension domains. II
L. J.
Ratliff
117-141
Abstract: Four related subjects are investigated: (1) If (L, N) is a locality over a local domain (R, M) such that $N \cap R = M$, and if there exists an integral extension domain of L which has a maximal chain of prime ideals of length n (for short, a mcpil n), then there exists an integral extension domain of R which has a mcpil $n - {\text{trd}}\;L/R + {\text{trd}}(L/N)/(R/M)$. A refinement of the altitude inequality follows from this. (2) A condition for the converse of (1) to hold is given. (3) The class of local domains R such that there exists an integral extension domain of R which has a mcpil n if and only if there exists a mcpil n in R is studied. (4) Two new equivalences for the existence of mcpil n in an integral extension domain of a local domain are given.
The spectral theory of posets and its applications to $C\sp*$-algebras
A. H.
Dooley
143-155
Abstract: This paper uses methods from the spectral theory of partially ordered sets to clarify and extend some recent results concerning approximately finite-dimensional $ {C^\ast}$-algebras. An extremely explicit description is obtained of the Jacobson topology on the primitive ideal space, and it is shown that this topology has a basis of quasi-compact open sets. In addition, the main results of [4] are proved using only elementary means.
Local Fatou theorem and area theorem for symmetric spaces of rank one
A.
Korányi;
R. B.
Putz
157-168
Abstract: The classical results for the unit disc mentioned in the title are extended to harmonic functions on symmetric spaces of rank one.
$Z\sb{p}$actions on symplectic manifolds
R. J.
Rowlett
169-177
Abstract: A bordism classification is studied for periodic maps of prime period p preserving a symplectic structure on a smooth manifold. In sharp contrast to the corresponding oriented bordism, this theory contains nontrivial ptorsion even when p is odd. Calculation gives an upper limit on the size of this p-torsion.
Partially ordered linear algebras with multiplicative diagonal map
Taen Yu
Dai;
Ralph
DeMarr
179-187
Abstract: The diagonal of the product of two triangular matrices is the product of the diagonals of each matrix. This idea is used to characterize partially ordered linear algebras which have order properties similar to an algebra of real triangular matrices.
The primitive lifting problem in the equivalence problem for transitive pseudogroup structures: a counterexample
Pierre
Molino
189-192
Abstract: A transitive Lie pseudogroup ${\Gamma _M}$ on M is a primitive extension of $ {\Gamma _N}$ if ${\Gamma _N}$ is the quotient of ${\Gamma _M}$ by an invariant fibration $\pi :M \to N$ and if the pseudogroup induced by ${\Gamma _M}$ on the fiber of $\pi$ is primitive. In the present paper an example of this situation is given with the following property (counterexample to the primitive lifting property): the equivalence theorem is true for almost- ${\Gamma _N}$-structures but false for almost- ${\Gamma _M}$-structures.
An application of functional operator models to dissipative scattering theory
Daniel Alan
Bondy
1-43
Abstract: An abstract framework for dissipative scattering theory is developed and then applied to two systems previously considered by P. Lax and R. S. Phillips. Results relating the poles and zeroes of the scattering matrix to the spectra of the infinitesimal generators A (which generates the semigroup formed by mapping initial data into solution data at time t) and B (which generates a ``local'' semigroup) are proven. In particular these results are shown to follow from the fact that the characteristic function of A (appropriately defined) and the scattering matrix combine to form the characteristic function of B.
Sets definable over finite fields: their zeta-functions
Catarina
Kiefe
45-59
Abstract: Sets definable over finite fields are introduced. The rationality of the logarithmic derivative of their zeta-function is established, an application of purely algebraic content is given. The ingredients used are a result of Dwork on algebraic varieties over finite fields and model-theoretic tools.
Consistency results concerning supercompactness
Telis K.
Menas
61-91
Abstract: A general framework for proving relative consistency results with regard to supercompactness is developed. Within this framework we prove the relative consistency of the assertion that every set is ordinal definable with the statement asserting the existence of a supercompact cardinal. We also generalize Easton's theorem; the new element in our result is that our forcing conditions preserve supercompactness.
On almost bounded functions
Ruth
Miniowitz
93-102
Abstract: New results are presented with regard to the ``almost bounded functions'' introduced by Goodman [2], including a theorem which contains a proof of Goodman's conjecture for a particular case.
The Remez exchange algorithm for approximation with linear restrictions
Bruce L.
Chalmers
103-131
Abstract: This paper demonstrates a Remez exchange algorithm applicable to approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions. As special cases are included the notion of restricted derivatives approximation (examples of which are monotone and convex approximation and restricted range approximation) and the notion of approximation with restrictions at poised Birkhoff data (examples of which are bounded coefficients approximation, $\varepsilon $-interpolator approximation, and polynomial approximation with restrictions at Hermite and ``Ferguson-Atkinson-Sharma'' data and pyramid matrix data). Furthermore the exchange procedure is completely simplified in all the cases of approximation with restrictions at poised Birkhoff data. Also results are obtained in the cases of general linear restrictions where the Haar condition prevails. In the other cases (e.g., monotone approximation) the exchange in general requires essentially a matrix inversion, although insight into the exchange is provided and partial alternation results are obtained which lead to simplifications.
Hypoelliptic convolution equations in $K'_p$, $p>1$
G.
Sampson;
Z.
Zieleźny
133-154
Abstract: We consider convolution equations in the space $\exp (k\vert x{\vert^p})$ for some constant k. Our main result is a complete characterization of hypoelliptic convolution operators in ${K'_p}$ in terms of their Fourier transforms.
Systems-conjugate and focal points of fourth order nonselfadjoint differential equations
Sui Sun
Cheng
155-165
Abstract: Systems-conjugate points have been studied by John Barrett in relation to selfadjoint fourth order differential equations of the form $({p_2}u'')'' + {p_0}u = 0$. This paper extends his results to the general nonselfadjoint fourth order differential equation via a system of second order equations.
Comparison of eigenvalues associated with linear differential equations of arbitrary order
R. D.
Gentry;
C. C.
Travis
167-179
Abstract: Existence and comparison theorems for eigenvalues of $ (k,n - k)$-focal point and $ (k,n - k)$-conjugate point problems are proved for a class of nth order linear differential equations for arbitrary n.
Sufficient conditions for an operator-valued Feynman-Kac formula
Michael Dale
Grady
181-203
Abstract: Let E be a locally compact, second countable Hausdorff space and let $X(t)$ be a Markov process with state space E. Sufficient conditions are given for the existence of a solution to the initial value problem, $ \partial u/\partial t = Au + V(x) \cdot u,u(0) = f$, where A is the infinitesimal generator of the process X on a certain Banach space and for each $x \in E,V(x)$ is the infinitesimal generator of a $ {C_0}$ contraction semigroup on another Banach space.
On the local stability of differential forms
Martin
Golubitsky;
David
Tischler
205-221
Abstract: In this paper we determine which germs of differential s-forms on an n-manifold are stable (in the sense of Martinet). We show that when $s \ne 1$ or when $s = 1$ and $n \leqslant 4$ Martinet had found almost all of the possible examples. The most interesting result states that for certain generic singularities of 1-forms on 4-manifolds an infinite dimensional moduli space occurs in the classification of the 1-forms with this given singularity type up to equivalence by pull-back via a diffeomorphism.
Periodic homeomorphisms of $3$-manifolds fibered over $S\sp{1}$
Jeffrey L.
Tollefson
223-234
Abstract: Two problems concerning periodic homeomorphisms of 3-manifolds are considered. The first is that of obtaining systems of incompressible surfaces invariant under a given involution. The second problem is the realization by a periodic homeomorphism of an element of finite order in the mapping class group of a 3-manifold. Solutions to both problems are obtained in certain instances.
Hilbert transforms associated with plane curves
Alexander
Nagel;
Stephen
Wainger
235-252
Abstract: Let $(t,\gamma (t))$ be a plane curve. Set ${H_\gamma }f(x,y) =$ p.v.$ \;\smallint f(x - t,y - \gamma (t))dt/t$ for $f \in C_0^\infty ({R^2})$. For a large class of curves, the authors prove ${\left\Vert {{H_\gamma }f} \right\Vert _p} \leqslant {A_p}{\left\Vert f \right\Vert _p},5/3 < p < 5/2$. Various examples are given to show that some condition on the curve $ (t,\gamma (t))$ is necessary.
Smith theory for$p$-groups
James A.
Maiorana
253-266
Abstract: When a p-group G acts on a manifold, the behavior of the cohomology of the subgroups of G singles out a special collection of fixed point sets of these subgroups. A bound on the size of the spaces in this collection is derived using equivariant cohomology. For a special class of nonabelian p-groups this bound is strong enough to require that certain fixed point sets must vanish. Application of this bound to a linear representation of G yields a lower bound for the cohomology of G.
A quasi-Anosov diffeomorphism that is not Anosov
John
Franks;
Clark
Robinson
267-278
Abstract: In this note, we give an example of a diffeomorphism f on a three dimensional manifold M such that f has a property called quasi- Anosov but such that f does not have a hyperbolic structure (is not Anosov). Mañé has given a method of extending f to a diffeomorphism g on a larger dimensional manifold V such that g has a hyperbolic structure on M as a subset of V. This gives a counterexample to a question of M. Hirsch.
A Banach algebra of functions with bounded $n$th differences
John T.
Daly;
Philip B.
Downum
279-294
Abstract: Several characterizations are given for the Banach algebra of $ (n - 1)$-times continuously differentiable functions whose $(n - 1)$st derivative satisfies a bounded Lipschitz condition. The structure of the closed primary ideals is investigated and spectral synthesis is shown to be satisfied.
On bounded functions satisfying averaging conditions. II
Rotraut Goubau
Cahill
295-304
Abstract: Let $S(f)$ denote the subspace of ${L^\infty }({R^n})$ consisting of those real valued functions f for which $\displaystyle \mathop {\lim }\limits_{r \to 0} \frac{1}{{\vert B(x,r)\vert}} {\int} _{B(x,r)}f(y)dy = f(x)$ for all x in ${R^n}$ and let $L(f)$ be the subspace of $S(f)$ consisting of the approximately continuous functions. A number of results concerning the existence of functions in $S(f)$ and $L(f)$ with special properties are obtained. The extreme points of the unit balls of both spaces are characterized and it is shown that $L(f)$ is not a dual space. As a preliminary step, it is shown that if E is a ${G_\delta }$ set of measure 0 in ${R^n}$, then the complement of E can be decomposed into a collection of closed sets in a particularly useful way.
Boundary value problems for functional differential equations with $L\sp{2}$ initial functions
G. W.
Reddien;
G. F.
Webb
305-321
Abstract: Existence results are given for boundary value problems for vector systems of functional differential equations with $ {L^2}$ initial functions. The proofs are essentially constructive and lead to computational methods in important cases.
On nullity distributions
Sin Leng
Tan
323-335
Abstract: The nullity concept of Riemannian manifolds is extended to affine manifolds. Results obtained by Chern and Kuiper and Maltz on Riemannian manifolds are generalized to affine manifolds. A structure theorem for affine symmetric spaces is obtained. Finally, the nullity concept is generalized to study the partial integrability of certain geometric structures.
On the second Hankel determinant of areally mean $p$-valent functions
J. W.
Noonan;
D. K.
Thomas
337-346
Abstract: In this paper we determine the growth rate of the second Hankel determinant of an areally mean p-valent function. This result both extends and unifies previously known results concerning this problem.
Existence theorems for parametric problems in the calculus of variations and approximation
Robert M.
Goor
347-365
Abstract: In this paper, we investigate the parametric growth condition which arises in connection with existence theorems for parametric problems of the calculus of variations. In particular, we study conditions under which the length of a curve is dominated in a suitable sense by its ``cost". We show that we may restrict our attention to local growth conditions on a particular set. Then we link the growth conditions to a certain approximation problem on this set. Finally, we prove that under suitable topological restrictions related to dimension theory, the local and global problems can be solved.
Spherical distributions on Lie groups and $C\sp{\infty }$ vectors
R.
Penney
367-384
Abstract: Given a Lie group G (not necessarily unimodular) and a subgroup K of G (not necessarily compact), it is shown how to associate with every finite-dimensional unitary irreducible representation $ \delta$ of K a class of distributions analogous to the class of spherical functions of height $\delta$ familiar from the unimodular-maximal compact case. The two concepts agree as nearly as possible. A number of familiar theorems are generalized to our situation. As an application we obtain a generalization of the Frobenius reciprocity theorem and of Plancherel's theorem to arbitrary induced representations of Lie groups.
Necessary and sufficient conditions for the derivation of integrals of $L\sb{\psi }$-functions
C. A.
Hayes
385-395
Abstract: It has been shown recently that a necessary and sufficient condition for a derivation basis to derive the $\mu $-integrals of all functions in $ {L^{(q)}}(\mu )$, where $1 < q < + \infty$, and $ \mu$ is a $\sigma $-finite measure, is that the basis possess Vitali-like covering properties, with covering families having arbitrarily small ${L^{(p)}}(\mu )$-overlap, where ${p^{ - 1}} + {q^{ - 1}} = 1$. The corresponding theorem for the case $ p = 1,q = + \infty$ was established by R. de Possel in 1936. The present paper extends these results to more general dual Orlicz spaces. Under suitable restrictions on the dual Orlicz functions $\Phi$ and $\Psi$, it is shown that a necessary and sufficient condition for a basis to derive the $\mu $-integrals of all functions in ${L_\Psi }(\mu )$ is that the basis possess Vitali-like covering families whose ${L_\Phi }(\mu )$-overlap is arbitrarily small. Certain other conditions relating ${L_\Phi }(\mu )$-strength and derivability are also discussed.
Compact nilmanifold extensions of ergodic actions
Robert J.
Zimmer
397-406
Abstract: We study extensions of dymanical systems defined by co-cycles into nilpotent Lie groups.
Singularities in the nilpotent scheme of a classical group
Wim
Hesselink
1-32
Abstract: If $(X,x)$ is a pointed scheme over a ring k, we introduce a (generalized) partition $ {\text{ord}}(x,X/k)$. If G is a reductive group scheme over k, the existence of a nilpotent subscheme $N(G)$ of $ {\text{Lie}}(G)$ is discussed. We prove that ${\text{ord}}(x,N(G)/k)$ characterizes the orbits in $ N(G)$, their codimension and their adjacency structure, provided that G is $G{l_n}$, or $S{p_n}$ and $1/2 \in k$. For $S{O_n}$ only partial results are obtained. We give presentations of some singularities of $ N(G)$. Tables for its orbit structure are added.
Classification theory of abelian groups. I. Balanced projectives
R. B.
Warfield
33-63
Abstract: We introduce in this paper a class of Abelian groups which includes the torsion totally projective groups and those torsion-free groups which are direct sums of groups of rank one. Characterizations of the groups in this class are given, and a complete classification theorem, in terms of additive numerical invariants, is proved.
Interposition and lattice cones of functions
Jörg
Blatter;
G. L.
Seever
65-96
Abstract: A lattice cone of functions on a set X is a convex cone of bounded real-valued functions on X which contains the constants and which is closed under the lattice operations. Our principal results concern the relation between closed lattice cones on a set X and certain binary relations, called inclusions, on the power set of X. These results are applied to interposition problems, Császár compactifications of quasi-proximity spaces, the compactification of Nachbin's completely regular ordered topological spaces, and a problem in best approximation.
Centralisers of $C\sp{\infty }$ diffeomorphisms
Boyd
Anderson
97-106
Abstract: It is shown that if F is a hyperbolic contraction of $ {R^n}$, coordinates may be chosen so that not only is F a polynomial mapping, but so is any diffeomorphism which commutes with F. This implies an identity principle for diffeomorphisms ${G_1}$ and ${G_2}$ commuting with an arbitrary Morse-Smale diffeomorphism F of a compact manifold M: if $ {G_1},{G_2} \in Z(F)$, then ${G_1} = {G_2}$ on an open subset of $M \Rightarrow {G_1} \equiv {G_2}$ on M. Finally it is shown that under a certain linearisability condition at the saddles of F, $Z(F)$ is in fact a Lie group in its induced topology.
Holomorphic convexity of compact sets in analytic spaces and the structure of algebras of holomorphic germs
William R.
Zame
107-127
Abstract: Let $(X,{\mathcal{O}_X})$ be a reduced analytic space and let K be a compact, holomorphically convex subset of X. It is shown that analogs of Cartan's Theorems A and B are valid for coherent analytic sheaves on K. This result is applied to the study of the algebra of germs on K of functions holomorphic near K. In particular, characterizations of finitely generated ideals, prime ideals and homomorphisms are obtained.
Principal co-fiber bundles
Elyahu
Katz
129-141
Abstract: Principal co-fiber bundles are defined in the category of topological groups. They are Eckmann-Hilton duals of principal fiber bundles. A classification theorem is provided as well as an example which almost represents the most general case.
On the blocks of $GL(n,q)$. I
Jørn B.
Olsson
143-156
Abstract: A study is made of the distribution of the ordinary irreducible characters of $ {\text{GL}}(n,q)$ into p-blocks for primes different from the characteristic. The paper gives a description of all possible defect groups for $p \ne 2$ and their normalizers. Various other results are obtained, including a classification of the blocks of defect 0.
Algebras of functions on semitopological left-groups
John F.
Berglund;
Paul
Milnes
157-178
Abstract: We consider various algebras of functions on a semitopological left-group $S = X \times G$, the direct product of a left-zero semigroup X and a group G. In §1 we examine various analogues to the theorem of Eberlein that a weakly almost periodic function on a locally compact abelian group is uniformly continuous. Several appealing conjectures are shown by example to be false. In the second section we look at compactifications of products $S \times T$ of semitopological semigroups with right identity and left identity, respectively. We show that the almost periodic compactification of the product is the product of the almost periodic compactifications, thus generalizing a result of deLeeuw and Glicksberg. The weakly almost periodic compactification of the product is not the product of the weakly almost periodic compactifications except in restrictive circumstances; for instance, when T is a compact group. Finally, as an application, we define and study analytic weakly almost periodic functions and derive the theorem, analogous to a classical theorem about almost periodic functions, that an analytic function which is weakly almost periodic on a single line is analytic weakly almost periodic on a whole strip.
On Kolmogorov's inequalities $\tilde{f}_p \leq C_p$, $f_1$, $0<p<1$
Burgess
Davis
179-192
Abstract: Let $\mu$ be a signed measure on the unit circle A of the complex plane satisfying $\vert\mu \vert(A) < \infty$, where $\vert\mu \vert(A)$ is the total variation of $ \mu$, and let $\tilde \mu$ be the conjugate function of $ \mu$. A theorem of Kolmogorov states that for each real number p between 0 and 1 there is an absolute constant $ {C_p}$ such that ${({(2\pi )^{ - 1}}\smallint _0^{2\pi }\vert\tilde \mu ({e^{i\theta }}){\vert^p}d\theta )^{1/p}} \leqslant {C_p}\vert\mu \vert(A)$. Here it is shown that measures putting equal and opposite mass at points directly opposite from each other on the unit circle, and no mass any place else, are extremal for all of these inequalities, that is, if $\nu$ is one of these measures the number ${({(2\pi )^{ - 1}}\smallint _0^{2\pi }\vert\tilde \nu ({e^{i\theta }}){\vert^p}d\theta )^{1/p}}/\vert\nu \vert(A)$ is the smallest possible value for $ {C_p}$. These constants are also the best possible in the analogous Hilbert transform inequalities. The proof is based on probability theory.
A Paley-Wiener theorem for locally compact abelian groups
Gunar E.
Liepins
193-210
Abstract: Extending the Paley-Wiener theorem to locally compact Abelian groups involves both finding a suitable Laplace transform and a suitable analogue for analytic functions. The Laplace transform is defined in terms of complex characters, and differentiability is defined with use of one-parameter subgroups. The resulting theorem is much as conjectured by Mackey [7],($^{1}$) the major differences being that the theorem is very much an ${L^2}$ theorem and that the problem exhibits a surprising finite dimensional nature.
$T$ measure of Cartesian product sets. II
Lawrence R.
Ernst
211-220
Abstract: It is proven that there exists a subset A of Euclidean 2-space such that the 2-dimensional T measure of the Cartesian product of an interval of unit length and A is less than the 1-dimensional T measure of A. In a previous paper it was shown that there exists a subset of Euclidean 2-space such that the reverse inequality holds. T measure is the first measure of its type for which it has been shown that both of these relations are possible.
Universal properties of Prym varieties with an application to algebraic curves of genus five
Leon
Masiewicki
221-240
Abstract: It is proved that every morphism of a curve with an involution into an Abelian variety, anticommuting with the involution, factors through the associated Prym variety. This result is used to show that Jacobians of curves of genus five arise as Prym varieties associated to a certain class of curves.
Asymmetric maximal ideals in $M(G)$
Sadahiro
Saeki
241-254
Abstract: Let G be a nondiscrete LCA group, $M(G)$ the measure algebra of G, and $ {M_0}(G)$ the closed ideal of those measures in $M(G)$ whose Fourier transforms vanish at infinity. Let $ {\Delta _G},{\Sigma _G}$ and ${\Delta _0}$ be the spectrum of $ M(G)$, the set of all symmetric elements of $ {\Delta _G}$, and the spectrum of ${M_0}(G)$, respectively. In this paper this is shown: Let $\Phi$ be a separable subset of $M(G)$. Then there exist a probability measure $\tau$ in ${M_0}(G)$ and a compact subset X of $ {\Delta _0}\backslash {\Sigma _G}$ such that for each $\vert c\vert \leqslant 1$ and each $\displaystyle \nu \in \Phi \;{\text{Card}}\;\{ f \in X:\hat \tau (f) = c\;{\text{and}}\;\vert\hat \nu (f)\vert = r(\nu )\} \geqslant {2^{\text{c}}}.$ Here $ r(\nu ) = \sup \{ \vert\hat \nu (f)\vert:f \in {\Delta _G}\backslash \hat G\}$. As immediate consequences of this result, we have (a) every boundary for ${M_0}(G)$ is a boundary for $M(G)$ (a result due to Brown and Moran), (b) ${\Delta _G}\backslash {\Sigma _G}$ is dense in $ {\Delta _G}\backslash \hat G$, (c) the set of all peak points for $M(G)$ is $\hat G$ if G is $\sigma$-compact and is empty otherwise, and (d) for each $\mu \in M(G)$ the set $ \hat \mu ({\Delta _0}\backslash {\Sigma _G})$ contains the topological boundary of $ \hat \mu ({\Delta _G}\backslash \hat G)$ in the complex plane.
The Mackey problem for the compact-open topology
Robert F.
Wheeler
255-265
Abstract: Let ${C_c}(T)$ denote the space of continuous real-valued functions on a completely regular Hausdorff space T, endowed with the compact-open topology. Well-known results of Nachbin, Shirota, and Warner characterize those T for which ${C_c}(T)$ is bornological, barrelled, and infrabarrelled. In this paper the question of when $ {C_c}(T)$ is a Mackey space is examined. A convex strong Mackey property (CSMP), intermediate between infrabarrelled and Mackey, is introduced, and for several classes of spaces (including first countable and scattered spaces), a necessary and sufficient condition on T for $ {C_c}(T)$ to have CSMP is obtained.
Finite groups with prime $p$ to the first power
Zon I
Chang
267-288
Abstract: Earlier D. G. Higman classified the finite groups of order n, such that n is divisible by 3 to the first power, with the assumption that the centralizer $ {C_G}(X)$ of X, where X is a subgroup of order 3, is a cyclic trivial intersection set of even order 3s. In this paper the theorem is generalized to include all prime numbers greater than 3. With an additional assumption: $\vert{N_G}(X):{C_G}(X)\vert = 2$, we have proved that one of the following holds for these groups, hereafter designated as G: (A) G is isomorphic to ${L_2}(q)$, where $q = 2ps \pm 1$; (B) there exists a normal subgroup ${G_0}$ of odd index in G, and a normal subgroup N of ${G_0}$ of index 2 such that $G = N\langle \sigma \rangle $ where ${C_G}(X) = X \times \langle \sigma \rangle$.
Effective lower bounds for some linear forms
T. W.
Cusick
289-301
Abstract: It is proved that if $1, \alpha ,\beta $ are numbers, linearly independent over the rationals, in a real cubic number field, then given any real number $d \geqslant 2$, for any integers ${x_0},{x_1},{x_2}$ such that $\vert{\text{norm}}({x_0} + \alpha {x_1} + \beta {x_2})\vert \leqslant d$, there exist effectively computable numbers $c > 0$ and $k > 0$ depending only on $\alpha$ and $\beta$ such that $\vert{x_1}{x_2}\vert{(\log \vert{x_1}{x_2}\vert)^{k\log d}}\vert{x_0} + \alpha {x_1} + \beta {x_2}\vert > c$ holds whenever $ {x_1}{x_2} \ne 0$. It would be of much interest to remove the dependence on d in the exponent of $\log \vert{x_1}{x_2}\vert$, for then, among other things, one could deduce, for cubic irrationals, a stronger and effective form of Roth's Theorem.
The module of indecomposables for finite $H$-spaces
Richard
Kane
303-318
Abstract: The module of indecomposables obtained from the $\bmod p$ cohomology of a finite H-space is studied for p odd. General structure theorems are obtained, first, regarding the possible even dimensions in which this module can be nonzero and, secondly, regarding how the Steenrod algebra acts on the module.
An approximation theory for generalized Fredholm quadratic forms and integral-differential equations
J.
Gregory;
G. C.
Lopez
319-335
Abstract: An approximation theory is given for a very general class of elliptic quadratic forms which includes the study of 2nth order (usually in integrated form), selfadjoint, integral-differential equations. These ideas follows in a broad sense from the quadratic form theory of Hestenes, applied to integral-differential equations by Lopez, and extended with applications for approximation problems by Gregory. The application of this theory to a variety of approximation problem areas in this setting is given. These include focal point and focal interval problems in the calculus of variations/optimal control theory, oscillation problems for differential equations, eigenvalue problems for compact operators, numerical approximation problems, and finally the intersection of these problem areas. In the final part of our paper our ideas are specifically applied to the construction and counting of negative vectors in two important areas of current applied mathematics: In the first case we derive comparison theorems for generalized oscillation problems of differential equations. The reader may also observe the essential ideas for oscillation of many nonsymmetric (indeed odd order) ordinary differential equation problems which will not be pursued here. In the second case our methods are applied to obtain the ``Euler-Lagrange equations'' for symmetric tridiagonal matrices. In this significant new result (which will allow us to reexamine both the theory and applications of symmetric banded matrices) we can construct in a meaningful way, negative vectors, oscillation vectors, eigenvectors, and extremal solutions of classical problems as well as faster more efficient algorithms for the numerical solution of differential equations. In conclusion it appears that many physical problems which involve symmetric differential equations are more meaningful presented as integral differential equations (effects of friction on physical processes, etc.). It is hoped that this paper will provide the general theory and present examples and methods to study integral differential equations.
Weak bases and metrization
Harold W.
Martin
337-344
Abstract: Several weak base (in the sense of A. V. Arhangel'skiĭ) metrization theorems are established, including a weak base generalization of the Nagata-Smirnov Metrization Theorem.
Frobenius calculations of Picard groups and the Birch-Tate-Swinnerton-Dyer conjecture
Raymond T.
Hoobler
345-352
Abstract: Let $Y \subset {{\text{P}}^m}$ be a subvariety of codimension d defined by an ideal I in char $p > 0$ with $ {H^1}(Y,\mathcal{O}( - 1)) = 0$. If t is an integer greater than ${\log _p}(d)$ and ${H^i}(Y,{I^n}/{I^{n + 1}}) = 0$ for $n > > 0$ and $i = 1,2$, then ${\text{Pic}}(Y)$ is an extension of a finite p-primary group of exponent at most ${p^t}$ by $ Z[\mathcal{O}(1)]$ and $ \dim Y < p$ and $ p \ne 2$, then the B-T-SD conjecture holds for cycles of codimension 1. These results are proved by studying the etale cohomology of the Frobenius neighborhoods of Y in ${{\text{P}}^m}$.
$SK\sb{1}$ of $n$ lines in the plane
Leslie G.
Roberts
353-365
Abstract: We calculate $ S{K_1}(A)$ where A is the coordinate ring of the reduced affine variety consisting of n straight lines in the plane.
Sobolev inequalities for weight spaces and supercontractivity
Jay
Rosen
367-376
Abstract: For $\phi \in {C^2}({{\mathbf{R}}^n})$ with $ \phi (x) = a\vert x{\vert^{1 + s}}$ for $\vert x\vert \geqslant {x_0},a,s > 0$, define the measure $d\mu = \exp ( - 2\phi ){d^n}x$ on ${{\mathbf{R}}^n}$. We show that for any $k \in {{\mathbf{Z}}^ + }$ \begin{displaymath}\begin{array}{*{20}{c}} {\int {\vert f{\vert^2}\vert\lg(\vert... ...mu )}){\vert^{2sk/(s + 1)}}} } \right\}} \end{array} \end{displaymath} As a consequence we prove ${e^{ - t{\nabla ^\ast} \cdot \nabla }}:{L_q}({{\mathbf{R}}^n},d\mu ) \to {L_p}({{\mathbf{R}}^n},d\mu ),p,q \ne 1,\infty$, is bounded for all $ t > 0$.
Characterizations of continua in which connected subsets are arcwise connected
E. D.
Tymchatyn
377-388
Abstract: The purpose of this paper is to give several characterizations of the continua in which all connected subsets are arcwise connected. The methods used are those developed by B. Knaster and K. Kuratowski, G. T. Whyburn and the author. These methods depend on Bernstein's decomposition of a topologically complete metric space into totally imperfect sets and on Whyburn's theory of local cutpoints. Some properties of connected sets in finitely Suslinian spaces are obtained. Two questions raised by the author are answered. Several partial results of Whyburn are obtained as corollaries of the main result.
Stability theorems in shape and pro-homotopy
David A.
Edwards;
Ross
Geoghegan
389-403
Abstract: Conditions are given under which a topological space has the pointed shape of a CW complex. These are derived from analogous conditions in pro-homotopy.
Erratum to: ``The involutions on homotopy spheres and their gluing diffeomorphisms'' (Trans. Amer. Math. Soc. {\bf 215} (1976), 363--391)
Chao Chu
Liang
405
The translational hull of a topological semigroup
J. A.
Hildebrant;
J. D.
Lawson;
D. P.
Yeager
251-280
Abstract: This paper is concerned with three aspects of the study of topological versions of the translational hull of a topological semigroup. These include topological properties, applications to the general theory of topological semigroups, and techniques for computing the translational hull. The central result of this paper is that if S is a compact reductive topological semigroup and its translational hull $ \Omega (S)$ is given the topology of continuous convergence (which coincides with the topology of pointwise convergence and the compact-open topology in this case), then $\Omega (S)$ is again a compact topological semigroup. Results pertaining to extensions of bitranslations are given, and applications of these together with the central result to semigroup compactifications and divisibility are presented. Techniques for determining the translational hull of certain types of topological semigroups, along with numerous examples, are set forth in the final section.
The infinitesimal stability of semigroups of expanding maps
Richard
Sacksteder
281-288
Abstract: The concept of ${C^\infty }$ infinitesimal stability for representations of a semigroup by ${C^\infty }$ maps is defined. In the case of expanding linear maps of the torus ${T^d}$ it is shown that certain algebraic conditions assure such stability.
Uniqueness criteria for solutions of singular boundary value problems
D. R.
Dunninger;
Howard A.
Levine
289-301
Abstract: In this paper we consider the equation $u:(0,T) \to D(A) \subset B$ is a Banach space valued function taking values in a dense subdomain $ D(A)$ of the Banach space B. Here A is a closed (possibly unbounded) linear operator on $D(A)$ while k is a real constant. The differential equation is an abstract Euler-Poisson-Darboux equation. We give necessary and sufficient conditions on the point spectrum of A to insure uniqueness of the strong solution $u \equiv 0$ as well as sufficient conditions on the point spectrum to insure uniqueness of weak solutions. u is only required to satisfy (a) $t \to {0^ + }$ if $k > 1$, (b) $t \to {0^ + },0 < k \leqslant 1$, (c) $t \to {0^ + },k < 0$. The operator A need not possess a complete set of eigenvectors nor need one have a backward uniqueness theorem available for (1) for the Cauchy final value problem. Our techniques extend to the n-axially symmetric abstract equation $\displaystyle \sum\limits_{i = 1}^n {[{\partial ^2}u/\partial t_i^2 + ({k_i}/{t_i})\partial u/\partial {t_i}] + Au = 0.}$ ($ 2$) The proofs rest upon an application of the Hahn-Banach Theorem and the consequent separation properties of ${B^\ast}$, the dual of B, as well as the completeness properties of the eigenfunctions of certain Bessel equations associated with (1).
Deformations of formal embeddings of schemes
Miriam P.
Halperin
303-321
Abstract: A family of isolated singularities of k-varieties will be here called equisingular if it can be simultaneously resolved to a family of hypersurfaces embedded in nonsingular spaces which induce only locally trivial deformations of pairs of schemes over local artin k-algebras. The functor of locally trivial deformations of the formal embedding of an exceptional set has a versal object in the sense of Schlessinger. When the exceptional set ${X_0}$ is a collection of nonsingular curves meeting normally in a nonsingular surface X, the moduli correspond to Laufer's moduli of thick curves. When X is a nonsingular scheme of finite type over an algebraically closed field k and ${X_0}$ is a reduced closed subscheme of X, every deformation of $(X,{X_0})$ to $ k[\varepsilon ]$ such that the deformation of ${X_0}$ is locally trivial, is in fact a locally trivial deformation of pairs.
On chain varieties of linear algebras
V. A.
Artamonov
323-338
Abstract: In the present paper we study varieties of linear k-algebras over a commutative associative Noetherian ring k with 1, whose subvarieties form a chain. We describe these varieties in terms of identities in the following cases: residually nilpotent varieties, varieties of alternative, Jordan and $( - 1,1)$-algebras.
Applications of extreme point theory to classes of multivalent functions
David J.
Hallenbeck;
Albert E.
Livingston
339-359
Abstract: Extreme points of the closed convex hulls of several classes of multivalent functions are determined. These are then used to determine the precise bounds on the coefficients of a function majorized by or subordinate to a function in any of the classes. ${L^q}$ means are also discussed and subordination theorems are considered. The classes we consider are generalizations of the univalent starlike, convex and close-to-convex functions in addition to others.
Presentations of $3$-manifolds arising from vector fields
Peter
Percell
361-377
Abstract: A method is given for constructing a smooth, closed, orientable 3-manifold from the information contained in a combinatorial object called an abstract intersection sequence. An abstract intersection sequence of length n is just a cyclic ordering of the set $\{ \pm 1, \ldots , \pm n\} $ plus a map $\nu :\{ 1, \ldots ,n\} \to \{ \pm 1\}$. It is shown that up to diffeomorphism every closed, connected, orientable 3-manifold can be constructed by the method. This is proved by showing that compact, connected, orientable 3-manifolds with boundary the 2-sphere admit vector fields of a certain type. The intersection sequences arise as descriptions of the vector fields.
Reductivity and the automorphism group of locally compact groups
Dong Hoon
Lee
379-389
Abstract: In this paper, we study reductivity of locally compact groups and its effect on the automorphism group and generalize the classical results on the automorphism group of analytic semisimple groups on the one hand and of compact groups on the other.
Competitive processes and comparison differential systems
G. S.
Ladde
391-402
Abstract: Sufficient conditions are given for stability and nonnegativity of solutions of a system of differential equations, in particular, of comparison differential equations. Finally, it has been shown that the comparison differential equations represent the mathematical models for competitive processes in biological, physical and social sciences.
Closed $3$-manifolds with no periodic maps
Frank
Raymond;
Jeffrey L.
Tollefson
403-418
Abstract: Examples of closed, orientable, aspherical 3-manifolds are constructed on which every action of a finite group is trivial.
Dyadic methods in the measure theory of numbers
R. C.
Baker
419-432
Abstract: Some new theorems in metric diophantine approximation are obtained by dyadic methods. We show for example that if ${m_1},{m_2}, \ldots$, are distinct integers with ${m_n} = O({n^p})$ then ${\Sigma _{n \leqslant N}}e({m_n}x) = O({N^{1 - q}})$ except for a set of x of Hausdorff dimension at most $ (p + 4q - 1)/(p + 2q)$; and that for any sequence of intervals ${I_1},{I_2}, \ldots$ in [0, 1) the number of solutions of $\{ {x^n}\} \in {I_n}\;(n \leqslant N)$ is a.e. asymptotic to ${\Sigma _{n \leqslant N}}\vert{I_n}\vert(x > 1)$.
On representations of the group $SU(n,1)$
Hrvoje
Kraljević
433-448
Abstract: A natural bijection is established between the set of equivalence classes of irreducible unitary representations of the group $G = SU(n,1)$, which are not induced from a proper parabolic subgroup, and the set of equivalence classes of irreducible representations of a maximal compact subgroup.
Asymptotically autonomous multivalued differential equations
James P.
Foti
449-452
Abstract: The asymptotic behavior of solutions of the perturbed autonomous multivalued differential equation
Group extensions and cohomology for locally compact groups. III
Calvin C.
Moore
1-33
Abstract: We shall define and develop the properties of cohomology groups ${H^n}(G,A)$ which can be associated to a pair (G, A) where G is a separable locally compact group operating as a topological transformation group of automorphisms on the polonais abelian group A. This work extends the results in [29] and [30], and these groups are to be viewed as analogues of the Eilenberg-Mac Lane groups for discrete G and A. Our cohomology groups in dimension one are classes of continuous crossed homomorphisms, and in dimension two classify topological group extensions of G by A. We characterize our cohomology groups in all dimensions axiomatically, and show that two different cochain complexes can be used to construct them. We define induced modules and prove a version of Shapiro's lemma which includes as a special case the Mackey imprimitivity theorem. We show that the abelian groups $ {H^n}(G,A)$ are themselves topological groups in a natural way and we investigate this additional structure.
Group extensions and cohomology for locally compact groups. IV
Calvin C.
Moore
35-58
Abstract: In this paper we shall apply the cohomology groups constructed in [14] to a variety of problems in analysis. We show that cohomology classes admit direct integral decompositions, and we obtain as a special case a new proof of the existence of direct integral decompositions of unitary representations. This also leads to a Frobenius reciprocity theorem for induced modules, and we obtain splitting theorems for direct integrals of tori analogous to known results for direct sums. We also obtain implementation theorems for groups of automorphisms of von Neumann algebras. We show that the splitting group topology on the two-dimensional cohomology groups agrees with other naturally defined topologies and we find conditions under which this topology is ${T_2}$. Finally we resolve several questions left open concerning splitting groups in a previous paper [13].
Inclusions and noninclusion of spaces of convolution operators
Michael G.
Cowling;
John J. F.
Fournier
59-95
Abstract: Let G be an infinite, locally compact group. Denote the space of convolution operators, on G, of strong type $ (p,q)$ by $L_p^q(G)$. It is shown that, if $\vert 1/q - 1/2\vert < \vert 1/p - 1/2\vert$, then $L_q^q(G)$ is not included in $L_p^p(G)$. This result follows from estimates on the norms, in these spaces, of Rudin-Shapiro measures. The same method leads to a simple example of a convolution operator that is of strong type (q, q) for all q in the interval $(p,p')$ but is not of restricted weak type (p, p) or of restricted weak type $ (p',p')$. Other statements about noninclusion among the spaces $L_p^q(G)$ also follow from various assumptions about G. For instance, if G is unimodular, but not compact, $1 \leqslant p,q,r,s \leqslant \infty$, with $p \leqslant q$, and $\min (s,r') < \min (q,p')$, then $ L_p^q(G)$ is not included in $L_r^s(G)$. Using Zafran's multilinear interpolation theorem for the real method, it is shown that, if $ 1 < p < 2$, then there exists a convolution operator on G that is of weak type (p, p) but not of strong type (p, p); it is not known whether such operators exist when $p > 2$, but it is shown that if $p \ne 1,2,\infty$, then there exists a convolution operator that is of restricted weak type (p, p) but is not of weak type (p, p). Many of these results also hold for the spaces of operators that commute with left translation rather than right translation. Further refinements are presented in three appendices.
The regularity of the locally integrable and continuous solutions of nonlinear functional equations
Halina
Światak
97-118
Abstract: The purpose of this paper is to show a general method which allows one to find all the continuous (and sometimes also all the locally integrable) solutions of functional equations by considering solutions of class ${C^m}$. One can do it if one is assured that all the continuous (or all the locally integrable) solutions of a given equation are functions of class $ {C^m}$ or ${C^\infty }$. Such a property is characteristic for the solutions $ f:{R^n} \to R$ of the equations $\displaystyle \sum\limits_{i = 1}^k {{a_i}(x,t)f({\phi _i}(x,t)) = F(x,f({\lambda _1}(x)), \ldots ,f({\lambda _s}(x))) + b(x,t),}$ ($ \ast$) where $x \in {R^n},t \in {R^r},n \geqslant 1,r \geqslant 1$ and where the functions ${\phi _i}:{R^{n + r}} \to {R^n},{\lambda _j}:{R^n} \to {R^n},{a_i}:{R^{n + r}} \to R,b:{R^{n + r}} \to R,F:{R^{n + s}} \to R$ satisfy some regularity assumptions and the assumptions which guarantee that an equation obtained by differentiating $(\ast)$ and fixing t is of constant strength, hypoelliptic at a point ${x_0}$. A general theorem, concerning the regularity of the continuous and locally integrable solutions f of $(\ast)$, is formulated and proved by the reduction to the corresponding problem for the distributional solutions of linear partial differential equations.
Dualities for equational classes of Brouwerian algebras and Heyting algebras
Brian A.
Davey
119-146
Abstract: This paper focuses on the equational class $ {{\mathbf{S}}_n}$ of Brouwerian algebras and the equational class ${{\mathbf{L}}_n}$ of Heyting algebras generated by an n-element chain. Firstly, duality theories are developed for these classes. Next, the projectives in the dual categories are determined, and then, by applying the dualities, the injectives and absolute subretracts in $ {{\mathbf{S}}_n}$ and ${{\mathbf{L}}_n}$ are characterized. Finally, free products and the finitely generated free algebras in $ {{\mathbf{S}}_n}$ and ${{\mathbf{L}}_n}$ are described.
Homomorphisms of commutative cancellative semigroups into nonnegative real numbers
Mohan S.
Putcha;
Takayuki
Tamura
147-157
Abstract: Let S be a commutative cancellative semigroup and ${T_0}$ be a cofinal subsemigroup of S. Let ${h_0}$ be a homomorphism of ${T_0}$ into the semigroup of nonnegative real numbers under addition. We prove that Kobayashi's condition [2] is necessary and sufficient for $ {h_0}$ to be extended to S. Further, we find a necessary and sufficient condition in order that the extension be unique. Related to this, the ``boundedness condition'' is introduced. For further study, several examples are given.
Two extremal problems
Andrew
Harrington;
Marvin
Ortel
159-167
Abstract: The class H consists of all functions f which are analytic in the disk D and satisfy $\displaystyle \int\limits_D {\int {\vert f(z)\vert dx\;dy} < \infty .}$ Linear extremal problems for functionals of the type $\Lambda(f) = \smallint {\smallint _D}f(z)\phi (z)\;dx\;dy,\phi \in {L^\infty }(D),f \in H$, are studied.
Linking forms and maps of odd prime order
J. P.
Alexander;
G. C.
Hamrick;
J. W.
Vick
169-185
Abstract: A differentiable orientation preserving map of odd prime period on a closed oriented differentiable manifold gives rise to two invariants taking values in a Witt group of bilinear forms. One is globally defined in terms of the rational cohomology of the manifold and the other is locally defined in terms of the fixed point set and its normal bundle. We show that these two invariants are, in fact, equal and apply this result to relate the structure of the manifold to that of the fixed point set and the quotient space.
Multipliers of $L\sp p\sb E$. I
Daniel M.
Oberlin
187-198
Abstract: Let X be an abelian group, the character group of a compact group G. For a subset E of X let $ L_E^p$ be the subspace of E-spectral functions in ${L^p}(G)$. We show that if X is infinite and $1 \leqslant p < 2$, then E can be chosen so that not every multiplier of $\widehat{L_E^p}$ extends to a multiplier of $\widehat{{L^p}}(G)$.
Admissible and singular translates of measures on vector spaces
Alan
Gleit;
Joel
Zinn
199-211
Abstract: We provide a general setting for studying admissible and singular translates of measures on linear spaces. We apply our results to measures on $D[0,1]$. Further, we show that in many cases convex, balanced, bounded, and complete subsets of the admissible translates are compact. In addition, we generalize Sudakov's theorem on the characterization of certain quasi-invariant sets to separable reflexive spaces which have the Central Limit Property.
Docility at infinity and compactifications of ANR's
R. B.
Sher
213-224
Abstract: Various conditions of contractibility and extensibility at $ \infty$ for locally compact metric spaces are studied. These are shown to be equivalent if the space under consideration is an absolute neighborhood retract (ANR) and an ANR satisfying them is called docile at $\infty$. Docility at $\infty$ is invariant under proper homotopy domination. The ANR X is docile at $ \infty$ if and only if FX (the Freudenthal compactification of X) is an ANR and $FX - X$ is unstable in FX; the inclusion of X into FX is a homotopy equivalence.
Monotonicity, convexity and symmetric derivates
Clifford E.
Weil
225-237
Abstract: If the first lower symmetric derivate of a continuous function is nonnegative, then it is nondecreasing. If the second lower symmetric derivate of a continuous function is nonnegative, then it is convex. In this paper it is shown that if continuity is replaced by Baire one, Darboux in each of these, then the resulting statements are true.
Every weak proper homotopy equivalence is weakly properly homotopic to a proper homotopy equivalence
David A.
Edwards;
Harold M.
Hastings
239-248
Abstract: We prove that every weak proper homotopy equivalence of $ \sigma$-compact, locally compact Hausdorff spaces is weakly properly homotopic to a proper homotopy equivalence.
Erratum to: ``Isolated invariant sets for flows on vector bundles'' (Trans. Amer. Math. Soc. {\bf 203} (1975), 359--390)
James F.
Selgrade
249
Abstract: Let (F, f) be a flow on a vector bundle (E, M). If f is minimal on M, then the exponential growth rate intervals associated with the factors of an invariant splitting of E, corresponding to a Morse decomposition of PE, are disjoint intervals.
Differentiable representations. I. Induced representations and Frobenius reciprocity
Johan F.
Aarnes
1-35
Abstract: In this paper we give the construction of the adjoint and the co-adjoint of the restriction functor in the category of differentiable G-modules, where G is a Lie group. Stated in terms of representation theory this means that two types of induced representations are introduced, both differing from the classical definition of differentiably induced representation given by Bruhat. The Frobenius reciprocity theorem is shown to hold. The main part of the paper is devoted to obtaining suitable realizations of the spaces of the induced representations. It turns out that they may be given as $ {E_K}(G,E)$ and
Norming $C(U)$ and related algebras
B. E.
Johnson
37-58
Abstract: The first result of the paper is that the question of defining a submultiplicative seminorm on the commutative unital $ {C^\ast}$ algebra $C(\Omega )$ is equivalent to that of putting a nontrivial submultiplicative seminorm on the algebra of infinitesimals in some nonstandard model of C. The extent to which the existence of such a norm on one $C(\Omega )$ implies the existence for others is investigated. Using the continuum hypothesis it is shown that the algebras of infinitesimals are isomorphic and that if such an algebra has a submultiplicative norm (or, equivalently, seminorm) then, for any totally ordered field $ \mathfrak{k}$ containing R, the R-algebra of infinitesimals in $\mathfrak{k}$ has a norm. A result of Allan is extended to show that in the particular case when $\mathfrak{k}$ is a certain field of Laurent series in several (possibly infinitely many) unknowns then the infinitesimals have a submultiplicative seminorm.
$T\sp{3}$-actions on simply connected $6$-manifolds. I
Dennis
McGavran
59-85
Abstract: We are concerned with ${T^3}$-actions on simply connected 6-manifolds $ {M^6}$. As in the codimension two case, there exists, under certain restrictions, a cross-section. Unlike the codimension two case, the orbit space need not be a disk and there can be finite stability groups. C. T. C. Wall has determined (Invent. Math. 1 (1966), 355-374) a complete set of invariants for simply connected 6-manifolds with ${H_\ast}({M^6})$ torsion-free and ${\omega _2}({M^6}) = 0$. We establish sufficient conditions for these two requirements to be met when M is a ${T^3}$-manifold. Using surgery and connected sums, we compute the invariants for manifolds satisfying these conditions. We then construct a $ {T^3}$-manifold $ {M^6}$ with invariants different than any well-known manifold. This involves comparing the trilinear forms (defined by Wall) for two different manifolds.
Some topics on equilibria
Ezio
Marchi
87-102
Abstract: In the present paper we introduce a proof for the existence of equilibrium points of a certain nonbilinear problem by using the Knaster-Kuratowski-Mazurkiewicz theorem, which turns out to be somewhat efficient for studies related to n-person games. As an application of this result, by embedding an n-person game in the ``cooperative'' set of action the existence of an equilibrium point in the strict noncooperative case and more general cases is obtained.
Differential equations on closed subsets of a Banach space
V.
Lakshmikantham;
A. Richard
Mitchell;
Roger W.
Mitchell
103-113
Abstract: The problem of existence of solutions to the initial value problem $f \in C[[{t_0},{t_0} + a] \times F,E]$, F is a locally closed subset of a Banach space E is considered. Nonlinear comparison functions and dissipative type conditions in terms of Lyapunov-like functions are employed. A new comparison theorem is established which helps in surmounting the difficulties that arise in this general setup.
Solution of Hallam's problem on the terminal comparison principle for ordinary differential inequalities
Giovanni
Vidossich
115-132
Abstract: We solve affirmatively the open problem raised by Hallam [3] and we apply this result to classical differential inequalities as well as to get existence and uniqueness theorems for the terminal value problem for ordinary differential equations.
Nonlinear differential inequalities and functions of compact support
Ray
Redheffer
133-157
Abstract: This paper is concerned with strongly nonlinear (and possibly degenerate) elliptic partial differential equations in unbounded regions. To broaden the class of problems for which solutions exist, the equation and boundary conditions are expressed by use of set-valued functions; this involves no technical complications. The concept of ``solution'' is so formulated that existence is needed only in bounded regions. Uniform boundedness is first established, and compactness of support is then deduced by a comparison argument, similar to that in recent work of Brezis, but simpler in detail. The central problems here are not associated with the comparison argument, but with the nonlinearities. Our hypotheses are given only when $ \vert{\text{grad}}\;u\vert$ is small, so that the minimal surface operator (for example) is just as tractable as the Laplacian. Further nonlinearity is allowed by the use of the Bernstein-Serrin condition on the quadratic form, and by a suitably generalized version of the Meyers-Serrin concept of essential dimension. Although the boundary can have corners, we allow nonlinear boundary conditions of mixed type. Counterexamples show that certain seemingly ad hoc distinctions are in fact necessary to the truth of the theorems.
The integral closure of a Noetherian ring
James A.
Huckaba
159-166
Abstract: Let R be a commutative ring with identity and let $ R'$ denote the integral closure of R in its total quotient ring. The basic question that this paper is concerned with is: What finiteness conditions does the integral closure of a Noetherian ring R possess? Unlike the integral domain case, it is possible to construct a Noetherian ring R of any positive Krull dimension such that $ R'$ is non-Noetherian. It is shown that if $ \dim R \leqslant 2$, then every regular ideal of $R'$ is finitely generated. This generalizes the situation that occurs in the integral domain case. In particular, it generalizes Nagata's Theorem for two-dimensional Noetherian domains.
Topological measure theory for double centralizer algebras
Robert A.
Fontenot
167-184
Abstract: The classes of tight, $\tau$-additive, and $\sigma$-additive linear functionals on the double centralizer algebra of a ${C^\ast}$-algebra A are defined. The algebra A is called measure compact if all three classes coincide. Several theorems relating the existence of certain types of approximate identities in A to measure compactness of A are proved. Next, permanence properties of measure compactness are studied. For example, the $ {C^\ast}$-algebra tensor product of two measure compact $ {C^\ast}$-algebras is measure compact. Next, the question of weak-star metrizability of the positive cone in the space of tight measures is considered. In the last part of the paper, another topology is defined and is used to study the relationship of measure compactness of A and the property that the strict topology is the Mackey topology in the pairing of $M(A)$ with the tight functionals on $ M(A)$. Also, in the last section of the paper is an extension of a result of Glickberg about finitely additive measures on pseudocompact topological spaces.
On the integrability of Jacobi fields on minimal submanifolds
D. S. P.
Leung
185-194
Abstract: Let M be a minimal submanifold of a Riemannian manifold. It is proved that every Jacobi field on M is locally the deformation vector field along M of some one-parameter families of minimal submanifolds. This fact follows from a theorem on nonlinear elliptic systems which is also proved in this paper. The related global problems are also discussed briefly.
The law of infinite cardinal addition is weaker than the axiom of choice
J. D.
Halpern;
Paul E.
Howard
195-204
Abstract: We construct a permutation model of set theory with urelements in which $ {C_2}$ (the choice principle restricted to families whose elements are unordered pairs) is false but the principle, ``For every infinite cardinal m, $2m = m$'' is true. This answers in the negative a question of Tarski posed in 1924. We note in passing that the choice principle restricted to well-ordered families of finite sets is also true in the model.
Hypersurfaces of order two
Tibor
Bisztriczky
205-233
Abstract: A hypersurface ${S^{n - 1}}$ of order two in the real projective n-space is met by every straight line in maximally two points; cf. [1, p. 391]. We develop a synthetic theory of these hypersurfaces inductively, basing it upon a concept of differentiability. We define the index and the degree of degeneracy of an ${S^{n - 1}}$ and classify the ${S^{n - 1}}$ in terms of these two quantities. Our main results are (i) the reduction of the theory of the ${S^{n - 1}}$ to the nondegenerate case; (ii) the Theorem (A.5.11) that a nondegenerate ${S^{n - 1}}$ of positive index must be a quadric and (iii) a comparison of our theory with Marchaud's discussion of ``linearly connected'' sets; cf. [3].
Generalized super-solutions of parabolic equations
Neil A.
Eklund
235-242
Abstract: Let L be a linear, second order parabolic operator in divergence form and let Q be a bounded cylindrical domain in ${E^{n + 1}}$. Super-solutions of $ Lu = 0$ are defined and generalized to three equivalent forms. Generalized super-solutions are shown to satisfy a minimum principle and form a lattice.
Relativization of the theory of computational complexity
Nancy Ann
Lynch;
Albert R.
Meyer;
Michael J.
Fischer
243-287
Abstract: The axiomatic treatment of the computational complexity of partial recursive functions initiated by Blum is extended to relatively computable functions (as computed, for example, by Turing machines with oracles). Relativizations of several results of complexity theory are carried out. A recursive relatedness theorem is proved, showing that any two relative complexity measures are related by a fixed recursive function. This theorem allows proofs of results for all measures to be obtained from proofs for a particular measure. Complexity-determined reducibilities are studied. Truth-table and primitive recursive reducibilities are proved to be reducibilities of this type. The concept of a set ``helping'' the computation of a function (by causing a saving in resource when used as an oracle in the computation of the function) is formalized. Basic properties of the helping relation are given, including nontransitivity and bounds on the amount of help certain sets can provide. Several independence results (results about sets that do not help each other's computation) are proved; they are subrecursive analogs to degrees-of-unsolvability theorems with proofs using diagonalization and priority arguments. In particular, the existence of a ``universally-helped set'' is discussed; partial results are obtained in both directions. The deepest result in the paper is a finite-injury priority argument (without an apparent recursive bound on the number of injuries) which produces sets preserving an arbitrary lower bound on the complexity of any given set.
Exact colimits and fixed points
John
Isbell;
Barry
Mitchell
289-298
Abstract: In this paper we shall give details of some work sketched in [6] on the exactness of the functor colim: $ {\text{Ab}}^\mathcal{\text{C}} \to {\text{Ab}}$. We shall also investigate the connection between this work and a paper of J. Adámek and J. Reiterman [1] characterizing those categories $ \mathcal{\text{C}}$ with the property that every endomorphism of an indecomposable functor $ \mathcal{\text{C}} \to$ Sets has a fixed point. Exactness of colim implies the fixed point property, and in some cases (such as when $ \mathcal{\text{C}}$ has only finitely many objects) both conditions turn out to be equivalent to the components of $\mathcal{\text{C}}$ being filtered. We do not expect that the two conditions are equivalent in general, although we have no example. However the category of finite ordinals and order preserving injections is an example of a connected, nonfiltered category relative to which colim is exact. This was conjectured by Mitchell, and is proved by Isbell in [5].
Favard's solution is the limit of $W\sp{k,p}$-splines
C. K.
Chui;
P. W.
Smith;
J. D.
Ward
299-305
Abstract: The purpose of this paper is to affirm a conjecture of C. de Boor, namely: The ${W^{k,p}}$-splines converge in ${W^{k,r}}[a,b]$ for all $r,1 \leqslant r < \infty$, to the Favard solution as p tends to infinity.
The Lebesgue decomposition for group-valued set functions
Tim
Traynor
307-319
Abstract: A Lebesgue-type decomposition is obtained for finitely additive set functions on a ring with values in topological groups. Corresponding results for Fréchet-Nikodým topologies are included. This generalizes Darst's result for real-valued set functions and a result of Drewnowski.
The multiplicity function of a local ring
James
Hornell
321-341
Abstract: Let A be a local ring with maximal ideal m. Let $f \in A$, and define ${\mu _A}(f)$ to be the multiplicity of the A-module $A/Af$ with respect to m. Under suitable conditions ${\mu _A}(fg) = {\mu _A}(f) + {\mu _A}(g)$. The relationship of ${\mu _A}$ to reduction of A, normalization of A and a quadratic transform of A is studied. It is then shown that there are positive integers ${n_1}, \ldots ,{n_s}$ and rank one discrete valuations $ {v_1}, \ldots ,{v_s}$ of A centered at m such that $ {\mu _A}(f) = {n_1}{v_1}(f) + \cdots + {n_s}{v_s}(f)$ for all regular elements f of A.
The fixed-point property of $(2m-1)$-connected $4m$-manifolds
S. Y.
Husseini
343-359
Abstract: Suppose that M is a $(2m - 1)$-connected smooth and compact manifold of dimension 4m. Assume that its intersection pairing is positive definite, and denote its signature by $\sigma$. Two notions are introduced. The first is that of a $ (\xi ,\lambda )$-map $ f:M \to M$ where $\xi \in K(M)$ and $\lambda$ an integer. It describes the concept of f preserving $\xi$ up to multiplication by $ \lambda$ outside a point. The second notion is that of $\xi$ being sufficiently asymmetric. It describes in terms of the Chern class of $\xi$ the concept that the restrictions of $ \xi$ to the 2m-spheres realizing a basis for ${H_{2m}}(M;Z)$ are sufficiently different so that no map which preserves $\xi$ can move the spheres among themselves. One proves that $ (\xi ,\lambda )$-maps with $ \xi$ being sufficiently asymmetric have fixed points, except possibly when $\sigma = 2$. On taking $\xi$ to be the complexification of the tangent bundle of M, one sees that mainfolds with sufficiently asymmetric tangent structures have the fixed point property with respect to a family of maps which includes diffeomorphisms. The question of the existence of $(\xi ,\lambda )$-maps as well as the question of the preservation of the fixed-point property under products are also discussed.
Decomposability of homotopy lens spaces and free cyclic group actions on homotopy spheres
Kai
Wang
361-371
Abstract: Let $\rho$ be a linear ${Z_n}$ action on ${{\text{C}}^m}$ and let $\rho$ also denote the induced ${Z_n}$ action on $ {S^{2p - 1}} \times {D^{2q}},{D^{2p}} \times {S^{2q - 1}}$ and ${S^{2p - 1}} \times {S^{2q - 1}}$ where $ p = [m/2]$ and $ q = m - p$. A free differentiable ${Z_n}$ action $({\Sigma ^{2m - 1}},\mu )$ on a homotopy sphere is $ \rho$-decomposable if there is an equivariant diffeomorphism $ \Phi$ of $ ({S^{2p - 1}} \times {S^{2q - 1}},\rho )$ such that $({\Sigma ^{2m - 1}},\mu )$ is equivalent to $(\Sigma (\Phi ),A(\Phi ))$ where $ \Sigma (\Phi ) = {S^{2p - 1}} \times {D^{2q}}{ \cup _\Phi }{D^{2p}} \times {S^{2q - 1}}$ and $A(\Phi )$ is a uniquely determined action on $\Sigma (\Phi )$ such that $A(\Phi )\vert{S^{2p - 1}} \times {D^{2q}} = \rho$ and $A(\Phi )\vert{D^{2p}} \times {S^{2q - 1}} = \rho$. A homotopy lens space is $\rho$-decomposable if it is the orbit space of a $ \rho$-decomposable free $ {Z_n}$ action on a homotopy sphere. In this paper, we will study the decomposabilities of homotopy lens spaces. We will also prove that for each lens space $ {L^{2m - 1}}$, there exist infinitely many inequivalent free $ {Z_n}$ actions on ${S^{2m - 1}}$ such that the orbit spaces are simple homotopy equivalent to $ {L^{2m - 1}}$.
One-parameter groups of isometries on Hardy spaces of the torus
Earl
Berkson;
Horacio
Porta
373-391
Abstract: The strongly continuous one-parameter groups of isometries on ${H^p}$ of the torus $(1 \leqslant p < \infty ,p \ne 2)$, as well as their generators, are classified and concretely described.
Weakly normal filters and irregular ultrafilters
A.
Kanamori
393-399
Abstract: For a filter over a regular cardinal, least functions and the consequent notion of weak normality are described. The following two results, which make a basic connection between the existence of least functions and irregularity of ultrafilters, are then proved: Let U be a uniform ultrafilter over a regular cardinal $ \kappa$. (a) If $\kappa = {\lambda ^ + }$, then U is not $(\lambda ,{\lambda ^ + })$-regular iff U has a least function f such that $\{ \xi < {\lambda ^ + }\vert{\text{cf}}(f(\xi )) = \lambda \} \in U$. (b) If $\omega \leqslant \mu < \kappa$ and U is not $(\omega ,\mu )$-regular, then U has a least function.
Free topological groups and dimension
Charles
Joiner
401-418
Abstract: For a completely regular space X we denote by $F(X)$ and $A(X)$ the free topological group of X and the free Abelian topological group of X, respectively, in the sense of Markov and Graev. Let X and Y be locally compact metric spaces with either $A(X)$ topologically isomorphic to $A(Y)$ or $F(X)$ topologically isomorphic to $F(Y)$. We show that in either case X and Y have the same weak inductive dimension. To prove these results we use a Fundamental Lemma which deals with the structure of the topology of $ F(X)$ and $A(X)$. We give other results on the topology of $F(X)$ and $A(X)$ and on the position of X in $ F(X)$ and $A(X)$.
Topologically defined classes of going-down domains
Ira J.
Papick
1-37
Abstract: Let R be an integral domain. Our purpose is to study GD (going-down) domains which arise topologically; that is, we investigate how certain going-down assumptions on R and its overrings relate to the topological space $ {\text{Spec}}(R)$. Many classes of GD domains are introduced topologically, and a systematic study of their behavior under homomorphic images, localization and globalization, integral change of rings, and the ``$D + M$ construction'' is undertaken. Also studied, is the algebraic and topological relationships between these newly defined classes of GD domains.
Differential games with Lipschitz control functions and applications to games with partial differential equations
Emmanuel Nicholas
Barron
39-76
Abstract: In §1 we formulate a differential game when the dynamics is the inhomogeneous heat equation. In §2 we state the basic theory of differential games when the controls must choose uniformly Lipschitz control functions. We then prove some general theorems for the case when the controls may choose any measurable control functions. These theorems hold for games with any dynamics. In §3 we apply our theory developed to our particular example and in §4 we prove the existence of value for games with partial differential equations.
On analytic independence
S. S.
Abhyankar;
T. T.
Moh
77-87
Abstract: This article examines the concept of ``analytic independence". Several illustrative examples have been included. The main results are Theorems 1-4 which state the relations between analytic independence and the degree of field extensions, transcendence degree, order of poles and ``gap'' respectively.
Symmetrizable and related spaces
Peter W.
Harley;
R. M.
Stephenson
89-111
Abstract: A study is made of a family of spaces which contains the symmetrizable spaces as well as many of the well-known examples of perfectly normal spaces.
Limit properties of Poisson kernels of Siegel domains of type II
Lawrence J.
Dickson
113-131
Abstract: The results of [1] concerning tight $C_0^\ast$ limit of the Poisson kernel of a tube domain, as its parameter converges to a point on the cone boundary, are extended under certain hypotheses to Siegel domains of type II. In the case where the domain is polytopic, almost everywhere convergence of the ${L^p}$ Poisson integral to its boundary values is obtained. Examples and further conjectures conclude the paper.
On the existence of compact metric subspaces with applications to the complementation of $c\sb{0}$
William H.
Chapman;
Daniel J.
Randtke
133-148
Abstract: A topological space X has property $\sigma - {\text{CM}}$ if for every countable family F of continuous scalar valued functions on X there is a compact metrizable subspace M of X such that $ f(X) = f(M)$ for every f in F. Every compact metric space, every weakly compact subset of a Banach space and every closed ordinal space has property $\sigma - {\text{CM}}$. Every continuous image of an arbitrary product of spaces having property $\sigma - {\text{CM}}$ also has property $\sigma - {\text{CM}}$. If X has property $ \sigma - {\text{CM}}$, then every copy of ${c_0}$ in $C(X)$ is complemented in $C(X)$. If a locally convex space E belongs to the variety of locally convex spaces generated by the weakly compactly generated Banach spaces, then every copy of ${c_0}$ in E is complemented in E.
The \v Cech cohomology of movable and $n$-movable spaces
James
Keesling
149-167
Abstract: In this paper the Čech cohomology of movable and n-movable spaces is studied. Let X be a space and let $ {H^k}(X)$ denote the k-dimensional Čech cohomology of X with integer coefficients based on the numerable covers of X. Then if X is movable, there is a subgroup E of ${H^k}(X)$ which is the union of all the algebraically compact subgroups of ${H^k}(X)$. Furthermore, $ {H^k}(X)/E$ is an ${\aleph _1}$-free abelian group. If X is an n-movable space, then it is shown that this structure holds for ${H^k}(X)$ for $0 \leqslant k \leqslant n$ and may be false for $k \geqslant n + 1$. If X is an ${\text{LC}^{n - 1}}$ paracompactum, then X is known to be n-movable. However, in this case and in the case that X is an ${\text{LC}^{n - 1}}$ compactum a stronger structure theorem is proved for ${H^k}(X)$ for $0 \leqslant k \leqslant n - 1$ than that stated above. Using these results examples are given of n-movable continua that are not shape equivalent to any ${\text{LC}^{n - 1}}$ paracompactum.
Some examples in shape theory using the theory of compact connected abelian topological groups
James
Keesling
169-188
Abstract: In previous papers the author has studied the shape of compact connected abelian topological groups. This study has led to a number of theorems and examples in shape theory. In this paper a theorem is proved concerning the Čech homology of compact connected abelian topological groups. This theorem together with the author's previous results are then used to study the movability of general compact Hausdorff spaces. In the theory of shape for compact metric spaces, a number of significant theorems have been proved for movable compact metric spaces. Among these are a theorem of Hurewicz type due to K. Kuperberg, a Whitehead type theorem due to Moszyńska, and a theorem concerning the exactness of the Čech homology sequence for movable compact metric pairs due to Overton. In this paper examples are constructed which show that these theorems do not generalize to arbitrary movable compact Hausdorff spaces without additional assumptions.
Classification of simply connected four-dimensional $RR$-manifolds
Gr.
Tsagas;
A.
Ledger
189-210
Abstract: Let (M, g) be a Riemannian manifold. We assume that there is a mapping $s:M \to I(M)$, where $I(M)$ is the group of isometries of (M, g), such that ${s_x} = s(x),\forall x \in M$, has x as a fixed isolated point, then (M, g) is called a Riemannian s-manifold. If the tensor field S on M defined by the relation ${S_x} = {(d{s_x})_x},\forall x \in M$, is differentiable and invariant by each isometry $ {s_x}$, then the manifold (M, g) is called a regularly s-symmetric Riemannian manifold. The aim of the present paper is to classify simply connected four-dimensional regularly s-symmetric Riemannian manifolds.
A study of graph closed subsemigroups of a full transformation semigroup
R. G.
Biggs;
S. A.
Rankin;
C. M.
Reis
211-223
Abstract: Let ${T_X}$ be the full transformation semigroup on the set X and let S be a subsemigroup of ${T_X}$. We may associate with S a digraph $ g(S)$ with X as set of vertices as follows: $i \to j \in g(S)$ iff there exists $\alpha \in S$ such that $\alpha (i) = j$. Conversely, for a digraph G having certain properties we may assign a semigroup structure, $S(G)$, to the underlying set of G. We are thus able to establish a ``Galois correspondence'' between the subsemigroups of ${T_X}$ and a particular class of digraphs on X. In general, S is a proper subsemigroup of $S \cdot g(S)$.
Equivariant stable homotopy and framed bordism
Czes
Kosniowski
225-234
Abstract: This paper gives an elementary proof of the result that equivariant stable homotopy is the same as equivariant framed bordism.
Yoneda products in the Cartan-Eilenberg change of rings spectral sequence with applications to ${\rm BP}\sb\ast ({\rm BO}(n))$
Ronald
Ming
235-252
Abstract: Yoneda product structure is defined on a Cartan-Eilenberg change of rings spectral sequence. Application is made to a factorization theorem for the ${E_2}$-term of the Adams spectral sequence for Brown-Peterson homology of the classifying spaces $ BO(n)$.
Concordances of Hilbert cube manifolds
T. A.
Chapman
253-268
Abstract: The main result of this paper asserts that homotopy groups of concordances of compact Hilbert cube manifolds are isomorphic to homotopy groups of concordances of compact finite-dimensional piecewise-linear manifolds. This enables us to apply some finite-dimensional results to obtain some new information about homotopy groups of homeomorphism groups of compact Hilbert cube manifolds. Our approach also yields a much shorter proof of the local contractibility of the homeomorphism group of any compact Hilbert cube manifold.
Simplicial triangulation of noncombinatorial manifolds of dimension less than $9$
Martin
Scharlemann
269-287
Abstract: Necessary and sufficient conditions are given for the simplicial triangulation of all noncombinatorial manifolds in the dimension range $5 \leqslant n \leqslant 7$, for which the integral Bockstein of the combinatorial triangulation obstruction is trivial. A weaker theorem is proven in case $n = 8$. The appendix contains a proof that a map between PL manifolds which is a TOP fiber bundle can be made a PL fiber bundle.
Complex space forms immersed in complex space forms
H.
Nakagawa;
K.
Ogiue
289-297
Abstract: We determine all the isometric immersions of complex space forms into complex space forms. Our result can be considered as the local version of a well-known result of Calabi.
Partitions of large multipartites with congruence conditions. I
M. M.
Robertson;
D.
Spencer
299-322
Abstract: Let $ p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})$ be the number of partitions of $({n_1}, \ldots ,{n_j})$ where, for $1 \leqslant l \leqslant j$, the lth component of each part belongs to the set ${A_l} = \bigcup\nolimits_{h(l) = 1}^{q(l)} {\{ {a_{lh(l)}} + Mv :v = 0,1,2, \ldots \} }$ and $M,q(l)$ and the $ {a_{lh(l)}}$ are positive integers such that $0 < {a_{l1}} < \cdots < {a_{lq(l)}} \leqslant M$. Asymptotic expansions for $ p({n_1}, \ldots ,{n_j}:{A_1}, \ldots ,{A_j})$ are derived, when the ${n_l} \to \infty$ subject to the restriction that ${n_1} \cdots {n_j} \leqslant n_l^{j + 1 - \in }$ for all l, where $\in$ is any fixed positive number. The case $ M = 1$ and arbitrary j was investigated by Robertson [10] while several authors between 1940 and 1960 investigated the case $ j = 1$ for different values of M.
Maximal orders and reflexive modules
J. H.
Cozzens
323-336
Abstract: If R is a maximal two-sided order in a semisimple ring and $ {M_R}$ is a finite dimensional torsionless faithful R-module, we show that $m = {\text{End}_R}\;{M^\ast}$ is a maximal order. As a consequence, we obtain the equivalence of the following when ${M_R}$ is a generator: 1. M is R-reflexive. 2. $k = {\text{End}}\;{M_R}$ is a maximal order. 3. $k = {\text{End}_R}\;{M^\ast}$ where $ {M^\ast} = {\hom _R}(M,R)$. When R is a prime maximal right order, we show that the endomorphism ring of any finite dimensional, reflexive module is a maximal order. We then show by example that R being a maximal order is not a property preserved by k. However, we show that $k = {\text{End}}\;{M_R}$ is a maximal order whenever ${M_R}$ is a maximal uniform right ideal of R, thereby sharpening Faith's representation theorem for maximal two-sided orders. In the final section, we show by example that even if $R = {\text{End}_k}V$ is a simple pli (pri)-domain, k can have any prescribed right global dimension $\geqslant 1$, can be right but not left Noetherian or neither right nor left Noetherian.
Breadth two topological lattices with connected sets of irreducibles
J. W.
Lea
337-345
Abstract: Breadth two topological lattices with connected sets of irreducible elements are characterized by these sets.
On the zeros of Dirichlet $L$-functions. III
Akio
Fujii
347-349
Abstract: It is shown that the ordinates of the zeros of the Riemann zeta function are uniformly distributed. Similar results pertain to zeros of L-functions.
Infinite-dimensional Whitehead and Vietoris theorems in shape and pro-homotopy
David A.
Edwards;
Ross
Geoghegan
351-360
Abstract: In Theorem 3.3 and Remark 3.4 conditions are given under which an infinite-dimensional Whitehead theorem holds in pro-homotopy. Applications to shape theory are given in Theorems 1.1, 1.2, 4.1 and 4.2.
A Hausdorff measure inequality
Lawrence R.
Ernst;
Gerald
Freilich
361-368
Abstract: We prove that the Hausdorff $(m + k)$-measure of a product set is no less than the product of the Hausdorff m-measure of the (measurable) first component set in ${{\mathbf{R}}^m}$ and the (finite) Hausdorff k-measure of the second component in ${{\mathbf{R}}^n}$.
On the trivial extension of equivalence relations on analytic spaces
Kunio
Takijima;
Tetsutaro
Suzuki
369-377
Abstract: In this paper, we shall consider the problem: let X be a (reduced) analytic space and A a nowhere dense analytic set in X. And let R be a proper equivalence relation on A such that the quotient space $ A/R$ is an analytic space, and $\tilde R$ the trivial extension of R to X. Then, is $ X/\tilde R$ an analytic space? To this, we have three sufficient conditions. Moreover, using this result we shall extend Satz 1 of H. Kerner [8].
Asymptotic relations for partitions
L. B.
Richmond
379-385
Abstract: Asymptotic relations are obtained for the number ${p_A}(n)$ of partitions of the integer n into summands from a set A of integers. The set A is subject to certain conditions; however the only arithmetic condition is that A have property ${P_k}$ of Bateman and Erdös. A conjecture of Bateman and Erdös concerning the kth differences of ${p_A}(n)$ may be verified using these asymptotic relations.
Extreme points of univalent functions with two fixed points
Herb
Silverman
387-395
Abstract: Univalent functions of the form $f(z) = {a_1}z - \Sigma _{n = 2}^\infty {a_n}{z^n}$, where $ {a_n} \geqslant 0$, are considered. We examine the subclasses for which $f({z_0}) = {z_0}$ or
Asymptotic equipartition of energy for differential equations in Hilbert space
Jerome A.
Goldstein;
James T.
Sandefur
397-406
Abstract: Of concern are second order differential equations of the form $ (d/dt - i{f_1}(A))(d/dt - i{f_2}(A))u = 0$. Here A is a selfadjoint operator and $ {f_1},{f_2}$ are real-valued Borel functions on the spectrum of A. The Cauchy problem for this equation is governed by a certain one parameter group of unitary operators. This group allows one to define the energy of a solution; this energy depends on the initial data but not on the time t. The energy is broken into two parts, kinetic energy $K(t)$ and potential energy $P(t)$, and conditions on A, ${f_1},{f_2}$ are given to insure asymptotic equipartition of energy: $ {\lim _{t \to \pm \infty }}K(t) = {\lim _{t \to \pm \infty }}P(t)$ for all choices of initial data. These results generalize the corresponding results of Goldstein for the abstract wave equation ${d^2}u/d{t^2} + {A^2}u = 0$. (In this case, ${f_1}(\lambda ) \equiv \lambda ,{f_2}(\lambda ) \equiv - \lambda$.)
The $2$-transitive permutation representations of the finite Chevalley groups
Charles W.
Curtis;
William M.
Kantor;
Gary M.
Seitz
1-59
Abstract: The permutation representations in the title are all determined, and no surprises are found to occur.
Mean convergence of Fourier series on compact Lie groups
Robert J.
Stanton
61-87
Abstract: The main result is an ${L^p}$ mean convergence theorem for the partial sums of the Fourier series of a class function on a compact semi-simple Lie group. A central element in the proof is a Lie group-Lie algebra analog of the theorems in classical Fourier analysis that allow one to pass back and forth between multiplier operators for Fourier series in several variables and multiplier operators for the Fourier transform in Euclidean space. To obtain the ${L^p}$ mean convergence theorem, the theory of the Hilbert transform with weight function is needed.
Reversible diffeomorphisms and flows
Robert L.
Devaney
89-113
Abstract: We generalize the classical notion of reversibility of a mechanical system. The generic qualitative properties of symmetric orbits of such systems are studied using transversality theory. In particular, we prove analogues of the closed orbit, Liapounov, and homoclinic orbit theorems for R-reversible systems.
On polar relations of abstract homogeneous polynomials
Neyamat
Zaheer
115-131
Abstract: In this paper we generalize, to vector spaces over algebraically closed fields of characteristic zero, two well-known classical results due to Laguerre and Grace, concerning, respectively, the relative location of the zeros of a complex-valued polynomial and its polar-derivative and the relative location of the zeros of two apolar polynomials. Vector space analogues of their results were generalized, to a certain degree, by Hörmander, Marden, and Zervos. Our results in this paper further generalize their results and, in the complex plane, improve upon those of Laguerre and Grace. Besides, the present treatment unifies their completely independent approaches into an improved and more systematic and abstract theory. We have also shown that our results are best possible in the sense that they cannot be further generalized in certain directions.
On the structure of certain subalgebras of a universal enveloping algebra
Bertram
Kostant;
Juan
Tirao
133-154
Abstract: The representation theory of a semisimple group G, from an algebraic point of view, reduces to determining the finite dimensional representation of the centralizer ${U^\mathfrak{k}}$ of the maximal compact subgroup K of G in the universal enveloping algebra U of the Lie algebra $ \mathfrak{g}$ of G. The theory of spherical representations has been determined in this way since by a result of Harish-Chandra $ {U^\mathfrak{k}}$ modulo a suitable ideal I is isomorphic to the ring of Weyl group W invariants $U{(\mathfrak{a})^W}$ in a suitable polynomial ring $U(\mathfrak{a})$. To deal with the general case one must determine the image of ${U^\mathfrak{k}}$ in $U(\mathfrak{k}) \otimes U(\mathfrak{a})$, where $\mathfrak{k}$ is the Lie algebra of K. We prove that if W is replaced by the Kunze-Stein intertwining operators $\tilde W$ then $ {U^\mathfrak{k}}$ suitably localized and completed is indeed isomorphic to $U(\mathfrak{k}) \otimes U{(\mathfrak{a})^{\tilde W}}$ suitably localized and completed.
Hermite series as boundary values
G. G.
Walter
155-171
Abstract: The relations between Hermite series expansions of functions and tempered distributions on the real axis and holomorphic or harmonic functions or generalizations of them in the upper half plane are studied. The Hermite series expansions of $ {H^2}$ functions are characterized in terms of their coefficients. Series of analytic representations of Hermite functions, series of Hermite functions of the second kind, and combined series of Hermite functions of the first and second kind are investigated. The functions to which these series converge in the upper half plane are shown to approach (in various ways) the distributions or functions whose Hermite series have the same coefficients.
Analytic hypoellipticity of certain second-order evolution equations with double characteristics
Mario
Tosques
173-196
Abstract: The present article establishes the analytic hypoellipticity (Definition 1.2) of a class of abstract evolution equations of order two, with double characteristics, under the hypothesis that the coefficients are analytic (in a suitable sense; see §2). The noteworthy feature of the main result (Theorem 4.1) is that analytic hypoellipticity holds whenever hypoellipticity does, even when one of the asymptotic eigenvalues $ {c^j}(A)$ fails to be elliptic of order one.
Piecewise monotone interpolation and approximation with Muntz polynomials
Eli
Passow;
Louis
Raymon;
Oved
Shisha
197-205
Abstract: The possibility (subject to certain restrictions) of solving the following approximation and interpolation problem with a given set of ``Muntz polynomials'' on a real interval is demonstrated: (i) approximation of a continuous function by a ``copositive'' Muntz polynomial; (ii) approximation of a continuous function by a ``comonotone'' Muntz polynomial; (iii) approximation of a continuous function with a monotone kth difference by a Muntz polynomial with a monotone kth derivative; (iv) interpolation by piecewise monotone Muntz polynomials--i. e., polynomials that are monotone on each of the intervals determined by the points of interpolation. The strong interrelationship of these problems is shown implicitly in the proofs.
Certain continua in $S\sp{n}$ of the same shape have homeomorphic complements
Vo Thanh
Liem
207-217
Abstract: As a consequence of Theorem 1 of this paper, we see that if X and Y are globally 1-alg continua in ${S^n}\;(n \geqslant 5)$ having the shape of the real projective space ${P^k}\;(k \ne 2,2k + 2 \leqslant n)$, then ${S^n} - X \approx {S^n} - Y$. (For ${P^1} = {S^1}$, this establishes the last case of such a result for spheres.) We also show that if X and Y are globally 1-alg continua in ${S^n},n \geqslant 6$, which have the shape of a codimension $ \geqslant 3$, closed, $0 < (2m - n + 1)$-connected, PL-manifold $ {M^m}$, then ${S^n} - X \approx {S^n} - Y$.
Positive definite measures with applications to a Volterra equation
Olof J.
Staffans
219-237
Abstract: We study the asymptotic behavior of the solutions of the nonlinear Volterra integrodifferential equation
Tauberian theorems for a positive definite form, with applications to a Volterra equation
Olof J.
Staffans
239-259
Abstract: We study the relation between the condition $\displaystyle \mathop {\sup }\limits_{T > 0} \int_{[0,T]} {\bar \varphi (t)} \int_{[t - T,t]} {\varphi (t - s)\;dv (s)\;dt < \infty }$ and the asymptotic behavior of the bounded function $\varphi$ when $\nu$ is a positive definite measure. Earlier we have proved that if $\nu$ is strictly positive definite and $\varphi$ satisfies a tauberian condition, then $ \varphi (t) \to 0$ as $t \to \infty$. Here we characterize the spectrum of the limit set of $\varphi$ in the case when $\nu$ is not strictly positive definite. Applying this theory to a nonlinear Volterra equation we get some new results on the asymptotic behavior of its bounded solutions.
Pointwise bounded approximation and analytic capacity of open sets
Steven
Jacobson
261-283
Abstract: We examine the semi-additivity question for analytic capacity by studying the relation between the capacities of bounded open sets and their closures.
Hyperfinite extensions of bounded operators on a separable Hilbert space
L. C.
Moore
285-295
Abstract: Let H be a separable Hilbert space and Ĥ the nonstandard hull of H with respect to an ${\aleph _1}$-saturated enlargement. Let S be a $^\ast$-finite dimensional subspace of $^\ast H$ such that the corresponding hyperfinite dimensional subspace Ŝ of Ĥ contains H. If T is a bounded operator on H, then an extension  of T to Ŝ where  is obtained from an internal $^\ast$-linear operator on S is called a hyperfinite extension of T. It is shown that T has a compact (selfadjoint) hyperfinite extension if and only if T is compact (selfadjoint). However T has a normal hyperfinite extension if and only if T is subnormal. The spectrum of a hyperfinite extension  equals the point spectrum of Â, and if T is quasitriangular, A can be chosen so that the spectrum of  equals the spectrum of T. A simple proof of the spectral theorem for bounded selfadjoint operators is given using a hyperfinite extension.
Principal quadratic functionals
E. C.
Tomastik
297-309
Abstract: The general theory of the existence of a minimum limit in the fixed and variable end point problems of singular quadratic functionals of n dependent variables is developed, generalizing the one dimensional results of Marston Morse, Walter Leighton and A. D. Martin and completing a phase of the n-dimensional theory initiated recently by the author.
Mean convergence of generalized Walsh-Fourier series
Wo Sang
Young
311-320
Abstract: Paley proved that Walsh-Fourier series converges in ${L^p}(1 < p < \infty )$. We generalize Paley's result to Fourier series with respect to characters of countable direct products of finite cyclic groups of arbitrary orders.
Some results on orientation preserving involutions
David E.
Gibbs
321-332
Abstract: The bordism of orientation preserving differentiable involutions is studied by use of the signature-like invariant ${\text{ab}}: {\mathcal{O}_\ast}({Z_2}) \to {W_0}({Z_2};Z)$. The equivariant Witt ring ${W_0}({Z_2};Z)$ is calculated and is shown to be isomorphic under ab to the effective part of ${\mathcal{O}_4}({Z_2})$. Modulo 2 relations are established between the representation of the involution on $ {H^{2k}}({M^{4k}};Z)/{\operatorname{torsion}}$ and $ {\chi _0}(F)$ and ${\chi _2}(F)$, where $ {\chi _i}(F)$ is the Euler characteristic of those components of the fixed point set with dimensions congruent to i modulo 4. For manifolds of dimension $4k + 2$, it is shown that $ {\chi _0}(F) \equiv {\chi _2}(F) \equiv 0\;(\bmod 2)$. Finally the ideal ${E_0}({Z_2};Z)$ consisting of those elements of $ {W_0}({Z_2};Z)$ admitting a representative of type II is determined.
Total mean curvature of immersed surfaces in $E\sp{m}$
Bang-yen
Chen
333-341
Abstract: Total mean curvature and value-distribution of mean curvature for certain pseudo-umbilical surfaces are studied.
Slowly varying functions in the complex plane
Monique
Vuilleumier
343-348
Abstract: Let f be analytic and have no zeros in $\vert\arg z\vert < \alpha \leqslant \pi$; f is called slowly varying if, for every $ \lambda > 0,f(\lambda z)/f(z) \to 1$ uniformly in $\vert\arg z\vert \leqslant \beta < \alpha$, when $\vert z\vert \to \infty$. One shows that f is slowly varying if and only if $\vert\arg z\vert \leqslant \beta < \alpha$, when $\vert z\vert \to \infty$.
Cohomology of finite covers
Allan
Calder
349-352
Abstract: For a finite dimensional CW-complex, X, and $q > 1$, it is shown that the qth Čech cohomology group based on finite open covers of X, $H_f^q(X)$, is naturally isomorphic to ${H^q}(X)$, the qth Čech cohomology of X (i.e. based on locally finite covers), and for reasonable X, ${H^1}(X)$ can be obtained algebraically from $ H_f^1(X)$.
Isotropic transport process on a Riemannian manifold
Mark A.
Pinsky
353-360
Abstract: We construct a canonical Markov process on the tangent bundle of a complete Riemannian manifold, which generalizes the isotropic scattering transport process on Euclidean space. By inserting a small parameter it is proved that the transition semigroup converges to the Brownian motion semigroup provided that the latter preserves the class $ {C_0}$. The special case of a manifold of negative curvature is considered as an illustration.
Representations of the $l\sp{1}$-algebra of an inverse semigroup
Bruce A.
Barnes
361-396
Abstract: In this paper the star representations on Hilbert space of the $ {l^1}$-algebra of an inverse semigroup are studied. It is shown that the set of all irreducible star representations form a separating family for the ${l^1}$-algebra. Then specific examples of star representations are constructed, and some theory of star representations is developed for the $ {l^1}$-algebra of a number of the most important examples of inverse semigroups.
Boundary value problems for second order differential equations in convex subsets of a Banach space
Klaus
Schmitt;
Peter
Volkmann
397-405
Abstract: Let E be a real Banach space, C a closed, convex subset of E and $f:[0,1] \times E \times E \to E$ be continuous. Let ${u_0},{u_1} \in C$ and consider the boundary value problem $u:[0,1] \to C$.
Erratum to: ``Homology with multiple-valued functions applied to fixed points'' (Trans. Amer. Math. Soc. {\bf 213} (1975), 407--427)
Richard
Jerrard
406
On the torus theorem and its applications
C. D.
Feustel
1-43
Abstract: In this paper, we prove the torus theorem and that manifolds in a certain class of 3-manifolds with toral boundary are determined by their fundamental groups alone. Both of these results were reported by F. Waldhausen. We also give an extension of Waldhausen's generalization of the loop theorem.
On the torus theorem for closed $3$-manifolds
C. D.
Feustel
45-57
Abstract: In this paper we give the appropriate generalization of the torus theorem to closed, sufficiently large, irreducible, orientable 3-manifolds.
The multiplicative behavior of $\mathcal{H}$
Pierre Antoine
Grillet
59-86
Abstract: Various results are given describing the product of two $\mathcal{H}$-classes in an arbitrary semigroup in terms of groups and homomorphisms.
Adjoint abelian operators on $L\sp{p}$ and $C(K)$
Richard J.
Fleming;
James E.
Jamison
87-98
Abstract: An operator A on a Banach space X is said to be adjoint abelian if there is a semi-inner product $[ \cdot , \cdot ]$ consistent with the norm on X such that $ [Ax,y] = [x,Ay]$ for all $ x,y \in X$. In this paper we show that every adjoint abelian operator on $ C(K)$ or $ {L^p}(\Omega ,\Sigma ,\mu )\;(1 < p < \infty ,p \ne 2)$ is a multiple of an isometry whose square is the identity and hence is of the form $Ax( \cdot ) = \lambda \alpha ( \cdot )(x \circ \phi )( \cdot )$ where $\alpha$ is a scalar valued function with $\alpha ( \cdot )\alpha \circ \phi ( \cdot ) = 1$ and $\phi$ is a homeomorphism of K (or a set isomorphism in case of ${L^p}(\Omega ,\Sigma ,\mu ))$ with $\phi \circ \phi =$ identity (essentially).
Bernoulli convolutions and differentiable functions
R.
Kaufman
99-104
Abstract: Bernoulli convolutions, similar in structure to convolutions with a constant ratio, are considered in relation to differentiable transformations. A space of functions on the Cantor set leads to highly singular measures that nevertheless resemble absolutely continuous measures sufficiently to control their Fourier-Stieltjes transforms.
Convolution equations for vector-valued entire functions of nuclear bounded type
Thomas A. W.
Dwyer
105-119
Abstract: Given two complex Banach spaces E and F, convolution operators ``with scalar coefficients'' are characterized among all convolution operators on the space $ {H_{Nb}}(E';F)$. THEOREM 2. Solutions of homogeneous convolution equations of scalar type can be approximated in
Simplicial geometry and transportation polytopes
Ethan D.
Bolker
121-142
Abstract: The classical transportation problem is the study of the set of nonnegative matrices with prescribed nonnegative row and column sums. It is aesthetically satisfying and perhaps potentially useful to study more general higher dimensional rectangular arrays whose sums on some subarrays are specified. We show how such problems can be rewritten as problems in homology theory. That translation explains the appearance of bipartite graphs in the study of the classical transportation problem. In our generalization, higher dimensional cell complexes occur. That is why the general problem requires a substantial independent investigation of simplicial geometry, the name given to the class of theorems on the geometry of a cell complex which depend on a particular cellular decomposition. The topological invariants of the complex are means, not ends. Thus simplicial geometry attempts to do for complexes what graph theory does for graphs. The dual title of this paper indicates that we shall spend as much time studying simplicial geometry for its own sake as applying the results to transportation problems. Our results include formulas for inverting the boundary operator of an acyclic cell complex, and some information on the number of such subcomplexes of a given complex.
Coincidence index and multiplicity
B.
Laloux;
J.
Mawhin
143-162
Abstract: This paper is devoted to the extension, in the frame of coincidence degree theory in normed spaces, of the concept of Leray-Schauder index of an isolated fixed point. The generalization includes basic properties of the coincidence index, Krasnosel'skiĭ type theorems for the case of noninvertible linear part and a Leray-Schauder's type formula relating the index and spectral theory in the linear case. This last problem needs the introduction of the concept of characteristic value for some couples of linear mappings and of its multiplicity.
A special integral and a Gronwall inequality
Burrell W.
Helton
163-181
Abstract: This paper considers a special integral $(I)\smallint _a^b(fdg + H)$ which is a subdivision-refinement-type limit of the approximating sum $\displaystyle \sum\limits_1^n {\{ f({t_i})[g({x_i}) - g({x_{i - 1}})] + H({x_{i - 1}},{x_i})\} ,}$ where ${x_{i - 1}} < {t_i} < {x_i}$. The author shows, with appropriate restrictions, that $(I)\smallint _a^b(fdg + H)$ exists if and only if $\displaystyle (R)\smallint _x^y(fdg + H - {A^ - }) = (L)\smallint _x^y(fdg + H + {A^ + })$ for $a \leqslant x < y \leqslant b$, where $ A(p,q) = [f(q) - f(p)][g(q) - g(p)],{A^ - }(p,q) = A({q^ - },q)$ and ${A^ + }(p,q) = A(p,{p^ + })$. Furthermore, if either of the equivalent statements is true, then all the integrals are equal. These equivalent statements are used to prove an integration-by-parts theorem and to solve a Gronwall inequality involving this special integral. Product integrals are used in the solution of the Gronwall inequality.
The semilattice tensor product of distributive lattices
Grant A.
Fraser
183-194
Abstract: We define the tensor product $A \otimes B$ for arbitrary semilattices A and B. The construction is analogous to one used in ring theory (see [4], [7], [8]) and different from one studied by A. Waterman [12], D. Mowat [9], and Z. Shmuely [10]. We show that the semilattice $A \otimes B$ is a distributive lattice whenever A and B are distributive lattices, and we investigate the relationship between the Stone space of $A \otimes B$ and the Stone spaces of the factors A and B. We conclude with some results concerning tensor products that are projective in the category of distributive lattices.
Compactifications of spaces of functions and integration of functionals
L. Š.
Grinblat
195-223
Abstract: For a locally compact space there exists a compactification such that all its points are effectively describable, namely, Alexandroff's onepoint compactification. The effective construction of compactifications for numerous standard separable metric spaces is already a very nontrivial problem. We propose a method of compactification which enables us to effectively construct compactifications of some spaces of functions (for example, of a ball in ${L_p}( - \infty ,\infty )$). It will be shown that the study of compactifications of spaces of functions is of principal importance in the theory of integration of functionals and in limit theorems for random processes.
Existence of periodic solutions of nonlinear differential equations
R.
Kannan
225-236
Abstract: The nonlinear differential equation $ x'' = f(t,x(t))$, f being $2\pi$-periodic in t, is considered for the existence of $2\pi$-periodic solutions. The equation is reduced to an equivalent system of two Hammerstein equations. The case of nonlinear perturbation at resonance is also discussed.
Real prime flows
H. B.
Keynes;
D.
Newton
237-255
Abstract: In this paper, we construct examples of real-type prime flows and study these examples in detail. General properties of prime flows are studied, with emphasis on proximality conditions and properties of automorphisms. Examples of prime flows which are not POD are shown to exist, and results analogous to number-theoretic properties, such as a ``unique factorisation'' theorem, are shown to hold for prime flows.
On the connectedness of homomorphisms in topological dynamics
D.
McMahon;
T. S.
Wu
257-270
Abstract: Let (X, T) be a minimal transformation group with compact Hausdorff phase space. We show that if $\phi :(X,T) \to (Y,T)$ is a distal homomorphism and has a structure similar to the structure Furstenberg derived for distal minimal sets, then for T belonging to a class of topological groups T, the homomorphism $X \to X/S(\phi )$ has connected fibers, where $S(\phi )$ is the relativized equicontinuous structure relation. The class T is defined by Sacker and Sell as consisting of all groups T with the property that there is a compact set $K \subseteq T$ such that T is generated by each open neighborhood of K. They show that for such T, a distal minimal set which is a finite-to-one extension of an almost periodic minimal set is itself an almost periodic minimal set. We provide an example that shows that the restriction on T cannot be dropped. As one of the preliminaries to the above we show that given $\phi :(X,T) \to (Y,T)$, the relation $Rc(\phi )$ induced by the components in the fibers relative to $\phi$, i.e.,
Light open and open mappings on manifolds. II
John J.
Walsh
271-284
Abstract: Sufficient conditions are given for the existence of light open mappings between p.l. manifolds. In addition, it is shown that a mapping f from a p.l. manifold ${M^m},m \geqslant 3$, to a polyhedron Q is homotopic to an open mapping of M onto Q iff the index of $ {f_\ast}({\pi _1}(M))$ in ${\pi _1}(Q)$ is finite. Finally, it is shown that an open mapping of ${M^m}$ onto a p.l. manifold ${N^n},n \geqslant m \geqslant 3$, can be approximated by a light open mapping of M onto N.
A local spectral theory for operators. V. Spectral subspaces for hyponormal operators
Joseph G.
Stampfli
285-296
Abstract: In the first part of the paper we show that the local resolvent of a hyponormal operator satisfies a rather stringent growth condition. This result enables one to show that under a mild restriction, hyponormal operators satisfy Dunford's C condition. This in turn leads to a number of corollaries concerning invariant subspaces. In the second part we consider the local spectrum of a subnormal operator. The third section is concerned with the study of quasi-triangular hyponormal operators.
$L$-functions of a quadratic form
T.
Callahan;
R. A.
Smith
297-309
Abstract: Let Q be a positive definite integral quadratic form in n variables, with the additional property that the adjoint form ${Q^\dag }$ is also integral. Using the functional equation of the Epstein zeta function, we obtain a symmetric functional equation of the L-function of Q with a primitive character $\omega \bmod q$ (additive or multiplicative) defined by $ \Sigma \omega (Q({\text{x}}))Q{({\text{x}})^{ - s}},\operatorname{Re} (s) > n/2$, where the summation extends over all $ {\text{x}} \in {Z^n},{\text{x}} \ne 0$; our result does not depend upon the usual restriction that q be relatively prime to the discriminant of Q, but rather on a much milder restriction.
Hypoellipticity of certain degenerate elliptic boundary value problems
Yakar
Kannai
311-328
Abstract: The concept of hypoellipticity for degenerate elliptic boundary value problems is defined, and its relation with the hypoellipticity of certain pseudo-differential operators on the boundary is discussed (for second order equations). A theorem covering smoothness of solutions of boundary value problems such as $ a(x)\partial u/\partial n + b(x)u = f(x)$ for the Laplace equation is proved. An almost complete characterization of hypoelliptic boundary value problems for elliptic second order equations in two dimensions is given via analysis of hypoelliptic pseudo-differential operators in one variable.
Generalized Kloosterman sums and the Fourier coefficients of cusp forms
L. Alayne
Parson
329-350
Abstract: Certain generalized Kloosterman sums connected with congruence subgroups of the modular group and suitably restricted multiplier systems of half-integral degree are studied. Then a Fourier coefficient estimate is obtained for cusp forms of half-integral degree on congruence subgroups of the modular group and the Hecke groups $G(\sqrt 2 )$ and $ G(\sqrt 3 )$.
On the jump of an $\alpha $-recursively enumerable set
Richard A.
Shore
351-363
Abstract: We discuss the proper definition of the jump operator in $\alpha $-recursion theory and prove a sample theorem: There is an incomplete $\alpha$-r.e. set with jump $0''$ unless there is precisely one nonhyperregular $\alpha$-r.e. degree. Thus we have a theorem in the first order language of Turing degrees with the jump which fails to generalize to all admissible $ \alpha$.
Continuous cohomology for compactly supported vectorfields on $R\sp{n}$
Steven
Shnider
364-377
Abstract: In this paper we study the Gelfand-Fuks cohomology of the Lie algebra of compactly supported vectorfields on ${{\mathbf{R}}^n}$ and establish the degeneracy of a certain spectral sequence at the ${E_1}$ level. We apply this result to the study of another spectral sequence introduced by Resetnikov for the cohomology of the algebra of vectorfields on ${S^n}$.
Characteristic classes for the deformation of flat connections
Huei Shyong
Lue
379-393
Abstract: In this paper, we study the secondary characteristic classes derived from flat connections. Let M be a differential manifold with flat connection ${\omega _0}$. If f is a diffeomorphism of M, then ${\omega _1} = {f^\ast}{\omega _0}$ is another flat connection. Denote by $\alpha$ the difference of these two connections. Then $\alpha$ and its exterior covariant derivative $ D\alpha$ are both tensorial forms on M. To each invariant polynomial $ \varphi$ of $ {\text{GL}}(n,{\text{R}})$, where $n = \dim M,\varphi (\alpha ;D\alpha )$ is a globally defined form on M. The class $ \{ \varphi (\alpha ;D\alpha )\} \in H(M;{\text{R}})$ for $\deg \varphi > 1$ gives rise to an obstruction of the deformability from $ {\omega _0}$ to ${\omega _1}$. In particular, we prove that $ ( + )$ and $( - )$ connections, in the sense of E. Cartan, cannot be deformed to each other.
Estimates for the $\bar \partial $-Neumann operator in weighted Hilbert spaces
Sidney L.
Hantler
395-406
Abstract: Estimates for the $\bar \partial$ operator are used to derive estimates for the Neumann operator in weighted Hilbert spaces. The technique is similar to that used to prove regularity of solutions of elliptic partial differential equations. A priori estimates are first obtained for smooth compactly supported forms and these estimates are then extended by suitable approximation results. These estimates are applied to give new bounds for the reproducing kernels in the subspaces of entire functions.
Lie group representations and harmonic polynomials of a matrix variable
Tuong
Ton-That
1-46
Abstract: The first part of this paper deals with problems concerning the symmetric algebra of complex-valued polynomial functions on the complex vector space of n by k matrices. In this context, a generalization of the so-called ``classical separation of variables theorem'' for the symmetric algebra is obtained. The second part is devoted to the study of certain linear representations, on the above linear space (the symmetric algebra) and its subspaces, of the complex general linear group of order k and of its subgroups, namely, the unitary group, and the real and complex special orthogonal groups. The results of the first part lead to generalizations of several well-known theorems in the theory of group representations. The above representation, of the real special orthogonal group, which arises from the right action of this group on the underlying vector space (of the symmetric algebra) of matrices, possesses interesting properties when restricted to the Stiefel manifold. The latter is defined as the orbit (under the action of the real special orthogonal group) of the n by k matrix formed by the first n row vectors of the canonical basis of the k-dimensional real Euclidean space. Thus the last part of this paper is involved with questions in harmonic analysis on this Stiefel manifold. In particular, an interesting orthogonal decomposition of the complex Hilbert space consisting of all square-integrable functions on the Stiefel manifold is also obtained.
Inseparable finite solvable groups
Homer
Bechtell
47-60
Abstract: A finite group is called inseparable if the only proper normal subgroup over which it splits is the identity element. The E-residual, for the formation E of groups in which all Sylow subgroups are elementary abelian, appears to control the action of splitting. In this article, inseparable solvable groups are identified that have a metacyclic Fitting subgroup and the E-residual a p-group.
Generalized and classical solutions of the nonlinear stationary Navier-Stokes equations
Victor L.
Shapiro
61-79
Abstract: New regularity results in domains of Euclidean 3-space are established for the generalized solutions of the nonlinear stationary Navier-Stokes equations in terms of Dini criteria on the external force.
Units and one-sided units in regular rings
Gertrude
Ehrlich
81-90
Abstract: A ring R is unit regular if for every $a \in R$, there is a unit $x \in R$ such that $axa = a$, and one-sided unit regular if for every $a \in R$, there is a right or left invertible element $x \in R$ such that $axa = a$. In this paper, unit regularity and one-sided unit regularity are characterized within the lattice of principal right ideals of a regular ring R (Theorem 3). If M is an A-module and $R = {\text{End}_A}$ M is a regular ring, then R is unit regular if and only if complements of isomorphic summands of M are isomorphic, and R is one-sided unit regular if and only if complements of isomorphic summands of M are comparable with respect to the relation ``is isomorphic to a submodule of'' (Theorem 2). A class of modules is given for whose endomorphism rings it is the case that regularity in conjunction with von Neumann finiteness is equivalent to unit regularity. This class includes all abelian torsion groups and all nonreduced abelian groups with regular endomorphism rings.
Some inequalities for polynomials with a prescribed zero
Q. I.
Rahman;
G.
Schmeisser
91-103
Abstract: Let $f(z)$ be a polynomial of degree n. Given that $f(z)$ has a zero on the circle $\vert z\vert = \rho \;(0 < \rho < \infty )$, we estimate $\vert f(0)\vert$ and $\displaystyle {({(2\pi )^{ - 1}}\int _0^{2\pi }\vert f({e^{i\theta }}){\vert^2}d\theta )^{1/2}}$ in terms of $ {\max _{\vert z\vert = 1}}\vert f(z)\vert$. We also consider some other related problems.
The quasi-orbit space of continuous $C\sp{\ast} $-dynamical systems
Hiroshi
Takai
105-113
Abstract: Let $(A,G,\alpha )$ be a separable continuous $ {C^\ast}$-dynamical system. Suppose G is amenable and $\alpha$ is free on the dual  of A. Then the quasi-orbit space $ {({\text{Prim}}\;A/\alpha )^ \sim }$ of the primitive ideal space Prim A of A by $\alpha$ is homeomorphic to the induced primitive ideal space which is dense in the primitive ideal space Prim $ {C^\ast}(A;\alpha )$ of the ${C^\ast}$-crossed product ${C^\ast}(A;\alpha )$ of A by $\alpha$.
Some special decompositions of $E\sp{3}$
Charles D.
Bass
115-130
Abstract: A great deal of attention has been given to the question: which upper semicontinuous decompositions of ${E^3}$ into pointlike continua give ${E^3}$. It has recently been determined that some decompositions of ${E^3}$ into points and straight line segments give decomposition spaces which are topologically distinct from ${E^3}$. In this paper we apply a new condition to the set of nondegenerate elements of a decomposition which enables one to conclude that the resulting decomposition space is homeomorphic to $ {E^3}$.
An order topology in ordered topological vector spaces
Lyne H.
Carter
131-144
Abstract: An order topology $ \Omega$ that can be defined on any partially-ordered space has as its closed sets those that contain the (o)-limits of all their (o)-convergent nets. In this paper we study the situation in which a topological vector space with a Schauder basis is ordered by the basis cone. In a Fréchet space $(E,\tau )$, we obtain necessary and sufficient conditions both for $ \tau \subset \Omega$ and for $\tau = \Omega $. Characterizations of (o)- and $\Omega$-convergence and of $\Omega$-closed sets are obtained. The equality of the order topology with the strong topology in certain dual Banach spaces is related to weak sequential completeness through the concept of a shrinking basis.
Convolution, differential equations, and entire functions of exponential type
Dale H.
Mugler
145-187
Abstract: The Whittaker-Shannon interpolation formula, or ``cardinal series", is a special case of the more general linear integro-differential equation with constant complex coefficients $\Sigma _{n = 0}^\infty {a_n}{f^{(n)}}(z) = \smallint f(z - t)d\mu (t)$ where the integral is taken over the whole real line with respect to the measure $\mu$. In this study, I show that many of these equations provide representations for particular classes of entire functions of exponential type. That is, every function in the class satisfies the equation and conversely every solution of the equation is a member of the class of functions. When the measure in the convolution integral above is chosen to be discrete, a particular form of the above type of equation is an equation of periodicity $f(z) = f(z + \tau )$. Following an extensive treatment of the general equation written above, the study concludes by offering a generalization in terms of these convolution equations of a classical theorem in complex analysis concerning periodic entire functions.
Cylindric algebras of first-order languages
Dale
Myers
189-202
Abstract: We show when two countable first-order languages have isomorphic cylindric algebras.
On the topological extension to the boundary of biholomorphic maps in $C\sp{n}$
R. Michael
Range
203-216
Abstract: Let $F:{D_1} \to {D_2}$ be a biholomorphic map between bounded domains in $ {{\mathbf{C}}^n}$ with piecewise smooth strictly pseudoconvex boundaries. It is shown that F is Hölder continuous of some positive order, and hence F extends to a homeomorphism of the closures of the domains. This generalizes recent results of G. M. Henkin and N. Vormoor for domains with smooth strictly pseudoconvex boundary.
Cartan subspaces of symmetric Lie algebras
J.
Lepowsky;
G. W.
McCollum
217-228
Abstract: A symmetric Lie algebra is defined, following J. Dixmier, to be a Lie algebra $\mathfrak{g}$ with a decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ such that $ \mathfrak{k}$ is a subalgebra of $ \mathfrak{g},[\mathfrak{k},\mathfrak{p}] \subset \mathfrak{p}$ and $[\mathfrak{p},\mathfrak{p}] \subset \mathfrak{k}$. A definition of Cartan subspace of a symmetric Lie algebra is given, and a theory is presented which parallels the standard theory of Cartan subalgebras of Lie algebras, and which generalizes the classical results for real and complex semisimple symmetric Lie algebras.
A generalization of H. Weyl's ``unitary trick''
J.
Lepowsky
229-236
Abstract: H. Weyl's ``unitary trick'' is generalized to the context of semisimple symmetric Lie algebras with Cartan subspaces, over fields of characteristic zero. As an illustration of its usefulness, the result is used to transfer to characteristic zero an important theorem in the representation theory of real semisimple Lie algebras.
Linear factorization of conical polynomials over certain nonassociative algebras
J.
Lepowsky
237-248
Abstract: Conical polynomials are defined as certain polynomials in quadratic elements of the universal enveloping algebra of a semisimple symmetric Lie algebra over a field of characteristic zero. These polynomials were used in an earlier paper to describe the conical vectors in certain induced modules. Here it is shown that when the base field is extended to a certain type of nonassociative algebra, the conical polynomials can be factored ``linearly". One such nonassociative algebra is discussed in detail--an (alternative) composition algebra intimately related to the structure of the Lie algebra and studied earlier by B. Kostant in the context of real semisimple Lie algebras. The linear factorization leads in a later paper to an extension of the earlier work on conical vectors in induced modules.
General position maps for topological manifolds in the ${2\over 3}$rds range
Jerome
Dancis
249-266
Abstract: For each proper map f of a topological m-manifold M into a topological q-manifold Q, $m \leqslant (2/3)q - 1/3$, we build an approximating map g such that the set of singularities S of g is a locally finite simplicial $(2m - q)$-complex locally tamely embedded in M, $ g(S)$ is another locally finite complex $g\vert:S \twoheadrightarrow g(S)$ is a piecewise linear map and g is a locally flat embedding on the complement of S. Furthermore if $f\vert\partial M$ is a locally flat embedding then we construct g so that it agrees with f on $\partial M$ even when $f(\partial M)$ meets $\operatorname{Int} Q \cap f({\operatorname{Int}}\;M)$. In addition we present two other general position lemmas. Also, we show that given two codimension $\geqslant 3$ locally flat topological submanifolds M and V of a topological manifold Q, $\dim \;M + \dim \;V - \dim \;Q \leqslant 3$, then we can move M so that M and V are transverse in Q.
Almost isolated spectral parts and invariant subspaces
C. R.
Putnam
267-277
Abstract: Let T be an operator with spectrum $ \sigma (T)$ on a Hilbert space. A compact subset E of $\sigma (T)$ is called a disconnected part of $\sigma (T)$ if, for some bounded open set A, E is the closure of $ \sigma (T) \cap A$ and $\sigma (T) - E$ is the union of the isolated parts of $\sigma (T)$ lying completely outside the closure of A. The set E is called an almost isolated part of $ \sigma (T)$ if, in addition, the set A can be chosen so as to have a rectifiable boundary $ \partial A$ on which the subset $ \sigma (T) \cap \partial A$ has arc length measure 0. The following results are obtained. If T is subnormal and if E is a disconnected part of $ \sigma (T)$ then there exists a reducing subspace $ \mathfrak{M}$ of T for which $ \sigma (T\vert\mathfrak{M}) = E$. If ${T^\ast}$ is hyponormal and if E is an almost isolated part of $ \sigma (T)$ then there exists an invariant subspace $ \mathfrak{M}$ of T for which $ \sigma (T\vert\mathfrak{M}) = E$. An example is given showing that if T is arbitrary then an almost isolated part of $\sigma (T)$ need not be the spectrum of the restriction of T to any invariant subspace.
An index theorem for $p$-adic differential operators
A.
Adolphson
279-293
Abstract: A system of first order linear differential operators satisfying conditions arising naturally in geometry (rational function coefficients, regular singularities, non-Liouville exponents) is considered. It is shown that the index of the system on certain spaces of holomorphic functions can be calculated by restricting to a subspace of rational functions. This is applied to obtain an explicit formula for the index of a single kth order linear differential operator.
A product formula for generalizations of the Kervaire invariant
Edgar H.
Brown
295-311
Abstract: Formulas are developed for the Arf invariant of the product of two manifolds in terms of invariants of the factors. If the Wu orientations are carefully chosen the formula is $ \sigma (M \times N) = \sigma (M)\sigma (N)$.
The generalized Fredholm operators
Kung Wei
Yang
313-326
Abstract: Let X, Y be Banach spaces over either the real field or the complex field. A continuous linear operator will be called a generalized Fredholm operator if $ T(X)$ is closed in Y, and Ker T and Coker T are reflexive Banach spaces. A theory similar to the classical Fredholm theory exists for the generalized Fredholm operators; and the similarity brings out the correspondence: Reflexive Banach spaces $\leftrightarrow$ finite-dimensional spaces, weakly compact operators $\leftrightarrow$ compact operators, generalized Fredholm operators $ \leftrightarrow$ Fredholm operators, Tauberian operators with closed range $\leftrightarrow $ semi-Fredholm operators.
On the uniqueness of solutions to hyperbolic boundary value problems
C. C.
Travis
327-336
Abstract: The paper is concerned with the uniqueness of solutions to non-well-posed hyperbolic boundary value problems. Both regular and singular boundary value problems are considered. For the singular problem a class of boundary conditions is considered that has not appeared in the literature before in connection with this problem.
Piecewise linear bundles in the metastable range
Kenneth C.
Millett
337-350
Abstract: For numerable vector bundles a nonzero section determines a unique trivial line subbundle containing the section and this subbundle is a direct summand of the bundle. The main result, a consequence of concordance-isotopy theory, states that in the metastable range a nonzero section to a piecewise linear ${{\mathbf{R}}^n}$ bundle determines a unique trivial line subbundle and that this is the best possible result. This fact is then compared with the known failure of the summand property below the stable range.
Signature of links
Louis H.
Kauffman;
Laurence R.
Taylor
351-365
Abstract: Let L be an oriented tame link in the three sphere ${S^3}$. We study the Murasugi signature, $\sigma (L)$, and the nullity, $\eta (L)$. It is shown that $\sigma (L)$ is a locally flat topological concordance invariant and that $\eta (L)$ is a topological concordance invariant (no local flatness assumption here). Known results about the signature are re-proved (in some cases generalized) using branched coverings.
Some locally convex spaces of continuous vector-valued functions over a completely regular space and their duals
A.
Katsaras
367-387
Abstract: The strict, superstrict and the $ {\beta _F}$ topologies are defined on a space A of continuous functions from a completely regular space into a Banach space E. Properties of these topologies are discussed and the corresponding dual spaces are identified with certain spaces of operator-valued measures. In case E is a Banach lattice, A becomes a lattice under the pointwise ordering and the strict and superstrict duals of A coincide with the spaces of all $\tau$-additive and all $\sigma$-additive functionals on A respectively.
Finite groups as isometry groups
D.
Asimov
388-390
Abstract: We show that given any finite group G of cardinality $k + 1$, there is a Riemannian sphere ${S^{k - 1}}$ (imbeddable isometrically as a hypersurface in $ {{\mathbf{R}}^k}$) such that its full isometry group is isomorphic to G. We also show the existence of a finite metric space of cardinality $k(k + 1)$ whose full isometry group is isomorphic to G.
Erratum to: ``A constructive ergodic theorem'' (Trans. Amer. Math. Soc. {\bf 164} (1972), 115--137)
J. A.
Nuber
393
Geodesics in piecewise linear manifolds
David A.
Stone
1-44
Abstract: A simplicial complex M is metrized by assigning to each simplex $a \in {\mathbf{M}}$ a linear simplex ${a^\ast}$ in some Euclidean space ${{\mathbf{R}}^k}$ so that face relations correspond to isometries. An equivalence class of metrized complexes under the relation generated by subdivisions and isometries is called a metric complex; it consists primarily of a polyhedron M with an intrinsic metric $ {\rho _{\mathbf{M}}}$. This paper studies geodesics in metric complexes. Let $P \in {\mathbf{M}}$; then the tangent space ${T_P}({\mathbf{M}})$ is canonically isometric to an orthogonal product of cones from $ P,{{\mathbf{R}}^k} \times {\nu _P}({\mathbf{M}})$; once k is as large as possible. $ {\nu _P}({\mathbf{M}})$ is called the normal geometry at P in M. Let $P\bar X$ be a tangent direction at P in $ {\nu _P}({\mathbf{M}})$. I define numbers $ {\kappa _ + }(P\bar X)$ and $ {\kappa _ - }(P\bar X)$, called the maximum and minimum curvatures at P in the direction $P\bar X$. THEOREM. Let M be a complete, simply-connected metric complex which is a p.l. n-manifold without boundary. Assume $ {\kappa _ + }(P\bar X) \leqslant 0$ for all $P \in {\mathbf{M}}$ and all $ P\bar X \subseteq {\nu _P}({\mathbf{M}})$. Then M is p.l. isomorphic to $ {{\mathbf{R}}^n}$. This is analogous to a well-known theorem for smooth manifolds by E. Cartan and J. Hadamard. THEOREM (ROUGHLY). Let M be a complete metric complex which is a p.l. n-manifold without boundary. Assume (1) there is a number $\kappa \geqslant 0$ such that $ {\kappa _ - }(P\bar X) \geqslant \kappa$ whenever P is in the $ (n - 2)$-skeleton of M and whenever $ P\bar X \subseteq {\nu _P}({\mathbf{M}})$; (2) the simplexes of M are bounded in size and shape. Then M is compact. This is analogous to a weak form of a well-known theorem of S. B. Myers for smooth manifolds.
Espaces fibr\'es lin\'eaires faiblement n\'egatifs sur un espace complexe
Vincenzo
Ancona
45-61
Abstract: Let F be a coherent sheaf over a compact reduced complex space $X,L($F$)$ the linear fibre space associated with F, ${S^k}($F$)$ the kth symmetric power of F. We show that if the zero-section of $L($F$)$ is exceptional, then ${H^r}(X,$E${ \otimes _{{O_X}}}{S^k}($F$ )) = 0$ for every coherent sheaf E on X and for $r \geqslant 1$ and sufficiently large k. Using this result, we deduce moreover that Supp F is a Moišezon space.
Evolution system approximations of solutions to closed linear operator equations
Seaton D.
Purdom
63-79
Abstract: With S a linearly ordered set with the least upper bound property, with g a nonincreasing real-valued function on S, and with A a densely defined dissipative linear operator, an evolution system M is developed to solve the modified Stieljes integral equation $M(s,t)x = x + A((L)\smallint _S^t dgM( \cdot ,t)x)$. An affine version of this equation is also considered. Under the hypothesis that the evolution system associated with the linear equation is strongly (resp. weakly) asymptotically convergent, an evolution system is used to approximate strongly (resp. weakly) solutions to the closed operator equation $ Ay = - z$.
Balanced subgroups of abelian groups
Roger H.
Hunter
81-98
Abstract: The balanced subgroups of Fuchs are generalised to arbitrary abelian groups. Projectives and injectives with respect to general balanced exact sequences are classified; a new class of groups is introduced in order to classify these projectives.
The sets that are scissor congruent to an unbounded convex subset of the plane
Sydell Perlmutter
Gold
99-117
Abstract: It is shown that an unbounded convex plane body is scissor congruent to the union of a congruent body with a finite number of arbitrary topological discs. It is proved that 'is scissor congruent to' is an equivalence relation. Thus two unbounded convex plane bodies are scissor congruent if and only if the union of one with a finite number of topological discs is scissor congruent to the other.
Universally torsionless and trace modules
Gerald S.
Garfinkel
119-144
Abstract: We study, over an arbitrary ring R, a class of right modules intermediate between the projective and the flat content modules. Over the ring of rational integers these modules are the locally free abelian groups. Over any commutative ring they are the modules which remain torsionless under all scalar extensions. They each possess a certain separability property exactly when R is left semihereditary. We define M to be universally torsionless if the natural map $M \otimes A \to {\operatorname{Hom}}({M^\ast},A)$ is monic for all left modules A. We give various equivalent conditions for M to be universally torsionless, one of which is that M is a trace module, i.e. that $x \in M \cdot {M^\ast}(x)$ for all $x \in M$. We show the countably generated such modules are projective. Chase showed that rings over which products of projective or flat modules are also, respectively, projective or flat have other interesting properties and that they are characterized by certain left ideal theoretical conditions. We show similar results hold when the trace or content properties are preserved by products.
Automorphisms of ${\rm GL}\sb{n}(R)$
Bernard R.
McDonald
145-159
Abstract: Let R be a commutative ring and S a multiplicatively closed subset of R having no zero divisors. The pair $\langle R,S\rangle$ is said to be stable if the ring of fractions of R, ${S^{ - 1}}R$, defined by S is a ring for which all finitely generated projective modules are free. For a stable pair $ \langle R,S\rangle$ assume 2 is a unit in R and V is a free R-module of dimension $ \geqslant 3$. This paper examines the action of a group automorphism of $ GL(V)$ (the general linear group) on the elementary matrices relative to a basis B of V. In the case that R is a local ring, a Euclidean domain, a connected semilocal ring or a Dedekind domain whose quotient field is a finite extension of the rationals, we obtain a description of the action of the automorphism on all elements of $GL(V)$.
Geometry of Banach spaces of functions associated with concave functions
Paul
Hlavac;
K.
Sundaresan
161-189
Abstract: Let $(X,\Sigma ,\mu )$ be a positive measure space, and $ \phi$ be a concave nondecreasing function on ${R^ + } \to {R^ + }$ with $\phi (0) = 0$. Let $ {N_\phi }(R)$ be the Lorentz space associated with the function $\phi$. In this paper a complete characterization of the extreme points of the unit ball of ${N_\phi }(R)$ is provided. It is also shown that the space $ {N_\phi }(R)$ is not reflexive in all nontrivial cases, thus generalizing a result of Lorentz. Several analytical properties of spaces ${N_\phi }(R)$, and their abstract analogues ${N_\phi }(E)$, are obtained when E is a Banach space.
Cell-like closed-$0$-dimensional decompositions of $R\sp{3}$ are $R\sp{4}$ factors
Robert D.
Edwards;
Richard T.
Miller
191-203
Abstract: It is proved that the product of a cell-like closed-0-dimensional upper semicontinuous decomposition of ${R^3}$ with a line is ${R^4}$. This establishes at once this feature for all the various dogbone-inspired decompositions of ${R^3}$. The proof makes use of an observation of L. Rubin that the universal cover of a wedge of circles admits a 1-1 immersion into the wedge crossed with $ {R^1}$.
Some $C\sp{\ast} $-alegebras with a single generator
Catherine L.
Olsen;
William R.
Zame
205-217
Abstract: This paper grew out of the following question: If X is a compact subset of ${C^n}$, is $C(X) \otimes {{\mathbf{M}}_n}$ (the $ {C^\ast}$-algebra of $n \times n$ matrices with entries from $ C(X)$) singly generated? It is shown that the answer is affirmative; in fact, $ A \otimes {{\mathbf{M}}_n}$ is singly generated whenever A is a $ {C^\ast}$-algebra with identity, generated by a set of $ n(n + 1)/2$ elements of which $n(n - 1)/2$ are selfadjoint. If A is a separable ${C^\ast}$-algebra with identity, then $A \otimes K$ and $ A \otimes U$ are shown to be singly generated, where K is the algebra of compact operators in a separable, infinite-dimensional Hilbert space, and U is any UHF algebra. In all these cases, the generator is explicitly constructed.
Essential embeddings of annuli and M\"obius bands in $3$-manifolds
James W.
Cannon;
C. D.
Feustel
219-239
Abstract: In this paper we give conditions when the existence of an ``essential'' map of an annulus or Möbius band into a 3-manifold implies the existence of an ``essential'' embedding of an annulus or Möbius band into that 3-manifold. Let $ {\lambda _1}$ and ${\lambda _2}$ be disjoint simple ``orientation reversing'' loops in the boundary of a 3-manifold M and A an annulus. Let $f:(A,\partial A) \to (M,\partial M)$ be a map such that ${f_\ast}:{\pi _1}(A) \to {\pi _1}(M)$ is monic and $f(\partial A) = {\lambda _1} \cup {\lambda _2}$. Then we show that there is an embedding $g:(A,\partial A) \to (M,\partial M)$ such that $g(\partial A) = {\lambda _1} \cup {\lambda _2}$.
Fixed point theorems for mappings satisfying inwardness conditions
James
Caristi
241-251
Abstract: Let X be a normed linear space and let K be a convex subset of X. The inward set, ${I_K}(x)$, of x relative to K is defined as follows: ${I_K}(x) = \{ x + c(u - x):c \geqslant 1,u \in K\}$. A mapping $T:K \to X$ is said to be inward if $Tx \in {I_K}(x)$ for each $x \in K$, and weakly inward if Tx belongs to the closure of ${I_K}(x)$ for each $x \in K$. In this paper a characterization of weakly inward mappings is given in terms of a condition arising in the study of ordinary differential equations. A general fixed point theorem is proved and applied to derive a generalization of the Contraction Mapping Principle in a complete metric space, and then applied together with the characterization of weakly inward mappings to obtain some fixed point theorems in Banach spaces.
On the topology of a compact inverse Clifford semigroup
D. P.
Yeager
253-267
Abstract: A description of the topology of a compact inverse Clifford semigroup S is given in terms of the topologies of its subgroups and that of the semilattice X of idempotents. It is further shown that the category of compact inverse Clifford semigroups is equivalent to a full subcategory of the category whose objects are inverse limit preserving functors $F:X \to G$, where X is a compact semilattice and G is the category of compact groups and continuous homomorphisms, and where a morphism from $ F:X \to G$ to $ G:Y \to G$ is a pair $(\varepsilon ,w)$ such that $\varepsilon$ is a continuous homomorphism of X into Y and w is a natural transformation from F to $ G\varepsilon$. Simpler descriptions of the topology of S are given in case the topology of X is first countable and in case the bonding maps between the maximal subgroups of S are open mappings.
Composition series and intertwining operators for the spherical principal series. II
Kenneth D.
Johnson
269-283
Abstract: In this paper, we consider the connected split rank one Lie group of real type ${F_4}$ which we denote by $F_4^1$. We first exhibit $F_4^1$ as a group of operators on the complexification of A. A. Albert's exceptional simple Jordan algebra. This enables us to explicitly realize the symmetric space $F_4^1/{\text{Spin}}(9)$ as the unit ball in ${{\mathbf{R}}^{16}}$ with boundary ${S^{15}}$. After decomposing the space of spherical harmonics under the action of ${\text{Spin}}(9)$, we obtain the matrix of a transvection operator of $F_4^1{\text{/Spin}}(9)$ acting on a spherical principal series representation. We are then able to completely determine the Jordan Holder series of any spherical principal series representation of $ F_4^1$.
A property for inverses in a partially ordered linear algebra
Taen Yu
Dai;
Ralph
DeMarr
285-292
Abstract: We consider a Dedekind $\sigma$-complete partially ordered linear algebra A which has the following property: if $x \in A$ and $1 \leqslant x$, then $- u \leqslant {x^{ - 1}}$, where $u = {u^2}$. This property is used to show that A must be commutative. We also show that A is the direct sum of two algebras, each of which behaves like an algebra of real-valued functions.
Generalized inverses and spectral theory
Constantin
Apostol;
Kevin
Clancey
293-300
Abstract: The concept of a generalized spectral projection associated with a subset in the semi-Fredholm domain of a bounded operator on a Hilbert space is introduced. These generalized spectral projections possess many of the desirable properties of spectral projections associated with spectral sets. In particular, generalized spectral projections are used to separate finite sets of singular points from the semi-Fredholm domain.
On the Stone-\v Cech compactification of the space of closed sets
John
Ginsburg
301-311
Abstract: For a topological space X, we denote by ${2^X}$ the space of closed subsets of X with the finite topology. If X is normal and $ {T_1}$, the map $F \to {\text{cl}_{\beta X}}F$ is an embedding of ${2^X}$ onto a dense subspace of ${2^{\beta X}}$, and, in this way, we regard ${2^{\beta X}}$ as a compactification of $ {2^X}$. This paper is motivated by the following question. When can ${2^{\beta X}}$ be identified as the Stone-Čech compactification of ${2^X}$? In [11], J. Keesling states that $ \beta ({2^X}) = {2^{\beta X}}$ implies ${2^X}$ is pseudocompact. We give a proof of this result and establish the following partial converse. If ${2^X} \times {2^X}$ is pseudocompact, then $ \beta ({2^X}) = {2^{\beta X}}$. A corollary of this theorem is that $\beta ({2^X}) = {2^{\beta X}}$ when X is ${\aleph _0}$-bounded.
Equivariant bordism and Smith theory. IV
R. E.
Stong
313-321
Abstract: This paper analyzes two types of characteristic numbers defined for manifolds with ${Z_4}$ action, showing their relation and that neither suffices to detect ${Z_4}$ equivariant bordism. This extends work of Bix who had given examples not detected by one type of number.
Homogeneous manifolds with negative curvature. I
Robert
Azencott;
Edward N.
Wilson
323-362
Abstract: This paper solves the problem of determining which Lie groups act simply transitively on a Riemannian manifold with negative curvature. The results obtained extend those of Heintze for the case of strictly negative curvature. Using results of Wolf and Heintze, it is established that every connected, simply connected, homogeneous manifold M with negative curvature admits a Lie group S acting simply transitively by isometries and every group with this property must be solvable. Formulas for the curvature tensor on M are established and used to show that the Lie algebra of any such group S must satisfy a number of structural conditions. Conversely, given a Lie algebra $\mathfrak{s}$ satisfying these conditions and any member of an easily constructed family of inner products on $\mathfrak{s}$, a metric deformation argument is used to obtain a modified inner product which gives rise to a left invariant Riemannian structure with negative curvature on the associated simply connected Lie group.
Involutions on homotopy spheres and their gluing diffeomorphisms
Chao Chu
Liang
363-391
Abstract: Let $hS({P^{2n + 1}})$ denote the set of equivalence classes of smooth fixed-point free involutions on $ (2n + 1)$-dimensional homotopy spheres. Browder and Livesay defined an invariant $ \sigma ({\Sigma ^{2n + 1}},T)$ for each $({\Sigma ^{2n + 1}},T) \in hS({P^{2n + 1}})$, where $\sigma \in Z$ if n is odd, $\sigma \in {Z_2}$ if n is even. They showed that for $n \geqslant 3,\sigma ({\Sigma ^{2n + 1}},T) = 0$ if and only if $ ({\Sigma ^{2n + 1}},T)$ admits a codim 1 invariant sphere. For any $({\Sigma ^{2n + 1}},T)$, there exists an A-equivariant diffeomorphism f of ${S^n} \times {S^n}$ such that $({\Sigma ^{2n + 1}},T) = ({S^n} \times {D^{n + 1}},A){ \cup _f}({D^{n + 1}} \times {S^n},A)$, where A denotes the antipodal map. Let $ \beta (f) = \sigma ({\Sigma ^{2n + 1}},T)$. In the case n is odd, we can show that the Browder-Livesay invariant is additive: $\beta (fg) = \beta (f) + \beta (g)$. But if n is even, then there exists f and g such that $\beta (gf) = \beta (g) + \beta (f) \ne \beta (fg)$. Let $ {D_0}({S^n} \times {S^n},A)$ be the group of concordance classes of A-equivariant diffeomorphisms which are homotopic to the identity map of ${S^n} \times {S^n}$. We can prove that ``For $ n \equiv 0,1,2 \bmod 4, hS({P^{2n + 1}})$ is in 1-1 correspondence with a subgroup of $ {D_0}({S^n} \times {S^n},A)$. As an application of these theorems, we demonstrated that ``Let $ \Sigma _0^{8k + 3}$ denote the generator of $ b{P_{8k + 4}}$. Then the number of $ (\Sigma _0^{8k + 3},T)$'s with $\sigma (\Sigma _0^{8k + 3},T) = 0$ is either 0 or equal to the number of $ ({S^{8k + 3}},T)$'s with $ \sigma ({S^{8k + 3}},T) = 0$, where ${S^{8k + 3}}$ denotes the standard sphere".
An asymptotic formula for an integral in starlike function theory
R. R.
London;
D. K.
Thomas
393-406
Abstract: The paper is concerned with the integral $\displaystyle H = \int _0^{2\pi }\vert f{\vert^\sigma }\vert F{\vert^\tau }{(\operatorname{Re} F)^\kappa }\;d\theta$ in which f is a function regular and starlike in the unit disc, $F = zf'/f$, and the parameters $\sigma ,\tau ,\kappa$ are real. A study of H is of interest since various well-known integrals in the theory, such as the length of $f(\vert z\vert = r)$, the area of $f(\vert z\vert \leqslant r)$, and the integral means of f, are essentially obtained from it by suitably choosing the parameters. An asymptotic formula, valid as $r \to 1$, is obtained for H when f is a starlike function of positive order $ \alpha$, and the parameters satisfy $\alpha \sigma + \tau + \kappa > 1,\tau + \kappa \geqslant 0,\kappa \geqslant 0,\sigma > 0$. Several easy applications of this result are made; some to obtaining old results, two others in proving conjectures of Holland and Thomas.
Normal structure of the one-point stabilizer of a doubly-transitive permutation group. I
Michael E.
O’Nan
1-42
Abstract: Let G be a doubly-transitive permutation group on a finite set X and x a point of X. Let ${N^x}$ be a normal subgroup of $ {G_x}$, the subgroup fixing x, such that ${N^x}$ is a T.I. set and not semiregular on $X - x$. Then, $ PSL(n,q) \subseteq G \subseteq P\Gamma L(n,q)$. Geometrical consequences of this result are also obtained.
Normal structure of the one-point stabilizer of a doubly-transitive permutation group. II
Michael E.
O’Nan
43-74
Abstract: The main result is that the socle of the point stabilizer of a doubly-transitive permutation group is abelian or the direct product of an abelian group and a simple group. Under certain circumstances, it is proved that the lengths of the orbits of a normal subgroup of the one point stabilizer bound the degree of the group. As a corollary, a fixed nonabelian simple group occurs as a factor of the socle of the one point stabilizer of at most finitely many doubly-transitive groups.
Inclusion maps of $3$-manifolds which induce monomorphisms of fundamental groups
Jože
Vrabec
75-93
Abstract: The main result is the following ``duality'' theorem. Let M be a 3-manifold, P a compact and connected polyhedral 3-submanifold of $\int M$, and X a compact and connected polyhedron in $\int P$. If ${\pi _1}(X) \to {\pi _1}(P)$ is onto, then $ {\pi _1}(M - P) \to {\pi _1}(M - X)$ is one-to-one. Some related results are proved, for instance: we can allow P to be noncompact if also X satisfies a certain noncompactness condition: if M lies in a 3-manifold W with $ {H_1}(W) = 0$, then the condition that $ {\pi _1}(X) \to {\pi _1}(P)$ is onto can be replaced by the weaker one that ${H_1}(X) \to {H_1}(P)$ is onto.
Partition theorems related to some identities of Rogers and Watson
Willard G.
Connor
95-111
Abstract: This paper proves two general partition theorems and several special cases of each with both of the general theorems based on four q-series identities originally due to L. J. Rogers and G. N. Watson. One of the most interesting special cases proves that the number of partitions of an integer n into parts where even parts may not be repeated, and where odd parts occur only if an adjacent even part occurs is equal to the number of partitions of n into parts $ \equiv \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 7 \pmod 20$. The companion theorem proves that the number of partitions of an integer n into parts where even parts may not be repeated, where odd parts $> 1$ occur only if an adjacent even part occurs, and where 1's occur arbitrarily is equal to the number of partitions of n into parts $\equiv \pm 1, \pm 2, \pm 5, \pm 6, \pm 8, \pm 9 \pmod 20$.
Equisingular deformations of Puiseux expansions
Augusto
Nobile
113-135
Abstract: Parametrizations of a deformation (over a complete local ring) of an irreducible algebroid curve are studied. With these parametrizations another definition of equisingularity (equivalent to the known ones) is given, by using their characteristic numbers. These methods are also used in the complex analytic case. The following applications of these techniques are given: a proof of the existence of a versal equisingular deformation in the complex analytic case; a proof that an equisingular formal family of branches is determined by its $\nu $-truncation ($ \nu$ large enough, depending only on the characteristic of the special fiber).
${\rm BV}$-functions, positive-definite functions and moment problems
P. H.
Maserick
137-152
Abstract: Let S be a commutative semigroup with identity 1 and involution. A complex valued function f on S is defined to be positive definite if ${\Pi _j}{\Delta _j}f(1) \geqslant 0$ where the ${\Delta _j}$'s belong to a certain class of linear sums of shift operators. For discrete groups the positive definite functions defined herein are shown to be the classically defined positive definite functions. An integral representation theorem is proved and necessary and sufficient conditions for a function to be the difference of two positive-definite functions, i.e. a BV-function, are given. Moreover the BV-function defined herein agrees with those previously defined for semilattices, with respect to the identity involution. Connections between the positive-definite functions and completely monotonic functions are discussed along with applications to moment problems.
Weakly smooth continua
Lewis
Lum
153-167
Abstract: We define and investigate a class of continua called weakly smooth. Smooth dendroids, weakly smooth dendroids, generalized trees, and smooth continua are all examples of weakly smooth continua. We generalize characterizations of the above mentioned examples to weakly smooth continua. In particular, we characterize them as compact Hausdorff spaces which admit a quasi order satisfying certain properties.
On the double suspension homomorphism
Mark
Mahowald
169-178
Abstract: This paper studies the family of unstable Adams spectral sequences, $E_2^{s,t}({S^{2n + 1}})$. The main results deal with the range of filtrations for which these groups stabilize and for which the groups $ E_2^{s,t}({\Omega ^2}{S^{2n + 1}},{S^{2n - 1}})$ stabilize.
A codimension theorem for pseudo-Noetherian rings
Kenneth
McDowell
179-185
Abstract: M. Auslander and M. Bridger have shown that the depth of a Noetherian local ring is the sum of the Gorenstein dimension and the depth of any given nonzero finitely generated module of finite Gorenstein dimension. In this paper it is demonstrated that this result remains true when suitably interpreted for the class of coherent rings herein entitled pseudo-Noetherian rings. This class contains, among others, all Noetherian rings and valuation domains as well as non-Noetherian local rings of infinite depth.
The Gaussian law and the law of the iterated logarithm for lacunary sets of characters
E.
Dudley
187-214
Abstract: Salem and Zygmund showed that the Gaussian law holds for Hadamard sequences of real numbers while Mary Weiss proved a similar result for the law of the iterated logarithm. In the present paper, the author obtains corresponding results for lacunary sets of characters of an arbitrary infinite compact abelian group. It is shown that the laws are best satisfied for a certain class of lacunary sets but that modified results apply to more general classes.
Infinite convolutions on locally compact Abelian groups and additive functions
Philip
Hartman
215-231
Abstract: Let ${\mu _1},{\mu _2}, \ldots $ be regular probability measures on a locally compact Abelian group G such that $ \mu = {\mu _1} \ast {\mu _2} \ast \cdots = \lim {\mu _1} \ast \cdots \ast {\mu _n}$ exists (and is a probability measure). For arbitrary G, we derive analogues of the Lévy theorem on the existence of an atom for $\mu$ and of the ``pure theorems'' of Jessen, Wintner and van Kampen (dealing with discrete ${\mu _1},{\mu _2}, \ldots$) in the case $G = {R^d}$. These results are applied to the asymptotic distribution $\mu$ of an additive function $f:{Z_ + } \to G$ after generalizing the Erdös-Wintner result $(G = {R^1})$ which implies that $ \mu$ is an infinite convolution of discrete probability measures.
The bracket ring of a combinatorial geometry. II. Unimodular geometries
Neil L.
White
233-248
Abstract: The bracket ring of a combinatorial geometry G is a ring of generalized determinants which acts as a universal coordinatization object for G. Our main result is the characterization of a unimodular geometry as a binary geometry such that the radical of the bracket ring is a prime ideal. This implies that a unimodular geometry has a universal coordinatization over an integral domain, which domain we construct explicitly using multisets. An ideal closely related to the radical, the coordinatizing radical, is also defined and proved to be a prime ideal for every binary geometry. To prove these results, we use two major preliminary theorems, which are of interest in their own right. The first is a bracket-theoretic version of Tutte's Homotopy Theorem for Matroids. We then prove that any two coordinatizations of a binary geometry over a given field are projectively equivalent.
Adjoint groups, regular unipotent elements and discrete series characters
G. I.
Lehrer
249-260
Abstract: It is shown that if G is a finite Chevalley group or twisted type over a field of characteristic p and U is a maximal p-subgroup of G then any nonlinear irreducible character of U vanishes on regular elements. For groups of adjoint type the linear content of the restriction to U of a discrete series character J of G is calculated and it is deduced that J takes the value 0 or ${( - 1)^s}$ on regular elements of U $ (s = {\text{rank}}\;G)$.
The stability problem in shape, and a Whitehead theorem in pro-homotopy
David A.
Edwards;
Ross
Geoghegan
261-277
Abstract: Theorem 3.1 is a Whitehead theorem in pro-homotopy for finite-dimensional pro-complexes. This is used to obtain necessary and sufficient algebraic conditions for a finite-dimensional tower of complexes to be pro-homotopy equivalent to a complex (§4) and for a finite-dimensional compact metric space to be pointed shape equivalent to an absolute neighborhood retract (§5).
Spectral analysis of finite convolution operators
Richard
Frankfurt
279-301
Abstract: In this paper the similarity problem for operators of the form $( \ast )\;T:f(x) \to \smallint _0^xk(x - t)f(t)dt$ on $ {L^2}(0,1)$ is studied. Let $K(z) = \smallint _0^1\;k(t){e^{itz}}dt$. A function $C(z)$ is called a symbol for T if $ C(z)$ can be written in the form $ C(z) = K(z) + {e^{iz}}G(z)$, where $G(z)$ is a function bounded and analytic in a half plane $y > \delta $, for some real number $ \delta$. Under suitable restrictions, it is shown that two operators of the form $( \ast )$ will be similar if they possess symbols which are asymptotically close together as $z \to \infty$ in some half plane $y > \delta$.
Differential geometry on simplicial spaces
Michael A.
Penna
303-323
Abstract: A simplicial space M is a separable Hausdorff topological space equipped with an atlas of linearly related charts of varying dimension; for example every polyhedron is a simplicial space in a natural way. Every simplicial space possesses a natural structure complex of sheaves of piecewise smooth differential forms, and the homology of the corresponding de Rham complex of global sections is isomorphic to the real cohomology of M. A cosimplicial bundle is a continuous surjection $ \xi :E \to M$ from a topological space E to a simplicial space M which satisfies certain criteria. There is a category of cosimplicial bundles which contains a subcategory of vector bundles. To every simplicial space M a cosimplicial bundle $\tau (M)$ over M is associated; $ \tau (M)$ is the cotangent object of M since there is an isomorphism between the module of global piecewise smooth one-forms on M and sections of $\tau (M)$.
Two applications of twisted wreath products to finite soluble groups
Trevor O.
Hawkes
325-335
Abstract: The group construction sometimes known as the twisted wreath product is used here to answer two questions in the theory of finite, soluble groups: first to show that an arbitrary finite, soluble group may be embedded as a subgroup of a group whose upper nilpotent series is a chief series; second to construct an A-group whose Carter subgroup is ``small'' relative to its nilpotent length.
On a Galois theory for inseparable field extensions
John N.
Mordeson
337-347
Abstract: Heerema has developed a Galois theory for fields L of characteristic $p \ne 0$ in which the Galois subfields K are those for which $L/K$ is normal, modular and, for some nonnegative integer $e,K({L^{{p^{e + 1}}}})/K$ is separable. The related automorphism groups G are subgroups of a particular group A of automorphisms on $L[x]/{x^{{p^e} + 1}}L[x]$ where x is an indeterminate over L. For $H \subseteq G$ Galois subgroups of A, we give a necessary and sufficient condition for H to be G-invariant. An extension of a result of the classical Galois theory is also given as is a necessary and sufficient condition for every intermediate field of $L/K$ to be Galois where K is a Galois subfield of L.
On angular momentum Helmholtz theorems and cohomology of Lie algebras
Henrik
Stetkaer
349-374
Abstract: Helmholtz' 2nd theorem (that every vector field on ${{\mathbf{R}}^3}$ with vanishing curl is gradient of a function) can be viewed as a statement about the group of translations of $ {{\mathbf{R}}^3}$. We prove similar theorems for other Lie transformation groups, in particular for semidirect products of abelian and compact semisimple groups. Using Hodge theory we also obtain results analogous to the 1st Helmholtz theorem, but only for compact Lie transformation groups.
Ruelle's operator theorem and $g$-measures
Peter
Walters
375-387
Abstract: We use g-measures to give a proof of a convergence theorem of Ruelle. The method of proof is used to gain information about the ergodic properties of equilibrium states for subshifts of finite type.
$o$-weakly compact mappings of Riesz spaces
P. G.
Dodds
389-402
Abstract: A characterization is given of linear mappings from a Riesz space to a Banach space which map order intervals to relatively weakly compact sets. The characterization is based on recent results of Burkinshaw and Fremlin. A number of applications are made to known results concerning weakly compact mappings and to results in the theory of Banach space-valued measures.
Diffeomorphisms obtained from endomorphisms
Louis
Block
403-413
Abstract: It is shown that if f is a differentiable map of a compact manifold, and the singularities of f satisfy a certain condition, then there is a diffeomorphism (of a different manifold) whose orbit structure is closely related to that of f. This theorem is then used to extend several results on the orbit structure of diffeomorphisms to the noninvertible case.
On the structure of $S$ and $C(S)$ for $S$ dyadic
James
Hagler
415-428
Abstract: A dyadic space S is defined to be a continuous image of ${\{ 0,1\} ^\mathfrak{m}}$ for some infinite cardinal number $ \mathfrak{m}$. We deduce Banach space properties of $C(S)$ and topological properties of S. For example, under certain cardinality restrictions on $\mathfrak{m}$, we show: Every dyadic space of topological weight $ \mathfrak{m}$ contains a closed subset homeomorphic to ${\{ 0,1\} ^\mathfrak{m}}$. Every Banach space X isomorphic to an $\mathfrak{m}$ dimensional subspace of $ C(S)$ (for S dyadic) contains a subspace isomorphic to ${l^1}(\Gamma )$ where $\Gamma$ has cardinality $ \mathfrak{m}$.
Extension of Fourier $L\sp{p}---L\sp{q}$ multipliers
Michael G.
Cowling
1-33
Abstract: By $M_p^q(\Gamma )$ we denote the space of Fourier ${L^p} - {L^q}$ multipliers on the LCA group $ \Gamma$. K. de Leeuw [4] (for $\Gamma = {R^a}$), N. Lohoué [16] and S. Saeki [19] have shown that if ${\Gamma _0}$ is a closed subgroup of $\Gamma$, and $\phi$ is a continuous function in $M_p^p(\Gamma )$, then the restriction $ {\phi _0}$ of $ \phi$ to ${\Gamma _0}$ is in $M_p^p({\Gamma _0})$, and ${\left\Vert {{\phi _0}} \right\Vert _{M_p^p}} \leqslant {\left\Vert \phi \right\Vert _{M_p^p}}$. We answer here a natural question arising from this result: we show that every continuous function $ \psi$ in $M_p^p(\Gamma )$ is the restriction to ${\Gamma _0}$ of a continuous $M_p^p(\Gamma )$ function whose norm is the same as that of $\psi$. A Figà-Talamanca and G. I. Gaudry [8] proved this with the extra condition that ${\Gamma _0}$ be discrete: our technique develops their ideas. An extension theorem for $M_p^q({\Gamma _0})$ is obtained: this complements work of Gaudry [11] on restrictions of $M_p^q(\Gamma )$-functions to ${\Gamma _0}$.
Functions which are restrictions of $L\sp{p}$-multipliers
Michael G.
Cowling
35-51
Abstract: Raouf Doss has given a sufficient condition for a measurable function $ \phi$ on a measurable subset $\Lambda$ of an LCA group $\Gamma$ to be the restriction (l.a.e.) to $ \Lambda$ of the Fourier transform of a bounded measure, i.e., a Fourier multiplier of type (1, 1). We generalise Doss' theorem, and prove that, if the measurable function $ \phi$ on $\Lambda$ is approximable on finite subsets of $\Lambda$ by trigonometric polynomials which are Fourier multipliers of type (p, p) on $ \Gamma$ of norms no greater than C, then $\phi$ is equal locally almost everywhere to the restriction to $\Lambda$ of a Fourier multiplier of type (p, p) and norm no greater than C.
On a varietal structure of algebras
R. D.
Giri
53-60
Abstract: Shafaat introduced two successive generalisations of the variety of algebras: namely the semivariety and the quasivariety. We study a slightly more generalised concept which we call a pseudovariety.
An improved version of the noncompact weak canonical Schoenflies theorem
W. R.
Brakes
61-69
Abstract: The main result of this paper is that any proper collared embedding of ${R^{n - 1}}$ in ${R^n}$ can be extended to a homeomorphism of $ {R^n}$ such that the extension depends continuously on the original embedding in a stronger sense than previously known. Analogous results are proved for proper embeddings of $ {R^k}$ in ${R^n}$ (with the usual homotopy conditions when $k = n - 2$). An alternative proof of the usual compact form of the weak canonical Schoenflies theorem is also obtained.
Binary digit distribution over naturally defined sequences
D. J.
Newman;
Morton
Slater
71-78
Abstract: In a previous paper the first author showed that multiples of 3 prefer to have an even number of ones in their binary digit expansion. In this paper it is shown that in some general classes of naturally defined sequences, the probability that a member of a particular sequence has an even number of ones in its binary expansion is $ 1/2$.
Monads defined by involution-preserving adjunctions
Paul H.
Palmquist
79-87
Abstract: Consider categories with involutions which fix objects, functors which preserve involution, and natural transformations. In this setting certain natural adjunctions become universal and, thereby, become constructible from abstract data. Although the formal theory of monads fails to apply and the Eilenberg-Moore category fails to fit, both are successfully adapted to this setting, which is a 2-category. In this 2-category, each monad (= triple = standard construction) defined by an adjunction is characterized by a pair of special equations. Special monads have universal adjunctions which realize them and have both underlying Frobenius monads and adjoint monads. Examples of monads which do (respectively, do not) satisfy the special equations arise from finite monoids (= semigroups with unit) which are (respectively, are not) groups acting on the category of linear transformations between finite dimensional Euclidean (= positive definite inner product) spaces over the real numbers. More general situations are exposed.
Circle actions on homotopy spheres not bounding spin manifolds
Reinhard
Schultz
89-98
Abstract: Smooth circle actions are constructed on odd-dimensional homotopy spheres that do not bound spin manifolds. Examples are given in every dimension for which exotic spheres of the described type exist.
On $\pi \sb{3}$ of a finite $H$-space
J. R.
Hubbuck;
R.
Kane
99-105
Abstract: The third homotopy group of a finite H-space is shown to have no torsion.
On stable noetherian rings
Zoltán
Papp
107-114
Abstract: A ring R is called stable if every localizing subcategory of $_R{\text{M}}$ is closed under taking injective envelopes. In this paper the stable noetherian rings are characterized in terms of the idempotent kernel functors of $ _R{\text{M}}$ (O. Goldman [5]). The stable noetherian rings, the classical rings (Riley [11]) and the noetherian rings ``with sufficiently many two-sided ideals'' (Gabriel [4]) are compared and their relationships are studied. The close similarity between the commutative noetherian rings and the stable noetherian rings is also pointed out in the results.
M\"untz-Sz\'asz theorem with integral coefficients. II
Le Baron O.
Ferguson;
Manfred
von Golitschek
115-126
Abstract: The classical Müntz-Szász theorem concerns uniform approximation on [0, 1] by polynomials whose exponents are taken from a sequence of real numbers. Under mild restrictions on the exponents or the interval, the theorem remains valid when the coefficients of the polynomials are taken from the integers.
The residue calculus in several complex variables
Gerald Leonard
Gordon
127-176
Abstract: Let W be a complex manifold and V an analytic variety. Then homology classes in $W - V$ which bound in V, called the geometric residues, are studied. In fact, a long exact sequence analogous to the Thom-Gysin sequence for nonsingular V is formed by a geometric construction. A geometric interpretation of the Leray spectral sequence of the inclusion of $W - V \subset V$ is also given. If the complex codimension of V is one, then one shows that each cohomology class of $W - V$ can be represented by a differential form of the type $\theta \wedge \lambda + \eta$ where $ \lambda$ is the kernel associated to V and $\theta \vert V$ is the Poincaré residue of this class.
A decomposition for certain real semisimple Lie groups
H. Lee
Michelson
177-193
Abstract: For a class of real semisimple Lie groups, including those for which G and K have the same rank, Kostant introduced the decomposition $G = K{N_0}K$, where $ {N_0}$ is a certain abelian subgroup of N, and conjectured that the Jacobian of the decomposition with respect to Haar measure, as well as the spherical functions, would be polynomial in the canonical coordinates of $ {N_0}$. We compute here the Jacobian, which turns out to be polynomial precisely when the equality of ranks is satisfied. We also compute those spherical functions which restrict to polynomials on ${N_0}$.
A Galois theory for a class of inseparable field extensions
R. L.
Davis
195-203
Abstract: The structure of the group of rank n higher derivations in a field K is discussed and a characterization of its Galois subgroups is given. This yields a Galois type correspondence between these subgroups and the subfields over which K is purely inseparable, finite dimensional and modular.
$Z$-sets in ANR's
David W.
Henderson
205-216
Abstract: (1) Let A be a closed Z-set in an ANR X. Let $\mathcal{F}$ be an open cover of X. Then there is a homotopy inverse $f:X \to X - A$ to the inclusion $X - A \to X$ such that f and both homotopies are limited by $ \mathcal{F}$. (2) If, in addition, X is a manifold modeled on a metrizable locally convex TVS, F, such that F is homeomorphic to ${F^\omega }$, then there is a homotopy $j:X \times I \to X$ limited by $\mathcal{F}$ such that the closure (in X) of $ j(X \times \{ 1\} )$ is contained in $X - A$.
The hyperspace of the closed unit interval is a Hilbert cube
R. M.
Schori;
J. E.
West
217-235
Abstract: Let X be a compact metric space and let ${2^X}$ be the space of all nonvoid closed subsets of X topologized with the Hausdorff metric. For the closed unit interval I the authors prove that ${2^I}$ is homeomorphic to the Hilbert cube ${I^\infty }$, settling a conjecture of Wojdyslawski that was posed in 1938. The proof utilizes inverse limits and near-homeomorphisms, and uses (and developes) several techniques and theorems in infinite-dimensional topology.
Ergodic theorems for the asymmetric simple exclusion process
Thomas M.
Liggett
237-261
Abstract: Consider the infinite particle system on the integers with the simple exclusion interaction and one-particle motion determined by $p(x,x + 1) = p$ and $p(x,x - 1) = q$ for $x \in Z$, where $p + q = 1$ and $p > q$. If $\mu$ is the initial distribution of the system, let ${\mu _t}$ be the distribution at time t. The main results determine the limiting behavior of ${\mu _t}$ as $t \to \infty$ for simple choices of $\mu$. For example, it is shown that if $ \mu$ is the pointmass on the configuration in which all sites to the left of the origin are occupied, while those to the right are vacant, then the system converges as $t \to \infty$ to the product measure on ${\{ 0,1\} ^Z}$ with density $ {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$. For the proof, an auxiliary process is introduced which is of interest in its own right. It is a process on the positive integers in which particles move according to the simple exclusion process, but with the additional feature that there can be creation and destruction of particles at one. Ergodic theorems are proved for this process also.
Embeddings and immersions of manifolds in Euclidean space
David R.
Bausum
263-303
Abstract: The problem of computing the number of embeddings or immersions of a manifold in Euclidean space is treated from a different point of view than is usually taken. Also, a theorem dealing with the existence of an embedding of $ {M^m}$ in ${R^{2m - 2}}$ is given.
A formula for the tangent bundle of flag manifolds and related manifolds
Kee Yuen
Lam
305-314
Abstract: A formula is given for the tangent bundle of a flag manifold G in terms of canonically defined vector bundles over G. The formula leads to a unified proof of some parallelizability theorems of Stiefel manifolds. It can also be used to deduce some immersion theorems for flag manifolds.
Heegaard splittings of branched coverings of $S\sp{3}$
Joan S.
Birman;
Hugh M.
Hilden
315-352
Abstract: This paper concerns itself with the relationship between two seemingly different methods for representing a closed, orientable 3-manifold: on the one hand as a Heegaard splitting, and on the other hand as a branched covering of the 3-sphere. The ability to pass back and forth between these two representations will be applied in several different ways: 1. It will be established that there is an effective algorithm to decide whether a 3-manifold of Heegaard genus 2 is a 3-sphere. 2. We will show that the natural map from 6-plat representations of knots and links to genus 2 closed oriented 3-manifolds is injective and surjective. This relates the question of whether or not Heegaard splittings of closed, oriented 3-manifolds are ``unique'' to the question of whether plat representations of knots and links are ``unique". 3. We will give a counterexample to a conjecture (unpublished) of W. Haken, which would have implied that ${S^3}$ could be identified (in the class of all simply-connected 3-manifolds) by the property that certain canonical presentations for ${\pi _1}{S^3}$ are always ``nice". The final section of the paper studies a special class of genus 2 Heegaard splittings: the 2-fold covers of ${S^3}$ which are branched over closed 3-braids. It is established that no counterexamples to the ``genus 2 Poincaré conjecture'' occur in this class of 3-manifolds.
Picard's theorem and Brownian motion
Burgess
Davis
353-362
Abstract: Properties of the paths of two dimensional Brownian motion are used as the basis of a proof of the little Picard theorem and its analog for complex valued functions, defined on simply connected n dimensional manifolds, which map certain diffusions into Brownian motion.
The absolute continuity of phase operators
J.
Dombrowski;
G. H.
Fricke
363-372
Abstract: This paper studies the spectral properties of a class of operators known as phase operators which originated in the study of harmonic oscillator phase. Ifantis conjectured that such operators had no point spectrum. It was later shown that certain phase operators were, in fact, absolutely continuous and that all phase operators at least had an absolutely continuous part. The present work completes the discussion by showing that all phase operators are absolutely continuous.
On Dedekind's problem: the number of isotone Boolean functions. II
D.
Kleitman;
G.
Markowsky
373-390
Abstract: It is shown that $ \psi (n)$, the size of the free distributive lattice on n generators (which is the number of isotone Boolean functions on subsets of an n element set), satisfies $\displaystyle \psi (n) \leqslant {2^{(1 + O(\log \;n/n))\left( {\begin{array}{*{20}{c}} n {[n/2]} \end{array} } \right)}}.$ This result is an improvement by a factor $ \sqrt n$ in the 0 term of a previous result of Kleitman. In the course of deriving the main result, we analyze thoroughly the techniques used here and earlier by Kleitman, and show that the result in this paper is ``best possible'' (up to constant) using these techniques.
The predual theorem to the Jacobson-Bourbaki theorem
Moss
Sweedler
391-406
Abstract: Suppose $ R\xrightarrow{\varphi }S$ is a map of rings. S need not be an R algebra since R may not be commutative. Even if R is commutative it may not have central image in S. Nevertheless the ring structure on S can be expressed in terms of two maps $\displaystyle S{ \otimes _R}S\xrightarrow{{({s_1} \otimes {s_2} \to {s_1}{s_2})}}S,\quad R\xrightarrow{\varphi }S,$ which satisfy certain commutative diagrams. Reversing all the arrows leads to the notion of an R-coring. Suppose R is an overing of B. Let $ {C_B} = R{ \otimes _B}R$. There are maps \begin{displaymath}\begin{array}{*{20}{c}} {{C_B} = R{ \otimes _B}R\xrightarrow{... ...{r_1} \otimes {r_2} \to {r_1}{r_2})}}R.} \end{array} \end{displaymath} These maps give ${C_B}$ an R-coring structure. The dual $^\ast{C_B}$ is naturally isomorphic to the ring ${\text{End}_{{B^ - }}}R$ of B-linear endomorphisms of R considered as a left B-module. In case B happens to be the subring of R generated by 1, we write ${C_{\text{Z}}}$. Then $^\ast{C_{\text{Z}}}$ is $ {\text{End}_{\text{Z}}}R$, the endomorphism ring of R considered as an additive group. This gives a clue how certain R-corings correspond to subrings of R and subrings of $ {\text{End}_{\text{Z}}}R$, both major ingredients of the Jacobson-Bourbaki theorem. $1 \otimes 1$ is a ``grouplike'' element in the R-coring $ {C_{\text{Z}}}$ (and should be thought of as a generic automorphism of R). Suppose R is a division ring and B a subring which is a division ring. The natural map ${C_{\text{Z}}} \to {C_B}$ is a surjective coring map. Conversely if $ {C_{\text{Z}}}\xrightarrow{\pi }D$ is a (surjective) coring map then $\pi (1 \otimes 1)$ is a grouplike in D and $\{ r \in R\vert r\pi (1 \otimes 1) = \pi (1 \otimes 1)r\}$ is a subring of R which is a division ring. This gives a bijective correspondence between the quotient corings of $C{ \otimes _{\text{Z}}}C$ and the subrings of R which are division rings. We show how the Jacobson-Bourbaki correspondence is dual to the above correspondence.
Homology with multiple-valued functions applied to fixed points
Richard
Jerrard
407-427
Abstract: Certain multiple-valued functions (m-functions) are defined and a homology theory based upon them is developed. In this theory a singular simplex is an m-function from a standard simplex to a space and an m-function from one space to another induces a homomorphism of homology modules. In a family of functions ${f_x}:Y \to Y$ indexed by $x \in X$ the fixed points of ${f_x}$ are taken to be the images at x of a multiple-valued function $\phi :X \to Y$. In certain circumstances $ \phi$ is an m-function, giving information about the behavior of the fixed points of ${f_x}$ as x varies over X. These facts are applied to self-maps of products of compact polyhedra and the question of whether such a product has the fixed point property for continuous functions is essentially reduced to the question of whether one of its factors has the fixed point property for m-functions. Some light is thrown on the latter problem by using the homology theory to prove a Lefschetz fixed point theorem for m-functions.
Erratum to: ``Regular overrings of regular local rings'' (Trans. Amer. Math. Soc. {\bf 171} (1972), 291--300)
Judith
Sally
429
Integrability of infinite sums of independent vector-valued random variables
Naresh C.
Jain;
Michael B.
Marcus
1-36
Abstract: Let B be a normed vector space (possibly a Banach space, but it could be more general) and $ \{ {X_n}\}$ a sequence of B-valued independent random variables on some probability space. Let $ {S_n} = \Sigma _{j = 1}^n{X_j},M = {\sup _n}\vert{S_n}\vert$ and $S = {\lim _n}{S_n}$ is norm, whenever it exists. Assuming that S exists or $M < \infty$ a.s. and given certain nondecreasing functions $\varphi$, we find conditions in terms of the distributions of $\left\Vert {{X_n}} \right\Vert$ such that $E(\varphi (M))$ or $E(\varphi (\left\Vert S \right\Vert))$ is finite. Let $\{ {u_n}\}$ be a sequence of elements in B and $ \{ {\varepsilon _n}\}$ a sequence of independent, identically distributed random variables such that $P\{ {\varepsilon _1} = 1\} = P\{ {\varepsilon _1} = - 1\} = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$. We prove some comparison theorems which generalize the following contraction principle of Kahane: If $\{ {\lambda _n}\}$ is a bounded sequence of scalars, then $ \Sigma {\varepsilon _n}{u_n}$ converges in norm a.s. (or is bounded a.s.) implies the corresponding conclusion for the series $\Sigma {\lambda _n}{\varepsilon _n}{u_n}$. Some generalizations of this contraction principle have already been carried out by Hoffmann-Jørgensen. All these earlier results are subsumed by ours. Applications of our results are made to Gaussian processes, random Fourier series and other random series of functions.
Well-ordering of certain numerical polynomials
William Yu
Sit
37-45
Abstract: An ordering is introduced in the set of numerical polynomials (in one variable) which is then shown to induce a well-ordering on a certain subset of numerical polynomials, namely those which occur as the differential dimension polynomials of differential algebraic varieties, or equivalently, those which come from initial subsets of ${{\text{N}}^m}$.
Finite projections in tensor product von Neumann algebras
George A.
Elliott
47-60
Abstract: The work of Bures, Moore, Takenouchi, Hill and Størmer on the type classification of infinite tensor products of factors is extended to the nonfactor case.
Integration of functions with values in locally convex Suslin spaces
G. Erik F.
Thomas
61-81
Abstract: The main purpose of the paper is to give some easily applicable criteria for summability of vector valued functions with respect to scalar measures. One of these is the following: If E is a quasi-complete locally convex Suslin space (e.g. a separable Banach or Fréchet space), $\sigma (E,H)$. f is actually Pettis summable for the given topology. (Thus any E-valued function for which the integrals over measurable subsets can be reasonably defined as elements of E is Pettis summable.) A class of ``totally summable'' functions, generalising the Bochner integrable functions, is introduced. For these Fubini's theorem, in the case of a product measure, and the differentiation theorem, in the case of Lebesgue measure, are valid. It is shown that weakly summable functions with values in the spaces
The structure of semiprimary and Noetherian hereditary rings
John
Fuelberth;
James
Kuzmanovich
83-111
Abstract: In the first portion of this paper a structure theorem for semiprimary hereditary rings is given in terms of $M \times M$ ``triangular'' row-finite matrices over a division ring D. This structure theorem differs from previous theorems of this type in that the representation is explicit in terms of matrices over a division ring. In the second portion of this paper we are able to apply the results of Gordon and Small to obtain a structure theorem for semihereditary and left hereditary rings which are left orders in a semiprimary ring. In the case of the left hereditary rings, the representation is explicit in terms of matrices over left hereditary Goldie prime rings and their respective classical left quotient rings. As an application we obtain, by a different method, a non-Noetherian generalization of a result of Chatters which states that a two-sided hereditary Noetherian ring is a ring direct sum of an Artinian ring and a semiprime ring.
Symplectic homogeneous spaces
Shlomo
Sternberg
113-130
Abstract: In this paper we make various remarks, mostly of a computational nature, concerning a symplectic manifold X on which a Lie group G acts as a transitive group of symplectic automorphisms. The study of such manifolds was initiated by Kostant [4] and Souriau [5] and was recently developed from a more general point of view by Chu [2]. The first part of this paper is devoted to reviewing the Kostant, Souriau, Chu results and deriving from them a generalization of the Cartan conjugacy theorem. In the second part of this paper we apply these results to Lie algebras admitting a generalized (k, p) decomposition.
Minimal complementary sets
Gerald
Weinstein
131-137
Abstract: Let G be a group on which a measure m is defined. If $A,B \subset G$ we define $A \oplus B = C = \{ c\vert c = a + b,a \in A,b \in B\}$. By $ {A_k} \subset G$ we denote a subset of G consisting of k elements. Given ${A_k}$ we define $s({A_k}) = \inf m\{ B\vert B \subset G,{A_k} \oplus B = G\}$ and ${c_k} = {\sup _{{A_k} \subset G}}s({A_k})$. Theorems 1, 2, and 3 deal with the problem of determining $ {c_k}$. In the dual problem we are given B, $m(B) > 0$, and required to find minimal A such that $A \oplus B = G$ or, sometimes, $m(A \oplus B) = m(G)$. Theorems 5 and 6 deal with this problem.
The Wedderburn principal theorem for generalized alternative algebras. I
Harry F.
Smith
139-148
Abstract: A generalized alternative ring I is a nonassociative ring R in which the identities $(wx,y,z) + (w,x,(y,z)) - w(x,y,z) - (w,y,z)x;((w,x),y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z$; and $(x,x,x)$ are identically zero. Let A be a finite-dimensional algebra of this type over a field F of characteristic $\ne 2,3$. Then it is established that (1) A cannot be a nodal algebra, and (2) the standard Wedderburn principal theorem is valid for A.
Inequalities for a complex matrix whose real part is positive definite
Charles R.
Johnson
149-154
Abstract: Denote the real part of $A \in {M_n}(C)$ by $ H(A) = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}(A + {A^\ast})$. We provide dual inequalities relating $H({A^{ - 1}})$ and $ H{(A)^{ - 1}}$ and an identity between two functions of A when A satisfies $H(A) > 0$. As an application we give an inequality (for matrices A satisfying $H(A) > 0$) which generalizes Hadamard's determinantal inequality for positive definite matrices.
Equivariant homology theories on $G$-complexes
Stephen J.
Willson
155-171
Abstract: A definition is given for a ``cellular'' equivariant homology theory on G-complexes. The definition is shown to generalize to G-complexes with prescribed isotropy subgroups. A ring I is introduced to deal with the general definition. One obtains a universal coefficient theorem and studies the universal coefficients.
On the existence of a universal germ of deformations for elliptic pseudogroup structures on compact manifolds
Suresh H.
Moolgavkar
173-197
Abstract: The purpose of this paper is to prove the existence of a versal germ of deformations for elliptic pseudogroup structures on compact manifolds. Under suitable restrictions, the versal germ is shown to be universal.
$C\sp*$-algebras with Hausdorff spectrum
John W.
Bunce;
James A.
Deddens
199-217
Abstract: By the spectrum of a ${C^\ast}$-algebra we mean the set of unitary equivalence classes of irreducible representations equipped with the hull-kernel topology. We are concerned with characterizing the $ {C^\ast}$-algebras with identity which have Hausdorff spectrum. We characterize the ${C^\ast}$-algebras with identity and bounded representation dimension which have Hausdorff spectrum. Our results are more natural when the $ {C^\ast}$-algebra is singly generated. For singly generated $ {C^\ast}$-algebras with unbounded representation dimension, we reduce the problem to the case when the generator is an infinite direct sum of irreducible finite scalar matrices, and we have partial results in this case.
Semigroups with a dense subgroup
W. S.
Owen
219-228
Abstract: The purpose of this paper is twofold. First, it is shown that the ideal structure of a semigroup with dense subgroup is closely related to its transformation group structure. That is, if a left orbit through a given point is locally compact, then the members of this orbit are precisely those elements which generate the same left ideal as the given point. Secondly, the author gives a number of theorems which have as their goal the establishment of a natural product structure near a nonzero idempotent. Specifically the work of F. Knowles [11] is improved upon to include (1) the possibility of a nonconnected group; (2) the possibility of a nonsimply connected orbit; and (3) the case in which the boundary of the group is more than a single orbit.
Singularity subschemes and generic projections
Joel
Roberts
229-268
Abstract: Corresponding to a morphism $f:V \to W$ of algebraic varieties (such that $\dim (V) \leqslant \dim (W)$), we construct a family of subschemes $S_1^{(q)}(f) \subset V$. When V and W are nonsingular, the $S_1^{(q)},q \geqslant 1$, induce a filtration of the set of closed points $x \in V$ such that the tangent space map $ d{f_x}:T{(V)_x} \to T{(W)_{f(x)}}$ has rank $ = \dim (V) - 1$. We prove that if V is a suitably embedded nonsingular projective variety and $\pi :V \to {{\mathbf{P}}^m}$ is a generic projection, then the $ S_1^{(q)}(f)$ and certain fibre products of several of the $S_1^{(q)}(f)$ are either empty or smooth and of the smallest possible dimension, except in cases where $ q + 1$ is divisible by the characteristic of the ground field. We apply this result to describe explicitly the ring homomorphisms ${\pi ^\ast}:{\hat{\mathcal{O}}_{{{\mathbf{P}}^m}\pi (x)}} \to {\hat{\mathcal{O}}_{V,x}}$ and (when $ m \geqslant r + 1$) to study the local structure of the image
The spectral sequence of a finite group extension stops
Leonard
Evens
269-277
Abstract: It is proved that if G is a finite group, H a normal subgroup, and A a finitely generated G-module, then both the cohomology and homology spectral sequences for the group extension stop in a finite number of stops. A lemma about $ {\operatorname{Tor}}(M,N)$ as a module over $R \otimes S$ is proved. Two spectral sequences of Hochschild and Serre are shown to be the same.
On properties of the approximate Peano derivatives
Bruce S.
Babcock
279-294
Abstract: The notion of kth approximate Peano differentiation not only generalizes kth ordinary differentiation but also kth Peano differentiation and kth ${L_p}$ differentiation. Recently, M. Evans has shown that a kth approximate Peano derivative at least shares with these other derivatives the property of belonging to Baire class one. In this paper the author extends the properties possessed by a kth approximate Peano derivative by showing that it is like the above derivatives in that it also possesses the following properties: Darboux, Denjoy, Zahorski, and a new property stronger than the Zahorski property, Property Z.
Units and periodic Jacobi-Perron algorithms in real algebraic number fields of degree $3$
Leon
Bernstein
295-306
Abstract: It is not known whether or not the Jacobi-Perron Algorithm of a vector in $ {R_{n - 1}},n \geqslant 3$, whose components are algebraic irrationals, always becomes periodic. The author enumerates, from his previous papers, a few infinite classes of real algebraic number fields of any degree for which this is the case. Periodic Jacobi-Perron Algorithms are important, because they can be applied, inter alia, to calculate units in the corresponding algebraic number fields. The main result of this paper is expressed in the following theorem: There are infinitely many real cubic fields $ Q(w),{w^3}$ cubefree, a and T natural numbers, such that the Jacobi-Perron Algorithm of the vector $(w,{w^2})$ becomes periodic; the length of the primitive preperiod is four, the length of the primitive period is three; a fundamental unit of $ Q(w)$ is given by $e = {a^3}T + 1 - aw$.
On sum-free subsequences
S. L. G.
Choi;
J.
Komlós;
E.
Szemerédi
307-313
Abstract: A subsequence of a sequence of n distinct integers is said to be sum-free if no integer in it is the sum of distinct integers in it. Let $f(n)$ denote the largest quantity so that every sequence of n distinct integers has a sum-free subsequence consisting of $f(n)$ integers. In this paper we strengthen previous results by Erdös, Choi and Cantor by proving $\displaystyle {(n\;\log \;n/\log \;\log \;n)^{{\raise0.5ex\hbox{$\scriptstyle 1... ....15em \lower0.25ex\hbox{$\scriptstyle 2$}}}} \ll f(n) \ll n{(\log \;n)^{ - 1}}.$
On commutators of singular integrals and bilinear singular integrals
R. R.
Coifman;
Yves
Meyer
315-331
Abstract: ${L^p}$ estimates for multilinear singular integrals generalizing Calderón's commutator integral are obtained. The methods introduced involve Fourier and Mellin analysis.
Stable equivalence for some categories with radical square zero
Idun
Reiten
333-345
Abstract: For certain abelian categories with radical square zero, containing artin rings with radical square zero as a special case, we give a way of constructing hereditary abelian categories stably equivalent to them, i.e. such that their categories modulo projectives are equivalent categories.
On finite Hilbert transforms
Kevin F.
Clancey
347-354
Abstract: Let E be a bounded measurable subset of the real line. The finite Hilbert transform is the operator ${H_E}$ defined on one of the spaces ${L^p}(E)(1 < p < \infty )$ by $\displaystyle {H_E}f(x) = {(\pi i)^{ - 1}}\int_E {f(t){{(t - x)}^{ - 1}}\;dt;}$ here, the singular integral is interpreted as a Cauchy principal value. The main result establishes that for ${H_E}$ to be Fredholm on ${L^p}(E)$, when $p \ne 2$, it is necessary and sufficient that E be equal almost everywhere to a finite union of intervals. The sufficiency of this condition was established in 1960 by H. Widom. In the case where E is not a finite union of intervals and $p < 2$ it is shown that the operator ${H_E}$ has an infinite dimensional null space. The method of proof is constructive.
On some real hypersurfaces of a complex projective space
Masafumi
Okumura
355-364
Abstract: A principal circle bundle over a real hypersurface of a complex projective space $C{P^n}$ can be regarded as a hypersurface of an odd-dimensional sphere. From this standpoint we can establish a method to translate conditions imposed on a hypersurface of $C{P^n}$ into those imposed on a hypersurface of ${S^{2n + 1}}$. Some fundamental relations between the second fundamental tensor of a hypersurface of $C{P^n}$ and that of a hypersurface of ${S^{2n + 1}}$ are given.
The trigonometric Hermite-Birkhoff interpolation problem
Darell J.
Johnson
365-374
Abstract: The classical Hermite-Birkhoff interpolation problem, which has recently been generalized to a special class of Haar subspaces, is here considered for trigonometric polynomials. It is shown that a slight weakening of the result (conservativity and Pólya conditions) established for those special Haar subspaces also holds for trigonometric polynomials after one rephrases the statement of the problem, the underlying assumptions, and the result itself appropriately to reflect the inherent differences between algebraic polynomials (which the special class of Haar subspaces essentially are) and the periodic trigonometric polynomials. Furthermore, simple necessary and sufficient conditions for poisedness of one-rowed incidence matrices analogous to the Pólya conditions for two-rowed incidence matrices in the algebraic version are proved, and an elementary necessary condition for the poisedness of an arbitrary (trigonometric) incidence matrix stated.
Existence and uniqueness theorems for Riemann problems
Tai Ping
Liu
375-382
Abstract: In [2] the author proposed the entropy condition (E) and solved the Riemann problem for general $2 \times 2$ conservation laws ${u_t} + f{(u,v)_x} = 0,{v_t} + g{(u,v)_x} = 0$, under the assumptions that the system is hyperbolic, and ${f_u} \geqslant 0$ and ${g_v} \leqslant 0$. The purpose of this paper is to extend the above results to a much wider class of $2 \times 2$ conservation laws. Instead of assuming that $ {f_u} \geqslant 0$ and ${g_v} \leqslant 0$, we assume that the characteristic speed is not equal to the shock speed of different family. This assumption is motivated by the works of Lax [1] and Smoller [4].
Smooth locally convex spaces
John
Lloyd
383-392
Abstract: The main theorem is Let E be a separable (real) Fréchet space with a nonseparable strong dual. Then E is not strongly $D_F^1$-smooth. It follows that if X is uncountable, locally compact, $\sigma $-compact, metric space, then $C(X)$ (with the topology of compact convergence) does not have a class of seminorms which generate its topology and are Fréchet differentiable (away from their null-spaces).
On all kinds of homogeneous spaces
Gerald S.
Ungar
393-400
Abstract: Several open questions on homogeneous spaces are answered. A few of the results are: (1) An n-homogeneous metric continuum, which is not the circle, is strongly n-homogeneous. (2) A 2-homogeneous metric continuum is locally connected. (3) If X is a homogeneous compact metric space or a homogeneous locally compact, locally connected separable metric space, then X is a coset space. (4) If G is a complete separable metric topological group with is n-connected, then G is locally n-connected.
Iterated nontangential limits
K.
Gowrisankaran
401-402
Abstract: For functions f in the Nevanlinna class of ${U^n}$ it is proved that the iterated nontangential limits are equal up to a set of measure zero.
On the fixed point indices and Nielsen numbers of fiber maps on Jiang spaces
Jingyal
Pak
403-415
Abstract: Let $T = \{ E,P,B\}$ be a locally trivial fiber space, where E, B and $ {P^{ - 1}}(b)$ for each $b \in B$ are compact, connected ANR's (absolute neighborhood retracts). If $ f:E \to E$ is a fiber (preserving) map then f induces ${f_b}:{P^{ - 1}}(b) \to {P^{ - 1}}(b)$ for each $b \in B$ such that $ Pf = f'P$. If E, B and ${P^{ - 1}}(b)$ for each $ b \in B$ satisfy the Jiang condition then $i:{P^{ - 1}}(b) \to E$ induces a monomorphism ${i_\char93 }:{\pi _1}({P^{ - 1}}(b)) \to {\pi _1}(E)$ and $f'$ induces a fixed point free homomorphism $T = \{ E,P,CP(n)\}$ be a principal torus bundle over an n-dimensional complex projective space $ CP(n)$. If $f:E \to E$ is a fiber map such that for some $b \in CP(n),{f_b}:{P^{ - 1}}(b) \to {P^{ - 1}}(b)$ is homotopic to a fixed point free map, then there exists a map $g:E \to E$ homotopic to f and fixed point free.
Addendum to: ``Behnke-Stein theorem for analytic spaces'' (Trans. Amer. Math. Soc. {\bf 199} (1974), 317--326)
Alessandro
Silva
417-418
Abstract: A very simple argument shows that Theorem 3.1 in my paper Behnke-Stein theorem for analytic spaces, (these Transactions, 199 (1974), pp. 317 326) is enough, via a Narasimhan result, to obtain information about the torsion of the homology groups of a Runge pair of Stein spaces.
Commuting and topological densities and liftings
Trevor J.
McMinn
1-22
Abstract: For fairly general conditions on a measure space, a group of bijections on the space and a topology on the space, densities and lifttings commuting with members of the group and with topologies finer than the given topology are obtained.
A generalization of Jarn\'\i k's theorem on Diophantine approximations to Ridout type numbers
I.
Borosh;
A. S.
Fraenkel
23-38
Abstract: Let s be a positive integer, $c > 1,{\mu _0}, \ldots ,{\mu _s}$ reals in [0, 1], $ \sigma = \Sigma _{i = 0}^s\;{\mu _i}$, and t the number of nonzero $ {\mu _i}$. Let ${\Pi _i}\;(i = 0, \ldots ,s)$ be $s + 1$ disjoint sets of primes and S the set of all $(s + 1)$-tuples of integers $({p_0}, \ldots ,{p_s})$ satisfying $\vert p_i^\ast\vert \leq c\vert{p_i}{\vert^{{\mu _i}}}$, and all prime factors of $ {p'_i}$ are in ${\Pi _i},i = 0, \ldots ,s$. Let $\lambda > 0$ if $t = 0,\lambda > \sigma /\min (s,t)$ otherwise, ${E_\lambda }$ the set of all real s-tuples $ ({\alpha _1}, \ldots ,{\alpha _s})$ satisfying $ \vert{\alpha _i} - {p_i}/{p_0}\vert < p_0^{ - \lambda }\;(i = 1, \ldots ,s)$ for an infinite number of $({p_0}, \ldots ,{p_s}) \in S$. The main result is that the Hausdorff dimension of $ {E_\lambda }$ is $\sigma /\lambda$. Related results are obtained when also lower bounds are placed on the $p_i^\ast$. The case $s = 1$ was settled previously (Proc. London Math. Soc. 15 (1965), 458-470). The case $ {\mu _i} = 1\;(i = 0, \ldots ,s)$ gives a well-known theorem of Jarník (Math. Z. 33 (1931), 505-543).
The structure of inseparable field extensions
William C.
Waterhouse
39-56
Abstract: The goal of this paper is to introduce some structural ideas into the hitherto chaotic subject of infinite inseparable field extensions. The basic discovery is that the theory is closely related to the well-developed study of primary abelian groups. This analogy undoubtedly has implications beyond those included here. We consider only modular extensions, which are the inseparable equivalent of galois extensions. §§2 and 3 develop the theory of pure independence, basic subfields, and tensor products of simple extensions. The following sections are devoted to Ulm invariants and their computation; the existence of nonzero invariants of arbitrary index is proved by means of a theorem which furnishes an actual connection between primary groups and inseparable fields. The final section displays some complications in the field extensions not occurring in abelian groups.
Topological properties of subanalytic sets
Robert M.
Hardt
57-70
Abstract: The stratification of a semianalytic or subanalytic set (that is, a set which locally is the proper analytic image of some semianalytic set) leads easily, by consecutive projections in Euclidean space, to a CW decomposition. In the category of subanalytic sets and continuous maps with subanalytic graphs, theories of slicing, intersection, and homology result through use of the topological chains defined by subanalytic sets.
On differentials of the first kind and theta constants for certain congruence subgroups
A. J.
Crisalli
71-84
Abstract: Let $\Gamma (8)$ denote the principal congruence subgroup of level 8 and let $ \Gamma (16,32)$ denote the subgroup of $ \Gamma (16)$ satisfying $ cd \equiv ab \equiv 0\;(\bmod 32)$. We are dealing only with the elliptic modular case. Consider the spaces of cusp forms of weight 2 (differentials of the first kind) with respect to these groups. It is proved that these spaces are generated by certain monomials of theta constants of degree 4.
Extensions and liftings of positive linear mappings on Banach lattices
Heinrich P.
Lotz
85-100
Abstract: Let F be a closed sublattice of a Banach lattice G. We show that any positive linear mapping from F into ${L^1}(\mu )$ or $C(X)$ for a Stonian space X has a positive norm preserving extension to G. A dual result for positive norm preserving liftings is also established. These results are applied to obtain extension and lifting theorems for order summable and majorizing linear mappings. We also obtain some partial results concerning positive extensions and liftings of compact linear mappings.
Submersive and unipotent group quotients among schemes of a countable type over a field $k$
Paul
Cherenack
101-112
Abstract: An algebraic group G is called submersive if every quotient in affine schemes ${c^G}:{\text{Spec}}\;A \to {\text{Spec}}\;{A^G}$ which is surjective is also submersive. We prove that every unipotent group is submersive. Suppose G is submersive. We show that if ${c^G}({\text{Spec}}\;A)$ is open in ${\text{Spec}}\;{A^G}$ or if some restrictions on the action of G on A are made, $ {c^G}$ is a topological quotient. A criterion for semisimplicity of points is extended to the case where G is unipotent. Finally, applications of the theory are provided.
Conversion from nonstandard to standard measure spaces and applications in probability theory
Peter A.
Loeb
113-122
Abstract: Let $(X,\mathcal{A},\nu )$ be an internal measure space in a denumerably comprehensive enlargement. The set X is a standard measure space when equipped with the smallest standard $\sigma$-algebra $ \mathfrak{M}$ containing the algebra $ \mathcal{A}$, where the extended real-valued measure $\mu$ on $ \mathfrak{M}$ is generated by the standard part of $\nu$. If f is $ \mathcal{A}$-measurable, then its standard part $^0f$ is $ \mathfrak{M}$-measurable on X. If f and $\mu$ are finite, then the $\nu $-integral of f is infinitely close to the $\mu$-integral of $^0f$. Applications include coin tossing and Poisson processes. In particular, there is an elementary proof of the strong Markov property for the stopping time of the jth event and a construction of standard sample functions for Poisson processes.
Results on sums of continued fractions
James L.
Hlavka
123-134
Abstract: Let $F(m)$ be the (Cantor) set of infinite continued fractions with partial quotients no greater than m and let $ F(m) + F(n) = \{ \alpha + \beta :\alpha \in F(m),\beta \in F(n)\}$. We show that $F(3) + F(4)$ is an interval of length 1.14 ... so every real number is the sum of an integer, an element of $F(3)$ and an element of $F(4)$. Similar results are given for $F(2) + F(7),F(2) + F(2) + F(4),F(2) + F(3) + F(3)$ and $ F(2) + F(2) + F(2) + F(2)$. The techniques used are applicable to any Cantor sets in R for which certain parameters can be evaluated.
On the Calkin algebra and the covering homotopy property
John B.
Conway
135-142
Abstract: Let $\mathcal{H}$ be a separable Hilbert space, $\mathcal{B}(\mathcal{H})$ the bounded operators on $\mathcal{H},\mathcal{K}$ the ideal of compact operators, and $\pi$ the natural map from $\mathcal{B}(\mathcal{H})$ onto the Calkin algebra $ \mathcal{B}(\mathcal{H})/\mathcal{K}$. Suppose X is a compact metric space and $\Phi :C(X) \times [0,1] \to \mathcal{B}(\mathcal{H})/\mathcal{K}$ is a continuous function such that $\Phi ( \cdot ,t)$ is a $ \ast$-isomorphism for each t and such that there is a $ \ast$-isomorphism $\psi :C(X) \to \mathcal{B}(\mathcal{H})$ with $\pi \psi ( \cdot ) = \Phi ( \cdot ,0)$. It is shown in this paper that if X is a simple Jordan curve, a simple closed Jordan curve, or a totally disconnected metric space then there is a continuous map $\Psi :C(X) \times [0,1] \to \mathcal{B}(\mathcal{H})$ such that $ \pi \Psi = \Phi$ and $ \Psi ( \cdot ,0) = \psi ( \cdot )$. Furthermore if X is the disjoint union of two spaces that both have this property, then X itself has this property.
On imbedding finite-dimensional metric spaces
Stephen Leon
Lipscomb
143-160
Abstract: The classical imbedding theorem in dimension theory gives a nice topological characterization of separable metric spaces of finite covering dimension. The longstanding problem of obtaining an analogous theorem for the nonseparable case is solved.
Commutative regular rings with integral closure
L.
Lipshitz
161-170
Abstract: First order conditions are given which are necessary for a commutative regular ring to have a prime integrally closed extension. If the ring is countable these conditions are also sufficient.
Immersions of complex hypersurfaces
Stanley R.
Samsky
171-184
Abstract: The varieties ${V^n}(d) = \{ [{z_0}, \ldots ,{z_n}] \in C{P^n}:z_0^d + \cdots + z_n^d = 0,d > 0\}$ form a class of manifolds containing the complex projective spaces. Maps from ${V^n}(d)$ to ${V^k}(e)$ are partially characterized by a ``degree". We prove some nonimmersion results which are phrased in terms of this degree, and which generalize the results of S. Feder [4] on complex projective spaces.
Duality theory for locally compact groups with precompact conjugacy classes. I. The character space
Terje
Sund
185-202
Abstract: Let G be a locally compact group, and let $\mathcal{X}(G)$ consist of the nonzero extreme points of the set of continuous, G-invariant, positive definite functions f on G such that $f(e) \leq 1$. $ \mathcal{X}(G)$ is called the character space, and is given the topology of uniform convergence on compacta. The purpose of the present paper is to extend the main results from the duality theory of abelian groups and [Z] groups to the class of ${[FC]^ - }$ groups (i.e., groups with precompact conjugacy classes), letting $\mathcal{X}(G)$ play the role of the character group in the abelian theory. Some of our theorems are only proved for the class ${[FD]^ - }\;( \subset {[FC]^ - })$. If $G \in {[FC]^ - }$ then $\mathcal{X}(G) \approx \mathcal{X}(H)$ where H is a certain $ {[FIA]^ - }$ quotient group. Hence there is no loss of generality to study character spaces of $ {[FIA]^ - }$ groups.
Extensions of maps as fibrations and cofibrations
Frank
Quinn
203-208
Abstract: Suppose $ f:X \to Y$ is a map of 1-connected spaces. In the ``stable'' range, roughly where the connectivity of Y exceeds the homology, or homotopy, dimension of X, it is well known that f can be extended as a cofibration $C \to X \to Y$, or respectively a fibration $X \to Y \to B$. A criterion is given for the existence of such extensions in a less restrictive ``metastable'' range. A main result is that if f is at least 2-connected and 2 con $Y \geq \dim Y - 1,\dim X$, then f extends as a cofibration if and only if the map $ (1 \times f)\Delta :X \to (X \times Y)/X$ factors through f.
Strongly prime rings
David
Handelman;
John
Lawrence
209-223
Abstract: A ring R is (right) strongly prime (SP) if every nonzero twosided ideal contains a finite set whose right annihilator is zero. Examples are domains, prime Goldie rings and simple rings; however, this notion is asymmetric and a right but not left SP ring is exhibited. All SP rings are prime, and every prime ring may be embedded in an SP ring. SP rings are nonsingular, and a regular SP ring is simple; since faithful rings of quotients of SP rings are SP, the complete ring of quotients of an SP ring is simple. All SP rings are coefficient rings for some primitive group ring (a generalization of a result proved for domains by Formanek), and this was the initial motivation for their study. If the group ring RG is SP, then R is SP and G contains no nontrivial locally finite normal subgroups. Coincidentally, SP rings coincide with the ATF rings of Rubin, and so every SP ring has a unique maximal proper torsion theory, and (0) and R are the only torsion ideals.($^{1}$) A list of questions is appended.
Endomorphism rings and direct sums of torsion free abelian groups
D. M.
Arnold;
E. L.
Lady
225-237
Abstract: Properties of abelian groups related to a given finite rank torsion free abelian group A are analyzed in terms of End (A), the endomorphism ring of A. This point of view gives rise to generalizations of some classical theorems by R. Baer and examples of pathological direct sum decompositions of finite rank torsion free abelian groups.
Fixed points in representations of categories
J.
Adámek;
J.
Reiterman
239-247
Abstract: Fixed points of endomorphisms of representations, i.e. functors into the category of sets, are investigated. A necessary and sufficient condition on a category K is given for each of its indecomposable representations to have the fixed point property. The condition appears to be the same as that found by Isbell and Mitchell for Colim: ${\text{Ab}^K} \to {\text{Ab}}$ to be exact. A well-known theorem on mappings of Katětov and Kenyon is extended to transformations of functors.
On coverings and hyperalgebras of affine algebraic groups
Mitsuhiro
Takeuchi
249-275
Abstract: Over an algebraically closed field of characteristic zero, the universal group covering of a connected affine algebraic group, if such exists, can be constructed canonically from its Lie algebra only. In particular the isomorphism classes of simply connected affine algebraic groups are in 1-1 correspondence with the isomorphism classes of finite dimensional Lie algebras of some sort. We shall consider the counterpart of these results (due to Hochschild) in case of a positive characteristic, replacing the Lie algebra by the ``hyperalgebra". We show that the universal group covering of a connected affine algebraic group scheme can be constructed canonically from its hyperalgebra only and hence, in particular, that the category of simply connected affine algebraic group schemes is equivalent to a subcategory of the category of hyperalgebras of finite type which contains all the semisimple hyperalgebras.
On the two sheeted coverings of conics by elliptic curves
R. E.
MacRae
277-287
Abstract: Let K be the field of algebraic functions on an elliptic curve that can be described by an equation of the form ${y^2} = f(x)$ where $f(x)$ is a quartic polynomial over a field k. Moreover, assume that the Riemann surface for K contains no points rational over k. When k is the field of real numbers it is well known that K may also be expressed as a quadratic extension of a function field $L = k(u,v)$ of algebraic functions on a conic whose Riemann surface also contains no points rational over k. We extend this result to p-adic ground fields k. Moreover, we describe the various subfields of index two and genus zero (conic subfields) in terms of the k-rational points on the Jacobian of K. This is done for arbitrary ground fields. In particular, the embedding of the projective class group of K (over k) is seen to describe exactly those conic subfields that possess k-rational points.
Unions of Hilbert cubes
Raymond Y. T.
Wong;
Nelly
Kroonenberg
289-297
Abstract: This paper gives a partial solution to the problem whether the union of two Hilbert cubes is a Hilbert cube if the intersection is a Hilbert cube and a Z-set in one of them. Our results imply West's Intermediate Sum Theorem on Hilbert cube factors. Also a technique is developed to obtain Z-sets as limits of Z-sets.
The spinor genus of quaternion orders
Gordon L.
Nipp
299-309
Abstract: Let D be a global domain whose quotient field F does not have characteristic 2, let $ \mathfrak{A}$ be a quaternion algebra over F, and let $\mathfrak{D}$ be an order on $\mathfrak{A}$ over D. A right $\mathfrak{D}$-module M which is simultaneously a lattice on $ \mathfrak{A}$ over D is said to be right $ \mathfrak{D}$-generic if there exists $\alpha \in \mathfrak{A},N(\alpha ) \ne 0$, such that ${\alpha ^{ - 1}}M \in {\operatorname{gen}}\;\mathfrak{D}$. Our main result is that every right $\mathfrak{D}$-generic module is cyclic if and only if every class in the spinor genus of $\mathfrak{D}$ represents a unit in D. One consequence is that $ \mathfrak{D}$ is in a spinor genus of one class if and only if $\mathfrak{D}$-generic modules are cyclic and $\mathfrak{D}$ represents every unit represented by its spinor genus. In addition, it is shown that a necessary and sufficient condition that an integral ternary lattice L be in a spinor genus of one class is that every right $ {\mathfrak{D}_L}$-generic pair be equivalent to a two-sided $ {\mathfrak{D}_L}$-generic pair, where $ {\mathfrak{D}_L}$ is the quaternion order associated with L.
Some open mapping theorems for marginals
Larry Q.
Eifler
311-319
Abstract: Let S and T be compact Hausdorff spaces and let $ P(S),P(T)$ and $P(S \times T)$ denote the collection of probability measures on S, T and $S \times T$, respectively. Given a probability measure $\mu$ on $S \times T$, set $\pi \mu = (\alpha ,\beta )$ where $\alpha$ and $\beta$ are the marginals of $\mu$ on S and T. We prove that the mapping $\pi :P(S \times T) \to P(S) \times P(T)$ is norm open and $ {\text{weak}^\ast}$ open. An analogous result for ${L_1}(X \times Y,\mu \times \nu )$ where $ (X,\mu )$ and $ (Y,\nu )$ are probability spaces is established.
Semifree actions on homotopy spheres
Kai
Wang
321-337
Abstract: In this paper, we study the semifree ${Z_m}$ actions on homotopy sphere pairs. We show that in some cases the equivariant normal bundle to the fixed point set is equivariantly stably trivial. We compute the rank of the torsion free part of the group of semifree actions on homotopy sphere pairs in some cases. We also show that there exist infinitely many semifree ${Z_{4s}}$ actions on even dimensional homotopy sphere pairs.
The oscillation of an operator on $L\sp{p}$
George R.
Barnes;
Robert
Whitley
339-351
Abstract: We introduce and discuss the oscillation of an operator T mapping ${L^p}(S,\Sigma ,\mu )$ into a Banach space. We establish results relating the oscillation, a ``local norm", to the norm of the operator. Also using the oscillation we define a generalization of the Fredholm operators T with index $\kappa (T) < \infty$ and a corresponding perturbation class which contains the compact operators.
Mutual existence of product integrals in normed rings
Jon C.
Helton
353-363
Abstract: Definitions and integrals are of the subdivision-refinement type, and functions are from $R \times R$ to N, where R denotes the set of real numbers and N denotes a ring which has a multiplicative identity element represented by 1 and a norm $\vert \cdot \vert$ with respect to which N is complete and $\vert 1\vert = 1$. If G is a function from $R \times R$ to N, then $G \in O{M^\ast}$ on [a, b] only if (i) $_x{\Pi ^y}(1 + G)$ exists for $a \leq x < y \leq b$ and (ii) if $\varepsilon > 0$, then there exists a subdivision D of [a, b] such that, if $\{ {x_i}\} _{i = 0}^n$ is a refinement of D and $0 \leq p < q \leq n$, then $\displaystyle \left\vert{}_{x_{p}}\prod ^{x_q} (1 + G) - \prod\limits_{i = p + 1}^q {(1 + {G_i})} \right\vert < \varepsilon ;$ and $G \in O{M^ \circ }$ on [a, b] only if (i) $_x{\Pi ^y}(1 + G)$ exists for $a \leq x < y \leq b$ and (ii) the integral $ \smallint _a^b\vert 1 + G - \Pi (1 + G)\vert$ exists and is zero. Further, $G \in O{P^ \circ }$ on [a, b] only if there exist a-subdivision D of [a, b] and a number B such that, if $\{ {x_i}\} _{i = 0}^n$ is a refinement of D and $0 < p \leq q \leq n$, then $\vert\Pi _{i = p}^q(1 + {G_i})\vert < B$. If F and G are functions from $R \times R$ to N, $F \in O{P^ \circ }$ on [a, b], each of $ {\lim _{x,y \to {p^ + }}}F(x,y)$ and ${\lim _{x,y \to {p^ - }}}F(x,y)$ exists and is zero for $p \in [a,b]$, each of ${\lim _{x \to {p^ + }}}F(p,x),{\lim _{x \to {p^ - }}}F(x,p),{\lim _{x \to {p^ + }}}G(p,x)$ and $ {\lim _{x \to {p^ - }}}G(x,p)$ exists for $ p \in [a,b]$, and G has bounded variation on [a, b], then any two of the following statements imply the other: (1) $F + G \in OM^\ast$ on [a, b], (2) $F \in OM^\ast$ on [a, b], and (3) $G \in OM^\ast$ on [a, b]. In addition, with the same restrictions on F and G, any two of the following statements imply the other: (1) $F + G \in OM^\circ$ on [a, b], (2) $F \in OM^\circ$ on [a, b], and (3) $G \in OM^\circ$ on [a, b]. The results in this paper generalize a theorem contained in a previous paper by the author [Proc. Amer. Math. Soc. 42 (1974), 96-103]. Additional background on product integration can be obtained from a paper by B. W. Helton [Pacific J. Math. 16 (1966), 297-322].
Characteristic principal bundles
Harvey A.
Smith
365-375
Abstract: Characteristic principal bundles are the duals of commutative twisted group algebras. A principal bundle with locally compact second countable (Abelian) group and base space is characteristic iff it supports a continuous eigenfunction for almost every character measurably in the characters, also iff it is the quotient by Z of a principal E-bundle for every E in $ {\operatorname{Ext}}(G,Z)$ and a measurability condition holds. If a bundle is locally trivial, n.a.s.c. for it to be such a quotient are given in terms of the local transformations and Čech cohomology of the base space. Although characteristic G-bundles need not be locally trivial, the class of characteristic G-bundles is a homotopy invariant of the base space. The isomorphism classes of commutative twisted group algebras over G with values in a given commutative $ {C^\ast}$-algebra A are classified by the extensions of G by the integer first Čech cohomology group of the maximal ideal space of A.
On subnormal operators
Mehdi
Radjabalipour
377-389
Abstract: Let T be the adjoint of a subnormal operator defined on a Hilbert space H. For any closed set $\delta$, let ${X_T}(\delta ) = \{ x \in H$: there exists an analytic function ${f_x}:{\text{C}}\backslash \delta \to H$ such that $ (z - T){f_x}(z) \equiv x\}$. It is shown that T is decomposable (resp. normal) if ${X_T}(\partial {G_\alpha })$ is closed (resp. if ${X_T}(\partial {G_\alpha }) = \{ 0\} )$ for a certain family $ \{ {G_\alpha }\}$ of open sets. Some of the results are extended to the case that T is the adjoint of the restriction of a spectral or decomposable operator to an invariant subspace.
On K\=omura's closed-graph theorem
Michael H.
Powell
391-426
Abstract: Let $(\alpha )$ be a property of separated locally convex spaces. Call a locally convex space $E[\mathcal{J}]$ an $ (\bar \alpha )$-space if $\mathcal{J}$ is the final topology defined by ${\{ {u_i}:{E_i}[{\mathcal{J}_i}] \to E\} _{i \in I}}$, where each ${E_i}[{\mathcal{J}_i}]$ is an $(\alpha )$-space. Then, for each locally convex space $ E[\mathcal{J}]$, there is a weakest $(\bar \alpha )$-topology on E stronger that $ \mathcal{J}$, denoted $ {\mathcal{J}^{\bar \alpha }}$. Kōmura's closed-graph theorem states that the following statements about a locally convex space $ E[\mathcal{J}]$ are equivalent: (1) For every $(\alpha )$-space F and every closed linear map $ u: F \to E[\mathcal{J}]$, u is continuous. (2) For every separated locally convex topology $ {\mathcal{J}_0}$ on E, weaker than $ \mathcal{J}$, we have $\mathcal{J} \subset \mathcal{J}_0^{\bar \alpha }$. Much of this paper is devoted to amplifying Kōmura's theorem in special cases, some well-known, others not. An entire class of special cases, generalizing Adasch's theory of infra-(s) spaces, is established by considering a certain class of functors, defined on the category of locally convex spaces, each functor yielding various notions of ``completeness'' in the dual space.
Gelfond's method for algebraic independence
W. Dale
Brownawell
1-26
Abstract: This paper extends Gelfond's method for algebraic independence to fields $K$ with transcendence type $\leqslant \tau$. The main results show that the elements of a transcendence basis for $K$ and at least two more numbers from a prescribed set are algebraically independent over $Q$. The theorems have a common hypothesis: $\{ {\alpha _1}, \ldots ,{\alpha _M}\} ,\{ {\beta _1}, \ldots ,{\beta _N}\}$ are sets of complex numbers, each of which is $Q$-linearly independent. THEOREM A. If $(2\tau - 1) < MN$, then at least two of the numbers ${\alpha _i},{\beta _j},\exp ({\alpha _i}{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N$, are algebraically dependent over $ K$. THEOREM B. If $ 2\tau (M + N) \leqslant MN + M$, then at least two of the numbers $ {\alpha _i},\exp ({\alpha _i},{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N$, are algebraically dependent over $ K$. THEOREM C. If $ 2\tau (M + N) \leqslant MN$, then at least two of the numbers $ 1 \leqslant i \leqslant M,1 \leqslant j \leqslant N$, are algebraically dependent over $K$.
The subgroups of a tree product of groups
J.
Fischer
27-50
Abstract: Let $G = {\Pi ^ \ast }({A_i};{U_{jk}} = {U_{kj}})$ be a tree product with $H$ a subgroup of $G$. By extending the technique of using a rewriting process we show that $H$ is an HNN group whose base is a tree product with vertices of the form $x{A_i}{x^{ - 1}} \cap H$. The associated subgroups are contained in vertices of the base, and both the associated subgroups of $H$ and the edges of its base are of the form $y{U_{jk}}{y^{ - 1}} \cap H$. The $ x$ and $y$ are certain double coset representatives for $ G\bmod (H,{A_i})$ and $G\bmod (H,{U_{jk}})$, respectively, and the elements defined by the free part of $ H$ are specified. More precise information about $H$ is given when $H$ is either indecomposable or $ H$ satisfies a nontrivial law. Introducing direct tree products, we use our subgroup theorem to prove that if each edge of $G$ is contained in the center of its two vertices then the cartesian subgoup of $ G$ is a free group. We also use our subgroup theorem in proving that if each edge of $G$ is a finitely generated subgroup of finite index in both of its vertices and some edge is a proper subgroup of both its vertices then $G$ is a finite extension of a free group iff the orders of the ${A_i}$ are uniformly bounded.
Hypercontractive semigroups and Sobolev's inequality
George F.
Feissner
51-62
Abstract: If $H \geqslant 0$ is the generator of a hypercontractive semigroup (HCSG), it is known that ${(H + 1)^{ - 1/2}}$ is a bounded operator from $ {L^p}$ to ${L^p},1 \leqslant p \leqslant \infty$. We prove that ${(H + 1)^{ - 1/2}}$ is bounded from ${L^2}$ to the Orlicz space ${L^2}{\text{ I}}{{\text{n}}^ + }L$, basing the proof on the uniform semiboundedness of the operator $H + V$, for suitable $V$. We also prove by an interpolation argument, that ${(H + 1)^{ - 1/2}}$ is bounded from $ {L^p}$ to ${L^p}{\text{ I}}{{\text{n}}^ + }L,2 \leqslant p < \infty$. Another interpolation argument shows that $ {(H + 1)^{ - 1/2}}$ is bounded from $ {L^p}{({\text{I}}{{\text{n}}^ + }L)^m}$ to $ {L^p}{({\text{I}}{{\text{n}}^ + }L)^{m + 1}}$ and $m$ a positive integer. Finally, we identify the topological duals of the spaces mentioned above.
Duality for Hadamard products with applications to extremal problems for functions regular in the unit disc
Stephan
Ruscheweyh
63-74
Abstract: Let $A$ be the set of functions regular in the unit disc $ \mathcal{U}$ and $ {A_0}$ the set of all functions $f \in A$ which satisfy $f(0) = 1$. For $V \subset {A_0}$ define the dual set $ {V^ \ast } = \{ f \in {A_0}\vert f \ast g \ne 0{\text{ for all }}g \in V,z \in \mathcal{U}\} ,{V^{ \ast \ast }} = {({V^ \ast })^ \ast }$. Here $f \ast g$ denotes the Hadamard product. THEOREM. Let $V \subset {A_0}$ have the following properties: (i) $V$ is compact, (ii) $f \in V$ implies $f(xz) \in V$ for all $\vert x\vert \leqslant 1$. Then $ \lambda (V) = \lambda ({V^{ \ast \ast }})$ for all continuous linear functionals $\lambda$ on $A$. This theorem has many applications to functions in $A$ which are defined by properties like bounded real part, close-to-convexity, univalence etc.
On minimal immersions of $S\sp{2}$ into $S\sp{2m}$
Jo ao Lucas Marquês
Barbosa
75-106
Abstract: The study of minimal immersions of the $2$-sphere into the standard $n$-sphere of the euclidean space has been better accomplished by associating to each such immersion a certain holomorphic curve. This has been done in several ways in the literature. In the present paper we explore this technique applying some knowledge about the topological and analytical invariants of the particular set of holomorphic curves used to obtain further results. Some new examples are provided, a beginning of a general description of such immersions is given and a rigidity theorem is proved.
Topological dynamics and $C\sp{\ast} $-algebras
William L.
Green
107-121
Abstract: If $G$ is a group of automorphisms of a ${C^ \ast }$-algebra $A$ with identity, then $G$ acts in a natural way as a transformation group on the state space $S(A)$ of $A$. Moreover, this action is uniformly almost periodic if and only if $G$ has compact pointwise closure in the space of all maps of $A$ into $A$. Consideration of the enveloping semigroup of $ (S(A),G)$ shows that, in this case, this pointwise closure $\bar G$ is a compact topological group consisting of automorphisms of $A$. The Haar measure on $\bar G$ is used to define an analogue of the canonical center-valued trace on a finite von Neumann algebra. If $A$ possesses a sufficiently large group $ {G_0}$ of inner automorphisms such that $ (S(A),{G_0})$ is uniformly almost periodic, then $A$ is a central $ {C^ \ast }$-algebra. The notion of a uniquely ergodic system is applied to give necessary and sufficient conditions that an approximately finite dimensional $ {C^ \ast }$-algebra possess exactly one finite trace.
On automorphism groups and endomorphism rings of abelian $p$-groups
Jutta
Hausen
123-128
Abstract: Let $A$ be a noncyclic abelian $ p$-group where $p \geqslant 5$, and let $ {p^\infty }A$ be the maximal divisible subgroup of $A$. It is shown that $ A/{p^\infty }A$ is bounded and nonzero if and only if the automorphism group of $ A$ contains a minimal noncentral normal subgroup. This leads to the following connection between the ideal structure of certain rings and the normal structure of their groups of units: if the noncommutative ring $R$ is isomorphic to the full ring of endomorphisms of an abelian $p$-group, $ p \geqslant 5$, then $ R$ contains minimal twosided ideals if and only if the group of units of $ R$ contains minimal noncentral normal subgroups.
The similarity orbit of a normal operator
L. A.
Fialkow
129-137
Abstract: If $N$ is a bounded normal operator on a separable Hilbert space $ \mathcal{H}$, let $\mathcal{S}(N)$ denote the similarity orbit of $ N$ in $L(\mathcal{H})$ and let ${\mathcal{S}_k}(N)$ denote the set of all compact perturbations of elements of $\mathcal{S}(N)$. It is proved that $ \mathcal{S}(N)({\mathcal{S}_K}(N))$ is norm closed in $L(\mathcal{H})$ if and only if the spectrum (essential spectrum) of $N$ is finite. If the essential spectrum of $ N$ is infinite and $ M$ is a normal operator whose spectrum is connected and contains that of $ N$, then $M$ is in the closure of $\mathcal{S}(N)$. If the spectrum of $N$ is connected, this result characterizes the normal elements of the closure of $\mathcal{S}(N)$. A normal operator is similar to a nonquasidiagonal operator if and only if its essential spectrum contains more than two points.
The zeroes of nonnegative holomorphic curvature operators
A. M.
Naveira;
C.
Fuertes
139-147
Abstract: Here, we study the structure of points in a holomorphic Grassmann's submanifold where the holomorphic sectional curvature assumes its minimum and maximum. For spaces of nonnegative holomorphic sectional curvature we study the set of points on which it assumes the value zero. We show that the minimum and maximum sets of holomorphic sectional curvature are the intersections of a holomorphic Grassmann's submanifold with linear complex holomorphic subspaces of type (1, 1).
A classification theorem for abelian $p$-groups
R. B.
Warfield
149-168
Abstract: A new class of Abelian $p$-groups, called $S$-groups, is studied, and the groups in this class are classified in terms of cardinal invariants. The class of $S$-groups includes Nunke's totally projective $ p$-groups. The invariants consist of the Ulm invariants (which Hill has shown can be used to classify the totally projective groups) together with a new sequence of invariants indexed by limit ordinals which are not cofinal with $ \omega$. The paper includes a fairly complete discussion of dense isotype subgroups of totally projective $p$-groups, including necessary and sufficient conditions for two of them to be congruent under the action of an automorphism of the group. It also includes an extension of Ulm's theorem to a class of mixed modules over a discrete valuation ring.
Concerning first countable spaces. III
G. M.
Reed
169-177
Abstract: The primary purpose of this paper is (1) to provide a ``real'' example of a regular first countable ${T_1}$-space which has no dense developable subspace and (2) to provide a new technique for producing Moore spaces which fail to have dense metrizable subspaces. Related results are established which produce new examples of noncompletable Moore spaces and which show that each regular hereditary $M$-space with a $ {G_\delta }$-diagonal has a dense metrizable subspace.
A property of finite $p$-groups with trivial multiplicator
Michael R.
Jones
179-183
Abstract: A sufficient condition for a finite $2$-generator $p$-group to have nontrivial multiplicator is given. To show that this result is best possible, a finite $2$-group with trivial multiplicator is constructed.
On the variety of manifolds without conjugate points
Robert
Gulliver
185-201
Abstract: The longest geodesic segment in a convex ball of a riemannian manifold, where the convexity is ensured by an upper bound on sectional curvatures, is the diameter. This and related results are demonstrated and applied to show that there exist manifolds with sectional curvatures of both signs but with-out conjugate points.
On the construction of the Bockstein spectral sequence
Jerrold
Siegel
203-224
Abstract: The Bockstein spectral sequence is developed from a direct limit construction. This is shown to clarify its relation to certain associated structures, in particular the divided power operations. Finally, the direct limit construction is used to study the problem of enumerating the Bockstein spectral sequences over a given simple $R$-module.
Two weight function norm inequalities for the Poisson integral
Benjamin
Muckenhoupt
225-231
Abstract: Let $f(x)$ denote a complex valued function with period $2\pi$, let $\displaystyle {P_r}(f,x) = \frac{1}{{2\pi }}\int_{ - \pi }^\pi {\frac{{(1 - {r^2})f(y)dy}}{{1 - 2r\cos (x - y) + {r^2}}}}$ be the Poisson integral of $f(x)$ and let $\vert I\vert$ denote the length of an interval $ I$. For $1 \leqslant p < \infty $ and nonnegative $ U(x)$ and $V(x)$ with period $2\pi$ it is shown that there is a $ C$, independent of $ f$, such that $\displaystyle \mathop {\sup }\limits_{0 \leqslant r < 1} \int_{ - \pi }^\pi {\v... ...,x){\vert^p}U(x)dx \leqslant C\int_{ - \pi }^\pi {\vert f(x){\vert^p}V(x)dx} }$ if and only if there is a $ B$ such that for all intervals $I$ $\displaystyle \left[ {\frac{1}{{\vert I\vert}}\int_I {U(x)dx} } \right]{\left[ ... ...t I\vert}}\int_I {{{[V(x)]}^{ - 1/(p - 1)}}dx} } \right]^{p - 1.}} \leqslant B.$ Similar results are obtained for the nonperiodic case and in the case where $U(x)dx$ and $V(x)dx$ are replaced by measures.
Polynomials related to the Bessel functions
F. T.
Howard
233-248
Abstract: In this paper we examine the polynomials ${W_n}(a)$ defined by means of $\displaystyle - 4{e^{xa}}{[x({e^x} - 1) - 2({e^x} + 1)]^{ - 1}} = \sum\limits_{n = 0}^\infty {{W_n}(a){x^n}/n!} .$ These polynomials are closely related to the zeros of the Bessel function of the first kind of index --3/2, and they are in some ways analogous to the Bernoulli and Euler polynomials. This analogy is discussed, and the real and complex roots of ${W_n}(a)$ are investigated. We show that if $n$ is even then $ {W_n}(a) > 0$ for all $ a$, and if $n$ is odd then ${W_n}(a)$ has only the one real root $ a = 1/2$. Also we find upper and lower bounds for all $b$ such that $ {W_n}(a + bi) = 0$. The problem of multiple roots is discussed and we show that if $n \equiv 0,1,5,8$ or 9 $(\bmod\; 12)$, then ${W_n}(a)$ has no multiple roots. Finally, if $ n \equiv 0,1,2,5,6$ or 8 $(\bmod \; 12)$, then ${W_n}(a)$ has no factor of the form ${a^2} + ca + d$ where $c$ and $ (\bmod\; 12)$ are integers.
The center of an order with finite global dimension
Mark
Ramras
249-257
Abstract: Let $\Lambda$ be a quasi-local ring of global dimension $n < \infty $. Assume that its center $ R$ is a noetherian domain, that $\Lambda$ is finitely generated torsion-free as an $ R$-module, and that $ R$ is an $R$-direct summand of $\Lambda$. Then $R$ is integrally closed in its quotient field $K$ and Macauley of dimension $n$. Furthermore, when $n = 2,\Lambda$ is a maximal $R$-order in the central simple $ K$-algebra $\Lambda { \otimes _R}K$. This extends an earlier result of the author, in which $R$ was assumed to have global dimension 2. Examples are given to show that in the above situation $ R$ can have infinite global dimension.
Newman's theorem in the Riemannian category
L. N.
Mann;
J. L.
Sicks
259-266
Abstract: In 1931 M. H. A. Newman proved that if $M$ is a connected topological manifold with metric $d$, there exists a number $\varepsilon > 0$, depending only upon $ M$ and $d$, such that every compact Lie group $ G$ acting effectively on $ M$ has at least one orbit of diameter at least $ \varepsilon$. Aside from isolated results nothing appears to be known about $\varepsilon$. In order to learn more about the invariant $ \varepsilon$, attention is restricted here to groups of isometries on a Riemannian manifold. It is found that the invariant $\varepsilon$ of $M$ is connected with the notion of convexity introduced by J. H. C. Whitehead in 1932.
Sobolev-Galpern equations of order $n+2$ in $R\sp{m}\times R$, $m\geq 2$
V. R. Gopala
Rao
267-278
Abstract: Equations with mixed time and space derivatives play an important role in several branches of physics. Here we establish existence and uniqueness results for such equations. In addition, we also prove a regularity result which employs a regularity result for nonhomogeneous elliptic equations whose proof is also included.
Localization and sheaf reflectors
J.
Lambek;
B. A.
Rattray
279-293
Abstract: Given a triple $(S,\eta ,\mu )$ on a category $\mathcal{A}$ with equalizers, one can form a new triple whose functor $Q$ is the equalizer of $\eta S$ and $S\eta$. Fakir has studied conditions for $ Q$ to be idempotent, that is, to determine a reflective subcategory of $\mathcal{A}$. Here we regard $S$ as the composition of an adjoint pair of functors and give several new such conditions. As one application we construct a reflector in an elementary topos $ \mathcal{A}$ from an injective object $I$, taking $ S = {I^{{I^{( - )}}}}$. We show that this reflector preserves finite limits and that the sheaf reflector for a topology in $\mathcal{A}$ can be obtained in this way. We also show that sheaf reflectors in functor categories can be obtained from a triple of the form $S = {I^{( - ,I)}},I$ injective, which we studied in a previous paper. We deduce that the opposite of a sheaf subcategory of a functor category is tripleable over Sets.
On bilateral derivates and the derivative
K. M.
Garg
295-329
Abstract: In this paper we prove a new result on the monotonicity of a function in terms of its bilateral derivates, and obtain from it extensions of several existing results on such derivates and the derivative of a function. Let $ f:R \to R$, where $ R$ denotes the set of real numbers. If its lower derivate $\underline{D} f > 0$ at a nonmeager set of points, we prove $f$ to be ``adequately'' increasing in some interval, viz. even the function $ f(x) - \alpha x$ is increasing for some $ \alpha > 0$. When $ f$ is nowhere adequately monotone, it follows that there exists a residual set of points where $f$ has a zero ``median'' derivate, i.e. either $ D\_f \leqslant 0 \leqslant {D^ - }f$ or ${D_ + }f \leqslant 0 \leqslant {D^ + }f$. These results remain valid for functions defined on an arbitrary set $ X \subset R$ under a mild continuity hypothesis, e.g. the absence of ordinary discontinuity at the unilateral limit points of $ X$. The last result leads to a new version of A. P. Morse's theorem for median derivates, and this in turn yields an improved version of the Goldowski-Tonelli theorem. We also obtain some necessary and sufficient conditions for a function to be nondecreasing, and extensions of the mean-value theorem and the Denjoy and other properties of a derivative. If $f:X \to R$, where $ X \subset R$, and both the derivates of $f$ are finite at a set of points that is not meager in $X$, then $f$ is further proved to satisfy the Lipschitz condition on some portion of $X$. When $f$ has a finite derivate almost everywhere and $ X$ has a finite measure, it is shown that $f$ can be made Lipschitz by altering its values on a set with arbitrarily small measure. Some results on singular functions are also strengthened. The results and the methods of this paper further provide extensions of some results of Young, Tolstoff, Kronrod, Zahorski, Brudno, Fort, Hájek, Filipczak, Neugebauer and Lipiński on derivates and the derivability of a function.
On the generation of one-relator groups
Stephen J.
Pride
331-364
Abstract: This paper is concerned with obtaining information about the Nielsen equivalence classes and $T$-systems of certain two-generator HNN groups, and in particular of certain two-generator one-relator groups. The theorems presented here extend results of the author appearing in the Proceedings of the Second International Conference on the Theory of Groups. In particular it is shown here that if $G = \langle a,t;{a^{{\alpha _1}}}E_r^{ - 1}{a^{{\beta _1}}}{E_r} \cdots {a^{{a_s}}}E_r^{ - 1}{a^{{B_s}}}{E_r})$ where the $ {\alpha _j}$ are positive, the ${\beta _i}$ are nonzero, ${E_r}$ has the form $[{a^{{ \in _1}}},[{a^{{ \in _2}}},[ \cdots ,[{a^{{ \in _r}}},t] \cdots ]]]$ with $ \vert{ \in _1}\vert = \vert{ \in _2}\vert = \cdots = \vert{ \in _r}\vert = 1$, then in a large number of cases $G$ has one Nielsen equivalence class. Similar results are also obtained for certain groups with more than one relator. A fair proportion of the paper is given to developing a method for reducing pairs of elements in HNN groups. This method has some of the features of Nielsen's reduction theorem for free groups. One other interesting result obtained here is that a one-relator group with torsion which has one $ T$-system is Hopfian. The early part of the paper is discursive. It contains most of the known results concerning $T$-systems of one-relator groups, and highlights several open problems, some of which have been raised by other authors.
Topological extension properties
R. Grant
Woods
365-385
Abstract: It is known that if a topological property $ \mathcal{P}$ of Tychonoff spaces is closed-hereditary, productive, and possessed by all compact $ \mathcal{P}$-regular spaces, then each $ \mathcal{P}$-regular space $ X$ is a dense subspace of a space $ {\gamma _\mathcal{P}}X$ with $\mathcal{P}$ such that if $Y$ has $ \mathcal{P}$ and $ f:X \to Y$ is continuous, then $f$ extends continuously to ${f^\gamma }:{\gamma _\mathcal{P}}X \to Y$. Such topological properties are called extension properties; $ {\gamma _\mathcal{P}}X$ is called the maximal $ \mathcal{P}$-extension of $ X$. In this paper we study the relationships between pairs of extension properties and their maximal extensions. A basic tool is the concept of $ \mathcal{P}$-pseudocompactness, which is studied in detail (a $\mathcal{P}$-regular space $X$ is $ \mathcal{P}$-pseudocompact if $ {\gamma _\mathcal{P}}X$ is compact). A classification of extension properties is attempted, and several means of constructing extension properties are studied. A number of examples are considered in detail.
Nonlinear techniques for linear oscillation problems
Zeev
Nehari
387-406
Abstract: It is shown that for differential equations of the form ${y^{(n)}} + py = 0$ there exist associated sets of systems of nonlinear equations which play a role similar to that of the ordinary Riccati equation in the case $n = 2$. In particular, the existence of continuous solutions of the nonlinear system is equivalent to the absence of certain types of oscillatory solutions of the linear equation. If $ p$ is of constant sign, the coefficients of the ``Riccati systems'' are all nonnegative, and the resulting positivity and monotonicity properties make it possible to obtain explicit oscillation criteria for the original equation.
Fourier analysis on the sphere
Thomas O.
Sherman
1-31
Abstract: A new approach to harmonic analysis on the unit sphere in ${{\mathbf{R}}^{d + 1}}$ is given, closer in form to Fourier analysis on $ {{\mathbf{R}}^d}$ than the usual development in orthonormal polynomials. Singular integrals occur in the transform formulae. The results generalize to symmetric space.
Ramsey graphs and block designs. I
T. D.
Parsons
33-44
Abstract: This paper establishes a connection between a certain class of Ramsey numbers for graphs and the class of symmetric block designs admitting a polarity. The main case considered here relates the projective planes over Galois fields to the Ramsey numbers $R({C_4},{K_{1,n}}) = f(n)$. It is shown that, for every prime power $q,f({q^2} + 1) = {q^2} + q + 2$, and $f({q^2}) = {q^2} + q + 1$.
Minimal covers and hyperdegrees
Stephen G.
Simpson
45-64
Abstract: Every hyperdegree at or above that of Kleene's $O$ is the hyperjump and the supremum of two minimal hyperdegrees (Theorem 3.1). There is a nonempty $\Sigma _1^1$ class of number-theoretic predicates each of which has minimal hyperdegree (Theorem 4.7). If $V = L$ or a generic extension of $L$, then there are arbitrarily large hyperdegrees which are not minimal over any hyperdegree (Theorems 5.1, 5.2). If ${O^\char93 }$ exists, then there is a hyperdegree such that every larger hyperdegree is minimal over some hyperdegree (Theorem 5.4). Several other theorems on hyperdegrees and degrees of nonconstructibility are presented.
Global dimension of differential operator rings. II
K. R.
Goodearl
65-85
Abstract: The aim of this paper is to find the global homological dimension of the ring of linear differential operators $R[{\theta _1}, \ldots ,{\theta _u}]$ over a differential ring $R$ with $u$ commuting derivations. When $R$ is a commutative noetherian ring with finite global dimension, the main theorem of this paper (Theorem 21) shows that the global dimension of $R[{\theta _1}, \ldots ,{\theta _u}]$ is the maximum of $k$ and $q + u$, where $q$ is the supremum of the ranks of all maximal ideals $ M$ of $R$ for which $R/M$ has positive characteristic, and $ k$ is the supremum of the sums $ rank(P) + diff\;dim(P)$ for all prime ideals $P$ of $R$ such that $R/P$ has characteristic zero. [The value $diff\;dim(P)$ is an invariant measuring the differentiability of $P$ in a manner defined in §3.] In case we are considering only a single derivation on $R$, this theorem leads to the result that the global dimension of $ R[\theta ]$ is the supremum of gl $dim(R)$ together with one plus the projective dimensions of the modules $R/J$, where $J$ is any primary differential ideal of $ R$. One application of these results derives the global dimension of the Weyl algebra in any degree over any commutative noetherian ring with finite global dimension.
Ramsey theorems for multiple copies of graphs
S. A.
Burr;
P.
Erdős;
J. H.
Spencer
87-99
Abstract: If $G$ and $H$ are graphs, define the Ramsey number $ r(G,H)$ to be the least number $p$ such that if the edges of the complete graph $ {K_p}$ are colored red and blue (say), either the red graph contains $ G$ as a subgraph or the blue graph contains $H$. Let $mG$ denote the union of $m$ disjoint copies of $G$. The following result is proved: Let $ G$ and $H$ have $k$ and $l$ points respectively and have point independence numbers of $i$ and $j$ respectively. Then $N - 1 \leqslant r(mG,nH) \leqslant N + C$, where $ N = km + ln - min(mi,mj)$ and where $C$ is an effectively computable function of $ G$ and $H$. The method used permits exact evaluation of $r(mG,nH)$ for various choices of $G$ and $H$, especially when $m = n$ or $G = H$. In particular, $r(m{K_3},n{K_3}) = 3m + 2n$ when $m \geqslant n,m \geqslant 2$.
A characterization of manifolds
Louis F.
McAuley
101-107
Abstract: The purpose of this paper is (1) to give a proof of one general theorem characterizing certain manifolds and (2) to illustrate a technique which should be useful in proving various theorems analogous to the one proved here. Theorem. Suppose that $ f:X \Rightarrow [0,1]$, where $X$ is a compactum, and that $f$ has the properties: (1) for $0 \leqslant x < 1/2,{f^{ - 1}}(x) = {S^n} \cong {M_0}$, (2) $ {f^{ - 1}}(1/2) \cong {S^n}$ with a tame (or flat) $k$-sphere ${S^k}$ shrunk to a point, (3) for $ 1/2 < x \leqslant 1,{f^{ - 1}}(x) \cong$ a compact connected $n$-manifold ${M_1} \cong {S^{n - (k + 1)}} \times {S^{k + 1}}$ (a spherical modification of $ {M_0}$ of type $ k$), and (4) there is a continuum $C$ in $X$ such that (letting ${C_x} = {f^{ - 1}}(x) \cap C$) (a) $0 \leqslant x < 1/2,{C_x} \cong {S^k}$, (b) $ {C_{1/2}} = \{ p\}$ a point, (c) for $1/2 < x \leqslant 1$, and (d) each of $ f\vert(X - C),f\vert{f^{ - 1}}[0,1/2)$, and $ f\vert{f^{ - 1}}(1/2,1]$ is completely regular. Then $X$ is homeomorphic to a differentiable $(n + 1)$-manifold $M$ whose boundary is the disjoint union of ${\bar M_0}$ and $ {\bar M_1}$ where ${M_i} = {\bar M_i},i = 0,1$.
Isotropic immersions and Veronese manifolds
T.
Itoh;
K.
Ogiue
109-117
Abstract: An $n$-dimensional Veronese manifold is defined as a minimal immersion of an $n$-sphere of curvature $ n/2(n + 1)$ into an $\{ n(n + 3)/2 - 1\}$-dimensional unit sphere. The purpose of this paper is to give some characterizations of a Veronese manifold in terms of isotropic immersions.
Projective limits in harmonic analysis
William A.
Greene
119-142
Abstract: A treatment of induced transformations of measures and measurable functions is presented. Given a diagram $\varphi :G \to H$ in the category of locally compact groups and continuous proper surjective group homomorphisms, functors are produced which on objects are given by $G \to {L^2}(G),{L^1}(G)$, $ M(G),W(G)$, denoting, resp., the ${L^2}$-space, ${L^1}$-algebra, measure algebra, and von Neu mann algebra generated by left regular representation of $ {L^1}$ on ${L^2}$. All functors but but the second are shown to preserve projective limits; by example, the second is shown not to do so. The category of Hilbert spaces and linear transformations of norm $\leqslant 1$ is shown to have projective limits; some propositions on such limits are given. Also given is a type and factor characterization of projective limits in the category of $ {W^ \ast }$-algebras and surjective normal $\ast$-algebra homomorphisms.
Sheaves of $H$-spaces and sheaf cohomology
James M.
Parks
143-156
Abstract: The concept of a sheaf of $H$-spaces is introduced and, using the Čech technique, a cohomology theory is defined in which the cohomology ``groups'' are $H$-spaces. The corresponding axioms of Cartan [3] for this theory are verified and other properties of the theory are investigated.
On rearrangements of Vilenkin-Fourier series which preserve almost everywhere convergence
J. A.
Gosselin;
W. S.
Young
157-174
Abstract: It is known that the partial sums of Vilenkin-Fourier series of $ {L^q}$ functions $ (q > 1)$ converge a.e. In this paper we establish the ${L^2}$ result for a class of rearrangements of the Vilenkin-Fourier series, and the $ {L^q}$ result $(1 < q < 2)$ for a subclass of rearrangements. In the case of the Walsh-Fourier series, these classes include the Kaczmarz rearrangement studied by L. A. Balashov. The ${L^2}$ result for the Kaczmarz rearrangement was first proved by K. H. Moon. The techniques of proof involve a modification of the Carleson-Hunt method and estimates on maximal functions of the Hardy-Littlewood type that arise from these rearrangements.
Measures associated with Toeplitz matrices generated by the Laurent expansion of rational functions
K. Michael
Day
175-183
Abstract: Let ${T_n}(a) = ({a_{i - j}})_{i,j = 0}^n$ be the finite Toeplitz matrices generated by the Laurent expansion of an arbitrary rational function, and let ${\sigma _n} = \{ {\lambda _{n0}}, \ldots ,{\lambda _{nn}}\}$ be the corresponding sets of eigenvalues of ${T_n}(f)$. Define a sequence of measures ${\alpha _n},{\alpha _n}(E) = {(n + 1)^{ - 1}}{\Sigma _{{\lambda _{ni}} \in E}}1,{\lambda _{ni}} \in {\sigma _n}$, and $E$ a set in the $\lambda$-plane. It is shown that the weak limit $ \alpha$ of the measures ${\alpha _n}$ is unique and possesses at most two atoms, and the function $f$ which give rise to atoms are identified.
On bounded elements of linear algebraic groups
Kwan-Yuk Claire
Sit
185-198
Abstract: Let $F$ be a local field of characteristic zero and ${\text{G }}$ a connected algebraic group defined over $F$. Let $G$ be the locally compact group of $F$-rational points. One characterizes the group $B(G)$ of $g \in G$ whose conjugacy class is relatively compact. For instance, if $ {\text{G}}$ is $ F$-split or reductive without anisotropic factors then $B(G)$ is the center of $G$. If $H$ is a closed subgroup of $G$ such that $G/H$ has finite volume, then the centralizer of $ H$ in $G$ is contained in $B(G)$. If, moreover, $H$ is the centralizer of some $ x \in G$ then $ G/H$ is compact.
Interpolation and uniqueness results for entire functions
James D.
Child
199-209
Abstract: Let $K[\Omega ]$ denote the collection of entire functions of exponential type whose Borel transforms are analytic on ${\Omega ^c}$ (the complement of the simply connected domain taken relative to the sphere). Let $ f$ be in $K[\Omega ]$ and set ${L_n}(f) = {(2\pi i)^{ - 1}}\smallint \Gamma {g_n}(\lambda )F(\lambda )d\lambda (n = 0,1, \ldots )$ where $F$ is the Borel transform of $f,\Gamma \subset \Omega$ is a simple closed contour chosen so that $F$ is analytic outside and on $\Gamma$ and each ${g_n}$ is in $ H(\Omega )$ (the collection of functions analytic on $\Omega$). In what follows read 'the sequence of linear functionals $ \{ {L_n}(f)\}$' wherever the sequence of functions ' $ \{ {g_n}\}$' appears. Let $ T$ denote a continuous linear operator from $ H(\Omega )$ to $H(\Lambda )$ where $\Lambda$ is also a simply connected domain. The topologies on $ H(\Omega )$ and $H(\Lambda )$ are those of uniform convergence on compact subsets of $\Omega$ (resp. $\Lambda$). The purpose of this paper is to consider uniqueness preserving operators, i.e., operators $T$ which have the property that $K[\Lambda ]$ is a uniqueness class for $\{ T({g_n})\}$ whenever $ K[\Omega ]$ is a uniqueness class for $\{ {g_n}\}$, and to examine interpolation preserving operators, i.e., operators $ T$ which have the property that $K[\Lambda ]$ interpolates the sequence of complex numbers $ \{ {b_n}\}$ relative to $\{ T({g_n})\}$ whenever $ K[\Omega ]$ interpolates $\{ {b_n}\}$ relatives to $\{ {g_n}\}$. Once some classes of uniqueness preserving operators and some classes of interpolation preserving operators have been found, we proceed to obtain new uniqueness and interpolation results from our knowledge of these operators and from previously known uniqueness and interpolation results. Operators which multiply by analytic functions and some differential operators are considered. Composition operators are studied and the results are used to extend the interpolation results for sequences of functions of the form $ \{ {[W(\zeta )]^n}\}$ where $W$ is analytic and univalent on a simply connected domain.
Some $H\sp{\infty }$-interpolating sequences and the behavior of certain of their Blaschke products
Max L.
Weiss
211-223
Abstract: Let $f$ be a strictly increasing continuous real function defined near ${0^ + }$ with $k,f(\theta + kf(\theta ))/f(\theta ) \to 1/$ as $\theta \to {0^ + }$. The curve in the open unit disc with corresponding representation $1 - r = f(\theta )$ is called a $K$-curve. Several analytic and geometric conditions are obtained for $K$-curves and $K$-functions. This provides a framework for some rather explicit results involving parts in the closure of $K$-curves, $ {H^\infty }$-interpolating sequences lying on $K$-curves and the behavior of their Blaschke products. In addition, a sequence of points in the disc tending upper tangentially to 1 with moduli increasing strictly to 1 and arguments decreasing strictly to 0 is proved to be interpolating if and only if the hyperbolic distance between successive points remains bounded away from zero.
The Boolean space of orderings of a field
Thomas C.
Craven
225-235
Abstract: It has been pointed out by Knebusch, Rosenberg and Ware that the set $ X$ of all orderings on a formally real field can be topologized to make a Boolean space (compact, Hausdorff and totally disconnected). They have called the sets of orderings $W(a) = \{ < {\text{ in }}X\vert a < 0\}$ the Harrison subbasis of $X$. This subbasis is closed under symmetric difference and complementation. In this paper it is proved that, given any Boolean space $X$, there exists a formally real field $ F$ such that $ X$ is homeomorphic to the space of orderings on $F$. Also, an example is given of a Boolean space and a basis of clopen sets closed under symmetric difference and complementation which cannot be the Harrison subbasis of any formally real field.
Analytic continuation, envelopes of holomorphy, and projective and direct limit spaces
Robert
Carmignani
237-258
Abstract: For a Riemann domain $\Omega$, a connected complex manifold where $n(n = dimension)$ globally defined functions form a local system of coordinates at every point, and an arbitrary holomorphic function $f$ in $\Omega$, the ``Riemann surface'' ${\Omega _f}$, a maximal holomorphic extension Riemann domain for $f$, is formed from the direct limit of a sequence of Riemann domains. Projective limits are used to construct an envelope of holomorphy for $ \Omega$, a maximal holomorphic extension Riemann domain for all holomorphic functions in $\Omega$, which is shown to be the projective limit space of the ``Riemann surfaces'' ${\Omega _f}$. Then it is shown that the generalized notion of envelope of holomorphy of an arbitrary subset of a Riemann domain can also be characterized in a natural way as the projective limit space of a family of ``Riemann surfaces".
Conditions for the existence of contractions in the category of algebraic spaces
Joseph
Mazur
259-265
Abstract: Artin's conditions for the existence of a new contraction of an algebraic subspace are changed. The new conditions are more applicable in special cases, especially when the subspace has a conormal bundle.
On some classes of multivalent starlike functions
Ronald J.
Leach
267-273
Abstract: Classes of multivalent functions analogous to certain classes of univalent starlike functions are defined and studied. Estimates on coefficients and distortion are made, using a variety of techniques.
Holomorphic functions on nuclear spaces
Philip J.
Boland
275-281
Abstract: The space $\mathcal{H}(E)$ of holomorphic functions on a quasi-complete nuclear space is investigated. If $\mathcal{H}(E)$ is endowed with the compact open topology, it is shown that $\mathcal{H}(E)$ is nuclear if and only if $ E'$ (continuous dual of $ E$) is nuclear. If $ E$ is a $\mathcal{D}FN$ (dual of a Fréchet nuclear space) and $F$ is a closed subspace of $E$, then the restriction mapping $\mathcal{H}(E) \to \mathcal{H}(F)$ is a surjective strict morphism.
On differential rings of entire functions
A. H.
Cayford;
E. G.
Straus
283-293
Abstract: Consider an entire function $f$ which is a solution of the differential equation $R = {\mathbf{C}}$ and of finite exponential order in case $ R = {\mathbf{C}}[z]$. We use this result to prove a conjecture made in [2] that entire functions of order $\rho < s$, all of whose derivatives at $ s$ points are integers in an imaginary quadratic number field, must be solutions of linear differential equations with constant coefficients and therefore of order $\leqslant 1$.
On sums over Gaussian integers
D. G.
Hazlewood
295-310
Abstract: The object of this paper is to give asymptotic estimates for some number theoretic sums over Gaussian integers. As a consequence of general estimates, asymptotic estimates with explicit error terms for the number of Gaussian integers with only ``large'' prime factors and for the number of Gaussian integers with only ``small'' prime factors are given.
On the representation of lattices by modules
George
Hutchinson
311-351
Abstract: For a commutative ring $R$ with unit, a lattice $L$ is ``representable by $R$-modules'' if $L$ is embeddable in the lattice of submodules of some unitary left $R$-module. A procedure is given for generating an infinite first-order axiomatization of the class of all lattices representable by $ R$-modules. Each axiom is a universal Horn formula for lattices. The procedure for generating the axioms is closely related to the ring structure, and is ``effective'' in the sense that many nontrivial axioms can be obtained by moderate amounts of computation.
Product integral techniques for abstract hyperbolic partial differential equations
J. W.
Spellmann
353-365
Abstract: Explicit and implicit product integral techniques are used to represent a solution $U$ to the abstract system: ${U_{12}}(x,y) = AU(x,y);U(x,0) = p = U(0,y)$. The coefficient $A$ is a closed linear transformation defined on a dense subspace $D(A)$ of the Banach space $X$ and the point $p$ in $D(A)$ satisfies the condition that $ \vert\vert{A^i}p\vert\vert < {S^i}{(i!)^{3/2}}$ for all integers $i \geqslant 0$ and some $S > 0$. The implicit technique is developed under the additional assumption that $A$ generates a strongly continuous semigroup of bounded linear transformations on $ X$. Both methods provide representations for the ${J_0}$ Bessel function.
$a\sp*$-closures of lattice-ordered groups
Roger
Bleier;
Paul
Conrad
367-387
Abstract: A convex $ l$-subgroup of an $ l$-group $G$ is closed if it contains the join of each of its subsets that has a join in $ G$. An extension of $ G$ which preserves the lattice of closed convex $l$-subgroups of $G$ is called an $ {a^ \ast }$-extension of $G$. In this paper we consider ${a^ \ast }$-extensions and ${a^ \ast }$-closures of $G$.
The uniqueness of the one-dimensional paraboson field
Steven
Robbins
389-397
Abstract: A paraboson analog of the one-dimensional boson field is discussed and a uniqueness result similar to a result of Putnam is obtained. It is shown that the paraboson operators must be unbounded.
Compactness properties of topological groups. III
S. P.
Wang
399-418
Abstract: Compactness properties of topological groups and finiteness of Haar measure on homogeneous spaces are studied. Some concrete structure theorems are presented.
Monotone and open mappings on manifolds. I
John J.
Walsh
419-432
Abstract: Sufficient conditions are given for the existence of open mappings from a p. 1. manifold ${M^m},m \geqslant 3$, onto a polyhedron $ Q$. In addition, it is shown that a mapping $f$ from $ {M^m},m \geqslant 3$, to $ Q$ is homotopic to a monotone mapping of $M$ onto $Q$ iff ${f_ \ast }:{\pi _1}(M) \to {\pi _1}(Q)$ is onto. Finally, it is shown that a monotone mapping of ${M^m},m \geqslant 3$, onto $Q$ can be approximated by a monotone open mapping of $M$ onto $Q$.
A generalisation of supersoluble groups
R. J.
Haggarty
433-441
Abstract: A $p$-soluble group $G$ belongs to the class $F(n,p)$ whenever the ranks of the $ p$-chief factors of $ G$ divide $n$ and $G$ has order coprime to $n$. A group in $F(n,p)$ is characterised by the embedding of its maximal subgroups. Whenever ${N_1}$ and ${N_2}$ are normal subgroups of $G$, of coprime indices in $ G$, which lie in $ F(n,p)$, then $ G$ lies in $F(n,p)$ also. $F(n)$ denotes the intersection, taken over all primes $p$, of the classes $F(n,p)$. Simple groups all of whose proper subgroups lie in $F(n)$ are determined. Given an integer $ n > 2$, there exist an integer $m$ with the same prime divisors as $n$ and a soluble group $G$ such that $G$ lies in $F(m)$ but $G$ does not possess a Sylow tower. (We may take $ m = n$ provided that $ n$ is not a multiple of 1806.) Furthermore, when $n$ is odd, an example of a soluble group $ G$, all of whose proper subgroups lie in $F(n)$ but $G$ has no Sylow tower, is given.
Erratum to: ``Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics'' (Trans. Amer. Math. Soc. {\bf 186} (1973), 259--272, 1974)
Thomas G.
Kurtz
442
Weighted shifts and covariance algebras
Donal P.
O’Donovan
1-25
Abstract: The ${C^ \ast }$-algebras generated by bilateral and unilateral shifts are studied in terms of certain covariance algebras. This enables one to obtain an answer to the question of when such shifts are G.C.R., or not, or even when they are N.G.C.R.. In addition these shifts are classified to within algebraic equivalence.
A stability theorem for minimum edge graphs with given abstract automorphism group
Donald J.
McCarthy;
Louis V.
Quintas
27-39
Abstract: Given a finite abstract group $ \mathcal{G}$, whenever $ n$ is sufficiently large there exist graphs with $n$ vertices and automorphism group isomorphic to $\mathcal{G}$. Let $(\mathcal{G},n)$ denote the minimum number of edges possible in such a graph. It is shown that for each $\mathcal{G}$ there always exists a graph $ M$ such that for $ n$ sufficiently large, $ e(\mathcal{G},n)$ is attained by adding to $M$ a standard maximal component asymmetric forest. A characterization of the graph $M$ is given, a formula for $e(\mathcal{G},n)$ is obtained (for large $ n$), and the minimum edge problem is re-examined in the light of these results.
Induced automorphisms on Fricke characters of free groups
Robert D.
Horowitz
41-50
Abstract: The term character in this paper will denote the character of a group element under a general or indeterminate representation of the group in the special linear group of $2 \times 2$ matrices with determinant 1; the properties of characters of this type were first studied by R. Fricke in the late nineteenth century. Theorem 1 determines the automorphisms of a free group which leave the characters invariant. In a previous paper it was shown that the character of each element in the free group $ {F_n}$ of finite rank $ n$ can be identified with an element of a certain quotient ring of the commutative ring of polynomials with integer coefficients in $ {2^n} - 1$ indeterminates. It follows that any automorphism of $ {F_n}$ induces in a natural way an automorphism on this quotient ring. Corollary 1 shows that for $n \geqslant 3$ the group of induced automorphisms of ${F_n}$ is isomorphic to the group of outer automorphism classes of ${F_n}$. The possibility is thus raised that the induced automorphisms may be useful in studying the structure of this group. Theorem 2 gives a characterization for the group of induced automorphisms of $ {F_2}$ in terms of an invariant polynomial.
Some one-sided theorems on the tail distribution of sample sums with applications to the last time and largest excess of boundary crossings
Y. S.
Chow;
T. L.
Lai
51-72
Abstract: In this paper, we prove certain one-sided Paley-type inequalities and use them to study the convergence rates for the tail probabilities of sample sums. We then apply our results to find the limiting moments and the limiting distribution of the last time and the largest excess of boundary crossings for the sample sums, generalizing the results previously obtained by Robbins, Siegmund and Wendel. Certain one-sided limit theorems for delayed sums are also obtained and are applied to study the convergence rates of tail probabilities.
Necessary conditions for isomorphism of Lie algebras of Block
John B.
Jacobs
73-79
Abstract: Two algebras of Block, $\mathcal{L}(G,\delta ,f)$ and
Rings with idempotents in their nuclei
Michael
Rich
81-90
Abstract: Let $R$ be a prime nonassociative ring. If the set of idempotents of $R$ is a subset of the nucleus of $R$ or of the alternative nucleus of $ R$ then it is shown that $ R$ is respectively an associative or an alternative ring. Also if $R$ has one idempotent $\ne 0,1$ which is in the Jordan nucleus or in the noncommutative Jordan nucleus then it is shown that $R$ is respectively a Jordan or a noncommutative Jordan ring.
On the extension of mappings in Stone-Weierstrass spaces
Anthony J.
D’Aristotle
91-101
Abstract: N. Veličko generalized the well-known result of A. D. Taĭmanov on the extension of continuous functions by showing that Taĭmanov's theorem holds when $ Y$ (the image space) is $ H$-closed and Urysohn and the mapping $f$ is weakly $\theta$-continuous. We obtain, in a more direct fashion, an even stronger generalization of this theorem. We proceed to show that the class of all SW spaces is not reflective in the category of all completely Hausdorff spaces and continuous mappings. However, an epi-reflective situation is achieved by suitably enlarging the class of admissible morphisms. We conclude by establishing a number of results about SW extension spaces.
Nearness structures and proximity extensions
M. S.
Gagrat;
W. J.
Thron
103-125
Abstract: Proximity, contiguity and nearness structures are here studied from a unified point of view. In the discussion the role that grills can play in the theory is emphasized. Nearness structures were recently introduced by Herrlich and Naimpally. Thron pointed out the importance of grills in proximity theory. Nearness structures $v$ are then used to generate proximity extensions $ (\phi ,({X^v},{\Pi ^v}))$ of a given LO-proximity space $(X,\Pi )$, where ${\Pi _v} = \Pi$. Finally, the relation of the extensions $ (\phi ,({X^v},{\Pi ^v}))$ to arbitrary extensions $(i,(Y,{\Pi ^ \ast }))$ is investigated.
An embedding theorem for matrices of commutative cancellative semigroups
James
Streilein
127-140
Abstract: In this paper it is shown that each semigroup which is a matrix of commutative cancellative semigroups has a ``quotient semigroup'' which is a completely simple semigroup with abelian maximal subgroups. This result is proved by explicitly constructing the quotient semigroup. The paper also gives necessary and sufficient conditions for a semigroup of the type being considered in the paper to be isomorphic to a Rees matrix semigroup over a commutative cancellative semigroup. Several special cases and examples are also briefly discussed.
Polar sets and Palm measures in the theory of flows
Donald
Geman;
Joseph
Horowitz
141-159
Abstract: Given a flow $({\theta _t}),t$ real, over a probability space $ \Omega$, we prove that certain measures on $\Omega$ (viewed as the state space of the flow) decompose uniquely into a Palm measure $Q$ which charges no ``polar set'' and a measure supported by a polar set. Considering the continuous and discrete parts of the additive functional corresponding to $Q$, we find that $Q$ further decomposes into a measure charging no ``semipolar set'' and a measure supported by one. As a consequence, Palm measures are exactly those which neglect sets which the flow neglects, and polar sets are exactly those neglected by every Palm measure. Finally, we characterize various properties, such as predictability and continuity, of an additive functional in terms of its Palm measure. These results further illuminate the role played by supermartingales in the theory of flows, as pointed by J. de Sam Lazaro and P. A. Meyer.
Group presentations and formal deformations
Perrin
Wright
161-169
Abstract: Formal deformations (expansions and collapses) of dimension $\leqslant 3$ among $2$-dimensional polyhedra are explained in terms of a certain collection of operations on finite group presentations. The results are valid for any simple homotopy type of $2$-dimensional polyhedra, and simplifications are possible within the simply connected simple homotopy types.
Convergent subsequences from sequences of functions
James L.
Thornburg
171-192
Abstract: Let $\{ {y_k}\}$ be a sequence of functions, $ {y_k} \in {\Pi _{s \in S}}{E_s}$ where $S$ is a nonempty subset of the $l$-dimensional Euclidean space and $ {E_s}$ is an ordered vector space with positive cone ${K_s}$. If ${y_k} \in {\Pi _{s \in S}}{E_s}$, sufficient conditions are given that $ \{ {y_k}\}$ have a subsequence $\{ {h_k}\}$ such that for each $ t \in S$ the sequence $\{ {h_k}(t)\}$ is monotone for $k$ sufficiently large, depending on $t$. If each ${E_s}$ is an ordered topological vector space, sufficient conditions are given that $\{ {y_k}\}$ has a subsequence $\{ {h_k}\}$ such that for every $t \in S$ the sequence $\{ {h_k}(t)\}$ is either monotone for $ k$ sufficiently large depending on $t$, or else the sequence $\{ {h_k}(t)\}$ is convergent. If $ {E_s} = B$ for each $ s$ and $B$ a Banach space then a definition of bounded variation is given so that if $\{ {y_k}\}$ is uniformly norm bounded and the variation of the functions ${y_k}$ is uniformly bounded then there is a convergent subsequence $ \{ {h_k}\}$ of $\{ {y_k}\}$. In the case ${E_s} = E$ for each $s \in S$ and $E$ is such that bounded monotone sequences converge then the given conditions imply the existence of a subsequence $\{ {h_k}\}$ of $\{ {y_k}\}$ which converges for each $ t \in S$.
On the Harish-Chandra homomorphism
J.
Lepowsky
193-218
Abstract: Using the Iwasawa decomposition $ \mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$ of a real semisimple Lie algebra $ \mathfrak{g}$, Harish-Chandra has defined a now-classical homomorphism from the centralizer of $ \mathfrak{k}$ in the universal enveloping algebra of $ \mathfrak{g}$ into the enveloping algebra $ \mathcal{A}$ of $\mathfrak{a}$. He proved, using analysis, that its image is the space of Weyl group invariants in $\mathcal{A}$. Here the weaker fact that the image is contained in this space of invariants is proved ``purely algebraically". In fact, this proof is carried out in the general setting of semisimple symmetric Lie algebras over arbitrary fields of characteristic zero, so that Harish-Chandra's result is generalized. Related results are also obtained.
Conical vectors in induced modules
J.
Lepowsky
219-272
Abstract: Let $\mathfrak{g}$ be a real semisimple Lie algebra with Iwasawa decomposition $ \mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$, and let $\mathfrak{m}$ be the centralizer of $\mathfrak{a}$ in $ \mathfrak{k}$. A conical vector in a $ \mathfrak{g}$-module is defined to be a nonzero $\mathfrak{m} \oplus \mathfrak{n}$-invariant vector. The $ \mathfrak{g}$-modules which are algebraically induced from one-dimensional $(\mathfrak{m} \oplus \mathfrak{a} \oplus \mathfrak{n})$-modules on which the action of $\mathfrak{m}$ is trivial have ``canonical generators'' which are conical vectors. In this paper, all the conical vectors in these $ \mathfrak{g}$-modules are found, in the special case $\dim \mathfrak{a} = 1$. The conical vectors have interesting expressions as polynomials in two variables which factor into linear or quadratic factors. Because it is too difficult to determine the conical vectors by direct computation, metamathematical ``transfer principles'' are proved, to transfer theorems about conical vectors from one Lie algebra to another; this reduces the problem to a special case which can be solved. The whole study is carried out for semisimple symmetric Lie algebras with splitting Cartan subspaces, over arbitrary fields of characteristic zero. An exposition of the Kostant-Mostow double transitivity theorem is included.
The generalized Martin's minimum problem and its applications in several complex variables
Shozo
Matsuura
273-307
Abstract: The objectives of this paper are to generalize the Martin's ${\mathfrak{L}^2}$-minimum problem under more general additional conditions given by bounded linear functionals in a bounded domain $D$ in ${C^n}$ and to apply this problem to various directions. We firstly define the new $i$th biholomorphically invariant Kähler metric and the $i$th representative domain $(i = 0,1,2, \ldots )$, and secondly give estimates on curvatures with respect to the Bergman metric and investigate the asymptotic behaviors via an $ A$-approach on the curvatures about a boundary point having a sort of pseudoconvexity. Further, we study (i) the extensions of some results recently obtained by K. Kikuchi on the Ricci scalar curvature, (ii) a minimum property on the reproducing subspace-kernel in $ \mathfrak{L}_{(m)}^2(D)$, and (iii) an extension of the fundamental theorem of K. H. Look.
Uniqueness and $\alpha $-capacity on the group $2\sp{\omega }$
William R.
Wade
309-315
Abstract: We introduce a class of Walsh series $ \mathcal{J}_\alpha ^ +$ for each $0 < \alpha < 1$ and show that a necessary and sufficient condition that a closed set $E \subseteq {2^\omega }$ be a set of uniqueness for $\mathcal{J}_\alpha ^ +$ is that the $\alpha $-capacity of $ E$ be zero.
Pointwise bounds on eigenfunctions and wave packets in $N$-body quantum systems. III
Barry
Simon
317-329
Abstract: We provide a number of bounds of the form $ \vert\psi \vert \leqslant O(\exp ( - \alpha \vert x{\vert^\alpha })),\alpha > 1$, for ${L^2}$-eigenfunctions $\psi$ of $- \Delta + V$ with $V \to \infty$ rapidly as $\vert x\vert \to \infty $. Our strongest results assert that if $\vert V(x)\vert \geqslant c{x^{2m}}$ near infinity, then $\vert\psi (x)\vert \leqslant {D _\varepsilon }\exp ( - {(c - \varepsilon )^{1/2}}{(m + 1)^{ - 1}}{x^{m + 1}})$, and if $\vert V(x)\vert \leqslant c{x^{2m}}$ neat infinity, then for the ground state eigenfunction, $\Omega ,\Omega (x) \geqslant {E _\varepsilon }\exp ( - {(c + \varepsilon )^{1/2}}{(m + 1)^{ - 1}}{x^{m + 1}})$.
Uniqueness of commuting compact approximations
Richard B.
Holmes;
Bruce E.
Scranton;
Joseph D.
Ward
330-340
Abstract: Let $H$ be an infinite dimensional complex Hilbert space, and let $ \mathcal{B}(H)$ (resp. $\mathcal{C}(H)$) be the algebra of all bounded (resp. compact) linear operators on $H$. It is well known that every $T \in \mathcal{B}(H)$ has a best approximation from the subspace $ \mathcal{C}(H)$. The purpose of this paper is to study the uniqueness problem concerning the best approximation of a bounded linear operator by compact operators. Our criterion for selecting a unique representative from the set of best approximants is that the representative should commute with $ T$. In particular, many familiar operators are shown to have zero as a unique commuting best approximant.
On semisimple commutative semigroups
B. D.
Arendt
341-351
Abstract: This paper presents an application of radical theory to the structure of commutative semigroups via their semilattice decomposition. Maximal group congruences and semisimplicity are characterized for certain classes of commutative semigroups and $N$-semigroups.
Similarity of quadratic forms and isomorphism of their function fields
Adrian R.
Wadsworth
352-358
Abstract: This paper considers the question: Given anisotropic quadratic forms $ Q$ and $Q'$ over a field $K$ (char $K \ne 2$), if their function fields are isomorphic must $Q$ and $Q'$ be similar? It is proved that the answer is yes if $Q$ is a Pfister form or the pure part of a Pfister form, or a $4$-dimensional form. The argument for Pfister forms and their pure parts does not generalize because these are the only anisotropic forms which attain maximal Witt index over their function fields. To handle the $ 4$-dimensional case the following theorem is proved: If $Q$ and $Q'$ are two $4$-dimensional forms over $K$ with the same determinant $ d$, then $Q$ and $Q'$ are similar over $K$ iff they are similar over $K[\sqrt d ]$. The example of Pfister neighbors suggests that quadratic forms arguments are unlikely to settle the original question for other kinds of forms.
Finite groups with Sylow 2-subgroups of class two. I
Robert
Gilman;
Daniel
Gorenstein
1-101
Abstract: In this paper we classify finite simple groups whose Sylow 2-subgroups have nilpotency class two.
Finite groups with Sylow 2-subgroups of class two. II
Robert
Gilman;
Daniel
Gorenstein
103-126
Abstract: In this paper we classify finite simple groups whose Sylow 2-subgroups have nilpotence class two.
Convergence and divergence of series conjugate to a convergent multiple Fourier series
J. Marshall
Ash;
Lawrence
Gluck
127-142
Abstract: In this note we consider to what extent the classical theorems of Plessner and Kuttner comparing the set of convergence of a trigonometric series with that of the conjugate trigonometric series can be generalized to higher dimensions. We show that if a function belongs to ${L^p},p > 1$, of the $2$-torus, then the convergence (= unrestricted rectangular convergence) of the Fourier series on a set implies its three conjugate functions converge almost everywhere on that set. That this theorem approaches the best possible may be seen from two examples which show that the dimension may not be increased to 3, nor the required power of integrability be decreased to 1. We also construct a continuous function having a boundedly divergent Fourier series of power series type and an a.e. circularly convergent double Fourier series whose $y$-conjugate diverges circularly a.e. Our $ {L^p}$ result depends on a theorem of L. Gogöladze (our proof is included for the reader's convenience), work of J. M. Ash and G. Welland on $(C,1,0)$ summability, and on a result deducing the boundedness of certain partial linear means from convergence of those partial means. The construction of the counterexamples utilizes examples given by C. Fefferman, J. Marcinkiewicz, A. Zygmund, D. Menšov, and the present authors' earlier work.
Generalization of right alternative rings
Irvin Roy
Hentzel;
Giulia Maria
Piacentini Cattaneo
143-161
Abstract: We study nonassociative rings $R$ satisfying the conditions (1) $ (ab,c,d) + (a,b,[c,d]) = a(b,c,d) + (a,c,d)b$ for all $a,b,c,d \in R$, and (2) $ (x,x,x) = 0$ for all $x \in R$. We furthermore assume weakly characteristic not 2 and weakly characteristic not 3. As both (1) and (2) are consequences of the right alternative law, our rings are generalizations of right alternative rings. We show that rings satisfying (1) and (2) which are simple and have an idempotent $\ne 0, \ne 1$, are right alternative rings. We show by example that $(x,e,e)$ may be nonzero. In general, $(a,b,c) + (b,c,a) + (c,a,b) = 0$ for all $a,b,c \in R$. We generate the Peirce decomposition. If $R'$ has no trivial ideals contained in its center, the table for the multiplication of the summands is associative, and the nucleus of $R'$ contains ${R'_{10}} + {R'_{01}}$. Without the assumption on ideals, the table for the multiplication need not be associative; however, if the multiplication is defined in the most obvious way to force an associative table, the new ring will still satisfy (1), (2), (3).
The zeros of holomorphic functions in strictly pseudoconvex domains
Lawrence
Gruman
163-174
Abstract: We determine a sufficient condition on a positive divisor in certain strictly pseudoconvex domains in ${{\mathbf{C}}^n}$ such that there exists a function in the Nevanlinna class which determines the divisor.
Families of holomorphic maps into Riemann surfaces
Theodore J.
Barth
175-187
Abstract: In analogy with the Hartogs theorem that separate analyticity of a function implies analyticity, it is shown that a separately normal family of holomorphic maps from a polydisk into a Riemann surface is a normal family. This contrasts with examples of discontinuous separately analytic maps from a bidisk into the Riemann sphere. The proof uses a theorem on pseudoconvexity of normality domains, which is proved via the following convergence criterion: a sequence $ \{ {f_j}\}$ of holomorphic maps from a complex manifold into a Riemann surface converges to a nonconstant holomorphic map if and only if the sequence $\{ f_j^{ - 1}\}$ of set-valued maps, defined on the Riemann surface, converges to a suitable set-valued map. Extending Osgood's theorem, it is also shown that a separately analytic map (resp. a separately normal family of holomorphic maps) from a polydisk into a hyperbolic complex space is analytic (resp. normal).
A generalized topological measure theory
R. B.
Kirk;
J. A.
Crenshaw
189-217
Abstract: The theory of measures in a topological space, as developed by V. S. Varadarajan for the algebra ${C^b}$ of bounded continuous functions on a completely regular topological space, is extended to the context of an arbitrary uniformly closed algebra $ A$ of bounded real-valued functions. Necessary and sufficient conditions are given for ${A^ \ast }$ to be represented in the natural way by a space of regular finitely-additive set functions. The concepts of additivity and tightness for these set functions are considered and some remarks about weak convergence are made.
Boundary behavior of the Carath\'eodory and Kobayashi metrics on strongly pseudoconvex domains in $C\sp{n}$ with smooth boundary
Ian
Graham
219-240
Abstract: The Carathéodory and Kobayashi distance functions on a bounded domain $G$ in $ {{\mathbf{C}}^n}$ have related infinitesimal forms. These are the Carathéodory and Kobayashi metrics. They are denoted by $ F(z,\xi )$ (length of the tangent vector $\xi$ at the point $z$). They are defined in terms of holomorphic mappings, from $G$ to the unit disk for the Carathéodory metric, and from the unit disk to $G$ for the Kobayashi metric. We consider the boundary behavior of these metrics on strongly pseudoconvex domains in $ {{\mathbf{C}}^n}$ with $ {C^2}$ boundary. $ \xi$ is fixed and $ z$ is allowed to approach a boundary point ${z_0}$. The quantity $F(z,\xi )d(z,\partial G)$ is shown to have a finite limit. In addition, if $\xi$ belongs to the complex tangent space to the boundary at ${z_0}$, then this first limit is zero, and $ {(F(z,\xi ))^2}d(z,\partial G)$ has a (nontangential) limit in which the Levi form appears. We prove an approximation theorem for bounded holomorphic functions which uses peak functions in a novel way. The proof was suggested by N. Kerzman. This theorem is used here in studying the boundary behavior of the Carathéodory metric.
Fiber preserving equivalence
Richard T.
Miller
241-268
Abstract: We give a theory of fibered regular neighborhoods based on a remarkable property of simplicial fibered projections. All the usual properties of regular neighborhoods are retained. Using Millett's fibered general position, together with the regular neighborhoods, we prove THEOREM. The simplicial space of codimension 4 PL embeddings of a complex into a PL manifold is locally contractible at each point of the space of topological embeddings.
Cone complexes and PL transversality
Clint
McCrory
269-291
Abstract: A definition of PL transversality is given, using the orderreversing duality on partially ordered sets. David Stone's theory of stratified polyhedra is thereby simplified; in particular, the symmetry of blocktransversality is proved. Also, polyhedra satisfying Poincaré duality are characterized geometrically.
Finitary imbeddings of certain generalized sample spaces
Marie A.
Gaudard;
Robert J.
Weaver
293-307
Abstract: A generalized sample space each of whose subspaces has as its logic an orthomodular poset is called an HD sample space. In this paper it is shown that any HD sample space may be imbedded in a natural way in a generalized sample space which is HD and at the same time admits a full set of dispersion free weight functions.
Approximate isometries on finite dimensional Banach spaces
Richard D.
Bourgin
309-328
Abstract: A map $ T:{{\mathbf{E}}_1} \to {{\mathbf{E}}_2}$ ( $ {{\mathbf{E}}_1},{{\mathbf{E}}_2}$ Banach spaces) is an $\epsilon$-isometry if $ \vert\;\vert\vert T(X) - T(Y)\vert\vert - \vert\vert X - Y\vert\vert\;\vert \leqslant \epsilon$ whenever $X,Y \in {{\mathbf{E}}_1}$. The problem of uniformly approximating such maps by isometries was first raised by Hyers and Ulam in 1945 and subsequently studied for special infinite dimensional Banach spaces. This question is here broached for the class of finite dimensional Banach spaces. The only positive homogeneous candidate isometry $ U$ approximating a given $ \epsilon$-isometry $ T$ is defined by the formal limit $U(X) = {\lim _{r \to \infty }}{r^{ - 1}}T(rX)$. It is shown that, whenever $T:{\mathbf{E}} \to {\mathbf{E}}$ is a surjective $\epsilon$-isometry and $ {\mathbf{E}}$ is a finite dimensional Banach space for which the set of extreme points of the unit ball is totally disconnected, then this limit exists. When ${\mathbf{E}} = \ell _1^k( = k$ - dimensional$\; {\ell _1})$ a uniform bound of uniform approximation is obtained for surjective $\epsilon $-isometries by isometries; this bound varies linearly in $\epsilon$ and with ${k^3}$.
The subclass algebra associated with a finite group and subgroup
John
Karlof
329-341
Abstract: Let $G$ be a finite group and let $H$ be a subgroup of $G$. If $g \in G$, then the set ${E_g} = \{ hg{h^{ - 1}}\vert h \in H\}$ is the subclass of $G$ containing $g$ and $ {\Sigma _{x \in {E_g}}}x$ is the subclass sum containing $g$. The algebra over the field of complex numbers generated by these subclass sums is called the subclass algebra (denoted by $S$) associated with $ G$ and $H$. The irreducible modules of $ S$ are demonstrated, and results about Schur algebras are used to develop formulas relating the irreducible characters of $S$ to the irreducible characters of $ G$ and $H$.
Quantum logic and the locally convex spaces
W. John
Wilbur
343-360
Abstract: An important theorem of Kakutani and Mackey characterizes an infinite dimensional real (complex) Hilbert space as an infinite dimensional real (complex) Banach space whose lattice of closed subspaces admits an orthocomplementation. This result, also valid for quaternionic spaces, has proved useful as a justification for the unique role of Hilbert space in quantum theory. With a like application in mind, we present in the present paper a number of characterizations of real and complex Hilbert space in the class of locally convex spaces. One of these is an extension of the Kakutani-Mackey result from the infinite dimensional Banach spaces to the class of all infinite dimensional complete Mackey spaces. The implications for the foundations of quantum theory are discussed.
$I$-rings
W. K.
Nicholson
361-373
Abstract: A ring $ R$, possibly with no identity, is called an ${I_0}$-ring if each one-sided ideal not contained in the Jacobson radical $J(R)$ contains a nonzero idempotent. If, in addition, idempotents can be lifted modulo $J(R),R$ is called an $I$-ring. A survey of when these properties are inherited by related rings is given. Maximal idempotents are examined and conditions when $ {I_0}$-rings have an identity are given. It is shown that, in an ${I_0}$-ring $R$, primitive idempotents are local and primitive idempotents in $R/J(R)$ can always be lifted. This yields some characterizations of ${I_0}$-rings $R$ such that $R/J(R)$ is primitive with nonzero socle. A ring $ R$ (possibly with no identity) is called semiperfect if $R/J(R)$ is semisimple artinian and idempotents can be lifted modulo $J(R)$. These rings are characterized in several new ways: among them as ${I_0}$-rings with no infinite orthogonal family of idempotents, and as ${I_0}$-rings $R$ with $R/J(R)$ semisimple artinian. Several other properties are derived. The connection between $ {I_0}$-rings and the notion of a regular module is explored. The rings $ R$ which have a regular module $M$ such that $J(R) = \operatorname{ann} (M)$ are studied. In particular they are ${I_0}$-rings. In addition, it is shown that, over an ${I_0}$-ring, the endomorphism ring of a regular module is an ${I_0}$-ring with zero radical.
On the inverse problem of Galois theory
J.
Kovacic
375-390
Abstract: Let $k$ be a field, $F$ a finite subfield and $G$ a connected solvable algebraic matric group defined over $F$. Conditions on $G$ and $k$ are given which ensure the existence of a Galois extension of $k$ with group isomorphic to the $F$-rational points of $G$.
Functions of vanishing mean oscillation
Donald
Sarason
391-405
Abstract: A function of bounded mean oscillation is said to have vanishing mean oscillation if, roughly speaking, its mean oscillation is locally small, in a uniform sense. In the present paper the class of functions of vanishing mean oscillation is characterized in several ways. This class is then applied to answer two questions in analysis, one involving stationary stochastic processes satisfying the strong mixing condition, the other involving the algebra ${H^\infty } + C$.
Erratum to ``Transversally parallelizable foliations of codimension two''
Lawrence
Conlon
406
Extending closed plane curves to immersions of the disk with $n$ handles
Keith D.
Bailey
1-24
Abstract: Let $f:S \to E$ be a normal curve in the plane. The extensions of $f$ to immersions of the disk with $n$ handles $({T_n})$ can be determined as follows. A word for $f$ is constructed using the definitions of Blank and Marx and a combinatorial structure, called a $ {T_n}$-assemblage, is defined for such words. There is an immersion extending $ f$ to ${T_n}$ iff the tangent winding number of $f$ is $1 - 2n$ and $f$ has a ${T_n}$-assemblage. For each $n$, a canonical curve $ {f_n}$ with a topologically unique extension to ${T_n}$ is described (${f_0}$ = Jordan curve). Any extendible curve with the minimum number $(2n + 2\;{\text{for}}\;n > 0)$ of self-intersections is equivalent to ${f_n}$.
Interpolation properties of generalized perfect splines and the solutions of certain extremal problems. I
Samuel
Karlin
25-66
Abstract: The existence of generalized perfect splines satisfying certain interpolation and/or moment conditions are established. In particular, the existence of ordinary perfect splines obeying boundary and interpolation conditions is demonstrated; precise criteria for the uniqueness of such interpolatory perfect splines are indicated. These are shown to solve a host of variational problems in certain Sobolev spaces.
One-sided congruences on inverse semigroups
John
Meakin
67-82
Abstract: By the kernel of a one-sided (left or right) congruence $\rho$ on an inverse semigroup $ S$, we mean the set of $ \rho$-classes which contain idempotents of $S$. We provide a set of independent axioms characterizing the kernel of a one-sided congruence on an inverse semigroup and show how to reconstruct the one-sided congruence from its kernel. Next we show how to characterize those partitions of the idempotents of an inverse semigroup $S$ which are induced by a one-sided congruence on $ S$ and provide a characterization of the maximum and minimum one-sided congruences on $S$ inducing a given such partition. The final two sections are devoted to a study of indempotent-separating one-sided congruences and a characterization of all inverse semigroups with only trivial full inverse subsemigroups. A Green-Lagrange-type theorem for finite inverse semigroups is discussed in the fourth section.
Iterated integrals, fundamental groups and covering spaces
Kuo Tsai
Chen
83-98
Abstract: Differential $ 1$-forms are integrated iteratedly along paths in a differentiable manifold $ X$. The purpose of this article is to consider those iterated integrals whose value along each path depends only on the homotopy class of the path. The totality of such integrals is shown to be dual, in an appropriate sense, to the ``maximal'' residually torsion free nilpotent quotient of the fundamental group $ {\pi _1}(X)$. Taken as functions on the universal covering space $ \tilde X$, these integrals separate points of $\tilde X$ if and only if $ {\pi _1}(X)$ is residually torsion free nilpotent.
Groups of free involutions of homotopy $S\sp{[n/2]}\times S\sp{[(n+1)/2]}$'s
H. W.
Schneider
99-136
Abstract: Let $M$ be an oriented $ n$-dimensional manifold which is homotopy equivalent to ${S^l} \times {S^{n - l}}$, where $l$ is the greatest integer in $ n/2$. Let $Q$ be the quotient manifold of $ M$ by a fixed point free involution. Associated to each such $Q$ are a unique integer $k\bmod {2^{\varphi (l)}}$, called the type of $ Q$, and a cohomology class $ \omega$ in ${H^1}(Q;{Z_2})$ which is the image of the generator of the first cohomology group of the classifying space for the double cover of $Q$ by $M$. Let ${I_n}(k)$ be the set of equivalence classes of such manifolds $Q$ of type $k$ for which $ {\omega ^{l + 1}} = 0$, where two such manifolds are equivalent if there is a diffeomorphism, orientation preserving if $k$ is even, between them. It is shown in this paper that if $n \geq 6$, then ${I_n}(k)$ can be given the structure of an abelian group. The groups ${I_8}(k)$ are partially calculated for $ k$ even.
On the fixed point set of a compact transformation group with some applications to compact monoids
Karl Heinrich
Hofmann;
Michael
Mislove
137-162
Abstract: Under various special additional hypotheses we prove that the fixed point set of the group of inner automorphisms of a compact connected monoid with zero is connected.
On bounded functions satisfying averaging conditions. I
Rotraut Goubau
Cahill
163-174
Abstract: Let $R(T)$ be the space of real valued ${L^\infty }$ functions defined on the unit circle $ C$ consisting of those functions $f$ for which $li{m_{h \to 0}}(1/h)\int_\theta ^{\theta + h} {f({e^{it}})dt = f({e^{i\theta }})}$ for every ${e^{i\theta }}$ in $C$. The extreme points of the unit ball of $R(T)$ are found and the extreme points of the unit ball of the space of all bounded harmonic functions in the unit disc which have non-tangential limit at each point of the unit circle are characterized. We show that if $g$ is a real valued function in ${L^\infty }(C)$ and if $K$ is a closed subset of $ \{ {e^{i\theta }}\vert li{m_{h \to 0}}(1/h)\int_\theta ^{\theta + h} {g({e^{it}})dt = g({e^{i\theta }})\} } $, then there is a function in $R(T)$ whose restriction to $K$ is $g$. If $E$ is a $ {G_\delta }$ subset of $ C$ of measure 0 and if $ F$ is a closed subset of $ C$ disjoint from $ E$, there is a function of norm 1 in $R(T)$ which is on $E$ and 1 on $F$. Finally, we show that if $E$ and $F$ are as in the preceding result, then there is a function of norm 1 in $ {H^\infty }$ (unit disc) the modulus of which has radial limit along every radius, which has radial limit of modulus 1 at each point of $F$ and radial limit 0 at each point of $E$.
Weakly almost periodic functions and almost convergent functions on a group
Ching
Chou
175-200
Abstract: Let $G$ be a locally compact group, $ UC(G)$ the space of bounded uniformly continuous complex functions on $G,{C_0}(G)$ the subspace of $UC(G)$ consisting of functions vanishing at infinity. Let $W(G)$ be the space of weakly almost periodic functions on $G$ and ${W_0}(G)$ the space of functions in $W(G)$ such that their absolute values have zero invariant mean. If $G$ is amenable let $F(G)$ be the space of almost convergent functions in $UC(G)$ and ${F_0}(G)$ the space of functions in $F(G)$ such that their absolute values are almost convergent to zero. The inclusive relations among the above-mentioned spaces are studied. It is shown that if $G$ is noncompact and satisfies certain conditions, e.g. $G$ is nilpotent, then each of the quotient Banach spaces $ UC(G)/W(G),{W_0}(G)/{C_0}(G),{F_0}(G)/{W_0}(G)$ contains a linear isometric copy of ${l^\infty }$. On the other hand, an example of a noncompact group $G$ is given which satisfies the condition that $ {C_0}(G) = {W_0}(G)$.
On $h$-local integral domains
Willy
Brandal
201-212
Abstract: Related to the question of determining the integral domains with the property that finitely generated modules are a direct sum of cyclic submodules is the question of determining when an integral domain is $h$-local, especially for Bezout domains. Presented are ten equivalent conditions for a Prüfer domain with two maximal ideals not to be $h$-local. If $R$ is an integral domain with quotient field $ Q$, if every maximal ideal of $R$ is not contained in the union of the rest of the maximal ideals of $R$, and if $Q/R$ is an injective $R$-module, then $R$ is $h$-local; and if in addition $R$ is a Bezout domain, then every finitely generated $R$-module is a direct sum of cyclic submodules. In particular if $R$ is a semilocal Prüfer domain with $ Q/R$ an injective $ R$-module, then every finitely generated $R$-module is a direct sum of cyclic submodules.
The Radon-Nikodym property in conjugate Banach spaces
Charles
Stegall
213-223
Abstract: We characterize conjugate Banach spaces ${X^\ast }$ having the Radon-Nikodym Property as those spaces such that any separable subspace of $ X$ has a separable conjugate. Several applications are given.
Toeplitz matrices generated by the Laurent series expansion of an arbitrary rational function
K. Michael
Day
224-245
Abstract: Let ${T_n}(f) = ({a_{i - j}})_{i,j = 0}^n$ be the finite Toeplitz matrices generated by the Laurent expansion of an arbitrary rational function. An identity is developed for $\det ({T_n}(f) - \lambda )$ which may be used to prove that the limit set of the eigenvalues of the $ {T_n}(f)$ is a point or consists of a finite number of analytic arcs.
Weak maps of combinatorial geometries
Dean
Lucas
247-279
Abstract: Weak maps of combinatorial geometries are studied, with particular emphasis on rank preserving weak bijections. Equivalent conditions for maps to be reversed under duality are given. It is shown that each simple image (on the same rank) of a binary geometry $G$ is of the form $ G/F \oplus F$ for some subgeometry $F$ of $G$. The behavior of invariants under mappings is studied. The Tutte polynomial, Whitney numbers of both kinds, and the Möbius function are shown to behave systematically under rank preserving weak maps. A weak map lattice is presented and, through it, the lattices of elementary images and preimages of a fixed geometry are studied.
Partitions of unity and a closed embedding theorem for $(C\sp{p},b\sp*)$-manifolds
Richard E.
Heisey
281-294
Abstract: Many manifolds of fiber bundle sections possess a natural atlas $\{ ({U_\alpha },{\phi _\alpha })\}$ such that the transition maps ${\phi _\beta }\phi _\alpha ^{ - 1}$, in addition to being smooth, are continuous with respect to the bounded weak topology of the model. In this paper we formalize the idea of such manifolds by defining $({C^p},{b^\ast })$-manifolds, $({C^p},{b^\ast })$-morphisms, etc. We then show that these manifolds admit $({C^p},{b^\ast })$-partitions of unity subordinate to certain open covers and that they can be embedded as closed $ ({C^p},{b^\ast })$-submanifolds of their model. A corollary of our work is that for any Banach space $B$, the conjugate space ${B^\ast }$ admits smooth partitions of unity subordinate to covers by sets open in the bounded weak-$ \ast$ topology.
Manifolds modelled on $R\sp{\infty }$ or bounded weak-* topologies
Richard E.
Heisey
295-312
Abstract: Let $ {R^\infty } = \mathop {\lim {R^n}}\limits_ \to$, and let ${B^ \ast }({b^ \ast })$ denote the conjugate, ${B^ \ast }$, of a separable, infinite-dimensional Banach space with its bounded weak-$\ast$ topology. We investigate properties of paracompact, topological manifolds $ M,N$ modelled on $ F$, where $F$ is either $ {R^\infty }$ or ${B^ \ast }({b^ \ast })$. Included among our results are that locally trivial bundles and microbundles over $M$ with fiber $F$ are trivial; there is an open embedding $M \to M \times F$; and if $M$ and $N$ have the same homotopy type, then $M \times F$ and $ N \times F$ are homeomorphic. Also, if $U$ is an open subset of ${B^ \ast }({b^ \ast })$, then $U \times {B^ \ast }({b^ \ast })$ is homeomorphic to $U$. Thus, two open subsets of ${B^ \ast }({b^ \ast })$ are homeomorphic if and only if they have the same homotopy type. Our theorems about $ {B^ \ast }({b^ \ast })$-manifolds, $ {B^ \ast }({b^ \ast })$ as above, immediately yield analogous theorems about $ B(b)$-manifolds, where $ B(b)$ is a separable, reflexive, infinite-dimensional Banach space with its bounded weak topology.
Spaces of vector measures
A.
Katsaras
313-328
Abstract: Let ${C_{rc}} = {C_{rc}}(X,E)$ denote the space of all continuous functions $f$, from a completely regular Hausdorff space $ X$ into a locally convex space $E$, for which $f(X)$ is relatively compact. As it is shown in [8], the uniform dual ${C'_{rc}}$ of ${C_{rc}}$ can be identified with a space $ M(B,E')$ of $E'$-valued measures defined on the algebra of subsets of $X$ generated by the zero sets. In this paper the subspaces of all $\sigma$-additive and all $\tau $-additive members of $ M(B,E')$ are studied. Two locally convex topologies $\beta$ and $ {\beta _1}$ are considered on ${C_{rc}}$. They yield as dual spaces the spaces of all $\tau$-additive and all $\sigma$-additive members of $M(B,E')$ respectively. In case $ E$ is a locally convex lattice, the $\sigma$-additive and $\tau$-additive members of $M(B,E')$ correspond to the $\sigma $-additive and $ \tau$-additive members of $ {C_{rc}}$ respectively.
Absolutely continuous functions on idempotent semigroups in the locally convex setting
A.
Katsaras
329-337
Abstract: Let $E$ be a locally convex space and let $ T$ be a semigroup of semicharacters on an idempotent semigroup. It is shown that there exists an isomorphism between the space of $ E$-valued functions on $ T$ and the space of all $ E$-valued finitely additive measures on a certain algebra of sets. The space of all $E$-valued functions on $T$ which are absolutely continuous with respect to a positive definite function $F$ is identified with the space of all $E$-valued measures which are absolutely continuous with respect to the measure ${m_F}$ corresponding to $ F$. Finally a representation is given for the operators on the set of all $ E$-valued finitely additive measures on an algebra of sets which are absolutely continuous with respect to a positive measure.
On the action of $\Theta \sp{n}$. I
H. E.
Winkelnkemper
339-346
Abstract: We prove two theorems about the inertia groups of closed, smooth, simply-connected $n$-manifolds. Theorem A shows that, in certain dimensions, the special inertia group, unlike the full inertia group, can never be equal to ${\Theta ^n}$; Theorem B shows, in $ \operatorname{dimensions} \equiv 3\bmod 4$, how to construct explicit closed $ n$-manifolds $ {M^n}$ such that $\Theta (\partial \pi )$ is contained in the inertia group of ${M^n}$.
Wild spheres in $E\sp{n}$ that are locally flat modulo tame Cantor sets
Robert J.
Daverman
347-359
Abstract: Kirby has given an elementary geometric proof showing that if an $ (n - 1)$-sphere $ \Sigma$ in Euclidean $ n$-space ${E^n}$ is locally flat modulo a Cantor set that is tame relative to both $\Sigma$ and ${E^n}$, then $\Sigma$ is locally flat. In this paper we illustrate the sharpness of the result by describing a wild $ (n - 1)$-sphere $ \Sigma$ in ${E^n}$ such that $\Sigma$ is locally flat modulo a Cantor set $C$ and $C$ is tame relative to ${E^n}$. These examples then are used to contrast certain properties of embedded spheres in higher dimensions with related properties of spheres in ${E^3}$. Rather obviously, as Kirby points out in [11], his result cannot be weak-ened by dismissing the restriction that the Cantor set be tame relative to ${E^n}$. It is well known that a sphere in $ {E^n}$ containing a wild (relative to ${E^n}$) Cantor set must be wild. Consequently the only variation on his work that merits consideration is the one mentioned above. The phenomenon we intend to describe also occurs in $3$-space. Alexander's horned sphere [1] is wild but is locally flat modulo a tame Cantor set. In fact, at one spot methods used here parallel those used to construct that example. However, other properties of $ 3$-space are strikingly dissimilar to what can be derived from the higher dimensional examples constructed here, for, as discussed in §2, natural analogues to some important results concerning locally flat embeddings in ${E^3}$ are false.
Simultaneous approximation of additive forms
Ming Chit
Liu
361-373
Abstract: Let $X = ({x_1}, \cdots ,{x_s})$ be a vector of $ s$ real components and ${f_i}(X) = \sum\nolimits_{j = 1}^s {{\theta _{ij}}x_j^k} (k = 2,3, \cdots ;i = 1, \cdots ,R) R$ additive forms, where ${\theta _{ij}}$ are arbitrary real numbers. The author obtains some results on the simultaneous approximation of $ \vert\vert{f_i}(X)\vert\vert$, where $\vert\vert t\vert\vert$ means the distance from $ t$ to the nearest integer.
Amalgamated products of semigroups: the embedding problem
Gérard
Lallement
375-394
Abstract: A necessary and sufficient condition for a semigroup amalgam to be embeddable is given. It is in the form of a countable set of equational implications with existential quantifiers. Furthermore it is shown that no finite set of equational implications can serve as a necessary and sufficient condition. Howie's sufficient condition (see [5]) is derived as a consequence of our main theorem.
Holomorphic functions with growth conditions
Bent E.
Petersen
395-406
Abstract: Let $P$ be a $p \times q$ matrix of polynomials in $n$ complex variables. If $\Omega$ is a domain of holomorphy in ${{\mathbf{C}}^n}$ and $u$ is a $q$-tuple of holomorphic functions we show that the equation $Pv = Pu$ has a solution $v$ which is a holomorphic $q$-tuple in $\Omega$ and which satisfies an ${L^2}$ estimate in terms of $ Pu$. Similar results have been obtained by Y.-T. Siu and R. Narasimhan for bounded domains and by L. Höormander for the case $ \Omega = {{\mathbf{C}}^n}$.
Applications of a theorem of L\'evy to Boolean terms and algebras
Jonathan
Stavi
1-36
Abstract: The paper begins with a short proof of the Gaifman-Hales theorem and the solution of a problem of Gaifman about the depth and length of Boolean terms. The main results are refinements of the following theorem: Let $\kappa$ be regular, ${\aleph _1} \leq \kappa \leq \infty$. A $ < \kappa$-complete Boolean algebra on $ {\aleph _0}$ generators, which are restricted by just one countably long equation, is either atomic with $ \leq {\aleph _0}$ atoms or isomorphic to the free $< \kappa$-complete Boolean algebra on ${\aleph _0}$ generators. The main tools are a Skolem-Löwenheim type theorem of Azriel Lévy and a coding of Borel sets and Borel-measurable functions by Boolean terms.
Location of the zeros of a polynomial relative to certain disks
R. C.
Riddell
37-45
Abstract: The zeros of the complex polynomial $P(z) = {z^n} + \Sigma {\alpha _i}{z^{n - 1}}$ are studied under the assumption that some $\vert{\alpha _k}\vert$ is large in comparison with the other $\vert{\alpha _i}\vert$. It is shown under certain conditions that $P(z)$ has $n - k$ zeros in $ \vert z\vert \leq {m_ - }$ and $k$ zeros in $ \vert z\vert \geq {m_ + }$, where ${m_ - } < {m_ + } \leq \vert{\alpha _k}{\vert^{1/k}}$; and under suitably strengthened conditions, one of the $k$ zeros of larger modulus is shown to lie in each of the $k$ disks $\vert z - {( - {\alpha _k})^{1/k}}\vert \leq R$, where ${m_ - } + R < \vert{\alpha _k}{\vert^{1/k}}$.
Hall-Higman type theorems. II
T. R.
Berger
47-69
Abstract: This paper continues the investigations of this series. Suppose $ {\mathbf{K}} =$ GF${\text{(}}r{\text{)}}$ is a field for a prime $ r;G$-is a nilpotent; $ V$ is a nonsingular symplectic space with form $g$; and $V$ is a faithful irreducible ${\mathbf{K}}[G]$-module where $G$ fixes the form $g$. This paper describes completely the structure of $G$ and its representation upon $ V$ when $G$ is symplectic primitive. This latter condition is described in §4 and is a primitivity condition.
Stable thickenings in the homotopy category
R. L.
Chazin
71-77
Abstract: This paper extends the result that the set of stable thickenings of a simply-connected complex $K$ are in 1-1 correspondence with $[K,BQ](Q = {\rm O},$PL$,{\text{or TOP)}}$. which holds in the smooth, PL, and topological categories, to the homotopy category.
A construction of Lie algebras by triple systems
W.
Hein
79-95
Abstract: A construction of Lie algebras by means of special unital representations of Jordan algebras on a certain kind of triple systems is given which generalizes the construction due to Freudenthal, Faulkner and Koecher.
Fourier coefficients of Eisenstein series of one complex variable for the special linear group
A.
Terras
97-114
Abstract: The Eisenstein series in question are generalizations of Epstein's zeta function, whose Fourier expansions generalize the formula of Selberg and Chowla (for the binary quadratic form case of Epstein's zeta function). The expansions are also analogous to Siegel's calculation of the Fourier coefficients of Eisenstein series for the symplectic group. The only ingredients not appearing in Siegel's formula are the Bessel functions of matrix argument studied by Herz. These functions generalize the modified Bessel function of the second kind appearing in the Selberg-Chowla formula.
Trees of homotopy types of 2-dimensional ${\rm CW}$ complexes. II
Micheal N.
Dyer;
Allan J.
Sieradski
115-125
Abstract: A $\pi$-complex is a finite, connected $ 2$-dimensional CW complex with fundamental group $\pi$. The tree HT$(\pi )$ of homotopy types of $\pi $-complexes has width $ \leq N$ if there is a root $ Y$ of the tree such that, for any $\pi$-complex $X,X \vee ( \vee _{i = 1}^NS_i^2)$ lies on the stalk generated by $Y$. Let $\pi$ be a finite abelian group with torsion coefficients $ {\tau _1}, \cdots ,{\tau _n}$. The main theorem of this paper asserts that width HT$ (\pi ) \leq n(n - 1)/2$. This generalizes the results of [4].
$S$-operations in representation theory
Evelyn Hutterer
Boorman
127-149
Abstract: For $G$ a group and ${\text{A} ^G}$ the category of $G$-objects in a category A$$, a collection of functors, called ``$S$-operations,'' is introduced under mild restrictions on A$$. With certain assumptions on A$ $ and with $G$ the symmetric group $ {S_k}$, one obtains a unigeneration theorem for the Grothendieck ring formed from the isomorphism classes of objects in ${\text{A} ^{{S_k}}}$. For A = finite-dimensional vector spaces over $C$, the result says that the representation ring $ R({S_k})$ is generated, as a $\lambda$-ring, by the canonical $ k$-dimensional permutation representation. When A = finite sets, the $ S$-operations are called ``$ \beta$-operations,'' and the result says that the Burnside ring $B({S_k})$ is generated by the canonical $ {S_k}$-set if $ \beta$-operations are allowed along with addition and multiplication.
Classification of $3$-manifolds with certain spines
Richard S.
Stevens
151-166
Abstract: Given the group presentation $ \varphi = \left\langle {a,b\backslash {a^m}{b^n},{a^p}{b^q}} \right\rangle$ with $ m,n,p,q \ne 0$, we construct the corresponding $2$-complex $ {K_\varphi }$. We prove the following theorems. THEOREM 7. ${K_\varphi }$ is a spine of a closed orientable $ 3$-manifold if and only if (i) $ \vert m\vert = \vert p\vert = 1$ or $ \vert n\vert = \vert q\vert = 1$, or (ii) $(m,p) = (n,q) = 1$. THEOREM 10. If $ M$ is a closed orientable $ 3$-manifold having ${K_\varphi }$ as a spine and $\lambda = \vert mq - np\vert$ then $ M$ is a lens space ${L_{\lambda ,k}}$ where $(\lambda ,k) = 1$ except when $\lambda = 0$ in which case $M = {S^2} \times {S^1}$.
$p$-factorable operators
C. V.
Hutton
167-180
Abstract: Several classes of operators on Banach spaces, defined by certain summability conditions on the $k$th approximation numbers, are introduced and studied. Characterizations of these operators in terms of tensor-product representations are obtained. The relationship between these operators and other classes of operators introduced by various authors is studied in some detail.
Some theorems on $({\rm CA})$ analytic groups
David
Zerling
181-192
Abstract: An analytic group $ G$ is called $ (CA)$ if the group of inner automorphisms of $G$ is closed in the Lie group of all (bicontinuous) automorphisms of $G$. We show that each non-$(CA)$ analytic group $G$ can be written as a semidirect product of a $(CA)$ analytic group and a vector group. This decomposition yields a natural dense immersion of $ G$ into a $(CA)$ analytic group $ H$, such that each automorphism of $G$ can be extended to an automorphism of $ H$. This immersion and extension property enables us to derive a sufficient condition for the normal part of a semidirect product decomposition of a $(CA)$ analytic group to be $(CA)$.
Smooth $Z\sb{p}$-actions on spheres which leave knots pointwise fixed
D. W.
Sumners
193-203
Abstract: The paper produces, via handlebody construction, a family of counterexamples to the generalized Smith conjecture; that is, for each pair of integers $(n,p)$ with $n \geq 2$ and $p \geq 2$ there are infinitely many knots $({S^{n + 2}},k{S^n})$ which admit smooth semifree ${Z_p}$-actions (fixed on the knotted submanifold $ k{S^n}$ and free on the complement $ ({S^{n + 2}} - k{S^n}))$. This produces previously unknown $ {Z_p}$-actions on $({S^4},k{S^2})$ for $p$ even, the one case not covered by the work of C. H. Giffen. The construction is such that all of the knots produced are equivariantly null-cobordant. Another result is that if a knot admits ${Z_p}$ -actions for all $ p$, then the infinite cyclic cover of the knot complement is acyclic, and thus leads to an unknotting theorem for $ {Z_p}$-actions.
Asymptotic enumeration of partial orders on a finite set
D. J.
Kleitman;
B. L.
Rothschild
205-220
Abstract: By considering special cases, the number ${P_n}$ of partially ordered sets on a set of $ n$ elements is shown to be $ (1 + O(1/n)){Q_n}$, where $ {Q_n}$ is the number of partially ordered sets in one of the special classes. The number ${Q_n}$ can be estimated, and we ultimately obtain $\displaystyle {P_n} = \left( {1 + O\left( {\frac{1}{n}} \right)} \right)\left( ... ...{{2^i} - 1} \right)}^j}{{\left( {{2^j} - 1} \right)}^{n - i - j}}} } } \right).$
Order summability of multiple Fourier series
G. E.
Peterson;
G. V.
Welland
221-246
Abstract: Jurkat and Peyerimhoff have characterized monotone Fouriereffective summability methods as those which are stronger than logarithmic order summability. Here the analogous result for double Fourier series is obtained assuming unrestricted rectangular convergence. It is also shown that there is a class of order summability methods, which are weaker than any Cesàro method, for which the double Fourier series of any $f \in L$ is restrictedly summable almost everywhere. Finally, it is shown that square logarithmic order summability has the localization property for exponentially integrable functions.
Generalized gradients and applications
Frank H.
Clarke
247-262
Abstract: A theory of generalized gradients for a general class of functions is developed, as well as a corresponding theory of normals to arbitrary closed sets. It is shown how these concepts subsume the usual gradients and normals of smooth functions and manifolds, and the subdifferentials and normals of convex analysis. A theorem is proved concerning the differentiability properties of a function of the form $ \max \{ g(x,u):u \in U\}$. This result unifies and extends some theorems of Danskin and others. The results are then applied to obtain a characterization of flow-invariant sets which yields theorems of Bony and Brezis as corollaries.
Hausdorff $m$ regular and rectifiable sets in $n$-space
Pertti
Mattila
263-274
Abstract: The purpose of this paper is to prove the following theorem: If $ E$ is a subset of Euclidean $n$-space and if the $m$-dimensional Hausdorff density of $E$ exists and equals one ${H^m}$ almost everywhere in $E$, then $E$ is countably $({H^m},m)$ rectifiable. Here ${H^m}$ is the $m$-dimensional Hausdorff measure. The proof is a generalization of the proof given by J. M. Marstrand in the special case $ n = 3,m = 2$.
Three local conditions on a graded ring
Jacob
Matijevic
275-284
Abstract: Let $R = {\Sigma _{i \in Z}}{R_i}$ be a commutative graded Noetherian ring with unit and let $A = {\Sigma _{i \in Z}}{A_i}$ be a finitely generated graded $R$ module. We show that if we assume that $ {A_M}$ is a Cohen Macaulay $ {R_M}$ module for each maximal graded ideal $M$ of $R$, then ${A_P}$ is a Cohen Macaulay ${R_P}$ module for each prime ideal $ P$ of $R$. With $A = R$ we show that the same is true with Cohen Macaulay replaced by regular and Gorenstein, respectively.
Splitting isomorphisms of mapping tori
Terry C.
Lawson
285-294
Abstract: Necessary and sufficient conditions involving invertible cobordisms are given for two mapping tori to be isomorphic. These are used to give conditions under which a given isomorphism ${M_f} \to {N_g}$ is pseudoisotopic to an isomorphism which sends $M$ to $N$. An exact sequence for the group of pseudoisotopy classes of automorphisms of $M \times {S^1}$ is derived. The principal tools are an imbedding technique due to C. T. C. Wall as well as arguments involving invertible cobordisms. Applications and examples are given, particularly for manifolds of higher dimension where the $s$-cobordism theorem is applied.
The geometric dimension of some vector bundles over projective spaces
Donald M.
Davis;
Mark E.
Mahowald
295-315
Abstract: We prove that in many cases the geometric dimension of the $ p$-fold Whitney sum $ p{H_k}$ of the Hopf bundle $ {H_k}$ over quaternionic projective space $Q{P^k}$ is the smallest $n$ such that for all $i \leq k$ the reduction of the $i$th symplectic Pontryagin class of $p{H_k}$ to coefficients ${\pi _{4i - 1}}(({\text{R}}{P^\infty }/{\text{R}}{P^{n - 1}})\Lambda bo)$ is zero, where bo is the spectrum for connective KO-theory localized at 2. We immediately obtain new immersions of real projective space $ {\text{R}}{P^{4k + 3}}$ in Euclidean space if the number of 1's in the binary expansion of $k$ is between 5 and 8.
Continua in which all connected subsets are arcwise connected
E. D.
Tymchatyn
317-331
Abstract: Let $X$ be a metric continuum such that every connected subset of $X$ is arcwise connected. Some facts concerning the distribution of local cutpoints of $X$ are obtained. These results are used to prove that $X$ is a regular curve.
Further generalizations of the Nehari inequalities
Duane W.
DeTemple
333-340
Abstract: Inequalities of the Nehari type are obtained for bounded univalent functions on the unit disc, including a form which depends upon the parameters $a$ and $d$, where $d = f(a)$.
Stiefel-Whitney homology classes and bordism
Ethan
Akin
341-359
Abstract: We develop the theory of $\bmod 2$ Stiefel-Whitney homology classes for Euler polyhedra. We then describe a simple method of obtaining p.1. bordism theories. Finally, we define the ungraded bordism theory of Euler spaces and show that it is isomorphic to ordinary total homology.
Some polynomials defined by generating relations
H. M.
Srivastava;
R. G.
Buschman
360-370
Abstract: In an attempt to present a unified treatment of the various polynomial systems introduced from time to time, new generating functions are given for the sets of polynomials $ \{ S_{n,q}^{(\alpha ,\beta )}(\lambda ;x)\}$ and $\{ T_{n,q}^{(\alpha ,\beta )}(\lambda ;x)\}$, defined respectively by (6) and (29) below, and for their natural generalizations in several complex variables. This paper also indicates relevant connections of the results derived here with different classes of generating relations which have appeared recently in the literature.
Exotic singular structures on spheres
Norman
Levitt
371-388
Abstract: It is shown how the category of PL-manifolds may be obtained from the smooth category by an iterative procedure, viz., first form singular smooth manifolds where smooth seven-spheres are allowed as links. Then, in the new category one has obtained, kill all eight-spheres in similar fashion. Repeating this process ad infinitum (but requiring only finitely many stages in each dimension), one obtains the category of PL-manifolds. By taking care that the set of ``singular'' points is always given enough structure, it is seen that this iterative process corresponds to a skeletal filtration of $ BPL \bmod BO$. Also, a geometric interpretation of the Hurewicz map $ {\pi _ \ast }(BPL,BO) \to {H_ \ast }(BPL,BO)$ is inferred.
$\Phi $-like holomorphic functions in ${\bf C}\sp{n}$ and Banach spaces
Kenneth R.
Gurganus
389-406
Abstract: In a recent paper, L. Brickman introduced the concept of $\Phi$-like holomorphic functions as a complete generalization of starlike and spirallike functions of a single complex variable. In the present paper, the author extends this work to locally biholomorphic mappings of several complex variables and then to locally biholomorphic mappings defined in an arbitrary Banach space. Complete characterizations of univalency and starlikeness of locally biholomorphic maps in general Banach spaces are obtained.
On smoothing compact measure spaces by multiplication
Dorothy
Maharam
1-39
Abstract: Let $\mu$ be a regular Borel probability measure on a compact Hausdorff space $S$, and let $m$ be Lebesgue measure on the unit interval $ I$. It is proved that the measure-theoretic product $(S,\mu ) \times ({I^w},{m^w})$, where $ w$ is a large enough cardinal and ${m^w}$ denotes product Lebesgue measure, is ``pseudo-isometric'' to $ ({I^w},{m^w})$. Here a pseudo-isometry $\phi$ is a point-isometry except that, instead of $\phi (A)$ being measurable for every measurable $A$, it is required only that $A$ differ by a null set from a set with measurable image. If instead $\mu$ is a Baire probability measure and $ S$ is a Baire subset of $ {I^w}$, then $(S,\mu ) \times ({I^w},{m^w})$ is point-isometric to $ ({I^w},{m^w})$. Finally it is shown that (roughly speaking) continuous maps can be ``smoothed'' into projection maps (to within pseudo-isometries) by multiplication by suitable projection maps.
On the Alder polynomials and a new generalization of the Rogers-Ramanujan identities
George E.
Andrews
40-64
Abstract: In 1954, H. L. Alder obtained certain analytic generalizations of the Rogers-Ramanujan identities that involved a sequence of polynomials subsequently named the Alder polynomials. The object here is to obtain a new representation of the Alder polynomials from which a new generalization of the Rogers-Ramanujan identities is easily deduced.
Splitting an $\alpha $-recursively enumerable set
Richard A.
Shore
65-77
Abstract: We extend the priority method in $\alpha$-recursion theory to certain arguments with no a priori bound on the required preservations by proving the splitting theorem for all admissible $\alpha$. THEOREM: Let $C$ be a regular $\alpha$-r.e. set and $D$ be a nonrecursive $\alpha$-r.e. set. Then there are regular $ \alpha$-r.e. sets $ A$ and $B$ such that $ A \cup B = C,A \cap B = \phi ,A,B{ \leq _\alpha }C$ and such that $D$ is not $\alpha$-recursive in $A$ or $B$. The result is also strengthened to apply to $ { \leq _{c\alpha }}$, and various corollaries about the structure of the $ \alpha$ and $ c\alpha$ recursively enumerable degrees are proved.
On $J$-convexity and some ergodic super-properties of Banach spaces
Antoine
Brunel;
Louis
Sucheston
79-90
Abstract: Given two Banach spaces $F\vert\vert$ and $X\vert\vert\,\vert\vert$, write $F{\text{ fr }}X{\text{ iff}}$ for each finite-dimensional subspace $F'$ of $F$ and each number $ \varepsilon > 0$, there is an isomorphism $V$ of $F'$ into $X$ such that $\vert\vert x\vert - \vert\vert Vx\vert\vert\vert \leq \varepsilon$ for each $x$ in the unit ball of $F'$. Given a property ${\mathbf{P}}$ of Banach spaces, $X$ is called super- $ {\mathbf{P}}{\text{ iff }}F{\text{ fr }}X$ implies $F$ is $ {\mathbf{P}}$. Ergodicity and stability were defined in our articles On $ B$-convex Banach spaces, Math. Systems Theory 7 (1974), 294-299, and C. R. Acad. Sci. Paris Ser. A 275 (1972), 993, where it is shown that super-ergodicity and super-stability are equivalent to super-reflexivity introduced by R. C. James [Canad. J. Math. 24 (1972), 896-904]. $Q$-ergodicity is defined, and it is proved that super-$Q$-ergodicity is another property equivalent with super-reflexivity. A new proof is given of the theorem that $ J$-spaces are reflexive [Schaffer-Sundaresan, Math. Ann. 184 (1970), 163-168]. It is shown that if a Banach space $X$ is $B$-convex, then each bounded sequence in $ X$ contains a subsequence $ ({y_n})$ such that the Cesàro averages of $ {( - 1)^i}{y_i}$ converge to zero.
Rings of continuous functions with values in a topological field
George
Bachman;
Edward
Beckenstein;
Lawrence
Narici;
Seth
Warner
91-112
Abstract: Let $F$ be a complete topological field. We undertake a study of the ring $C(X,F)$ of all continuous $ F$-valued functions on a topological space $X$ whose topology is determined by $C(X,F)$, in that it is the weakest making each function in $C(X,F)$ continuous, and of the ring ${C^\ast }(X,F)$ of all continuous $F$-valued functions with relatively compact range, where the topology of $X$ is similarly determined by ${C^\ast }(X,F)$. The theory of uniform structures permits a rapid construction of the appropriate generalizations of the Hewitt realcompactification of $X$ in the former case and of the Stone-Čech compactification of $X$ in the latter. Most attention is given to the case where $F$ and $X$ are ultraregular; in this case we determine conditions on $F$ that permit a development parallel to the classical theory where $F$ is the real number field. One example of such conditions is that the cardinality of $ F$ be nonmeasurable and that the topology of $F$ be given by an ultrametric or a valuation. Measure-theoretic interpretations are given, and a nonarchimedean analogue of Nachbin and Shirota's theorem concerning the bornologicity of $ C(X)$ is obtained.
Conservative dynamical systems involving strong forces
William B.
Gordon
113-135
Abstract: We consider conservative dynamical systems associated with potentials $ V$ which have singularities at a set $ S:V(x) \to - \infty$ as $x \to S$. It is shown that various ``action'' integrals satisfy Condition $ C$ of Palais and Smale provided that the potential satisfy a certain strong force (SF) condition. Hence, e.g., we establish the existence in SF systems of periodic trajectories which wind around $S$ and have arbitrary given topological (homotopy) type and which have arbitrary given period, and also periodic trajectories which make arbitrarily tight loops around $S$. Similar results are also obtained concerning the existence of trajectories which wind around $ S$ and join two given points. The SF condition is shown to be closely related to the completeness (in the riemannian sense) of certain Jacobi metrics associated with the potential $ V$, and this fact permits the use of the standard results of riemannian geometry in the analysis of SF systems. The SF condition excludes the gravitational case, and the action integrals do not satisfy the Palais-Smale condition in the gravitational case. The Jacobi metrics associated with gravitational potentials are not complete. For SF systems there exist trajectories which join two given points and make arbitrarily tight loops around $S$, and this is not the case in the gravitational two body problem. On the other hand, for SF systems any smooth family of $\lambda$-periodic trajectories ($\lambda$ fixed) is bounded away from $ S$, and this also is not the case for gravitational systems. Thus the definition of the SF condition is ``well motivated", and leads to the disclosure of certain differences between the behavior of SF systems and gravitational (and other weak force) systems.
The Brauer group of graded Azumaya algebras. II. Graded Galois extensions
Lindsay N.
Childs
137-160
Abstract: This paper continues the study of the Brauer group ${B_\phi }(R,G)$ of $G$-graded Azumaya $R$-algebras begun in [5]. A group $ {\operatorname{Galz} _\phi }(R,G)$ of graded Galois extensions is constructed which always contains, and often equals, the cokernel of ${B_\phi }(R,G)$ modulo the usual Brauer group of $R$. Sufficient conditions for equality are found. The structure of ${\operatorname{Galz} _\phi }(R,G)$ is studied, and ${\operatorname{Galz} _\phi }(R,{(Z/{p^e}Z)^r})$ is computed. These results are applied to give computations of a Brauer group of dimodule algebras constructed by F. W. Long.
Symmetries of spherical harmonics
Roberto De Maria Nunes
Mendes
161-178
Abstract: Let $G$ be a group of linear transformations of $ {R^n}$ and ${H_k}(G)$ the vector space of spherical harmonics invariant under $G$. The Pálya function is the formal power series ${\Sigma _{k \geq 0}}{t^k}\dim {H_k}(G)$. In this paper, after classifying all closed subgroups of $O(4)$, we compute the Pólya functions for these groups. These functions have recently proved to be of interest in quantum mechanics and elementary particle physics.
The radicals of a semigroup
Rebecca
Slover
179-195
Abstract: This paper investigates various radicals and radical congruences of a semigroup. A strongly prime ideal is defined. It is shown that the nil radical of a semigroup is the intersection of all strongly prime ideals of the semigroup. Furthermore, a semigroup with zero element is nil if and only if it has no strongly prime ideals. We investigate the question of when the left and right radical congruence relations of various radicals are equal. Some theorems analogous to theorems concerning the radicals of rings are also proved.
Topological dynamics on $C\sp *$-algebras
Diane
Laison;
Gary
Laison
197-205
Abstract: Dynamical properties of a group of homeomorphisms of a compact Hausdorff space $X$ can be interpreted in terms of the commutative $ {C^\ast }$-algebra $ C(X)$. We investigate a noncommutative topological dynamics extending dynamical concepts to the context of a group of automorphisms on a general ${C^\ast }$-algebra with unit. Such concepts as minimality, almost periodicity, and point-wise almost periodicity are extended to this situation. Theorems are obtained extending commutative dynamical results and relating the noncommutative dynamics to the transformation groups induced on the state space and the weak* closure of the pure states. We show, for example, that the group acts almost periodically on the $ {C^\ast }$-algebra if and only if each of these induced transformation groups is almost periodic.
A canonical form for symmetric and skew-symmetric extended symplectic modular matrices with applications to Riemann surface theory
Robert
Zarrow
207-227
Abstract: The (extended) symplectic modular group $({ \wedge _n}){\Gamma _n}$ is the set of all $2n \times 2n$ integer matrices $M$ such that $\displaystyle (M{J^t}M = \pm J),M{J^t}M = J,J = \left[ {\begin{array}{*{20}{c}} 0 & I { - I} & 0 \end{array} } \right],$ $I$ being the $n \times n$ identity matrix. Let ${S_n} = \{ M \in { \wedge _n} - {\Gamma _n}\vert M = - {}^tM\}$ and ${T_n} = \{ M \in { \wedge _n} - {\Gamma _n}\vert M = {}^tM\}$. We say $M \sim N$ if there exists $K \in {\Gamma _n}$ such that $M = KN{}^tK$. This defines an equivalence relation on each of these sets separately and we obtain a canonical form for this equivalence. We use this canonical form to study two types of Riemann surfaces which are conformally equivalent to their conjugates and obtain characterizations of their period matrices. We also obtain characterizations of the symplectic matrices which the conformal equivalence induces on the first homology group. One type of surface dealt with is the symmetric Riemann surfaces, i.e. those surfaces which have a conjugate holomorphic self-map of order 2. The other type of surface studied we we call pseudo-symmetric surfaces. These are the hyperelliptic surfaces with the property that the sheet interchange is the square of a conjugate holomorphic automorphism.
On the $2$-realizability of $2$-types
Micheal N.
Dyer
229-243
Abstract: A $2$-type is a triple $(\pi ,{\pi _2},k)$, where $\pi$ is a group, ${\pi _2}$ a $\pi$-module and $k \in {H^3}(\pi ,{\pi _2})$. The following question is studied: When is a $2$-type $ (\pi ,{\pi _2},k)$ realizable by $2$-dimensional CW-complex $X$ such that the $2$-type $({\pi _1}X,{\pi _2}X,k(X))$ is equivalent to $(\pi ,{\pi _2},k)$? A long list of necessary conditions is given (2.2). One necessary and sufficient condition (3.1) is proved, provided $ \pi$ has the property that stably free, finitely generated $\pi$-modules are free. ``Stable'' $ 2$-realizability is characterized (4.1) in terms of the Wall invariant of [15]. Finally, techniques of [5] are used to extend C. T. C. Wall's Theorem F of [15] to a space $X$ which is dominated by a finite CW-complex of dimension 2, provided ${\pi _1}X$ is finite cyclic. Under these conditions $X$ has the homotopy type of a finite $2$-complex if and only if the Wall invariant vanishes.
Integration of ordinary linear differential equations by Laplace-Stieltjes transforms
James
D’Archangelo;
Philip
Hartman
245-266
Abstract: Let $R$ be a constant $N \times N$ matrix and $g(t)$ an $N \times N$ matrix of functions representable as absolutely convergent Laplace-Stieltjes transforms for $t > 0$. The paper gives sufficient conditions for certain solutions of the system
Equations with constant coefficients invariant under a group of linear transformations
André
Cerezo
267-298
Abstract: If $P$ is a linear differential operator on $ {{\mathbf{R}}^n}$ with constant coefficients, which is invariant under a group $ G$ of linear transformations, it is not true in general that the equation $ Pu = f$ always has a $ G$-invariant solution $ u$ for a $G$-invariant $f$. We elucidate here the particular case of a ``big'' group $G$, and we count the invariant solutions when they exist (see Corollary 28 and Theorems 32, 33). The case, of special interest, of the wave equation and the Lorentz group is covered (Corollary 27). The theory of hyperfunctions provides the frame for the work.
On a class of new inequalities
Daniel T.
Shum
299-341
Abstract: Inequalities of considerable interest are associated with the names of Beesack, Benson, Boyd, Calvert, Das, Hardy, Hua, Opial, Wong and Yang. In this note an elementary method used in a recent paper by Benson will be further investigated. The resultant new class of inequalities will bring a great number of inequalities--such as inequalities of Hardy's and those of Opial's--under one roof, so to speak.
Generalized quantifiers and compact logic
Saharon
Shelah
342-364
Abstract: We solve a problem of Friedman by showing the existence of a logic stronger than first-order logic even for countable models, but still satisfying the general compactness theorem, assuming e.g. the existence of a weakly compact cardinal. We also discuss several kinds of generalized quantifiers.
Obstruction theory with coefficients in a spectrum
Mark
Mahowald;
Robert
Rigdon
365-384
Abstract: In this paper an obstruction theory with coefficients in a spectrum is developed. An idea of orientability of a fiber bundle with respect to a spectrum is introduced and for bundles orientable with respect to the spectrum a resolution is produced which corresponds to a modified Postnikov tower in the classical case.
Surgery on a curve in a solid torus
J. P.
Neuzil
385-406
Abstract: We consider the following surgery question: If a regular neighborhood of a polyhedral knot in a solid torus is removed and then sewn back differently, what manifold results? We consider two classes of knots, torus knots and what we call doubly twisted knots. We obtain some related results on surgery on knots in ${S^3}$.
Modular constructions for combinatorial geometries
Tom
Brylawski
1-44
Abstract: R. Stanley, in an investigation of modular flats in geometries (Algebra Universalis 1-2 (1971), 214-217), proved that the characteristic polynomial $\chi (x)$ of a modular flat $x$ divides the characteristic polynomial $ \chi (G)$ of a geometry $ G$. In this paper we identify the quotient: THEOREM. If $x$ is a modular flat of $G,\chi (G)/\chi (x) = \chi (\overline {{T_x}} (G))/(\lambda - 1)$, where $\overline {{T_x}} (G)$ is the complete Brown truncation of $G$ by $x$. (The lattice of $\overline {{T_x}} (G)$ consists of all flats containing $x$ and all flats disjoint from $x$, with the induced order from $ G$.) We give many characterizations of modular flats in terms of their lattice properties as well as by means of a short-circuit axiom and a modular version of the MacLane-Steinitz exchange axiom. Modular flats are shown to have many of the useful properties of points and distributive flats (separators) in addition to being much more prevalent. The theorem relating the chromatic polynomials of two graphs and the polynomial of their vertex join across a common clique generalizes to geometries: THEOREM. Given geometries $G$ and $H$, if $x$ is a modular flat of $G$ as well as a subgeometry of $ H$, then there exists a geometry $ P = {P_x}(G,H)$ which is a pushout in the category of injective strong maps and such that $\chi (P) = \chi (G)\chi (H)/\chi (x)$. The closed set structure, rank function, independent sets, and lattice properties of $P$ are characterized. After proving a modular extension theorem we give applications of our results to Crapo's single element extension theorem, Crapo's join operation, chain groups, unimodular geometries, transversal geometries, and graphs.
The automorphism group of a compact group action
W. D.
Curtis
45-54
Abstract: This paper contains results on the structure of the group, $\operatorname{Diff} _G^r(M)$, of equivariant $ {C^r}$-diffeomorphisms of a free action of the compact Lie group $G$ on $M$. $\operatorname{Diff} _G^r(M)$ is shown to be a locally trivial principal bundle over a submanifold of ${\operatorname{Diff} ^r}(X),X$ the orbit manifold. The structural group of this bundle is ${E^r}(G,M)$, the set of equivariant $ {C^r}$-diffeomorphisms which induce the identity on $X$. $ {E^r}(G,M)$ is shown to be a submanifold of ${\operatorname{Diff} ^r}(M)$ and in fact a Banach Lie group $(r < \infty )$.
Function algebras and flows. IV
Paul S.
Muhly
55-66
Abstract: The automorphisms of the algebra $ \mathfrak{A}$ of analytic functions associated with a flow (without periodic orbits) are completely determined. This result extends earlier work of Arens who determined the automorphisms of $ \mathfrak{A}$ when the flow is almost periodic. The Choquet boundary of the maximal ideal space of $ \mathfrak{A}$ is also determined under the hypothesis that the flow has no fixed points.
Duality theories for metabelian Lie algebras. II
Michael A.
Gauger
67-75
Abstract: In this paper I have replaced one of the axioms given in my Duality theory for metabelian Lie algebras (Trans. Amer. Math. Soc. 187 (1974), 89-102) concerning duality theories by a considerably more natural assumption which yields identical results--a uniqueness theorem.
Automorphisms of commutative rings
H. F.
Kreimer
77-85
Abstract: Let $B$ be a commutative ring with 1, let $ G$ be a finite group of automorphisms of $B$, and let $A$ be the subring of $G$-invariant elements of $B$. For any separable $A$-subalgebra $A'$ of $B$, the following assertions are proved: (1) $ A'$ is a finitely generated, protective $A$-module; (2) for each prime ideal $p$ of $A$, the rank of ${A'_p}$ over ${A_p}$ does not exceed the order of $G$; (3) there is a finite group $ H$ of automorphisms of $ B$ such that $ A'$ is the subring of $ H$-invariant elements of $ B$. If, in addition, $ A'$ is $G$-stable, then every automorphism of $ A'$ over $A$ is the restriction of an automorphism of $B$, and ${\operatorname{Hom} _A}(A',A')$ is generated as a left $A'$-module by those automorphisms of $ A'$ which are the restrictions of elements of $G$.
Under the degree of some finite linear groups. II
Harvey I.
Blau
87-96
Abstract: Let $G$ be a finite group with a cyclic Sylow $ p$-subgroup for some prime $p \geq 13$. Assume that $ G$ is not of type $ {L_2}(p)$, and that $ G$ has a faithful indecomposable modular representation of degree $d \leq p$. Some known lower bounds for $d$ are improved, in case the center of the group is trivial, as a consequence of results on the degrees $ \pmod p$ of irreducible Brauer characters in the principal $p$-block.
${\rm PL}$ involutions of $S\sp{1}\times S\sp{1}\times S\sp{1}$
Kyung Whan
Kwun;
Jeffrey L.
Tollefson
97-106
Abstract: We prove that the $ 3$-dimensional torus $ {S^1} \times {S^1} \times {S^1}$ admits exactly nine nonequivalent PL involutions. With the exception of the four fixed point free ones, the involutions may be distinguished by their fixed point sets: (1) eight points, (2) two simple closed curves, (3) four simple closed curves, (4) one torus, (5) two tori.
Asymptotic values of modulus $1$ of Blaschke products
K. K.
Leung;
C. N.
Linden
107-118
Abstract: A sufficient condition is found for each subproduct of a Blaschke product to have an asymptotic value of modulus 1 along a prescribed arc of a specified type in the unit disc. The condition obtained is found to be necessary in the case of further restrictions of the arc, and the two results give rise to a necessary and sufficient condition for the existence of $ {T_\gamma }$-limits of modulus 1 for Blaschke products.
Asymptotic values of modulus $1$ of functions in the unit ball of $H\sp{\infty }$
Kar Koi
Leung
119-128
Abstract: The main purpose of this paper is to prove a theorem concerning a necessary and sufficient condition for an inner function to have a limiting value of modulus 1 along an arc inside the unit disc, terminating at a point of the unit circle.
Quasi-equivalence classes of normal representations for a separable $C\sp{\ast} $-algebra
Herbert
Halpern
129-140
Abstract: It is shown that the set of quasi-equivalence classes of normal representations of a separable ${C^\ast }$-algebra is a Borel subset of the quasi-dual with the Mackey Borel structure and forms a standard Borel space in the induced Borel structure. It is also shown that the set of factor states which induce normal representations forms a Borel set of the space of factor states with the ${w^\ast }$-topology and that this set has a Borel transversal.
On the dimension of varieties of special divisors
R. F.
Lax
141-159
Abstract: Let ${T_g}$ denote the Teichmüller space and let $V$ denote the universal family of Teichmüller surfaces of genus $g$ Let $ V_{{T_g}}^{(n)}$ denote the $n$th symmetric product of $V$ over ${T_g}$ and let $J$ denote the family of Jacobians over $ {T_g}$. Let $ f:V_{{T_g}}^{(n)} \to$ J$$ be the natural relativization over $ {T_g}$ of the classical map defined by integrating holomorphic differentials. Let $\displaystyle u:{f^\ast }\Omega _{\text{J} /{T_g}}^1 \to \Omega _{V_{{T_g}/{T_g}}^{(n)}}^1$ be the map induced by $f$. We define $G_n^r$ to be the analytic subspace of $V_{{T_g}}^{(n)}$ defined by the vanishing of ${ \wedge ^{n - r + 1}}u$. Put $\tau = (r + 1)(n - r) - rg$. We show that $G_n^1 - G_n^2$, if nonempty, is smooth of pure dimension $3g - 3 + \tau + 1$. From this result, we may conclude that, for a generic curve $X$, the fiber of $ G_n^1 - G_n^2$ over the module point of $X$, if nonempty, is smooth of pure dimension $ \tau + 1$, a classical assertion. Variational formulas due to Schiffer and Spencer and Rauch are employed in the study of $ G_n^r$.
Stable positive definite functions
K. R.
Parthasarathy;
K.
Schmidt
161-174
Abstract: This paper investigates the stability of positive definite functions on locally compact groups under one parameter groups of automorphisms. As an application of this it is shown that the only probability distributions on $ {R^n}$ which are stable under the full automorphism group GL$(n,R)$ of ${R^n}$ are the nondegenerate Gaussian distributions. It is furthermore shown that there are no nondegenerate probability distributions of $ {R^n}$ which are stable under $ {\text{SL}}(n,R)$.
On constructing least squares solutions to two-point boundary value problems
John
Locker
175-183
Abstract: For an $ n$th order linear boundary value problem $Lf = {g_0}$ in the Hilbert space ${L^2}[a,b]$, a sequence of approximate solutions is constructed which converges to the unique least squares solution of minimal norm. The method is practical from a computational viewpoint, and it does not require knowing the null spaces of the differential operator $ L$ or its adjoint ${L^ \ast }$.
The factorization and representation of lattices
George
Markowsky
185-200
Abstract: For a complete lattice $L$, in which every element is a join of completely join-irreducibles and a meet of completely meet-irreducibles (we say $L$ is a jm-lattice) we define the poset of irreducibles $P(L)$ to be the poset (of height one) $ J(L) \cup M(L)(J(L)$ is the set of completely join-irreducibles and $ M(L)$ is the set of completely meet-irreducibles) ordered as follows: $a{ < _{P(L)}}b$ if and only if $a \in J(L),b \in M(L)$, and $a \nleqslant { _L}b$. For a jm-lattice $L$, the automorphism groups of $ L$ and $P(L)$ are isomorphic, $L$ can be reconstructed from $ P(L)$, and the irreducible factorization of $L$ can be gotten from the components of $ P(L)$. In fact, we can give a simple characterization of the center of a jm-lattice in terms of its separators (or unions of connected components of $P(L)$). Thus $P(L)$ extends many of the properties of the poset of join-irreducibles of a finite distributive lattice to the class of all jm-lattices. We characterize those posets of height 1 which are $P(L)$ for some jm-lattice $ L$. We also characterize those posets of height 1 which are $P(L)$ for a completely distributive jm-lattice, as well as those posets which are $ P(L)$ for some geometric lattice $L$. More generally, if $L$ is a complete lattice, many of the above arguments apply if we use ``join-spanning'' and ``meet-spanning'' subsets of $L$, instead of $J(L)$ and $M(L)$. If $L$ is an arbitrary lattice, the same arguments apply to ``join-generating'' and ``meet-generating'' subsets of $L$.
The regular ring and the maximal ring of quotients of a finite Baer $\sp{\ast} $-ring
Ernest S.
Pyle
201-213
Abstract: Necessary and sufficient conditions are obtained for extending the involution of a Baer $\ast$-ring to its maximal ring of quotients. Berberian's construction of the regular ring of a Baer $ \ast$-ring is generalized and this ring is identified (under suitable hypotheses) with the maximal ring of quotients.
Analytic structure in some analytic function algebras
William R.
Zame
215-226
Abstract: A complete description is given of the analytic structure of maximal dimension in the spectra of a wide class of concrete function algebras generated by analytic functions. A connection is also given with point derivations on such algebras.
An extremal length problem on a bordered Riemann surface
Jeffrey Clayton
Wiener
227-245
Abstract: Partition the contours of a compact bordered Riemann surface $ R'$ into four disjoint closed sets ${\alpha _0},{\alpha _1},{\alpha _2}$ and $ \gamma$ with ${\alpha _0}$ and $ {\alpha _1}$ nonempty. Let $ F$ denote the family of all locally rectifiable $1$-chains in ${\alpha _0}$ to $ {\alpha _1}$. The extremal length problem on $R'$ considers the existence of a real-valued harmonic function $u$ on $R'$ which is 0 on $ {\alpha _0},1$ on ${\alpha _1}$, a constant on each component $ {\nu _k}$ of ${\alpha _2}$ with ${\smallint _{{\nu _k}}}^ \ast du = 0$ and $^ \ast du = 0$ along $\gamma$ such that the extremal length of $ F$ is equal to the reciprocal of the Dirichlet integral of $u$, that is, $ \partial \bar R \subset S$. We consider the extremal length problem on $ \bar R$ (as a subset of $ S$) when ${\alpha _0},{\alpha _1}$, and ${\alpha _2}$ are relatively closed subarcs of $\partial \bar R$ and when ${\alpha _0},{\alpha _1}$ and ${\alpha _2}$ are closed subsets of $ \partial S = (S - \bar R) \cup \partial \bar R$.
Fuchsian manifolds
Su Shing
Chen
247-256
Abstract: Recently Eberlein and O'Neill have investigated Riemannian manifolds of negative sectional curvature. For visibility manifolds, they have obtained a classification into three types: parabolic, axial and fuchsian. Fundamental groups of fuchsian manifolds of finite type will be investigated. The main theorem is that isometry groups of certain (not necessarily compact) fuchsian manifolds are finite. Fundamental groups of fuchsian manifolds of finite type are not amenable. The spectral radius of the random matrix of the fundamental group of a compact Riemannian manifold of negative sectional curvature is less than one.
Conjugacy correspondences: a unified view
L.
McLinden
257-274
Abstract: As preparation for a duality theory for saddle programs, a partial conjugacy correspondence is developed among equivalence classes of saddle functions. Three known conjugacy correspondences, including Fenchel's correspondence among convex functions and Rockafellar's extension of it to equivalence classes of saddle functions, are shown to be degenerate special cases. Additionally, two new correspondences are brought to light as further special cases. Various questions are answered concerning the lower and upper closures and effective domain of the resulting equivalence class, as well as the effect the correspondence has on the related purely convex function and the subdifferential mapping.
On entire functions of fast growth
S. K.
Bajpai;
G. P.
Kapoor;
O. P.
Juneja
275-297
Abstract: Let $\displaystyle (\ast )\quad f(z) = \sum\limits_{n = 0}^\infty {{a_n}{z^{{\lambda _n}}}}$ be a transcendental entire function. Set $\displaystyle M(r) = \mathop {max}\limits_{\vert z\vert = r} \vert f(z)\vert,\q... ...(r) = \mathop {\max }\limits_{n \geq 0} \{ \vert{a_n}\vert{r^{{\lambda _n}}}\}$ and $\displaystyle N(r) = \mathop {\max }\limits_{n \geq 0} \{ {\lambda _n}\vert m(r) = \vert{a_n}\vert{r^{{\lambda _n}}}\} .$ Sato introduced the notion of growth constants, referred in the present paper as $ {S_q}$-order $ \lambda$ and $ {S_q}$-type $T$, which are generalizations of concepts of classical order and type by defining $\displaystyle (\ast \ast )\quad \lambda = \mathop {\lim }\limits_{r \to \infty } \sup ({\log ^{[q]}}M(r)\vert\log r)$ and if $0 < \lambda < \infty $, then $\displaystyle (\ast \ast \ast )\quad T = \mathop {\lim }\limits_{r \to \infty } \sup ({\log ^{[q - 1]}}M(r)\vert{r^\lambda })$ for $q = 2,3,4, \cdots$ where ${\log ^{[0]}}x = x$ and ${\log ^{[q]}}x = \log ({\log ^{[q - 1]}}x)$. Sato has also obtained the coefficient equivalents of $(\ast \ast )$ and $(\ast \ast \ast )$ for the entire function $ (\ast )$ when ${\lambda _n} = n$. It is noted that Sato's coefficient equivalents of $\lambda$, and $T$ also hold true for $(\ast )$ if $n$'s are replaced by $ {\lambda _n}$'s in his coefficient equivalents. Analogous to $(\ast \ast )$ and $ (\ast \ast \ast )$ lower $ {S_q}$-order $ v$ and lower $ {S_q}$-type $t$ for entire function $f(z)$ are introduced here by defining $\displaystyle v = \mathop {\lim }\limits_{r \to \infty } \inf ({\log ^{[q]}}M(r)\vert\log r)$ and if $0 < \lambda < \infty$ then $\displaystyle t = \mathop {\lim }\limits_{r \to \infty } \inf ({\log ^{[q - 1]}}M(r)\vert{r^\lambda }),\quad q = 2,3,4, \cdots .$ For the case $q = 2$, these notions are due to Whittakar and Shah respectively. For the constant $v$, two complete coefficient characterizations have been found which generalize the earlier known results. For $t$ coefficient characterization only for those entire functions for which the consecutive principal indices are asymptotic is obtained. Determination of a complete coefficient characterization of $ t$ remains an open problem. Further ${S_q}$-growth and lower ${S_q}$-growth numbers for entire function $ f(z)$ we defined \begin{displaymath}\begin{array}{*{20}{c}} \delta \mu \end{array} = \matho... ...\inf } \end{array} ({\log ^{[q - 1]}}N(r)\vert{r^\lambda }),\end{displaymath} for $ q = 2,3,4, \cdots$ and $0 < \lambda < \infty$. Earlier results of Juneja giving the coefficients characterization of $\delta$ and $\mu$ are extended and generalized. A new decomposition theorem for entire functions of $ {S_q}$-regular growth but not of perfectly ${S_q}$-regular growth has been found.
Right orders in full linear rings
K. C.
O’Meara
299-318
Abstract: In this paper a right order $R$ in an infinite dimensional full linear ring is characterized as a ring satisfying: (1) $ R$ is meet-irreducible (with zero right singular ideal) and contains uniform right ideals; (2) the closed right ideals of $ R$ are right annihilator ideals, and each such right ideal is essentially finitely generated; (3) $R$ possesses a reducing pair (i.e. a pair $({\beta _1},{\beta _2})$ of elements for which ${\beta _1}R,{\beta _2}R$ and $\beta _1^r + \beta _2^r$ are large right ideals of $ R$); (4) for each $a \in R$ with ${a^l} = 0,aR$ contains a regular element of $ R$. A second characterization of $R$ is also given. This is in terms of the right annihilator ideals of $R$ which have the same (uniform) dimension as $ {R_R}$. The problem of characterizing right orders in (infinite dimensional) full linear rings was posed by Carl Faith. The Goldie theorems settled the finite dimensional case.
On the Frattini subgroups of generalized free products and the embedding of amalgams
R. B. J. T.
Allenby;
C. Y.
Tang
319-330
Abstract: In this paper we shall prove a basic relation between the Frattini subgroup of the generalized free product of an amalgam $\mathfrak{A} = (A,B;H)$ and the embedding of $\mathfrak{A}$ into nonisomorphic groups, namely, if $ \mathfrak{A}$ can be embedded into two non-isomorphic groups ${G_1} = \langle A,B\rangle $ and ${G_2} = \langle A,B\rangle $ then the Frattini subgroup of $G = {(A \ast B)_H}$ is contained in $H$. We apply this result to various cases. In particular, we show that if $A,B$ are locally solvable and $H$ is infinite cyclic then $ \Phi (G)$ is contained in $ H$.
Two model theoretic proofs of R\"uckert's Nullstellensatz
Volker
Weispfenning
331-342
Abstract: Rückert's Nullstellensatz for germs of analytic functions and its analogue for germs of real analytic functions are proved by a combination of nonstandard analysis with a model theoretic transfer principle. It is also shown that Rückert's Nullstellensatz is constructive essentially relative to the Weierstrass preparation theorem.
Torus actions on a cohomology product of three odd spheres
Christopher
Allday
343-358
Abstract: The main purpose of this paper is to describe how a torus group may act on a space, $X$, whose rational cohomology ring is isomorphic to that of a product of three odd-dimensional spheres, in such a way that the fixed point set is nonempty, and $X$ is not totally nonhomologous to zero in the associated $X$-bundle, $ {X_T} \to {B_T}$. In the first section of the paper some general results on the cohomology theory of torus actions are established. In the second section the cohomology theory of the above type of action is described; and in the third section the results of the first two sections are used to prove a Golber formula for such actions, which, under certain conditions, bears an interesting interpretation in terms of rational homotopy.
Isolated invariant sets for flows on vector bundles
James F.
Selgrade
359-390
Abstract: This paper studies isolated invariant sets for linear flows on the projective bundle associated to a vector bundle, e.g., the projective tangent flow to a smooth flow on a manifold. It is shown that such invariant sets meet each fiber, roughly in a disjoint union of linear subspaces. Isolated invariant sets which are intersections of attractors and repellers (Morse sets) are discussed. We show that, over a connected chain recurrent set in the base space, a Morse filtration gives a splitting of the projective bundle into a direct sum of invariant subbundles. To each factor in this splitting there corresponds an interval of real numbers (disjoint from those for other factors) which measures the exponential rate of growth of the orbits in that factor. We use these results to see that, over a connected chain recurrent set, the zero section of the vector bundle is isolated if and only if the flow is hyperbolic. From this, it follows that if no equation in the hull of a linear, almost periodic differential equation has a nontrivial bounded solution then the solution space of each equation has a hyperbolic splitting.
Sequential convergence in the order duals of certain classes of Riesz spaces
P. G.
Dodds
391-403
Abstract: Several results of Hahn-Vitali-Saks type are given for sequences in the order dual of an Archimedean Riesz space with separating order dual. The class of Riesz spaces considered contains those which are Dedekind $\sigma $-complete, or have the projection property or have an interpolation property introduced by G. L. Seever. The results depend on recent work of O. Burkinshaw and some results of uniform boundedness type.
Acknowledgement of priority: ``Asymptotic behavior of solutions of linear stochastic differential systems'' (Trans. Amer. Math. Soc. {\bf 181} (1973), 1--22)
Avner
Friedman;
Mark A.
Pinsky
405
Binomial enumeration on dissects
Michael
Henle
1-39
Abstract: The Mullin-Rota theory of binomial enumeration is generalized to an abstract context and applied to rook polynomials, order invariants of posets, Tutte invariants of combinatorial geometries, cycle indices and symmetric functions.
Local Jordan algebras
Marvin E.
Camburn
41-50
Abstract: A local Jordan algebra $\mathfrak{J}$ is a unital quadratic Jordan algebra in which $\operatorname{Rad} \mathfrak{J}$ is a maximal ideal, $ \mathfrak{J}/\operatorname{Rad} \mathfrak{J}$ satisfies the DCC, and ${ \cap _k}\operatorname{Rad} {\mathfrak{J}^{(k)}} = 0$ where ${K^{(n + 1)}} = {U_K}(n){K^{(n)}}$. We show that the completion of a local Jordan algebra is also local Jordan, and if $ \mathfrak{J}$ is a complete local Jordan algebra over a field of characteristic not 2, then either (1) $ \mathfrak{J}$ is a complete completely primary Jordan algebra, (2) $\mathfrak{J} \cong {\mathfrak{J}_1} \oplus {\mathfrak{J}_2} \oplus S$ where each ${\mathfrak{J}_i}$ is a completely primary local Jordan algebra, or (3) $\mathfrak{J} \cong \mathfrak{H}({D_n},{J_a})$ where $(D,j)$ is either a not associative alternative algebra with involution or a complete semilocal associative algebra with involution.
Decomposable braids as subgroups of braid groups
H.
Levinson
51-55
Abstract: The group of all decomposable $3$-braids is the commutator subgroup of the group ${I_3}$ of all $3$-braids which leave strand positions invariant. The group of all $2$-decomposable $4$-braids is the commutator subgroup of $ {I_4}$, and the group of all decomposable $4$-braids is explicitly characterized as a subgroup of the second commutator subgroup of ${I_4}$.
Selection theorems and the reduction principle
A.
Maitra;
B. V.
Rao
57-66
Abstract: In this paper we attempt a unification of several selection theorems in the literature. It is proved that the existence of ``nice'' selectors in a certain class of selection problems is essentially equivalent to the fact that certain families of sets satisfy a weak version of Kuratowski's reduction principle. Various special cases are discussed.
Manifolds with no periodic homeomorphisms
Edward M.
Bloomberg
67-78
Abstract: An analysis of the ends of the universal cover of the connected sum of two aspherical manifolds leads to the construction of a class of closed manifolds with no nontrivial periodic homeomorphisms.
The bracket ring of a combinatorial geometry. I
Neil L.
White
79-95
Abstract: The bracket ring is a ring of generalized determinants, called brackets, constructed on an arbitrary combinatorial geometry $G$. The brackets satisfy several familiar properties of determinants, including the syzygies, which are equivalent to Laplace's expansion by minors. We prove that the bracket ring is a universal coordinatization object for $G$ in two senses. First, coordinatizations of $ G$ correspond to homomorphisms of the ring into fields, thus reducing the study of coordinatizations of $G$ to the determination of the prime ideal structure of the bracket ring. Second, $G$ has a coordinatization-like representation over its own bracket ring, which allows an interesting generalization of some familiar results of linear algebra, including Cramer's rule. An alternative form of the syzygies is then derived and applied to the problem of finding a standard form for any element of the bracket ring. Finally, we prove that several important relations between geometries, namely orthogonality, subgeometry, and contraction, are directly reflected in the structure of the bracket ring.
Nil and power-central polynomials in rings
Uri
Leron
97-103
Abstract: A polynomial in noncommuting variables is vanishing, nil or central in a ring, $R$, if its value under every substitution from $ R$ is 0, nilpotent or a central element of $R$, respectively. THEOREM. If $ R$ has no nonvanishing multilinear nil polynomials then neither has the matrix ring ${R_n}$. THEOREM. Let $R$ be a ring satisfying a polynomial identity modulo its nil radical $N$, and let $f$ be a multilinear polynomial. If $ f$ is nil in $ R$ then $f$ is vanishing in $R/N$. Applied to the polynomial $ xy - yx$, this establishes the validity of a conjecture of Herstein's, in the presence of polynomial identity. THEOREM. Let $m$ be a positive integer and let $F$ be a field containing no $m$th roots of unity other than 1. If $ f$ is a multilinear polynomial such that for some $ n > 2{f^m}$ is central in ${F_n}$, then $f$ is central in ${F_n}$. This is related to the (non)existence of noncrossed products among $ {p^2}$-dimensional central division rings.
On symmetrically distributed random measures
Olav
Kallenberg
105-121
Abstract: A random measure $ \xi$ defined on some measurable space $ (S,\mathcal{S})$ is said to be symmetrically distributed with respect to some fixed measure $\omega$ on $S$, if the distribution of $(\xi {A_1}, \cdots ,\xi {A_k})$ for $k \in N$ and disjoint $ {A_1}, \cdots ,{A_k} \in \mathcal{S}$ only depends on $(\omega {A_1}, \cdots ,\omega {A_k})$. The first purpose of the present paper is to extend to such random measures (and then even improve) the results on convergence in distribution and almost surely, previously given for random processes on the line with interchangeable increments, and further to give a new proof of the basic canonical representation. The second purpose is to extend a well-known theorem of Slivnyak by proving that the symmetrically distributed random measures may be characterized by a simple invariance property of the corresponding Palm distributions.
Presentations of $n$-knots
C.
Kearton
123-140
Abstract: The method of critical level embeddings is used to generalize the technique of knot presentations from the classical case to the case of $n$-knots. For $n > 3$, it is shown that an $n$-knot with algebraically simple complement has a correspondingly simple presentation.
Blanchfield duality and simple knots
C.
Kearton
141-160
Abstract: The method of presentation for $n$-knots is used to classify simple $ (2q - 1)$-knots, $ q > 3$, in terms of the Blanchfield duality pairing. As a corollary, we characterize the homology modules and pairings which can arise from classical knots.
$\omega $-cohesive sets
Barbara F.
Ryan
161-171
Abstract: We define and investigate $\omega$-cohesiveness, a strong notion of indecomposability for subsets of the integers and their isols. This notion says, for example, that if $X$ is the isol of an $\omega $-cohesive set then, for any integer $n$ implies that, for some integer $k, \cdot (\begin{array}{*{20}{c}} {X - k} n \end{array} ) \leq Y$ or $Z$. From this it follows that if $f(x) \in {T_1}$, the collection of almost recursive combinatorial polynomials, then the predecessors of $ {f_\Lambda }(X)$ are limited to isols $ {g_\Lambda }(X)$ where $g(X) \in {T_1}$. We show existence of $\omega $-cohesive sets. And we show that the isol of an $\omega$-cohesive set is an $n$-order indecomposable isol as defined by Manaster. This gives an alternate proof to one half of Ellentuck's theorem showing a simple algebraic difference between the isols and cosimple isols. In the last section we study functions of several variables when applied to isols of $\omega$-cohesive sets.
Comparison theorems for bounded solutions of $\triangle u=Pu$
Moses
Glasner
173-179
Abstract: Let $P$ and $Q$ be ${C^1}$ densities on a hyperbolic Riemann surface $ R$. A characterization of isomorphisms between the spaces of bounded solutions of $\Delta u = Pu$ and $\Delta u = Qu$ on $R$ in terms of the Wiener harmonic boundary is given.
Weakly starlike meromorphic univalent functions
Richard J.
Libera;
Albert E.
Livingston
181-191
Abstract: A weakly starlike meromorphic univalent function is one of the form $f(z) = - \rho zg(z){[(z - \rho )(1 - \rho z)]^{ - 1}}$ for $0 < \rho < 1$ and $g(z)$ a meromorphic starlike function. The behavior of coefficients and growth of this class of functions and of a subset are studied.
$H$-closed extensions. II
Jack R.
Porter;
Charles
Votaw
193-209
Abstract: The internal structure and external properties (in terms of other $ H$-closed extensions) of the Fomin extension $\sigma X$ of a Hausdorff space $X$ are investigated. The relationship between $\sigma X$ and the Stone-Čech compactification of the absolute of $X$ is developed and used to prove that a $ \sigma X$-closed subset of $ \sigma X\backslash X$ is compact and to show the existence of a Tychonoff space $Y$ such that $ \sigma X\backslash X$ is homeomorphic to $ \beta Y\backslash Y$. The sequential closure of $X$ in $\sigma X$ is shown to be $X$. It is known that $\sigma X$ is not necessarily projectively larger than any other strict $H$-closed extension of $X$; a necessary and sufficient condition is developed to determine when a $H$-closed extension of $X$ is projectively smaller then $\sigma X$. A theorem by Magill is extended by showing that the sets of $\theta$-isomorphism classes of $H$-closed extensions of locally $ H$-closed spaces $ X$ and $Z$ are lattice isomorphic if and only if $ \sigma X\backslash X$ and $ \sigma Z\backslash Z$ are homeomorphic. Harris has characterized those simple Hausdorff extensions of $X$ which are subextensions of the Katětov extension. Characterizations of Hausdorff (not necessarily simple) extensions of $X$ which are subextensions of $ H$-closed extensions $ \theta$-isomorphic and $ S$-equivalent to the Katětov extension are presented.
Weighted $L\sp{2}$ approximation of entire functions
Devora
Wohlgelernter
211-219
Abstract: Let $S$ be the space of entire functions $ f(z)$ such that $\vert\vert f(z)\vert{\vert^2} = \smallint \smallint \vert f(z){\vert^2}dm(z)$, where $ m$ is a positive measure defined on the Borel sets of the complex plane. Write $ dm(z) = K(z)d{A_z} = K(r,\theta )dAz$. Theorem 1. If $\ln {\inf _\theta }K(r,\theta )$ is asymptotic to $\ln {\sup_\theta} K(r,\theta )$ (together with other mild restrictions) then polynomials are dense in $S$. Theorem 2. Let $K(z) = {e^{ - \phi (z)}}$ where $ \phi (z)$ is a convex function of $z$ such that all exponentials belong to $ S$. Then polynomials are dense in $S$.
Primitive ideals of twisted group algebras
Otha L.
Britton
221-241
Abstract: E. Effros and F. Hahn have conjectured that if $(G,Z)$ is a second countable locally compact transformation group, with $G$ amenable, then every primitive ideal of the associated ${C^\ast }$-algebra arises as the kernel of an irreducible representation induced from a stability subgroup. Results of Effros and Hahn concerning this conjecture are extended to include the twisted group algebra $ {L^1}(G,A;T,\alpha )$, where $A$ is a separable type I ${C^\ast }$-algebra.
Fixed points of pointwise almost periodic homeomorphisms on the two-sphere
W. K.
Mason
243-258
Abstract: A homeomorphism $ f$ of the two-sphere $ {S^2}$ onto itself is defined to be pointwise almost periodic (p.a.p.) if the collection of orbit closures forms a decomposition of $ {S^2}$. It is shown that if $f:{S^2} \to {S^2}$ is p.a.p. and orientation-reversing then the set of fixed points of $ f$ is either empty or a simple closed curve; if $f:{S^2} \to {S^2}$ is p.a.p. orientation-preserving and has a finite number of fixed points, then $ f$ is shown to have exactly two fixed points.
The theory of countable analytical sets
Alexander S.
Kechris
259-297
Abstract: The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd $n$ a largest countable $\Pi _n^1$ set of reals, ${\mathcal{C}_n}$ (this is also true for $ n$ even, replacing $ \Pi _n^1$ by $\Sigma _n^1$ and has been established earlier by Solovay for $n = 2$ and by Moschovakis and the author for all even $n > 2$). The internal structure of the sets ${\mathcal{C}_n}$ is then investigated in detail, the point of departure being the fact that each ${\mathcal{C}_n}$ is a set of $\Delta _n^1$-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, $ \omega$-models of analysis, higher-level analogs of the constructible universe, inductive definability, etc.
One-dimensional polyhedral irregular sets of homomorphisms of $3$-manifolds
L. S.
Husch;
W. H.
Row
299-323
Abstract: Examples are given to show that there exist homeomorphisms of open $ 3$-manifolds whose sets of irregular points are wildly embedded one-dimensional polyhedra. The main result of the paper is that a one-dimensional polyhedral set of irregular points can fail to be locally tame on, at most, a discrete subset of the set of points of order greater than one. Necessary and sufficient conditions are given so that the set of irregular points is locally tame at each point.
Fatou properties of monotone seminorms on Riesz spaces
Theresa K. Y. Chow
Dodds
325-337
Abstract: A monotone seminorm $ \rho$ on a Riesz space $ L$ is called $ \sigma$-Fatou if $ \rho ({u_n}) \uparrow \rho (u)$ holds for every $u \in {L^ + }$ and sequence $\{ {u_n}\}$ in $L$ satisfying $0 \leq {u_n} \uparrow u$. A monotone seminorm $ \rho$ on $L$ is called strong Fatou if $\rho ({u_v}) \uparrow \rho (u)$ holds for every $u \in {L^ + }$ and directed system $\{ {u_v}\}$ in $L$ satisfying $0 \leq {u_v} \uparrow u$. In this paper we determine those Riesz spaces $L$ which have the property that, for any monotone seminorm $\rho$ on $L$, the largest strong Fatou seminorm $ {\rho _m}$ majorized by $ \rho$ is of the form: ${\rho _m}(f) = \inf \{ {\sup _v}\rho ({u_v}):0 \leq {u_v} \uparrow \vert f\vert\}$]> for <![CDATA[$f \in L$. We discuss, in a Riesz space $L$, the condition that a monotone seminorm $\rho$ as well as its Lorentz seminorm $ {\rho _L}$ is $ \sigma$-Fatou in terms of the order and relative uniform topologies on $ L$. A parallel discussion is also given for outer measures on Boolean algebras.
A family of countably compact $P\sb{\ast}$-hypergroups
Charles F.
Dunkl;
Donald E.
Ramirez
339-356
Abstract: An infinite compact group is necessarily uncountable, by the Baire category theorem. A compact ${P_\ast }$-hypergroup, in which the product of two points is a probability measure, is much like a compact group, having an everywhere supported invariant measure, an orthogonal system of characters which span the continuous functions in the uniform topology, and a multiplicative semigroup of positive-definite functions. It is remarkable that a compact ${P_\ast }$-hypergroup can be countably infinite. In this paper a family of such hypergroups, which include the algebra of measures on the $ p$-adic integers which are invariant under the action of the units (for $p = 2,3,5, \cdots )$) is presented. This is an example of the symmetrization technique. It is possible to give a nice characterization of the Fourier algebra in terms of a bounded-variation condition, which shows that the usual Banach algebra questions about the Fourier algebra, such as spectral synthesis, and Helson sets have easily determinable answers. Helson sets are finite, each closed set is a set of synthesis, the maximal ideal space is exactly the underlying hypergroup, and the functions that operate are exactly the Lip 1 functions.
Results on measures of irreducibility and full indecomposability
D. J.
Hartfiel
357-368
Abstract: This paper develops a notion of $k$th measure of irreducibility and $ k$th measure of full indecomposability. The combinatorial properties of these notions, as well as relationships between these notions, are explored. The results are then used in converting results on positive matrices into results on nonnegative matrices.
Converses to the $\Omega $-stability and invariant lamination theorems
Allan
Gottlieb
369-383
Abstract: In 1967 Smale proved that for diffeomorphisms on closed smooth manifolds, Axiom ${\text{A}}$ and no cycles are sufficient conditions for $\Omega$-stability and asserted the analogous theorem for vectorfields. Pugh and Shub have supplied a proof of the latter. Since then a major problem in dynamical systems has been Smale's conjecture that Axiom ${\text{A}}$ (resp.
Flat analytic extensions
Ana M. D.
Viola-Prioli
385-404
Abstract: This paper is concerned, in the first place, with the conditions to be imposed on an ideal $I$ of the power series ring in one indeterminate $ A[[x]]$ ($A$ noetherian) in order that the analytic extension $B = A[[x]]/I$ be a flat $ A$-module. Also the relationship between the projectivity and finiteness of $B$ is found when the content of $I$ (the ideal of $A$ generated by the coefficients of all power series in $I$) equals $A$. A generalization of this result to the power series ring in any finite number of indeterminates is obtained when $A$ is local, noetherian of Krull $\dim \geq 1$, and under certain restrictions on $ I$, for the global case but only for domains. Finally, a contribution to the problem of the finiteness of $I$ when $A[[x]]/I$ is a flat analytic extension is given for $A$ a local ring, not necessarily noetherian.
Abstract: Some theorems in the Benson-Curtis paper [1] were stated subject to possible exceptions in type ${E_7}$, corresponding to the two irreducible characters of the Weyl group of degree 512. An argument due to T. A. Springer($^{1}$) shows that these cases actually are exceptions to the theorems, and also that there are four exceptional cases in type ${E_8}$ (whose possible existence was overlooked in the original version of the paper), corresponding to the characters of the Weyl group of degree 4096.
Cloverleaf representations of simply connected $3$-manifolds
Edwin E.
Moise
1-30
Abstract: Let $M$ be a triangulated $3$-manifold satisfying the hypothesis of the Poincaré Conjecture. In the present paper it is shown that there is a finite linear graph $ {K_1}$ in the $ 3$-sphere, with exactly two components, and a finite linear graph ${K_2}$ in $M$, such that when the components of the graphs $ {K_i}$ are regarded as points, the resulting hyperspaces are homeomorphic. ${K_2}$ satisfies certain conditions which imply that each component of ${K_2}$ is contractible in $M$. Thus the conclusion of the theorem proved here is equivalent to the hypothesis of the Poincaré Conjecture.
Modular and distributive semilattices
Joe B.
Rhodes
31-41
Abstract: A modular semilattice is a semilattice $S$ in which $w \geq$ implies that there exist $x,y \in S$ such that $x \geq a,y \geq b$ and $x \wedge y = x \wedge w$. This is equivalent to modularity in a lattice and in the semilattice of ideals of the semilattice, and the condition implies the Kurosh-Ore replacement property for irreducible elements in a semilattice. The main results provide extensions of the classical characterizations of modular and distributive lattices by their sublattices: A semilattice $S$ is modular if and only if each pair of elements of $ S$ has an upper bound in $ S$ and there is no retract of $S$ isomorphic to the nonmodular five lattice. A semilattice is distributive if and only if it is modular and has no retract isomorphic to the nondistributive five lattice.
Associated and perspective simplexes
Leon
Gerber
43-55
Abstract: A set of $n + 1$ lines in $ n$-space such that any $({\text{n}} - 2)$-dimensional flat which meets $n$ of the lines also meets the remaining line is said to be an associated set of lines. Two Simplexes are associated if the joins of corresponding vertices are associated. A simple criterion is given for simplexes to be associated and an analogous one for Simplexes to be perspective. These are used to give a brief proof of the following generalization of the theorem of Pappus. Let $ \mathcal{A}$ and $\mathcal{B}$ be $n$-simplexes and let $p$ be a permutation on the vertices of $\mathcal{B}$. If $ \mathcal{A}$ and $\mathcal{B}$ are associated (respectively perspective) and $ \mathcal{A}$ and $\mathcal{B}p$ are associated (perspective) then $\mathcal{A}$ and $\mathcal{B}{p^k}$ are associated (perspective) for any integer $k$. Very short proofs are given of extensions to $ n$-dimensions of many theorems from Neuberg's famous Memoir sur le Tétraèdre, such as: the altitudes of a simplex are associated.
The structure of nested spaces
T. B.
Muenzenberger;
R. E.
Smithson
57-87
Abstract: The structure of nested spaces is studied in this paper using such tools as branches, chains, partial orders, and rays in the context of semitrees. A classification scheme for various kinds of acyclic spaces is delineated in terms of semitrees. Several families of order compatible topologies for semitrees are investigated, and these families are grouped in a spectrum (inclusion chain) of topologies compatible with the semitree structure. The chain, interval, and tree topologies are scrutinized in some detail. Several topological characterizations of semitrees with certain order compatible topologies are also derived.
A $5-r$ uniqueness theorem
Jessie Ann
Engle
89-104
Abstract: A Borel-regular Carathéodory outer measure $\Lambda$ on a separable metric space $X$, where $\Lambda$ is invariant with respect to a family $ H$ of homeomorphisms from $ X$ onto $X$, is unique if $\Lambda$ satisfies a $5$ - $r$ condition at one point in $X$ and if $H$ satisfies Condition I, a condition much weaker than, but related to, the invariance of distance under $ H$.
$k$-regular elements in semisimple algebraic groups
Peter P.
Andre
105-124
Abstract: In this paper, Steinberg's concept of a regular element in a semisimple algebraic group defined over an algebraically closed field is generalized to the concept of a $k$-regular element in a semisimple algebraic group defined over an arbitrary field of characteristic zero. The existence of semisimple and unipotent $ k$-regular elements in a semisimple algebraic group defined over a field of characteristic zero is proved. The structure of all $ k$-regular unipotent elements is given. The number of minimal parabolic subgroups containing a $k$-regular element is given. The number of conjugacy classes of $R$-regular unipotent elements is given, where $ R$ is the real field. The number of conjugacy classes of ${Q_p}$-regular unipotent elements is shown to be finite, where ${Q_p}$ is the field of $p$-adic numbers.
Theta constants and cusp forms
A. J.
Crisalli
125-132
Abstract: For principal congruence subgroups of levels 2 and 4 a basis for their cusp forms consisting of monomials of theta constants is displayed. Some conditions for the vanishing of Poincaré series of these groups are found.
On centralizers of generalized uniform subgroups of locally compact groups
Kwan-Yuk Law
Sit
133-146
Abstract: Let $G$ be a locally compact group and $ H$ a closed subgroup of $ G$ such that the homogeneous space $G/H$ admits a finite invariant measure. Let $ {Z_G}(H)$ be the centralizer of $H$ in $G$. It is shown that if $G$ is connected then ${Z_G}(H)$ modulo its center is compact. If $ G$ is only assumed to be locally connected it is shown that the commutator subgroup of ${Z_G}(H)$ has compact closure. Consequences of these results are found for special classes of groups, such as Lie groups. An example of a totally disconnected group $G$ is given to show that the results for $ {Z_G}(H)$ need not hold if $ G$ is not connected or locally connected.
Deformations of group actions
Allan L.
Edmonds
147-160
Abstract: Let $G$ be a finite group and $M$ be a compact piecewise linear (PL) manifold. Define a PL $G$-isotopy to be a level-preserving PL action of $G$ on $M \times [0,1]$. In this paper PL $ G$-isotopies are studied and PL $G$-isotopic actions (which need not be equivalent) are characterized.
A homotopy theory of pro-spaces
Jerrold W.
Grossman
161-176
Abstract: The category of towers of spaces, $\ldots \to {X_{s + 1}} \to {X_s} \to \ldots \to {X_0}$, viewed as pro-spaces, appears to be useful in the study of the relation between homology and homotopy of nonsimply connected spaces. We show that this category admits the structure of a closed model category, in the sense of Quillen; notions of fibration, cofibration, and weak equivalence are defined and shown to satisfy fundamental properties that the corresponding notions satisfy in the category of spaces. This enables one to develop a ``homotopy theory'' for pro-spaces.
On the decomposition of tensor products of principal series representations for real-rank one semisimple groups
Robert Paul
Martin
177-211
Abstract: Let $G$ be a connected semisimple real-rank one Lie group with finite center. It is shown that the decomposition of the tensor product of two representations from the principal series of $G$ consists of two pieces, ${T_c}$ and ${T_d}$, where ${T_c}$ is a continuous direct sum with respect to Plancherel measure on $\hat G$ of representations from the principal series only, occurring with explicitly determined multiplicities, and ${T_d}$ is a discrete sum of representations from the discrete series of $G$, occurring with multiplicities which are, for the present, undetermined.
Teichm\"uller spaces and representability of functors
Michael
Engber
213-226
Abstract: The Teichmëller space of compact Riemann surfaces with punctures is exhibited as the object representing a certain functor. This extends the work of Grothendieck, who treated the unpunctured case. The relationship between the two cases is exploited to obtain specific results on the connection between the two Teichmüller spaces.
Large models of countable height
Harvey
Friedman
227-239
Abstract: Eery countable transitive model $M$ of ZF (without choice) has an ordinal preserving extension satisfying ZF, of power ${ \sqsupset _{M \cap On}}$. An application to infinitary logic is given.
On the analytic continuation of the Minakshisundaram-Pleijel zeta function for compact Riemann surfaces
Burton
Randol
241-246
Abstract: A formula is derived for the Minakshisundaram-Pleijel zeta function in the half-plane Re $s < 0$.
Identities involving the coefficients of a class of Dirichlet series. VII
Bruce C.
Berndt
247-261
Abstract: Let $a(n)$ be an arithmetical function, and consider the Riesz sum $ {A_\rho }(x) = {\Sigma _{n \leq x}}a(n){(x - n)^\rho }$. For $a(n)$ belonging to a certain class of arithmetical functions, $ {A_\rho }(x)$ can be expressed in terms of an infinite series of Bessel functions. K. Chandrasekharan and R. Narasimhan have established this identity for the widest known range of $ \rho$. Their proof depends upon equi-convergence theory of trigonometric series. An alternate proof is given here which uses only the classical theory of Bessel functions.
${\rm PI}$-algebras satisfying identities of degree $3$
Abraham A.
Klein
263-277
Abstract: A method of classification of PI-algebras over fields of characteristic 0 is described and applied to algebras satisfying polynomial identities of degree 3. Two algebras satisfying the same identities of degree 3 are considered in the same class. For the degree 3 all the possible classes are obtained. In each case the identities of degree 4 that can be deduced from those of degree 3 have been obtained by means of a computer. These computations have made it possible to obtain-except for three cases-all the identities of higher degrees. It turns out that except for a finite number of cases an algebra satisfying an identity of degree 3 is either nilpotent of order 4, or commutative of order 4, namely the product of 4 elements of the algebra is a symmetric function of its factors.
$K$-groups generated by $K$-spaces
Eric C.
Nummela
279-289
Abstract: A $K$-group $G$ with identity $e$ is said to be generated by the $ K$-space $X$ if $X$ is a subspace of $G$ containing $e,X$ algebraically generates $G$, and the canonical morphism from the Graev free $K$-group over $(X,e)$ on-to $G$ is a quotient morphism. An internal characterization of the topology of such a group $ G$ is obtained, as well as a sufficient condition that a subgroup $H$ of $G$ be generated by a subspace $Y$ of $H$. Several illuminating examples are provided.
Relationships between $\lambda $-nuclearity and pseudo-$\mu $-nuclearity
William B.
Robinson
291-303
Abstract: It is shown that for any Köthe space $\lambda$, $\lambda$-nuclearity coincides with pseudo- $\lambda {\lambda ^ \times }$-nuclearity. More particular results, including a Grothendieck-Pietsch criterion for $\lambda$-nuclearity of sequence spaces, are given for Köthe spaces which are regular.
Subcategories of uniform spaces
Michael D.
Rice
305-314
Abstract: The problem of embedding a topological space as a closed subspace of a product of members from a given family has received considerable attention in the past twenty years, while the corresponding problem in uniform spaces has been largely ignored. In this paper we initiate the study of the closed uniform subspaces of products of metric spaces. In §1 we introduce the functor $m$, which is used in §2 to characterize the closed subspaces of products of metric spaces and separable metric spaces, and the closed subspaces of powers of the open unit interval $(0, 1)$. In §3 we obtain various descriptions of the functor $d$ which associates to each uniform space a closed subspace of a product of metric spaces and establish the equation $md = dm$. This leads to a characterzation of the completeness of $euX$, the uniform space generated by the countable $ u$-uniform covers, in terms of the completeness of $uX$ and a countable intersection property on Cauchy filters.
A sufficient condition for hyperbolicity of partial differential operators with constant coefficient principal part
Joseph L.
Dunn
315-327
Abstract: Let $P$ be a differential operator with principal part ${P_m}$, and suppose that ${P_m}$ has constant coefficients and is hyperbolic. It is shown that the condition for hyperbolicity of $P$ when $P$ has constant coefficients, namely, that $ P$ is weaker than $ {P_m}$ is also a sufficient condition for hyperbolicity in the case where $P$ does not have constant coefficients. Some generalizations are also made to the case where $P$ is a square matrix of differential operators.
Solvable groups and quadratic forms
Richard
Tolimieri
329-345
Abstract: A solvable Lie group and a unitary representation are constructed from a given binary quadratic form. The multiplicity of this representation is related to the arithmetic of the form.
Polynomials and the limit point condition
Robert M.
Kauffman
347-366
Abstract: An $n$th order, possibly nonselfadjoint, ordinary differential expression $L$ is said to be in the limit point condition if the maximal operator ${L_M}$ in $ {L_2}[0,\infty )$ is an $ n$-dimensional extension of the minimal operator ${L_0}$. If range ${L_0}$ is closed, this definition is equivalent to the assertion that nullity ${L_M} +$ nullity${({L^ + })_M} = n$, where ${L^ + }$ is the formal adjoint of $ L$. It also implies that any operator $T$ such that ${L_0} \subseteq T \subseteq {L_M}$ is the restriction of ${L_M}$ to a set of functions described by a boundary condition at zero. In this paper, we discuss the question of when differential expressions involving complex polynomials in selfadjoint expressions are in the limit point condition.
The algebraic EHP sequence
William M.
Singer
367-382
Abstract: Let $A$ be the dual of the $\bmod - 2$ Steenrod algebra. If $ M,N$, are graded unstable $ A$-comodules, one can define and compute the derived functors ${\text{Coext} _A}(M,N)$ using unstable injective resolutions of $N$. Bousfield and Curtis have shown that these unstable Coext groups can be fit into a long exact ``EHP sequence", an algebraic analogue of the EHP sequence of homotopy theory. Our object in the present paper is to study the relationship between the $E,H$, and $P$ homomorphisms and the composition pairing $ {\text{Coext} _A}(N,R) \otimes {\text{Coext} _A}(M,N) \to {\text{Coext} _A}(M,R)$. Among our results is a formula that measures the failure of the composition product to commute.
Bounds in piecewise linear topology
L. B.
Treybig
383-405
Abstract: The following types of results are obtained: Given a polyhedral $ 2$-sphere $P$ with rectilinear triangulation $ T$ lying in the interior of a solid tetrahedron $G$ in ${E^3}$, then there is a simplicial isotopy $f:G \times [0,1] \to G$ taking $ P$ onto a tetrahedron so that for $t$ in $ [0,1],f(x,t) = x$ on Bd$(G)$ and ${f_t}$ is affine on each element of the triangulation $ S$ of $G$, where card $(S)$ is a known function of card $ (T)$. Also, given (1) $ P$ as above, (2) polyhedral disks ${D_1}$ and ${D_2}$, where Bd$({D_1}) =$ Bd$({D_2}) \subset P$ and $ \operatorname{Int} ({D_1}) \cup \operatorname{Int} ({D_2}) \subset \operatorname{Int} (P)$ and (3) a triangulation $T$ of ${D_1} \cup {D_2} \cup P$, then analogous results are found for a simplicial isotopy $f$ which is fixed on $P$ and takes ${D_1}$ onto ${D_2}$. Given $G$ as above and a piecewise linear homeomorphism $h:G \to G$ which is fixed on ${\text{Bd(G)}}$ and affine on each $ r \in R$, then analogous bounds are found for a simplicial isotopy $f:G \times [0,1] \to G$ so that ${f_0}(x) = x$ and $ {f_1}(r) = h(r)$ for all $ r$ in $R$. In the second half of this paper the normal surface and normal equation theory of Haken is briefly explained and extended slightly. Bounds are found in connection with nontrivial integer entried solutions of normal equations. Also bounds are found for the number of Simplexes used in triangulating normal surfaces associated with certain solutions of the extended normal equations.
Modules over coproducts of rings
George M.
Bergman
1-32
Abstract: Let ${R_0}$ be a skew field, or more generally, a finite product of full matrix rings over skew fields. Let ${({R_\lambda })_{\lambda \in \Lambda }}$ be a family of faithful ${R_0}$rings (associative unitary rings containing $ {R_0}$) and let $ R$ denote the coproduct ("free product") of the $ {R_\lambda }$ as $ {R_0}$-rings. An easy way to obtain an $R$-module $M$ is to choose for each $\lambda \in \Lambda \cup \{ 0\}$ an ${R_\lambda }$-module $ {M_\lambda }$, and put $M = \oplus {M_\lambda }{ \otimes _{{R_\lambda }}}R$. Such an $M$ will be called a ``standard'' $R$-module. (Note that these include the free $R$-modules.) We obtain results on the structure of standard $R$-modules and homomorphisms between them, and hence on the homological properties of $R$. In particular: (1) Every submodule of a standard module is isomorphic to a standard module. (2) If $M$ and $N$ are standard modules, we obtain simple criteria, in terms of the original modules ${M_\lambda },{N_\lambda }$, for $N$ to be a homomorphic image of $ M$, respectively isomorphic to a direct summand of $M$, respectively isomorphic to $M$. (3) We find that r gl$\dim R = {\sup _\Lambda }($r gl$\dim {R_\lambda })$ if this is > 0, and is 0 or 1 in the remaining case.
Coproducts and some universal ring constructions
George M.
Bergman
33-88
Abstract: Let $R$ be an algebra over a field $ k$, and $P,Q$ be two nonzero finitely generated projective $R$-modules. By adjoining further generators and relations to $R$, one can obtain an extension $S$ of $R$ having a universal isomorphism of modules, $i:P{ \otimes _R}S \cong Q{ \otimes _R}S$. We here study this and several similar constuctions, including (given a single finitely generated projective $ R$-module $P$) the extension $S$ of $R$ having a universal idempotent module-endomorphism $ e:P \otimes S \to P \otimes S$, and (given a positive integer $n$) the $k$-algebra $S$ with a universal $k$-algebra homomorphism of $R$ into its $n \times n$ matrix ring, $f:R \to {\mathfrak{m}_n}(S)$. A trick involving matrix rings allows us to reduce the study of each of these constructions to that of a coproduct of rings over a semisimple ring $ {R_0}$ ( $( = k \times k \times k,k \times k$, and $ k$ respectively in the above cases), and hence to apply the theory of such coproducts. As in that theory, we find that the homological properties of the construction are extremely good: The global dimension of $S$ is the same as that of $R$ unless this is 0, in which case it can increase to 1, and the semigroup of isomorphism classes of finitely generated projective modules is changed only in the obvious fashion; e.g., in the first case mentioned, by the adjunction of the relation $[P] = [Q]$. These results allow one to construct a large number of unusual examples. We discuss the problem of obtaining similar results for some related constructions: the adjunction to $R$ of a universal inverse to a given homomorphism of finitely generated projective modules, $ f:P \to Q$, and the formation of the factor-ring $R/{T_P}$ by the trace ideal of a given finitely generated projective $R$-module $P$ (in other words, setting $P = 0$). The idea for a category-theoretic generalization of the ideas of the paper is also sketched.
Topological semigroups and representations. I
James C. S.
Wong
89-109
Abstract: Let $S$ be a topological semigroup (separately continuous multiplication) with identity and $W(S)$ the Banach space of all weakly almost periodic functions on $S$. It is well known that if $S = G$ is a locally compact group, then $ W(G)$ always has a (unique) invariant mean. In other words, there exists $m \in W{(G)^ \ast }$ such that $\vert\vert m\vert\vert = m(1) = 1$ and $m({l_s}f) = m({r_s}f) = m(f)$ for any $s \in G,f \in W(G)$ where ${l_s}f(t) = f(st)$ and ${r_s}f(t) = f(ts),t \in S$ The main purpose of this paper is to present several characterisations (functional analytic and algebraic) of the existence of a left (right) invariant mean on $W(S)$ In particular, we prove that $ W(S)$ has a left (right) invariant mean iff a certain compact topological semigroup $p{(S)^\omega }$ (to be defined) associated with $S$ contains a right (left) zero. Other results in this direction are also obtained.
Jordan rings with involution
Seong Nam
Ng
111-139
Abstract: Let $J$ be a Jordan ring with involution $ \ast$ in which $ 2x = 0$ implies $x = 0$ and in which $ 2J = J$. Let the set $ S$ of symmetric elements of $J$ be periodic and let $N$ be the Jacobson radical of $J$. Then ${N^2} = 0$ and $J/N$ is a subdirect sum of $\ast$-simple Jordan rings of the following types (1) a periodic field, (2) a direct sum of two simple periodic Jordan rings with exchange involution, (3) a $3 \times 3$ or $ 4 \times 4$ Jordan matrix algebra over a periodic field, (4) a Jordan algebra of a nondegenerate symmetric bilinear form on a vector space over a periodic field.
Join-irreducible cross product varieties of groups
James J.
Woeppel
141-148
Abstract: Let $ \mathfrak{U},\mathfrak{B}$ be varieties of groups which have finite coprime exponents, let $ \mathfrak{U}$ be metabelian and nilpotent with ``small'' nilpotency class, and let $ \mathfrak{B}$ be abelian. The product variety $\mathfrak{U}\mathfrak{B}$ is shown to be join-irreducible if and only if $ \mathfrak{U}$ is join-irreducible. This is done by obtaining a simple description for the critical groups generating $\mathfrak{U}\mathfrak{B}$ when $\mathfrak{U}$ is join-irreducible and finding a word which is not a law in $ \mathfrak{U}\mathfrak{B}$ but is a law in every proper subvariety of $ \mathfrak{U}\mathfrak{B}$
Extending continuous linear functionals in convergence vector spaces
S. K.
Kranzler;
T. S.
McDermott
149-168
Abstract: Let $(E,\tau )$ be a convergence vector space, $ M$ a subspace of $ E$, and $\varphi$ a linear functional on $ M$ continuous in the induced convergence structure. Sufficient and sometimes necessary conditions are given that (1) $\varphi$ has a continuous linear extension to the $\tau$-adherence $\bar M$ of$M$; (2) $\varphi$ has a continuous linear extension to $E$; (3) $\bar M$ is $\tau$-closed; (4) every $\tau$-closed convex subset of $E$ is
Homeomorphisms between Banach spaces
Roy
Plastock
169-183
Abstract: We consider the problem of finding precise conditions for a map $ F$ between two Banach spaces $X,Y$ to be a global homeomorphism. Using methods from covering space theory we reduce the global homeomorphism problem to one of finding conditions for a local homeomorphism to satisfy the ``line lifting property.'' This property is then shown to be equivalent to a limiting condition which we designate by $ (L)$. Thus we finally show that a local homeomorphism is a global homeomorphism if and only if $(L)$ is satisfied. In particular we show that if a local homeomorphism is (i) proper (Banach-Mazur) or (ii)
Subbundles of the tangent bundle
R. E.
Stong
185-197
Abstract: This paper studies pairs $(M,\xi )$ where $M$ is a closed manifold and $\xi$ is a $k$-dimensional subbundle of the tangent bundle of $ M$ in terms of cobordism.
On certain sequences of integers
K.
Thanigasalam
199-205
Abstract: Let the sequence $\{ {k_i}\}$ satisfy $2 \leqslant {k_1} \leqslant {k_2} \leqslant \cdots$. Then, under certain conditions satisfied by $\{ {k_i}\}$, it is shown that there exists an integer $s$ such that the sequence of integers of the form $ x_1^{{k_1}} + \cdots + x_s^{{k_s}}$ has positive density. Also, some special sequences having positive densities are constructed.
Semigroups of scalar type operators on Banach spaces
Ahméd Ramzy
Sourour
207-232
Abstract: The main result is that if $ \{ T(t):t \geqslant 0\}$ is a strongly continuous semigroup of scalar type operators on a weakly complete Banach space $X$ and if the resolutions of the identity for $T(t)$ are uniformly bounded in norm, then the infinitesimal generator is scalar type. Moreover, there exists a countably additive spectral measure $K( \cdot )$ such that $T(t) = \smallint \exp (\lambda t)dK(\lambda )$, for $t \geqslant 0$. This is a direct generalization of the well-known theorem of Sz.-Nagy about semigroups of normal operators on a Hilbert space. Similar spectral representations are given for representations of locally compact abelian groups and for semigroups of unbounded operators. Connections with the theory of hermitian and normal operators on Banach spaces are established. It is further shown that $ R$ is the infinitesimal generator of a semigroup of hermitian operators on a Banach space if and only if iR is the generator of a group of isometries.
Linear control problems with total differential equations without convexity
M. B.
Suryanarayana
233-249
Abstract: Neustadt type existence theorems are given for optimal control problems described by Dieudonné-Rashevsky type total differential equations which are linear in the state variable. The multipliers from the corresponding conjugate problem are used to obtain an integral representation for the functional which in turn is used in conjunction with a Lyapunov type theorem on convexity of range of integrals to derive the existence of a usual solution from that of a generalized solution, which thus needs no convexity. Existence of optimal solutions is also proved in certain cases using an implicit function theorem along with the sufficiency of the maximum principle for optimality in the case of linear systems. Bang bang type controls are shown to exist when the system is linear in the control variable also.
Right-bounded factors in an LCM domain
Raymond A.
Beauregard
251-266
Abstract: A right-bounded factor is an element in a ring that generates a right ideal which contains a nonzero two-sided ideal. Right-bounded factors in an LCM domain are considered as a generalization of the theory of two-sided bounded factors in an atomic $2$-fir, that is, a weak Bezout domain satisfying the acc and dcc for left factors. Although some elementary properties are valid in a more general context most of the main results are obtained for an LCM domain satisfying $ ({\text{M}})$ and the dcc for left factors; the condition $({\text{M}})$ is imposed to insure that prime factorizations are unique in an appropriate sense. The right bound ${b^ \ast }$ of a right bounded element $ b$ is considered in general, then in case $b$ is a prime, and finally in case $ b$ is indecomposable. The effect of assuming that right bounds are two-sided is also considered.
Deficiency sets and bounded information reducibilities
Leonard P.
Sasso
267-290
Abstract: For recursively enumerable sets $A$ and $H$ of natural numbers $H$ is a deficiency set of $A$ if there is a one-one, recursive function $ f$ with $A = \operatorname{Rng} (f)$ and $H = \{ i:(\exists j)[i < j \mathrel{\&} f(j) < f(i)]\}$. The relation between recursively enumerable sets and their deficiency sets under bounded information reducibilities (i.e. weak truth table, truth table, bounded truth table, many-one, and one-one reducibility) is investigated.
The multiplicative Cousin problem and a zero set for the Nevanlinna class in the polydisc
Sergio E.
Zarantonello
291-313
Abstract: Let $\Omega$ be a polydomain in ${{\mathbf{C}}^n}$, the Nevanlinna class $N(\Omega )$ consists of all holomorphic functions $ f$ in $\Omega$ such that ${\log ^ + }\vert f\vert$ has an $n$-harmonic majorant in $\Omega$. Let ${U^n}$ be the open unit polydisc $\{ z \in {{\mathbf{C}}^n}:\vert{z_1}\vert < 1, \cdots ,\vert{z_n}\vert < 1\}$. THEOREM 1. Given an open covering ${({\Omega _\alpha })_{\alpha \in A}}$ of the closure $ {\bar U^n}$ of the polydisc, consisting of polydomains, and for each $\alpha \in A$ a function ${f_\alpha } \in N({\Omega _\alpha } \cap {U^n})$ such that for all $\alpha ,\beta \in A,{f_\alpha }f_\beta ^{ - 1}$ is an invertible element of $ N({\Omega _\alpha } \cap {\Omega _\beta } \cap {U^n})$. There exists a function $ F \in N({U^n})$ such that for all $\alpha \in A,Ff_\alpha ^{ - 1}$ is an invertible element of $ N({\Omega _\alpha } \cap {U^n})$. This result enables us to find the following sufficient condition for the zero sets of $ N({U^n})$: THEOREM 2. Let $f$ be a holomorphic function in ${U^n},n \geqslant 2$. If there exists a constant $ 0 < r < 1$ and a continuous function $ n:[r,1) \to [r,1)$ such that $\displaystyle \vert{z_n}\vert \leqslant n\left( {\frac{{\vert{z_1}\vert + \cdots + \vert{z_{n - 1}}\vert}}{{n - 1}}} \right)$ for all points $({z_1}, \cdots ,{z_n})$ satisfying $\vert{z_1}\vert > r, \cdots ,\vert{z_n}\vert > r$ and $f({z_1}, \cdots ,{z_n}) = 0$, then $f$ has the same zeros as some function $F \in N({U^n})$. In the above if ${\overline {\lim } _{\lambda \to 1}}n(x) < 1$, then $Z(f)$ is a Rudin variety in which case there is a bounded holomorphic function with the same zeros as $ f$.
A local result for systems of Riemann-Hilbert barrier problems
Kevin F.
Clancey
315-325
Abstract: The Riemann-Hilbert barrier problem (for $n$ pairs of functions) $\displaystyle G{\Phi ^ + } = {\Phi ^ - } + g$ is investigated for the square integrable functions on a union of analytic Jordan curves $C$ bounding a domain in the complex plane. In the special case, where at each point ${t_0}$ of $C$ the symbol $G$ has at most two essential cluster values $ {G_1}({t_0}),{G_2}({t_0})$, then the condition $ \det [(1 - \lambda ){G_1}({t_0}) + \lambda {G_2}({t_0})] \ne 0$, for all $ {t_0}$ in $C$ and all $\lambda (0 \leqslant \lambda \leqslant 1)$, implies the Riemann-Hilbert operator is Fredholm. In the case, where for some ${t_0}$ in $C$ and some $ {\lambda _0}(0 \leqslant {\lambda _0} \leqslant 1),\det [(1 - {\lambda _0}){G_1}({t_0}) + {\lambda _0}{G_2}({t_0})] = 0$, the Riemann-Hilbert operator is not Fredholm. An application is given to systems of singular integral equation on ${L^2}(E)$, where $E$ is a measurable subset of $C$.
Bordism invariants of intersections of submanifolds
Allen
Hatcher;
Frank
Quinn
327-344
Abstract: This paper characterizes certain geometric intersection problems in terms of bordism obstructions. These obstructions give a setting in which to study such things as parametrized $ h$-cobordisms (pseudoisotopy), and surgery above the middle dimension and on fibrations, where such intersection problems arise.
The space of conjugacy classes of a topological group
Dennis
Daluge
345-353
Abstract: The space $ {G^\char93 }$ of conjugacy classes of a topological group $G$ is the orbit space of the action of $ G$ on itself by inner automorphisms. For a class of connected and locally connected groups which includes all analytic $[Z]$-groups, the universal covering space of ${G^\char93 }$ may be obtained as the space of conjugacy classes of a group which is locally isomorphic with $G$, and the Poincaré group of ${G^\char93 }$ is found to be isomorphic with that of $G/G'$, the commutator quotient group. In particular, it is shown that the space ${G^\char93 }$ of a compact analytic group $ G$ is simply connected if and only if $G$ is semisimple. The proof of this fact has not appeared in the literature, even though more specialized methods are available for this case.
The $p$-class in a dual $B\sp{\ast} $-algebra
Pak Ken
Wong
355-368
Abstract: In this paper, we introduce and study the class ${A_p}(0 < p \leqslant \infty )$ in a dual ${B^ \ast }$-algebra $A$. We show that, for $1 \leqslant p \leqslant \infty ,{A_p}$ is a dual ${A^ \ast }$-algebra which is a dense two-sided ideal of $A$. If $1 < p < \infty$, we obtain that $ {A_p}$ is uniformly convex and hence reflexive. We also identify the conjugate space of $ {A_p}(1 \leqslant p < \infty )$. This is a generalization of the class $ {C_p}$ of compact operators on a Hilbert space.
A continuity property with applications to the topology of $2$-manifolds
Neal R.
Wagner
369-393
Abstract: A continuity property is proved for variable simply connected domains with locally connected boundaries. This theorem provides a link between limits of conformal mappings and of retractions. Applications are given to the space of retractions of a compact $2$-manifold ${M^2}$, where it is shown that the space of deformations retractions is contractible and the space of nullhomotopic retrac tions has the same homotopy type as ${M^2}$. Other applications include a proof that the space of retracts of ${M^2}$ (with a natural quotient topology) is an absolute neighborhood retract, and a type of global solution to the Dirichlet problem.
Existence and stability for partial functional differential equations
C. C.
Travis;
G. F.
Webb
395-418
Abstract: The existence and stability properties of a class of partial functional differential equations are investigated. The problem is formulated as an abstract ordinary functional differential equation of the form $du(t)/dt = Au(t) + F({u_t})$, where $ A$ is the infinitesimal generator of a strongly continuous semigroup of linear operators $ T(t),t \geqslant 0$, on a Banach space $X$ and $F$ is a Lipschitz operator from $C = C([ - r,0];X)$ to $X$. The solutions are studied as a semigroup of linear or nonlinear operators on $C$. In the case that $F$ has Lipschitz constant $L$ and $\vert T(t)\vert \leqslant {e^{\omega t}}$, then the asymptotic stability of the solutions is demonstrated when $\omega + L < 0$. Exact regions of stability are determined for some equations where $ F$ is linear.
Interpolation polynomials which give best order of approximation among continuously differentiable functions of arbitrary fixed order on $[-1,\,+1]$
A. K.
Varma
419-426
Abstract: The object of this paper is to show that there exists a polynomial $ {P_n}(x)$ of degree $\leqslant 2n - 1$ which interpolates a given function exactly at the zeros of $n$th Tchebycheff polynomial and for which $\vert\vert f - {P_n}\vert\vert \leqslant {C_k}{w_k}(1/n,f)$ where $ {w_k}(1/n,f)$ is the modulus of continuity of $f$ of $k$th order.
Sets of multiplicity and differentiable functions. II
Robert
Kaufman
427-435
Abstract: The stability of certain sets of multiplicity is studied with reference to special classes of differentiable functions. Kronecker sets are produced as examples of instability. The most difficult theorem uses probability theory and an estimation of Kolmogoroff's $ \varepsilon$-entropy in a certain space of functions.
Spectral order preserving matrices and Muirhead's theorem
Kong Ming
Chong
437-444
Abstract: In this paper, a characterization is given for matrices which preserve the Hardy-Littlewood-Pólya spectral order relation $ \prec$ for $n$-vectors in ${R^n}$. With this characterization, a new proof is given for the classical Muirhead theorem and some Muirhead-type inequalities are obtained. Moreover, sufficient conditions are also given for matrices which preserve the Hardy-Littlewood-Pólya weak spectral order relation $ \prec\prec$.
Erratum to ``On the uniform convergence of quasiconformal mappings''
Bruce
Palka
445-445
Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers)
Marian Boykan
Pour-el
1-28
Abstract: Our aim is to study computability from the viewpoint of the analog computer. We present a mathematical definition of an analog generable function of a real variable. This definition is formulated in terms of a simultaneous set of nonlinear differential equations possessing a ``domain of generation.'' (The latter concept is explained in the text.) Our definition includes functions generated by existing general-purpose analog computers. Using it we prove two theorems which provide a characterization of analog generable functions in terms of solutions of algebraic differential polynomials. The characterization has two consequences. First we show that there are entire functions which are computable (in the sense of recursive analysis) but which cannot be generated by any analog computer in any interval--e.g. $1/\Gamma (x)$ and $\Sigma _{n = 1}^\infty ({x^n}/{n^{({n^3})}})$. Second we note that the class of analog generable functions is very large: it includes special functions which arise as solutions to algebraic differential polynomials. Although not all computable functions are analog generable, a kind of converse holds. For entire functions, $f(x) = \Sigma _{i = 0}^\infty {b_i}{x^i}$, the theorem takes the following form. If $f(x)$ is analog generable on some closed, bounded interval then there is a finite number of $ {b_k}$ such that, on every closed bounded interval, $f(x)$ is computable relative to these $ {b_k}$. A somewhat similar theorem holds if $f$ is not entire. Although the results are stated and proved for functions of a real variable, they hold with minor modifications for functions of a complex variable.
Perturbed semigroup limit theorems with applications to discontinuous random evolutions
Robert P.
Kertz
29-53
Abstract: For $\varepsilon > 0$ small, let ${U^\varepsilon }(t)$ and $S(t)$ be strongly continuous semigroups of linear contractions on a Banach space $L$ with infinitesimal operators $A(\varepsilon )$ and $B$ respectively, where $ A(\varepsilon ) = {A^{(1)}} + {\varepsilon A^{(2)}} + o()$ as $\varepsilon \to 0$. Let $\{ B(u);u \geqslant 0\} $ be a family of linear operators on $L$ satisfying $ B(\varepsilon ) = B + {\varepsilon \Pi ^{(1)}} + {\varepsilon ^2}{\Pi ^{\varepsilon (2)}} + o({\varepsilon ^2})$ as $\varepsilon \to 0$. Assume that $A(\varepsilon ) + {\varepsilon ^{ - 1}}B()$ is the infinitesimal operator of a strongly continuous contraction semigroup ${T_\varepsilon }(t)$ on $L$ and that for each $f \in L,{\lim _{\lambda \to 0}}\lambda \int_0^\infty {{e^{ - \lambda t}}} S(t)fdt \equiv Pf$ exists. We give conditions under which ${T_\varepsilon }(t)$ converges as $\to 0$ to the semigroup generated by the closure of $P({A^{(1)}} + {\Pi ^{(1)}})$ on $\mathcal{R}(P) \cap \mathcal{D}({A^{(1)}}) \cap \mathcal{D}({\Pi ^{(1)}})$. If $P({A^{(1)}} + {\Pi ^{(1)}})f = 0,Bh = - ({A^{(1)}} + {\Pi ^{(1)}})f$, and we let $ \hat Vf = P({A^{(1)}} + {\Pi ^{(1)}})h$, then we show that ${T_\varepsilon }(t/\varepsilon )f$ converges as $ \varepsilon \to 0$ to the strongly continuous contraction semigroup generated by the closure of ${V^{(2)}} + \hat V$. From these results we obtain new limit theorems for discontinuous random evolutions and new characterizations of the limiting infinitesimal operators of the discontinuous random evolutions. We apply these results in a model for the approximation of physical Brownian motion and in a model of the content of an infinite capacity dam.
Asymptotic properties of $U$-statistics
Raymond N.
Sproule
55-64
Abstract: Let $r$ be a fixed positive integer. A $ U$-statistic $ {U_n}$ is an average of a symmetric measurable function of $r$ arguments over a random sample of size $ n$. Such a statistic may be expressed as an average of independent and identically distributed random variables plus a remainder term. We develop a Kolmogorov-like inequality for this remainder term as well as examine some of its (a.s.) convergence properties. We then relate these properties to the $U$-statistic. In addition, the asymptotic normality of ${U_N}$, where $N$ is a positive integer-valued random variable, is established under certain conditions.
Chapman-Enskog-Hilbert expansion for the Ornstein-Uhlenbeck process and the approximation of Brownian motion
Richard S.
Ellis
65-74
Abstract: Let $(x(t),\upsilon (t))$ denote the joint Ornstein-Uhlenbeck position-velocity process. Special solutions of the backward equation of this process are studied by a technique used in statistical mechanics. This leads to a new proof of the fact that, as $\varepsilon \downarrow 0,\varepsilon x(t/{\varepsilon ^2})$ tends weakly to Brownian motion. The same problem is then considered for $ \upsilon (t)$ belonging to a large class of diffusion processes.
A density property and applications
Richard J.
O’Malley
75-87
Abstract: An unexpected metric density property of a certain type of ${F_\sigma }$ set is proven. An immediate application is a characterization of monotone functions similar to a well-known result by Zygmund. Several corollaries of this characterization are given as well as a simple proof of a theorem due to Tolstoff.
The Riemann problem for general $2\times 2$ conservation laws
Tai Ping
Liu
89-112
Abstract: The Riemann Problem for a system of hyperbolic conservation laws of form $\displaystyle (1)\quad \begin{array}{*{20}{c}} {{u_t} + f{{(u,\upsilon )}_x} = 0,} {{\upsilon _t} + g{{(u,\upsilon )}_x} = 0} \end{array}$ with arbitrary initial constant states $\displaystyle (2)\quad ({u_0}(x),{v_0}(x)) = \left\{ {\begin{array}{*{20}{c}} {... ...},{v_l}),\quad x < 0,} {({u_r},{v_r}),\quad x > 0,} \end{array} } \right.$ is considered. We assume that ${f_\upsilon } < 0,{g_u} < 0$. Let ${l_i}({r_i})$ be the left (right) eigenvectors of $dF \equiv d(f,g)$ for eigenvalues ${\lambda _1} < {\lambda _2}$. Instead of assuming the usual convexity condition $d{\lambda _i}({r_i}) \ne 0,i = 1,2$ we assume that $d{\lambda _i}({r_i}) = 0$ on disjoint union of $ 1$-dim manifolds in the $(u,\upsilon )$ plane. Oleinik's condition (E) for single equation is extended to system (1); again call this new condition (E). Our condition (E) implies Lax's shock inequalities and, in case $d{\lambda _i}({r_i}) \ne 0$, the two are equivalent. We then prove that there exists a unique solution to the Riemann Problem (1) and (2) in the class of shocks, rarefaction waves and contact discontinuities which satisfies condition (E).
The oriented bordism of $Z\sb{2\sp{k}}$ actions
E. R.
Wheeler
113-121
Abstract: Let ${R_2}$ be the subring of the rationals given by ${R_2} = Z[1/2]$. It is shown that for $G = {Z_{{2^k}}}$ the bordism group of orientation preserving $G$ actions on oriented manifolds tensored with ${R_2}$ is a free ${\Omega _ \ast } \otimes {R_2}$ module on even dimensional generators (where ${\Omega _ \ast }$ is the oriented bordism ring).
Examples of nonsolvable partial differential equations
Robert
Rubinstein
123-129
Abstract: Two examples of nonsolvable partial differential operators with multiple characteristics are presented. They illustrate the possibility that certain terms in the principal part may play no role in determining the solvability properties of the operator. This situation cannot occur for simple characteristics, where solvability is determined by the principal part.
An isomorphism and isometry theorem for a class of linear functionals
William D. L.
Appling
131-140
Abstract: Suppose $ U$ is a set, ${\mathbf{F}}$ is a field of subsets of $U$ and ${\mathfrak{p}_{AB}}$ is the set of all real-valued, finitely additive functions defined on ${\mathbf{F}}$. Two principal notions are considered in this paper. The first of these is that of a subset of $ {\mathfrak{p}_{AB}}$, defined by certain closure properties and called a $ C$-set. The second is that of a collection $ \mathcal{C}$ of linear transformations from $ {\mathfrak{p}_{AB}}$ into $ {\mathfrak{p}_{AB}}$ with special boundedness properties. Given a $ C$-set $M$ which is a linear space, an isometric isomorphism is established from the dual of $ M$ onto the set of all elements of $ \mathcal{C}$ with range a subset of $M$. As a corollary it is demonstrated that the above-mentioned isomorphism and isometry theorem, together with a previous representation theorem of the author (J. London Math. Soc. 44 (1969), pp. 385-396), imply an analogue of a dual representation theorem of Edwards and Wayment (Trans. Amer. Math. Soc. 154 (1971), pp. 251-265). Finally, a ``pseudo-representation theorem'' for the dual of ${\mathfrak{p}_{AB}}$ is demonstrated.
Generation of analytic semigroups by strongly elliptic operators
H. Bruce
Stewart
141-162
Abstract: Strongly elliptic operators realized under Dirichlet boundary conditions in unbounded domains are shown to generate analytic semigroups in the topology of uniform convergence. This fact is applied to initial-boundary value problems for temporally homogeneous and temporally inhomogeneous parabolic equations.
Semigroups of operators on locally convex spaces
V. A.
Babalola
163-179
Abstract: Let $X$ be a complex Hausdorff locally convex topological linear space and $L(X)$ the family of all continuous linear operators on $X$. This paper discusses the generation and perturbation theory for ${C_0}$ semigroups $\{ S(\xi ):\xi \geqslant 0\} \subset L(X)$ such that for each continuous seminorm $p$ on $X$ there exist a positive number ${\sigma _p}$ and a continuous seminorm $ q$ on $X$ with $p(S(\xi )x) \leqslant {e^{^\sigma {p^\xi }}}q(x)$ for all $ \xi \geqslant 0$ and $ x \in X$. These semigroups are studied by means of a realization of $ X$ as a projective limit of Banach spaces, using certain naturally-defined operators and ${C_0}$ semigroups on these Banach spaces to connect the present results to the classical Hille-Yosida-Phillips theory.
Conjugate points, triangular matrices, and Riccati equations
Zeev
Nehari
181-198
Abstract: Let $A$ be a real continuous $n \times n$ matrix on an interval $\Gamma$, and let the $n$-vector $x$ be a solution of the differential equation $ x' = Ax$ on $\Gamma$. If $[\alpha ,\beta ] \in \Gamma ,\beta$ is called a conjugate point of $\alpha$ if the equation has a nontrivial solution vector $x = ({x_1},{\kern 1pt} \ldots ,{x_n})$ such that $ {x_1}(\alpha ) = \ldots = {x_k}(\alpha ) = {x_{k + 1}}(\beta ) = \ldots = {x_n}(\beta ) = 0$ for some $k \in [1,n - 1]$. It is shown that the absence on $ ({t_1},{t_2})$ of a point conjugate to ${t_1}$ with respect to the equation $x' = Ax$ is equivalent to the existence on $ ({t_1},{t_2})$ of a continuous matrix solution $L$ of the nonlinear differential equation $L({t_1}) = I$, where ${[B]_{{\tau _0}}}$ denotes the matrix obtained from the $n \times n$ matrix $B$ by replacing the elements on and above the main diagonal by zeros. This nonlinear equation--which may be regarded as a generalization of the Riccati equation, to which it reduces for $n = 2$--can be used to derive criteria for the presence or absence of conjugate points on a given interval.
Killing vector fields and harmonic forms
Edward T.
Wright
199-202
Abstract: The paper is concerned with harmonic $(p,q)$-forms on compact Kähler manifolds which admit Killing vector fields with discrete zero sets. Let $ {h^{p,q}}$ denote the dimension of the space of harmonic $(p,q)$-forms. The main theorem states that ${h^{p,q}} = 0,p \ne q$.
A Pr\"ufer transformation for the equation of the vibrating beam
D. O.
Banks;
G. J.
Kurowski
203-222
Abstract: In this paper, the oscillatory properties of the eigenfunctions of an elastically constrained beam are studied. The method is as follows. The eigenfunction and its first three derivatives are considered as a four-dimensional vector, $ (u,u',pu'',(pu'')')$. This vector is projected onto two independent planes and polar coordinates are introduced in each of these two planes. The resulting transformation is then used to study the oscillatory properties of the eigenfunctions and their derivatives in a manner analogous to the use of the Prüfer transformation in the study of second order Sturm-Liouville systems. This analysis yields, for a given set of boundary conditions, the number of zeros of each of the derivatives, $u',pu'',(pu'')'$ and the relation of these zeros to the $n - 1$ zeros of the $n$th eigenfunction. The method also can be used to establish comparison theorems of a given type.
M\"obius transformations of the disc and one-parameter groups of isometries of $H\sp{p}$
Earl
Berkson;
Robert
Kaufman;
Horacio
Porta
223-239
Abstract: Let $\{ {T_t}\}$ be a strongly continuous one-parameter group of isometries in ${H^p}(1 \leqslant p < \infty ,p \ne 2)$ with unbounded generator. There is a uniquely determined one-parameter group $\{ {\phi _t}\} ,t \in {\mathbf{R}}$, of Möbius transformations of the (open) disc $D$ corresponding to $\{ {T_t}\}$ by way of Forelli's theorem. The interplay between $\{ {T_t}\}$ and $\{ {\phi _t}\}$ is studied, and the spectral properties of the generator $A$ of $\{ {T_t}\}$ are analyzed in this context. The nature of the set $S$ of common fixed points of the functions $ {\phi _t}$ plays a crucial role in determining the behavior of $A$. The spectrum of $A$, which is a subset of $i{\mathbf{R}}$, can be a discrete set, a translate of $i{{\mathbf{R}}_ + }$ or of $i{{\mathbf{R}}_ - }$, or all of $i{\mathbf{R}}$. If $S$ is not a doubleton subset of the unit circle, $\{ {T_t}\}$ can be extended to a holomorphic semigroup of ${H^p}$-operators, the semigroup being defined on a half-plane. The treatment of $\{ {T_t}\}$ is facilitated by developing appropriate properties of one-parameter groups of Möbius transformations of $D$. In particular, such groups are in one-to-one correspondence (via an initial-value problem) with the nonzero polynomials $q$, of degree at most 2, such that Re$[\bar zq(z)] = 0$ for all unimodular $ z$. A has an explicit description (in terms of the polynomial corresponding to $\{ {\phi _t}\} $) as a differential operator.
Series inequalities involving convex functions
Christopher O.
Imoru
241-252
Abstract: The object of this paper is to extend some recent generalizations of Petrović's inequality by Vasić and others. We shall also use our technique to obtain some results which have interesting applications in the theory of Fourier series as well as the theory of approximations.
``Image of a Hausdorff arc'' is cyclically extensible and reducible
J. L.
Cornette
253-267
Abstract: It is shown that a Hausdorff continuum $S$ is the continuous image of an arc (respectively arcwise connected) if and only if each cyclic element of $S$ is the continuous image of an arc (respectively, arcwise connected). Also, there is given an analogue to the metric space cyclic chain approximation theorem of G. T. Whyburn which applies to locally connected Hausdorff continua.
On the constructibility of prime characteristic periodic associative and Jordan rings
J. A.
Loustau
269-279
Abstract: The object of this paper is to show that any periodic associative ring of prime characteristic can be embedded in a periodic associative ring of prime characteristic which is constructible from a relatively complemented, distributive lattic and a family of periodic fields. Further, it will be proved that any periodic Jordan ring of prime characteristic is also embeddable in a periodic Jordan ring which is constructible from a lattice of the above type and a family of periodic Jordan rings of a symmetric bilinear form.
The category of generalized Lie groups
Su Shing
Chen;
Richard W.
Yoh
281-294
Abstract: We consider the category $\Gamma$ of generalized Lie groups. A generalized Lie group is a topological group $G$ such that the set $LG = Hom({\mathbf{R}},G)$ of continuous homomorphisms from the reals $ {\mathbf{R}}$ into $ G$ has certain Lie algebra and locally convex topological vector space structures. The full subcategory ${\Gamma ^r}$ of $r$-bounded ($r$ positive real number) generalized Lie groups is shown to be left complete. The class of locally compact groups is contained in $\Gamma$. Various properties of generalized Lie groups $G$ and their locally convex topological Lie algebras $ LG = Hom({\mathbf{R}},G)$ are investigated.
On the asymptotic distribution of eigenvalues for semi-elliptic operators
Akira
Tsutsumi;
Chung Lie
Wang
295-315
Abstract: This paper is focused on the asymptotic distribution of eigenvalues for semielliptic operators under weaker smoothness assumptions on coefficients of operators than those of F. E. Browder [3] and Y. Kannai [8] by applying the method of Maruo-Tanabe [9].
Behnke-Stein theorem for analytic spaces
Alessandro
Silva
317-326
Abstract: The notion of $ q$-Runge pair is extended to reduced complex analytic spaces. A necessary and sufficient condition for a pair of $n$-dimensional analytic spaces to be an $ n$-Runge pair is proved and it is shown that this result extends a Behnke-Stein theorem when $n = 1$. A topological property of $q$-Runge pairs of spaces is also proved.
The unitary representations of the generalized Lorentz groups
Ernest A.
Thieleker
327-367
Abstract: For $n \geqslant 2$, let $G(n)$ denote the two-fold covering group of ${\text{SO} _e}(1,n)$. In case $n \geqslant 3,G(n)$ is isomorphic to $\operatorname{Spin} (1,n)$ and is simply connected. In a previous paper we determined all the irreducible quasi-simple representations of these groups, up to infinitesimal equivalence. The main purpose of the present paper is to determine which of these representations are unitarizable. Thus, with the aid of some results of Harish-Chandra and Nelson we determine all the irreducible unitary representations of $G(n)$, up to unitary equivalence. One by-product of our analysis is the explicit construction of the infinitesimal equivalences, which are known to exist from our previous work, between the various subquotient representations and certain subrepresentations in the nonirreducible cases of the nonunitary principal series representations of $G(n)$.
Interpolating sequences in polydisks
Eric P.
Kronstadt
369-398
Abstract: Let ${D^n}$ be the unit polydisk in ${{\mathbf{C}}^n}$, $A$ be a uniform algebra, ${H^\infty }({D^n},A)$, the space of bounded analytic $A$ valued functions on ${D^n}$, $ {l^\infty }A$, the space of bounded sequences of elements in $A$. A sequence, $ S = \{ {a_i}\} _{i = 1}^\infty \subset {D^n}$ will be called an interpolating sequence with respect to $A$ if the map $T:{H^\infty }({D^n},A) \to {l^\infty }A$ given by $ T(f) = \{ f({a_i})\} _{i = 1}^\infty$ is surjective. In 1958, L. Carleson showed that for $n = 1,S$ is interpolating w.r.t. ${\mathbf{C}}$ iff $S$ satisfies a certain zero-one interpolation property called uniform separation. We generalize this result to cases where $n > 1$ and $ A \ne {\mathbf{C}}$. In particular, we show that if $S \subset {D^n}$ is uniformly separated and $S \subset {W_1} \times {W_2} \times \cdots \times {W_n}$ (where each ${W_j}$ is a region in $D$ lying between two circular arcs which intersect twice on the boundary of $D$) then $S$ is an interpolating sequence w.r.t. any uniform algebra. If $ S \subset {D^n}$ is uniformly separated and $S \subset D \times {W_2} \times \cdots \times {W_n}$ then $S$ is interpolating w.r.t. ${\mathbf{C}}$. Other examples and generalizations of interpolating sequences are discussed.
Smooth complex projective space bundles and $B{\rm U}(n)$
R. Paul
Beem
399-411
Abstract: Smooth fiberings with complex projective and Dold manifold fibers are studied and a bordism classification for even complex projective space bundles is given. The $ {Z_2}$-cohomology of $B\tilde U(n)$ is computed with its Steenrod algebra action.
Semi-isotopies and the lattice of inner ideals of certain quadratic Jordan algebras
Jerome M.
Katz
413-427
Abstract: The concept of isotopy plays an extremely important role in the structure theory of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals. We take Koecher's characterization of isotopy and use it as the basis of a definition of semi-isotopy. It is clear that semi-isotopies induce, in a natural way, automorphisms of the lattice of inner ideals. We concern ourselves with the converse problem; namely, if $\eta$ is a semilinear bijection of a quadratic Jordan algebra such that $\eta$ induces an automorphism of the lattice of inner ideals, is $\eta$ necessarily a semi-isotopy? We answer the above question in the affirmative for a large class of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals (said class includes all such algebras of capacity at least three over fields of characteristic unequal to two). Moreover, we prove that the only such maps which induce the identity automorphism on the lattice are the scalar multiplications.
Erratum to: ``Essential spectrum for a Hilbert space operator'' (Trans. Amer. Math. Soc. {\bf 163} (1972), 437--445)
Richard
Bouldin
429
Irregularities of distribution. VIII
Wolfgang M.
Schmidt
1-22
Abstract: If ${x_1},{x_2} \ldots$ is a sequence in the unit interval $0 \leqslant x \leqslant 1$ and if $S$ is a subinterval, write $C(n,S)$ for the number of elements among ${x_1}, \ldots ,{x_n}$ which lie in $ S$, minus $n$ times the length of $S$. For a well distributed sequence, $ C(n,S)$ as a function of $ n$ will be small. It is shown that the lengths of the intervals $S$ for which $C(n,S)(n = 1,2, \ldots )$ is bounded form at most a countable set.
Consistency theorems for almost convergence
G.
Bennett;
N. J.
Kalton
23-43
Abstract: The concept of almost convergence of a sequence of real or complex numbers was introduced by Lorentz, who developed a very elegant theory. The purpose of the present paper is to continue Lorentz's investigations and obtain consistency theorems for almost convergence; this is achieved by studying certain locally convex topological vector spaces.
Cauchy problems for certain Isaacs-Bellman equations and games of survival
Robert J.
Elliott;
Nigel J.
Kalton
45-72
Abstract: Two person zero sum differential games of survival are considered; these terminate as soon as the trajectory enters a given closed set $F$, at which time a cost or payoff is computed. One controller, or player, chooses his control values to make the payoff as large as possible, the other player chooses his controls to make the payoff as small as possible. A strategy is a function telling a player how to choose his control variable and values of the game are introduced in connection with there being a delay before a player adopts a strategy. It is shown that various values of the differential game satisfy dynamic programming identities or inequalities and these results enable one to show that if the value functions are continuous on the boundary of $ F$ then they are continuous everywhere. To discuss continuity of the values on the boundary of $F$ certain comparison theorems for the values of the game are established. In particular if there are sub- and super-solutions of a related Isaacs-Bellman equation then these provide upper and lower bounds for the appropriate value function. Thus in discussing value functions of a game of survival one is studying solutions of a Cauchy problem for the Isaacs-Bellman equation and there are interesting analogies with certain techniques of classical potential theory.
Ordinary differential operators under Stieltjes boundary conditions
Richard C.
Brown;
Allan M.
Krall
73-92
Abstract: The operator $\int_0^1 {d\nu (t)y(t) = 0}$, is studied in $\mathcal{L}_n^p(0,1),1 \leqslant p < \infty$. It is shown that ${L_p}$ has a dense domain; hence there exists a dual operator $L_q^ +$ operating on $\mathcal{L}_n^q(0,1)$. After finding $L_q^ +$ we show that both $ {L_p}$ and $L_q^ +$ are Fredholm operators. This implies some elementary results concerning the spectrum and states of ${L_p}$. Finally two eigenfunction expansions are derived.
Linear transformations on matrices
D. Ž.
Djoković
93-106
Abstract: The real orthogonal group $O(n)$, the unitary group $U(n)$ and the symplectic group ${\text{sp(}}n{\text{)}}$ are embedded in a standard way in the real vector space of $n \times n$ real, complex and quaternionic matrices, respectively. Let $F$ be a nonsingular real linear transformation of the ambient space of matrices such that $F(G) \subset G$ where $G$ is one of the groups mentioned above. Then we show that either $ F(x) = a\sigma (x)b$ or $ F(x) = a\sigma ({x^\ast })b$ where $a,b \in G$ are fixed, ${x^\ast }$ is the transpose conjugate of the matrix $x$ and $\sigma$ is an automorphism of reals, complexes and quaternions, respectively.
Entire vectors and holomorphic extension of representations
Richard
Penney
107-121
Abstract: Let $G$ be a connected, simply connected real Lie group and let $U$ be a representation of $G$ in a complete, locally convex, topological vector space $\mathcal{J}$. If $G$ is solvable, it can be canonically embedded in its complexification ${G_c}$. A vector $v \in \mathcal{J}$ is said to be entire for $U$ if the map $ g \to {U_g}v$ of $ G$ into $\mathcal{J}$ is holomorphically extendible to ${G_c}$. The space of entire vectors is an invariant subspace of the space of analytic vectors. $ U$ is said to be holomorphically extendible iff the space of entire vectors is dense. In this paper we consider the question of existence of holomorphic extensions We prove Theorem. A unitary representation $ U$ is holomorphically extendible to ${G_C}$ iff $G$ modulo the kernel of $U$ is type $R$ in the sense of Auslander-Moore [1]. In the process of proving the above results, we develop several interesting characterizations of entire vectors which generalize work of Goodman for solvable Lie groups and we prove a conjecture of Nelson concerning the relationship between infinitesimal representations of Lie algebras and representations of the corresponding Lie groups.
On the structure of ideals of the dual algebra of a coalgebra
David E.
Radford
123-137
Abstract: The weak-* topology is seen to play an important role in the study of various finiteness conditions one may place on a coalgebra $C$ and its dual algebra $ {C^ \ast }$. Here we examine the interplay between the topology and the structure of ideals of ${C^ \ast }$. The basic theory has been worked out for the commutative and almost connected cases (see [2]). Our basic tool for reducing to the almost connected case is the classical technique of lifting idempotents. Any orthogonal set of idempotents modulo a closed ideal of $\operatorname{Rad} {C^\ast }$ can be lifted. This technique is particularly effective when $C = {C_1}$. The strongest results we obtain concern ideals of $C_1^ \ast$. Using the properties of idempotents we show that ${C_1} = \Sigma_{x,y} {{C_x}\Lambda {C_y}}$ where ${C_x}$ and ${C_y}$ run over the simple subcoalgebras of $ C$. Our first theorem states that a coalgebra $C$ is locally finite and ${C_0}$ is reflexive if and only if every cofinite ideal of ${C^ \ast }$ contains a finitely generated dense ideal. We show in general that a cofinite ideal $I$ which contains a finitely generated dense ideal is not closed. (In fact either equivalent condition of the theorem does not imply $C$ reflexive.) The preceding statement is true if $C = {C_1}$, or more importantly if $ I \supset \operatorname{Rad} {C^\ast }$ and $ {C^ \ast }/I$ is algebraic. The second theorem characterizes the closure of an ideal with cofinite radical which also contains a finitely generated dense ideal.
The Wedderburn principal theorem for a generalization of alternative algebras
Harry F.
Smith
139-154
Abstract: A generalized alternative ring I is a nonassociative ring $ R$ in which the identities $ (wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x;([w,x],y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z$; and $(x,x,x)$ are identically zero. It is here demonstrated that if $A$ is a finite-dimensional algebra of this type over a field $F$ of characteristic # 2, 3, then $A$ a nilalgebra implies $ A$ is nilpotent. A generalized alternative ring II is a nonassociative ring $ R$ in which the identities $(wx,y,z) + (w,x,[y,z]) - w(x,y,z) - (w,y,z)x$ and $ (x,y,x)$ are identically zero. Let $A$ be a finite-dimensional algebra of this type over a field $F$ of characteristic # 2. Then it is here established that (1) $A$ a nilalgebra implies $A$ is nilpotent; (2) $A$ simple with no nonzero idempotent other than 1 and $F$ algebraically closed imply $A$ itself is a field; and (3) the standard Wedderburn principal theorem is valid for $ A$.
Initial-boundary value problems for hyperbolic systems in regions with corners. II
Stanley
Osher
155-175
Abstract: In the previous paper in this series we obtained conditions equivalent to the validity of certain energy estimates for a general class of hyperbolic systems in regions with corners. In this paper we examine closely the phenomena which occur near the corners if these conditions are violated. These phenomena include: the development of strong singularities (lack of existence), travelling waves which pass unnoticed through the corner (lack of uniqueness), existence and uniqueness if and only if additional conditions are imposed at the corner, and weak solutions which are not strong solutions. We also systematically analyze the conditions for certain important problems. We discuss the physical and computational significance of these results.
Parabolic It\^o equations
Robert
Marcus
177-190
Abstract: A parabolic Itô equation is an equation of the form $\displaystyle (\partial u/\partial t)(t,\omega ) = Lu(t,\omega ) + f(u(t,\omega )) + \alpha (t,\omega ),\quad u(0) = {u_0},{u_0},u \in H.$ $H$ is a Hilbert space with scalar product $u \cdot \upsilon$ and norm $\vert \cdot \vert$. $L$ is a linear time-independent negative-definite operator from $H$ to $H$. $f$ is a Lipschitz continuous operator from $ H$ to $H$. $ \alpha (t,\omega )$ is a white noise process in $H$. Under suitable technical conditions the following results are obtained: I. A unique nonanticipating solution of (1) exists with ${\sup _t}E\{ \vert u{\vert^2}\} < \infty$. II. $u(t,\omega ) = R(t,\omega ) + V(t,\omega )$ where $R(t,\omega )$ is a stationary process and $\displaystyle \mathop {\lim }\limits_{t \to \infty } E\{ \vert V(t,\omega ){\vert^2}\} = 0.$ III. If $L$ is selfadjoint and $f$ is the gradient of a smooth functional then an explicit expression is found for the stationary density of $R(t,\omega )$ on $H$. IV. For the equation $ (\partial u/\partial t)(t,\omega ) = Lu(t,\omega ) + f(u(t,\omega )) + \varepsilon \alpha (t,\omega )$ an asymptotic expansion in $\varepsilon$ is proven which holds uniformly in $ t$.
Recurrent random walk of an infinite particle system
Frank
Spitzer
191-199
Abstract: Let $p(x,y)$ be the transition function for a symmetric irreducible recurrent Markov chain on a countable set $S$. Let ${\eta _t}$ be the infinite particle system on $ S$ moving according to simple exclusion interaction with the one particle motion determined by $p$. Assume that $p$ is such that any two particles moving independently on $S$ will sooner or later meet. Then it is shown that every invariant measure for ${\eta _t}$ is a convex combination of Bernoulli product measures $ {\mu _\alpha }$ on ${\{ 0,1\} ^s}$ with density $ 0 \leqslant \alpha = \mu [\eta (x) = 1] \leqslant 1$. Ergodic theorems are proved concerning the convergence of the system to one of the ${\mu _\alpha }$.
A characterization of the invariant measures for an infinite particle system with interactions. II
Thomas M.
Liggett
201-213
Abstract: Let $p(x,y)$ be the transition function for a symmetric, irreducible Markov chain on the countable set $ S$. Let $\eta (t)$ be the infinite particle system on $S$ with the simple exclusion interaction and one-particle motion determined by $p$. The present author and Spitzer have determined all of the invariant measures of $\eta (t)$, and have obtained ergodic theorems for $\eta (t)$, under two different sets of assumptions. In this paper, these problems are solved in the remaining case.
Oscillation criteria and growth of nonoscillatory solutions of even order ordinary and delay-differential equations
R.
Grimmer
215-228
Abstract: A number of results are presented on oscillation and growth of nonoscillatory solutions of the differential equation $ {x^{(n)}}(t) + f(t,x(t)) = 0$. It is shown that a nonoscillatory solution satisfies a first-order integral inequality while its $ (n - 1)$st derivative satisfies a first-order differential inequality. By applying the comparison principle, results are obtained by analyzing the two associated first-order scalar differential equations. In the last section it is shown that these results can be easily extended to delay-differential equations.
Geometry of Lebesgue-Bochner function spaces---smoothness
I. E.
Leonard;
K.
Sundaresan
229-251
Abstract: There exist real Banach spaces $E$ such that the norm in $E$ is of class $ {C^\infty }$ away from zero; however, for any $p,1 \leqslant p \leqslant \infty$, the norm in the Lebesgue-Bochner function space ${L_p}(E,\mu )$ is not even twice differentiable away from zero. The main objective of this paper is to give a complete determination of the order of differentiability of the norm function in this class of Banach spaces.
On vector measures
J.
Diestel;
B.
Faires
253-271
Abstract: The four sections of this paper treat four different but somewhat related topics in the theory of vector measures. In §1 necessary and sufficient conditions for a Banach space $X$ to have the property that bounded additive $ X$-valued maps on $ \sigma$-algebras be strongly bounded are presented, namely, $X$ can contain no copy of ${l_\infty }$. The next two sections treat the Jordan decomposition for measures with values in $ {L_1}$-spaces on ${c_0}(\Gamma )$ spaces and criteria for integrability of scalar functions with respect to vector measures. In particular, a different proof is presented for a result of D. R. Lewis to the effect that scalar integrability implies integrability is equivalent to noncontainment of ${c_0}$. The final section concerns the Radon-Nikodym theorem for vector measures. A generalization of a result due to E. Leonard and K. Sundaresan is given, namely, if a Banach space $X$ has an equivalent very smooth norm (in particular, a Fréchet differentiable norm) then its dual has the Radon-Nikodym property. Consequently, a $C(\Omega )$ space which is a Grothendieck space (weak-star and weak-sequential convergence in dual coincide) cannot be renormed smoothly. Several open questions are mentioned throughout the paper.
On sums and products of unbounded operators in Hilbert space
M. J. J.
Lennon
273-285
Abstract: The characteristic matrices (in the sense of Stone) of the sum and product of two closed linear operators in Hilbert space are found in terms of the characteristic matrix of each operator. From these, necessary and sufficient conditions for the domain of the sum or product to be dense are found, and a new simple condition for the density of the domain of the sum is proved. The ideas developed are applied to the direct integral decomposition of closed linear operators.
An intermediate theory for a purely inseparable Galois theory
James K.
Deveney
287-295
Abstract: Let $K$ be a finite dimensional purely inseparable modular extension of $F$, and let $L$ be an intermediate field. This paper is concerned with an intermediate theory for the Galois theory of purely inseparable extensions using higher derivations [4]. If $L$ is a Galois intermediate field and $ M$ is the field of constants of all higher derivations on $L$ over $F$, we prove that every higher derivation on $ L$ over $F$ extends to $K$ if and only if $K = L{ \otimes _M}J$ for some field $J$. Similar to classical Galois theory the distinguished intermediate fields are those which are left invariant under a standard generating set for the group of all rank $t$ higher derivations on $ K$ over $F$. We prove: $L$ is distinguished if and only if $L$ is $M$-homogeneous (4.9).
Majorization-subordination theorems for locally univalent functions. III
Douglas Michael
Campbell
297-306
Abstract: A quantitative majorization-subordination result of Goluzin and Tao Shah for univalent functions is generalized to ${\mathfrak{n}_\alpha }$, the linear invariant family of locally univalent functions of finite order $ \alpha$. If $ f(z)$ is subordinate to $ F(z)$ in the open unit disc, ${\mathfrak{n}_\alpha },1.65 \leqslant \alpha < \infty$, then $f'(z)$ is majorized by $F'(z)$ in $ \vert z\vert \leqslant (\alpha + 1) - {({\alpha ^2} + 2\alpha )^{1/2}}$. The result is sharp.
Laplace operators and the $\mathfrak{h}$ module structure of certain cohomology groups
Floyd L.
Williams
1-57
Abstract: Let $\mathfrak{n}$ be the maximal nilpotent ideal of a Borel subalgebra of a complex semisimple Lie algebra $\mathfrak{g}$. Under the adjoint action $ \mathfrak{n},\mathfrak{g}/\mathfrak{n}$, and $ \mathfrak{n}$) are $\mathfrak{n}$ modules. Laplace operators for these three modules are computed by techniques which extend those introduced by B. Kostant in [6]. The kernels of these operators are then determined and, in view of the existence of a Hodge decomposition, the detailed structure of the first degree cohomology groups of $\mathfrak{n}$ with coefficients in $ \mathfrak{n},\mathfrak{g}/\mathfrak{n}$, and $\mathfrak{h}$ of $\mathfrak{g}$.
Periodic solutions of $x'' + g(x) + \mu h(x) = 0$
G. J.
Butler;
H. I.
Freedman
59-74
Abstract: Necessary and sufficient conditions for $ x'' + f(x) = 0$ to admit at least one nontrivial periodic solution are given. The results are applied to
Modules over quadratic and quaternion rings and transformations of quadratic forms
Bart
Rice
75-86
Abstract: A study is made of transformations carrying certain quadratic and quaternary quadratic forms into multiples of themselves, and it is shown how these are related to the study of modules over quadratic and quaternion rings. Special automorphic transformations of n-ary quadratic forms may also exhibit a structure like those in the quadratic and quaternary cases.
Finite groups with a proper $2$-generated core
Michael
Aschbacher
87-112
Abstract: H. Bender's classification of finite groups with a strongly embedded subgroup is an important tool in the study of finite simple groups. This paper proves two theorems which classify finite groups containing subgroups with similar but somewhat weaker embedding properties. The first theorem, classifying the groups of the title, is useful in connection with signalizer functor theory. The second theorem classifies a certain subclass of the class of finite groups possessing a permutation representation in which some involution fixes a unique point.
Formal groups and Hopf algebras over discrete rings
Robert A.
Morris;
Bodo
Pareigis
113-129
Abstract: A theory of formal schemes and groups over abitrary rings is presented. The flat formal schemes in this theory have coalgebras of distributions which behave in the usual way. Frobenius and Verschiebung maps are studied.
Subspaces of the nonstandard hull of a normed space
C. Ward
Henson;
L. C.
Moore
131-143
Abstract: Normed spaces which are isomorphic to subspaces of the nonstandard hull of a given normed space are characterized. As a consequence it is shown that a normed space is B-convex if and only if the nonstandard hull contains no subspace isomorphic to ${l_1}$ and a Banach space is super-reflexive if and only if the nonstandard hull is reflexive. Also, embeddings of second dual spaces into the nonstandard hull are studied. In particular, it is shown that the second dual space of a normed space E is isometric to a complemented subspace of the nonstandard hull of E.
Symplectic homogeneous spaces
Bon Yao
Chu
145-159
Abstract: It is proved in this paper that for a given simply connected Lie group G with Lie algebra $ \mathfrak{g}$, every left-invariant closed 2-form induces a symplectic homogeneous space. This fact generalizes the results in [7] and [12] that if ${H^1}(\mathfrak{g}) = {H^2}(\mathfrak{g}) = 0$, then the most general symplectic homogeneous space covers an orbit in the dual of the Lie algebra $\mathfrak{g}$. A one-to-one correspondence can be established between the orbit space of equivalent classes of 2-cocycles of a given Lie algebra and the set of equivalent classes of simply connected symplectic homogeneous spaces of the Lie group. Lie groups with left-invariant symplectic structure cannot be semisimple; hence such groups of dimension four have to be solvable, and connected unimodular groups with left-invariant symplectic structure are solvable [4].
Sufficient sets for some spaces of entire functions
Dennis M.
Schneider
161-180
Abstract: B. A. Taylor [13] has shown that the lattice points in the plane form a sufficient set for the space of entire functions of order less than two. We obtain a generalization of this result to functions of several variables and to more general spaces of entire functions. For example, we prove that if $S \subset {{\mathbf{C}}^n}$ such that $d(z,S) \leq \operatorname{const}\vert z{\vert^{1 - \rho /2}}$ for all $z \in {{\mathbf{C}}^n}$, then S is a sufficient set for the space of entire functions on ${{\mathbf{C}}^n}$ of order less than $ \rho$. The proof involves estimating the growth rate of an entire function from its growth rate on S. We also introduce the concept of a weakly sufficient set and obtain sufficient conditions for a set to be weakly sufficient. We prove that sufficient sets are weakly sufficient and that certain types of effective sets [8] are weakly sufficient.
Polyanalytic functions with exceptional values
P.
Krajkiewicz
181-210
Abstract: Let $ f(z) = \sum\nolimits_{k = 0}^n {{{\bar z}^k}{f_k}(z)} $ where the functions ${f_0},{f_1}, \cdots ,{f_n}$ are analytic on some annular neighborhood A of the point $\infty$ and $ {f_n} \equiv 1$ on A and z denotes the complex conjugate of z. If f does not vanish on A it is shown that the functions ${f_0},{f_1}, \cdots ,{f_{n - 1}}$ have a nonessential isolated singularity at the point infinity.
Spectrum and direct integral
Edward A.
Azoff
211-223
Abstract: Let $T = \smallint _Z^ \oplus T(\mathcal{E})$ be a direct integral of Hilbert space operators, and equip the collection $\mathcal{C}$ of compact subsets of C with the Hausdorff metric topology. Consider the [set-valued] function sp which associates with each $\mathcal{E} \in Z$ the spectrum of $T(\mathcal{E})$. The main theorem of this paper states that sp is measurable. The relationship between $\sigma (T)$ and $\{ \sigma (T(\mathcal{E}))\}$ is also examined, and the results applied to the hyperinvariant subspace problem. In particular, it is proved that if $ \sigma (T(\mathcal{E}))$ consists entirely of point spectrum for each $\mathcal{E} \in Z$, then either T is a scalar multiple of the identity or T has a hyperinvariant subspace; this generalizes a theorem due to T. Hoover.
Compact sets definable by free $3$-manifolds
W. H.
Row
225-244
Abstract: Shape conditions are given that force a compactum (i.e., a compact metric space) embedded in the interior of a nonclosed, piecewise-linear 3-manifold to have arbitrarily close, compact, polyhedral neighborhoods each component of which is a 3-manifold with free fundamental group (i.e., to be definable by free 3-manifolds). For compact, connected ANR's these conditions reduce to the criterion of having a free fundamental group. Additional conditions are given that insure definability by handlebodies or cubes-with-handles. An embedding of Menger's universal 1-dimensional curve in Euclidean 3-space is shown to have the property that all tame surfaces, separating in 3-space a fixed pair of points, cannot be adjusted (by a small space homeomorphism) to intersect the embedded curve in a 0-dimensional set.
Nonattainability of a set by a diffusion process
Avner
Friedman
245-271
Abstract: Consider a system of n stochastic differential equations $ d\xi = b(\xi )dt + \sigma (\xi )dw$. Let M be a k-dimensional submanifold in $ {R^n},k \leq n - 1$. For $x \in M$, denote by $ d(x)$ the rank of $\sigma {\sigma ^ \ast }$ restricted to the linear space of all normals to M at x. It is proved that if $ d(x) \geq 2$ for all $x \in M$, then $ \xi (t)$ does not hit M at finite time, given $\xi (0) \notin M$, i.e., M is nonattainable. The cases $ d(x) \geq 1,d(x) \geq 0$ are also studied.
Constructions in algebra
A.
Seidenberg
273-313
Abstract: It is shown how to construct a primary decomposition and to find the associated prime ideals of a given ideal in a polynomial ring. This is first done from a classical, and then from a strictly constructivist, point of view.
Commutative twisted group algebras
Harvey A.
Smith
315-326
Abstract: A twisted group algebra $ {L^1}(A,G;T,\alpha )$ is commutative iff A and G are, T is trivial and $\alpha$ is symmetric: $\alpha (\gamma ,g) = \alpha (g,\gamma )$. The maximal ideal space ${L^1}(A,G;\alpha )\hat \emptyset$ of a commutative twisted group algebra is a principal $G\hat \emptyset$ bundle over $A\hat \emptyset$. A class of principal $G\hat \emptyset$ bundles over second countable locally compact M is defined which is in 1-1 correspondence with the (isomorphism classes of) ${C_\infty }(M)$-valued commutative twisted group algebras on G. If G is finite only locally trivial bundles can be such duals, but in general the duals need not be locally trivial.
Wreath products and existentially complete solvable groups
D.
Saracino
327-339
Abstract: It is known that the theory of abelian groups has a model companion but that the theory of groups does not. We show that for any fixed $n \geq 2$ the theory of groups solvable of length $ \leq n$ has no model companion. For the metabelian case $(n = 2)$ we prove the stronger result that the classes of finitely generic, infinitely generic, and existentially complete metabelian groups are all distinct. We also give some algebraic results on existentially complete metabelian groups.
General theory of the factorization of ordinary linear differential operators
Anton
Zettl
341-353
Abstract: The problem of factoring the general ordinary linear differential operator $Ly = {y^{(n)}} + {p_{n - 1}}{y^{(n - 1)}} + \cdots + {p_0}y$ into products of lower order factors is studied. The factors are characterized completely in terms of solutions of the equation $Ly = 0$ and its adjoint equation ${L^ \ast }y = 0$. The special case when L is formally selfadjoint of order $n = 2k$ and the factors are of order k and adjoint to each other reduces to a well-known result of Rellich and Heinz: $L = {Q^ \ast }Q$ if and only if there exist solutions ${y_1}, \cdots ,{y_k}$ of $Ly = 0$ satisfying $W({y_1}, \cdots ,{y_k}) \ne 0$ and $[{y_i};{y_j}] = 0$ for $i,j = 1, \cdots ,k$; where [ ; ] is the Lagrange bilinear form of L.
Coherent extensions and relational algebras
Marta C.
Bunge
355-390
Abstract: The notion of a lax adjoint to a 2-functor is introduced and some aspects of it are investigated, such as an equivalent definition and a corresponding theory of monads. This notion is weaker than the notion of a 2-adjoint (Gray) and may be obtained from the latter by weakening that of 2functor and replacing the adjointness equations by adding 2-cells satisfying coherence conditions. Lax monads are induced by and resolve into lax adjoint pairs, the latter via 2-categories of lax algebras. Lax algebras generalize the relational algebras of Barr in the sense that a relational algebra for a monad in Sets is precisely a lax algebra for the lax monad induced in Rel. Similar considerations allow us to recover the T-categories of Burroni as well. These are all examples of lax adjoints of the ``normalized'' sort and the universal property they satisfy can be expressed by the requirement that certain generalized Kan extensions exist and are coherent. The most important example of relational algebras, i.e., topological spaces, is analysed in this new light also with the purpose of providing a simple illustration of our somewhat involved constructions.
Prime and search computability, characterized as definability in certain sublanguages of constructible $L\sb{\omega }{}\sb{1,\omega }$
Carl E.
Gordon
391-407
Abstract: The prime computable (respectively, search computable) relations of an arbitrary mathematical structure are shown to be those relations R such that both R and its complement are definable by disjunctions of recursively enumerable sets of quantifier free (respectively, existential) formulas of the first order language for the structure. The prime and search computable functions are also characterized in terms of recursive sequences of terms and formulas of this language.
Complementation for right ideals in generalized Hilbert algebras
John
Phillips
409-417
Abstract: Let $\mathfrak{A}$ be a generalized Hilbert algebra and let $ \mathcal{J}$ be a closed right ideal of $ \mathfrak{A}$. Let ${\mathcal{J}^ \bot }$ denote the pre-Hilbert space orthogonal complement of $\mathcal{J}$ in $ \mathfrak{A}$. The problem investigated in this paper is: for which algebras $\mathfrak{A}$ is it true that $ \mathfrak{A} = \mathcal{J} \oplus {\mathcal{J}^ \bot }$ for every closed right ideal $ \mathcal{J}$ of $\mathfrak{A}$? In the case that $\mathfrak{A}$ is achieved, a slightly stronger property is characterized and these characterizations are then used to investigate some interesting examples.
The homotopy type of the space of diffeomorphisms. I
Dan
Burghelea;
Richard
Lashof
1-36
Abstract: A new proof is given of the unpublished results of Morlet on the relation between the homeomorphism group and the diffeomorphism group of a smooth manifold. In particular, the result $ {\operatorname{Diff}}({D^n},\partial ) \simeq {\Omega ^{n + 1}}({\text{Top}_n}/{O_n})$ is obtained. The main technique is fibrewise smoothing.
The homotopy type of the space of diffeomorphisms. II
Dan
Burghelea;
Richard
Lashof
37-50
Abstract: The result (proved in Part I) that $ {\operatorname{Diff}}({D^n},\partial ) \simeq {\Omega ^{n + 1}}({\text{PL}_n}/{O_n})$ is used to compute some new homotopy of $ {\operatorname{Diff}}({D^n},\partial {D^n})$. The relation between smooth and PL pseudo-isotopy is explored. Known and new results on the homotopy of $ {\text{PL}_n}$ are summarized.
Mayer-Vietoris sequences and Brauer groups of nonnormal domains
L. N.
Childs
51-67
Abstract: Let R be a Noetherian domain with finite integral closure $ \bar R$. We study the map from the Brauer group of $R,B(R)$, to $B(\bar R)$: first, by embedding $B(R)$ into the Čech etale cohomology group ${H^2}(R,U)$ and using a Mayer-Vietoris sequence for Čech cohomology of commutative rings; second, via Milnor's theorem from algebraic K-theory. We apply our results to show, i.e., that if R is a domain with quotient field K a global field, then the map from $B(R)$ to $B(K)$ is 1-1.
Location of the zeros of polynomials with a prescribed norm
Q. I.
Rahman;
G.
Schmeisser
69-78
Abstract: For monic polynomials ${f_n}(z)$ of degree n with prescribed $ {L^p}$ norm $(1 \leq p \leq \infty )$ on the unit circle or supremum norm on the unit interval we determine bounded regions in the complex plane containing at least $k(1 \leq k \leq n)$ zeros. We deduce our results from some new inequalities which are similar to an inequality of Vicente Gonçalves and relate the zeros of a polynomial to its norm.
The inertial aspects of Stein's condition $H-C\sp{\ast} HC\gg O$
Bryan E.
Cain
79-91
Abstract: To each bounded operator C on the complex Hilbert space $\mathcal{H}$ we associate the vector space ${\mathcal{K}_C}$ consisting of those $x \in \mathcal{H}$ for which ${C^n}x \to 0$ as $n \to \infty$. We let $\alpha (C)$ denote the dimension of the closure of $ {\mathcal{K}_C}$ and we set $\beta (C) = \dim (\mathcal{K}_C^ \bot )$. Our main theorem states that if H is Hermitian and if $H - {C^ \ast }HC$ is positive and invertible then $\alpha (C) \leq \pi (H),\beta (C) = \nu (H)$, and $ \beta (C) \geq \delta (H)$ where $(\pi (H),\nu (H),\delta (H))$ is the inertia of H. (That is, $\pi (H) = \dim \;({\text{Range}}\;E[(0,\infty )])$) where E is the spectral measure of H; $ \nu (H) = \pi ( - H)$; and $\delta (H) = \dim ({\operatorname{Ker}}\;H)$.) We also show (l) that in general no stronger conclusion is possible, (2) that, unlike previous inertia theorems, our theorem allows 1 to lie in $\sigma (C)$, the spectrum of C, and (3) that the main inertial results associated with the hypothesis that $ \operatorname{Re} (HA)$ is positive and invertible can be derived from our theorem. Our theorems (1) characterize C in the extreme cases that either $ \pi (H) = 0$ or $\nu (H) = 0$, and (2) prove that $ \alpha (C) = \pi (H),\beta (C) = \nu (H),\delta (H) = 0$ if either $1 \notin \sigma (C)$ or $\beta (C) < \infty $.
Any infinite-dimensional Fr\'echet space homeomorphic with its countable product is topologically a Hilbert space
Wesley E.
Terry
93-104
Abstract: In this paper we will prove that any infinite-dimensional Fréchet space homeomorphic with its own countable product is topologically a Hilbert space. This will be done in two parts. First we will prove the result for infinite-dimensional Banach spaces, and then we will show that the result for Fréchet spaces follows as a corollary.
On the completion of Hausdorff locally solid Riesz spaces
Charalambos D.
Aliprantis
105-125
Abstract: In this paper we consider Hausdorff locally solid Riesz spaces $ (L,\tau )$ and we denote by $(\hat L,\hat \tau )$ the Hausdorff topological completion of $(L,\tau )$. It is proved that $(\hat L,\hat \tau )$ is a Hausdorff locally solid Riesz space containing L as a Riesz subspace. We study the properties of $(L,\tau )$ which are inherited by $(\hat L,\hat \tau )$.
Maximal quotients of semiprime PI-algebras
Louis Halle
Rowen
127-135
Abstract: J. Fisher [3] initiated the study of maximal quotient rings of semiprime PI-rings by noting that the singular ideal of any semiprime Pi-ring R is 0; hence there is a von Neumann regular maximal quotient ring $Q(R)$ of R. In this paper we characterize $Q(R)$ in terms of essential ideals of C = cent R. This permits immediate reduction of many facets of $Q(R)$ to the commutative case, yielding some new results and some rapid proofs of known results. Direct product decompositions of $Q(R)$ are given, and $Q(R)$ turns out to have an involution when R has an involution.
Involutions preserving an ${\rm SU}$ structure
R. J.
Rowlett
137-147
Abstract: Bordism theories $ S{U_ \ast }({Z_2},all)$ for SU-manifolds with involution and $S{U_ \ast }({Z_2},free)$ for SU-manifolds with free involution are defined. The latter is studied by use of the SU-bordism spectral sequence of $B{Z_2}$, and the orders of the spheres ${S^{4n + 3}}$ with antipodal action are determined. It is shown that $ S{U_{2k}}({Z_2},free) \to S{U_{2k}}({Z_2},all)$ is monic, and that an element of $ S{U_{2k}}({Z_2},all)$ bounds as a unitary involution if and only if it is a multiple of the nonzero class $\alpha \in S{U_1}$.
Algebras over absolutely flat commutative rings
Joseph A.
Wehlen
149-160
Abstract: Let A be a finitely generated algebra over an absolutely flat commutative ring. Using sheaf-theoretic techniques, it is shown that the weak Hochschild dimension of A is equal to the supremum of the Hochschild dimension of ${A_x}$ for x in the decomposition space of R. Using this fact, relations are obtained among the weak Hochschild dimension of A and the weak global dimensions of A and ${A^e}$. It is also shown that a central separable algebra is a biregular ring which is finitely generated over its center. A result of S. Eilenberg concerning the separability of A modulo its Jacobson radical is extended. Finally, it is shown that every homomorphic image of an algebra of weak Hochschild dimension 1 is a type of triangular matrix algebra.
Fixed point iterations using infinite matrices
B. E.
Rhoades
161-176
Abstract: Let E be a closed, bounded, convex subset of a Banach space $X,f:E \to E$. Consider the iteration scheme defined by ${\bar x_0} = {x_0} \in E,{\bar x_{n + 1}} = f({x_n}),{x_n} = \Sigma _{k = 0}^n{a_{nk}}{\bar x_k},\;n \geq 1$, where A is a regular weighted mean matrix. For particular spaces X and functions f we show that this iterative scheme converges to a fixed point of f.
Products of initially $m$-compact spaces
R. M.
Stephenson;
J. E.
Vaughan
177-189
Abstract: The main purpose of this paper is to give several theorems and examples which we hope will be of use in the solution of the following problem. For an infinite cardinal number $\mathfrak{m}$, is initial $ \mathfrak{m}$-compactness preserved by products? We also give some results concerning properties of Stone-Čech compactifications of discrete spaces.
New criteria for freeness in abelian groups. II
Paul
Hill
191-201
Abstract: A new criterion is established for an abelian group to be free. The criterion is in terms of an ascending chain of free subgroups and is dependent upon a new class of torsion-free groups. The result leads to the construction, for each positive integer n, of a group ${G_n}$ of cardinality ${\aleph _n}$ that is not free but is ${\aleph _n}$-free. A conjecture in infinitary logic concerning free abelian groups is also verified.
The slimmest geometric lattices
Thomas A.
Dowling;
Richard M.
Wilson
203-215
Abstract: The Whitney numbers of a finite geometric lattice L of rank r are the numbers ${W_k}$ of elements of rank k and the coefficients ${w_k}$ of the characteristic polynomial of L, for $0 \leq k \leq r$. We establish the following lower bounds for the $ {W_k}$ and the absolute values $ w_k^ + = {( - 1)^k}{w_k}$ and describe the lattices for which equality holds (nontrivially) in each case: $\displaystyle {W_k} \geq \left( {\begin{array}{*{20}{c}} r & - & ... ... \left( {\begin{array}{*{20}{c}} r k \end{array} } \right),$ where $n = {W_1}$ is the number of points of L.
On certain convex sets in the space of locally schlicht functions
Y. J.
Kim;
E. P.
Merkes
217-224
Abstract: Let $H = H{(^ \ast },[ + ])$ denote the real linear space of locally schlicht normalized functions in $ \vert z\vert < 1$ as defined by Hornich. Let K and C respectively be the classes of convex functions and the close-to-convex functions. If $M \subset H$ there is a closed nonempty convex set in the $\alpha \beta$-plane such that for $(\alpha ,\beta )$ in this set ${\alpha ^ \ast }f[ + ]{\beta ^ \ast }g \in C$ (in K) whenever f, $g \in M$. This planar convex set is explicitly given when M is the class K, the class C, and for other classes. Some consequences of these results are that K and C are convex sets in H and that the extreme points of C are not in K.
On the zeros of Dirichlet $L$-functions. I
Akio
Fujii
225-235
Abstract: A mean value theorem for $ \arg \;L({\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.... ...e 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}} + it,\chi )$ is established. This yields mean estimates for the number of zeros of $L(s,\chi )$ in small boxes.
Locally $e$-fine measurable spaces
Zdeněk
Frolík
237-247
Abstract: Hyper-Baire sets and hyper-cozero sets in a uniform space are introduced, and it is shown that for metric-fine spaces the property ``every hypercozero set is a cozero set'' is equivalent to several much stronger properties like being locally e-fine (defined in §1), or having locally determined precompact part (introduced in §2). The metric-fine spaces with these additional properties form a coreflective subcategory of uniform spaces; the coreflection is explicitly described. The theory is applied to measurable uniform spaces. It is shown that measurable spaces with the additional properties mentioned above are coreflective and the coreflection is explicitly described. The two coreflections are not metrically determined.
An internal characterization of paracompact $p$-spaces
R. A.
Stoltenberg
249-263
Abstract: The purpose of this paper is to characterize paracompact p-spaces in terms of spaces with refining sequences $ \bmod\;k$. A space X has a refining sequence $\bmod\;k$ if there exists a sequence $\{ {\mathcal{G}_n}\vert n \in N\}$ of open covers for X such that $\cap _{n = 1}^\infty {\text{St}}(C,{\mathcal{G}_n}) = P_C^1$ is compact for each compact subset C of X and ${\text{\{ St}}{(C,{\mathcal{G}_n})^ - }\vert n \in N\}$ is a neighborhood base for $ P_C^1$. If $P_C^1 = C$ for each compact subset C of X then X is metrizable. On the other hand if we restrict the set C to the family of finite subsets of X in the above definition then we have a characterization for strict p-spaces. Moreover, in this case, if $P_C^1 = C$ for all such sets then X is developable. Thus the concept of a refining sequence $ \bmod\;k$ is natural and it is helpful in understanding paracompact p-spaces.
Analytic equivalence among simply connected domains in $C(X)$
Hugh E.
Warren
265-288
Abstract: This work considers analytic equivalence within the analytic function theory for commutative Banach algebras which was introduced by E. R. Lorch. Necessary conditions of a geometric nature are given for simply connected domains in $ C(X)$. These show that there are a great many equivalence classes. In some important cases, as when one domain is the unit ball, the given conditions are also sufficient. The main technique is the association of a simply connected domain in $C(X)$ with a family of Riemann surfaces over the plane.
$\omega $-linear vector fields on manifolds
William
Perrizo
289-312
Abstract: The classical study of a flow near a fixed point is generalized by composing, at each point in the manifold, the flow derivative with a parallel translation back along the flow. Circumstances under which these compositions form a one-parameter group are studied. From the point of view of the linear frame bundle, the condition is that the canonical lift commute with its horizontal part (with respect to some metric connection). The connection form applied to the lift coincides with the infinitesimal generator of the one-parameter group. Analysis of this matrix provides dynamical information about the flow. For example, if such flows are equicontinuous, they have uniformly bounded derivatives and therefore the enveloping semigroup is a Lie transformation group. Subclasses of ergodic, minimal, and weakly mixing flows with integral invariants are determined according to the eigenvalues of the matrices. Such examples as Lie algebra flows, infinitesimal affine transformations, and the geodesic flows on manifolds of constant negative curvature are examined.
Global dimension of tiled orders over a discrete valuation ring
Vasanti A.
Jategaonkar
313-330
Abstract: Let R be a discrete valuation ring with maximal ideal $\mathfrak{m}$ and the quotient field K. Let $\Lambda = ({\mathfrak{m}^{{\lambda _{ij}}}}) \subseteq {M_n}(K)$ be a tiled R-order, where $ {\lambda _{ij}} \in {\mathbf{Z}}$ and $ {\lambda _{ii}} = 0$ for $1 \leq i \leq n$. The following results are proved. Theorem 1. There are, up to conjugation, only finitely many tiled R-orders in $ {M_n}(K)$ of finite global dimension. Theorem 2. Tiled R-orders in ${M_n}(K)$ of finite global dimension satisfy DCC. Theorem 3. Let $\Lambda \subseteq {M_n}(R)$ and let $ \Gamma$ be obtained from $\Lambda$ by replacing the entries above the main diagonal by arbitrary entries from R. If $\Gamma$ is a ring and if gl $\dim \;\Lambda < \infty $, then gl $\dim \;\Gamma < \infty$. Theorem 4. Let $\Lambda$ be a tiled R-order in $ {M_4}(K)$. Then gl $\dim \;\Lambda < \infty$ if and only if $\Lambda$ is conjugate to a triangular tiled R-order of finite global dimension or is conjugate to the tiled R-order $ \Gamma = ({\mathfrak{m}^{{\lambda _{ij}}}}) \subseteq {M_4}(R)$, where $ {\gamma _{ii}} = {\gamma _{1i}} = 0$ for all i, and ${\gamma _{ij}} = 1$ otherwise.
${\rm SU}(n)$ actions on differentiable manifolds with vanishing first and second integral Pontrjagin classes
Edward A.
Grove
331-350
Abstract: In this paper we determine the connected component of the identity of the isotropy subgroups of a given action of ${\text{SU}}(n)$ on a connected manifold whose first and second integral Pontrjagin classes are zero and whose dimension is less than ${n^2} - 8n/3 - 1$.
Groups, semilattices and inverse semigroups. II
D. B.
McAlister
351-370
Abstract: An inverse semigroup is called proper if the equations $ae = e = {e^2}$ together imply ${a^2} = a$. In a previous paper, with the same title, the author proved that every inverse semigroup is an idempotent separating homomorphic image of a proper inverse semigroup. In this paper a structure theorem is given for all proper inverse semigroups in terms of partially ordered sets and groups acting on them by order automorphisms. As a consequence of these two theorems, and Preston's construction for idempotent separating congruences on inverse semigroups, one can give a structure theorem for all inverse semigroups in terms of groups and partially ordered sets.
An induction principle for spectral and rearrangement inequalitities
Kong Ming
Chong
371-383
Abstract: In this paper, expressions of the form $f \prec g$ or $f \prec \prec g$ (where $ \prec$ and $\prec \prec$ denote the Hardy-Littlewood-Pólya spectral order relations) are called spectral inequalities. Here a general induction principle for spectral and rearrangement inequalities involving a pair of n-tuples in ${R^n}$ as well as their decreasing and increasing rearrangements is developed. This induction principle proves that such spectral or rearrangement inequalities hold iff they hold for the case when $n = 2$, and that, under some mild conditions, this discrete result can be generalized to include measurable functions with integrable positive parts. A similar induction principle for spectral and rearrangement inequalities involving more than two measurable functions is also established. With this induction principle, some well-known spectral or rearrangement inequalities are obtained as particular cases and additional new results given.
Axisymmetric harmonic interpolation polynomials in ${\bf R}\sp{N}$
Morris
Marden
385-402
Abstract: Corresponding to a given function $ F(x,\rho )$ which is axisymnetric harmonic in an axisymmetric region $ \Omega \subset {{\text{R}}^3}$ and to a set of $n + 1$ circles ${C_n}$ in an axisymmetric subregion $A \subset \Omega$, an axisymmetric harmonic polynomial $ {\Lambda _n}(x,\rho ;{C_n})$ is found which on the ${C_n}$ interpolates to $ F(x,\rho )$ or to its partial derivatives with respect to x. An axisymmetric subregion $B \subset \Omega$ is found such that ${\Lambda _n}(x,\rho ;{C_n})$ converges uniformly to $F(x,\rho )$ on the closure of B. Also a ${\Lambda _n}(x,\rho ;{x_0},{\rho _0})$ is determined which, together with its first n partial derivatives with respect to x, coincides with $F(x,\rho )$ on a single circle $({x_0},{\rho _0})$ in $\Omega$ and converges uniformly to $F(x,\rho )$ in a closed torus with $({x_0},{\rho _0})$ as central circle.
Density of parts of algebras on the plane
Anthony G.
O’Farrell
403-414
Abstract: We study the Gleason parts of a uniform algebra A on a compact subset of the plane, where it is assumed that for each point $x \in {\text{C}}$ the functions in A which are analytic in a neighborhood of x are uniformly dense in A. We prove that a part neighborhood N of a nonpeak point x for A satisfies a density condition of Wiener type at $x:\Sigma _{n = 1}^{ + \infty }{2^n}C({A_n}(x)\backslash N) < + \infty$, and if A admits a pth order bounded point derivation at x, then N satisfies a stronger density condition: $\Sigma _{n = 1}^{ + \infty }{2^{(p + 1)n}}C({A_n}(x)\backslash N) < + \infty$. Here C is Newtonian capacity and ${A_n}(x)$ is $ \{ z \in {\text{C}}:{2^{ - n - 1}} \leq \vert z - x\vert \leq {2^{ - n}}\}$. These results strengthen and extend Browder's metric density theorem. The relation with potential theory is examined, and analogous results for the algebra ${H^\infty }(U)$ are obtained as corollaries.
Analytic capacity, H\"older conditions and $\tau $-spikes
Anthony G.
O’Farrell
415-424
Abstract: We consider the uniform algebra $R(X)$, for compact $X \subset {\text{C}}$, in relation to the condition ${I_{p + \alpha }} = \Sigma _1^\infty {2^{(p + \alpha + 1)n}}\gamma ({A_n}(x)\backslash X) < + \infty$, where $0 \leq p \in {\mathbf{Z}},0 < \alpha < 1,\gamma$ is analytic capacity, and ${A_n}(x)$ is the annulus $\{ z \in {\text{C}}:{2^{ - n - 1}} < \vert z - x\vert < {2^{ - n}}\}$. We introduce the notion of $ \tau$-spike for $ \tau > 0$, and show that $ {I_{p + \alpha }} = + \infty$ implies x is a $ p + \alpha$-spike. If $\mathop X\limits^ \circ$ satisfies a cone condition at x, and ${I_{p + \alpha }} < + \infty$, we show that the pth derivatives of the functions in $R(X)$ satisfy a uniform Hölder condition at x for nontangential approach. The structure of the set of non-$\tau$-spikes is examined and the results are applied to rational approximation. A geometric question is settled.
Unitary measures on LCA groups
Lawrence
Corwin
425-430
Abstract: A unitary measure on a locally compact Abelian (LCA) group G is a complex measure whose Fourier transform is of absolute value 1 everywhere. The problem of finding all such measures is known to be closely related to that of finding all invertible measures on G. In this paper, we find all unitary measures when G is the circle or a discrete group. If G is a torsion-free discrete group, the characterization generalizes a theorem of Bohr.
Generalized hypercomplex function theory
Robert P.
Gilbert;
Gerald
Hile
1-29
Abstract: Lipman Bers and Ilya Vekua extended the concept of an analytic function by considering the distributional solutions of elliptic systems of two equations with two unknowns and two independent variables. These solutions have come to be known as generalized (or pseudo) analytic functions. Subsequently, Avron Douglis introduced an algebra and a class of functions which satisfy (classically) the principal part of an elliptic system of 2r equations with 2r unknowns and two independent variables. In Douglis' algebra these systems of equations can be represented by a single ``hypercomplex'' equation. Solutions of such equations are termed hyperanalytic functions. In this work, the class of functions studied by Douglis is extended in a distributional sense much in the same way as Bers and Vekua extended the analytic functions. We refer to this extended class of functions as the class of generalized hyperanalytic functions.
Separable topological algebras. I
Michael J.
Liddell
31-59
Abstract: Let A be a complete topological algebra with identity and B a subalgebra of the center of A. A notion of relative topological tensor product ${\hat \otimes _B}$ for topological A modules and the resultant relative homology theory are introduced. Algebras of bidimension zero in this sense are called separable relative to B. Structure theorems are proved for such algebras under various topological assumptions on the algebra and its maximal ideal space.
$T$-faithful subcategories and localization
John A.
Beachy
61-79
Abstract: For any additive functor from a category of modules into an abelian category there is a largest Giraud subcategory for which the functor acts faithfully on homomorphisms into the subcategory. It is the largest Giraud subcategory into which the functor reflects exact sequences, and under certain conditions it is just the largest Giraud subcategory on which the functor acts faithfully. If the functor is exact and has a right adjoint, then the subcategory is equivalent to the quotient category determined by the kernel of the functor. In certain cases, the construction can be applied to a Morita context in order to obtain a recent theorem of Mueller. Similarly, the functor defines a certain reflective subcategory and an associated radical, which is a torsion radical in case the functor preserves monomorphisms. Certain results concerning this radical, when defined by an adjoint functor, can be applied to obtain two theorems of Morita on balanced modules.
An asymptotic formula in adele Diophantine approximations
Melvin M.
Sweet
81-96
Abstract: In this paper an asymptotic formula is found for the number of solutions of a system of linear Diophantine inequalities defined over the ring of adeles of an algebraic number field. The theorem proved is a generalization of results of S. Lang and W. Adams.
First order differential closures of certain partially ordered fields
Joseph E.
Turcheck
97-114
Abstract: First order algebraic differential equations (a.d.e.'s) are considered in the setting of an abstract differential field with an abstract order relation, whose properties mirror those of the usual asymptotic dominance relations of analysis. An abstract existence theorem, for such equations, is proved by constructing an extension of both the differential field and the abstract order relation. As a consequence, a first order differential closure theorem, for those differential fields with order relations which we consider, is obtained. The closure theorem has corollaries which are important to the asymptotic theory of a.d.e.'s and have application to a.d.e.'s with coefficients meromorphic in a sector of the complex plane.
Linear ordinary differential equations with Laplace-Stieltjes transforms as coefficients
James
D’Archangelo
115-145
Abstract: The n-dimensional differential system $n \times n$ complex matrix and $ A(t)$ is an $n \times n$ matrix whose entries $a(t)$ are complex valued functions which are representable as absolutely convergent Laplace-Stieltjes transforms, $\smallint _0^\infty {e^{ - st}}d\alpha (s)$, for $t > 0$. The determining functions, $\alpha (s)$, are C valued, locally of bounded variation on $ [0,\infty )$, continuous from the right, and $\alpha ( + 0) = \alpha (0) = 0$. Sufficient conditions on the determining functions are found which assure the existence of solutions of certain specified forms involving absolutely convergent Laplace-Stieltjes transforms for $t > 0$ and which behave asymptotically like certain solutions of the nonperturbed equation $\Pi _{i = 1}^m{(D - {r_i})^{e(i)}}z + \Sigma _{j = 0}^{n - 1}{a_j}(t){D^j}z = 0$, where ${r_i} \in {\mathbf{C}}$ and the ${a_j}(t)$ are like $a(t)$ above for $t > 0$.
Harmonic analysis and centers of group algebras
J.
Liukkonen;
R.
Mosak
147-163
Abstract: The purpose of this paper is to present some results of harmonic analysis on the center of the group algebra $Z({L^1}(G))$ where G is a locally compact group. We prove that $ Z({L^1}(G))$ is a regular, Tauberian, symmetric Banach $^\ast$-algebra and contains a bounded approximate identity. Wiener's generalized Tauberian theorem is therefore applicable to $ Z({L^1}(G))$. These results complement those of I. E. Segal relating to the group algebra of locally compact abelian and compact groups. We also prove that if G contains a compact normal subgroup K such that G/K is abelian, then $ Z({L^1}(G))$ satisfies the condition of Wiener-Ditkin, so that any closed set in its maximal ideal space whose boundary contains no perfect subset is a set of spectral synthesis. We give an example of a general locally compact group for which $Z({L^1}(G))$ does not satisfy the condition of Wiener-Ditkin.
Almost equicontinuous transformation groups
Ping Fun
Lam
165-200
Abstract: A class of transformation groups called strictly almost equicontinuous transformation groups is studied. Manifolds which carry such transformation groups are determined. Applications to related classes are obtained.
Pairs of domains where all intermediate domains are Noetherian
Adrian R.
Wadsworth
201-211
Abstract: For Noetherian integral domains R and T with $R \subseteq T,(R,T)$ is called a Noetherian pair (NP) if every domain A, $R \subseteq A \subseteq T$, is Noetherian. When $\dim R = 1$ (Krull dimension) it is shown that the only NP's are those given by the Krull-Akizuki Theorem. For $\dim R \geq 2$, there is another type of NP besides the finite integral extension, namely $(R,\tilde R)$ where $\tilde R = \bigcap {\{ {R_P}\vert{\text{rk}}\;P \geq 2\} }$. Further, for every NP (R, T) with $\dim R \geq 2$ there is an integral NP extension B of R with $T \subseteq \tilde B$. In all known examples B can be chosen to be a finite integral extension of R. For such NP's it is shown that the NP relation is transitive. T may itself be an infinite integral extension R, though, and an example of this is given. It is unknown exactly which infinite integral extensions are NP's.
Henselian valued fields with prescribed value group and residue field
Linda
Hill
213-222
Abstract: A class of fields supplementary to the inertia field of a given henselian valued field is used to construct extensions of that field having prescribed value group and residue field. The extensions so-constructed are characterized, and their number investigated.
On the functional equation $f\sp{2}=e\sp{2\phi\sb{1}}+e\sp{2\phi\sb{2}}+e\sp{2\phi\sb{3}} $ and a new Picard theorem
Mark
Green
223-230
Abstract: By analogy with E. Borel's reduction of the classical Picard theorem to an analytic statement about linear relations among exponentials of entire functions, a new Picard theorem is proved by considering the functional relation ${f^2} = {e^{2{\phi _1}}} + {e^{2{\phi _2}}} + {e^{2{\phi _3}}}$ for entire functions. The analytic techniques used are those of Nevanlinna theory.
On a problem of Gronwall for Bazilevi\v c functions
John L.
Lewis
231-242
Abstract: Let $B(\alpha ,\beta ),\alpha $ positive, $\beta$ real, denote the class of normalized univalent Bazilevič functions in $K = \{ z:\vert z\vert < 1\}$ of type $\alpha ,\beta$. Let $B = { \cup _{\alpha ,\beta }}B(\alpha ,\beta )$. Let $\alpha ,0 \leq \alpha \leq 2$, and $\alpha ,0 < \alpha < \infty$, be fixed and suppose that $ f(z) = z + a{z^2} + \cdots$ is in $B(\alpha ,0)$. In this paper for given ${z_0} \in K$, the author finds a sharp upper bound for $\vert f({z_0})\vert$. Also, a sharp asymptotic bound is obtained for ${(1 - r)^2}{\max _{\vert z\vert = r}}\vert f(z)\vert$. Finally, a sharp asymptotic bound is found for ${(1 - r)^2}{\max _{\vert z\vert = r}}\vert f(z)\vert$ when f is in B with second coefficient a.
The convertibility of ${\rm Ext}\sp{n}\sb{R}(-,\,A)$
James L.
Hein
243-264
Abstract: Let R be a commutative ring and $\operatorname{Mod} (R)$ the category of R-modules. Call a contravariant functor $ F:\operatorname{Mod} (R) \to \operatorname{Mod} (R)$ convertible if for every direct system $ \{ {X_\alpha }\}$ in $ \operatorname{Mod} (R)$ there is a natural isomorphism $ \gamma :F(\mathop {\lim }\limits_ \to {X_\alpha }) \to \mathop {\lim }\limits_ \leftarrow F({X_\alpha })$. If A is in $\operatorname{Mod} (R)$ and n is a positive integer then ${\text{Ext}}_R^n( - ,A)$ is not in general convertible. The purpose of this paper is to study the convertibility of Ext, and in so doing to find out more about Ext as well as the modules A that make $ {\text{Ext}}_R^n( - ,A)$ convertible for all n. It is shown that $ {\text{Ext}}_R^n( - ,A)$ is convertible for all A having finite length and all n. If R is Noetherian then A can be Artinian, and if R is semilocal Noetherian then A can be linearly compact in the discrete topology. Characterizations are studied and it is shown that if A is a finitely generated module over the semilocal Noetherian ring R, then ${\text{Ext}}_R^1( - ,A)$ is convertible if and only if A is complete in the J-adic topology where J is the Jacobson radical of R. Morita-duality is characterized by the convertibility of ${\text{Ext}}_R^1( - ,R)$ when R is a Noetherian ring, a reflexive ring or an almost maximal valuation ring. Applications to the vanishing of Ext are studied.
Homogeneity and extension properties of embeddings of $S\sp{1}$ in $E\sp{3}$
Arnold C.
Shilepsky
265-276
Abstract: Two properties of embeddings of simple closed curves in ${E^3}$ are explored in this paper. Let $ {S^1}$ be a simple closed curve and $ f({S^1}) = S$ an embedding of ${S^1}$ in ${E^3}$. The simple closed curve S is homogeneously embedded or alternatively f is homogeneous if for any points p and q of S, there is an automorphism h of $ {E^3}$ such that $ h(S) = S$ and $ h(p) = q$. The embedding f or the simple closed curve S is extendible if any automorphism of S extends to an automorphism of ${E^3}$. Two classes of wild simple closed curves are constructed and are shown to be homogeneously embedded. A new example of an extendible simple closed curve is constructed. A theorem of H. G. Bothe about extending orientation-preserving automorphisms of a simple closed curve is generalized.
Fields of constants of integral derivations on a $p$-adic field
Henry W.
Thwing;
Nickolas
Heerema
277-290
Abstract: Let ${K_0}$ be a p-adic subfield of a p-adic field K with residue fields ${k_0} \subset k$. If ${K_0}$ is algebraically closed in K and k is finitely generated over ${k_0}$ then ${K_0}$ is the subfield of constants of an analytic derivation on K or equivalently, ${K_0}$ is the invariant subfield of an inertial automorphism of K. If (1) $ {k_0}$ is separably algebraically closed in k, (2) $[k_0^{{p^{ - 1}}} \cap k:{k_0}] < \infty$, and (3) k is not algebraic over $ {k_0}$ then there exists a p-adic subfield ${K_0}$ over ${k_0}$ which is algebraically closed in K. All subfields over ${k_0}$ are algebraically closed in K if and only if ${k_0}$ is algebraically closed in k. Every derivation on k trivial on ${k_0}$ lifts to a derivation on K trivial on ${K_0}$ if k is separable over $ {k_0}$. If k is finitely generated over ${k_0}$ the separability condition is necessary. Applications are made to invariant fields of groups of inertial automorphisms on p-adic fields and of their ramification groups.
Some mapping theorems
R. C.
Lacher
291-303
Abstract: Various mapping theorems are proved, culminating in the following result for mappings f from a closed $ (2k + 1)$-manifold M to another, N: If ``almost all'' point-inverses of f are strongly acyclic in dimensions less than k and if ``almost all'' point-inverses of f have Euler characteristic equal to one, then all but finitely many point-inverses are totally acyclic. (Here ``almost all'' means ``except on a zero-dimensional set in N".) More can be said when $k = 1$: If f is a monotone map between closed 3-manifolds and if the Euler characteristic of almost-all point-inverses is one, then all but finitely many point-inverses of f are cellular in M; consequently M is the connected sum of N and some other closed 3-manifold and f is homotopic to a spine map. Other results include an acyclicity criterion using the idea of ``nonalternating'' mapping and the following result for PL maps $\phi$ between finite polyhedra X and Y: If the Euler characteristic of each point-inverse of $\phi$ is the integer c then $\chi (X) = c\chi (Y)$.
Cohomology of nilradicals of Borel subalgebras
George F.
Leger;
Eugene M.
Luks
305-316
Abstract: Let $\mathfrak{N}$ be the maximal nilpotent ideal in a Borel subalgebra of a complex simple Lie algebra. The cohomology groups $ {H^1}(\mathfrak{N},\mathfrak{N}),{H^1}(\mathfrak{N},{\mathfrak{N}^\ast})$ and the $\mathfrak{N}$-invariant symmetric bilinear forms on $\mathfrak{N}$ are determined. The main result is the computation of $ {H^2}(\mathfrak{N},\mathfrak{N})$.
Regular elements in rings with involution
Charles
Lanski
317-325
Abstract: The purpose of this paper is to determine when a symmetric element, regular with respect to other symmetries, is regular in the ring. This result is true for simple rings, for prime rings with either Goldie chain condition, and for semiprime Goldie rings. Examples are given to show that these results are the best that can be hoped for.
Extremal problems in classes of analytic univalent functions with quasiconformal extensions
J. Olexson
McLeavey
327-343
Abstract: This work solves many of the classical extremal problems posed in the class of functions $ {\Sigma _{K(\rho )}}$, the class of functions in $\Sigma$ with $K(\rho )$-quasiconformal extensions into the interior of the unit disk where $K(\rho )$ is a piecewise continuous function of bounded variation on $ [r,1],0 \leq r < 1$. The approach taken is a variational technique and results are obtained through a limiting procedure. In particular, sharp estimates are given for the Golusin distortion functional, the Grunsky quadratic form, the first coefficient, and the Schwarzian derivative. Some extremal problems in $ {S_{K(\rho )}}$, the subclass of functions in S with $K(\rho )$-quasiconformal extensions to the exterior of the unit disk, are also solved.
A $2$-sphere in $E\sp{3}$ with vertically connected interior is tame
J. W.
Cannon;
L. D.
Loveland
345-355
Abstract: A set X in $ {E^3}$ is said to have vertical number n if the intersection of each vertical line with X contains at most n components. The set X is said to have vertical order n if each vertical line intersects X in at most n points. A set with vertical number 1 is said to be vertically connected. We prove that a 2-sphere in ${E^3}$ with vertically connected interior is tame. This result implies as corollaries several previously known taming theorems involving vertical order and vertical number along with several more general and previously unknown results.
On a certain sum in number theory. II
Břetislav
Novák
357-364
Abstract: We derive ``exact order'' of the function $\displaystyle \sum\limits_{k \leq \sqrt x } {{k^\rho }{{\min }^\beta }\left( {\frac{{\sqrt x }}{k},\frac{1}{{{P_k}}}} \right)}.$ Here $ \rho$ and $\beta$ are nonnegative real numbers and, for given real ${\delta _1},{\delta _2}, \cdots ,{\delta _r},{P_k} = {\max _j}\langle k{\delta _j}\rangle$ where $\langle t\rangle$, for real t, denotes distance of t from the nearest integer. Using our results, we obtain the solution of the basic problem in the theory of lattice points with weight in rational many-dimensional ellipsoids.
Square integrable differentials on Riemann surfaces and quasiconformal mappings
Carl David
Minda
365-381
Abstract: If $ K{(f)^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}$ where $ K(f)$ is the maximal dilatation of f. In addition, f defines an isomorphism of the square integrable harmonic differentials and some important subspaces are preserved. It is shown that not all important subspaces are preserved. The relationship of this to other work is investigated; in particular, the connection with the work of Nakai on the isomorphism of Royden algebras induced by a quasiconformal mapping is explored. Finally, the induced isomorphisms are applied to the classification theory of Riemann surfaces to show that various types of degeneracy are quasiconformally invariant.
Semigroups over trees
M. W.
Mislove
383-400
Abstract: A semigroup over a tree is a compact semigroup S such that $\mathcal{H}$ is a congruence on S and $S/\mathcal{H}$ is an abelian tree with idempotent endpoints. Each such semigroup is characterized as being constructible from cylindrical subsemigroups of S and the tree $ S/\mathcal{H}$ in a manner similar to the construction of the hormos. Indeed, the hormos is shown to be a particular example of the construction given herein when $S/\mathcal{H}$ is an I-semigroup. Several results about semigroups whose underlying space is a tree are also established as lemmata for the main results.
The closure of the space of homeomorphisms on a manifold
William E.
Haver
401-419
Abstract: The space, $ \bar H(M)$, of all mappings of the compact manifold M onto itself which can be approximated arbitrarily closely by homeomorphisms is studied. It is shown that $\bar H(M)$ is homogeneous and weakly locally contractible. If M is a compact 2-manifold without boundary, then $\bar H(M)$ is shown to be locally contractible.
On free products of finitely generated abelian groups
Anthony M.
Gaglione
421-430
Abstract: Let the group G be a free product of a finite number of finitely generated abelian groups. Let $G'$ be its commutator subgroup. It is proven here that the ``quasi-G-simple'' commutators, defined below, are free generators of $ G'$.
A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact mappings
W. V.
Petryshyn;
P. M.
Fitzpatrick
1-25
Abstract: We define and study the properties of a topological degree for ultimately compact, multivalued vector fields defined on the closures of open subsets of certain locally convex topological vector spaces. In addition to compact mappings, the class of ultimately compact mappings includes condensing mappings, generalized condensing mappings, perturbations of compact mappings by certain Lipschitz-type mappings, and others. Using this degree we obtain fixed point theorems and mapping theorems.
Local decay of solutions of conservative first order hyperbolic systems in odd dimensional space
James V.
Ralston
27-51
Abstract: This paper deals with symmetric hyperbolic systems, $\partial u/\partial t = Lu$, where L is equal to the homogeneous, constant coefficient operator ${L_0}$ for $\vert x\vert > R$. Under the hypothesis that L has simple null bicharacteristics and these propagate to infinity, local decay of solutions and completeness of the wave operators relating solutions of $ \partial u/\partial t = Lu$ and solutions of $\partial u/\partial t = {L_0}u$ are established. Results of this type for elliptic L are due to Lax and Phillips. The proof here is based, in part, on a new estimate of the regularity of the $ {L^2}$-solutions of the equation $Lu + (i\lambda + \varepsilon )u = g$ for smooth g with support in $\vert x\vert \leq R$.
Equivariant endomorphisms of the space of convex bodies
Rolf
Schneider
53-78
Abstract: We consider maps of the set of convex bodies in d-dimensional Euclidean space into itself which are linear with respect to Minkowski addition, continuous with respect to Hausdorff metric, and which commute with rigid motions. Examples constructed by means of different methods show that there are various nontrivial maps of this type. The main object of the paper is to find some reasonable additional assumptions which suffice to single out certain special maps, namely suitable combinations of dilatations and reflections, and of rotations if $d = 2$. For instance, we determine all maps which, besides having the properties mentioned above, commute with affine maps, or are surjective, or preserve the volume. The method of proof consists in an application of spherical harmonics, together with some convexity arguments.
Transversally parallelizable foliations of codimension two
Lawrence
Conlon
79-102
Abstract: We study framed foliations such that the framing of the normal bundle can be chosen to be invariant under the linear holonomy of each leaf. In codimension one there is a strong structure theory for such foliations due, e.g., to Novikov, Sacksteder, Rosenberg, Moussu. An analogous theory is developed here for the case of codimension two.
Generators for $A(\Omega )$
N.
Sibony;
J.
Wermer
103-114
Abstract: We consider a bounded domain $\Omega$ in $ {{\mathbf{C}}^n}$ and the Banach algebra $ A(\Omega )$ of all continuous functions on $ \bar \Omega$ which are analytic in $\Omega$. Fix $ {f_1}, \ldots ,{f_k}$ in $A(\Omega )$. We say they are a set of generators if $A(\Omega )$ is the smallest closed subalgebra containing the ${f_i}$. We restrict attention to the case when $ \Omega$ is strictly pseudoconvex and smoothly bounded and the ${f_i}$ are smooth on $\bar \Omega$. In this case, Theorem 1 below gives conditions assuring that a given set ${f_i}$ is a set of generators.
Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups
Ronald E.
Kutz
115-129
Abstract: Theorem. Let R be a commutative Noetherian ring with identity. Let M = $M = ({c_{ij}})$ be an s by s symmetric matrix with entries in R. Let I the be ideal of $t + 1$ by $t + 1$ minors of M. Suppose that the grade of I is as large as possible, namely, gr $I = g = s(s + 1)/2 - st + t(t - 1)/2$. Then I is a perfect ideal, so that $R/I$ is Cohen Macaulay if R is. Let G be a linear algebraic group acting rationally on $R = K[{x_1}, \ldots ,{x_n}]$. Hochster has conjectured that if G is reductive, then $ {R^G}$ is Cohen-Macaulay, where ${R^G}$ denotes the ring of invariants of the action of G. The above theorem provides a special case of this conjecture. For $G = O(t,K)$, the orthogonal group, and K a field of characteristic zero, the above yields: Corollary. For R and G as above, ${R^G}$ is Cohen-Macaulay for an appropriate action of G. In order to obtain these results it was necessary to prove a more general form of the theorem stated above, which in turn yields a more general form of the corollary.
Square-integrable representations and the Mackey theory
Terje
Sund
131-139
Abstract: It is the purpose of this paper to clarify the relationship between the square-integrable irreducible representations of a 2nd countable unimodular locally compact group G and a closed normal subgroup N using the Mackey theory relating the dual spaces $\hat G$ and $\hat N$.
Pseudo-boundaries and pseudo-interiors in Euclidean spaces and topological manifolds
Ross
Geoghegan;
R. Richard
Summerhill
141-165
Abstract: The negligibility theorems of infinite-dimensional topology have finite-dimensional analogues. The role of the Hilbert cube $ {I^\omega }$ is played by euclidean n-space ${E^n}$, and for any nonnegative integer $k < n$, k-dimensional dense $ {F_\sigma }$-subsets of $ {E^n}$ exist which play the role of the pseudo-boundary of ${I^\omega }$. Their complements are $(n - k - 1)$-dimensional dense ${G_\delta }$ pseudo-interiors of $ {E^n}$. Two kinds of k-dimensional pseudo-boundaries are constructed, one from universal compacta, the other from polyhedra. All the constructions extend to topological manifolds.
A nonlinear Boltzmann equation in transport theory
C. V.
Pao
167-175
Abstract: The method of successive approximations is used to show the existence of a unique solution to a model of a nonlinear Boltzmann equation under the homogeneous boundary and typical initial conditions. An explicit formula in terms of the prescribed functions for the calculation of an approximate solution and its error estimate are given. This formula reveals an interesting analogy between the initial-boundary value problem of the Boltzmann equation and the Cauchy problem for ordinary differential equations. Numerical results for approximate solutions of the problem can be computed by using a computer. The linear Boltzmann equation is considered as a special case and a similar formula for the calculation of approximate solutions is included.
Regularly varying functions and convolutions with real kernels
G. S.
Jordan
177-194
Abstract: Let $\phi$ be a positive, measurable function and k a real-valued function on $(0,\infty ),k \in {L^1}(dt/t)$. We give conditions on $\phi$ and k sufficient to deduce the regular variation of $\phi$ from the assumption that $\displaystyle \alpha = \mathop {\lim }\limits_{x \to \infty } \frac{1}{{\phi (x... ...\frac{x}{2}} \right)} \;\frac{{dt}}{t}\;{\text{exits}}\;(\alpha \ne 0,\infty ).$ The general theorems extend in certain ways results of other authors and yield a new theorem on the relation between the radial growth and zero-distribution of those entire functions which are canonical products of nonintegral order with negative zeros.
On Borel mappings and Baire functions
R. W.
Hansell
195-211
Abstract: This paper studies conditions under which classes of Borel mappings (i.e., mappings such that the inverse image of open sets are Borel sets) coincide with certain classes of Baire functions (i.e., functions which belong to the smallest family containing the continuous functions and closed with respect to pointwise limits). Generalizations of the classical Lebesgue-Hausdorff and Banach theorems are obtained for the class of mappings which we call ``$\sigma$-discrete". These results are then applied to the problem of extending Borel mappings over Borel sets, and generalizations of the theorems of Lavrentiev and Kuratowski are obtained.
Parabolic equations associated with the number operator
M. Ann
Piech
213-222
Abstract: We study existence and uniqueness of solutions of the Cauchy problem for $\dot u = Nu$ where N is the number operator on abstract Wiener space.
Algebras of analytic operator valued functions
Kenneth O.
Leland
223-239
Abstract: This paper proves and generalizes the following characterization of the algebra $A(K,K)$ of complex analytic functions on open subsets of the complex plane K into K to the case of algebras of functions on a real Euclidean space E into a real Banach algebra B. Theorem. Let $F(K,K)$ be the algebra of all continuous functions on open subsets of K into K, and F a subalgebra of $ F(K,K)$ with nonconstant elements such that ${ \cup _{f \in F}}$ range $f = K,F$ is closed under uniform convergence on compact sets and domain transformations of the form $z \to {z_0} + z\sigma ,z,{z_0},\sigma \in K$. Then F is $F(K,K)$ or $A(K,K)$ or $\bar A(K,K) = \{ \bar f;f \in A(K,K)\}$. In the general case conditions on B are studied that insure that either F contains an embedment of $ F(R,R)$ and thus contains quite arbitrary continuous functions or that the elements of F are analytic and F can be expressed as a direct sum of algebras ${F_1}, \ldots ,{F_n}$ such that for $i = 1, \ldots ,n$, there exist complexifications $ {M_i}$ of E and $ {N_i}$ of ${\cup _{f \in {F_i}}}$ range f, such that with respect to ${M_i}$ and ${N_i}$ the elements of ${F_i}$ are complex differentiable.
Group algebras whose simple modules are injective
Daniel R.
Farkas;
Robert L.
Snider
241-248
Abstract: Let F be either a field of char 0 with all roots of unity or a field of char $p > 0$. Let G be a countable group. Then all simple $F[G]$-modules are injective if and only if G is locally finite with no elements of order char F and G has an abelian subgroup of finite index.
Rational points of commutator subgroups of solvable algebraic groups
Amassa
Fauntleroy
249-275
Abstract: Let G be a connected algebraic group defined over a field k. Denote by $G(k)$ the group of k-rational points of G. Suppose that A and B are closed subgroups of G defined over k. Then $ [A,B](k)$ is not equal to $[A(k),B(k)]$ in general. Here [A,B] denotes the group generated by commutators $ab{a^{ - 1}}{b^{ - 1}},a \in A,b \in B$. We say that a field of k of characteristic p is p-closed if given any additive polynomial $f(x)$ in $k[x]$ and any element c in k, there exists an element $\alpha$ in k such that $f(\alpha ) = c$. Theorem 1. Let G be a connected solvable algebraic group defined over the p-closed field k. Let A and B be closed connected subgroups of G, which are also defined over k, and suppose A normalizes B. $Then\;[A,B]\;(k) = [A(K),B(K)]$. 2. If G, A and B are as above and k is only assumed to be perfect then there exists a finite extension $ {k_0}$ of k such that if K is the maximal p-extension of $ {k_0}$, then $[A,B](K) = [A(K),B(K)]$.
Power residues and nonresidues in arithmetic progressions
Richard H.
Hudson
277-289
Abstract: Let k be an integer $\geq 2$ and p a prime such that ${v_k}(p) = (k,p - 1) > 1$. Let $bn + c(n = 0,1, \ldots ;b \geq 2,1 \leq c < b,(b,p) = (c,p) = 1)$ be an arithmetic progression. We denote the smallest kth power nonresidue in the progression $bn + c$ by $ g(p,k,b,c)$, the smallest quadratic residue in the progression $bn + c$ by $ {r_2}(p,b,c)$, and the nth smallest prime kth power nonresidue by $ {g_n}(p,k),n = 0,1,2, \ldots$. If $C(p)$ is the multiplicative group consisting of the residue classes $\bmod\;p$, then the kth powers $ \bmod\;p$ form a multiplicative subgroup, ${C_k}(p)$. Among the ${v_k}(p)$ cosets of ${C_k}(p)$ denote by T the coset to which c belongs (where c is the first term in the progression $bn + c)$, and let $ h(p,k,b,c)$ denote the smallest number in the progression $bn + c$ which does not belong to T so that $ h(p,k,b,c)$ is a natural generalization of $ g(p,k,b,c)$. We prove by purely elementary methods that $h(p,k,b,c)$ is bounded above by $ {2^{7/4}}{b^{5/2}}{p^{2/5}} + 3{b^3}{p^{1/5}} + {b^2}$ if p is a prime for which either b or $p - 1$ is a kth power nonresidue. The restriction on b and $p - 1$ may be lifted if $p > {({g_1}(p,k))^{7.5}}$. We further obtain a similar bound for ${r_2}(p,b,c)$ for every prime p, without exception, and we apply our results to obtain a bound of the order of ${p^{2/5}}$ for the nth smallest prime kth power nonresidue of primes which are large relative to $ \Pi _{j = 1}^{n - 1}{g_j}(p,k)$.
On the construction of split-face topologies
Alan
Gleit
291-299
Abstract: We give a general theorem to facilitate the construction of interesting examples of split-face topologies of compact, convex sets.
An extension of Weyl's lemma to infinite dimensions
Constance M.
Elson
301-324
Abstract: A theory of distributions analogous to Schwartz distribution theory is formulated for separable Banach spaces, using abstract Wiener space techniques. A distribution T is harmonic on an open set U if for any test function f on U, $T(\Delta f) = 0$, where $\Delta f$ fis the generalized Laplacian of f. We prove that a harmonic distribution on U can be represented as a unique measure on any subset of U which is a positive distance from $ {U^C}$. In the case where the space is finite dimensional, it follows from Weyl's lemma that the measure is in fact represented by a ${C^\infty }$ function. This functional representation cannot be expected in infinite dimensions, but it is shown that the measure has smoothness properties analogous to infinite differentiability of functions.
Analytic domination with quadratic form type estimates and nondegeneracy of ground states in quantum field theory
Alan D.
Sloan
325-336
Abstract: We present a theorem concerning the analytic domination by a semi-bounded selfadjoint operator H of another linear operator A which requires only the quadratic form type estimates $\displaystyle \left\Vert {{H^{ - 1/2}}({{({\text{ad}}\;A)}^n}H){H^{ - 1/2}}u} \right\Vert \leq {c_n}\left\Vert u \right\Vert$ instead of the norm estimates $\displaystyle \left\Vert {{{({\text{ad}}\;A)}^n}Hu} \right\Vert \leq {c_n}\left\Vert {Hu} \right\Vert$ usually required for this type of theorem. We call the new estimates ``quadratic form type", since they are sometimes equivalent to $\displaystyle \vert({({\text{ad}}\;A)^n}Hu,u)\vert \leq {c_n}\vert(Hu,u)\vert.$ The theorem is then applied with H the Hamiltonian for the spatially cutoff boson field model with real, bounded below, even ordered polynomial self-interaction in one space dimension and $A = \pi (g)$, the conjugate momentum to the free field. When the underlying Hilbert space of this model is represented as $ {L^2}(Q,dq)$ where dq is a probability measure on Q, the spectrum of the von Neumann algebra generated by bounded functions of certain field operators, then ${e^{ - tH}}$ maximizes support in the sense that ${e^{ - tH}}f$ is nonzero almost everywhere whenever f is not identically zero.
On the tensor product of $W\sp{\ast} $ algebras
Bruce B.
Renshaw
337-347
Abstract: We develop the algebra underlying the reduction theory of von Neumann in the language and spirit of Sakai's abstract ${W^ \ast }$ algebras, and using the maximum spectrum of an abelian von Neumann algebra rather than a measure-theoretic surrogate. We are thus enabled to obtain the basic fact of the von Neumann theory as a special case of a weaker general decomposition theorem, valid without separability or type restrictions, and adapted to comparison with Wright's theory in the finite case.
Shape theory and compact connected abelian topological groups
James
Keesling
349-358
Abstract: Let C denote the category of compact Hausdorff spaces and continuous maps. Let $S:C \to SC$ denote the functor of shape in the sense of Holsztyński from C to the shape category SC determined by the homotopy functor $ H:C \to HC$ from C to the homotopy category HC. Let A, B, and D denote compact connected abelian topological groups. In this paper it is shown that if G is a morphism in the shape category from A to B, then there is a unique continuous homomorphism $g:A \to B$ such that $S(g) = G$. This theorem is used in a study of shape properties of continua which support an abelian topological group structure. The following results are shown: (1) The spaces A and B are shape equivalent if and only if $A \simeq B$. (2) The space A is movable if and only if A is locally connected. (3) The space A shape dominates $B,S(A) \geq S(B)$, if and only if there is a D such that $A \simeq B \times D$. (4) The fundamental dimension of A is the same as the dimension of $A,{\text{Sd}}(A) = \dim A$. In an Appendix it is shown that the Holsztyński approach to shape and the approach of Mardešić and Segal using ANR-systems are equivalent. Thus, the results apply to either theory and to the Borsuk theory in the metrizable case.
Extremal problems of distance geometry related to energy integrals
Ralph
Alexander;
Kenneth B.
Stolarsky
1-31
Abstract: Let K be a compact set, $ \mathcal{M}$ a prescribed family of (possibly signed) Borel measures of total mass one supported by K, and f a continuous real-valued function on $K \times K$. We study the problem of determining for which $\mu \in \mathcal{M}$ (if any) the energy integral $I(K,\mu ) = \smallint_K {\smallint_K {f(x,y)d\mu (x)d\mu (y)} }$ is maximal, and what this maximum is. The more symmetry K has, the more we can say; our results are best when K is a sphere. In particular, when $ \mathcal{M}$ is atomic we obtain good upper bounds for the sums of powers of all $ (_2^n)$ distances determined by n points on the surface of a sphere. We make use of results from Schoenberg's theory of metric embedding, and of techniques devised by Pólya and Szegö for the calculation of transfinite diameters.
Invariant measures and growth conditions
Joseph Max
Rosenblatt
33-53
Abstract: Let G be a finitely-generated group acting on a set X and let A be a nonempty subset of X. If G has polynomial growth then there exists a finitely-additive G-invariant positive extended real-valued measure $\mu$ defined on all subsets of X such that $\mu (A) = 1$. When G is solvable, it has polynomial growth if and only if it does not contain a free subsemigroup on two generators. If G contains a free subsemigroup S on two generators, then G has exponential growth and there does not exist a measure $\mu$ as above with G acting on itself by multiplication and $A = S$.
Some thin sets in discrete abelian groups
Ron C.
Blei
55-65
Abstract: Let $\Gamma$ be a discrete abelian group, and $E \subset \Gamma $. For $F \subset E$, we say that $F \in \mathcal{P}(E)$, if for all $\Lambda$, finite subsets of $ \Gamma ,0 \notin \Lambda ,\Lambda + F \cap F$ is finite. Having defined the Banach algebra, $\tilde A(E) = c(E) \cap B(E)$, we prove the following: (i) $ E \subset \Gamma$ is a Sidon set if and only if every $F \in \mathcal{P}(E)$ is a Sidon set; (ii) $E \in \mathcal{P}(\Gamma )$ is a Sidon set if and only if $\tilde A(E) = A(E)$.
Inequalities for polynomials with a prescribed zero
A.
Giroux;
Q. I.
Rahman
67-98
Abstract: Inequalities for the derivative and for the maximum modulus on a larger circle of a polynomial with a given zero on the unit circle are obtained in terms of its degree and maximum modulus on the unit circle; examples are given to show that these are sharp with respect to the degree (best constants are not known). Inequalities for $ {L^p}$ norms, in particular ${L^2}$ norms, are also derived. Also certain functions of exponential type are considered and similar inequalities are obtained for them. Finally, the problem of estimating ${P_n}(r)$ (with $0 < r < 1$) given $ {P_n}(1) = 0$ is taken up.
Multiplier transformations on compact Lie groups and algebras
Robert S.
Strichartz
99-110
Abstract: Let G be a semisimple compact Lie group and $Tf = \sum \phi (m){d_{m \chi m}} \ast f$ a bi-invariant operator on ${L^2}(G)$, where ${\chi _m}$ and ${d_m}$ are the characters and dimensions of the irreducible representations of G, which are indexed by a lattice of points m in the Lie algebra $\mathfrak{G}$ in a natural way. If $ \Phi$ is a bounded ad-invariant function on $ \mathfrak{G}$ and $\displaystyle \phi {\text{(}}m{\text{) = }}\Phi {\text{(}}m{\text{ + }}\beta {\text{)}}\quad{\text{or}}$ ($\ast$) $\displaystyle \phi {\text{(}}m{\text{) = }}\int_G {\Phi (m + \beta - {\text{ad}}\;g\beta )dg}$ ($ \ast \ast$) $ \beta$ being half the sum of the positive roots, then various properties of T are related to properties of the Fourier multiplier transformation on $ \mathfrak{G}$ with multiplier $\Phi$. These properties include boundedness on $ {L^1}$, uniform boundedness on ${L^p}$ of a family of operators, and, in the special case $ G = {\text{SO}}(3)$, boundedness in ${L^p}$ for ad-invariant functions with $1 \leq p < 3/2$.
Somewhere locally flat codimension one manifolds with $1-{\rm ULC}$ complements are locally flat
T. M.
Price;
C. L.
Seebeck
111-122
Abstract: The purpose of this paper is to prove a taming theorem for a codimension one manifold that is locally flat at some point and has 1-ULC complement. We also prove that any two sufficiently close locally flat embeddings of a codimension one manifold are ambient isotopic. Since this paper was first submitted, R. Daverman has shown that, given any point on a codimension one manifold with 1-ULC complement, some neighborhood of that point lies on a codimension one sphere that is locally flat at some points and has 1-ULC complement. Hence the two papers combined prove that a codimension one manifold is locally flat if and only if its complement is 1-ULC.
Almost complex structures on complex projective spaces
Alan
Thomas
123-132
Abstract: In this paper we classify the almost complex structures on a complex projective space as roots of a certain polynomial equation.
A relation between $K$-theory and cohomology
Alan
Thomas
133-142
Abstract: It is well known that for X a CW-complex, $K(X)$ and ${H^{{\text{ev}}}}(X)$ are isomorphic modulo finite groups, although the ``isomorphism'' is not natural. The purpose of this paper is to improve this result for X a finite CW-complex.
Equisingular deformations of plane algebroid curves
Jonathan M.
Wahl
143-170
Abstract: We construct a formal versal equisingular deformation of a plane algebroid curve (in characteristic zero), and show it is smoothly embedded in the whole deformation space of the singularity. Closer analysis relates equisingular deformations of the curve to locally trivial deformations of a certain (nonreduced) projective curve. Finally, we prove that algebraic ${\pi _1}$ of the complement of a plane algebroid curve remains constant during formal equisingular deformation.
The undecidability of the word problems for projective geometries and modular lattices
L.
Lipshitz
171-180
Abstract: We show that the restricted word problems for finite-dimensional projective geometries and finite modular lattices and the word problem for modular lattices are undecidable.
A generalization of the ${\rm cos} \pi \,\rho $ theorem
Albert
Baernstein
181-197
Abstract: Let f be an entire function, and let $\beta$ and $\lambda$ be positive numbers with $\beta \leq \pi$ and $\beta \lambda < \pi$. Let $E(r) = \{ \theta :\log \vert f(r{e^{i\theta }})\vert > \cos \beta \lambda \log M(r)\}$. It is proved that either there exist arbitrarily large values of r for which $E(r)$ contains an interval of length at least $ 2\beta$, or else ${\lim _{r \to \infty }}{r^{ - \lambda }}\log M(r,f)$ exists and is positive or infinite. For $\beta = \pi$ this is Kjellberg's refinement of the cos $\pi \rho$ theorem.
Bounded mean oscillation and regulated martingales
Carl
Herz
199-215
Abstract: In the martingale context, the dual Banach space to ${H_1}$ is BMO in analogy with the result of Charles Fefferman [4] for the classical case. This theorem is an easy consequence of decomposition theorems for ${H_1}$-martingales which involve the notion of $ {L_p}$-regulated $ {L_1}$-martingales where $1 < p \leq \infty$. The strongest decomposition theorem is for $p = \infty$, and this provides full information about BMO. The weaker $p = 2$ decomposition is fundamental in the theory of martingale transforms.
On a Wedderburn principal theorem for the flexible algebras
Robert A.
Chaffer
217-229
Abstract: A strictly power-associative algebra A over a field K is said to have a Wedderburn decomposition if there is a subalgebra S of A such that $A = S + N$, where N is the nil radical of A, and $S = A - N$. A Wedderburn principal theorem for a class of algebras is a theorem which asserts that the algebras A, in the class, with $A - N$ separable have Wedderburn decompositions. It is known that there is no such theorem for the class of noncommutative Jordan algebras. A partial result in this direction is the following theorem. Theorem. Let A be a strictly power-associative, flexible algebra over a field F with characteristic not 2 or 3, with $A - N$ separable and such that $A = {A_1} \oplus {A_2} \oplus \cdots \oplus {A_n}$ where each ${A_i}$. has $ {A_i} - {N_i}$ simple and has more than two pairwise orthogonal idempotents. Then $A = S + N$ where S is a subalgebra of A.
Product of ring varieties and attainability
Awad A.
Iskander
231-238
Abstract: The class of all rings that are Everett extensions of a ring in a variety $ \mathfrak{U}$ by a ring in a variety $ \mathfrak{B}$ is a variety $ \mathfrak{U} \cdot \mathfrak{B}$. With respect to this operation the set of all ring varieties is a partially ordered groupoid (under inclusion), that is not associative. A variety is idempotent iff it is the variety of all rings, or generated by a finite number of finite fields. No families of polynomial identities other than those equivalent to $x = x$ or $x = y$ are attainable on the class of all rings or on the class of all commutative rings.
The genera of edge amalgamations of complete bigraphs
Seth R.
Alpert
239-247
Abstract: If G and H are graphs, then $G \vee H$ is defined to be a graph obtained by identifying some edge of G with some edge of H. It is shown that for all m, n, p, and q the genus $g({K_{m,n}} \vee {K_{p,q}})$ is either $ g({K_{m,n}}) + g({K_{p,q}})$ or else $g({K_{m,n}}) + g({K_{p,q}}) - 1$. The latter value is attained if and only if both ${K_{m,n}}$ and ${K_{p,q}}$ are critical in the sense that the deletion of any edge results in a graph whose genus is one less than the genus of the original graph.
The group of $PL$-homeomorphisms of a compact $PL$-manifold is an $1\sp{f}\sb{2}$-manifold
James
Keesling;
David C.
Wilson
249-256
Abstract: In this paper it is shown that if M is a compact PL-manifold and ${H_{PL}}(M)$ is the group of PL-homeomorphisms of M onto itself, then $ {H_{PL}}(M)$ is an $ l_2^f$-manifold. Here $ {l_2}$ is the Hilbert space of all real-valued square-summable sequences and $l_2^f = \{ ({x_i}) \in {l_2}:{x_i} = 0$ for almost all i.
On totally real submanifolds
Bang-yen
Chen;
Koichi
Ogiue
257-266
Abstract: Complex analytic submanifolds and totally real submanifolds are two typical classes among all submanifolds of an almost Hermitian manifold. In this paper, some characterizations of totally real submanifolds are given. Moreover some classifications of totally real submanifolds in complex space forms are obtained.
A general representation theorem for analytic solutions of first-order algebraic differential equations in sectors
Steven B.
Bank
267-289
Abstract: In this paper, we obtain precise asymptotic representations for a broad class of solutions of first-order algebraic differential equations whose coefficients belong to a certain type of function field.
Irreducible congruences over ${\rm GF}(2)$
C. B.
Hanneken
291-301
Abstract: In characterizing and determining the number of conjugate sets of irreducible congruences of degree m belonging to $ GF(p)$ relative to the group $G(p)$ of linear fractional transformations with coefficients belonging to the same field, the case $p = 2$ has been consistently excluded from considerations. In this paper we consider the special case $p = 2$ and determine the number of conjugate sets of m-ic congruences belonging to $ GF(2)$ relative to $ G(2)$.
Riesz points of the spectrum of an element in a semisimple Banach algebra
Lynn D.
Pearlman
303-328
Abstract: Let A be a semisimple Banach algebra with unit element and let $ {S_A}$ denote the socle of A. For an element y in A, let ${L_y}[{R_y}]$ denote the operator of left [right] multiplication by y on A. The operational calculus and A. E. Taylor's theory of the ascent $\alpha (T)$ and descent $ \delta (T)$ of an operator T on A are used to show that the following conditions on a number $\lambda$ in the spectrum of an element x in A are all equivalent. (1) $\lambda$ is a pole of the resolvent mapping $z \to {(z - x)^{ - 1}}$ and the spectral idempotent f, for x at $ \lambda$ is in $ {S_A}$. (2) $\lambda - x - c$ is invertible in A for some c in the closure of ${S_A}$ such that $cx = xc$. (3) $ \lambda - x$ is invertible modulo the closure of ${S_A}$ and $ 0 < \alpha ({L_{(\lambda - x)}}) = \delta ({L_{(\lambda - x)}}) < \infty$. (4) $ \lambda - x$ is invertible modulo the closure of ${S_A}$ and $0 < \alpha ({R_{(\lambda - x)}}) = \delta ({R_{(\lambda - x)}}) = \alpha ({L_{(\lambda - x)}}) = \delta ({L_{(\lambda - x)}}) < \infty$. Such numbers $\lambda$ are called Riesz points. An element x is called a Riesz element of A if it is topologically nilpotent modulo the closure of ${S_A}$. It is shown that x is a Riesz element if and only if every nonzero number in the spectrum of x is a Riesz point.
Characterization of privileged polydomains
Yum Tong
Siu
329-357
Abstract: This paper gives a number of equivalent conditions for a bounded polydomain to be privileged with respect to a coherent analytic sheaf in the sense of Douady. One of the equivalent conditions is in terms of the homological codimensions of the sheaf at the boundary of the polydomain. In the case of a polydisc, this condition about homological codimensions coincides with a conjecture of Douady. The other equivalent conditions concern some weaker concepts of privilegedness and the existence of privileged sets at the boundary.
A finitely additive generalization of the Fichtenholz-Lichtenstein theorem
George Edward
Sinclair
359-374
Abstract: Let $\mu$ and $\nu$ be bounded, finitely additive measures on algebras over sets X and Y, respectively. Conditions are determined for a bounded function $ f:X \times Y \to {\mathbf{R}}$, without assuming bimeasurability, so that the iterated integrals $\smallint_X {\smallint_Y {fd\mu d\mu } }$ and $\smallint_Y {\smallint_X {fd\mu d\nu } }$ exist and are equal. This result is then used to construct a product algebra and finitely additive product measure for $ \mu$ and $\nu$. Finally, a simple Fubini theorem with respect to this product algebra and product measure is established.
Primitive elements and one relation algebras
Catherine
Aust
375-387
Abstract: Let F be a free algebra in a variety V. An element p of F is called primitive if it is contained in some free generating set for F. In 1936, J. H. C. Whitehead proved that a group with generators $ {g_1}, \ldots ,{g_n}$ and one relation $r = 1$ is free if and only if the relator r is primitive in the free group on ${g_1}, \ldots ,{g_n}$. In tnis paper, tne question of whether there is an analogous theorem for other varieties is considered. A necessary and sufficient condition that a finitely generated, one relation algebra be free is proved for any Schreier variety of nonassociative linear algebras and for any variety defined by balanced identities. An identity $ u({x_1}, \ldots ,{x_n}) = v({x_1}, \ldots ,{x_n})$ is called balanced if each of u and v has the same length and number of occurrences of each ${x_i}$. General sufficiency conditions that a finitely generated, one relation algebra be free are given, and all of the known results analogous to the Whitehead theorem are shown to be equivalent to a general necessary condition. Also an algebraic proof of Whitehead's theorem is outlined to suggest the line of argument for other varieties.
Contracting spaces of maps on the countable direct limit of a space
Richard E.
Heisey
389-411
Abstract: We give conditions sufficient to imply the contractibility of the space of maps, with compact-open topology, on the countable direct limit of a space. Applying these conditions we obtain the following: Let F be the conjugate of a separable infinite-dimensional Banach space with bounded weak-$^\ast$ topology, or the countable direct limit of the real line. Then there is a contraction of the space of maps on F which simultaneously contracts the subspaces of open maps, embeddings, closed embeddings, and homeomorphisms. Corollaries of our work are that any homeomorphism on F, F as above, is invertibly isotopic to the identity, and the general linear group of the countable direct limit of lines is contractible.
Conditions for the absolute continuity of two diffusions
Steven
Orey
413-426
Abstract: Consider two diffusion processes on the line. For each starting point x and each finite time t, consider the measures these processes induce in the space of continuous functions on [0, t]. Necessary and sufficient conditions on the generators are found for the induced measures to be mutually absolutely continuous for each x and t. If the first process is Brownian motion, the second one must be Brownian motion with drift $b(x)$, where $b(x)$ is locally in ${L_2}$ and satisfies a certain growth condition at $\pm \infty$.
Waring's problem for twenty-two biquadrates
Henry E.
Thomas
427-430
Abstract: That every natural number is the sum of at most twenty-two biquadrates is proven by ascent from machine results on sums of six fourth powers.
Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu\sb{tt} = -Au + \mathcal{F}(u)$
Howard A.
Levine
1-21
Abstract: For the equation in the title, let P and A be positive semidefinite operators (with P strictly positive) defined on a dense subdomain $D \subseteq H$, a Hilbert space. Let D be equipped with a Hilbert space norm and let the imbedding be continuous. Let $\mathcal{F}:D \to H$ be a continuously differentiable gradient operator with associated potential function $\mathcal{G}$. Assume that $ (x,\mathcal{F}(x)) \geq 2(2\alpha + 1)\mathcal{G}(x)$ for all $x \in D$ and some $\alpha > 0$. Let $ E(0) = \tfrac{1}{2}[({u_0},A{u_0}) + ({v_0},P{v_0})]$ where ${u_0} = u(0),{v_0} = {u_t}(0)$ and $u:[0,T) \to D$ be a solution to the equation in the title. The following statements hold: If $ \mathcal{G}({u_0}) > E(0)$, then ${\lim _{t \to {T^ - }}}(u,Pu) = + \infty$ for some $T < \infty $. If $ ({u_0},P{v_0}) > 0,0 < E(0) - \mathcal{G}({u_0}) < \alpha {({u_0},P{v_0})^2}/4(2\alpha + 1)({u_0},P{u_0})$ and if u exists on $ [0,\infty )$, then (u,Pu) grows at least exponentially. If $({u_0},P{v_0}) > 0$ and $\alpha {({u_0},P{v_0})^2}/4(2\alpha + 1)({u_0},P{u_0}) \leq E(0) - \mathcal{G}({u_0}) < \tfrac{1}{2}{({u_0},P{v_0})^2}/({u_0},P{u_0})$ and if the solution exists on $[0,\infty )$, then (u,Pu) grows at least as fast as ${t^2}$. A number of examples are given.
A divergent weighted orthonormal series of broken line Franklin functions
Coke S.
Reed
23-28
Abstract: The purpose of this paper is to define a differentiable function F and an inner product on the space of continuous functions on [0,1] in such a way that the Fourier expansion of F obtained by orthonormalizing the broken line Franklin functions according to this inner product is divergent.
A theory of Stiefel harmonics
Stephen S.
Gelbart
29-50
Abstract: An explicit theory of special functions is developed for the homogeneous space $ SO(n)/SO(n - m)$ generalizing the classical theory of spherical harmonics. This theory is applied to describe the decomposition of the Fourier operator on $n \times m$ matrix space in terms of operator valued Bessel functions of matrix argument. Underlying these results is a hitherto unnoticed relation between certain irreducible representations of $SO(n)$ and the polynomial representations of $GL(m,{\mathbf{C}})$.
Lower semicontinuity of integral functionals
Leonard D.
Berkovitz
51-57
Abstract: It is shown that the integral functional $I(y,z) = {\smallint _G}f(t,y(t),z(t))d\mu$ is lower semicontinuous on its domain with respect to the joint strong convergence of $ {y_k} \to y$ in $ {L_p}(G)$ and the weak convergence of ${z_k} \to z$ in ${L_p}(G)$, where $1 \leq p \leq \infty$ and $1 \leq q \leq \infty$, under the following conditions. The function $f:(t,x,w) \to f(t,x,w)$ is measurable in t for fixed (x, w), is continuous in (x, w) for a.e. t, and is convex in w for fixed (t, x).
Sequence of regular finitely additive set functions
J. D.
Stein
59-66
Abstract: The purpose of this paper is to deduce versions of Phillips' lemma and the Vitali-Hahn-Saks theorem, with weaker conditions placed on the set functions and convergence conditions than is usually required.
Monotone decompositions of continua not separated by any subcontinua
Eldon J.
Vought
67-78
Abstract: Let M be a compact, metric continuum that is separated by no subcontinuum. If such a continuum has a monotone, upper semicontinuous decomposition, the elements of which have void interior and for which the quotient space is a simple closed curve, then it is said to be of type
Symmetric jump processes
Martin L.
Silverstein
79-96
Abstract: We use the theory of Dirichlet spaces to construct symmetric Markov processes of pure jump type and to identify the Lévy measures for these processes. Particular attention is paid to lattice and hard sphere systems which interact through speed change and exclusion.
Interpolation in a classical Hilbert space of entire functions
Robert M.
Young
97-114
Abstract: Let H denote the Paley-Wiener space of entire functions of exponential type $\pi$ which belong to ${L^2}( - \infty ,\infty )$ on the real axis. A sequence $\{ {\lambda _n}\}$ of distinct complex numbers will be called an interpolating sequence for H if $ TH \supset {l^2}$, where T is the mapping defined by $Tf = \{ f({\lambda _n})\} $. If in addition $\{ {\lambda _n}\}$ is a set of uniqueness for H, then $ \{ {\lambda _n}\}$ is called a complete interpolating sequence. The following results are established. If $ \operatorname{Re} ({\lambda _{n + 1}}) - \operatorname{Re} ({\lambda _n}) \geq \gamma > 1$ and if the imaginary part of ${\lambda _n}$ is sufficiently small, then $\{ {\lambda _n}\}$ is an interpolating sequence. If $\vert\operatorname{Re} ({\lambda _n}) - n\vert \leq L \leq (\log 2)/\pi \;( - \infty < n < \infty )$ and if the imaginary part of ${\lambda _n}$ is uniformly bounded, then $\{ {\lambda _n}\}$ is a complete interpolating sequence and $\{ {e^{i{\lambda _n}t}}\} $ is a basis for ${L^2}( - \pi ,\pi )$. These results are used to investigate interpolating sequences in several related spaces of entire functions of exponential type.
Hereditary $QI$-rings
Ann K.
Boyle
115-120
Abstract: We consider in this paper rings in which every quasi-injective right R-module is injective. These rings will be referred to as right QI-rings. For a hereditary ring, this is equivalent to the condition that R be noetherian and a right V-ring. We also consider rings in which proper cyclic right R-modules are injective. These are right QI-rings which are either semisimple or right hereditary, right Ore domains in which indecomposable injective right R-modules are either simple or isomorphic to the injective hull of $ {R_R}$.
Countable box products of ordinals
Mary Ellen
Rudin
121-128
Abstract: The countable box product of ordinals is examined in the paper for normality and paracompactness. The continuum hypothesis is used to prove that the box product of countably many $ \sigma$-compact ordinals is paracompact and that the box product of another class of ordinals is normal. A third class trivially has a nonnormal product.
Representing measures and topological type of finite bordered Riemann surfaces
David
Nash
129-138
Abstract: A finite bordered Riemann surface $ \mathcal{R}$ with s boundary components and interior genus g has first Betti number $r = 2g + s - 1$. Let a be any interior point of $ \mathcal{R}$ and $ {e_a}$ denote evaluation at a on the usual hypo-Dirichlet algebra associated with $ \mathcal{R}$. We establish some connections between the topological and, more strongly, the conformal type of $\mathcal{R}$ and the geometry of ${\mathfrak{M}_a}$ the set of representing measures for ${e_a}$. For example, we show that if ${\mathfrak{M}_a}$ has an isolated extreme point, then $\mathcal{R}$ must be a planar surface. Several questions posed by Sarason are answered through exhausting the possibilities for the case $r = 2$.
Linear operators and vector measures
J. K.
Brooks;
P. W.
Lewis
139-162
Abstract: Compact and weakly compact operators on function spaces are studied. Those operators are characterized by properties of finitely additive set functions whose existence is guaranteed by Riesz representation theorems.
Global residues and intersections on a complex manifold
James R.
King
163-199
Abstract: This paper is the study of a class of forms $\eta$ on a complex manifold V which are smooth on $V - W$ and have poles of kernel type on a complex submanifold W of codimension d; such a form is one whose pull-back to the monoidal transform of V along W has a logarithmic pole. A global existence theorem is proved which asserts that any smooth form $\varphi$ on W of filtration s (no (p, q) components with $p < s$) is the residue of a form $\eta$ of filtration $s + d$ such that $d\eta$ is smooth on V. This result is used to construct global kernels for $\bar \partial$ which establish similar global existence theorems for W with singularities. We then establish formulas connecting intersection and wedge product on the d-cohomology theory of Dolbeault which preserve the Hodge filtration. A number of results are also proved on the integrability of ${f^\ast}\eta$ where f is a rather general holomorphic map.
K\"ahler differentials and differential algebra in arbitrary characteristic
Joseph
Johnson
201-208
Abstract: Let L and K be differential fields with L an extension of K. It is shown how the module of Kähler differentials $ \Omega _{L/K}^1$ can be used to ``linearize'' properties of a differential field extension $L/K$. This is done without restriction on the characteristic p and yields a theory which for $p \ne 0$ is no harder than the case $p = 0$. As an application a new proof of the Ritt basis theorem is given.
The geometry of flat Banach spaces
R. E.
Harrell;
L. A.
Karlovitz
209-218
Abstract: A Banach space is flat if the girth of its unit ball is 4 and if the girth is achieved by some curve. (Equivalently, its unit ball can be circumnavigated along a centrally symmetric path whose length is 4.) Some basic geometric properties of flat Banach spaces are given. In particular, the term flat is justified.
Analytic equivalence in the disk algebra
Hugh E.
Warren
219-226
Abstract: The notion of analytically equivalent domains can be extended from the complex plane to commutative Banach algebras with identity. In $C(X)$ a domain equivalent to the unit ball must have a boundary that is in a certain sense continuous. This paper shows that in the disk algebra ``continuous'' must be replaced with ``analytic.'' These results set limits in the classical Riemann mapping theorem on how smoothly the mapping can respond to changes in the domain being mapped.
Groups, semilattices and inverse semigroups. I, II
D. B.
McAlister
227-244
Abstract: An inverse semigroup S is called proper if the equations $ea = e = {e^2}$ together imply ${a^2} = a$ for each a, $a,e \in S$. In this paper a construction is given for a large class of proper inverse semigroups in terms of groups and partially ordered sets; the semigroups in this class are called P-semigroups. It is shown that every inverse semigroup divides a P-semigroup in the sense that it is the image, under an idempotent separating homomorphism, of a full subsemigroup of a P-semigroup. Explicit divisions of this type are given for $\omega $-bisimple semigroups, proper bisimple inverse semigroups, semilattices of groups and Brandt semigroups.
A probabilistic approach to $H\sp{p}(R\sp{d})$
D.
Stroock;
S. R. S.
Varadhan
245-260
Abstract: The relationship between ${H^p}({R^d}),1 \leq p < \infty$, and the integrability of certain functionals of Brownian motion is established using the connection between probabilistic and analytic notions of functions with bounded mean oscillation. An application of this relationship is given in the derivation of an interpolation theorem for operators taking $ {H^1}({R^d})$ to ${L^1}({R^d})$.
Weighted norm inequalities for fractional integrals
Benjamin
Muckenhoupt;
Richard
Wheeden
261-274
Abstract: The principal problem considered is the determination of all nonnegative functions, $V(x)$, such that $\left\Vert{T_\gamma }f(x)V(x)\right\Vert _q \leq C\left\Vert f(x)V(x)\right\Vert _p$ where the functions are defined on ${R^n},0 < \gamma < n,1 < p < n/\gamma ,1/q = 1/p - \gamma /n$, C is a constant independent of f and ${T_\gamma }f(x) = \smallint f(x - y)\vert y{\vert^{\gamma - n}}dy$. The main result is that $ V(x)$ is such a function if and only if $\displaystyle {\left( {\frac{1}{{\vert Q\vert}}\int_Q {{{[V(x)]}^q}dx} } \right... ...{\frac{1}{{\vert Q\vert}}\int_Q {{{[V(x)]}^{ - p'}}dx} } \right)^{1/p'}} \leq K$ where Q is any n dimensional cube, $\vert Q\vert$ denotes the measure of Q, $q = \infty$ and a weighted version of the Sobolev imbedding theorem are also proved.
$H\sp{r,}\,\sp{\infty }(R)$- and $W\sp{r,\infty }(R)$-splines
Philip W.
Smith
275-284
Abstract: Let E be a subset of R the real line and $f:E \to R$. Necessary and sufficient conditions are derived for $\inf (\left\Vert{D^r}x\right\Vert _{{L^\infty }}:x{\vert _E} = f)$ to have a solution. When restricted to quasi-uniform partitions E, necessary and sufficient conditions are derived for the solution to be in ${L^\infty }$. For finite partitions E it is shown that a solution to the $ {L^\infty }$ infimum problem can be obtained by solving $\inf (\left\Vert{D^r}x\right\Vert _{{L^p}}:x{\vert _E} = f)$ and letting p go to infinity. In this way it was discovered that solutions to the $ {L^\infty }$ problem could be chosen to be piecewise polynomial (of degree r or less). The solutions to the ${L^p}$ problem are called $ {H^{r,p}}$-splines and were studied extensively by Golomb in [3].
Convex hulls and extreme points of some families of univalent functions
D. J.
Hallenbeck
285-292
Abstract: The closed convex hull and extreme points are obtained for the functions which are convex, starlike, and close-to-convex and in addition are real on $( - 1,1)$. We also obtain this result for the functions which are convex in the direction of the imaginary axis and real on $( - 1,1)$. Integral representations are given for the hulls of these families in terms of probability measures on suitable sets. We also obtain such a representation for the functions $f(z)$ analytic in the unit disk, normalized and satisfying $\alpha < 1$. These results are used to solve extremal problems. For example, the upper bounds are determined for the coefficients of a function subordinate to some function satisfying
The norm of the $L\sp{p}$-Fourier transform on unimodular groups
Bernard
Russo
293-305
Abstract: We discuss sharpness in the Hausdorff Young theorem for unimodular groups. First the functions on unimodular locally compact groups for which equality holds in the Hausdorff Young theorem are determined. Then it is shown that the Hausdorff Young theorem is not sharp on any unimodular group which contains the real line as a direct summand, or any unimodular group which contains an Abelian normal subgroup with compact quotient as a semidirect summand. A key tool in the proof of the latter statement is a Hausdorff Young theorem for integral operators, which is of independent interest. Whether the Hausdorff Young theorem is sharp on a particular connected unimodular group is an interesting open question which was previously considered in the literature only for groups which were compact or locally compact Abelian.
A summation formula involving $\sigma (N)$
C.
Nasim
307-317
Abstract: The ${L^2}$ theories are known of the summation formula involving ${\sigma _k}(n)$, the sum of the kth power of divisors of n, as coefficients, for all k except $k = 1$. In this paper, techniques are used to overcome the extra convergence difficulty of the case $k = 1$, to establish a symmetric formula connecting the sums of the form $ \sum {\sigma _1}(n){n^{ - 1/2}}f(n)$ and $\sum {{\sigma _1}} (n){n^{ - 1/2}}g(n)$, where $ f(x)$ and $g(x)$ are Hankel transforms of each other.
The concordance diffeomorphism group of real projective space
Robert
Wells
319-337
Abstract: Let ${P_r}$ be r-dimensional real projective space with r odd, and let ${\pi _0}{\text{Diff}^ + }:{P_r}$ be the group of orientation preserving diffeomorphisms $ {P_r} \to {P_r}$ factored by the normal subgroup of those concordant (= pseudoisotopic) to the identity. The main theorem of this paper is that for $r \equiv 11 \bmod 16$ the group ${\pi _0}{\text{Diff}^ + }:{P_r}$ is isomorphic to the homotopy group ${\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}})$, where $ k = d{2^L} - r - 1$ with $L \geq \varphi ((r + 1)/2)$ and $d{2^L} \geq r + 1$. The function $\varphi$ is denned by $\varphi (l) = \{ i\vert < i \leq l,i \equiv 0,1,2,4 \bmod (8)\}$. The theorem is proved by introducing a cobordism version of the mapping torus construction; this mapping torus construction is a homomorphism $t:{\pi _0}{\text{Diff}^ + }:{P_r} \to {\Omega _{r + 1}}(v)$ for $ r \equiv 11 \bmod 16$ and ${\Omega _{r + 1}}(v)$ a suitable Lashof cobordism group. It is shown that t is an isomorphism onto the torsion subgroup ${\Omega _{r + 1}}(v)$, and that this subgroup is isomorphic to ${\pi _{r + 1 + k}}({P_\infty }/{P_{k - 1}})$ as above. Then one reads off from Mahowald's tables of ${\pi _{n + m}}({P_\infty }/{P_{m - 1}})$ that $ {\pi _0}{\text{Diff}^ + }:{P_{11}} = {Z_2}$ and ${\pi _0}{\text{Diff}^ + }:{P_{27}} = 6{Z_2}$.
Fields of fractions for group algebras of free groups
Jacques
Lewin
339-346
Abstract: Let KF be the group algebra over the commutative field K of the free group F. It is proved that the field generated by KF in any Mal'cev-Neumann embedding for KF is the universal field of fractions $ U(KF)$ of KF. Some consequences are noted. An example is constructed of an embedding $ KF \subset D$ into a field D with $ D\;\not\simeq\;U(KF)$. It is also proved that the generalized free product of two free groups can be embedded in a field.
Further results on prime entire functions
Fred
Gross;
Chung Chun
Yang
347-355
Abstract: Let H denote the set of all the entire functions $f(z)$ of the form: $f(z) \equiv h(z){e^{p(z)}} + k(z)$ where $ p(z)$ is a nonconstant polynomial of degree m, and $h(\nequiv\;0)$, $k(\nequiv$ constant) are two entire functions of order less than m. In this paper, a necessary and sufficient condition for a function in H to be a prime is established. Several generalizations of known results follow. Some sufficient conditions for primeness of various subclasses of H are derived. The methods used in the proofs are based on Nevanlinna's theory of meromorphic functions and some elementary facts about algebraic functions.
van Kampen's theorem for $n$-stage covers
J. C.
Chipman
357-370
Abstract: A version of van Kampen's theorem is obtained for covers whose members do not share a common point and whose pairwise intersection need not be connected.
Expansion of entire functions of several complex variables having finite growth
P. K.
Kamthan;
Manjul
Gupta
371-382
Abstract: We consider the space $\chi$ of entire functions of two complex variables having a finite nonzero order point and type, equip it with the natural locally convex topology, such that $\chi$ becomes a Fréchet space. Apart from finding the characterization of continuous linear functionals, linear transformations on $ \chi$, we have obtained the necessary and sufficient conditions for a double sequence in $\chi$ to be a proper base.
Many-one reducibility within the Turing degrees of the hyperarithmetic sets $H\sb{a}(x)$
G. C.
Nelson
1-44
Abstract: Spector [13] has proven that the hyperarithmetic sets $ {H_a}(x)$ and $ {H_b}(x)$ have the same Turing degree iff $\vert a\vert = \vert b\vert$. Y. Moschovakis has proven that the sets ${H_a}(x)$ under many-one reducibility for $\vert a\vert = \gamma$ and $a \in \mathcal{O}$ have nontrivial reducibility properties if $\gamma$ is not of the form $\alpha + 1$ or $ \alpha + \omega$ for any ordinal a. In particular, he proves that there are chains of order type ${\omega _1}$ and incomparable many-one degrees within these Turing degrees. In Chapter II, we extend this result to show that any countable partially ordered set can be embedded in the many-one degrees within these Turing degrees. In Chapter III, we prove that if $\gamma$ is also not of the form $\alpha + {\omega ^2}$ for some ordinal a, then there is no minimal many-one degree of the form $ {H_a}(x)$ in this Turing degree, answering a question of Y. Moschovakis posed in [8]. In fact, we prove that given ${H_a}(x)$ there are ${H_b}(x)$ and ${H_c}(x)$ both many-one reducible to ${H_a}(x)$ with incomparable many-one degrees, $ \vert a\vert = \vert b\vert = \vert c\vert = \gamma $.
On Fourier transforms
C.
Nasim
45-51
Abstract: If $f(x)$ and $g(x)$ satisfy the equations $\displaystyle g(x) = \frac{d}{{dx}}\int _0^\infty \frac{1}{t}f(t){k_1}(xt)dt,\quad f(x) = \frac{d}{{dx}}\int _0^\infty \frac{1}{t}g(t){k_1}(xt)dt,$ then we call f and g a pair of ${k_1}$-transforms, where $\displaystyle {k_1} = \frac{1}{{2\pi i}}\int _{{\raise0.5ex\hbox{$\scriptstyle ... ...r0.25ex\hbox{$\scriptstyle 2$}} + i\infty }\frac{{K(s)}}{{1 - s}}{x^{1 - s}}ds.$ In this paper alternative sets of conditions are established for f and g to be ${k_1}$-transform provided $K(s)$ is decomposable in a special way. These conditions involve simpler functions, which replace the kernel ${k_1}(x)$. Results are proved for the function spaces ${L^2}$. The necessary and sufficient conditions are established for the two functions to be self-reciprocal. Conditions are given for generating pairs of transforms for a given kernel. Two examples are given at the end to illustrate the methods and the advantage of the results.
Splitting the tangent bundle
Wolf
Iberkleid
53-59
Abstract: We determine those unoriented cobordism classes which can be realized by a manifold whose tangent bundle splits into a sum of real (complex) line bundles.
Free products of topological groups which are $k\sb{\omega }$-spaces
Edward T.
Ordman
61-73
Abstract: Let G and H be topological groups and $G \ast H$ their free product topologized in the manner due to Graev. The topological space $G \ast H$ is studied, largely by means of its compact subsets. It is established that if G and H are ${k_\omega }$-spaces (respectively: countable CW-complexes) then so is $G \ast H$. These results extend to countably infinite free products. If G and H are ${k_\omega }$-spaces, $G \ast H$ is neither locally compact nor metrizable, provided G is nondiscrete and H is nontrivial. Incomplete results are obtained about the fundamental group $ \pi (G \ast H)$. If $ {G_1}$ and ${H_1}$ are quotients (continuous open homomorphic images) of G and H, then ${G_1} \ast {H_1}$ is a quotient of $ G \ast H$.
Interpolation between $H\sp{p}$ spaces: the real method
C.
Fefferman;
N. M.
Rivière;
Y.
Sagher
75-81
Abstract: The interpolation spaces in the Lions-Peetre method between $ {H^p}$ spaces, $0 < p < \infty$, are calculated.
Analytic centers and analytic diameters of planar continua
Steven
Minsker
83-93
Abstract: This paper contains some basic results about analytic centers and analytic diameters, concepts which arise in Vitushkin's work on rational approximation. We use Carathéodory's theorem to calculate $ \beta (K,z)$ in the case in which K is a continuum in the complex plane. This leads to the result that, if $g:\Omega (K) \to \Delta (0;1)$ is the normalized Riemann map, then $\beta (g,0)/\gamma (K)$ is the unique analytic center of K and $\beta (K) = \gamma (K)$. We also give two proofs of the fact that $\beta (g,0)/\gamma (K) \in {\text{co}}\;(K)$. We use Bieberbach's and Pick's theorems to obtain more information about the geometric location of the analytic center. Finally, we obtain inequalities for $\beta (E,z)$ for arbitrary bounded planar sets E.
Norm inequalities for the Littlewood-Paley function $g\sp{\ast} \sb{\lambda }$
Benjamin
Muckenhoupt;
Richard L.
Wheeden
95-111
Abstract: Weighted norm inequalities for ${L^p}$ and ${H^p}$ are derived for the Littlewood-Paley function $g_\lambda ^ \ast $. New results concerning the boundedness of this function are obtained, by a different method of proof, even in the unweighted case. The proof exhibits a connection between $g_\lambda ^\ast$ and a maximal function for harmonic functions which was introduced by C. Fefferman and E. M. Stein. A new and simpler way to determine the behavior of this maximal function is given.
Free $S\sp{1}$ actions and the group of diffeomorphisms
Kai
Wang
113-127
Abstract: Let ${S^1}$ act linearly on ${S^{2p - 1}} \times {D^{2q}}$ and ${D^{2p}} \times {S^{2q - 1}}$ and let $f:{S^{2p - 1}} \times {S^{2q - 1}} \to {S^{2p - 1}} \times {S^{2q - 1}}$ be an equivariant diffeomorphism. Then there is a well-defined ${S^1}$ action on ${S^{2p - 1}} \times {D^{2q}}{ \cup _f}{D^{2p}} \times {S^{2q - 1}}$. An ${S^1}$ action on a homotopy sphere is decomposable if it can be obtained in this way. In this paper, we will apply surgery theory to study in detail the set of decomposable actions on homotopy spheres.
Pointwise differentiability and absolute continuity
Thomas
Bagby;
William P.
Ziemer
129-148
Abstract: This paper is concerned with the relationships between ${L_p}$ differentiability and Sobolev functions. It is shown that if f is a Sobolev function with weak derivatives up to order k in ${L_p}$, and $0 \leq l \leq k$, then f has an ${L_p}$ derivative of order l everywhere except for a set which is small in the sense of an appropriate capacity. It is also shown that if a function has an ${L_p}$ derivative everywhere except for a set small in capacity and if these derivatives are in $ {L_p}$, then the function is a Sobolev function. A similar analysis is applied to determine general conditions under which the Gauss-Green theorem is valid.
Complex approximation for vector-valued functions with an application to boundary behaviour
Leon
Brown;
P. M.
Gauthier;
W.
Seidel
149-163
Abstract: This paper deals with the qualitative theory of uniform approximation by holomorphic functions. The first theorem is an extension to vector-valued mappings of N. U. Arakélian's theorem on uniform holomorphic approximation on closed sets. Our second theorem is on asymptotic approximation and yields, as in the scalar case, applications to cluster sets.
Extensions of the $v$-integral
J. R.
Edwards;
S. G.
Wayment
165-184
Abstract: In Representations for transformations continuous in the BV norm [J. R. Edwards and S. G. Wayment, Trans. Amer. Math. Soc. 154 (1971), 251-265] the $\nu$-integral is defined over intervals in $ {E^1}$ and is used to give a representation for transformations continuous in the BV norm. The functions f considered therein are real valued or have values in a linear normed space X, and the transformation $ T(f)$ is real or has values in a linear normed space Y. In this paper the $ \nu$-integral is extended in several directions: (1) The domain space to (a) $ {E^n}$, (b) an arbitrary space S, a field $\Sigma$ of subsets of S and a bounded positive finitely additive set function $\mu$ on $\Sigma$ (in this setting the function space is replaced by the space of finitely additive set functions which are absolutely continuous with respect to $ \mu$); (2) the function space to (a) bounded continuous, (b) $ {C_c}$, (c) ${C_0}$, (d) C with uniform convergence on compact sets; (3) range space X for the functions and Y for the transformation to topological vector spaces (not necessarily convex); (4) when X and Y are locally convex spaces, then a representation for transformations on a $ {C_1}$-type space of continuously differentiable functions with values in X is given.
The continuity of Arens' product on the Stone-\v Cech compactification of semigroups
Nicholas
Macri
185-193
Abstract: A discrete semigroup is said to have the compact semigroup property (c.s.p.) [the compact semi-semigroup property (c.s.s.p.)] if the multiplication Arens' product, on its Stone-Čech compactification, is jointly [separately] ${w^ \ast }$-continuous. We obtain an algebraic characterization of those semigroups which have c.s.p. by characterizing algebraically their almost periodic subsets. We show that a semigroup has c.s.p. if and only if each of its subsets is almost periodic. This characterization is employed to prove that for a cancellation semigroup to have c.s.p., it is necessary and sufficient that each of its countable subsets be almost periodic. We answer in the negative a heretofore open question--is c.s.p. equivalent to c.s.s.p.
Entire vectors and holomorphic extension of representations. II
Richard
Penney
195-207
Abstract: Let G be a connected, simply connected Lie group and let $ {G_c}$ be its complexification. Let U be a unitary representation of G. The space of vectors v at which U is holomorphically extendible to ${G_c}$ is denoted $\mathcal{H}_\infty ^\omega (U)$. In [9] we characterized those U for which $\mathcal{H}_\infty ^\omega $ is dense. In the present work we study $\mathcal{H}_\infty ^\omega $ as a topological vector space, proving e.g., that $\mathcal{H}_\infty ^\omega $ is a Montel space if U is irreducible and G is nilpotent. We prove a representation theorem for
An approach to fixed-point theorems on uniform spaces
E.
Tarafdar
209-225
Abstract: Diaz and Metcalf [2] have some interesting results on the set of successive approximations of a self mapping which is either a nonexpansion or a contraction on a metric space with respect to the set of fixed points of the mapping. We have extended most of these results to a Hausdorff uniform space. We have also proved a Banach's contraction mapping principle on a complete Hausdorff uniform space and indicated some applications in locally convex linear topological spaces.
Semicellularity, decompositions and mappings in manifolds
Donald
Coram
227-244
Abstract: If X is an arbitrary compact set in a manifold, we give algebraic criteria on X and on its embedding to determine that X has an arbitrarily small, closed neighborhood each component of which is a p-connected, piecewise linear manifold which collapses to a q-dimensional subpolyhedron from some p and q. This property generalizes cellularity. The criteria are in terms of UV properties and Alexander-Spanier cohomology. These criteria are then applied to decide when the components of a given compact set in a manifold are elements of a decomposition such that the quotient space is the n-sphere. Conversely, algebraic criteria are given for the point inverses of a map between manifolds to have arbitrarily small neighborhoods of the type mentioned above; these criteria are considerably weaker than for an arbitrary compact set.
On homeomorphisms of infinite-dimensional bundles. I
Raymond Y. T.
Wong
245-259
Abstract: In this paper we present several aspects of homeomorphism theory in the setting of fibre bundles modeled on separable infinite-dimensional Hilbert (Fréchet) spaces. We study (homotopic) negligibility of subsets, separation of sets, characterization of subsets of infinite-deficiency and extending homeomorphisms; in an essential way they generalize previously known results for manifolds. An important tool is a lemma concerning the lifting of a map to the total space of a bundle whose image misses a certain closed subset presented as obstruction; from this we are able to obtain a result characterizing all subsets of infinite deficiency (for bundles) by their restriction to each fibre. Other results then follow more or less routinely by employing the rather standard methods of infinite-dimensional topology.
On homeomorphisms of infinite dimensional bundles. II
T. A.
Chapman;
R. Y. T.
Wong
261-268
Abstract: This paper presents some aspects of homeomorphism theory in the setting of (fibre) bundles modeled on separable Hilbert manifolds and generalizes results previously established. The main result gives a characterization of subsets of infinite deficiency in a bundle by means of their restriction to the fibres, from which we are able to prove theorems of the following types: (a) mapping replacement, (b) separation of sets, (c) negligibility of subsets, and (d) extending homeomorphisms.
On homeomorphisms of infinite dimensional bundles. III
T. A.
Chapman;
R. Y. T.
Wong
269-276
Abstract: In this paper we continue the study of homeomorphisms and prove an analogue of the homeomorphism extension theorem for bundles modeled on Hilbert cube manifolds; thus we generalize previous results for Q-manifolds (Anderson-Chapman). This analogy, as in the case of manifolds, requires a consideration of proper maps and proper homotopies. The approach to the present problem is similar to that considered in our previous papers. Bear in mind several distinct difficulties occur in our setting.
$\lambda $ connected plane continua
Charles L.
Hagopian
277-287
Abstract: A continuum M is said to be $ {\mathbf{\lambda }}$ connected if any two distinct points of M can be joined by a hereditarily decomposable continuum in M. Recently this generalization of arcwise connectivity has been related to fixed point ptoblems in the plane. In particular, it is known that every ${\mathbf{\lambda }}$ connected nonseparating plane continuum has the fixed point property. The importance of arcwise connectivity is, to a considerable extent, due to the fact that it is a continuous invariant. To show that $ {\mathbf{\lambda }}$ connectivity has a similar feature is the primary purpose of this paper. Here it is proved that if M is a ${\mathbf{\lambda }}$ connected continuum and f is a continuous function of M into the plane, then $f(M)$ is $ {\mathbf{\lambda }}$ connected. It is also proved that every semiaposyndetic plane continuum is $ {\mathbf{\lambda }}$ connected.
Homeomorphisms of a certain cube with holes
Donald
Myers
289-299
Abstract: For some manifolds the group of isotopy classes of self-homeomorphisms is known. In this paper this group is computed for a well-known cube with two holes. Two related manifolds are defined and the groups of isotopy classes on these manifolds are given without proof. One of these cubes with holes is such that every homeomorphism is isotopic to the identity.
Rational approximation on product sets
Otto B.
Bekken
301-316
Abstract: Our object here is to study pointwise bounded limits, decomposition of orthogonal measures and distance estimates for $R({K_1} \times {K_2})$ where $ {K_1}$ and ${K_2}$ are compact sets in the complex plane.
Weighted join semilattices and transversal matroids
Richard A.
Brualdi
317-328
Abstract: We investigate join-semilattices in which each element is assigned a nonnegative weight in a strictly increasing way. A join-subsemilattice of a Boolean lattice is weighted by cardinality, and we give a characterization of these in terms of the notion of a spread. The collection of flats with no coloops (isthmuses) of a matroid or pregeometry, partially ordered by set-theoretic inclusion, forms a join-semilattice which is weighted by rank. For transversal matroids these join-semilattices are isomorphic to join-subsemilattices of Boolean lattices. Using a previously obtained characterization of transversal matroids and results on weighted join-semilattices, we obtain another characterization of transversal matroids. The problem of constructing a transversal matroid whose join-semilattice of flats is isomorphic to a given join-subsemilattice of a Boolean lattice is then investigated.
Generalized almost periodicity in groups
Henry W.
Davis
329-352
Abstract: A module of almost periodic functions on a group is closed with respect to a quite general seminorm. The new space of functions is characterized in terms of the internal properties of its members. This yields new characterizations of Besicovitch and Weyl almost periodic functions in a variety of group-theoretic settings. Eberlein's theorem that weakly almost periodic functions on the real line are Weyl almost periodic is extended to locally compact groups.
On deformations of homomorphisms of locally compact groups
Dong Hoon
Lee
353-361
Abstract: The rigidity of homomorphisms of compactly generated locally compact groups into Lie groups is investigated.
A proof that $\mathcal{H}^2$ and $\mathcal{T}^2$ are distinct measures
Lawrence R.
Ernst
363-372
Abstract: It is proven that there exists a subset E of ${{\mathbf{R}}^3}$ such that the two-dimensional $\mathcal{J}$ measure of E is less than its two-dimensional Hausdorff measure. E is the image under the usual isomorphism of $ {\mathbf{R}} \times {{\mathbf{R}}^2}$ onto $ {{\mathbf{R}}^3}$ of the Cartesian product of $\{ x: - 4 \leq x \leq 4\} $ and a Cantor type subset of $ {{\mathbf{R}}^2}$; the latter term in this product is the intersection of a decreasing sequence, every member of which is the union of certain closed circular disks.
Construction of automorphic forms and integrals
Douglas
Niebur
373-385
Abstract: It is well known that modular forms of positive dimension have Fourier coefficients given by certain infinite series involving Kloostermann sums and the modified Bessel function of the first kind. In this paper a functional equation which characterizes all such Fourier series is found. It is also shown that these Fourier series have a construction similar to that of Poincaré series of negative dimension.
The Fredholm spectrum of the sum and product of two operators
Jack
Shapiro;
Morris
Snow
387-393
Abstract: Let $C(X)$ denote the set of closed operators with dense domain on a Banach space X, and $ L(X)$ the set of all bounded linear operators on X. Let ${\mathbf{\Phi }}(X)$ denote the set of all Fredholm operators on X, and ${\sigma _{\mathbf{\Phi }}}(A)$ the set of all complex numbers $ {\mathbf{\lambda }}$ such that $ ({\mathbf{\lambda }} - A) \notin {\mathbf{\Phi }}(X)$. In this paper we establish conditions under which $ {\sigma _{\mathbf{\Phi }}}(A + B) \subseteq {\sigma _{\mathbf{\Phi }}}(A) + {\... ...} ) \subseteq {\sigma _{\mathbf{\Phi }}}(A) \cdot {\sigma _{\mathbf{\Phi }}}(B)$, and ${\sigma _\Phi }(AB) \subseteq {\sigma _\Phi }(A){\sigma _\Phi }(B)$.
Spectral orders, uniform integrability and Lebesgue's dominated convergence theorem
Kong Ming
Chong
395-404
Abstract: Using the 'spectral' order relations $\prec$ and $ \prec\prec$ introduced by Hardy, Littlewood and Pólya, we characterize the uniform integrability of a family of integrable functions. We also prove an extension and a 'converse' of the classical Lebesgue's dominated convergence theorem in terms of the 'spectral' orders $\prec$ and $ \prec\prec$.
Limit theorems for variational sums
William N.
Hudson;
Howard G.
Tucker
405-426
Abstract: Limit theorems in the sense of a.s. convergence, convergence in $ {L_1}$-norm and convergence in distribution are proved for variational series. In the first two cases, if g is a bounded, nonnegative continuous function satisfying an additional assumption at zero, and if $\{ X(t),0 \leq t \leq T\} $ is a stochastically continuous stochastic process with independent increments, with no Gaussian component and whose trend term is of bounded variation, then the sequence of variational sums of the form $\Sigma _{k = 1}^ng(X({t_{nk}}) - X({t_{n,k - 1}}))$ is shown to converge with probability one and in ${L_1}$-norm. Also, under the basic assumption that the distribution of the centered sum of independent random variables from an infinitesimal system converges to a (necessarily) infinitely divisible limit distribution, necessary and sufficient conditions are obtained for the joint distribution of the appropriately centered sums of the positive parts and of the negative parts of these random variables to converge to a bivariate infinitely divisible distribution. A characterization of all such limit distributions is obtained. An application is made of this result, using the first theorem, to stochastic processes with (not necessarily stationary) independent increments and with a Gaussian component.
Correction to: ``The separable closure of some commutative rings'' (Trans. Amer. Math. Soc. {\bf 170} (1972), 109--124)
Andy R.
Magid
427-430
A combinatorial approach to the diagonal $N$-representability problem
Mark Laurance
Yoseloff
1-41
Abstract: The problem considered is that of the diagonal N-representability of a pth-order reduced density matrix, $p \geq 2$, for a system of N identical fermions or bosons. A finite number M of allowable single particle states is assumed. The problem is divided into three cases, namely: Case I. $M = N + p$ ; Case II. $M < N + p$; Case III. $M > N + p$. Using the theory of polyhedral convex cones, a complete set of necessary and sufficient conditions is first found for Case I. This solution is then employed to find such conditions for Case II. For Case III, two algorithms are developed to generate solutions for the problem, and examples of the usage of these algorithms are given.
Classical quotient rings
Robert C.
Shock
43-48
Abstract: Throughout R is a ring with right singular ideal $ Z(R)$. A right ideal K of R is rationally closed if $ {x^{ - 1}}K = \{ y \in R:xy \in K\}$ is not a dense right ideal for all $x \in R - K$. A ring R is a Cl-ring if R is (Goldie) right finite dimensional, $ R/Z(R)$ is semiprime, $ Z(R)$ is rationally closed, and $Z(R)$ contains no closed uniform right ideals. These rings include the quasi-Frobenius rings as well as the semiprime Goldie rings. The commutative Cl-rings have Cl-classical quotient rings. The injective ones are congenerator rings. In what follows, R is a Cl-ring. A dense right ideal of R contains a right nonzero divisor. If R satisfies the minimum condition on rationally closed right ideals then R has a classical Artinian quotient ring. The complete right quotient ring Q (also called the Johnson-Utumi maximal quotient ring) of R is a Cl-ring. If R has the additional property that bR is dense whenever b is a right nonzero divisor, then Q is classical. If Q is injective, then Q is classical.
A bounded difference property for classes of Banach-valued functions
Wilbur P.
Veith
49-56
Abstract: Let $A(G,E)$ denote the set of functions f from a Hausdorff topological group G to a Banach space E such that the range of f is relatively compact in E and $\phi \circ f$ is in $A(G,C)$ for each $\phi$ in the dual of E, where $ A(G,C)$ is a translation-invariant ${C^\ast}$ algebra of bounded, continuous, complex-valued functions on G with respect to the supremum norm and complex conjugation. $A(G,E)$ has the bounded difference property if whenever $F:G \to E$ is a bounded function such that ${\Delta _t}F(x) = F(tx) - F(x)$ is in $ A(G,E)$ for each t in G, then F is also an element of $ A(G,E)$. A condition on $ A(G,C)$ and a condition on E are given under which $A(G,E)$ has the bounded difference property. The condition on $A(G,C)$ is satisfied by both the class of almost periodic functions and the class of almost automorphic functions.
Square integrable representations of semisimple Lie groups
Juan A.
Tirao
57-75
Abstract: Let D be a bounded symmetric domain. Let G be the universal covering group of the identity component ${A_0}(D)$ of the group of all holomorphic diffeomorphisms of D onto itself. In this case, any G-homogeneous vector bundle $E \to D$ admits a natural structure of G-homogeneous holomorphic vector bundles. The vector bundle $E \to D$ must be holomorphically trivial, since D is a Stein manifold. We exhibit explicitly a holomorphic trivialization of $E \to D$ by defining a map $ \Phi :G \to {\text{GL}}(V)$ (V being the fiber of the vector bundle) which extends the classical ``universal factor of automorphy'' for the action of ${A_0}(D)$ on D. Then, we study the space H of all square integrable holomorphic sections of $E \to D$. The natural action of G on H defines a unitary irreducible representation of G. The representations obtained in this way are square integrable over $ G/Z$ (Z denotes the center of G) in the sense that the absolute values of their matrix coefficients are in ${L_2}(G/Z)$.
Indecomposable polytopes
Walter
Meyer
77-86
Abstract: The space of summands (with respect to vector addition) of a convex polytope in n dimensions is studied. This space is shown to be isomorphic to a convex pointed cone in Euclidean space. The extreme rays of this cone correspond to similarity classes of indecomposable polytopes. The decomposition of a polytope is described and a bound is given for the number of indecomposable summands needed. A means of determining indecomposability from the equations of the bounding hyperplanes is given.
Goldie-like conditions on Jordan matrix rings
Daniel J.
Britten
87-98
Abstract: In this paper Goldie-like conditions are put on a Jordan matrix ring $J = H({R_n},{\gamma _a})$ which are necessary and sufficient for R to be a $\ast$-prime Goldie ring or a Cayley-Dickson ring. Existing theory is then used to obtain a Jordan ring of quotients for J.
Class numbers of real quadratic number fields
Ezra
Brown
99-107
Abstract: This article is a study of congruence conditions, modulo powers of two, on class number of real quadratic number fields $Q(\sqrt d )$, for which d has at most three distinct prime divisors. Techniques used are those associated with Gaussian composition of binary quadratic forms.
Injective modules and localization in noncommutative Noetherian rings
Arun Vinayak
Jategaonkar
109-123
Abstract: Let $\mathfrak{S}$ be a semiprime ideal in a right Noetherian ring R and $ \mathcal{C}(\mathfrak{S}) = \{ c \in R\vert[c + \mathfrak{S}$ regular in $R/\mathfrak{S}\} $. We investigate the following two conditions: $ ({\text{A}})\;\mathcal{C}(\mathfrak{S})$ is a right Ore set in R. $ ({\text{B}})\;\mathcal{C}(\mathfrak{S})$ is a right Ore set in R and the right ideals of $ {R_{\mathfrak{S}}}$, the classical right quotient ring of R w.r.t. $ \mathcal{C}(\mathfrak{S})$ are closed in the $ J({R_{\mathfrak{S}}})$-adic topology. The main results show that conditions (A) and (B) can be characterized in terms of the injective hull of the right R-module $R/\mathfrak{S}$. The J-adic completion of a semilocal right Noetherian ring is also considered.
Primitive satisfaction and equational problems for lattices and other algebras
Kirby A.
Baker
125-150
Abstract: This paper presents a general method of solving equational problems in all equational classes of algebras whose congruence lattices are distributive, such as those consisting of lattices, relation algebras, cylindric algebras, orthomodular lattices, lattice-ordered rings, lattice-ordered groups, Heyting algebras, other lattice-ordered algebras, implication algebras, arithmetic rings, and arithmetical algebras.
An algebraic property of the \v Cech cohomology groups which prevents local connectivity and movability
James
Keesling
151-162
Abstract: Let C denote the category of compact Hausdorff spaces and $H:C \to HC$ be the homotopy functor. Let $S:C \to SC$ be the functor of shape in the sense of Holsztyński for the projection functor H. Let X be a continuum and ${H^n}(X)$ denote n-dimensional Čech cohomology with integer coefficients. Let ${A_x} = {\text{char}}\;{H^1}(X)$ be the character group of ${H^1}(X)$ considering ${H^1}(X)$ as a discrete group. In this paper it is shown that there is a shape morphism $F \in {\text{Mor}_{SC}}(X,{A_X})$ such that ${F^\ast}:{H^1}({A_X}) \to {H^1}(X)$ is an isomorphism. It follows from the results of a previous paper by the author that there is a continuous mapping $f:X \to {A_X}$ such that $S(f) = F$ and thus that ${f^\ast}:{H^1}({A_X}) \to {H^1}(X)$ is an isomorphism. This result is applied to show that if X is locally connected, then ${H^1}(X)$ has property L. Examples are given to show that X may be locally connected and $ {H^n}(X)$ not have property L for $n > 1$. The result is also applied to compact connected topological groups. In the last section of the paper it is shown that if X is compact and movable, then for every integer n, $ {H^n}(X)/{\operatorname{Tor}}\;{H^n}(X)$ has property L. This result allows us to construct peano continua which are nonmovable. An example is given to show that ${H^n}(X)$ itself may not have property L even if X is a finite-dimensional movable continuum.
Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients
Jean-Pierre
Gossez
163-205
Abstract: Variational boundary value problems for quasilinear elliptic systems in divergence form are studied in the case where the nonlinearities are nonpolynomial. Monotonicity methods are used to derive several existence theorems which generalize the basic results of Browder and Leray-Lions. Some features of the mappings of monotone type which arise here are that they act in nonreflexive Banach spaces, that they are unbounded and not everywhere defined, and that their inverse is also unbounded and not everywhere defined.
Koebe sequences of arcs and normal meromorphic functions
Stephen
Dragosh
207-222
Abstract: Let f be a normal meromorphic function in the unit disk. An estimate for the growth of the modulus of f on a Koebe sequence of arcs is obtained; the estimate is in terms of the order of normality of f. An immediate consequence of the estimate is the following theorem due to F. Bagemihl and W. Seidel: A nonconstant normal meromorphic function has no Koebe values. Another consequence is that each level set of a nonconstant normal meromorphic function cannot contain a Koebe sequence of arcs provided the order of normality of f is less than a certain positive constant ${C^\ast}$.
A Siegel formula for orthogonal groups over a function field
Stephen J.
Haris
223-231
Abstract: We obtain a Siegel formula for a quadratic form over a function field, by establishing the convergence of the corresponding Eisenstein-Siegel series directly, then via the Hasse principle, that of the associated Poisson formula.
Conditions for a TVS to be homeomorphic with its countable product
Wesley E.
Terry
233-242
Abstract: C. Bessaga has given conditions for a Banach space to be homeomorphic with its countable product. In this paper, we extend and generalize these results to complete metric topological vector spaces by using infinite dimensional techniques.
A new characterization of tame $2$-spheres in $E\sp{3}$
Lawrence R.
Weill
243-252
Abstract: It is shown in Theorem 1 that a 2-sphere S in ${E^3}$ is tame from $A = {\text{Int}}\;S$ if and only if for each compact set $F \subset A$ there exists a 2-sphere $ S'$ with complementary domains $A' = {\text{Int}}\;S',B' = {\text{Ext}}\;S'$, such that $F \subset A' \subset \overline {A'} \subset A$ and for each $x \in S'$ there exists a path in $\rho (F,S)$ which runs from x to a point $ y \in S$. Furthermore, the theorem holds when A is replaced by B, $ A'$ by $B',B'$ by $A'$, and Int by Ext. Two applications of this characterization are given. Theorem 2 states that a 2-sphere is tame from the complementary domain C if for arbitrarily small $ \varepsilon > 0$, S has a metric $ \varepsilon$-envelope in C which is a 2-sphere. Theorem 3 answers affirmatively the following question: Is a 2-sphere $S \subset {E^3}$ tame in ${E^3}$ if there exists an $\varepsilon > 0$ such that if a, $ b \in S$ satisfy $\rho (a,b) < \varepsilon$, then there exists a path in S of spherical diameter $\rho (a,b)$ which connects a and b?
Standard polynomials in matrix algebras
Louis H.
Rowen
253-284
Abstract: Let ${M_n}(F)$ be an $n \times n$ matrix ring with entries in the field F, and let ${S_k}({X_1}, \ldots ,{X_k})$ be the standard polynomial in k variables. Amitsur-Levitzki have shown that ${S_{2n}}({X_1}, \ldots ,{X_{2n}})$ vanishes for all specializations of ${X_1}, \ldots ,{X_{2n}}$ to elements of $ {M_n}(F)$. Now, with respect to the transpose, let $ M_n^ - (F)$ be the set of antisymmetric elements and let $M_n^ + (F)$ be the set of symmetric elements. Kostant has shown using Lie group theory that for n even ${S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})$ vanishes for all specializations of ${X_1}, \ldots ,{X_{2n - 2}}$ to elements of $M_n^ - (F)$. By strictly elementary methods we have obtained the following strengthening of Kostant's theorem: ${S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})$ vanishes for all specializations of ${X_1}, \ldots ,{X_{2n - 2}}$ to elements of $M_n^ - (F)$, for all n. ${S_{2n - 1}}({X_1}, \ldots ,{X_{2n - 1}})$ vanishes for all specializations of ${X_1}, \ldots ,{X_{2n - 2}}$ to elements of $M_n^ - (F)$ and of ${X_{2n - 1}}$ to an element of $M_n^ + (F)$, for all n. $ {S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})$ vanishes for all specializations of $ {X_1}, \ldots ,{X_{2n - 3}}$ to elements of $ M_n^ - (F)$ and of ${X_{2n - 2}}$ to an element of $M_n^ + (F)$, for n odd. These are the best possible results if F has characteristic 0; a complete analysis of the problem is also given if F has characteristic 2.
On dynamical systems with the specification property
Karl
Sigmund
285-299
Abstract: A continuous transformation T of a compact metric space X satisfies the specification property if one can approximate distinct pieces of orbits by single periodic orbits with a certain uniformity. There are many examples of such transformations which have recently been studied in ergodic theory and statistical mechanics. This paper investigates the relation between Tinvariant measures and the frequencies of T-orbits. In particular, it is shown that every invariant measure (and even every closed connected subset of such measures) has generic points, but that the set of all generic points is of first category in X. This generalizes number theoretic results concerning decimal expansions and normal numbers.
Ergodic measure preserving transformations with quasi-discrete spectrum
James B.
Robertson
301-311
Abstract: It is shown that an ergodic measure preserving transformation with quasi-discrete spectrum is conjugate to: (a) the skew-product of an ergodic measure preserving transformation with discrete spectrum and a measurable family of totally ergodic measure preserving transformations with quasi-discrete spectrum; (b) a factor of the direct product of an ergodic measure preserving transformation with discrete spectrum and a totally ergodic measure preserving transformation with quasi-discrete spectrum. Sufficient conditions are given to insure that an ergodic measure preserving transformation with quasi-discrete spectrum is conjugate to the direct product of an ergodic measure preserving transformation with discrete spectrum and a totally ergodic measure preserving transformation with quasi-discrete spectrum.
Convolution operators on $G$-holomorphic functions in infinite dimensions
Philip J.
Boland;
Seán
Dineen
313-323
Abstract: For a complex vector space E, let ${H_G}(E)$ denote the space of G (Gateaux)-holomorphic functions on $E\;(f:E \to C$ is G-holomorphic if the restriction of f to every finite dimensional subspace of E is holomorphic in the usual sense). The most natural topology on ${H_G}(E)$ is that of uniform convergence on finite dimensional compact subsets of E. A convolution operator A on ${H_G}(E)$ is a continuous linear mapping $ A:{H_G}(E) \to {H_G}(E)$ such that A commutes with translations. The concept of a convolution operator generalizes that of a differential operator with constant coefficients. We prove that if A is a convolution operator on $ {H_G}(E)$, then the kernel of A is the closed linear span of the exponential polynomials contained in the kernel. In addition, we show that any nonzero convolution operator on $ {H_G}(E)$ is a surjective mapping.
Finite extensions of minimal transformation groups
Robert J.
Sacker;
George R.
Sell
325-334
Abstract: In this paper we shall study homomorphisms $p:W \to Y$ on minimal transformation groups. We shall prove, in the case that W and Y are metrizable, that W is a finite (N-to-1) extension of Y if and only if W is an N-fold covering space of Y and p is a covering map. This result places no further restrictions on the acting group. We shall then use this characterization to investigate the question of lifting an equicontinuous structure from Y to W. We show that, under very weak restrictions on the acting group, this lifting is always possible when W is a finite extension of Y.
Singular perturbations for systems of linear partial differential equations
A.
Livne;
Z.
Schuss
335-343
Abstract: We consider the system of linear partial differential equations $ \varepsilon {A^{ij}}u_{ij}^\varepsilon + {B^i}u_i^\varepsilon + C{u^\varepsilon } = f$ where ${A^{ij}},{B^i}$ are symmetric $m \times m$ matrices and -- C is a sufficiently large positive definite matrix. We prove that under suitable conditions ${\left\Vert {{u^\varepsilon } - u} \right\Vert _{{L^2}}} \leq c\surd \varepsilon {\left\Vert f \right\Vert _{{H^1}}}$ where u is the solution of a suitable boundary value problem for the system ${B^i}{u_i} + Cu = f$.
Property $SUV\sp{\infty }$ and proper shape theory
R. B.
Sher
345-356
Abstract: A class of spaces called the $ SU{V^\infty }$ spaces has arisen in the study of a possibly noncompact variant of cellularity. These spaces play a role in this new theory analogous to that of the $ U{V^\infty }$ spaces in cellularity theory. Herein it is shown that the locally compact metric space X is an $SU{V^\infty }$ space if and only if there exists a tree T such that X and T have the same proper shape. This result is then used to classify the proper shapes of the $SU{V^\infty }$ spaces, two such being shown to have the same proper shape if and only if their end-sets are homeomorphic. Also, a possibly noncompact analog of property $U{V^n}$, called $SU{V^n}$, is defined and it is shown that if X is a closed connected subset of a piecewise linear n-manifold, then X is an $ SU{V^n}$ space if and only if X is an $ SU{V^\infty }$ space. Finally, it is shown that a locally finite connected simplicial complex is an $ SU{V^\infty }$ space if and only if all of its homotopy and proper homotopy groups vanish.
Global dimension of tiled orders over commutative noetherian domains
Vasanti A.
Jategaonkar
357-374
Abstract: Let R be a commutative noetherian domain and $ \Lambda = ({\Lambda _{ij}}) \subseteq {M_n}(R)$ be a tiled R-order. The main result of this paper is the following Theorem. Let gl $\dim R = d < \infty$ and $\Lambda$ a triangular tiled R-order (i.e., $ {\Lambda _{ij}} = R$ whenever $i \leq j$). Then the following three conditions are equivalent: (1) gl $\dim \Lambda < \infty $; (2) ${\Lambda _{i,i - 1}} = R$ or gl $ \dim \;(R/{\Lambda _{i,i - 1}}) < \infty$, whenever $2 \leq i \leq n$; (3) gl $\dim \Lambda \leq d(n - 1)$. If $d = 1$ or 2 then the upper bound in the above theorem is best possible. We give a sufficient condition for an arbitrary tiled R-order $\Lambda$ to be of finite global dimension.
Semigroups with midunits
Janet E.
Ault
375-384
Abstract: A semigroup has a midunit u if $aub = ab$ for all a and b. It is the purpose of this paper to explore semigroups with midunits, and to find an analog to the group of units. In addition, two constructions will be given which produce some semigroups with midunits, and an abstract characterization will be made of certain of these semigroups.
Countable unions of totally projective groups
Paul
Hill
385-391
Abstract: Let the p-primary abelian group G be the set-theoretic union of a countable collection of isotype subgroups $ {H_n}$ of countable length. We prove that if ${H_n}$ is totally projective for each n, then G must be totally projective. In particular, an ascending sequence of isotype and totally projective subgroups of countable length leads to a totally projective group. The result generalizes and complements a number of theorems appearing in various articles in the recent literature. Several applications of the main result are presented.
On structure spaces of ideals in rings of continuous functions
David
Rudd
393-403
Abstract: A ring of continuous functions is a ring of the form $C(X)$, the ring of all continuous real-valued functions on a completely regular Hausdorff space X. With each ideal I of $ C(X)$, we associate certain subalgebras of $C(X)$, and discuss their structure spaces. We give necessary and sufficient conditions for two ideals in rings of continuous functions to have homeomorphic structure spaces.
Isometries of $\sp{\ast} $-invariant subspaces
Arthur
Lubin
405-415
Abstract: We consider families of increasing $^\ast$-invariant subspaces of ${H^2}(D)$, and from these we construct canonical isometrics from certain ${L^2}$ spaces to ${H^2}$. We give necessary and sufficient conditions for these maps to be unitary, and discuss the relevance to a problem concerning a concrete model theory for a certain class of operators.
Regular self-injective rings with a polynomial identity
Efraim P.
Armendariz;
Stuart A.
Steinberg
417-425
Abstract: This paper studies maximal quotient rings of semiprime P. I.-rings; such rings are regular, self-injective and satisfy a polynomial identity. We show that the center of a regular self-injective ring is regular self-injective; this enables us to establish that the center of the maximal quotient ring of a semiprime P. I.-ring R is the maximal quotient ring of the center of R, as well as some other relationships. We give two decompositions of a regular self-injective ring with a polynomial identity which enable us to show that such rings are biregular and are finitely generated projective modules over their center.
Erratum to: ``Weakly almost periodic functionals on the Fourier algebra'' (Trans. Amer. Math. Soc. {\bf 185} (1973), 501--514)
Charles F.
Dunkl;
Donald E.
Ramirez
427
Some $p$-groups of maximal class
R. J.
Miech
1-47
Abstract: This paper deals with the construction of some p-groups of maximal class.
Counting commutators
R. J.
Miech
49-61
Abstract: Let G be a group generated by x and y, ${G_2}$ be the commutator subgroup of G, and ${G_1}$ be the group generated by y and $ {G_2}$. This paper contains explicit expansions of $ {y^{{x^m}}}$ modulo [ ${G_2},{G_2},{G_2}$] and ${(xy)^m}$ modulo [ $ {G_1},{G_1},{G_1}$]. The motivation for these results stem from the p-groups of maximal class, for a large number of these groups have $[{G_1},{G_1},{G_1}] = 1$.
Fixed point theorems for certain classes of semigroups of mappings
Mo Tak
Kiang
63-76
Abstract: Fixed point theorems for commuting semigroups of self-mappings are considered in this paper. A generalization of the classical Markov-Kakutani theorem is first given. This is followed by a fixed point theorem for commutative semigroups of continuous asymptotically-nonexpansive self-mappings on a weakly compact, convex subset of a strictly convex Banach space.
Explicit class field theory for rational function fields
D. R.
Hayes
77-91
Abstract: Developing an idea of Carlitz, I show how one can describe explicitly the maximal abelian extension of the rational function field over $ {{\mathbf{F}}_q}$ (the finite field of q elements) and the action of the idèle class group via the reciprocity law homomorphism. The theory is closely analogous to the classical theory of cyclotomic extensions of the rational numbers.
Minimal sequences in semigroups
Mohan S.
Putcha
93-106
Abstract: In this paper we generalize a result of Tamura on $\delta$-indecomposable semigroups. Based on this, the concept of a minimal sequence between two points, and from a point to another, is introduced. The relationship between two minimal sequences between the same points is studied. The rank of a semigroup S is defined to be the supremum of the lengths of the minimal sequences between points in S. The semirank of a semigroup S is defined to be the supremum of the lengths of the minimal sequences from a point to another in S. Rank and semirank are further studied.
On the structure of the set of solutions of equations involving $A$-proper mappings
P. M.
Fitzpatrick
107-131
Abstract: Let X and Y be Banach spaces having complete projection schemes (say, for example, they have Schauder bases). We consider various properties of mappings $T:D \subset X \to Y$ which are either Approximation-proper (A-proper) or the uniform limit of such mappings. In §1 general properties, including those of the generalized topological degree, of such mappings are discussed. In §2 we give sufficient conditions in order that the solutions of an equation involving a nonlinear mapping be a continuum. The conditions amount to requiring that the generalized topological degree not vanish, and that the mapping involved be the uniform limit of well structured mappings. We devote §3 to proving a result connecting the topological degree of an A-proper Fréchet differentiable mapping to the degree of its derivative. Finally, in §4, various Lipschitz-like conditions are discussed in an A-proper framework, and constructive fixed point and surjectivity results are obtained.
Subgroups of groups of central type
Kathleen M.
Timmer
133-161
Abstract: Let $\lambda$ be a linear character on the center Z of a finite group Z of a finite group H, such that (1) ${\lambda ^H} = \sum\nolimits_{i = 1}^p {{\phi _i}(1){\phi _i}}$ where the ${\phi _i}$'s are inequivalent irreducible characters on H of the same degree, and (2) if $\sum\nolimits_{i = 1}^p {{m_i}{\phi _i}(x) = 0}$ for some $x \in H$ and nonnegative integers ${m_i}$, then either ${\phi _i}(x) = 0$ for all i or ${m_i} = {m_j}$ for all i, j. The object of the paper is to describe finite groups which satisfy conditions (1) and (2) in terms of the multiplication of the group. If S is a p Sylow subgroup of the group H, and $R = S \cdot Z$, then H satisfies conditions (1) and (2) if and only if (a) $\{ x \in H:{x^{ - 1}}{h^{ - 1}}xh \in Z \Rightarrow \lambda ({x^{ - 1}}{h^{ - 1}}xh) = 1,h \in H\} /Z$ consists of elements of order a power of p in $H/Z$, and these elements form p conjugacy classes of $H/Z$, and (b) the elements of $ \{ x \in R:{x^{ - 1}}{r^{ - 1}}xr \in Z \Rightarrow \lambda ({x^{ - 1}}{r^{ - 1}}xr) = 1,r \in R\} /Z$ form p conjugacy classes of $R/Z$.
Lattices of topological extensions
John
Mack;
Marlon
Rayburn;
Grant
Woods
163-174
Abstract: For completely regular Hausdorff spaces, we consider topological properties P which are akin to compactness in the sense of Herrlich and van der Slot and satisfy the equivalent of Mrowka's condition (W). The algebraic structure of the family of tight extensions of X (which have P and contain no proper extension with that property) is studied. Where X has P locally but not globally, the relations between the complete lattice $ {P^ \ast }(X)$ of those tight extensions which are above the maximal one-point extension and the topology of the P-reflection are investigated and conditions found under which ${P^\ast}(X)$ characterizes $\gamma X - X$. The results include those of Magill on the lattice of compactifications of a locally compact space, and other applications are considered.
Equivariant method for periodic maps
Wu Hsiung
Huang
175-183
Abstract: The notion of coherency with submanifolds for a Morse function on a manifold is introduced and discussed in a general way. A Morse inequality for a given periodic transformation which compares the invariants called qth Euler numbers on fixed point set and the invariants called qth Lefschetz numbers of the transformations is thus obtained. This gives a fixed point theorem in terms of qth Lefschetz number for arbitrary q.
Asymptotic solutions of linear Volterra integral equations with singular kernels
J. S. W.
Wong;
R.
Wong
185-200
Abstract: Volterra integral equations of the form $a(t) \in C(0,\infty ) \cap {L_1}(0,1)$. Explicit asymptotic forms are obtained for the solutions, when the kernels $a(t)$ have a specific asymptotic representation.
Rank $r$ solutions to the matrix equation $XAX\sp{T}=C,\,A$ alternate, over ${\rm GF}(2\sp{y})$
Philip G.
Buckhiester
201-209
Abstract: Let ${\text{GF}}(q)$ denote a finite field of characteristic two. Let ${V_n}$ denote an n-dimensional vector space over $ {\text{GF}}(q)$. An $n \times n$ symmetric matrix A over ${\text{GF}}(q)$ is said to be an alternate matrix if A has zero diagonal. Let A be an $n \times n$ alternate matrix over ${\text{GF}}(q)$ and let C be an $s \times s$ symmetric matrix over ${\text{GF}}(q)$. By using Albert's canonical forms for symmetric matrices over fields of characteristic two, the number $ N(A,C,n,s,r)$ of $s \times n$ matrices X of rank r over $ {\text{GF}}(q)$ such that $XA{X^T} = C$ is determined. A symmetric bilinear form on $ {V_n} \times {V_n}$ is said to be alternating if $ f(x,x) = 0$, for each x in ${V_n}$. Let f be such a bilinear form. A basis $({x_1}, \ldots ,{x_\rho },{y_1}, \ldots ,{y_\rho }),n = 2\rho$, for ${V_n}$ is said to be a symplectic basis for $ {V_n}$ if $f({x_i},{x_j}) = f({y_i},{y_j}) = 0$ and $ f({x_i},{y_j}) = {\delta _{ij}}$, for each i, $j = 1,2, \ldots ,\rho$. In determining the number $N(A,C,n,s,r)$, it is shown that a symplectic basis for any subspace of ${V_n}$, can be extended to a symplectic basis for $ {V_n}$. Furthermore, the number of ways to make such an extension is determined.
The structure of completely regular semigroups
Mario
Petrich
211-236
Abstract: The principal result is a construction of completely regular semigroups in terms of semilattices of Rees matrix semigroups and their translational hulls. The main body of the paper is occupied by considerations of various special cases based on the behavior of either Green's relations or idempotents. The influence of these special cases on the construction in question is studied in considerable detail. The restrictions imposed on Green's relations consist of the requirement that some of them be congruences, whereas the restrictions on idempotents include various covering conditions or the requirement that they form a subsemigroup.
Volumes of images of varieties in projective space and in Grassmannians
H.
Alexander
237-249
Abstract: If V is a complex analytic subvariety of pure dimension k in the unit ball in $ {{\mathbf{C}}^n}$ which does not contain the origin, then the 2k-volume of V equals the measure computed with multiplicity of the set of $(n - k)$-complex subspaces through the origin which meet V. The measure of this set computed without multiplicity is a smaller quantity which is nevertheless bounded below by a number depending only on the distance from V to the origin. As an application we characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. The complex analysis which we shall need will be developed in the context of uniform algebras.
Approximation of analytic functions on compact sets and Bernstein's inequality
M. S.
Baouendi;
C.
Goulaouic
251-261
Abstract: The characterization of analytic functions defined on a compact set K in $ {{\mathbf{R}}_N}$ by their polynomial approximation is possible if and only if K satisfies some ``Bernstein type inequality", estimating any polynomial P in some neighborhood of K using the supremum of P on K. Some criterions and examples are given. Approximation by more general sets of analytic functions is also discussed.
Galois theory for fields $K/k$ finitely generated
Nickolas
Heerema;
James
Deveney
263-274
Abstract: Let K be a field of characteristic $p \ne 0$. A subgroup G of the group $ {H^t}(K)$ of rank t higher derivations $ (t \leq \infty )$ is Galois if G is the group of all d in $ {H^t}(K)$ having a given subfield h in its field of constants where K is finitely generated over h. We prove: G is Galois if and only if it is the closed group (in the higher derivation topology) generated over K by a finite, abelian, independent normal iterative set F of higher derivations or equivalently, if and only if it is a closed group generated by a normal subset possessing a dual basis. If $t < \infty$ the higher derivation topology is discrete. M. Sweedler has shown that, in this case, h is a Galois subfield if and only if $ K/h$ is finite modular and purely inseparable. Also, the characterization of Galois groups for $t < \infty $ is closely related to the Galois theory announced by Gerstenhaber and and Zaromp. In the case $t = \infty$, a subfield h is Galois if and only if $K/h$ is regular. Among the applications made are the following: (1) ${ \cap _n}h({K^{{p^n}}})$ is the separable algebraic closure of h in K, and (2) if $K/h$ is algebraically closed, $K/h$ is regular if and only if $K/h({K^{{p^n}}})$ is modular for $n > 0$.
Quasi-bounded and singular functions
Maynard
Arsove;
Heinz
Leutwiler
275-302
Abstract: A general formulation is given for the concepts of quasi-bounded and singular functions, thereby extending to a much broader class of functions the concepts initially formulated by Parreau in the harmonic case. Let $\Omega$ be a bounded Euclidean region. With the underlying space taken as the class $\mathcal{M}$ of all nonnegative functions u on $\Omega$ admitting superharmonic majorants, an operator S is introduced by setting Su equal to the regularization of the infimum over $\lambda \geq 0$ of the regularized reduced functions for $ {(u - \lambda )^ + }$. Quasi-bounded and singular functions are then defined as those u for which $Su = 0$ and $Su = u$, respectively. A development based on properties of the operator S leads to a unified theory of quasi-bounded and singular functions, correlating earlier work of Parreau (1951), Brelot (1967), Yamashita(1968), Heins (1969), and others. It is shown, for example, that a nonnegative function u on $ \Omega$ is quasi-bounded if and only if there exists a nonnegative, increasing, convex function $\varphi$ on $ [0,\infty ]$ such that $\varphi (x)/x \to + \infty$ as $x \to \infty$ and $ \varphi \circ u$ admits a superharmonic majorant. Extensions of the theory are made to the vector lattice generated by the positive cone of functions u in $ \mathcal{M}$ satisfying $ Su \leq u$.
Differentiability of solutions to hyperbolic initial-boundary value problems
Jeffrey B.
Rauch;
Frank J.
Massey
303-318
Abstract: This paper establishes conditions for the differentiability of solutions to mixed problems for first order hyperbolic systems of the form $ (\partial /\partial t - \sum {A_j}\partial /\partial {x_j} - B)u = F$ on $[0,T] \times \Omega ,Mu = g$ on $[0,T] \times \partial \Omega ,u(0,x) = f(x),x \in \Omega$. Assuming that ${\mathcal{L}_2}$ a priori inequalities are known for this equation, it is shown that if $F \in {H^s}([0,T] \times \Omega ),g \in {H^{s + 1/2}}([0,T] \times \partial \Omega ),f \in {H^s}(\Omega )$ satisfy the natural compatibility conditions associated with this equation, then the solution is of class ${C^p}$ from [0, T] to $ {H^{s - p}}(\Omega ),0 \leq p \leq s$. These results are applied to mixed problems with distribution initial data and to quasi-linear mixed problems.
Canonical forms and principal systems for general disconjugate equations
William F.
Trench
319-327
Abstract: It is shown that the disconjugate equation (1) $Lx \equiv (1/{\beta _n})(d/dt) \cdot (1/{\beta _{n - 1}}) \cdots (d/dt)(1/{\beta _1})(d/dt)(x/{\beta _0}) = 0$ , a $< t < b$, where ${\beta _i} > 0$, and (2) ${\beta _i} \in C(a,b)$, can be written in essentially unique canonical forms so that ${\smallint ^b}{\beta _i}dt = \infty ({\smallint _a}{\beta _i}dt = \infty )$ for $1 \leq i \leq n - 1$. From this it follows easily that (1) has solutions ${x_1}, \ldots ,{x_n}$ which are positive in (a, b) near $b(a)$ and satisfy ${\lim _{t \to b}} - {x_i}(t)/{x_j}(t) = 0({\lim _{t \to a}} + {x_i}(t)/{x_j}(t) = \infty )$ for $ 1 \leq i < j \leq n$. Necessary and sufficient conditions are given for (1) to have solutions $ {y_1}, \ldots ,{y_n}$ such that $ {\lim _{t \to b}} - {y_i}(t)/{y_j}(t) = {\lim _{t \to a}} + {y_j}(t)/{y_i}(t) = 0$ for $ 1 \leq i < j \leq n$. Using different methods, P. Hartman, A. Yu. Levin and D. Willett have investigated the existence of fundamental systems for (1) with these properties under assumptions which imply the stronger condition
Pairs of compacta and trivial shape
Sibe
Mardešić
329-336
Abstract: Let (X, Y, A), ${\text{sh}}\;X = {\text{sh}}\;X' = 0$ and ${\text{sh}}(X,A) = {\text{sh}}(X',A')$, which in view of an example of Borsuk shows that for compact metric pairs the ANR-system approach to shapes differs from the Borsuk approach.
An equiconvergence theorem for a class of eigenfunction expansions
C. G. C.
Pitts
337-350
Abstract: A recent result of Muckenhoupt concerning the convergence of the expansion of an arbitrary function in terms of the Hermite series of orthogonal polynomials is generalised to a class of orthogonal expansions which arise from an eigenfunction problem associated with a second-order linear differential equation.
Fundamental constants for rational functions
S. J.
Poreda;
E. B.
Saff;
G. S.
Shapiro
351-358
Abstract: Suppose R is a rational function with n poles all of which lie inside $\Gamma$, a closed Jordan curve. Lower bounds for the uniform norm of the difference $R - p$ on $\Gamma$, where p is any polynomial, are obtained (in terms of the norm of R on $\Gamma$). In some cases these bounds are independent of $\Gamma$ as well as R and p. Some related results are also given.
A multiplier theorem for Fourier transforms
James D.
McCall
359-369
Abstract: A function f analytic in the upper half-plane ${\Pi ^ + }$ is said to be of class ${E_p}({\Pi ^ + })(0 < p < \infty )$ if there exists a constant C such that $\smallint _{ - \infty }^\infty \vert f(x + iy){\vert^p}dx \leq C < \infty $ for all $y > 0$. These classes are an extension of the ${H_p}$ spaces of the unit disc U. For f belonging to ${E_p}({\Pi ^ + })(0 < p \leq 2)$, there exists a Fourier transform f with the property that $f(z) = 2{(\pi )^{ - 1}}\smallint _0^\infty \hat f(t){e^{izt}}dt$. This makes it possible to give a definition for the multiplication of ${E_p}({\Pi ^ + })(0 < p \leq 2)$ into ${L_q}(0,\infty )$ that is analogous to the multiplication of ${H_p}(U)$ into ${l_q}$. In this paper, we consider the case $ 0 < p < 1$ and $ p \leq q$ and derive a necessary and sufficient condition for multiplying ${E_p}({\Pi ^ + })$ into ${L_q}(0,\infty )$.
Weakly almost periodic and uniformly continuous functionals on the Fourier algebra of any locally compact group
Edmond E.
Granirer
371-382
Abstract: We define for any locally compact group G, the space of bounded uniformly continuous functionals on Ĝ, $UCB(\hat G)$, in the context of P. Eymard [Bull. Soc. Math. France 92 (1964), 181-236. MR 37 #4208] (for notations see next section). For $u \in A(G)$ let $ {u^ \bot } = \{ \phi \in VN(G);\phi [A(G)u] = 0\}$. Theorem. If for some norm separable subspace $X \subset VN(G)$ and some positive definite $0 \ne u \in A(G),UCB(\hat G) \subset$ norm closure $ [W(\hat G) + X + {u^ \bot }]$ then G is discrete. If G is discrete then $UCB(\hat G) \subset AP(\hat G) \subset W(\hat G)$.
Spaces homeomorphic to $(2\sp{a})\sb{a}$. II
H. H.
Hung;
S.
Negrepontis
1-30
Abstract: Topological characterizations and properties of the spaces ${({2^\alpha })_\alpha }$, where $\alpha$ is an infinite regular cardinal, are studied; the principal interest lying in the significance that these spaces have in questions of existence of ultrafilters (or of elements of the Stone-Čech compactification of spaces) with special properties. The main results are (a) the characterization theorem of the spaces ${({2^\alpha })_\alpha }$ in terms of a simple set of conditions, and (b) the $\alpha$-Baire category property of ${({2^\alpha })_\alpha }$ and the stability of the class of spaces homeomorphic to ${({2^\alpha })_\alpha }$ (or to ${({\alpha ^\alpha })_\alpha }$) when taking intersections of at most $\alpha$ open and dense subsets of ${({2^\alpha })_\alpha }$. Among the applications of these results are the following. Assuming ${\alpha ^ + } = {2^\alpha }$, the class of spaces homeomorphic to ${({2^{({\alpha ^ + })}})_{{\alpha ^ + }}}$ includes the space of uniform ultrafilters on $ \alpha$ with the ${P_{{\alpha ^ + }}}$-topology ${(U(\alpha ))_{{\alpha ^ + }}}$, its subspaces of good ultrafilters and/or Rudin-Keisler minimal ultrafilters. Assuming ${\omega ^ + } = {2^\omega }$ (or in some cases only Martin's axiom), the class of spaces homeomorphic to $ {({2^{({\omega ^ + })}})_{{\omega ^ + }}}$ includes the following: The space $ {(\beta X\backslash X)_{{\omega ^ + }}}$ where X is a noncompact locally compact realcompact space such that $\vert C(X)\vert \leq {2^\omega }$ and its subspaces of $ {P_{{\omega ^ + }}}$-points of $ \beta X\backslash X$ and/or (if X is in addition a metric space without isolated elements) the remote points. In particular the existence of good and/or Rudin-Keisler minimal ultrafilters and the existence of P-points and/or remote points follows always from a Baire category type of argument.
Moufang loops of small order. I
Orin
Chein
31-51
Abstract: The main result of this paper is the determination of all nonassociative Moufang loops of orders $\leq 31$. Combinatorial type methods are used to consider a number of cases which lead to the discovery of 13 loops of the type in question and prove that there can be no others. All of the loops found are isomorphic to all of their loop isotopes, are solvable, and satisfy both Lagrange's theorem and Sylow's main theorem. In addition to finding the loops referred to above, we prove that Moufang loops of orders p, ${p^2}$, ${p^3}$ or pq (for p and q prime) must be groups. Finally, a method is found for constructing nonassociative Moufang loops as extensions of nonabelian groups by the cyclic group of order 2.
$p$-absolutely summing operators and the representation of operators on function spaces
John William
Rice
53-75
Abstract: We introduce a class of p-absolutely summing operators which we call p-extending. We show that for a logmodular function space $A(K)$, an operator $ T:A(K) \to X$ is p-extending if and only if there exists a probability measure $\mu$ on K such that T extends to an isometry $ T:{A^p}(K,\mu ) \to X$. We use this result to give necessary and sufficient conditions under which a bounded linear operator is isometrically equivalent to multiplication by z on a space $ {L^p}(K,\mu )$ and certain Hardy spaces $ {H^p}(K,\mu )$.
On existence and uniqueness for a new class of nonlinear partial differential equations using compactness methods and differential difference schemes
Theodore E.
Dushane
77-96
Abstract: We prove existence and uniqueness results for the following Cauchy problem in the half plane $t \geq 0:{u_t} + {(f(u))_x} + {u_{xxx}} = {g_1}(u){u_{xx}} + {g_2}(u){({u_x})^2} + p(u),u(x,0) = {u_0}(x)$, where $u = u(x,t)$ and the subscripts indicate partial derivatives. We require that f, ${g_1}$, ${g_2}$, and p be sufficiently smooth and satisfy $f(u) = {u^{2n + 1}},{g_1}(u) = {u^{2m}},{g_2}(u) = - {u^{2r + 1}}$, and $p(u) = - {u^{2s + 1}}$, for n, m, r, and s nonnegative integers. To obtain a global solution in time, we perturb the equation by $- \epsilon ({u_{xxxx}} - {(f(u))_{xx}})$. The perturbed equation is solved locally (in time) and this solution is extended to a global solution by means of a priori estimates on the ${H^s}$ (of space) norms of the local solution. These estimates require the use of new nonlinear functionals. We then obtain the solution to the original equation as a limit of solutions to the perturbed equation as $\epsilon$ tends to zero using the standard techniques. For the related periodic problem, for which we require $ u(x + 2\pi ,t) = u(x,t)$ for all $t \geq 0$, we also obtain existence and uniqueness results. We prove existence for this problem via similar techniques to the nonperiodic case. We then consider differential difference schemes for the periodic initial value problem and show that we may obtain the solution as the limit of solutions to an appropriate scheme.
Recapturing $H\sp{2}$-functions on a polydisc
D. J.
Patil
97-103
Abstract: Let ${U^2}$ be the unit polydisc and $ {T^2}$ its distinguished boundary. If $E \subset {T^2}$ is a set of positive measure and the restriction to E of a function f in $ {H^2}({U^2})$ is given then an algorithm to recapture f is developed.
On proper homotopy theory for noncompact $3$-manifolds
E. M.
Brown;
T. W.
Tucker
105-126
Abstract: Proper homotopy groups analogous to the usual homotopy groups are defined. They are used to prove, modulo the Poincaré conjecture, that a noncompact 3-manifold having the proper homotopy type of a closed product $F \times [0,1]$ or a half-open product $F \times [0,1)$ where F is a 2-manifold is actually homeomorphic to $F \times [0,1]$ or $F \times [0,1)$, respectively. By defining a concept for noncompact manifolds similar to boundary-irreducibility, a well-known result of Waldhausen concerning homotopy and homeomorphism type of compact 3-manifolds is extended to the noncompact case.
Wave equations with finite velocity of propagation
Stephen J.
Berman
127-148
Abstract: If B is a selfadjoint translation-invariant operator on the space ${L^2}$ of complex-valued functions on n-dimensional Euclidean space which are square-summable with respect to Lebesgue measure, then the wave equation ${d^2}F/d{t^2} + {B^2}F = 0$ has the solution $ F(t) = (\cos tB)f + ((\sin tB)/B)g$, for f and g in ${L^2}$. In the classical case in which $- {B^2}$ is the Laplacian, this solution has finite velocity of propagation in the sense that (letting supp denote support of a function) ${\text{supp}}\;F(t) \subset ({\text{supp}}\;f \cup {\text{supp}}\;g) + {K_t}$ for all f and g and some compact set ${K_t}$ independent of f and g. We show that a converse holds, namely, if $\cos \;tB$ has finite velocity of propagation (that is, if $ {\text{supp}}\,(\cos tB)f \subset {\text{supp}}\;f + {K_t}$ for all f and some compact ${K_t}$) for three values of t whose reciprocals are independent over the rationals, then $ {B^2}$ must be a second order differential operator. If Euclidean space is replaced by a locally compact abelian group which does not contain the real line as a subgroup, then $ \cos \;tB$ has finite velocity of propagation for all t if and only if it is convolution with a distribution ${T_t}$ such that all ${T_t}$ are supported on a compact open subgroup. Problems of a similar nature are discussed for compact connected abelian groups and for the nonabelian group $ {\text{SL}}(2,{\mathbf{R}})$.
Uniformly distributed sequences in locally compact groups. I
Leonora
Benzinger
149-165
Abstract: We investigate the notion of uniformly distributed sequences in locally compact groups. Our main result is the following: A locally compact group G possesses a uniformly distributed sequence if and only if it possesses a sequence whose homomorphic images are dense in each of the compact quotients of G.
Uniformly distributed sequences in locally compact groups. II
Leonora
Benzinger
167-178
Abstract: We consider the following question. When is there a compactification $ {G_0}$ of a locally compact group G (recall that a compact group $ {G_0}$ is a compactification of G if there is a continuous homomorphism $ \phi :G \to {G_0}$ so that $ \phi (G)$ is dense in G) with continuous homomorphism $\phi :G \to {G_0}$ with the property that $\{ {g_\nu }\}$ is uniformly distributed in G if and only if $\{ \phi ({g_\nu })\}$ is uniformly distributed in $ {G_0}$? Such a compactification ${G_0}$ is called a D-compactification of G. We obtain a solution to this problem and thereby generalize to locally compact groups some results of Berg, Rajagopalan, and Rubel concerning D-compactifications of locally compact abelian groups.
Representations and classifications of stochastic processes
Dudley Paul
Johnson
179-197
Abstract: We show that to every stochastic process X one can associate a unique collection $(\Phi ,{\Phi _ + },T(t),E(U),{p^\ast})$ consisting of a linear space $\Phi$, on which is defined a linear functional ${p^ \ast }$, together with a convex subset ${\Phi _ + }$ which is invariant under the semigroup of operators $T(t)$ and the resolution of the identity $ E(U)$. The joint distributions of X, there being one version for each $\phi \in {\Phi _ + }$, are then given by $\displaystyle {P_\phi }(X({t_1}) \in {U_1}, \cdots ,X({t_1} + \cdots + {t_n}) \in {U_n}) = {p^ \ast }E({U_n})T({t_n}) \cdots E({U_1})T({t_1})\phi .$ To each $\phi$ contained in the extreme points ${\Phi _{ + + }}$ of $ {\Phi _ + }$ and each time t we find a probability measure $P_t^ \ast (\phi, \cdot )$ on ${\Phi _{ + + }}$ such that $T(t)\phi = {\smallint _{{\Phi _{ + + }}}}\psi P_t^ \ast (\phi ,d\psi )$. $P_t^ \ast$ is the transition probability function of a temporally homogeneous Markov process ${X^ \ast }$ on $ {\Phi _{ + + }}$ for which there exists a function f such that $X = f({X^ \ast })$. We show that in a certain sense ${X^ \ast }$ is the smallest of all Markov processes Y for which there exists a function g with $X = g(Y)$. We then apply these results to a class of stochastic process in which future and past are independent given the present and the conditional distribution, on the past, of a collection of random variables in the future.
Nonlinear approximation in uniformly smooth Banach spaces
Edward R.
Rozema;
Philip W.
Smith
199-211
Abstract: John R. Rice [Approximation of functions. Vol. II, Addison-Wesley, New York, 1969] investigated best approximation from a nonlinear manifold in a finite dimensional, smooth, and rotund space. The authors define the curvature of a manifold by comparing the manifold with the unit ball of the space and suitably define the ``folding'' of a manifold. Rice's Theorem 11 extends as follows: Theorem. Let X be a uniformly smooth Banach space, and $ F:{R^n} \to X$ be a homeomorphism onto $M = F({R^n})$. Suppose $\nabla F(a)$ exists for each a in X, $\nabla F$ is continuous as a function of a, and $ \nabla F(a) \cdot {R^n}$ has dimension n. Then, if M has bounded curvature, there exists a neighborhood of M each point of which has a unique best approximation from M. A variation theorem was found and used which locates a critical point of a differentiable functional defined on a uniformly rotund space Y. [See M. S. Berger and M. S. Berger, Perspectives in nonlinearity, Benjamin, New York, 1968, p. 58ff. for a similar result when $Y = {R^n}$.] The paper is concluded with a few remarks on Chebyshev sets.
Symmetric norm ideals and relative conjugate ideals
Norberto
Salinas
213-240
Abstract: In this paper some aspects of the algebraic structure of the ring of all bounded linear operators on an infinite dimensional separable complex Hilbert space are discussed. In particular, a comparison criterion for maximal and minimal norm ideals is established. Also, a general notion of the conjugate of an ideal relative to another ideal is studied and some questions concerning joins and intersections of ideals are solved.
Semirings and $T\sb{1}$ compactifications. I
Douglas
Harris
241-258
Abstract: With each infinite cardinal ${\omega _\mu }$ is associated a topological semiring $ {{\mathbf{F}}_\mu }$, whose underlying space is finite complement topology on the set of all ordinals less than ${\omega _\mu }$, and whose operations are the natural sum and natural product defined by Hessenberg. The theory of the semirings $ {C_\mu }(X)$ of maps from a space X into ${{\mathbf{F}}_\mu }$ is developed in close analogy with the theory of the ring $C(X)$ of continuous real-valued functions; the analogy is not on the surface alone, but may be pursued in great detail. With each semiring a structure space is associated; the structure space of ${C_\mu }(X)$ for sufficiently large ${\omega _\mu }$ will be the Wallman compactification of X. The classes of ${\omega _\mu }$-entire and ${\omega _\mu }$-total spaces, which are respectively analogues of realcompact and pseudocompact spaces, are examined, and an $ {\omega _\mu }$-entire extension analogous to the Hewitt realcompactification is constructed with the property (not possessed by the Wallman compactification) that every map between spaces has a unique extension to their ${\omega _\mu }$-entire extensions. The semiring of functions of compact-small support is considered, and shown to be related to the locally compact-small spaces in the same way that the ring of functions of compact support is related to locally compact spaces.
A discontinuous intertwining operator
Allan M.
Sinclair
259-267
Abstract: If T and R are continuous linear operators on Banach spaces X and Y with the spectrum of R countable, we obtain necessary and sufficient conditions on the pair T, R that imply the continuity of every linear operator S from X into Y satisfying $ST = RS$.
Local norm convergence of states on the zero time Bose fields
Ola
Bratteli
269-280
Abstract: For a sequence of vector states on the Boson Fock space which are norm convergent on the Newton-Wigner local algebras, conditions are given which guarantee norm convergence on the relativistic local algebras also. These conditions are verified for the cutoff physical vacuum states of the $P{(\phi )_2}$ field theory, and yield a simplification of the proof of the locally normal property of the physical vacuum in that theory.
A partition property characterizing cardinals hyperinaccessible of finite type
James H.
Schmerl
281-291
Abstract: Let ${\mathbf{P}}(n,\alpha )$ be the class of infinite cardinals which have the following property: Suppose for each $\nu < \kappa $ that ${C_\nu }$ is a partition of ${[\kappa ]^n}$ and card $({C_\nu }) < \kappa$; then there is $X \subset \kappa$ of length $\alpha$ such that for each $\nu < \kappa$, the set $X - (\nu + 1)$ is ${C_\nu }$-homogeneous. In this paper the classes ${\mathbf{P}}(n,\alpha )$ are studied and a nearly complete characterization of them is given. A principal result is that ${\mathbf{P}}(n + 2,n + 5)$ is the class of cardinals which are hyperinaccessible of type n.
A matrix representation for associative algebras. I
Jacques
Lewin
293-308
Abstract: Let F be a mixed free algebra on a set X over the field K. Let U, V be two ideals of F, and $ \{ \delta (x),(x \in X)\}$ a basis for a free $(F/U,F/V)$-bimodule T. Then the map $x \to (\begin{array}{*{20}{c}} {x + V} & 0 {\delta (x)} & {x + U} \end{array} )$ induces an injective homomorphism $ F/UV \to (\begin{array}{*{20}{c}} {F/V} & 0 T & {F/U} \end{array} )$. If $F/U$ and $F/V$ are embeddable in matrices over a commutative algebra, so is $F/UV$. Some special cases are investigated and it is shown that a PI algebra with nilpotent radical satisfies all identities of some full matrix algebra.
A matrix representation for associative algebras. II
Jacques
Lewin
309-317
Abstract: The results of part I of this paper are applied to show that if F is a free algebra over the field K and W is a subset of F which is algebraically independent modulo the commutator ideal [F, F], then W again generates a free algebra. On the way a similar theorem is proved for algebras that are free in the variety of K-algebras whose commutator ideal is nilpotent of class n. It is also shown that if L is a Lie algebra with universal enveloping algebra F, and U, V are ideals of L, then $FUF \cdot FVF \cap L = [U \cap V,U \cap V]$. This is used to extend the representation theorem of part I to free Lie algebras.
$K\sb{1}$ of a curve of genus zero
Leslie G.
Roberts
319-326
Abstract: We determine the structure of the vector bundles on a curve of genus zero and calculate the ``universal determinant'' $ {K_1}$ of such a curve.
Ultrafilter mappings and their Dedekind cuts
Andreas
Blass
327-340
Abstract: Let D be an ultrafilter on the set N of natural numbers. To each function $p:N \to N$ and each ultrafilter E that is mapped to D by p, we associate a Dedekind cut in the ultrapower D-prod N. We characterize, in terms of rather simple closure conditions, the cuts obtainable in this manner when various restrictions are imposed on E and p. These results imply existence theorems, some known and some new, for various special kinds of ultrafilters and maps.
Maximal $\alpha $-r.e. sets
Manuel
Lerman
341-386
Abstract: Various generalizations of maximal sets from ordinary recursion theory to recursion theory on admissible ordinals are considered. A justification is given for choosing one of these definitions as superior to the rest. For all the definitions considered to be reasonable, a necessary and sufficient condition for the existence of such maximal $ \alpha$-r.e. sets is obtained.
Some combinatorial principles
Jussi
Ketonen
387-394
Abstract: We extend some large cardinal axioms of Jensen to weakly inaccessible cardinals. Related problems regarding the saturatedness of certain filters are also studied.
Representation of functions as limits of martingales
Charles W.
Lamb
395-405
Abstract: In this paper we show that if $(\Omega ,\mathcal{F},P)$ is a probability space and if ${\{ \mathcal{F}{_n}\} _{n \geq 1}}$ is an increasing sequence of sub-$\sigma$-fields of $ \mathcal{F}$ which satisfy an additional condition, then every real valued, $ {\mathcal{F}_\infty }$-measurable function f can be written as the a.e. limit of a martingale ${\{ {f_n},{\mathcal{F}_n}\} _{n \geq 1}}$. The case where f takes values in the extended real line is also studied. A construction is given of a ``universal'' martingale ${\{ {f_n},{\mathcal{F}_n}\} _{n \geq 1}}$ such that any $ {\mathcal{F}_\infty }$-measurable function is the a.e. limit of a suitably chosen subsequence.
Symmetric integro-differential-boundary problems
Hyman J.
Zimmerberg
407-417
Abstract: Necessary and sufficient conditions for a linear vector integro-differential-boundary problem to be symmetric (selfadjoint) are developed, and then applied to obtain canonical forms of such symmetric problems. Moreover, the formulation of the integro-boundary conditions herein yields a simplification of one of the conditions for selfadjointness of a differential-boundary operator previously announced.
Spectra of polar factors of hyponormal operators
C. R.
Putnam
419-428
Abstract: An investigation is made of the interdependence and properties of the spectrum of a hyponormal operator T and of the spectra, and absolutely continuous spectra, of the factors in a polar factorization of T when the latter exists.
On Sylow $2$-subgroups with no normal Abelian subgroups of rank $3$, in finite fusion-simple groups
Anne R.
Patterson
1-67
Abstract: Let T be any finite 2-group which has a normal four-group but has no normal Abelian subgroup of rank 3, and assume T is not the dihedral group of order 8. If T is a Sylow 2-subgroup of a finite fusion-simple group G, it follows (Thompson) from Glauberman's ${Z^ \ast }$-theorem that T has exactly one normal four-group, say W. This paper establishes what isomorphism types of T can so occur under the hypothesis that ${{\mathbf{N}}_G}(T) = T{{\mathbf{C}}_G}(T)$ and the three nonidentity elements of W are not all G-conjugate. All T arrived at in this paper are known to so occur. The reason for this hypothesis is that the similar situation for T with a normal four-group and no normal Abelian subgroup of rank 3, where T is a Sylow 2-subgroup of a finite simple group G but without the above hypothesis, had been analyzed earlier by the author (under her maiden name, MacWilliams; Trans. Amer. Math. Soc. 150 (1970), 345-408).
$B$-convexity and reflexivity in Banach spaces
Dean R.
Brown
69-76
Abstract: A proof of James that uniformly nonsquare spaces are reflexive is extended in part to B-convex spaces. A condition sufficient for non-B-convexity and related conditions equivalent to non-B-convexity are given. The following theorem is proved: A Banach space is B-convex if each subspace with basis is B-convex.
$P$-convexity and $B$-convexity in Banach spaces
Dean R.
Brown
77-81
Abstract: Two properties of B-convexity are shown to hold for P-convexity: (1) Under certain conditions, the direct sum of two P-convex spaces is P-convex. (2) A Banach space is P-convex if each subspace having a Schauder decomposition into finite dimensional subspaces is P-convex.
On a compactness property of topological groups
S. P.
Wang
83-88
Abstract: A density theorem of semisimple analytic groups acting on locally compact groups is presented.
Duality theories for metabelian Lie algebras
Michael A.
Gauger
89-102
Abstract: This paper is concerned with duality theories for metabelian (2-step nilpotent) Lie algebras. A duality theory associates to each metabelian Lie algebra N with cod $ {N^2} = g$, another such algebra ${N_D}$ satisfying ${({N_D})_D} \cong N,{N_1} \cong {N_2}$ if and only if $ {({N_1})_D} \cong {({N_2})_D}$, and if $ \dim \,N = g + p$ then $ \dim \,{N_D} = g + (_2^g) - p$. The obvious benefit of such a theory lies in its reduction of the classification problem.
A topology for a lattice-ordered group
R. H.
Redfield
103-125
Abstract: Let G be an arbitrary lattice-ordered group. We define a topology on G, called the $ \mathcal{J}$-topology, which is a group and lattice topology for G and which is preserved by cardinal products. The $\mathcal{J}$-topology is the interval topology on totally ordered groups and is discrete if and only if G is a lexico-sum of lexico-extensions of the integers. We derive necessary and sufficient conditions for the $ \mathcal{J}$-topology to be Hausdorff, and we investigate $\mathcal{J}$-topology convergence.
Structure theory for equational classes generated by quasi-primal algebras
Robert W.
Quackenbush
127-145
Abstract: Quasi-primal algebras (which include finite simple polyadic and cylindric algebras) were introduced by A. S. Pixley. In this paper equational classes generated hy quasi-primal algebras are investigated with respect to the following concepts: the congruence extension property, the amalgamation property and the amalgamation class, weak injectives and weak injective hulls, the standard semigroup of operators. A brief discussion of monadic algebras is included to illustrate the results of the paper.
The theory of $Q$-rings
Eben
Matlis
147-181
Abstract: An integral domain R with quotient field Q is defined to be a Q-ring if $\operatorname{Ext}_R^1(Q,R) \cong Q$. It is shown that R is a Q-ring if and only if there exists an R-module A such that $ {\operatorname{Hom}_R}(A,R) = 0$ and $\operatorname{Ext}_R^1(A,R) \cong Q$. If A is such an R-module and $t(A)$ is its torsion submodule, then it is proved that $A/t(A)$ necessarily has rank one. There are only three kinds of Q-rings, namely, $ {Q_0}{\text{-}},{Q_1}{\text{-}}$, or ${Q_2}$-rings. These are described by the fact that if R is a Q-ring, then $ K = Q/R$ can only have 0, 1, or 2 proper h-divisible submodules. If H is the completion of R in the R-topology, then R is one of the three kinds of Q-rings if and only if $H{ \otimes _R}Q$ is one of the three possible kinds of 2-dimensional commutative Q-algebras. Examples of all three kinds of Q-rings are produced, and the behavior of Q-rings under ring extensions is examined. General conditions are given for a ring not to be a Q-ring. As an application of the theory, necessary and sufficient conditions are found for the integral closure of a non-complete Noetherian domain to be a complete discrete valuation ring.
Weak compactness in the order dual of a vector lattice
Owen
Burkinshaw
183-201
Abstract: A sequence $\{ {x_n}\}$ in a vector lattice E will be called an l'-sequence if there exists an x in E such that $\Sigma _{k = 1}^n\vert{x_k}\vert \leq x$ for all n. Denote the order dual of E by ${E^b}$. For a set $A \subset {E^b}$, let $ {\left\Vert \cdot \right\Vert _{{A^ \circ }}}$ denote the Minkowski functional on E defined by its polar ${A^ \circ }$ in E. A set $A \subset {E^b}$ will be called equi-l'-continuous on E if $\lim {\left\Vert {{x_n}} \right\Vert _{{A^ \circ }}} = 0$ for each l'-sequence $ \{ {x_n}\}$ in E. The main objective of this paper will be to characterize compactness in ${E^b}$ in terms of the order structure on E and ${E^b}$. In particular, the relationship of equi-l'-continuity to compactness is studied. §2 extends to ${E^{\sigma c}}$ the results in Kaplan [8] on vague compactness in ${E^C}$. Then this is used to study vague convergence of sequences in ${E^b}$.
Disjoint meromorphic functions and nonoscillatory differential systems
D.
Aharonov;
M.
Lavie
203-216
Abstract: Conditions satisfied by disjoint meromorphic functions are obtained. These results are applied to nonoscillatory differential systems and disfocal differential equations.
The $p$-adic hull of abelian groups
A.
Mader
217-229
Abstract: In this paper we define ``p-adic hull'' for p-reduced groups K. The p-adic hull ${K^P}$ of K is a module over the ring P of p-adic integers containing K and satisfying certain additional properties. The notion is investigated and then used to prove some known and some new theorems on $\operatorname{Ext}(K,T)$ and $\operatorname{Hom}(K,T)$ for K torsion-free and T a reduced p-group.
Elementary divisor rings and finitely presented modules
Max D.
Larsen;
William J.
Lewis;
Thomas S.
Shores
231-248
Abstract: Throughout, rings are commutative with unit and modules are unital. We prove that R is an elementary divisor ring if and only if every finitely presented module over R is a direct sum of cyclic modules, thus providing a converse to a theorem of Kaplansky and answering a question of Warfield. We show that every Bezout ring with a finite number of minimal prime ideals is Hermite. So, in particular, semilocal Bezout rings are Hermite answering affirmatively a question of Henriksen. We show that every semihereditary Bezout ring is Hermite. Semilocal adequate rings are characterized and a partial converse to a theorem of Henriksen is established.
Smooth partitions of unity on manifolds
John
Lloyd
249-259
Abstract: This paper continues the study of the smoothness properties of (real) topological linear spaces. First, the smoothness results previously obtained about various important classes of locally convex spaces, such as Schwartz spaces, are improved. Then, following the ideas of Bonic and Frampton, we use these results to give sufficient conditions for the existence of smooth partitions of unity on manifolds modelled on topological linear spaces.
Selfadjoint algebras of unbounded operators. II
Robert T.
Powers
261-293
Abstract: Unbounded selfadjoint representations of $^\ast$-algebras are studied. It is shown that a selfadjoint representation of the enveloping algebra of a Lie algebra can be exponentiated to give a strongly continuous unitary representation of the simply connected Lie group if and only if the representation preserves a certain order structure. This result follows from a generalization of a theorem of Arveson concerning the extensions of completely positive maps of ${C^ \ast }$-algebras. Also with the aid of this generalization of Arveson's theorem it is shown that an operator $ \overline {\pi (A)}$ is affiliated with the commutant $\mathcal{A}$, with $A = {A^ \ast } \in \mathcal{A}$, if and only if $\pi$ preserves a certain order structure associated with A and $ \mathcal{A}$. This result is then applied to obtain a characterization of standard representations of commutative $^\ast $-algebras in terms of an order structure.
Some positive trigonometric sums
Richard
Askey;
John
Steinig
295-307
Abstract: Vietoris found an interesting generalization of the classical inequality $\Sigma _{k = 1}^n(\sin \,k\theta /k) > 0,0 < \theta < \pi$. A simplified proof is given for his inequality and his similar inequality for cosine series. Various new results which follow from these inequalities include improved estimates for the location of the zeros of a class of trigonometric polynomials and new positive sums of ultraspherical polynomials which extend an old inequality of Fejér. Both of Vietoris' inequalities are special cases of a general problem for Jacobi polynomials, and a summary is given of known results on this problem.
Extensions of normal immersions of $S\sp{1}$ into $R\sp{2}$
Morris L.
Marx
309-326
Abstract: Suppose that $f:{S^1} \to {R^2}$ is an immersion, i.e., a $ {C^1}$ map such that $ f'$ is never zero. We call f normal if there are only finitely many self-intersections and these are transverse double points. A normal immersion f can be topologically determined by a finite number of combinatorial invariants. Using these invariants it is possible to give considerable information about extensions of f to $ {D^2}$. In this paper we give a new set of invariants, inspired by the work of S. Blank, to solve several problems concerning the existence of certain kinds of extensions. The problems solved are as follows: (1) When does f have a light open extension $ F:{D^2} \to {R^2}$? (Recall that light means ${F^{ - 1}}(y)$ is totally disconnected for all y and open means F maps open sets of the interior of ${D^2}$ to open sets of ${R^2}$.) Because of the work of Stoïlow, the question is equivalent to the following: when does there exist a homeomorphism $h:{S^1} \to {S^1}$, such that fh has an analytic extension to ${D^2}$? (2) Suppose that $F:{D^2} \to {R^2}$ is light, open, sense preserving, and, at each point of ${S^1}$, F is a local homeomorphism. At each point of the interior of ${D^2}$, F is locally topologically equivalent to the power mapping ${z^m}$ on $ {D^2},m \geq 1$. Points where $m > 1$ are called branch points and $ m - 1$ is the multiplicity of the point. There are only a finite number of branch points. The problem is to discover the minimum number of branch points of any properly interior extension of f. Also we can ask what multiplicities can arise for extensions of a given f. (3) Given a normal f, find the maximum number of properly interior extensions of f that are pairwise inequivalent. Since each immersion of the disk is equivalent to a local homeomorphism, the problem of immersion extensions is a special case of this. It is Blank's solution of the immersion problem that prompted this paper.
Topological dynamics and group theory
Shmuel
Glasner
327-334
Abstract: We prove, using notions and techniques of topological dynamics, that a nonamenable group contains a finitely-generated subgroup of exponential growth. We also show that a group which belongs to a certain class, defined by means of topological dynamical properties, always contains a free subgroup on two generators.
Isolated singularities for solutions of the nonlinear stationary Navier-Stokes equations
Victor L.
Shapiro
335-363
Abstract: The notion for (u, p) to be a distribution solution of the nonlinear stationary Navier-Stokes equations in an open set is defined, and a theorem concerning the removability of isolated singularities for distribution solutions in the punctured open ball $B(0,{r_0}) - \{ 0\}$ is established. This result is then applied to the classical situation to obtain a new theorem for the removability of isolated singularities. In particular, in two dimensions this gives a better than expected result when compared with the theory of removable isolated singularities for harmonic functions.
Egoroff properties and the order topology in Riesz spaces
Theresa K. Y. Chow
Dodds
365-375
Abstract: In this paper we prove that, for a Riesz space L, the order closure of each subset of L coincides with its pseudo order closure if and only if the order closure of each convex subset of L coincides with its pseudo order closure; moreover, each of these statements is equivalent to the strong Egoroff property. For Archimedean Riesz spaces, similar results hold for the relative uniform topology.
One-parameter semigroups holomorphic away from zero
Melinda W.
Certain
377-389
Abstract: Suppose T is a one-parameter semigroup of bounded linear operators on a Banach space, strongly continuous on $[0,\infty )$. It is known that $\lim {\sup _{x \to 0}}\vert T(x) - I\vert < 2$ implies T is holomorphic on $(0,\infty )$. Theorem I is a generalization of this as follows: Suppose $M > 0,0 < r < s$, and $\rho$ is in (1,2). If $\vert{(T(h) - I)^n}\vert \leq M{\rho ^n}$ whenever nh is in $[r,s],n = 1,2, \cdots ,h > 0$, then there exists $b > 0$ such that T is holomorphic on $ [b,\infty )$. Theorem II shows that, in some sense, $b \to 0$ as $r \to 0$. Theorem I is an application of Theorem III: Suppose $ M > 0,0 < r < s,\rho$ is in (1,2), and f is continuous on $[ - 4s,4s]$. If $\vert\sum\nolimits_{q = 0}^n {(\mathop n\limits_q ){{( - 1)}^{n - q}}f(t + qh)\vert \leq M{\rho ^n}}$ whenever nh is in $ [r,s],n = 1,2, \cdots ,h > 0,[t,t + nh] \subset [ - 4s,4s]$, then f has an analytic extension to an ellipse with center zero. Theorem III is a generalization of a theorem of Beurling in which the inequality on the differences is assumed for all nh. An example is given to show the hypothesis of Theorem I does not imply T holomorphic on $ (0,\infty )$.
The existence, characterization and essential uniqueness of solutions of $L\sp{\infty }$ extremal problems
S. D.
Fisher;
J. W.
Jerome
391-404
Abstract: Let $I = (a,b)$ be an interval in R and let ${H^{n,\infty }}$ consist of those real-valued functions f such that $ {f^{(n - 1)}}$ is absolutely continuous on I and ${f^{(n)}} \in {L^\infty }(I)$. Let L be a linear differential operator of order n with leading coefficient $1,a = {x_1} < \cdots < {x_m} = b$ be a partition of I and let the linear functionals $ {L_{ij}}$ on ${H^{n,\infty }}$ be given by $\displaystyle {L_{ij}}f = \sum\limits_{v = 0}^{n - 1} {a_{ij}^{(v)}{f^{(v)}}({x_i}),\quad j = 1, \cdots ,{k_i},i = 1, \cdots ,m,}$ where $1 \leq {k_i} \leq n$ and the ${k_i}$ n-tuples $ (a_{ij}^{(0)}, \cdots ,a_{ij}^{(n - 1)})$ are linearly inde pendent. Let $ {r_{ij}}$ be prescribed real numbers and let $U = \{ f \in {H^{n,\infty }}:{L_{ij}}f = {r_{ij}},j = 1, \cdots ,{k_i},i = 1, \cdots ,m\}$. In this paper we consider the extremal problem $\displaystyle {\left\Vert {Ls} \right\Vert _{{L^\infty }}} = \alpha = \inf \{ {\left\Vert {Lf} \right\Vert _{{L^\infty }}}:f \in U\} .$ ($\ast$) We show that there are, in general, many solutions to $( \ast )$ but that there is, under certain consistency assumptions on L and the ${L_{ij}}$, a fundamental (or core) interval of the form $({x_i},{x_{i + {n_0}}})$ on which all solutions to $ ( \ast )$ agree; $ {n_0}$ is determined by the ${k_i}$ and satisfies $ {n_0} \geq 1$. Further, if s is any solution to $( \ast )$ then on $({x_i},{x_{i + {n_0}}}),\vert Ls\vert = \alpha$ a.e. Further, we show that there is a uniquely determined solution $ {s_ \ast }$ to $ ( \ast )$, found by minimizing ${\left\Vert {Lf} \right\Vert _{{L^\infty }}}$ over all subintervals $({x_j},{x_{j + 1}}),j = 1, \cdots ,m - 1$, with the property that $ \vert L{s_ \ast }\vert$ is constant on each subinterval $({x_j},{x_{j + 1}})$ and $ L{s_ \ast }$ is a step function with at most $n - 1$ discontinuities on $({x_j},{x_{j + 1}})$. When $L = {D^n},{s_ \ast }$ is a piecewise perfect spline. Examples show that the results are essentially best possible.
Jordan algebras and connections on homogeneous spaces
Arthur A.
Sagle
405-427
Abstract: We use the correspondence between G-invariant connections on a reductive homogeneous space $ G/H$ and certain nonassociative algebras to explicitly compute the pseudo-Riemannian connections in terms of a Jordan algebra J of endomorphisms. It is shown that if G and H are semisimple Lie groups, then J is a semisimple Jordan algebra. Also a general method for computing examples of J is given.
Erratum to ``Oscillation, continuation, and uniqueness of solutions of retarded differential equations'' (Trans. Amer. Math. Soc. {\bf 179} (1973), 193--209)
T.
Burton;
R.
Grimmer
429
Uncomplemented $C(X)$-subalgebras of $C(X)$
John Warren
Baker
1-15
Abstract: In this paper, the uncomplemented subalgebras of the Banach algebra $ C(X)$ which are isometrically and algebraically isomorphic to $C(X)$ are investigated. In particular, it is shown that if X is a 0-dimensional compact metric space with its $\omega$th topological derivative ${X^{(\omega )}}$ nonempty, then there is an uncomplemented subalgebra of $C(X)$ isometrically and algebraically isomorphic to $ C(X)$. For each ordinal $\alpha \geq 1$, a class ${\mathcal{C}_\alpha }$ of homeomorphic 0-dimensional uncountable compact metric spaces is introduced. It is shown that each uncountable 0-dimensional compact metric space contains an open-and-closed subset which belongs to some $ {\mathcal{C}_\alpha }$.
The values of exponential polynomials at algebraic points. I
Carlos Julio
Moreno
17-31
Abstract: A strengthening of Siegel's proof of the Hermite-Lindemann Theorem is given. The results are used to obtain lower bounds for the values of exponential polynomials at algebraic points. The question of how well the root of an exponential polynomial can be approximated by algebraic numbers is considered, and a lower bound is obtained for the absolute value of the difference between a root of the exponential polynomial and an algebraic number.
Archimedean-like classes of lattice-ordered groups
Jorge
Martinez
33-49
Abstract: Suppose $\mathcal{C}$ denotes a class of totally ordered groups closed under taking subgroups and quotients by o-homomorphisms. We study the following classes: (1) $ {\text{Res}}(\mathcal{C})$, the class of all lattice-ordered groups which are subdirect products of groups in $\mathcal{C}$; (2) $ {\text{Hyp}}(\mathcal{C})$, the class of lattice-ordered groups in $ {\text{Res}}(\mathcal{C})$ having all their l-homomorphic images in $ {\text{Res}}(\mathcal{C})$; Para $ (\mathcal{C})$, the class of lattice-ordered groups having all their principal convex l-subgroups in $ {\text{Res}}(\mathcal{C})$. If $\mathcal{C}$ is the class of archimedean totally ordered groups then Para $(\mathcal{C})$ is the class of archimedean lattice-ordered groups, $ {\text{Res}}(\mathcal{C})$ is the class of subdirect products of reals, and $ {\text{Hyp}}(\mathcal{C})$ consists of all the hyper archimedean lattice-ordered groups. We show that under an extra (mild) hypothesis, any given representable lattice-ordered group has a unique largest convex l-subgroup in $ {\text{Hyp}}(\mathcal{C})$; this socalled hyper- $ \mathcal{C}$-kernel is a characteristic subgroup. We consider several examples, and investigate properties of the hyper- $\mathcal{C}$-kernels. For any class $\mathcal{C}$ as above we show that the free lattice-ordered group on a set X in the variety generated by $ \mathcal{C}$ is always in $ {\text{Res}}(\mathcal{C})$. We also prove that $ {\text{Res}}(\mathcal{C})$ has free products.
Relative projectivity, the radical and complete reducibility in modular group algebras
D. C.
Khatri
51-63
Abstract: If $H \leq G$ and every G-module is H-projective then (G, H) is a projective pairing. If Rad $FG \subseteq ({\text{Rad}}\;FH)FG$ then (G, H) is said to have property p. A third property considered is that for each irreducible H-module the induced G-module be completely reducible. It is shown that these three are equivalent properties in many interesting cases. Also examples are given to show that they are, in general, independent of each other.
Structure theorems for certain topological rings
James B.
Lucke;
Seth
Warner
65-90
Abstract: A Hausdorff topological ring B is called centrally linearly compact if the open left ideals form a fundamental system of neighborhoods of zero and B is a strictly linearly compact module over its center. A topological ring A is called locally centrally linearly compact if it contains an open, centrally linearly compact subring. For example, a totally disconnected (locally) compact ring is (locally) centrally linearly compact, and a Hausdorff finite-dimensional algebra with identity over a local field (a complete topological field whose topology is given by a discrete valuation) is locally centrally linearly compact. Let A be a Hausdorff topological ring with identity such that the connected component c of zero is locally compact, A/c is locally centrally linearly compact, and the center C of A is a topological ring having no proper open ideals. A general structure theorem for A is given that yields, in particular, the following consequences: (1) If the additive order of each element of A is infinite or squarefree, then $A = {A_0} \times D$ where ${A_0}$ is a finite-dimensional real algebra and D is the local direct product of a family $({A_\gamma })$ of topological rings relative to open subrings $ ({B_\gamma })$, where each $({A_\gamma })$ is the cartesian product of finitely many finite-dimensional algebras over local fields. (2) If A has no nonzero nilpotent ideals, each $ {A_\gamma }$ is the cartesian product of finitely many full matrix rings over division rings that are finite dimensional over their centers, which are local fields. (3) If the additive order of each element of A is infinite or squarefree and if C contains an invertible, topologically nilpotent element, then A is the cartesian product of finitely many finite-dimensional algebras over R, C, or local fields.
Extremal structure in operator spaces
M.
Sharir
91-111
Abstract: (a) A characterization of extreme operators (in the unit ball of operators) between ${L^1}$-spaces is given, together with other related properties, (b) A general theorem of Kreĭn-Milman type for the unit ball of operator spaces is proved, and is applied to operators between ${L^1}$-spaces and to operators into C-spaces.
The module decomposition of $I(\bar A/A)$
Klaus G.
Fischer
113-128
Abstract: Let A and B be scalar rings with B an A-algebra. The B-algebra ${D^n}(B/A) = I(B/A)/{I^n}(B/A)$ is universal for n-truncated A-Taylor series on B. In this paper, we consider the $\bar A$ module decomposition of ${D^n}(\bar A/A)$ with a view to classifying the singularity A which is assumed to be the complete local ring of a point on an algebraic curve at a one-branch singularity. We assume that $A/M = k < A$ and that k is algebraically closed with no assumption on the characteristic. We show that ${D^n}(\bar A/A) = I(\bar A/A)$ for n large and that the decomposition of $ I(\bar A/A)$ as a module over the P.I.D. $\bar A$ is completely determined by the multiplicity sequence of A. The decomposition is displayed and a length formula for $ I(\bar A/A)$ developed. If B is another such ring, where $\bar B = \bar A = k[[t]]$, we show that $ I(\bar A/A) \cong I(\bar B/B)$ as $k[[t]]$ modules if and only if the multiplicity sequence of A is equal to the multiplicity sequence of B. If $ A < B < \bar A$, then $ I(\bar A/A) \cong I(\bar B/B)$ as $\bar A = \bar B$ modules if and only if the Arf closure of A and B coincide. This is equivalent to the existence of an algebra isomorphism between $ I(\bar A/A)$ and $I(\bar B/B)$.
Chebyshev constant and Chebyshev points
Susan L.
Friedman
129-139
Abstract: Using $ \lambda$th power means in the case $\lambda \geq 1$, it is proven that the Chebyshev constant for any compact set in $ {R_n}$, real Euclidean n-space, is equal to the radius of the spanning sphere. When $\lambda > 1$, the Chebyshev points of order m for all $m \geq 1$ are unique and coincide with the center of the spanning sphere. For the case $\lambda = 1$, it is established that Chebyshev points of order m for a compact set E in $ {R_2}$ are unique if and only if the cardinality of the intersection of E with its spanning circle is greater than or equal to three.
Complex Lindenstrauss spaces with extreme points
B.
Hirsberg;
A. J.
Lazar
141-150
Abstract: We prove that a complex Lindenstrauss space whose unit ball has at least one extreme point is isometric to the space of complex valued continuous affine functions on a Choquet simplex. If X is a compact Hausdorff space and $A \subset {C_{\text{C}}}(X)$ is a function space then A is a Lindenstrauss space iff A is selfadjoint and Re A is a real Lindenstrauss space.
Critical point theory for nonlinear eigenvalue problems with indefinite principal part
Melvyn S.
Berger
151-169
Abstract: A study of the nontrivial solutions of the operator equation
Ergodicity of the Cartesian product
Elias G.
Flytzanis
171-176
Abstract: ${h_1}$ is an ergodic conservative transformation on a $\sigma$-finite measure space and ${h_2}$ is an ergodic measure preserving transformation on a finite measure space. We study the point spectrum properties of ${h_1} \times {h_2}$. In particular we show ${h_1} \times {h_2}$ is ergodic if and only if ${h_1} \times {h_2}$ have no eigenvalues in common other than the eigenvalue 1. The conditions on ${h_1},{h_2}$ stated above are in a sense the most general for the validity of this result.
The constrained coefficient problem for typically real functions
George B.
Leeman
177-189
Abstract: Let $- 2 \leq c \leq 2$. In this paper we find the precise upper and lower bounds on the nth Taylor coefficient ${a_n}$ of functions $ f(z) = z + c{z^2} + \Sigma _{k = 3}^\infty {a_k}{z^k}$ typically real in the unit disk for $n = 3,4, \cdots $. In addition all the extremal functions are identified.
Symmetric completions and products of symmetric matrices
Morris
Newman
191-201
Abstract: We show that any vector of n relatively prime coordinates from a principal ideal ring R may be completed to a symmetric matrix of $ {\text{SL}}(n,R)$, provided that $n \geq 4$. The result is also true for $n = 3$ if R is the ring of integers Z. This implies for example that if F is a field, any matrix of ${\text{SL}}(n,F)$ is the product of a fixed number of symmetric matrices of $ {\text{SL}}(n,F)$ except when $n = 2$, $F = {\text{GF}}(3)$, which is a genuine exception.
The Veech structure theorem
Robert
Ellis
203-218
Abstract: The main result is the proof of the Veech structure theorem for point-distal flows without the assumption that the distal points form a residual set. This allows one to conclude that, in the case of metrizable flows, if there is one distal point then there is a residual set of such points.
Denjoy-type flows on orientable $2$-manifolds of higher genus
Carl S.
Hartzman
219-227
Abstract: The author generalizes A. Denjoy's theory of flows on a torus to compact orientable 2-manifolds of higher genus. Natural extensions of A. Denjoy's hypotheses are made and necessary conditions that a flow satisfy the new hypotheses are given.
Extending cell-like maps on manifolds
B. J.
Ball;
R. B.
Sher
229-246
Abstract: Let X be a closed subset of a manifold M and ${G_0}$ be a cell-like upper semicontinuous decomposition of X. We consider the problem of extending ${G_0}$ to a cell-like upper semicontinuous decomposition G of M such that $M/G \approx M$. Under fairly weak restrictions (which vanish if $M = {E^n}$ or ${S^n}$ and $n \ne 4$ we show that such a G exists if and only if the trivial extension of ${G_0}$, obtained by adjoining to $ {G_0}$ the singletons of $ M - X$, has the desired property. In particular, the nondegenerate elements of Bing's dogbone decomposition of ${E^3}$ are not elements of any cell-like upper semicontinuous decomposition G of ${E^3}$ such that ${E^3}/G \approx {E^3}$. Call a cell-like upper semicontinuous decomposition G of a metric space X simple if $ X/G \approx X$ and say that the closed set Y is simply embedded in X if each simple decomposition of Y extends trivially to a simple decomposition of X. We show that tame manifolds in ${E^3}$ are simply embedded and, with some additional restrictions, obtain a similar result for a locally flat k-manifold in an m-manifold $(k,m \ne 4)$. Examples are given of an everywhere wild simply embedded simple closed curve in $ {E^3}$ and of a compact absolute retract which embeds in ${E^3}$ yet has no simple embedding in ${E^3}$.
Weighted Grothendieck subspaces
Jo ao B.
Prolla;
Silvio
Machado
247-258
Abstract: Let V be a family of nonnegative upper semicontinuous functions on a completely regular Hausdorff space X. For a locally convex Hausdorff space E, let $C{V_\infty }(X;E)$ be the corresponding Nachbin space, that is, the vector space of all continuous functions f from X into E such that vf vanishes at infinity for all $v \in V$, endowed with the topology given by the seminorms of the type $f\vert \to \sup \{ v(x)p(f(x));x \in X\}$, where $v \in V$ and p is a continuous seminorm on E. Given a vector subspace L of $C{V_\infty }(X;E)$, the set of all pairs $ x,y \in X$ such that either $0 = {\delta _x}\vert L = {\delta _y}\vert L$ or there is $t \in R,t \ne 0$, such that $0 \ne {\delta _x}\vert L = t{\delta _y}\vert L$, is an equivalence relation, denoted by $ {G_L}$, and we define for $ (x,y) \in {G_L},g(x,y) = 0$ or t, accordingly. The subsets $K{S_L}$, resp. $W{S_L}$, where $g(x,y) \geq 0$, resp. $g(x,y) \in \{ 0,1\}$, are likewise equivalence relations. The G-hull (resp. KS-hull, WS-hull) of L is the vector subspace $\{ f \in C{V_\infty }(X;E);f(x) = g(x,y)f(y)$ for all $(x,y) \in {G_L}\;({\text{resp}}.\;K{S_L},W{S_L})\}$ and L is a G-space (resp. KS-space, WS-space) if its G-hull (resp. KS-hull, WS-hull) is contained in its closure. This paper is devoted to the characterization, by invariance properties, of the G-spaces resp. KS-spaces and WS-spaces of a given Nachbin space $C{V_\infty }(X;E)$. As an application we derive an infinite-dimensional Weierstrass polynomial approximation theorem, and a Tietze extension theorem for Banach space valued compact mappings.
Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics
Thomas G.
Kurtz
259-272
Abstract: Let ${A_1},{A_2}, \cdots$ be dissipative sets that generate semigroups of nonlinear contractions ${T_1}(t),{T_2}(t) \cdots $ Conditions are given on $\{ {A_n}\}$ which imply the existence of a limiting semigroup T(t). The results include types of convergence besides strong convergence. As an application, it is shown that solutions of the pair of equations $\displaystyle {u_t} = - \alpha {u_x} + {\alpha ^2}({v^2} - {u^2})$ and $\displaystyle {v_t} = \alpha {v_x} + {\alpha ^2}({u^2} - {v^2}),$ $\alpha$ a constant, approximate the solutions of $\displaystyle {u_t} = 1/4({d^2}/d{x^2})\,\log \,u$ as $ \alpha$ goes to infinity.
Joint measures and cross-covariance operators
Charles R.
Baker
273-289
Abstract: Let ${H_1}$ (resp., ${H_2}$) be a real and separable Hilbert space with Borel $\sigma$-field $ {\Gamma _1}$ (resp., ${\Gamma _2}$), and let $ ({H_1} \times {H_2},{\Gamma _1} \times {\Gamma _2})$ be the product measurable space generated by the measurable rectangles. This paper develops relations between probability measures on $({H_1} \times {H_2},{\Gamma _1} \times {\Gamma _2})$, i.e., joint measures, and the projections of such measures on $ ({H_1},{\Gamma _1})$ and $({H_2},{\Gamma _2})$. In particular, the class of all joint Gaussian measures having two specified Gaussian measures as projections is characterized, and conditions are obtained for two joint Gaussian measures to be mutually absolutely continuous. The cross-covariance operator of a joint measure plays a major role in these results and these operators are characterized.
Zero-one laws for Gaussian measures on Banach space
Charles R.
Baker
291-308
Abstract: Let $\mathcal{B}$ be a real separable Banach space, $ \mu$ a Gaussian measure on the Borel $\sigma$-field of $ \mathcal{B}$, and ${B_\mu }[\mathcal{B}]$ the completion of the Borel $\sigma$-field under $\mu$. If $G \in {B_\mu }[\mathcal{B}]$ is a subgroup, we show that $ \mu (G) = 0$ or 1, a result essentially due to Kallianpur and Jain. Necessary and sufficient conditions are given for $\mu (G) = 1$ for the case where G is the range of a bounded linear operator. These results are then applied to obtain a number of 0-1 statements for the sample function properties of a Gaussian stochastic process. The zero-one law is then extended to a class of non-Gaussian measures, and applications are given to some non-Gaussian stochastic processes.
A Kurosh subgroup theorem for free pro-$\mathcal{C}$-products of pro-$\mathcal{C}$-groups
Dion
Gildenhuys;
Luis
Ribes
309-329
Abstract: Let $\mathcal{C}$ be a class of finite groups, closed under finite products, subgroups and homomorphic images. In this paper we define and study free pro- $\mathcal{C}$-products of pro- $\mathcal{C}$-groups indexed by a pointed topological space. Our main result is a structure theorem for open subgroups of such free products along the lines of the Kurosh subgroup theorem for abstract groups. As a consequence we obtain that open subgroups of free pro- $\mathcal{C}$-groups on a pointed topological space, are free pro- $ \mathcal{C}$-groups on (compact, totally disconnected) pointed topological spaces.
Asymptotic stability and spiraling properties for solutions of stochastic equations
Avner
Friedman;
Mark A.
Pinsky
331-358
Abstract: We consider a system of Itô equations in a domain in ${R^d}$. The boundary consists of points and closed surfaces. The coefficients are such that, starting for the exterior of the domain, the process stays in the exterior. We give sufficient conditions to ensure that the process converges to the boundary when $t \to \infty $. In the case of plane domains, we give conditions to ensure that the process ``spirals"; the angle obeys the strong law of large numbers.
Dirichlet problem for degenerate elliptic equations
Avner
Friedman;
Mark A.
Pinsky
359-383
Abstract: Let ${L_0}$ be a degenerate second order elliptic operator with no zeroth order term in an m-dimensional domain G, and let $L = {L_0} + c$. One divides the boundary of G into disjoint sets ${\Sigma _1},{\Sigma _2},{\Sigma _3};{\Sigma _3}$ is the noncharacteristic part, and on ${\Sigma _2}$ the ``drift'' is outward. When c is negative, the following Dirichlet problem has been considered in the literature: $Lu = 0$ in G, u is prescribed on $ {\Sigma _2} \cup {\Sigma _3}$. In the present work it is assume that $c \leq 0$. Assuming additional boundary conditions on a certain finite number of points of ${\Sigma _1}$, a unique solution of the Dirichlet problem is established.
On the structure of semigroups which are unions of groups
R. J.
Warne
385-401
Abstract: We characterize semigroups S which are unions of groups as generalized Schreier products of groups, semilattices of right zero semigroups, and semilattices of left zero semigroups. We then give several specializations of this result utilizing Schreier products, semidirect products, and direct products.
Conditions under which disks are $P$-liftable
Edythe P.
Woodruff
403-418
Abstract: A generalization of the concept of lifting of an n-cell is studied. In the usual upper semicontinuous decomposition terminology, let S be a space, $S/G$ be the decomposition space, and the projection mapping be $ P:S \to S/G$ . A set $X \subset S/G$ if $X'$ is homeomorphic to X and $ P(X')$ is X. Examples are given in which the union of two P-liftable sets does not P-lift. We prove that if compact 2-manifolds A and B each P-lift, their union is a disk in ${E^3}/G$, their intersection misses the singular points of the projection mapping, and the intersection of the singular points with the union of A and B is a 0-dimensional set, then the union of A and B does P-lift. Even if a disk D does not P-lift, it is proven that for $ \epsilon > 0$ there is a P-liftable disk $\epsilon$-homeomorphic to D, provided that the given decomposition of ${E^3}$ is either definable by 3-cells, or the set of nondegenerate elements is countable and $ {E^3}/G$ is homeomorphic to ${E^3}$. With further restrictions on the decomposition, this P-liftable disk can be chosen in such a manner that it agrees with D on the singular points of D.
Abstract homotopy theory and generalized sheaf cohomology
Kenneth S.
Brown
419-458
Abstract: Cohomology groups ${H^q}(X,E)$ are defined, where X is a topological space and E is a sheaf on X with values in Kan's category of spectra. These groups generalize the ordinary cohomology groups of X with coefficients in an abelian sheaf, as well as the generalized cohomology of X in the usual sense. The groups are defined by means of the ``homotopical algebra'' of Quillen applied to suitable categories of sheaves. The study of the homotopy category of sheaves of spectra requires an abstract homotopy theory more general than Quillen's, and this is developed in Part I of the paper. Finally, the basic cohomological properties are proved, including a spectral sequence which generalizes the Atiyah-Hirzebruch spectral sequence (in generalized cohomology theory) and the ``local to global'' spectral sequence (in sheaf cohomology theory).
Central idempotent measures on compact groups
Daniel
Rider
459-479
Abstract: Let G be a compact group with dual object $\Gamma = \Gamma (G)$ and let $M(G)$ be the convolution algebra of regular finite Borel measures on G. The author has characterized the central idempotent measures on certain G, including the unitary groups, in terms of the hypercoset structure of $\Gamma$. The characterization also says that, on certain G, a central idempotent measure is a sum of such measures each of which is absolutely continuous with respect to the Haar measure of a closed normal subgroup. The main result of this paper is an extension of this characterization to products of certain groups. The known structure of connected groups and a recent result of Ragozin on connected simple Lie groups will then show that the characterization is valid for connected groups. On the other hand, a simple example will show it is false in general for non-connected groups. This characterization was done by Cohen for abelian groups and the proof borrows extensively from Amemiya and Itô's simplified proof of Cohen's result.
On the existence of invariant measures for piecewise monotonic transformations
A.
Lasota;
James A.
Yorke
481-488
Abstract: A class of piecewise continuous, piecewise ${C^1}$ transformations on the interval [0, 1] is shown to have absolutely continuous invariant measures.
Diffeomorphisms homotopic to the identity
Edward C.
Turner
489-498
Abstract: In this paper, an inductive procedure for describing the group of isotopy classes of null homotopic diffeomorphisms of a manifold is developed --this process depends on the handlebody structure of the manifold. This group is also shown to be finitely generated and in many cases abelian.
Harmonic analysis for functors on categories of Banach spaces of distributions
Thomas
Donaldson
1-82
Abstract: This paper develops a theory of harmonic analysis (Fourier series, approximation, convolution, and singular integrals) for a general class of Banach function or distribution spaces. Continuity of singular convolution operators and convergence of trigonometric series is shown with respect to the norms of all the spaces in this class; the maximal class supporting a theory of the type developed in this paper is characterized (for other classes other theories exist). Theorems are formulated in category language throughout. However only elementary category theory is needed, and for most results the notions of functor and natural mapping are sufficient.
Homology of the classical groups over the Dyer-Lashof algebra
Stanley O.
Kochman
83-136
Abstract: The action of the Dyer-Lashof algebra is computed on the homology of the infinite classical groups (including Spin), their classifying spaces, their homogeneous spaces, Im J, B Im J and BBSO. Some applications are given while applications by other authors appear elsewhere.
The calculation of penetration indices for exceptional wild arcs
James M.
McPherson
137-149
Abstract: A new class of wild arcs is defined, the class of ``exceptional'' arcs, which is a subclass of the class of arcs whose only wildpoint is an endpoint. This paper then uses geometric techniques to calculate the penetration indices of these exceptional arcs.
On coefficient means of certain subclasses of univalent functions
F.
Holland;
J. B.
Twomey
151-163
Abstract: Let $\mathcal{R}$ denote the class of regular functions whose derivatives have positive real part in the unit disc $\gamma$ and let $ \mathcal{S}$ denote the class of functions starlike in $\gamma$. In this paper we investigate the rates of growth of the means ${s_n}(\lambda ) = {n^{ - 1}}\Sigma _1^n\vert{a_k}{\vert^\lambda }(0 < \lambda \leq 1)$ and ${t_n}(\lambda ) = {n^{ - 1}}\Sigma _1^n{k^\lambda }\vert{a_k}{\vert^\lambda }\;(\lambda > 0)$ as $n \to + \infty$ for bounded $f(z) = \Sigma _1^\infty {a_k}{z^k} \in \mathcal{R} \cup \mathcal{S}$. It is proved, for example, that the estimate ${t_n}(\lambda ) = o(1){(\log n)^{ - \alpha (\lambda )}}(n \to + \infty )$, where $\alpha (\lambda ) = \lambda /2$ for $0 < \lambda < 2$ and $\alpha (\lambda ) = 1$ for $\lambda \geq 2$, holds for such functions f, and that it is best possible for each fixed $\lambda > 0$ within the class $\mathcal{R}$ and for each fixed $\lambda \geq 2$ within the class $\mathcal{S}$. It is also shown that the inequality ${s_n}(1) = o(1){n^{ - 1}}{(\log n)^{1/2}}$, which holds for all bounded univalent functions, cannot be improved for bounded $ f \in \mathcal{R}$. The behavior of $ {t_n}(\lambda )$ as $n \to + \infty$ when ${a_k} \geq 0(k \geq 1)$ and $\lambda \geq 1$ is also examined.
Multipliers for certain convolution measure algebras
Charles Dwight
Lahr
165-181
Abstract: Let ($A,\ast$) be a commutative semisimple convolution measure algebra with structure semigroup $ \Gamma$, and let S denote a commutative locally compact topological semigroup. Under the assumption that A possesses a weak bounded approximate identity, it is shown that there is a topological embedding of the multiplier algebra $ \mathcal{M}(A)$ of A in $M(\Gamma )$. This representation leads to a proof of the commutative case of Wendel's theorem for $A = {L_1}(G)$, where G is a commutative locally compact topological group. It is also proved that if ${l_1}(S)$ has a weak bounded approximate identity of norm one, then $ \mathcal{M}({l_1}(S))$ is isometrically isomorphic to ${l_1}(\Omega (S))$, where $\Omega (S)$ is the multiplier semigroup of S. Likewise, if S is cancellative, then $ \mathcal{M}({l_1}(S))$ is isometrically isomorphic to ${l_1}(\Omega (S))$. An example is provided of a semigroup S for which ${l_1}(\Omega (S))$ is isomorphic to a proper subset of $ \mathcal{M}({l_1}(S))$.
Stability properties of a class of attractors
Jorge
Lewowicz
183-198
Abstract: Let A be an attractor of an analytical dynamical system defined in ${R^n} \times R$. The class of attractors considered in this paper consists of those A which remain stable as invariant subsets of the complex extension of the flow to ${C^n} \times R$. If A is a critical point or a closed orbit, these are the elementary or generic attractors. It is shown that such an A is always a submanifold of ${R^n}$ and that there exists a Lie group acting on A and containing the given flow as a one parameter dense subgroup; as a consequence, some necessary and sufficient conditions for an analytical dynamical system to have an attracting generic periodic motion are given. It is also shown that for any flow $ {C^1}$-close to the given one, there is a unique retraction of a neighbourhood of A onto a submanifold of $ {R^n}$ homeomorphic to A that commutes with the flow.
Representations of Jordan triples
Ottmar
Loos
199-211
Abstract: Some standard results on representations of quadratic Jordan algebras are extended to Jordan triples. It is shown that the universal envelope of a finite-dimensional Jordan triple is finite-dimensional, and that it is nilpotent if the Jordan triple is radical. A permanence principle and a duality principle are proved which are useful in deriving identities.
On recurrent random walks on semigroups
T. C.
Sun;
A.
Mukherjea;
N. A.
Tserpes
213-227
Abstract: Let $\mu$ be a regular Borel probability measure on a locally compact semigroup S and consider the right (resp. left) random walk on $D = \overline {{\text{U}}{F^n}} ,F = {\text{Supp}}\;\mu$, with transition function ${P^n}(x,B) \equiv {\mu ^n}({x^{ - 1}}B)\;({\text{resp}}.\;{\mu ^n}(B{x^{ - 1}}))$. These Markov chains can be represented as $ {Z_n} = {X_1}{X_2} \cdots {X_n}\;({\text{resp}}.\;{S_n} = {X_n}{X_{n - 1}} \cdots {X_1}),\;{X_i}$'s independent $\mu $-distributed with values in S defined on an infinite-sequence space $ (\Pi _1^\infty {S_i},P),{S_i} = S$ for all i. Let $ {R_r}\;({\text{resp}}.\;{R_t}) = \{ x \in D;{P_x}({Z_n}({S_n}) \in {N_x}\;{\text{i.o.}}) = 1$ for all neighborhoods $ {N_x}$ of x} and ${R'_r}({R'_t}) = \{ x \in D;P({Z_n}({S_n}) \in {N_x}\;{\text{i.o.}}) = 1$ for all ${N_x}$ of x}. Let S be completely simple ( $ = E \times G \times F$, usual Rees product) in the results (1), (2), (3), (4), (5) below: (1) $x \in {R_r}\;iff\;\Sigma \;{\mu ^n}({x^{ - 1}}{N_x}) = \infty $ for all neighborhoods $ {N_x}$ of $ x\;iff\;\Sigma \;{\mu ^n}({N_x}) = \infty$ for all ${N_x}$ of x. (2) Either ${R_r} = {R_t} = \emptyset$ or ${R_r} = {R_t} = D =$ also completely simple. (3) If the group factor G is compact, then there are recurrent values and we have ${R_r} = {R_t} = D =$ completely simple. (4) $R' = R = K = $ the kernel of S. These results extend previously known results of Chung and Fuchs and Loynes.
On strictly cyclic algebras, $\mathcal{P}$-algebras and reflexive operators
Domingo A.
Herrero;
Alan
Lambert
229-235
Abstract: An operator algebra $\mathfrak{A} \subset \mathcal{L}(\mathcal{X})$ (the algebra of all operators in a Banach space $\mathcal{X}$ over the complex field C) is called a ``strictly cyclic algebra'' (s.c.a.) if there exists a vector $ {x_0} \in \mathcal{X}$ such that $ \mathfrak{A}({x_0}) = \{ A{x_0}:A \in \mathfrak{A}\} = \mathcal{X};{x_0}$ is called a ``strictly cyclic vector'' for $\mathfrak{A}$. If, moreover, ${x_0}$ separates elements of $\mathfrak{A}$ (i.e., if $A \in \mathfrak{A}$ and $ A{x_0} = 0$, then $ A = 0$), then $\mathfrak{A}$ is called a ``separated s.c.a." $\mathfrak{A}$ is a $ \mathcal{P}$-algebra if, given ${x_1}, \ldots ,{x_n} \in \mathcal{X}$, there exists $ {x_0} \in \mathcal{X}$ such that $\left\Vert {A{x_j}} \right\Vert \leq \left\Vert {A{x_0}} \right\Vert$, for all $A \in \mathfrak{A}$ and for $j = 1, \ldots ,n$. Among other results, it is shown that if the commutant $\mathfrak{A}$ is an s.c.a., then $\mathfrak{A}$ is a $ \mathcal{P}$-algebra and the strong and the uniform operator topology coincide on $ \mathfrak{A}$; these results are specialized for the case when $\mathfrak{A}$ and
Statistical mechanics on a compact set with $Z^{v}$ action satisfying expansiveness and specification
David
Ruelie
237-251
Abstract: We consider a compact set $\Omega$ with a homeomorphism (or more generally a $ {{\mathbf{Z}}^\nu }$ action) such that expansiveness and Bowen's specification condition hold. The entropy is a function on invariant probability measures. The pressure (a concept borrowed from statistical mechanics) is defined as function on $ \mathcal{C}(\Omega )$--the real continuous functions on $\Omega$. The entropy and pressure are shown to be dual in a certain sense, and this duality is investigated.
The law of the iterated logarithm for Brownian motion in a Banach space
J.
Kuelbs;
R.
Lepage
253-264
Abstract: Strassen's version of the law of the iterated logarithm is proved for Brownian motion in a real separable Banach space. We apply this result to obtain the law of the iterated logarithm for a sequence of independent Gaussian random variables with values in a Banach space and to obtain Strassen's result.
Extreme points for some classes of univalent functions
W.
Hengartner;
G.
Schober
265-270
Abstract: Monotonicity properties are given for extreme points in classes of normalized analytic and univalent mappings of an arbitrary domain. For the familiar class of normalized univalent mappings of the unit disk, extreme points f are shown to have the remarkable property that $f/z$ is univalent.
Triangular representations of splitting rings
K. R.
Goodearl
271-285
Abstract: The term ``splitting ring'' refers to a nonsingular ring R such that for any right R-module M, the singular submodule of M is a direct summand of M. If R has zero socle, then R is shown to be isomorphic to a formal triangular matrix ring $\left( {\begin{array}{*{20}{c}} A & 0 B & C \end{array} } \right)$, where A is a semiprime ring, C is a left and right artinian ring, and $_C{B_A}$ is a bimodule. Also, necessary and sufficient conditions are found for such a formal triangular matrix ring to be a splitting ring.
Multiplicative structure of generalized Koszul complexes
Eugene H.
Gover
287-307
Abstract: A multiplicative structure is defined for the generalized Koszul complexes $K({ \wedge ^p}f)$ associated with the exterior powers of a map $f:{R^m} \to {R^n}$ where R is a commutative ring and $m \geq n$. With this structure $K({ \wedge ^n}f)$ becomes a differential graded R-algebra over which each $K({ \wedge ^p}f),1 \leq p \leq n$, is a DG right $K({ \wedge ^n}f)$-module. For $f = 0$ and $n > 1$, the multiplication and all higher order Massey operations of $K({ \wedge ^n}f)$ are shown to be trivial. When R is noetherian local, $K({ \wedge ^n}f)$ is used to define a class of local rings which includes the local complete intersections. The rings obtained for $n > 1$ are Cohen-Macaulay but not Gorenstein. Their Betti numbers and Poincaré series are computed but these do not characterize the rings.
Isomorphisms of the lattice of inner ideals of certain quadratic Jordan algebras
Jerome M.
Katz
309-329
Abstract: The inner ideals play a role in the theory of quadratic Jordan algebras analogous to that played by the one-sided ideals in the associative theory. In particular, the simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals play a role analogous to that of the simple artinian algebras in the associative theory. In this paper, we investigate the automorphism group of the lattice of inner ideals of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals. For the case $ \mathfrak{H}(\mathfrak{A}{,^ \ast })$ where $(\mathfrak{A}{,^ \ast })$ is a simple artinian algebra with hermitian involution, we show that the automorphism group of the lattice of inner ideals is isomorphic to the group of semilinear automorphisms of $\mathfrak{A}$. For the case $ \mathfrak{H}({\mathfrak{Q}_n}{,^ \ast })$ where $ \mathfrak{Q}$ is a split quaternion algebra, we obtain only a partial result. For the cases $J = \mathfrak{H}({\mathfrak{O}_3})$ and $J = {\text{Jord}}(Q,1)$ for $\mathfrak{O}$ an octonion algebra, $(Q,1)$ a nondegenerate quadratic form with base point of Witt index at least three and J finite dimensional, it is shown that every automorphism of the lattice of inner ideals is induced by a norm semisimilarity. Finally, we determine conditions under which two algebras of the type under consideration can have isomorphic lattices of inner ideals.
Hermitian operators and one-parameter groups of isometries in Hardy spaces
Earl
Berkson;
Horacio
Porta
331-344
Abstract: Call an operator A with domain and range in a complex Banach space X hermitian if and only if iA generates a strongly continuous one-parameter group of isometries on X. Hermitian operators in the Hardy spaces of the disc $({H^p},1 \leq p \leq \infty )$ are investigated, and the following results, in particular, are obtained. For $1 \leq p \leq \infty ,p \ne 2$, the bounded hermitian operators on ${H^p}$ are precisely the trivial ones--i.e., the real scalar multiples of the identity operator. Furthermore, as pointed out to the authors by L. A. Rubel, there are no unbounded hermitian operators in ${H^\infty }$. To each unbounded hermitian operator in the space ${H^p},1 \leq p < \infty ,p \ne 2$, there corresponds a uniquely determined one-parameter group of conformal maps of the open unit disc onto itself. Such unbounded operators are classified into three mutually exclusive types, an operator's type depending on the nature of the common fixed points of the associated group of conformal maps. The hermitian operators falling into the classification termed ``type (i)'' have compact resolvent function and one-dimensional eigenmanifolds which collectively span a dense linear manifold in $ {H^p}$.
Almost everywhere convergence of Vilenkin-Fourier series
John
Gosselin
345-370
Abstract: It is shown that the partial sums of Vilenkin-Fourier series of functions in $ {L^q}(G),q < 1$, converge almost everywhere, where G is a zero-dimensional, compact abelian group which satisfies the second axiom of countability and for which the dual group X has a certain bounded subgroup structure. This result includes, as special cases, the Walsh-Paley group $ {2^w}$, local rings of integers, and countable products of cyclic groups for which the orders are uniformly bounded.
The strong law of large numbers when the mean is undefined
K. Bruce
Erickson
371-381
Abstract: Let ${S_n} = {X_1} + \cdots + {X_n}$ where $\{ {X_n}\}$ are i.i.d. random variables with $EX_1^ \pm = \infty$. An integral test is given for each of the three possible alternatives $\lim ({S_n}/n) = + \infty $ a.s.; $\lim ({S_n}/n) = - \infty $ a.s.; $\lim \sup ({S_n}/n) = + \infty$ and $\lim \inf ({S_n}/n) = - \infty$ a.s. Some applications are noted.
Neocontinuous Mikusi\'nski operators
Carl C.
Hughes;
Raimond A.
Struble
383-400
Abstract: A class of Mikusiński-type operators in several variables, called neocontinuous operators, is studied. These particular operators are closely affiliated with Schwartz distributions on ${R^k}$ and share certain continuity properties with them. This affiliation is first of all revealed through a common algebraic view of operators and distributions as homomorphic mappings and a new representation theory, and is then characterized in terms of continuity properties of the mappings. The traditional procedures of the operational calculus apply to the class of neocontinuous operators. Moreover, the somewhat vague association of operational and distributional solutions of partial differential equations is replaced by the decisive representation concept, thus illustrating the appropriateness of the study of neocontinuous operators.
Perturbations of nonlinear differential equations
R. E.
Fennell;
T. G.
Proctor
401-411
Abstract: Scalar and vector comparison techniques are used to study the comparative asymptotic behavior of the systems (1)
Convex hulls and extreme points of families of starlike and convex mappings
L.
Brickman;
D. J.
Hallenbeck;
T. H.
Macgregor;
D. R.
Wilken
413-428
Abstract: The closed convex hull and extreme points are obtained for the starlike functions of order $\alpha$ and for the convex functions of order $ \alpha$. More generally, this is determined for functions which are also k-fold symmetric. Integral representations are given for the hulls of these and other families in terms of probability measures on suitable sets. These results are used to solve extremal problems. For example, the upper bounds are determined for the coefficients of a function subordinate to or majorized by some function which is starlike of order $ \alpha$. Also, the lower bound on $\operatorname{Re} \{ f(z)/z\}$ is found for each $z\;(\vert z\vert < 1)$ where f varies over the convex functions of order $ \alpha$ and $\alpha \geq 0$.
Cauchy problem and analytic continuation for systems of first order elliptic equations with analytic coefficients
Chung Ling
Yu
429-443
Abstract: Let a, b, c, d, f, g be analytic functions of two real variables x, y in the $z = x + iy$ plane. Consider the elliptic equation (M) $\partial u/\partial x - \partial v/\partial y = au + bv + f,\partial u/\partial y + \partial v/\partial x = cu + dv + g$. The following areas will be investigated: (1) the integral respresentations for solutions of (M) to the boundary $\partial G$ of a simply connected domain G; (2) reflection principles for (M) under nonlinear analytic boundary conditions; (3) the sufficient conditions for the nonexistence and analytic continuation for the solutions of the Cauchy problem for (M).
Square integrable representations of nilpotent groups
Calvin C.
Moore;
Joseph A.
Wolf
445-462
Abstract: We study square integrable irreducible unitary representations (i.e. matrix coefficients are to be square integrable mod the center) of simply connected nilpotent Lie groups N, and determine which such groups have such representations. We show that if N has one such square integrable representation, then almost all (with respect to Plancherel measure) irreducible representations are square integrable. We present a simple direct formula for the formal degrees of such representations, and give also an explicit simple version of the Plancherel formula. Finally if $\Gamma$ is a discrete uniform subgroup of N we determine explicitly which square integrable representations of N occur in ${L_2}(N/\Gamma )$, and we calculate the multiplicities which turn out to be formal degrees, suitably normalized.
Appell polynomials and differential equations of infinite order
J. D.
Buckholtz
463-476
Abstract: Let $ \Phi (z) = \Sigma _0^\infty {\beta _j}{z^j}$ have radius of convergence $r\;(0 < r < \infty )$ and no singularities other than poles on the circle $\vert z\vert = r$. A complete solution is obtained for the infinite order differential equation $( \ast )\;\Sigma _0^\infty {\beta _j}{u^{(j)}}(z) = g(z)$. It is shown that $(\ast)$ possesses a solution if and only if the function g has a polynomial expansion in terms of the Appell polynomials generated by $\Phi$. The solutions of $( \ast )$ are expressed in terms of the coefficients which appear in the Appell polynomial expansions of g. An alternate method of solution is obtained, in which the problem of solving $( \ast )$ is reduced to the problem of finding a solution, within a certain space of entire functions, of a finite order linear differential equation with constant coefficients. Additionally, differential operator techniques are used to study Appell polynomial expansions.
Mechanical systems with symmetry on homogeneous spaces
Ernesto A.
Lacomba
477-491
Abstract: The geodesic flow on a homogeneous space with an invariant metric can be naturally considered within the framework of Smale's mechanical systems with symmetry. In this way we have at our disposal the whole method of Smale for studying such systems. We prove that certain sets
A geometrical characterization of Banach spaces with the Radon-Nikodym property
Hugh B.
Maynard
493-500
Abstract: A characterization of Banach spaces having the Radon-Nikodym property is obtained in terms of a convexity requirement on all bounded subsets. In addition a Radon-Nikodym theorem, utilizing this convexity property, is given for the Bochner integral and it is easily shown that this theorem is equivalent to the Phillips-Metivier Radon-Nikodym theorem as well as all the standard Radon-Nikodym theorems for the Bochner integral.
Weakly almost periodic functionals on the Fourier algebra
Charles F.
Dunkl;
Donald E.
Ramirez
501-514
Abstract: The theory of weakly almost periodic functional on the Fourier algebra is herein developed. It is the extension of the theory of weakly almost periodic functions on locally compact abelian groups to the duals of compact groups. The complete direct product of a countable collection of nontrivial compact groups furnishes an important example for some of the constructions.
Complete topologies on spaces of Baire measure
R. B.
Kirk
1-29
Abstract: Let X be a completely regular Hausdorff space, let L be the linear space of all finite linear combinations of the point measures on X and let ${M_\sigma }$ denote the space of Baire measures on X. The following is proved: If ${M_\sigma }$ is endowed with the topology of uniform convergence on the uniformly bounded, equicontinuous subsets of ${C^b}(X)$, then $ {M_\sigma }$ is a complete locally convex space in which L is dense and whose dual is ${C^b}(X)$, provided there are no measurable cardinals. A complete description of the situation in the presence of measurable cardinals is also given. Let ${M_C}$ be the subspace of $ {M_\sigma }$ consisting of those measures which have compact support in the realcompactification of X. The following result is proved: If ${M_C}$ is endowed with the topology of uniform convergence on the pointwise bounded and equicontinuous subsets of $C(X)$, then ${M_C}$ is a complete locally convex space in which L is dense and whose dual is $C(X)$, provided there are no measurable cardinals. Again the situation if measurable cardinals exist is described completely. Let M denote the Banach dual of ${C^b}(X)$. The following is proved: If M is endowed with the topology of uniform convergence on the norm compact subsets of ${C^b}(X)$, then M is a complete locally convex space in chich L is dense. It is also proved that ${M_\sigma }$ is metrizable if and only if X is discrete and that the metrizability of either $ {M_C}$ or M is equivalent to X being finite. Finally the following is proved: If ${M_C}$ has the Mackey topology for the pair $({M_C},C(X))$, then ${M_C}$ is complete and L is dense in ${M_C}$.
Products of decompositions of $E\sp{n}$
Brian J.
Smith
31-41
Abstract: In this paper we give a sufficient condition for the existence of a homeomorphism $h:{E^m}/G \times {E^n}/H \to {E^{m + n}}$, where G and H are u.s.c. decompositions of Euclidean space. This condition is then shown to hold for a wide class of examples in which the decomposition spaces ${E^m}/G$ and ${E^n}/H$ may fail to be Euclidean.
Prime ideals and sheaf representation of a pseudo symmetric ring
Gooyong
Shin
43-60
Abstract: Almost symmetric rings and pseudo symmetric rings are introduced. The classes of symmetric rings, of almost symmetric rings, and of pseudo symmetric rings are in a strictly increasing order. A sheaf representation is obtained for pseudo symmetric rings, similar to the cases of symmetric rings, semiprime rings, and strongly harmonic rings. Minimal prime ideals of a pseudo symmetric ring have the same characterization, due to J. Kist, as for the commutative case. A characterization is obtained for a pseudo symmetric ring with a certain right quotient ring to have compact minimal prime ideal space, extending a result due to Mewborn.
Restricting a Schauder basis to a set of positive measure
James
Shirey
61-71
Abstract: Let $\{ {f_n}\}$ be an orthonormal system of functions on [0, 1] containing a subsystem $\{ {f_{{n_k}}}\}$ for which (a) ${f_{{n_k}}} \to 0$ weakly in ${L_2}$, and (b) given $E \subset [0,1]$, $\vert E\vert > 0$, $ {\operatorname{Lim}}\;{\operatorname{Inf}}{\smallint _E}\vert{f_{{n_k}}}(x)\vert dx > 0$. There then exists a subsystem $\{ {g_n}\}$ of $ \{ {f_n}\}$ such that for any set E as above, the linear span of $\{ {g_n}\}$ in ${L_1}(E)$ is not dense. For every set E as above, there is an element of ${L_p}(E)$, $ 1 < p < \infty$, whose Walsh series expansion converges conditionally and an element of ${L_1}(E)$ whose Haar series expansion converges conditionally.
Commutative torsion theory
Paul-Jean
Cahen
73-85
Abstract: This paper links several notions of torsion theory with commutative concepts. The notion of dominant dimension [H. H. Storrer, Torsion theories and dominant dimensions, Appendix to Lecture Notes in Math., vol. 177, Springer-Verlag, Berlin and New York, 1971. MR 44 #1685.] is shown to be very close to the notion of depth. For a commutative ring A and a torsion theory such that the primes of A, whose residue field is torsion-free, form an open set U of the spectrum of A, Spec A, a concrete interpretation of the module of quotients is given: if M is an A-module, its module of quotients $Q(M)$ is isomorphic to the module of sections $\tilde M(U)$, of the quasi-coherent module $ \tilde M$ canonically associated to M. In the last part it is proved that the (T)-condition of Goldman is satisfied [O. Goldman, Rings and modules of quotients, J. Algebra 13 (1969), 10-47. MR 39 #6914.] if and only if the set of primes, whose residue field is torsion-free, is an affine subset of Spec A, together with an extra condirion. The extra, more technical, condition is always satisfied over a Noetherian ring, in this case also it is classical that the (T)-condition of Goldman means that the localization functor Q is exact. This gives a new proof to Serre's theorem [J.-P. Serre, Sur la cohomologie des variétés algébriques, J. Math. Pures Appl. (9) 36 (1957), 1-16. MR 18,765.]. As an application, the affine open sets of a regular Noetherian ring are also characterized.
Smoothness of certain metric projections on Hilbert space
Richard B.
Holmes
87-100
Abstract: A study is made of differential properties of the distance function and the metric projection defined by a closed convex subset of Hilbert space. The former mapping is also considered within the context of more general Banach spaces.
Measurable transformations on compact groups
J. R.
Choksi
101-124
Abstract: For an arbitrary finite Baire measure $\mu$ on an arbitrary compact group G, it is shown that every automorphism of the measure algebra of $\mu$ can be induced by an invertible completion Baire measurable point transformation of G. If $\mu$ is Haar measure, the point transformation is completion Borel measurable.
Topological entropy for noncompact sets
Rufus
Bowen
125-136
Abstract: For $f:X \to X$ continuous and $Y \subset X$ a topological entropy $ h(f,Y)$ is defined. For X compact one obtains results generalizing known theorems about entropy for compact Y and about Hausdorff dimension for certain $Y \subset X = {S^1}$ . A notion of entropy-conjugacy is proposed for homeomorphisms.
On the uniform convergence of quasiconformal mappings
Bruce
Palka
137-152
Abstract: Let D be a domain in extended Euclidean n-space with ``smooth'' boundary and let $ \{ {f_j}\}$ be a sequence of K-quasiconformal mappings of D into $ {R^n}$ which converges uniformly on compact sets in D to a quasiconformal mapping. This paper considers the question: When does the sequence $ \{ {f_j}\}$ converge uniformly on all of D? Geometric conditions on the domains ${f_j}(D)$ are given which are sufficient and, in many cases, necessary for uniform convergence. The particular case where D is the unit ball in ${R^n}$ is examined to obtain analogues to classical convergence theorems for conformal mappings in the plane.
Fixed point structures
T. B.
Muenzenberger;
R. E.
Smithson
153-173
Abstract: A fixed point structure is a triple $ (X,\mathcal{P},\mathcal{F})$ where X is a set, $ \mathcal{P}$ a collection of subsets of X, and $ \mathcal{F}$ a family of multifunctions on X into itself together with a set of axioms which insure that each member of $\mathcal{F}$ has a fixed point. A fixed point structure for noncontinuous multifunctions on semitrees is established that encompasses fixed point theorems of Wallace-Ward and Young-Smithson as well as new fixed point theorems for partially ordered sets and closed stars in real vector spaces. Also two other fixed point structures are presented that subsume fixed point theorems of Tarski-Ward-Smithson on semilattices and, more generally, partially ordered sets. Also the Davis-Ward converse to this last fixed point theorem is obtained.
Analytic functions characterized by their means on an arc
Chin Hung
Ching;
Charles K.
Chui
175-183
Abstract: It is known that a function f, holomorphic in the open unit disc U with ${C^{1 + \varepsilon }}$ boundary data for some $\varepsilon > 0$, is uniquely determined by its arithmetic means over equally spaced points on $\partial U$. By using different techniques, we weaken the hypothesis ${C^{1 + \varepsilon }}(\partial U)$ to functions with ${L^p}$ derivatives, $1 < p \leq \infty$. We also prove that a function is determined by its averages over an arc K if f is holomorphic in a neighborhood of $\bar U$, and that this result is false for some functions f in $A \cap {C^\infty }(\bar U)$. On the other hand, we can capture a $ A \cap {C^2}(\bar U)$ function from its means and shifted means on K.
Wandering out to infinity of diffusion processes
Avner
Friedman
185-203
Abstract: Let $\xi (t)$ be a diffusion process in $ {R^n}$, given by $ d\xi = b(\xi )dt + \sigma (\xi )dw$. Conditions are given under which either $\vert\xi (t)\vert \to \infty$ as $t \to \infty$ with probability 1, or $ \xi (t)$ visits any neighborhood at a sequence of times increasing to infinity, with probability 1. The results are obtained both in case (i) $\sigma (x)$ is nondegenerate, and (ii) $\sigma (x)$ is degenerate at a finite number of points and hypersurfaces.
On the singular boundary value problem for elliptic equations
Kazunari
Hayashida
205-221
Abstract: The operator $\mathcal{L}$ is elliptic and of second order in a domain $\Omega$ in ${R^N}$. We consider the following boundary value problem: $ \mathcal{L}u = f$ in $ \Omega$ and $\mathcal{B}u = 0$ on $ \partial \Omega$, where $ \mathcal{B} = ad/dn + \beta$ (d/dn is the conormal derivative on $\partial \Omega$). The coefficient $\alpha$ is assumed to be nonnegative. However, $\alpha$ may vanish partly on $\partial \Omega$. Then the regularity of the weak solutions for the above problem is shown by the variational method.
Finite- and infinite-dimensional representation of linear semisimple groups
James
Lepowsky;
Nolan R.
Wallach
223-246
Abstract: Every representation in the nonunitary principal series of a noncompact connected real semisimple linear Lie group G with maximal compact subgroup K is shown to have a K-finite cyclic vector. This is used to give a new proof of Harish-Chandra's theorem that every member of the nonunitary principal series has a (finite) composition series. The methods of proof are based on finite-dimensional G-modules, concerning which some new results are derived. Further related results on infinite-dimensional representations are also obtained.
Cross-sections of symplectic Stiefel manifolds
François
Sigrist;
Ulrich
Suter
247-259
Abstract: The cross-section problem for the symplectic Stiefel manifolds is solved, using the now-proved Adams conjecture.
The commutant of analytic Toeplitz operators
James A.
Deddens;
Tin Kin
Wong
261-273
Abstract: In this paper we study the commutant of an analytic Toeplitz operator. For $ \phi \;\;{H^\infty }$, let $\phi = \chi F$ be its inner-outer factorization. Our main result is that if there exists $ \lambda \;\epsilon \;{\text{C}}$ such that X factors as $ \chi = {\chi _1}{\chi _2} \cdots {\chi _n}$, each ${\chi _i}$ an inner function, and if $F - \lambda$ is divisible by each $ {\chi _i}$, then $\chi (z) = {z^n},n \geq 1$, then $ \phi \;\epsilon {H^\infty }$ is univalent then $\{ {T_\phi }\} ' = \{ {T_z}\} '$. We are also able to prove that if the inner factor of $ \phi$ is $\chi (z) = {z^n},n \geq 1$, then $ \{ {T_\phi }\} ' = \{ {T_{{z^s}}}\} '$ where s is a positive integer maximal with respect to the property that ${z^n}$ and $F(z)$ are both functions of ${z^s}$. We conclude by raising six questions.
Universal generators for varieties of nuclear spaces
B.
Rosenberger
275-290
Abstract: It is shown that a product of several copies of $\Lambda ({\beta ^\phi })$ is a universal $ \phi$-nuclear space if the power series space $\Lambda ({\beta ^\phi })$ with $\beta _k^\phi = - \log ({\phi ^{ - 1}}(1/\sqrt {k + 1} )),k\;\epsilon \;\{ 0,1,2, \cdots \}$, is $\phi$-nuclear; here $\phi = [0,\infty ) \to [0,\infty )$ is a continuous, strictly increasing subadditive function with $\phi (0) = 0$. In case $\Lambda ({\beta ^\phi })$ is not $\phi $-nuclear the sequence space $ \Lambda (l_\phi ^ + )$ is a $\phi$-nuclear space with the property that every $ \phi$-nuclear space is isomorphic to a subspace of a product of $\Lambda (l_\phi ^ + )$ if ${\lim\;\sup _{t \to 0}}{(\phi (t))^{ - 1}}\phi (\sqrt t ) < \infty $.
Absolute convergence of series of Fourier coefficients
James R.
McLaughlin
291-316
Abstract: In this article the author unifies and generalizes practically all known sufficiency results for absolute convergence of series of Fourier coefficients that are given in terms of the integrated modulus of continuity, best approximation, or bounded pth variation. This is done for the trigonometric, Walsh, Haar, Franklin, and related systems as well as general orthonormal systems. Many of the original proofs of previous results relied upon special properties of the trigonometric, Haar, and other systems and were done independently of one another. Also, several authors have proved results which at the time they believed to be generalizations of past results, but are, in fact, corollaries of them. The present author will expose underlying principles and illustrate their usefulness.
A metric characterization of cells
Ellard
Nunnally
317-325
Abstract: We examine finite dimensional compact convex metric spaces each having the property that the union of two line segments in the space, having more than one point in common, is a line segment. The question has been asked (Borsuk; Bing) whether each such space is a cell. The answer is yes if the dimension of the space is $\leq 2$ (Lelek and Nitka) or 3 (Rolfsen). Here we provide an affirmative answer for arbitrary finite dimension provided the space has the additional property that the join of any point to any line segment in the space is a convex set.
Decomposition theorems of Riemannian manifolds
Pyng
Wang
327-341
Abstract: Given two complementary orthogonal parallel foliations on a complete connected Riemannian manifold M, a necessary and sufficient condition for the direct product of the two leaves through a point m being a covering manifold of M is obtained. It is shown that the direct product of the two leaves through m of the two foliations is a Riemannian covering of M if the two leaves are regular at m. Moreover, if one of the foliations is regular and the intersection of the two leaves through m contains only the point m, then M is isometric to the direct product of the two leaves.
The representation of norm-continuous multipliers on $L\sp{\infty }$-spaces
Gregory A.
Hively
343-353
Abstract: If G is a group and ${\mathcal{L}^\infty }(G,\mathcal{S})$ is an appropriate space of bounded measurable functions on G, a representation is obtained for the algebra of norm-continuous multipliers on ${\mathcal{L}^\infty }(G,\mathcal{S})$ as an algebra of bounded additive set functions on G. If G is a locally compact group, a representation of the norm-continuous multipliers on the quotient space ${\mathcal{L}^\infty }(G)$ is obtained in terms of a quotient algebra of bounded additive set functions on G.
Pseudo-differential estimates for linear parabolic operators
David
Ellis
355-371
Abstract: In recent papers, S. Kaplan and D. Ellis have used singular integral operator theory, multilinear interpolation and forms of the classical ``energy inequality'' to obtain results for linear parabolic operators. For higher order linear parabolic operators the local estimates were globalized by a Gårding type partition of unity. In the present paper it is shown how the theory of pseudo-differential operators can be used to study linear parabolic operators without recourse to multilinear interpolation. We also prove that the Gårding type partition of unity is square summable in the Sobolev type spaces ${H^S}$ and $ {\mathcal{K}^{r,S}}$.
Convolution equations and harmonic analysis in spaces of entire functions
D. G.
Dickson
373-385
Abstract: If H is the topological space of functions analytic in the simply connected open set $\Omega$ of the plane with the topology of compact convergence, its dual may be identified with the space E of functions of exponential type whose Borel transforms have their singularities in $ \Omega$. For f in H and $\phi$ in E, $ (f \ast \phi )(z) \equiv \left\langle {f,{\phi _z}} \right\rangle$ where $ {\phi _z}$ is the z-translate of $\phi$. If $ f{\nequiv}0$ in any component of $ \Omega ,f \ast \phi = 0$ if and only if $\phi$ is a finite linear combination of monomial-exponentials $ {z^p} \exp (\omega z)$ where $\omega$ is a zero of f in $\Omega$ of order at least $ p + 1$. For such f and $\psi$ in E, $f \ast \phi = \psi$ is solved explicitly for $ \phi$. If E is assigned its strong dual topology and $\tau (\phi )$ is the closed linear span in E of the translates of $\phi$, then $ \tau (\phi )$ is a finite direct sum of closed subspaces spanned by monomial-exponentials. Each closed translation invariant subspace of E is the kernel of a convolution mapping $\phi \to f \ast \phi$; there is a one-to-one correspondence between such subspaces and the closed ideals of H with the correspondence that of annihilators.
On arbitrary sequences of isomorphisms in $R\sp{m}\rightarrow R\sp{m}$
Charles C.
Pugh
387-400
Abstract: In this paper a new, clean proof of an algebraic theorem needed in ordinary differential equations is presented. The theorem involves the existence and uniqueness of a ``complete splitting'' for some subsequence of an arbitrary sequence of isomorphisms of Euclidean m-space. In the positive-definite case, a complete splitting is a limit condition on eigenspaces and eigenvalues.
On the points of Weierstrass in dimensions greater than one
Roy H.
Ogawa
401-417
Abstract: In this paper, the classical concept of Weierstrass points on a Riemann surface is generalized to the consideration of similar points associated with a holomorphic vector bundle E over a compact complex manifold M. These points are invariants of the pair (E, M). The study of these general Weierstrass points is then initiated in this paper by deriving some results about the relationship of the points to singular sets of holomorphic mappings of the manifold to Grassmann spaces associated with the vector space of sections of the vector bundle. The accessibility of the point sets are demonstrated with some examples.
Stability of foliations
Harold I.
Levine;
Michael
Shub
419-437
Abstract: Let X be a compact manifold and let k be an integer. It is shown that the set of homeomorphism conjugacy classes of germs at X of foliations of codimension k and the set of homeomorphism conjugacy classes of (holonomy) representations of ${\prod _1}(X)$ in the group of germs at 0 of 0-fixed self-diffeomorphisms of ${{\text{R}}^k}$ are homeomorphic when given appropriate topologies. Stable foliation germs and stable holonomy representations correspond under this homeomorphism. It is shown that there are no stable foliation germs at a toral leaf if the dimension of the torus is greater than one.
The periods of Eichler integrals for Kleinian groups
Hiroki
Sato
439-456
Abstract: We shall give period relations and inequalities for Eichler integrals for Kleinian groups $\Gamma$ which have simply connected components of of the region of discontinuity of $ \Gamma$. These are a generalization of those for abelian integrals. By using the period inequality, we shall give an alternate proof of a result of Kra.
A free boundary problem connected with the optimal stopping problem for diffusion processes
Daniel B.
Kotlow
457-478
Abstract: This paper deals with a free boundary problem for a parabolic equation in one space variable which arises from the problem of selecting an optimal stopping strategy for the diffusion process connected with the equation. It is shown that a solution of the free boundary problem yields the solution of a minimum problem concerning supersolutions of the parabolic equation as well as the solution of the optimal stopping problem. Theorems regarding the existence, uniqueness, regularity, and approach to the steady state of solutions of the free boundary problem are established.
A completely mitotic nonrecursive r.e. degree
Richard E.
Ladner
479-507
Abstract: A nonrecursive r.e. degree d is constructed that has the property that every r.e. set of degree d is mitotic. The degree d has several other interesting properties including the property that any two r.e. sets of degree d are weak truth table equivalent.
Erratum to ``The nonstandard theory of topological vector spaces'' (Trans. Amer. Math. Soc. {\bf 172} (1972), 405--435)
C. Ward
Henson;
L. C.
Moore
509
Existence theorems for matroid designs
H. Peyton
Young
1-35
Abstract: A study is made of matroids in which the hyperplanes have equal cardinality. Fundamental constructions of such matroids are exhibited, and existence theorems are proved for large parametric classes of them.
Branched structures and affine and projective bundles on Riemann surfaces
Richard
Mandelbaum
37-58
Abstract: A classification for analytic branched G-structures on a Riemann surface M is provided by means of a map $ {\phi _G}$, into the moduli spaces of flat G-bundles on $M.\;(G = {\text{GA}}(1,{\text{C}})$ or $ {\text{PL}}(1,{\text{C}}).)$ Conditions are determined under which ${\phi _G}$ is injective and these conditions are related to the total branching order of the G-structures. A decomposition of the space of analytic branched G-structures into a disjoint union of analytic varieties is exhibited and it is shown that ${\phi _G}$ is is fact holomorphic on each such variety.
Torsion in $K$-theory and the Bott maps
Albert T.
Lundell
59-85
Abstract: The nonstable Bott maps
Generalized semigroups of quotients
C. V.
Hinkle
87-117
Abstract: For S a semigroup with 0 and ${M_S}$ a right S-set, certain classes of sub S-sets called right quotient filters are defined. A study of these right quotient filters is made and examples are given including the classes of intersection large and dense sub S-sets respectively. The general semigroup of right quotients Q corresponding to a right quotient filter on a semigroup S is developed and basic properties of this semigroup are noted. A nonzero regular semigroup S is called primitive dependent if each nonzero right ideal of S contains a 0-minimal right ideal of S. The theory developed in the paper enables us to characterize all primitive dependent semigroups having singular congruence the identity in terms of subdirect products of column monomial matrix semigroups over groups.
Lattice points and Lie groups. I
Robert S.
Cahn
119-129
Abstract: Assume that G is a compact semisimple Lie group and $\mathfrak{G}$ its associated Lie algebra. It is shown that the number of irreducible representations of G of dimension less than or equal to n is asymptotic to $ k{n^{a/b}}$, where a = the rank of $ \mathfrak{G}$ and b = the number of positive roots of $\mathfrak{G}$.
Lattice points and Lie groups. II
Robert S.
Cahn
131-137
Abstract: Let C be the Casimir operator on a compact, simple, simply connected Lie group G of dimension n. The number of eigenvalues of C, counted with their multiplicities, of absolute value less than or equal to t is asymptotic to $k{t^{n/2}},\;k$ a constant. This paper shows the error of this estimate to be $O({t^{2b + a(a - 1)/(a + 1)}})$; where a = rank of G and $b = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}(n - a)$.
Involutions on $S\sp{1}\times S\sp{2}$ and other $3$-manifolds
Jeffrey L.
Tollefson
139-152
Abstract: This paper exploits the following observation concerning involutions on nonreducible 3-manifolds: If the dimension of the fixed point set of a PL involution is less than or equal to one then there exists a pair of disjoint 2-spheres that do not bound 3-cells and whose union is invariant under the given involution. The classification of all PL involutions of $ {S^1} \times {S^2}$ is obtained. In particular, ${S^1} \times {S^2}$ admits exactly thirteen distinct PL involutions (up to conjugation). It follows that there is a unique PL involution of the solid torus ${S^1} \times {D^2}$ with 1-dimensional fixed point set. Furthermore, there are just four fixed point free ${Z_{2k}}$-actions and just one fixed point free ${Z_{2k + 1}}$-action on ${S^1} \times {S^2}$ for each positive integer k (again, up to conjugation). The above observation is also used to obtain a general description of compact, irreducible 3-manifolds that admit two-sided embeddings of the projective plane.
Remarks on global hypoellipticity
Stephen J.
Greenfield;
Nolan R.
Wallach
153-164
Abstract: We study differential operators D which commute with a fixed normal elliptic operator E on a compact manifold M. We use eigenfunction expansions relative to E to obtain simple conditions giving global hypoellipticity. These conditions are equivalent to D having parametrices in certain spaces of functions or distributions. An example is given by M = compact Lie group and and E = Casimir operator, with D any invariant differential operator. The connections with global subelliptic estimates are investigated.
Solvable groups having system normalizers of prime order
Gary M.
Seitz
165-173
Abstract: Let G be a solvable group having system normalizer D of prime order. If G has all Sylow groups abelian then we prove that $ l(G) = l({C_G}(D)) + 2$, provided $l(G) \geq 3$ (here $l(H)$ denotes the nilpotent length of the solvable group H). We conjecture that the above result is true without the condition on abelian Sylow subgroups. Other special cases of the conjecture are handled.
Multilinear identities of the matrix ring
Uri
Leron
175-202
Abstract: Let V be a vector space over a field F of zero characteristic, which is acted upon by the symmetric group. Systems of generators for V are constructed, which have special symmetry and skew symmetry properties. This is applied to prove that every multilinear polynomial identity of degree $2n + 1$ which holds in the matrix ring ${F_n}(n > 2)$ is a consequence of the standard identity ${s_{2n}}$. The notions of rigid and semirigid sequences of matrices are defined and treated.
Almost maximal integral domains and finitely generated modules
Willy
Brandal
203-222
Abstract: We present a class of integral domains with all finitely generated modules isomorphic to direct sums of cyclic modules. This class contains all previously known examples (i.e., the principal ideal domains and the almost maximal valuation rings) and, by an example, at least one more domain. The class consists of the integral domains satisfying (1) every finitely generated ideal is principal (obviously a necessary condition) and (2) every proper homomorphic image of the domain is linearly compact. We call an integral domain almost maximal if it satisfies (2). This is one of eleven conditions which, for valuation rings, is equivalent of E. Matlis' ``almost maximal.'' An arbitrary integral domain R is almost maximal if and only if it is h-local and $ {R_M}$ is almost maximal for every maximal ideal M of R. Finally, equivalent conditions for a Prüfer domain to be almost maximal are studied, and in the process some conjectures of E. Matlis are answered.
Sommes de Ces\`aro et multiplicateurs des d\'eveloppements en harmoniques sph\'eriques
Aline
Bonami;
Jean-Louis
Clerc
223-263
Abstract: Nous établissons une inégalité entre les sommes de Cesáro et la fonction maximale associées á une fonction définie sur la sphére, et nous en déduisons divers résultats de convergence en norme ${L^p}$, convergence presque partout, localisation des développements en harmoniques sphériques, ainsi qu'un théorème de multiplicateurs qui généralise le théorème classique de Marcinkiewicz sur les séries trigonométriques. La même étude est faite pour les développements suivant les polynômes ultrasphériques. Nous montrons de plus que les sommes partielles du développement en harmoniques sphériques d'une fonction de ${L^p}({\Sigma _n}),p \ne 2$, ne convergent pas forcément en norme.
Parametrizations of analytic varieties
Joseph
Becker
265-292
Abstract: Let V be an analytic subvariety of an open subset $\Omega$ of ${{\text{C}}^n}$ of pure dimension r; for any $p \in V$, there exists an $n - r$ dim plane T such that ${\pi _T}:V \to {{\text{C}}^r}$, the projection along T to $ {{\text{C}}^r}$, is a branched covering of finite sheeting order $\mu (V,p,T)$ in some neighborhood of V about p. ${\pi _T}$ is called a global parametrization of V if ${\pi _T}$ has all discrete fibers, e.g. ${\dim _p}V \cap (T + p) = 0$ for all $ p \in V$. Theorem. $B = \{ (p,T) \in V \times G(n - r,n)\vert{\dim _p}V \cap (T + p) > 0\}$ is an analytic set. If $ {\pi _2}:V \times G \to G$ is the natural projection, then ${\pi _2}(B)$ is a negligible set in G. Theorem. $\{ (p,T) \in V \times G\vert\mu (V,p,T) \geq k\}$ is an analytic set. For each $p \in V$, there is a least $ \mu (V,p)$ and greatest $m(V,p)$ sheeting multiplicity over all $ T \in G$. If $ \Omega$ is Stein, V is the locus of finitely many holomorphic functions but its ideal in $ \mathcal{O}(\Omega )$ is not necessarily finitely generated. Theorem. If $\mu (V,p)$ is bounded on V, then its ideal is finitely generated.
Discrete $\omega $-sequences of index sets
Louise
Hay
293-311
Abstract: We define a discrete $\omega$-sequence of index sets to be a sequence $ {\{ \theta {A_n}\} _{n \geq 0}}$, of index sets of classes of recursively enumerable sets, such that for each n, $\theta {A_{n + 1}}$ is an immediate successor of $\theta {A_n}$ in the partial order of degrees of index sets under one-one reducibility. The main result of this paper is that if S is any set to which the complete set K is not Turing-reducible, and $ {A^S}$ is the class of recursively enumerable subsets of S, then $\theta {A^S}$ is at the bottom of c discrete $ \omega$-sequences. It follows that every complete Turing degree contains c discrete $\omega$-sequences.
Nilpotent-by-finite groups with isomorphic finite quotients
P. F.
Pickel
313-325
Abstract: Let $\mathcal{F}(G)$ denote the set of isomorphism classes of finite homomorphic images of a group G. We say that groups G and H have isomorphic finite quotients if $\mathcal{F}(G) = \mathcal{F}(H)$. Let $\mathcal{H}$ denote the class of finite extensions of finitely generated nilpotent groups. In this paper we show that if G is in $\mathcal{H}$, then the groups H in $\mathcal{H}$ for which $\mathcal{F}(G) = \mathcal{F}(H)$ lie in only finitely many isomorphism classes.
Slicing and intersection theory for chains modulo $\nu $ associated with real analytic varieties
Robert M.
Hardt
327-340
Abstract: In a real analytic manifold a k dimensional (real) analytic chain is a locally finite sum of integral multiples of chains given by integration over certain k dimensional analytic submanifolds (or strata) of some k dimensional real analytic variety. In this paper, for any integer $\nu \geq 2$, the concepts and results of [6] on the continuity of slicing and the intersection theory for analytic chains are fully generalized to the modulo $\nu$ congruence classes of such chains.
Curvature tensors in Kaehler manifolds
Malladi
Sitaramayya
341-353
Abstract: Curvature tensors of Kaehler type (or type K) are defined on a hermitian vector space and it has been proved that the real vector space $ {\mathcal{L}_K}(V)$ of curvature tensors of type K on V is isomorphic with the vector space of sym metric endomorphisms of the symmetric product of ${V^ + }$, where ${V^{\text{C}}} = {V^ + } \oplus {V^ - }$ (Theorem 3.6). Then it is shown that ${\mathcal{L}_K}(V)$ admits a natural orthogonal decomposition (Theorem 5.1) and hence every $L \in {\mathcal{L}_K}(V)$ is expressed as $L = {L_1} + {L_W} + {L_2}$. These components are explicitly determined and then it is observed that ${L_W}$ is a certain formal tensor introduced by Bochner. We call ${L_W}$ the Bochner-Weyl part of L and the space of all these ${L_W}$ is called the Weyl subspace of $ {\mathcal{L}_K}(V)$.
Schur multipliers of finite simple groups of Lie type
Robert L.
Griess
355-421
Abstract: This paper presents results on Schur multipliers of finite groups of Lie type. Specifically, let p denote the characteristic of the finite field over which such a group is defined. We determine the p-part of the multiplier of the Chevalley groups ${G_2}(4),{G_2}(3)$ and ${F_4}(2)$ the Steinberg variations; the Ree groups of type ${F_4}$ and the Tits simple group
On factorized groups
David C.
Buchthal
423-430
Abstract: The effect on a finite group G by the imposition of the condition that G is factorized by each of its maximal subgroups has been studied by Huppert, Deskins, Kegel, and others. In this paper, the effect on G brought about by the condition that G is factorized by a normalizer of a Sylow p-subgroup for each $p \in \pi (G)$ is studied. Through an extension of a classical theorem of Burnside, it is shown that certain results in the case where the factors are maximal subgroups continue to hold under the new conditions. Definite results are obtained in the case where the supplements of the Sylow normalizers are cyclic groups of prime power order or are abelian Hall subgroups of G.
Finite groups with nicely supplemented Sylow normalizers
David
Perin
431-435
Abstract: This paper considers finite groups G whose Sylow normalizers are supplemented by groups D having a cyclic Hall $2'$-subgroup. G is solvable and all odd order composition factors of G are cyclic. If $S \in {\text{Syl}_2}(D)$ is cyclic, dihedral, semidihedral, or generalized quaternion, then G is almost super-solvable.
On sequences containing at most $3$ pairwise coprime integers
S. L. G.
Choi
437-440
Abstract: It has been conjectured by Erdös that the largest number of natural numbers not exceeding n from which one cannot select $k + 1$ pairwise coprime integers, where $k \geq 1$ and $n \geq {p_k}$, with $ {p_k}$ denoting the kth prime, is equal to the number of natural numbers not exceeding n which are multiples of at least one of the first k primes. It is known that the conjecture holds for k = 1 and 2. In this paper we establish the truth of the conjecture for k = 3.
Mielnik's probability spaces and characterization of inner product spaces
C. V.
Stanojevic
441-448
Abstract: A characterization of inner product spaces is given in terms of Mielnik's probability function. The generalized parallelogram law is related to the functional equation $f + f \circ g = 1$.
Bicohomology theory
Donovan H.
Van Osdol
449-476
Abstract: Given a triple T and a cotriple G on a category $\mathcal{D}$ such that T preserves group objects in $ \mathcal{D}$, let P and M be in $ \mathcal{D}$ with M an abelian group object. Applying the ``hom functor'' $\mathcal{D}( - , - )$ to the (co)simplicial resolutions ${G^ \ast }P$ and ${T^ \ast }M$ yields a double complex $ \mathcal{D}({G^ \ast }P,{T^ \ast }M)$. The nth homology group of this double complex is denoted $ {H^n}(P,M)$, and this paper studies ${H^0}$ and ${H^1}$. When $ \mathcal{D}$ is the category of bialgebras arising from a triple, cotriple, and mixed distributive law, a complete description of $ {H^0}$ and ${H^1}$ is given. The applications include a solution of the singular extension problem for sheaves of algebras.
Bessel series expansions of the Epstein zeta function and the functional equation
Audrey A.
Terras
477-486
Abstract: For the Epstein zeta function of an n-ary positive definite quadratic form, $n - 1$ generalizations of the Selberg-Chowla formula (for the binary case) are obtained. Further, it is shown that these $n - 1$ formulas suffice to prove the functional equation of the Epstein zeta function by mathematical induction. Finally some generalizations of Kronecker's first limit formula are obtained.
Irreducible representations of the $C\sp{\ast} $-algebra generated by a quasinormal operator
John W.
Bunce
487-494
Abstract: For A a quasinormal operator on Hilbert space, we determine the irreducible representations of ${C^ \ast }(A)$, the $ {C^ \ast }$-algebra generated by A and the identity. We also explicitly describe the topology on the space of unitary equivalence classes of irreducible representations of ${C^ \ast }(A)$ and calculate the regularized transform of $ {C^ \ast }(A)$, thus exhibiting an isomorphic copy of ${C^ \ast }(A)$.
On the isotropic group of a homogeneous polynomial
Siu Ming
Ho
495-498
Abstract: Let G be the linear group leaving a homogeneous polynomial of degree k fixed. The author shows that either the polynomial is a polynomial in fewer than the assigned number of variables or that the $(k - 1)$st prolongation of G is 0. The author also shows that this result is optimal.
Hull subordination and extremal problems for starlike and spirallike mappings
Thomas H.
MacGregor
499-510
Abstract: Let $\mathfrak{F}$ be a compact subset of the family $\mathcal{A}$ of functions analytic in $\Delta = \{ z:\;\vert z\vert < 1\}$, and let $\mathcal{L}$ be a continuous linear operator of order zero on $ \mathcal{A}$. We show that if the extreme points of the closed convex hull of $\mathcal{F}$ is the set $\{ {f_0}(xz)\} (\vert x\vert = 1)$, then $\mathcal{L}(f)$ is hull subordinate to $\mathcal{L}({f_0})$ in $\Delta$. This generalizes results of R. M. Robinson corresponding to families $\mathcal{F}$ of functions that are subordinate to $(1 + z)/(1 - z)$ or to $1/{(1 - z)^2}$. Families $ \mathcal{F}$ to which this theorem applies are discussed and we identify each such operator $ \mathcal{L}$ with a suitable sequence of complex numbers. Suppose that $ \Phi$ is a nonconstant entire function and that $ 0 < \vert{z_0}\vert < 1$. We show that the maximum of $ \operatorname{Re} \{ \Phi [\log (f({z_0})/{z_0})]\}$ over the class of starlike functions of order a is attained only by the functions $f(z) = z/{(1 - xz)^{2 - 2\alpha }},\;\vert x\vert = 1$. A similar result is obtained for spirallike mappings. Both results generalize a theorem of G. M. Golusin corresponding to the family of starlike mappings.
Coterminal families and the strong Markov property
A. O.
Pittenger;
C. T.
Shih
1-42
Abstract: Let ${E_\Delta }$ be a compact metric space and assume that a strong Markov process X is defined on ${E_\Delta }$. Under the assumption that X has right continuous paths with left limits, it is shown that a version of the strong Markov property extends to coterminal families, a class of random times which can be visualized as last exit times before t from a fixed subset of $ {E_\Delta }$. Since the random times are not Markov times, the conditioning $ \sigma$-field and the new conditional probabilities must be defined. If X is also assumed to be nearly quasileft continuous, i.e. branching points are permitted, two different conditionings are possible--one on the ``past'' of the random time and one on the ``past plus present"--and two different conditional probabilities must be defined.
On the congruence lattice characterization theorem
William A.
Lampe
43-60
Abstract: A simplified proof is given for the theorem characterizing the congruence lattice of a universal algebra.
Decomposition theories for abelian categories
Joe W.
Fisher;
Harvey
Wolff
61-69
Abstract: Both the classical approach to decomposition theories and Fisher's technique of constructing decomposition theories from radical functions are extended to and exploited in the context of abelian categories. These two different approaches to decomposition theories for abelian categories intertwine in one theorem from which flows necessary and sufficient conditions for the existence of the tertiary, primary, and Bourbaki's $\mathcal{P}$-primary decomposition theories.
A fixed point theorem, a perturbed differential equation, and a multivariable Volterra integral equation
David Lowell
Lovelady
71-83
Abstract: A fixed point theorem is obtained for an equation of the form $u = T[p,f + G[u]]$. This theorem is then applied to a functionally perturbed ordinary differential equation of the form \begin{displaymath}\begin{array}{*{20}{c}} {{u_1}(s,t) = {g_1}(s,t) + \int_0^s {... ...),{u_2}(x,t),{u_3}(x,t)} \right]} \;dx,} \end{array} \end{displaymath} and, as a corollary to this, a differential equation of the form \begin{displaymath}\begin{array}{*{20}{c}} {\frac{{{\partial ^2}}}{{\part... ... {u(s,0) = \phi (s),u(0,t) = \psi (t).} \end{array} \end{displaymath} These last two equations are set in a Banach space so as to allow applications to integrodifferential equations such as \begin{displaymath}\begin{array}{*{20}{c}} {\frac{{{\partial ^2}}}{{\partial s\p... ...u(s,0,z) = \sigma (s,z),u(0,t,z) = \tau (t,z).} \end{array} \end{displaymath}
Inverse semigroups which are separated over a subsemigroup
D. B.
McAlister
85-117
Abstract: An inverse semigroup T is separated over a subsemigroup S if T is generated, as an inverse semigroup, by S and for each $ a,b,\epsilon S$ there exists $ x\;\epsilon \;Sa \cap Sb$ such that ${a^{ - 1}}a{b^{ - 1}}b = {x^{ - 1}}x$ and dually for right ideals. For example, if T is generated as an inverse semigroup by a semigroup S whose principal left and right ideals form chains under inclusion, then T is separated over S. In this paper we investigate the structure of inverse semigroups T which are separated over subsemigroups S.
Invariant differential operators on a real semisimple Lie algebra and their radial components
Mohsen
Pazirandeh
119-131
Abstract: Let $S({\mathfrak{g}_C})$ be the symmetric algebra over the complexification $ {\mathfrak{g}_C}$ of the real semisimple Lie algebra $ \mathfrak{g}$. For $u\;\epsilon \;S({\mathfrak{g}_C}),\partial (u)$ is the corresponding differential operator on $ \mathfrak{g}$. $ \mathcal{D}(\mathfrak{g})$ denotes the algebra generated by $\partial (S({\mathfrak{g}_C}))$ and multiplication by polynomials on ${\mathfrak{g}_C}$. For any open set $U \subset \mathfrak{g},{\text{Diff}}(U)$ is the algebra of differential operators with ${C^\infty }$-coefficients on U. Let $ \mathfrak{h}$ be a Cartan subalgebra of $ \pi = {\Pi _{\alpha \epsilon P}}\alpha$, P some positive system of roots. Let $\mathfrak{g}$. Harish-Chandra showed that, for each $D\;\epsilon \;{\text{Diff}}(W)$, there is a unique differential operator $(Df){\left\vert {_\mathfrak{h}' = \delta {'_\mathfrak{h}}(D)(f} \right\vert _\mathfrak{h}})$ for all G-invariant $f\epsilon \;{C^\infty }(W)$, and that if $D\;\epsilon \mathcal{D}(\mathfrak{h})$, then $\bar D\epsilon \mathcal{D}(\mathfrak{g})$. In particular $\overline {\partial (u)} = \partial (u{\vert _\mathfrak{h}}),u\;\epsilon \;S({\mathfrak{g}_C})$ and invariant. We prove these results by different, yet simpler methods. We reduce evaluation of ${\pi ^{ - 1}} \circ \delta '(D) \circ \pi \;\epsilon \;\mathcal{D}(\mathfrak{h})$ using properties of derivations $E \to \left[ {\partial (u),E} \right]$ of $ \mathcal{D}(\mathfrak{g})$ induced by $\partial (u)\;(u\;\epsilon \;S({\mathfrak{g}_C}))$ and of the algebra of polynomials on ${\mathfrak{h}_C}$ invariant under the Weyl group.
Completely seminormal operators with boundary eigenvalues
Kevin
Clancey
133-143
Abstract: For $f \in {L^2}(E)$ we consider the singular integral operator $ {T_E}f(s) = sf(s) + {\pi ^{ - 1}}{\smallint _E}f(t){(t - s)^{ - 1}}dt$. These singular integral operators are a special case of operators acting on a Hilbert space with one dimensional self-commutator. We discover generalized eigenfunctions of the equation ${T_E}f = 0$ and, for $p < 2$, we will give an ${L^p}(E)$ solution of the equation ${T_E}f = {\chi _E}$. The main result of the paper is an example of a nonzero ${L^2}(E)$ solution of ${T_E}f = 0$, with $ \lambda = 0$ a boundary point of the spectrum of ${T_E}$.
A Wedderburn theorem for alternative algebras with identity over commutative rings
W. C.
Brown
145-158
Abstract: In this paper, we study alternative algebras ${\mathbf{\Lambda }}$ over a commutative, associative ring R with identity. When ${\mathbf{\Lambda }}$ is finitely generated as an R-module, we define the radical J of ${\mathbf{\Lambda }}$. We show that matrix units and split Cayley algebras can be lifted from ${\mathbf{\Lambda }}/J$ to ${\mathbf{\Lambda }}$ when R is a Hensel ring. We also prove the following Wedderburn theorem: Let $ {\mathbf{\Lambda }}$ be an alternative algebra over a complete local ring R of equal characteristic. Suppose ${\mathbf{\Lambda }}$ is finitely generated as an R-module, and ${\mathbf{\Lambda }}/J$ is separable over $ \bar R$ ($\bar R$ the residue class field of R). Then there exists an $\bar R$-subalgebra S of $ {\mathbf{\Lambda }}$ such that $ S + J = {\mathbf{\Lambda }}$ and $ S \cap J = 0$.
The automorphism group of an abelian $p$-group and its normal $p$-subgroups
Jutta
Hausen
159-164
Abstract: Let ${\mathbf{\Gamma }}$ be the automorphism group of a nonelementary reduced abelian p-group, $p \geq 5$. It is shown that every noncentral normal subgroup of ${\mathbf{\Gamma }}$ contains a noncentral normal subgroup $ {\mathbf{\Delta }}$ of ${\mathbf{\Gamma }}$ such that ${{\mathbf{\Delta }}^p} = 1$. Furthermore, every cyclic normal subgroup of ${\mathbf{\Gamma }}$ is contained in the center of ${\mathbf{\Gamma }}$.
Existence of sum and product integrals
Jon C.
Helton
165-174
Abstract: Functions are from $R \times R$ to R, where R represents the set of real numbers. If c is a number and either (1) $ \smallint _a^b{G^2}$ exists and $\smallint _a^bG$ exists, (2) $\smallint _a^bG$ exists and $_a{{\mathbf{\Pi }}^b}(1 + G)$ exists and is not zero or (3) each of $_a{{\mathbf{\Pi }}^b}(1 + G)$ and $_a{\Pi ^b}(1 - G)$ exists and is not zero, then $\smallint _a^bcG$ exists, $\smallint _a^b\vert cG - \smallint cG\vert = 0{,_x}{{\mathbf{\Pi }}^y}(1 + cG)$ exists for $a \leq x < y \leq b$ and $ \smallint _a^b\vert 1 + cG - {\mathbf{\Pi }}(1 + cG)\vert = 0$. Furthermore, if H is a function such that ${\lim _{x \to {p^ - }}}H(x,p),{\lim _{x \to {p^ + }}}H(p,x),{\lim _{x,y \to {p^ - }}}H(x,y)$ and $ {\lim _{x,y \to {p^ + }}}H(x,y)$ exist for each $p \in [a,b],n \geq 2$ is an integer, and G satisfies either (1), (2) or (3) of the above, then $\smallint _a^bH{G^n}$ exists, $\smallint _a^b\vert H{G^n} - \smallint H{G^n}\vert = 0{,_x}{{\mathbf{\Pi }}^y}(1 + H{G^n})$ exists for $a \leq x < y \leq b$ and $ \smallint _a^b\vert 1 + H{G^n} - {\mathbf{\Pi }}(1 + H{G^n})\vert = 0$.
Overrings of commutative rings. III. Normal pairs
Edward D.
Davis
175-185
Abstract: A pair of integral domains (A, B) is a normal (resp., QR-) pair provided that A is a subring of B and all intermediate rings are normal in B (resp., rings of quotients of A). The special case of B the field of fractions of A (e.g., Prüfer domains and Dedekind domains with torsion class group) has been studied in detail. It is shown that any domain A possesses a unique overring B maximal with respect to forming a normal (resp., QR-) pair with A. An explicit description of this overring and all the intermediate rings in terms of localizations A is obtained, and further details are provided in the presence of a noetherian-like condition on A. In addition, the ``overring'' characterizations of Prüfer domains are extended to ``intermediate ring'' characterizations of normal pairs.
On the asymptotic behavior of nonlinear wave equations
Robert T.
Glassey
187-200
Abstract: Positive energy solutions of the Cauchy problem for the equation $\square u = {m^2}u + F(u)$ are considered. With $ G(u) = \smallint _0^uF(s)ds$, it is proven that $G(u)$ must be nonnegative in order for uniform decay and the existence of asymptotic ``free'' solutions to hold. When $G(u)$ is nonnegative and satisfies a growth restriction at infinity, the kinetic and potential energies (with m = 0) are shown to be asymptotically equal. In case $F(u)$ has the form $ \vert u{\vert^{p - 1}}u$, scattering theory is shown to be impossible if $1 < p \leq 1 + 2{n^{ - 1}}\;(n \geq 2)$.
New criteria for freeness in abelian groups
Paul
Hill
201-209
Abstract: A new criterion is established for an abelian group to be free. The criterion applies to an ascending chain of free subgroups. The result is used to construct groups that are almost free but not free. In particular, we construct examples that show that the class of free abelian groups is not definable in the logical language ${L_{\infty \kappa }}$ if $\kappa \leq {\aleph _2}$. In doing so, we take advantage of a recent theorem of P. Eklof.
The Cauchy problem for Douglis-Nirenberg elliptic systems of partial differential equations
Richard J.
Kramer
211-225
Abstract: Several partial answers are given to the question: Suppose U is a solution of the Douglis-Nirenberg elliptic system $LU = F$ where F is analytic and L has analytic coefficients. If $U = 0$ in some appropriate sense on a hyperplane (or any analytic hypersurface) must U vanish identically? One answer follows from introducing a so-called formal Cauchy problem for Douglis-Nirenberg elliptic systems and establishing existence and uniqueness theorems. A second Cauchy problem, in some sense a more natural one, is discussed for an important subclass of the Douglis-Nirenberg elliptic systems. The results in this case give a second partial answer to the original question. The methods of proof employed are largely algebraic. The systems are reduced to systems to which the Cauchy-Kowalewski theorem applies.
Homeomorphisms of Hilbert cube manifolds
T. A.
Chapman
227-239
Abstract: It is shown in this paper that the homeomorphism group of any compact Hubert cube manifold is locally contractible. The proof uses some standard infinite-dimensional techniques along with an infinite-dimensional version of the torus-homeomorphism idea which was used by Edwards and Kirby to establish a corresponding finite-dimensional result.
On the Green's function for the biharmonic equation in an infinite wedge
Joseph B.
Seif
241-260
Abstract: The Green's function for the biharmonic equation in an infinite angular wedge is considered. The main result is that if the angle a is less than ${a_1} \cong 0.812\pi$, then the Green's function does not remain positive; in fact it oscillates an infinite number of times near zero and near $\infty$. The method uses a number of transformations of the problem including the Fourier transform. The inversion of the Fourier transform is accomplished by means of the calculus of residues and depends on the zeros of a certain transcendental function. The distribution of these zeros in the complex plane gives rise to the determination of the angle $ {a_1}$. A general expression for the asymptotic behavior of the solution near zero and near infinity is obtained. This result has the physical interpretation that if a thin elastic plate is deflected downward at a point, the resulting shape taken by the plate will have ripples which protrude above the initial plane of the plate.
Existence of solutions of ordinary differential equations with generalized boundary conditions
Stephen R.
Bernfeld;
V.
Lakshmikantham
261-274
Abstract: An investigation of the existence of solutions of the nonlinear boundary value problem $x' = f(t,x,y),y' = g(t,x,y),AV(a,x(a),y(a)) + BW(a,x(a),y(a)) = {C_1},CV(b,x(b),y(b)) + DW(b,x(b),y(b)) = {C_2}$, is made. Here we assume $ g,f:[a,b] \times {R^p} \times {R^q} \to {R^p}$ are continuous, and $ V,W:[a,b] \times {R^p} \times {R^q} \to R$ are continuous and locally Lipschitz. The main techniques used are the theory of differential inequalities and Lyapunov functions.
Some highly homogeneous groups
Larry
Dornhoff
275-301
Abstract: We study finite rank 4 permutation groups which are solvable and imprimitive, with a regular normal subgroup. This means determining those groups N with a solvable automorphism group A which has only three orbits on the nonidentity elements of N.
Inequalities associated with regular and singular problems in the calculus of variations
J. S.
Bradley;
W. N.
Everitt
303-321
Abstract: An inequality of the form $- \infty < a < b \leq \infty$. The inequality is first established for functions in the domain of an operator in the Hilbert function space $ {L^2}(a,b)$ that is associated with the differential equation $- (py')' + qy = \lambda y$, and the number $ {\mu _0}$ in the inequality is the smallest number in the spectrum of this operator. An approximation theorem is given that allows the inequality to be established for the larger set of functions D. An extension of some classical results from the calculus of variations and some spectral theory is then used to give necessary and sufficient conditions for equality and to show that the constant $ {\mu _0}$ is best possible. Certain consequences of these conclusions are also discussed.
Fixed point theorems for various classes of $1$-set-contractive and $1$-ball-contractive mappings in Banach spaces
W. V.
Petryshyn
323-352
Abstract: Let X be a real Banach space, D a bounded open subset of X, and $\bar D$ the closure of D. In §1 of this paper we establish a general fixed point theorem (see Theorem 1 below) for 1-set-contractions and 1-ball-contractions $ T:\bar D \to X$ under very mild conditions on T. In addition to classical fixed point theorems of Schauder, Leray and Schauder, Rothe, Kransnoselsky, Altman, and others for T compact, Theorem 1 includes as special cases the earlier theorem of Darbo as well as the more recent theorems of Sadovsky, Nussbaum, Petryshyn, and others (see §1 for further contributions and details) for T k-set-contractive with $k < 1$, condensing, and 1-set-contractive. In §§2, 3, 4, and 5 of this paper Theorem 1 is used to deduce a number of known, as well as some new, fixed point theorems for various special classes of mappings (e.g. mappings of contractive type with compact or completely continuous perturbations, mappings of semicontractive type introduced by Browder, mappings of pseudo-contractive type, etc.) which have been recently extensively studied by a number of authors and, in particular, by Browder, Krasnoselsky, Kirk, and others (see §1 for details),
On rationally convex hulls
Richard F.
Basener
353-381
Abstract: For a compact set $X \subseteq {{\mathbf{C}}^n}$, let ${h_r}(X)$ denote the rationally convex hull of X; let $ {\mathbf{\Delta }}$ denote the closed unit disk in C; and, following Wermer, for a compact set S such that $\partial {\mathbf{\Delta }} \subseteq S \subseteq {\mathbf{\Delta }}$ let ${X_S} = S \times S \cap \partial {{\mathbf{\Delta }}^2}$. It is shown that $\displaystyle {h_r}({X_S}) = \{ (z,w) \in S \times S\vert{u_S}{(z)^ + }{u_S}(w) \leq 1\}$ where ${u_S}$ is a function on S which, in the case when S is smoothly bounded, is specified by requiring $ {u_S}{\vert _{\partial {\mathbf{\Delta }}}} = 0,{u_S}{\vert _{\partial S\backslash \partial {\mathbf{\Delta }}}} = 1$ and ${u_S}{\vert _{\operatorname{int} S}}$ harmonic. In particular this provides a precise description of ${h_r}(X)$ for certain sets $X \subseteq {{\mathbf{C}}^2}$ with the property that $ {h_r}(X) \ne X$, but $ {h_r}(X)$ does not contain analytic structure (as Wermer demonstrated, there are S for which $X = {X_S}$ has these properties). Furthermore, it follows that whenever ${h_r}({X_S}) \ne {X_S}$ then there is a Gleason part of ${h_r}({X_S})$ for the algebra $ R({X_S})$ with positive four-dimensional measure. In fact, the Gleason part of any point $(z,w) \in {h_r}({X_S}) \cap \operatorname{int} {{\mathbf{\Delta }}^2}$ such that ${u_S}(z) + {u_S}(w) < 1$ has positive four-dimensional measure. A similar idea is then used to construct a compact rationally convex set $ Y \subseteq {{\mathbf{C}}^2}$ such that each point of Y is a peak point for $ R(Y)$ even though $R(Y) \ne C(Y)$; namely, $Y = {\tilde X_T} = \{ (z,w) \in {{\mathbf{C}}^2}\vert z \in T,\vert w\vert = \sqrt {1 - \vert z{\vert^2}} \}$ where T is any compact subset of $\operatorname{int} {\mathbf{\Delta }}$ having the property that $ R(T) \ne C(T)$ even though there are no nontrivial Jensen measures for $ R(T)$. This example is more concrete than the original example of such a uniform algebra which was discovered by Cole. It is possible to show, for instance, that $ R({\tilde X_T})$ is not even in general locally dense in $C({\tilde X_T})$, a possibility which had been suggested by Stuart Sidney. Finally, smooth examples (3-spheres in $ {{\mathbf{C}}^6}$) with the same pathological properties are obtained from $ {X_S}$ and ${\tilde X_T}$.
Limit properties of Poisson kernels of tube domains
Lawrence J.
Dickson
383-401
Abstract: If certain local boundary conditions hold near $P \in \partial \Gamma$, the Poisson kernel belonging to a proper cone $\Gamma \subset {{\mathbf{R}}^n}$ converges to a tight $C_0^\ast$ limit as its parameter converges admissibly to P in $\Gamma$. This limit can be identified with a lower-dimensional Poisson kernel. The result always works for polytopic and ``rounded'' cones; for these, a result on the decrease at infinity is obtained which in fact implies convergence almost everywhere in the appropriate sense of the Poisson integral to certain of its boundary values.
Visitations of ruled sums
Leonard E.
Baum;
H. H.
Stratton
403-430
Abstract: Let $\{ {X_i}\}$ be a sequence of independent identically distributed random variables and for $D \subseteq {I^ + }$ let ${S_D} = {\Sigma _{i \in D}}{X_i}$. A rule $ (\;)$ is a mapping $ {I^ + } \to {2^{{I^ + }}}:\forall n\vert(n)\vert = n$ and ${S_{(\;)}} = \{ {S_{(n)}}\} $ is its associated ruled sum. Ruled sums generalize ordinary sums $ {S_n}$. Indeed, all a.e. results for ${S_n}$ can be investigated for $ {S_{(n)}}$ frequently requiring different methods and sometimes yielding different conclusions. In a previous paper we studied strong laws of large numbers and the law of the iterated logarithm. In this paper we study infinite visitation. Under suitable hypotheses on the basic distribution function F of the ${X_i}$ we show that, for all rules $(\;),{S_{(n)}}$ visits each integer infinitely often a.e. in the lattice case (or has all points of the real line as accumulation points in the nonlattice case). In fact we obtain a ``rate of visitation.'' There follows extensions of the Pólya theorem on encounters in the plane and 3-space from random walks to these ruled sums. Some equivalence relations and partial orderings on rules are defined. For normal variables this leads to an extension of the previously mentioned result for ruled sums of the type of the iterated logarithm law.
Perfect open and distinguishable multivalued maps
Eric John
Braude
431-441
Abstract: It is shown that perfect open multivalued maps preserve $\mathcal{Z}$-analytic sets (which include compact zero sets) as well as other objects of descriptive set theory. The concept of ``distinguishability", introduced by Frolík, is applied to multivalued maps, yielding a new class of such maps with similar preservation properties. That the projection of a compact zero set is a zero set is one corollary, and another is a generalized $ {\mathcal{G}_\delta }$ diagonal metrization theorem.
Inner product modules over $B\sp{\ast} $-algebras
William L.
Paschke
443-468
Abstract: This paper is an investigation of right modules over a $ {B^\ast}$-algebra B which posses a B-valued ``inner product'' respecting the module action. Elementary properties of these objects, including their normability and a characterization of the bounded module maps between two such, are established at the beginning of the exposition. The case in which B is a $ {W^\ast}$-algebra is of especial interest, since in this setting one finds an abundance of inner product modules which satisfy an analog of the self-duality property of Hilbert space. It is shown that such self-dual modules have important properties in common with both Hilbert spaces and $ {W^\ast}$-algebras. The extension of an inner product module over B by a $ {B^\ast}$-algebra A containing B as a $^\ast$-subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.
On Green's function of an $n$-point boundary value problem
K. M.
Das;
A. S.
Vatsala
469-480
Abstract: The Green's function ${g_n}(x,s)$ for an n-point boundary value problem, $ {y^{(n)}}(x) = 0,y({a_1}) = y({a_2}) = \cdots = y({a_n}) = 0$ is explicitly given. As a tool for discussing $\operatorname{sgn} g_n(x,s)$ on the square $ [{a_1},{a_n}] \times [{a_1},{a_n}]$, some results about polynomials with coefficients as symmetric functions of a's are obtained. It is shown that $\displaystyle \int_{{a_1}}^{{a_n}} {\vert{g_n}(x,s)\vert ds}$ is a suitable polynomial in x. Applications to n-point boundary value problems and lower bounds for ${a_m}\;(m \geq n)$ are included.
On the integrals of Perron type
P. S.
Bullen;
C. M.
Lee
481-501
Abstract: An abstract derivate system is defined axiomatically, and then a naturally corresponding Perron integral theory is developed, unifying all the existing integral theories of Perron type of first order. A new scale of approximately mean-continuous integrals and a new scale of symmetric Cesàro-Perron integrals are obtained as examples of the general theory. Also, the MZ-integral and the SCP-integral are proved to be equivalent.
Bernstein approximation problem for differentiable functions and quasi-analytic weights
Guido
Zapata
503-509
Abstract: The Bernstein problem for differentiable functions is considered. Sufficient conditions in order that a decreasing family of weights be fundamental are given. Some of these conditions are also related to the concept of quasi-analytic weight.
Asymptotic behavior of solutions of linear stochastic differential systems
Avner
Friedman;
Mark A.
Pinsky
1-22
Abstract: Following Kasminski, we investigate asymptotic behavior of solutions of linear time-independent Itô equations. We first give a sufficient condition for asymptotic stability of the zero solution. Then in dimension 2 we determine conditions for spiraling at a linear rate. Finally we give applications to the Cauchy problem for the associated parabolic equation by the use of a tauberian theorem.
Perfect mappings and certain interior images of $M$-spaces
J. M.
Worrell;
H. H.
Wicke
23-35
Abstract: The main theorems of this paper show that certain conditions (called ${\lambda _c},{\lambda _b},{\beta _c}$, and ${\beta _b}$) are invariant, in the presence of ${T_0}$-regularity, under the application of closed continuous peripherally compact mappings. Interest in these conditions lies in the fact that they may be used to characterize certain regular ${T_0}$ open continuous images of some classes of $M$-spaces in the sense of K. Morita, and in the fact that they are preserved by open continuous mappings with certain appropriate additional conditions. For example, the authors have shown that a regular $ {T_0}$-space is an open continuous image of a paracompact Čech complete space if and only if the space satisfies condition ${\lambda _b}$ [Pacific J. Math. 37 (1971), 265-275]. Moreover, in the same paper it is shown that if a completely regular ${T_0}$-space satisfies condition ${\lambda _b}$ then any ${T_0}$ completely regular open continuous image of it also satisfies $ {\lambda _b}$. These results together with the results of the present paper and certain known results lead to the following theorem: The smallest subclass of the class of regular $ {T_0}$-spaces which contains all paracompact Čech complete spaces and which is closed with respect both to the application of perfect mappings and to the application of open continuous mappings preserving ${T_0}$-regularity is the subclass satisfying condition ${\lambda _b}$. Similar results are obtained for the regular ${T_0}$-spaces satisfying ${\lambda _c},{\beta _b}$, and ${\beta _c}$. The other classes of $M$-spaces involved are the regular $ {T_0}$ complete $ M$-spaces (i.e., spaces which are quasi-perfect preimages of complete metric spaces), ${T_2}$ paracompact $M$-spaces, and regular ${T_0}M$-spaces. In the last two cases besides the inferiority of the mappings the notion of uniform $ \lambda$-completeness, which generalizes compactness of a mapping, enters. (For details see General Topology and Appl. 1 (1971), 85-100.) The proofs are accomplished through the use of two basic lemmas on closed continuous mappings satisfying certain additional conditions.
$\alpha_T$ is finite for $\aleph_1$-categorical $T$
John T.
Baldwin
37-51
Abstract: Let $T$ be a complete countable ${\aleph _1}$-categorical theory. Definition. If $\mathcal{A}$ is a model of $T$ and $A$ is a $1$-ary formula in $ L(\mathcal{A})$ then $ A$ has rank 0 if $A(\mathcal{A})$ is finite. $A(\mathcal{A})$ has rank $n$ degree $m$ iff for every set of $m + 1$ formulas $ {B_1}, \cdots ,{B_{m + 1}} \in {S_1}(L(\mathcal{A}))$ which partition $A(\mathcal{A})$ some ${B_i}(\mathcal{A})$ has rank $\leqslant n - 1$. Theorem. If $ T$ is ${\aleph _1}$-categorical then for every $\mathcal{A}$ a model of $T$ and every $A \in {S_1}(L(\mathcal{A})),A(\mathcal{A})$ has finite rank. Corollary. ${\alpha _T}$ is finite. The methods derive from Lemmas 9 and 11 in ``On strongly minimal sets'' by Baldwin and Lachlan. $ {\alpha _T}$ is defined in ``Categoricity in power'' by Michael Morley.
Upper bounds for vertex degrees of planar $5$-chromatic graphs
Lee W.
Johnson
53-59
Abstract: Upper bounds are given for the degrees of vertices in planar $ 5$-chromatic graphs. Some inequalities are derived for irreducible graphs which restrict the type of planar graphs that can be irreducible.
Moment and BV-functions on commutative semigroups
P. H.
Maserick
61-75
Abstract: A general notion of variation of functions on an arbitrary commutative semigroup with identity is introduced. The concept includes Hausdorff's for the additive semigroup of nonnegative integers as well as the more recent notions introduced for semilattices. An abstract moment problem is formulated and solved.
Subgroups of free products with amalgamated subgroups: A topological approach
J. C.
Chipman
77-87
Abstract: The structure of an arbitrary subgroup of the limit of a group system is shown to be itself the limit of a group system, the elements of which can be described in terms of subgroups of the original group system.
On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities
Alan
Landman
89-126
Abstract: We consider a holomorphic family $ {\{ {V_t}\} _{t \in D}}$ of projective algebraic varieties ${V_t}$ parametrized by the unit disc $D = \{ t \in {\mathbf{C}}:\vert t\vert < 1\}$ and where ${V_t}$ is smooth for $t \ne 0$ but ${V_0}$ may have arbitrary singularities. Displacement of cycles around a path $t = {t_0}{e^{i\theta }}(0 \leqslant \theta \leqslant 2\pi )$ leads to the Picard-Lefschetz transformation $T:{H_\ast }({V_{{t_0}}},{\mathbf{Z}}) \to {H_\ast }({V_{{t_0}}},{\mathbf{Z}})$ on the homology of a smooth ${V_{t0}}$. We prove that the eigenvalues of $T$ are roots of unity and obtain an estimate on the elementary divisors of $T$. Moreover, we give a global inductive procedure for calculating $T$ in specific examples, several of which are worked out to illustrate the method.
Regularity properties of the element of closest approximation
Harold S.
Shapiro
127-142
Abstract: Given an element $f \epsilon {L^p}(T),1 < p < \infty$, and a closed translation invariant subspace $S$ of ${L^p}(T)$, we investigate the regularity (smoothness) properties of the element of $S$ which is closest to $f$. The regularity of this element is in general less than that of $f$. The problem reveals a surprising connection with a hitherto unstudied class of extremal Fourier multipliers.
Wreath products and representations of degree one or two
J. M.
Bateman;
Richard E.
Phillips;
L. M.
Sonneborn
143-153
Abstract: ${\mathcal{S}_2}$ denotes all groups $G$ that possess an ascending invariant series whose factors are one- or two-generated Abelian groups. We are interested in the ptoblem (1): For which nontrivial groups $A$ and $B$ is $A$ wr $B$ in $ {\mathcal{S}_2}?$ (1) has been completely solved by D. Parker in the case where $ A$ and $B$ are finite of odd order. Parker's results are partially extended here to cover groups of even order. Our answer to (1) is complete in the case where $ A$ is a finite $ 2$-group: If $ A$ is a finite $ 2$-group, $A$ wr $B$ is in $ {\mathcal{S}_2}$ iff $ B$ is finite and $B/{O_2}(B)$ is isomorphic to a subgroup of a dihedral group of an elementary $3$-group. If $A$ is not a $2$-group, we offer only necessary conditions on $ B$. Problem (1) is closely related to Problem (2): If $F$ is a prime field or the integers, which finite groups $B$ have all their irreducible representations over $F$ of degrees one or two? It is shown that all finite $B$ which satisfy (2) are ${\mathcal{S}_2}$ groups; in particular all such $ B$ are solvable.
On derived functors of limit
Dana May
Latch
155-163
Abstract: If $\mathcal{A}$ is a cocomplete category with enough projectives and $ {\mathbf{C}}$ is a $\downarrow$-finite small category, then there is a spectral sequence which shows that the cardinality of ${\mathbf{C}}$ and colimits over finite initial subcategories ${\mathbf{C}}$ are determining factors for computation of derived functors of colimit. Applying a recent result of Mitchell to this spectral sequence we show that if the cardinality of $ {\mathbf{C}}$ is at most $\aleph _{n}$, and the flat dimension of ${\Delta ^ \ast }Z$ (constant diagram of type $ {{\mathbf{C}}^{{\text{op}}}}$ with value $Z$) is $k$, then the derived functors of $ {\lim _{\mathbf{C}}}:\mathcal{A}{b^{\mathbf{C}}} \to \mathcal{A}b$ vanish above dimension $n + 1 + k$.
Adjoining inverses to commutative Banach algebras
Béla
Bollobás
165-174
Abstract: Let $A$ be a commutative unital Banach algebra. Suppose $G \subset A$ is such that $\vert\vert a\vert\vert \leqslant \vert\vert ga\vert\vert$ for all $g \in G,a \in A$. Two questions are considered in the paper. Does there exist a superalgebra $ B$ of $A$ in which every $g \in G$ is invertible? Can one always have also $ \vert\vert{g^{ - 1}}\vert\vert \leqslant 1$ if $g \in G$? Arens proved that if $G = \{ g\}$ then there is an algebra containing ${g^{ - 1}}$, with $\vert\vert{g^{ - 1}}\vert\vert \leqslant 1$. In the paper it is shown that if $G$ is countable $B$ exists, but if $G$ is uncountable, this is not necessarily so. The answer to the second question is negative even if $G$ consists of only two elements.
Differential geometric structures on principal toroidal bundles
David E.
Blair;
Gerald D.
Ludden;
Kentaro
Yano
175-184
Abstract: Under an assumption of regularity a manifold with an $f$-structure satisfying certain conditions analogous to those of a Kähler structure admits a fibration as a principal toroidal bundle ovet a Kähler manifold. In some natural special cases, additional information about the bundle space is obtained. Finally, curvature relations between the bundle space and the base space are studied.
A law of iterated logarithm for stationary Gaussian processes
Pramod K.
Pathak;
Clifford
Qualls
185-193
Abstract: In this article the following results are established. Theorem A. Let $\{ X(t):0 \leqslant t < \infty \}$ be a stationary Gaussian process with continuous sample functions and $ E[X(t)] \equiv 0$. Suppose that the covariance function $r(t)$ satisfies the following conditions. (a) $r(t) = 1 - \vert t{\vert^\alpha }H(t) + o(\vert t{\vert^\alpha }H(t))$ as $t \to 0$, where $0 < \alpha \leqslant 2$ and $H$ varies slowly at zero, and (b) $ r(t) = O(1/\log t)$ as $t \to \infty$ Then for any nondecreasing positive function $\phi (t)$ defined on $ [a,\infty )$ with $ \phi (\infty ) = \infty ,P[X(t) > \phi (t)$ i.o. for some sequence ${t_n} \to \infty ] = 0or1$ according as the integral $I(\phi ) = \int_a^\infty {g(\phi (t))\phi {{(t)}^{ - 1}}\exp ( - {\phi ^2}(t)/2)dt}$ is finite or infinite, where $g(x) = 1/_\sigma ^{ \sim - 1}(1/x)$ is a regularly varying function with exponent $2/\alpha$ and $_\sigma ^{ \sim 2}(t) = 2\vert t{\vert^\alpha }H(t)$. Theorem C. Let $\{ {X_n}:n \geqslant 1\} $ be a stationary Gaussian sequence with zero mean and unit variance. Suppose that its covariance function satisfies, for some $\gamma > 0,r(n) = O(1/{n^\gamma })\;as\;n \to \infty$. Let $\{ \phi (n):n \geqslant 1\}$ be a nondecreasing sequence of positive numbers with ${\lim _{n \to \infty }}\phi (n) = \infty$; suppose that $\Sigma (1/\phi (n))\exp ( - {\phi ^2}(n)/2) = \infty$. Then $\displaystyle \mathop {\lim }\limits_{n \to \infty } \sum\limits_{1 \leq k \leq n} {{I_k}} /\sum\limits_{1 \leq k \leq n} {E[{I_k}] = 1\quad a.s.,}$ where $ {I_k}$ denotes the indicator function of the event $\{ {X_k} > \phi (k)\} $.
Free $Z\sb{8}$ actions on $S\sp{3}$
Gerhard X.
Ritter
195-212
Abstract: This paper is devoted to the problem of classifying periodic homeomorphisms which act freely on the $3$-sphere. The main result is the classification of free period eight actions and a generalization to free actions whose squares are topologically equivalent to orthogonal transformations. The result characterizes those $3$-manifolds which have the $3$-sphere as universal covering space and the cyclic group of order eight as fundamental group.
On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. I
Chung Wu
Ho
213-233
Abstract: Let $K$ be a proper rectilinear triangulation of a $2$-simplex $S$ in the plane and $L(K)$ be the space of all homeomorphisms of $ S$ which are linear on each simplex of $K$ and are fixed on Bd$ (S)$. The author shows in this paper that $L(K)$ with the compact open topology is simply-connected. This is a generalization of a result of S. S. Cairns in 1944 that the space $L(K)$ is pathwise connected. Both results will be used in Part II of this paper to show that $ {\pi _0}({L_2}) = {\pi _1}({L_2}) = 0$ where ${L_n}$ is a space of p.l. homeomorphisms of an $ n$-simplex, a space introduced by $ {\mathbf{R}}$. Thom in his study of the smoothings of combinatorial manifolds.
On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. II
Chung Wu
Ho
235-243
Abstract: In his study of the smoothings of p. l. manifolds, R. Thom considered the homotopy groups of a certain space ${L_n}$ of p.l. homeomorphisms on an $ n$-simplex. N. H. Kuiper showed in 1965 that the higher homotopy groups of $ {L_n}$ were in general nontrivial. The main result in this paper is that $ {\pi _0}({L_2}) = {\pi _1}({L_2}) = 0$. The proof of this result is based on a theorem of S. S. Cairns in 1944 on the deformation of rectilinear complexes in ${R^2}$ and a theorem established in Part I of this paper.
Appell polynomial expansions and biorthogonal expansions in Banach spaces
J. D.
Buckholtz
245-272
Abstract: Let $\{ {p_k}\} _0^\infty$ denote the sequence of Appell polynomials generated by an analytic function $ \phi$ with the property that the power series for $\theta = 1/\phi$ has a larger radius of convergence than the power series for $\phi$. The expansion and uniqueness properties of $\{ {p_k}\}$ are determined completely. In particular, it is shown that the only convergent $\{ {p_k}\}$ expansions are basic series, and that there are no nontrivial representations of 0. An underlying Banach space structure of these expansions is also studied.
Global regularity for $\bar \partial $ on weakly pseudo-convex manifolds
J. J.
Kohn
273-292
Abstract: Let $M'$ be a complex manifold and let $\overline \partial$-closed $ (p,q)$-form $\alpha$, which is ${C^\infty }$ on $ \overline M$ and which is cohomologous to zero on $M$ then for every $m$ there exists a $(p,q - 1)$-form ${u_{(m)}}$ which is ${C^m}$ on $ \overline M$ such that $ \overline \partial {u_{(m)}} = \alpha$.
Fourier analysis on linear metric spaces
J.
Kuelbs
293-311
Abstract: Probability measures on a real complete linear metric space $E$ are studied via their Fourier transform on $E'$ provided $E$ has the approximation property and possesses a real positive definite continuous function $ \Phi (x)$ such that $\vert\vert x\vert\vert > \epsilon$ implies $ \Phi (0) - \Phi (x) > c(\epsilon)$ where $ c(\epsilon) > 0$. In this setting we obtain conditions on the Fourier transforms of a family of tight Borel probabilities which yield tightness of the family of measures. This then is applied to obtain necessary and sufficient conditions for a complex valued function on $E'$ to be the Fourier transform of a tight Borel probability on $E$. An extension of the Levy continuity theorem as given by $ {\text{L}}$. Gross for a separable Hilbert space is obtained for such metric spaces. We also prove that various Orlicz-type spaces are in the class of spaces to which our results apply. Finally we apply our results to certain Orlicz-type sequence spaces and obtain conditions sufficient for tightness of a family of probability measures in terms of uniform convergence of the Fourier transforms on large subsets of the dual. We also obtain a more explicit form of Bochner's theorem for these sequence spaces. The class of sequence spaces studied contains the $ {l_p}$ spaces $(0 < p \leqslant 2)$ and hence these results apply to separable Hilbert space.
Two-norm spaces and decompositions of Banach spaces. II
P. K.
Subramanian;
S.
Rothman
313-327
Abstract: Let $X$ be a Banach space, $Y$ a closed subspace of ${X^\ast }$. One says $X$ is $Y$-reflexive if the canonical imbedding of $ X$ onto ${Y^\ast }$ is an isometry and $ Y$-pseudo reflexive if it is a linear isomorphism onto. If $ X$ has a basis and $ Y$ is the closed linear span of the corresponding biorthogonal functionals, necessary and sufficient conditions for $ X$ to be $Y$-pseudo reflexive are due to I. Singer. To every $B$-space $X$ with a decomposition we associate a canonical two-norm space ${X_s}$ and show that the properties of ${X_s}$, in particular its $ \gamma$-completion, may be exploited to give different proofs of Singer's results and, in particular, to extend them to $ B$-spaces with decompositions. This technique is then applied to a study of direct sum of $B$-spaces with respect to a BK space. Necessary and sufficient conditions for such a space to be reflexive are obtained.
Intersections of quasi-local domains
Bruce
Prekowitz
329-339
Abstract: Let $R = \bigcap {{V_i}}$ be an intersection of quasi-local domains with a common quotient field $ K$. Our goal is to find conditions on the ${V_i}$'s in order to get some or all of $ {V_i}$'s to be localizations of $R$. We show for example that if ${V_1}$ is a $1$-dimensional valuation domain and if ${V_1} \nsupseteq {V_2}$, then both ${V_1}$ and ${V_2}$ are localizations of $R = {V_1} \cap {V_2}$.
Deforming cohomology classes
John J.
Wavrik
341-350
Abstract: Let $\pi :X \to S$ be a flat proper morphism of analytic spaces. $\pi$ may be thought of as providing a family of compact analytic spaces, ${X_s}$, parametrized by the space $S$. Let $ \mathcal{F}$ be a coherent sheaf on $X$ flat over $S$. $ \mathcal{F}$ may be thought of as a family of coherent sheaves, ${\mathcal{F}_s}$, on the family of spaces $ {X_s}$. Let $o \in S$ be a fixed point, ${\xi _o} \in Hq({X_o},{\mathcal{F}_o})$. In this paper, we consider the problem of extending $ {\xi _o}$ to a cohomology class $\xi \in Hq({\pi ^{ - 1}}(U),\mathcal{F})$ where $ U$ is some neighborhood of $ o$ in $S$. Extension problems of this type were first considered by P. A. Griffiths who obtained some results in the case in which the morphism $\pi$ is simple and the sheaf $\mathcal{F}$ is locally free. We obtain generalizations of these results without the restrictions. Among the applications of these results is a necessary and sufficient condition for the existence of a space of moduli for a compact manifold. This application was discussed in an earlier paper by the author. We use the Grauert ``direct image'' theorem, the theory of Stein compacta, and a generalization of a result of M. Artin on solutions of analytic equations to reduce the problem to an algebraic problem. In §2 we discuss obstructions to deforming ${\xi _o}$; in §3 we show that if no obstructions exist, ${\xi _o}$ may be extended; in §4 we give a useful criterion for no obstructions; and in §5 we discuss some examples.
Improbability of collisions in Newtonian gravitational systems. II
Donald G.
Saari
351-368
Abstract: It is shown that the set of initial conditions leading to collision in the inverse square force law has measure zero. For the inverse $q$ force law the behavior of binary collisions for $1 < q < 3$ and the behavior of any collision for $q = 1$ is developed. This information is used to show that collisions are improbable in the inverse $ q$ force law where $q < 17/7$ and that binary collisions are improbable for $q < 3$.
Successive remainders of the Newton series
G. W.
Crofts;
J. K.
Shaw
369-383
Abstract: If $f$ is analytic in the open unit disc $ D$ and $\lambda$ is a sequence of points in $ D$ converging to 0, then $ f$ admits the Newton series expansion $f(z) = f({\lambda _1}) + \sum\nolimits_{n = 1}^\infty {\Delta _\lambda ^nf({\lambda _{n + 1}})(z - {\lambda _1})(z - {\lambda _2}) \cdots (z - {\lambda _n})}$, where $ \Delta _\lambda ^nf(z)$ is the $n$th divided difference of $f$ with respect to the sequence $\lambda$. The Newton series reduces to the Maclaurin series in case ${\lambda _n} \equiv 0$. The present paper investigates relationships between the behavior of zeros of the normalized remainders $\Delta _\lambda ^kf(z) = \Delta _\lambda ^kf({\lambda _{k + 1}}) + \sum\nolimi... ...bda ^nf({\lambda _{n + 1}})(z - {\lambda _{k + 1}}) \cdots (z - {\lambda _n})}$ of the Newton series and zeros of the normalized remainders $\sum\nolimits_{n = k}^\infty {{a_n}{z^{n - k}}}$ of the Maclaurin series for $f$. Let $ {C_\lambda }$ be the supremum of numbers $c > 0$ such that if $f$ is analytic in $D$ and each of $\Delta _\lambda ^kf(z),\;0 \leqslant k < \infty$, has a zero in $ \vert z\vert \leqslant c$, then $f \equiv 0$. The corresponding constant for the Maclaurin series ( ${C_\lambda }$, where ${\lambda _n} \equiv 0$) is called the Whittaker constant for remainders and is denoted by $W$. We prove that ${C_\lambda } \geqslant W$, for all $\lambda$, and, moreover, ${C_\lambda } = W$ if $\lambda \in {l_1}$. In obtaining this result, we prove that functions $f$ analytic in $D$ have expansions of the form $f(z) = \sum\nolimits_{n = 0}^\infty {\Delta _\lambda ^nf({z_n}){C_n}(z)}$, where $\vert{z_n}\vert \leqslant W$, for all $n$, and ${C_n}(z)$ is a polynomial of degree $ n$ determined by the conditions $\Delta _\lambda ^j{C_k}({z_j}) = {\delta _{jk}}$.
Functions automorphic on large domains
David A.
James
385-400
Abstract: For a discontinuous group $\Gamma \subset {\text{SL}}(2,R)$, Poincaré produced a corresponding nonconstant automorphic form, meromorphic on the open upper half plane ${\Pi ^ + }$. When the domain of meromorphicity grows larger than $ {\Pi ^ + }$, the type of group which can support an automorphic form is restricted, and the corresponding forms are generally quite simple. A complete analysis of this phenomenon is presented, with examples which show results are best possible.
The degree of approximation by Chebyshevian splines
R.
DeVore;
F.
Richards
401-418
Abstract: This paper studies the connections between the smoothness of a function and its degree of approximation by Chebyshevian splines. This is accomplished by proving companion direct and inverse theorems which give a characterization of smoothness in terms of degree of approximation. A determination of the saturation properties is included.
Generalized Laplacians and multiple trigonometric series
M. J.
Kohn
419-428
Abstract: V. L. Shapiro gave a $k$-variable analogue for Riemann's theorem on formal integration of trigonometric series. This paper derives Shapiro's results with weaker conditions on the coefficients of the series and extends the results to series which are Bochner-Riesz summable of larger order.
The polynomial identities of the Grassmann algebra
D.
Krakowski;
A.
Regev
429-438
Abstract: By using the theory of codimensions the $T$-ideal of polynomial identities of the Grassmann (exterior) algebra is computed.
Quadratic expressions in a free boson field
Abel
Klein
439-456
Abstract: Quadratic expressions in a massive spinless free Boson field are treated by an appropriate extension of the method of second quantization. A certain class of these second quantized operators is shown to generate semigroups that act on a suitable scale of ${L_p}$-spaces, obtained through the diagonalization of the field at a fixed time, in a particularly regular fashion. The techniques are developed first in an abstract setting, and then applied to the neutral scalar free field. The locally correct generator of Lorentz transformations for $ P{(\varphi )_2}$ is studied in detail, and essential selfadjointness is shown. These techniques are also used to solve explicitly the ${({\varphi ^2})_n}$ model.
The lattice triple packing of spheres in Euclidean space
G. B.
Purdy
457-470
Abstract: We say that a lattice $\Lambda$ in $n$-dimensional Euclidean space ${E_n}$ provides a $k$-fold packing for spheres of radius 1 if, when open spheres of radius 1 are centered at the points of $\Lambda$, no point of space lies in more than $ k$ spheres. The multiple packing constant $ \Delta _k^{(n)}$ is the smallest determinant of any lattice with this property. In the plane, the first three multiple packing constants $\Delta _2^{(2)},\Delta _3^{(2)}$, and $\Delta _4^{(2)}$ are known, due to the work of Blundon, Few, and Heppes. In ${E_3},\Delta _2^{(3)}$ is known, because of work by Few and Kanagasabapathy, but no other multiple packing constants are known. We show that $ \Delta _3^{(3)} \leqslant 8\sqrt {38} /27$ and give evidence that $\Delta _3^{(3)} = 8\sqrt {38} /27$. We show, in fact, that a lattice with determinant $8\sqrt {38} /27$ gives a local minimum of the determinant among lattices providing a $3$-fold packing for the unit sphere in ${E_3}$.
Boundary values in the four color problem
Michael O.
Albertson;
Herbert S.
Wilf
471-482
Abstract: Let $G$ be a planar graph drawn in the plane so that its outer boundary is a $k$-cycle. A four coloring of the outer boundary $ \gamma$ is admissible if there is a four coloring of $G$ which coincides with $\gamma$ on the boundary. If $\psi$ is the number of admissible boundary colorings, we show that the 4CC implies $\psi \geqslant 3 \cdot {2^k}$ for $k = 3, \cdots ,6$. We conjecture this to be true for all $k$ and show $\psi$ is $\geqslant c{((1 + {5^{1/2}})/2)^k}$. A graph is totally reducible (t.r.) if every boundary coloring is admissible. There are triangulations of the interior of a $k$-cycle which are t.r. for anv $k$. We investigate a class of graphs called annuli, characterize t.r. annuli and show that annuli satisfy the above conjecture.
Product integral solutions for hereditary systems
James A.
Reneke
483-493
Abstract: Hereditary systems which satisfy a Lipschitz condition are solved in terms of product integrals. Realizations of this type of hereditary system are provided from functional differential and integral equations.
Topological spaces and lattices of lower semicontinuous functions
M. C.
Thornton
495-506
Abstract: Lower semicontinuous real-valued functions on a space $X$ form a conditionally complete distributive lattice $L(X)$. Those lattices which can be represented as $ L(X)$ for some $ X$ are characterized algebraically. All spaces producing isomorphic lattices ate determined. The class of spaces which are determined by their function lattices is introduced.
Erratum to ``Martingales of strongly measurable Pettis integrable functions''
J. J.
Uhl
507
Erratum to ``Entropy for group endomorphisms and homogeneous spaces''
R.
Bowen
509-510
The Picard group of noncommutative rings, in particular of orders
A.
Fröhlich
1-45
Abstract: The structure of the Picard group of not necessarily commutative rings, and specifically of orders, and its relation to the automorphism group are studied, mainly with arithmetic applications in mind.
Higher derivations and field extensions
R. L.
Davis
47-52
Abstract: Let $K$ be a field having prime characteristic $ p$. The following conditions on a subfield $k$ of $K$ are equivalent: (i) ${ \cap _n}{K^{{p^n}}}(k) = k$ and $K/k$ is separable. (ii) $ k$ is the field of constants of an infinite higher derivation defined in $ K$. (iii) $k$ is the field of constants of a set of infinite higher derivations defined in $ K$. If $K/k$ is separably generated and $ k$ is algebraically closed in $K$, then $k$ is the field of constants of an infinite higher derivation in $K$. If $K/k$ is finitely generated then $ k$ is the field of constants of an infinite higher derivation in $K$ if and only if $K/k$ is regular.
Optimal arcs and the minimum value function in problems of Lagrange
R. Tyrrell
Rockafellar
53-83
Abstract: Existence theorems are proved for basic problems of Lagrange in the calculus of variations and optimal control theory, in particular problems for arcs with both endpoints fixed. Emphasis is placed on deriving continuity and growth properties of the minimum value of the integral as a function of the endpoints of the arc and the interval of integration. Control regions are not required to be bounded. Some results are also obtained for problems of Bolza. Conjugate convex functions and duality are used extensively in the development, but the problems themselves are not assumed to be especially ``convex". Constraints are incorporated by the device of allowing the Lagrangian function to be extended-real-valued. This necessitates a new approach to the question of what technical conditions of regularity should be imposed that will not only work, but will also be flexible and general enough to meet the diverse applications. One of the underlying purposes of the paper is to present an answer to this question.
Dual spaces of groups with precompact conjugacy classes
John R.
Liukkonen
85-108
Abstract: We show that a second countable locally compact type I group with a compact invariant neighborhood of the identity is CCR, and has a Hausdorff dual if and only if its conjugacy classes are precompact. We obtain sharper results if the group is almost connected or has a fundamental system of invariant neighborhoods of the identity. Along the way we show that for a locally compact abelian group $ A$ and a group $ B$ of topological group automorphisms of $A, A$ has small $B$ invariant neighborhoods at 1 if and only if $\hat A$ has precompact orbits under the dual actions of $B$.
The structure of Dedekind cardinals
Erik
Ellentuck
109-125
Abstract: Semantic criteria are given for provability in set theory without the axiom of choice of positive sentences about the Dedekind cardinals. These criteria suggest that Dedekind cardinals (as well as general cardinals) have an internal structure.
Maximal regular right ideal space of a primitive ring. II
Kwangil
Koh;
Hang
Luh
127-141
Abstract: If $R$ is a ring, let $X(R)$ be the set of maximal regular right ideals of $R$. For each nonempty subset $E$ of $R$, define the hull of $ E$ to be the set $ \{ I \epsilon\, X(R)\vert E \subseteq I\}$ and the support of $ E$ to be the complement of the hull of $E$. Topologize $X(R)$ by taking the supports of right ideals of $ R$ as a subbase. If $ R$ is a right primitive ring, then $X(R)$ is homeomorphic to an open subset of a compact space $X({R^\char93 })$ of a right primitive ring ${R^\char93 }$, and $X(R)$ is a discrete space if and only if $ X(R)$ is a compact Hausdorff space if and only if either $R$ is a finite ring or a division ring. Call a closed subset $F$ of $X(R)$ a line if $F$ is the hull of $I \cap J$ for some two distinct elements $ I$ and $J$ in $X(R)$. If $R$ is a semisimple ring, then every line contains an infinite number of points if and only if either $ R$ is a division ring or $ R$ is a dense ring of linear transformations of a vector space of dimension two or more over an infinite division ring such that every pair of simple (right) $R$-modules are isomorphic.
Inversion formulae for the probability measures on Banach spaces
G.
Gharagoz Hamedani;
V.
Mandrekar
143-169
Abstract: Let $B$ be a real separable Banach space, and let $\mu$ be a probability measure on $\mathcal{B}(B)$, the Borel sets of $B$. The characteristic functional (Fourier transform) $\phi$ of $\mu$, defined by $\phi (y) = \int_B {\exp \{ i(y,x)\} d\mu (x)\;}$ for $y \in {B^\ast }$ (the topological dual of $B$), uniquely determines $\mu$. In order to determine $\mu$ on $ \mathcal{B}(B)$, it suffices to obtain the value of $\int_B {G(s)d\mu (s)}$ for every real-valued bounded continuous function $G$ on $B$. Hence an inversion formula for $\mu$ in terms of $\phi$ is obtained if one can uniquely determine the value of $\int_B {G(s)d\mu (s)}$ for all real-valued bounded continuous functions $G$ on $B$ in terms of $\phi$ and $G$. The main efforts of this paper will be to prove such inversion formulae of various types. For the Orlicz space $ {E_\alpha }$ of real sequences we establish inversion formulae (Main Theorem II) which properly generalize the work of L. Gross and derive as a corollary the extension of the Main Theorem of L. Gross to ${E_\alpha }$ spaces (Corollary 2.2.12). In Part I we prove a theorem (Main Theorem I) which is Banach space generalization of the Main Theorem of L. Gross by reinterpreting his necessary and sufficient conditions in terms of convergence of Gaussian measures. Finally, in Part III we assume our Banach space to have a shrinking Schauder basis to prove inversion formulae (Main Theorem III) which express the measure directly in terms of $\phi$ and $G$ without the use of extension of $\phi$ as required in the Main Theorems I and II. Furthermore this can be achieved without using the Lévy Continuity Theorem and we hope that one can use this theorem to obtain a direct proof of the Lévy Continuity Theorem.
Orbits of families of vector fields and integrability of distributions
Héctor J.
Sussmann
171-188
Abstract: Let $D$ be an arbitrary set of ${C^\infty }$ vector fields on the ${C^\infty }$ manifold $M$. It is shown that the orbits of $D$ are $ {C^\infty }$ submanifolds of $M$, and that, moreover, they are the maximal integral submanifolds of a certain $ {C^\infty }$ distribution $ {P_D}$. (In general, the dimension of ${P_D}(m)$ will not be the same for all $ m \in M$.) The second main result gives necessary and sufficient conditions for a distribution to be integrable. These two results imply as easy corollaries the theorem of Chow about the points attainable by broken integral curves of a family of vector fields, and all the known results about integrability of distributions (i.e. the classical theorem of Frobenius for the case of constant dimension and the more recent work of Hermann, Nagano, Lobry and Matsuda). Hermann and Lobry studied orbits in connection with their work on the accessibility problem in control theory. Their method was to apply Chow's theorem to the maximal integral submanifolds of the smallest distribution $\Delta$ such that every vector field $X$ in the Lie algebra generated by $ D$ belongs to $ \Delta$ (i.e. $X(m) \in \Delta (m)$ for every $m \in M$). Their work therefore requires the additional assumption that $\Delta$ be integrable. Here the opposite approach is taken. The orbits are studied directly, and the integrability of $\Delta$ is not assumed in proving the first main result. It turns out that $\Delta$ is integrable if and only if $\Delta = {P_D}$, and this fact makes it possible to derive a characterization of integrability and Chow's theorem. Therefore, the approach presented here generalizes and unifies the work of the authors quoted above.
Hermitian vector bundles and value distribution for Schubert cycles
Michael J.
Cowen
189-228
Abstract: R. Bott and S. S. Chern used the theory of characteristic differential forms of a holomorphic hermitian vector bundle to study the distribution of zeroes of a holomorphic section. In this paper their methods are extended to study how often a holomorphic mapping into a Grassmann manifold hits Schubert cycles of fixed type.
Extension of Loewner's capacity theorem
Raimo
Näkki
229-236
Abstract: Analogues of a well-known theorem of Loewner concerning conformal capacity of a space ring are given in the case of an arbitrary domain.
Constructing isotopes on noncompact $3$-manifolds
Marianne S.
Brown
237-263
Abstract: We consider the question ``When are two homeomorphisms of a noncompact $3$-manifold onto itself isotopic?'' Roughly, the answer is when they are homotopic to each othet. More precisely, this paper deals with the question for irreducible $3$-manifolds which either have an infinite hierarchy or have such a hierarchy after the removal of a compact set. Manifolds having the first property are called end-irreducible; the others are called eventually endirreducible. There are two results fot each type of manifold depending on whether the homotopy between the two homeomorphisms sends the boundary of the manifold into itself or not.
Variational problems within the class of solutions of a partial differential equation
Robert
Delver
265-289
Abstract: The subject of this paper is the optimization of a multiple integral over a domain $G$ of a function, containing as arguments the independent variables, the unknown function and its partial derivatives up to order $l$, within the class of all sufficiently smooth solutions in $G$ of a given partial differential equation of order greater than or equal to $2l$. Necessary conditions in the form of a boundary value problem are derived. A physical application occurs in the control with boundary and initial conditions of a process in $G$ that is described by a specific partial differential equation.
On Lagrangian groups
J. F.
Humphreys;
D. L.
Johnson
291-300
Abstract: We study the class $\mathcal{L}$ of Lagrangian groups, that is, of finite groups $G$ possessing a subgroup of index $n$ for each factor $n$ of $\vert G\vert$. These groups and their analogues were considered by McLain in [4] and the object of the present work is to extend the results in this article. We study the classes $(G) = \{ H\vert G \times H \in \mathcal{L}\}$ and also the closure of $ \mathcal{L}$ under wreath products. We also consider the two classes $\mathfrak{X}$ and $ \mathfrak{Y}$ introduced in [2] and [4] respectively.
On symmetric orders and separable algebras
T. V.
Fossum
301-314
Abstract: Let $K$ be an algebraic number field, and let $\Lambda$ be an $R$-order in a separable $K$-algebra $A$, where $R$ is a Dedekind domain with quotient field $ K$; let $\Delta$ denote the center of $ \Lambda$. A left $ \Lambda$-lattice is a finitely generated left $\Lambda$-module which is torsion free as an $ R$-module. For left $ \Lambda$-modules $ M$ and $N, \operatorname{Ext} _\Lambda ^1(M,N)$ is a module over $\Delta$. In this paper we examine ideals of $ \Delta$ which are the annihilators of $\operatorname{Ext} _\Lambda ^1(M,\_)$ for certain classes of left $\Lambda$-lattices $M$ related to the central idempotents of $ A$, and we compute these ideals explicitly if $\Lambda$ is a symmetric $R$-algebra. For a group algebra, these ideals determine the defect of a block. We then compare these annihilator ideals with another set of ideals of $ \Delta$ which are closely related to the homological different of $\Lambda$, and which in a sense measure deviation from separability. Finally we show that, for $\Lambda$ to be separable over $ R$, it is necessary and sufficient that $\Lambda$ is a symmetric $R$-algebra, $\Delta$ is separable over $R$, and the center of each localization of $\Lambda$ at the maximal ideals of $R$ maps onto the center of its residue class algebra.
A characteristic zero non-Noetherian factorial ring of dimension three
John
David
315-325
Abstract: This paper shows the previously unknown existence of a finite dimensional non-Noetherian factorial ring in characteristic zero. The example, called ``$J$", contains a field of characteristic zero and is contained in a pure transcendental extension of degree three of that field. $J$ is seen to be an ascending union of polynomial rings and degree functions are introduced on each of the polynomial rings. These are the basic facts that enable it to be seen that two extensions of $ J$ are Krull. One of these extensions is a simple one and the other is a localization of $J$ at a prime ideal $P$. In the case of the latter extension, it is necessary to show that the intersection of the powers of $P$ is zero. As $J$ is the intersection of these two extensions, a theorem of Nagata is all that is needed to show then that $ J$ is factorial. It is easily proved that $J$ is non-Noetherian once it is known to be factorial.
$P$-commutative Banach $\sp{\ast} $-algebras
Wayne
Tiller
327-336
Abstract: Let $A$ be a complex $^ \ast $-algebra. If $ f$ is a positive functional on $A$, let ${I_f} = \{ x \in A:f(x^ \ast x) = 0\}$ be the corresponding left ideal of $A$. Set $ P = \cap {I_f}$, where the intersection is over all positive functionals on $ A$. Then $A$ is called $P$-commutative if $ xy - yx \in P$ for all $ x,y \in A$. Every commutative $^ \ast$-algebra is $P$-commutative and examples are given of noncommutative $^ \ast$-algebras which are $ P$-commutative. Many results are obtained for $P$-commutative Banach $^ \ast$-algebras which extend results known for commutative Banach $^ \ast$-algebras. Among them are the following: If $ {A^2} = A$, then every positive functional on $A$ is continuous. If $A$ has an approximate identity, then a nonzero positive functional on $A$ is a pure state if and only if it is multiplicative. If $A$ is symmetric, then the spectral radius in $ A$ is a continuous algebra seminorm.
Scattering theory for hyperbolic systems with coefficients of Gevrey type
William L.
Goodhue
337-346
Abstract: Using the techniques developed by P. D. Lax and R. S. Phillips, qualitative results on the location of the poles of the scattering matrix for symmetric, hyperbolic systems are obtained. The restrictions placed on the system are that the coefficient matrices be of Gevrey type and that the bicharacteristic rays tend to infinity.
Almost spherical convex hypersurfaces
John Douglas
Moore
347-358
Abstract: Let $M$ be a smooth compact hypersurface with positive sectional curvatures in $n$-dimensional euclidean space. This paper gives a sufficient condition for $M$ to lie in the spherical shell bounded by concentric spheres of radius $1 - \epsilon$ and $1 + \epsilon$. This condition is satisfied, in the case where $n = 3$, if the Gaussian curvature or the mean curvature of $M$ is sufficiently pointwise close to one.
$C\sp{2}$-preserving strongly continuous Markovian semigroups
W. M.
Priestley
359-365
Abstract: Let $X$ be a compact $ {C^2}$-manifold. Let $\{ {P^t}\}$ be a Markovian semigroup on $ C(X)$. The semigroup's infinitesimal generator $A$, with domain $ \mathcal{D}$, is defined by $Af = {\lim _{t \to 0}}{t^{ - 1}}({P^t}f - f)$, whenever the limit exists in $\vert\vert\;\vert\vert$. Theorem. Assume that $\{ {P^t}\}$ preserves $ {C^2}$-functions and that the restriction of $ \{ {P^t}\}$ to $ {C^2}(X) \subset \mathcal{D}$ and $A$ is a bounded operator from $ C(X),\vert\vert\;\vert\vert$. From the conclusion is obtained a representation of $Af \cdot (x)$ as an integrodifferential operator on ${C^2}(X)$. The representation reduces to that obtained by Hunt [Semi-groups of measures on Lie groups, Trans. Amer. Math. Soc. 81 (1956), 264-293] in case $X$ is a Lie group and ${P^t}$ commutes with translations. Actually, a stronger result is proved having the above theorem among its corollaries.
Quasiconformal mappings and sets of finite perimeter
James C.
Kelly
367-387
Abstract: Let $D$ be a domain in ${R^n},n \geqslant 2,f$ a quasiconformal mapping on $ D$. We give a definition of bounding surface of codimension one lying in $ D$, and show that, given a system $\Sigma$ of such surfaces, the image of the restriction of $f$ to ``almost every'' surface is again a surface. Moreover, on these surfaces, $f$ takes $ {H^{n - 1}}$ (Hausdorff $ (n - 1)$-dimensional) null sets to ${H^{n - 1}}$ null sets. ``Almost every'' surface is given a precise meaning via the concept of the module of a system of measures, a generalization of the concept of extremal length.
Homeomorphisms with polyhedral irregular sets. I
P. F.
Duvall;
L. S.
Husch
389-406
Abstract: Homeomorphisms on open manifolds with polyhedral irregular sets are studied. For high dimensions, necessary and sufficient conditions for certain codimension three irregular sets to be tame are given. Several examples of homeomorphisms with wild irregular sets are given.
A Laurent expansion for solutions to elliptic equations
Reese
Harvey;
John C.
Polking
407-413
Abstract: Let $P(\xi )$ be a homogeneous elliptic polynomial of degree $m$. Let $E$ be a fundamental solution for the partial differential operator $P(D)$. Suppose $\Omega$ is a neighborhood of 0 in ${{\mathbf{R}}^n}$. Suppose $f \in {C^\infty }(\Omega \sim \{ 0\} )$ satisfies $P(D)f = 0$ in $ \Omega \sim \{ 0\}$. It is shown that there is a differential operator $ H(D)$ (perhaps of infinite order) and a function $g \in {C^\infty }(\Omega )$ satisfying $ P(D)g = 0$ in $ \Omega$, such that $f = H(D)E + g$ in $\Omega \sim \{ 0\}$. This analog of the Laurent expansion for $f$ is made unique by requiring that the Cauchy principal value of $H(D)E$ be equal to $H(D)E$.
Solutions of partial differential equations with support on leaves of associated foliations
E. C.
Zachmanoglou
415-421
Abstract: Suppose that the linear partial differential operator $P(x,D)$ has analytic coefficients and that it can be written in the form $P(x,D) = R(x,D)S(x,D)$ where $S(x,D)$ is a polynomial in the homogeneous first order operators ${A_1}(x,D), \cdots ,{A_r}(x,D)$. Then in a neighborhood of any point ${x^0}$ at which the principal part of $ S(x,D)$ does not vanish identically, there is a solution of $P(x,D)u = 0$ with support the leaf through $ {x^0}$ of the foliation induced by the Lie algebra generated by ${A_1}(x,D), \cdots ,{A_r}(x,D)$. This result yields necessary conditions for hypoellipticity and uniqueness in the Cauchy problem. An application to second order degenerate elliptic operators is also given.
Obstructions to embedding $n$-manifolds in $(2n-1)$-manifolds
J. W.
Maxwell
423-435
Abstract: Suppose $f:({M^n},\partial {M^n}) \to ({Q^{2n - 1}},\partial {Q^{2n - 1}})$ is a proper PL map between PL manifolds $ {M^n}$ and ${Q^{2n - 1}}$ of dimension $n$ and $2n - 1$ respectively, $M$ compact. J. F. P. Hudson has shown that associated with each such map $f$ that is an embedding on $\partial M$ is an element $\bar \alpha (f)$ in $ {H_1}(M;{Z_2})$ when $ n$ is odd and an element $\bar \beta (f)$ in ${H_1}(M;Z)$ when $n$ is even. These elements are invariant under a homotopy relative to $ \partial M$. We show that, under slight additional assumptions on $ M,Q$ and $f,f$ is homotopic to an embedding if and only if $ \bar \alpha (f) = 0$ for $ n$ odd and $\bar \beta (f) = 0$ for $n$ even. This result is used to give a sufficient condition for extending an embedding $ f:\partial {M^n} \to \partial {B^{2n - 1}}$ ( $ {B^{2n - 1}}$ denotes $ (2n - 1)$-dimensional ball) to an embedding $ F:({M^n},\partial {M^n}) \to ({B^{2n - 1}},\partial {B^{2n - 1}})$.
On the Arens products and certain Banach algebras
Pak Ken
Wong
437-448
Abstract: In this paper, we study several problems in Banach algebras concerned with the Arens products.
Multipliers and linear functionals for the class $N\sp{+}$
Niro
Yanagihara
449-461
Abstract: Multipliers for the classes ${H^p}$ are studied recently by several authors, see Duren's book, Theory of ${H^p}$ spaces, Academic Press, New York, 1970. Here we consider corresponding problems for the class ${N^ + }$ of holomorphic functions in the unit disk such that $\displaystyle \mathop {\lim }\limits_{r \to 1} \int_0^{2\pi } {{{\log }^ + }} \... ...= \int_0^{2\pi } {{{\log }^ + }\vert f({e^{i\theta }})\vert} d\theta < \infty .$ Our results are: 1. ${N^ + }$ is an $F$-space in the sense of Banach with the distance function $\displaystyle \rho (f,g) = \frac{1}{{2\pi }}\int_0^{2\pi } {\log (1 + \vert f({e^{i\theta }}) - g({e^{i\theta }})\vert)} d\theta .$ 2. A complex sequence $\Lambda = \{ {\lambda _n}\}$ is a multiplier for ${N^ + }$ into ${H^q}$ for a fixed $ q,0 < q < \infty$, if and only if ${\lambda _n} = O(\exp [ - c\sqrt n ])$ for a positive constant $c$. 3. A continuous linear functional $ \phi$ on the space $ {N^ + }$ is represented by a holomorphic function $g(z) = \Sigma {b_n}{z^n}$ which satisfies $ {b_n} = O(\exp [ - c\sqrt n ])$ for a positive constant $c$. Conversely, such a function $ g(z) = \Sigma {b_n}{z^n}$ defines a continuous linear functional on the space $ {N^ + }$.
Weak compactness in locally convex spaces
D. G.
Tacon
463-474
Abstract: The notion of weak compactness plays a central role in the theory of locally convex topological vector spaces. However, in the statement of many theorems, completeness of the space, or at least quasi-completeness of the space in the Mackey topology is an important assumption. In this paper we extend the concept of weak compactness in a general way and obtain a number of useful particular cases. If we replace weak compactness by these generalized notions we can drop the completeness assumption from the statement of many theorems; for example, we generalize the classical theorems of Eberlein and Kreĭn. We then consider generalizations of semireflexivity and reflexivity and characterize these properties in terms of our previous ideas as well as in terms of known concepts. In most of the proofs we use techniques of nonstandard analysis.
Closed hulls in infinite symmetric groups
Franklin
Haimo
475-484
Abstract: Let $\operatorname{Sym} M$ be the symmetric group of an infinite set $M$. What is the smallest subgroup of $\operatorname{Sym} M$ containing a given element if the subgroup is subject to the further condition that it is also the automorphism group of some finitary algebra on $M$? The structures of such closed hulls are related to the disjoint-cycle decompositions of the given elements. If the closed hull is not just the cyclic subgroup on the given element then it is nonminimal as a closed hull and is represented as a subdirect product of finite cyclic groups as well as by a quotient group of a group of infinite sequences. We determine the conditions under which it has a nontrivial primary component for a given prime $p$ and show that such components must be bounded abelian groups.
Heegaard splittings of homology $3$-spheres
Dean A.
Neumann
485-495
Abstract: We investigate properties of Heegaard splittings of closed $ 3$-manifolds which are known for simply-connected manifolds and which might provide the basis for a general test for simple-connectivity. Our results are negative: each property considered is shown to hold in a wider class of manifolds.
Barycenters, pinnacle points, and denting points
Surjit Singh
Khurana
497-503
Abstract: Some properties of probability measures, on closed convex bounded sets in locally convex spaces, having barycenters are obtained. Also some geometric and measure-theoretic characterizations of pinnacle points are given, and a result about denting points is proved.
Addendum to: ``Differential-boundary operators''
Allan M.
Krall
505
Abstract: The proof of a lemma and the statement of another were omitted from an earlier paper. This corrects that omission.
Addendum to: ``Modular representations of metabelian groups'' (Trans. Amer. Math. Soc. {\bf 169}(1972), 389--399)
B. G.
Basmaji
507-508
Algebraic extensions of difference fields
Peter
Evanovich
1-22
Abstract: An inversive difference field $ \mathcal{K}$ is a field K together with a finite number of automorphisms of K. This paper studies inversive extensions of inversive difference fields whose underlying field extensions are separable algebraic. The principal tool in our investigations is a Galois theory, first developed by A. E. Babbitt, Jr. for finite dimensional extensions of ordinary difference fields and extended in this work to partial difference field extensions whose underlying field extensions are infinite dimensional Galois. It is shown that if $ \mathcal{L}$ is a finitely generated separable algebraic inversive extension of an inversive partial difference field $\mathcal{K}$ and the automorphisms of $\mathcal{K}$ commute on the underlying field of $\mathcal{K}$ then every inversive subextension of $\mathcal{L}/\mathcal{K}$ is finitely generated. For ordinary difference fields the paper makes a study of the structure of benign extensions, the group of difference automorphisms of a difference field extension, and two types of extensions which play a significant role in the study of difference algebra: monadic extensions (difference field extensions $\mathcal{L}/\mathcal{K}$ having at most one difference isomorphism into any extension of $\mathcal{K}$) and incompatible extensions (extensions $ \mathcal{L}/\mathcal{K},\mathcal{M}/\mathcal{K}$ having no difference field compositum).
On embedding set functions into covariance functions
G. D.
Allen
23-33
Abstract: We consider any continuous hermitian kernel $\mathcal{P} \times \mathcal{P}$ where $\mathcal{P}$ is the prering of intervals of [0,1]. Conditions on M are given to find an interval covariance function $K(\Delta ,\Delta ') = M(\Delta ,\Delta ')$ for all nonoverlapping $\Delta$ and $\Delta '$ in $ \mathcal{P}$. The problem is solved by first treating finite hermitian matrices A and finding a positive definite matrix B so that ${b_{ij}} = {a_{ij}},i \ne j$, so that tr B is minimized. Using natural correspondence between interval covariance functions and stochastic processes, a decomposition theorem is derived for stochastic processes of bounded quadratic variation into an orthogonal process and a process having minimal quadratic variation.
The $L^p$ norm of sums of translates of a function
Kanter
Marek
35-47
Abstract: For p not an even integer, $p > 0$, we prove that knowledge of the $ {L^p}$ norm of all linear combinations of translates of a real valued function in $ {L^p}(R)$ determines the function up to translation and multiplication by $ \pm 1$.
On the convergence of best uniform deviations
S. J.
Poreda
49-59
Abstract: If a function f is continuous on a closed Jordan curve $ \Gamma$ and meromorphic inside $\Gamma$, then the polynomials of best uniform approximation to f on $\Gamma$ converge interior to $\Gamma$. Furthermore, the limit function can in each case be explicitly determined in terms of the mapping function for the interior of $ \Gamma$. Applications and generalizations of this result are also given.
Extensions of functions and spaces
Giovanni
Viglino
61-69
Abstract: We investigate, for a given map $\varphi$ from a topological space X to a topological space Y (denoted by $[X,\varphi ,Y]$), those triples $[E,\Phi ,Y]$ where E is an extension of X and $\Phi$ extends $\varphi$ to E. A maximal such extension, similar to the Katětov extension of a topological space, is examined.
Summability of Jacobi series
Richard
Askey
71-84
Abstract: The positivity of some Cesàro mean is proven for Jacobi series $ \Sigma {a_n}P_n^{(\alpha ,\beta )}(x),\alpha ,\beta \geqq - \tfrac{1}{2}$. This has applications to the mean convergence of Lagrange interpolation at the zeros of Jacobi polynomials. The positivity of the $(C,\alpha + \beta + 2)$ means is conjectured and proven for some $ (\alpha ,\beta )$. One consequence of this conjecture would be the complete monotonicity of ${x^{ - c}}{({x^2} + 1)^{ - c}},c \geqq 1$.
Groups of diffeomorphisms and their subgroups
Hideki
Omori
85-122
Abstract: This paper has two purposes. The first is to prove the existence of a normal coordinate with respect to a connection defined on the group of diffeomorphisms of a closed manifold, relating to an elliptic complex. The second is to prove a Frobenius theorem with respect to a right invariant distribution defined on the group of diffeomorphisms of a closed manifold, relating to an elliptic complex. Consequently, the group of all volume preserving diffeomorphisms and the group of all symplectic diffeomorphisms are Fréchet Lie groups.
Absolutely summing and dominated operators on spaces of vector-valued continuous functions
Charles
Swartz
123-131
Abstract: A. Pietsch has shown that the class of dominated linear operators on $C(S)$ coincides with the class of absolutely summing operators. If the space $C(S)$ is replaced by ${C_X}(S)$, where X is a Banach space, this is no longer the case. However, any absolutely summing operator is always dominated, and the classes of operators coincide exactly when X is finite dimensional. A characterization of absolutely summing operators on $ {C_X}(S)$ is given.
Fundamental theory of contingent differential equations in Banach space
Shui Nee
Chow;
J. D.
Schuur
133-144
Abstract: For a contingent differential equation that takes values in the closed, convex, nonempty subsets of a Banach space E, we prove an existence theorem and we investigate the extendability of solutions and the closedness and continuity properties of solution funnels. We consider first a space E that is separable and reflexive and then a space E with a separable second dual space. We also consider the special case of a point-valued or ordinary differential equation.
The Rudin-Keisler ordering of $P$-points
Andreas
Blass
145-166
Abstract: The Stone-Čech compactification $ \beta \omega$ of the discrete space $\omega$ of natural numbers is weakly ordered by the relation ``D is the image of E under the canonical extension $\beta f:\beta \omega \to \beta \omega$ of some map $f:\omega \to \omega$.'' We shall investigate the structure, with respect to this ordering, of the set of P-points of $\beta \omega - \omega $.
The existence and uniqueness of nonstationary ideal incompressible flow in bounded domains in $R\sb{3}$
H. S. G.
Swann
167-180
Abstract: It is shown here that the mixed initial-boundary value problem for the Euler equations for ideal flow in bounded domains of ${R_3}$ has a unique solution for a small time interval. The existence of a solution is shown by converting the equations to an equivalent system involving the vorticity and applying Schauder's fixed point theorem to an appropriate mapping.
Euclidean $n$-space modulo an $(n-1)$-cell
J. L.
Bryant
181-192
Abstract: This paper, together with another paper by the author titled similarly, provides a complete answer to a conjecture raised by Andrews and Curtis: if D is a k-cell topologically embedded in euclidean n-space ${E^n}$, then ${E^n}/D \times {E^1}$ is homeomorphic to ${E^{n + 1}}$. Although there is at present only one case outstanding ( $ n \geqslant 4$ and $ k = n - 1$), the proof we give here works whenever $n \geqslant 4$. We resolve this conjecture (for $n \geqslant 4$) by proving a stronger result: if $Y \times {E^1} \approx {E^{n + 1}}$ and if D is a k-cell in Y, then $Y/D \times {E^1} \approx {E^{n + 1}}$. This theorem was proved by Glaser for $k \leqslant n - 2$ and has as a corollary: if K is a collapsible polyhedron topologically embedded in ${E^n}$, then ${E^n}/K \times {E^1} \approx {E^{n + 1}}$. Our method of proof uses radial engulfing and a well-known procedure devised by Bing.
Oscillation, continuation, and uniqueness of solutions of retarded differential equations
T.
Burton;
R.
Grimmer
193-209
Abstract: In this paper we present a number of results on continuation and uniqueńess of solutions of the n-dimensional system $ \tau (t) \geq 0$. We then give some necessary, some sufficient, and some necessary and sufficient conditions for oscillation of solutions of the second order equation $x'' + a(t)f(x(t - \tau (t))) = 0$.
A Stong-Hattori spectral sequence
David Copeland
Johnson
211-225
Abstract: Let ${G_ \ast }(\;)$ be the Adams summand of connective K-theory localized at the prime p. Let $B{P_\ast}(\;)$ be Brown-Peterson homology for that prime. A spectral sequence is constructed with $ {E^2}$ term determined by ${G_ \ast }(X)$ and whose ${E^\infty }$ terms give the quotients of a filtration of $B{P_ \ast }(X)$ where X is a connected spectrum. A torsion property of the differentials implies the Stong-Hattori theorem.
Isotopic unknotting in $F\times I$
C. D.
Feustel
227-238
Abstract: This paper is essentially a generalization of Unknotting in ${M^2} \times I$, by E. M. Brown. The major results in this paper concern the existence of ambient isotopies of unknotted arcs (families of arcs) properly embedded in $F \times I$.
The Cauchy problem for degenerate parabolic equations with discontinuous drift
Edward D.
Conway
239-249
Abstract: The coefficient of the gradient is allowed to be discontinuous but is assumed to satisfy a ``one-sided'' Lipschitz condition. This condition insures the pathwise uniqueness of the underlying Markov process which in turn yields the existence of a unique stable generalized solution of the parabolic equation. If the data is Lipschitz continuous, then so is the solution.
Properties of fixed-point sets of nonexpansive mappings in Banach spaces
Ronald E.
Bruck
251-262
Abstract: Let C be a closed convex subset of the Banach space X. A subset F of C is called a nonexpansive retract of C if either $F = \emptyset$ or there exists a retraction of C onto F which is a nonexpansive mapping. The main theorem of this paper is that if $T:C \to C$ is nonexpansive and satisfies a conditional fixed point property, then the fixed-point set of T is a nonexpansive retract of C. This result is used to generalize a theorem of Belluce and Kirk on the existence of a common fixed point of a finite family of commuting nonexpansive mappings.
Higher dimensional generalizations of the Bloch constant and their lower bounds
Kyong T.
Hahn
263-274
Abstract: A higher dimensional generalization of the classical Bloch theorem depends in an essential way on the ``boundedness'' of the family of holomorphic mappings considered. In this paper the author considers two types of such ``bounded'' families and obtains explicit lower bounds of the generalized Bloch constants of these families on the hyperball in the space $ {{\mathbf{C}}^n}$ in terms of universal constants which characterize the families.
Some stable results on the cohomology of the classical infinite-dimensional Lie algebras
Victor
Guillemin;
Steven
Shnider
275-280
Abstract: In this paper we compute the cohomology of various classical infinite-dimensional Lie algebras generalizing results of Gel'fand-Fuks for the Lie algebra of all formal power series vector fields.
Concerning the shapes of finite-dimensional compacta
Ross
Geoghegan;
R. Richard
Summerhill
281-292
Abstract: It is shown that two ``tamely'' embedded compacta of dimension $ \leq k$ lying in ${E^n}(n \geq 2k + 2)$ have the same (Borsuk) shape if and only if their complements are homeomorphic. In particular, two k-dimensional closed submanifolds of $ {E^{2k + 2}}$ have the same homotopy type if and only if their complements are homeomorphic.
On the classification of metabelian Lie algebras
Michael A.
Gauger
293-329
Abstract: The classification of 2-step nilpotent Lie algebras is attacked by a generator-relation method. The main results are in low dimensions or a small number of relations.
The measure algebra of a locally compact hypergroup
Charles F.
Dunkl
331-348
Abstract: A hypergroup is a locally compact space on which the space of finite regular Borel measures has a convolution structure preserving the probability measures. This paper deals only with commutative hypergroups. §1 contains definitions, a discussion of invariant measures, and a characterization of idempotent probability measures. §2 deals with the characters of a hypergroup. §3 is about hypergroups, which have generalized translation operators (in the sense of Levitan), and subhypergroups of such. In this case the set of characters provides much information. Finally §4 discusses examples, such as the space of conjugacy classes of a compact group, certain compact homogeneous spaces, ultraspherical series, and finite hypergroups.
Localization, homology and a construction of Adams
Aristide
Deleanu;
Peter
Hilton
349-362
Abstract: In recent papers, the authors have developed the technique of using Kan extensions to obtain extensions of homology and cohomology theories from smaller to larger categories of topological spaces. In the present paper, it is shown that the conditions imposed there to guarantee that the Kan extension of a cohomology theory is again a cohomology theory in fact also imply that the Kan extension commutes with stabilization. A construction, due to Adams, for completing a space with respect to a homology theory by using categories of fractions is generalized to triangulated categories, and it is shown that, for any family of primes P, the Adams completion of a space X with respect to the homology theory ${\tilde H_ \ast }( - ;{{\mathbf{Z}}_P})$ is the localization of X at P in the sense of Sullivan. Using this, the Kan extension of the restriction of a homology theory to the category of spaces having P-torsion homotopy groups is determined.
Dual operations on saddle functions
L.
McLinden
363-381
Abstract: Dual operations on convex functions play a central role in the analysis of constrained convex optimization problems. Our aim here is to provide tools for a similar analysis of constrained concave-convex minimax problems. Two pairs of dual operations on convex functions, including addition and infimal convolution, are extended to saddle functions. For the resulting saddle functions much detailed information is given, including subdifferential formulas. Also, separable saddle functions are defined and some basic facts about them established.
On unaveraged convergence of positive operators in Lebesgue space
H.
Fong;
L.
Sucheston
383-397
Abstract: Let T be a power-bounded positive conservative operator on $ {L_1}$ of a $\sigma $-finite measure space. Let e be a bounded positive function invariant under the operator adjoint to T. Theorem. (1) $\smallint \vert{T^n}f\vert \cdot \;e \to 0$ implies (2) $\smallint \vert{T^n}f\vert \to 0$. If T and all its powers are ergodic, and T satisfies an abstract Harris condition, then(l) holds by the Jamison-Orey theorem for all integrable f with $ \smallint f \cdot e = 0$, and hence also (2) holds for such f. A new proof of the Jamison-Orey theorem is given, via the 'filling scheme'. For discrete measure spaces this is due to Donald Ornstein, Proc. Amer. Math. Soc. 22 (1969), 549-551. If T is power-bounded, conservative and ergodic, and $0 < {f_0} = T{f_0}$, then $ {f_0}\; \cdot \;e \in {L_1}$ implies $ {f_0} \in {L_1}$, hence (2) implies that ${T^n}f$ converges for each $f \in {L_1}$. Theorem. Let T be a positive conservative contraction on $ {L_1}$; then the class of functions $\{ f - Tf,f \in L_1^ + \} $ is dense in the class of functions $\{ f - Tf,f \in {L_1}\} $.
Differential equations on closed subsets of a Banach space
R. H.
Martin
399-414
Abstract: In this paper the problem of existence of solutions to the initial value problem $A:[a,b) \times D \to E$ is continuous, D is a closed subset of a Banach space E, and $z \in D$. With a dissipative type condition on A, we establish sufficient conditions for this initial value problem to have a solution. Using these results, we are able to characterize all continuous functions which are generators of nonlinear semigroups on D.
Envelopes of holomorphy and holomorphic convexity
Robert
Carmignani
415-431
Abstract: This paper is primarily a study of generalized notions of envelope of holomorphy and holomorphic convexity for special (algebraically restricted) subsets of ${{\mathbf{C}}^n}$ and in part for arbitrary subsets of $ {{\mathbf{C}}^n}$. For any special set S in ${{\mathbf{C}}^n}$, we show that every function holomorphic in a neighborhood of S not only can be holomorphically continued but also holomorphically extended to a neighborhood in ${{\mathbf{C}}^n}$ of a maximal set $\tilde{S}$, the ``envelope of holomorphy'' of S, which is also a special set of the same type as S. Formulas are obtained for constructing $\tilde{S}$ for any special set S. ``Holomorphic convexity'' is characterized for these special sets. With one exception, the only topological restriction on these special sets is connectivity. Examples are given which illustrate applications of the theorems and help to clarify the concepts of ``envelope of holomorphy'' and ``holomorphic convexity."
A characterization of the invariant measures for an infinite particle system with interactions
Thomas M.
Liggett
433-453
Abstract: Let $p(x,y)$ be the transition function for a symmetric, irreducible, transient Markov chain on the countable set S. Let ${\eta _t}$ be the infinite particle system on S with the simple exclusion interaction and one-particle motion determined by p. A characterization is obtained of all the invariant measures for $ {\eta _t}$ in terms of the bounded functions on S which are harmonic with respect to $p(x,y)$. Ergodic theorems are proved concerning the convergence of the system to an invariant measure.
The \v Silov boundary of $M\sb{0}(G)$
William
Moran
455-464
Abstract: Let G be a locally compact abelian group and let ${M_0}(G)$ be the convolution algebra consisting of those Radon measures on G whose Fourier-Stieltjes transforms vanish at infinity. It is shown that the Šilov boundary of ${M_0}(G)$ is a proper subset of the maximal ideal space of ${M_0}(G)$. The measures constructed to prove this theorem are also used to obtain a stronger result for the full measure algebra $M(G)$.
On the quasi-simple irreducible representations of the Lorentz groups
Ernest
Thieleker
465-505
Abstract: For $n \geq 2$, let $G(n)$ denote the generalized homogeneous Lorentz group of an $n + 1$-dimensional real vector space; that is, $ G(n)$ is the identity component of the orthogonal group of a real quadratic form of index $( + , - - \ldots - )$. Let $ \hat G(n)$ denote a two-fold covering group of $G(n)$, and let $ \hat S(n)\hat M(n)$ be a parabolic subgroup of $\hat G(n)$. We consider the induced representations of $\hat G(n)$, induced by the finite-dimensional irreducible representations of $\hat S(n)\hat M(n)$. By an extension of the methods used in a previous paper, we determine precise criteria for the topological irreducibility of these representations. Moreover, in the exceptional cases when these representations fail to be irreducible, we determine the irreducible subrepresentations of these induced representations. By means of some general results of Harish-Chandra together with the main results of this paper, we obtain a complete classification, up to infinitesimal equivalence, of the quasi-simple irreducible representations of the groups $\hat G(n)$.
The genera of amalgamations of graphs
Seth R.
Alpert
1-39
Abstract: If $p \leq m$, n then ${K_m}{ \vee _{{K_p}}}{K_n}$ is the graph obtained by identify ing a copy of ${K_p}$ contained in ${K_m}$ with a copy of ${K_p}$ contained in ${K_n}$ . It is shown that for all integers $p \leq m$, n the genus $ g({K_m}{ \vee _{{K_p}}}{K_n})$ of $ {K_m}{ \vee _{{K_p}}}{K_n}$ is less than or equal to $g({K_m}) + g({K_n})$. Combining this fact with the lower bound obtained from the Euler formula, one sees that for $2 \leq p \leq 5,g({K_m}{ \vee _{{K_p}}}{K_n})$ is either $ g({K_m}) + g({K_n})$ or else $ g({K_m}) + g({K_n}) - 1$. Except in a few special cases, it is determined which of these values is actually attained.
Asymptotic abelianness of infinite factors
M. S.
Glaser
41-56
Abstract: Studying Pukánszky's type III factor, ${M_2}$, we show that it does not have the property of asymptotic abelianness and discuss how this property is related to property L. We also prove that there are no asymptotic abelian ${\text{II}_\infty }$ factors. The extension (by ampliation) of central sequences in a finite factor, N, to $ M \otimes N$ is shown to be central. Also, we give two examples of the reduction (by equivalence) of a central sequence in $M \otimes N$ to a sequence in N. Finally, applying the definition of asymptotic abelianness of $ {C^\ast}$-algebras to $ {W^\ast}$-algebras leads to the conclusion that all factors satisfying this property are abelian.
Geodesic flows on negatively curved manifolds. II
Patrick
Eberlein
57-82
Abstract: Let M be a complete Riemannian manifold with sectional curvature $K \leq 0$, SM the unit tangent bundle of M, ${T_t}$ the geodesic flow on SM and $ \Omega \subseteq SM$ the set of nonwandering points relative to ${T_t}$. ${T_t}$ is topologically mixing (respectively topologically transitive) on SM if for any open sets 0, U of SM there exists $A > 0$ such that $\left\vert t \right\vert \geq A$ implies $ {T_t}(O) \cap U \ne \emptyset$ (respectively there exists $t\;\varepsilon \;R$ such that ${T_t}(O) \cap U \ne \emptyset$). For each vector $ v\;\varepsilon \;SM$ we define stable and unstable sets $ {W^s}(v),{W^{ss}}(v),{W^u}(v)$ and $ {W^{uu}}(v)$, and we relate topological mixing (respectively topological transitivity) of ${T_t}$ to the existence of a vector $v\; \in \;SM$ such that $ {W^{ss}}(v)$ (respectively $ {W^s}(v)$) is dense in SM. If M is a Visibility manifold (implied by $K \leq c < 0$) and if $\Omega = SM$ then ${T_t}$ is topologically mixing on SM. Let $ {S_n} =$ {Visibility manifolds M of dimension n such that $ {T_t}$ is topologically mixing on SM}. For each $n \geq 2$, ${S_n}$ is closed under normal (Galois) Riemannian coverings. If $ M\; \in \;{S_n}$ we classify { $ v\; \in \;SM:\;{W^{ss}}(v)$ is dense in SM}, and M is compact if and only if this set = SM. We also consider the case where $\Omega$ is a proper subset of SM.
Algebraic cohomology of topological groups
David
Wigner
83-93
Abstract: A general cohomology theory for topological groups is described, and shown to coincide with the theories of C. C. Moore [12] and other authors. We also recover some invariants from algebraic topology.
Alternating Chebyshev approximation
Charles B.
Dunham
95-109
Abstract: An approximating family is called alternating if a best Chebyshev approximation is characterized by its error curve having a certain number of alternations. The convergence properties of such families are studied. A sufficient condition for the limit of best approximation on subsets to converge uniformly to the best approximation is given: it is shown that this is often (but not always) a necessary condition. A sufficient condition for the Chebyshev operator to be continuous is given: it is shown that this is often (but not always) a necessary condition.
Decomposable braids and linkages
H.
Levinson
111-126
Abstract: An n-braid is called k-decomposable if and only if the removal of k arbitrary strands results in a trivial $(n - k)$-braid. k-decomposable n-linkages are similarly defined. All k-decomposable n-braids are generated by an explicit geometric process, and so are all k-decomposable n-linkages. The latter are not always closures of k-decomposable n-braids. Many examples are given.
Elementary properties of free groups
George S.
Sacerdote
127-138
Abstract: In this paper we show that several classes of elementary properties (properties definable by sentences of a first order logic) of groups hold for all nonabelian free groups. These results are obtained by examining special embeddings of these groups into one another which preserve the properties in question.
Iterated limits in $N\sp{\ast} (U\sp{n})$
Carl Stephen
Davis
139-146
Abstract: It is shown that if f is in $ {N^\ast}({U^n})$, then the iterated limits of f are almost everywhere independent of the order of iteration. In fact, the iterated limit and the radial limit are equal almost everywhere.
Free products of von Neumann algebras
Wai Mee
Ching
147-163
Abstract: A new method of constructing factors of type ${\text{II}_1}$, called free product, is introduced. It is a generalization of the group construction of factors of type $ {\text{II}_1}$ when the given group is a free product of two groups. If $ {A_1}$ and ${A_2}$ are two von Neumann algebras with separating cyclic trace vectors and ortho-unitary bases, then the free product ${A_1} \ast {A_2}$ of ${A_1}$ and ${A_2}$ is a factor of type ${\text{II}_1}$ without property $\Gamma$.
Surjective stability in dimension $0$ for $K\sb{2}$ and related functors
Michael R.
Stein
165-191
Abstract: This paper continues the investigation of generators and relations for Chevalley groups over commutative rings initiated in [14]. The main result is that if A is a semilocal ring generated by its units, the groups $L({\mathbf{\Phi }},A)$ of [14] are generated by the values of certain cocycles on ${A^\ast} \times {A^\ast}$. From this follows a surjective stability theorem for the groups $L({\mathbf{\Phi }},A)$, as well as the result that $ L({\mathbf{\Phi }},A)$ is the Schur multiplier of the elementary subgroup of the points in A of the universal Chevalley-Demazure group scheme with root system ${\mathbf{\Phi }}$, if ${\mathbf{\Phi }}$ has large enough rank. These results are proved via a Bruhat-type decomposition for a suitably defined relative group associated to a radical ideal. These theorems generalize to semilocal rings results of Steinberg for Chevalley groups over fields, and they give an effective tool for computing Milnor's groups ${K_2}(A)$ when A is semilocal.
Cobordism invariants, the Kervaire invariant and fixed point free involutions
William
Browder
193-225
Abstract: Conditions are found which allow one to define an absolute version of the Kervaire invariant in ${Z_2}$ of a $ {\text{Wu - }}(q + 1)$ oriented 2q-manifold. The condition is given in terms of a new invariant called the spectral cobordism invariant. Calculations are then made for the Kervaire invariant of the n-fold disjoint union of a manifold M with itself, which are then applied with $M = {P^{2q}}$, the real protective space. These give examples where the Kervaire invariant is not defined, and other examples where it has value $1 \in {{\mathbf{Z}}_2}$. These results are then applied to construct examples of smooth fixed point free involutions of homotopy spheres of dimension $4k + 1$ with nonzero desuspension obstruction, of which some Brieskorn spheres are examples (results obtained also by Berstein and Giffen). The spectral cobordism invariant is also applied directly to these examples to give another proof of a result of Atiyah-Bott. The question of which values can be realized as the sequence of Kervaire invariants of characteristic submanifolds of a smooth homotopy real projective space is discussed with some examples. Finally a condition is given which yields smooth embeddings of homotopy $ {P^m}$'s in ${R^{m + k}}$ (which has been applied by E. Rees).
Kernels in dimension theory
J. M.
Aarts;
T.
Nishiura
227-240
Abstract: All spaces are metrizable. A conjecture of de Groot states that a weak inductive dimension theory beginning with the class of compact spaces will characterize those spaces which can be extended to a compact space by the adjunction of a set of dimension not exceeding n. Nagata has proposed a variant of this conjecture as a means of finding insights into the original conjecture. (See Internat. Sympos. on Extension Theory, Berlin, 1967, pp. 157-161.) The proposed variant replaces compact with $ \sigma$-compact. The present paper concerns a study of strong inductive dimension theory beginning with an arbitrary class of spaces. The study is motivated by the above two conjectures. It indicates that a theory of kernels is a more natural by-product of inductive theory than a theory of extensions. An example has resulted which, with the aid of the developed theory and the Baire category theorem, resolves the second conjecture in the negative. The original conjecture is still unresolved. It is also shown that the notion of kernels results in a further generalization of Lelek's form of the dimension lowering map theorem (Colloq. Math. 12 (1964), 221-227. MR 31 #716).
Equivariant cobordism and duality
Edward C.
Hook
241-258
Abstract: We consider equivariant cobordism theory, defined by means of an equivariant Thorn spectrum; in particular, we investigate the relationship between this theory and the more geometric equivariant bordism theory, showing that there is a Poincaré-Lefschetz duality theorem which is valid in this setting.
The trace-class of a full Hilbert algebra
Michael R. W.
Kervin
259-270
Abstract: The trace-class of a full Hilbert algebra A is the set $ \tau (A) = \{ xy\vert x \in A,y \in A\}$. This set is shown to be a $\ast$-ideal of A, and possesses a norm $\tau$ defined in terms of a positive hermitian linear functional on $\tau (A)$. The norm $\tau$ is in general both incomplete and not an algebra norm, and is also not comparable with the Hilbert space norm $ \left\Vert\right\Vert$ on $ \tau (A)$. However, a one-sided ideal of $\tau (A)$ is closed with respect to one norm if and only if it is closed with respect to the other. The topological dual of $\tau (A)$ with respect to the norm $\tau$ is isometrically isomorphic to the set of left centralizers on A.
Multiplicities of second order linear recurrences
Ronald
Alter;
K. K.
Kubota
271-284
Abstract: A second order linear recurrence is a sequence $ \{ {a_n}\}$ of integers satisfying a ${a_{n + 2}} = M{a_{n + 1}} - N{a_n}$ where N and M are fixed integers and at least one $ {a_n}$ is nonzero. If k is an integer, then the number $m(k)$ of solutions of ${a_n} = k$ is at most 3 (respectively 4) if ${M^2} - 4N < 0$ and there is an odd prime $q \ne 3$ (respectively q = 3) such that $ q\vert M$ and $ q\nmid kN$. Further $M = {\sup _k}{\;_{{\text{integer}}}}m(k)$ is either infinite or $\leq 5$ provided that either (i) $(M,N) = 1$ or (ii) $6\nmid N$.
Submanifolds and a pinching problem on the second fundamental tensors
Masafumi
Okumura
285-291
Abstract: This paper gives a sufficient condition for a submanifold of a Riemannian manifold of nonnegative constant curvature to be totally umbilical. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.
Local and asymptotic approximations of nonlinear operators by $(k\sb{1},\,\ldots k\sb{N})$-homogeneous operators
R. H.
Moore;
M. Z.
Nashed
293-305
Abstract: Notions of local and asymptotic approximations of a nonlinear mapping F between normed linear spaces by a sum of N $ {k_i}$-homogeneous operators are defined and investigated. It is shown that the approximating operators inherit from F properties related to compactness and measures of noncompactness. Nets of equi-approximable operators with collectively compact (or bounded) approximates, which arise in approximate solutions of integral and operator equations, are studied with particular reference to pointwise (or weak convergence) properties. As a by-product, the well-known result that the Fréchet (or asymptotic) derivative of a compact operator is compact is generalized in several directions and to families of operators.
Infinite particle systems
Sidney C.
Port;
Charles J.
Stone
307-340
Abstract: We consider a system of denumerably many particles that are distributed at random according to a stationary distribution P on some closed subgroup X of Euclidean space. We assume that the expected number of particles in any compact set is finite. We investigate the relationship between P and the distribution Q of particles as viewed from a particle selected ``at random'' from some set. The distribution Q is called the tagged particle distribution. We give formulas for computing P in terms of Q and Q in terms of P and show that, with the appropriate notion of convergences, $ {P_n} \to P$ implies ${Q_n} \to Q$ and vice versa. The particles are allowed to move in an appropriate translation invariant manner and we show that the tagged particle distribution Q' at a later time 1 is the same as the distribution of particles at time 1 as viewed from a particle selected ``at random'' from those initially in some set. We also show that Q' is the same as the distribution of particles at time 1 as viewed from a particle selected at random from those at the origin, when initially the particles are distributed according to Q. The one-dimensional case is treated in more detail. With appropriate topologies, we show that in this case there is a homeomorphism between the collection of stationary distributions P and tagged particle distributions Q. A stationary spacings distribution ${Q_0}$ related to Q is introduced, and we show that with the appropriate topology the map taking Q to ${Q_0}$ is a homeomorphism. Explicit expressions are found for all these maps and their inverses. The paper concludes by using the one-dimensional results to find stationary distributions for a class of motions of denumerably many unit intervals and to establish criteria for convergence to one of these distributions.
Additive set functions on lattices of sets
Gene A.
DeBoth
341-355
Abstract: This paper is concerned with properties of additive set functions defined on lattices of sets. Extensions of results of Brunk and Johansen, Darst, Johansen, and Uhl are obtained. Two fundamental approximation properties for lattices of sets (established in another paper) permit us to translate the setting and consider countably additive set functions defined on sigma lattices of sets. Thereby results for countably additive set functions defined on sigma lattices of sets are used to obtain alternate derivations and extensions of Darst's results for additive set functions defined on lattices of sets, i.e., we consider the Radon-Nikodym derivative, conditional expectation, and martingale convergence for lattices of sets.
A nonlinear optimal control minimization technique
Russell D.
Rupp
357-381
Abstract: Hestenes' method of multipliers is applied to a nonlinear optimal control problem. This requires that a differentially constrained problem be embedded in a family of unconstrained problems so as to preserve standard sufficiency criteria. Given an initial estimate of the Lagrange multipliers, a convergent sequence of arcs is generated. They are minimizing with respect to members of the above family, and their limit is the solution to the differentially constrained problem.
Decreasing rearrangements and doubly stochastic operators
Peter W.
Day
383-392
Abstract: In this paper generalizations to measurable functions on a finite measure space $ (X,\Lambda ,\mu )$ of some characterizations of the Hardy-Littlewood-Pólya preorder relation $\prec$ are considered. Let $\rho$ be a saturated, Fatou function norm such that ${L^\infty } \subset {L^\rho } \subset {L^1}$, and let $ {L^\rho }$ be universally rearrangement invariant. The following equivalence is shown to hold for all $f \in {L^\rho }$ iff $(X,\Lambda ,\mu )$ is nonatomic or discrete: $g \prec f$ iff g is in the $\rho$-closed convex hull of the set of all rearrangements of f. Finally, it is shown that $g \prec f \in {L^1}$ iff g is the image of f by a doubly stochastic operator.
On the zeros of power series with Hadamard gaps-distribution in sectors
I Lok
Chang
393-400
Abstract: We give a sufficient condition for a power series with Hadamard gaps to assume every complex value infinitely often in every sector of the unit disk.
Some integral inequalities with applications to the imbedding of Sobolev spaces defined over irregular domains
R. A.
Adams
401-429
Abstract: This paper examines the possibility of extending the Sobolev Imbedding Theorem to certain classes of domains which fail to have the ``cone property'' normally required for that theorem. It is shown that no extension is possible for certain types of domains (e.g. those with exponentially sharp cusps or which are unbounded and have finite volume), while extensions are obtained for other types (domains with less sharp cusps). These results are developed via certain integral inequalities which generalize inequalities due to Hardy and to Sobolev, and are of some interest in their own right. The paper is divided into two parts. Part I establishes the integral inequalities; Part II deals with extensions of the imbedding theorem. Further introductory information may be found in the first section of each part.
On fibering of cobordism classes
R. E.
Stong
431-447
Abstract: This paper studies the problem posed by Conner and Floyd of finding which cobordism classes are represented by the total space of a fibering with prescribed base or fiber.
Equivariant bordism of maps
R. E.
Stong
449-457
Abstract: This note computes the bordism classification of equivariant maps between closed manifolds with action of a cyclic group of prime order.
Exit properties of stochastic processes with stationary independent increments
P. W.
Millar
459-479
Abstract: Let $\{ {X_t},t \geq 0\}$ be a real stochastic process with stationary independent increments. For $x > 0$, define the exit time ${T_x}$ from the interval $( - \infty ,x]$ by ${T_x} = \inf \{ t > 0:{X_t} > x\}$. A reasonably complete solution is given to the problem of deciding precisely when ${P^0}\{ {X_{{T_x}}} = x\} > 0$ and precisely when $ {P^0}\{ {X_{{T_x}}} = x\} = 0$. The solution is given in terms of parameters appearing in the Lévy formula for the characteristic function of ${X_t}$. A few applications of this result are discussed.
Monotonically normal spaces
R. W.
Heath;
D. J.
Lutzer;
P. L.
Zenor
481-493
Abstract: This paper begins the study of monotone normality, a common property of linearly ordered spaces and of Borges' stratifiable spaces. The concept of monotone normality is used to give necessary and sufficient conditions for stratifiability of a ${T_1}$-space, to give a new metrization theorem for p-spaces with $ {G_\delta }$-diagonals, and to provide an easy proof of a metrization theorem due to Treybig. The paper concludes with a list of examples which relate monotone normality to certain familiar topological properties.
Generalized Dedekind eta-functions and generalized Dedekind sums
Bruce C.
Berndt
495-508
Abstract: A transformation formula under modular substitutions is derived for a very large class of generalized Eisenstein series. The result also gives a transformation formula for generalized Dedekind eta-functions. Various types of Dedekind sums arise, and reciprocity laws are established.
Analytic wave front sets for solutions of linear differential equations of principal type
Karl Gustav
Andersson
1-27
Abstract: The propagation of analyticity for solutions u of $P(x,D)u = f$ is studied, in terms of wave front sets, for a large class of differential operators $P = P(x,D)$ of principal type. In view of a theorem by L. Hörmander [9], the results obtained imply rather precise results about the surjectivity of the mapping $P:{C^\infty }(\Omega ) \to {C^\infty }(\Omega )$.
Some results on the length of proofs
R. J.
Parikh
29-36
Abstract: Given a theory T, let $\vdash _T^kA$ mean ``A has a proof in T of at most k lines". We consider a formulation $P{A^\ast}$ of Peano arithmetic with full induction but addition and multiplication being ternary relations. We show that ${ \vdash ^k}A$ is decidable for $P{A^\ast}$ and hence $P{A^\ast}$ is closed under a weak $ \omega$-rule. An analogue of Gödel's theorem on the length of proofs is an easy corollary.
Primary ideals in rings of analytic functions
R. Douglas
Williams
37-49
Abstract: Let A be the ring of all analytic functions on a connected, noncompact Riemann surface. We use the valuation theory of the ring A as developed by N. L. Alling to analyze the structure of the primary ideals of A. We characterize the upper and lower primary ideals of A and prove that every nonprime primary ideal of A is either an upper or a lower primary ideal. In addition we give some necessary and sufficient conditions for certain ideals of A to be intersections of primary ideals.
Reductions of ideals in commutative rings
James H.
Hays
51-63
Abstract: All rings considered in this paper are commutative, associative, and have an identity. If A and B are ideals in a ring, then B is a reduction of A if $B \subseteq A$ and if $B{A^n} = {A^{n + 1}}$ for some positive integer n. An ideal is basic if it has no reductions. These definitions were considered in local rings by Northcott and Rees; this paper considers them in more general rings. Basic ideals in Noetherian rings are characterized to the extent that they are characterized in local rings. It is shown that elements of the principal class generate a basic ideal in a Noetherian ring. Prüfer domains do not have the basic ideal property, that is, there may exist ideals which are not basic; however, a characterization of Prüfer domains can be given in terms of basic ideals. A domain is Prüfer if and only if every finitely generated ideal is basic.
Differentials on quotients of algebraic varieties
Carol M.
Knighten
65-89
Abstract: The relations between differentials invariant with respect to a finite group acting on a variety and the differentials on the quotient variety are studied. If the quotient map is unramified in codimension 1 we have an isomorphism for Zariski differentials, but not in general for Käahler differentials. Necessary and sufficient conditions for isomorphism of the Zariski differentials are given when the finite group acts linearly. Examples illustrate the scope of the theorems and some open problems.
Jacobson structure theory for Hestenes ternary rings
Robert Allan
Stephenson
91-98
Abstract: The principal results are an extension of the density theorem to Hestenes ternary rings and a characterization of primitive ternary rings.
Classes of automorphisms of free groups of infinite rank
Robert
Cohen
99-120
Abstract: This paper is concerned with finding classes of automorphisms of an infinitely generated free group F which can be generated by ``elementary'' Nielsen transformations. Two different notions of ``elementary'' Nielsen transformations are explored. One leads to a classification of the automorphisms generated by these transformations. The other notion leads to the subgroup B of $ {\operatorname{Aut}}(F)$ consisting of the ``bounded length'' automorphisms of F. We prove that the class of ``bounded 3-length'' automorphisms ${B_3}$ and the class of ``elementary simultaneous'' Nielsen transformations generate the same subgroup of ${\operatorname{Aut}}(F)$. We show that for the class T of automorphisms of ``2 occurring generators", the groups generated by $T \cap B$ and the ``elementary simultaneous'' Nielsen transformations are identical. These results lead to the conjecture that B is generated by the ``elementary simultaneous Nielsen transformations". A study is also made of the subgroup S of the ``triangular automorphisms'' of ${F_\infty }$, the free group on a countably infinite set of free generators. It is found that a ``triangular automorphism'' may be factored into ``splitting automorphisms'' of $ {F_\infty }$, which may be viewed as the ``elementary'' automorphisms of S.
Logic and invariant theory. I. Invariant theory of projective properties
Walter
Whiteley
121-139
Abstract: This paper initiates a series of papers which will reexamine some problems and results of classical invariant theory, within the framework of modern first-order logic. In this paper the notion that an equation is of invariant significance for the general linear group is extended in two directions. It is extended to define invariance of an arbitrary first-order formula for a category of linear transformations between vector spaces of dimension n. These invariant formulas are characterized by equivalence to formulas of a particular syntactic form: homogeneous formulas in determinants or ``brackets". The fuller category of all semilinear transformations is also introduced in order to cover all changes of coordinates in a projective space. Invariance for this category is investigated. The results are extended to cover invariant formulas with both covariant and contravariant vectors. Finally, Klein's Erlanger Program is reexamined in the light of the extended notion of invariance as well as some possible geometric categories.
On a theorem of Chern
J. H.
Sampson
141-153
Abstract: A new proof is given for Chern's theorem showing that the Laplace operator for differential forms commutes with decomposition of forms associated with G-structures admitting a suitable connection. An analogous result is proved for symmetric tensor fields, and an application is made to determine all harmonic symmetric fields on a compact space of constant negative curvature. Vector-valued forms are also discussed.
Asymptotic properties of Gaussian random fields
Clifford
Qualls;
Hisao
Watanabe
155-171
Abstract: In this paper we study continuous mean zero Gaussian random fields $ X(p)$ with an N-dimensional parameter and having a correlation function $\rho (p,q)$ for which $1 - \rho (p,q)$ is asymptotic to a regularly varying (at zero) function of the distance ${\text{dis}}\;(p,q)$ with exponent $0 < \alpha \leq 2$. For such random fields, we obtain the asymptotic tail distribution of the maximum of $X(p)$ and an asymptotic almost sure property for $ X(p)$ as $\vert p\vert \to \infty $. Both results generalize ones previously given by the authors for $N = 1$.
Isomorphism of simple Lie algebras
B. N.
Allison
173-190
Abstract: Let $\mathcal{L}$ and $ \mathcal{L}$ and $ \mathcal{L}$ is also studied. In particular, a result about this kernel in the rank one reduced case is proved. This result is then used to prove a conjugacy theorem for the simple summands of the anisotropic kernel in the general reduced case. The results and methods of this paper are rational in the sense that they involve no extension of the base field.
Closed countably generated structures in $C(X)$
B.
Roth
191-197
Abstract: Let $C(X)$ be the space of continuous real or complex valued functions on a compact space X with the sup norm topology. In the present paper, the subalgebras, vector lattices, and vector lattice ideals of $ C(X)$ which are closed and countably generated are characterized.
Oriented and weakly complex bordism algebra of free periodic maps
Katsuyuki
Shibata
199-220
Abstract: Free cyclic actions on a closed oriented (weakly almost complex, respectively) manifold which preserve the orientation (weakly complex structure) are considered from the viewpoint of equivariant bordism theory. The author gives an explicit presentation of the oriented bordism module structure and multiplicative structure of all orientation preserving (and reversing) free involutions. The odd period and weakly complex cases are also determined with the aid of the notion of formal group laws. These results are applied to a nonexistence problem for certain equivariant maps.
Self-dual axioms for many-dimensional projective geometry
Martinus
Esser
221-236
Abstract: Proposed and compared are four equivalent sets R, S, T, D of self-dual axioms for projective geometries, using points, hyperplanes and incidence as primitive elements and relation. The set R is inductive on the number of dimensions. The sets S, T, D all include the axiom ``on every n points there is a plane", the dual of this axiom, one axiom on the existence of a certain configuration, and one or several axioms on the impossibility of certain configurations. These configurations consist of $(n + 1)$ points and $(n + 1)$ planes for sets S, T, but of $ (n + 2)$ points and $ (n + 2)$ planes for set D. Partial results are obtained by a preliminary study of self-dual axioms for simplicial spaces (spaces which may have fewer than 3 points per line).
Interpolation between consecutive conjugate points of an $n$th order linear differential equation
G. B.
Gustafson
237-255
Abstract: The interpolation problem ${x^{(n)}} + {P_{n - 1}}{x^{(n - 1)}} + \cdots + {P_0}x = 0$, $ {x^{(i)}}({t_j}) = 0,i = 0, \cdots ,{k_j} - 1,j = 0, \cdots ,m$, is studied on the conjugate interval $ [a,{\eta _1}(a)]$. The main result is that there exists an essentially unique nontrivial solution of the problem almost everywhere, provided $ {k_1} + \cdots + {k_m} \geq n$, and cer tain other inequalities are satisfied, with $a = {t_0} < {t_1} < \cdots < {t_m} = {\eta _1}(a)$. In particular, this paper corrects the results of Azbelev and Caljuk (Mat. Sb. 51 (93) (1960), 475-486; English transl., Amer. Math. Soc. Transl. (2) 42 (1964), 233-245) on third order equations, and shows that their results are correct almost everywhere.
Fatou theorems for eigenfunctions of the invariant differential operators on symmetric spaces
H. Lee
Michelson
257-274
Abstract: On a Riemannian symmetric space of noncompact type we introduce a generalization of the Poisson kernel which may be used to generate simultaneous eigenfunctions of the invariant differential operators with eigenvalues not necessarily zero. We investigate the boundary behavior of our generalized Poisson integrals, extending to them many of the Fatou-type theorems known for harmonic functions.
On the principal series of ${\rm Gl}\sb{n}$ over$p$-adic fields
Roger E.
Howe
275-286
Abstract: The entire principal series of $G = G{l_n}(F)$, for a p-adic field F, is analyzed after the manner of the analysis of Bruhat and Satake for the spherical principal series. If K is the group of integral matrices in $G{l_n}(F)$, then a ``principal series'' of representations of K is defined. It is shown that precisely one of these occurs, and only once, in a given principal series representation of G. Further, the spherical function algebras attached to these representations of K are all shown to be abelian, and their explicit spectral decomposition is accomplished using the principal series of G. Computation of the Plancherel measure is reduced to MacDonald's computation for the spherical principal series, as is computation of the spherical functions themselves.
On the character of Weil's representation
Roger E.
Howe
287-298
Abstract: The importance of certain representations of symplectic groups, usually called Weil representations, for the general problem of finding representations of certain group extensions is made explicit. Some properties of the character of Weil's representation for a finite symplectic group are given and discussed, again in the context of finding representations of group extensions. As a by-product, the structure of anisotropic tori in symplectic groups is given.
Krull dimension in power series rings
Jimmy T.
Arnold
299-304
Abstract: Let R denote a commutative ring with identity. If there exists a chain ${P_0} \subset {P_1} \subset \cdots \subset {P_n}$ of $n + 1$ prime ideals of R, where ${P_n} \ne R$, but no such chain of $n + 2$ prime ideals, then we say that R has dimension n. The power series ring $R[[X]]$ may have infinite dimension even though R has finite dimension.
A second quadrant homotopy spectral sequence
A. K.
Bousfield;
D. M.
Kan
305-318
Abstract: For each cosimplicial simplicial set with basepoint, the authors construct a homotopy Spectral sequence generalizing the usual spectral sequence for a second quadrant double chain complex. For such homotopy spectral sequences, a uniqueness theorem and a general multiplicative pairing are established. This machinery is used elsewhere to show the equivalence of various unstable Adams spectral sequences and to construct for them certain composition pairings and Whitehead products.
Pairings and products in the homotopy spectral sequence
A. K.
Bousfield;
D. M.
Kan
319-343
Abstract: Smash and composition pairings, as well as Whitehead products are constructed in the unstable Adams spectral sequence; and these pairings and products are described homologically on the ${E_2}$. level. In the special case of the Massey-Peterson spectral sequence, the composition action is given homologically by the Yoneda product, while the Whitehead product vanishes. It is also shown that the unstable Adams spectral sequence over the rationals, with its Whitehead products, is given by the primitive elements in the rational cobar spectral sequence.
$k$-parameter semigroups of measure-preserving transformations
Norberto Angel
Fava
345-352
Abstract: An individual ergodic theorem is proved for semigroups of measure-preserving transformations depending on k real parameters, which generalizes N. Wiener's ergodic theorem.
A fixed point theorem-free approach to weak almost periodicity
William A.
Veech
353-362
Abstract: In this paper we present a generalization of the Eberlein, de Leeuw and Glicksberg decomposition theorem for weakly almost periodic functions which does not rely on any fixed point theorem for its proof. A generalization of the Ryll-Nardzewski fixed point theorem is given.
Comparison of eigenvalues for linear differential equations of order $2n$
Curtis C.
Travis
363-374
Abstract: An abstract eigenvalue comparison theorem is proven for $ {u_0}$-positive linear operators in a Banach space equippped with a cone of ``nonnegative'' elements. This result is then applied to certain linear differential equations of order 2n in order to obtain eigenvalue comparison theorems of an ``integral type."
Induced flows
Karl
Petersen;
Leonard
Shapiro
375-390
Abstract: The construction of induced transformations is considered in the setting of topological dynamics. Sufficient conditions are given for induced flows to be topologically weakly mixing, and it is proved that Toeplitz flows and certain Sturmian flows satisfy these conditions and give rise to new and easily constructed classes of flows which have entropy zero and are uniquely ergodic, minimal, and topologically weakly mixing. An example is given of a weakly mixing minimal flow which is not topologically strongly mixing.
Positive approximants
Richard
Bouldin
391-403
Abstract: Let $T = B + iC$ with $B = {B^\ast},C = {C^\ast}$ and let $\delta (T)$ denote the the distance of T to the set of nonnegative operators. We find upper and lower bounds for $ \delta (T)$. We prove that if P is any best approximation for T among nonnegative operators then $P \leq B + {({(\delta (T))^2} - {C^2})^{{\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}$. Provided $B \geq 0$ or T is normal we characterize those T which have a unique best approximation among the nonnegative operators. If T is normal we characterize its best approximating nonnegative operators which commute with it. We characterize those T for which the zero operator is the best approximating nonnegative operator.
$k$-congruence orders for $E\sb{k}$
Grattan P.
Murphy
405-412
Abstract: This paper generalizes the notion of congruence order for metric spaces to k-metric (k-dimensional metric) spaces. The k-congruence order of $ {E_k}$ with respect to the class of oriented semi k-metric spaces is determined. An example shows that this result is sharp.
An invariance principle for a class of $d$-dimensional polygonal random functions
Luis G.
Gorostiza
413-445
Abstract: A class of random functions is formulated, which represent the motion of a point in d-dimensional Euclidean space $(d \geq 1)$ undergoing random changes of direction at random times while maintaining constant speed. The changes of direction are determined by random orthogonal matrices that are irreducible in the sense of not having an almost surely invariant nontrivial subspace if $d \geq 2$, and not being almost surely nonnegative if $d = 1$. An invariance principle stating that under certain conditions a sequence of such random functions converges weakly to a Gaussian process with stationary and independent increments is proved. The limit process has mean zero and its covariance matrix function is given explicitly. It is shown that when the random changes of direction satisfy an appropriate condition the limit process is Brownian motion. This invariance principle includes central limit theorems for the plane, with special distributions of the random times and direction changes, that have been proved by M. Kac, V. N. Tutubalin and T. Watanabe by methods different from ours. The proof makes use of standard methods of the theory of weak convergence of probability measures, and special results due to P. Billingsley and B. Rosén, the main problem being how to apply them. For this, renewal theoretic techniques are developed, and limit theorems for sums of products of independent identically distributed irreducible random orthogonal matrices are obtained.
Probabilistic recursive functions
Irwin
Mann
447-467
Abstract: The underlying question considered in this paper is whether or not the purposeful introduction of random elements, effectively governed by a probability distribution, into a calculation may lead to constructions of number-theoretic functions that are not available by deterministic means. A methodology for treating this question is developed, using an effective mapping of the space of infinite sequences over a finite alphabet into itself. The distribution characterizing the random elements, under the mapping, induces a new distribution. The property of a distribution being recursive is defined. The fundamental theorem states that recursive distributions induce only recursive distributions. A function calculated by any probabilistic means is called $\psi$-calculable. For a class of such calculations, these functions are recursive. Relative to Church's thesis, this leads to an extension of that thesis: Every $\psi$-effectively calculable function is recursive. In further development, a partial order on distributions is defined through the concept of ``inducing.'' It is seen that a recursive atom-free distribution induces any recursive distribution. Also, there exist distributions that induce, but are not induced by, any recursive distribution. Some open questions are mentioned.
$\theta $-modular bands of groups
C.
Spitznagel
469-482
Abstract: The class of $ \theta$-modular bands of groups is defined by means of a type of modularity condition on the lattice of congruences on a band of groups. The main result characterizes $\theta $-modularity as a condition on the multiplication in the band of groups. This result is then applied to the classes of normal bands of groups and orthodox bands of groups.
Boundary representations on $C\sp{\ast} $-algebras with matrix units
Alan
Hopenwasser
483-490
Abstract: Let $\mathcal{A}$ be a ${C^\ast}$-algebra with unit, let $\mathcal{S}$ be a linear subspace of $\mathcal{A} \otimes {M_n}$ which contains the natural set of matrix units and which generates $\mathcal{A}$ as a ${C^\ast}$-algebra. Let $ \mathcal{J}$ be the subset of $\mathcal{A}$ consisting of entries of matrices in $ \mathcal{S}$. Then the boundary representations of $\mathcal{A} \otimes {M_n}$ relative to $\mathcal{S}$ are parametrized by the boundary representations of $ \mathcal{A}$ relative to $\mathcal{J}$. Also, a nontrivial example is given of a subalgebra of a ${C^\ast}$-algebra which possesses exactly one boundary representation.
Sets of formulas valid in finite structures
Alan L.
Selman
491-504
Abstract: A function $\mathcal{V}$ is defined on the set of all subsets of $ \omega$ so that for each set K, the value, ${\mathcal{V}_K}$, is the set of formulas valid in all structures of cardinality in K. An analysis is made of the dependence of ${\mathcal{V}_K}$ on K, For any set K, let $ {\text{d}}(K)$ be the Kleene-Post degree to which K belongs. It is easily seen that for all infinite sets K, ${\text{d}}({\mathcal{V}_{K \vee J}}) = {\text{d}}({\mathcal{V}_K}) \vee {\text{d}}({\mathcal{V}_J})$, and use this to prove that, for any two degrees a and b, ${\text{d}}(K) = {\text{a}}$ and $ {\text{d}}({\mathcal{V}_K}) = b$. Various similar results are also included.
Erratum to: ``A general class of factors of $E\sp{4}$''
Leonard R.
Rubin
505
Erratum to: ``Functional analytic properties of extremely amenable semigroups''
Edmond E.
Granirer
507
Algebraic results on representations of semisimple Lie groups
J.
Lepowsky
1-44
Abstract: Let G be a noncompact connected real semisimple Lie group with finite center, and let K be a maximal compact subgroup of G. Let $ \mathfrak{g}$ and $\mathfrak{k}$ denote the respective complexified Lie algebras. Then every irreducible representation $ \pi$ of $\mathfrak{g}$ which is semisimple under $\mathfrak{k}$ and whose irreducible $\mathfrak{k}$-components integrate to finite-dimensional irreducible representations of K is shown to be equivalent to a subquotient of a representation of $ \mathfrak{g}$ belonging to the infinitesimal nonunitary principal series. It follows that $\pi$ integrates to a continuous irreducible Hilbert space representation of G, and the best possible estimate for the multiplicity of any finite-dimensional irreducible representation of $\mathfrak{k}$ in $\pi$ is determined. These results generalize similar results of Harish-Chandra, R. Godement and J. Dixmier. The representations of $\mathfrak{g}$ in the infinitesimal nonunitary principal series, as well as certain more general representations of $ \mathfrak{g}$ on which the center of the universal enveloping algebra of $\mathfrak{g}$ acts as scalars, are shown to have (finite) composition series. A general module-theoretic result is used to prove that the distribution character of an admissible Hilbert space representation of G determines the existence and equivalence class of an infinitesimal composition series for the representation, generalizing a theorem of N. Wallach. The composition series of Weylgroup-related members of the infinitesimal nonunitary principal series are shown to be equivalent. An expression is given for the infinitesimal spherical functions associated with the nonunitary principal series. In several instances, the proofs of the above results and related results yield simplifications as well as generalizations of certain results of Harish-Chandra.
On the determination of irreducible modules by restriction to a subalgebra
J.
Lepowsky;
G. W.
McCollum
45-57
Abstract: Let $\mathcal{B}$ be an algebra over a field, $\mathcal{A}$ a subalgebra of $\mathcal{B}$, and $\alpha$ an equivalence class of finite dimensional irreducible $ \mathcal{A}$-modules. Under certain restrictions, bijections are established between the set of equivalence classes of irreducible $ \mathcal{B}$-modules containing a nonzero $\alpha$-primary $ \mathcal{A}$-submodule, and the sets of equivalence classes of all irreducible modules of certain canonically constructed algebras. Related results had been obtained by Harish-Chandra and R. Godement in special cases. The general methods and results appear to be useful in the representation theory of semisimple Lie groups.
Hankel transforms and GASP
Stanton
Philipp
59-72
Abstract: The inversion of the classical Hankel transform is considered from three viewpoints. The first approach is direct, and a theorem is given which allows inversion in the (C, 1) sense under fairly weak hypotheses. The second approach is via Abel summability, and it is shown that inversion is possible if it is known that the Hankel transform is Abel summable and if certain critical growth conditions are satisfied. The third approach rests on the observation that Abel means of Hankel transforms satisfy a variant of the GASP equation in two arguments. In this setting the inversion problem becomes a boundary value problem for GASP in a quadrant of the plane with boundary values on one of the axes; a uniqueness theorem for this problem is proved which is best possible in several respects.
Free vector lattices
Roger D.
Bleier
73-87
Abstract: An investigation into the algebraic properties of free objects in the category of vector lattices is carried out. It is shown that each ideal of a free vector lattice is a cardinal (direct) sum of indecomposable ideals, and that there are no nonzero proper characteristic ideals. Questions concerning injective and surjective endomorphisms are answered. Moreover, for finitely generated free vector lattices it is shown that the maximal ideals are precisely those which are both prime and principal. These results are preceded by an efficient review of the known properties of free vector lattices. The applicability of the theory to abelian lattice-ordered groups is discussed in a brief appendix.
Bibasic sequences and norming basic sequences
William J.
Davis;
David W.
Dean;
Bor Luh
Lin
89-102
Abstract: It is shown that every infinite dimensional Banach space X contains a basic sequence $({x_n})$ having biorthogonal functionals $({f_n}) \subset {X^\ast}$ such that $ ({f_n})$ is also basic. If $ [{f_n}]$ norms $ [{x_n}]$ then $ ({f_n})$ is necessarily basic. If $[{f_n}]$ norms $[{x_n}]$ then $[{x_n}]$ norms $[{f_n}]$. In order that $[{f_n}]$ norms $[{x_n}]$ it is necessary and sufficient that the operators ${S_n}x = \Sigma _1^n{f_i}(x){x_i}$ be uniformly bounded. If $[{f_n}]$ norms $[{x_n}]$ then ${X^\ast}$ has a complemented subspace isomorphic to $ {[{x_n}]^\ast}$. Examples are given to show that $({f_n})$ need not be basic and, if $({f_n})$ is basic, still $[{f_n}]$ need not norm $[{x_n}]$.
On weighted norm inequalities for the Lusin area integral
Carlos
Segovia;
Richard L.
Wheeden
103-123
Abstract: It is shown that the Lusin area integral for the unit circle is a bounded operator on any weighted ${L^p}$ space, $1 < p < \infty$, on which the conjugate function is a bounded operator. Results are also proved for the case $0 < p \leq 1$.
Codominant dimension of rings and modules
Gary L.
Eerkes
125-139
Abstract: Expanding Nakayama's original concept of dominant dimension, Tachikawa, Müller and Kato have obtained a number of results pertaining to finite dimensional algebras and more generally, rings and their modules. The purpose of this paper is to introduce and examine a categorically dual notion, namely, codominant dimension. Special attention is given to the question of the relation between the codominant and dominant dimensions of a ring. In particular, we show that the two dimensions are equivalent for artinian rings. This follows from our main result that for a left perfect ring R the dominant dimension of each projective left R-module is greater than or equal to n if and only if the codominant dimension of each injective left R-module is greater than or equal to n. Finally, for computations, we consider generalized uniserial rings and show that the codominant dimension, or equivalently, dominant dimension, is a strict function of the ring's Kupisch sequence.
Initial-boundary value problems for hyperbolic systems in regions with corners. I
Stanley
Osher
141-164
Abstract: In recent papers Kreiss and others have shown that initial-boundary value problems for strictly hyperbolic systems in regions with smooth boundaries are well-posed under uniform Lopatinskiĭ conditions. In the present paper the author obtains new conditions which are necessary for existence and sufficient for uniqueness and for certain energy estimates to be valid for such equations in regions with corners. The key tool is the construction of a symmetrizer which satisfies an operator valued differential equation.
On manifolds with the homotopy type of complex projective space
Bruce
Conrad
165-180
Abstract: It is known that in every even dimension greater than four there are infinitely many nonhomeomorphic smooth manifolds with the homotopy type of complex projective space. In this paper we provide an explicit construction of homotopy complex projective spaces. Our initial data will be a manifold X with the homotopy type of $ {\mathbf{C}}{{\mathbf{P}}^3}$ and an embedding ${\gamma _3}:{S^5} \to {S^7}$ . A homotopy 7-sphere ${\Sigma ^7}$ is constructed and an embedding ${\gamma _4}:{\Sigma ^7} \to {S^9}$ may be chosen. The procedure continues inductively until either an obstruction or the desired dimension is reached; in the latter case the final obstruction is the class of ${\Sigma ^{2n - 1}}$ in ${\Theta _{2n - 1}}$. Should this obstruction vanish, the final choice is of a diffeomorphism $ {\gamma _n}:{\Sigma ^{2n - 1}} \to {S^{2n - 1}}$. There results a manifold, denoted $(X,{\gamma _3}, \cdots ,{\gamma _{n - 1}},{\gamma _n})$, with the homotopy type of $ {\mathbf{C}}{{\mathbf{P}}^n}$. We describe the obstructions encountered, but are able to evaluate only the primary ones. It is shown that every homotopy complex projective space may be so constructed, and in terms of this construction, necessary and sufficient conditions for two homotopy complex projective spaces to be diffeomorphic are stated.
Groups whose homomorphic images have a transitive normality relation
Derek J. S.
Robinson
181-213
Abstract: A group G is a T-group if $H \triangleleft K \triangleleft G$ implies that $H \triangleleft G$, i.e. normality is transitive. A just non-T-group (JNT-group) is a group which is not a T-group but all of whose proper homomorphic images are T-groups. In this paper all soluble JNT-groups are classified; it turns out that these fall into nine distinct classes. In addition all soluble $ JN\bar T$-groups and all finite $JN\bar T$-groups are determined; here a group G is a $\bar T$-group if $H \triangleleft K \triangleleft L \leq G$ implies that $H \triangleleft L$. It is also shown that a finitely generated soluble group which is not a T-group has a finite homomorphic image which is not a T-group.
Inverse limits on graphs and monotone mappings
J. W.
Rogers
215-225
Abstract: In 1935, Knaster gave an example of an irreducible continuum (i.e. compact connected metric space) K which can be mapped onto an arc so that each point-preimage is an arc. The continuum K is chainable (or arc-like). In this paper it is shown that every one-dimensional continuum M is a continuous image, with arcs as point-preimages, of some one-dimensional continuum $ M'$. Moreover, if M is G-like, for some collection G of graphs, then $M'$ can be chosen to be G-like. A corollary is that every chainable continuum is a continuous image, with arcs as point-inverses, of a chainable (and hence, by a theorem of Bing, planar) continuum. These investigations give rise to the study of certain special types of inverse limit sequences on graphs.
Weighted norm inequalities for the conjugate function and Hilbert transform
Richard
Hunt;
Benjamin
Muckenhoupt;
Richard
Wheeden
227-251
Abstract: The principal problem considered is the determination of all non-negative functions $W(x)$ with period $2\pi$ such that $\displaystyle \int_{ - \pi }^\pi {\vert\tilde f(\theta ){\vert^p}W(\theta )\;d\... ...a \leq C} \;\int_{ - \pi }^\pi {\vert f(\theta ){\vert^p}W(\theta )\;d\theta }$ where $1 < p < \infty$, f has period $2\pi$, C is a constant independent of f, and $\tilde f$ is the conjugate function defined by $\displaystyle \tilde f(\theta ) = \mathop {\lim }\limits_{\varepsilon \to {0^ +... ...rt\phi \vert \leq \pi } {\frac{{f(\theta - \phi )\;d\phi }}{{2\tan \phi /2}}.}$ The main result is that $W(x)$ is such a function if and only if $\displaystyle \left[ {\frac{1}{{\vert I\vert}}\int_I {W(\theta )\;d\theta } } \... ...ert}}\int_I {{{[W(\theta )]}^{ - 1/(p - 1)}}d\theta } } \right]^{p - 1}} \leq K$ where I is any interval, $ \vert I\vert$ denotes the length of I and K is a constant independent of I. Various related problems are also considered. These include weak type results, the nonperiodic case, the discrete case, an application to weighted mean convergence of Fourier series, and an estimate for one of the functions in the Fefferman and Stein decomposition of functions of bounded mean oscillation.
The modulus of the boundary values of bounded analytic functions of several variables
Chester Alan
Jacewicz
253-261
Abstract: One necessary condition and one sufficient condition are given in order that a nonnegative function be the modulus of the boundary values of a bounded analytic function on the polydisc. As a consequence, a weak version of a theorem of F. Riesz is generalized to several variables. For special classes of functions several conditions are given which are equivalent to a function's being the modulus of the boundary values of a bounded analytic function. Finally, an algebraic structure is provided for these special classes of functions.
On the asymptotic representation of analytic solutions of first-order algebraic differential equations in sectors
Steven
Bank
263-283
Abstract: In this paper, we treat first-order algebraic differential equations whose coefficients belong to a certain type of function field. (Our results include as a special case, the case when the coefficients are rational functions.) In our main result, we obtain precise asymptotic representations for a broad class of solutions of such equations.
A necessary and sufficient condition for a ``sphere'' to separate points in euclidean, hyperbolic, or spherical space
J. E.
Valentine;
S. G.
Wayment
285-295
Abstract: The purpose of this paper is to give conditions wholly and explicitly in terms of the mutual distances of $n + 3$ points in n-space which are necessary and sufficient for two of the points to lie in the same or different components of the space determined by the sphere which is determined by $n + 1$ of the points. Thus in euclidean space we prove that if the cofactor $[{p_i}{p_j}^2]$ of the element ${p_i}{p_j}^2\;(i \ne j)$ in the determinant $ \vert{p_i}{p_j}^2\vert(i,j = 0,1, \cdots ,n + 2)$ is nonzero then ${p_i},{p_j}$ lie in the same or different components of ${E_n} - \Omega $ (where $\Omega$ denotes the sphere or hyperplane containing the remaining $n + 1$ points) if and only if $\operatorname{sgn} [{p_i}{p_j}^2] = {( - 1)^n}$ or ${( - 1)^{n + 1}}$, respectively. In hyperbolic space the result is: if the cofactor $[{\sinh ^2}\;{p_i}{p_j}/2]$ of the element ${\sinh ^2}\;{p_i}{p_j}/2\;(i \ne j)$ in the determinant $\vert{\sinh ^2}\;{p_i}{p_j}/2\vert(i,j = 0,1, \cdots ,n + 1)$ is nonzero then ${p_i},{p_j}$ lie in the same or different components of ${H_n} - \Omega $ (where $\Omega$ denotes the hyperplane, sphere, horosphere, or one branch of an equidistant surface containing the remaining $n + 1$ points) if and only if $\operatorname{sgn} [{\sinh ^2}\;{p_i}{p_j}/2] = {( - 1)^n}$ or $ {( - 1)^{n + 1}}$, respectively. For spherical space we obtain: if the cofactor $ [{\sin ^2}\;{p_i}{p_j}/2]$ of the element ${\sin ^2}\;{p_i}{p_j}/2\;(i \ne j)$ in the determinant $\vert{\sin ^2}\;{p_i}{p_j}/2\vert(i,j = 0,1, \cdots ,n + 2)$ is nonzero then ${p_i},{p_j}$ lie in the same or different components of ${S_n} - \Omega $ (where $\Omega$ denotes the sphere containing the remaining $n + 1$ points which may be an $(n - 1)$ dimensional subspace) if and only if $ \operatorname{sgn} [{\sin ^2}\;{p_i}{p_j}/2] = {( - 1)^n}$ or ${( - 1)^{n + 1}}$ respectively.
Locally $B\sp{\ast} $-equivalent algebras. II
Bruce A.
Barnes
297-303
Abstract: Let A be a locally ${B^\ast}$-equivalent Banach $ ^\ast$-algebra. Then A possesses a unique norm $\vert \cdot \vert$ with the property that $\vert{a^\ast}a\vert = \vert a{\vert^2}$ for all $a \in A$. Let B be the ${B^\ast}$-algebra which is the completion of A in the norm $\vert \cdot \vert$. In this paper it is shown that there exists a closed ${B^\ast}$-equivalent $^\ast$-ideal of A which contains the maximal GCR ideal of B. In particular, when B is a GCR algebra, then $A = B$.
On monotone matrix functions of two variables
Harkrishan
Vasudeva
305-318
Abstract: The theory of monotone matrix functions has been developed by K. Loewner; he first gives some necessary and sufficient conditions for a function to be a monotone matrix function of order n, and then, as a result of further deep investigations including questions of interpolation he arrives at the following criterion: A real-valued function $f(x)$ defined in (a, b) is monotone of arbitrary high order n if and only if it is analytic in (a, b), can be analytically continued onto the entire upper half-plane, and has there a nonnegative imaginary part. The problem of monotone operator functions of two real variables has recently been considered by A. Koranyi. He has generalized Loewner's theorem on monotone matrix functions of arbitrary high order n to two variables. We seek a theory of monotone matrix functions of two variables analogous to that developed by Loewner and show that a complete analogue to Loewner's theory exists in two dimensions.
The support of Mikusi\'nski operators
Thomas K.
Boehme
319-334
Abstract: A class of Mikusiński operators, called regular operators, is studied. The class of regular operators is strictly smaller than the class of all operators, and strictly larger than the class of all distributions with left bounded support. Regular operators have local properties. Lions' theorem of supports holds for regular operators with compact support. The fundamental solution to the Cauchy-Riemann equations is not regular, but the fundamental solution to the heat equation in two dimensions is regular and has support on a half-ray.
Purely inseparable, modular extensions of unbounded exponent
Linda Almgren
Kime
335-349
Abstract: Let K be a purely inseparable extension of a field k of characteristic $p \ne 0$. Sweedler has shown in [2, p. 403] that if K over k is of finite exponent, then K is modular over k if and only if K can be written as the tensor product of simple extensions of k. This paper grew out of an attempt to find an analogue to this theorem if K is of unbounded exponent over k. The definition of a simple extension is extended to include extensions of the form $ k[x,{x^{1/p}},{x^{1/{p^2}}}, \cdots ][{x^{1/{p^\infty }}}]$. If K is the tensor product of simple extensions, then K is modular. The converse, however, is not true, as several counterexamples in §4 illustrate. Even if we restrict $ [k:{k^p}] < \infty$, the converse is still shown to be false. Given K over k modular, we construct a field $\cap _{i = 1}^\infty k{K^{{p^i}}} \otimes M( = Q)$ that always imbeds in K where M is the tensor product of simple extensions in the old sense. In general $K \ne Q$. For K to be the tensor product of simple extensions, we need $K = Q$, and $ \cap _{i = 1}^\infty k{K^{{p^i}}} = k( \cap _{i = 1}^\infty {K^{{p^i}}})$. If for some finite N, $k{K^{{p^N}}} = k{K^{{p^{N + 1}}}}$, then we have (by Theorem 11) that $K = Q$. This finiteness condition guarantees that M is of finite exponent. Should $ \cap _{i = 1}^\infty k{K^{{p^i}}} = k$, then we would have the condition of Sweedler's original theorem. The counterexamples in §4 will hopefully be useful to others interested in unbounded exponent extensions. Of more general interest are two side theorems on modularity. These state that any purely inseparable field extension has a unique minimal modular closure, and that the intersection of modular extensions is again modular.
Monads of infinite points and finite product spaces
Frank
Wattenberg
351-368
Abstract: The notion of ``monad'' is generalized to infinite (i.e. non-near-standard) points in arbitrary nonstandard models of completely regular topological spaces. The behaviour of several such monad systems in finite product spaces is investigated and we prove that for paracompact spaces X such that $X \times X$ is normal, the covering monad $ \mu$ satisfies $\mu (x,y) = \mu (x) \times \mu (y)$ whenever x and y have the same ``order of magnitude.'' Finally, monad systems, in particular non-standard models of the real line, R, are studied and we show that in a minimal nonstandard model of R exactly one monad system exists and, in fact, $\mu (x) = \{ x\} $ if x is infinite.
On a functional calculus for decomposable operators and applications to normal, operator-valued functions
Frank
Gilfeather
369-383
Abstract: Whenever $A = {\smallint _\Lambda } \oplus A(\lambda )\mu (d\lambda )$ is a decomposable operator on a direct integral $H = {\smallint _\Lambda } \oplus H(\lambda )\mu (d\lambda )$ of Hilbert spaces and f is a function analytic on a neighborhood of $ \sigma (A)$, then we obtain that $ f(A(\lambda ))$ is defined almost everywhere and $f(A)(\lambda ) = f(A(\lambda ))$ almost everywhere. This relationship is used to study operators A, on a separable Hilbert space, for which some analytic function A is a normal operator. Two main results are obtained. Let f be an analytic function on a neighborhood of the spectrum of an operator A. If $ f''(z) \ne 0$ for all z in the spectrum of A and if $ f(A)$ is a normal operator, then A is similar to a binormal operator. It is known that a binormal operator is unitarily equivalent to the direct sum of a normal and a two by two matrix of commuting normal operators. As above if $ f(A)$ is normal and in addition, $ f(z) - {\zeta _0}$ has at most two roots counted to their multiplicity for each ${\zeta _0}$ in the spectrum of N, then A is a binormal operator.
Local finite cohesion
W. C.
Chewning
385-400
Abstract: Local finite cohesion is a new condition which provides a general topological setting for some useful theorems. Moreover, many spaces, such as the product of any two nondegenerate generalized Peano continua, have the local finite cohesion property. If X is a locally finitely cohesive, locally compact metric space, then the complement in X of a totally disconnected set has connected quasicomponents; connectivity maps from X into a regular ${T_1}$ space are peripherally continuous; and each connectivity retract of X is locally connected. Local finite cohesion is weaker than finite coherence [4], although these conditions are equivalent among planar Peano continua. Local finite cohesion is also implied by local cohesiveness [l2] in locally compact ${T_2}$ spaces, and a converse holds if and only if the space is also rim connected. Our study answers a question of Whyburn about local cohesiveness.
Cells and cellularity in infinite-dimensional normed linear spaces
R. A.
McCoy
401-410
Abstract: Certain concepts such as cells, cellular sets, point-like sets, and decomposition spaces are studied and related in normed linear spaces. The relationships between these concepts in general resemble somewhat the corresponding relationships in Euclidean space.
Embedding rings with a maximal cone and rings with an involution in quaternion algebras
Carl W.
Kohls;
William H.
Reynolds
411-419
Abstract: Sufficient conditions are given for an algebra over a totally ordered field F to be isomorphic to a subring of the algebra of quaternions over the real closure of F. These conditions include either the requirement that the nonnegative scalars form a maximal cone in the algebra, or that the algebra have an involution such that the scalars are the only symmetric elements. For many matrix algebras, the cone requirement alone is imposed.
Prehomogeneous vector spaces and varieties
Frank J.
Servedio
421-444
Abstract: An affine algebraic group G over an algebraically closed field k of characteristic 0 is said to act prehomogeneously on an affine variety W over k if G has a (unique) open orbit $o(G)$ in W. When W is the variety of points of a vector space V, $G \subseteq GL(V)$ and G acts prehomogeneously and irreducibly on V (We say an irreducibly prehomogeneous pair (G, V).), the following conditions are shown to be equivalent: 1. the existence of a nonconstant semi-invariant P in $ k[V] \cong S({V^\ast})$, 2. $(G',V)$ is not a prehomogeneous pair ($ G'$ is the commutator subgroup of G, a semisimple closed subgroup of G.), 3. if $ X \in o(G)$, then $ B \subseteq G$ acting prehomogeneously on W'' is shown to be sufficient for $G\backslash W$, the set of G-orbits in the affine variety W to be finite. These criteria are then applied to a class of irreducible prehomogeneous pairs (G, V) for which $ G'$ is simple and three further conjectures, one due to Mikio Sato, are stated.
Boundary values of solutions of elliptic equations satisfying $H\sp{p}$ conditions
Robert S.
Strichartz
445-462
Abstract: Let A be an elliptic linear partial differential operator with ${C^\infty }$ coefficients on a manifold ${\mathbf{\Omega }}$ with boundary ${\mathbf{\Gamma }}$. We study solutions of $Au = \sigma$ which satisfy the ${H^p}$ condition that ${\sup _{0 < t < 1}}{\left\Vert {u( \cdot ,t)} \right\Vert _p} < \infty$, where we have chosen coordinates in a neighborhood of ${\mathbf{\Gamma }}$ of the form ${\mathbf{\Gamma }} \times [0,1]$ with ${\mathbf{\Gamma }}$ identified with $ t = 0$. If A has a well-posed Dirichlet problem such solutions may be characterized in terms of the Dirichlet data $ u( \cdot ,0) = {f_0},{(\partial /\partial t)^j}u( \cdot ,0) = {f_j},j = 1, \cdots ,m - 1$ as follows: $ {f_0} \in {L^p}$ (or $\mathfrak{M}$ if $p = 1$) and $ {f_j} \in {\mathbf{\Lambda }}( - j;p,\infty ),j = 1, \cdots ,m$ . Here ${\mathbf{\Lambda }}$ denotes the Besov spaces in Taibleson's notation. If $m = 1$ then u has nontangential limits almost everywhere.
Banach spaces whose duals contain $l\sb{1}(\Gamma )$ with applications to the study of dual $L\sb{1}(\mu )$ spaces
C.
Stegall
463-477
Abstract: THEOREM I. If E is a separable Banach space such that $E'$ has a complemented subspace isomorphic to ${l_1}({\mathbf{\Gamma }})$ with ${\mathbf{\Gamma }}$ uncountable then $ E'$ contains a complemented, $ M({\mathbf{\Delta }})$, the Radon measures on the Cantor set. THEOREM II. If E is a separable Banach space such that $ E'$ has a subspace isomorphic to ${l_1}({\mathbf{\Gamma }})$ with ${\mathbf{\Gamma }}$ uncountable, then E contains a subspace isomorphic to ${l_1}$, THEOREM III. Let E be a Banach space. The following are equivalent: (i) $ E'$ is isomorphic to ${l_1}({\mathbf{\Gamma }})$; (ii) every absolutely summing operator on E is nuclear; (iii) every compact, absolutely summing operator on E is nuclear; (iv) if X is a separable subspace of E, then there exists a subspace Y such that $ X \subseteq Y \subseteq E$ and $Y'$ is isomorphic to ${l_1}$. THEOREM IV. If E is a $ {\mathcal{L}_\infty }$ space then (i) $E'$ is isomorphic to ${l_1}({\mathbf{\Gamma }})$ for some set ${\mathbf{\Gamma }}$ or (ii) $ E'$ contains a complemented subspace isomorphic to $M({\mathbf{\Delta }})$. COROLLARY. If E is a separable $ {\mathcal{L}_\infty }$ space, then $E'$ is (i) finite dimensional, or (ii) isomorphic to ${l_1}$, or (iii) isomorphic to $ M({\mathbf{\Delta }})$. COROLLARY. If $ {L_1}(\mu )$ is isomorphic to the conjugate of a separable Banach space, then ${L_1}(\mu )$ is isomorphic to $ {l_1}$ or $M({\mathbf{\Delta }})$.
Infinite compositions of M\"obius transformations
John
Gill
479-487
Abstract: A sequence of Möbius transformations $\{ {t_n}\} _{n = 1}^\infty $, which converges to a parabolic or elliptic transformation t, may be employed to generate a second sequence $\{ {T_n}\} _{n = 1}^\infty$ by setting $ {T_n} = {t_1} \circ \cdots \circ {t_n}$. The convergence behavior of $\{ {T_n}\}$ is investigated and the ensuing results are shown to apply to continued fractions which are periodic in the limit.
Uniqueness of Haar series which are $(C,\,1)$ summable to Denjoy integrable functions
William R.
Wade
489-498
Abstract: A Haar series $\Sigma \;{a_k}{\chi _k}$ satisfies Condition H if ${a_k}{\chi _k}/k \to 0$ uniformly as $k \to \infty$. We show that if such a series is (C, 1) summable to a Denjoy integrable function f, except perhaps on a countable subset of [0, l], then that series must be the Denjoy-Haar Fourier series of f.
On the multiplicative completion of certain basic sequences in $L\sp{p},$ $1<p<\infty $
Ben-Ami
Braun
499-508
Abstract: Boas and Pollard proved that given any basis $\{ {f_n}\} _{n = 1}^\infty $ for ${L^2}(E)$ one can delete the first k basis elements and then find a bounded measurable function M such that $\{ M{f_n}\} _{n = k + 1}^\infty$ is total in $ {L^2}(E)$, that is, the closure of the linear span of the set $\{ M{f_n}:n \geq k + 1\} $ is ${L^2}(E)$. We improve this result by weakening the hypothesis to accept bases of ${L^p}(E),1 < p < \infty$, and strengthening the conclusion to read serially total, that is, given any $f \in {L^2}(E)$ one can find a sequence of reals $\{ {a_n}\} _{n = k + 1}^\infty$ such that $ \Sigma _{n = k + 1}^\infty {a_n}M{f_n}$ converges to f in the norm. We also show that certain infinite deletions are possible.
Tensor products of group algebras
J. E.
Kerlin
1-36
Abstract: Let C be a commutative Banach algebra. A commutative Banach algebra A is a Banach C-algebra if A is a Banach C-module and $c \cdot (aa') = (c \cdot a)a'$ for all ${A_1}, \cdots ,{A_n}$ are commutative Banach C-algebras, then the C-tensor product ${A_1}{ \otimes _C} \cdots { \otimes _C}{A_n} \equiv D$ is defined and is a commutative Banach C-algebra. The maximal ideal space ${\mathfrak{M}_D}$ of D is identified with a closed subset of $ {\mathfrak{M}_{{A_1}}} \times \cdots \times {\mathfrak{M}_{{A_n}}}$ in a natural fashion, yielding a generalization of the Gelbaum-Tomiyama characterization of the maximal ideal space of ${A_1}{ \otimes _\gamma } \cdots { \otimes _\gamma }{A_n}$. If $C = {L^1}(K)$ and ${A_i} = {L^1}({G_i})$, for LCA groups K and $ {G_i},i = 1, \cdots ,n$, then the ${L^1}(K)$-tensor product D of $ {L^1}({G_1}), \cdots ,{L^1}({G_n})$ is uniquely written in the form $D = N \oplus {D_e}$, where N and $ {D_e}$ are closed ideals in D, ${L^1}(K) \cdot N = \{ 0\} $, and ${D_e}$ is the essential part of D, i.e. $ {D_e} = {L^1}(K) \cdot D$. Moreover, if $ {D_e} \ne \{ 0\}$, then $ {D_e}$ is isometrically $ {L^1}(K)$-isomorphic to ${L^1}({G_1}{ \otimes _K} \cdots { \otimes _K}{G_n})$, where $ {G_1}, \cdots ,{G_n}$ is a K-tensor product of ${G_1}, \cdots ,{G_n}$ with respect to naturally induced actions of K on ${G_1}, \cdots ,{G_n}$. The above theorems are a significant generalization of the work of Gelbaum and Natzitz in characterizing tensor products of group algebras, since here the algebra actions are arbitrary. The Cohen theory of homomorphisms of group algebras is required to characterize the algebra actions between group algebras. Finally, the space of multipliers $ {\operatorname{Hom}_{{L^1}(K)}}({L^1}(G),{L^\infty }(H))$ is characterized for all instances of algebra actions of ${L^1}(K)$ on ${L^1}(G)$ and ${L^1}(H)$, generalizing the known result when $ K = G = H$ and the module action is given by convolution.
Distance estimates and pointwise bounded density
A. M.
Davie;
T. W.
Gamelin;
J.
Garnett
37-68
Abstract: Let U be a bounded open subset of the complex plane, and let H be a closed subalgebra of ${H^\infty }(U)$, the bounded analytic functions on U. If E is a subset of $\partial U$, let ${L_E}$ be the algebra of all bounded continuous functions on U which extend continuously to E, and set $ {H_E} = H \cap {L_E}$. This paper relates distance estimates of the form $d(h,H) = d(h,{H_E})$, for all $h \in {L_E}$, to pointwise bounded density of $ {H_E}$ in H. There is also a discussion of the linear space $H + {L_E}$, which turns out often to be a closed algebra.
On arithmetical classifications of inaccessable cardinals and their applications
Géza
Fodor;
Attila
Máté
69-99
Abstract: Lately several authors, among them Fodor, Gaifman, Hanf, Keisler, Lévy and Tarski, dug out an interesting and unduly forgotten operation of Mahlo that, loosely speaking, from a sequence of ordinals discards all those that are easy to locate in this sequence. The purpose of these authors was to invent strengthenings and schemes for repetitions of this and similar operations and to study the properties of cardinals that can be discarded in this way when started with a specific class; for example, the class of all inaccessible cardinals. Our attempt here is to consider such schemes for repetitions of operations that can in a sense be described in an arithmetical way, which might also be called constructive; our investigations are akin to the problem of constructive description of possibly large segments of, say, the set of all countable ordinals. Some applications of our classifications scheme are exhibited, questions ranging from definability of inaccessible cardinals in terms of sets of lower ranks to incompactness theorems in infinitary languages. The paper is concluded with an algebraic-axiomatic type study of our scheme.
Modular permutation representations
L. L.
Scott
101-121
Abstract: A modular theory for permutation representations and their centralizer rings is presented, analogous in several respects to the classical work of Brauer on group algebras. Some principal ingredients of the theory are characters of indecomposable components of the permutation module over a p-adic ring, modular characters of the centralizer ring, and the action of normalizers of p-subgroups P on the fixed points of P. A detailed summary appears in [15]. A main consequence of the theory is simplification of the problem of computing the ordinary character table of a given centralizer ring. Also, some previously unsuspected properties of permutation characters emerge. Finally, the theory provides new insight into the relation of Brauer's theory of blocks to Green's work on indecomposable modules.
The existence of ${\rm Irr}(X)$
M. W.
Mislove
123-140
Abstract: If X is a compact totally ordered space, we obtain the existence of an irreducible semigroup with idempotents X, ${\text{Irr}}(X)$, with the property that any irreducible semigroup with idempotents X is the idempotent separating surmorphic image of ${\text{Irr}}(X)$. Furthermore, it is shown that the Clifford-Miller endomorphism on ${\text{Irr}}(X)$ is an injection when restricted to each $ \mathcal{H}$-class of ${\text{Irr}}(X)$. A construction technique for noncompact semigroups is given, and some results about the structure of such semigroups are obtained.
Infinite matroids
Samuel S.
Wagstaff
141-153
Abstract: Matroids axiomatize the related notions of dimension and independence. We prove that if S is a set with k matroid structures, then S is the union of k subsets, the ith of which is independent in the ith matroid structure, iff for every (finite) subset A of S, $\vert A\vert$ is not larger than the sum of the dimensions of A in the k matroids. A matroid is representable if there is a dimension-preserving imbedding of it in a vector space. A matroid is constructed which is not the union of finitely many representable matroids. It is shown that a matroid is representable iff every finite subset of it is, and that if a matroid is representable over fields of characteristic p for infinitely many primes p, then it is representable over a field of characteristic 0. Similar results for other kinds of representation are obtained.
The growth of subuniform ultrafilters
S.
Negrepontis
155-165
Abstract: Some of the results on the topology of spaces of uniform ultrafilters are applied to the space $\Omega ({\alpha ^ + })$ of subuniform ultrafilters (i.e., the set of ultrafilters which are $ \alpha$-uniform but not ${\alpha ^ + }$-uniform) on ${\alpha ^ + }$ when $\alpha$ is a regular cardinal. The main object is to find for infinite cardinals $\alpha$, such that $\alpha = {\alpha ^{\underbar{a}}}$, a topological property that separates the space $\beta (\Omega ({\alpha ^ + }))\backslash \Omega ({\alpha ^ + })$ (the growth of $\Omega ({\alpha ^ + })$) from the space $U({\alpha ^ + })$ of uniform ultrafilters on ${\alpha ^ + }$. Property ${\Phi _\alpha }$ fulfils this rôle defined for a zero-dimensional space X by the following condition: every nonempty closed subset of X of type at most $\alpha$ is not contained in the uniform closure of a family of $\alpha$ pairwise disjoint nonempty open-and-closed subsets of X. The ``infinitary'' properties of $ \Omega ({\alpha ^ + })$, as they are measured by $ {\Phi _\alpha }$, are more closely related to those of $ U(\alpha )$ than to those of $U({\alpha ^ + })$. A consequence of this topological separation is that the growth of $\Omega ({\alpha ^ + })$ is not homeomorphic to $U({\alpha ^ + })$ and, in particular, that $ \Omega ({\alpha ^ + })$ is not ${C^ \ast }$-embedded in the space $\Sigma ({\alpha ^ + })$ of $\alpha $-uniform ultrafilters on ${\alpha ^ + }$. These results are related to, and imply easily, the Aronszajn-Specker theorem: if $\alpha = \alpha^{\underbar{a}}$ then ${\alpha ^ + }$ is not a ramifiable cardinal. It seems possible that similar questions on the ${C^ \ast }$-embedding of certain spaces of ultrafilters depend on (and imply) results in partition calculus.
Cylindric algebras and algebras of substitutions
Charles
Pinter
167-179
Abstract: Several new formulations of the notion of cylindric algebra are presented. The class $ C{A_\alpha }$ of all cylindric algebras of degree $\alpha$ is shown to be definitionally equivalent to a class of algebras in which only substitutions (together with the Boolean $+ , \cdot$, and $-$) are taken to be primitive operations. Then $C{A_\alpha }$ is shown to be definitionally equivalent to an equational class of algebras in which only substitutions and their conjugates (together with $+ , \cdot$, and $- $) are taken to be primitive operations.
A characterization of $U\sb{3}(2\sp{n})$ by its Sylow $2$-subgroup
Robert L.
Griess
181-186
Abstract: We determine all the finite groups having a Sylow 2-subgroup isomorphic to that of $ {U_3}({2^n}),n \geq 3$. In particular, the only such simple groups are the ${U_3}({2^n})$.
Boundedly complete $M$-bases and complemented subspaces in Banach spaces
William J.
Davis;
Ivan
Singer
187-194
Abstract: Subsequences of boundedly complete M-bases need not be boundedly complete. An example of a somewhat reflexive space is given whose dual and one of whose factors fail to be somewhat reflexive. A geometric description of boundedly complete M-bases is given which is equivalent to the definitions of V. D. Milman and W. B. Johnson. Finally, certain M-bases for separable spaces give rise to proper complemented subspaces.
Averaging operators in $C(S)$ and lower semicontinuous sections of continuous maps
Seymour Z.
Ditor
195-208
Abstract: For certain kinds of compact Hausdorff spaces S, necessary and sufficient topological conditions are provided for determining if there exists a norm 1 projection of $ C(S)$ onto any given separable selfadjoint subalgebra A, the conditions being in terms of the decomposition that A induces on S. In addition, for arbitrary S and selfadjoint closed subalgebra A of $ C(S)$, some results on lower bounds for the norms of projections of $ C(S)$ onto A are obtained. An example is given which shows that the greatest lower bound of the projection norms need not be attained.
Class numbers of totally imaginary quadratic extensions of totally real fields
Judith S.
Sunley
209-232
Abstract: Let K be a totally real algebraic number field. This paper provides an effective constant $C(K,h)$ such that every totally imaginary quadratic extension L of K with ${h_L} = h$ satisfies $\vert{d_L}\vert < C(K,h)$ with at most one possible exception. This bound is obtained through the determination of a lower bound for $L(1,\chi )$ where $\chi$ is the ideal character of K associated to L. Results of Rademacher concerning estimation of L-functions near $s = 1$ are used to determine this lower bound. The techniques of Tatuzawa are used in the proof of the main theorem.
Projective groups of degree less than $4p/3$ where centralizers have normal Sylow $p$-subgroups
J. H.
Lindsey
233-247
Abstract: This paper proves the following theorem: Theorem 1. Let $ \bar G$ be a finite primitive complex projective group of degree n with a Sylow p-subgroup $\bar P$ of order greater than p for p prime greater than five. Let $n \ne p,n < 4p/3$, and if $p = 7,n \leqslant 8$. Then $p \equiv 1 \pmod 4,\bar P$ is a trivial intersection set, and for some nonidentity element $ \bar x\;in\;\bar G,C(\bar x)$ does not have a normal Sylow p-subgroup.
The structure of $n$-uniform translation Hjelmslev planes
David A.
Drake
249-282
Abstract: Affine or projective Hjelmslev planes are called 1-uniform (also strongly 1-uniform) if they are finite customary affine or projective planes. If $n > 1$, an n-uniform affine or projective Hjelmslev plane is a (finite) Hjelmslev plane $\mathfrak{A}$ with the following property: for each point P of $ \mathfrak{A}$, the substructure $^{n - 1}P$ of all neighbor points of P is an $(n - 1)$-uniform affine Hjelmslev plane. Associated with each point P is a sequence of neighborhoods $ ^1P \subset {\;^2}P \subset \cdots \subset {\;^n}P = \mathfrak{A}$. For $i < n,{\;^i}P$ is an i-uniform affine Hjelmslev plane under the induced incidence relation (for some parallel relation). Hjelmslev planes are called strongly n-uniform if they are n-uniform and possess one additional property; the additional property is designed to assure that the planes have epimorphic images which are strongly $(n - 1)$-uniform. Henceforth, assume that $ \mathfrak{A}$ is a strongly n-uniform translation (affine) Hjelmslev plane. Let $ {{(^i}P)^ \ast }$ denote the incidence structure $^iP$ together with the parallel relation induced therein by the parallel relation holding in $\mathfrak{A}$. Then for all positive integers $ i \leq n$ and all points P and Q of $ \mathfrak{A}$, ${{(^i}P)^ \ast }$ and ${{(^i}Q)^\ast}$ are isomorphic strongly i-uniform translation Hjelmslev planes. Let $^i\mathfrak{A}$ denote this common i-uniform plane; $ {{(^i}\mathfrak{A})_j}$, denote the ``quotient'' of $^i\mathfrak{A}$ modulo $^j\mathfrak{A}$. The invariant $ r = {p^x}$ of $\mathfrak{A}$ is the order of the ordinary translation plane ${{(^n}\mathfrak{A})_{n - 1}}$. Then the translation group of $ \mathfrak{A}$ is an abelian group with 2xk cyclic summands, k an integer $\leq n$; one calls k the width of $ \mathfrak{A}$. If $0 \leq j < i \leq n$, then ${{(^i}\mathfrak{A})_j}$ is a strongly $(i - j)$-uniform translation Hjelmslev plane; if also $j \geq k,{{(^i}\mathfrak{A})_j}$ and ${{(^{i - k}}\mathfrak{A})_{j - k}}$ are isomorphic. Then if $\mathfrak{A}(i)$ denotes ${{(^i}\mathfrak{A})_{i - 1}},\mathfrak{A}(1), \cdots ,\mathfrak{A}(n)$ is a periodic sequence of ordinary translation planes (all of order r) whose period is divisible by k. It is proved that if ${T_1}, \cdots ,{T_k}$ is an arbitrary sequence of translation planes with common order and if $ n \geq k$, then there exists a strongly n-uniform translation Hjelmslev plane $ \mathfrak{A}$ of width k such that $\mathfrak{A}(i) \cong {T_i}$ for $i \leq k$. The proof of this result depends heavily upon a characterization of the class of strongly n-uniform translation Hjelmslev planes which is given in this paper. This characterization is given in terms of the constructibility of the n-uniform planes from the $ (n - 1)$-uniform planes by means of group congruences.
Embedding theorems and generalized discrete ordered abelian groups
Paul
Hill;
Joe L.
Mott
283-297
Abstract: Let G be a totally ordered commutative group. For each nonzero element $g \in G$, let $L(g)$ denote the largest convex subgroup of G not containing g. Denote by $U(g)$ the smallest convex subgroup of G that contains g. The group G is said to be generalized discrete if $U(g)/L(g)$ is order isomorphic to the additive group of integers for all $g \ne 0$ in G. This paper is principally concerned with the structure of generalized discrete groups. In particular, the problem of embedding a generalized discrete group in the lexicographic product of its components, $U(g)/L(g)$, is studied. We prove that such an embedding is not always possible (contrary to statements in the literature). However, we do establish the validity of this embedding when G is countable. In case F is o-separable as well as countable, the structure of G is completely determined.
The Brauer group of graded Azumaya algebras
L. N.
Childs;
G.
Garfinkel;
M.
Orzech
299-326
Abstract: We study G-graded Azumaya R-algebras for R a commutative ring and G a finite abelian group, and a Brauer group formed by such algebras. A short exact sequence is obtained which relates this Brauer group with the usual Brauer group of R and with a group of graded Galois extensions of R. In case G is cyclic a second short exact sequence describes this group of graded Galois extensions in terms of the usual group of Galois extensions of R with group G and a certain group of functions on $ {\text{Spec}}(R)$.
Steenrod squares in spectral sequences. I
William M.
Singer
327-336
Abstract: We define two kinds of Steenrod operations on the spectral sequence of a bisimplicial coalgebra. We show these operations compatible with the differentials of the spectral sequence, and with the Steenrod squares defined on the cohomology of the total complex. We give a general rule for computing the operations on ${E_2}$.
Steenrod squares in spectral sequences. II
William M.
Singer
337-353
Abstract: We apply the results of the previous paper to three special cases. We obtain Steenrod operations on the change-of-rings spectral sequence, on the Eilenberg-Moore spectral sequence for the cohomology of classifying spaces, and on the Serre spectral sequence.
Integral representation of functions and distributions positive definite relative to the orthogonal group
A. E.
Nussbaum
355-387
Abstract: A continuous function f on an open ball B in ${R^N}$ is called positive definite relative to the orthogonal group $O(N)$ if f is radial and $\smallint \smallint f(x - y)\phi (x)\overline {\phi (y)} \;dx\;dy \geq 0$ for all radial $\phi \in C_0^\infty (B/2)$. It is shown that f is positive definite in B relative to $O(N)$ if and only if f has an integral representation $f(x) = \smallint {e^{ix \cdot t}}d{\mu _1}(t) + \smallint {e^{x \cdot t}}d{\mu _2}(t)$, where $ {\mu _1}$ and $ {\mu _2}$ are bounded, positive, rotation invariant Radon measures on $ {R^N}$ and ${\mu _2}$ may be taken to be zero if, in addition to f being positive definite relative to $O(N),\smallint \smallint f(x - y)( - \Delta \phi )(x)\phi (y)\;dx\;dy \geq 0$ for all radial $\phi \in C_0^\infty (B/2)$. Both conditions are satisfied if f is a radial positive definite function in B. Thus the theorem yields as a special case Rudin's theorem on the extension of radial positive definite functions. The result is extended further to distributions.
On functions positive definite relative to the orthogonal group and the representation of functions as Hankel-Stieltjes transforms
A. Edward
Nussbaum
389-408
Abstract: To every continuous function f on an interval $0 \leq x < a(0 < a \leq \infty )$ and every positive number $\nu$ associate the kernel $\displaystyle f(x,y) = \int_0^\pi {f({{({x^2} + {y^2} - 2xy\;\cos \;\theta )}^{1/2}}){{(\sin \;\theta )}^{2\nu - 1}}d\theta ,\quad 0 < x,y < a/2.}$ Let $ \Omega (z) = \Gamma (\nu + {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\ke... ...iptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}(z)$, where $ {J_{\nu - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}(z)$ is the Bessel function of index $\nu - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$. It is shown that f has an integral representation $f(x) = \smallint_{ - \infty }^\infty {\Omega (x\sqrt \lambda )d\gamma (\lambda )}$, where $\gamma$ is a finite, positive Radon measure on R, if and only if the kernel $f(x,y)$ is positive definite. If $\nu = (N - 1)/2$, where N is an integer $ \geq 2$, this condition is equivalent to ${f_N}(x) = f(\vert x\vert),\;x \in {R^N},\;\vert x\vert < \alpha$, is positive definite relative to the orthogonal group $O(N)$. The results of this investigation extend the preceding one of the author on functions positive definite relative to the orthogonal group. In particular they yield the result of Rudin on the extensions of radial positive definite functions.
Space-time processes, parabolic functions and one-dimensional diffusions
Tze Leung
Lai
409-438
Abstract: In this paper, we study the properties of the space-time process and of parabolic functions associated with a Markov process. Making use of these properties and the asymptotic behavior of the first passage probabilities near the boundary points, we prove certain theorems concerning when $ u(X(t),t)$ is a martingale, where $X(t)$ is a conservative regular one-dimensional diffusion with inaccessible boundaries. A characterization of the class of parabolic functions associated with classical diffusions is also obtained.
A generalized operational calculus developed from Fredholm operator theory
Jack
Shapiro;
Martin
Schechter
439-467
Abstract: Let A be a closed operator on the Banach space X. We construct an operator, \begin{displaymath}\begin{array}{*{20}{c}} {(\lambda - A){{R'}_\lambda }(A) = I ... ... {{{R'}_\lambda }(A)(\lambda - A) = I + {F_2}} \end{array} \end{displaymath} where ${F_1}$ and ${F_2}$ are bounded finite rank operators. $\lambda \in {\Phi _A}$ except for at most a countable set containing no accumulation point in ${\Phi _A}$. Let ${\sigma _\Phi }(A)$ be the complement of $ {\Phi _A}$, and let ${\sigma _\Phi }(A)$ and at $(\infty )$. We then use the operator, $ N(A - \lambda )$.
Projections and approximate identities for ideals in group algebras
Teng-Sun
Liu;
Arnoud
van Rooij;
Ju Kwei
Wang
469-482
Abstract: For a locally compact group G with property $({{\text{P}}_1})$, if there is a continuous projection of ${L^1}(G)$ onto a closed left ideal I, then there is a bounded right approximate identity in I. If I is further 2-sided, then I has a 2-sided approximate identity. The converse is proved for ${w^ \ast }$-closed left ideals. Let G be further abelian and let I be a closed ideal in ${L^1}(G)$. The condition that I has a bounded approximate identity is characterized in a number of ways which include (1) the factorability of I, (2) that the hull of I is in the discrete coset ring of the dual group, and (3) that I is the kernel of a closed element in the discrete coset ring of the dual group.
On the regularity of the Riemann function for hyperbolic equations
William L.
Goodhue
483-490
Abstract: In an earlier paper, A. Friedman demonstrated that the Riemann function for a strictly hyperbolic system with Gevrey coefficients was locally Gevrey of some higher order except along the bicharacteristics. By representing the Riemann function in terms of a progressing wave expansion, this result is extended beyond caustics.
An approximation theorem for biholomorphic functions on $D\sp{n}$
Joseph A.
Cima
491-497
Abstract: Let F be a biholomorphic mapping of the polydisk ${D^n}$ into ${{\mathbf{C}}^n}$. We construct a sequence of polynomial mappings $ \{ {P_j}\}$ such that each ${P_j}$ is subordinate to $ {P_{j + 1}}$, each $ {P_j}$ is subordinate to F and the ${P_j}$ converge uniformly on compacta to F. The polynomials ${P_j}$ are biholomorphic.
Lower semicontinuity of parametric integrals
Edward
Silverman
499-508
Abstract: It has been known for a long time that the usual two-dimensional parametric integrals in three-space are lower semicontinuous with respect to uniform convergence. In an earlier paper we saw that an easy argument extends this result to all parametric integrals generated by simply-convex integrands, with no restrictions on the dimension of the surfaces or the containing space. By using these techniques again, and generalizing to surfaces a result concerning convergent sequences of closed curves we show that a parametric integral generated by a parametric integrand which is convex in the Jacobians is lower semicontinuous with respect to uniform convergence provided all of the functions lie in a bounded subset of the Sobolev space $H_s^1$ where $s + 1$ exceeds the dimension of the parametric integral.
Equational bases and nonmodular lattice varieties
Ralph
McKenzie
1-43
Abstract: This paper is focused on equational theories and equationally defined varieties of lattices which are not assumed to be modular. It contains both an elementary introduction to the subject and a survey of open problems and recent work. The concept of a ``splitting'' of the lattice of lattice theories is defined here for the first time in print. These splittings are shown to correspond bi-uniquely with certain finite lattices, called ``splitting lattices". The problems of recognizing whether a given finite lattice is a splitting lattice, whether it can be embedded into a free lattice, and whether a given interval in a free lattice is atomic are shown to be closely related and algorithmically solvable. Finitely generated projective lattices are characterized as being those finitely generated lattices that can be embedded into a free lattice.
Some remarks concerning the varieties generated by the diamond and the pentagon
S. D.
Comer;
D. X.
Hong
45-54
Abstract: In 1945 M. P. Schützenberger exhibited two identities. He asserted that one provided an equational base for the diamond $ {M_3}$ and the other a base for the pentagon ${N_5}$. Recently Ralph McKenzie produced another equational base for ${N_5}$. In the present paper the authors modify McKenzie's idea to verify Schützenberger's assertion for ${M_3}$. They also show Schützenberger's claim about ${N_5}$ is false.
Topologies of closed subsets
Louis
Narens
55-76
Abstract: In this paper various topologies on closed subsets of a topological space are considered. The interrelationships between these topologies are explored, and several applications are given. The methods of proof as well as some intrinsic definitions assume a familiarity with A. Robinson's nonstandard analysis. E. Michael (Topologies of spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152-182), K. Kuratowski (Topology, Vols. I and II, Academic Press, New York, 1968), L. Vietoris (Berichezweiter Ordnung, Monatsh. Math.-Phys. 33 (1923), 49-62), and others have considered methods of putting topologies on closed subsets of a topological space. These topologies have the property that if the underlying topological space is compact then the topology of closed subsets is also compact. In general, however, these topologies of closed subsets are not compact. In this paper, a topology of closed subsets of a topological space is constructed that is always compact. This topology is called the compact topology and has many pleasant features. For closed subsets of compact Hausdorff spaces, this topology agrees with Vietoris' topology. For arbitrary spaces, there are interesting connections between the compact topology and topological convergence of subsets, including generalized versions of the Bolzano-Weierstrass theorem.
Real length functions in groups
Nancy
Harrison
77-106
Abstract: This paper is a study of the structure of a group G equipped with a 'length' function from G to the nonnegative real numbers. The properties that we require this function to satisfy are derived from Lyndon's work on groups with integer-valued functions. A real length function is a function which assigns to each $g \in G$ a nonnegative real number $\vert g\vert$ such that the following axioms are satisfied: \begin{displaymath}\begin{array}{*{20}{c}} {{{\text{A}}_0}.\quad \vert x\vert < ... ... \geq m\;{\text{imply}}\;c(x,z) \geq m.} \end{array} \end{displaymath} In this paper structure theorems are obtained for the cases when G is abelian and when G can be generated by two elements. We first prove that if G is abelian, then G is isomorphic to a subgroup of the additive group of the real numbers. Then we introduce a reduction process based on a generalized notion of Nielsen transformation. We apply this reduction process to finite sets of elements of G. We prove that if G can be generated by two elements, then G is either free or abelian.
Smooth embeddings of homologically similar manifolds
Dennis M.
Roseman
107-126
Abstract: We consider the situation where we have two smooth n-manifolds $N \subseteq M$ with ${H_\ast }(M,N) = 0$ and show that given a smooth embedding of N into some manifold Q we may, under suitable conditions, extend this to embeddings of M into Q, $Q \times I$, or $Q \times {I^2}$ (where I is the unit interval). We can apply these results to obtain smooth embeddings of homologically k-connected manifolds into $(2n - k + 1)$-dimensional euclidian space.
Normed convex processes
Stephen M.
Robinson
127-140
Abstract: We show that several well-known results about continuous linear operators on Banach spaces can be generalized to the wider class of convex processes, as defined by Rockafellar. In particular, the open mapping theorem and the standard bound for the norm of the inverse of a perturbed linear operator can be extended to convex processes. In the last part of the paper, these theorems are exploited to prove results about the stability of solution sets of certain operator inequalities and equations in Banach spaces. These results yield quantitative bounds for the displacement of the solution sets under perturbations in the operators and/or in the right-hand sides. They generalize the standard results on stability of unique solutions of linear operator equations.
Quasi-complemented algebras
T.
Husain;
Pak-Ken
Wong
141-154
Abstract: In this paper we introduce a class of algebras which we call quasi-complemented algebras. A structure and representation theory is developed. We also study the uniformly continuous quasi-complementors on ${B^\ast}$-algebras.
Cones and Vietoris-Begle type theorems
D. G.
Bourgin
155-183
Abstract: Infinite cone constructions are exploited to yield diverse generalizations of the Vietoris-Begle theorem for paracompact spaces and Abelian group sheaves. The constructions suggest natural space, map classifications designated as almost p-solid. The methods are extended to upper semicontinuous closed multivalued maps and homotopies and culminate in a disk fixed point theorem for possibly nonacyclic point images.
The Lefschetz fixed point theorem for noncompact locally connected spaces
R. J.
Knill
185-198
Abstract: Leray's notion of convexoid space is localized and used to show that if $ f:M \to M$ is a relatively compact map on a locally convex manifold M, and f has no fixed points then its Lefschetz trace is zero. A similar theorem holds for certain adjunction spaces $ Y{ \cup _g}Z$ where Y is Q-simplicial and Z is locally convexoid. A number of other properties of locally convexoid spaces are derived; for example, any neighborhood retract of a locally convexoid space is locally convexoid.
Near central automorphisms of abelian torsion groups
Jutta
Hausen
199-215
Abstract: This paper is concerned with the normal structure of the automorphism group $A(T)$ of an abelian torsion group T. The concept of the near center of a group is introduced in order to determine all subgroups of $A(T)$ the centralizer of which has finite index. Consequences are the fact that the finite normal subgroups of $A(T)$ are nilpotent if T is a primary group of infinite rank, and that every normal torsion subgroup of $A(T)$ is contained in the center of $A(T)$ if T is divisible.
Instability in ${\rm Diff}\sp{r}$ $(T\sp{3})$ and the nongenericity of rational zeta functions
Carl P.
Simon
217-242
Abstract: In the search for an easily-classified Baire set of diffeomorphisms, all the studied classes have had the property that all maps close enough to any diffeomorphism in the class have the same number of periodic points of each period. The author constructs an open subset U of $ {\text{Diff}^r}({T^3})$ with the property that if f is in U there is a g arbitrarily close to f and an integer n such that ${f^n}$ and ${g^n}$ have a different number of fixed points. Then, using the open set U, he illustrates that having a rational zeta function is not a generic property for diffeomorphisms and that $\Omega $-conjugacy is an ineffective means for classifying any Baire set of diffeomorphisms.
Singular homology as a derived functor
G. S.
Rinehart
243-256
Abstract: A general theory of relative derived tunctors is applied to the category of topological spaces to obtain singular homology and cohomology, verify the Eilenberg-Streenrod axioms, and show that singular and simplicial theory agree.
Semi-$p$-functions
J. F. C.
Kingman
257-273
Abstract: A generalisation of the theory of p-functions which applies, for instance, to the diagonal elements of one-parameter semigroups of infinite matrices which satisfy no boundedness condition.
Algebras of analytic germs
William R.
Zame
275-288
Abstract: Let S be a Stein-Riemann domain with global local coordinates $ {\sigma _1}, \cdots ,{\sigma _n}$. Let X be a compact subset of S. Denote by $ \mathcal{O}(X)$ the algebra of germs on X of functions analytic near X. A subalgebra of $ \mathcal{O}(X)$ containing the germs of ${\sigma _1}, \cdots ,{\sigma _n}$ and the constants is stable if it is closed under differen tiation with respect to the coordinates ${\sigma _1}, \cdots ,{\sigma _n}$. In this paper the relationship of a stable algebra to its spectrum is investigated. In general, there is no natural imbedding of the spectrum into a Stein manifold. We give necessary and sufficient conditions that such an imbedding exists, and show that a stable algebra whose spectrum admits such an imbedding has a simple description. More generally, we show that a stable algebra is determined by its spectrum. This leads to certain approximation theorems.
A pair of indices for function spaces on the circle
Colin
Bennett
289-304
Abstract: We give here some of the basic properties of the classes $\{ {\Phi _r}\}$, $\{ {\Psi _r}\} , - 1 < r < 1$, of dilation operators acting in rearrangement-invariant spaces $\mathfrak{X}$ on the circle It is shown that to each space $ \mathfrak{X}$ there correspond two numbers $\xi ,\eta$, called indices, which satisfy $0 \leq \eta \leq \xi \leq 1$; these numbers represent the rate of growth or decay of $\left\Vert {{\Psi _r}} \right\Vert$ as $r \to \pm 1$. By using the operators $ {\Psi _r}$ to obtain estimates for certain averaging operators ${A_\gamma }$, we are able to show that the indices $(\xi ,\eta )$ coincide with the Boyd indices $(\alpha ,\beta )$. As a consequence, we obtain a Marcinkiewicz-type interpolation theorem for rearrangement-invariant spaces on the circle.
Constructive proof of Hilbert's theorem on ascending chains
A.
Seidenberg
305-312
Abstract: In a previous note it was shown that if a bound $f(i)$ is placed on the degrees of the elements in some basis of an ideal ${A_i}$ in the polynomial ring $k[{X_1}, \cdots ,{X_n}]$ over an explicitly given field $k,i = 0,1,2, \cdots$, then a bound can be (and was) constructed for the length of a strictly ascending chain ${A_0} < {A_1} < \cdots$. This result is now obtained using a strictly finitist argument. A corollary is a finitist version of Hilbert's theorem on ascending chains.
Strictly regular elements in Freudenthal triple systems
J. C.
Ferrar
313-331
Abstract: Strictly regular elements play a role in the structure theory of Freudenthal triple systems analogous to that played by idempotents in nonassociative algebras with identity. In this paper we study the coordinatization of reduced triple systems relative to a connected pair of strictly regular elements and use the explicit form of strictly regular elements in terms of the coordinatization to prove uniqueness of the coordinatizing Jordan algebra, as well as several generalizations of known results regarding groups of transformations related to triple systems. Finally, we classify forms of a particularly important triple system (the representation module for the Lie algebra ${E_7}$) over finite, p-adic or real fields.
The $L\sp{p}$ behavior of eigenfunction expansions
Harold E.
Benzinger
333-344
Abstract: We investigate the extent to which the eigenfunction expansions arising from a large class of two-point boundary value problems behave like Fourier series expansions in the norm of $ {L^p}(0,1),1 < p < \infty$. We obtain our results by relating Green's function to the Hilbert transform.
Adequate ultrafilters of special Boolean algebras
S.
Negrepontis
345-367
Abstract: In his paper Good ideals in fields of sets Keisler proved, with the aid of the generalized continuum hypothesis, the existence of countably incomplete, ${\beta ^ + }$-good ultrafilters on the field of all subsets of a set of (infinite) cardinality $ \beta$. Subsequently, Kunen has proved the existence of such ultrafilters, without any special set theoretic assumptions, by making use of the existence of certain families of large oscillation. In the present paper we succeed in carrying over the original arguments of Keisler to certain fields of sets associated with the homogeneous-universal (and more generally with the special) Boolean algebras. More specifically, we prove the existence of countably incomplete, a-good ultrafilters on certain powers of the a-homogeneous-universal Boolean algebras of cardinality a and on the a-completions of the a-homogeneous-universal Boolean algebras of cardinality a, where $a = a^{[unk]} > \omega$. We then develop a method that allows us to deal with the special Boolean algebras of cardinality $a = 2^{[unk]}$. Thus, we prove the existence of an ultrafilter p (which will be called adequate) on certain powers $\mathcal{S}_\alpha ^\delta $ of the special Boolean algebra $ {\mathcal{S}_\alpha }$ of cardinality a, and the existence of a specializing chain $\{ {\mathcal{C}_\beta }:\beta < \alpha \}$ for $ {\mathcal{S}_\alpha }$, such that $\mathcal{C}_\beta ^\delta \cap p$ is ${\beta ^ + }$-good and countably incomplete for $\beta < \alpha$. The corresponding result on the existence of adequate ultrafilters on certain completions of the special Boolean algebras is more technical. These results do not use any part of the generalized continuum hypothesis.
A generalization of univalent functions with bounded boundary rotation
Edward J.
Moulis
369-381
Abstract: This paper introduces a class of functions which generalizes both those functions $f(z)$ with bounded boundary rotation and those functions for which $zf'(z)$ is a-spirallike. A simple variational formula for this class is derived and used to determine sufficient conditions for the univalency of functions there in. Various representations for these functions are given, and these are used to derive another condition for univalence; this one is the best known so far in the subclass consisting of functions $f(z)$ for which $zf'(z)$ is a-spirallike. Bounds on the modulus of the Schwarzian derivative are also derived; these are sharp in the subclass of functions having bounded boundary rotation.
$Q$-composable properties, semigroups and ${\rm CM}$-homomorphisms
A. R.
Bednarek;
K. D.
Magill
383-398
Abstract: A certain type of topological property is investigated. To each such property and each topological space satisfying various conditions there is associated, in a natural way, a semigroup of relations. The nonconstant, union and symmetry preserving homomorphisms from one such semigroup into another are completely determined and this results in a topological version of the Clifford-Miller Theorem on endomorphisms of the full binary relation semigroup on a set.
Equicontinuity and indivisibility in transformation groups
Ping-fun
Lam
399-424
Abstract: A notion called indivisibility for transformation groups is studied in some detail. The proofs of some theorems announced in [10] are given. Certain types of nonequicontinuous sets which do not disconnect their metric continuum phase spaces and which were not known to exist previously are exhibited.
Rings which are almost polynomial rings
Paul
Eakin;
James
Silver
425-449
Abstract: If A is a commutative ring with identity and B is a unitary A-algebra, B is locally polynomial over A provided that for every prime p of A, $ {B_p} = B{ \otimes _A}{A_p}$ is a polynomial ring over ${A_p}$. For example, the ring $Z[\{ X/{p_i}\} _{i = 1}^\infty ]$, where $ \{ {p_i}\} _{i = 1}^\infty$ is the set of all primes of Z, is locally polynomial over Z, but is not a polynomial ring over Z. If B is locally polynomial over A, the following results are obtained, B is faithfully flat over A. If A is an integral domain, so is B. If $ \mathfrak{a}$ is any ideal of A, then $ B/\mathfrak{a}B$ is locally polynomial over $ A/\mathfrak{a}$. If p is any prime of A, then pB is a prime of B. If B is a Krull ring, so is A and the class group of B is isomorphic to the class group of A . If A is a Krull ring and B is contained in an affine domain over A, then B is a Krull ring. If A is a noetherian normal domain and B is contained in an affine ring over A, then B is a normal affine ring over A. If M is a module over a ring A, the content of an element x of M over A is defined to be the smallest ideal ${A_x}$ of A such that x is in $ {A_x}M$. A module is said to be a content module over A if ${A_x}$ exists for every x in M. M is a content module over A if and only if arbitrary intersections of ideals of A extend to M. Projective modules are content modules. If B is locally polynomial over a Dedekind domain A, then B is a content module over A if and only if B is Krull.
Valuations, primes and irreducibility in polynomial rings and rational function fields
Ron
Brown
451-488
Abstract: The set of extensions of the valuation v on a linearly compact (i.e. maximal) field F to the polynomial ring $ F[x]$ is shown to depend only on the value group and residue class field of v. By a method related to Mac Lane's construction of (rank one) valuations on polynomial rings, a determining invariant is associated with each such extension, called its ``signature". Very roughly, a signature is a pair of sequences, one in the algebraic closure of the residue class field of v and one in the divisible closure of the value group of v. Signatures are also associated with various mathematical objects by means of the extensions of the above sort which naturally arise from them. For example, the set of nonconstant monic irreducible polynomials in $F[x]$, the set of all finite Harrison primes of the polynomial ring of a global field, and the set of equivalence classes of valuations on the field of rational functions over a global field are each shown to be bijective with a simple set of signatures. Moreover, these objects are studied by means of their associated signatures. For example, necessary and sufficient conditions for irreducibility in $ F[x]$ are given, independent of the language of signatures.
Generalized eigenfunctions and real axis limits of the resolvent
N. A.
Derzko
489-506
Abstract: Let $(\mathcal{H},( \cdot , \cdot ))$ be a Hilbert space and A, E be a selfadjoint operator and corresponding spectral measure in $ \mathcal{H}\;(A = \smallint \lambda E(d\lambda ))$. It is known that for a suitable positive subspace ${\mathcal{H}_ + } \subset \mathcal{H}$ and measure $ \rho$ the generalized eigenfunctions $\displaystyle {\phi _{\lambda ,f}} = \mathop {\lim }\limits_{h \to 0} \frac{{E(... ...hop {\lim }\limits_{\Delta \to \lambda } \frac{{E(\Delta )f}}{{\rho (\Delta )}}$ exist in $ {\mathcal{H}_ - }$, the corresponding negative space, for $\rho$-almost every $\lambda$ and $f \in {\mathcal{H}_ + }$. It is shown that for each $ \lambda$ the ${\phi _{\lambda ,f}}$ form a pre-Hilbert space $ {\mathcal{H}_\lambda }$ using the natural inner product $ {({\phi _f},{\phi _g})_\lambda } = {\lim _{\Delta \to \lambda }}((E(\Delta )f,g)/\rho (\Delta ))$, and that $ \left\Vert \phi \right\Vert - \leq C{\left\Vert \phi \right\Vert _\lambda }$. Furthermore, if $\{ \phi (\lambda ,\alpha )\}$ is a suitably chosen basis for $ {\mathcal{H}_\lambda }, - \infty < \lambda < \infty$, then one obtains the eigenfunction expansion suggested by $\displaystyle (f,g) = \int {\rho (d\lambda )\;\sum\limits_{\alpha ,\beta } {(f,... ...)){\sigma _{\alpha \beta }}(\lambda )\overline{(g,\phi (\lambda ,\beta )).}} }$ . Finally it is shown that, for a suitable function $w(\varepsilon ,\lambda ),{\phi _{\lambda ,f}}$ is given by ${\lim _{\varepsilon \downarrow 0}}w(\varepsilon ,\lambda )[R(\lambda - i\varepsilon ) - R(\lambda + i\varepsilon )]f$, where $R(z) = {(z - A)^{ - 1}}$.
Erratum to: ``On character sums and power residues''
Karl K.
Norton
507
Mean Ces\`aro summability of Laguerre and Hermite series
Eileen L.
Poiani
1-31
Abstract: The primary purpose of this paper is to prove inequalities of the type $\vert\vert{\sigma _n}(f,x)W(x)\vert{\vert _p} \leqslant C\vert\vert f(x)W(x)\vert{\vert _p}$ where ${\sigma _n}(f,x)$ is the $ n$th $(C,1)$ mean of the Laguerre or Hermite series of $f, W(x)$ is a suitable weight function of particular form, $C$ is a constant independent of $ f(x)$ and $n$, and the norm is taken over $(0,\infty )$ in the Laguerre case and $( - \infty ,\infty )$ in the Hermite case for $1 \leqslant p \leqslant \infty$. Both necessary and sufficient conditions for these inequalities to remain valid are determined. For $p < \infty$ and $f(x)W(x) \in {L^p}$, mean summability results showing that $ \mathop {\lim }\nolimits_{n \to \infty } \vert\vert[{\sigma _n}(f,x) - f(x)]W(x)\vert{\vert _p} = 0$ are derived by use of the appropriate density theorems. Detailed proofs are presented for the Laguerre expansions, and the analogous results for Hermite series follow as corollaries.
$\Pi \sp{0}\sb{1}$ classes and degrees of theories
Carl G.
Jockusch;
Robert I.
Soare
33-56
Abstract: Using the methods of recursive function theory we derive several results about the degrees of unsolvability of members of certain $\Pi _1^0$ classes of functions (i.e. degrees of branches of certain recursive trees). As a special case we obtain information on the degrees of consistent extensions of axiomatizable theories, in particular effectively inseparable theories such as Peano arithmetic, ${\mathbf{P}}$. For example: THEOREM 1. If a degree $ {\mathbf{a}}$ contains a complete extension of $ {\mathbf{P}}$, then every countable partially ordered set can be embedded in the ordering of degrees $\leqslant {\mathbf{a}}$. (This strengthens a result of Scott and Tennenbaum that no such degree ${\mathbf{a}}$ is a minimal degree.) THEOREM 2. If $ {\mathbf{T}}$ is an axiomatizable, essentially undecidable theory, and if $ \{ {{\mathbf{a}}_n}\}$ is a countable sequence of nonzero degrees, then ${\mathbf{T}}$ has continuum many complete extensions whose degrees are pairwise incomparable and incomparable with each ${{\mathbf{a}}_n}$. THEOREM 3. There is a complete extension $ {\mathbf{T}}$ of ${\mathbf{P}}$ such that no nonrecursive arithmetical set is definable in $ {\mathbf{T}}$. THEOREM 4. There is an axiomatizable, essentially undecidable theory $ {\mathbf{T}}$ such that any two distinct complete extensions of ${\mathbf{T}}$ are Turing incomparable. THEOREM 5. The set of degrees of consistent extensions of ${\mathbf{P}}$ is meager and has measure zero.
Automorphisms of $\omega \sb{1}$-trees
Thomas J.
Jech
57-70
Abstract: The number of automorphisms of a normal $ {\omega _1}$-tree $ T$, denoted by $\sigma (T)$, is either finite or ${2^{{\aleph _0}}} \leqslant \sigma (T) \leqslant {2^{{\aleph _1}}}$. Moreover, if $ \sigma (T)$ is infinite then $\sigma {(T)^{{\aleph _0}}} = \sigma (T)$. Moreover, if $T$ has no Suslin subtree then $\sigma (T)$ is finite or $\sigma (T) = {2^{{\aleph _0}}}$ or $\sigma (T) = {2^{{\aleph _1}}}$. It is consistent that there is a Suslin tree with arbitrary precribed $\sigma (T)$ between ${2^{{\aleph _0}}}$ and ${2^{{\aleph _1}}}$, subject to the restriction above; e.g. $ {2^{{\aleph _0}}} = {\aleph _1},{2^{{\aleph _1}}} = {\aleph _{324}}$ and $\sigma (T) = {\aleph _{17}}$. We prove related results for Kurepa trees and isomorphism types of trees. We use Cohen's method of forcing and Jensen's techniques in $L$.
Iterated fine limits and iterated nontangential limits
Kohur
Gowrisankaran
71-92
Abstract: Let ${\Omega _k},k = 1{\text{ to }}n$, be harmonic spaces of Brelot and ${u_k} > 0$ harmonic functions on ${\Omega _k}$. For each $w$ in a class of multiply superharmonic functions it is shown that the iterated fine limits of $[w/{u_1} \cdots {u_n}]$ exist up to a set of measure zero for the product of the canonical measures corresponding to ${u_k}$ and are independent of the order of iteration. This class contains all positive multiply harmonic functions on the product of ${\Omega _k}$'s. For a holomorphic function $ f$ in the Nevanlinna class of the polydisc ${U^n}$, it is shown that the $n$th iterated fine limits exist and equal almost everywhere on ${T^n}$ the $n$th iterated nontangential limits of $ f$, for any fixed order of iteration. It is then deduced that, with the exception of a set of measure zero on ${T^n}$, the absolute values of the different iterated limits of $f$ are equal. It is also shown that the $ n$th iterated nontangential limits are equal almost everywhere on ${T^n}$ for any $f$ in $ {N_1}({U^n})$.
The exceptional subset of a $C\sb{0}$-contraction
Domingo A.
Herrero
93-115
Abstract: Let $T$ be a ${C_0}$-operator acting on a (complex separable) Hilbert space $ \mathcal{K}$; i.e., $ T$ is a contraction on $\mathcal{K}$ and it satisfies the equation $ q(T) = 0$ for some inner function $q$, where $q(T)$ is defined in the sense of the functional calculus of B.Sz.-Nagy and C. Foiaş. Among all those inner functions $q$ there exists a unique minimal function $ p$ defined by the conditions: (1) $p(T) = 0$; (2) if $q(T) = 0$, then $p$ divides $q$. A vector $ F \in \mathcal{K}$ is called exceptional if there exists an inner function $r$ such that $r(T)F = 0$, but $p$ does not divide $r$. The existence of nonexceptional vectors plays a very important role in the theory of $ {C_0}$-operators. The main result of this paper says that nonexceptional vectors actually exist; moreover, the exceptional subset of a $ {C_0}$-operator is a topologically small subset of $ \mathcal{K}$.
Analytically invariant and bi-invariant subspaces
Domingo Antonio
Herrero;
Norberto
Salinas
117-136
Abstract: The purpose of this paper is to call attention to some interesting weakly closed algebras related to a bounded linear operator $ T$ acting on a Banach space $\mathfrak{X}$ and their associated lattices of invariant subspaces, namely, the algebras generated by the polynomials and by the rational functions in $ T$, and the commutant and the double-commutant of $T$. The relationship between those algebras and their lattices, as well as the ones corresponding to the operators induced by $T$ on an invariant subspace (restriction), or on the quotient space $ \mathfrak{X}/\mathfrak{M}$ (where $ \mathfrak{M}$ is an invariant subspace of a given type) are analyzed. Several results relative to the decomposition of invariant subspaces and the topological structure of the lattices (under the ``gap-between-subspaces'' metric topology) are also considered.
Localizations of HNP rings
James
Kuzmanovich
137-157
Abstract: In this paper it is shown that every hereditary Noetherian prime ring is the intersection of a hereditary Noetherian prime ring having no invertible ideals with a bounded hereditary Noetherian prime ring in which every nonzero two-sided ideal contains an invertible two-sided ideal. Further, it is shown that this intersection corresponds to a decomposition of torsion modules over such a ring; if $R$ is an HNP ring with enough invertible ideals, then this decomposition coincides with that of Eisenbud and Robson. If $M$ is a maximal invertible ideal of $ R$ where $R$ is as above, then an overring of $ R$ is constructed which is a localization of $R$ at $M$ in a ``classical sense"; that is, it is a ring of quotients with respect to a multiplicatively closed set of regular elements satisfying the Ore conditions. The localizations are shown to have nonzero radical and are also shown to satisfy a globalization theorem. These localizations are generalizations of ones constructed by A. V. Jategaonkar for HNP rings with enough invertible ideals.
On the classification of simple antiflexible algebras
Mahesh Chandra
Bhandari
159-181
Abstract: In this paper, we begin a classification of simple totally antiflexible algebras (finite dimensional) over splitting fields of characteristic $\ne 2,3$. For such an algebra $A$ let $P$ be the largest associative ideal in $ {A^ + }$ and let $ N$ be the radical of $ P$. We say that $ A$ is of type $ (m,n)$ if $N$ is nilpotent of class $ m$ with $\dim A = n$. Define ${N_i} = {N_{i - 1}} \cdot N,{N_1} = N$, then $ A$ is said to be of type $ (m,n,{d_1},{d_2}, \cdots ,{d_q})$ if $A$ is of type $(m,n),\dim ({N_i} - {N_{i - 1}}) = {d_i}$ for $1 \leqslant i \leqslant q$ and $\dim ({N_i} - {N_{i + 1}}) = 1$ for $ q < i < m$. We then determine all nodal simple totally antiflexible algebras of types $(n,n),(n - k,n,k + 1),(n - 2,n)$ (over fields of characteristic $\ne 2,3$) and of type (3, 6) (over the field of complex numbers). We also give preliminary results for nodal simple totally antiflexible algebras of type $(n - k,n,k,2)$ and of type $(m,n,{d_1}, \cdots ,{d_q})$ in general with $m > 2$ (the case $m = 2$ has been classified by D. J. Rodabaugh).
On bounded oscillation and asymptotic expansion of conformal strip mappings
Arthur E.
Obrock
183-201
Abstract: Relations between the boundary parameters ${\phi _ - },{\phi _ + }$ of a strip $S = \{ {\phi _ - }(x) < y < {\phi _ + }(x)\}$ and the values $f(x)$ of its canonical conformal mapping onto a horizontal strip $H = \{ \vert\upsilon \vert < 1\}$ are studied. Bounded oscillation $({\max _y}\operatorname{Re} f(x + iy) - {\min _y}\operatorname{Re} f(x + iy) = \omega (x) = O(1))$ is characterized in terms of ${\phi _ - },{\phi _ + }$. A formal series expansion $ \upsilon = \sum {y^m}{a_{m,n}}(x)$ is derived for the solution to the Dirichlet problem on $S$ and its partial sums are used to obtain formulas for the asymptotic expansion of $f$ in terms of ${\phi _ + },{\phi _ - }$.
On the genus of a group
Arthur T.
White
203-214
Abstract: The genus of a group is defined to be the minimum genus for any Cayley color graph of the group. All finite planar groups have been determined, but little is known about the genus of finite nonplanar groups. In this paper two families of toroidal groups are presented; the genus is calculated for certain abelian groups; and upper bounds are given for the genera of the symmetric and alternating groups and for some hamiltonian groups.
Actions of groups of order $pq$
Connor
Lazarov
215-230
Abstract: We study the bordism group of stably complex $G$-manifolds in the case where $G$ is a metacyclic group of order $ pq$ and $p$ and $q$ are distinct primes. This bordism group is a module over the complex bordism ring and we compute the projective dimension of this module. We develop some techniques necessary for the study of this module in case $ G$ is a more general metacyclic group.
The equivariant Plateau problem and interior regularity
H. Blaine
Lawson
231-249
Abstract: Let $M \subset {{\text{R}}^n}$ be a compact submanifold of Euclidean space which is invariant by a compact group $G \subset SO(n)$. When $\dim (M) = n - 2$, it is shown that there always exists a solution to the Plateau problem for $M$ which is invariant by $G$ and, furthermore, that uniqueness of this solution among $G$-invariant currents implies uniqueness in general. This result motivates the subsequent study of the Plateau problem for $M$ within the class of $G$-invariant integral currents. It is shown that this equivariant problem reduces to the study of a corresponding Plateau problem in the orbit space ${\text{R}}/G$ where, for ``big'' groups, questions of uniqueness and regularity are simplified. The method is then applied to prove that for a constellation of explicit manifolds $M$, the cone $ C(M) = \{ tx;x \in M$ and $ 0 \leqslant t \leqslant 1\}$ is the unique solution to the Plateau problem for $ M$, (Thus, there is no hope for general interior regularity of solutions in codimension one.) These manifolds include the original examples of type $ {S^n} \times {S^n} \subset {{\text{R}}^{2n + 2}},n \geqslant 3$, due to Bombieri, DeGiorgi, Giusti and Simons. They also include a new example in $ {{\text{R}}^8}$ and examples in $ {{\text{R}}^n}$ for $n \geqslant 10$ with any prescribed Betti number nonzero.
A class of representations of the full linear group. II
Stephen
Pierce
251-262
Abstract: Let $V$ be an $n$-dimensional vector space over complex numbers $ C$. Let $W$ be the $m$th tensor product of $V$. If $T \in {\operatorname{Hom} _C}(V,V)$, let ${ \otimes ^m}T \in {\operatorname{Hom} _C}(W,W)$ be the $m$th tensor product of $T$. The homomorphism $T \to { \otimes ^m}T$ is a representation of the full linear group $ {\text{G}}{{\text{L}}_n}(C)$. If $H$ is a subgroup of the symmetric group $ {S_m}$, and $\chi$ a linear character on $ H$, let $V_\chi ^m(G)$ be the subspace of $W$ consisting of all tensors symmetric with respect to $H$ and $\chi$. Then $ V_\chi ^m(H)$ is invariant under ${ \otimes ^m}T$. Let $K(T)$ be the restriction of ${ \otimes ^m}T$ to $ V_\chi ^m(H)$. For $ n$ large compared with $ m$ and for $H$ transitive, we determine all cases when the representation $T \to K(T)$ is irreducible.
On the null-spaces of elliptic partial differential operators in $R\sp{n}$
Homer F.
Walker
263-275
Abstract: The objective of this paper is to generalize the results of Lax and Phillips [4] and Walker [6] to include elliptic partial differential operators of all orders whose coefficients approach constant values at infinity with a certain swiftness. An example is given of an elliptic operator having an infinite-dimensional null-space whose coefficients slowly approach constant limiting values.
Asymptotic behavior of linear integrodifferential systems
Viorel
Barbu;
Stanley I.
Grossman
277-288
Abstract: We consider the system $n \times n$ matrices. System $({\text{L)}}$ generates a semigroup given by ${T_t}f(s) = y(t + s;f)$ for $f$ bounded, continuous and having a finite limit at $- \infty$. Under hypotheses concerning the roots of $\det (\lambda I - A - \hat B(\lambda ))$, where $\hat B(\lambda )$ is the Laplace transform, various results about the asymptotic behavior of $y(t)$ are derived, generally after invoking the Hille-Yosida theorem. Two typical results are Theorem 1. If $B(t) \in {L^1}[0,\infty )$ and $ {(\lambda I - A - \hat B(\lambda ))^{ - 1}}$ exists for $\operatorname{Re} \lambda > 0$, then for every $\epsilon > 0$, there is an ${M_{\epsilon}}$ such that $\vert\vert{T_t}f\vert\vert \leqslant {M_{\epsilon}}{e^{\epsilon t}}\vert\vert f\vert\vert$. Theorem 2. If ${(\lambda I - A - \hat B(\lambda ))^{ - 1}}$ exists for $\operatorname{Re} \lambda > - \alpha (\alpha > 0)$ and if $B(t){e^{\alpha t}} \in {L^1}[0,\infty )$, then the solution to $ ({\text{L)}}$ is exponentially asymptotically stable.
On the semisimplicity of group rings of solvable groups
C. R.
Hampton;
D. S.
Passman
289-301
Abstract: Let $K[G]$ denote the group ring of $ G$ over the field $ K$ of characteristic $p > 0$. An interesting unsolved problem is to find necessary and sufficient conditions on $G$ for $K[G]$ to be semisimple. Even the special case in which $G$ is assumed to be a solvable group is still open. In this paper we prove a number of theorems which may be of use in this special case.
Closed subgroups of lattice-ordered permutation groups
Stephen H.
McCleary
303-314
Abstract: Let $G$ be an $l$-subgroup of the lattice-ordered group $A(\Omega )$ of order-preserving permutations of a chain $\Omega$; and in this abstract, assume for convenience that $G$ is transitive. Let $ \bar \Omega$ denote the completion by Dedekind cuts of $\Omega$. The stabilizer subgroups $ {G_{\bar \omega }} = \{ g \epsilon G\vert\bar \omega g = \bar \omega \} ,\bar \omega \epsilon \bar \Omega$, will be used to characterize certain subgroups of $G$ which are closed (under arbitrary suprema which exist in $G$). If $\Delta$ is an $o$-block of $G$ (a nonempty convex subset such that for any $g \epsilon G$, either $\Delta g = \Delta$ or $\Delta g \cap \Delta$ is empty), and if $ \bar \omega = \sup \Delta ,{G_\Delta }$ will denote $ \{ g \epsilon G\vert\Delta g = \Delta \} = {G_{\bar \omega }}$; and the $ o$-block system $\tilde \Delta$ consisting of the translates $ \Delta g$ of $ \Delta$ will be called closed if $ {G_\Delta }$ is closed. When the collection of $o$-block systems is totally ordered (by inclusion, viewing the systems as congruences), there is a smallest closed system $ \mathcal{C}$, and all systems above $ \mathcal{C}$ are closed. $\mathcal{C}$ is the trivial system (of singletons) iff $G$ is complete (in $A(\Omega )$). $ {G_{\bar \omega }}$ is closed iff $\bar \omega$ is a cut in $\mathcal{C}$ i.e., $ \bar \omega$ is not in the interior of any $\Delta \epsilon \mathcal{C}$. Every closed convex $l$-subgroup of $G$ is an inter-section of stabilizers of cuts in $\mathcal{C}$. Every closed prime subgroup $ \ne G$ is either a stabilizer of a cut in $ \mathcal{C}$, or else is minimal and is the intersection of a tower of such stabilizers. $ L(\mathcal{C}) = \cap \{ {G_\Delta }\vert\Delta \epsilon \mathcal{C}\}$ is the distributive radical of $G$, so that $G$ acts faithfully (and completely) on $\mathcal{C}$ iff $G$ is completely distributive. Every closed $ l$-ideal of $G$ is $ L(\mathcal{D})$ for some system $ \mathcal{D}$. A group $ G$ in which every nontrivial $o$-block supports some $1 \ne g \epsilon G$ (e.g., a generalized ordered wreath product) fails to be complete iff $G$ has a smallest nontrivial system $\tilde \Delta$ and the restriction ${G_\Delta }\vert\Delta$ is $o$-$2$-transitive and lacks elements $\ne 1$ of bounded support. These results about permutation groups are used to show that if $H$ is an abstract $l$-group having a representing subgroup, its closed $l$-ideals form a tower under inclusion; and that if $\{ {K_\lambda }\}$ is a Holland kernel of a completely distributive abstract $l$-group $H$, then so is the set of closures $\{ K_\lambda ^ \ast \} $, so that if $ H$ has a transitive representation as a permutation group, it has a complete transitive representation.
Some results on parafree groups
Yael
Roitberg
315-339
Abstract: We obtain some theorems concerning parafree groups in certain varieties, which are analogs of corresponding theorems about free groups in these varieties. Our principal results are: (1) A normal subgroup $N$ of a parafree metabelian group $ P$ of rank $\geqslant 2$ such that $N \cdot {\gamma _2}P$ has infinite index in $ P$ is not finitely generated unless it is trivial. (2) If $x$ and $y$ are elements of a parafree group $ P$ in any variety containing the variety of all metabelian groups which are independent modulo $ {\gamma _2}P$, then the commutator $[x,y]$ is not a proper power.
$0\leq X\sp{2}\leq X$
Ralph
Gellar
341-352
Abstract: This paper studies the structure of elements $X$ satisfying $0 \leqslant {X^2} \leqslant X$ in a Dedekind $ \sigma$-complete partially ordered real linear algebra. The lollipop-shaped possible spectrum of $X$ had been described previously. Three basic example types are described, each with possible spectrum a characteristic part of the lollipop and the possibility of splitting $X$ into a sum of these types is considered. The matrix case is scrutinized. There are applications to operator theory. Contributions to the theory of convergence in partially ordered algebras are developed for technical purposes.
On a variation of the Ramsey number
Gary
Chartrand;
Seymour
Schuster
353-362
Abstract: Let $c(m,n)$ be the least integer $p$ such that, for any graph $G$ of order $p$, either $G$ has an $m$-cycle or its complement $\bar G$ has an $n$-cycle. Values of $c(m,n)$ are established for $m,n \leqslant 6$ and general formulas are proved for $ c(3,n),c(4,n)$, and $ c(5,n)$.
A new class of functions of bounded index
S. M.
Shah;
S. N.
Shah
363-377
Abstract: Entire functions of strongly bounded index have been defined and it is shown that functions of genus zero and having all negative zeros satisfying a one sided growth condition belong to this class.
Automorphisms of ${\rm GL}\sb{n}(R),\,R$ a local ring
J.
Pomfret;
B. R.
McDonald
379-388
Abstract: Let $R$ denote a commutative local ring with maximal ideal $m$ and residue field $k = R/m$. In this paper we determine the group automorphisms of the general linear group $G{L_n}(R)$ when $n \geqslant 3$ and the characteristic of $ k$ is not 2.
Simple groups of order $2\sp{a}3\sp{b}5\sp{c}7\sp{d}p$
Leo J.
Alex
389-399
Abstract: Let $ {\operatorname{PSL}}(n,q)$ denote the projective special linear group of degree $n$ over $ {\text{GF}}(q)$, the field with $q$ elements. The following theorem is proved. Theorem. Let $G$ be a simple group of order ${2^a}{3^b}{5^c}{7^d}p,a > 0,p$ an odd prime. If the index of a Sylow $p$-subgroup of $G$ in its normalizer is two, then $G$ is isomorphic to one of the groups, $ {\operatorname{PSL}}(2,5),{\operatorname{PSL}}(2,7),{\operatorname{PSL}}(2,9),... ...\operatorname{PSL}}(2,25),{\operatorname{PSL}}(2,27),{\operatorname{PSL}}(2,81)$, and $ {\operatorname{PSL}}(3,4)$.
The study of commutative semigroups with greatest group-homomorphism
Takayuki
Tamura;
Howard B.
Hamilton
401-419
Abstract: This paper characterizes commutative semigroups which admit a greatest group-homomorphism in various ways. One of the important theorems is that a commutative semigroup $ S$ has a greatest group-homomorphic image if and only if for every $a \in S$ there are $b,c \in S$ such that $abc = c$. Further the authors study a relationship between $S$ and a certain cofinal subsemigroup and discuss the structure of commutative separative semigroups which have a greatest group-homomorphic image.
Functions and integrals
J.
Malone
421-447
Abstract: In §2 a mapping of nonnegative functions is defined to be an integral if it has the following properties: $I(f) \geqslant 0,I(f) < \infty$ for some $f$, if $f \leqslant g$ then $I(f) \leqslant I(g),I(f + f) = 2I(f),I(\sum\nolimits_{n = 1}^\infty {{g_n}) \leqslant \sum\nolimits_{n = 1}^\infty {I({g_n})} }$. Given an integral $ I$ a nonnegative function $ f$ is defined to be a measurable function if $I(f + g) = I(f) + I(g)$ for all nonnegative functions $g$. If $f,g,({g_n})_{n = 1}^\infty $ are measurable functions then the following functions are measurable: $ f + g,af$ for all $a \geqslant 0,\sum\nolimits_{n = 1}^\infty {{g_n},f - g}$ if $ f - g \geqslant 0$ and $I(g) < \infty$; also $\sum\nolimits_{n = 1}^\infty {I({g_n}) = I(\sum\nolimits_{n = 1}^\infty {{g_n})} }$. An example shows if $f,g$ are measurable functions then $\max \{ f,g\}$ may fail to be a measurable function. If an integral has the property that if $ f,g$ are measurable functions then $\max \{ f,g\} $ is a measurable function, then the following functions are also measurable: $ \min \{ f,g\} ,\vert f - g\vert,\sup {g_n}$ and under certain conditions ${\lim _{n \to \infty }}\sup {g_n},\inf {g_n},{\lim _{n \to \infty }}\inf {g_n}$ whenever $ ({g_n})_{n = 1}^\infty$ is a sequence of measurable functions. A theorem similar to Lebesgue's dominated convergence theorem is shown to hold. In §1 the Lebesgue integral, which does not in general have the properties required to be an integral as defined in §2, is used to obtain an integral $ {\mathbf{U}}$ which does. If $\mu$ is an outer measure and ${\mathfrak{M}_\mu }$ is the $\sigma $-algebra of $ \mu$-measurable sets then the set of measurable functions defined in §2 for the integral $ {\mathbf{U}}$ contains the usual set of $ {\mathfrak{M}_\mu }$-measurable functions. $ {\mathbf{U}}$ has the property that if $f$ is a $ {\mathfrak{M}_\mu }$-measurable function and if $ \int_X {fd\mu }$ denotes the Lebesgue integral of $f$ on a set $X$ then $\int_X {fd\mu = {{\mathbf{U}}_X}fd\mu }$. In §3 it is shown that an integral $ I$ defined on a set $ X$ induces an outer measure $\mu$. If $\mu$ is a regular outer measure, a representation theorem holds for $I$: if $f$ is a nonnegative function and ${\mathbf{U}}$ is the integral of §1 then $I(f) = {U_X}fd\mu$. Regardless of whether or not the outer measure $\mu$ is regular a similar theorem can be obtained: if $f$ is a nonnegative ${\mathfrak{M}_\mu }$-measurable function then $ I(f) = {{\mathbf{U}}_X}fd\mu$. The relationship between $\mu $-measurable sets and measurable functions is explored.
A Radon-Niko\'ym theorem for operator-valued measures
Hugh B.
Maynard
449-463
Abstract: The purpose of this paper is to obtain a characterization of indefinite integrals of vector-valued functions with respect to countably additive operator-valued measures with finite variation. This result is then specialized to several simpler situations.
Canonical neighborhoods for topologically embedded polyhedra
Robert
Craggs
465-490
Abstract: D. R. McMillan has shown that in any neighborhood of a compact two sided surface in a $3$-manifold there is a closed neighborhood of the surface which is the sum of a solid homeomorphic to the cartesian product of the surface with the unit interval and some small disjoint cubes-with-handles each of which intersects the cartesian product in a disk on its boundary. In the present paper the author generalizes this notion of canonical neighborhood so that it applies to topological embeddings of arbitrary polyhedra in $3$-manifolds. This is done by replacing the cartesian products by small regular neighborhoods of polyhedral approximations to the topological embeddings.
On trigonometric series associated with separable, translation invariant subspaces of $L\sp{\infty }(G)$
Ron C.
Blei
491-499
Abstract: $G$ denotes a compact abelian group, and $ \Gamma$ denotes its dual. Our main result is that every non-Sidon set $E \subset \Gamma$ contains a non-Sidon set $ F$ such that $L_F^\infty (G) = { \oplus _l}1_{i = 1}^\infty {C_{{F_i}}}(G)$, where the ${F_i}$'s are finite, mutually disjoint, and $ \cup _{i = 1}^\infty {F_i} = F$.
A proof that $\mathcal{C}^2$ and $\mathcal{T}^2$ are distinct measures
Lawrence R.
Ernst
501-508
Abstract: We prove that there exists a nonempty family $X$ of subsets of $ {{\text{R}}^3}$ such that the two-dimensional Carathéodory measure of each member of $X$ is less than its two-dimensional $\mathcal{T}$ measure. Every member of $ X$ is the Cartesian product of 3 copies of a suitable Cantor type subset of ${\text{R}}$.
Characteristic classes of real manifolds immersed in complex manifolds
Hon Fei
Lai
1-33
Abstract: Let $M$ be a compact, orientable, $ k$-dimensional real differentiaable manifold and $N$ an $n$-dimensional complex manifold, where $k \geq n$. Given an immersion $\iota :M \to N$, a point $x \in M$ is called an RC-singular point of the immersion if the tangent space to $ \iota (M)$ at $ \iota (x)$ contains a complex subspace of dimension $> k - n$. This paper is devoted to the study of the cohomological properties of the set of RC-singular points of an immersion. When $k = 2n - 2$, the following formula is obtained: $\displaystyle \Omega (M) + \sum\limits_{r = 0}^{n - 1} {\tilde \Omega } {(\iota )^{n - r - 1}}{\iota ^ \ast }{c_r}(N) = 2{t^ \ast }DK,$ where $\Omega (M)$ is the Euler class of $M,\widetilde\Omega (\iota )$ is the Euler class of the normal bundle of the immersion, ${c_r}(N)$ are the Chern classes of $ N$, and ${t^ \ast }DK$ is a cohomology class of degree $ 2n - 2$ in $M$ whose value on the fundamental class of $M$ gives the algebraic number of RC-singular points of $\iota$. Various applications are discussed. For $ n \leq k \leq 2n - 2$, it is shown that, as long as dimensions allow, all Pontrjagin classes and the Euler class of $M$ are carried by subsets of the set of RC-singularities of an immersion $\iota :M \to {{\text{C}}^n}$.
Topological properties of paranormal operators on Hilbert space
Glenn R.
Luecke
35-43
Abstract: Let $B(H)$ be the set of all bounded endomorphisms (operators) on the complex Hilbert space $H.T \in B(H)$ is paranormal if $ \vert\vert{(T - zI)^{ - 1}}\vert\vert = 1/d(z,\sigma (T))$ for all $z \notin \sigma (T)$ where $d(z,\sigma (T))$ is the distance from $z$ to $ \sigma (T)$, the spectrum of $T$. If $ \mathcal{P}$ is the set of all paranormal operators on $H$, then $ \mathcal{P}$ contains the normal operators, $ \mathfrak{N}$, and the hyponormal operators; and $ \mathcal{P}$ is contained in $\mathcal{L}$, the set of all $T \in B(H)$ such that the convex hull of $\sigma (T)$ equals the closure of the numerical range of $T$. Thus, $\mathfrak{N} \subseteq \mathcal{P} \subseteq \mathcal{L} \subseteq B(H)$. Give $B(H)$ the norm topology. The main results in this paper are (1) $ \mathfrak{N},\mathcal{P}$, and $\mathcal{L}$ are nowhere dense subsets of $ B(H)$ when $\dim H \geq 2$, (2) $ \mathfrak{N},\mathcal{P}$, and $\mathcal{L}$ are arcwise connected and closed, and (3) $ \mathfrak{N}$ is a nowhere dense subset of $ \mathcal{P}$ when $\dim H = \infty$.
Global stability in $n$-person games
Louis J.
Billera
45-56
Abstract: A class of bargaining sets, including the bargaining set $\mathfrak{M}_1^{(i)}$ and the kernel, is treated with regard to studying the tendency to reach stability from unstable points. A known discrete procedure is extended, and these results are applied to derive global stability properties for the solutions of certain differential equations. These differential equations are given in terms of the demand functions which define the bargaining sets, and the set of critical points is precisely the bargaining set in question.
Finite commutative subdirectly irreducible semigroups
Phillip E.
McNeil
57-67
Abstract: This paper is devoted to completing the solution to the problem of constructing all finite commutative subdirectly irreducible semigroups. Those semigroups of this type which were formerly unknown are realized as certain permutation group extensions of nilpotent semigroups. The results in this paper extend the efforts in this area by G. Thierrin and B. M. Schein.
Flexible algebras of degree two
Joseph H.
Mayne
69-81
Abstract: All known examples of simple flexible power-associative algebras of degree two are either commutative or noncommutative Jordan. In this paper we construct an algebra which is partially stable but not commutative and not a noncommutative Jordan algebra. We then investigate the multiplicative structure of those algebras which are partially stable over an algebraically closed field of characteristic $p \ne 2,3,5$. The results obtained are then used to develop conditions under which such algebras must be commutative.
Extreme invariant means without minimal support
Lonnie
Fairchild
83-93
Abstract: Let $S$ be a left amenable semigroup. We show that if $S$ has a subset satisfying a certain condition, then there is an extreme left invariant mean on $ S$ whose support is not a minimal closed invariant subset of $\beta S$. Then we show that all infinite solvable groups and countably infinite locally finite groups have such subsets.
Lefschetz duality and topological tubular neighbourhoods
F. E. A.
Johnson
95-110
Abstract: We seek an analogue for topological manifolds of closed tubular neighbourhoods (for smooth imbeddings) and closed regular neighbourhoods (for piecewise linear imbeddings). We succeed when the dimension of the ambient manifold is at least six. The proof uses topological handle theory, the results of Siebenmann's thesis, and a strong version of the Lefschetz Duality Theorem which yields a duality formula for Wall's finiteness obstruction.
The action of the automorphism group of $F\sb{2}$ upon the $A\sb{6}$- and ${\rm PSL}(2,\,7)$-defining subgroups of $F\sb{2}$
Daniel
Stork
111-117
Abstract: In this paper is described a graphical technique for determining the action of the automorphism group ${\Phi _2}$, of the free group $ {F_2}$ of rank 2 upon those normal subgroups of ${F_2}$ with quotient groups isomorphic to $ G$, where $G$ is a group represented faithfully as a permutation group. The procedure is applied with $G = {\text{PSL}}(2,7)$ and $ {A_6}$ (the case $G = {A_5}$ having been treated in an earlier paper) with the following results: Theorem 1. ${\Phi _2}$, acts upon the 57 subgroups of $ {F_2}$ with quotient isomorphic to $ {\text{PSL}}(2,7)$ with orbits of lengths 7, 16, 16, and 18. The action of ${\Phi _2}$ is that of ${A_{16}}$ in one orbit of length 16, and of symmetric groups of appropriate degree in the other three orbits. Theorem 2. $ {\Phi _2}$, acts upon the 53 subgroups of ${F_2}$ with quotients isomorphic to ${A_6}$ with orbits of lengths 10, 12, 15, and 16. The action is that of full symmetric groups of appropriate degree in all orbits.
The group of homeomorphisms of a solenoid
James
Keesling
119-131
Abstract: Let $X$ be a topological space. An $ n$-mean on $X$ is a continuous function $\mu :{X^n} \to X$ which is symmetric and idempotent. In the first part of this paper it is shown that if $ X$ is a compact connected abelian topological group, then $X$ admits an $n$-mean if and only if $ {H^1}(X,Z)$ is $ n$-divisible where ${H^m}(X,Z)$ is $m$-dimensional Čech cohomology with integers $ Z$ as coefficient group. This result is used to show that if ${\Sigma _a}$ is a solenoid and $ \operatorname{Aut} ({\Sigma _a})$ is the group of topological group automorphisms of ${\Sigma _a}$, then $\operatorname{Aut} ({\Sigma _a})$ is algebraically ${Z_2} \times G$ where $G$ is $ \{ 0\} ,{Z^n}$, or $\oplus _{i = 1}^\infty Z$. For a given ${\Sigma _a}$, the structure of $ \operatorname{Aut} ({\Sigma _a})$ is determined by the $n$-means which $ {\Sigma _a}$. admits. Topologically, $\operatorname{Aut} ({\Sigma _a})$ is a discrete space which has two points or is countably infinite. The main result of the paper gives the precise topological structure of the group of homeomorphisms $G({\Sigma _a})$ of a solenoid ${\Sigma _a}$ with the compact open topology. In the last section of the paper it is shown that $G({\Sigma _a})$ is homeomorphic to ${\Sigma _a} \times {l_2} \times \operatorname{Aut} ({\Sigma _a})$ where ${l_2}$ is separable infinite-dimensional Hilbert space. The proof of this result uses recent results in infinite-dimensional topology and some techniques using flows developed by the author in a previous paper.
Cohomology of sheaves of holomorphic functions satisfying boundary conditions on product domains
Alexander
Nagel
133-141
Abstract: This paper considers sheaves of germs of holomorphic functions which satisfy certain boundary conditions on product domains in $ {{\mathbf{C}}^n}$. Very general axioms for boundary behavior are given. This includes as special cases ${L^p}$ boundary behavior, $1 \leq p \leq \infty$; continuous boundary behavior; differentiable boundary behavior of order $m,0 \leq m \leq \infty$, with an additional Hölder condition of order $\alpha ,0 \leq \alpha \leq 1$, on the $m$th derivatives. A fine resolution is constructed for those sheaves considered, and the main result of the paper is that all higher cohomology groups for these sheaves are zeŕo.
Zeros of entire functions in several complex variables
Richard A.
Kramer
143-160
Abstract: A geometric condition on the zero set of an entire function $ f$ in ${{\mathbf{C}}^N}(N \geq 1)$ is presented which is both necessary and sufficient for $ f$ to have the same zeros as some polynomial in $ {{\mathbf{C}}^N}$.
Asymptotic behavior of transforms of distributions
E. O.
Milton
161-176
Abstract: In this paper final and initial value type Abelian theorems for Laplace and Fourier transforms of certain types of distributions are obtained. The class of distributions under consideration contains the singular distributions. Thus we generalize the results previously obtained by A. H. Zemanian in two ways: we add Fourier transforms to those considered, and we also deal with a larger class of distributions.
Groups of linear operators defined by group characters
Marvin
Marcus;
James
Holmes
177-194
Abstract: Some of the recent work on invariance questions can be regarded as follows: Characterize those linear operators on $\operatorname{Hom} (V,V)$ which preserve the character of a given representation of the full linear group. In this paper, for certain rational characters, necessary and sufficient conditions are described that ensure that the set of all such operators forms a group $\mathfrak{L}$. The structure of $\mathfrak{L}$ is also determined. The proofs depend on recent results concerning derivations on symmetry classes of tensors.
Maximal ideals in the group algebra of an extension, with applications
Arnold J.
Insel
195-206
Abstract: Let $A$ be a closed central subgroup of $ E$ (all groups are locally compact and second countable). Let $G = E/A$. For each $a\in \hat A$, the dual of $A$, a multiplication is introduced with respect to which the Banach space ${L^1}(G)$ is a Banach algebra, denoted by ${L^1}(G,a(\sigma ))$. A one-to-one correspondence is established between the maximal closed (right, left, $ 2$-sided) ideals of the group algebra ${L^1}(E)$ and the totality of maximal closed (right, left, $2$-sided) ideals of ${L^1}(G,a(\sigma ))$, where $a$ varies over $\hat A$. Applications include a bound for the spectral norm of an element of ${L^1}(E)$ and the representation of a continuous positive definite function on $E$ as an integral (a 'Bochner' theorem).
Varieties of linear topological spaces
J.
Diestel;
Sidney A.
Morris;
Stephen A.
Saxon
207-230
Abstract: This paper initiates the formal study of those classes of locally convex spaces which are closed under the taking of arbitrary subspaces, separated quotients, cartesian products and isomorphic images. Well-known examples include the class of all nuclear spaces and the class of all Schwartz spaces.
Matrix rings over polynomial identity rings
Elizabeth
Berman
231-239
Abstract: We prove that if $ A$ is an algebra over a field with at least $k$ elements, and $A$ satisfies ${x^k} = 0$, then ${A_n}$, the ring of $n$-by-$n$ matrices over $A$, satisfies ${x^q} = 0$, where $ q = k{n^2} + 1$. Theorem 1.3 generalizes this result to rings: If $A$ is a ring satisfying ${x^k} = 0$, then for all $n$, there exists $q$ such that ${A_n}$ satisfies ${x^q} = 0$. Definitions. A checkered permutation of the first $n$ positive integers is a permutation of them sending even integers into even integers. The docile polynomial of degree $n$ is $\displaystyle \prod\limits_{i = 1}^p {D({x_{i1}}, \cdots ,{x_{ik}}){u_i},}$ athewhere the sum is over all checkered permutations $f$ of the first $k$ positive integers. The docile product polynomial of degree $k,p$is $\displaystyle \prod\limits_{i = 1}^p {D({x_{i1}}, \cdots ,{x_{ik}}){u_i},}$ where the $x$'s and $u$'s are noncommuting variables. Theorem 2.1. Any polynomial identity algebra over a field of characteristic 0 satisfies a docile product polynomial identity. Theorem 2.2. If $ A$ is a ring satisfying the docile product polynomial identity of degree $ 2k,p$, and $n$ is a positive integer, and $q = 2{k^2}{n^2} + 1$; then ${A_n}$ satisfies a product of $ p$ standard identities, each of degree $q$.
Group rings, matrix rings, and polynomial identities
Elizabeth
Berman
241-248
Abstract: This paper studies the question, if $R$ is a ring satisfying a polynomial identity, what polynomial identities are satisfied by group rings and matrix rings over $R$? Theorem 2.6. If $R$ is an algebra over a field with at least $ q$ elements, and $ R$ satisfies $ {x^q} = 0$, and $ G$ is a group with an abelian subgroup of index $k$, then the group ring $R(G)$ satisfies ${x^t} = 0$, where $t = q{k^2} + 2$. Theorem 3.2. If $R$ is a ring satisfying a standard identity, and $G$ is a finite group, then $R(G)$ satisfies a standard identity. Theorem 3.4. If $R$ is an algebra over a field, and $R$ satisfies a standard identity, then the $k$-by-$k$ matrix ring ${R_k}$ satisfies a standard identity. Each theorem specifies the degree of the polynomial identity.
Amalgamations of lattice ordered groups
Keith R.
Pierce
249-260
Abstract: The author considers the problem of determining whether certain classes of lattice ordered groups ($l$-groups) have the amalgamation property. It is shown that the classes of abelian totally ordered groups ($o$-groups) and abelian $l$-groups have the property, but that the class of $l$-groups does not. However, under certain cardinality restrictions one can find an $l$-group which is the ``product'' of $ l$-groups with an amalgamated subgroup whenever (a) the $l$-subgroup is an Archimedian $ o$-group, or (b) the $ l$-subgroup is a direct product of Archimedian $o$-groups and the $l$-groups are representable. This yields a new proof that any $l$-group is embeddable in a divisible $ l$-group, and implies that any $l$-group is embeddable in an $l$-group in which any two positive elements are conjugate.
Embedding theorems and quasi-linear elliptic boundary value problems for unbounded domains
Melvyn S.
Berger;
Martin
Schechter
261-278
Abstract: The Sobolev-Kondrachov embedding and compactness theorems are extended to cover general unbounded domains, by introducing appropriate weighted ${L_p}$ norms. These results are then applied to the Dirichlet problem for quasi-linear elliptic partial differential equations and isoperimetric variational problems defined on general unbounded domains in ${{\mathbf{R}}^N}$.
Horn classes and reduced direct products
Richard
Mansfield
279-286
Abstract: Boolean-valued model theory is used to give a direct proof that an $E{C_\Delta }$ model class closed under reduced direct products can be characterized by a set of Horn sentences. Previous proofs by Keisler and Galvin used either the G. C H. or involved axiomatic set theory.
The access theorem for subharmonic functions
R.
Hornblower;
E. S.
Thomas
287-297
Abstract: A chain from a point ${z_0}$ of the open unit disk $\Delta$ to the boundary of $ \Delta$ is a set $\Gamma = \cup \{ {\gamma _n}\vert n = 0,1,2, \cdots \}$ where the $ {\gamma _n}$ are compact, connected subsets of $ \Delta ,{z_0}$ is in ${\gamma _0},{\gamma _n}$ meets ${\gamma _{n + 1}}$ and the ${\gamma _n}$ approach the boundary of $\Delta$. The following ``Access Theorem'' is proved: If $u$ is subharmonic in $\Delta ,{z_0}$ is a point of $\Delta$ and $ M < u({z_0})$, then there is a chain from ${z_0}$ to the boundary of $\Delta$ on which $u \geq M$ and on which $u$ tends to a limit. A refinement, in which the chain is a polygonal arc, is established, and an example is constructed to show that the theorem fails if $M = u({z_0})$ even for bounded, continuous subharmonic functions.
Ultrafilters and independent sets
Kenneth
Kunen
299-306
Abstract: Independent families of sets and of functions are used to prove some theorems about ultrafilters. All of our results are well known to be provable from some form of the generalized continuum hypothesis, but had remained open without such an assumption. Independent sets are used to show that the Rudin-Keisler ordering on ultrafilters is nonlinear. Independent functions are used to prove the existence of good ultrafilters.
Group actions on spin manifolds
G.
Chichilnisky
307-315
Abstract: A generalization of the theorem of V. Bargmann concerning unitary and ray representations is obtained and is applied to the general problem of lifting group actions associated to the extension of structure of a bundle. In particular this is applied to the Poincaré group $\mathcal{P}$ of a Lorentz manifold $M$. It is shown that the topological restrictions needed to lift an action in $\mathcal{P}$ are more stringent than for actions in the proper Poincaré group $\mathcal{P}_ \uparrow ^ + $. Similar results hold for the Euclidean group of a Riemannian manifold.
Stability of group representations and Haar spectrum
Robert
Azencott;
William
Parry
317-327
Abstract: If $U$ and $V$ are commuting unitary representations of locally compact abelian groups $S$ and $T$, new representations of $S$ (perturbations of $U$) can be obtained from composition with images of $ U$ in $V$. If most of these representations are equivalent to $U,U$ is said to be $V$ stable. We investigate conditions which, together with stability, ensure that $U$ has (uniform) Haar spectrum. The principal applications are to dynamical systems which possess auxiliary groups with respect to which motion is stable.
Split and minimal abelian extensions of finite groups
Victor E.
Hill
329-337
Abstract: Criteria for an abelian extension of a group to split are given in terms of a Sylow decomposition of the kernel and of normal series for the Sylow subgroups. An extension is minimal if only the entire extension is carried onto the given group by the canonical homomorphism. Various basic results on minimal extensions are given, and the structure question is related to the case of irreducible kernels of prime exponent. It is proved that an irreducible modular representation of ${\text{SL}}(2,p)$ or $ {\text{PSL}}(2,p)$ for $ p$ prime and $ \geq 5$ afford a minimal extension with kernel of exponent $p$ only when the representation has degree 3, i.e., when the kernel has order ${p^3}$.
On Ess\'en's generalization of the Ahlfors-Heins theorem
John L.
Lewis
339-345
Abstract: Recently, Essén has proven a generalization of the Ahlfors-Heins Theorem. In this paper we use Essén's Theorem to obtain a different generalization of the Ahlfors-Heins Theorem.
Extensions of holomorphic maps
Peter
Kiernan
347-355
Abstract: Several generalizations of the big Picard theorem are obtained. We consider holomorphic maps $f$ from $X - A$ into $M \subset Y$. Under various assumptions on $X,A$, and $M$ we show that $f$ can be extended to a holomorphic or meromorphic map of $X$ into $Y$.
Multipliers on modules over the Fourier algebra
Charles F.
Dunkl;
Donald E.
Ramirez
357-364
Abstract: Let $G$ be an infinite compact group and $ \hat G$ its dual. For $1 \leq p < \infty ,{\mathfrak{L}^p}(\hat G)$ is a module over ${\mathfrak{L}^1}(\hat G) \cong A(G)$, the Fourier algebra of $G$. For $ 1 \leq p,q < \infty$, let $ {\mathfrak{M}_{p,q}} = {\operatorname{Hom} _{A(G)}}({\mathfrak{L}^p}(\hat G),{\mathfrak{L}^q}(\hat G))$. If $G$ is abelian, then ${\mathfrak{M}_{p,p}}$ is the space of ${L^P}(\hat G)$-multipliers. For $1 \leq p < 2$ and $p'$ the conjugate index of $p$, $\displaystyle A(G) \cong {\mathfrak{M}_{1,1}} \subset {\mathfrak{M}_{p,p}} = {\mathfrak{M}_{p',p'}} \subsetneqq {\mathfrak{M}_{2,2}} \cong {L^\infty }(G).$ Further, the space $ {\mathfrak{M}_{p,p}}$ is the dual of a space called $\mathcal{A}_p$, a subspace of ${\mathcal{C}_0}(\hat G)$. Using a method of J. F. Price we observe that $\displaystyle \cup \{ {\mathfrak{M}_{q,q}}:1 \leq q < p\} \subsetneqq {\mathfrak{M}_{p,p}} \subsetneqq \cap \{ {\mathfrak{M}_{q,q}}:p < q < 2\}$ (where $1 < p < 2$). Finally, $ {\mathfrak{M}_{q,p}} = \{ 0\}$ for $ 1 \leq p < q < \infty$.
On Lie's theorem in operator algebras
F.-H.
Vasilescu
365-372
Abstract: This work contains some algebraic results concerning infinite dimensional Lie algebras, as well as further statements within a topological background. Natural generalizations of the notion of radical, solvable and semisimple Lie algebra are introduced. The last part deals with variants of a Lie's theorem in operator algebras.
Integral representation theorems in topological vector spaces
Alan H.
Shuchat
373-397
Abstract: We present a theory of measure and integration in topological vector spaces and generalize the Fichtenholz-Kantorovich-Hildebrandt and Riesz representation theorems to this setting, using strong integrals. As an application, we find the containing Banach space of the space of continuous $p$-normed space-valued functions. It is known that Bochner integration in $p$-normed spaces, using Lebesgue measure, is not well behaved and several authors have developed integration theories for restricted classes of functions. We find conditions under which scalar measures do give well-behaved vector integrals and give a method for constructing examples.
Complete multipliers
J. S.
Byrnes
399-403
Abstract: We investigate whether the completeness of a complete orthonormal sequence for $ {L^2}( - \pi ,\pi )$ is preserved if the sequence is perturbed by multiplying a portion of it by a fixed function. For the particular sequence $\{ {(2\pi )^{ - 1/2}}{e^{inx}}\}$ we show that given any $\psi \in {L^\infty }( - \pi ,\pi )$, except $ \psi = 0$ a.e., there is a nontrivial portion of $\{ {(2\pi )^{ - 1/2}}{e^{inx}}\}$ which will maintain completeness under this perturbation.
The nonstandard theory of topological vector spaces
C. Ward
Henson;
L. C.
Moore
405-435
Abstract: In this paper the nonstandard theory of topological vector spaces is developed, with three main objectives: (1) creation of the basic nonstandard concepts and tools; (2) use of these tools to give nonstandard treatments of some major standard theorems; (3) construction of the nonstandard hull of an arbitrary topological vector space, and the beginning of the study of the class of spaces which results.
On the nonstandard representation of measures
C. Ward
Henson
437-446
Abstract: In this paper it is shown that every finitely additive probability measure $ \mu$ on $S$ which assigns 0 to finite sets can be given a nonstandard representation using the counting measure for some $^ \ast$-finite subset $F$ of $^ \ast S$. Moreover, if $\mu$ is countably additive, then $ F$ can be chosen so that $\displaystyle \int {fd\mu } = {\text{st( }}\frac{1}{{\vert\vert F\vert\vert}}\sum _{p \in F} ^\ast f(p))$ for every $\mu$-integrable function $f$. An application is given of such representations. Also, a simple nonstandard method for constructing invariant measures is presented.
Hyponormal operators having real parts with simple spectra
C. R.
Putnam
447-464
Abstract: Let ${T^ \ast }T - T{T^ \ast } = D \geq 0$ and suppose that the real part of $T$ has a simple spectrum. Then $D$ is of trace class and $\pi$ trace$(D)$ is a lower bound for the measure of the spectrum of $T$. This latter set is specified in terms of the real and imaginary parts of $T$. In addition, the spectra are determined of self-adjoint singular integral operators on ${L^2}(E)$ of the form $ A(x)f(x) + \Sigma {b_j}(x)H[f{\bar b_j}](x)$, where $E \ne ( - \infty ,\infty ),A(x)$ is real and bounded, $ \Sigma \vert{b_j}(x){\vert^2}$ is positive and bounded, and $H$ denotes the Hilbert transform.
Piecewise monotone polynomial approximation
D. J.
Newman;
Eli
Passow;
Louis
Raymon
465-472
Abstract: Given a real function $f$ satisfying a Lipschitz condition of order 1 on $ [a,b]$, there exists a sequence of approximating polynomials $\{ {P_n}\}$ such that the sequence ${E_n} = \vert\vert{P_n} - f\vert\vert$ (sup norm) has order of magnitude $1/n$ (D. Jackson). We investigate the possibility of selecting polynomials ${P_n}$ having the same local monotonicity as $ f$ without affecting the order of magnitude of the error. In particular, we establish that if $f$ has a finite number of maxima and minima on $ [a,b]$ and $S$ is a closed subset of $ [a,b]$ not containing any of the extreme points of $f$, then there is a sequence of polynomials $ {P_n}$ such that $ {E_n}$ has order of magnitude $1/n$ and such that for $n$ sufficiently large ${P_n}$ and $f$ have the same monotonicity at each point of $S$. The methods are classical.
Homological dimensions of stable homotopy modules and their geometric characterizations
T. Y.
Lin
473-490
Abstract: Projective dimensions of modules over the stable homotopy ring are shown to be either 0, 1 or $\infty$; weak dimensions are shown to be 0 or $ \infty$. Also geometric charactetizations are obtained for projective dimensions 0, 1 and weak dimension 0. The geometric characterizations are interesting; for projective modules they are about the cohomology of geometric realization; while for flat modules they are about homology. This shows that the algebraic duality between ``projective'' and ``flat'' is strongly connected with the topological duality between ``cohomology'' and ``homology". Finally, all the homological numerical invariants of the stable homotopy ring--the so-called finitistic dimensions--are completely computed except the one on injective dimension.
Vector valued absolutely continuous functions on idempotent semigroups
Richard A.
Alò;
André
de Korvin;
Richard J.
Easton
491-500
Abstract: In this paper the concept of vector valued, absolutely continuous functions on an idempotent semigroup is studied. For $ F$ a function of bounded variation on the semigroup $S$ of semicharacters with values of $F$ in the Banach space $X$, let $A = {\text{AC}}(S,X,F)$ be all those functions of bounded variation which are absolutely continuous with respect to $F$. A representation theorem is obtained for linear transformations from the space $A$ to a Banach space which are continuous in the BV-norm. A characterization is also obtained fot the collection of functions of $A$ which are Lipschitz with respect to $ F$. With regards to the new integral being utilized it is shown that all absolutely continuous functions are integrable.
Zero points of Killing vector fields, geodesic orbits, curvature, and cut locus
Walter C.
Lynge
501-506
Abstract: Let $(M,g)$ be a compact, connected, Riemannian manifold. Let $X$ be a Killing vector field on $M$. $f = g(X,X)$ is called the length function of $X$. Let $D$ denote the minimum of the distances from points to their cut loci on $M$. We derive an inequality involving $ f$ which enables us to prove facts relating $D$, the zero ponts of $X$, orbits of $X$ which are closed geodesics, and, applying theorems of Klingenberg, the curvature of $M$. Then we use these results together with a further analysis of $f$ to describe the nature of a Killing vector field in a neighborhood of an isolated zero point.
Erratum to: ``Automorphisms of group extensions''
Charles
Wells
507
Erratum to: ``Oscillation of an operator''
Robert
Whitley
507
Erratum to ``The $(\phi^{2n})_2$ field Hamiltonian for complex coupling constant
Lon
Rosen;
Barry
Simon
508
Extremal length and harmonic functions on Riemann surfaces
Carl David
Minda
1-22
Abstract: Expressions for several conformally invariant pseudometrics on a Riemann surface $R$ are given in terms of three new forms of reduced extremal distance. The pseudometrics are defined by means of various subclasses of the set of all harmonic functions on $R$ having finite Dirichlet integral. The reduced extremal distance between two points is defined on $R$, on the Alexandroff one-point compactification of $R$ and on the Kerékjártó-Stoïlow compactification of $R$. These reduced extremal distances are computed in terms of harmonic functions having specified singularities and boundary behavior. The key to establishing this connection with harmonic functions is a general theorem dealing with extremal length on a compact bordered Riemann surface and its extensions to noncompact bordered surfaces. These results are used to obtain new tests for degeneracy in the classification theory of Riemann surfaces. Finally, some of the results are illustrated for a hyperbolic simply connected Riemann surface.
Applications of bundle map theory
Daniel Henry
Gottlieb
23-50
Abstract: This paper observes that the space of principal bundle maps into the universal bundle is contractible. This fact is added to I. M. James' Bundle map theory which is slightly generalized here. Then applications yield new results about actions on manifolds, the evaluation map, evaluation subgroups of classifying spaces of topological groups, vector bundle injections, the Wang exact sequence, and $H$-spaces.
The Hochschild homology of complete intersections
Klaus
Wolffhardt
51-66
Abstract: Let $\tilde{R}$ be the algebra of all convergent (or of all strictly convergent) power series in $ N$ variables over a commutative field $K$ of characteristic 0 with a valuation, e.g. $ \tilde{R} = K[{X_1}, \cdots ,{X_N}]$. With each $K$-algebra $R \cong \tilde{R}/\mathfrak{a}$ we associate a bigraded $R$-algebra $E$. By the powers of $ \mathfrak{a}$ a filtration of the Poincaré complex of $\tilde{R}$ is induced, and $E$ is the first term of the corresponding spectral sequence. If $ \mathfrak{a}$ is generated by a prime sequence in $ \tilde{R},R$ is called a complete intersection, and $E$--with an appropriate simple grading--is isomorphic to the Hochschild homology of $R$. The result is applied to hypersurfaces.
Differentiable structures on function spaces
Nishan
Krikorian
67-82
Abstract: A ${C^s}$ differentiable manifold structure is constructed for spaces of maps from a compact ${C^r}$ manifold $M$ to a ${C^{r + s}}$ manifold $N$. The method (1) is inspired by Douady; (2) does not require any additional structure on $ N$ (such as sprays); (3) includes the case when $N$ is an analytic manifold and concludes that the mapping space is also an analytic manifold; (4) can be used to treat all the classical mapping spaces ($ {C^r}$ functions, $ {C^r}$ functions with Hölder conditions, and Sobolev functions). Several interesting aspects of these manifolds are investigated such as their tangent spaces, their behavior with respect to functions, and realizations of Lie group structures on them. Differentiable structures are also exhibited for spaces of compact maps with noncompact domain.
The tangential Cauchy-Riemann complex on spheres
G. B.
Folland
83-133
Abstract: This paper investigates the ${\overline \partial _b}$ complex of Kohn and Rossi on the unit sphere in complex $n$-space (considered as the boundary of the unit ball). The methods are Fourier-analytic, exploiting the fact that the unitary group $U(n)$ acts homogeneously on the complex. We decompose the spaces of sections into irreducible components under the action of $U(n)$ and compute the action of ${\overline \partial _b}$ on each irreducible piece. We then display the connection between the ${\overline \partial _b}$ complex and the Dolbeault complexes of certain line bundles on complex projective space. Precise global regularity theorems for ${\overline \partial _b}$ are proved, including a Sobolev-type estimate for norms related to ${\overline \partial _b}$. Finally, we solve the $\overline \partial$-Neumann problem on the unit ball and obtain a proof by explicit calculations of the noncoercive nature of this problem.
Banach space bifurcation theory
David
Westreich
135-156
Abstract: We consider the bifurcation problem for the nonlinear operator equation $x = \lambda Lx + T(\lambda ,x,y)$ in a real Banach space $X$. Here $ {\lambda _0}$ is an eigenvalue of the bounded linear operator $L,X = N(I - {\lambda _0}L) \oplus R(I - {\lambda _0}L),T \in {C^1}$ and $T$ is of higher order in $x$. New techniques are developed to simplify the solution of the bifurcation problem. When ${\lambda _0}$ is a simple eigenvalue, ${\lambda _0}$ is shown to be a bifurcation point of the homogeneous equation (i.e. $ y \equiv 0$) with respect to 0. All solutions near $({\lambda _0},0)$ are shown to be of the form $(\lambda (\epsilon),x(\epsilon)),0 \leqslant \vert\epsilon\vert < {\epsilon_0},\lambda (\epsilon)$ and $ x(\epsilon)$ are continuous and $ \lambda (\epsilon)$ and $x(\epsilon)$ are in ${C^n}$ or real analytic as $T$ is in $ {C^{n + 1}}$ or is real analytic. When $T$ is real analytic and $\lambda (\epsilon){\lambda _0}$ then there are at most two solution branches, and each branch is an analytic function of $\lambda$ for $\lambda \ne {\lambda _0}$. If $T$ is odd and analytic, for each $\lambda \in ({\lambda _0} - \delta ,{\lambda _0})$ (or $\lambda \in ({\lambda _0},{\lambda _0} + \delta )$) there exist two nontrivial solutions near 0 and there are no solutions near 0 for $ \lambda \in ({\lambda _0},{\lambda _0} + \delta )$ (or $\lambda \in ({\lambda _0} - \delta ,{\lambda _0})$). We then demonstrate that in each sufficiently small neighborhood of a solution of the homogeneous bifurcation problem there are solutions of the nonhomogeneous equation (i.e. $y \not\equiv 0$) depending continuously on a real parameter and on $y$. If $ {\lambda _0}$ is an eigenvalue of odd multiplicity we prove it is a point of bifurcation of the homogeneous equation. With a strong restriction on the projection of $T$ onto the null space of $I - {\lambda _0}L$ we show ${\lambda _0}$ is a bifurcation point of the homogeneous equation when $ {\lambda _0}$ is a double eigenvalue. Counterexamples to some of our results are given when the hypotheses are weakened.
Topological types of polynomial differential equations
L.
Markus
157-178
Abstract: Consider a first order system of real ordinary differential equations, with polynomial coefficients, having no critical points in the number space ${R^n}$. Two such differential systems are called topologically equivalent in case there exists a homeomorphism of ${R^n}$ onto itself carrying the sensed (not parametrized) solutions of the first system onto the solution family of the second system. Let ${B^n}(m)$ be the cardinal number of topological equivalence classes for systems in $ {R^n}$ with polynomial coefficients of degree at most $m$. The author proves that ${B^2}(m)$ is finite and obtains explicit upper and lower bounds in terms of $m$. Also examples are given to show that ${B^n}(m)$ is noncountable for $n \geqslant 3$ and $ m \geqslant 6$.
Symmetrization of distributions and its application. II. Liouville type problem in convolution equations
Kuang-ho
Chen
179-194
Abstract: The symmetrization of distributions corresponding to a bounded $n - 1$ dimensional ${C^\infty }$-submanifold of a ${C^\infty }$-manifold is constructed. This device reduces the consideration of distributions in $ {R^n}$ to the one of distributions in ${R^1}$, i.e. the symmetrized distributions. Using the relation between the inverse Fourier transform of a symmetrized distribution and the one of the original (nonsymmetrized) distribution, we determine the rate of decay at infinity of solutions to a general convolution equation necessary to assure uniqueness. Using a result in the division problem for distributions, we achieve the following result: If $u \in C({R^n})$ is a solution of the convolution equation $S \ast u = f,f \in \mathcal{D}({R^n})$, with some suitable $ u \in \mathcal{D}({R^n})$, provided $u$ decays sufficiently fast at infinity.
Inductive limits of finite dimensional $C\sp{\ast} $-algebras
Ola
Bratteli
195-234
Abstract: Inductive limits of ascending sequences of finite dimensional ${C^ \ast }$-algebras are studied. The ideals of such algebras are classified, and a necessary and sufficient condition for isomorphism of two such algebras is obtained. The results of Powers concerning factor states and representations of UHF-algebras are generalized to this case. A study of the current algebra of the canonical anticommutation relations is then being made.
A decomposition for combinatorial geometries
Thomas H.
Brylawski
235-282
Abstract: A construction based on work by Tutte and Grothendieck is applied to a decomposition on combinatorial pregeometries in order to study an important class of invariants. The properties of this Tutte decomposition of a pregeometry into a subgeometry $G\backslash e$ and contraction $G/e$ is explored in a categorically integrated view using factored strong maps. After showing that direct sum decomposition distributes over the Tutte decomposition we construct a universal pair $(R,t)$ where $R$ is a free commutative ring with two generators corresponding to a loop and an isthmus; and $t$, the Tutte polynomial assigns a ring element to each pregeometry. Evaluations of $ t(G)$ give the Möbius function, characteristic polynomial, Crapo invariant, and numbers of subsets, bases, spanning and independent sets of $G$ and its Whitney dual. For geometries a similar decomposition gives the same information as the chromatic polynomial throwing new light on the critical problem. A basis is found for all linear identities involving Tutte polynomial coefficients. In certain cases including Hartmanis partitions one can recover all the Whitney numbers of the associated geometric lattice $L(G)$ from $t(G)$ and conversely. Examples and counterexamples show that duals, minors, connected pregeometries, series-parallel networks, free geometries (on which many invariants achieve their upper bounds), and lower distributive pregeometries are all characterized by their polynomials. However, inequivalence, Whitney numbers, and representability are not always invariant. Applying the decomposition to chain groups we generalize the classical two-color theorem for graphs to show when a geometry can be imbedded in binary affine space. The decomposition proves useful also for graphical pregeometries and for unimodular (orientable) pregeometries in the counting of cycles and co-boundaries.
Radial limit sets on the torus
Laurence D.
Hoffmann
283-290
Abstract: Let ${U^N}$ denote the unit polydisc and $ {T^N}$ the unit torus in the space of $N$ complex variables. A subset $A$ of ${T^N}$ is called an (RL)-set (radial limit set) if to each positive continuous function $ \rho$ on ${T^N}$, there corresponds a function $ f$ in ${H^\infty }({U^N})$ such that the radial limit $\vert f{\vert^ \ast }$ of the absolute value of $ f$ equals $\rho$, a.e. on ${T^N}$ and everywhere on $A$. If $N > 1$, the question of characterizing (RL)-sets is open, but two positive results are obtained. In particular, it is shown that ${T^N}$ contains an (RL)-set which is homeomorphic to a cartesian product $K \times {T^{N - 1}}$, where $K$ is a Cantor set. Also, certain countable unions of ``parallel'' copies of ${T^{N - 1}}$ are shown to be (RL)-sets in $ {T^N}$. In one variable, every subset of $T$ is an (RL)-set; in fact, there is always a zero-free function $f$ in $ {H^\infty }(U)$ with the required properties. It is shown, however, that there exist a circle $ A \subset {T^2}$ and a positive continuous function $\rho$ on ${T^2}$ to which correspond no zero-free $ f$ in ${H^\infty }({U^2})$ with $\vert f{\vert^ \ast } = \rho$ a.e. on $ {T^2}$ and everywhere on $ A$.
Regular overrings of regular local rings
Judith
Sally
291-300
Abstract: The local factorization theorem of Zariski and Abhyankar characterizes all $ 2$-dimensional regular local rings which lie between a given $2$-dimensional regular local ring $ R$ and its quotient field as finite quadratic transforms of $R$. This paper shows that every regular local ring $R$ of dimension $n > 2$ has infinitely many minimal regular local overrings which cannot be obtained by a monoidal transform of $R$. These overrings are localizations of rings generated over $R$ by certain quotients of elements of an $ R$-sequence. Necessary and sufficient conditions are given for this type of extension of $R$ to be regular.
Irreducible representations of the $C\sp{\ast} $-algebra generated by an $n$-normal operator
John W.
Bunce;
James A.
Deddens
301-307
Abstract: For $A$ an $n$-normal operator on Hilbert space, we determine the irreducible representations of ${C^ \ast }(A)$, the $ {C^ \ast }$-algebra generated by $A$ and the identity. For $A$ a binormal operator, we determine an explicit description of the topology on the space of unitary equivalence classes of irreducible representations of ${C^ \ast }(A)$.
Automorphisms of a free associative algebra of rank $2$. II
Anastasia J.
Czerniakiewicz
309-315
Abstract: Let $R$ be a commutative domain with 1. $ R\langle x,y\rangle$ stands for the free associative algebra of rank 2 over $R;R[\tilde x,\tilde y]$ is the polynomial algebra over $R$ in the commuting indeterminates $ \tilde x$ and $ \tilde y$. We prove that the map Ab$: \operatorname{Aut} (R\langle x,y\rangle ) \to \operatorname{Aut} (R[\tilde x,\tilde y])$ induced by the abelianization functor is a monomorphism. As a corollary to this statement and a theorem of Jung [5], Nagata [7] and van der Kulk [8]* that describes the automorphisms of $ F[\tilde x,\tilde y]$ ($ F$ a field) we are able to conclude that every automorphism of $F\langle x,y\rangle$ is tame (i.e. a product of elementary automorphisms).
On the evaluation of Brewer's character sums
Reinaldo E.
Giudici;
Joseph B.
Muskat;
Stanley F.
Robinson
317-347
Abstract: A decade ago in this journal B. W. Brewer defined a sequence of polynomials ${V_n}(x,1)$ and for $n = 4$ and 5 evaluated $\displaystyle \sum\limits_{x = 1}^p {{}_\chi ({V_n}(x,1))},$ $\chi$ the nonprincipal quadratic character of the prime $p$, in closed form. A. L Whiteman derived these results by means of cyclotomy. Brewer subsequently defined ${V_n}(x,Q)$. This paper applies cyclotomy to the more general polynomials and provides evaluations for several more values of $n$. Relevant quadratic decompositions of primes are studied.
Some asymptotic fixed point theorems
Roger D.
Nussbaum
349-375
Abstract: By an asymptotic fixed point theorem we mean a theorem in functional analysis in which the existence of fixed points of a map $ f$ is established with the aid of assumptions on the iterates ${f^n}$ of $f$. We prove below some new theorems of this type, and we obtain as corollaries results of F. E. Browder, G. Darbo, R. L. Frum-Ketkov, W. A. Horn and others. We also state a number of conjectures about fixed point theorems at the end of the paper.
The finiteness of $I$ when ${\it R}[{\it X}]/{\it I}$ is flat
Jack
Ohm;
David E.
Rush
377-408
Abstract: Let $R$ be a commutative ring with identity, let $X$ be an indeterminate, and let $I$ be an ideal of the polynomial ring $ R[X]$. Let $\min I$ denote the set of elements of $I$ of minimal degree and assume henceforth that $ \min I$ contains a regular element. Then $R[X]/I$ is a flat $R$-module implies $I$ is a finitely generated ideal. Under the additional hypothesis that $R$ is quasi-local integrally closed, the stronger conclusion that $I$ is principal holds. (An example shows that the first statement is no longer valid when $\min I$ does not contain a regular element.) Let $c(I)$ denote the content ideal of $I$, i.e. $c(I)$ is the ideal of $R$ generated by the coefficients of the elements of $I$. A corollary to the above theorem asserts that $ R[X]/I$ is a flat $ R$-module if and only if $ I$ is an invertible ideal of $R[X]$ and $c(I) = R$. Moreover, if $R$ is quasi-local integrally closed, then the following are equivalent: (i) $R[X]/I$ is a flat $R$-module; (ii) $R[X]/I$ is a torsion free $R$-module and $c(I) = R$; (iii) $I$ is principal and $c(I) = R$. Let $\xi$ denote the equivalence class of $ X$ in $R[X]/I$, and let $\langle 1,\xi , \cdots ,{\xi ^t}\rangle$ denote the $R$-module generated by $1,\xi , \cdots ,{\xi ^t}$. The following statements are also equivalent: (i) $\langle 1,\xi , \cdots ,{\xi ^t}\rangle$ is flat for all $ t \geqslant 0$; (ii) $ \langle 1,\xi , \cdots ,{\xi ^t}\rangle$ is flat for some $t \geqslant 0$ for which $1,\xi , \cdots ,{\xi ^t}$ are linearly dependent over $R$; (iii) $I = ({f_1}, \cdots ,{f_n}),{f_i} \in \min I$, and $c(I) = R$; (iv) $ c(\min I) = R$. Moreover, if $R$ is integrally closed, these are equivalent to $ R[X]/I$ being a flat $ R$-module. A certain symmetry enters in when $\xi$ is regular in $R[\xi ]$, and in this case (i)-(iv) are also equivalent to the assertion that $R[\xi ]$ and $R[1/\xi ]$ are flat $R$-modules.
Analytic capacity and approximation problems
A. M.
Davie
409-444
Abstract: We consider some problems concerning analytic capacity as a set function, which are relevant to approximation problems for analytic functions on plane sets. In particular we consider the question of semiadditivity of capacity. We obtain positive results in some special cases and give applications to approximation theory. In general we establish some equivalences among various versions of the semiadditivity question and certain questions in approximation theory.
Approximation on disks
Kenneth John
Preskenis
445-467
Abstract: Let $D$ be a closed disk in the complex plane, $ f$ a complex valued continuous function on $D$ and $ {R_f}(D) =$ the uniform closure on $D$ of rational functions in $z$ and $f$ which are finite. Among other results we obtain the following. Theorem. If $f$ is of class ${C^1}$ in a neighborhood of $D$ and $\vert{f_{\bar z}}\vert > \vert{f_z}\vert$ everywhere (i.e., $f$ is an orientation reversing immersion of $ D$ in the plane), then ${R_f}(D) = C(D)$. Theorem. Let $ f$ be a polynomial in $ z$ and $\bar z$. If for each a in $D,f - \Sigma {(j!)^{ - 1}}{D^j}f(a){(z - a)^j} = {(\bar z - \bar a)^k}g$ with $\vert{g_{\bar z}}\vert > \vert{g_z}\vert$ at he zeros of $g$ in $D$ where $Df = {f_z}$, then ${R_f}(D) = C(D)$. Corollary. Let $ f$ be a polynomial in $ z$ and $\bar z$ and let $ \vert{f_{z\bar z}}(0)\vert < \vert{f_{\bar z\,\bar z}}(0)\vert/2$. Then there exists an $r > 0$ such that, for $D = (\vert z\vert \leqslant r),{R_f}(D) = C(D)$. The proofs of the theorems use measures and the conditions involved in the theorems are independent of each other. Concerning the corollary, results of E. Bishop and G. Stolzenberg show that ${f_{\bar z}}(0) = 0$ and $\vert{f_{\bar z\,\bar z}}(0)\vert < \vert{f_{z\bar z}}(0)\vert$, then there exists no $ r$ such that ${R_f}(D) = C(D)$ where $D = (\vert z\vert \leqslant r)$. Let $F = ({f_1}, \cdots ,{f_n})$ be a map on $B$ = unit polydisk in ${{\mathbf{C}}^n}$ with values in ${{\mathbf{C}}^n},{P_F}$ = uniform closure on $ B$ of polynomials in ${z_1}, \cdots {z_n},{f_1}, \cdots ,{f_n}$. Theorem. If $F$ is of class ${C^1}$ in a neighborhood of $B,{F_{\bar z}}$ is invertible and if for each $a = ({a_1}, \cdots ,{a_n})$ in $ B$, there exist complex constants $\{ {c_j}\} ,\{ {d_{ij}}\} ,i,j = 1, \cdots ,n$, such that $\Sigma {c_j}({z_j} - {a_j})({f_j}(z) - {f_j}(a)) + \Sigma {d_{ij}}({z_i} - {a_i})({z_j} - {a_j})$ has positive real part for all $z \ne a$, then $\{ (\zeta ,F(\zeta )):\zeta \in B\}$ is a polynomially convex set. Corollary. If $ F = (f,g)$ where $ f(z,w) = \bar z + cz\bar z + d{\bar z^2} + q\bar zw,g(z,w) = \bar w + sw\bar w + t{\bar w^2} + p\bar wz$ and the coefficients satisfy $ \vert\bar c + d\vert + \vert d\vert + \vert q\vert < 1$ and $\vert\bar s + t\vert + \vert t\vert + \vert p\vert < 1$, then $ {P_F} = C(B)$. Corollary. If $ F(z) = \bar z + R(z)$ where $ R = ({R_1}, \cdots ,{R_n})$ is of class ${C^2}$ and satisfies the Lipschitz condition $\vert R(\zeta ) - R(\eta )\vert \leqslant k\vert\zeta - \eta\vert$ with $k < 1$, then $ {P_F} = C(B)$. This last corollary is a result of Hörmander and Wermer. The proof of the theorem uses methods from several complex variables.
Analytic continuation of Eisenstein series
Joseph
Lewittes
469-490
Abstract: The classical Eisenstein series are essentially of the form $ \operatorname{Im} z > 0,{r_1},{r_2}$ rational and $s$ an integer $> 2$. In this paper we show that if $s$ is taken to be complex the series, with ${r_1},{r_2}$ any real numbers, defines an analytic function of $(z,s)$ for $\operatorname{Im} z > 0,\operatorname{Re} s > 2$. Furthermore this function has an analytic continuation over the entire $s$ plane, exhibted explicitly by a convergent Fourier expansion. A formula for the transformation of the function when $z$ is subjected to a modular transformation is obtained and the special case of $s$ an integer is studied in detail.
Boundary links and an unlinking theorem
M. A.
Gutiérrez
491-499
Abstract: This paper gives a homotopic theoretic criterion for a higher dimensional link to be trivial.
Infinite games with imperfect information
Michael
Orkin
501-507
Abstract: We consider an infinite, two person zero sum game played as follows: On the $n$th move, players $A,B$ select privately from fixed finite sets, ${A_n},{B_n}$, the result of their selections being made known before the next selection is made. After an infinite number of selections, a point in the associated sequence space, $\Omega$, is produced upon which $B$ pays $A$ an amount determined by a payoff function defined on $\Omega$. In this paper we extend a result of Blackwell and show that if the payoff function is the indicator function of a set in the Boolean algebra generated by the $ {G_\delta }$'s (with respect to a natural topology on $\Omega$) then the game in question has a value.
Class groups of integral group rings
I.
Reiner;
S.
Ullom
1-30
Abstract: Let $\Lambda$ be an $R$-order in a semisimple finite dimensional $ K$-algebra, where $ K$ is an algebraic number field, and $R$ is the ring of algebraic integers of $ K$. Denote by $C(\Lambda )$ the reduced class group of the category of locally free left $\Lambda$-lattices. Choose $\Lambda = ZG$, the integral group ring of a finite group $G$, and let $\Lambda '$ be a maximal $Z$-order in $QG$ containing $\Lambda$. There is an epimorphism $ D(\Lambda )$ be the kernel of this epimorphism; the groups $D(\Lambda ),C(\Lambda )$ and
Second-order time degenerate parabolic equations
Margaret C.
Waid
31-55
Abstract: We study the degenerate parabolic operator $Lu = \sum\nolimits_{i,j = 1}^n {{a^{ij}}{u_{{x_j}{x_j}}}} + \sum\nolimits_{i = 1}^n {{b^i}{u_{{x_i}}}} - c{u_t} + du$ where the coefficients of $ L$ are bounded, real-valued functions defined on a domain $D = \Omega \times (0,T] \subset {R^{n + 1}}$. Classically, $ c(x,t) \equiv 1$ or, equivalently, $ c(x,t) \geq \eta > 0$ for all $(x,t) \in \bar D$. We assume only that $ c$ is non-negative. We prove weak maximum principles and Harnack inequalities. Assume that ${a^{ij}}$ is constant, the coefficients of $ L$ and $f$ and their derivatives with respect to time are uniformly Hölder continuous (exponent $\alpha$) in $ \bar D,\bar D$ has sufficiently nice boundary, $c > 0$ on the normal boundary of $D$, $\psi \in {\bar C_{z + \alpha}}$, and $ L\psi = f$ on $ \partial B = \partial (\bar D \cap \{ t = 0\} )$. Then there exists a unique solution $u$ of the first initial-boundary value problem $Lu = f,u = \psi $ on $\bar B + (\partial B \times [0,T])$; and, furthermore, $u \in {\bar C_{2 + \alpha}}$. All results require proofs that differ substantially from the classical ones.
Stone's topology for pseudocomplemented and bicomplemented lattices
P. V.
Venkatanarasimhan
57-70
Abstract: In an earlier paper the author has proved the existence of prime ideals and prime dual ideals in a pseudocomplemented lattice (not necessarily distributive). The present paper is devoted to a study of Stone's topology on the set of prime dual ideals of a pseudocomplemented and a bicomplemented lattice. If $\hat L$ is the quotient lattice arising out of the congruence relation defined by $a \equiv b \Leftrightarrow {a^ \ast } = {b^ \ast }$ in a pseudocomplemented lattice $ L$, it is proved that Stone's space of prime dual ideals of $\hat L$ is homeomorphic to the subspace of maximal dual ideals of $L$.
Approximation in operator algebras on bounded analytic functions
M. W.
Bartelt
71-83
Abstract: Let $B$ denote the algebra of bounded analytic functions on the open unit disc in the complex plane. Let $(B,\beta )$ denote $B$ endowed with the strict topology $ \beta$. In 1956, R. C. Buck introduced $ [\beta :\beta ]$, the algebra of all continuous linear operators from $(B,\beta )$ into $ (B,\beta )$. This paper studies the algebra $ [\beta :\beta ]$ and some of its subalgebras, in the norm topology and in the topology of uniform convergence on bounded subsets. We also study a special class of operators, the translation operators. For $\phi$ an analytic map of the open unit disc into itself, the translation operator ${U_\phi }$ is defined on $B$ by $ {U_\phi }f(x) = f(\phi x)$. In particular we obtain an expression for the norm of the difference of two translation operators.
Approximating embeddings of polyhedra in codimension three
J. L.
Bryant
85-95
Abstract: Let $P$ be a $p$-dimensional polyhedron and let $Q$ be a PL $q$-manifold without boundary. (Neither is necessarily compact.) The purpose of this paper is to prove that, if $q - p \geqslant 3$, then any topological embedding of $P$ into $Q$ can be pointwise approximated by PL embeddings. The proof of this theorem uses the analogous result for embeddings of one PL manifold into another obtained by Černavskiĭ and Miller.
Primitive ideals of $C\sp{\ast} $-algebras associated with transformation groups
Elliot C.
Gootman
97-108
Abstract: We extend results of E. G. Effros and F. Hahn concerning their conjecture that if $(G,Z)$ is a second countable locally compact transformation group, with $G$ amenable, then every primitive ideal of the associated ${C^ \ast }$-algebra arises as the kernel of an irreducible representation induced from an isotropy subgroup. The conjecture is verified if all isotropy subgroups lie in the center of $G$ and either (a) the restriction of each unitary representation of $G$ to some open subgroup contains a one-dimensional subrepresentation, or (b) $G$ has an open abelian subgroup and orbit closures in $Z$ are compact and minimal.
The separable closure of some commutative rings
Andy R.
Magid
109-124
Abstract: The separable closure of a commutative ring with an arbitrary number of idempotents is defined and its Galois theory studied. Projective separable algebras over the ring are shown to be determined by the 'Galois groupoid' of the closure. The existence of the closure is demonstrated for certain rings.
Projective moduli and maximal spectra of certain quotient rings
Aron
Simis
125-136
Abstract: The projective modulus of a (commutative) ring is defined and a class of quotient rings is given for which the projective moduli are arbitrarily smaller than the dimension of the maximal spectra. Families of prime ideals of Towber and maximal type are introduced herein.
An eigenfunction expansion for a nonselfadjoint, interior point boundary value problem
Allan M.
Krall
137-147
Abstract: Under discussion is the vector system $ \sum\nolimits_{j = 0}^\infty {\vert\vert A\vert\vert < \infty }$. The eigenvalues for the system are known to be countable and approach $\infty$ in the complex plane in a series of well-defined vertical steps. For each eigenvalue there exists an eigenmanifold, generated by the residue of the Green's function. Further, since the Green's function vanishes near $\infty$ in the complex plane when the path toward $ \infty$ is horizontal, the Green's function can be expressed as a series of its residues. This in turn leads to two eigenfunction expansions, one for elements in the domain of the original system, another for elements in the domain of the adjoint system.
Extreme limits of compacta valued functions
T. F.
Bridgland
149-163
Abstract: Let $X$ denote a topological space and $\Omega (X)$ the space of all nonvoid closed subsets of $X$. Recent developments in analysis, especially in control theory, have rested upon the properties of the space $\Omega (X)$ where $X$ is assumed to be metric but not necessarily compact and with $ \Omega (X)$ topologized by the Hausdorff metric. For a continuation of these developments, it is essential that definitions of extreme limits of sequences in $ \Omega (X)$ be formulated in such a way that the induced limit is topologized by the Hausdorff metric. It is the purpose of this paper to present the formulation of such a definition and to examine some of the ramifications thereof. In particular, we give several theorems which embody ``estimates of Fatou'' for integrals of set valued functions.
Norm of a derivation on a von Neumann algebra
P.
Gajendragadkar
165-170
Abstract: A derivation on an algebra $ \mathfrak{A}$ is a linear function $\mathcal{D}:\mathfrak{A} \to \mathfrak{A}$ satisfying $\mathcal{D}(ab) = \mathcal{D}(a)b + a\mathcal{D}(b)$ for all $a,b$ in $ \mathfrak{A}$. If there exists an $a$ in $ \mathfrak{A}$ such that $\mathcal{D}(b) = ab - ba$ for $ b$ in $\mathfrak{A}$, then $ \mathcal{D}$ is called the inner derivation induced by $a$. If $ \mathfrak{A}$ is a von Neumann algebra, then by a theorem of Sakai [7], every derivation on $ \mathfrak{A}$ is inner. In this paper we compute the norm of a derivation on a von Neumann algebra. Specifically we prove that if $\mathfrak{A}$ is a von Neumann algebra acting on a separable Hilbert space $\mathcal{H},T$ is in $ \mathfrak{A}$, and ${\mathcal{D}_T}$ is the derivation induced by $ T$, then $ \vert\vert{\mathcal{D}_T}\vert\mathfrak{A}\vert\vert = 2\inf \{ \vert\vert T - Z\vert\vert,Z\;{\text{in}}\;{\text{centre}}\;\mathfrak{A}\} $.
Converse theorems and extensions in Chebyshev rational approximation to certain entire functions in $[\ast \ast \ast w(\ast \ast 0,\,+\infty )\ast \ast \ast w\ast \ast $
G.
Meinardus;
A. R.
Reddy;
G. D.
Taylor;
R. S.
Varga
171-185
Abstract: Recent interest in rational approximations to $ {e^{ - x}}$ in $[0, + \infty )$, arising naturally in numerical methods for approximating solutions of heat-conduction-type parabolic differential equations, has generated results showing that the best Chebyshev rational approximations to ${e^{ - x}}$, and to reciprocals of certain entire functions, have errors for the interval $[0, + \infty )$ which converge geometrically to zero. We present here some related converse results in the spirit of the work of S. N. Bernstein.
Solid $k$-varieties and Henselian fields
Gustave
Efroymson
187-195
Abstract: Let $k$ be a field with a nontrivial absolute value. Define property $( \ast )$ for $k$: Given any polynomial $f(x)$ in $k[x]$ with a simple root $\alpha$ in $k$; then if $g(x)$ is a polynomial near enough to $ f(x),g(x)$ has a simple root $\beta$ near $\alpha$. A characterization of fields with property $( \ast )$ is given. If $Y$ is an affine $k$-variety, $ Y \subset {\bar k^{(n)}}$, define $ {Y_k} = Y \cap {k^{(n)}}$. Define $Y$ to be solid if $I(Y) = I({Y_k})$ in $k[{x_1}, \cdots ,{x_n}]$. If $\pi :Y \to {\bar k^d}$ is a projection induced by Noether normalization, and if $k$ has property $( \ast )$, then $Y$ is a solid $k$-variety if and only if $\pi ({Y_k})$ contains a sphere in ${k^d}$. Using this characterization of solid $k$-varieties and Bertini's theorem, a dimension theorem is proven.
Operators on tensor products of Banach spaces
Takashi
Ichinose
197-219
Abstract: The present paper is a study of operators on tensor products of Banach spaces with the notion of maximal extensions introduced by G. Köthe such that the closure of a closable operator is its unique maximal extension. For a class of such operators the spectral mapping theorem is established. The results apply to the operator $A \otimes I + I \otimes B$ and give a new meaning to the method of separation of variables.
Invariant polynomials on Lie algebras of inhomogeneous unitary and special orthogonal groups
S. J.
Takiff
221-230
Abstract: The ring of invariant polynomials for the adjoint action of a Lie group on its Lie algebra is described for the inhomogeneous unitary and special orthogonal groups. In particular a new proof is given for the fact that this ring for the inhomogeneous Lorentz group is generated by two algebraically independent homogeneous polynomials of degrees two and four.
Non-Hopfian groups with fully invariant kernels. I
Michael
Anshel
231-237
Abstract: Let $\mathcal{L}$ consist of the groups $G(l,m) = (a,b;{a^{ - 1}}{b^l}a = {b^m})$ where $ \vert l\vert \ne 1 \ne \vert m\vert,lm \ne 0$ and $l,m$ are coprime. We characterize the endomorphisms of these groups, compute the centralizers of special elements and show that the endomorphism $a \to a,b \to {b^l}$ is onto with a nontrivial fully invariant kernel. Hence $G(l,m)$ is non-Hopfian in the'fully invariant sense.'
Remarks on the wave front of a distribution
Akiva
Gabor
239-244
Abstract: Basic facts about composition and multiplication of distributions as given in [1] are proved using the formulas for the wave front set of the image and pullback of distributions.
Regular semigroups satisfying certain conditions on idempotents and ideals
Mario
Petrich
245-267
Abstract: The structure of regular semigroups is studied (1) whose poset of idempotents is required to be a tree or to satisfy a weaker condition concerning the behavior of idempotents in different $\mathcal{D}$-classes, or (2) all of whose ideals are categorical or satisfy a variation thereof. For this purpose the notions of $D$-majorization of idempotents, where $ D$ is a $\mathcal{D}$-class, $ \mathcal{D}$-majorization, $\mathcal{D}$-categorical ideals, and completely semisimple semigroups without contractions are introduced and several connections among them are established. Some theorems due to G. Lallement concerning subdirect products and completely regular semigroups and certain results of the author concerning completely semisimple inverse semigroups are either improved or generalized.
Maximal regular right ideal space of a primitive ring
Kwangil
Koh;
Jiang
Luh
269-277
Abstract: If $R$ is a ring, let $X(R)$ be the set of maximal regular right ideals of $R$ and $ \mathfrak{L}(R)$ be the lattice of right ideals. For each $A \in \mathfrak{L}(R)$, define $ \operatorname{supp} (A) = \{ I \in X(R)\vert A \nsubseteq I\}$. We give a topology to $X(R)$ by taking $\{ \operatorname{supp} (A)\vert A \in \mathfrak{L}(R)\}$ as a subbase. Let $R$ be a right primitive ring. Then $X(R)$ is the union of two proper closed sets if and only if $R$ is isomorphic to a dense ring with nonzero socle of linear transformations of a vector space of dimension two or more over a finite field. $X(R)$ is a Hausdorff space if and only if either $R$ is a division ring or $R$ modulo its socle is a radical ring and $ R$ is isomorphic to a dense ring of linear transformations of a vector space of dimension two or more over a finite field.
Products of weakly-$\aleph $-compact spaces
Milton
Ulmer
279-284
Abstract: A space is said to be weakly- $ {\aleph _1}$ -compact (or weakly-Lindelöf) provided each open cover admits a countable subfamily with dense union. We show this property in a product space is determined by finite subproducts, and by assuming that ${2^{{\aleph _0}}} = {2^{{\aleph _1}}}$ we show the property is not preserved by finite products. These results are generalized to higher cardinals and two research problems are stated.
Invariant means on a class of von Neumann algebras
P. F.
Renaud
285-291
Abstract: For $G$ a locally compact group with associated von Neumann algebra $VN(G)$ we prove the existence of an invariant mean on $VN(G)$. This mean is shown to be unique if and only if $G$ is discrete.
Homology in varieties of groups. IV
C. R.
Leedham-Green;
T. C.
Hurley
293-303
Abstract: The study of homology groups ${\mathfrak{B}_n}(\Pi ,A),\mathfrak{B}$ a variety, $\Pi$ a group in $ \mathfrak{B}$, and $ A$ a suitable $ \Pi$-module, is continued. A 'Tor' is constructed which gives a better (but imperfect) approximation to these groups than a Tor previously considered. ${\mathfrak{B}_2}(\Pi ,Z)$ is calculated for various varieties $ \mathfrak{B}$ and groups $ \Pi$.
Infinite dimensional Jordan operators and Sturm-Liouville conjugate point theory
J. William
Helton
305-331
Abstract: This article concerns two simple types of bounded operators with real spectrum on a Hilbert space $H$. The purpose of this note is to suggest an abstract algebraic characterization for these operators and to point out a rather unexpected connection between such algebraic considerations and the classical theory of ordinary differential equations. Now some definitions. A Jordan operator has the form $S + N$ where $S$ is selfadjoint, ${N^2} = 0$, and $S$ commutes with $N$. A sub-Jordan operator is the restriction of a Jordan operator $J$ to an invariant subspace of $J$. A coadjoint operator $ T$ satisfies ${e^{ - is{T^ \ast }}}{e^{isT}} = I + {A_1}s + {A_2}{s^2}$ for some operators ${A_1}$ and ${A_2}$ or equivalently $ {T^{ \ast 3}} - 3{T^{ \ast 2}}T + 3{T^ \ast }{T^2} - {T^3} = 0$. The main results are Theorem A. An operator $ T$ is Jordan if and only if both $T$ and ${T^ \ast }$ are coadjoint. Theorem B. If $T$ is coadjoint, if $T$ has a cyclic vector, and if $\sigma (T) = [a,b]$, then $T$ is unitarily equivalent to ' multiplication by $x$' on a weighted Sobolev space of order 1 which is supported on $[a,b]$. Theorem C. If $ T$ is coadjoint and satisfies additional technical assumptions, then $ T$ is a sub-Jordan operator. Let us discuss Theorem C. Its converse, every sub-Jordan operator is coadjoint, is easy to prove. The proof of Theorem C consists of using Theorem B to reduce Theorem C to a question about ordinary differential equations which can be solved by an exacting application of the Jacobi conjugate point theorem for Sturm-Liouville operators. The author suspects that Theorem C is itself related to the conjugate point theorem.
Deformations of integrals of exterior differential systems
Dominic S. P.
Leung
333-358
Abstract: On any general solution of an exterior differential system $ I$, a system of linear differential equations, called the equations of variation of $I$, is defined. Let $ {\text{v}}$ be a vector field defined on a general solution of $I$ such that it satisfies the equations of variation and wherever it is defined, ${\text{v}}$ is either the zero vector or it is not tangential to the general solution. By means of some associated differential systems and the fundamental theorem of Cartan-Kähler theory, it is proved that, under the assumption of real analyticity, ${\text{v}}$ is locally the deformation vector field of a one-parameter family of general solutions of $I$. As an application, it is proved that, under the assumption of real analyticity, every Jacobi field on a minimal submanifold of a Riemannian manifold is locally the deformation vector field of a one-parameter family of minimal submanifolds.
Limit behavior of solutions of stochastic differential equations
Avner
Friedman
359-384
Abstract: Consider a system of $m$ stochastic differential equations $ d\xi = a(t,\xi )dt + \sigma (t,\xi )dw$. The limit behavior of $\xi (t)$, as $t \to \infty$, is studied. Estimates of the form $ E\vert\xi (t) - \bar \sigma w(t){\vert^2} = O({t^{1 - \delta }})$ are derived, and various applications are given.
On Knaster's conjecture
R. P.
Jerrard
385-402
Abstract: Knaster's conjecture is: given a continuous $g:{S^n} \to {E^m}$ and a set $\Delta$ of $n - m + 2$ distinct points $({q_1}, \ldots ,{q_{n - m + 2}})$ in ${S^n}$ there exists a rotation $r:{S^n} \to {S^n}$ such that $\displaystyle g(r({q_1})) = g(r({q_2})) = \cdots = g(r({q_{n - m + 2}})).$ We prove a stronger statement about a smaller class of functions. If $f:{S^n} \to {E^n}$ we write $f = ({f_1},{f_2}, \ldots ,{f_n})$ where ${f_i}:{S^n} \to {E^1}$, and put $ {F_i} = ({f_1}, \ldots ,{f_i}):{S^n} \to {E^i}$ so that ${F_n} = f$. The level surface of $ {F_i}$ in ${S^n}$ containing $x$ is ${l_i}(x) = \{ y \in {S^n}\vert{F_i}(x) = {F_i}(y)\}$. Theorem. Given an $(n + 1)$-frame $\Delta \subset {S^n}$ and a real-analytic function $f:{S^n} \to {E^n}$ such that each $ {l_i}(x)$ is either a point or a topological $(n - i)$-sphere, there exist at least ${2^{n - 1}}$ distinct rotations $r:{S^n} \to {S^n}$ such that $\displaystyle {f_i}(r({q_1})) = \cdots = {f_i}(r({q_{n - i + 2}})),\quad i = 1,2, \ldots ,n,$ for each rotation. It follows that for $m = 1,2, \ldots ,n$, $\displaystyle {F_m}(r({q_1})) = {F_m}(r({q_2})) = \cdots = {F_m}(r({q_{n - m + 2}})),$ so that the functions ${F_m}:{S^n} \to {E^m}$ satisfy Knaster's conjecture simultaneously. Given ${F_i}$, the definition of $f$ can be completed in many ways by choosing ${f_{i + 1}}, \cdots ,{f_n}$, each way giving rise to different rotations satisfying the Theorem. A suitable homotopy of $f$ which changes ${f_n}$ slightly will give locally a continuum of rotations $r$ each of which satisfies Knaster's conjecture for ${F_{n - 1}}$. In general there exists an $ (n - m)$-dimensional family of rotations satisfying Knaster's conjecture for $ {F_m}$.
The completion of an abelian category
H. B.
Stauffer
403-414
Abstract: Any category $\underline{A}$ can be embedded in its right completion $ \underline{\hat{A}}$. When $\underline{A}$ is small and abelian, this completion $ \underline{\hat{A}}$ is AB5 and the embedding is exact.
Piecewise linear critical levels and collapsing
C.
Kearton;
W. B. R.
Lickorish
415-424
Abstract: In this paper the idea of collapsing, and the associated idea of handle cancellation, in a piecewise linear manifold are used to produce a version of Morse theory for piecewise linear embeddings. As an application of this it is shown that, if $n > 2$, there exist triangulations of the $ n$-ball that are not simplicially collapsible.
Exponential decay of weak solutions for hyperbolic systems of first order with discontinuous coefficients
Hang Chin
Lai
425-436
Abstract: The weak solution of the Cauchy problem for symmetric hyperbolic systems with discontinuous coefficients in several space variables satisfying the fact that the coefficients and their first derivatives are bounded in the distribution sense is identically equal to zero if it is exponential decay.
Rings with property $D$
Eben
Matlis
437-446
Abstract: An integral domain is said to have property $ {\text{D}}$ if every torsion-free module of finite rank is a direct sum of modules of rank one. In recent papers the author has given partial solutions to the problem of finding all rings with this property. In this paper the author is finally able to show that an integrally closed integral domain has property $ {\text{D}}$ if and only if it is the intersection of at most two maximal valuation rings.
$m$-symplectic matrices
Edward
Spence
447-457
Abstract: The symplectic modular group $ \mathfrak{M}$ is the set of all $2n \times 2n$ matrices $M$ with rational integral entries, which satisfy $n \times n$ matrix. Let $m$ be a positive integer. Then the $2n \times 2n$ matrix $N$ is said to be $m$-symplectic if it has rational integral entries and if it satisfies $NJN' = mJ$. In this paper we consider canonical forms for $m$-symplectic matrices under left-multiplication by symplectic modular matrices (corresponding to Hermite's normal form) and under both left- and right-multiplication by symplectic modular matrices (corresponding to Smith's normal form). The number of canonical forms in each case is determined explicitly in terms of the prime divisors of $m$. Finally, corresponding results are stated, without proof, for 0-symplectic matrices; these are $2n \times 2n$ matrices $M$ with rational integral entries and which satisfy $ MJM' = M'JM = 0$.
Strictly irreducible $\sp{\ast} $-representations of Banach $\sp{\ast} $-algebras
Bruce A.
Barnes
459-469
Abstract: In this paper strictly irreducible $\ast$-representations of Banach $\ast $-algebras and the positive functionals associated with these representations are studied.
Exterior powers and torsion free modules over discrete valuation rings
David M.
Arnold
471-481
Abstract: Pure $R$-submodules of the $p$-adic completion of a discrete valuation ring $R$ with unique prime ideal $(p)$ (called purely indecomposable $ R$-modules) have been studied in detail. This paper contains an investigation of a new class of torsion free $R$-modules of finite rank (called totally indecomposable $R$-modules) properly containing the class of purely indecomposable $R$-modules of finite rank. Exterior powers are used to construct examples of totally indecomposable modules.
Sequences of convergence regions for continued fractions $K(a\sb{n}/1)$
William B.
Jones;
R. I.
Snell
483-497
Abstract: Sufficient conditions are given for convergence of continued fractions $K({a_n}/1)$ such that ${a_n} \in {E_n},n \geqslant 1$, where $\{ {E_n}\}$ is a sequence of element regions in the complex plane. The method employed makes essential use of a nested sequence of circular disks (inclusion regions), such that the $n$th disk contains the $n$th approximant of the continued fraction. This sequence can either shrink to a point, the limit point case, or to a disk, the limit circle case. Sufficient conditions are determined for convergence of the continued fraction in the limit circle case and these conditions are incorporated in the element regions ${E_n}$. The results provide new criteria for a sequence $\{ {E_n}\}$ with unbounded regions to be an admissible sequence. They also yield generalizations of certain twin-convergence regions.
The number of roots in a simply-connected $H$-manifold
Robert F.
Brown;
Ronald J.
Stern
499-505
Abstract: An $H$-manifold is a triple $(M,m,e)$ where $M$ is a compact connected triangulable manifold without boundary, $e \in M$, and $ m:M \times M \to M$ is a map such that $ m(x,e) = m(e,x) = x$ for all $x \in M$. Define $ {m_1}:M \to M$ to be the identity map and, for $ k \geqslant 2$, define ${m_k}:M \to M$ by ${m_k}(x) = m(x,{m_{k - 1}}(x))$. It is proven that if $(M,m,e)$ is an $H$-manifold, then $M$ is simply-connected if and only if given $k \geqslant 1$ there exists a multiplication $ m'$ on $M$ homotopic to $m$ such that $j \leqslant k$.
Erratum: ``Subordination principle and distortion theorems on holomorphic mappings in the space $C\sp{n}$'' (Trans. Amer. Math. Soc. {\bf 162} (1971), 327--336)
Kyong T.
Hahn
507-508
Plurisubharmonic functions and convexity properties for general function algebras
C. E.
Rickart
1-24
Abstract: A ``natural system'' consists of a Hausdorff space $\Sigma$ plus an algebra $\mathfrak{A}$ of complex-valued continuous functions on $\Sigma$ (which contains the constants and determines the topology in $\Sigma$) such that every continuous homomorphism of $\mathfrak{A}$ onto $ {\mathbf{C}}$ is given by an evaluation at a point of $\Sigma$ (compact-open topology in $\mathfrak{A}$). The prototype of a natural system is $ [{{\mathbf{C}}^n},\mathfrak{P}]$, where $ \mathfrak{P}$ is the algebra of polynomials on $ {{\mathbf{C}}^n}$. In earlier papers (Pacific J. Math. 18 and Canad. J. Math. 20), the author studied $ \mathfrak{A}$-holomorphic functions, which are generalizations of ordinary holomorphic functions in ${{\mathbf{C}}^n}$, and associated concepts of $\mathfrak{A}$-analytic variety and $ \mathfrak{A}$-holomorphic convexity in $\Sigma$. In the present paper, a class of extended real-valued functions, called $ \mathfrak{A}$-subharmonic functions, is introduced which generalizes the ordinary plurisubharmonic functions in ${{\mathbf{C}}^n}$. These functions enjoy many of the properties associated with plurisubharmonic functions. Furthermore, in terms of the $ \mathfrak{A}$-subharmonic functions, a number of convexity properties of ${{\mathbf{C}}^n}$ associated with plurisubharmonic functions can be generalized. For example, if $ G$ is an open $ \mathfrak{A}$-holomorphically convex subset of $\Sigma$ and $K$ is a compact subset of $G$, then the convex hull of $K$ with respect to the continuous $ \mathfrak{A}$-subharmonic functions on $G$ is equal to its hull with respect to the $ \mathfrak{A}$-holomorphic functions on $G$.
A characterization of $M$-spaces in the class of separable simplex spaces
Alan
Gleit
25-33
Abstract: We show that a separable simplex space is an $M$-space iff the arbitrary intersection of closed ideals is always an ideal.
Ascent, descent, and commuting perturbations
M. A.
Kaashoek;
D. C.
Lay
35-47
Abstract: In the present paper we investigate the stability of the ascent and descent of a linear operator $T$ when $T$ is subjected to a perturbation by a linear operator $C$ which commutes with $T$. The domains and ranges of $T$ and $C$ lie in some linear space $X$. The results are used to characterize the Browder essential spectrum of $T$. We conclude with a number of remarks concerning the notion of commutativity used in the present paper.
A comparison principle for terminal value problems in ordinary differential equations
Thomas G.
Hallam
49-57
Abstract: A comparison principle for a terminal value problem of an ordinary differential equation is formulated. Basic related topics such as the existence of maximal and minimal solutions of terminal value problems are investigated. The close relationship between the existence of the extremal solutions of a terminal value problem and the concept of asymptotic equilibrium of the differential equation is explored. Several applications of the terminal comparison principle are given.
On the ideal structure of Banach algebras
William E.
Dietrich
59-74
Abstract: For Banach algebras $ A$ in a class which includes all group and function algebras, we show that the family of ideals of $A$ with the same hull is typically quite large, containing ascending and descending chains of arbitrary length through any ideal in the family, and that typically a closed ideal of $A$ whose hull meets the Šilov boundary of $ A$ cannot be countably generated algebraically.
A sheaf-theoretic duality theory for cylindric algebras
Stephen D.
Comer
75-87
Abstract: Stone's duality between Boolean algebras and Boolean spaces is extended to a dual equivalence between the category of all $ \alpha$-dimensional cylindric algebras and a certain category of sheaves of such algebras. The dual spaces of important types of algebras are characterized and applications are given to the study of direct and subdirect decompositions of cylindric algebras.
Holomorphic maps into complex projective space omitting hyperplanes
Mark L.
Green
89-103
Abstract: Using methods akin to those of E. Borel and R. Nevanlinna, a generalization of Picard's theorem to several variables is proved by reduction to a lemma on linear relations among exponentials of entire functions. More specifically, it is shown that a holomorphic map from ${{\mathbf{C}}^m}$ to $ {{\mathbf{P}}_n}$ omitting $n + 2$ distinct hyperplanes has image lying in a hyperplane. If the map omits $n + 2$ or more hyperplanes in general position, the image will lie in a linear subspace of low dimension, being forced to be constant if the map omits $2n + 1$ hyperplanes in general position. The limits found for the dimension of the image are shown to be sharp.
Close-to-convex multivalent functions with respect to weakly starlike functions
David
Styer
105-112
Abstract: It is the object of this article to define close-to-convex multivalent functions in terms of weakly starlike multivalent functions. Six classes are defined, and shown to be equal. These generalize the class of close-to-convex functions developed by Livingston in the article, $p$-valent close-to-convex functions, Trans. Amer. Math. Soc. 115 (1965), 161-179.
Harmonic analysis on $F$-spaces with a basis
J.
Kuelbs;
V.
Mandrekar
113-152
Abstract: We establish Bochner's theorem and the Levy continuity theorem in the case that the underlying space is a real $F$-space with a basis, and then examine the infinitely divisible probability measures on a class of such spaces.
Wall manifolds with involution
R. J.
Rowlett
153-162
Abstract: Consider smooth manifolds $W$ with involution $t$ and a Wall structure described by a map $f:W \to {S^1}$ such that $ft = f$. For such objects we define cobordism theories ${\text{W}}_\ast ^I$ (in case $ W$ is closed, $ t$ unrestricted), ${\text{W}}_ \ast ^F$ (for $W$ closed, $t$ fixed-point free), and ${\text{W}}_ \ast ^{{\text{rel}}}$ ($ W$ with boundary, $ t$ free on $W$). We prove that there is an exact sequence $\displaystyle 0 \to {\text{W}}_ \ast ^I \to {\text{W}}_ \ast ^{{\text{rel}}} \to {\text{W}}_ \ast ^F \to 0.$ As a corollary, $ {\text{W}}_ \ast ^I$ imbeds in the cobordism of unoriented manifolds with involution. We also describe how ${\text{W}}_ \ast ^I$ determines the $ 2$-torsion in the cobordism of oriented manifolds with involution.
An algebra of distributions on an open interval
Harris S.
Shultz
163-181
Abstract: Let $(a,b)$ be any open subinterval of the reals which contains the origin and let $\mathfrak{B}$ denote the family of all distributions on $(a,b)$ which are regular in some interval $( \in ,0)$, where $ \in < 0$. Then $\mathfrak{B}$ is a commutative algebra: Multiplication is defined so that, when restricted to those distributions on $(a,b)$ whose supports are contained in $[0,b)$, it is ordinary convolution. Also, $\mathfrak{B}$ can be injected into an algebra of operators; this family of operators is a sequentially complete locally convex space. Since it preserves multiplication, this injection serves as a generalization (there are no growth restrictions) of the two-sided Laplace transformation.
A notion of capacity which characterizes removable singularities
Reese
Harvey;
John C.
Polking
183-195
Abstract: In this paper the authors define a capacity for a given linear partial differential operator acting on a Banach space of distributions. This notion has as special cases Newtonian capacity, analytic capacity, and AC capacity. It is shown that the sets of capacity zero are precisely those sets which are removable sets for the corresponding homogeneous equation. Simple properties of the capacity are derived and special cases examined.
Radon-Nikodym theorems for vector valued measures
Joseph
Kupka
197-217
Abstract: Let $\mu$ be a nonnegative measure, and let $ m$ be a measure having values in a real or complex vector space $V$. This paper presents a comprehensive treatment of the question: When is $m$ the indefinite integral with respect to $ \mu$ of a $V$ valued function $f?$ Previous results are generalized, and two new types of Radon-Nikodym derivative, the ``type $\rho$'' function and the ``strongly $\Gamma$ integrable'' function, are introduced. A derivative of type $\rho$ may be obtained in every previous Radon-Nikodym theorem known to the author, and a preliminary result is presented which gives necessary and sufficient conditions for the measure $m$ to be the indefinite integral of a type $\rho$ function. The treatment is elementary throughout, and in particular will include the first elementary proof of the Radon-Nikodym theorem of Phillips.
On antiflexible algebras
David J.
Rodabaugh
219-235
Abstract: In this paper we begin a classification of simple and semisimple totally antiflexible algebras (finite-dimensional) over splitting fields of char. $\ne 2,3$. For such an algebra $A$, let $P$ be the largest associative ideal in $ {A^ + }$ and let $ {N^ + }$ be the radical of $ P$. We determine all simple and semisimple totally antiflexible algebras in which $N \cdot N = 0$. Defining $ A$ to be of type $ (m,n)$ if ${N^ + }$ is nilpotent of class $ m$ with $\dim A = n$, we then characterize all simple nodal totally anti-flexible algebras (over fields of char. $\ne 2,3$) of types $(n,n)$ and $(n - 1,n)$ and give preliminary results for certain other types.
M\"untz-Szasz type approximation and the angular growth of lacunary integral functions
J. M.
Anderson
237-248
Abstract: We consider analogues of the Müntz-Szasz theorem, as in [15] and [4], for functions regular in an angle. This yields necessary and sufficient conditions for the existence of integral functions which are bounded in an angle and have gaps of a very regular nature in their power series expansion. In the case when the gaps are not so regular, similar results hold for formal power series which converge in the angle concerned.
A characterization of badly approximable functions
S. J.
Poreda
249-256
Abstract: Complex valued functions, continuous on a closed Jordan curve in the plane and having the property that all of their polynomials of best uniform approximation on that curve are identically zero are characterized in terms of their mapping properties on that curve.
Torsion differentials and deformation
D. S.
Rim
257-278
Abstract: Let $S$-scheme $X$ be a Schlessinger deformation of a curve $ {X_0}$ defined over a field $k$. In §§1 and 2, the dimension of the parameter space $S$, the relative differentials of $ X$ over $S$, and the fibres with singularity were studied, in case when ${X_0}$ is locally complete-intersection. In §3 we show that if $k$-scheme ${X_0}$ is a specialization of a smooth $ k$-scheme, then the punctured spectrum $\operatorname{Spex} ({O_{{X_{0,x}}}})$ has to be connected for every point $ x \in {X_0}$ such that $ \dim {O_{{X_{0,x}}}} \geqslant 2$. In turn we construct a rigid singularity on a surface. In the last section a few conjectures amplifying those of P. Deligne are made.
A method of symmetrization and applications. II
Dov
Aharonov;
W. E.
Kirwan
279-291
Abstract: In this paper we make use of a new method of symmetrization introduced in [1] to study various covering properties of univalent functions. More specifically, we introduce a generalization of a classical problem raised by Fekete and give a partial solution.
On the finitely generated subgroups of an amalgamated product of two groups
R. G.
Burns
293-306
Abstract: Sufficient conditions are found for the free product $G$ of two groups $A$ and $B$ with an amalgamated subgroup $U$ to have the properties (1) that the intersection of each pair of finitely generated subgroups of $G$ is again finitely generated, and (2) that every finitely generated subgroup containing a nontrivial subnormal subgroup of $G$ has finite index in $G$. The known results that Fuchsian groups and free products (under the obvious conditions on the factors) have properties (1) and (2) follow as instances of the main result.
Spaces of set-valued functions
David N.
O’Steen
307-315
Abstract: If $X$ and $Y$ are topological spaces, the set of all continuous functions from $X$ into $CY$, the space of nonempty, compact subsets of $ Y$ with the finite topology, contains a copy (with singleton sets substituted for points) of ${Y^X}$, the continuous point-valued functions from $ X$ into $Y$. It is shown that ${Y^X}$ is homeomorphic to this copy contained in ${(CY)^X}$ (where all function spaces are assumed to have the compact-open topology) and that, if $ X$ or $Y$ is $ {T_2},{(CY)^X}$ is homoemorphic to a subspace of $ {(CY)^{CX}}$. Further, if $ Y$ is ${T_2}$, then these images of $ {Y^X}$ and ${(CY)^X}$ are closed in ${(CY)^X}$ and $ {(CY)^{CX}}$ respectively. Finally, it is shown that, under certain conditions, some elements of ${X^Y}$ may be considered as elements of $ {(CY)^X}$ and that the induced $1$-$1$ function between the subspaces is open.
Complex structures on Riemann surfaces
Garo
Kiremidjian
317-336
Abstract: Let $X$ be a Riemann surface (compact or noncompact) with the property that the length of every closed geodesic is bounded away from zero. Then we show that sufficiently small complex structures on $ X$ can be described without making use of Schwarzian derivatives or the theory of quasiconformal mappings. Instead, we use methods developed in Kuranishi's work on the existence of locally complete families of deformations of compact complex manifolds. We introduce norms $\vert\quad {\vert _k}$ ($k$ a positive integer) on the space of ${C^\infty }(0,p)$-forms with values in the tangent bundle on $X$, which are similar to the usual Sobolev $\vert\vert\quad \vert{\vert _k}$-norms. (In the compact case $\vert\quad {\vert _k}$ is equivalent to $\vert\vert\quad \vert{\vert _k}$.) Then we prove that certain properties of $\vert\vert\quad \vert{\vert _k}$, crucial for Kuranishi's approach, are also satisfied by $\vert\quad {\vert _k}$.
Characteristic classes of stable bundles of rank $2$ over an algebraic curve
P. E.
Newstead
337-345
Abstract: Let $X$ be a complete nonsingular algebraic curve over $ {\mathbf{C}}$ and $ L$ a line bundle of degree 1 over $X$. It is well known that the isomorphism classes of stable bundles of rank 2 and determinant $L$ over $X$ form a nonsingular projective variety $ S(X)$. The Betti numbers of $S(X)$ are also known. In this paper we define certain distinguished cohomology classes of $S(X)$ and prove that these classes generate the rational cohomology ring. We also obtain expressions for the Chern character and Pontrjagin classes of $S(X)$ in terms of these generators.
Endomorphism rings of reduced torsion-free modules over complete discrete valuation rings
Wolfgang
Liebert
347-363
Abstract: The purpose of this paper is to find necessary and sufficient conditions that an abstract ring be isomorphic to the ring of all endomorphisms of a reduced torsion-free module over a (possibly noncommutative) complete discrete valuation ring.
Lattice-ordered injective hulls
Stuart A.
Steinberg
365-388
Abstract: It is well known that the injective hull of a lattice-ordered group ($ l$-group) $M$ can be given a lattice order in a unique way so that it becomes an $l$-group extension of $M$. This is not the case for an arbitrary $ f$-module over a partially ordered ring (po-ring). The fact that it is the case for any $l$-group is used extensively to get deep theorems in the theory of $l$-groups. For instance, it is used in the proof of the Hahn-embedding theorem and in the characterization of ${\aleph _a}$-injective $ l$-groups. In this paper we give a necessary and sufficient condition on the injective hull of a torsion-free $f$-module $M$ (over a directed essentially positive po-ring) for it to be made into an $f$-module extension of $M$ (in a unique way). An $f$-module is called an $i - f$-module if its injective hull can be made into an $f$-module extension. The class of torsion-free $ i - f$-modules is closed under the formation of products, sums, and Hahn products of strict $f$-modules. Also, an $l$-submodule and a torsion-free homomorphic image of a torsion-free $i - f$-module are $i - f$-modules. Let $R$ be an $f$-ring with zero right singular ideal whose Boolean algebra of polars is atomic. We show that $ R$ is a $qf$-ring (i.e., ${R_R}$ is an $i - f$-module) if and only if each torsion-free $ R - f$-module is an $ i - f$-module. There are no injectives in the category of torsion-free $ R - f$-modules, but there are ${\aleph _a}$-injectives. These may be characterized as the $f$-modules that are injective $R$-modules and ${\aleph _a}$-injective $l$-groups. In addition, each torsion-free $ f$-module over $ R$ can be embedded in a Hahn product of $l$-simple $Q(R) - f$-modules. We note, too, that a totally ordered domain has an $i - f$-module if and only if it is a right Ore domain.
Modular representations of metabelian groups
B. G.
Basmaji
389-399
Abstract: The irreducible modular representations, the blocks, and the defect groups of finite metabelian groups are determined. Also the dimensions of the principal indecomposable modules are computed.
Transverse cellular mappings of polyhedra
Ethan
Akin
401-438
Abstract: We generalize Marshall Cohen's notion of transverse cellular map to the polyhedral category. They are described by the following: Proposition. Let $f:K \to L$ be a proper simplicial map of locally finite simplicial complexes. The following are equivalent: (1) The dual cells of the map are all cones. (2) The dual cells of the map are homogeneously collapsible in $K$. (3) The inclusion of $L$ into the mapping cylinder of $ f$ is collared. (4) The mapping cylinder triad $({C_f},K,L)$ is homeomorphic to the product triad $(K \times I;K \times 1,K \times 0)$ rel $K = K \times 1$. Condition (2) is slightly weaker than ${f^{ - 1}}$(point) is homogeneously collapsible in $K$. Condition (4) when stated more precisely implies $f$ is homotopic to a homeomorphism. Furthermore, the homeomorphism so defined is unique up to concordance. The two major applications are first, to develop the proper theory of ``attaching one polyhedron to another by a map of a subpolyhedron of the former into the latter". Second, we classify when two maps from $ X$ to $Y$ have homeomorphic mapping cylinder triads. This property turns out to be equivalent to the equivalence relation generated by the relation $ f \sim g$, where $f,g:X \to Y$ means $f = gr$ for $r:X \to X$ some transverse cellular map.
Diffusion and Brownian motion on infinite-dimensional manifolds
Hui Hsiung
Kuo
439-459
Abstract: The purpose of this paper is to construct certain diffusion processes, in particular a Brownian motion, on a suitable kind of infinite-dimensional manifold. This manifold is a Banach manifold modelled on an abstract Wiener space. Roughly speaking, each tangent space ${T_x}$ is equipped with a norm and a densely defined inner product $g(x)$. Local diffusions are constructed first by solving stochastic differential equations. Then these local diffusions are pieced together in a certain way to get a global diffusion. The Brownian motion is completely determined by $g$ and its transition probabilities are proved to be invariant under ${d_g}$-isometries. Here ${d_g}$ is the almost-metric (in the sense that two points may have infinite distance) associated with $g$. The generalized Beltrami-Laplace operator is defined by means of the Brownian motion and will shed light on the study of potential theory over such a manifold.
Hypersurfaces of nonnegative curvature in a Hilbert space
Leo
Jonker
461-474
Abstract: In this paper we prove the following generalizations of known theorems about hypersurfaces in ${{\mathbf{R}}^n}$: Let $M$ be a hypersurface in a Hilbert space. (1) If on $ M$ the sectional curvature $K(\sigma )$ is nonnegative for every $ 2$-plane section $ \sigma$ and if $K(\sigma ) > 0$ for at least one $\sigma$, then $M$ is the boundary of a convex body. (2) If $K(\sigma ) = 0$ for all $\sigma$, then $M$ is a hypercylinder. The main tool in these theorems is Smale's infinite dimensional Sard's theorem.
The structure of pseudocomplemented distributive lattices. III. Injective and absolute subretracts
G.
Grätzer;
H.
Lakser
475-487
Abstract: Absolute subretracts are characterized in the classes $ {\mathcal{B}_n},n \leqslant \omega$. This is applied to describe the injectives in $ {\mathcal{B}_1}$ (due to R. Balbes and G. Grätzer) and ${\mathcal{B}_2}$.
Abelian subgroups of finite $p$-groups
Susan Claire
Dancs
489-493
Abstract: Information is obtained about the order of maximal abelian subgroups of central powers and crown products of finite $ p$-groups. This is used to construct groups with ``small'' maximal abelian subgroups.
A non-Noetherian factorial ring
John
David
495-502
Abstract: This paper supplies a counterexample to the conjecture that factorial implies Noetherian in finite Krull dimension. The example is the integral closure of a three-dimensional Noetherian ring, and is the union of Noetherian domains, which are proven to be factorial by means of derivation techniques.
A note on the geometric means of entire functions of several complex variables
P. K.
Kamthan
503-508
Abstract: Let $f({z_1}, \cdots ,{z_n})$ be an entire function of $n( \geqslant 2)$ complex variables. Recently Agarwal [Trans. Amer. Math. Soc. 151 (1970), 651-657] has obtained certain results involving geometric mean values of $ f$. In this paper we have constructed examples to contradict some of the results of Agarwal and have thereafter given improvements and modifications of his results.
Asymptotic formulae for the eigenvalues of a two-parameter ordinary differential equation of the second order
M.
Faierman
1-52
Abstract: We consider a two-point boundary value problem associated with an ordinary differential equation defined over the unit interval and containing the two parameters $\lambda$ and $\mu$. If for each real $\mu$ we denote the $m$th eigenvalue of our system by ${\lambda _m}(\mu )$, then it is known that ${\lambda _m}(\mu )$ is real analytic in $- \infty < \mu < \infty$. In this paper we concern ourselves with the asymptotic development of ${\lambda _m}(\mu )$ as $\mu \to \infty$, and indeed obtain such a development to an accuracy determined by the coefficients of our differential equation. With suitable conditions on the coefficients of our differential equation, the asymptotic formula for ${\lambda _m}(\mu )$ may be further developed using the methods of this paper. These results may be modified so as to apply to ${\lambda _m}(\mu )$ as $\mu \to - \infty$ if the coefficients of our differential equation are also suitably modified.
One-parameter inverse semigroups
Carl
Eberhart;
John
Selden
53-66
Abstract: This is the second in a projected series of three papers, the aim of which is the complete description of the closure of any one-parameter inverse semigroup in a locally compact topological inverse semigroup. In it we characterize all one-parameter inverse semigroups. In order to accomplish this, we construct the free one-parameter inverse semigroups and then describe their congruences.
Two Hilbert spaces in which polynomials are not dense
D. J.
Newman;
D. K.
Wohlgelernter
67-72
Abstract: Let $S$ be the Hilbert space of entire functions $f(z)$ such that $\vert\vert f(z)\vert{\vert^2} = \iint {\vert f(z){\vert^2}}dm(z)$, where $m$ is a positive measure defined on the Borel sets of the complex plane. Two Hilbert spaces are constructed in which polynomials are not dense. In the second example, our space is one which contains all exponentials and yet in which the exponentials are not complete. This is a somewhat surprising result since the exponentials are always complete on the real line.
Polynomials defined by generating relations
Hari Ballabh
Mittal
73-84
Abstract: Various authors have made efforts for finding new generating functions for known polynomial sets. In the present paper, by making use of the operator ${T_k} = x(k + xD)$, a number of generating functions and characterizations have been obtained for various polynomials in a systematic manner.
A representation theorem and approximation operators arising from inequalities involving differential operators
D.
Leviatan
85-99
Abstract: A representation of functions as integrals of a kernel $\psi (t;x)$, which was introduced by Studden, with respect to functions of bounded variation in $[0,\infty )$ is obtained whenever the functions satisfy some conditions involving the differential operators $(d/dt)\{ f(t)/{w_i}(t)\} ,i = 0,1,2, \ldots$. The results are related to the concepts of generalized completely monotonic functions and generalized absolutely monotonic functions in $ (0,\infty )$. Some approximation operators for the approximation of continuous functions in $ [0,\infty )$ arise naturally and are introduced; some sequence-to-function summability methods are also introduced.
Covering relations in the lattice of $T\sb{1}$-topologies
Roland E.
Larson;
W. J.
Thron
101-111
Abstract: A topology ${\mathcal{T}_1}$ is said to cover another topology $ {\mathcal{T}_2}$ if ${\mathcal{T}_2} \subsetneq {\mathcal{T}_1}$ and no other topology may be included between the two. In this paper, we characterize the relationship between a $ {T_1}$-topology and its covers. This characterization is used to prove that the lattice of ${T_1}$-topologies is both upper and lower semimodular. We also prove that the sublattice generated by the covers of a ${T_1}$-topology is isomorphic to the Boolean lattice of all subsets of the set of covers.
Semigroups satisfying variable identities. II
Mohan S.
Putcha;
Julian
Weissglass
113-119
Abstract: The concept of a semigroup satisfying an identity $xy = f(x,y)$ is generalized by considering identities in $n$-variables and letting the identity depend on the variables. The property of satisfying a ``variable identity'' is studied. Semigroups satisfying certain types of identities are characterized in terms of unions and semilattices of groups.
Smooth extensions in infinite dimensional Banach spaces
Peter
Renz
121-132
Abstract: If $B$ is $ {l_p}(\omega )$ or ${c_0}(\omega )$ we show $B$ has the following extension property. Any homeomorphism from a compact subset $M$ of $B$ into $B$ may be extended to a homeomorphism of $ B$ onto $B$ which is a $ {C^\infty }$ diffeomorphism on $B\backslash M$ to its image in $ B$. This is done by writing $B$ as a direct sum of closed subspaces $ {B_1}$ and ${B_2}$ both isomorphically isometric to $ B$ so that the natural projection of $K$ into ${B_1}$ along ${B_2}$ is one-to-one (see H. H. Corson, contribution in Symposium on infinite dimensional topology, Ann. of Math. Studies (to appear)). With $ K,B,{B_1}$ and $ {B_2}$ as above a homeomorphism of $B$ onto itself is constructed which leaves the $ {B_1}$-coordinates of points in $B$ unchanged, carries $K$ into ${B_1}$ and is a $ {C^\infty }$ diffeomorphic map on $ B\backslash K$. From these results the extension theorem may be proved by standard methods.
A local spectral theory for operators. III. Resolvents, spectral sets and similarity
J. G.
Stampfli
133-151
Abstract: Let $T$ be a bounded linear operator on a Hilbert space and assume $T$ has thin spectrum. When is $T$ similar to a normal operator? This problem is studied in a variety of situations and sufficient conditions are given in terms of characteristic functions, resolvents, spectral sets, and spectral resolutions. By contrast, the question ``When is $ T$ normal?'' has a relatively simple answer since in that case a necessary and sufficient condition can be given in terms of the resolvent alone.
The variation of singular cycles in an algebraic family of morphisms
Joel
Roberts
153-164
Abstract: (1) Let $g:{V^r} \to {W^m}(m \geqq r)$ be a morphism of nonsingular varieties over an algebraically closed field. Under certain conditions, one can define a cycle ${S_i}$ on $V$ with $\operatorname{Supp} ({S_i}) = \{ x\vert{\dim _{k(x)}}(\Omega _{X/Y}^1)(x) \geqq i\} $. The multiplicity of a component of ${S_i}$ can be computed directly from local equations for $g$. If $ {V^r} \subset {P^n}$, and if $g:V \to {P^m}$ is induced by projection from a suitable linear subspace of ${P^n}$, then ${S_1}$ is ${c_{m - r + 1}}(N \otimes \mathcal{O}( - 1))$, up to rational equivalence, where $N$ is the normal bundle of $V$ in ${P^n}$. (2) Let $f:X \to S$ be a smooth projective morphism of noetherian schemes, where $S$ is connected, and the fibres of $f$ are absolutely irreducible $ r$-dimensional varieties. For a geometric point $\eta :\operatorname{Spec} (k) \to S$, and a locally free sheaf $E$ on $X$, let ${X_\eta }$ be the corresponding geometric fibre, and ${E_\eta }$ the sheaf induced on ${X_\eta }$. If ${E_1}, \ldots ,{E_m}$ are locally free sheaves on $ X$, and if ${i_1} + \cdots + {i_m} = r$, then the degree of the zero-cycle $ {c_{{i_1}}}({E_{1\eta }}) \cdots {c_{{i_m}}}({E_{m\eta }})$ is independent of the choice of $\eta$. (3) The results of (1) and (2) are used to study the behavior under specialization of a closed subvariety ${V^r} \subset {P^n}$.
Abstract evolution equations and the mixed problem for symmetric hyperbolic systems
Frank J.
Massey
165-188
Abstract: In this paper we show that Kato's theory of linear evolution equations may be applied to the mixed problem for first order symmetric hyperbolic systems of partial differential equations.
Solvable fundamental groups of compact $3$-manifolds
Benny
Evans;
Louise
Moser
189-210
Abstract: A classification is given for groups which can occur as the fundamental group of some compact $3$-manifold. In most cases we are able to determine the topological structure of a compact $3$-manifold whose fundamental group is known to be solvable. Using the results obtained for solvable groups, we are able to extend some known results concerning nilpotent groups of closed $3$-manifolds to the more general class of compact $3$-manifolds. In the final section it is shown that each nonfinitely generated abelian group which occurs as a subgroup of the fundamental group of a $ 3$-manifold is a subgroup of the additive group of rationals.
The fix-points and factorization of meromorphic functions
Fred
Gross;
Chung-chun
Yang
211-219
Abstract: In this paper, we use the Nevanlinna theory of meromorphic functions and a result of Goldstein to generalize some known results in factorization and fixpoints of entire functions. Specifically, we prove (1) If $f$ and $g$ are nonlinear entire functions such that $ f(g)$ is transcendental and of finite order, then $f(g)$ has infinitely many fix-points. (2) If $ f$ is a polynomial of degree $\geqq 3$, and $g$ is an arbitrary transcendental meromorphic function, then $f(g)$ must have infinitely many fix-points. (3) Let $p(z),q(z)$ be any nonconstant polynomials, at least one of which is not $c$-even, and let $a$ and $b$ be any constants with $a$ or $b \ne 0$. Then $h(z) = q(z)\exp (a{z^2} + bz) + p(z)$ is prime.
On residually finite knot groups
E. J.
Mayland
221-232
Abstract: The residual finiteness of the class of groups of fibred knots, or those knot groups with finitely generated and, therefore, free commutator subgroups, has been known for some time. Using Baumslag's results on absolutely parafree groups, this paper extends the result to twist knots (Whitehead doubles of the trivial knot) and certain other classes of nonfibred knots whose minimal spanning surface has complement with free fundamental group. As a by-product more explicit finite representations, namely cyclic extensions of certain $p$-groups, are obtained for these knot groups and the groups of fibred knots. Finally composites of two such knots also have residually finite groups.
Absolute Tauberian constants for Ces\`aro means
Soraya
Sherif
233-241
Abstract: This paper is concerned with introducing two inequalities of the form $\sum\nolimits_{n = 0}^\infty {\vert{\tau _n}} - {a_n}\vert \leqq KA$ and ${\tau _n} = C_n^{(k)} - C_{n - 1}^{(k)},C_n^{(k)}$ denote the Cesàro transform of order $ k,K$ and $K'$ are absolute Tauberian constants, $A = \sum\nolimits_{n = 0}^\infty {\vert\Delta (n{a_n}} )\vert < \infty ,B = \s... ...ty {\vert\Delta ((1/n)\sum\nolimits_{v = 1}^{n - 1} {v{a_v}} } )\vert < \infty$ and $ \Delta {u_k} = {u_k} - {u_{k + 1}}$. The constants $K,K'$ will be determined.
A generalized area theorem for harmonic functions on hermitian hyperbolic space
Robert Byrne
Putz
243-258
Abstract: Let $D$ be the noncompact realization of hermitian hyperbolic space. We consider functions on $ D$ which are harmonic with respect to the Laplace-Beltrami operator. The principal result is a generalized area theorem which gives a necessary and sufficient condition for the admissible convergence of harmonic functions.
Equivariant bordism and Smith theory. III
R. E.
Stong
259-263
Abstract: The bordism classes of a manifold with involution is determined by means of Wu type classes in the Smith cohomology.
The asymptotic behavior of a Fourier transform and the localization property for eigenfunction expansions for some partial differential operators
Burton
Randol
265-271
Abstract: The asymptotic behavior of a certain Fourier transform is investigated, and the result is applied to obtain a localization theorem for elliptic operators on the torus.
Some degeneracy theorems for entire functions with values in an algebraic variety
James A.
Carlson
273-301
Abstract: In the first part of this paper we prove the following extension theorem. Let $P_q^ \ast$ be a $q$-dimensional punctured polycylinder, i.e. a product of disks and punctured disks. Let ${W_n}$ be a compact complex manifold such that the bundle of holomorphic $q$-forms is positive in the sense of Grauert. Let $ f:P_q^ \ast \to {W_n}$ be a holomorphic map whose Jacobian determinant does not vanish identically. Then $f$ extends as a rational map to the full polycylinder $ {P_q}$. In the second half of the paper we prove the following generalization of the little Picard theorem to several complex variables: Let $V \subset {P_n}$ be a hypersurface of degree $d \geqq n + 3$ whose singularities are locally normal crossings. Then any holomorphic map $f:{C^n} \to {P_n} - V$ has identically vanishing Jacobian determinant.
The connectedness of the collection of arc cluster sets
Peter
Lappan
303-310
Abstract: Let $f$ be a continuous complex-valued function defined on the unit disk and let $p$ be a boundary point of the disk. A very natural topology on the collection of all arc cluster sets of $f$ at the point $p$ has been investigated by Belna and Lappan [1] who proved that this collection is a compact set under certain suitable conditions. It is proved here that this collection is an arcwise connected set under the topology in question, but is not in general locally arcwise connected or even locally connected. It is also shown by example that it is generally not possible to map the real line onto the collection of arc cluster sets at $p$ in a continuous manner.
Bounded continuous functions on a completely regular space
F. Dennis
Sentilles
311-336
Abstract: Three locally convex topologies on $C(X)$ are introduced and developed, and in particular shown to coincide with the strict topology on locally compact $X$ and yield dual spaces consisting of tight, $ \tau$-additive and $ \sigma$-additive functionals respectively for completely regular $ X$.
On the H. Lewy extension phenomenon
Ricardo
Nirenberg
337-356
Abstract: We prove local approximation and extension theorems for ${C^\infty }$ submanifolds $M$ of ${C^n}$ (CR submanifolds). Under some conditions on $ M$, any smooth solution of the induced Cauchy-Riemann equations can be extended holomorphically to bigger (and sometimes open) sets.
Knots whose branched cyclic coverings have periodic homology
C. McA.
Gordon
357-370
Abstract: Let ${M_k}$ be the $k$-fold branched cyclic covering of a (tame) knot of $ {S^1}$ in ${S^3}$. Our main result is that the following statements are equivalent: (1) ${H_1}({M_k})$ is periodic with period $ n$, i.e. ${H_1}({M_k}) \cong {H_1}({M_{k + n}})$ for all $k$, (2) ${H_1}({M_k}) \cong {H_1}({M_{(k,n)}})$ for all $k$, (3) the first Alexander invariant of the knot, ${\lambda _1}(t) = {\Delta _1}(t)/{\Delta _2}(t)$, divides ${t^n} - 1$.
Integral decomposition of functionals on $C\sp{\ast} $-algebras
Herbert
Halpern
371-385
Abstract: The spectrum of the center of the weak closure of a ${C^ \ast }$-algebra with identity on a Hilbert space is mapped into a set of quasi-equivalence classes of representations of the $ {C^ \ast }$-algebra so that every positive $\sigma$-weakly continuous functional on the algebra can be written in a central decomposition as an integral over the spectrum of a field of states whose canonical representations are members of the respective quasi-equivalence classes except for a nowhere dense set. Various questions relating to disjointness of classes, factor classes, and uniformly continuous functionals are studied.
Classification of all hereditarily indecomposable circularly chainable continua
Lawrence
Fearnley
387-401
Abstract: In a recent paper the author has established an affirmative solution to a well-known and previously unsolved problem raised by R. H. Bing in 1951 concerning whether or not the pseudo-circle is topologically unique. Now in this present paper, as the natural culmination of the study initiated in the earlier paper, complete topological and mapping classification theorems are established for all hereditarily indecomposable circularly chainable continua. The principal topological classification results of this paper are the theorems that hereditarily indecomposable circularly chainable continua are characterized set-theoretically by their equivalence classes of fundamental sequences and are characterized algebraic-topologically by their Čech cohomology groups. These topological classification theorems are then used in establishing mapping classification theorems for all hereditarily indecomposable circularly chainable continua and in proving that the mapping hierarchy of hereditarily indecomposable circularly chainable continua constitutes a lattice. Among the consequences of the foregoing primary results of the paper are the additional theorems that two hereditarily indecomposable circularly chainable continua are topologically equivalent if and only if each of them is a continuous image of the other, and that every $ k$-adic pseudo-solenoid is topologically unique.
Orbits in a real reductive Lie algebra
L. Preiss
Rothschild
403-421
Abstract: The purpose of this paper is to give a classification of the orbits in a real reductive Lie algebra under the adjoint action of a corresponding connected Lie group. The classification is obtained by examining the intersection of the Lie algebra with the orbits in its complexification. An algebraic characterization of the minimal points in the closed orbits is also given.
Morita theorems for functor categories
D. C.
Newell
423-433
Abstract: We generalize the Morita theorems to certain functor categories using properties of adjoint functors.
Locally uniform spaces
James
Williams
435-469
Abstract: The axioms for a locally uniform space $(X,\mathcal{U})$ may be obtained by localizing the last axiom for a uniform space to obtain $ \forall x \in X,\forall U \in \mathcal{U},\exists V \in \mathcal{V}:(V \circ V)[x] \subseteq U[x]$. With each locally uniform space one may associate a regular topology, just as one associates a completely regular topology with each uniform space. The topologies of locally uniform spaces with nested bases may be characterized using Boolean algebras of regular open sets. As a special case, one has that locally uniform spaces with countable bases have pseudo-metrizable topologies. Several types of Cauchy filters may be defined for locally uniform spaces, and a major portion of the paper is devoted to a study and comparison of their properties. For each given type of Cauchy filter, complete spaces are those in which every Cauchy filter converges; to complete a space is to embed it as a dense subspace in a complete space. In discussing these concepts, it is convenient to make the mild restriction of considering only those locally uniform spaces $(X,\mathcal{V})$ in which each element of $\mathcal{V}$ is a neighborhood of the diagonal in $X \times X$ with respect to the relative topology; these spaces I have called NLU-spaces. With respect to the more general types of Cauchy filters, some NLU-spaces are not completable; this happens even though some completable NLU-spaces can still have topologies which are not completely regular. Examples illustrating these completeness situations and having various topological properties are obtained from a generalized construction. It is also shown that there is a largest class of Cauchy filters with respect to which each NLU-space has a completion that is also an NLU-space.
Nested sequences of local uniform spaces
James
Williams
471-481
Abstract: A locally uniform space is a pair of sets $ (X,\mathcal{V}),\mathcal{V}$ being a filter on $X \times X$ such that $\forall U \in \mathcal{V},\Delta (X) = \{ \langle x,x\rangle :x \in X\} \subse... ...,{U^{ - 1}} = \{ \langle y,x\rangle :\langle x,y\rangle \in U\} \in \mathcal{V}$, and $ \forall x \in X,\exists V \in \mathcal{V}:(V \circ V)[x] \subseteq U[x]$. We shall say that a sequence $\{ ({X_n},{\mathcal{V}_n}):n \in \omega \}$ is nested iff $ \forall n \in \omega ,{X_n} \subseteq {X_{n + 1}}$ and $ {\mathcal{V}_{n + 1}}\vert{X_n} = {\mathcal{V}_n}$. By a limit for a nested sequence $\{ ({X_n},{\mathcal{V}_n}):n \in \omega \}$, we shall mean any locally uniform space $ (X,\mathcal{V})$ such that $ X = \cup \{ {X_n}:n \in \omega \}$ and $ \forall n \in \omega ,\mathcal{V}\vert{X_n} = {\mathcal{V}_n}$. Our first task will be to consider when a nested sequence of locally uniform spaces has a limit; in order to do this, we shall introduce a weak generalization of pseudo-metric functions. We shall also show that, in contrast to locally uniform spaces, each nested sequence of uniform spaces has a limit. With each locally uniform space one can associate a regular relative topology in the obvious fashion. E. Hewitt and J. Novak have constructed regular spaces of the type on which every real-valued continuous function is constant; we shall use our results about limits to give a relatively simple general construction for producing locally uniform spaces which have this type of relative topology. The construction may be done in such a way that the spaces produced have several pleasant topological properties.
Approximation of strictly singular and strictly cosingular operators using nonstandard analysis
J. W.
Brace;
R. Royce
Kneece
483-496
Abstract: The strictly singular operators and the strictly cosingular operators are characterized by the manner in which they can be approximated by continuous linear operators of finite-dimensional range. We make use of linear convergence structures to obtain each class as limit points of the operators with finite-dimensional range. The construction of a nonstandard model makes it possible to replace convergence structures by topologies. Our nonstandard models are called nonstandard locally convex spaces.
Open mappings of the universal curve onto continuous curves
David C.
Wilson
497-515
Abstract: A criterion for the existence of an open mapping from one compact metric space onto another is established in this paper. This criterion is then used to establish the existence of a monotone open mapping of the universal curve onto any continuous curve and the existence of a light open mapping of the universal curve onto any nondegenerate continuous curve. These examples show that if $f$ is a monotone open or a light open mapping of one compact space $X$ onto another $Y$, then it will not necessarily be the case that $ \dim Y \leqq \dim X + k$, where $k$ is some positive integer.
Addendum to: ``On a problem of Tur\'an about polynomials with curved majorants''
Q. I.
Rahman
517-518
Erratum to: ``On entropy and generators of measure-preserving transformations''
Wolfgang
Krieger
519
Erratum to ``Improbability of collisions in Newtonian gravitational systems''
Donald Gene
Saari
521
On subgroups of $M\sb{24}$. I. Stabilizers of subsets
Chang
Choi
1-27
Abstract: In this paper we study the orbits of the Mathieu group ${M_{24}}$ on sets of n points, $1 \leqq n \leqq 12$. For $n \geqq 6,{M_{24}}$ is not transitive on these sets, so we may classify the sets into types corresponding to the orbits of ${M_{24}}$ and then show how to construct a set of each type from smaller sets. We determine the stabilizer of a set of each type and describe its representation on the 24 points. From the conclusions, the class of subgroups which are maximal among the intransitives of $ {M_{24}}$ can be read off. This work forms the first part of a study which yields, in particular, a complete list of the primitive representations of ${M_{24}}$.
On subgroups of $M\sb{24}$. II. The maximal subgroups of $M\sb{24}$
Chang
Choi
29-47
Abstract: In this paper we effect a systematic study of transitive subgroups of $ {M_{24}}$, obtaining 5 transitive maximal subgroups of ${M_{24}}$ of which one is primitive and four imprimitive. These results, along with the results of the paper, On subgroups of ${M_{24}}$. I, enable us to enumerate all the maximal subgroups of ${M_{24}}$. There are, up to conjugacy, nine of them. The complete list includes one more in addition to those listed by J. A. Todd in his recent work on $ {M_{24}}$. The two works were done independently employing completely different methods.
A geometry for $E\sb{7}$
John R.
Faulkner
49-58
Abstract: A geometry is defined by the 56-dimensional representation $\mathfrak{M}$ of a Lie algebra of type $ {E_7}$. Every collineation is shown to be induced by a semisimilarity of $\mathfrak{M}$, and the image of the automorphism group of $ \mathfrak{M}$ in the collineation group is shown to be simple.
A reciprocity theorem for ergodic actions
Kenneth
Lange
59-78
Abstract: An analogue of the Frobenius Reciprocity Theorem is proved for virtual groups over a locally compact separable group G. Specifically, an ergodic analytic Borel G-space $M(V\pi )$ is constructed from a virtual group V and a homomorphism $ \pi :V \to G$ of V into G. This construction proves to be functorial for the category of virtual groups over G; in fact, it is a left adjoint of the functor which takes an ergodic analytic Borel G-space T into the virtual group $T \times G$ together with projection $ \rho :T \times G \to G$ onto G. Examples such as Kakutani's induced transformation and flows under functions show the scope of this construction. A method for constructing the product of two virtual groups is also presented. Some of the structural properties of the product virtual group are deduced from those of the components. Finally, for virtual groups ${\pi _1}:{V_1} \to {G_1}$ and ${\pi _2}:{V_2} \to {G_2}$ over groups ${G_1}$ and ${G_2}$ respectively, the adjoint functor construction applied to ${\pi _1} \times {\pi _2}:{V_1} \times {V_2} \to {G_1} \times {G_2}$ is shown to give the product of the ${G_1}$-space derived from ${\pi _1}:{V_1} \to {G_1}$ and the $ {G_2}$-space derived from $ {\pi _2}:{V_2} \to {G_2}$, up to suitably defined isomorphism.
Some classes of flexible Lie-admissible algebras
Hyo Chul
Myung
79-88
Abstract: Let $\mathfrak{A}$ be a finite-dimensional, flexible, Lie-admissible algebra over a field of characteristic $\ne 2$. Suppose that ${\mathfrak{A}^ - }$ has a split abelian Cartan subalgebra $\mathfrak{H}$ which is nil in $\mathfrak{A}$. It is shown that if every nonzero root space of $ {\mathfrak{A}^ - }$ for $\mathfrak{H}$ is one-dimensional and the center of $ {\mathfrak{A}^ - }$ is 0, then $ \mathfrak{A}$ is a Lie algebra isomorphic to $ {\mathfrak{A}^ - }$. This generalizes the known result obtained by Laufer and Tomber for the case that ${\mathfrak{A}^ - }$ is simple over an algebraically closed field of characteristic 0 and $\mathfrak{A}$ is power-associative. We also give a condition that a Levi-factor of ${\mathfrak{A}^ - }$ be an ideal of $\mathfrak{A}$ when the solvable radical of ${\mathfrak{A}^ - }$ is nilpotent. These results yield some interesting applications to the case that ${\mathfrak{A}^ - }$ is classical or reductive.
Stochastic integral representation of multiplicative operator functionals of a Wiener process
Mark A.
Pinsky
89-104
Abstract: Let M be a multiplicative operator functional of (X, L) where X is a d-dimensional Wiener process and L is a separable Hilbert space. Sufficient conditions are given in order that M be equivalent to a solution of the linear Itô equation $\displaystyle M(t) = I + \sum\limits_{j = 1}^d {\int_0^t {M(s){B_j}} } (x(s))d{x_j}(s) + \int_0^t {M(s){B_0}(x(s))ds,}$ where ${B_0}, \ldots ,{B_d}$ are bounded operator functions on ${R^d}$. The conditions require that the equation $ T(t)f = E[M(t)f(x(t))]$ define a semigroup on $ {L^2}({R^d})$ whose infinitesimal generator has a domain which contains all linear functions of the coordinates $({x_1}, \ldots ,{x_d})$. The proof of this result depends on an a priori representation of the semigroup $T(t)$ in terms of the Wiener semigroup and a first order matrix operator. A second result characterizes solutions of the above Itô equation with $ {B_0} = 0$. A sufficient condition that M belong to this class is that $ E[M(t)]$ be the identity operator on L and that $M(t)$ be invertible for each $t > 0$. The proof of this result uses the martingale stochastic integral of H. Kunita and S. Watanabe.
Some remarks on quasi-analytic vectors
Paul R.
Chernoff
105-113
Abstract: Recently a number of authors have developed conditions of a generalized quasi-analytic nature which imply essential selfadjointness for semibounded symmetric operators in Hilbert space. We give a unified derivation of these results by reducing them to the basic theorems of Nelson and Nussbaum. In addition we present an extension of Nussbaum's quasi-analytic vector theorem to the setting of semigroups in Banach spaces.
Multipliers for spherical harmonic expansions
Robert S.
Strichartz
115-124
Abstract: Sufficient conditions are given for an operator on the sphere that commutes with rotations to be bounded in ${L^p}$. The conditions are analogous to those of Hörmander's well-known theorem on Fourier multipliers.
Hyperbolic limit sets
Sheldon E.
Newhouse
125-150
Abstract: Many known results for diffeomorphisms satisfying Axiom A are shown to be true with weaker assumptions. It is proved that if the negative limit set $ {L^ - }(f)$ of a diffeomorphism f is hyperbolic, then the periodic points of f are dense in ${L^ - }(f)$. A spectral decomposition theorem and a filtration theorem for such diffeomorphisms are obtained and used to prove that if $ {L^ - }(f)$ is hyperbolic and has no cycles, then f satisfies Axiom A, and hence is $\Omega$-stable. Examples are given where ${L^ - }(f)$ is hyperbolic, there are cycles, and f fails to satisfy Axiom A.
Geodesic flow in certain manifolds without conjugate points
Patrick
Eberlein
151-170
Abstract: A complete simply connected Riemannian manifold H without conjugate points satisfies the uniform Visibility axiom if the angle subtended at a point p by any geodesic $ \gamma$ of H tends uniformly to zero as the distance from p to $ \gamma$ tends uniformly to infinity. A complete manifold M is a uniform Visibility manifold if it has no conjugate points and if the simply connected covering H satisfies the uniform Visibility axiom. We derive criteria for the existence of uniform Visibility manifolds. Let M be a uniform Visibility manifold, SM the unit tangent bundle of M and $ {T_t}$ the geodesic flow on SM. We prove that if every point of SM is nonwandering with respect to ${T_t}$ then ${T_t}$ is topologically transitive on SM. We also prove that if $M'$ is a normal covering of M then $ {T_t}$ is topologically transitive on $SM'$ if ${T_t}$ is topologically transitive on SM.
Some theorems on the cos $\pi \lambda $ inequality
John L.
Lewis
171-189
Abstract: In this paper we consider subharmonic functions $u \leqq 1$ in the unit disk whose minimum modulus and maximum modulus satisfy a certain inequality. We show the existence of an extremal member of this class with largest maximum modulus. We then obtain an upper bound for the maximum modulus of this function in terms of the logarithmic measure of a certain set. We use this upper bound to prove theorems about subharmonic functions in the plane.
Subharmonic functions in certain regions
John L.
Lewis
191-201
Abstract: In a recent paper Hellsten, Kjellberg, and Norstad considered bounded subharmonic functions u in $\vert z\vert < 1$ which satisfy a certain inequality. They obtained an exact upper bound for the maximum modulus of u. We first show that this bound still holds when u satisfies less restrictive hypotheses. We then give an application of this result.
On character sums and power residues
Karl K.
Norton
203-226
Abstract: Sharp estimates are given for a double sum involving Dirichlet characters. These are applied to the problem of estimating certain sums whose values give a measure of the average distance between successive power residues to an arbitrary modulus. A particularly good result of the latter type is obtained when the modulus is prime.
Correspondence between Lie algebra invariant subspaces and Lie group invariant subspaces of representations of Lie groups
Joel
Zeitlin
227-242
Abstract: Let G be a Lie group with Lie algebra $ \mathfrak{g}$ and $\mathfrak{B} = \mathfrak{u}(\mathfrak{g})$, the universal enveloping algebra of $\mathfrak{g}$; also let U be a representation of G on H, a Hilbert space, with dU the corresponding infinitesimal representation of $\mathfrak{g}$ and $\mathfrak{B}$. For G semisimple Harish-Chandra has proved a theorem which gives a one-one correspondence between $ dU(\mathfrak{g})$ invariant subspaces and $U(G)$ invariant subspaces for certain representations U. This paper considers this theorem for more general Lie groups. A lemma is proved giving such a correspondence without reference to some of the concepts peculiar to semisimple groups used by Harish-Chandra. In particular, the notion of compactly finitely transforming vectors is supplanted by the notion of ${\Delta _f}$, the $\Delta$ finitely transforming vectors, for $ \Delta \in \mathfrak{B}$. The lemma coupled with results of R. Goodman and others immediately yields a generalization to Lie groups with large compact subgroup. The applicability of the lemma, which rests on the condition $\mathfrak{g}{\Delta _f} \subseteq {\Delta _f}$, is then studied for nilpotent groups. The condition is seen to hold for all quasisimple representations, that is representations possessing a central character, of nilpotent groups of class $\leqq 2$. However, this condition fails, under fairly general conditions, for $\mathfrak{g} = {N_4}$, the 4-dimensional class 3 Lie algebra. ${N_4}$ is shown to be a subalgebra of all class 3 $\mathfrak{g}$ and the condition is seen to fail for all $ \mathfrak{g}$ which project onto an algebra where the condition fails. The result is then extended to cover all $\mathfrak{g}$ of class 3 with general dimension 1. Finally, it is conjectured that $ \mathfrak{g}{\Delta _f} \subseteq {\Delta _f}$ for all quasisimple representations if and only if class $\mathfrak{g} \leqq 2$.
Temperatures in several variables: Kernel functions, representations, and parabolic boundary values
John T.
Kemper
243-262
Abstract: This work develops the notion of a kernel function for the heat equation in certain regions of $n + 1$-dimensional Euclidean space and applies that notion to the study of the boundary behavior of nonnegative temperatures. The regions in question are bounded between spacelike hyperplanes and satisfy a parabolic Lipschitz condition at points on the lateral boundary. Kernel functions (normalized, nonnegative temperatures which vanish on the parabolic boundary except at a single point) are shown to exist uniquely. A representation theorem for nonnegative temperatures is obtained and used to establish the existence of finite parabolic limits at the boundary (except for a set of heat-related measure zero).
Bounded linear operators on Banach function spaces of vector-valued functions
N. E.
Gretsky;
J. J.
Uhl
263-277
Abstract: Representations of bounded linear operators on Banach function spaces of vector-valued functions to Banach spaces are given in terms of operator-valued measures. Then spaces whose duals are Banach function spaces are characterized. With this last information, reflexivity of this type of space is discussed. Finally, the structure of compact operators on these spaces is studied, and an observation is made on the approximation problem in this context.
Orientation-preserving mappings, a semigroup of geometric transformations, and a class of integral operators
Antonio O.
Farias
279-289
Abstract: A Titus transformation $T = \langle \alpha ,v\rangle$ is a linear operator on the vector space of ${C^\infty }$ mappings from the circle into the plane given by ${C^\infty }$ function on the circle ${S^1}$. Let $\tau$ denote the semigroup generated by finite compositions of Titus transformations. A Titus mapping is the image by an element of $\tau$ of a degenerate curve, ${\alpha _0}{v_0}$, where $ {\alpha _0}$ is a ${C^\infty }$ function on ${S^1}$ and ${v_0}$ is fixed in the plane ${R^2}$. A $ {C^\infty }$ mapping $f:{S^1} \to {R^2}$ is called properly extendable if there is a $ {C^\infty }$ mapping $F:{D^ - } \to {R^2}$, D the open unit disk and ${D^ - }$ its closure, such that ${J_F} \geqq 0$ on $ D,{J_F} > 0$ near the boundary ${S^1}$ of ${D^ - }$ and $ F{\vert _{{s^1}}} = f$. A ${C^\infty }$ mapping $f:{S^1} \to {R^2}$ is called normal if it is an immersion with no triple points and all its double points are transversal. The main result of this paper can be stated: a normal mapping is extendable if and only if it is a Titus mapping. An application is made to a class of integral operators of the convolution type, $y(t) = - \smallint_0^{2\pi } {k(s)x(t - s)ds}$. It is proved that, under certain technical conditions, such an operator is topologically equivalent to Hilbert's transform of potential theory, $y(t) = \smallint_0^{2\pi } {\cot (s/2)x(t - s)ds}$, which gives the relation between the real and imaginary parts of the restriction to the boundary of a function holomorphic inside the unit disk.
Noetherian intersections of integral domains
William
Heinzer;
Jack
Ohm
291-308
Abstract: Let $D < R$ be integral domains having the same quotient field K and suppose that there exists a family $ {\{ {V_i}\} _{i \in I}}$ of 1-dim quasi-local domains having quotient field K such that $D = R \cap \{ {V_i}\vert i \in I\}$. The goal of this paper is to find conditions on R and the ${V_i}$ in order for D to be noetherian and, conversely, conditions on D in order for R and the ${V_i}$ to be noetherian. An important motivating case is when the set $ \{ {V_i}\}$ consists of a single element V and V is a valuation ring. It is shown, for example, in this case that (i) if V is centered on a finitely generated ideal of D, then V is noetherian and (ii) if V is centered on a maximal ideal of D, then D is noetherian if and only if R and V are noetherian.
Interpolation to analytic data on unbounded curves
Maynard
Thompson
309-318
Abstract: This paper provides a method for constructing a family of sets of points on the boundary (assumed suitably smooth) of an unbounded Jordan region in the complex plane which is useful for certain interpolation problems. It is proved that if these sets are used as nodes for Lagrange interpolation to analytic data, then the resulting polynomials converge in the region, and the limit function is related in a natural way to the boundary data. Subsidiary results include an approximate quadrature formula for slowly decreasing functions on an infinite interval.
The dominion of Isbell
Barry
Mitchell
319-331
Abstract: A well-known characterization of epimorphisms in the category of rings with identity is imitated to give a similar characterization of epimorphisms in the category of small pre-additive categories. From this one deduces Isbell's ``Zigzag Theorem'' concerning epimorphisms in Cat.
On the equivalence of multiplicity and the generalized topological degree
T.
O’Neil;
J. W.
Thomas
333-345
Abstract: In this paper we first extend the definition of the multiplicity (as defined by J. Cronin-Scanlon) of operators of the form $I + C + T$ to operators of the form $H + C + T$. We then show that the generalized topological degree (as defined by F. E. Browder and W. V. Petryshyn) of operators of the form $H + C + T$ is also defined. Finally, we show that when both the multiplicity and generalized topological degree of $H + C + T$ are defined, they are equal.
An equality for $2$-sided surfaces with a finite number of wild points
Michael D.
Taylor;
Harvey
Rosen
347-358
Abstract: Let S be a 2-sided surface in a 3-manifold that is wild from one side U at just m points. It is shown that the minimal genus possible for all members of a sequence of surfaces in U converging to S (where these surfaces each separate the same point from S in $U \cup S$) is equal to the sum of the genus of S and a certain multiple of the sum of m special topological invariants associated with the wild points. In this equality, the sum of these invariants is multiplied by just one of the numbers 0, 1, or 2, dependent upon the genus and orientability class of S and the value of m. As an application, an upper bound is given for the number of nonpiercing points that a 2-sided surface has with respect to one side.
Transversals to the flow induced by a differential equation on compact orientable $2$-dimensional manifolds
Carl S.
Hartzman
359-368
Abstract: Every treatment of the theory of differential equations on a torus uses the fact that given a differential equation on a torus of class ${C^k}$, there is a non-null-homotopic closed Jordan curve $\Gamma$ of class ${C^k}$ which is transverse to the trajectories of the differential equation that pass through points of $\Gamma$. Such a curve necessarily cannot separate the torus. Here, we prove that given a differential equation on an n-fold torus ${T_n}$ of class ${C^k}$, possessing only ``simple'' singularities of negative index there is a non-null-homotopic closed Jordan curve $\Gamma$ of class ${C^k}$ which is a transversal. The nonseparating property, however, does not follow immediately. For the particular case ${T_2}$, we prove the existence of such a transversal that does not separate ${T_2}$.
Martingales of strongly measurable Pettis integrable functions
J. J.
Uhl
369-378
Abstract: This paper deals with convergence theorems for martingales of strongly measurable Pettis integrable functions. First, a characterization of those martingales which converge in the Pettis norm is obtained. Then it is shown that a martingale which is convergent in the Pettis norm converges to its limit strongly in measure and, if the index set is the positive integers, it converges strongly almost everywhere to its limit. The second part of the paper deals with the strong measure and strong almost everywhere convergence of martingales which are not necessarily convergent in the Pettis norm. The resulting theorems here show that $ {L^1}$-boundedness can be considerably relaxed to a weaker control condition on the martingale by the use of some facts on finitely additive vector measures.
The envelope of holomorphy of Riemann domains over a countable product of complex planes
Mário C.
Matos
379-387
Abstract: This paper deals with the problem of constructing envelopes of holomorphy for Riemann domains over a locally convex space. When this locally convex space is a countable product of complex planes the existence of the envelope of holomorphy is proved and the domains of holomorphy are characterized.
Two theorems in the commutator calculus
Hermann V.
Waldinger
389-397
Abstract: Let $F = \langle a,b\rangle$. Let ${F_n}$ be the nth subgroup of the lower central series. Let p be a prime. Let $ {c_3} < {c_4} < \cdots < {c_z}$ be the basic commutators of dimension $ > 1$ but $< p + 2$. Let ${P_1} = (a,b),{P_m} = ({P_{m - 1}},b)$ for $m > 1$. Then $ (a,{b^p}) \equiv \prod\nolimits_{i = 3}^z {c_i^{{\eta _i}}\bmod {F_{p + 2}}}$. It is shown in Theorem 1 that the exponents $ {\eta _i}$ are divisible by p, except for the exponent of ${P_p}$ which $= 1$. Let the group $ \mathcal{G}$ be a free product of finitely many groups each of which is a direct product of finitely many groups of order p, a prime. Let $\mathcal{G}$-simple basic commutators'' of dimension $> 1$ defined below are free generators of
Oscillation and nonoscillation of solutions of generalized Emden-Fowler equations
C. V.
Coffman;
J. S. W.
Wong
399-434
Abstract: This paper treats the ordinary differential equation $y'' + yF({y^2},x) = 0,x > 0$ , where $yF({y^2},x)$ is continuous in (y, x) for $x > 0,\vert y\vert < \infty$, and $ F(t,x)$ is non-negative; the equation is assumed to be either of sublinear or superlinear type. Criteria are given for the equation to be oscillatory, to be nonoscillatory, to possess oscillatory solutions or to possess nonoscillatory solutions. An attempt has been made to unify the methods of treatment of the sublinear and superlinear cases. These methods consist primarily of comparison with linear equations and the use of ``energy'' functions. An Appendix treats the questions of continuability and uniqueness of solutions of the equation considered in the main text.
Locally $B\sp{\ast} $-equivalent algebras
Bruce A.
Barnes
435-442
Abstract: Let A be a Banach $^ \ast$-algebra. A is locally ${B^ \ast }$-equivalent if, for every selfadjoint element $t \in A$, the closed $^ \ast$-subalgebra of A generated by t is $^\ast$-isomorphic to a $ {B^ \ast }$-algebra. In this paper it is shown that when A is locally $ {B^\ast}$-equivalent, and in addition every selfadjoint element in A has at most countable spectrum, then A is $^ \ast$-isomorphic to a ${B^ \ast }$-algebra.
On modification theorems
Murali
Rao
443-450
Abstract: Given a right continuous family ${F_t}$ of complete $\sigma$-fields and a bounded right continuous family ${X_t}$ of random variables, we show in this paper that it is possible to modify the conditional expectations $ E({X_t}\vert{F_t})$ to be right continuous. When ${X_t} = X$, this reduces to a result of J. L. Doob.
Restricted mean values and harmonic functions
John R.
Baxter
451-463
Abstract: A function h defined on a region R in ${{\mathbf{R}}^n}$ will be said to possess a restricted mean value property if the value of the function at each point is equal to the mean value of the function over one open ball in R, with centre at that point. It is proved here that this restricted mean value property implies h is harmonic under certain conditions.
A sufficient condition for the lower semicontinuity of parametric integrals
Edward
Silverman
465-469
Abstract: We use simple convex functions and standard techniques in area theory to treat Morrey's extension of McShane's lower semicontinuity theorem for parametric integrals. This enables us to eliminate some technical hypotheses, simplify the proof and obtain a more general result.
On time-free functions
Gideon
Schwarz
471-478
Abstract: By regarding as equivalent any two real-valued functions of a real variable that can be obtained from each other by a monotone continuous transformation of the independent variable, time-free functions are defined. A convenient maximal invariant is presented, and applied to some time-free functional equations.
Mappings from $3$-manifolds onto $3$-manifolds
Alden
Wright
479-495
Abstract: Let f be a compact, boundary preserving mapping from the 3-manifold $ {M^3}$ onto the 3-manifold $ {N^3}$. Let ${Z_p}$ denote the integers mod a prime p, or, if $p = 0$, the integers. (1) If each point inverse of f is connected and strongly 1-acyclic over $ {Z_p}$, and if $ {M^3}$ is orientable for $p > 2$, then all but a locally finite collection of point inverses of f are cellular. (2) If the image of the singular set of f is contained in a compact set each component of which is strongly acyclic over ${Z_p}$, and if ${M^3}$ is orientable for $p \ne 2$, then ${N^3}$ can be obtained from ${M^3}$ by cutting out of $ \operatorname{Int} \;{M^3}$ a compact 3-manifold with 2-sphere boundary, and replacing it by a ${Z_p}$-homology 3-cell. (3) If the singular set of f is contained in a 0-dimensional set, then all but a locally finite collection of point inverses of f are cellular.
Polars and their applications in directed interpolation groups
A. M. W.
Glass
1-25
Abstract: In the study of l-groups, as in many other branches of mathematics, use is made of the concept of ``orthogonal elements". The purpose of this paper is to show that this concept can be extended to directed, interpolation groups and that most of the theorems in l-groups concerning polars hold in the more general setting of directed, interpolation groups. As consequences, generalisations of Holland's and Lorenzen's theorems are obtained and a result on o-simple abelian, directed, interpolation groups.
Skew products of dynamical systems
Eijun
Kin
27-43
Abstract: In 1950-1951, H. Anzai introduced a method of skew products of dynamical systems in connection with isomorphism problems in ergodic theory. There is a problem to give a necessary and sufficient condition under which an ergodic skew product dynamical system has pure point spectrum. For the special case, translations on the torus, he gave a partial answer for this question. However, this problem has been open in the general case. In the present paper, we generalize the notion of skew products proposed by Anzai and give a complete answer for this problem.
Automorphism groups of bounded domains in Banach spaces
Stephen J.
Greenfield;
Nolan R.
Wallach
45-57
Abstract: We prove a weak Schwarz lemma in Banach space and use it to show that in Hilbert space a Siegel domain of type II is not necessarily biholomorphic to a bounded domain. We use a strong Schwarz lemma of L. Harris to find the full group of automorphisms of the infinite dimensional versions of the Cartan domains of type I. We then show that all domains of type I are holomorphically inequivalent, and are different from k-fold products of unit balls $(k \geqq 2)$. Other generalizations and comments are given.
A class of regular functions containing spirallike and close-to-convex functions
M. R.
Ziegler
59-70
Abstract: A class of functions $\mathcal{A}$ is defined which contains the spirallike and close-to-convex functions. By decomposing $ \mathcal{A}$ into subclasses in a natural way, some basic properties of $\mathcal{A}$ and these subclasses are determined, including solutions to extremal problems; distortion theorems; coefficient inequalities; and the radii of convexity and close-to-convexity.
Complex structures on real product bundles with applications to differential geometry
Richard S.
Millman
71-99
Abstract: The purpose of this paper is to classify holomorphic principal fibre bundles which admit a smooth section (i.e. are real product bundles). This is accomplished if the structure group is solvable of type (E). In the general case, a sufficient condition is obtained for a real product bundle to be equivalent to the complex product bundle. A necessary and sufficient condition for the existence of a holomorphic connection on a real product bundle is also obtained. Using this criterion in the case where the structure group is abelian, a generalization of a theorem due to Atiyah (in the case the structure group is ${C^ \ast }$) is obtained.
On inverse scattering for the Klein-Gordon equation
Tomas P.
Schonbek
101-123
Abstract: A scattering operator $S = S(V)$ is set up for the Klein-Gordon equation $ \square u = {m^2}u(m > 0)$ perturbed by a linear potential $V = V(x)$ to $\square u = {m^2}u + Vu$. It is found that for each $R > 0$ there exists a constant $c(R)$ (of order $ {R^{2 - n}}$ as $R \to + \infty$, n = space dimension) such that if the ${L_1}$ and the ${L_q}$ norm of V and $V'$ are bounded by $S(V') \ne S(V)$ or $V' = V$. Here $q > n/2$, and $c(R)$ may also depend on q.
A Sturmian theorem for first order partial differential equations
Pui Kei
Wong
125-131
Abstract: A pair of first order partial differential equations is considered. The system is transformed into a single nonlinear scalar equation of the Riccati type from which some Wirtinger type integral inequalities for functions of several variables are derived. A comparison theorem for two such pairs of first order equations is then proved using the Wirtinger inequalities.
Current valued measures and Ge\"ocze area
Ronald
Gariepy
133-146
Abstract: If f is a continuous mapping of finite Geöcze area from a polyhedral region $X \subset {R^k}$ into ${R^n},2 \leqq k \leqq n$, then, under suitable hypotheses, one can associate with f, by means of the Cesari-Weierstrass integral, a current valued measure T over the middle space of f. In particular, if either $k = 2$ or the $k + 1$-dimensional Hausdorff measure of $f(X)$ is zero, then T is essentially the same as a current valued measure defined by H. Federer and hence serves to describe the tangential properties of f and the multiplicities with which f assumes its values. Further, the total variation of T is equal to the Geöcze area of f.
Extending congruences on semigroups
A. R.
Stralka
147-161
Abstract: The two main results are: (1) Let S be a semigroup which satisfies the relation $abcd = acbd$, let A be a subsemigroup of Reg S which is a band of groups and let $[\varphi ]$ be a congruence on A. Then $[\varphi ]$ can be extended to a congruence on S. (2) Let S be a compact topological semigroup which satisfies the relation $abcd = acbd$, let A be a closed subsemigroup of Reg S and let $[\varphi ]$ be a closed congruence on A such that $\dim \,\varphi (A)\vert\mathcal{H} = 0$. Then $[\varphi ]$ can be extended to a closed congruence on S.
Tangential limits of functions orthogonal to invariant subspaces
David
Protas
163-172
Abstract: For any inner function $\varphi$, let $ {M^ \bot }$ be the orthogonal complement of $ \varphi {H^2}$, in $ {H^2}$, where $ {H^2}$ is the usual Hardy space. The relationship between the tangential convergence of all functions in $ {M^ \bot }$ and the finiteness of certain sums and integrals involving $ \varphi$ is studied. In particular, it is shown that the tangential convergence of all functions in $ {M^ \bot }$ is a stronger condition than the tangential convergence of $ \varphi$, itself.
On the rank of a space
Christopher
Allday
173-185
Abstract: The rank of a space is defined as the dimension of the highest dimensional torus which can act almost-freely on the space. (By an almost-free action is meant one for which all the isotropy subgroups are finite.) This definition is shown to extend the classical definition of the rank of a Lie group. A conjecture giving an upper bound for the rank of a space in terms of its rational homotopy is investigated.
$G\sb{0}$ of a graded ring
Leslie G.
Roberts
187-195
Abstract: We consider the Grothendieck group ${G_0}$ of various graded rings, including ${G_0}(A_n^r)$ where A is a commutative noetherian ring, and $A_n^r$ is the A-subalgebra of the polynomial ring $A[{X_0}, \ldots ,{X_n}]$ generated by monomials of degree r. If A is regular, then ${G_0}(A_n^r)$ has a ring structure. The ideal class groups of these rings are also considered.
Cobordism Massey products
J. C.
Alexander
197-214
Abstract: The structure of Massey products is introduced into the bordism ring ${\Omega ^S}$ of manifolds with structure S and machinery is developed to investigate it. The product is changed to one in homotopy via the Pontrjagin-Thom map and methods for computation via the Adams spectral sequence are developed. To illustrate the methods, some products in ${\Omega ^{SU}}$ and $ {\Omega ^{Sp}}$ are computed.
A general class of factors of $E^4$
Leonard R.
Rubin
215-224
Abstract: In this paper we prove that any upper semicontinuous decomposition of $E^n$ which is generated by a trivial defining sequence of cubes with handles determines a factor of $ E^{n + 1}$. An important corollary to this result is that every 0-dimensional point-like decomposition of $E^3$ determines a factor of $E^4$. In our approach we have simplified the construction of the sequence of shrinking homeomorphisms by eliminating the necessity of shrinking sets piecewise in a collection of n-cells, the technique employed by R. H. Bing in the original result of this type.
A continuity theorem for Fuchsian groups
C. K.
Wong
225-239
Abstract: On a given Riemann surface, fix a discrete (finite or infinite) sequence of points $\{ {P_k}\} ,k = 1,2,3, \ldots ,$ and associate to each ${P_k}$ an ``integer'' ${\nu _k}$ (which may be $1,2,3, \ldots ,{\text{or}}\;\infty )$. This sequence of points and ``integers'' is called a ``signature'' on the Riemann surface. With only a few exceptions, a Riemann surface with signature can always be represented by a Fuchsian group. We investigate here the dependence of the group on the number ${\nu _k}$. More precisely, keeping the points ${P_k}$ fixed, we vary the numbers ${\nu _k}$ in such a way that the signature tends to a limit signature. We shall prove that the corresponding representing Fuchsian group converges to the Fuchsian group which corresponds to the limit signature.
Analytic sets as branched coverings
John
Stutz
241-259
Abstract: In this paper we study the relation between the tangent structure of an analytic set V at a point p and the local representation of V as a branched covering. A prototype for our type of result is the fact that one obtains a covering of minimal degree by projecting transverse to the Zariski tangent cone $ {C_3}(V,p)$. We show, for instance, that one obtains the smallest possible branch locus for a branched covering if one projects transverse to the cone $ {C_4}(V,p)$. This and similar results show that points where the various tangent cones ${C_i}(V,p),i = 4,5,6$, have minimal dimension give rise to the simplest branched coverings. This observation leads to the idea of ``Puiseux series normalization", generalizing the situation in one dimension. These Puiseux series allow us to strengthen some results of Hironaka and Whitney on the local structure of certain types of singularities.
On replacing proper Dehn maps with proper embeddings
C. D.
Feustel
261-267
Abstract: In this paper we develop algebraic and geometric conditions which imply that a given proper Dehn map can be replaced by an embedding. The embedding, whose existence is implied by our theorem, retains most of the algebraic and geometric properties required in the original proper Dehn map.
Zeros of partial sums and remainders of power series
J. D.
Buckholtz;
J. K.
Shaw
269-284
Abstract: For a power series $ f(z) = \Sigma _{k = 0}^\infty {a_k}{z^k}$ let ${s_n}(f)$ denote the maximum modulus of the zeros of the nth partial sum of f and let $ {r_n}(f)$ denote the smallest modulus of a zero of the nth normalized remainder $\Sigma _{k = n}^\infty {a_k}{z^{k - n}}$. The present paper investigates the relationships between the growth of the analytic function f and the behavior of the sequences $ \{ {s_n}(f)\}$ and $\{ {r_n}(f)\}$. The principal growth measure used is that of R-type: if $R = \{ {R_n}\}$ is a nondecreasing sequence of positive numbers such that $\lim ({R_{n + 1}}/{R_n}) = 1$, then the R-type of f is $ {\tau _R}(f) = \lim \sup \vert{a_n}{R_1}{R_2} \cdots {R_n}{\vert^{1/n}}$. We prove that there is a constant P such that $\displaystyle {\tau _R}(f)\lim \inf ({s_n}(f)/{R_n}) \leqq P\quad {\text{and}}\quad {\tau _R}(f)\lim \sup ({r_n}(f)/{R_n}) \geqq (1/P)$ for functions f of positive finite R-type. The constant P cannot be replaced by a smaller number in either inequality; P is called the power series constant.
The gliding humps technique for $FK$-spaces
G.
Bennett
285-292
Abstract: The gliging humps technique has been used by various authors to establish the existence of bounded divergent sequences in certain summability domains. The purpose of this paper is to extend these results and to obtain analogous ones for sequence spaces other than c and m. This serves to unify and improve many known results and to obtain several new ones--applications include extensions to theorems of Dawson, Lorentz-Zeller, Snyder-Wilansky and Yurimyae. Improving another result of Wilansky allows us to consider countable collections of sequence spaces--applications including the proof of a conjecture of Hill and Sledd and extensions to theorems of Berg and Brudno. A related result of Petersen is also considered and a simple proof using the Baire category theorem is given.
A class of complete orthogonal sequences of broken line functions
J. L.
Sox
293-296
Abstract: A class of orthonormal sets of continuous broken line functions is defined. Each member is shown to be complete in ${L_2}(0,1)$ and pointwise convergence theorems are obtained for the Fourier expansions relative to these sets.
A new characterization of the $F$ set of a rational function
Marilyn K.
Oba;
Tom S.
Pitcher
297-308
Abstract: In the early part of this century G. Julia and P. Fatou extensively studied the iteration of functions on the complex plane. More recently Hans Brolin reopened the investigation. In this paper, we are interested in the F set which is the set of points at which the family of iterates of a given rational function R is not normal and in a measure which is in some sense naturally imposed on the F set by the iterates of R. We construct a sequence of probability measures via the inverse functions of the iterates of R and almost any starting point. The measure of primary interest is the weak limit of such sequences. These weak limits are supported by F and have certain invariance properties. We establish that this weak limit measure is unique and is ergodic with respect to the transformation R on the F set for a large class of rational functions. In the course of the proof of uniqueness we develop expressions for the logarithmic potential function and for the energy integral of F. We also establish inequalities for the capacity of the F set which become equalities for the polynomial case.
A unified approach to uniform real approximation by polynomials with linear restrictions
Bruce L.
Chalmers
309-316
Abstract: Problems concerning approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions have been studied by several authors. This paper is an attempt to provide a unified approach to these problems. In particular, the notion of restricted derivatives approximation is seen to fit into the theory and includes as special cases the notions of monotone approximation and restricted range approximation. Also bounded coefficients approximation, $ \varepsilon$-interpolator approximation, and polynomial approximation with interpolation fit into our scheme.
Products of complexes and Fr\'echet spaces which are manifolds
James E.
West
317-337
Abstract: It is shown that if a locally finite-dimensional simplicial complex is given the ``barycentric'' metric, then its product with any Fréchet space X of suitably high weight is a manifold modelled on X, provided that X is homeomorphic to its countably infinite Cartesian power. It is then shown that if X is Banach, all paracompact X-manifolds may be represented (topologically) by such products.
Hereditary properties and maximality conditions with respect to essential extensions of lattice group orders
Jorge
Martinez
339-350
Abstract: An l-group will be denoted by the pair (G, P), where G is the group and P is the positive cone. The cone Q is an essential extension of P if every convex l-subgroup of (G, Q) is a convex l-subgroup of (G, P). Q is very essential over P if it is essential over P and for each $0 \ne x \in G$ and each Q-value D of x, there is a unique P-value C of x containing D. We seek conditions which are preserved by essential extensions; normal valuedness and the existence of a finite basis are so preserved. We then investigate l-groups which have the property that their positive cone has no proper very essential extensions. Q is a c-essential extension of P if Q is essential over P and every closed convex l-subgroup of (G, Q) is closed in (G, P). We show that a wreath product of totally ordered groups has no proper very c-essential extensions. We derive a sufficient condition for the nonexistence of such extensions in case the l-group has property (F): no positive element exceeds an infinite set of pairwise disjoint elements.
Segment-preserving maps of partial orders
Geert
Prins
351-360
Abstract: A bijective map from a partial order P to a partial order Q is defined to be segment-preserving if the image of every segment in P is a segment in Q. It is proved that a partial order P with 0-element admits nontrivial segment-preserving maps if and only if P is decomposable in a certain sense. By introducing the concept of ``strong'' segment-preserving maps further insight into the relations between segment-preserving maps and decompositions of partial orders is obtained.
Regular functions $f(z)$ for which $z f'(z)$ is $\alpha$-spiral
Richard J.
Libera;
Michael R.
Ziegler
361-370
Abstract: A function $ f(z) = z + \Sigma _{n = 2}^\infty {a_n}{z^n}$ regular in the open unit disk $\Delta = \{ z:\vert z\vert < 1\}$ is a (univalent) $\alpha$-spiral function for real $\alpha ,\vert\alpha \vert < \pi /2$, if $f(z) \in {\mathcal{F}_\alpha }$. A fundamental result of this paper shows that the transformation $\displaystyle {f_ \ast }(z) = \frac{{azf((z + a)/(1 + \bar az))}}{{f(a)(z + a){{(1 + \bar az)}^{{e^{ - 2i\alpha }}}}}}$ defines a function in $ {\mathcal{F}_\alpha }$ whenever $f(z)$ is in $ {\mathcal{F}_\alpha }$ and a is in $\Delta$. If $g(z)$ is regular in $ \Delta ,g(0) = 0$ and $ g'(0) = 1$, then $ g(z)$ is in ${\mathcal{G}_\alpha }$ if and only if $ zg'(z)$ is in ${\mathcal{F}_\alpha }$. The main result of the paper is the derivation of the sharp radius of close-to-convexity for each class ${\mathcal{G}_\alpha }$; it is given as the solution of an equation in r which is dependent only on $ \alpha$. (Approximate solutions of this equation were made by computer and these suggest that the radius of close-to-convexity of the class $\mathcal{G} = { \cup _\alpha }{\mathcal{G}_\alpha }$ is approximately $ .99097^{+}$.) Additional results are also obtained such as the radius of convexity of $ {\mathcal{G}_\alpha }$, a range of $\alpha$ for which $g(z)$ in $ {\mathcal{G}_\alpha }$ is always univalent is given, etc. These conclusions all depend heavily on the transformation cited above and its analogue for ${\mathcal{G}_\alpha }$.
Two methods of integrating Monge-Amp\`ere's equations. II
Michihiko
Matsuda
371-386
Abstract: Generalizing the notion of an integrable system given in the previous note [2], we shall define an integrable system of higher order, and obtain the following results: 1. A linear hyperbolic equation is solved by integrable systems of order n if and only if its $(n + 1)$th Laplace invariant $ {H_n}$ vanishes. 2. An equation of Laplace type is solved by integrable systems of the second order if and only if the transformed equation by the associated Imschenetsky transformation is solved by integrable systems of the first order.
Operator and dual operator bases in linear topological spaces
William B.
Johnson
387-400
Abstract: A net $\{ {S_d}:d \in D\}$ of continuous linear projections of finite range on a Hausdorff linear topological space V is said to be a Schauder operator basis--S.O.B. --(resp. Schauder dual operator basis--S.D.O.B.) provided it is pointwise bounded and converges pointwise to the identity operator on V, and ${S_e}{S_d} = {S_d}$ (resp. ${S_d}{S_e} = {S_d}$) whenever $e \geqq d$. S.O.B.'s and S.D.O.B.'s are natural generalizations of finite dimensional Schauder bases of subspaces. In fact, a sequence of operators is both a S.O.B. and S.D.O.B. iff it is the sequence of partial sum operators associated with a finite dimensional Schauder basis of subspaces. We show that many duality-theory results concerning Schauder bases can be extended to S.O.B.'s or S.D.O.B.'s. In particular, a space with a S.D.O.B. is semi-reflexive if and only if the S.D.O.B. is shrinking and boundedly complete. Several results on S.O.B.'s and S.D.O.B.'s were previously unknown even in the case of Schauder bases. For example, Corollary IV.2 implies that the strong dual of an evaluable space which admits a shrinking Schauder basis is a complete barrelled space.
Groups with finite dimensional irreducible representations
Calvin C.
Moore
401-410
Abstract: It will be shown that a locally compact group has a finite bound for the dimensions of its irreducible unitary representations if and only if it has a closed abelian subgroup of finite index. It will further be shown that a locally compact group has all of its irreducible representations of finite dimension if and only if it is a projective limit of Lie groups with the same property, and finally that a Lie group has this property if and only if it has a closed subgroup H of finite index such that H modulo its center is compact.
Diffusion semigroups on abstract Wiener space
M. Ann
Piech
411-430
Abstract: The existence of a semigroup of solution operators associated with a second order infinite dimensional parabolic equation of the form $\partial u/\partial t = {L_x}u$ was previously established by the author. The present paper investigates the relationship between ${L_x}$ and the infinitesimal generator $\mathcal{U}$ of the semigroup. In particular, it is shown that $ \mathcal{U}$ is the closure of ${L_x}$ in a natural sense. This in turn implies certain uniqueness results for both the semigroup and for solutions of the parabolic equation.
Uniformly bounded representations for the Lorentz groups
Edward N.
Wilson
431-438
Abstract: A family of uniformly bounded class 1 representations of the Lorentz groups is constructed. This family of representations includes, but is larger than, a similar family of representations constructed by Lipsman. The construction technique relies on a multiplicative analysis of various operators under a Mellin transform.
Categorical $W\sp{\ast} $-tensor product
John
Dauns
439-456
Abstract: If A and B are von Neumann algebras and $A\bar \otimes B$ denotes their categorical ${C^ \ast }$-tensor product with the universal property, then the von Neumann tensor product $ A\nabla B$ of A and B is defined as $\displaystyle A\nabla B = {(A\bar \otimes B)^{ \ast \ast }}/J,$ where $J \subset {(A\bar \otimes B)^{\ast \ast}}$ is an appropriate ideal. It has the universal property.
Simple modules and centralizers
John
Dauns
457-477
Abstract: A class of modules generalizing the simple ones is constructed. Submodule structure and centralizers of quotient modules are completely determined. The above class of modules is used to study the primitive ideal structure of the tensor products of algebras.
Slicing theorems for $n$-spheres in Euclidean $(n+1)$-space
Robert J.
Daverman
479-489
Abstract: This paper describes conditions on the intersection of an n-sphere $\Sigma$ in Euclidean $(n + 1)$-space $ {E^{n + 1}}$ with the horizontal hyperplanes of $ {E^{n + 1}}$ sufficient to determine that the sphere be nicely embedded. The results generally are pointed towards showing that the complement of $\Sigma$ is 1-ULC (uniformly locally 1-connected) rather than towards establishing the stronger property that $\Sigma$ is locally flat. For instance, the main theorem indicates that ${E^{n + 1}} - \Sigma$ is 1-ULC provided each non-degenerate intersection of $\Sigma$ and a horizontal hyperplane be an $ (n - 1)$-sphere bicollared both in that hyperplane and in $\Sigma$ itself $(n \ne 4)$.
An extension of the theorem of Hartogs
L. R.
Hunt
491-495
Abstract: Hartogs proved that every function which is holomorphic on the boundary of the unit ball in $ {C^n},n > 1$, can be extended to a function holomorphic on the ball itself. It is conjectured that a real k-dimensional $ {\mathcal{C}^\infty }$ compact submanifold of $ {C^n},k > n$, is extendible over a manifold of real dimension $(k + 1)$. This is known for hypersurfaces (i.e., $k = 2n - 1$) and submanifolds of real codimension 2. It is the purpose of this paper to prove this conjecture and to show that we actually get C-R extendibility.
On the automorphism group of a reduced primary Abelian group
Kai
Faltings
1-25
Abstract: The classical theorems concerning isomorphisms and automorphisms of full linear groups are generalized to reduced primary Abelian groups and their automorphism groups. Also, a duality theory for (not necessarily finite) reduced Abelian p-groups is presented.
Strong convergence of functions on K\"othe spaces
Gerald
Silverman
27-35
Abstract: Let $\Lambda$ be a rearrangement invariant Köthe space over a nondiscrete group G with Haar measure $\mu$. For a function $f \in \Lambda$ and relatively compact 0-neighborhood U in G the function $\displaystyle {T_U}f(x) = \frac{1}{{\mu (U)}} \cdot \int_{U + x} {f\,d\mu }$ is continuous and also belongs to $\Lambda$. The convergence ${T_U}f \to f$ (as $U \to 0$) for the strong Köthe topology on $ \Lambda$ is involved in establishing compactness criteria for subsets of a Köthe space. The main result of this paper is a necessary and sufficient condition for convergence ${T_U}f \to f$ in the strong topology on $ \Lambda$.
A constructive approach to the theory of stochastic processes
Yuen Kwok
Chan
37-44
Abstract: Some basic problems in probability theory will be considered with the constructive point of view. Among them are the construction of measurable stochastic processes from finite joint probabilities, and the construction of interesting random variables related to a given process. These random variables include (1) the first instant when a process has spent a definite length of time in a definite set, and (2) the value of another process at such an instant.
A method for shrinking decompositions of certain manifolds
Robert D.
Edwards;
Leslie C.
Glaser
45-56
Abstract: A general problem in the theory of decompositions of topological manifolds is to find sufficient conditions for the associated decomposition space to be a manifold. In this paper we examine a certain class of decompositions and show that the nondegenerate elements in any one of these decompositions can be shrunk to points via a pseudo-isotopy. It follows then that the decomposition space is a manifold homeomorphic to the original one. As corollaries we obtain some results about suspensions of homotopy cells and spheres, including a new proof that the double suspension of a Poincaré 3-sphere is a real topological 5-sphere.
Successive approximations in ordered vector spaces and global solutions of nonlinear Volterra integral equations
Terrence S.
McDermott
57-64
Abstract: Conditions are found under which a nonlinear operator in an ordered topological vector space will have a fixed point. This result is applied to study a nonlinear Volterra integral operator in the space of continuous, real valued functions on $ [0,\infty )$ equipped with the topology of uniform convergence on compact subsets. Two theorems on the global existence of solutions to the related Volterra integral equation as limits of successive approximations are proved in this manner.
The oscillation of an operator
Robert
Whitley
65-73
Abstract: Foiaş and Singer introduced the oscillation of a bounded linear operator mapping $C(S)$ into a Banach space. Using this concept we define a generalization of the Fredholm operators T with $\mathcal{K}(T) < \infty $ and a corresponding perturbation class which contains the weakly compact operators. We show that a bounded linear operator on c is a conservative summability matrix which sums every bounded sequence if and only if it has zero oscillation at infinity.
A Fredholm theory for a class of first-order elliptic partial differential operators in ${\bf R}\sp{n}$
Homer F.
Walker
75-86
Abstract: The objects of interest are linear first-order elliptic partial differential operators with domain ${H_1}({R^n};{C^k})$ in ${L_2}({R^n};{C^k})$, the first-order coefficients of which become constant and the zero-order coefficient of which vanishes outside a compact set in $ {R^n}$. It is shown that operators of this type are ``practically'' Fredholm in the following way: Such an operator has a finite index which is invariant under small perturbations, and its range can be characterized in terms of the range of an operator with constant coefficients and a finite index-related number of orthogonality conditions.
Characterization of precompact maps, Schwartz spaces and nuclear spaces
Dan
Randtke
87-101
Abstract: A general representation theorem for ``precompact'' seminorms on a locally convex space is proven. Using this representation theorem the author derives a representation theorem for precompact maps from one locally convex space into another, that is analogous to the spectral representation theorem for compact maps from one Hilbert space into another and that is applicable to a very extensive class of locally convex spaces. The author uses his representation theorem to derive new characterizations of Schwartz spaces and proves analogous results for nuclear and strongly nuclear spaces.
Set-valued measures
Zvi
Artstein
103-125
Abstract: A set-valued measure is a $\sigma$-additive set-function which takes on values in the nonempty subsets of a euclidean space. It is shown that a bounded and non-atomic set-valued measure has convex values. Also the existence of selectors (vector-valued measures) is investigated. The Radon-Nikodym derivative of a set-valued measure is a set-valued function. A general theorem on the existence of R.-N. derivatives is established. The techniques require investigations of measurable set-valued functions and their support functions.
Sets of uniqueness on the $2$-torus
Victor L.
Shapiro
127-147
Abstract: $ {H^{(J)}}$-sets are defined on the 2-torus and the following results are established: (1) ${H^{(J)}}$-sets are sets of uniqueness both for Abel summability and circular convergence of double trigonometric series; (2) a countable union of closed sets of uniqueness of type (A) (i.e., Abel summability) is also a set of uniqueness of type (A).
Principal local ideals in weighted spaces of entire functions
James J.
Metzger
149-158
Abstract: This paper deals with principal local ideals in a class of weighted spaces of entire functions of one variable. Let $\rho > 1$ and $q > 1$, and define ${E_I}[\rho ,q]$ (respectively, ${E_P}[\rho ,q]$) to be the space of all entire functions f of one variable which satisfy $\vert f(x + iy)\vert = O(\exp (A\vert x{\vert^\rho } + A\vert y{\vert^q}))$ for some (respectively, all) $A > 0$. It is shown that in each of the spaces $ {E_I}[\rho ,q]$ and ${E_P}[\rho ,q]$, the local ideal generated by any one function coincides with the closed ideal generated by the function. This result yields consequences for convolution on these spaces. It is also proved that when $\rho \ne q$ a division theorem fails to hold for either space $ {E_I}[\rho ,q]$ or ${E_P}[\rho ,q]$.
A representation theorem for functions holomorphic off the real axis
Albert
Baernstein
159-165
Abstract: Let f be holomorphic in the union of the upper and lower half planes, and let $p \in [1,\infty )$. We prove that there exists an entire function $\varphi$ and a sequence $\{ {f_n}\}$ in ${L^p}(R)$ satisfying $\left\Vert {{f_n}} \right\Vert _p^{1/n} \to 0$ such that $\displaystyle f(z) = \varphi (z) + \sum\limits_{n = 0}^\infty {\int_{ - \infty }^\infty {{{(t - z)}^{ - n - 1}}{f_n}(t)dt.} }$ This complements an earlier result of the author's on representation of function holomorphic outside a compact subset of the Riemann sphere. A principal tool in both proofs is the Köthe duality between the spaces of functions holomorphic on and off a subset of the sphere. A corollary of the present result is that each hyperfunction of one variable can be represented by a sum of Cauchy integrals over the real axis.
An energy inequality for higher order linear parabolic operators and its applications
David
Ellis
167-206
Abstract: A generalization of the classical energy inequality is obtained for evolution operators $(\partial /\partial t)I - H(t){\Lambda ^{2k}} - J(t)$, associated with higher order linear parabolic operators with variable coefficients. Here $H(t)$ and $J(t)$ are matrices of singular integral operators. The key to the result is an algebraic inequality involving matrices similar to the symbol of $H(t)$ having their eigenvalues contained in a fixed compact subset of the open left-half complex plane. Then a sharp estimate on the norms of certain imbedding maps is obtained. These estimates along with the energy inequality is applied to the Cauchy problem for higher order linear parabolic operators restricted to slabs in $ {R^{n + 1}}$.
Weighted norm inequalities for the Hardy maximal function
Benjamin
Muckenhoupt
207-226
Abstract: The principal problem considered is the determination of all nonnegative functions, $U(x)$, for which there is a constant, C, such that $\displaystyle \int_J {{{[{f^ \ast }(x)]}^p}U(x)dx \leqq C\int_J {\vert f(x){\vert^p}U(x)dx,} }$ where $1 < p < \infty$, J is a fixed interval, C is independent of f, and ${f^ \ast }$ is the Hardy maximal function, $\displaystyle {f^ \ast }(x) = \mathop {\sup }\limits_{y \ne x;y \in J} \frac{1}{{y - x}}\int_x^y {\vert f(t)\vert dt.}$ The main result is that $ U(x)$ is such a function if and only if $\displaystyle \left[ {\int_I {U(x)dx} } \right]{\left[ {\int_I {{{[U(x)]}^{ - 1/(p - 1)}}dx} } \right]^{p - 1}} \leqq K\vert I{\vert^p}$ where I is any subinterval of J, $\vert I\vert$ denotes the length of I and K is a constant independent of I. Various related problems are also considered. These include weak type results, the problem when there are different weight functions on the two sides of the inequality, the case when $p = 1$ or $ p = \infty$, a weighted definition of the maximal function, and the result in higher dimensions. Applications of the results to mean summability of Fourier and Gegenbauer series are also given.
Sequences having an effective fixed-point property
T. H.
Payne
227-237
Abstract: Let $\alpha$ be any function whose domain is the set N of all natural numbers. A subset B of N precompletes the sequence $\alpha$ if and only if for every partial recursive function (p.r.f.) $\psi$ there is a recursive function f such that $\alpha f$ extends $ \alpha \psi$ and $ f[N - \operatorname{Dom} \psi ] \subset B$. An object e in the range of $ \alpha$ completes $ \alpha$ if and only if ${\alpha ^{ - 1}}[\{ e\} ]$ precompletes $ \alpha$. The theory of completed sequences was introduced by A. I. Mal'cev as an abstraction of the theory of standard enumerations. In this paper several results are obtained by refining and extending his methods. It is shown that a sequence is precompleted (by some B) if and only if it has a certain effective fixed-point property. The completed sequences are characterized, up to a recursive permutation, as the composition $F\varphi$ of an arbitrary function F defined on the p.r.f.'s with a fixed standard enumeration $\varphi$ of the p.r.f.'s. A similar characterization is given for the precompleted sequences. The standard sequences are characterized as the precompleted indexings which satisfy a simple uniformity condition. Several further properties of completed and precompleted sequences are presented, for example, if B precompletes $\alpha$ and S and T are r.e. sets such that ${\alpha ^{ - 1}}[\alpha [S]] \ne N$ and $ {\alpha ^{ - 1}}[\alpha [T]] \ne N$, then $ B - (S \cup T)$ precompletes $\alpha$.
Upon a convergence result in the theory of the Pad\'e table
P.
Wynn
239-249
Abstract: The main theorem of this paper is the following: Let $ {M_\nu },{b_\nu }(\nu = 1,2, \ldots ,n)$ be two sets of finite positive real numbers, with $ {b_1} > {b_2} > \cdots > {b_n}$, and let $\sigma (\varsigma )$ be a bounded nondecreasing function for $a \leqq \varsigma \leqq b$ where $0 \leqq a \leqq b < {b_n}$; denote the Padé quotients derived from the series expansion of the function $\displaystyle f(z) = \sum\limits_{\nu = 1}^n {\frac{{{M_\nu }}}{{(1 + {b_\nu }z)}} + \int_a^b {\frac{{d\sigma (\varsigma )}}{{1 + z\varsigma }}} }$ in ascending powers of z by $\{ {R_{i,j}}(z)\} ;$ let $ \mathfrak{D}$ be the open disc $\vert z\vert < {b^{ - 1}}$ cut along the real segment $( - {b^{ - 1}}, - b_1^{ - 1}];$ define a progressive sequence of Padé quotients to be one in which the successor $ {R_{i'',j''}}(z)$ to ${R_{i',j'}}(z)$ is such that either $i'' > i'$ and ${R_{n + r,n + n' + r}}(z)$, where $ n'$ is a finite nonnegative integer and $ r = 0,1, \ldots ,$ converge uniformly for $ z \in \mathfrak{D}$ to $ f(z)$. From a theorem of de Montessus de Ballore the row sequence ${R_{n,n + r}}(z)(r = 0,1, \ldots )$ converges uniformly for $z \in \mathfrak{D}$ to $f(z)$. From a result of the author the backward diagonal sequences ${R_{n + r,2m - n - r}}(z)(r = 0,1, \ldots ,m - n)$ and ${R_{n + r,2m - n - r + 1}}(z)(r = 0,1, \ldots ,m - n + 1)$, where m is a finite positive integer, are, when z is real and positive, respectively monotonically decreasing and monotonically increasing. Hence the result of the theorem is true for the restricted progressive sequences in question when z is real and positive. Using the result of de Montessus de Ballore, and extending a result of Nevanlinna to the theory of the Padé table in question, it is shown that there exists a finite positive integer $ r'$ such that all quotients ${R_{n + r,n + r'' + r}}(r = 0,1, \ldots ;r'' = r',r' + 1, \ldots )$ are uniformly bounded for $ \mathfrak{D}$ from which points lying in the neighborhood of the negative real axis have been excluded. Thus, using the Stieltjes-Vitali theorem, all progressive sequences of Padé quotients taken from the latter double array converge uniformly for $z \in \mathfrak{D}';$ that this result also holds for values of $ z \in \mathfrak{D}$ lying in the neighborhood of the negative real axis (and not, therefore, belonging to ${R_{n + r,n}}(z)(r = 0,1, \ldots )$. The two partial results are then combined.
On the degrees and rationality of certain characters of finite Chevalley groups
C. T.
Benson;
C. W.
Curtis
251-273
Abstract: Let $\mathcal{S}$ be a system of finite groups with (B, N)-pairs, with Coxeter system (W, R) and set of characteristic powers $\{ q\}$ (see [4]). Let A be the generic algebra of the system, over the polynomial ring $\mathfrak{o} = Q[u]$. Let K be $Q(u)$, K an algebraic closure of K, and $ {\mathfrak{o}^ \ast }$ the integral closure of $ \mathfrak{o}$ in K. For the specialization $f:u \to q$ mapping $\mathfrak{o} \to Q$, let ${f^ \ast }:{\mathfrak{o}^ \ast } \to \bar Q$ be a fixed extension of f. For each irreducible character $\chi$ of the algebra $ {A^{\bar K}}$, there exists an irreducible character ${\zeta _{\chi ,{f^ \ast }}}$ of the group $ G(q)$ in the system corresponding to q, such that $({\zeta _{\chi ,{f^ \ast }}},1_{B(q)}^{G(q)}) > 0$, and $\chi \to {\zeta _{\chi ,{f^ \ast }}}$ is a bijective correspondence between the irreducible characters of ${A^{\bar K}}$ and the irreducible constituents of $ 1_{B(q)}^{G(q)}$. Assume almost all primes occur among the characteristic powers $ \{ q\}$. The first main result is that, for each $\chi$, there exists a polynomial ${d_\chi }(t) \in Q[t]$ such that, for each specialization $f:u \to q$, the degree ${\zeta _{\chi ,{f^ \ast }}}(1)$ is given by ${d_\chi }(q)$. The second result is that, with two possible exceptions in type ${E_7}$, the characters ${\zeta _{\chi ,{f^ \ast }}}$ are afforded by rational representations of $G(q)$.
On the asymptotic behaviour of nonnegative solutions of a certain integral inequality
Gunnar A.
Brosamler
275-289
Abstract: The asymptotic behaviour of nonnegative solutions of a certain integral inequality is discussed, in the framework of a probabilistic-potential theoretic boundary theory.
A generalized Weyl equidistribution theorem for operators, with applications
J. R.
Blum;
V. J.
Mizel
291-307
Abstract: The present paper is motivated by the observation that Weyl's equidistribution theorem for real sequences on a bounded interval can be formulated in a way which is also meaningful for sequences of selfadjoint operators on a Hilbert space. We shall provide general results on weak convergence of operator measures which yield this version of Weyl's theorem as a corollary. Further, by combining the above results with the von Neumann ergodic theorem, we will obtain a Cesàro convergence property, equivalently, an ``ergodic theorem", which is valid for all (projection-valued) spectral measures whose support is in a bounded interval, as well as for the more general class of positive operator-valued measures. Within the same circle of ideas we deduce a convergence property which completely characterizes those spectral measures associated with ``strongly mixing'' unitary transformations. The final sections are devoted to applications of the preceding results in the study of complex-valued Borel measures as well as to an extension of our results to summability methods other than Cesàro convergence. In particular, we obtain a complete characterization, in purely measure theoretic terms, of those complex measures on a bounded interval whose Fourier-Stieltjes coefficients converge to zero.
Schauder bases in the Banach spaces $C\sp{k}({\bf T}\sp{q})$
Steven
Schonefeld
309-318
Abstract: A Schauder basis is constructed for the space $ {C^k}({T^q})$ of k-times continuously differentiable functions on $ {T^q}$, the product of q copies of the one-dimensional torus. This basis has the property that is also a basis for the spaces $ {C^1}({T^q}),{C^2}({T^q}), \ldots ,{C^{k - 1}}({T^q})$, and an interpolating basis for $C({T^q})$.
A characterization of compact multipliers
Gregory F.
Bachelis;
Louis
Pigno
319-322
Abstract: Let G be a compact abelian group and $\varphi$ a complex-valued function defined on the dual $\Gamma$. The main result of this paper is that $ \varphi$ is a compact multiplier of type $(p,q),1 \leqq p < \infty $ and $1 \leqq q \leqq \infty$, if and only if it satisfies the following condition: Given $\varepsilon > 0$ there corresponds a finite set $K \subset \Gamma $ such that $\vert\sum {a_\gamma }{b_\gamma }\varphi (\gamma )\vert < \varepsilon$ whenever $P = \sum {a_\gamma }\gamma $ and $Q = \sum {b_\gamma }\gamma $ are trigonometric polynomials satisfying
The local behavior of principal and chordal principal cluster sets
John T.
Gresser
323-332
Abstract: Let K be the unit circle, and let f be a function whose domain is the open unit disk and whose range is a subset of the Riemann sphere. We define a set, called the boundary principal cluster set of f at ${\zeta _0} \in K$, which characterizes the behavior of the principal cluster sets of f at points $\zeta \in K$ which are near ${\zeta _0}$ and distinct from ${\zeta _0}$. It is shown that if f is continuous, then the principal and boundary principal cluster sets of f at $ {\zeta _0}$ are equal for nearly every point $ {\zeta _0} \in K$. A similar result holds for chordal principal cluster sets. Examples are provided that indicate directions in which the result cannot be improved. Some results concerning points that are accessible through sets which are unions of arcs are also presented.
Removable sets for pointwise solutions of elliptic partial differential equations
Jim
Diederich
333-352
Abstract: We prove that dense sets of zero newtonian capacity are removable for bounded generalized pointwise solutions of second order elliptic equations.
Relative imaginary quadratic fields of class number $1$ or $2$
Larry Joel
Goldstein
353-364
Abstract: Let K be a normal totally real algebraic number field. It is shown how to effectively classify all totally imaginary quadratic extensions of class number 1. Let K be a real quadratic field of class number 1, whose fundamental unit has norm $- 1$. Then it is shown how to effectively classify all totally imaginary quadratic extensions of class number 2.
The $(\phi\sp{2n})\sb{2}$ field Hamiltonian for complex coupling constant
Lon
Rosen;
Barry
Simon
365-379
Abstract: We consider hamiltonians ${H_\beta } = {H_0} + \beta {H_I}(g)$, where $ {H_0}$ is the hamiltonian of a free Bose field $\phi (x)$ of mass $m > 0$ in two-dimensional space-time, ${H_I}(g) = \smallint g(x):P(\phi (x)):dx$ where $ g \geqq 0$ is a spatial cutoff and P is an arbitrary polynomial which is bounded below, and the coupling constant $ \beta$ is in the cut plane, i.e. $\beta \ne$ negative real. We show that ${H_\beta }$ generates a semigroup with hypercontractive properties and satisfies higher order estimates of the form $\left\Vert {{H_0}{N^r}R_\beta ^s} \right\Vert < \infty$, where N is the number operator, ${R_\beta } = {({H_\beta } + b)^{ - 1}}$, r a positive integer, and $\beta$, s, and b are suitably chosen. For any $ 0 \leqq \Theta < \pi$, ${R_\beta }$ converges in norm to ${R_0}$ as $\vert\beta \vert \to 0$ with $\vert\arg \beta \vert \leqq \Theta$. Finally we discuss applications of these results and establish asymptotic series and Borel summability for various objects in the real $\beta$ theory.
$L\,\sb{p}$ derivatives and approximate Peano derivatives
Michael J.
Evans
381-388
Abstract: It is known that approximate derivatives and kth Peano derivatives share several interesting properties with ordinary derivatives. In this paper the author points out that kth ${L_p}$ derivatives also share these properties. Furthermore, a definition for a kth approximate Peano derivative is given which generalizes the notions of a kth Peano derivative, a kth $ {L_p}$ derivative, and an approximate derivative. It is then shown that a kth approximate Peano derivative at least shares the property of belonging to Baire class one with these other derivatives.
Trace algebras
R. P.
Sheets
389-423
Abstract: We give an algebraic unification for those mathematical structures which possess the abstract properties of finite-dimensional vector spaces: scalars, duality theories, trace functions, etc. The unifying concept is the ``trace algebra,'' which is a set with a ternary operation which satisfies certain generalized associativity and identity laws. Every trace algebra induces naturally an object which (even though no additive structure may be available) possesses a summation operator and inner product which obey the Fourier expansion and other familiar properties. We construct the induced object in great detail. The ultimate results of the paper are: a theorem which shows that the induced object of a ``well-behaved'' trace algebra determines it uniquely; and a theorem which shows that well-behaved trace algebras look, formally, like the trace algebras associated with finite-dimensional vector spaces.
Linearization for the Boltzmann equation
F. Alberto
Grünbaum
425-449
Abstract: In this paper we compare the nonlinear Boltzmann equation appearing in the kinetic theory of gases, with its linearized version. We exhibit an intertwining operator for the two semigroups involved. We do not assume from the reader any familiarity with Boltzmann's equation but rather start from scratch.
$2$-groups of normal rank $2$ for which the Frattini subgroup has rank $3$
Marc W.
Konvisser
451-469
Abstract: All finite 2-groups G with the following property are classified: Property. The Frattini subgroup of G contains an abelian subgroup of rank 3, but G contains no normal abelian subgroup of rank 3. The method of classification involves showing that if G is such a group, then G contains a normal abelian subgroup W isomorphic to ${Z_4} \times {Z_4}$, and that the centralizer C of W in G has an uncomplicated structure. The groups with the above property are then constructed as extensions of C.
Existence theorems for infinite particle systems
Thomas M.
Liggett
471-481
Abstract: Sufficient conditions are given for a countable sum of bounded generators of semigroups of contractions on a Banach space to be a generator. This result is then applied to obtain existence theorems for two classes of models of infinite particle systems. The first is a model of a dynamic lattice gas, while the second describes a lattice spin system.
Integrability of expected increments of point processes and a related random change of scale
F.
Papangelou
483-506
Abstract: Given a stationary point process with finite intensity on the real line R, denote by $N(Q)$ (Q Borel set in R) the random number of points that the process throws in Q and by ${\mathcal{F}_t}(t \in R)$ the $\sigma$-field of events that happen in $( - \infty ,t)$. The main results are the following. If for each partition $ \Delta = \{ b = {\xi _0} < {\xi _1} < \cdots < {\xi _{n + 1}} = c\}$ of an interval [b, c] we set ${S_\Delta }(\omega ) = \sum\nolimits_{\nu = 0}^n {E(N[{\xi _\nu },{\xi _{\nu + 1}})\vert{\mathcal{F}_{{\xi _\nu }}})}$ then ${\lim _\Delta }{S_\Delta }(\omega ) = W(\omega ,[b,c))$ exists a.s. and in the mean when ${\max _{0 \leqq \nu \leqq n}}({\xi _{\nu + 1}} - {\xi _\nu }) \to 0$ (the a.s. convergence requires a judicious choice of versions). If the random transformation $ t \Rightarrow W(\omega ,[0,1))$ of $ [0,\infty )$ onto itself is a.s. continuous (i.e. without jumps), then it transforms the nonnegative points of the process into a Poisson process with rate 1 and independent of ${\mathcal{F}_0}$. The ratio $ {\varepsilon ^{ - 1}}E(N[0,\varepsilon )\vert{\mathcal{F}_0})$ converges a.s. as $\varepsilon \downarrow 0$. A necessary and sufficient condition for its convergence in the mean (as well as for the a.s. absolute continuity of the function $W[0,t)$ on $ (0,\infty ))$ is the absolute continuity of the Palm conditional probability $ {P_0}$ relative to the absolute probability P on the $\sigma $-field ${\mathcal{F}_0}$. Further results are described in §1.
Stability theorems for some functional equations
R. C.
MacCamy;
J. S. W.
Wong
1-37
Abstract: Functional-differential equations of the form $\displaystyle \dot u(t) = - \int_0^t {A(t - \tau )g(u(\tau ))d\tau + f(t,u(t))}$ are considered. Here $ u(t)$ is to be an element of a Hilbert space $ \mathcal{H},A(t)$ a family of bounded symmetric operators on $\mathcal{H}$ and g an operator with domain in $\mathcal{H}$. g may be unbounded. A is called strongly positive if there exists a semigroup exp St, where S is symmetric and $(S\xi ,\xi ) \leqq - m{\left\Vert \xi \right\Vert^2},m > 0$, such that ${A^ \ast } = A - \exp$ St is positive, that is, $\displaystyle \mathop \int \nolimits_0^T \left( {v(t),\int_0^t {{A^\ast}(t - \tau )v(\tau )} } \right)d\tau \geqq 0,$ for all smooth $v(t)$. It is shown that if A is strongly positive, and g and f are suitably restricted, then any solution which is weakly bounded and uniformly continuous must tend weakly to zero. Examples are given of both ordinary and partial differential-functional equations.
Two point boundary value problems for nonlinear functional differential equations
Paul
Waltman;
James S. W.
Wong
39-54
Abstract: This paper is concerned with the existence of solutions of two point boundary value problems for functional differential equations. Specifically, we consider $L(t,{y_t}) = A(t)y(t)$, that is, when the reduced linear equation is an ordinary rather than a functional differential equation. Several examples are discussed to illustrate the results.
Inverse limits, entropy and weak isomorphism for discrete dynamical systems
James R.
Brown
55-66
Abstract: A categorical approach is taken to the study of a single measure-preserving transformation of a finite measure space and to inverse systems and inverse limits of such transformations. The questions of existence and uniqueness of inverse limits are settled. Sinai's theorem on generators is recast and slightly extended to say that entropy respects inverse limits, and various known results about entropy are obtained as immediate corollaries, e.g. systems with quasi-discrete or quasi-periodic spectrum have zero entropy. The inverse limit $\Phi$ of an inverse system $\{ {\Phi _\alpha }:\alpha \in J\}$ of dynamical systems is (1) ergodic, (2) weakly mixing, (3) mixing (of any order) iff each ${\Phi _\alpha }$ has the same property. Finally, inverse limits are used to lift a weak isomorphism of dynamical systems ${\Phi _1}$ and ${\Phi _2}$ to an isomorphism of systems ${\hat \Phi _1}$ and $ {\hat \Phi _2}$ with the same entropy.
Elementary statements over large algebraic fields
Moshe
Jarden
67-91
Abstract: We prove here the following theorems: A. If k is a denumerable Hilbertian field then for almost all $({\sigma _1}, \ldots ,{\sigma _e}) \in \mathcal{G}{({k_s}/k)^e}$ the fixed field of $\{ {\sigma _1}, \ldots ,{\sigma _e}\} ,{k_s}({\sigma _1}, \ldots ,{\sigma _e})$, has the following property: For any non-void absolutely irreducible variety V defined over $ {k_s}({\sigma _1}, \ldots ,{\sigma _e})$ the set of points of V rational over K is not empty. B. If E is an elementary statement about fields then the measure of the set of $\sigma \in \mathcal{G}(\tilde Q/Q)$ (Q is the field of rational numbers) for which E holds in $ \tilde Q(\sigma )$ is equal to the Dirichlet density of the set of primes p for which E holds in the field ${F_p}$ of p elements.
A note on quadratic Jordan algebras of degree $3$
M. L.
Racine
93-103
Abstract: McCrimmon has defined a class of quadratic Jordan algebras of degree 3 obtained from a cubic form, a quadratic mapping and a base point. The structure of such an algebra containing no absolute zero-divisor is determined directly. A simple proof of Springer's Theorem on isomorphism of reduced simple exceptional quadratic Jordan algebras is given.
Nonresidually finite one-relator groups
Stephen
Meskin
105-114
Abstract: The study of one-relator groups includes the connections between group properties and the form of the relator. In this paper we discuss conditions on the form ${u^{ - 1}}{v^l}u{v^m}$ which force the corresponding one-relator groups to be nonresidually finite, i.e. the intersection of the normal subgroups of finite index to be nontrivial. Moreover we show that these forms can be detected amongst the words of a free group.
A constructive ergodic theorem
J. A.
Nuber
115-137
Abstract: As discussed by Bishop, Birkhoff's Ergodic Theorem is not constructively valid. In this paper we present an hypothesis which is necessary and sufficient for the constructive almost everywhere convergence of the Césaro averages of the translates of an integrable function by a measure preserving transformation. In addition necessary and sufficient conditions are given for the limit function to be constructively integrable. Also we present a necessary and sufficient condition that the averages converge to a constant function and give an equivalent formulation of this condition for finite measure spaces. Several interesting examples are given which satisfy these conditions.
Slices of maps and Lebesgue area
William P.
Ziemer
139-151
Abstract: For a large class of k dimensional surfaces, S, it is shown that the Lebesgue area of S can be essentially expressed in terms of an integral of the $k - 1$ area of a family, F, of $k - 1$ dimensional surfaces that cover S. The family F is regarded as being composed of the slices of F. The definition of the $k - 1$ area of a surface restricted to one of its slices is formulated in terms of the theory developed by H. Federer, [F3].
The size function of abelian varieties
Allen
Altman
153-161
Abstract: The size function is defined for points in projective space over any field K, finitely generated field over Q, generalizing the height function for number fields. We prove that the size function on the K-rational points of an abelian variety is bounded by a quadratic function.
One-dimensional basic sets in the three-sphere
Joel C.
Gibbons
163-178
Abstract: This paper is a continuation of Williams' classification of one-dimensional attracting sets of a diffeomorphism on a compact manifold [Topology 6 (1967)]. After defining the knot presentation of a solenoid in ${S^3}$ and some knottheoretic preliminaries, we prove Theorem: If ${\sum _1},{h_1}$ and $ {\sum _2},{h_2}$ are shift classes of oriented solenoids admitting elementary presentations K, $K,{g_1}$ and K, $K,{g_2}$, resp., where $ {g_1}^ \ast = {({g_2}^ \ast )^t}:{H_1}(K) \to {H_1}(K)$, there is an Anosov-Smale diffeomorphism f of ${S^3}$ such that $\Omega (f)$ consists of a source ${\Lambda ^ - }$ and a sink ${\Lambda ^ + }$ for which ${\Lambda ^ + },f/{\Lambda ^ + }$ and $ {\Lambda ^ - },{f^{ - 1}}/{\Lambda ^ - }$ are conjugate, resp., to ${\sum _1},{h_1}$ and ${\sum _2},{h_2}$. (The author has proved [Proc. Amer. Math. Soc., to appear] that if f is an Anosov-Smale map of $ {S^3},\Omega (f)$ has dimension one, and contains no hyperbolic sets, then f has the above structure.) We also prove Theorem: there is a nonempty ${C^1}$-open set ${F_2}$ in the class of such diffeomorphisms for which $K = {S^1}$ and $ {g_1} = {g_2}$ is the double covering such that each f in ${F_2}$ defines a loop t in $ {S^3}$, stable up to $ {C^1}$ perturbations, for which at every x in t the generalized stable and unstable manifolds through x are tangent at x.
Local to global theorems in the theory of Hurewicz fibrations
James
Arnold
179-188
Abstract: This paper is concerned with the problem of showing a local fibration is a fibration. There are two kinds of local to global theorems proven. The first type of theorem considers local fibrations where local is in terms of closed covers of the base (e.g. the set of closed simplices of a polyhedron, the cones of a suspension). The second type of theorem deals with local in terms of open covers of the total space.
The structure of certain unitary representations of infinite symmetric groups
Arthur
Lieberman
189-198
Abstract: Let S be an infinite set, $\beta$ an infinite cardinal number, and ${G_\beta }(S)$ the group of those permutations of S whose support has cardinal number less than $ \beta$. If T is any nonempty set, ${S^T}$ is the set of functions from T to S. The canonical representation $\Lambda _\beta ^T$ of ${G_\beta }(S)$ on $ {L^2}({S^T})$ is the direct sum of factor representations. Factor representations of types ${{\text{I}}_\infty },{\text{II}_1}$, and $ {\text{II}_\infty }$ occur in this decomposition, depending upon S, $ \beta$, and T; the type $ {\text{II}_1}$ factor representations are quasi-equivalent to the left regular representation. Let ${G_\beta }(S)$ have the topology of pointwise convergence on S. $ {G_\beta }(S)$ is a topological group but is not locally compact. Every continuous representation of $ {G_\beta }(S)$ is the direct sum of irreducible representations. Let $ \Gamma$ be a nontrivial continuous irreducible representation of ${G_\beta }(S)$. Then $\Gamma$ is continuous iff $\Gamma$ is equivalent to a subrepresentation of $ \Lambda _\beta ^T$ for some nonempty finite set T iff there is a nonempty finite subset Z of S such that the restriction of $\Gamma$ to the subgroup of those permutations which leave Z pointwise fixed contains the trivial representation of this subgroup.
Weakly wandering vectors and weakly independent partitions
Ulrich
Krengel
199-226
Abstract: We first characterize continuous spectrum and purely discrete spectrum of an isometry U of a Hilbert space geometrically by the existence of a spanning system, resp. by the absence, of vectors with infinitely many orthogonal images under powers of U. We then characterize weak mixing and discrete spectrum of an invertible measure preserving transformation of a probability space in terms of the null sets of the space. Finally for two-fold weakly mixing transformations the result on isometries is strengthened by proving the density of the set of partitions with infinitely many mutually independent images in the set of all finite partitions.
Quotient sheaves and valuation rings
Joel
Cunningham
227-239
Abstract: In this paper a construction of a quotient sheaf of a sheaf of rings is given. This construction is analogous to the Utumi ring of quotients of a ring. For a valuation ring V, a sheaf of rings corresponding to V is introduced and its quotient sheaf is computed. It is shown that this quotient sheaf corresponds to the completion of V in case V is discrete rank one and that V is maximal if and only if its associated sheaf of rings is its own quotient sheaf.
Almost recursively enumerable sets
John W.
Berry
241-253
Abstract: An injective function on N, the nonnegative integers, taking values in N, is called almost recursive (abbreviated a.r.) if its inverse has a partial recursive extension. The range of an a.r. function f is called an almost recursively enumerable set in general; an almost recursive set if in addition f is strictly increasing. These are natural generalizations of regressive and retraceable sets respectively. We show that an infinite set is almost recursively enumerable iff it is point decomposable in the sense of McLaughlin. This leads us to new characterizations of certain classes of immune sets. Finally, in contrast to the regressive case, we show that a.r. functions and sets are rather badly behaved with respect to recursive equivalence.
On a convolution theorem for $L(p,q)$ spaces
A. P.
Blozinski
255-265
Abstract: The principal result of this paper is a proof of the Convolution Theorem based on the definition of a convolution operator as presented by E. M. Stein and R. O'Neil. Closely related are earlier versions and special cases of the Convolution Theorem, which are $L(p,q)$ analogues of an inequality of W. H. Young, given in papers by R. O'Neil, L. Y. H. Yap, R. Hunt, and B. Muckenhoupt and E. M. Stein.
An optimization problem for unitary and orthogonal representations of finite groups
D. Ž.
Djoković;
I. F.
Blake
267-274
Abstract: Let $G \to {\text{GL}}(V)$ be a faithful orthogonal representation of a finite group G acting in an Euclidean space V. For a unit vector x we choose $g \ne 1$ in G so that $\vert gx - x\vert$ is minimal and put $\delta (x) = \vert gx - x\vert$. We study the class of vectors x which maximize $\delta (x)$ and have the additional property that $ \vert gx - x\vert$ depends only on the conjugacy class of $g \in G$. For some special types of representations we are able to characterize completely this class of vectors.
The space of homeomorphisms on a compact two-manifold is an absolute neighborhood retract
R.
Luke;
W. K.
Mason
275-285
Abstract: The theorem mentioned in the title is proved.
Some open mapping theorems for measures
Seymour
Ditor;
Larry Q.
Eifler
287-293
Abstract: Given a compact Hausdorff space X, let $C(X)$ be the Banach space of continuous real valued functions on X with sup norm and let $M(X)$ be its dual considered as finite regular Borel measures on X. Let $U(X)$ denote the closed unit ball of $ M(X)$ and let $ P(X)$ denote the nonnegative measures in $M(X)$ of norm 1. A continuous map $ \varphi$ of X onto another compact Hausdorff space Y induces a natural linear transformation $\pi$ of $M(X)$ onto $M(Y)$ defined by setting $\pi (\mu )(g) = \mu (g \circ \varphi )$ for $\mu \in M(X)$ and $ g \in C(Y)$. It is shown that $\pi$ is norm open on $U(X)$ and on $ A \cdot P(X)$ for any subset A of the real numbers. If $\varphi$ is open, then $\pi$ is $ \mathrm{weak}^*$ open on $A \cdot P(X)$. Several examples are given which show that generalization in certain directions is not possible. The paper concludes with some remarks about continuous selections.
Tensor products of locally convex modules and applications to the multiplier problem
Roger
Rigelhof
295-307
Abstract: In this paper we present a representation theorem for the tensor product of locally convex modules. This theorem has a number of consequences in the study of the multiplier problem in harmonic analysis, and the remainder of the paper is devoted to these applications.
Local behaviour of solutions of stochastic integral equations
William J.
Anderson
309-321
Abstract: Let X denote the solution process of the stochastic equation $ dX(t) = a(X(t))dt + \sigma (X(t))dW(t)$. In this paper, conditions on $a( \cdot )$ and $ \sigma ( \cdot )$ are given under which the sample paths of X are differentiate at $t = 0$ with probability one. Variations of these results are obtained leading to a new uniqueness criterion for solutions of stochastic equations. If $\sigma ( \cdot )$ is Hölder continuous with exponent greater than $ \tfrac{1}{2}$ and $a( \cdot )$ satisfies a Lipschitz condition, it is shown that in the one-dimensional case the above equation has only one continuous solution.
Entropy-expansive maps
Rufus
Bowen
323-331
Abstract: Let $f:X \to X$ be a uniformly continuous map of a metric space. f is called h-expansive if there is an $ \varepsilon > 0$ so that the set $ {\Phi _\varepsilon }(x) = \{ y:d({f^n}(x),{f^n}(y)) \leqq \varepsilon$ for all $n \geqq 0$} has zero topological entropy for each $x \in X$. For X compact, the topological entropy of such an f is equal to its estimate using $\varepsilon :h(f) = h(f,\varepsilon )$. If X is compact finite dimensional and $ \mu$ an invariant Borel measure, then ${h_\mu }(f) = {h_\mu }(f,A)$ for any finite measurable partition A of X into sets of diameter at most $ \varepsilon$. A number of examples are given. No diffeomorphism of a compact manifold is known to be not h-expansive.
Mapping cylinder neighborhoods of one-complexes in four-space
J. L.
Bryant;
R. C.
Lacher
333-339
Abstract: We prove the following theorem: If K is a 1-complex topologically embedded in ${S^4}$, and if K has mapping cylinder neighborhoods in ${S^4}$ at almost all of its points, then K is tame. The proof uses engulfing and the theory of proper, one-acyclic mappings of 3-manifolds onto the real line.
A dominance theorem for partitioned Hermitian matrices
Russell
Merris
341-352
Abstract: Let $A = ({A_{ij}})$ be a partitioned positive semidefinite hermitian matrix, where ${A_{ij}}$ is n-square, $1 \leqq i,j \leqq m$. A class of ordered pairs of functions $ ({f_1},{f_2})$ is given such that $({f_1}({A_{ij}})) - ({f_2}({A_{ij}}))$ is positive semidefinite hermitian. Applications are given.
On the irreducibility of nonunitary induced representations of certain semidirect products
Ernest
Thieleker
353-369
Abstract: Let G be a connected Lie group which is a semidirect product of a compact subgroup K and a normal solvable subgroup S. Let $\Lambda$ be a character of S, and let ${M_\Lambda }$ be the stabilizer of $\Lambda$ in K. Let $[H,{\Lambda _\mu }]$ be a finite-dimensional irreducible representation of the subgroup $S{M_\Lambda }$ on the complex vector space H. In this paper we consider the induced representations of G on various Banach spaces, and study their topological irreducibility. The basic method used consists in studying the irreducibility of the Lie algebra representations which arise on the linear subspaces of K-finite vectors. The latter question then can be reduced to the problem of determining when certain modules over certain commutative algebras are irreducible. The method discussed in this paper leads to two theorems giving sufficient conditions on the character $\Lambda$ that the induced representations be topologically irreducible. The question of infinitesimal equivalence of various induced representations is also discussed.
A characterization of the group ${\rm U}\sb{3}\,(4)$
Richard
Lyons
371-387
Abstract: Let T be a Sylow 2-subgroup of the projective special unitary group ${U_3}(4)$, and let G be a finite group with Sylow 2-subgroups isomorphic to T. It is shown that if G is simple, then $G \cong {U_3}(4)$; if G has no proper normal subgroup of odd order or index, then $G \cong {U_3}(4)$ or T.
Cyclic vectors and irreducibility for principal series representations. II
Nolan R.
Wallach
389-396
Abstract: This paper is a continuation of the author's paper Cyclic vectors and irreducibility for principal series representations. In this paper the nonunitary principal series is studied. Using a theorem of Kostant, a sufficient condition is found for irreducibility of nonunitary principal series representations.
Asymptotic behavior of functions with bounded boundary rotation
James W.
Noonan
397-410
Abstract: For $k \geqq 2$ denote by ${V_k}$ the class of normalized functions, analytic in the unit disc, which have boundary rotation at most $k\pi$. Let ${a_n}$ be the nth Taylor coefficient of $f(z) \in {V_k}$. Let ${I_\lambda }(r,f)$ be the $\lambda$-integral mean of $f'(z)$ and $f(z)$ respectively. We determine asymptotic formulas for $f'(z)$, and these formulas are then applied to study the behavior of $\vert{a_n}\vert$ as $ n \to \infty$, and the behavior of ${I_\lambda }(r,f)$ as $r \to 1$.
A divergence theorem for Hilbert space
Victor
Goodman
411-426
Abstract: Let B be a real separable Banach space. A suitable linear imbedding of a real separable Hilbert space into B with dense range determines a probability measure on B which is known as abstract Wiener measure. In this paper it is shown that certain submanifolds of B carry a surface measure uniquely defined in terms of abstract Wiener measure. In addition, an identity is obtained which relates surface integrals to abstract Wiener integrals of functions associated with vector fields on regions in B. The identity is equivalent to the classical divergence theorem if the Hilbert space is finite dimensional. This identity is used to estimate the total measure of certain surfaces, and it is established that in any space B there exist regions whose boundaries have finite surface measure.
Weakly almost periodic functionals carried by hypercosets
Charles F.
Dunkl;
Donald E.
Ramirez
427-434
Abstract: For G a compact group and H a closed normal subgroup, we show that a weakly almost periodic (w.a.p.) linear functional on the Fourier algebra of G/H lifts to a w.a.p. linear functional on the Fourier algebra of G.
$C\sp{\ast} $-algebras generated by Fourier-Stieltjes transforms
Charles F.
Dunkl;
Donald E.
Ramirez
435-441
Abstract: For G a locally compact group and $\hat G$ its dual, let ${\mathcal{M}_d}(\hat G)$ be the ${C^ \ast }$-algebra generated by the Fourier-Stieltjes transforms of the discrete measures on G. We show that the canonical trace on $ {\mathcal{M}_d}(\hat G)$ is faithful if and only if G is amenable as a discrete group. We further show that if G is nondiscrete and amenable as a discrete group, then the only measures in ${\mathcal{M}_d}(\hat G)$ are the discrete measures, and also the sup and lim sup norms are identical on $ {\mathcal{M}_d}(\hat G)$. These results are extensions of classical theorems on almost periodic functions on locally compact abelian groups.
Associo-symmetric algebras
Raymond
Coughlin;
Michael
Rich
443-451
Abstract: Let A be an algebra over a field F satisfying $(x,x,x) = 0$ with a function $g:A \times A \times A \to F$ such that $(xy)z = g(x,y,z)x(yz)$ for all x, y, z in A. If $ g({x_1},{x_2},{x_3}) = g({x_{1\pi }},{x_{2\pi }},{x_{3\pi }})$ for all $ \pi$ in ${S_3}$ and all ${x_1},{x_2},{x_3}$ in A then A is called an associo-symmetric algebra. It is shown that a simple associo-symmetric algebra of degree $ > 2$ or degree $= 1$ over a field of characteristic $\ne 2$ is associative. In addition a finite-dimensional semisimple algebra in this class has an identity and is a direct sum of simple algebras.
Connections on semisimple Lie groups
Robert E.
Beck
453-460
Abstract: The plus and minus connections of Cartan and Schouten, which exist on any Lie group, have the following three properties: (1) the connection is left invariant, (2) the curvature of the connection is zero, (3) the set of maximal geodesics through the identity of the Lie group is equal to the set of one-parameter subgroups of the Lie group. It is shown that the plus and minus connections are the only ones with these properties on a real simple Lie group. On a real semisimple Lie group the connections with these properties are in one-to-one correspondence with the ways of choosing an ideal of the Lie algebra and then choosing a complementary subspace to it.
Integral operators and the compactness of induced representations
Robert C.
Busby;
Irwin
Schochetman;
Harvey A.
Smith
461-477
Abstract: Integral operators are investigated which are compositions of multiplication by bounded vector valued functions and convolutions with vector valued functions. All of the functions are based on a fixed locally compact group. Conditions are given under which certain of these operators are compact. As an application of these conditions we consider induced representations of twisted group algebras (these are generalizations of representations of groups induced from closed normal sub-groups in the sense of Mackey and include these as special cases) and we give necessary and sufficient conditions for these representations to be compact (that is, to consist entirely of compact operators).
Convolution operators on Lebesgue spaces of the half-line
Victor W.
Daniel
479-488
Abstract: In this paper we determine the lattice of closed invariant subspaces for certain convolution operators on Lebesgue spaces ${L^p}(d\sigma )$ where $\sigma$ is a suitable weighted measure on the half-line. We exploit the rather close relationship between convolution operators and the collection of right translation operators ${\{ {T_\lambda }\} _{\lambda \geqq 0}}$ on ${L^p}(d\sigma )$. We show that a convolution operator K and the collection $ {\{ {T_\lambda }\} _{\lambda \geqq 0}}$ have the same lattice of closed invariant subspaces provided the kernel k of K is a cyclic vector. The converse also holds if we assume in addition that the closed span of $ {\{ {T_\lambda }k\} _{\lambda \geqq 0}}$ is all of ${L^p}(d\sigma )$. We show that the lattice of closed right translation invariant subspaces of ${L^p}(d\sigma )$ is totally ordered by set inclusion whenever $\sigma$ has compact support. Thus in this case a convolution operator K is unicellular if and only if its kernel is a cyclic vector. Finally, we show for suitable weighted measures $\sigma$ on the half-line that the convolution operators on $ {L^p}(d\sigma )$ are Volterra.
Reflection principle for systems of first order elliptic equations with analytic coefficients
Chung Ling
Yu
489-501
Abstract: Let T be a simply connected domain of the $z = x + iy$ plane, whose boundary contains a portion $ \sigma$ of the x-axis. Also let $A(z,\zeta ),B(z,\zeta ),F(z,\zeta ),\alpha (z),\beta (z)$ and $\rho (z)$ be holomorphic functions for $z,\zeta \in T \cup \sigma \cup \bar T$, with $ \alpha (z) - i\beta (z) \ne 0$ for $z \in \bar T \cup \sigma ,\alpha (z) + i\beta (z) \ne 0$ for $z \in T \cup \sigma$. Furthermore, we assume that $ \alpha (x)$ and $ \beta (x)$ are real valued functions for $ x \in \sigma$. Our reflection principle states that for any solution $w = u + iv$ of an equation of the type $\partial w/\partial \bar z = A(z,\bar z)w + B(z,\bar z)\bar w + F(z,\bar z)$ in T under the boundary condition $\alpha (x)u + \beta (x)v = \rho (x)$ on $ \sigma ,w$ can be continued analytically across the x-axis, onto the entire mirror image $\bar T$.
Decompositions of $3$-manifolds and pseudo-isotopies
William
Voxman
503-508
Abstract: In this paper we construct pseudo-isotopies which realize certain cellular decompositions of 3-manifolds. In general we show that the pseudo-isotopy may be defined so as to leave points fixed outside of a given open set containing the nondegenerate elements of the decomposition. For nondegenerately continuous decompositions it is shown that the pseudo-isotopy does not move the nondegenerate elements far from their original positions.
Criteria for absolute convegence of Fourier series of functions of bounded variation
Ingemar
Wik
1-24
Abstract: The usual criteria for establishing that a function of bounded variation or an absolutely continuous function has an absolutely convergent Fourier series are given in terms of the modulus of continuity, the integrated modulus of continuity or conditions on the derivative. The relations between these criteria are investigated. A class of functions is constructed to provide counterexamples which show to what extent the existing theorems are best possible. In the case of absolutely continuous functions a few new criteria are given involving the variation of the given function. A couple of necessary and sufficient conditions are given for a class of absolutely continuous functions to have absolutely convergent Fourier series.
Sequences of divided powers in irreducible, cocommutative Hopf algebras
Kenneth
Newman
25-34
Abstract: In Hopf algebras with one grouplike element, M. E. Sweedler showed that over perfect fields, sequences of divided powers in cocommutative, irreducible Hopf algebras can be extended if certain ``coheight'' conditions are met. Here, we show that with a suitable generalization of ``coheight", Sweedler's theorem is true over nonperfect fields. (We also point out, that in one case Sweedler's theorem was false, and additional conditions must be assumed.) In the same paper, Sweedler gave a structure theorem for irreducible, cocommutative Hopf algebras over perfect fields. We generalize this theorem in both the perfect and nonperfect cases. Specifically, in the nonperfect case, while a cocommutative, irreducible Hopf algebra does not, in general, satisfy the structure theorem, the sub-Hopf algebra, generated by all sequences of divided powers, does. Some additional properties of this sub-Hopf algebra are also given, including a universal property.
On the summation formula of Voronoi
C.
Nasim
35-45
Abstract: A formula involving sums of the form $ \Sigma d(n)f(n)$ and $\Sigma d(n)g(n)$ is derived, where $ d(n)$ is the number of divisors of $n$, and $f(x),g(x)$ are Hankel transforms of each other. Many forms of such a formula, generally known as Voronoi's summation formula, are known, but we give a more symmetrical formula. Also, the reciprocal relation between $ f(x)$ and $g(x)$ is expressed in terms of an elementary kernel, the cosine kernel, by introducing a function of the class $ {L^2}(0,\infty )$. We use $ {L^2}$-theory of Mellin and Fourier-Watson transformations.
Diffeomorphic invariants of immersed circles
Roger F.
Verhey
47-63
Abstract: The intersection sequences of a normal immersion form a complete invariant for diffeomorphically equivalent normal immersions. Numerical invariants and inequalities on numerical invariants are obtained using intersection sequences.
Conformality and isometry of Riemannian manifolds to spheres
Chuan-chih
Hsiung;
Louis W.
Stern
65-73
Abstract: Suppose that a compact Riemannian manifold ${M^n}$ of dimension $n > 2$ admits an infinitesimal nonisometric conformal transformation $\upsilon$. Some curvature conditions are given for ${M^n}$ to be conformal or isometric to an $ n$-sphere under the initial assumption that $ {L_\upsilon }R = 0$, where ${L_\upsilon }$ is the operator of the infinitesimal transformation $\upsilon$ and $R$ is the scalar curvature of ${M^n}$. For some special cases, these conditions were given by Yano [10] and Hsiung [2].
Inverse $H$-semigroups and $t$-semisimple inverse $H$-semigroups
Mary Joel
Jordan
75-84
Abstract: An $H$-semigroup is a semigroup such that both its right and left congruences are two-sided. A semigroup is $t$-semisimple provided the intersection of all its maximal modular congruences is the identity relation. We prove that a semigroup is an inverse $H$-semigroup if and only if it is a semilattice of disjoint Hamiltonian groups. Using the set $E$ of idempotents of $S$ as the semilattice, we show that an inverse $ H$-semigroup $ S$ is $t$-semisimple if and only if for each pair of groups $ {G_e},{G_f}$, in the semilattice, with $f \geqq e$ in $E$, the homomorphism ${\varphi _{f,e}}$ on ${G_f}$, into ${G_e}$, defined by $a{\varphi _{f,e}} = ae$, is a monomorphism; and for each $e$ in $E$, for each $a \ne e$ in ${G_e}$, there exists a subsemigroup ${T_p}$ of $S$ such that $ a \notin {T_p}$ and, for each $f$ in $E$, $ {T_p} \cap {G_f} = {H_f}$, where ${H_f} = {G_f}$ or ${H_f}$ is a maximal subgroup of prime index $p$ in ${G_f}$.
$F$-minimal sets
Nelson G.
Markley
85-100
Abstract: An $F$-minimal set is the simplest proximal extension of an equicontinuous minimal set. It has one interesting proximal cell and all the points in this proximal cell are uniformly asymptotic. The Sturmian minimal sets are the best known examples of $F$-minimal sets. Our analysis of them is in terms of their maximal equicontinuous factors. Algebraically speaking $F$-minimal sets are obtained by taking an invariant *-closed algebra of almost periodic functions and adjoining some suitable functions to it. Our point of view is to obtain these functions from the maximal equicontinuous factor. In §3 we consider a subclass of $F$-minimal sets which generalize the classical Sturmian minimal sets, and in §4 we examine the class of minimal sets obtained by taking the minimal right ideal of an $F$-minimal set and factoring by a closed invariant equivalence relation which is smaller than the proximal relation.
Homology invariants of cyclic coverings with application to links
Y.
Shinohara;
D. W.
Sumners
101-121
Abstract: The main purpose of this paper is to study the homology of cyclic covering spaces of a codimension two link. The integral (rational) homology groups of an infinite cyclic cover of a finite complex can be considered as finitely generated modules over the integral (rational) group ring of the integers. We first describe the properties of the invariants of these modules for certain finite complexes related to the complementary space of links. We apply this result to the homology invariants of the infinite cyclic cover of a higher dimensional link. Further, we show that the homology invariants of the infinite cyclic cover detect geometric splittability of a link. Finally, we study the homology of finite unbranched and branched cyclic covers of a link.
The sign of Lommel's function
J.
Steinig
123-129
Abstract: Lommel's function ${s_{\mu ,\nu }}(x)$ is a particular solution of the differential equation ${s_{\mu ,\nu }}(x) > 0$ for $x > 0$, if $\mu = \tfrac{1}{2}$ and $\vert\nu \vert < \tfrac{1}{2}$, or if $\mu > \tfrac{1}{2}$ and $\vert\nu \vert \leqq \mu $. This includes earlier results of R. G. Cooke's. The sign of ${s_{\mu ,\nu }}(x)$ for other values of $ \mu$ and $\nu$ is also discussed.
Using additive functionals to embed preassigned distributions in symmetric stable processes
Itrel
Monroe
131-146
Abstract: Following Skorokhod, several authors in recent years have proposed methods to define a stopping time $T$ for Brownian motion $({X_t},{\mathcal{F}_t})$ such that ${X_T}$ will have some preassigned distribution. In this paper a method utilizing additive functionals is explored. It is applicable not only to Brownian motion but all symmetric stable processes of index $\alpha > 1$. Using this method one is able to obtain any distribution having a finite $\alpha - 1$ absolute moment. There is also a discussion of the problem of approximating symmetric stable processes with random walks.
A theorem of completeness for families of compact analytic spaces
John J.
Wavrik
147-155
Abstract: A sufficient condition is given for a family of compact analytic spaces to be complete. This condition generalizes to analytic spaces the Theorem of Completeness of Kodaira and Spencer [6]. It contains, as a special case, the rigidity theorem proved by Schuster in [11].
Approximation in the mean by analytic functions
Lars Inge
Hedberg
157-171
Abstract: Let $E$ be a compact set in the plane, let ${L^p}(E)$ have its usual meaning, and let $ L_a^p(E)$ be the subspace of functions analytic in the interior of $E$. The problem studied in this paper is whether or not rational functions with poles off $ E$ are dense in $ L_a^p(E)$ (or in $ {L^p}(E)$ in the case when $ E$ has no interior). For $1 \leqq p \leqq 2$ the problem has been settled by Bers and Havin. By a method which applies for $1 \leqq p < \infty$ we give new results for $p > 2$ which improve earlier results by Sinanjan. The results are given in terms of capacities.
Conjugacy separability of certain Fuchsian groups
P. F.
Stebe
173-188
Abstract: Let $G$ be a group. An element $g$ is c.d. in $G$ if and only if given any element $h$ of $G$, either it is conjugate to $h$ or there is a homomorphism $ \xi$ from $G$ onto a finite group such that $ \xi (g)$ is not conjugate to $\xi (h)$. Following A. Mostowski, a group is conjugacy separable or c.s. if and only if every element of the group is c.d. Let $F$ be a Fuchsian group, i.e. let $F$ be presented as $\displaystyle F = ({S_1}, \ldots ,{S_n},{a_1}, \ldots ,{a_{2r}},{b_1}, \ldots ,... ..._n}{a_1} \ldots {a_{2r}}a_1^{ - 1} \ldots a_{2r}^{ - 1}{b_1} \ldots {b_t} = 1).$ In this paper, we show that every element of infinite order in $F$ is c.d. and if $t \ne 0$ or $r \ne 0$, $F$ is c.s.
Measure algebras and functions of bounded variation on idempotent semigroups
Stephen E.
Newman
189-205
Abstract: Our main result establishes an isomorphism between all functions on an idempotent semigroup $S$ with identity, under the usual addition and multiplication, and all finitely additive measures on a certain Boolean algebra of subsets of $S$, under the usual addition and a convolution type multiplication. Notions of a function of bounded variation on $S$ and its variation norm are defined in such a way that the above isomorphism, restricted to the functions of bounded variation, is an isometry onto the set of all bounded measures. Our notion of a function of bounded variation is equivalent to the classical notion in case $S$ is the unit interval and the ``product'' of two numbers in $S$ is their maximum.
Discrete sufficient sets for some spaces of entire functions
B. A.
Taylor
207-214
Abstract: Let $E$ denote the space of all entire functions $f$ of exponential type (i.e. $\vert f(z)\vert = O(\exp (B\vert z\vert))$) for some $B > 0$). Let $ \mathcal{K}$ denote the space of all positive continuous functions $ k$ on the complex plane $ C$ with $\exp (B\vert z\vert) = O(k(z))$ for each $B > 0$. For $k \in \mathcal{K}$ and $ S \subset C$, let $ \vert\vert f\vert{\vert _{k,s}} = \sup \{ \vert f(z)\vert/k(z):z \in S\}$. We prove that the two families of seminorms $ {\{ \vert\vert\vert{\vert _{k,C}}\} _{k \in \mathcal{K}}}$ and $ {\{ \vert\vert\vert{\vert _{k,s}}\} _{k \in \mathcal{K}}}$, where $\displaystyle S = \{ n + im: - \infty < n,m < + \infty \}$ , determine the same topology on $E$.
Noncommutative Jordan division algebras
Kevin
McCrimmon
215-224
Abstract: The structure theory for noncommutative Jordan algebras with chain conditions leads to the following simple algebras: (I) division algebras, (II) forms of nodal algebras, (III) algebras of generic degree two, (IV) commutative Jordan matrix algebras, (V) quasi-associative algebras. The chain condition is always satisfied in a division algebra, hence does not serve as a finiteness restriction. Consequently, the general structure of noncommutative Jordan division algebras, even commutative Jordan division algebras, is unknown. In this paper we will classify those non-commutative Jordan division algebras which are forms of algebras of types (II)-(V); this includes in particular all the finite-dimensional ones.
Extreme points in a class of polynomials having univalent sequential limits
T. J.
Suffridge
225-237
Abstract: This paper concerns a class $ {\mathcal{P}_n}$ (defined below) of polynomials of degree less than or equal to $ n$ having the properties: each polynomial which is univalent in the unit disk and of degree $n$ or less is in $ {\mathcal{P}_n}$ and if $ \{ {P_{{n_k}}}\} _{k = 1}^\infty$ is a sequence of polynomials such that $ {P_{{n_k}}} \in {\mathcal{P}_{{n_k}}}$ and ${\lim _{k \to \infty }}{P_{{n_k}}} = f$ (uniformly on compact subsets of the unit disk) then $ f$ is univalent. The approach is to study the extreme points in ${\mathcal{P}_n}$ ( $P \in {\mathcal{P}_n}$ is extreme if $P$ is not a proper convex combination of two distinct elements of ${\mathcal{P}_n}$). Theorem 3 shows that if $ P \in {\mathcal{P}_n}$ is extreme then $P(z) = z + {a_2}{z^2} + \cdots + {a_n}{z^n},{a_n} = 1/n$, are dense in the class $S$ of normalized univalent functions. These polynomials have the very striking geometric property that the tangent line to the curve $P({e^{i\theta }})$, $0 \leqq \theta \leqq 2\pi $, turns at a constant rate (between cusps) as $\theta$ varies.
The local spectral behavior of completely subnormal operators
K. F.
Clancey;
C. R.
Putnam
239-244
Abstract: For any compact set $ X$, let $C(X)$ denote the continuous functions on $ X$ and $R(X)$ the functions on $X$ which are uniformly approximable by rational functions with poles off $X$. Let $A$ denote a subnormal operator having no reducing space on which it is normal. It is shown that a necessary and sufficient condition that $X$ be the spectrum of such an operator $ A$ is that $ R(X \cap \overline D ) \ne C(X \cap \overline D )$ whenever $D$ is an open disk intersecting $ X$ in a nonempty set.
Symmetric Massey products and a Hirsch formula in homology
Stanley O.
Kochman
245-260
Abstract: A Hirsch formula is proved for the singular chains of a second loop space and is applied to show that the symmetric Massey produce $ {\langle x\rangle ^p}$ is defined for $x$ an odd dimensional $\bmod p$ homology class of a second loop space with $ p$ an odd prime. ${\langle x\rangle ^p}$ is then interpreted in terms of the Dyer-Lashof and Browder operations.
Branched structures on Riemann surfaces
Richard
Mandelbaum
261-275
Abstract: Following results of Gunning on geometric realizations of projective structures on Riemann surfaces, we investigate more fully certain generalizations of such structures. We define the notion of a branched analytic cover on a Riemann surface $M$ (of genus $g$) and specialize this to the case of branched projective and affine structures. Establishing a correspondence between branched projective and affine structures on $M$ and the classical projective and affine connections on $M$ we show that if a certain linear homogeneous differential equation involving the connection has only meromorphic solutions on $M$ then the connection corresponds to a branched structure on $M$. Utilizing this fact we then determine classes of positive divisors on $M$ such that for each divisor $\mathfrak{D}$ in the appropriate class the branched structures having $ \mathfrak{D}$ as their branch locus divisor form a nonempty affine variety. Finally we apply some of these results to study the structures on a fixed Riemann surface of genus 2.
The $L\sp{1}$- and $C\sp{\ast} $-algebras of $[FIA]\sp{-}\sb{B}$ groups, and their representations
Richard D.
Mosak
277-310
Abstract: Let $G$ be a locally compact group, and $ B$ a subgroup of the (topologized) group $ \operatorname{Aut} (G)$ of topological automorphisms of $G$; $G$ is an $ [FIA]_B^ -$ group if $ B$ has compact closure in $ \operatorname{Aut} (G)$. Abelian and compact groups are $[FIA]_B^ -$ groups, with $B = I(G)$; the purpose of this paper is to generalize certain theorems about the group algebras and representations of these familiar groups to the case of general $ [FIA]_B^ -$ groups. One defines the set $ {\mathfrak{X}_B}$ of $ B$-characters to consist of the nonzero extreme points of the set of continuous positive-definite $B$-invariant functions $\phi$ on $G$ with $ \phi (1) \leqq 1$. ${\mathfrak{X}_B}$ is naturally identified with the set of pure states on the subalgebra of $B$-invariant elements of ${C^\ast}(G)$. When this subalgebra is commutative, this identification yields generalizations of known duality results connecting the topology of $G$ with that of $\hat G$. When $B = I(G),{\mathfrak{X}_B}$ can be identified with the structure spaces of ${C^\ast}(G)$ and ${L^1}(G)$, and one obtains thereby information about representations of $G$ and ideals in ${L^1}(G)$. When $G$ is an $ [FIA]_B^ -$ group, one has under favorable conditions a simple integral formula and a functional equation for the $ B$-characters. $ {L^1}(G)$ and ${C^\ast}(G)$ are ``semisimple'' in a certain sense (in the two cases $B = (1)$ and $B = I(G)$ this ``semisimplicity'' reduces to weak and strong semisimplicity, respectively). Finally, the $B$-characters have certain separation properties, on the level of the group and the group algebras, which extend to ${[SIN]_B}$ groups (groups which contain a fundamental system of compact $B$-invariant neighborhoods of the identity). When $B = I(G)$ these properties generalize known results about separation of conjugacy classes by characters in compact groups; for example, when $B = (1)$ they reduce to a form of the Gelfand-Raikov theorem about ``sufficiently many'' irreducible unitary representations.
A characterization of odd order extensions of the finite projective symplectic groups ${\rm PSp}(4,\,q)$
Morton E.
Harris
311-327
Abstract: In a recent paper, W. J. Wong characterized the finite projective symplectic groups $ {\text{PSp}}(4,q)$ where $ q$ is a power of an odd prime integer by the structure of the centralizer of an involution in the center of a Sylow $2$-subgroup of ${\text{PSp}}(4,q)$. In the present paper, finite groups which contain an involution in the center of a Sylow $2$-subgroup whose centralizer has a more general structure than in the ${\text{PSp}}(4,q)$ case are classified by showing them to be odd ordered extensions of ${\text{PSp}}(4,q)$.
Factoring functions on Cartesian products
N.
Noble;
Milton
Ulmer
329-339
Abstract: A function on a product space is said to depend on countably many coordinates if it can be written as a function defined on some countable subproduct composed with the projection onto that subproduct. It is shown, for $X$ a completely regular Hausdorff space having uncountably many nontrivial factors, that each continuous real-valued function on $ X$ depends on countably many coordinates if and only if $X$ is pseudo- $ {\aleph _1}$-compact. It is also shown that a product space is pseudo- ${\aleph _1}$-compact if and only if each of its finite subproducts is. (This fact derives from a more general theorem which also shows, for example, that a product satisfies the countable chain condition if and only if each of its finite subproducts does.) All of these results are generalized in various ways.
Regular modules
J.
Zelmanowitz
341-355
Abstract: In analogy to the elementwise definition of von Neumann regular rings an $ R$-module $M$ is called regular if given any element $m \in M$ there exists $f \in {\operatorname{Hom} _R}(M,R)$ with $ (mf)m = m$. Other equivalent definitions are possible, and the basic properties of regular modules are developed. These are applied to yield several characterizations of regular self-injective rings. The endomorphism ring $ E(M)$ of a regular module $ _RM$ is examined. It is in general a semiprime ring with a regular center. An immediate consequence of this is the recently observed fact that the endomorphism ring of an ideal of a commutative regular ring is again a commutative regular ring. Certain distinguished subrings of $E(M)$ are also studied. For example, the ideal of $ E(M)$ consisting of the endomorphisms with finite-dimensional range is a regular ring, and is simple when the socle of $ _RM$ is homogeneous. Finally, the self-injectivity of $E(M)$ is shown to depend on the quasi-injectivity of $ _RM$.
Some invariant $\sigma $-algebras for measure-preserving transformations
Peter
Walters
357-368
Abstract: For an invertible measure-preserving transformation $ T$ of a Lebesgue measure space $ (X,\mathcal{B},m)$ and a sequence $N$ of integers, a $T$-invariant partition ${\alpha _N}(T)$ of $ (X,\mathcal{B},m)$ is defined. The relationship of these partitions to spectral properties of $T$ and entropy theory is discussed and the behaviour of the partitions $ {\alpha _N}(T)$ under group extensions is investigated. Several examples are discussed.
A method of symmetrization and applications
W. E.
Kirwan
369-377
Abstract: In this paper we define a method of symmetrization for plane domains that includes as special cases methods of symmetrization considered by Szegö and by Marcus. We prove that under this method of symmetrization the mapping radius of a fixed point is not decreased. This fact is used to obtain some results concerning covering properties of Bieberbach-Eilenberg functions.
Quadratic extensions of linearly compact fields
Ron
Brown;
Hoyt D.
Warner
379-399
Abstract: A group valuation is constructed on the norm factor group of a quadratic extension of a linearly compact field, and the norm factor group is explicitly computed as a valued group. Generalizations and applications of this structure theory are made to cyclic extensions of prime degree, to square (and $p$th power) factor groups, to generalized quaternion algebras, and to quadratic extensions of arbitrary fields.
Convergence, uniqueness, and summability of multiple trigonometric series
J. Marshall
Ash;
Grant V.
Welland
401-436
Abstract: In this paper our primary interest is in developing further insight into convergence properties of multiple trigonometric series, with emphasis on the problem of uniqueness of trigonometric series. Let $E$ be a subset of positive (Lebesgue) measure of the $k$ dimensional torus. The principal result is that the convergence of a trigonometric series on $ E$ forces the boundedness of the partial sums almost everywhere on $E$ where the system of partial sums is the one associated with the system of all rectangles situated symmetrically about the origin in the lattice plane with sides parallel to the axes. If $E$ has a countable complement, then the partial sums are bounded at every point of $ E$. This result implies a uniqueness theorem for double trigonometric series, namely, that if a double trigonometric series converges unrestrictedly rectangularly to zero everywhere, then all the coefficients are zero. Although uniqueness is still conjectural for dimensions greater than two, we obtain partial results and indicate possible lines of attack for this problem. We carry out an extensive comparison of various modes of convergence (e.g., square, triangular, spherical, etc.). A number of examples of pathological double trigonometric series are displayed, both to accomplish this comparison and to indicate the ``best possible'' nature of some of the results on the growth of partial sums. We obtain some compatibility relationships for summability methods and finally we present a result involving the $ (C,\alpha ,0)$ summability of multiple Fourier series.
Essential spectrum for a Hilbert space operator
Richard
Bouldin
437-445
Abstract: Various notions of essential spectrum have been defined for densely defined closed operators on a Banach space. This paper shows that the theory for those notions of essential spectrum simplifies if the underlying space is a Hilbert space and the operator is reduced by its finite-dimensional eigenspaces. In that situation this paper classifies each essential spectrum in terms of the usual language for the spectrum of a Hilbert space operator. As an application this paper deduces the main results of several recent papers dealing with generalizations of the Weyl theorem.
On a problem of Tur\'an about polynomials with curved majorants
Q. I.
Rahman
447-455
Abstract: Let $\phi (x) \geqq 0$ for $- 1 \leqq x \leqq 1$. For a fixed ${x_0}$ in $[ - 1,1]$ what can be said for $ \vert{p_n}(x)\vert \leqq \phi (x)$ for $ - 1 \leqq x \leqq 1$? The case $\phi (x) = 1$ was considered by A. A. Markov and S. N. Bernstein. We investigate the problem when $ \phi (x) = {(1 - {x^2})^{1/2}}$. We also study the case $\phi (x) = \vert x\vert$ and the subclass consisting of polynomials typically real in $\vert z\vert < 1$.
Fully nuclear and completely nuclear operators with applications to $\mathcal{L}_1-$ and $\mathcal{L}_\infty$-spaces
C. P.
Stegall;
J. R.
Retherford
457-492
Abstract: This paper is devoted to a study of the conjecture of A. Grothendieck that if $E$ and $F$ are Banach spaces and all operators from $ E$ to $F$ are nuclear, then $E$ or $F$ must be finite dimensional. Two partial solutions are given to this conjecture (Chapters II and IV). In these chapters, operators we call fully nuclear and completely nuclear are introduced and studied. The principal result of these two chapters is that if $\mathcal{L}(E,F) = \operatorname{FN} (E,F)$ or $\mathcal{L}(E,F) = \operatorname{CN} (E,F)$ (and $E$ is isomorphic to a conjugate space or $ E'$ contains a reflexive subspace in the latter case) then one of $E$, $F$ is finite dimensional. Two new properties of Banach spaces are introduced in Chapter I. We call these properties ``sufficiently Euclidean'' and ``the two-series property". Chapter I provides the machinery for all the subsequent chapters. The principal part of the paper (Chapters II and V) is devoted to internal characterizations of the $ {\mathcal{L}_\infty }$ - and $ {\mathcal{L}_1}$-spaces of Lindenstrauss and Pełlczyhski. These characterizations are in terms of the behavior of various classes of operators from or into these spaces. As a by-product an apparently new characterization of Hilbert spaces is obtained. Finally, Chapter VI is a summary of the known characterizations of ${\mathcal{L}_1}$ - and ${\mathcal{L}_\infty }$ -spaces.
The class group of Dedekind domains
C. R.
Leedham-Green
493-500
Abstract: A new proof is given of Claborn's theorem, namely that every abelian group is the class group of a Dedekind domain. A variation of the proof shows that the Dedekind domain can be constructed to be a quadratic extension of a principal ideal ring; a Dedekind domain is also constructed that is unrelated in a certain sense to any principal ideal ring.
Homology in varieties of groups. I
C. R.
Leedham-Green
1-14
Abstract: Well-known techniques allow one to construct a (co-) homology theory relative to a variety. After two paragraphs which discuss the modules to be considered and the construction of the (co-) homology groups, we come to our main homological result, namely that the theory is not always equivalent to a Tor or Ext. In the fourth paragraph we prove our main group-theoretic result; two covering groups of a finite group generate the same variety ``up to exponent". Finally we produce a restricted version of the Künneth formula.
Homology in varieties of groups. II
C. R.
Leedham-Green
15-25
Abstract: The study of (co-) homology groups ${\mathfrak{B}_n}(\Pi ,A)$, $ {\mathfrak{B}^n}(\Pi ,A),\mathfrak{B}$ a variety, II a group in $\mathfrak{B}$, and A a suitable II-module, is pursued. They are compared with a certain Tor and Ext. The definition of the homology of an epimorphism due to Rinehart is shown to agree with that due to Barr and Beck (whenever both are defined). The edge effects of a spectral sequence are calculated.
Homology in varieties of groups. III
C. R.
Leedham-Green
27-33
Abstract: A spectral sequence is used to calculate approximately the homology groups ${\mathfrak{B}_2}(\Pi ,Z)$ as defined in the first paper in this series, for $\Pi$ a finitely generated abelian group and $\mathfrak{B}$ the variety of all nilpotent groups of class at most c.
Essential extensions of partial orders on groups
Jorge
Martinez
35-61
Abstract: Let (G, P) be an l-group and $ \mathcal{C}(P)$ be the lattice of convex l-subgroups of (G, P). We say that the l-cone Q is essential over P if $\mathcal{C}(Q)$ is contained in $\mathcal{C}(P)$. It is shown that for each nonzero x in G and each Q-value D of x, there is a P-value C of x containing D and no other Q-value of x. We specialize to those essential extensions for which the above C always depends uniquely on x and D; these are called very essential extensions. We show that if (G, P) is a representable l-group then P is the meet of totally ordered very essential extensions of P. Further we investigate connections between the existence of total very essential extensions and both representability and normal valuedness. We also study the role played by the various radicals in the theory. The same two classes of extensions are treated in the context of abelian Riesz groups. Similar questions about existence of such total orders are dealt with. The main result in this connection is that such total extensions always exist for finite valued pseudo lattice groups, and that the original cone is the meet of them.
Bergman minimal domains in several complex variables
Shigeo
Ozaki;
Sadao
Katô
63-69
Abstract: K. T. Hahn has obtained the inequality between the Jacobians of a biholomorphic mapping and a holomorphic automorphism of a Bergman minimal domain. This paper extends Hahn's result. Some inequalities concerning Jacobians of the mappings of minimal domains onto another minimal domain are considered, and an example is given.
Quasi-disjointness in ergodic theory
Kenneth
Berg
71-87
Abstract: We define and study a relationship, quasi-disjointness, between ergodic processes. A process is a measure-preserving transformation of a measure space onto itself, and ergodicity means that the space cannot be written as a disjoint union of invariant pieces, unless one of the pieces is of zero measure. We restrict our attention to spaces of total measure one which also satisfy additional regularity properties. In particular, the associated Hilbert space of square-summable functions is separable. A simple class of examples is given by translation by a fixed element on a compact Abelian metrizable group, such processes being known as Kronecker processes. We introduce the notion of a maximal common Kronecker factor (or quotient) process for two processes. Quasi-disjointness is a notion tied to the homomorphisms from two processes into their maximal common Kronecker factor, and reduces to a previous notion, disjointness, when that factor is trivial. We show that a substantial class of processes, the Weyl processes, are quasi-disjoint from every ergodic process. As a corollary, we show that a Weyl process and an ergodic process are disjoint if and only if they have no nontrivial Kronecker factor in common, or, equivalently, if they form an ergodic product. We give an example which suggests an analogous theory could be constructed in topological dynamics.
The structure of substitution minimal sets
Ethan M.
Coven;
Michael S.
Keane
89-102
Abstract: Substitutions of constant length on two symbols and their corresponding minimal dynamical systems are divided into three classes: finite, discrete and continuous. Finite substitutions give rise to uninteresting systems. Discrete substitutions yield strictly ergodic systems with discrete spectra, whose topological structure is determined precisely. Continuous substitutions yield strictly ergodic systems with partly continuous and partly discrete spectra, whose topological structure is studied by means of an associated discrete substitution. Topological and measure-theoretic isomorphisms are studied for discrete and continuous substitutions, and a complete topological invariant, the normal form of a substitution, is given.
Fredholm equations on a Hilbert space of analytic functions
Clasine
van Winter
103-139
Abstract: It is shown that the Hardy class $ {\mathfrak{H}^2}$ for the upper half-plane is equal to the set of functions $f[r\exp \,(i\phi )]$ which are analytic in the open half-plane and square-integrable with respect to r for $0 < \phi < \pi$. A function f is in $ {\mathfrak{H}^2}$ if and only if its Mellin transform with respect to r is a constant times $f(t)\exp \,(\phi t - i\phi /2)$, where f must belong to a certain $ {\mathfrak{L}^2}$-space. This result enables f in ${\mathfrak{H}^2}$ to be constructed from its boundary values on the positive real axis. A study is made of a class $ \mathfrak{N}$ consisting of integral operators K on ${\mathfrak{H}^2}$ having kernels $K(r,r',\phi )$ of operators in $\mathfrak{N}$ form a Hardy class ${\mathfrak{H}^2}(2)$ of functions of two variables, one complex and one real. A generalization leads to Hardy classes $ {\mathfrak{H}^2}(n)$ of functions of n variables. On ${\mathfrak{H}^2}(n)$, there is a class of operators $ \mathfrak{N}(n)$ whose kernels form a class $ {\mathfrak{H}^2}(2n)$. This formalism was developed with a view to the n-body problem in quantum mechanics. It is explained that the results on ${\mathfrak{H}^2}(n - 1)$ are instrumental in evaluating quantities which occur in the theory of n-particle scattering.
Spectral concentration and virtual poles. II
James S.
Howland
141-156
Abstract: Spectral concentration at an isolated eigenvalue of finite multiplicity of the selfadjoint operator ${H_\varepsilon } = {T_\varepsilon } + {A_\varepsilon }{B_\varepsilon }$ is shown to arise from a pole of an analytic continuation of ${A_\varepsilon }{({H_\varepsilon } - z)^{ - 1}}{B_\varepsilon }$. An application to quantum mechanical barrier penetration is given.
Constructive polynomial approximation on spheres and projective spaces.
David L.
Ragozin
157-170
Abstract: This paper contains constructive generalizations to functions defined on spheres and projective spaces of the Jackson theorems on polynomial approximation. These results, (3.3) and (4.6), give explicit methods of constructing uniform approximations to smooth functions on these spaces by polynomials, together with error estimates based on the smoothness of the function and the degree of the polynomial. The general method used exploits the fact that each space considered is the orbit of some compact subgroup, G, of an orthogonal group acting on a Euclidean space. For such homogeneous spaces a general result (2.1) is proved which shows that a G-invariant linear method of polynomial approximation to continuous functions can be modified to yield a linear method which produces better approximations to k-times differentiable functions. Jackson type theorems (3.4) are also proved for functions on the unit ball (which is not homogeneous) in a Euclidean space.
Some characterizations of the spaces $L\sp{1}(\mu )$
Kenneth L.
Pothoven
171-183
Abstract: Answers are given to the question of when the so-called hom and tensor functors in categories of Banach spaces preserve certain short exact sequences. The answers characterize the spaces of integrable, real-valued functions ${L^1}(\mu )$.
Dirichlet spaces and strong Markov processes
Masatoshi
Fukushima
185-224
Abstract: We show that there exists a suitable strong Markov process on the underlying space of each regular Dirichlet space. Potential theoretic concepts due to A. Beurling and J. Deny are then described in terms of the associated strong Markov process. The proof is carried out by developing potential theory for Dirichlet spaces and symmetric Ray processes and by using a method of transformation of underlying spaces.
On Cartan subalgebras of alternative algebras
D. M.
Foster
225-238
Abstract: In 1966, Jacobson introduced the notion of a Cartan subalgebra for finite-dimensional Jordan algebras with unity over fields of characteristic not 2. Since finite-dimensional Jordan, alternative, and Lie algebras are known to be related through their structure theories, it would seem logical that such an analogue would also exist for finite-dimensional alternative algebras. In this paper, we show that this is the case. Moreover, the linear transformation we define that plays the role in alternative algebras that ``ad ( )'' plays in Lie algebras is identical with that used in the Jordan theory, and can be used in the Lie case as well. Hence we define Cartan subalgebras relative to this linear transformation for finite-dimensional alternative, Jordan, and Lie algebras, and observe that in the Lie case, they coincide with the classical definition of a Cartan subalgebra.
Concerning $n$-mutual aposyndesis in products of continua
Leland E.
Rogers
239-251
Abstract: This paper is concerned with Cartesian products of regular Hausdorff continua and certain conditions on the factors that make the product n-mutually aposyndetic (given n distinct points, there are n disjoint subcontinua, each containing one of the points in its interior). It is proved that the product of any three regular Hausdorff continua is n-mutually aposyndetic for each $n \geqq 2$. Next, certain conditions on factors of products of two continua are shown to be sufficient for the product to be n-mutually aposyndetic. In connection with this, the concepts of n-semiaposyndesis and aposyndetic-terminal points are introduced. Finally, it is proved that the product of a simple closed curve (or any other ``super n-mutually aposyndetic'' continuum) with every compact Hausdorff continuum is n-mutually aposyndetic for each $n \geqq 2$.
Hermitian functionals on $B$-algebras and duality characterizations of $C\sp{\ast} $-algebras
Robert T.
Moore
253-265
Abstract: The hermitian functionals on a unital complex Banach algebra are defined here to be those in the real span of the normalized states (tangent functionals to the unit ball at the identity). It is shown that every functional f in the dual A' of A can be decomposed as $f = h + ik$, where h and k are hermitian functionals. Moreover, this decomposition is unique for every $f \in A'$ iff A admits an involution making it a ${C^\ast}$-algebra, and then the hermitian functionals reduce to the usual real or symmetric functionals. A second characterization of ${C^\ast}$-algebras is given in terms of the separation properties of the hermitian elements of A (real numerical range) as functionals on A'. The possibility of analogous theorems for vector states and matrix element functionals on operator algebras is discussed, and potential applications to the representation theory of locally compact groups are illustrated.
Improbability of collisions in Newtonian gravitational systems
Donald Gene
Saari
267-271
Abstract: It is shown that the set of initial conditions leading to a collision in finite time has measure zero.
The Baer sum functor and algebraic $K$-theory
Irwin S.
Pressman
273-286
Abstract: The Baer sum operation can be described in such a way that it becomes a functorial product on categories of exact sequences of a fixed length. This product is proven to be coherently associative and commutative. The Grothendieck groups and Whitehead groups of some of these categories are computed.
Bonded projections, duality, and multipliers in spaces of analytic functions
A. L.
Shields;
D. L.
Williams
287-302
Abstract: Let $\varphi$ and $\psi$ be positive continuous functions on $ [0,1)$ with $\varphi (r) \to 0$ as $r \to 1$ and $\smallint _0^1\psi (r)\;dr < \infty$. Denote by ${A_0}(\varphi )$ and ${A_\infty }(\varphi )$ the Banach spaces of functions f analytic in the open unit disc D with $ \vert f(z)\vert\varphi (\vert z\vert) = o(1)$ and $\vert f(z)\vert\varphi (\vert z\vert) = O(1),\vert z\vert \to 1$, respectively. In both spaces $\left\Vert f\right\Vert _\varphi = {\sup _D}\vert f(z)\vert\varphi (\vert z\vert)$. Let ${A^1}(\psi )$ denote the space of functions analytic in D with $\left\Vert f\right\Vert _\psi = \smallint {\smallint _D}\vert f(z)\vert\psi (\vert z\vert)\;dx\;dy < \infty$. The spaces ${A_0}(\varphi ),{A^1}(\psi )$, and ${A_\infty }(\varphi )$ are identified in the obvious way with closed subspaces of ${C_0}(D),{L^1}(D)$, and ${L^\infty }(D)$, respectively. For a large class of weight functions $ \varphi ,\psi$ which go to zero at least as fast as some power of $(1 - r)$ but no faster than some other power of $(1 - r)$, we exhibit bounded projections from $ {C_0}(D)$ onto ${A_0}(\varphi )$, from ${L^1}(D)$ onto $ {A^1}(\psi )$, and from ${L^\infty }(D)$ onto ${A_\infty }(\varphi )$. Using these projections, we show that the dual of $ {A_0}(\varphi )$ is topologically isomorphic to $ {A^1}(\psi )$ for an appropriate, but not unique choice of $\psi$. In addition, ${A_\infty }(\varphi )$ is topologically isomorphic to the dual of $ {A^1}(\psi )$. As an application of the above, the coefficient multipliers of $ {A_0}(\varphi ),{A^1}(\psi )$, and $ {A_\infty }(\varphi )$ are characterized. Finally, we give an example of a weight function pair $ \varphi ,\psi$ for which some of the above results fail.
Some splitting theorems for algebras over commutative rings
W. C.
Brown
303-315
Abstract: Let R denote a commutative ring with identity and Jacobson radical p. Let $ {\pi _0}:R \to R/p$ denote the natural projection of R onto $R/p$ and $ j:R/p \to R$ a ring homomorphism such that ${\Pi _0}j$ is the identity on $R/p$. We say the pair (R, j) has the splitting property if given any R-algebra A which is faithful, connected and finitely generated as an R-module and has $A/N$ separable over R, then there exists an $(R/p)$-algebra homomorphism $(R[[x]],1)$ with the splitting property. Two examples are given at the end of the paper which show that $R/p$ being integrally closed is necessary but not sufficient to guarantee (R, j) has the splitting property.
Equivariant bordism and Smith theory. II
R. E.
Stong
317-326
Abstract: This paper analyzes the homomorphism from equivariant bordism to Smith homology for spaces with an action of a finite group G.
Subordination principle and distortion theorems on holomorphic mappings in the space $C\sp{n}$
Kyong T.
Hahn
327-336
Abstract: Generalizing the notion of subordination principle in the complex plane to the space of several complex variables, we obtain various distortion theorems on holomorphic mappings of one bounded domain into another in terms of geometrical quantities of the domains and the Bergman metric furnished, thus obtaining a generalization of the Koebe-Faber distortion theorem among others.
Normal operations on quaternionic Hilbert spaces
K.
Viswanath
337-350
Abstract: Simple modifications of standard complex methods are used to obtain a spectral theorem, a functional calculus and a multiplicity theory for normal operators on quaternionic Hilbert spaces. It is shown that the algebra of all operators on a quaternionic Hilbert space is a real $ {C^\ast}$-algebra in which (a) every normal operator is unitarily equivalent to its adjoint and (b) every operator in the double commutant of a hermitian operator is hermitian. Unitary representations of locally compact abelian groups in quaternionic Hilbert spaces are studied and, finally, the complete structure theory of commutative von Neumann algebras on quaternionic Hilbert spaces is worked out.
Automorphisms of Siegel domains
O. S.
Rothaus
351-382
Abstract: This paper studies nonaffine biholomorphisms from one tube domain to a second. A sequel will carry out the same study for arbitrary Siegel domains. With the help of the Bergman kernel function, we can give an explicit form for such biholomorphisms; and with the use of structure theory for Jordan algebras, we can give an algebraic and geometric description of the nature of such tube domains.
Bounds on the ratio $n(r,\,a)/S(r)$ for meromorphic functions
Joseph
Miles
383-393
Abstract: Let f be a meromorphic function in the plane. We prove the existence of an absolute constant K such that if ${a_1},{a_2}, \ldots ,{a_q}$ are distinct elements of the Riemann sphere then $\lim {\inf _{r \to \infty }}\;(\Sigma _{j = 1}^q\vert n(r,{a_j})/S(r) - 1\vert) < K$. We show by example that in general no such bound exists for the corresponding upper limit. These results involving the unintegrated functionals of Nevanlinna theory are related to previous work of Ahlfors, Hayman and Stewart, and the author.
Locally univalent functions with locally univalent derivatives
Douglas Michael
Campbell
395-409
Abstract: S. M. Shah and S. Y. Trimble have discovered that the behavior of an analytic function $f(z)$ is strongly influenced by the radii of univalence of its derivatives ${f^{(n)}}(z)\;(n = 0,1,2, \ldots )$. In this paper many of Shah and Trimble's results are extended to large classes of locally univalent functions with locally univalent derivatives. The work depends on the concept of the $ {\mathcal{U}_\beta }$-radius of a locally univalent function that is introduced and developed in this paper. Ch. Pommerenke's definition of a linear invariant family of locally univalent functions and the techniques of that theory are employed in this paper. It is proved that the universal linear invariant families $ {\mathcal{U}_\alpha }$ are rotationally invariant. For fixed $f(z)$ in ${\mathcal{U}_\alpha }$, it is shown that the function $r \to {\text{order}}\;[f(rz)/r]\;(0 < r \leqq 1)\;$ is a continuous increasing function of r.
Inseparable splitting theory
Richard
Rasala
411-448
Abstract: If L is a purely inseparable field extension of K, we show that, for large enough extensions E of K, the E algebra $L{ \otimes _K}E$ splits to become a truncated polynomial algebra. In fact, there is a unique smallest extension E of K which splits $L/K$ and we call this the splitting field $S(L/K)$ of $L/K$. Now $L \subseteq S(L/K)$ and the extension $S(L/K)$ of K is also purely inseparable. This allows us to repeat the splitting field construction and obtain inductively a tower of fields. We show that the tower stabilizes in a finite number of steps and we study questions such as how soon must the tower stabilize. We also characterize in many ways the case when L is its own splitting field. Finally, we classify all K algebras A which split in a similar way to purely inseparable field extensions.
Martingale convergence to infinitely divisible laws with finite variances
B. M.
Brown;
G. K.
Eagleson
449-453
Abstract: Some results are obtained concerning the convergence in distribution of the row sums of a triangular array of certain dependent random variables. The form of dependence considered is that of martingales within rows, and the results are obtained under conditions which parallel those of the classical case of convergence in distribution, to infinitely divisible laws with bounded variances, of the row sums of elementary systems of independent random variables.
Symmetrization of distributions and its application
Kuang-ho
Chen
455-471
Abstract: Let P be a polynomial such that k of the $n - 1$ principal curvatures are different from zero at each point of $N(P) = \{ s \in {R^n}:P(s) = 0\} ;N(P)$ is assumed to be nonempty, bounded, and $n - 1$ dimensional. If $ {\text{Supp}}\;\varphi \subset {U^\delta } = \{ s \in {R^n}:\vert P(s)\vert < \delta \}$ with $\delta$ small and $\varphi \in C_c^\infty ({R^n})$, let ${\varphi ^\rho }$ be the integral of $\varphi$ over $N(P - q)$ if $q \in [ - \delta ,\delta ]$ and $ {\varphi ^\sigma }(s) = {\varphi ^\rho }(P(s))$ on $ {U^\delta }$ and $ = 0$ outside ${U^\delta }$. Then ${\varphi ^\sigma } \in C_c^\infty ({R^n})$. We define the symmetrization $ {v^\sigma }$ of a distribution v, with ${\text{Supp}}\;v \subset {U^\delta }$, in a natural way. Setting $u = {\mathcal{F}^{ - 1}}\{ v\}$ and $ {u_0} = {\mathcal{F}^{ - 1}}\{ {v^\sigma }\}$, we prove that ${u_0}$ is the integral of the product of u with some function $w(,)$ which depends only on P. This result is used to prove a Liouville type theorem for entire solutions of $ P( - i{D_x})u(x) = f(x)$, with $ f \in C_c^\infty ({R^n})$.
Errata to ``Mappings onto the plane''
Dix H.
Pettey
473
Erratum to ``Under the degree of some finite linear groups''
Harvey I.
Blau
475
Using flows to construct Hilbert space factors of function spaces
James
Keesling
1-24
Abstract: Let X and Y be metric spaces. Let $G(X)$ be the group of homeomorphisms of X with the compact open topology. The main result of this paper is that if X admits a nontrivial flow, then $G(X)$ is homeomorphic to $G(X) \times {l_2}$ where ${l_2}$ is separable infinite-dimensional Hilbert space. The techniques are applied to other function spaces with the same result. Two such spaces for which our techniques apply are the space of imbeddings of X into Y, $E(X,Y)$, and the space of light open mappings of X into (or onto) Y, LO (X, Y). Some applications of these results are given. The paper also uses flows to show that if X is the $\sin (1/x)$-curve, then $G(X)$ is homeomorphic to ${l_2} \times N$, where N is the integers.
Oscillation properties of two term linear differential equations
G. A.
Bogar
25-33
Abstract: The two term differential equations $ {L_n}[y] + py = 0$, where $ \rho _1^{ - 1}(t)p(t)$ on $[a,\infty )$ when ${L_n}[y]$ is disconjugate. By changing the integral conditions slightly we then prove that the equation has n linearly independent oscillatory solutions.
Higher dimensional knots in tubes
Yaichi
Shinohara
35-49
Abstract: Let K be an n-knot in the $(n + 2)$-sphere and V a tubular neighborhood of K. Let $L'$ be an n-knot contained in a tubular neighborhood $V'$ of a trivial n-knot and L the image of $L'$ under an orientation preserving diffeomorphism of $ V'$ onto V. The purpose of this paper is to show that the higher dimensional Alexander polynomial and the signature of the n-knot L are determined by those of K and $L'$.
Degree of symmetry of a homotopy real projective space
H. T.
Ku;
L. N.
Mann;
J. L.
Sicks;
J. C.
Su
51-61
Abstract: The degree of symmetry $N(M)$ of a compact connected differentiable manifold M is the maximum of the dimensions of the compact Lie groups which can act differentiably and effectively on it. It is well known that $N(M) \leqq \dim \; SO(m + 1)$, for an m-dimensional manifold, and that equality holds only for the standard m-sphere and the standard real projective m-space. W. Y. Hsiang has shown that for a high dimensional exotic m-sphere M, $ N(M) < {m^2}/8 + 1 < \left( {\frac{1}{4}} \right)\dim SO(m + 1)$, and that $ N(M) = {m^2}/8 + 7/8$ for some exotic m-spheres. It is shown here that the same results are true for exotic real projective spaces.
Well distributed sequences of integers
William A.
Veech
63-70
Abstract: Niven's notion of a uniformly distributed sequence of integers is generalized to well distribution, and two classes of integer sequences are studied in terms of this generalization.
Studies in the representation theory of finite semigroups
Yechezkel
Zalcstein
71-87
Abstract: This paper is a continuation of [14], developing the representation theory of finite semigroups further. The main result, Theorem 1.24, states that if the group of units U of a mapping semigroup (X, S) is multiply transitive with a sufficiently high degree of transitivity, then for certain irreducible characters $\chi$ of U, $\chi$ can be ``extended'' formally to an irreducible character of S. This yields a partial generalization of a well-known theorem of Frobenius on the characters of multiply-transitive groups and provides the first nontrivial explicit formula for an irreducible character of a finite semigroup. The paper also contains preliminary results on the ``spectrum'' (i.e., the set of ranks of the various elements) of a mapping semigroup.
Implicitly defined mappings in locally convex spaces
Terrence S.
McDermott
89-99
Abstract: Results on existence, uniqueness, continuity and differentiability of implicit functions in locally convex, linear topological spaces are obtained, and certain of these results are applied to obtain results on the existence and continuous dependence on parameters of global solutions for a nonlinear Volterra integral equation.
On the Wedderburn principal theorem for nearly $(1,\,1)$ algebras
T. J.
Miles
101-110
Abstract: A nearly $ (1,1)$ algebra is a finite dimensional strictly power-associative algebra satisfying the identity $(x,x,y) = (x,y,x)$ where the associator $(x,y,z) = (xy)z - x(yz)$. An algebra A has a Wedderburn decomposition in case A has a subalgebra $S \cong A - N$ with $A = S + N$ (vector space direct sum) where N denotes the radical (maximal nil ideal) of A. D. J. Rodabaugh has shown that certain classes of nearly $(1,1)$ algebras have Wedderburn decompositions. The object of this paper is to expand these classes. The main result is that a nearly $(1,1)$ algebra A containing 1 over a splitting field of characteristic not 2 or 3 such that A has no nodal subalgebras has a Wedderburn decomposition.
On the differentiability of arbitrary real-valued set functions. II.
Harvel
Wright;
W. S.
Snyder
111-122
Abstract: Let f be a real-valued function defined and finite on sets from a family $\mathcal{F}$ of bounded measurable subsets of Euclidean n-space such that if $T \in \mathcal{F}$, the measure of T is equal to the measure of the closure of T. An earlier paper [Trans. Amer. Math. Soc. 145 (1969), 439-454] considered the questions of finiteness and boundedness of the upper and lower regular derivates of f and of the existence of a unique finite derivative. The present paper is an extension of the earlier paper and considers the summability of the derivates. Necessary and sufficient conditions are given for each of the upper and lower derivates to be summable on a measurable set of finite measure. A characterization of the integral of the upper derivate is given in terms of the sums of the values of the function over finite collections of mutually disjoint sets from the family.
Repairing embeddings of $3$-cells with monotone maps of $E\sp{3}$
William S.
Boyd
123-144
Abstract: If ${S_1}$ is a 2-sphere topologically embedded in Euclidean 3-space ${E^3}$ and ${S_2}$ is the unit sphere about the origin, then there may not be a homeomorphism of ${E^3}$ onto itself carrying ${S_1}$ onto ${S_2}$. We show here how to construct a map f of ${E^3}$ onto itself such that $f\vert{S_1}$ is a homeomorphism of $ {S_1}$ onto ${S_2}$, $f({E^3} - {S_1}) = {E^3} - {S_2}$ and ${f^{ - 1}}(x)$ is a compact continuum for each point x in ${E^3}$. Similar theorems are obtained for 3-cells and disks topologically embedded in ${E^3}$.
Borel measurable mappings for nonseparable metric spaces
R. W.
Hansell
145-169
Abstract: The main object of this paper is the extension of part of the basic theory of Borel measurable mappings, from the ``classical'' separable metric case, to general metric spaces. Although certain results of the standard theory are known to fail in the absence of separability, we show that they continue to hold for the class of ``$\sigma $-discrete'' mappings. This class is shown to be quite extensive, containing the continuous mappings, all mappings with a separable range, and any Borel measurable mappings whose domain is a Borel subset of a complete metric space. The last result is a consequence of our Basic Theorem which gives a topological characterization of those collections which are the inverse image of an open discrete collection under a Borel measurable mapping. Such collections are shown to possess a strong type of $ \sigma$-discrete refinement. The properties of $\sigma$-discrete mappings together with the known properties of ``locally Borel'' sets allow us to extend, to general metric spaces, well-known techniques used for separable spaces. The basic properties of ``complex'' and ``product'' mappings, well known for separable spaces, are proved for general metric spaces for the class of $\sigma$-discrete mappings. A consequence of these is a strengthening of the basic theorem of the structure theory of nonseparable Borel sets due to A. H. Stone. Finally, the classical continuity properties of Borel measurable mappings are extended, and, in particular, a generalization of the famous theorem of Baire on the points of discontinuity of a mapping of class 1 is obtained.
A characterization of the equicontinuous structure relation
Robert
Ellis;
Harvey
Keynes
171-183
Abstract: The main result in the paper is to show that in a large class of minimal transformation groups (including those with abelian phase groups, and point-distal transformation groups), the equicontinuous structure relation is precisely the regionally proximal relation. The techniques involved enable one to recover and extend the previously known characterizations. Several corollaries are indicated, among which the most important is a new criterion (which is easily applicable) for the existence of a nontrivial equicontinuous image of a given transformation group.
The space of all self-homeomorphisms of a two-cell which fix the cell's boundary is an absolute retract
W. K.
Mason
185-205
Abstract: The theorem mentioned in the title is proved. A corollary of the title theorem is: any homeomorphism between two compact subsets of the function space mentioned in the title can be extended to a homeomorphism of the function space onto itself.
The bifurcation of solutions in Banach spaces
William S.
Hall
207-218
Abstract: Let $L:D \subset X \to D \subset {X^ \ast }$ be a densely defined linear map of a reflexive Banach space X to its conjugate ${X^\ast}$. Define M and $ {M^\ast}$ to be the respective null spaces of L and its formal adjoint $ {L^\ast}$. Let $f:X \to {X^\ast}$ be continuous. Under certain conditions on ${L^\ast}$ and f there exist weak solutions to $Lu = f(u)$ provided for each $w \in X,v(w) \in M$ can be found such that $f(v(w) + w)$ annihilates $ {M^ \ast }$. Neither M and ${M^\ast}$ nor their annihilators need be the ranges of continuous linear projections. The results have applications to periodic solutions of partial differential equations.
Prime entire functions
Fred
Gross
219-233
Abstract: Factorizations of various functions are discussed. Complete factorizations of certain classes of functions are given. In particular it is shown that there exist primes of arbitrary growth.
The powers of a maximal ideal in a Banach algebra and analytic structure
T. T.
Read
235-248
Abstract: Sufficient conditions are given for the existence of an analytic variety at an element $\phi$ of the spectrum of a commutative Banach algebra with identity. An associated graded algebra first considered by S. J. Sidney is used to determine the dimension of the analytic variety in terms of the closed powers of the maximal ideal which is the kernel of $\phi$.
Weighted norm inequalities for singular and fractional integrals
Benjamin
Muckenhoupt;
Richard L.
Wheeden
249-258
Abstract: Inequalities of the form ${\left\Vert {{{\left\vert x \right\vert}^\alpha }Tf} \right\Vert _q} \leqq C{\left\Vert {{{\left\vert x \right\vert}^\alpha }f} \right\Vert _p}$ are proved for certain well-known integral transforms, T, in ${E^n}$. The transforms considered include Calderón-Zygmund singular integrals, singular integrals with variable kernel, fractional integrals and fractional integrals with variable kernel.
The two-piece property and tight $n$-manifolds-with-boundary in $E\sp{n}$
Thomas F.
Banchoff
259-267
Abstract: The two-piece property for a set A is a generalization of convexity which reduces to the condition of minimal total absolute curvature if A is a compact 2-manifold. We show that a connected compact 2-manifold-with-boundary in ${E^2}$ has the TPP if and only if each component of the boundary has the TPP. The analogue of this result is not true in higher dimensions without additional conditions, and we introduce a stronger notion called k-tightness and show that an $(n + 1)$-manifold-with-boundary ${M^{n + 1}}$ embedded in $ {E^{n + 1}}$ is 0- and $ (n - 1)$-tight if and only if its boundary is 0- and $(n - 1)$-tight.
Local theory of complex functional differential equations
Robert J.
Oberg
269-281
Abstract: We consider the equation $g({z_0}) = {z_0}$. We classify fixed points $ {z_0}$ of g as attractive if $ f({z_0}) = {w_0}$. This solution depends continuously on ${w_0}$ and on the functions F and g. For ``most'' indifferent fixed points the initial-value problem has a unique solution. Around a repulsive fixed point a solution in general does not exist, though in exceptional cases there may exist a singular solution which disappears if the equation is subjected to a suitable small perturbation.
Almost-arithmetic progressions and uniform distribution
H.
Niederreiter
283-292
Abstract: In a recent paper, P. E. O'Neil gave a new criterion for uniform distribution modulo one in terms of almost-arithmetic progressions. We investigate the relation between almost-arithmetic progressions and uniformly distributed sequences from a quantitative point of view. An upper bound for the discrepancy of almost-arithmetic progressions is given which is shown to be best possible. Estimates for more general sequences are also obtained. As an application, we prove a quantitative form of Fejér's theorem on the uniform distributivity of slowly increasing sequences.
Monofunctors as reflectors
Claus Michael
Ringel
293-306
Abstract: In a well-powered and co-well-powered complete category $\mathcal{K}$ with weak amalgamations, the class M of all reflective subcategories with a monofunctor as reflector forms a complete lattice; the limit-closure of the union of any class of elements of M belongs to M. If $\mathcal{K}$ has injective envelopes, then the set-theoretical intersection of any class of elements of M belongs to M.
Singular integrals and fractional powers of operators
Michael J.
Fisher
307-326
Abstract: Recently R. Wheeden studied a class of singular integral operators, the hypersingular integrals, as operators from $L_p^\alpha (H)$ to ${L_p}(H);L_p^\alpha (H)$ is the range of the $ \alpha$th order Bessel potential operator acting on ${L_p}(H)$ with the inherited norm. The purposes of the present paper are to extend the known results on hypersingular integrals to complex indices, to extend these results to operators defined over a real separable Hilbert space, and to use Komatsu's theory of fractional powers of operators to show that the hypersingular integral operator $ {G^\alpha }$ is ${\smallint _H}{( - {A_y})^\alpha }f\,d\mu (y)$ when ${\mathop{\rm Im}\nolimits} (\alpha ) \ne 0$ or when ${\mathop{\Re}\nolimits} (\alpha )$ is not a positive integer where ${A_y}g$ is the derivative of g in the direction y. The case where ${\mathop{\rm Im}\nolimits} (\alpha ) = 0$ and $ {\mathop{\Re}\nolimits} (\alpha )$ is a positive integer is treated in a sequel to the present paper.
Functions of finite $\lambda $-type in several complex variables
Robert O.
Kujala
327-358
Abstract: If $ \lambda :{{\bf {R}}^ + } \to {{\bf {R}}^ + }$ is continuous and increasing then a meromorphic function f on ${C^k}$ is said to be of finite $ \lambda$-type if there are positive constants s, A, B and R such that $ {T_f}(r,s) \leqq A\lambda (Br)$ for all $r > R$ where $ {T_f}(r,s)$ is the characteristic of f. It is shown that if $\lambda (Br)/\lambda (r)$ is bounded for r sufficiently large and $B > 1$, then every meromorphic function of finite $\lambda$-type is the quotient of two entire functions of finite $\lambda$-type. This theorem is the result of a careful and detailed analysis of the relation between the growth of a function and the growth of its divisors. The central fact developed in this connection is: A nonnegative divisor $\nu$ on ${C^k}$ with $ \nu ({\bf {0}}) = 0$ is the divisor of an entire function of finite $ \lambda$-type if and only if there are positive constants A, B and R such that \begin{displaymath}\begin{array}{*{20}{c}} {{N_{\nu \vert\xi }}(r) \leqq A\lambd... ...ambda (Br){r^{ - p}} + A\lambda (Bs){s^{ - p}},} \end{array}\end{displaymath} for all $r \geqq s > R$, all unit vectors $\xi$ in ${C^k}$, and all natural numbers p. Here $ \nu \vert\xi$ represents the lifting of the divisor $\nu$ to the plane via the map $z \mapsto z\xi$ and ${N_{\nu \vert\xi }}$ is the valence function of that divisor. Analogous facts for functions of zero $ \lambda$-type are also presented.
The maximal ideal space of algebras of bounded analytic functions on infinitely connected domains
Michael Frederick
Behrens
359-379
Closure theorems with applications to entire functions with gaps
J. M.
Anderson;
K. G.
Binmore
381-400
Abstract: In this paper we consider questions of completeness for spaces of continuous functions on a half line which satisfy appropriate growth conditions. The results obtained have consequences in the theory of entire functions with gap power series. In particular we show that, under an appropriate gap hypothesis, the rate of growth of an entire function in the whole plane is determined by its rate of growth along any given ray.
Degenerate evolution equations in Hilbert space
Avner
Friedman;
Zeev
Schuss
401-427
Abstract: We consider the degenerate evolution equation ${c_1}(t)du/dt + {c_2}(t)A(t)u = f(t)$ in Hilbert space, where ${c_1} \geqq 0,{c_2} \geqq 0,{c_1} + {c_2} > 0;A(t)$ is an unbounded linear operator satisfying the usual conditions which ensure that there is a unique solution for the Cauchy problem $du/dt + A(t)u = f(t){\rm {in}}(0,T],u(0) = {u_0}$. We prove the existence and uniqueness of a weak solution, and differentiability theorems. Applications to degenerate parabolic equations are given.
$\sp{\ast} $-taming sets for crumpled cubes. I. Basic properties
James W.
Cannon
429-440
Abstract: Is a surface in a 3-manifold tame if it is tame modulo a tame set? This question was answered by the author through the introduction and characterization of taming sets. The purpose of this paper is to introduce and establish the basic properties of the more general and more flexible, but closely related, $^ \ast$-taming set.
$\sp{\ast} $-taming sets for crumpled cubes. II. Horizontal sections in closed sets
James W.
Cannon
441-446
Abstract: We prove that a closed subset X of ${E^3}$ is a $^ \ast$-taming set if no horizontal section of X has a degenerate component. This implies, for example, that a 2-sphere S in ${E^3}$ is tame if no horizontal section of S has a degenerate component. It also implies (less obviously) that a 2-sphere S in $ {E^3}$ is tame if it can be touched at each point from each side of S by a pencil.
$\sp{\ast} $-taming sets for crumpled cubes. III. Horizontal sections in $2$-spheres
James W.
Cannon
447-456
Abstract: We prove that a 2-sphere S in ${E^3}$ is tame if each horizontal section of S has at most four components. Since there are wild spheres in ${E^3}$ whose horizontal sections have at most five components, this result is, in a sense, best possible. Much can nevertheless be said, however, even if certain sections have more than five components; and we show that the wildness of a 2-sphere S in $ {E^3}$ is severely restricted by the requirement that each of the horizontal sections of S have at most finitely many components that separate S.
Endomorphism rings of torsionless modules
Arun Vinayak
Jategaonkar
457-466
Abstract: Let A be a right order in a semisimple ring $\Sigma ,{M_A}$ be a finite-dimensional torsionless right A-module and $ {\hat M_A}$ be the injective hull of M. J. M. Zelmanowitz has shown that $Q = {\rm {End}}\;{\hat M_A}$ is a semisimple ring and $S = {\rm {End}}\;{M_A}$ is a right order in Q. Further, if A is a two-sided order in $ \Sigma$ then S is a two-sided order in Q. We give a conceptual proof of this result. Moreover, we show that if A is a bounded order then so is S. The underlying idea of our proofs is very simple. Rather than attacking $S = {\rm {End}}\;{M_A}$ directly, we prove the results for $B = {\rm {End}}\;({M_A} \oplus {A_A})$. If $e:{M_A} \oplus {A_A} \to {M_A} \oplus {A_A}$ is the projection on M along ${A_A}$ then, of course, $S \cong eBe$ and it is easy to transfer the required information from B to S. The reason why it is any easier to look at B rather than S is that $ {M_A} \oplus {A_A}$ is a generator in $ \bmod$-$A$ and a Morita type transfer of properties from A to B is available. We construct an Artinian ring resp. Noetherian prime ring containing a right ideal whose endomorphism ring fails to be Artinian resp. Noetherian from either side.
A generalization of the strict topology
Robin
Giles
467-474
Abstract: The strict topology $ \beta$ on the space $ C(X)$ of bounded real-valued continuous functions on a topological space X was defined, for locally compact X, by Buck (Michigan Math. J. 5 (1958), 95-104). Among other things he showed that (a) $C(X)$ is $\beta$-complete, (b) the dual of $C(X)$ under the strict topology is the space of all finite signed regular Borel measures on X, and (c) a Stone-Weierstrass theorem holds for $\beta$-closed subalgebras of $ C(X)$. In this paper the definition of the strict topology is generalized to cover the case of an arbitrary topological space and these results are established under the following conditions on X: for (a) X is a k-space; for (b) X is completely regular; for (c) X is unrestricted.
Nonstandard analysis of dynamical systems. I. Limit motions, stability.
A. E.
Hurd
1-26
Abstract: The methods of nonstandard analysis are applied to the study of the qualitative theory of dynamical systems. The nonstandard notions connected with limiting behavior of motions (limit sets, etc.) are developed, and then applied to the study of stability theory, including stability of sets, attracting properties, first prolongations and stability of motions.
Representation of holomorphic functions by boundary integrals
Albert
Baernstein
27-37
Abstract: Let $K$ be a compact locally connected set in the plane and let $f$ be a function holomorphic in the extended complement of $K$ with $ f(\infty ) = 0$. We prove that there exists a sequence of measures $\{ {\mu _n}\}$ on $K$ satisfying ${\lim _{n \to \infty }}\vert\vert{\mu _n}\vert{\vert^{1/n}} = 0$ such that $f(z) = \sum\nolimits_{n = 0}^\infty {\int_K {{{(w - z)}^{ - n - 1}}d{\mu _n}(w)(z \in K)} }$. It follows from the proof that two topologies for the space of functions holomorphic on $K$ are the same. One of these is the inductive limit topology introduced by Köthe, and the other is defined by a family of seminorms which involve only the values of the functions and their derivatives on $ K$. A key lemma is an open mapping theorem for certain locally convex spaces. The representation theorem and the identity of the two topologies is false when $K$ is a compact subset of the unit circle which is not locally connected.
New existence theorems for nonlinear equations of Hammerstein type.
W. V.
Petryshyn;
P. M.
Fitzpatrick
39-63
Abstract: Let $X$ be a real Banach space, ${X^ \ast }$ its dual, $A$ a linear map of $X$ into ${X^ \ast }$ and $N$ a nonlinear map of ${X^ \ast }$ into $X$. Using the recent results of Browder and Gupta, Brezis, and Petryshyn, in this paper we study the abstract Hammerstein equation, $ w + ANw = 0$. Assuming suitable growth conditions on $N$, new existence results are obtained under the following conditions on $X,A$ and $N$. In §1: $X$ is reflexive, $A$ bounded with $ f(x) = (Ax,x)$ weakly lower semicontinuous, $N$ bounded and of type $ ($M$)$. In §2: $X$ is a general space, $ A$ angle-bounded, $ N$ pseudo-monotone. In §3: $X$ is weakly complete, $A$ strictly (strongly) monotone, $N$ bounded (unbounded) and of type $($M$)$. In §4: $X$ is a general space, $A$ is monotone and symmetric, $N$ is potential. In §5: $ X$ is reflexive and with Schauder basis, $ {X^ \ast }$ strictly convex, $N$ quasibounded and either monotone, or bounded and pseudo-monotone, or bounded and of type $($M$)$.
Maxima and high level excursions of stationary Gaussian processes
Simeon M.
Berman
65-85
Abstract: Let $X(t),t \geqq 0$, be a stationary Gaussian process with mean 0, variance 1 and covariance function $ r(t)$. The sample functions are assumed to be continuous on every interval. Let $r(t)$ be continuous and nonperiodic. Suppose that there exists $ \alpha , 0 < \alpha \leqq 2$, and a continuous, increasing function $g(t),t \geqq 0$, satisfying $\displaystyle (0.1)\quad \mathop {\lim }\limits_{t \to 0} \frac{{g(ct)}}{{g(t)}} = 1,\quad for\;every\;c > 0,$ such that $\displaystyle (0.2)\quad 1 - r(t) \sim g(\vert t\vert)\vert t{\vert^\alpha },\quad t \to 0.$ For $u > 0$, let $ v$ be defined (in terms of $ u$) as the unique solution of $\displaystyle (0.3)\quad {u^2}g(1/v){v^{ - \alpha }} = 1.$ Let $ {I_A}$ be the indicator of the event $A$; then $\displaystyle \int_0^T {{I_{[X(s) > u]}}ds}$ represents the time spent above $u$ by $ X(s),0 \leqq s \leqq T$. It is shown that the conditional distribution of $\displaystyle (0.4)\quad v\int_0^T {{I_{[X(s) > u]}}ds,}$ given that it is positive, converges for fixed $T$ and $u \to \infty $ to a limiting distribution ${\Psi _\alpha }$, which depends only on $ \alpha$ but not on $ T$ or $g$. Let $ F(\lambda )$ be the spectral distribution function corresponding to $ r(t)$. Let ${F^{(p)}}(\lambda )$ be the iterated $p$-fold convolution of $F(\lambda )$. If, in addition to (0.2), it is assumed that $\displaystyle (0.5)\quad {F^{(p)}}\;is\;absolutely\;continuous\;for\;some\;p > 0,$ then $ \max (X(s):0 \leqq s \leqq t)$, properly normalized, has, for $t \to \infty$, the limiting extreme value distribution $ \exp ( - {e^{ - x}})$. If, in addition to (0.2), it is assumed that $\displaystyle (0.6)\quad F(\lambda )\;is\; absolutely \;continuous\; with\; the\; derivative\; f(\lambda ),$ and $\displaystyle (0.7)\quad \mathop {\lim }\limits_{h \to 0} \log h\int_{ - \infty }^\infty {\vert f(\lambda } + h) - f(\lambda )\vert d\lambda = 0,$ then (0.4) has, for $ u \to \infty$ and $T \to \infty$, a limiting distribution whose Laplace-Stieltjes transform is $\displaystyle (0.8)\quad \exp [{\text{constant}}\int_0^\infty {} ({e^{ - \lambda \xi }} - 1)d{\Psi _\alpha }(x)],\quad \lambda > 0.$
$\Gamma $-compact maps on an interval and fixed points
William M.
Boyce
87-102
Abstract: We characterize the $ \Gamma$-compact continuous functions $f:X \to X$ where $X$ is a possibly-noncompact interval. The map $f$ is called $\Gamma$-compact if the closed topological semigroup $\Gamma (f)$ generated by $f$ is compact, or equivalently, if every sequence of iterates of $f$ under functional composition $(\ast)$ has a subsequence which converges uniformly on compact subsets of $X$. For compact $X$ the characterization is that the set of fixed points of $f\ast f$ is connected. If $X$ is noncompact an additional technical condition is necessary. We also characterize those maps $ f$ for which iterates of distinct orders agree ( $ \Gamma (f)$ finite) and state a result on common fixed points of commuting functions when one of the functions is $\Gamma$-compact.
Metric properties of transformations of $G$-spaces
R. K.
Thomas
103-117
Abstract: The measure-preserving transformation $T$ acts on a Lebesgue space $(M,\mathcal{B},\mu )$ which is also a $ G$-space for a compact separable group $G$. It is proved that if the factor-transformation on the space of $G$-orbits has completely positive entropy and a certain condition regarding the relations between the actions of $G$ and $T$ is satisfied, then $T$ weakly mixing implies $T$ has completely positive entropy.
The addition theorem for the entropy of transformations of $G$-spaces
R. K.
Thomas
119-130
Abstract: For a measure-preserving transformation $T$ which is a skew-product of a measure-preserving transformation $S$ and a topological group endomorphism $ \sigma$, it is shown that the entropy $h$ satisfies the following ``addition theorem": $ h(T) = h(S) + h(\sigma )$.
Bounds for solutions to a class of nonlinear integral equations
Robert H.
Martin
131-138
Abstract: In this paper a bound is obtained for solutions to a class of nonlinear Stieltjes integral equations. The class of equations under consideration have been previously studied by J. S. Mac Nerney. It is shown that the bound obtained here is in some sense a best bound, and in the linear case, reduces to a previous result of the author.
Identities involving the coefficients of a class of Dirichlet series. V
Bruce C.
Berndt
139-156
Abstract: We derive various forms of the Voronoï summation formula for a large class of arithmetical functions. These arithmetical functions are generated by Dirichlet series satisfying a functional equation with certain gamma factors. Using our theorems, we establish several arithmetical identities.
Identities involving the coefficients of a class of Dirichlet series. V, VI
Bruce C.
Berndt
157-167
Abstract: In 1949 Chowla and Selberg gave a very useful formula for the Epstein zeta-function associated with a positive definite binary quadratic form. Several generalizations of this formula are given here. The method of proof is new and is based on a theorem that we formerly proved for ``generalized'' Dirichlet series. An easy proof of Kronecker's second limit formula is also given.
Products with closed projections. II
N.
Noble
169-183
Abstract: Conditions under which some or all of the projections on a product space will be closed or $z$-closed are studied, with emphasis on infinite products. These results are applied to characterize normal products up to countably many factors, to characterize closed product maps up to finitely many factors, and to give conditions under which products will be countably compact, Lindelöf, paracompact, $\mathfrak{m} - \mathfrak{n}$-compact, etc. Generalizations of these results to $\mathfrak{n}$-products and box products are also given. Our easily stated results include: All powers of a ${T_1}$ space $X$ are normal if and only if $X$ is compact Hausdorff, all powers of a nontrivial closed map $p$ are closed if and only if $p$ is proper, the product of countably many Lindelöf $P$-spaces is Lindelöf; and the product of countably many countably compact sequential spaces is countably compact sequential.
Critical Markov branching processes with general set of types
H.
Hering
185-202
Abstract: This paper is concerned with the asymptotic behaviour of critical, quasi-positively regular Markov branching processes. Several results which have been established with restrictions on the set of types or on the parameter are proven on slightly different moment assumptions for an arbitrary set of types and continuous as well as discrete parameter. The methods employed are analytic and rest upon the properties of probability-generating functionals constructed from the given transition function.
Comparison and oscillation theorems for matrix differential inequalitites
E. S.
Noussair
203-215
Abstract: Strong comparison theorems of Sturm's type are established for systems of second order quasilinear elliptic partial differential equations. The technique used leads to new oscillation and nonoscillation criteria for such systems. Some criteria are deduced from a comparison theorem, and others are derived by a direct variational method. Some of our results constitute extensions of known theorems to nonselfadjoint quasilinear systems.
On the concentration and extension of cylinder measures
Alejandro D.
de Acosta
217-228
Abstract: It is shown that with respect to certain set-theoretic operations-- directed decreasing or even arbitrary intersections of certain families of convex, balanced, weakly closed sets--cylinder measures behave almost as regular Borel measures do. A refinement is proved when the cylinder measure satisfies a scalar concentration condition. These results are applied to obtain stronger versions of Prohorov's theorem and an extension theorem for complete Hausdorff locally convex spaces generalizing a result of Dudley, Feldman, and Le Cam.
Maximal and minimal topologies
Douglas E.
Cameron
229-248
Abstract: A topological space $ (X,T)$ with property ${\text{R}}$ is maximal $ {\text{R}}$ (minimal ${\text{R}}$) if $T$ is a maximal (minimal) element in the set $ {\text{R}}(X)$ of all topologies on the set $X$ having property $ {\text{R}}$ with the partial ordering of set inclusions. The properties of maximal topologies for compactness, countable compactness, sequential compactness, Bolzano-Weierstrass compactness, and Lindelöf are investigated and the relations between these spaces are investigated. The question of whether any space having one of these properties has a strictly stronger maximal topology is investigated. Some interesting product theorems are discussed. The properties of minimal topologies and their relationships are discussed for the quasi-$ P$, Hausdorff quasi-$ P$, and $P$ topologies.
Rings of invariant polynomials for a class of Lie algebras
S. J.
Takiff
249-262
Abstract: Let $G$ be a group and let $\pi :G \to GL(V)$ be a finite-dimensional representation of $G$. Then for $g \in G,\pi (g)$ induces an automorphism of the symmetric algebra $ S(V)$ of $V$. We let $ I(G,V,\pi )$ be the subring of $S(V)$ consisting of elements invariant under this induced action. If $G$ is a connected complex semisimple Lie group with Lie algebra $L$ and if Ad is the adjoint representation of $ G$ on $L$, then Chevalley has shown that $I(G,L,$Ad$)$ is generated by a finite set of algebraically independent elements. However, relatively little is known for nonsemisimple Lie groups. In this paper the author exhibits and investigates a class of nonsemisimple Lie groups $G$ with Lie algebra $L$ for which $ I(G,L,$Ad$)$ is also generated by a finite set of algebraically independent elements.
A theorem and a counterexample in the theory of semigroups of nonlinear transformations
Michael G.
Crandall;
Thomas M.
Liggett
263-278
Abstract: This paper studies the basic method in current use for constructively obtaining a generator from a given semigroup of nonlinear transformations on a Banach space. The method is shown to succeed in real two-dimensional Banach spaces and to fail in a particular three-dimensional example. Other results of independent interest are obtained. For example, it is shown that the concepts of ``maximal accretive'' and ``hyperaccretive'' (equivalently, $m$-accretive or hypermaximal accretive) coincide in ${R^n}$ with the maximum norm.
Relative types of points in $\beta N-N$
A. K.
Steiner;
E. F.
Steiner
279-286
Abstract: Using the concepts of type and relative type for points in $\beta N - N$, as introduced by W. Rudin, M. E. Rudin, and Z. Frolik, an inductive method is presented for constructing types. The relative types are described for points having these constructed types and a point in $\beta N - N$ is found which has exactly $ c$ relative types.
Construction of measures and integrals
B. S.
Thomson
287-296
Abstract: The systems introduced by R. Henstock and later by E. J. McShane to provide powerful generalizations of the Riemann integral are used to construct outer measures and upper integrals and to develop a Lebesgue type theory in quite general settings.
An uncountable collection of chainable continua
David P.
Bellamy
297-304
Abstract: We demonstrate the existence of an uncountable collection of chainable continua, no member of which can be mapped onto any other member. Path connectedness considerations and techniques involving compactifications play an important role in the construction and argument.
Semigroups that are the union of a group on $E\sp{3}$ and a plane
Frank
Knowles
305-325
Abstract: In Semigroups on a half-space, Trans. Amer. Math. Soc. 147 (1970), 1-53, Horne considers semigroups that are the union of a group $G$ and a plane $L$ such that $G \cup L$ is a three-dimensional half-space and $G$ is the interior. After proving a great many things about half-space semigroups, Horne introduces the notion of a radical and determines all possible multiplications in $L$ for a half-space semigroup with empty radical. (It turns out that $S$ has empty radical if and only if each $ G$-orbit in $L$ contains an idempotent.) An example is provided for each configuration in $ L$. However, no attempt was made to show that the list of examples actually exhausted the possibilities for a half-space semigroup without radical. Another way of putting this problem is to determine when two different semigroups can have the same maximal group. In this paper we generalize Horne's results, for a semigroup without zero, by showing that if $S$ is any locally compact semigroup in which $ L$ is the boundary of $ G$, then $S$ is a half-space. Moreover, we are able to answer completely, for semigroups without radical and without a zero, the question posed above. It turns out that, with one addition (which we provide), Horne's list of half-space semigroups without radical and without zero is complete.
Finitely generated nilpotent groups with isomorphic finite quotients
P. F.
Pickel
327-341
Abstract: Let $\mathcal{F}(G)$ denote the set of isomorphism classes of finite homomorphic images of a group $ G$. We say that groups $ G$ and $H$ have isomorphic finite quotients if $ \mathcal{F}(G) = \mathcal{F}(H)$. In this paper we show that if $G$ is a finitely generated nilpotent group, the finitely generated nilpotent groups $ H$ for which $ \mathcal{F}(G) = \mathcal{F}(H)$) lie in only finitely many isomorphism classes. This is done using some finiteness results from the theory of algebraic groups along with some heretofore unpublished results of A. Borel.
Real orthogonal representations of algebraic groups
Frank
Grosshans
343-352
Abstract: The purpose of this paper is to determine explicitly, nondegenerate real symmetric bilinear forms invariant under a real absolutely irreducible representation of a real semisimple algebraic group, $G$. If $G$ is split, we construct an extension ${G^ \ast }$ containing $G$ and those outer automorphisms of $ G$ fixing the highest weight of the representation. The representation is then extended to ${G^ \ast }$ and the form is described in terms of the character of this extension. The case of a nonsplit algebraic group is then reduced to the above. The corresponding problem for representations by matrices over the real quaternion division algebra is also considered using similar methods.
An asymptotic theorem for systems of linear differential equations.
Allen
Devinatz
353-363
Abstract: Asymptotic estimates are obtained for a complete linearly independent set of solutions of a linear system of differential equations of the form $n \times n$ matrix with $n$ distinct eigenvalues, $R(t)$ is an integrable matrix valued function on $(0,\infty )$ and $V(t)$ is an $n \times n$ matrix valued function having certain differentiability properties. The theorem that is obtained generalizes a theorem of N. Levinson, Duke Math. J. 15 (1948), 111-126.
An extension of the Weyl-von Neumann theorem to normal operators
I. David
Berg
365-371
Abstract: We prove that a normal operator on a separable Hilbert space can be written as a diagonal operator plus a compact operator. If, in addition, the spectrum lies in a rectifiable curve we show that the compact operator can be made Hilbert-Schmidt.
Complex Fourier analysis on a nilpotent Lie group
Roe
Goodman
373-391
Abstract: Let $G$ be a simply-connected nilpotent Lie group, with complexification $ {G_c}$. The functions on $ G$ which are analytic vectors for the left regular representation of $ G$ on ${L_2}(G)$ are determined in this paper, via a dual characterization in terms of their analytic continuation to ${G_c}$, and by properties of their ${L_2}$ Fourier transforms. The analytic continuation of these functions is shown to be given by the Fourier inversion formula. An explicit construction is given for a dense space of entire vectors for the left regular representation. In the case $G = R$ this furnishes a group-theoretic setting for results of Paley and Wiener concerning functions holomorphic in a strip.
Automorphisms of a free associative algebra of rank $2$. I
Anastasia J.
Czerniakiewicz
393-401
Abstract: Let ${R_{\left\langle 2 \right\rangle }} = R\left\langle {x,y} \right\rangle$ be the free associative algebra of rank 2, on the free generators $x$ and $y$, over $R$ ($R$ a field, a Euclidean domain, etc.). We prove that if $\varphi$ is an automorphism of $ {R_{\left\langle 2 \right\rangle }}$ that keeps $(xy - yx)$ fixed (up to multiplication by an element of $R$), then $\varphi$ is tame, i.e. it is a product of elementary automorphisms of ${R_{\left\langle 2 \right\rangle }}$. This follows from a more general result about endomorphisms of ${R_{\left\langle 2 \right\rangle }}$ via a theorem due to H. Jung [6] concerning automorphisms of a commutative and associative algebra of rank 2.
Uniqueness of solutions of the Dirichlet and Neumann problems for hyperbolic equations
Eutiquio C.
Young
403-409
Abstract: Conditions for uniqueness of solutions of the Dirichlet and Neumann problems are obtained for a singular hyperbolic equation involving a real parameter.
On the remainders of certain metric spaces
Kenneth D.
Magill
411-417
Abstract: Let $X$ be completely regular and Hausdorff. The family $ \mathcal{R}(X)$ of all those spaces which are remainders of $X$ in compactifications has attracted a considerable amount of interest. In this paper we determine, in a certain sense, the family $\mathcal{R}(X)$ for a fairly large number of spaces $X$.
When is $\mu \ast L\,\sb{1}$ closed?
I.
Glicksberg
419-425
Abstract: For a finite measure $\mu$ on a locally compact abelian group, we partially answer the question of when $\mu \ast {L_1}$ is closed in ${L_1}$.
Some new classes of kernels whose Fredholm determinants have order less than one
Dale W.
Swann
427-435
Abstract: Let $K(s,t)$ be a complex-valued $ {L_2}$ kernel on the square $a \leqq s,t \leqq b$ and $\{ {\lambda _v}\}$, perhaps empty, denote the set of finite characteristic values (f.c.v.) of $ K$, arranged according to increasing modulus. Such f.c.v. are complex numbers appearing in the integral equation $ {\phi _v}(s) = {\lambda _v}\int_a^b {K(s,t){\phi _v}(t)dt}$, where the ${\phi _v}(s)$ are nontrivial ${L_2}$ functions on $[a,b]$. Further let ${k_1} = \int_a^b {K(s,s)} $ be well defined so that the Fredholm determinant of $K,D(\lambda )$, exists, and let $\mu$ be the order of this entire function. It is shown that (1) if $K(s,t)$ is a function of bounded variation in the sense of Hardy-Krause, then $ \mu \leqq 1$; (2) if in addition to the assumption (1), $K(s,t)$ satisfies a uniform Lipschitz condition of order $ \alpha > 0$ with respect to either variable, then $\mu < 1$ and ${k_1} = {\Sigma _v}1/{\lambda _v}$; (3) if $ K(s,t)$ is absolutely continuous as a function of two variables and $ {\partial ^2}K/\partial s\partial t$ (which exists almost everywhere) belongs to class ${L_p}$ for some $p > 1$, then $\mu < 1$ and ${k_1} = {\Sigma _v}1/{\lambda _v}$. In (2) and (3), the condition $ {k_1} \ne 0$ implies $ K(s,t)$ possesses at least one f.c.v.
Expandibility and collectionwise normality
J. C.
Smith;
L. L.
Krajewski
437-451
Abstract: In 1958 M. Katětov proved that in a normal space $X,X$ is expandable if and only if $ X$ is collectionwise normal and countably paracompact. This result has since been used to answer many questions in various areas of general topology. In this paper Katětov's theorem is generalized for nonnormal spaces and various characterizations of collectionwise normality are shown. Results concerning metrization, paracompactness, sum theorems, product theorems, mapping theorems and $M$-spaces are then obtained as applications of these new theorems.
Linear representations of certain compact semigroups
D. R.
Brown;
Michael
Friedberg
453-465
Abstract: In this paper we initiate the study of representation theory of compact, not necessarily commutative, uniquely divisible semigroups. We show that a certain class of semigroups are all topologically isomorphic to real matrix semigroups. The proof utilizes a group embedding theorem and the standard results on homomorphisms of Lie groups into matrix groups.
On knots with nontrivial interpolating manifolds
Jonathan
Simon
467-473
Abstract: If a polygonal knot $ K$ in the $3$-sphere ${S^3}$ does not separate an interpolating manifold $ S$ for $K$, then $S - K$ does not carry the first homology of either closed component of ${S^3} - S$. It follows that most knots $ K$ with nontrivial interpolating manifolds have the property that a simply connected manifold cannot be obtained by removing a regular neighborhood of $K$ from ${S^3}$ and sewing it back differently.
Homomorphisms on groups and induced maps on certain algebras of measures
Charles F.
Dunkl;
Donald E.
Ramirez
475-485
Abstract: Suppose that $ \varphi$ is a continuous homomorphism of a locally compact group $G$ into another such group, $ H$, then $\varphi$ induces in a natural way a homomorphism $ {\varphi ^ \ast }$ of the measure algebra of $G$, called $M(G)$, into $M(H)$. The action of $ {\varphi ^ \ast }$ on the subspace ${M_0}(G)$ is studied in this paper. The space $ {M_0}(G)$ is the nonabelian analogue to the space of measures on a locally compact abelian group whose Fourier-Stieltjes transforms vanish at infinity, and is defined herein. We prove that if $\varphi$ is an open homomorphism then $ {\varphi ^ \ast }({M_0}(G)) \subset {M_0}(H)$. If $G$ and $H$ are abelian and $\varphi$ is not open, then ${\varphi ^ \ast }(M(G)) \cap {M_0}(H) = \{ 0\}$. The main tool for this theorem is the fact, proved herein, that $\varphi$ is open if and only if its adjoint, $\hat \varphi :\hat H \to \hat G$, is proper (where $\hat G,\hat H$ are the character groups of $G,H$ resp.). Further properties of ${M_0}(G)$ for abelian or compact groups $ G$ are derived.
Minimal convergence spaces
D. C.
Kent;
G. D.
Richardson
487-499
Abstract: We are primarily concerned with minimal $ {\text{P}}$ convergence spaces, where $ {\text{P}}$ is one of the following convergence space properties: Hausdorff, ${{\text{T}}_2}, \lambda$-regular, $ \lambda$-Urysohn, and first countable, $\lambda$ an infinite cardinal number. Our conclusions usually resemble the corresponding topological results, but with some deviations ; for instance, a minimal Hausdorff convergence space is always compact, whereas a countable minimal regular convergence space need not be compact.
Structure of the solution set of some first order differential equations of comparison type
T. G.
Hallam;
J. W.
Heidel
501-512
Abstract: The comparison principle is a powerful tool that has a wide variety of applications in ordinary differential equations. The results of this article describe the geometric structure of the solution space of some first order scalar differential equations that may arise in the comparison method. A quite general class of differential equations is found to have a similar solution set configuration as the differential equation of separable variable type. One of the main results establishes, under certain conditions, that there is a unique unbounded solution of the first order differential equation which exists on an interval of the form [ ${t_0},\infty$). Furthermore, this unbounded solution separates the solutions that are bounded on [ ${t_0},\infty $) from those that are not continuable to all $t > {t_0}$.
Existence and duality theorems for convex problems of Bolza
R. T.
Rockafellar
1-40
Abstract: The theory of conjugate convex functions is applied to a fundamental class of ``convex'' problems in the calculus of variations and optimal control. This class has many special properties which have not previously been exploited and for which the standard methods of approach are inadequate. Duality theorems are established which yield new results on the existence of optimal arcs, as well as necessary and sufficient conditions for optimality. These results have some relevance also to the study of ``nonconvex'' problems.
Extensions in certain topological algebraic categories
Robert C.
Busby
41-56
Abstract: Categories which we call ``sufficiently algebraic'' are defined, and for certain objects $A$ (called faithful) in such categories, and arbitrary objects $C$, we partially order the sets $\operatorname{Ext} (C,A)$ of extensions of $ A$ by $C$. We prove that the maximal elements in $\operatorname{Ext} (C,A)$ (with respect to this ordering) are in bijective correspondence with the morphisms from $C$ to a canonical object $O(A)$. If the short five lemma holds in the category, all extensions are maximal and therefore obtained in this way. As an application we compute extensions in certain categories of topological rings. In particular we investigate the possible extensions of one group algebra (of a locally compact group) by another in the category of Banach algebras with norm decreasing homomorphisms, and using some analytic tools we give conditions for the splitting of such extensions. Previous results of the author on extensions of ${C^ \ast }$-algebras are also included in this theory as a special case.
Integration theory on infinite-dimensional manifolds
Hui Hsiung
Kuo
57-78
Abstract: The purpose of this paper is to develop a natural integration theory over a suitable kind of infinite-dimensional manifold. The type of manifold we study is a curved analogue of an abstract Wiener space. Let $H$ be a real separable Hilbert space, $ B$ the completion of $ H$ with respect to a measurable norm and $i$ the inclusion map from $H$ into $B$. The triple $(i,H,B)$ is an abstract Wiener space. $B$ carries a family of Wiener measures. We will define a Riemann-Wiener manifold to be a triple $ (\mathcal{W},\tau ,g)$ satisfying specific conditions, $ \mathcal{W}$ is a $ {C^j}$-differentiable manifold $(j \geqq 3)$ modelled on $B$ and, for each $x$ in $\mathcal{W},\tau (x)$ is a norm on the tangent space $ {T_x}(\mathcal{W})$ of $\mathcal{W}$ at $x$ while $g(x)$ is a densely defined inner product on $ {T_x}(\mathcal{W})$. We show that each tangent space is an abstract Wiener space and there exists a spray on $\mathcal{W}$ associated with $g$. For each point $x$ in $ \mathcal{W}$ the exponential map, defined by this spray, is a ${C^{j - 2}}$-homeomorphism from a $ \tau (x)$-neighborhood of the origin in $ {T_x}(\mathcal{W})$ onto a neighborhood of $x$ in $ \mathcal{W}$. We thereby induce from Wiener measures of ${T_x}(\mathcal{W})$ a family of Borel measures ${q_t}(x, \cdot ),t > 0$, in a neighborhood of $x$. We prove that $ {q_t}(x, \cdot )$ and ${q_s}(y, \cdot )$, as measures in their common domain, are equivalent if and only if $t = s$ and $ {d_g}(x,y)$ is finite. Otherwise they are mutually singular. Here $ {d_g}$ is the almost-metric (in the sense that two points may have infinite distance) on $ \mathcal{W}$ determined by $ g$. In order to do this we first prove an infinite-dimensional analogue of the Jacobi theorem on transformation of Wiener integrals.
Conjugacy separability of the groups of hose knots
Peter F.
Stebe
79-90
Abstract: Let $G$ be a group. An element $g$ of $G$ is c.d. in $G$ if and only if, given any element $h$ of $G$, either $h$ is conjugate to $g$ or there is a homomorphism $\xi$ from $G$ onto a finite group such that $\xi (g)$ is not conjugate to $ \xi (h)$. Following A. Mostowski, a group is conjugacy separable or c.s. if and only if every element of the group is c.d. In this paper we show that the groups of hose knots are c.s.
On zonotopes
P.
McMullen
91-109
Abstract: In this paper is described a diagram technique for zonotopes, or vector sums of line segments, which is analogous to that of Gale diagrams for general polytopes, and central diagrams for centrally symmetric polytopes. The use of these new zonal diagrams leads to relationships between zonotopes with $n$ zones of dimensions $d$ and $n - d$, and enables one to enumerate all the combinatorial types of $d$-zonotopes with $n \leqq d + 2$ zones. The connexion between arrangements of hyperplanes in projective space and zonotopes leads to corresponding new results about arrangements.
Representation of vector valued nonlinear functions
Victor J.
Mizel;
K.
Sundaresan
111-127
Abstract: A representation theorem for ``additive'' nonlinear functional on spaces ${L^p}(\mu )$ is here extended to ``additive'' nonlinear functions from Lebesgue-Bochner function spaces $ L_E^p(\mu )$ ($ E$ a separable Banach space) into Banach spaces $B$. A counterexample is provided to show that the restriction to separable $E$ is essential.
Estimates for best approximation to rational functions
S. J.
Poreda
129-135
Abstract: Estimates for the deviation of certain rational functions and their polynomials of best uniform approximation on various sets are given. As a result, in some cases these deviation and polynomials are explicitly calculated. For example, the polynomials of best uniform approximation to the function $(\alpha z + \beta )/(z - a)(1 - \bar az),\vert a\vert \ne 1$, on the unit circle are given.
Value distribution of harmonic polynomials in several real variables.
Morris
Marden
137-154
Abstract: Using Bergman's integral operator method, the author studies an arbitrary axisymmetric harmonic polynomial $H(x,\rho )$ in ${R^3}$ and ${R^N}$ in relation to its associate polynomial $ h(\zeta )$ in $ C$. His results pertain to the value distributions and critical circles of $H(x,\rho )$ in certain cones; bounds on the gradient of an $H(x,\rho )$ assumed bounded in sphere ${x^2} + {\rho ^2} \leqq 1$; axisymmetric harmonic vectors. Corresponding results are also obtained for axisymmetric harmonic functions $F(x,\rho )$ with rational associate $f(\zeta )$.
A pointwise convergence theorem for sequences of continuous functions.
K.
Schrader
155-163
Abstract: Let $\{ {f_k}\}$ be a sequence of continuous real valued functions defined on an interval $I$ and $N$ a fixed nonnegative integer such that if ${f_k}(x) = {f_i}(x)$ for more than $N$ distinct values of $x \in I$ then ${f_{k}}(x) \equiv {f_i}(x)$ for $ x \in I$. It follows that there is a subsequence $ \{ {g_j}\}$ of $\{ {f_k}\}$ such that for each $x$ the subsequence $\{ {g_j}(x)\}$ is eventually monotone. Thus ${\lim _{j \to + \infty }}{g_j}(x) = f(x)$ exists for all $x$, where $f$ is an extended real valued function. If $\vert{f_k}(x)\vert$ is bounded for each $ x \in I$ then $ {\lim _{j \to + \infty }}{g_j}(x) = f(x)$ exists as a finite limit for all $ x \in I$. For $ N = 0$ this reduces to picking a monotone subsequence from a sequence of continuous functions whose graphs are pairwise disjoint.
Computability by probabilistic Turing machines
Eugene S.
Santos
165-184
Abstract: In the present paper, the definition of probabilistic Turing machines is extended to allow the introduction of relative computability. Relative computable functions, predicates and sets are discussed and their operations investigated. It is shown that, despite the fact that randomness is involved, most of the conventional results hold in the probabilistic case. Various classes of ordinary functions characterizable by computable random functions are introduced, and their relations are examined. Perhaps somewhat unexpectedly, it is shown that, in some sense, probabilistic Turing machines are capable of computing any given function. Finally, a necessary and sufficient condition for an ordinary function to be partially recursive is established via computable probabilistic Turing machines.
Limit theorems for measures on nonmetrizable locally compact abelian groups
David C.
Bossard
185-205
Abstract: In a recent book, Parthasarathy provides limit theorems for sums of independent random variables defined on a metrizable locally compact abelian group. These results make heavy use of the metric assumption. This paper consists of a reworking of certain results contained in Parthasarathy to see what can be done without the metric restriction. Among the topics considered are: necessary and sufficient conditions for a limit law to have an idempotent factor; the relationship between limits of compound Poisson laws and limits of sums of independent random variables; and a representation theorem for certain limit laws.
Interpolation theorems for the pairs of spaces $(L\sp{p},\,L\sp{\infty })$ and $(L\sp{1},\,L\sp{q})$
George G.
Lorentz;
Tetsuya
Shimogaki
207-221
Abstract: A Banach space $ Z$ has the interpolation property with respect to the pair $(X,Y)$ if each $T$, which is a bounded linear operator from $ X$ to $X$ and from $Y$ to $Y$, can be extended to a bounded linear operator from $ Z$ to $Z$. If $X = {L^p},Y = {L^\infty }$ we give a necessary and sufficient condition for a Banach function space $ Z$ on $(0,l),0 < l \leqq + \infty$, to have this property. The condition is that $g \prec {}^pf$ and $f \in Z$ should imply $g \in Z$; here $ g \prec {}^pf$ means that $ {g^{ \ast p}} \prec {f^{ \ast p}}$ in the Hardy-Littlewood-Pólya sense, while ${h^ \ast }$ denotes the decreasing rearrangement of the function $\vert h\vert$. If the norms $\vert\vert T\vert{\vert _X},\vert\vert T\vert{\vert _Y}$ are given, we can estimate $\vert\vert T\vert{\vert _Z}$. However, there is a gap between the necessary and the sufficient conditions, consisting of an unknown factor not exceeding ${\lambda _p},{\lambda _p} \leqq {2^{1/q}},1/p + 1/q = 1$. Similar results hold if $X = {L^1},Y = {L^q}$. For all these theorems, the complete continuity of $T$ on $Z$ is assured if $T$ has this property on $X$ or on $Y$, and if $Z$ satisfies a certain additional necessary and sufficient condition, expressed in terms of $\vert\vert{\sigma _a}\vert{\vert _Z},a > 0$, where ${\sigma _a}$ is the compression operator $ {\sigma _a}f(t) = f(at),0 \leqq t < l$.
How changing $D[[x]]$ changes its quotient field
Philip B.
Sheldon
223-244
Abstract: Let $D[[x]]$ be the ring of formal power series over the commutative integral domain $ D$. It is shown that changing $D[[x]]$ to $D[[x/a]]$ changes (i.e. increases) the quotient field by an infinite transcendence degree over the original field whenever $\cap _{i = 1}^\infty {a^i}D = 0$. From this it follows that if ${D_1}$ and ${D_2}$ are two distinct rings between the integers and the rational numbers, with ${D_1}$ contained in ${D_2}$, then the change in the ring of coefficients from $ {D_1}[[x]]$ to $D_{2}[[x]]$ again yields a change in the quotient fields by an infinite transcendence degree. More generally, it is shown that $D$ is completely integrally closed iff any increase in the ring of coefficients yields an increase in the quotient field of $D[[x]]$. Moreover, $D$ is a one-dimensional Prüfer domain iff any change in the ring of coefficients from one overring of $D$ to another overring of $D$ yields a change in the quotient field of the respective power series rings. Finally it is shown that many of the domain properties of interest are really properties of their divisibility groups, and some examples are constructed by first constructing the required divisibility groups.
Generally $p\sp{\alpha }$-torsion complete abelian groups
Paul F.
Dubois
245-255
Abstract: A generalized $ p$-primary cotorsion abelian group $G$ is a $ {p^\alpha }$-injective, that is satisfies ${p^\alpha }\operatorname{Ext} ( - ,G) = 0$, iff ${G_t}$ is $ {p^\alpha }$-injective in the category of torsion abelian groups. Such a torsion group is generally $ {p^\alpha }$-torsion complete, but an example shows that all its Ulm factors need not be complete. The injective properties of generally ${p^\alpha }$-torsion complete groups are investigated. They are an injectively closed class, and the corresponding class of sequences is the class of ${p^\alpha }$-pure sequences with split com-c pletion when $\alpha$ is ``accessible". Also, these groups are the $ {p^\alpha }$-high injectives.
Ramsey's theorem for $n$-parameter sets
R. L.
Graham;
B. L.
Rothschild
257-292
Abstract: Classes of objects called $n$-parameter sets are defined. A Ramsey theorem is proved to the effect that any partitioning into $ r$ classes of the $ k$-parameter subsets of any sufficiently large $n$-parameter set must result in some $ l$-parameter subset with all its $k$-parameter subsets in one class. Among the immediate corollaries are the lower dimensional cases of a Ramsey theorem for finite vector spaces (a conjecture of Rota), the theorem of van der Waerden on arithmetic progressions, a set theoretic generalization of a theorem of Schur, and Ramsey's Theorem itself.
A representation theorem for large and small analytic solutions of algebraic differential equations in sectors
Steven
Bank
293-305
Abstract: In this paper, we treat first-order algebraic differential equations whose coefficients belong to a certain type of function field. In the particular case where the coefficients are rational functions, our main result states that for any given sector $S$ in the plane, there exists a positive real number $N$, depending only on the equation and the angle opening of $S$, such that any solution $y(z)$, which is meromorphic in $ S$ and satisfies the condition $ {z^{ - N}}y \to \infty$ as $z \to \infty$ in $S$, must be of the form $\exp \int {c{z^m}(1 + o(1))}$ in subsectors, where $c$ and $m$ are constants. (From this, we easily obtain a similar representation for analytic solutions in $ S$, which are not identically zero, and for which $ {z^K}y \to 0$ as $z \to \infty$ in $S$, where the positive real number $K$ again depends only on the equation and the angle opening of $S$fs
Theorems on Cartan subalgebras like some on Carter subgroups
Ernest L.
Stitzinger
307-315
Abstract: We consider some results on the Cartan subalgebras of a solvable Lie algebra which are analogues to some results on Carter subgroups of a finite solvable group. Only solvable Lie algebras are considered here.
Grunsky-Nehari inequalities for a subclass of bounded univalent functions
D. W.
DeTemple
317-328
Abstract: Let ${D_1}$ be the class of regular analytic functions $F(z)$ in the disc $U = \{ z:\vert z\vert < 1\}$ for which $F(0) > 0,\vert F(z)\vert < 1$, and $F(z) + F(\zeta ) \ne 0$ for all $z,\zeta \in U$. Inequalities of the Grunsky-Nehari type are obtained for the univalent functions in $ {D_1}$, the proof being based on the area method. By subordination it is shown univalency is unnecessary for certain special cases of the inequalities. Employing a correspondence between $ {D_1}$ and the class $ {S_1}$ of bounded univalent functions, the results can be reinterpreted to apply to this latter class.
Some characterizations of $n$-dimensional $F$-spaces
M. J.
Canfell
329-334
Abstract: In this paper we obtain characterizations of an $n$-dimensional $F$-space in terms of the rings of continuous real-valued and complex-valued functions defined on the space. Motivation for these results is the work of Gillman and Henriksen on $U$-spaces ($F$-spaces of dimension 0) and $T$-spaces ($F$-spaces of dimension 0 or 1).
On isomorphisms between ideals in rings of continuous functions
David
Rudd
335-353
Abstract: A ring of continuous functions is a ring of the form $C(X)$, the ring of all continuous real-valued functions on a completely regular Hausdorff space $X$. For an arbitrary ideal $I$ in $C(X)$, the author shows that the maximal ideals of $ I$ are precisely the ideals of the form $I \cap M$, for some maximal ideal $M$ in $C(X)$ not containing $I$. The author shows that any ring isomorphism between ideals in any two rings of continuous functions preserves order, boundedness, and lattice structure; and he uses these results to obtain one of the main theorems: An isomorphism of a maximal ideal in $C(X)$ onto a maximal idea in $C(Y)$ can be extended to an isomorphism of $C(X)$ onto $C(Y)$. Another of the main theorems characterizes those isomorphisms between ${C^ \ast }(X)$ and $ {C^ \ast }(Y)$ (the subrings of bounded functions in $C(X)$ and $C(Y)$ respectively) which can be extended to isomorphisms between $C(X)$ and $C(Y)$. The author proves that, given any ideal $ I$ in $C(X)$, there exists a space $ X(I)$ so that the uniform closure of $I$ is isomorphic to a maximal ideal in $ C(X(I))$.
Area measure and Rad\'o's lower area
Togo
Nishiura
355-367
Abstract: The theory of Geöcze area for two-dimensional surfaces in threedimensional space had been essentially completed by the mid 1950's. The only hypothesis needed for all theorems in this case is the finiteness of the area. See [2] for an account of this theory. In the early 1960's, H. Federer established, in his paper [6], fundamental facts concerning his integral geometric area for higher dimensional area theory by employing the theory of normal and integral currents. These facts employ not only the finiteness of area as a basic hypothesis but certain other hypotheses as well. The extensions of Geöcze type area to higher dimensions also employ not only the finiteness of area but certain added hypotheses. These hypotheses are of such a nature as to allow the use of the theory of quasi-additivity [3], [11]. The present paper concerns these added hypotheses which play such an important part of higher-dimensional area theory of today. It is shown that Radó's lower area is the best Geöcze type area to describe these added hypotheses. That is, it is shown that the quasi-additivity hypotheses of Geöcze area in [11] imply the quasi-additivity hypotheses of lower area. Second, it is shown that the quasi-additivity hypotheses for lower area imply that the surface has the essential cylindrical property defined by J. Breckenridge in [5]. This essential cylindrical property is proved to be equivalent to the existence of area measures on the middle space of the mapping representing the surface. Finally, it is shown that the essential cylindrical property of a surface is equivalent to the quasi-additivity condition for lower area. Thus, an intrinsic property of the surface characterizes the quasi-additivity condition for the lower area of a surface.
Removable sets for pointwise subharmonic functions
Victor L.
Shapiro
369-380
Abstract: Pointwise subharmonic is defined in terms of the pointwise ${L^1}$ total derivative of order 2. The class $\mathcal{A}({x^ \ast },{r_ \ast })$ is introduced for the ball $ B({x^ \ast },{r_ \ast })$, and the following theorem is established: Let $ Q$ be a Borel set of Lebesgue measure zero contained in $B({x^ \ast },{r_ \ast })$. Then a necessary and sufficient condition that $Q$ be removable for pointwise subharmonic functions with respect to the class $\mathcal{A}({x^ \ast },{r_ \ast })$ is that $ Q$ be countable. It is also shown that the class $\mathcal{A}({x^ \ast },{r_ \ast })$ is in a certain sense best possible for the sufficiency of the above theorem.
Bounds on the nilpotency class of certain semidirect products
Larry
Morley
381-390
Abstract: Gilbert Baumslag has shown that the standard wreath product of $ A$ by $B$ is nilpotent if and only if $ A$ and $B$ are $p$-groups for the same prime $p, A$ is nilpotent of bounded exponent and $B$ is finite. L. Kaloujnine and Marc Krasner have shown that the standard (unrestricted) wreath product of $A$ by $B$ contains an isomorphic copy of every group $G$ which is an extension of $A$ by $B$. Thus it follows that any extension subject to the above condition on $A$ and $B$ is nilpotent. In this paper, the author gives an explicit characterization of the terms of the lower central series of a semidirect product $W$ of an abelian group by an arbitrary group. He then establishes a formula for an upper bound on the nilpotency class of $W$ when $W$ is a semidirect product of an abelian $ p$-group $X$ of bounded exponent by a finite $ p$-group $B$. This new bound is given in terms of the exponent of $X$ and the cycle structure of the factor groups of the lower central series of $B$.
Teichm\"uller inequalities without coefficient normalization
Arthur E.
Obrock
391-416
Abstract: Teichmüller's relation between the coefficients of extremal schlicht functions and quadratic differentials is extended. The coefficient normalization hypothesis in his theorem is dropped with the result that the new coefficient relations become more complex. This completes the partial result in this direction which is contained in Jenkins' General Coefficient Theorem. A modification of the version of the length-area method used by Teichmüller and Jenkins is introduced in our proof.
Equivariant bordism and Smith theory
R. E.
Stong
417-426
Abstract: The relationship between equivariant bordism and Smith homology theory on the category of pairs with involution is studied.
Some examples in degree of approximation by rational functions
D.
Aharonov;
J. L.
Walsh
427-444
Abstract: We exhibit examples of (1) series that converge more rapidly than any geometric series where the function represented has a natural boundary, (2) the convergence of a series with maximum geometric degree of convergence yet having limit points of poles of the series everywhere dense on a circumference in the complement of $E$, (3) a Padé table for an entire function whose diagonal has poles every-where dense in the plane and (4) a corresponding example for the table of rational functions of best approximation of prescribed type.
Inner ideals in quadratic Jordan algebras
Kevin
McCrimmon
445-468
Abstract: The inner ideals play a role in the theory of quadratic Jordan algebras analogous to that played by the one-sided ideals in the theory of associative algebras. In particular, the Jordan algebras with descending chain condition on inner ideals are intimately related to the Artinian associative algebras. In this paper we will completely characterize all inner ideals in the semisimple Jordan algebras with descending chain condition. It is well known that any left or right ideal $\mathfrak{B}$ in a semisimple Artinian $\mathfrak{A}$ is determined by an idempotent, $ \mathfrak{B} = \mathfrak{A}f$ or $\mathfrak{B} = e\mathfrak{A}$. We show that any inner ideal in the quadratic Jordan algebra ${\mathfrak{A}^ + }$ has the form $ \mathfrak{B} = e\mathfrak{A}f$, and if $ \mathfrak{A}$ has involution $^\ast$ the inner ideals of the Jordan algebra $\mathfrak{H}(\mathfrak{A}, ^ \ast )$ of $ ^ \ast$-symmetric elements are ``usually'' of the form $\mathfrak{B} = {e^ \ast }\mathfrak{H}e$. We also characterize the inner ideals in the Jordan algebras $ \mathfrak{J}(Q,c)$ or $ \mathfrak{J}(N,c)$ determined by a quadratic or cubic form.
Modules over Pr\"ufer domains
Syed M.
Fakhruddin
469-487
Abstract: We prove that the category of torsion-free modules over a Prüfer domain is equivalent to a certain category of diagrams. As applications we derive (i) a strong version of Baer-Kulikov-Kaplansky theorem for modules over a valuation ring, (ii) the determination of a complete set of invariants for modules of almost countable rank over an almost-maximal valuation ring, (iii) a new nonhomological proof of the characterization of a maximal valuation ring due to Matlis.
A separable postliminal $C\sp{\ast} $-algebra without maximal closed ideals
H.
Leptin
489-496
Abstract: Let $G$ be the free abelian group with a countable number of generators. We construct a separable locally compact $G$-transformation space $X$ without closed minimal invariant subsets, such that the corresponding $ {C^ \ast }$-algebra ${C^ \ast }(G,X)$ has the properties mentioned in the title. Using $X$ we also give an example of a transformation space $(G,Z)$ without closed minimal invariant subset, on which $G$ acts freely.
Rearrangements
Henryk
Minc
497-504
Abstract: It is shown that if ${a^{(t)}} = (a_1^{(t)},a_2^{(t)}, \ldots ,a_n^{(t)}),t = 1, \ldots ,m$, are nonnegative $ n$-tuples, then the maxima of $ \sum\nolimits_{i = 1}^n {a_i^{(1)}a_i^{(2)} \cdots a_i^{(m)}}$ of $\prod\nolimits_{i = 1}^n {{{\min }_t}(a_i^{(t)})}$ and of $ \Sigma _{i = 1}^n$ min $(a_i^{(t)})$, and the minima of $\prod\nolimits_{i = 1}^n {(a_i^{(1)} + a_i^{(2)} + } \cdots + a_i^{(m)})$, of $ \prod\nolimits_{i = 1}^n {{{\max }_t}(a_i^{(t)})}$ and of $ \sum\nolimits_{i = 1}^n {{{\max }_t}(a_i^{(t)})}$ are attained when the $ n$-tuples $ {a^{(1)}},{a^{(2)}}, \ldots ,{a^{(m)}}$ are similarly ordered. Necessary and sufficient conditions for equality are obtained in each case. An application to bounds for permanents of $ (0,1)$-matrices is given.
Analytic continuation of the series $\sum \,(m+nz)\sp{-s}$
Joseph
Lewittes
505-509
Abstract: The series $\Sigma {(m + nz)^{ - s}},m,n$ ranging over all integers except both zero, for $s$ an integer greater than two is well known from the theory of elliptic functions and modular forms. In this paper, we show that this series defines an analytic function $(z,s)$ for $\operatorname{Im} z > 0$ and $\operatorname{Re} s > 2$ which has an analytic continuation to all values of $s$. It is then shown that $G$ satisfies a functional equation under the transformation $z \to - 1/z$, and finally as an application some numerical results are obtained.
Errata to ``Hypergeometric operator series and related partial differential equations''
Louis R.
Bragg
511
Prime ideals in a large class of nonassociative rings
Paul J.
Zwier
257-271
Abstract: In this paper a definition is given for a prime ideal in an arbitrary nonassociative ring $N$ under the single restriction that for a given positive integer $s \geqq 2$, if $A$ is an ideal in $N$, then ${A^s}$ is also an ideal. ($N$ is called an $s$-naring.) This definition is used in two ways. First it is used to define the prime radical of $ N$ and the usual theorems ensue. Second, under the assumption that the $ s$-naring $N$ has a certain property $ (\alpha )$, the Levitzki radical $L(N)$ of $N$ is defined and it is proved that $L(N)$ is the intersection of those prime ideals $P$ in $N$ whose quotient rings are Levitzki semisimple. $ N$ has property $ (\alpha )$ if and only if for each finitely generated subring $A$ and each positive integer $ m$, there is an integer $ f(m)$ such that ${A^{f(m)}} \subseteq {A_m}$. (Here ${A_1} = {A^s}$ and ${A_{ m + 1}} = A_m^s$.) Furthermore, conditions are given on the identities an $s$-naring $N$ satisfies which will insure that $N$ satisfies $(\alpha )$. It is then shown that alternative rings, Jordan rings, and standard rings satisfy these conditions.
Locally noetherian commutative rings
William
Heinzer;
Jack
Ohm
273-284
Abstract: This paper centers around the theorem that a commutative ring $ R$ is noetherian if every $ {R_P},P$ prime, is noetherian and every finitely generated ideal of $ R$ has only finitely many weak-Bourbaki associated primes. A more precise local version of this theorem is also given, and examples are presented to show that the assumption on the weak-Bourbaki primes cannot be deleted nor replaced by the assumption that Spec $(R)$ is noetherian. Moreover, an alternative statement of the theorem using a variation of the weak-Bourbaki associated primes is investigated. The proof of the theorem involves a knowledge of the behavior of associated primes of an ideal under quotient ring extension, and the paper concludes with some remarks on this behavior in the more general setting of flat ring extensions.
Some analytic varieties in the polydisc and the M\"untz-Szasz problem in several variables
Simon
Hellerstein
285-292
Abstract: For $1 \leqq {p_1} < {p_2} < \infty$ and $n \geqq 2$ it is shown that there exists a sequence of monomials $\{ \prod _{j = 1}^nS_j^\lambda mj\}$ with ${\lambda _{mj}} \sim m$ for each $j = 1, \ldots ,n$ whose linear span is dense in $ {L^{{p_1}}}({I^n})$ but not in $ {L^{{p_2}}}({I^n})$ ($ {I^n}$ is the Cartesian product of $n$ copies of the closed unit interval $[0, 1]$). Construction of the examples is via duality, making use of suitable analytic varieties in the polydisc.
Embedding a partially ordered ring in a division algebra
William H.
Reynolds
293-300
Abstract: D. K. Harrison has shown that if a ring with identity has a positive cone that is an infinite prime (a subsemiring that contains 1 and is maximal with respect to avoiding -- 1), and if the cone satisfies a certain archimedean condition for all elements of the ring, then there exists an order isomorphism of the ring into the real field. Our main result shows that if Harrison's archimedean condition is weakened so as to apply only to the elements of the cone and if a certain centrality relation is satisfied, then there exists an order isomorphism of the ring into a division algebra that is algebraic over a subfield of the real field. Also, Harrison's result and a related theorem of D. W. Dubois are extended to rings without identity; in so doing, it is shown that order isomorphic subrings of the real field are identical.
Necessary conditions for stability of diffeomorphisms
John
Franks
301-308
Abstract: S. Smale has recently given sufficient conditions for a diffeomorphism to be $\Omega$-stable and conjectured the converse of his theorem. The purpose of this paper is to give some limited results in the direction of that converse. We prove that an $\Omega$-stable diffeomorphism $ f$ has only hyperbolic periodic points and moreover that if $p$ is a periodic point of period $ k$ then the $k$th roots of the eigenvalues of $ df_p^k$ are bounded away from the unit circle. Other results concern the necessity of transversal intersection of stable and unstable manifolds for an $\Omega$-stable diffeomorphism.
Inductive definitions and computability
Thomas J.
Grilliot
309-317
Abstract: Sets inductively defined with respect to ${\prod _0},{\Sigma _1}$, (nonmonotonic) ${\prod _1}$ and $ {\Sigma _2}$ predicates are characterized in terms of the four chief notions of abstract recursion.
The space of retractions of the $2$-sphere and the annulus
Neal R.
Wagner
319-329
Abstract: Given a manifold $ M$, there is an embedding $ \Lambda$ of $ M$ into the space of retractions of $M$, taking each point to the retraction of $ M$ to that point. Considering $\Lambda$ as a map into the connected component containing its image, we prove that $\Lambda$ is a weak homotopy equivalence for two choices of $M$, namely, the $2$-sphere and the annulus.
Sufficiency classes of ${\rm LCA}$ groups
331-338
Abstract: By the sufficiency class $S(H)$ of a locally compact Abelian (LCA) group $ H$ we shall mean the class of LCA groups $G$ having sufficiently many continuous homomorphisms into $H$ to separate the points of $G$. In this paper we determine the sufficiency classes of a number of LCA groups and indicate how these determinations may help to describe the structure of certain classes of LCA groups. In particular, we give a new proof of a theorem of Robertson which states that an LCA group is torsion-free if and only if its dual contains a dense divisible subgroup. We shall also derive some facts about the compact connected Abelian groups and a result about topological $p$-groups containing dense divisible subgroups. We conclude by giving a necessary condition for two LCA groups to have the same sufficiency class.
Characteristic subgroups of lattice-ordered groups
Richard D.
Byrd;
Paul
Conrad;
Justin T.
Lloyd
339-371
Abstract: Characteristic subgroups of an $l$-group are those convex $l$-subgroups that are fixed by each $ l$-automorphism. Certain sublattices of the lattice of all convex $l$-subgroups determine characteristic subgroups which we call socles. Various socles of an $ l$-group are constructed and this construction leads to some structure theorems. The concept of a shifting subgroup is introduced and yields results relating the structure of an $ l$-group to that of the lattice of characteristic subgroups. Interesting results are obtained when the $l$-group is characteristically simple. We investigate the characteristic subgroups of the vector lattice of real-valued functions on a root system and determine those vector lattices in which every $l$-ideal is characteristic. The automorphism group of the vector lattice of all continuous real-valued functions (almost finite real-valued functions) on a topological space (a Stone space) is shown to be a splitting extension of the polar preserving automorphisms by the ring automorphisms. This result allows us to construct characteristically simple vector lattices. We show that self-injective vector lattices exist and that an archimedean self-injective vector lattice is characteristically simple. It is proven that each $l$-group can be embedded as an $l$-subgroup of an algebraically simple $ l$-group. In addition, we prove that each representable (abelian) $ l$-group can be embedded as an $l$-subgroup of a characteristically simple representable (abelian) $l$-group.
The existence of solutions of abstract partial difference polynomials.
Irving
Bentsen
373-397
Abstract: A partial difference (p.d.) ring is a commutative ring $ R$ together with a (finite) set of isomorphisms (called transforming operators) of $R$ into $R$ which commute under composition. It is shown here that (contrary to the ordinary theory [R. M. Cohn, Difference algebra]) there exist nontrivial algebraically irreducible abstract p.d. polynomials having no solution and p.d. fields having no algebraically closed p.d. overfield. If $F$ is a p.d. field with two transforming operators, then the existence of a p.d. overfield of $F$ whose underlying field is an algebraic closure of that of $F$ is a necessary and sufficient condition for every nontrivial algebraically irreducible abstract p.d. polynomial $P$ in the p.d. polynomial ring $ F\{ {y^{(1)}},{y^{(2)}}, \ldots ,{y^{(n)}}\}$ to have a solution $\eta$ (in some p.d. overfield of $ F$) such that: $ \eta$ has $n - 1$ transformal parameters, $ \eta$ is not a proper specialization over $F$ of any other solution of $P$, and, if $Q$ is a p.d. polynomial whose indeterminates appear effectively in $P$ and $Q$ is annulled by $\eta$, then $Q$ is a multiple of $P.P$ has at most finitely many isomorphically distinct such solutions. Necessity holds if $ F$ has finitely many transforming operators.
Word problem for ringoids of numerical functions
A.
Iskander
399-408
Abstract: A. The composition ringoid of functions on (i) the positive integers, (ii) all integers, (iii) the reals and (iv) the complex numbers, do not satisfy any identities other than those satisfied by all composition ringoids. B. Given two words $u,\upsilon$ of the free ringoid, specific functions on the positive integers, ${f_1}, \ldots ,{f_k}$ can be described such that $u({f_1}, \ldots ,{f_k})$ and $\upsilon ({f_1}, \ldots ,{f_k})$, evaluated at 1, are equal iff $u = \upsilon$ is an identity of the free ringoid.
An approach to the polygonal knot problem using projections and isotopies
L. B.
Treybig
409-421
Abstract: The author extends earlier work of Tait, Gauss, Nagy, and Penney in defining and developing properties of (1) the boundary collection of a knot function, and (2) simple sequences of knot functions or boundary collections. The main results are (1) if two knot functions have isomorphic boundary collections then the knots they determine are equivalent, and (2) if two knot functions determine equivalent knots, then the given functions (their boundary collections) are the ends of a simple sequence of knot functions (boundary collections). Matrices are also defined for knot functions.
Concerning a bound problem in knot theory
L. B.
Treybig
423-436
Abstract: In a recent paper Treybig shows that if two knot functions $ f,g$ determine equivalent knots, then $f,g$ are the ends of a simple sequence $ x$ of knot functions. In an effort to bound the length of $x$ in terms of $f$ and $g$ (1) a bound is found for the moves necessary in moving one polyhedral disk onto another in the interior of a tetrahedron and (2) it is shown that two polygonal knots $K,L$ in regular position can ``essentially'' be embedded as part of the $1$-skeleton of a triangulation $T$ of a tetrahedron, where (1) all 3 cells which are unions of elements of $T$ can be shelled and (2) the number of elements in $T$ is determined by $K,L$. A number of ``counting'' lemmas are proved.
Bounded holomorphic functions of several complex variables. I
Dong Sie
Kim
437-443
Abstract: A domain of bounded holomorphy in a complex analytic manifold is a maximal domain for which every bounded holomorphic function has a bounded analytic continuation. In this paper, we show that this is a local property: if, for each boundary point of a domain, there exists a bounded holomorphic function which cannot be continued to any neighborhood of the point, then there exists a single bounded holomorphic function which cannot be continued to any neighborhood of the boundary points.
The product theorem for topological entropy
L. Wayne
Goodwyn
445-452
Whitehead products as images of Pontrjagin products
Martin
Arkowitz
453-463
Abstract: A method is given for computing higher order Whitehead products in the homotopy groups of a space $X$. If $X$ can be embedded in an $H$-space $E$ such that the pair $(E,X)$ has sufficiently high connectivity, then we prove that a higher order Whitehead product element in the homotopy of $X$ is the homomorphic image of a Pontrjagin product in the homology of $E$. The two main applications determine a higher order Whitehead product element in (1) ${\pi _ \ast }(B{U_t})$, the homotopy groups of the classifying space of the unitary group ${U_t}$, (2) the homotopy groups of a space with two nonvanishing homotopy groups.
Engulfing continua in an $n$-cell
Richard J.
Tondra
465-479
Abstract: In this paper it is shown that there exist open connected subsets ${D_1},{D_2}$, and ${D_3}$ of an $n$-cell $E$ such that, if $C$ is any proper compact connected subset of $ E$ and $C \subset U,U$ open, then there exists a homeomorphism $ h$ of $E$ onto itself such that $C \subset h({D_i}) \subset U$ for some $i,1 \leqq i \leqq 3$.
Quasiconformal mappings and Royden algebras in space
Lawrence G.
Lewis
481-492
Abstract: On every open connected set $G$ in Euclidean $n$-space ${R^n}$ and for every index $p > 1$, we define the Royden $ p$-algebra ${M_p}(G)$. We use results by F. W. Gehring and W. P. Ziemer to prove that two such sets $ G$ and $G'$ are quasiconformally equivalent if and only if their Royden $n$-algebras are isomorphic as Banach algebras. Moreover, every such algebra isomorphism is given by composition with a quasiconformal homeomorphism between $G$ and $G'$. This generalizes a theorem by M. Nakai concerning Riemann surfaces. In case $p \ne n$, the only homeomorphisms which induce an isomorphism of the $p$-algebras are the locally bi-Lipschitz mappings, and for $1 < p < n$, every such isomorphism arises this way. Under certain restrictions on the domains, these results extend to the Sobolev space $H_p^1(G)$ and characterize those homeomorphisms which preserve the $H_p^1$ classes.
Differentiable monotone maps on manifolds. II
P. T.
Church
493-501
Abstract: Let ${M^n}$ and ${N^n}$ be closed manifolds, and let $ G$ be any (nonzero) module. (1) If $ f:{M^3} \to {N^3}$ is $ {C^3}$ $G$-acyclic, then there is a closed $ {C^3}$ $3$-manifold ${K^3}$ such that $ {N^3}\char93 {K^3}$ is diffeomorphic to ${M^3}$, and $ {f^{ - 1}}(y)$ is cellular for all but at most $r$ points $ y \in {N^3}$, where $ r$ is the number of nontrivial $G$-cohomology $3$-spheres in the prime decomposition of $ {K^3}$. (2) If $f:{M^3} \to {M^3}$ or $f:{S^3} \to {M^3}$ is $G$-acyclic, then $f$ is cellular. In case $G$ is $Z$ or ${Z_p}$ ($p$ prime), results analogous to (1) and (2) in the topological category have been proved by Alden Wright. (3) If $ f:{M^n} \to {M^n}$ or $f:{S^n} \to {M^n}$ is real analytic monotone onto, then $f$ is a homeomorphism.
Invariant states
Richard H.
Herman
503-512
Abstract: States of a ${C^ \ast }$-algebra invariant under the action of a group of automorphisms of the ${C^ \ast }$-algebra are considered. It is shown that ``clustering'' states in the same part are equal and thus the same is true of extremal invariant states under suitable conditions. The central decomposition of an invariant state is considered and it is shown that the central measure is mixing if and only if the state satisfies a strong notion of clustering. Under transitivity of the central measure and some reasonable restrictions, the central decomposition is the ergodic decomposition of the state with respect to the isotropy subgroup.
Noncommutative Jordan rings
Kevin
McCrimmon
1-33
Abstract: Heretofore most investigations of noncommutative Jordan algebras have been restricted to algebras over fields of characteristic $\ne 2$ in order to make use of the passage from a noncommutative Jordan algebra $ \mathfrak{A}$ to the commutative Jordan algebra ${\mathfrak{A}^ + }$ with multiplication $x \cdot y = \frac{1}{2}(xy + yx)$. We have recently shown that from an arbitrary noncommutative Jordan algebra $ \mathfrak{A}$ one can construct a quadratic Jordan algebra ${\mathfrak{A}^ + }$ with a multiplication $ {U_x}y = x(xy + yx) - {x^2}y = (xy + yx)x - y{x^2}$, and that these quadratic Jordan algebras have a theory analogous to that of commutative Jordan algebras. In this paper we make use of this passage from $ \mathfrak{A}$ to ${\mathfrak{A}^ + }$ to derive a general structure theory for noncommutative Jordan rings. We define a Jacobson radical and show it coincides with the nil radical for rings with descending chain condition on inner ideals; semisimple rings with d.c.c. are shown to be direct sums of simple rings, and the simple rings to be essentially the familiar ones. In addition we obtain results, which seem to be new even in characteristic $ \ne 2$, concerning algebras without finiteness conditions. We show that an arbitrary simple noncommutative Jordan ring containing two nonzero idempotents whose sum is not 1 is either commutative or quasiassociative.
Isolated invariant sets and isolating blocks
C.
Conley;
R.
Easton
35-61
Structure of the semigroup of semigroup extensions
R. O.
Fulp;
J. W.
Stepp
63-73
Abstract: Let $B$ denote a compact semigroup with identity and $G$ a compact abelian group. Let $\operatorname{Ext} (B,G)$ denote the semigroup of extensions of $G$ by $B$. We show that $\operatorname{Ext} (B,G)$ is always a union of groups. We show that it is a semilattice whenever $ B$ is. In case $ B$ is also an abelian inverse semigroup with its subspace of idempotent elements totally disconnected, we obtain a determination of the maximal groups of a commutative version of $ \operatorname{Ext} (B,G)$ in terms of the extension functor of discrete abelian groups.
Generating and cogenerating structures
John A.
Beachy
75-92
Abstract: A functor $ T:\mathcal{A} \to \mathcal{B}$ acts faithfully on the right of a class of objects $ \mathcal{A}$ if it distinguishes morphisms out of objects of $T(f) \ne T(g))$. We define a full subcategory $ \mathcal{R}\mathcal{F}(T)$ such that $T$ acts faithfully on the right of the objects of $ \mathcal{R}\mathcal{F}(T)$. An object $ U \in \mathcal{A}$ is a generator if ${H^U}:\mathcal{A} \to \mathcal{E}ns$ is faithful, and if ${H^U}$ is not faithful, we may still consider $ \mathcal{R}\mathcal{F}({H^U})$. This gives rise to the notion of a generating structure. Cogenerating structures are defined dually, and various canonical generating and cogenerating structures are defined for the category of $ R$-modules. Relationships between these can be used in the homological classification of rings.
Algebraic models for probability measures associated with stochastic processes
B. M.
Schreiber;
T.-C.
Sun;
A. T.
Bharucha-Reid
93-105
Abstract: This paper initiates the study of probability measures corresponding to stochastic processes based on the Dinculeanu-Foiaş notion of algebraic models for probability measures. The main result is a general extension theorem of Kolmogorov type which can be summarized as follows: Let $\{ (X,{\mathcal{A}_i},{\mu _i}),i \in I\}$ be a directed family of probability measure spaces. Then there is an associated directed family of probability measure spaces $\{ (G,{\mathcal{B}_i},{v_i}),i \in I\}$ and a probability measure $ v$ on the $\sigma $-algebra $\mathcal{B}$ generated by the ${\mathcal{B}_i}$ such that (i) $v(B) = {v_i}(B),B \in {\mathcal{B}_i},i \in I$, and (ii) for each is $i \in I$ the spaces $(X,{\mathcal{A}_i},{\mu _i})$ and $ (G,{\mathcal{B}_i},{v_i})$ are conjugate. The importance of the main theorem is that under certain mild conditions there exists an embedding $ \psi :X \to G$ such that the induced measures ${v_i}$ on $G$ are extendable to $v$, although the measures ${\mu _i}$ on $X$ may not be extendable. Using the algebraic model formulation, the Kolmogorov extension property and the notion of a representation of a directed family of probability measure spaces are discussed.
Cyclic vectors and irreducibility for principal series representations.
Nolan R.
Wallach
107-113
Abstract: Canonical sets of cyclic vectors for principal series representations of semisimple Lie groups having faithful representations are found. These cyclic vectors are used to obtain estimates for the number of irreducible subrepresentations of a principal series representations. The results are used to prove irreducibility for the full principal series of complex semisimple Lie groups and for $SL(2n + 1,R),n \geqq 1$.
Involutory automorphisms of operator algebras
E. B.
Davies
115-142
Abstract: We develop the mathematical machinery necessary in order to describe systematically the commutation and anticommutation relations of the field algebras of an algebraic quantum field theory of the fermion type. In this context it is possible to construct a skew tensor product of two von Neumann algebras and completely describe its type in terms of the types of the constituent algebras. Mathematically the paper is a study of involutory automorphisms of ${W^\ast}$-algebras, of particular importance to quantum field theory being the outer involutory automorphisms of the type III factors. It is shown that each of the hyperfinite type III factors studied by Powers has at least two outer involutory automorphisms not conjugate under the group of all automorphisms of the factor.
The curvature of level curves
Dorothy Browne
Shaffer
143-150
Abstract: Sharp bounds are derived for the curvature of level curves of analytic functions in the complex plane whose logarithmic derivative has the representation $ c/(w - g(w))$, where $ g(w)$ is analytic for $ \vert w\vert > a$ and $\vert g(w)\vert \leqq a,c$ real. These results are applied in particular to lemniscates and sharpened for the level curves of lacunary polynomials. Extensions to the level curves of Green's function and rational functions are indicated.
Noncommutative Jordan algebras of capacity two
Kirby C.
Smith
151-159
Abstract: Let $J$ be a noncommutative Jordan algebra with 1. If $J$ has two orthogonal idempotents $e$ and $f$ such that $1 = e + f$ and such that the Peirce $1$-spaces of each are Jordan division rings, then $J$ is said to have capacity two. We prove that a simple noncommutative Jordan algebra of capacity two is either a Jordan matrix algebra, a quasi-associative algebra, or a type of quadratic algebra whose plus algebra is a Jordan algebra determined by a nondegenerate symmetric bilinear form.
Arcwise connectedness of semiaposyndetic plane continua
Charles L.
Hagopian
161-165
Abstract: In a recent paper, the author proved that if a compact plane continuum $ M$ contains a finite point set $F$ such that, for each point $x$ in $M - F,M$ is semi-locally-connected and not aposyndetic at $x$, then $M$ is arcwise connected. The primary purpose of this paper is to generalize that theorem. Semiaposyndesis, a generalization of semi-local-connectedness, is defined and arcwise connectedness is established for certain semiaposyndetic plane continua.
A priori estimates and unique continuation theorems for second order parabolic equations
Raymond
Johnson
167-177
Abstract: It is shown that solutions of second-order linear parabolic equations subject to global constraint satisfy an a priori estimate which has among its consequences that if a solution of such an equation vanishes on the characteristic $t = T$ and satisfies the global constraint, it vanishes identically.
Regular identities in lattices
R.
Padmanabhan
179-188
Abstract: An algebraic system $\mathfrak{A} = \langle A; + , \circ \rangle$ is called a quasilattice if the two binary operations + and $ ^\circ$ are semilattice operations such that the natural partial order relation determined by + enjoys the substitution property with respect to $^\circ$ and vice versa. An identity ``$f = g$'' in an algebra is called regular if the set of variables occurring in the polynomial $ f$ is the same as that in $ g$. It is called $ n$-ary if the number of variables involved in it is at the most $n$. In this paper we show that the class of all quasilattices is definable by means of ternary regular lattice identities and that these identities span the set of all regular lattice identities and that the arity of these defining equations is the best possible. From these results it is deduced that the class of all quasilattices is the smallest equational class containing both the class of all lattices and the class of all semilattices in the lattice of all equational classes of algebras of type $\langle 2,2\rangle$ and that the lattice of all equational classes of quasilattices is distributive.
On the order of a starlike function
F.
Holland;
D. K.
Thomas
189-201
Abstract: It is shown that if $f \in S$, the class of normalised starlike functions in the unit $\operatorname{disc} \Delta $, then $\displaystyle ({\text{i}})\quad \quad \mathop {\lim }\limits_{r \to 1 - } \frac... ...P_\lambda }(r)}}{{ - \log (1 - r)}} = \alpha \lambda {\text{ for }}\lambda > 0;$ $\displaystyle ({\text{ii}})\quad \quad \mathop {\lim }\limits_{r \to 1 - } \fra... ...}\vert{\vert _p}}}{{ - \log (1 - r)}} = \alpha p - 1{\text{ for }}\alpha p > 1;$ and
On the $C\sp*$-algebra of Toeplitz operators on the quarterplane
R. G.
Douglas;
Roger
Howe
203-217
Abstract: Using the device of the tensor product of $ {C^ \ast }$-algebras, we study the ${C^ \ast }$-algebra generated by the Toeplitz operators on the quarter-plane. We obtain necessary and sufficient conditions for such an operator to be Fredholm, but show in this case that not all such operators are invertible.
Operations in polyadic algebras
Aubert
Daigneault
219-229
Abstract: A new treatment of P. R. Halmos' theory of terms and operations in (locally finite) polyadic algebras (of infinite degree) is given that is considerably simpler than the original one.
Matching theory for combinatorial geometries
Martin
Aigner;
Thomas A.
Dowling
231-245
Abstract: Given two combinatorial (pre-) geometries and an arbitrary binary relation between their point sets, a matching is a subrelation which defines a bijection between independent sets of the geometries. The theory of matchings of maximum cardinality is developed in two directions, one of an algorithmic, the other of a structural nature. In the first part, the concept of an augmenting chain is introduced to establish as principal results a min-max type theorem and a generalized Marriage Theorem. In the second part, Ore's notion of a deficiency function for bipartite graphs is extended to determine the structure of the set of critical sets, i.e. those with maximum deficiency. The two parts of the investigation are then connected using the theory of Galois connections.
On the classification of symmetric graphs with a prime number of vertices
Chong-yun
Chao
247-256
Abstract: We determine all the symmetric graphs with a prime number of vertices. We also determine the structure of their groups.
The approximation of linear operators
J. W.
Brace;
P. J.
Richetta
1-21
Abstract: Let $L(E,F)$ be the vector space of all linear maps of $E$ into $F$. Consider a subspace $G$ of $L(E,F)$ such as all continuous maps. In $ G$ distinguish a subspace $ H$ of maps which are to be approximated by members of a smaller subspace $ N$ of $G$. Thus we always have $N \subset H \subset G \subset L(E,F)$. Then the approximation problem which we consider is to find a locally convex linear Hausdorff topology on $ G$ such that $H \subset \bar N,H = \bar N$ or the completion of $N$ is $H$. In the case where $E$ and $F$ are Banach spaces, we have approximation topologies for (i) all linear operators, (ii) all the continuous linear operators, (iii) all weakly compact operators, (iv) all completely continuous operators, (v) all compact operators, and (vi) certain subclasses of the strictly singular operators. Our method is that of considering members of $L(E,F)$ as linear forms on
Entire functions and M\"untz-Sz\'asz type approximation
W. A. J.
Luxemburg;
J.
Korevaar
23-37
Abstract: Let $[a,b]$ be a bounded interval with $a \geqq 0$. Under what conditions on the sequence of exponents $\{ {\lambda _n}\}$ can every function in ${L^p}[a,b]$ or $C[a,b]$ be approximated arbitrarily closely by linear combinations of powers ${x^\lambda }n$? What is the distance between ${x^\lambda }$ and the closed span ${S_c}({x^\lambda }n)$? What is this closed span if not the whole space? Starting with the case of $ {L^2}$, C. H. Müntz and O. Szász considered the first two questions for the interval $[0, 1]$. L. Schwartz, J. A. Clarkson and P. Erdös, and the second author answered the third question for $[0, 1]$ and also considered the interval $ [a,b]$. For the case of $ [0, 1]$, L. Schwartz (and, earlier, in a limited way, T. Carleman) successfully used methods of complex and functional analysis, but until now the case of $[a,b]$ had proved resistant to a direct approach of that kind. In the present paper complex analysis is used to obtain a simple direct treatment for the case of $[a,b]$. The crucial step is the construction of entire functions of exponential type which vanish at prescribed points not too close to the real axis and which, in a sense, are as small on both halves of the real axis as such functions can be. Under suitable conditions on the sequence of complex numbers $\{ {\lambda _n}\}$, the construction leads readily to asymptotic lower bounds for the distances ${d_k} = d\{ {x^{{\lambda _k}}},{S_c}({x^{{\lambda _n}}},n \ne k)\}$. These bounds are used to determine ${S_c}({x^{{\lambda _n}}})$ and to generalize a result for a boundary value problem for the heat equation obtained recently by V. J. Mizel and T. I. Seidman.
A variational method for functions of bounded boundary rotation
H. B.
Coonce
39-51
Abstract: Let $f$ be a function analytic in the unit disc, properly normalized, with bounded boundary rotation. There exists a Stieltjes integral representation for
Incompressible surfaces in knot spaces
Herbert C.
Lyon
53-62
Abstract: The following theorems are proved. Theorem 1. There exist infinitely many distinct, prime, Neuwirth knots, each of which has the property that its complement contains closed, incompressible surfaces of arbitrarily high genus. Theorem 2. There exists a genus one knot which has incompressible spanning surfaces of arbitrarily high genus.
On the complex bordism of Eilenbeg-Mac Lane spaces and connective coverings of ${\rm BU}$
Peter S.
Landweber
63-71
Abstract: Explicit computations show that the universal coefficient spectral sequence from complex bordism to integral homology collapses for the spectra $K(Z)$ and bu, and also for their $ \bmod p$ reductions. Moreover the complex bordism modules of these spectra have infinite projective dimension.
On the solvability of unit groups of group algebras
J. M.
Bateman
73-86
Abstract: Let FG be the group algebra of a finite group $G$ over a field $F$ of characteristic $p \geqq 0$; and let $ U$ be the group of units of FG. We prove that $U$ is solvable if and only if (i) every absolutely irreducible representation of $G$ at characteristic $p$ is of degree one or two and (ii) if any such representation is of degree two, then it is definable in $ F$ and $F = GF(2)$ or $GF(3)$. This result is translated into intrinsic group-theoretic and field-theoretic conditions on $G$ and $F$, respectively. Namely, if ${O_p}(G)$ is the maximum normal $ p$-subgroup of $ G$ and $L = G/{O_p}(G)$, then (i) $L$ is abelian, or (ii) $ F = GF(3)$ and $ L$ is a $2$-group with exactly
On the stability of the cohomology of complex structures
Tapio
Klemola
87-97
Abstract: Let $ \mathcal{V}\mathop \pi \limits_ \to M$ be a differentiable family of compact complex manifolds ${V_t} = {\pi ^{ - 1}}(t)$ on $M = \{ t \in {R^m}\vert\;\vert t\vert < 1\} ,\;\mathcal{B} \to \mathcal{V}$ a differentiable family of holomorphic vector bundles ${B_t} \to {V_t},t \in M$. In this paper we study conditions for the cohomology groups $H_{\bar \partial t}^{r,s}({B_t})$ to be constant in a neighborhood of $0 \in M$.
Deleted products of spaces which are unions of two simplexes
W. T.
Whitley
99-111
Abstract: If $X$ is a space, the deleted product space, ${X^ \ast }$, is $X \times X - D$, where $ D$ is the diagonal. If $ Y$ is a space and $ f$ is a continuous map from $X$ to $Y$, then $X_f^ \ast$ is the inverse image of ${Y^ \ast }$ under the map $f \times f$ taking $ X \times X$ into $Y \times Y$. In this paper, we investigate the following questions: ``What maps $f$ are such that $X_f^ \ast$ is homotopically equivalent to ${X^ \ast }$", and ``What maps $ f$ are such that $ X_f^ \ast$ is homotopically equivalent to $ f{(X)^ \ast }$?'' If $ X$ is the union of two nondisjoint simplexes and $f$ is a simplicial map from $X \times X$ such that $f\vert f(X)$ is one-to-one, we obtain necessary and sufficient conditions for $ X_f^ \ast$ and $f{(X)^ \ast }$ to be homotopically equivalent. If $ X$ is the union of nondisjoint simplexes $A$ and $B$ with $\dim B = 1 + \dim (A \cap B)$, we obtain necessary and sufficient conditions for $ {X^ \ast }$ and $ X_f^ \ast$ to be homotopically equivalent if $f$ is in the class of maps mentioned.
A two-stage Postnikov system where $E\sb{2}\not=E\sb{\infty }$ in the Eilenberg-Moore spectral sequence
Claude
Schochet
113-118
Abstract: Let $\Omega B \to PB \to B$ be the path fibration over the simply-connected space $B$, let $ \Omega B \to E \to X$ be the induced fibration via the map $f:X \to B$, and let $X$ and $B$ be generalized Eilenberg-Mac Lane spaces. G. Hirsch has conjectured that ${H^ \ast }E$ is additively isomorphic to ${\text{Tor}}_{H^ \ast B}({Z_2},{H^ \ast }X)$, where cohomology is with ${Z_2}$ coefficients. Since the Eilenberg-Moore spectral sequence which converges to ${H^ \ast }E$ has ${E_2} = {\text{Tor}_{H^ \ast B}}({Z_2},{H^ \ast }X)$, the conjecture is equivalent to saying $ {E_2} = {E_\infty }$. In the present paper we set $X = K({Z_2} + {Z_2},2),B = K({Z_2},4)$ and ${f^ \ast }i =$ the product of the two fundamental classes, and we prove that ${E_2} \ne {E_3}$, disproving Hirsch's conjecture. The proof involves the use of homology isomorphisms ${C^ \ast }X\mathop \to \limits^g \bar C({H^ \ast }\Omega X)\mathop \to \limits^h {H^ \ast }X$ developed by J. P. May, where $\bar C$ is the reduced cobar construction. The map $ g$ commutes with cup-$ 1$ products. Since the cup-$ 1$ product in $\bar C({H^ \ast }\Omega X)$ is well known, and since differentials in the spectral sequence correspond to certain cup-$1$ products, we may compute ${d_2}$ on specific elements of $ {E_2}$.
A theory of focal points and focal intervals for an elliptic quadratic form on a Hilbert space
John
Gregory
119-128
Abstract: The theory of focal points and conjugate points is an important part of the study of problems in the calculus of variations and control theory. Hestenes has shown that for many problems this theory may be studied by Hilbert space methods. In a previous paper the author has extended the theory of Hestenes to elliptic quadratic forms $J(x;\sigma )$ defined on $\mathfrak{A}(\sigma )$ where $\sigma$ is a member of the metric space $(\Sigma ,\rho )$ and $\mathfrak{A}(\sigma )$ denotes a closed subspace of $\mathfrak{A}$. A fundamental part of this extension is concerned with inequalities dealing with the signature $ s(\sigma )$ and nullity $n(\sigma )$ of $ J(x;\sigma )$ on $\mathfrak{A}(\sigma )$ where $\sigma$ is in a $\rho$ neighborhood of a fixed point ${\sigma _0}$ in $\Sigma$. The purpose of this paper is threefold. The first purpose is to show that the extended theory includes the focal point hypotheses of Hestenes. The second purpose is to obtain a focal point theory much like that of Hestenes. It is interesting to note that our theory is based only on the nonnegative integers $s(\lambda )$ and $ n(\lambda )$. This will facilitate later work on numerical calculations of focal points. Our final purpose is to obtain an abstract focal interval theory in which the usual focal points are (degenerate) focal intervals. While previous authors have considered specific problems, no general results for the focal interval case seem to be contained in the literature. An expression for the number of focal intervals on a subinterval
A class of complete orthogonal sequences of step functions
J. L.
Sox;
W. J.
Harrington
129-135
Abstract: A class of orthogonal sets of step functions is defined and each member is shown to be complete in $ {L_2}(0,1)$. Pointwise convergence theorems are obtained for the Fourier expansions relative to these sets. The classical Haar orthogonal set is shown to be a set of this class and the class itself is seen to be a subclass of the ``generalized Haar systems'' defined recently by Price.
A maximal function characterization of the class $H\sp{p}$
D. L.
Burkholder;
R. F.
Gundy;
M. L.
Silverstein
137-153
Abstract: Let $u$ be harmonic in the upper half-plane and $0 < p < \infty$. Then $u =$ Re$F$ for some analytic function $F$ of the Hardy class ${H^p}$ if and only if the nontangential maximal function of $u$ is in ${L^p}$. A general integral inequality between the nontangential maximal function of $u$ and that of its conjugate function is established.
$p$-solvable linear groups of finite order
David L.
Winter
155-160
Abstract: The purpose of this paper is to prove the following result. Theorem. Let $p$ be an odd prime and let $G$ be a finite $p$-solvable group. Assume that $G$ has a faithful representation of degree $n$ over a field of characteristic zero or over a perfect field of characteristic $ p$. Let $P$ be a Sylow $p$-subgroup of $G$ and let ${O_p}(G)$ be the maximal normal $p$-subgroup of $G$. Then $\vert P:{O_p}(G)\vert \leqq {p^{{\lambda _p}(n)}}$ where \begin{displaymath}\begin{array}{*{20}{c}} {{\lambda _p}(n) = \sum\limits_{i = 0... ...ight]} \quad if\;p\;is\;not\;a\;Fermat\;prime.} \end{array} \end{displaymath}
Locally flat imbeddings of topological manifolds in codimension three.
Glenn P.
Weller
161-178
Abstract: This paper presents an imbedding theorem for one topological manifold $ {M^n}$ in another topological manifold ${Q^q}$, provided that the codimension $(q - n)$ is at least three. The result holds even if the manifolds are of the recently discovered non-piecewise-linear type. Denote the boundaries of $ M$ and $Q$ by $\dot M$ and $\dot Q$ respectively. Suppose that $M$ is $2n - q$ connected and $Q$ is $2n - q + 1$ connected. It is then proved that any map $f:(M,\dot M) \to (Q,\dot Q)$ such that $ f\vert\dot M$ is a locally flat imbedding is homotopic relative to $\dot M$ to a proper locally flat imbedding $g:M \to Q$. It is also shown that if $M$ is closed and $2n - q + 1$ connected and $Q$ is $ 2n - q + 2$ connected, then any two homotopic locally flat imbeddings are locally flatly concordant.
Real vector bundles and spaces with free involutions
Allan L.
Edelson
179-188
Abstract: The functor $ KR(X)$, defined in [4], is a contravariant functor defined in the category of spaces with involutions. It is shown herein that this functor is classified by equivariant maps into the complex Grassmann manifold, which is given the involution induced by complex conjugation. For the case of free involutions it is shown that the classifying maps can be taken to lie outside the fixed point set of the Grassmann manifold. This fixed point set can be identified with the real Grassmann manifold. It is then shown that, for free involutions, $KR(X)$ is an invariant of the homotopy type of the orbit space $X$ modulo its involution. The multiplicative group of real line bundles, real in the sense of [4], is shown to be classified by equivariant maps into a quadric surface $Q$ in complex projective space. $Q$ carries a free involution and this classification is again valid for spaces with free involutions.
A correlation between ${\rm PSU}\sb{4}\,(3)$, the Suzuki group, and the Conway group
J. H.
Lindsey
189-204
Abstract: We shall use a six dimensional projective representation of $PS{U_4}(3)$ of order ${2^7}{3^6}5 \cdot 7$ to construct 12 and $ 24$-dimensional complex projective representations of the Suzuki and Conway groups, respectively, acting on the Leech lattice. The construction makes it easy to show that the Suzuki and Conway simple groups have outer automorphism groups of order two and one, respectively. Also, the simple Suzuki group contains $3 \cdot PS{U_4}(3) \cdot 2,{3^5} \cdot {M_{11}}$, and a group which is probably $ PS{U_5}(2)$, where $A \cdot B$ denotes an extension of the group $A$ by the group $B$.
Coreflective subcategories
Horst
Herrlich;
George E.
Strecker
205-226
Abstract: General morphism factorization criteria are used to investigate categorical reflections and coreflections, and in particular epi-reflections and monocoreflections. It is shown that for most categories with ``reasonable'' smallness and completeness conditions, each coreflection can be ``split'' into the composition of two mono-coreflections and that under these conditions mono-coreflective subcategories can be characterized as those which are closed under the formation of coproducts and extremal quotient objects. The relationship of reflectivity to closure under limits is investigated as well as coreflections in categories which have ``enough'' constant morphisms.
Multicoherence techniques applied to inverse limits
Sam B.
Nadler
227-234
Abstract: Sufficient conditions are given so that the multicoherence degree of continua is not raised when taking inverse limits. These results are then applied to inverse limits of special types of spaces.
Stochastic equations with discontinuous drift
Edward D.
Conway
235-245
Abstract: We study stochastic differential equations, $dx = adt + \sigma d\beta $ where $\beta$ denotes a Brownian motion. By relaxing the definition of solutions we are able to prove existence theorems assuming only that $ a$ is measurable, $ \sigma$ is continuous and that both grow linearly at infinity. Nondegeneracy is not assumed. The relaxed definition of solution is an extension of A. F. Filippov's definition in the deterministic case. When $\sigma$ is constant we prove one-sided uniqueness and approximation theorems under the assumption that $ a$ satisfies a one-sided Lipschitz condition.
A pairing of a class of evolution systems with a class of generators.
J. V.
Herod
247-260
Abstract: Suppose that $ S$ is a Banach space and that $A$ and $M$ are functions such that if $x$ and $y$ are numbers, $x \geqq y$, and $P$ is in $S$ then each of $M(x,y)P$ and $A(y,P)$ is in $S$. This paper studies the relation $\displaystyle M(x,y)P = P + \int_x^y {A(t,M(t,y)P)dt.}$ Classes OM and OA will be described and a correspondence will be established which pairs members of the two classes which are connected as $ M$ and $A$ are by the relation indicated above.
Euclidean $(q+r)$-space modulo an $r$-plane of collapsible $p$-complexes
Leslie C.
Glaser
261-278
Abstract: The following general decomposition result is obtained: Suppose ${K^p}(p \geqq 1)$ is a finite collapsible $ p$-complex topologically embedded as a subset of a separable metric space $ {X^q}$ where, for some $ r \geqq 1,{X^q} \times {E^r}$ is homeomorphic to Euclidean $(q + r)$-space ${E^{q + r}}$. Then the Cartesian product of the quotient space $ {X^q}/{K^p}$ with $ {E^r}$ is topologically ${E^{q + r}}$ provided that $q \geqq 3$ and, for each simplex $ {\Delta ^k} \in {K^p},({X^q} \times {E^r},{\Delta ^k} \times ({[0,1]^{r - 1}} \times 0))$ is homeomorphic, as pairs, to $\displaystyle ({E^{q + r}},{[0,1]^{k + r - 1}} \times (0, \ldots ,0)).$ It is known that this condition is satisfied if $q - p \geqq 2$ and $q + r \geqq 5$. This result implies that if $ {K^k}$ is a finite collapsible $k$-complex topologically embedded as a subset of Euclidean $n$-space ${E^n}$, then the Cartesian product of the quotient space $ {E^n}/{K^k}$ with $ {E^1}$ is topologically ${E^{n + 1}}$ provided either (i) $n \leqq 3$, (ii) $n - k \geqq 2$, or (iii) each simplex of $ {K^k}$ is flat in ${E^{n + 1}}$.
Asymptotic behavior of solutions of hyperbolic inequalities
Amy C.
Murray
279-296
Abstract: This paper discusses the asymptotic behavior of ${C^2}$ solutions $u = u(t,{x_1}, \ldots ,{x_v})$ of the inequality (1) $\vert Lu\vert \leqq {k_1}(t,x)\vert u\vert + {k_2}(t,x)\vert\vert{\nabla _u}\vert\vert$, in domains in $(t,x)$-space which grow unbounded in $x$ as $t \to \infty$. The operator $ L$ is a second order hyperbolic operator with variable coefficients. The main results establish the maximum rate of decay of nonzero solutions of (1). This rate depends on the asymptotic behavior of $ {k_1},{k_2}$, and the time derivatives of the coefficients of $ L$.
Mappings onto the plane
Dix H.
Pettey
297-309
Abstract: In this paper, we show that if $X$ is a connected, locally connected, locally compact topological space and $f$ is a 1-1 mapping of $X$ onto ${E^2}$, then $f$ is a homeomorphism. Using this result, we obtain theorems concerning the compactness of certain mappings onto ${E^2}$.
$G$-structures on spheres
Peter
Leonard
311-327
Abstract: ${G_n}$ denotes one of the classical groups $SO(n),SU(n)$ or $Sp(n)$ and $H$ a closed connected subgroup of ${G_n}$. We ask whether the principal bundle ${G_n} \to {G_{n + 1}} \to {G_{n + 1}}/{G_n}$ admits a reduction of structure group to $H$. If $n$ is even and ${G_n}$ is $SO(n)$ or $SU(n)$ or if $n \not\equiv 11\bmod 12$ and ${G_n}$ is $Sp(n)$, we prove that there are no such reductions unless $ n = 6,{G_6} = SO(6)$ and $ H = SU(3)$ or $ U(3)$. In the remaining cases we consider the problem for $H$ maximal. We divide the maximal subgroups into three main classes: reducible, nonsimple irreducible and simple irreducible. We find a necessary and sufficient condition for reduction to a reducible maximal subgroup and prove that there are no reductions to the nonsimple irreducible maximal subgroups. The remaining case is unanswered.
Integral representations for continuous linear operators in the setting of convex topological vector spaces
J. R.
Edwards;
S. G.
Wayment
329-345
Abstract: Suppose $ X$ and $Y$ are locally convex Hausdorff spaces, $ H$ is arbitrary and $ \Sigma$ is a ring of subsets of $H$. The authors prove the analog of the theorem stated in [Abstract 672-372, Notices Amer. Math. Soc. 17 (1970), 188] in this setting. A theory of extended integration on function spaces with Lebesgue and non-Lebesgue type convex topologies is then developed. As applications, integral representations for continuous transformations into $Y$ for the following function spaces $ F$ (which have domain $ H$ and range $ X$) are obtained: (1) $ H$ and $\Sigma$ are arbitrary, $\tau$ is a convex topology on the simple functions over $\Sigma ,K$ is a set function on $\Sigma$ with values in $L[X,Y]$, and $F$ is the Lebesgue-type space generated by $ K$; (2) $H$ is a normal space and $ F$ is the space of continuous functions each of whose range is totally bounded, with the topology of uniform convergence; (3) $ H$ is a locally compact Hausdorff space, $F$ is the space of continuous functions of compact support with the topology of uniform convergence; (4) $H$ is a locally compact Hausdorff space and $ F$ is the space of continuous functions with the topology of uniform convergence on compact subsets. In the above $X$ and $Y$ may be replaced by topological Hausdorff spaces under certain additional compensating requirements.
The volume of tubes in complex projective space
Robert A.
Wolf
347-371
Abstract: A formula for the volume of a tube about a compact complex submanifold of complex projective space is derived.
The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in $R\sb{3}$
H. S. G.
Swann
373-397
Abstract: It is shown here that a unique solution to the Navier-Stokes equations exists in ${R_3}$ for a small time interval independent of the viscosity and that the solutions for varying viscosities converge uniformly to a function that is a solution to the equations for ideal flow in ${R_3}$. The existence of the solutions is shown by transforming the Navier-Stokes equations to an equivalent system solvable by applying fixed point methods with estimates derived from using semigroup theory.
Meta-analytic functions
M. S.
Krishna Sastry
399-415
On embedding polyhedra and manifolds
Krešo
Horvatić
417-436
Abstract: It is well known that every $n$-polyhedron PL embeds in a Euclidean $ (2n + 1)$-space, and that for PL manifolds the result can be improved upon by one dimension. In the paper are given some sufficient conditions under which the dimension of the ambient space can be decreased. The main theorem asserts that, for there to exist an embedding of the $ n$-polyhedron $ X$ into $2n$-space, it suffices that the integral cohomology group ${H^n}(X - \operatorname{Int} A) = 0$ for some $n$-simplex $A$ of a triangulation of $X$. A number of interesting corollaries follow from this theorem. Along the line of manifolds the known embedding results for PL manifolds are extended over a larger class containing various kinds of generalized manifolds, such as triangulated manifolds, polyhedral homology manifolds, pseudomanifolds and manifolds with singular boundary. Finally, a notion of strong embeddability is introduced which allows us to prove that some class of $n$-manifolds can be embedded into a $ (2n - 1)$-dimensional ambient space.
The embeddability of a semigroup---Conditions common to Mal'cev and Lambek
George C.
Bush
437-448
Abstract: Two systems of conditions--due to Mal'cev and to Lambek--are known to be necessary and sufficient for a semigroup to be embeddable in a group. This paper shows by means of an example that the conditions common to the two systems are not sufficient to guarantee embeddability.
On mean-periodicity. II
Edwin J.
Akutowicz
449-457
Abstract: This paper is devoted to the problem of representing all solutions of certain homogeneous convolution equations through series of exponential polynomials. This representation is sought in the dual space $\mathcal{M}$, the latter consisting of entire functions satisfying growth conditions in horizontal directions. The space $ \mathcal{M}$ is a Fréchet space, which fact permits a simpler and more thorough treatment than that given in the paper [1]. The technique used here is based upon a method developed by L. Ehrenpreis [5] and V. P. Palamodov [3] in the theory of differential equations with constant coefficients. We map the Fourier transform space $\mathcal{F}\mathcal{M}$ into a space of sequences, $\mathcal{M}'$. The crucial point is to identify the quotient space $\mathcal{F}\mathcal{M}/\ker \rho$.
The logarithmic limit-set of an algebraic variety
George M.
Bergman
459-469
Abstract: Let $C$ be the field of complex numbers and $ V$ a subvariety of ${(C - \{ 0\} )^n}$. To study the ``exponential behavior of $V$ at infinity", we define $V_\infty ^{(a)}$ as the set of limitpoints on the unit sphere ${S^{n - 1}}$ of the set of real $ n$-tuples $ ({u_x}\log \vert{x_1}\vert, \ldots ,{u_x}\log \vert{x_n}\vert)$, where $x \in V$ and $ {u_x} = {(1 + \Sigma {(\log \vert{x_i}\vert)^2})^{ - 1/2}}$. More algebraically, in the case of arbitrary base-field $k$ we can look at places ``at infinity'' on $V$ and use the values of the associated valuations on ${X_1}, \ldots ,{X_n}$ to construct an analogous set $ V_\infty ^{(b)}$. Thirdly, simply by studying the terms occurring in elements of the ideal $I$ defining $V$, we define another closely related set, $V_\infty ^{(c)}$. These concepts are introduced to prove a conjecture of A. E. Zalessky on the action of $GL(n,Z)$ on $k[X_1^{ \pm 1}, \ldots ,X_n^{ \pm 1}]$, then studied further. It is shown among other things that $V_\infty ^{(b)} = V_\infty ^{(c)} \supseteq$ (when defined) $V_\infty ^{(a)}$. If a certain natural conjecture is true, then equality holds where we wrote ``$\supseteq$", and the common set ${V_\infty } \subseteq {S^{n - 1}}$ is a finite union of convex spherical polytopes.
Weak topologies on subspaces of $C(S)$
Joel H.
Shapiro
471-479
Abstract: Let $S$ be a locally compact Hausdorff space, $E$ a linear subspace of $C(S)$. It is shown that the unit ball of $ E$ is compact in the strict topology if and only if both of the following two conditions are satisfied: (1) $E$ is the Banach space dual of $M(S)/{E^0}$ in the integration pairing, and (2) the bounded weak star topology on $E$ coincides with the strict topology. This result is applied to several examples, among which are ${l^\infty }$ and the space of bounded analytic functions on a plane region.
On the existence of strongly series summable Markuschevich bases in Banach spaces
William B.
Johnson
481-486
Abstract: The main result is: Let $X$ be a complex separable Banach space. If the identity operator on ${X^ \ast }$ is the limit in the strong operator topology of a uniformly bounded net of linear operators of finite rank, then $X$ admits a strongly series summable Markuschevich basis.
On the existence of trivial intersection subgroups
Mark P.
Hale
487-493
Abstract: Let $G$ be a transitive nonregular permutation group acting on a set $X$, and let $H$ be the subgroup of $G$ fixing some element of $X$. Suppose each nonidentity element of $ H$ fixes exactly $ b$ elements of $ X$. If $b = 1,G$ is a Frobenius group, and it is well known that $H$ has only trivial intersection with its conjugates. If $b > 1$, it is shown that this conclusion still holds, provided $H$ satisfies certain natural conditions. Applications to the study of Hall subgroups and certain simple groups related to Zassenhaus groups are given.
A rank theorem for coherent analytic sheaves
Günther
Trautmann
495-498
Abstract: Let $S$ be an analytic subvariety in $ {C^n}$ and $\mathcal{F}$ a coherent analytic sheaf on $ {C^n}$, such that $\mathcal{F}$ is locally free on ${C^n} - S$ and $ \Gamma (U,\mathcal{F}) = \Gamma (U - S,\mathcal{F})$ for every open set $U \subset {C^n}$. It is shown that $\mathcal{F}$ is locally free everywhere, if codh$ \mathcal{F} \geqq n - 1$ and $\dim S +$ rank$\mathcal{F} \leqq n - 2$.
Addenda to ``A variational problem related to an optimal filter problem with self-correlated noise''
Leonard D.
Berkovitz;
Harry
Pollard
499-504
Abstract: The explicit solution is given of a nonclassical variational problem that is related to an optimal filter problem.
Errata to ``General product measures''
E. O.
Elliott;
A. P.
Morse
505-506
Errata to ``Concerning arcwise connectedness and the existence of simple closed curves in plane continua''
Charles L.
Hagopian
507-509
Errata to ``Endomorphism rings generated by units''
Paul
Hill
511
A new basis for uniform asymptotic solution of differential equations containing one or several parameters
Gilbert
Stengle
1-43
Some immersion theorems for manifolds
A. Duane
Randall
45-58
Abstract: In this paper we obtain several results on immersing manifolds into Euclidean spaces. For example, a spin manifold $ {M^n}$ immerses in ${R^{2n - 3}}$ for dimension $n \equiv 0\bmod 4$ and n not a power of 2. A spin manifold ${M^n}$ immerses in $ {R^{2n - 4}}$ for $n \equiv 7\bmod 8$ and $n > 7$. Let ${M^n}$ be a 2-connected manifold for $n \equiv 6\bmod 8$ and $n > 6$ such that $ {H_3}(M;Z)$ has no 2-torsion. Then M immerses in ${R^{2n - 5}}$ and embeds in ${R^{2n - 4}}$. The method of proof consists of expressing k-invariants in Postnikov resolutions for the stable normal bundle of a manifold by means of higher order cohomology operations. Properties of the normal bundle are used to evaluate the operations.
Almost locally tame $2$-manifolds in a $3$-manifold
Harvey
Rosen
59-71
Abstract: Several conditions are given which together imply that a 2-manifold M in a 3-manifold is locally tame from one of its complementary domains, U, at all except possibly one point. One of these conditions is that certain arbitrarily small simple closed curves on M can be collared from U. Another condition is that there exists a certain sequence ${M_1},{M_2}, \ldots$ of 2-manifolds in U converging to M with the property that each unknotted, sufficiently small simple closed curve on each $ {M_i}$ is nullhomologous on ${M_i}$. Moreover, if each of these simple closed curves bounds a disk on a member of the sequence, then it is shown that M is tame from $U(M \ne {S^2})$. As a result, if U is the complementary domain of a torus in ${S^3}$ that is wild from U at just one point, then U is not homeomorphic to the complement of a tame knot in ${S^3}$.
Supports of continuous functions
Mark
Mandelker
73-83
Abstract: Gillman and Jerison have shown that when X is a realcompact space, every function in $C(X)$ that belongs to all the free maximal ideals has compact support. A space with the latter property will be called $\mu$-compact. In this paper we give several characterizations of $\mu$-compact spaces and also introduce and study a related class of spaces, the $\psi$-compact spaces; these are spaces X with the property that every function in $ C(X)$ with pseudocompact support has compact support. It is shown that every realcompact space is $\psi$-compact and every $\psi$-compact space is $\mu$-compact. A family $ \mathcal{F}$ of subsets of a space X is said to be stable if every function in $C(X)$ is bounded on some member of $\mathcal{F}$. We show that a completely regular Hausdorff space is realcompact if and only if every stable family of closed subsets with the finite intersection property has nonempty intersection. We adopt this condition as the definition of realcompactness for arbitrary (not necessarily completely regular Hausdorff) spaces, determine some of the properties of these realcompact spaces, and construct a realcompactification of an arbitrary space.
Semigroups on finitely floored spaces
John D.
McCharen
85-89
Abstract: This paper is concerned with certain aspects of acyclicity in a compact connected topological semigroup, and applications to the admissibility of certain multiplications on continua. The principal result asserts that if S is a semigroup on a continuum, finitely floored in dimension 2, then $S = ESE$ implies $S = K$.
Convex hulls of some classical families of univalent functions
L.
Brickman;
T. H.
MacGregor;
D. R.
Wilken
91-107
Abstract: Let S denote the functions that are analytic and univalent in the open unit disk and satisfy $f(0) = 0$ and $f'(0) = 1$. Also, let K, St, ${S_R}$, and C be the subfamilies of S consisting of convex, starlike, real, and close-to-convex mappings, respectively. The closed convex hull of each of these four families is determined as well as the extreme points for each. Moreover, integral formulas are obtained for each hull in terms of the probability measures over suitable sets. The extreme points for each family are particularly simple; for example, the Koebe functions $f(z) = z/{(1 - xz)^2},\vert x\vert = 1$ , are the extreme points of cl co St. These results are applied to discuss linear extremal problems over each of the four families. A typical result is the following: Let J be a ``nontrivial'' continuous linear functional on the functions analytic in the unit disk. The only functions in St. that satisfy $ \operatorname{Re} \,J(f) = \max \;\{ \operatorname{Re} \;J(g):g \in St\}$ are Koebe functions and there are only a finite number of them.
Representations of metabelian groups realizable in the real field
B. G.
Basmaji
109-118
Abstract: A necessary and sufficient condition is found such that all the nonlinear irreducible representations of a metabelian group are realizable in the real field, and all such groups with cyclic commutator subgroups are determined.
Conjugacy separability of certain free products with amalgamation
Peter F.
Stebe
119-129
Abstract: Let G be a group. An element g of G is called conjugacy distinguished or c.d. in G if and only if given any element h of G either h is conjugate to g or there is a homomorphism $ \xi$ from G onto a finite group such that $\xi (h)$ and $\xi (g)$ are not conjugate in $\xi (G)$. Following A. Mostowski, a group G is conjugacy separable or c.s. if and only if every element of G is c.d. in G. In this paper we prove that every element conjugate to a cyclically reduced element of length greater than 1 in the free product of two free groups with a cyclic amalgamated subgroup is c.d. We also prove that a group formed by adding a root of an element to a free group is c.s.
Stationary isotopies of infinite-dimensional spaces
Raymond Y. T.
Wong
131-136
Abstract: Let X denote the Hilbert cube or any separable infinite-dimensional Fréchet space. It has been shown that any two homeomorphisms f, g of X onto itself is isotopic to each other by means of an invertible-isotopy on X. In this paper we generalize the above results to the extent that if f, g are K-coincident on X (that is, $f(x) = g(x)$ for $x \in K$), then the isotopy can be chosen to be K-stationary provided K is compact and has property-Z in X. The main tool of this paper is the Stable Homeomorphism Extension Theorem which generalizes results of Klee and Anderson.
On the inertia group of a product of spheres
Reinhard
Schultz
137-153
Abstract: In this paper it is proved that the smooth connected sum of a product of ordinary spheres with an exotic combinatorial sphere is never diffeomorphic to the original product. This result is extended and compared to certain related examples.
The prime radical in special Jordan rings
T. S.
Erickson;
S.
Montgomery
155-164
Abstract: If R is an associative ring, we consider the special Jordan ring $ {R^ + }$, and when R has an involution, the special Jordan ring S of symmetric elements. We first show that the prime radical of R equals the prime radical of $ {R^ + }$, and that the prime radical of R intersected with S is the prime radical of S. Next we give an elementary characterization, in terms of the associative structure of R, of primeness of S. Finally, we show that a prime ideal of R intersected with S is a prime Jordan ideal of S.
Positive one-relator groups
Gilbert
Baumslag
165-183
Abstract: A group G which can be defined by a single relation in which there are no negative exponents is residually solvable. If G is also torsion-free then it is locally indicable and hence its integral group ring has no zero divisors.
Some iterated logarithm results related to the central limit theorem.
R. J.
Tomkins
185-192
Abstract: An iterated logarithm theorem is presented for sequences of independent, not necessarily bounded, random variables, the distribution of whose partial sums is related to the standard normal distribution in a particular manner. It is shown that if a sequence of independent random variables satisfies the Central Limit Theorem with a sufficiently rapid rate of convergence, then the law of the iterated logarithm holds. In particular, it is demonstrated that these results imply several known iterated logarithm results, including Kolmogorov's celebrated theorem.
Commutators, $C\sp{k}$-classification, and similarity of operators
Shmuel
Kantorovitz
193-218
Abstract: We generalize the results of our recent paper, The $ {C^k}$-classification of certain operators in ${L_p}$. II, to the abstract setting of a pair of operators satisfying the commutation relation $[M,N] = {N^2}$.
Expanding gravitational systems
Donald G.
Saari
219-240
Abstract: In this paper we obtain a classification of motion for Newtonian gravitational systems as time approaches infinity. The basic assumption is that the motion survives long enough to be studied, i.e., the solution exists in the interval $(0,\infty )$. From this classification it is possible to obtain a sketch of the evolving Newtonian universe.
An algebraic classification of noncompact $2$-manifolds
Martin E.
Goldman
241-258
Tensor products of polynomial identity algebras
Elizabeth
Berman
259-271
Abstract: We investigate matrix algebras and tensor products of associative algebras over a commutative ring R with identity, such that the algebra satisfies a polynomial identity with coefficients in R. We call A a P. I. algebra over R if there exists a positive integer n and a polynomial f in n noncommuting variables with coefficients in R, not annihilating A, such that for all ${a_1}, \ldots ,{a_n}$ in A, $f({a_1}, \ldots ,{a_n}) = 0$. We call A a P-algebra if f is homogeneous with at least one coefficient of 1. We define the docile identity, a polynomial identity generalizing commutativity, in that if A satisfies a docile identity, then for all n, ${A_n}$, the set of n-by-n matrices over A, satisfies a standard identity. We similarly define the unitary identity, which generalizes anticommutativity. Claudio Procesi and Lance Small recently proved that if A is a P. I. algebra over a field, then for all n, ${A_n}$ satisfies some power of a standard identity. We generalize this result to P-algebras over commutative rings with identity. It follows that if A is a P-algebra, A satisfies a power of the docile identity.
On the signature of knots and links
Yaichi
Shinohara
273-285
Abstract: In 1965, K. Murasugi introduced an integral matrix M of a link and defined the signature of the link by the signature of $M + M'$. In this paper, we study some basic properties of the signature of links. We also describe the effect produced on the signature of a knot contained in a solid torus by a further knotting of the solid torus.
A strong duality theorem for separable locally compact groups
John
Ernest
287-307
Abstract: We obtain a duality theorem for separable locally compact groups, where the group is regained from the set of factor unitary representations. Loosely stated, the group is isomorphic to the group of nonzero bounded, operator valued maps on the set of factor representations, which preserve unitary equivalence, direct sums, and tensor products. The axiom involving tensor products is formulated in terms of direct integral theory. The topology of G may be regained from the irreducible representations alone. Indeed a sequence $\{ {x_i}\}$ in G, converges to x in G if and and only if $ \pi ({x_i})$ converges strongly to $\pi (x)$ for each irreducible representation $ \pi$ of G. This result supplies the missing topological part of the strong duality theorem of N. Tatsuuma for type I separable locally compact groups (based on irreducible representations). Our result also generalizes this Tatsuuma strong duality theorem to the nontype I case.
Generalized interpolation spaces
Vernon
Williams
309-334
Abstract: In this paper we introduce the notion of ``generalized'' interpolation space, and state and prove a ``generalized'' interpolation theorem. This apparently provides a foundation for an axiomatic treatment of interpolation space theory, for subsequently we show that the ``mean'' interpolation spaces of Lions-Peetre, the ``complex'' interpolation spaces of A. P. Calderón, and the ``complex'' interpolation spaces of M. Schechter are all generalized interpolation spaces. Furthermore, we prove that each of the interpolation theorems established respectively for the above-mentioned interpolation spaces is indeed a special case of our generalized interpolation theorem. In §III of this paper we use the generalized interpolation space concept to state and prove a ``generalized'' duality theorem. The very elegant duality theorems proved by Calderón, Lions-Peetre and Schechter, respectively, are shown to be special cases of our generalized duality theorem. Of special interest here is the isolation by the general theorem of the need in each of the separate theorems for certain ``base'' spaces to be duals of others. At the close of §II of this paper we employ our generalized interpolation theorem ``structure'' to construct new interpolation spaces which are neither complex nor mean spaces.
The structure of pseudocomplemented distributive lattices. I. Subdirect decomposition
H.
Lakser
335-342
Abstract: In this paper all subdirectly irreducible pseudocomplemented distributive lattices are found. This result is used to establish a Stone-like representation theorem conjectured by G. Grätzer and to find all equational subclasses of the class of pseudocomplemented distributive lattices.
The structure of pseudocomplemented distributive lattices. II. Congruence extension and amalgamation
G.
Grätzer;
H.
Lakser
343-358
Abstract: This paper continues the examination of the structure of pseudocomplemented distributive lattices. First, the Congruence Extension Property is proved. This is then applied to examine properties of the equational classes ${\mathcal{B}_n}, - 1 \leqq n \leqq \omega$, which is a complete list of all the equational classes of pseudocomplemented distributive lattices (see Part I). The standard semigroups (i.e., the semigroup generated by the operators H, S, and P) are described. The Amalgamation Property is shown to hold iff $n \leqq 2$ or $n = \omega$. For $3 \leqq n < \omega ,{\mathcal{B}_n}$ does not satisfy the Amalgamation Property; the deviation is measured by a class Amal $ ({\mathcal{B}_n})( \subseteq {\mathcal{B}_n})$. The finite algebras in Amal $ ({\mathcal{B}_n})$ are determined.
Algebras of iterated path integrals and fundamental groups
Kuo-tsai
Chen
359-379
Abstract: A method of iterated integration along paths is used to extend deRham cohomology theory to a homotopy theory on the fundamental group level. For every connected ${C^\infty }$ manifold $ \mathfrak{M}$ with a base point p, we construct an algebra $ {\pi ^1} = {\pi ^1}(\mathfrak{M},p)$ consisting of iterated integrals, whose value along each loop at p depends only on the homotopy class of the loop. Thus ${\pi ^1}$ can be taken as a commutative algebra of functions on the fundamental group ${\pi _1}(\mathfrak{M})$, whose multiplication induces a comultiplication ${\pi ^1} \to {\pi ^1} \otimes {\pi ^1}$, which makes ${\pi ^1}$ a Hopf algebra. The algebra ${\pi ^1}$ relates the fundamental group to analysis of the manifold, and we obtain some analytical conditions which are sufficient to make the fundamental group nonabelian or nonsolvable. We also show that ${\pi ^1}$ depends essentially only on the differentiable homotopy type of the manifold. The second half of the paper is devoted to the study of structures of algebras of iterated path integrals. We prove that such algebras can be constructed algebraically from the following data: (a) the commutative algebra A of $ {C^\infty }$ functions on $\mathfrak{M}$; (b) the A-module M of ${C^\infty }$ 1-forms on $ \mathfrak{M}$; (c) the usual differentiation $d:A \to M$; and (d) the evaluation map at the base point p, $ \varepsilon :A \to K$, K being the real (or complex) number field.
Infinite nodal noncommutative Jordan algebras; differentiably simple algebras
D. R.
Scribner
381-389
Abstract: The first result is that any differentiably simple algebra of the form $A = F1 + R$, for R a proper ideal, 1 the identity element, and F the base field, must be a subalgebra of a (commutative associative) power series algebra over F, and is truncated if the characteristic is not zero. Moreover the algebra A contains the polynomial subalgebra generated by the indeterminates and identity of the power series algebra. This is used to prove that if A is any simple flexible algebra of the form $A = F1 + R$, R an ideal of ${A^ + }$, then ${A^ + }$ is a subalgebra of a power series algebra and multiplication in A is determined by certain elements ${c_{ij}}$ in A as in $\displaystyle fg = f \cdot g + \frac{1}{2}\sum {\frac{{\partial f}}{{\partial {x_i}}} \cdot \frac{{\partial g}}{{\partial {x_j}}} \cdot {c_{ij}},}$ where $ {c_{ij}} = - {c_{ji}}$ and ``$\cdot$'' is the multiplication in $ {A^ + }$. This applies in particular to simple nodal noncommutative Jordan algebras (of characteristic not 2). These results suggest a method of constructing noncommutative Jordan algebras of the given form. We have done this with the restriction that the ${c_{ij}}$ lie in F1. The last result is that if A is a finitely generated simple noncommutative algebra of characteristic 0 of this form, then Der (A) is an infinite simple Lie algebra of a known type.
Summability in amenable semigroups
Peter F.
Mah
391-403
Abstract: A theory of summability is developed in amenable semigroups. We give necessary and (or) sufficient conditions for matrices to be almost regular, almost Schur, strongly regular, and almost strongly regular. In particular, when the amenable semigroup is the additive positive integers, our theorems yield those results of J. P. King, P. Schaefer and G. G. Lorentz for some of the matrices mentioned above.
Theory of random evolutions with applications to partial differential equations
Richard
Griego;
Reuben
Hersh
405-418
Abstract: The selection from a finite number of strongly continuous semigroups by means of a finite-state Markov chain leads to the new notion of a random evolution. Random evolutions are used to obtain probabilistic solutions to abstract systems of differential equations. Applications include one-dimensional first order hyperbolic systems. An important special case leads to consideration of abstract telegraph equations and a generalization of a result of Kac on the classical n-dimensional telegraph equation is obtained and put in a more natural setting. In this connection a singular perturbation theorem for an abstract telegraph equation is proved by means of a novel application of the classical central limit theorem and a representation of the solution for the limiting equation is found in terms of a transformation formula involving the Gaussian distribution.
Harmonic analysis on central topological groups
Siegfried
Grosser;
Martin
Moskowitz
419-454
The Brauer-Wall group of a commutative ring
Charles
Small
455-491
Abstract: Let k be a commutative ring (with 1). We work with k-algebras with a grading $\bmod\;2$, and with graded modules over such algebras. Using graded notions of tensor product, commutativity, and morphisms, we construct an abelian group ${\rm {BW}}(k)$ whose elements are suitable equivalence classes of Azumaya k-algebras. The consruction generalizes, and is patterned on, the definition of the Brauer group $ {\rm {Br}}(k)$ given by Auslander and Goldman. $ {\rm {Br}}(k)$ is in fact a subgroup of $ {\rm {BW}}(k)$, and we describe the quotient as a group of graded quadratic extensions of k.
Characteristic spheres of free differentiable actions of $S\sp{1}$ and $S\sp{3}$ on homotopy spheres
Hsu-tung
Ku;
Mei-chin
Ku
493-504
Spatially induced groups of automorphisms of certain von Neumann algebras
Robert R.
Kallman
505-515
Abstract: This paper gives an affirmative solution, in a large number of cases, to the following problem. Let $ \mathcal{R}$ be a von Neumann algebra on the Hilbert space $\mathcal{H}$, let G be a topological group, and let $a \to \varphi (a)$ be a homomorphism of G into the group of $^ \ast $-automorphisms of $\mathcal{R}$. Does there exist a strongly continuous unitary representation $a \to U(a)$ of G on $\mathcal{H}$ such that each $U(a)$ induces $ \varphi (a)$?
Some transplantation theorems for the generalized Mehler transform and related asymptotic expansions
Susan
Schindler
257-291
Abstract: Let $P_{ - 1/2 + ix}^m(z)$ be the associated Legendre function of order $m$ and degree $- 1/2 + ix$. We give, here, two integral transforms ${G^m}$ and ${H^m}$, arising naturally from the generalized Mehler transform, which is induced by $P_{ - 1/2 + ix}^m(\cosh y)$, such thatb ${H^m}{G^m}$ = Identity (formally). We show that if $1 < p < \infty , - 1/p < \alpha < 1 - 1/p,m \leqq 1/2$ or $ m = 1,2, \ldots ,$ then $ \vert\vert{G^m}f\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert\hat f\vert{\vert _{p,\alpha }}$ and $ \vert\vert{H^m}f\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert\hat f\vert{\vert _{p,\alpha }}$, where $^ \wedge$ denotes the Fourier cosine transform. We also prove that ${G^m}f,{H^m}f$ exist as limits in ${L^{p,\alpha }}$ of partial integrals, and we prove inequalities equivalent to the above pair: $\vert\vert{G^m}\hat f\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert f\vert{\vert _{p,\alpha }}$ and $ \vert\vert{H^m}\hat f\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert f\vert{\vert _{p,\alpha }}$. These we dualize to $ \vert\vert{({H^m}f)^ \wedge }\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert f\vert{\vert _{p,\alpha }}$, and $\vert\vert{({G^m}f)^ \wedge }\vert{\vert _{p,\alpha }} \leqq A_{p,\alpha }^m\vert\vert f\vert{\vert _{p,\alpha }}$. ${G^m}$ and ${H^m}$ are given by ${G^m}(f;y) = \int_0^\infty {f(x){K^m}(x,y)dx}$ and ${H^m}(f;x) = \int_0^\infty {f(y){K^m}(x,y)dy\;} (0 \leqq y < \infty )$, where $\displaystyle {K^m}(x,y) = \vert\Gamma (1/2 - m - ix)/\Gamma ( - ix)/{(\sinh y)^{1/2}}P_{ - 1/2 + ix}^m(\cosh y).$ The principal method of proving the inequalities involves getting asymptotic expansions for ${K^m}(x,y)$; these are in terms of sines and cosines for large $y$, and in terms of Bessel functions for $ y$ small. Then we can use Fourier and Hankel multiplier theorems. The main consequences of our results are the typical ones for transplantation theorems: mean convergence and multiplier theorems. They can easily be restated in terms of the more usual Mehler transform pair $\displaystyle g(y) = \int_0^\infty {f(x){P_{ - 1/2 + ix}}(y)dx}$ and $ f(x) = {\pi ^{ - 1}}x\sinh \pi x \cdot \Gamma (1/2 - m + ix)\Gamma (1/2 - m - ix)\int_0^\infty {g(y){P_{ - 1/2 + ix}}(y)dy.}$
Bifunctors and adjoint pairs
J. Fisher
Palmquist;
David C.
Newell
293-303
Abstract: We use a definition of tensor products of functors to generalize some theorems of homological algebra. We show that adjoint pairs of functors between additive functor categories correspond to bifunctors and that composition of such adjoint pairs corresponds to the tensor product of the bifunctors. We also generalize some homological characterizations of finitely generated projective modules to characterizations of small projectives in a functor category. We apply our results to adjoint pairs arising from satellites and from a functor on the domain categories.
Some examples in topology
S. P.
Franklin;
M.
Rajagopalan
305-314
Abstract: §1 is concerned with variations on the theme of an ordinal compactification of the integers. Several applications are found, yielding, for instance, an example previously known only modulo the continuum hypothesis, and a counter-example to a published assertion. §2 is concerned with zero-one sequences and §3 with spaces built from sequential fans. Of two old problems of Čech, one is solved and one partly solved. Since the sections are more or less independent, each will have its own introduction. Sequential spaces form the connecting thread, although not all the examples are concerned with them.
Contractions on $L\sb{1}$-spaces
M. A.
Akcoglu;
A.
Brunel
315-325
Abstract: It is shown that a linear contraction on a complex ${L_1}$-space can be represented in terms of its linear modulus. This result is then used to give a direct proof of Chacon's general ratio ergodic theorem.
On embeddings with locally nice cross-sections
J. L.
Bryant
327-332
Abstract: A $k$-dimensional compactum ${X^k}$ in euclidean space ${E^n}(n - k \geqq 3)$ is said to be locally nice in $ {E^n}$ if ${E^n} - {X^k}$ is $1$-ULC. In this paper we prove a general theorem which implies, in particular, that ${X^k}$ is locally nice in $ {E^n}$ if the intersection of ${X^k}$ with each horizontal hyperplane of $ {E^n}$ is locally nice in the hyperplane. From known results we obtain immediately that a $k$-dimensional polyhedron $P$ in ${E^n}$ ( $n - k \geqq 3$ and $n \geqq 5$) is tame in ${E^n}$ if each $({E^{n - 1}} \times \{ w\} ) - P(w \in {E^1})$ is $1$-ULC. However, by strengthening our general theorem in the case $n = 4$, we are able to prove this result for $ n = 4$ as well. For example, an arc $A$ in ${E^4}$ is tame if each horizontal cross-section of $ A$ is tame in the cross-sectional hyperplane (that is, lies in an arc that is tame in the hyperplane).
Certain dense embeddings of regular semigroups
Mario
Petrich
333-343
Abstract: In a previous paper, the author has introduced a number of homomorphisms of an arbitrary semigroup into the translational hull of certain Rees matrix semigroups or orthogonal sums thereof. For regular semigroups, it is proved here that all of these homomorphisms have the property that the image is a densely embedded subsemigroup, i.e., is a densely embedded ideal of its idealizer, and that the corresponding Rees matrix semigroups are regular. Several of these homomorphisms are 1-1, in each case they furnish a different dense embedding of an arbitrary regular semigroup into the translational hull of a regular Rees matrix semigroup or orthogonal sums thereof. A new representation for regular semigroups is introduced.
Maximal orders over regular local rings
Mark
Ramras
345-352
Abstract: In this paper various sufficient conditions are given for the maximality of an $R$-order in a finite-dimensional central simple $K$-algebra, where $R$ is a regular local ring whose quotient field is $ K$. Stronger results are obtained when we assume the dimension of $R$ to be three. This work depends upon earlier results of this author [5] for regular local rings of dimension two, and the fundamental work of Auslander and Goldman [1] for dimension one.
Partial orders on the types in $\beta N$
Mary Ellen
Rudin
353-362
Abstract: Three partial orders on the types of points in $\beta N$ are defined and studied in this paper. Their relation to the types of points in $\beta N - N$ is also described.
Regularity conditions in nonnoetherian rings
T.
Kabele
363-374
Abstract: We show that properties of $R$-sequences and the Koszul complex which hold for noetherian local rings do not hold for nonnoetherian local rings. For example, we construct a local ring with finitely generated maximal ideal such that ${\text{hd} _R}M < \infty $ but $M$ is not generated by an $ R$-sequence. In fact, every element of $M - {M^2}$ is a zero divisor. Generalizing a result of Dieudonné, we show that even in local (nonnoetherian) integral domains a permutation of an $R$-sequence is not necessarily an $ R$-sequence.
Partitions with a restriction on the multiplicity of the summands
Peter
Hagis
375-384
Abstract: Using the circle dissection method, a convergent series and several asymptotic formulae are obtained for $p(n,t)$, the number of partitions of the positive integer $n$ in which no part may be repeated more than $ t$ times.
On a generalization of alternative and Lie rings
Erwin
Kleinfeld
385-395
Abstract: Alternative as well as Lie rings satisfy all of the following four identities: (i) $({x^2},y,z) = x(x,y,z) + (x,y,z)x$, (ii) $ (x,{y^2},z) = y(x,y,z) + (x,y,z)y$, (iii) $(x,y,{z^2}) = z(x,y,z) + (x,y,z)z$, (iv) $(x,x,x) = 0$, where the associator $(a,b,c)$ is defined by $(a,b,c) = (ab)c - a(bc)$. If $R$ is a ring of characteristic different from two and satisfies (iv) and any two of the first three identities, then it is shown that a necessary and sufficient condition for $R$ to be alternative is that whenever $a,b,c$ are contained in a subring $ S$ of $R$ which can be generated by two elements and whenever $ {(a,b,c)^2} = 0$, then $(a,b,c) = 0$.
A construction of Lie algebras from a class of ternary algebras
John R.
Faulkner
397-408
Abstract: A class of algebras with a ternary composition and alternating bilinear form is defined. The construction of a Lie algebra from a member of this class is given, and the Lie algebra is shown to be simple if the form is nondegenerate. A characterization of the Lie algebras so constructed in terms of their structure as modules for the three-dimensional simple Lie algebra is obtained in the case the base ring contains 1/2. Finally, some of the Lie algebras are identified; in particular, Lie algebras of type ${E_8}$ are obtained.
Nonlinear evolution equations and product stable operators on Banach spaces
G. F.
Webb
409-426
Abstract: The method of product integration is used to obtain solutions to the time dependent Banach space differential equation $ [0,\infty )$ to the set of nonlinear operators from the Banach space $ X$ to itself and $ u$ is a function from $[0,\infty )$ to $X$. The main requirements placed on $A$ are that $A$ is $m$-dissipative and product stable on its domain. Applications are given to a linear partial differential equation, to nonlinear dissipative operators in Hilbert space, and to continuous, $ m$-dissipative, everywhere defined operators in Banach spaces.
On conformal maps of infinitely connected Dirichlet regions
V. C.
Williams
427-453
Abstract: Let $D$ be a plane region of arbitrary connectivity $( > 1)$ for which the Dirichlet problem is solvable. There exists a conformal map of $D$ onto a region bounded by two level loci of $H$, a nontrivial harmonic measure. $H$ is essentially the difference of two logarithmic potentials. The two measures involved are mutually singular probability measures. Further properties of these measures, and of $ H$, are derived. The special case in which $D$ is of connectivity 2 is the classical theorem which states that an annular region is conformally equivalent to a region bounded by two circles. The case in which $D$ is of finite connectivity was treated by J. L. Walsh in 1956. A similar generalization of the Riemann mapping theorem is also established. Finally, converses of the above results are also valid.
Regular representations of Dirichlet spaces
Masatoshi
Fukushima
455-473
Abstract: We construct a regular and a strongly regular Dirichlet space which are equivalent to a given Dirichlet space in the sense that their associated function algebras are isomorphic and isometric. There is an appropriate strong Markov process called a Ray process on the underlying space of each strongly regular Dirichlet space.
Common partial transversals and integral matrices
R. A.
Brualdi
475-492
Abstract: Certain packing and covering problems associated with the common partial transversals of two families $\mathfrak{A}$ and $ \mathfrak{B}$ of subsets of a set $E$ are investigated. Under suitable finitary restrictions, necessary and sufficient conditions are obtained for there to exist pairwise disjoint sets ${F_1}, \ldots ,{F_t}$ where each $ {F_i}$ is a partial transversal of $ \mathfrak{A}$ with defect at most $p$ and a partial transversal of $\mathfrak{B}$ with defect at most $q$. We also prove that (i) $E = \cup _{i = 1}^t{T_i}$ where each $ {T_i}$ is a common partial transversal of $ \mathfrak{A}$ and $\mathfrak{B}$ if and only if (ii)
Systems of division problems for distributions
B.
Roth
493-504
Abstract: Suppose $ {({f_{ij}})_{1 \leqq i,j \leqq p}}$ is a $p \times p$ matrix of real-valued infinitely (respectively $m$-times continuously) differentiable functions on an open subset $\Omega$ of ${R^n}$. Then ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ maps the space of $p$-tuples of distributions on $ \Omega$ (respectively distributions of order $\leqq m$ on $\Omega$) into itself. In the present paper, the $p \times p$ matrices ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ for which this mapping is onto are characterized in terms of the zeros of the determinant of ${({f_{ij}})_{1 \leqq i,j \leqq p}}$ when the $ {f_{ij}}$ are infinitely differentiable on $ \Omega \subset {R^1}$ and when the ${f_{ij}}$ are $m$-times continuously differentiable on $\Omega \subset {R^n}$. Finally, partial results are obtained when the ${f_{ij}}$ are infinitely differentiable on $\Omega \subset {R^n}$ and extensions are made to $p \times q$ systems of division problems for distributions.
Zero divisors in Noetherian-like rings
E. Graham
Evans
505-512
Abstract: The zero divisors of $R/I$ for every ideal $I$ of a Noetherian ring is a finite union of primes. We take this property as a definition and study the class of rings so defined. Such rings are stable under localization and quotients. They are not stable under integral closure and are highly unstable under polynomial adjunction. The length of maximal $R$ sequences is well defined on them. In this paper all rings are commutative with unit and all modules are unitary.
Rings defined by $\mathcal{R}$-sets and a characterization of a class of semiperfect rings
Robert
Gordon
1-17
Completely $0$-simple semirings
Mireille Poinsignon
Grillet;
Pierre-Antoine
Grillet
19-33
Abstract: A completely $( - 0)$ simple semiring is a semiring $R$ which is $( - 0)$ simple and is the union of its $( - 0)$ minimal left ideals and the union of its $( - 0)$ minimal right ideals. Structure results are obtained for such semirings. First the multiplicative semigroup of $R$ is completely $( - 0)$ simple; for any $ \mathcal{H}$-class $H( \ne 0),H( \cup \{ 0\} )$ is a subsemiring. If furthermore $R$ has a zero but is not a division ring, and if $(H \cup \{ 0\} , + )$ has a completely simple kernel for some $H$ as above (for instance, if $R$ is compact or if the $\mathcal{H}$-classes are finite), then (i) $ (R, + )$ is idempotent; (ii) $R$ has no zero divisors, additively or multiplicatively. Additional results are given, concerning the additive $ \mathcal{J}$-classes of $ R$ and also $( - 0)$ minimal ideals of semirings in general.
Spheroidal decompostions of $E\sp{4}$
J. P.
Neuzil
35-64
Abstract: This paper investigates a generalization to ${E^4}$ of the notion of toroidal decomposition of $ {E^3}$. A certain type of this kind of upper semicontinuous decomposition is shown to be shrinkable and hence yield $ {E^4}$ as its decomposition space.
A generlization of Feit's theorem
J. H.
Lindsey
65-75
Abstract: This paper is part of a doctoral thesis at Harvard University. The title of the thesis is Finite linear groups in six variables. Using the methods of this paper, I believe that I can prove that if $p$ is a prime greater than five with $p \equiv - 1\pmod 4$, and $G$ is a finite group with faithful complex representation of degree smaller than $4p/3$ for $p > 7$ and degree smaller than 9 for $p = 7$, then $G$ has a normal $p$-subgroup of index in $G$ divisible at most by ${p^2}$. These methods are particularly effective when there is nontrivial intersection of $ p$-Sylow subgroups. In fact, if the current work people are doing on the trivial intersection case can be extended, it should be possible to show that, for $p$ a prime and $G$ a finite group with a faithful complex representation of degree less than $ 3(p - 1)/2,G$ has a normal $ p$-subgroup of index in $ G$ divisible at most by $ {p^2}$. (It may be possible to show that the index is divisible at most by $ p$ if the representation is primitive and has degree unequal to $p$.)
Stiefel-Whitney numbers of quaternionic and related manifolds
E. E.
Floyd
77-94
Abstract: There is considered the image of the symplectic cobordism ring $\Omega _\ast^{SP}$ in the unoriented cobordism ring $ {N_\ast }$. A polynomial subalgebra of ${N_\ast }$ is exhibited, with all generators in dimensions divisible by 16, such that the image is contained in the polynomial subalgebra. The methods combine the $K$-theory characteristic numbers as used by Stong with the use of the Landweber-Novikov ring.
Under the degree of some finite linear groups
Harvey I.
Blau
95-113
Abstract: Let $G$ be a finite group with a cyclic Sylow $ p$-subgroup $P$ for some prime $p \geqq 13$. Assume that $G$ is not of type ${L_2}(p)$, and that $G$ has a faithful indecomposable modular representation of degree $d \leqq p$. This paper offers several improvements of the known bound $ d \geqq (7p)/10 - 1/2$. In particular, $ d \geqq 3(p - 1)/4$. Other bounds are given relative to the order of the center of $G$ and the index of the centralizer of $ P$ in its normalizer.
On the injective hulls of semisimple modules
Jeffrey
Levine
115-126
Abstract: Let $R$ be a ring. Let $T = { \oplus _{i \in I}}E(R/{M_i})$ and $W = \prod\nolimits_{i \in I} {E(R/{M_i})}$, where each ${M_i}$ is a maximal right ideal and $E(A)$ is the injective hull of $ A$ for any $R$-module $A$. We show the following: If $R$ is (von Neumann) regular, $ E(T) = T$ iff ${\{ R/{M_i}\} _{i \in I}}$ contains only a finite number of nonisomorphic simple modules, each of which occurs only a finite number of times, or if it occurs an infinite number of times, it is finite dimensional over its endomorphism ring. Let $ R$ be a ring such that every cyclic $R$-module contains a simple. Let ${\{ R/{M_i}\} _{i \in I}}$ be a family of pairwise nonisomorphic simples. Then $E({ \oplus _{i \in I}}E(R/{M_i})) = \prod\nolimits_{i \in I} {E(R/{M_i})} $. In the commutative regular case these conditions are equivalent. Let $ R$ be a commutative ring. Then every intersection of maximal ideals can be written as an irredundant intersection of maximal ideals iff every cyclic of the form $ R/\bigcap\nolimits_{i \in I} {{M_i}}$, where ${\{ {M_i}\} _{i \in I}}$ is any collection of maximal ideals, contains a simple. We finally look at the relationship between a regular ring $R$ with central idempotents and the Zariski topology on spec $R$.
The family of all recursively enumerable classes of finite sets
T. G.
McLaughlin
127-136
Abstract: We prove that if $ P(x)$ is any first-order arithmetical predicate which enumerates the family Fin of all r.e. classes of finite sets, then $P(x)$ must reside in a level of the Kleene hierarchy at least as high as $\prod _3^0 - \Sigma _3^0$. (It is more easily established that some of the predicates $P(x)$ which enumerate Fin do lie in $\prod _3^0 - \Sigma _3^0$.)
Topologies for $2\sp{x}$; set-valued functions and their graphs
Louis J.
Billera
137-147
Abstract: We consider the problem of topologizing ${2^X}$, the set of all closed subsets of a topological space $X$, in such a way as to make continuous functions from a space $Y$ into ${2^X}$ precisely those functions with closed graphs. We show there is at most one topology with this property, and if $X$ is a regular space, the existence of such a topology implies that $X$ is locally compact. We then define the compact-open topology for ${2^X}$, which has the desired property for locally compact Hausdorff $X$. The space ${2^X}$ with this topology is shown to be homeomorphic to a space of continuous functions with the well-known compact-open topology. Finally, some additional properties of this topology are discussed.
Compact functors and their duals in categories of Banach spaces
Kenneth L.
Pothoven
149-159
Abstract: In a recent paper, B. S. Mityagin and A. S. Shvarts list many problems concerning functors and dual functors in categories of Banach spaces. Included in these problems is the question: What properties characterize compact functors? The purpose of this paper is to give partial answers to that question. Partial characterizations are given in terms of what are called Fredholm functors and finite rank functors. Affirmative answers are also given to two other questions of Mityagin and Shvarts. They are (1) If a functor is compact, is its dual compact? (2) If a natural transformation is compact, is its dual compact?
Jordan algebras with minimum condition
David L.
Morgan
161-173
Abstract: Let $J$ be a Jordan algebra with minimum condition on quadratic ideals over a field of characteristic not 2. We construct a maximal nil ideal $R$ of $J$ such that $J/R$ is a direct sum of a finite number of ideals each of which is a simple Jordan algebra. $R$ must have finite dimension if it is nilpotent and this is shown to be the case whenever $ J$ has ``enough'' connected primitive orthogonal idempotents.
Perturbations of solutions of Stieltjes integral equations
David Lowell
Lovelady
175-187
Abstract: Using multiplicative integration in two ways, formulae for solutions to perturbed Stieltjes integral equations are found in terms of unperturbed solutions. These formulae are used to obtain bounds on the difference between the perturbed solution and the unperturbed solution. The formulae are also used to explicitly solve, in terms of product integrals, a linear equation subject to nonlinear interface conditions.
Automorphisms of group extensions
Charles
Wells
189-194
Abstract: If $1 \to G\mathop \to ^\iota \Pi \mathop \to ^\eta 1$ is a group extension, with $ \iota$ an inclusion, any automorphism $\varphi$ of Let $\overline \alpha :\Pi \to $ Out $G$ be the homomorphism induced by the given extension. A pair $ (\sigma ,\tau ) \in {\rm {Aut }}\Pi \times {\rm {Aut }}G$ is called compatible if $\sigma$ fixes $ \ker \overline \alpha$, and the automorphism induced by $\sigma$ on $ \Pi \overline \alpha$ is the same as that induced by the inner automorphism of Out $G$ determined by $\tau$. Let $C < {\rm {Aut }}\Pi \times {\rm {Aut }}G$ be the group of compatible pairs. Let ${\rm {Aut (}}E;G{\rm {)}}$ denote the group of automorphisms of $E$ fixing $G$. The main result of this paper is the construction of an exact sequence $\displaystyle 1 \to Z_\alpha ^1(\Pi ,ZG) \to \operatorname{Aut} (E;G) \to C \to H_\alpha ^2(\Pi ,ZG).$ The last map is not surjective in general. It is not even a group homomorphism, but the sequence is nevertheless ``exact'' at $C$ in the obvious sense.
On $L\sp{p}$ estimates for integral transforms
T.
Walsh
195-215
Abstract: In a recent paper R. S. Strichartz has extended and simplified the proofs of a few well-known results about integral operators with positive kernels and singular integral operators. The present paper extends some of his results. An inequality of Kantorovič for integral operators with positive kernel is extended to kernels satisfying two mixed weak ${L^p}$ estimates. The ``method of rotation'' of Calderón and Zygmund is applied to singular integral operators with Banach space valued kernels. Another short proof of the fractional integration theorem in weighted norms is given. It is proved that certain sufficient conditions on the exponents of the $ {L^p}$ spaces and weight functions involved are necessary. It is shown that the integrability conditions on the kernel required for boundedness of singular integral operators in weighted ${L^p}$ spaces can be weakened. Some implications for integral operators in ${R^n}$ of Young's inequality for convolutions on the multiplicative group of positive real numbers are considered. Throughout special attention is given to restricted weak type estimates at the endpoints of the permissible intervals for the exponents.
Cubes with knotted holes
R. H.
Bing;
J. M.
Martin
217-231
Abstract: The statement that a knot $K$ has Property ${\rm {P}}$ means that (1) if $C$ is a cube with a
Endomorphism rings of projective modules
Roger
Ware
233-256
Abstract: The object of this paper is to study the relationship between certain projective modules and their endomorphism rings. Specifically, the basic problem is to describe the projective modules whose endomorphism rings are (von Neumann) regular, local semiperfect, or left perfect. Call a projective module regular if every cyclic submodule is a direct summand. Thus a ring is a regular module if it is a regular ring. It is shown that many other equivalent ``regularity'' conditions characterize regular modules. (For example, every homomorphic image is flat.) Every projective module over a regular ring is regular and a number of examples of regular modules over nonregular rings are given. A structure theorem is obtained: every regular module is isomorphic to a direct sum of principal left ideals. It is shown that the endomorphism ring of a finitely generated regular module is a regular ring. Conversely, over a commutative ring a projective module having a regular endomorphism ring is a regular module. Examples are produced to show that these results are the best possible in the sense that the hypotheses of finite generation and commutativity are needed. An application of these investigations is that a ring $R$ is semisimple with minimum condition if and only if the ring of infinite row matrices over $ R$ is a regular ring. Next projective modules having local, semiperfect and left perfect endomorphism rings are studied. It is shown that a projective module has a local endomorphism ring if and only if it is a cyclic module with a unique maximal ideal. More generally, a projective module has a semiperfect endomorphism ring if and only if it is a finite direct sum of modules each of which has a local endomorphism ring.
A combinatorial model for series-parallel networks
Thomas H.
Brylawski
1-22
Abstract: The category of pregeometries with basepoint is defined and explored. In this category two important operations are extensively characterized: the series connection $S(G,H)$, and the parallel connection $ P(G,H) = \tilde S(\tilde G,\tilde H)$; and the latter is shown to be the categorical direct sum. For graphical pregeometries, these notions coincide with the classical definitions. A pregeometry F is a nontrivial series (or parallel) connection relative to a basepoint p iff the deletion $F\backslash p$ (contraction $F/p$) is separable. Thus both connections are n-ary symmetric operators with identities and generate a free algebra. Elements of the subalgebra $A[{C_2}]$ generated by the two point circuit are defined as series-parallel networks, and this subalgebra is shown to be closed under arbitrary minors. Nonpointed series-parallel networks are characterized by a number of equivalent conditions: 1. They are in $A[{C_2}]$ relative to some point. 2. They are in $A[{C_2}]$ relative to any point. For any connected minor K of three or more points: 3. K is not the four point line or the lattice of partitions of a four element set. 4. K or $ \tilde K$ is not a geometry. 5. For any point e in K, $K\backslash e$ or $K/e$ is separable. Series-parallel networks can also be characterized in a universally constructed ring of pregeometries generalized from previous work of W. Tutte and A. Grothendieck. In this Tutte-Grothendieck ring they are the pregeometries for which the Crapo invariant equals one. Several geometric invariants are directly calculated in this ring including the complexity and the chromatic polynomial. The latter gives algebraic proofs of the two and three color theorems.
A Boolean algebra of regular closed subsets of $\beta X-X$
R. Grant
Woods
23-36
Abstract: Let X be a locally compact, $\sigma$-compact, noncompact Hausdorff space. Let $\beta X$ denote the Stone-Čech compactification of X. Let $R(X)$ denote the Boolean algebra of all regular closed subsets of the topological space X. We show that the map $A \to ({\text{cl}_{\beta X}}A) - X$ is a Boolean algebra homomorphism from $R(X)$ into $ R(\beta X - X)$. Assuming the continuum hypothesis, we show that if X has no more than $ {2^{{\aleph _0}}}$ zero-sets, then the image of a certain dense subalgebra of $ R(X)$ under this homomorphism is isomorphic to the Boolean algebra of all open-and-closed subsets of $ \beta N - N$ (N denotes the countable discrete space). As a corollary, we show that there is a continuous irreducible mapping from $\beta N - N$ onto $\beta X - X$. Some theorems on higher-cardinality analogues of Baire spaces are proved, and these theorems are combined with the previous result to show that if S is a locally compact, $\sigma $-compact noncompact metric space without isolated points, then the set of remote points of $\beta S$ (i.e. those points of $\beta S$ that are not in the $ \beta S$-closure of any discrete subspace of S) can be embedded densely in $\beta N - N$.
Ternary rings
W. G.
Lister
37-55
Abstract: We characterize those additive subgroups of rings which are closed under the triple ring product, then discuss their imbeddings in rings, their representation in terms of two types of modules, a radical theory, the structure of those which satisfy a minimum condition for certain ideals, and finally the classification of those which are simple ternary algebras over an algebraically closed or real closed field.
The method of least squares for boundary value problems
John
Locker
57-68
Abstract: The method of least squares is used to construct approximate solutions to the boundary value problem $\tau f = {g_0},{B_i}(f) = 0$ for $i = 1, \ldots ,k$, on the interval [a, b], where $\tau$ is an nth order formal differential operator, ${g_0}(t)$ is a given function in ${L^2}[a,b]$, and ${B_1}, \ldots ,{B_k}$ are linearly independent boundary values. Letting $ {H^n}[a,b]$ denote the space of all functions $f(t)$ in $ {C^{n - 1}}[a,b]$ with ${f^{(n - 1)}}$ absolutely continuous on [a, b] and ${f^{(n)}}$ in $ {L^2}[a,b]$, a sequence of functions ${\xi _i}(t)\;(i = 1,2, \ldots )$ in ${H^n}[a,b]$ is constructed satisfying the boundary conditions and a completeness condition. Assuming the boundary value problem has a solution, the approximate solutions $ {f_i}(t) = \Sigma _{j = 1}^ia_j^i{\xi _j}(t)\;(i = 1,2, \ldots )$ are constructed; the coefficients $a_j^i$ are determined uniquely from the system of equations $\displaystyle \sum\limits_{j = 1}^i {(\tau {\xi _j},\tau {\xi _l})a_j^i = ({g_0},\tau {\xi _l}),\quad l = 1, \ldots ,i,}$ where (f, g) denotes the inner product in ${L^2}[a,b]$. The approximate solutions are shown to converge to a solution of the boundary value problem, and error estimates are established.
The theory of $p$-spaces with an application to convolution operators.
Carl
Herz
69-82
Abstract: The class of p-spaces is defined to consist of those Banach spaces B such that linear transformations between spaces of numerical ${L_p}$-functions naturally extend with the same bound to B-valued ${L_p}$-functions. Some properties of p-spaces are derived including norm inequalities which show that 2-spaces and Hilbert spaces are the same and that p-spaces are uniformly convex for $1 < p < \infty$. An ${L_q}$-space is a p-space iff $p \leqq q \leqq 2$ or $p \geqq q \geqq 2$; this leads to the theorem that, for an amenable group, a convolution operator on $ {L_p}$ gives a convolution operator on ${L_q}$ with the same or smaller bound. Group representations in p-spaces are examined. Logical elementarity of notions related to p-spaces are discussed.
On some starlike and convex functions
G. M.
Shah
83-91
Abstract: In this paper we study functions of the form $\smallint _0^z(g(t)/\Pi _{k = 1}^n{(1 - t{z_k})^{{\alpha _k}}})$ for $\vert z\vert < 1$ and show under what conditions such a function is convex, convex in one direction and hence univalent in $\vert z\vert < 1$. We also study the functions $ g(z)$ where $g(0) = 1,g(z) \ne 0$ and
Analytic-function bases for multiply-connected regions
Victor
Manjarrez
93-103
Abstract: Let E be a nonempty (not necessarily bounded) region of finite connectivity, whose boundary consists of a finite number of nonintersecting analytic Jordan curves. Work of J. L. Walsh is utilized to construct an absolute basis $ ({Q_n},n = 0, \pm 1, \pm 2, \ldots )$ of rational functions for the space $ H(E)$ of functions analytic on E, with the topology of compact convergence; or the space $ H({\text{Cl}}\;(E))$ of functions analytic on $ {\text{Cl}}\;(E)$ = the closure of E, with an inductive limit topology. It is shown that $\Sigma _{n = 0}^\infty {Q_n}(z){Q_{ - n - 1}}(w) = 1/(w - z)$, the convergence being uniform for z and w on suitable subsets of the plane. A sequence $({P_n},n = 0, \pm 1, \pm 2, \ldots )$ of elements of $ H(E)$ (resp. $H({\text{Cl}}\;(E))$) is said to be absolutely effective on E(resp. $ {\text{Cl}}\;(E)$) if it is an absolute basis for $H(E)$ (resp. $ H({\text{Cl}}\;(E))$) and the coefficients arise by matrix multiplication from the expansion of $({Q_n})$. Conditions for absolute effectivity are derived from W. F. Newns' generalization of work of J. M. Whittaker and B. Cannon. Moreover, if $({P_n},n = 0,1,2, \ldots )$ is absolutely effective on a certain simply-connected set associated with E, the sequence is extended to an absolutely effective basis $({P_n},n = 0, \pm 1, \pm 2, \ldots )$ for $ H(E)$ (or $H({\text{Cl}}\;(E))$) such that $\Sigma _{n = 0}^\infty {P_n}(z){P_{ - n - 1}}(w) = 1/(w - z)$. This last construction applies to a large class of orthogonal polynomials.
Lie-admissible, nodal, noncommutative Jordan algebras
D. R.
Scribner
105-111
Abstract: The main theorem is that if A is a central simple flexible algebra, with an identity, of arbitrary dimension over a field F of characteristic not 2, and if A is Lie-admissible and ${A^ + }$ is associative, then
Boundary conditions in the infinite interval and some related results.
Rao V.
Govindaraju
113-128
Abstract: The number of square-integrable solutions of a real, selfadjoint differential equation are determined using exclusively the elementary theory of matrices. Boundary conditions in the infinite interval are given a simple format and a relation between any two selfadjoint boundary conditions is deduced. Finally a lemma due to Titchmarsh, which forms the basis of eigenfunction expansions, is generalized.
Semiprimary hereditary algebras
Abraham
Zaks
129-135
Abstract: Let $\Sigma$ be a semiprimary k-algebra, with radical M. If $\Sigma$ admits a splitting then $ {\dim _k}\Sigma /M \leqq {\dim _k}\Sigma$. The residue algebra $\Sigma /{M^2}$ is finite (cohomological) dimensional if and only if all residue algebras are finite dimensional. If $ {\dim _k}\Sigma = 1$ then all residue algebras are finite dimensional.
On Spencer's cohomology theory for linear partial differential operators
Joseph
Johnson
137-149
Abstract: Let D be a linear partial differential operator between vector bundles on a differentiable manifold X of dimension n. Let $ \mathcal{D}$ be the sheaf of germs of differentiable functions on X. For every $h \in Z$ a spectral sequence ${(^h}{E^{pq}})$ is associated to D. When D satisfies appropriate regularity conditions these spectral sequences degenerate for all sufficiently large h and $^hE_2^{p0}$ is the pth Spencer cohomology for D. One can compute $^hE_2^{pq}$ as the cohomology at ${\Lambda ^p}{T^\ast}{ \otimes _\mathcal{O}}{R_{h - p,q}}$ of a complex $\displaystyle 0 \to {R_{hq}} \to {\Lambda ^1}{T^\ast}{ \otimes _\mathcal{O}}{R_... ... \to \cdots \to {\Lambda ^n}{T^\ast}{ \otimes _\mathcal{O}}{R_{h - n,q}} \to 0.$ When q = 0 this complex coincides with the usual (first) Spencer complex for D. These results give a generalization of Spencer's theory. The principal importance of this generalization is that it greatly clarifies the role played by homological algebra in the theory of overdetermined systems of linear partial differential equations.
Functorial characterizations of Pontryagin duality
David W.
Roeder
151-175
Abstract: Let $\mathcal{L}$ be the category of locally compact abelian groups, with continuous homomorphisms as morphisms. Let $\chi :\mathcal{L} \to \mathcal{L}$ denote the contravariant functor which assigns to each object in $\mathcal{L}$ its character group and to each morphism its adjoint morphism. The Pontryagin duality theorem is then the statement that $\chi \circ \chi$ is naturally equivalent to the identity functor in $ \mathcal{L}$. We characterize $\chi$ by giving necessary and sufficient conditions for an arbitrary contravariant functor $ \varphi :\mathcal{L} \to \mathcal{L}$ to be naturally equivalent to $\chi$. A sequence of morphisms is called proper exact if it is exact in the algebraic sense and is composed of morphisms each of which is open considered as a function onto its image. A pseudo-natural transformation between two functors in $\mathcal{L}$ differs from a natural transformation in that the connecting maps are not required to be morphisms in $\mathcal{L}$. We study and classify pseudo-natural transformations in $ \mathcal{L}$ and use this to prove that (R denotes the real numbers) $ \varphi$ is naturally equivalent to $\chi$ if and only if the following three statements are all true: (1) $ \varphi (R)$ is isomorphic to R, (2) $\varphi$ takes short proper exact sequences to short proper exact sequences, and (3) $\varphi$ takes inductive limits of discrete groups to projective limits and takes projective limits of compact groups to inductive limits. From this we prove that $\varphi$ is naturally equivalent to $\chi$ if and only if $\varphi$ is a category equivalence.
Torus invariance for the Clifford algebra. I
Michael C.
Reed
177-183
Abstract: A problem in Quantum Field Theory leads to the study of a representation of the torus, ${T^3}$, as automorphisms of the infinite dimensional Clifford algebra. It is shown that the irreducible product representations of the Clifford algebra fall into two categories: the discrete representations where the automorphisms are unitarily implementable, and all the others in which the automorphisms are not implementable and which cannot even appear as subrepresentations of larger representations in which the automorphisms are implementable.
Conjugates in prime rings
Charles
Lanaki
185-192
Abstract: Let R be a prime ring with identity, center $Z \ne GF(2)$, and a nonidentity idempotent. If R is not finite and if $x \in R - Z$, then x has infinitely many distinct conjugates in R. If R has infinitely many Z-independent elements then $x \in R - Z$ has infinitely many Z-independent conjugates.
A Galois theory for inseparable field extensions
Nickolas
Heerema
193-200
Abstract: A Galois theory is obtained for fields k of characteristic $p \ne 0$ in which the Galois subfields h are those for which k/h is normal, modular, and for some nonnegative integer r, $h({k^{{p^{r + 1}}}})/h$ is separable. The related automorphism groups G are subgroups of the group A of automorphisms $\alpha$ on $k[\bar X] = k[X]/{X^{{p^{r + 1}}}}k[X]$, X an indeterminate, such that $\alpha (\bar X) = \bar X$. A subgroup G of A is Galois if and only if G is a semidirect product of subgroups ${G_k}$ and ${G_0}$, where ${G_k}$ is a Galois group of automorphisms on k (classical separable theory) and ${G_0}$ is a Galois group of rank $ {p^r}$ higher derivations on k (Jacobson-Davis purely inseparable theory). Implications of certain invariance conditions on a Galois subgroup of a Galois group are also investigated.
On $N$-parameter families and interpolation problems for nonlinear ordinary differential equations
Philip
Hartman
201-226
Abstract: Let $y = ({y_0}, \ldots ,{y_{N - 1}})$. This paper is concerned with the existence of solutions of a system of ordinary differential equations $(^\ast )\,{y_0}({t_j}) = {c_j}$ for $j = 1, \ldots$, N and ${t_1} < \cdots < {t_N}$. It is shown that, under suitable conditions, the assumption of uniqueness for all such problems and of ``local'' solvability (i.e., for $ {t_1}, \ldots ,{t_N}$ on small intervals) implies the existence for arbitrary ${t_1}, \ldots ,{t_N}$ and ${c_1}, \ldots ,{c_N}$. A result of Lasota and Opial shows that, in the case of a second order equation for $ {y_0}$, the assumption of uniqueness suffices, but it will remain undecided if the assumption of ``local'' solvability can be omitted in general. More general interpolation conditions involving N points, allowing coincidences, are also considered. Part I contains the statement of the principal results for interpolation problems and those proofs depending on the theory of differential equations. Actually, the main theorems are consequences of results in Part II dealing with ``N-parameter families'' and ``N-parameter families with pseudoderivatives.'' A useful lemma states that if F is a family of continuous functions $\{ {y^0}(t)\}$ on an open interval (a, b), then F is an N-parameter family (i.e., contains a unique solution of the interpolation conditions $(^\ast)$ for arbitrary ${t_1} < \cdots < {t_N}$ on (a, b) and $ {c_1}, \ldots ,{c_N}$) if and only if (i) $ {y^0},{z^0} \in F$ implies $ {y^0} - {z^0} \equiv 0$ or ${y^0} - {z^0}$ has at most N zeros; (ii) the set $ \Omega \equiv \{ ({t_1}, \ldots ,{t_N},{y^0}({t_1}), \ldots ,{y^0}({t_N})):a < {t_1} < \cdots < {t_N} < b$ and ${y^0} \in F\}$ is open in ${R^{2N}}$; (iii) ${y^1},{y^2}, \ldots , \in F$ and the inequalities $ {y^n}(t) \leqq {y^{n + 1}}(t)$ for $n = 1,2, \ldots $ or ${y^n}(t) \geqq {y^{n + 1}}(t)$ for $n = 1,2, \ldots$ on an interval $[\alpha ,\beta ] \subset (a,b)$ imply that either $ {y^0}(t) = \lim {y^n}(t)$ exists on (a, b) and ${y^0} \in F$ or $\lim \vert{y^n}(t)\vert = \infty$ on a dense set of (a, b); and finally, (iv) the set $ S(t) = \{ {y^0}(t):{y^0} \in F\}$ is not bounded from above or below for $ a < t < b$. The notion of pseudoderivatives permits generalizations to interpolation problems involving some coincident points.
On some solutions to the Klein-Gordon equation related to an integral of Sonine
Stuart
Nelson
227-237
Abstract: An integral due to Sonine is used to obtain an expansion for special solutions $W(x,t)$ of the Klein-Gordon equation. This expansion is used to estimate the ${L_p}$ norms $\left\Vert W( \cdot ,t)\right\Vert _p$ as $t \to \infty$. These estimates yield results on the time decay of a fairly wide class of solutions to the Klein-Gordon equation.
Integrally closed subrings of an integral domain
Robert
Gilmer;
Joe
Mott
239-250
Abstract: Let D be an integral domain with identity having quotient field K. This paper gives necessary and sufficient conditions on D in order that each integrally closed subring of D should belong to some subclass of the class of integrally closed domains; some of the subclasses considered are the completely integrally closed domains, Prüfer domains, and Dedekind domains.
Representations for transformations continuous in the ${\rm BV}$ norm
J. R.
Edwards;
S. G.
Wayment
251-265
Abstract: Riemann and Lebesgue-type integrations can be employed to represent operators on normed function spaces whose norms are not stronger than sup-norm by $T(f) = \smallint f\,d\mu $ where $\mu$ is determined by the action of T on the simple functions. The real-valued absolutely continuous functions on [0, 1] are not in the closure of the simple functions in the BV norm, and hence such an integral representation of an operator is not obtainable. In this paper the authors develop a v-integral whose structure depends on fundamental functions different than simple functions. This integral is as computable as the Riemann integral. By using these fundamental functions, the authors are able to obtain a direct, analytic representation of the linear functionals on AC which are continuous in the BV norm in terms of the v-integral. Further, the v-integral gives a characterization of the dual of AC in terms of the space of fundamentally bounded set functions which are convex with respect to length. This space is isometrically isomorphically identified with the space of Lipschitz functions anchored at zero with the norm given by the Lipschitz constant, which in turn is isometrically isomorphic to ${L^\infty }$. Hence a natural identification exists between the classical representation and the one given in this paper. The results are extended to the vector setting.
Conditions on an operator implying ${\rm Re}\,\sigma (T)=\sigma ({\rm Re}\,T)$
S. K.
Berberian
267-272
Abstract: It is shown that the equation of the title is valid for certain classes of not necessarily normal operators (including Toeplitz operators, and operators whose spectrum is a spectral set), and a new proof is given of C. R. Putnam's theorem that it is valid for seminormal operators.
Generic stability properties of periodic points
K. R.
Meyer
273-277
Abstract: A classification of the periodic points of a generic area-perserving diffeomorphism which depends on a parameter is given. The stability properties of each periodic point in the classification is decided.
Strongly locally setwise homogeneous continua and their homeomorphism groups
Beverly L.
Brechner
279-288
The closed ideals in a function algebra
Charles M.
Stanton
289-300
Abstract: We give a new method of determining the closed ideals in the algebra of functions continuous on a finite Riemann surface and analytic in its interior. Our approach is based on Ahlfors' mapping of a finite Riemann surface onto the unit disc.
Compactness properties of topological groups
S. P.
Wang
301-314
Abstract: In a paper of R. Baer and later in a paper of B. H. Neumann, finiteness properties of groups have been studied. In the present paper, we develop their analogous notions in topological groups and even sharpen some of their results.
On the derived quotient module
C. N.
Winton
315-321
Abstract: With every R-module M associate the direct limit of $ {\operatorname{Hom}_R}(D,M)$ over the dense right ideals of R, the derived quotient module $ \mathcal{D}(M)$ of M. $ \mathcal{D}(M)$ is a module over the complete ring of right quotients of R. Relationships between $ \mathcal{D}(M)$ and the torsion theory of Gentile-Jans are explored and functorial properties of $ \mathcal{D}$ are discussed. When M is torsion free, results are given concerning rational closure, rational completion, and injectivity.
Barycenters of measures on certain noncompact convex sets
Richard D.
Bourgin
323-340
Abstract: Each norm closed and bounded convex subset K of a separable dual Banach space is, according to a theorem of Bessaga and Pełczynski, the norm closed convex hull of its extreme points. It is natural to expect that this theorem may be reformulated as an integral representation theorem, and in this connection we have examined the extent to which the Choquet theory applies to such sets. Two integral representation theorems are proved and an example is included which shows that a sharp result obtains for certain noncompact sets. In addition, the set of extreme points of K is shown to be $ \mu$-measurable for each finite regular Borel measure $\mu$, hence eliminating certain possible measure-theoretic difficulties in proving a general integral representation theorem. The last section is devoted to the study of a class of extreme points (called pinnacle points) which share important geometric properties with extreme points of compact convex sets in locally convex spaces. A uniqueness result is included for certain simplexes all of whose extreme points are pinnacle points.
Extensions of locally compact abelian groups. I
Ronald O.
Fulp;
Phillip A.
Griffith
341-356
Abstract: This paper is concerned with the development of a (discrete) group-valued functor Ext defined on $\mathcal{L} \times \mathcal{L}$ where $\mathcal{L}$ is the category of locally compact abelian groups such that, for A and B groups in $ \mathcal{L}$, Ext (A, B) is the group of all extensions of B by A. Topological versions of homological lemmas are proven to facilitate the proof of the existence of such a functor. Various properties of Ext are obtained which include the usual long exact sequence which connects Hom to Ext. Along the way some applications are obtained one of which yields a slight improvement of one of the Noether isomorphism theorems. Also the injectives and projectives of the category of locally compact abelian totally disconnected groups are obtained. They are found to be necessarily discrete and hence are the same as the injectives and projectives of the category of discrete abelian groups. Finally we obtain the structure of those connected groups C of $\mathcal{L}$ which are direct summands of every G in $ \mathcal{L}$ which contains C as a component.
Extensions of locally compact abelian groups. II
Ronald O.
Fulp;
Phillip A.
Griffith
357-363
Abstract: It is shown that the extension functor defined on the category $\mathcal{L}$ of locally compact abelian groups is right-exact. Actually $ {\text{Ext}^n}$ is shown to be zero for all $n \geqq 2$. Various applications are obtained which deal with the general problem as to when a locally compact abelian group is the direct product of a connected group and a totally disconnected group. One such result is that a locally compact abelian group G has the property that every extension of G by a connected group in $ \mathcal{L}$ splits iff $G = {(R/Z)^\sigma } \oplus {R^n}$ for some cardinal $\sigma$ and positive integer n.
Groups of embedded manifolds
Max K.
Agoston
365-375
Abstract: This paper defines a group $ \theta ({M^n},{\nu _\varphi })$ which generalizes the group of framed homotopy n-spheres in $ {S^{n + k}}$. Let $ {M^n}$ be an arbitrary 1-connected manifold satisfying a weak condition on its homology in the middle dimension and let ${\nu _\varphi }$ be the normal bundle of some imbedding $ \varphi :{M^n} \to {S^{n + k}}$, where $ 2k \geqq n + 3$. Then $ \theta ({M^n},{\nu _\varphi })$ is the set of h-cobordism classes of triples $ (F,{V^n},f)$, where $ F:{S^{n + k}} \to T({\nu _\varphi })$ is a map which is transverse regular on M, $ {V^n} = {F^{ - 1}}({M^n})$, and $f = F\vert{V^n}$ is a homotopy equivalence. ( $ T({\nu _\varphi })$ is the Thom complex of $ {\nu _\varphi }$.) There is a natural group structure on $\theta ({M^n},{\nu _\varphi })$, and $ \theta ({M^n},{\nu _\varphi })$ fits into an exact sequence similar to that for the framed homotopy n-spheres.
Periodic points and measures for Axiom $A$ diffeomorphisms
Rufus
Bowen
377-397
Dense sigma-compact subsets of infinite-dimensional manifolds
T. A.
Chapman
399-426
Abstract: In this paper four classes of separable metric infinite-dimensional manifolds are studied; those which are locally the countable infinite product of lines, those which are locally open subsets of the Hubert cube, and those which are locally one of two dense sigma-compact subsets of the Hilbert cube. A number of homeomorphism, product, characterization, and embedding theorems are obtained concerning these manifolds.
Weakening a theorem on divided powers
Moss E.
Sweedler
427-428
Abstract: We show that if a Hopf algebra has finite dimensional primitives and a primitive lies in arbitrarily long finite sequences of divided powers then it lies in an infinite sequence of divided powers.
Differential-boundary operators
Allan M.
Krall
429-458
Abstract: Differential-boundary systems occur naturally as adjoints for ordinary differential systems involving integral boundary conditions. In this paper such systems are generalized so that the adjoint system has the same form as the original. Interior boundary points are introduced and removed, and the integrals, used in the boundary conditions, are also removed. Selfadjoint systems are classified, and an eigenfunction expansion is derived. Finally, nonselfadjoint systems are discussed and again, an eigenfunction expansion is derived.
The influence on a finite group of the cofactors and subcofactors of its subgroups
Larry R.
Nyhoff
459-491
Abstract: The effect on a finite group G of imposing a condition $ \theta$ on its proper subgroups has been studied by Schmidt, Iwasawa, Itô, Huppert, and others. In this paper, the effect on G of imposing $\theta$ on only the cofactor $H/{\text{cor}_G}\;H$ (or more generally, the subcofactor $ H/{\text{scor}_G}\;H$) of certain subgroups H of G is investigated, where $ {\text{cor}_G}\;H\;({\text{scor}_G}\;H)$ is the largest G-normal (G-subnormal) subgroup of H. It is shown, for example, that if (a) $H/{\text{scor}_G}\;H$ is p-nilpotent for all self-normalizing $H < G$, or if (b) $H/{\text{scor}_G}\;H$ is p-nilpotent for all abnormal $H < G$ and p is odd or the p-Sylows of G are abelian, then in either case, G has a normal p-subgroup P for which G/P is p-nilpotent. Results of this type are also derived for $\theta =$ nilpotent, nilpotent of class $\leqq n$, solvable of derived length $\leqq n,\sigma $-Sylow-towered, supersolvable. In some cases, additional structure in G is obtained by imposing $\theta$ not only on these ``worst'' parts of the ``bad'' subgroups of G (from the viewpoint of normality), but also on the ``good'' subgroups, those which are normal in G or are close to being normal in that their cofactors are small. Finally, this approach is in a sense dualized by an investigation of the influence on G of the outer cofactors of its subgroups. The consideration of nonnormal outer cofactors is reduced to that of the usual cofactors. The study of normal outer cofactors includes the notion of normal index of maximal subgroups, and it is proved, for example, that G is p-solvable iff the normal index of each abnormal maximal subgroup of G is a power of p or is prime to p.
Topological properties of analytically uniform spaces
C. A.
Berenstein;
M. A.
Dostál
493-513
Abstract: In the first part of the article we study certain topological properties of analytically uniform spaces (AU-spaces, cf. L. Ehrenpreis, Fourier transforms in several complex variables, Interscience, New York, 1970). In particular we prove that AU-spaces and their duals are always nuclear. From here one can easily obtain some important properties of these spaces, such as the Fourier type representation of elements of a given AU-space, etc. The second part is devoted to one important example of AU-space which was not investigated in the aforementioned monograph: the scale of Beurling spaces ${\mathcal{D}_\omega }$ and ${\hat{\mathcal{D}}_\omega }$. This shows that the spaces of Beurling distributions are AU-spaces. Moreover, it leads to some interesting consequences and new problems.
The components of the automorphism group of a Jordan algebra
S. Robert
Gordon
1-52
Abstract: Let $\mathfrak{F}$ be a semisimple Jordan algebra over an algebraically closed field $\Phi$ of characteristic zero. Let $ G$ be the automorphism group of $ \mathfrak{F}$ and $ \Gamma$ the structure groups of $ \mathfrak{F}$. General results on $G$ and $\Gamma$ are given, the proofs of which do not involve the use of the classification theory of simple Jordan algebras over $\Phi$. Specifically, the algebraic components of the linear algebraic groups $G$ and $\Gamma$ are determined, and a formula for the number of components in each case is given. In the course of this investigation, certain Lie algebras and root spaces associated with $ \mathfrak{F}$ are studied. For each component ${G_i}$ of $G$, the index of $G$ is defined to be the minimum dimension of the $ 1$-eigenspace of the automorphisms belonging to ${G_i}$. It is shown that the index of ${G_i}$ is also the minimum dimension of the fixed-point spaces of automorphisms in $ {G_i}$. An element of $ G$ is called regular if the dimension of its $1$-eigenspace is equal to the index of the component to which it belongs. It is proven that an automorphism is regular if and only if its $1$-eigenspace is an associative subalgebra of $ \mathfrak{F}$. A formula for the index of each component ${G_i}$ is given. In the Appendix, a new proof is given of the fact that the set of primitive idempotents of a simple Jordan algebra over $ \Phi$ is an irreducible algebraic set.
A generalization of the Steenrod classification theorem to $H$-spaces.
Byron
Drachman
53-88
On the domain of normality of an attractive fixpoint
P.
Bhattacharyya
89-98
Abstract: It is proved that an entire function of order less than $\frac{1}{2}$ has no unbounded immediate domains of attraction for any of its fixpoints. Estimates for the growth of functions with large infinite domains of attraction (e.g. including half planes) are obtained. It is shown that an entire function mapping an infinite domain into itself has polynomial growth in such domains.
Ordered inverse semigroups
Tôru
Saitô
99-138
Abstract: In this paper, we consider two questions: one is to characterize the structure of ordered inverse semigroups and the other is to give a condition in order that an inverse semigroup is orderable. The solution of the first question is carried out in terms of three types of mappings. Two of these consist of mappings of an $\mathcal{R}$-class onto an $\mathcal{R}$-class, while one of these consists of mappings of a principal ideal of the semilattice $ E$ constituted by idempotents onto a principal ideal of $E$. As for the second question, we give a theorem which extends a well-known result about groups that a group $G$ with the identity $e$ is orderable if and only if there exists a subsemigroup $P$ of $G$ such that $P \cup {P^{ - 1}} = G,P \cap {P^{ - 1}} = \{ e\}$ and $ xP{x^{ - 1}} \subseteqq P$ for every $x \in G$.
A generalized dual for a $C\sp*$-algebra
Herbert
Halpern
139-156
Abstract: Let $\mathcal{A}$ be a $ {C^ \ast }$-algebra, let $\mathcal{B}$ be its enveloping von Neumann algebra, and let $ \mathcal{F}$ be the center of $\mathcal{B} $. Let ${\mathcal{B}_ \sim }$ be the set of all $ \sigma$-weakly continuous $\mathcal{F}$-module homomorphisms of the $\mathcal{F}$-module $ \mathcal{B}$ into $\mathcal{F}$ and let ${\mathcal{A}^ \sim }$ be the set of all restrictions to $\mathcal{A}$ of elements of ${\mathcal{B}_ \sim }$. Then $\mathcal{A}$ is classified as CCR, GCR, and NGCR in terms of certain naturally occurring topologies on $ {\mathcal{A}^ \sim }$.
Nonlinear mappings in locally convex spaces
Terrence S.
McDermott
157-165
Abstract: A notion of local linear approximation is defined for a nonlinear mapping, $f$, defined on one locally convex linear topological space with values in another. By use of this notion, theorems on the local solvability of the equation $y = f(x)$ and on the existence of a local inverse for $f$ are obtained. The continuity and linear approximability of the inverse are discussed. In addition, a relationship between the notion of linear approximation used in the paper and the notion of Fréchet differentiability is shown in the case the intervening spaces are Banach spaces.
Irreducible congruences of prime power degree
C. B.
Hanneken
167-179
Abstract: The number of conjugate sets of irreducible congruences of degree $ m$ belonging to $GF(p),p > 2$, relative to the group $G$ of linear fractional transformations with coefficients belonging to the same field has been determined for $m \leqq 8$. In this paper the irreducible congruences of prime power degree ${q^\alpha },q > 2$, are considered and the number of conjugate sets relative to $G$ is determined.
Principal homogeneous spaces and group scheme extensions
William C.
Waterhouse
181-189
Abstract: Suppose $ G$ is a finite commutative group scheme over a ring $R$. Using Hopf-algebraic techniques, S. U. Chase has shown that the group of principal homogeneous spaces for $G$ is isomorphic to $\operatorname{Ext} (G',{G_m})$ vanishes, and from this derives a more general form of Chase's theorem. Our Ext will be in the usual (fpqc) topology, and we show why this gives the same group. We also give an explicit isomorphism and indicate how it is related to the existence of a normal basis.
Classification of generalized Witt algebras over algebraically closed fields
Robert Lee
Wilson
191-210
Abstract: Let $\Phi$ be a field of characteristic $p > 0$ and $m,{n_1}, \ldots ,{n_m}$ be integers $\geqq 1$. A Lie algebra $W(m:{n_1}, \ldots ,{n_m})$ over $ \Phi$ is defined. It is shown that if $\Phi$ is algebraically closed then $W(m:{n_1}, \ldots ,{n_m})$ is isomorphic to a generalized Witt algebra, that every finite-dimensional generalized Witt algebra over $\Phi$ is isomorphic to some $W(m:{n_1}, \ldots ,{n_m})$, and that $W(m:{n_1}, \ldots ,{n_m})$ is isomorphic to $ W(s:{r_1}, \ldots ,{r_s})$ if and only if $m = s$ and ${r_i} = {n_{\sigma (i)}}$ for $1 \leqq i \leqq m$ where $\sigma$ is a permutation of $\{ 1, \ldots ,m\}$. This gives a complete classification of the finite-dimensional generalized Witt algebras over algebraically closed fields. The automorphism group of $W(m:{n_1}, \ldots ,{n_m})$ is determined for $ p > 3$.
Approximations and representations for Fourier transforms
Raouf
Doss
211-221
Abstract: $G$ is a locally compact abelian group with dual $\Gamma$. If $p(\gamma ) = \sum\nolimits_1^N {{a_n}({x_n},\gamma )}$ is a trigonometric polynomial, its capacity, by definition is $\Sigma \vert{a_n}\vert$. The main theorem is: Let $ \varphi$ be a measurable function defined on the measurable subset $ \Lambda$ of $ \Gamma$. If $ \varphi$ can be approximated on finite sets in $\Lambda$ by trigonometric polynomials of capacity at most $C$ (constant), then $\varphi = \hat \mu$, locally almost everywhere on $ \Lambda$, where $ \mu$ is a regular bounded measure on $G$ and $ \vert\vert\mu \vert\vert \leqq C$.
Existence and uniqueness of solutions of boundary value problems for two dimensional systems of nonlinear differential equations
Paul
Waltman
223-234
Abstract: The paper considers the nonlinear system $x' = f(t,x,y),y' = g(t,x,y)$ with linear and nonlinear two point boundary conditions. With a Lipschitz condition, an interval of uniqueness for linear boundary conditions is determined using a comparison theorem. A corresponding existence theorem is established. Under the assumption of uniqueness, a general existence theorem is established for quite general nonlinearities in the functions and in the boundary conditions. Examples are provided. The results extend previous work on second order scalar differential equations.
Linear ordinary differential equations with boundary conditions on arbitrary point sets
Michael
Golomb;
Joseph
Jerome
235-264
Abstract: Boundary-value problems for differential operators $\Lambda$ of order $2m$ which are the Euler derivatives of quadratic functionals are considered. The boundary conditions require the solution $F$ to coincide with a given function $f \in {\mathcal{H}_L}(R)$ at the points of an arbitrary closed set $B$, to satisfy at the isolated points of $ B$ the knot conditions of $ 2m$-spline interpolations, and to lie in $ {\mathcal{H}_L}(R)$. Existence of solutions (called ``$\Lambda$-splines knotted on $B$") is proved by consideration of the associated variational problem. The question of uniqueness is treated by decomposing the problem into an equivalent set of problems on the disjoint intervals of the complement of $B'$, where $B'$ denotes the set of limit points of $ B$. It is also shown that $ \Lambda$, considered as an operator from $ {\mathcal{L}_2}(R)$ to $ {\mathcal{L}_2}(R)$), with appropriately restricted domain, has a unique selfadjoint extention $ {\Lambda _B}$ if one postulates that the domain of $ {\Lambda _B}$ contains only functions of $ {\mathcal{H}_L}(R)$ which vanish on $ B.I + {\Lambda _B}$ has a bounded inverse which serves to solve the inhomogeneous equation $\Lambda F = G$ with homogeneous boundary conditions. Approximations to the $ \Lambda$-splines knotted on $B$ are constructed, consisting of $\Lambda $-splines knotted on finite subsets ${B_n}$ of $B$, with $\cup {B_n}$ dense in $B$. These approximations $ {F_n}$ converge to $ F$ in the sense of ${\mathcal{H}_L}(R)$.
Quadratic Jordan algebras and cubing operations
Kevin
McCrimmon
265-278
Abstract: In this paper we show how the Jordan structure can be derived from the squaring and cubing operations in a quadratic Jordan algebra, and give an alternate axiomatization of unital quadratic Jordan algebras in terms of operator identities involving only a single variable. Using this we define nonunital quadratic Jordan algebras and show they can be imbedded in unital algebras. We show that a noncommutative Jordan algebra $ \mathfrak{A}$ (over an arbitrary ring of scalars) determines a quadratic Jordan algebra $ {\mathfrak{A}^ + }$.
Representations of quadratic Jordan algebras
Kevin
McCrimmon
279-305
Abstract: Although representations do not play as much of a role in the theory of Jordan algebras as they do in the associative or Lie theories, they are important in considering Wedderburn splitting theorems and other applications. In this paper we develop a representation theory for quadratic Jordan algebras over an arbitrary ring of scalars, generalizing the usual theory for linear Jordan algebras over a field of characteristic $\ne 2$. We define multiplication algebras and representations, characterize these abstractly as quadratic specializations, and relate them to bimodules. We obtain first and second cohomology groups with the usual properties. We define a universal object for quadratic specializations and show it is finite dimensional for a finite-dimensional algebra. The most important examples of quadratic representations, those obtained from commuting linear representations, are discussed and examples are given of new ``pathological'' representations which arise only in characteristic 2.
Representations of free metabelian $\mathcal{D}_\pi$-groups
John F.
Ledlie
307-346
Abstract: For $\pi$ a set of primes, a ${\mathcal{D}_\pi }$-group is a group $G$ with the property that, for every element $g$ in $G$ and every prime $p$ in $\pi ,g$ has a unique $p$th root in $G$. Two faithful representations of free metabelian $ {\mathcal{D}_\pi }$-groups are established: the first representation is inside a suitable power series algebra and shows that free metabelian $ {\mathcal{D}_\pi }$-groups are residually torsion-free nilpotent; the second is in terms of two-by-two matrices and is analogous to W. Magnus' representation of free metabelian groups using two-by-two matrices. In a subsequent paper [12], these representations will be used to derive several properties of free metabelian ${\mathcal{D}_\pi }$-groups.
$\sigma $-finite invariant measures on infinite product spaces
David G. B.
Hill
347-370
Abstract: A necessary and sufficient condition in terms of Hellinger integrals is established for the existence of a $\sigma $-finite invariant measure on an infinite product space. Using this it is possible to construct a wide class of transformations on the unit interval which have no $\sigma$-finite invariant measure equivalent to Lebesgue measure. This class includes most of the previously known examples of such transformations.
Generalizations of ${\rm QF}-3$ algebras
R. R.
Colby;
E. A.
Rutter
371-386
Abstract: This paper consists of three parts. The first is devoted to investigating the equivalence and left-right symmetry of several conditions known to characterize finite dimensional algebras which have a unique minimal faithful representation-- QF-$3$ algebras--in the class of left perfect rings. It is shown that the following conditions are equivalent and imply their right-hand analog: $ R$ contains a faithful $ \Sigma$-injective left ideal, $R$ contains a faithful $II$-projective injective left ideal; the injective hulls of projective left $R$-modules are projective, and the projective covers of injective left $R$-modules are injective. Moreover, these rings are shown to be semi-primary and to include all left perfect rings with faithful injective left and right ideals. The second section is concerned with the endomorphism ring of a projective module over a hereditary or semihereditary ring. More specifically we consider the question of when such an endomorphism ring is hereditary or semihereditary. In the third section we establish the equivalence of a number of conditions similar to those considered in the first section for the class of hereditary rings and obtain a structure theorem for this class of hereditary rings. The rings considered are shown to be isomorphic to finite direct sums of complete blocked triangular matrix rings each over a division ring.
On branch loci in Teichm\"uller space
W. J.
Harvey
387-399
Abstract: The branch locus of the ramified covering of the space of moduli of Fuchsian groups with fixed presentation by the corresponding Teichmüller space is decomposed into a union of Teichmüller spaces, each characterised by a description of the action of the conformal self-mappings admitted by the underlying Riemann surfaces. Equivalence classes of subloci under the action of the modular group are studied, and counted in certain simple cases. One may compute as a result the number of conjugacy classes of elements of prime order in the mapping class group of closed surfaces.
Entropy for group endomorphisms and homogeneous spaces
Rufus
Bowen
401-414
Abstract: Topological entropy $ {h_d}(T)$ is defined for a uniformly continuous map on a metric space. General statements are proved about this entropy, and it is calculated for affine maps of Lie groups and certain homogeneous spaces. We compare ${h_d}(T)$ with measure theoretic entropy $ h(T)$; in particular $h(T) = {h_d}(T)$ for Haar measure and affine maps $ T$ on compact metrizable groups. A particular case of this yields the well-known formula for $h(T)$ when $T$ is a toral automorphism.
On higher-dimensional fibered knots
J. J.
Andrews;
D. W.
Sumners
415-426
Abstract: The geometrical properties of a fibration of a knot complement over $ {S^1}$ are used to develop presentations for the homotopy groups as modules over the fundamental group. Some homotopy groups of spun and twist-spun knots are calculated.
Central separable algebras with purely inseparable splitting rings of exponent one
Shuen
Yuan
427-450
Abstract: Classical Galois cohomological results for purely inseparable field extensions of exponent one are generalized here to commutative rings of prime characteristic.
Integer-valued entire functions
Raphael M.
Robinson
451-468
Abstract: The theory of integer-valued entire functions is organized in an improved fashion. Detailed results are proved when the indicator diagram is a line segment. For the first time, a method is developed for treating completely integer-valued functions with an unsymmetrical growth pattern.
Embedding of abelian subgroups in $p$-groups
Marc W.
Konvisser
469-481
Abstract: Research concerning the embedding of abelian subgroups in $p$-groups generally has proceeded in two directions; either considering abelian subgroups of small index (cf. J. L. Alperin, Large abelian subrgoups of $p$-groups, Trans. Amer. Math. Soc. 117 (1965), 10-20) or considering elementary abelian subgroups of small order (cf. B. Huppert, Endliche Gruppen. I, Springer-Verlag, Berlin, 1967, p. 303). The following new theorems extend these results: Theorem A. Let $G$ be a $p$-group and $M$ a normal subgroup of $G$. (a) If $M$ contains an abelian subgroup of index $ p$, then $M$ contains an abelian subgroup of index $ p$ which is normal in $ G$. (b) If $p \ne 2$ and $M$ contains an abelian subgroup of index $ {p^2}$, then $ M$ contains an abelian subgroup of index ${p^2}$ which is normal in $G$. Theorem B. Let $G$ be a $p$-group, $p \ne 2, M$ a normal subgroup of $ G$, and let $k$ be 2, 3, 4, or 5. If $M$ contains an elementary abelian subgroup of order ${p^k}$, then $M$ contains an elementary abelian subgroup of order ${p^k}$ which is normal in $G$.
The convergence of rational functions of best approximation to the exponential function
E. B.
Saff
483-493
Abstract: The object of the paper is to establish convergence throughout the entire complex plane of sequences of rational functions of prescribed types which satisfy a certain degree of approximation to the function $a{e^{yz}}$ on the disk $\vert z\vert \leqq \rho $. It is assumed that the approximating rational functions have a bounded number of free poles. Estimates are given for the degree of best approximation to the exponential function by rational functions of prescribed types. The results obtained in the paper imply that the successive rows of the Walsh array for $a{e^{yz}}$ on $ \vert z\vert \leqq \rho$ converge uniformly to $a{e^{yz}}$ on each bounded subset of the plane.
Gelfand theory of pseudo differential operators and hypoelliptic operators
Michael E.
Taylor
495-510
Abstract: This paper investigates an algebra $ \mathfrak{A}$ of pseudo differential operators generated by functions $a(x) \in {C^\infty }({R^n}) \cap {L^\infty }({R^n})$ such that $ {D^\alpha }a(x) \to 0$ as $\vert x\vert \to \infty$, if $\vert\alpha \vert \geqq 1$, and by operators $q(D){Q^{ - 1/2}}$ where $q(D) < P(D),Q = I + P{(D)^ \ast }P(D)$, and $ P(D)$ is hypoelliptic. It is proved that such an algebra has compact commutants, and the maximal ideal space of the commutative ${C^ \ast }$ algebra $ \mathfrak{A}/J$ is investigated, where $J$ consists of the elements of $\mathfrak{A}$ which are compact. This gives a necessary and sufficient condition for a differential operator $ q(x,D):{\mathfrak{B}_2}_{,\tilde P} \to {L^2}$ to be Fredholm. (Here and in the rest of this paragraph we assume that the coefficients of all operators under consideration satisfy the conditions given on $a(x)$ in the first sentence.) It is also proved that if $p(x,D)$ is a formally selfadjoint operator on $ {R^n}$ which has the same strength as $P(D)$ uniformly on ${R^n}$, then $p(x,D)$ is selfadjoint, with domain $ {\mathfrak{B}_{2,\tilde P}}({R^n})$, and semibounded, if $n \geqq 2$. From this a Gårding type inequality for uniformly strongly formally hypoelliptic operators and a global regularity theorem for uniformly formally hypoelliptic operators are derived. The familiar local regularity theorem is also rederived. It is also proved that a hypoelliptic operator $ p(x,D)$ of constant strength is formally hypoelliptic, in the sense that for any $ {x_0}$, the constant coefficients operator $ p({x_0},D)$ is hypoelliptic.
Metabelian $p$-groups of maximal class
R. J.
Miech
331-373
Abstract: This paper deals with the classification of the metabelian $p$-groups of maximal class and order $ {p^n}$ where $ p$ is odd and, roughly, $n \geqq 2p$.
On the cohomology of stable two stage Postnikov systems
John R.
Harper
375-388
Abstract: We study the cohomology of certain fibre spaces. The spaces are the total spaces of stable two stage Postnikov systems. We study their cohomology as Hopf algebras over the Steenrod algebra. The first theorem determines the cohomology as a Hopf algebra over the ground field, the algebra structure being known previously. The second theorem relates the action of the Steenrod algebra to the Hopf algebra structure and other available structures. The work is in the direction of explicit computations of these structures but is not quite complete with regard to the action of the Steenrod algebra. The ideas of Massey and Peterson [7], Mem. Amer. Math. Soc. No. 74, are used extensively, and $\bmod 2$ cohomology is used throughout.
A surface in $E\sp{3}$ is tame if it has round tangent balls
L. D.
Loveland
389-397
Abstract: R. H. Bing has asked if a $2$-sphere $S$ in ${E^3}$ is tame when it is known that for each point $ p$ in $S$ there exist two round balls which are tangent to each other at $p$ and which lie, except for $p$, in opposite complementary domains of $S$. The main result in this paper is that Bing's question has an affirmative answer.
The dual topology for the principal and discrete series on semisimple groups
Ronald L.
Lipsman
399-417
Abstract: For a locally compact group $G$, the dual space $\hat G$ is the set of unitary equivalence classes of irreducible unitary representations equipped with the hull-kernel topology. We prove three results about $ \hat G$ in the case that $ G$ is a semisimple Lie group: (1) the irreducible principal series forms a Hausdorff subspace of $\hat G$; (2) the ``discrete series'' of square-integrable representations does in fact inherit the discrete topology from $\hat G$; (3) the topology of the reduced dual ${\hat G_r}$, that is the support of the Plancherel measure, is computed explicitly for split-rank 1 groups.
Spectral representation of certain one-parametric families of symmetric operators in Hilbert space
A. E.
Nussbaum
419-429
Abstract: It is proved that if a one-parameter family of symmetric operators acting in a Hilbert space has the semigroup property on a dense linear manifold and is weakly continuous, then the operators are essentially selfadjoint and permute in the sense of permuting spectral projections of the selfadjoint extensions. It follows from this that the operators have a joint spectral integral representation.
Functional analytic properties of topological semigroups and $n$-extreme amenability
Anthony To-ming
Lau
431-439
Abstract: Let $S$ be a topological semigroup, $ \operatorname{LUC} (S)$ be the space of left uniformly continuous functions on $ S$, and $\Delta (S)$ be the set of multiplicative means on $ \operatorname{LUC} (S)$. If $( \ast )\operatorname{LUC} (S)$ has a left invariant mean in the convex hull of $ \Delta (S)$, we associate with $S$ a unique finite group $G$ such that for any maximal proper closed left translation invariant ideal $I$ in $\operatorname{LUC} (S)$, there exists a linear isometry mapping $\operatorname{LUC} (G)/I$ one-one onto the set of bounded real functions on $G$. We also generalise some recent results of T. Mitchell and E. Granirer. In particular, we show that $ S$ satisfies $ ( \ast )$ iff whenever $ S$ is a jointly continuous action on a compact hausdorff space $ X$, there exists a nonempty finite subset $F$ of $X$ such that $sF = F$ for all $s \in S$. Furthermore, a discrete semigroup $ S$ satisfies $ ( \ast )$ iff whenever $\{ {T_s};s \in S\}$ is an antirepresentation of $S$ as linear maps from a norm linear space $ X$ into $X$ with $ \vert\vert{T_s}\vert\vert \leqq 1$ for all $s \in S$, there exists a finite subset $\sigma \subseteq S$ such that the distance (induced by the norm) of $x$ from $ {K_X} =$ linear span of $\{ x - {T_s}x;x \in X,s \in S\}$ in $ X$ coincides with distance of $O(\sigma ,x) = \{ (1/\vert\sigma \vert)\sum\nolimits_{a \in \sigma } {{T_{at}}(x);t \in S\} }$ from 0 for all $x \in X$.
Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions
Donald P.
Ballou
441-460
Abstract: This paper is concerned with the existence of weak solutions to certain nonlinear hyperbolic Cauchy problems. A condition on the curves of discontinuity is used which guarantees uniqueness in the class of piecewise smooth weak solutions. The method of proof is geometric in nature and is constructive in the manner of A. Douglis and Wu Cho-Chün; that is, for certain types of initial data the method of characteristics is employed to construct piecewise smooth weak solutions. A limiting process is then used to obtain existence for bounded, measurable initial data. The solutions in some cases exhibit interesting, new phenomena. For example, a certain class of initial data having one jump gives rise to a solution having a curving contact discontinuity which does not enter the region of intersecting characteristics.
A generalization of parallelism in Riemannian geometry, the $C\sp{\omega }$ case
Alan B.
Poritz
461-494
Abstract: The concept of parallelism along a curve in a Riemannian manifold is generalized to parallelism along higher dimensional immersed submanifolds in such a way that the minimal immersions are self parallel and hence correspond to geodesics. Let $g:N \to M$ be a (not necessarily isometric) immersion of Riemannian manifolds. Let $G:T(N) \to T(M)$ be a tangent bundle isometry along $g$, that is, $G$ covers $g$ and maps fibers isometrically. By mimicing the construction used for isometric immersions, it is possible to define the mean curvature vector field of $ G.G$ is said to be parallel along $g$ if this vector field vanishes identically. In particular, minimal immersions have parallel tangent maps. For curves, it is shown that the present definition reduces to the definition of Levi-Civita. The major effort is directed toward generalizations, in the real analytic case, of the two basic theorems for parallelism. On the one hand, the existence and uniqueness theorem for a geodesic in terms of data at a point extends to the well-known existence and uniqueness of a minimal immersion in terms of data along a codimension one submanifold. On the other hand, the existence and uniqueness theorem for a parallel unit vector field along a curve in terms of data at a point extends to a local existence and uniqueness theorem for a parallel tangent bundle isometry in terms of mixed initial and partial data. Since both extensions depend on the Cartan-Kahler Theorem, a procedure is developed to handle both proofs in a uniform manner using fiber bundle techniques.
Decomposing manifolds into homologically equivalent submanifolds
J. Scott
Downing
495-501
On the structure of primary abelian groups of countable Ulm type
Doyle O.
Cutler
503-518
Abstract: In this paper we will give structure theorems for abelian $p$-groups of countable Ulm type utilizing the notion of high subgroup introduced by John M. Irwin and its generalization, $N$-high subgroup, introduced by Irwin and E. A. Walker. The general technique employed is to give conditions under which automorphisms of these subgroups extend to automorphisms of the group.
On the solutions of a class of linear selfadjoint differential equations
Larry R.
Anderson;
A. C.
Lazer
519-530
Abstract: Let $L$ be a linear selfadjoint ordinary differential operator with coefficients which are real and sufficiently regular on $( - \infty ,\infty )$. Let ${A^ + }({A^ - })$ denote the subspace of the solution space of $Ly = 0$ such that $y \in {A^ + }(y \in {A^ - })$ iff ${D^k}y \in {L^2}[0,\infty )({D^k}y \in {L^2}( - \infty ,0])$ for $k = 0,1, \ldots ,m$ where $ 2m$ is the order of $ L$. A sufficient condition is given for the solution space of $Ly = 0$ to be the direct sum of $ {A^ + }$ and $ {A^ - }$. This condition which concerns the coefficients of $ L$ reduces to a necessary and sufficient condition when these coefficients are constant. In the case of periodic coefficients this condition implies the existence of an exponential dichotomy of the solution space of $Ly = 0$.
Actions of the torus on $4$-manifolds. I
Peter
Orlik;
Frank
Raymond
531-559
Abstract: Smooth actions of the $2$-dimensional torus group $SO(2) \times SO(2)$ on smooth, closed, orientable $ 4$-manifolds are studied. A cross-sectioning theorem for actions without finite nontrivial isotropy groups and with either fixed points or orbits with isotropy group isomorphic to $ SO(2)$ yields an equivariant classification for these cases. This classification is made numerically specific in terms of orbit invariants. A topological classification is obtained for actions on simply connected $4$-manifolds. It is shown that such a manifold is an equivariant connected sum of copies of complex projective space $ C{P^2}, - C{P^2}$ (reversed orientation), $ {S^2} \times {S^2}$ and the other oriented ${S^2}$ bundle over ${S^2}$. The latter is diffeomorphic (but not always equivariantly diffeomorphic) to $C{P^2}\char93 - C{P^2}$. The connected sum decomposition is not unique. Topological actions on topological manifolds are shown to reduce to the smooth case. In an appendix certain results are extended to torus actions on orientable $4$-dimensional cohomology manifolds.
Group algebra modules. III
S. L.
Gulick;
T.-S.
Liu;
A. C. M.
van Rooij
561-579
Abstract: Let $\Gamma$ be a locally compact group and $ K$ a Banach space. The left ${L^1}(\Gamma )$ module $K$ is by definition absolutely continuous under the composition $\ast$ if for $k \in K$ there exist ${L^1}(\Gamma )$ module--the main object we study. If $ Y \subseteq X$ is measurable, let ${L_Y}$ consist of all functions in ${L^1}(X)$ vanishing outside $ Y$. For $\Omega \subseteq \Gamma $ not locally null and $ B$ a closed linear subspace of $K$, we observe the connection between the closed linear span (denoted ${L_\Omega } \ast B$) of the elements $f \ast k$, with $f \in {L_\Omega }$ and $k \in B$, and the collection of functions of $ B$ shifted by elements in $ \Omega$. As a result, a closed linear subspace of ${L^1}(X)$ is an ${L_Z}$ for some measurable $Z \subseteq X$ if and only if it is closed under pointwise multiplication by elements of ${L^\infty }(X)$. This allows the theorem stating that if $ \Omega \subseteq \Gamma$ and $Y \subseteq X$ are both measurable, then there is a measurable subset $Z$ of $X$ such that ${L_\Omega } \ast {L_Y} = {L_Z}$. Under certain restrictions on $\Gamma$, we show that this $Z$ is essentially open in the (usually stronger) orbit topology on $X$. Finally we prove that if $ \Omega$ and $ Y$ are both relatively sigma-compact, and if also ${L_\Omega } \ast {L_Y} \subseteq {L_Y}$, then there exist ${\Omega _1}$ and ${Y_1}$ locally almost everywhere equal to $\Omega$ and $Y$ respectively, such that ${\Omega _1}{Y_1} \subseteq {Y_1}$; in addition we characterize those $\Omega$ and $Y$ for which ${L_\Omega } \ast {L_\Omega } = {L_\Omega }$ and ${L_\Omega } \ast {L_Y} = {L_Y}$.
Group algebra modules. IV
S. L.
Gulick;
T.-S.
Liu;
A. C. M.
van Rooij
581-596
Abstract: Let $\Gamma$ be a locally compact group, $ \Omega$ a measurable subset of $\Gamma$, and let $ {L_\Omega }$ denote the subspace of $ {L^1}(\Gamma )$ consisting of all functions vanishing off $\Omega$. Assume that ${L_\Omega }$ is a subalgebra of ${L^1}(\Gamma )$. We discuss the collection ${\Re _\Omega }(K)$ of all module homomorphisms from ${L_\Omega }$ into an arbitrary Banach space $ K$ which is simultaneously a left $ {L^1}(\Gamma )$ module. We prove that $ {\Re _\Omega }(K) = {\Re _\Omega }({K_0}) \oplus {\Re _\Omega }({K_{\text{abs} }})$, where ${K_0}$ is the collection of all $k \in K$ such that $fk = 0$, for all $f \in {L^1}(\Gamma )$, and where ${K_{\text{abs} }}$ consists of all elements of $ K$ which can be factored with respect to the module composition. We prove that ${\Re _\Omega }({K_0})$ is the collection of linear continuous maps from ${L_\Omega }$ to ${K_0}$ which are zero on a certain measurable subset of $X$. We reduce the determination of $ {\Re _\Omega }({K_{\text{abs} }})$ to the determination of $ {\Re _\Gamma }({K_{\text{abs} }})$. Denoting the topological conjugate space of $K$ by ${K^ \ast }$, we prove that $ {({K_{\text{abs} }})^ \ast }$ is isometrically isomorphic to ${\Re _\Omega }({K^ \ast })$. Finally, we discuss module homomorphisms $R$ from $ {L_\Omega }$ into $ {L^1}(X)$ such that for each $ f \in {L_\Omega },Rf$ vanishes off $Y$.
Semigroups through semilattices
J. H.
Carruth;
Jimmie D.
Lawson
597-608
Abstract: Presented in this paper is a method of constructing a compact semigroup $S$ from a compact semilattice $X$ and a compact semigroup $ T$ having idempotents contained in $X$. The notions of semigroups (straight) through chains and (straight) through semilattices are introduced. It is shown that the notion of a semigroup through a chain is equivalent to that of a generalized hormos. Universal objects are obtained in several categories including the category of clans straight through a chain and the category of clans straight through a semilattice relative to a chain. An example is given of a nonabelian clan $S$ with abelian set of idempotents $E$ such that $S$ is minimal (as a clan) about $E$.
Cohomology of $F$-groups
Peter
Curran
609-621
Abstract: Let $G$ be a group of Möbius transformations and $V$ the space of complex polynomials of degree $ \leqq$ some fixed even integer. Using the action of $G$ on $V$ defined by Eichler, we compute the dimension of the cohomology space $ {H^1}(G,V)$, first for $ G$ an arbitrary $ F$-group (a generalization of Fuchsian group) and then for the free product of finitely many $F$-groups. These results extend those which Eichler obtained in a 1957 paper, where a correspondence was established between elements of ${H^1}(G,V)$ and cusp forms on $G$.
On automorphism groups of $C\sp*$-algebras
Mi-soo Bae
Smith
623-648
Closedness of coboundary modules of analytic sheaves
Yum-tong
Siu;
Günther
Trautmann
649-658
Abstract: Suppose $ A$ is a subvariety of a complex space $X$ and $ \mathcal{J}$ is a coherent analytic sheaf on $X$. It is shown that, if the analytic sheaf $ \mathcal{H}_{A}^{\nu}(\mathcal{J})$ of local cohomology is coherent for $0 \leqq \nu \leqq q$, then for $0 \leqq \nu \leqq q$ the local cohomology group $\mathcal{H}_{A}^{\nu}(X, \mathcal{J})$ with its natural topology is Hausdorff and hence is a Fréchet space.
A formula for semigroups, with an application to branching diffusion processes
Stanley A.
Sawyer
1-38
Abstract: A Markov process $P = \{ {x_t}\}$ proceeds until a random time $ \tau$, where the distribution of $\tau$ given $P$ is $ \exp ( - {\phi _t})$ for finite additive functional $\{ {\phi _t}\}$, at which time it jumps to a new position given by a substochastic kernel $K({x_\tau },A)$. A new time $\tau '$ is defined, the process again jumps at a time $(P,\{ {\phi _t}\} ,K)$.
Koszul resolutions
Stewart B.
Priddy
39-60
Abstract: Resolutions, which generalize the classical Koszul resolutions, are constructed for a large class of augmented algebras including the Steenrod algebra and the universal enveloping algebras. For each such algebra $A$, an explicit differential algebra $\bar K\ast (A)$ is described such that (1) $\bar K^ \ast(A)$ is a small quotient algebra of the cobar complex and (2) the homology of $\bar K^ \ast(A)$ is the cohomology algebra $H ^ \ast (A)$. The resolution of May for restricted Lie algebras in characteristic 2 is retrieved and a simple derivation of the resolution of Kan et al. of the Steenrod algebra is given.
On the ideal structure of $C(X)$
William E.
Dietrich
61-77
Abstract: The ideal structure of $C(X)$, the algebra of continuous functions from a completely regular Hausdorff space $X$ to the scalars is analyzed by examining for fixed $A \subset \beta X$ (the Stone-Čech compactification of $X$) the structure of the quotient ${I^A}/{F^A}$, where $ {I^A}[{F^A}]$ is the ideal of maps $f \in C(X)$ for which $\displaystyle A \subset {\text{cl} _{\beta X}}Z(f)\quad [A \subset {\operatorname{int} _{\beta X}}{\text{cl} _{\beta X}}Z(f)].$ Unless it vanishes, $ {I^A}/{F^A}$ has no minimal or maximal ideals, and its Krull dimension is infinite. If $J$ is an ideal of $C(X)$ strictly between ${F^A}$ and ${I^A}$, there are ideals $ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{J} $ and $\bar J$ of $C(X)$ for which ${F^A} \subset \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{J} \subset J \subset \bar J \subset {I^A}$ with all inclusions proper. For $K \subset C(X)$, let $Z(K) = \cap \{ {\text{cl} _{\beta X}}Z(f):f \in K\}$. If $ J \subsetneqq I$ are ideals of $C(X)$ with $ Z(J) = Z(I)$ and if $ I$ is semiprime, there is an ideal $K$ strictly between $J$ and $I$. If $I$ and $J$ are $Z$-ideals, $K$ can be chosen to be of the form $P \cap I, P$ a prime ideal of $C(X)$. The maximal ideals of a semiprime ideal $I$ of $C(X)$ are of the form ${I^q} \cap I,q \in \beta X - Z(I)$. If $A \subset \beta X$ is closed, ${I^A}$ is a finitely generated ideal iff $ A$ is open.
On generalized commuting order of automorphisms with quasi-discrete spectrum
Nobuo
Aoki
79-97
Existence and uniqueness of fixed-points for semigroups of affine maps.
Robert E.
Huff
99-106
Abstract: The Day fixed-point theorem is extended to include both existence and uniqueness. For uniqueness of fixed-points, continuity for pointwise limits of a semigroup of continuous affine maps is needed ; necessary and sufficient conditions for this are obtained and compared with the stronger condition of equicontinuity. The comparison is between, on the one hand, the above condition, separate continuity, and weak compactness, and, on the other hand, equicontinuity, joint continuity, and strong compactness. An extension of the Kakutani fixed-point theorem results. Also as a corollary, known necessary and sufficient conditions for continuity of the convolution operation are obtained.
A collection of sequence spaces
J. R.
Calder;
J. B.
Hill
107-118
Abstract: This paper concerns a collection of sequence spaces we shall refer to as ${d_\alpha }$ spaces. Suppose $ \alpha = ({\alpha _1},{\alpha _2}, \ldots )$ is a bounded number sequence and ${\alpha _i} \ne 0$ for some $i$. Suppose $ \mathcal{P}$ is the collection of permutations on the positive integers. Then ${d_\alpha }$ denotes the set to which the number sequence $x = ({x_1},{x_2}, \ldots )$ belongs if and only if there exists a number $ k > 0$ such that $\displaystyle h_\alpha(x) = \operatorname{lub}_{p \in \mathcal{P}} \sum\limits_{i = 1}^\infty \vert x_{F(i)} \alpha_i\vert < k.$ $h_\alpha$ is a norm on $d_\alpha$ and $ (d_\alpha, h_\alpha)$ is complete. We classify the $ {d_\alpha }$ spaces and compare them with ${l_1}$ and $m$. Some of the $ {d_\alpha }$ spaces are shown to have a semishrinking basis that is not shrinking. Further investigation of the bases in these spaces yields theorems concerning the conjugate space properties of ${d_\alpha }$. We characterize the sequences $\beta$ such that, given $\alpha ,{d_\beta }, = {d_\alpha }$. A class of manifolds in the first conjugate space of ${d_\alpha }$ is examined. We establish some properties of the collection of points in the first conjugate space of a normed linear space $S$ that attain their maximum on the unit ball in $S$. The effect of renorming ${c_0}$ and ${l_1}$ with $ {h_\alpha }$ and related norms is studied in terms of the change induced on this collection of functionals.
Jacobi's bound for the order of systems of first order differential equations
Barbara A.
Lando
119-135
Abstract: Let ${A_1}, \ldots ,{A_n}$ be a system of differential polynomials in the differential indeterminates ${y^{(1)}}, \ldots ,{y^{(n)}}$, and let $\mathcal{M}$ be an irreducible component of the differential variety $\mathcal{M}({A_1}, \ldots ,{A_n})$. If $\dim \mathcal{M} = 0$, there arises the question of securing an upper bound for the order of $\mathcal{M}$ in terms of the orders ${r_{ij}}$ of the polynomials $ {A_i}$ in ${y^{(j)}}$. It has been conjectured that the Jacobi number $\displaystyle J = J({r_{ij}}) = \max \{\sum\limits_{i = 1}^n {{r_{i{j_i}}}} :{j_1}, \ldots ,{j_n}{\text{ is a permutation of 1,}} \ldots ,n \}$ provides such a bound. In this paper $J$ is obtained as a bound for systems consisting of first order polynomials. Differential kernels are employed in securing the bound, with the theory of kernels obtained in a manner analogous to that of difference kernels as given by R. M. Cohn.
Flat modules over commutative noetherian rings
Wolmer V.
Vasconcelos
137-143
Abstract: In this work we study flat modules over commutative noetherian rings under two kinds of restriction: that the modules are either submodules of free modules or that they have finite rank. The first ones have nicely behaved annihilators: they are generated by idempotents. Among the various questions related to flat modules of finite rank, emphasis is placed on discussing conditions implying its finite generation, as for instance, (i) over a local ring, a flat module of constant rank is free, and (ii) a flat submodule of finite rank of a free module is finitely generated. The rank one flat modules already present special problems regarding its endomorphism ring; in a few cases it is proved that they are flat over the base ring. Finally, a special class of flat modules--unmixed--is discussed, which have, so to speak, its source of divisibility somewhat concentrated in the center of its endomorphism ring and thus resemble projective modules over flat epimorphic images of the base ring.
Compact directed spaces
L. E.
Ward
145-157
Abstract: A directed space is a partially ordered topological space in which each two elements have a common predecessor. It is a consequence of a theorem of A. D. Wallace that a compact directed space is acyclic if each of its principal ideals is acyclic. This result is extended by considering the situation where at most finitely many principal ideals are not acyclic. It turns out that some of the elements which generate nonacyclic principal ideals must be maximal and that the $p$th cohomology group of the space must contain the $ p$th cohomology group of such a principal ideal as a direct summand. In the concluding sections it is shown that these spaces can be made acyclic by dividing out a closed ideal which contains all of the nonacyclic principal ideals, and some results on the acyclicity properties of minimal partial orders on compact spaces are proved.
Geometry associated with semisimple flat homogeneous spaces
Takushiro
Ochiai
159-193
Abstract: Our object is Cartan connections with semisimple flat homogeneous spaces as standard spaces. We study these from the viewpoint of $G$-structures of second order. This allows us especially to treat classical projective and conformal connections in the unifying manner. We also consider its application to the problem of certain geometric transformations.
Weak convergence of conditioned sums of independent random vectors.
Thomas M.
Liggett
195-213
Abstract: Conditions are given for the weak convergence of processes of the form $ ({{\mathbf{X}}_n}(t)\vert{{\mathbf{X}}_n}(1) \in {E^n})$ to tied-down stable processes, where $ ({{\mathbf{X}}_n}(t)$ is constructed from normalized partial sums of independent and identically distributed random vectors which are in the domain of attraction of a multidimensional stable law. The conditioning events are defined in terms of subsets ${E^n}$ of ${R^d}$ which converge in an appropriate sense to a set of measure zero. Assumptions which the sets $ {E^n}$ must satisfy include that they can be expressed as disjoint unions of ``asymptotically convex'' sets. The assumptions are seen to hold automatically in the special case in which $ {E^n}$ is taken to be a ``natural'' neighborhood of a smooth compact hypersurface in ${R^d}$.
The spectrum of partial differential operators on $L\sp{p}\,(R\sp{n})$
Franklin T.
Iha;
C. F.
Schubert
215-226
Abstract: The purpose of this paper is to prove that if the polynomial $ P(\xi )$ associated with a partial differential operator $P$ on $ {L^p}({R^n})$, with constant coefficients, has the growth property, $\vert P(\xi ){\vert^{ - 1}} = O(\vert\xi {\vert^{ - r}}),\vert\xi \vert \to \infty $ for some $r > 0$, then the spectrum of $ P$ is either the whole complex plane or it is the numerical range of $ P(\xi )$; and if $ P(\xi )$ has some additional property (all the coefficients of $ P(\xi )$ being real, for example), then the spectrum of $P$ is the numerical range for those $ p$ sufficiently close to 2. It is further shown that the growth property alone is not sufficient to ensure that the spectrum of $ P$ is the numerical range of $P(\xi )$.
On the generalized Blaschke condition
P. S.
Chee
227-231
Abstract: In an earlier paper, the author has shown that the generalized Blaschke condition is satisfied by bounded holomorphic functions in the polydisc or the ball in ${C^N}$. By essentially the same method, it is shown in the present paper that the same condition is satisfied by larger classes of functions, the Nevanlinna classes.
The local limit theorem and some related aspects of super-critical branching processes
Krishna B.
Athreya;
Peter
Ney
233-251
Abstract: Let $\{ {Z_n}:n = 0,1,2, \ldots \}$ be a Galton-Watson branching process with offspring p.g.f. $ f(s) = \Sigma _0^\infty {p_j}{s^j}$. Assume (i) $\Sigma _1^\infty {j^2}{p_j} < \infty$ and (iii) ${Z_n}{m^{ - n}}$ with ${Z_0} = 1,{w^{(i)}}(x)$ the $i$-fold convolution of $ w(x),{P_n}(i,j) = P({Z_n} = j\vert{Z_0} = i),{\delta _0} = (\log \gamma _0^{ - 1}){(\log m)^{ - 1}}$ and ${\beta _0} = {m^{{\delta _0}/(3 + {\delta _0})}}$. Then for any $0 < \beta < {\beta _0}$ and $i$ we can find a constant $C = C(i,\beta )$ such that $\displaystyle \vert{m^n}{P_n}(i,j) - {w^{(i)}}({m^{ - n}}j)\vert \leqq C[\beta _0^{ - n}{({m^{ - n}}j)^{ - 1}} + {\beta ^{ - n}}]$ for all $j \geqq 1$. Applications to the boundary theory of the associated space time process are also discussed.
From immersions to embeddings of smooth manifolds
Francis X.
Connolly
253-271
Existence and convergence of probability measures in Banach spaces.
Alejandro D.
de Acosta
273-298
Abstract: Theorems of the Bochner-Sazonov type are proved for Banach spaces with a basis. These theorems give sufficient conditions of a topological nature under which a positive definite function is the characteristic functional of a probability measure. The conditions are, in a certain natural sense, best possible. Central limit theorems of the Lindeberg type for triangular systems of random variables taking values in a Banach space with a basis are obtained. Applications to ${l_p}$ and $C[0,1]$ are given.
Logarithmic convexity, first order differential inequalities and some applications
Howard Allen
Levine
299-320
Abstract: Let, for $t \in [0,T)(T < \infty ),D(t)$ be a dense linear subspace of a Hilbert space $H$, and let $M(t)$ and $N(t)$ be linear operators (possibly unbounded) mapping $ D(t)$ into $H$. Let $f:[0,T) \times H \to H$. We give sufficient conditions on $M,N$ and $f$ in order to insure uniqueness and stability of solutions to $\displaystyle (1)\quad M(t)du/dt = N(t)u + f(t,u),\quad u(0)\;$given$\displaystyle .$ This problem is not in general well posed in the sense of Hadamard. We cite some examples of (1) from the literature. We also give some examples of the problem $\displaystyle (2)\quad M(t)\frac{{{d^2}u}}{{d{t^2}}} = N(t)u + f\left( {t,u,\frac{{du}}{{dt}}} \right),\quad u(0),\frac{{du}}{{du}}(0)\;$prescribed$\displaystyle ,$ for which questions of uniqueness and stability were discussed in a previous paper.
Some strict inclusions between spaces of $L\sp{p}$-multipliers
J. F.
Price
321-330
Abstract: Suppose that the Hausdorff topological group $G$ is either compact or locally compact abelian and that ${C_c}$ denotes the set of continuous complex-valued functions on $G$ with compact supports. Let $L_p^q$ denote the restrictions to $ {C_c}$ of the continuous linear operators from ${L^p}(G)$ into ${L^q}(G)$ which commute with all the right translation operators. When $1 \leqq p < q \leqq 2$ or $2 \leqq q < p \leqq \infty$ it is known that $\displaystyle (1)\quad L_p^p \subset L_q^q.$ The main result of this paper is that the inclusion in (1) is strict unless $G$ is finite. In fact it will be shown, using a partly constructive proof, that when $G$ is infinite $\displaystyle \bigcup\limits_{1 \leqq q < p} {L_q^q \subsetneqq } L_p^p \subsetneqq \bigcap\limits_{p < q \leqq 2} {L_q^q}$ for $1 < p < 2$, with the first inclusion remaining strict when $p = 2$ and the second inclusion remaining strict when $p = 1$. (Similar results also hold for $2 \leqq p \leqq \infty $.) When $ G$ is compact, simple relations will also be developed between idempotent operators in $L_p^q$ and lacunary subsets of the dual of $ G$ which will enable us to find necessary conditions so that inclusion (1) is strict even if, for example, $L_p^p$ and $L_q^q$ are replaced by the sets of idempotent operators in $L_p^p$ and $L_q^q$ respectively.
Positive Clifford semigroups on the plane
Reuben W.
Farley
353-369
Abstract: This work is devoted to a preliminary investigation of positive Clifford semigroups on the plane. A positive semigroup is a semigroup which has a copy of the nonnegative real numbers embedded as a closed subset in such a way that 0 is a zero and 1 is an identity. A positive Clifford semigroup is a positive semigroup which is the union of groups. In this work it is shown that if S is a positive Clifford semigroup on the plane, then each group in S is commutative. Also, a necessary and sufficient condition is given in order that S be commutative, and an example is given of such a semigroup which is, in fact, not commutative. In addition, both the number and the structure of the components of groups in S is determined. Finally, it is shown that S is the continuous isomorphic image of a semilattice of groups.
Infinite general linear groups over rings
George
Maxwell
371-375
Abstract: We are interested in the normal subgroups of the group G of automorphisms of a free module of infinite type over a commutative ring A. To this end, we introduce a certain ``elementary'' subgroup E of G and find that the subgroups of G normalised by E are exactly those which lie in congruence layers determined by the ideals of A. The normal subgroups are thus to be found in such layers.
Continuity of Gaussian processes
M. B.
Marcus;
L. A.
Shepp
377-391
Abstract: We give a proof of Fernique's theorem that if X is a stationary Gaussian process and ${\sigma ^2}(h) = E{(X(h) - X(0))^2}$ then X has continuous sample paths provided that, for some $\varepsilon > 0,\sigma (h) \leqq \psi (h),0 \leqq h \leqq \varepsilon $, where $\psi$ is any increasing function satisfying $\displaystyle \int_0^\varepsilon {\frac{{\psi (h)}}{{h{{(\log(1/h))}^{1/2}}}}} dh < \infty .$ ($ \ast$) We prove the partial converse that if $ \sigma (h) \geqq \psi (h),0 \leqq h \leqq \varepsilon $ and $\psi$ is any increasing function not satisfying $( ^\ast )$ then the paths are not continuous. In particular, if $\sigma$ is monotonic we may take $\psi = \sigma$ and $(^\ast)$ is then necessary and sufficient for sample path continuity. Our proof is based on an important lemma of Slepian. Finally we show that if $ \sigma$ is monotonie and convex in $ [0,\varepsilon ]$ then $ \sigma (h){(\log \,1/h)^{1/2}} \to 0$ as $h \to 0$ iff the paths are incrementally continuous, meaning that for each monotonic bounded sequence $t = {t_1},{t_2}, \ldots ,X({t_{n + 1}}) - X({t_n}) \to 0$, w.p.l.
The genus of repeated cartesian products of bipartite graphs
Arthur T.
White
393-404
Abstract: With the aid of techniques developed by Edmonds, Ringel, and Youngs, it is shown that the genus of the cartesian product of the complete bipartite graph $ {K_{2m,2m}}$ with itself is $1 + 8{m^2}(m - 1)$. Furthermore, let $ Q_1^{(s)}$ be the graph $ {K_{s,s}}$ and recursively define the cartesian product $ Q_n^{(s)} = Q_{n - 1}^{(s)} \times {K_{s,s}}$ for $n \geqq 2$. The genus of $Q_n^{(s)}$ is shown to be $1 + {2^{n - 3}}{s^n}(sn - 4)$, for all n, and s even; or for $n > 1$, and $s = 1 \;$ or$\; 3$. The graph $Q_n^{(1)}$ is the 1-skeleton of the n-cube, and the formula for this case gives a result familiar in the literature. Analogous results are developed for repeated cartesian products of paths and of even cycles.
Measures with bounded convolution powers
Bertram M.
Schreiber
405-431
Abstract: For an element x in a Banach algebra we study the condition $\displaystyle \mathop {\sup }\limits_{n \geqq 1} \left\Vert {{x^n}} \right\Vert < \infty .$ ($ 1$) Although our main results are obtained for the algebras $M(G)$ of finite complex measures on a locally compact abelian group, we begin by considering the question of bounded powers from the point of view of general Banach-algebra theory. We collect some results relating to (1) for an element whose spectrum lies in the unit disc D and has only isolated points on $\partial D$. There follows a localization theorem for commutative, regular, semisimple algebras A which says that whether or not (1) is satisfied for an element $x \in A$ with spectral radius 1 is determined by the behavior of its Gelfand transform $\hat x$ on any neighborhood of the points where $\vert\hat x\vert = 1$. We conclude the general theory with remarks on the growth rates of powers of elements not satisfying (1). After some applications of earlier results to the algebras $M(G)$, we prove our main theorem. Namely, we obtain strong necessary conditions on the Fourier transform for a measure to satisfy (1). Some consequences of this theorem and related results follow. Via the generalization of a result of G. Strang, sufficient conditions for (1) to hold are obtained for functions in ${L^1}(G)$ satisfying certain differentiability conditions. We conclude with the result that, for a certain class $ \mathcal{G}$ of locally compact groups containing all abelian and all compact groups, a group $ G \in \mathcal{G}$ has the property that every function in ${L^1}(G)$ with spectral radius one satisfies (1) if and only if G is compact and abelian.
Orbits of the automorphism group of the exceptional Jordan algebra.
John R.
Faulkner
433-441
Abstract: Necessary and sufficient conditions for two elements of a reduced exceptional simple Jordan algebra $\Im$ to be conjugate under the automorphism group $\mathrm{Aut} \Im $ of $\Im$ are obtained. It was known previously that if $\Im$ is split, then such elements are exactly those with the same minimum polynomial and same generic minimum polynomial. Also, it was known that two primitive idempotents are conjugate under $\mathrm{Aut} \Im$ if and only if they have the same norm class. In the present paper the notion of norm class is extended and combined with the above conditions on the minimum and generic minimum polynomials to obtain the desired conditions for arbitrary elements of $ \Im$.
On topologically invariant means on a locally compact group
Ching
Chou
443-456
Abstract: Let $\mathcal{M}$ be the set of all probability measures on $\beta N$. Let G be a locally compact, noncompact, amenable group. Then there is a one-one affine mapping of $ \mathcal{M}$ into the set of all left invariant means on ${L^\infty }(G)$. Note that $\mathcal{M}$ is a very big set. If we further assume G to be $\sigma$-compact, then we have a better result: The set $\mathcal{M}$ can be embedded affinely into the set of two-sided topologically invariant means on ${L^\infty }(G)$. We also give a structure theorem for the set of all topologically left invariant means when G is $\sigma$-compact.
A spectral sequence for the homotopy of nice spaces
A. K.
Bousfield;
E. B.
Curtis
457-479
The plane is not compactly generated by a free mapping
S. A.
Andrea
481-498
Abstract: Let X denote the plane, or the closed half-plane, and let $ T:X \to X$ be a self homeomorphism which preserves orientation and has no fixed points. It is proved that, if A is any compact subset of X, then there exists an unbounded connected subset B of X which does not meet $ {T^n}(A)$ for any integer n.
Homology of deleted products of one-dimensional spaces
Arthur H.
Copeland;
C. W.
Patty
499-510
Abstract: The object of this paper is to investigate the homology of deleted products of finitely triangulated one-dimensional spaces. By direct calculation, we obtain upper bounds for the two-dimensional Betti numbers, and, using a rather small system of topological types of spaces appearing as subspaces of the space under consideration, we obtain lower bounds for these Betti numbers. We demonstrate that, in general, the two-dimensional Betti numbers are larger than they were originally thought to be.
Isotopy invariants in quasigroups
Etta
Falconer
511-526
Abstract: The purpose of this paper is to investigate quasigroup and loop identities that are invariant under isotopy. The varieties of quasigroups and the varieties of loops that are closed under isotopy form isomorphic lattices. Some methods of generating isotopically closed varieties of loops are given.
On the symmetric cube of a sphere
Jack
Ucci
527-549
$k$-groups and duality
N.
Noble
551-561
Tensor product bases and tensor diagonals
J. R.
Holub
563-579
Abstract: Let X and Y denote Banach spaces with bases $({x_i})$ and $({y_i})$, respectively, and let $X{ \otimes _\varepsilon }Y$ and $X{ \otimes _\pi }Y$ denote the completion in the $\varepsilon$ and $\pi$ crossnorms of the algebraic tensor product $X \otimes Y$. The purpose of this paper is to study the structure of the tensor product spaces $X{ \otimes _\varepsilon }Y$ and $X{ \otimes _\pi }Y$ through a consideration of the properties of the tensor product basis $({x_i} \otimes {y_j})$ for these spaces and the tensor diagonal $({x_i} \otimes {y_i})$ of such bases.
Continua for which the set function $T$ is continuous
David P.
Bellamy
581-587
Abstract: The set-valued set function T has been studied extensively as an aid to classifying metric and Hausdorff continua. It is a consequence of earlier work of the author with H. S. Davis that T, considered as a map from the hyperspace of closed subsets of a compact Hausdorff space to itself, is upper semicontinuous. We show that in a continuum for which T is actually continuous (in the exponential, or Vietoris finite, topology) semilocal connectedness implies local connectedness, and raise the question of whether any nonlocally connected continuum for which T is continuous must be indecomposable.
Metrizability of compact convex sets
H. H.
Corson
589-596
Abstract: It is proved that a compact convex set is metrizable if the set of extreme points is the continuous image of a complete separable metric space.
Close isotopies on piecewise-linear manifolds
Richard T.
Miller
597-628
Locally convex topological lattices
Albert R.
Stralka
629-640
Abstract: The main theorem of this paper is: Suppose that L is a topological lattice of finite breadth n. Then L can be embedded in a product of n compact chains if and only if L is locally convex and distributive. With this result it is then shown that the concepts of metrizability and separability are equivalent for locally convex, connected, distributive topological lattices of finite breadth.
Homological dimension and cardinality
B. L.
Osofsky
641-649
Abstract: Let $\{ e(i)\vert i \in \mathcal{I}\}$ be an infinite set of commuting idempotents in a ring R with 1 such that $\displaystyle \prod\limits_{\alpha = 0}^n {e({i_\alpha })\prod\limits_{\beta = n + 1}^m {(1 - e({i_\beta })) \ne 0} }$ for $\{ {i_\alpha }\vert \leqq \alpha \leqq n\} \cap \{ {i_\beta }\vert n + 1 \leqq \beta \leqq m\} = \emptyset$. Let I be the right ideal generated by these idempotents. We show that the projective dimension of I is $n < \infty $ if and only if the cardinality of $I = {\aleph _n}$. As a consequence, a countable direct product of fields has global dimension $k + 1$ if and only if ${2^{{\aleph _0}}} = {\aleph _k}$. The same is true for a full linear ring on a countable dimensional vector space over a field of cardinality at most ${2^{{\aleph _0}}}$. On the other hand, if $ {2^{{\aleph _0}}} > {\aleph _\omega }$, then any right and left self-injective ring which is not semi-perfect, any ring containing an infinite direct product of subrings, any ring containing the endomorphism ring of a countable direct sum of modules, and many quotient rings of such rings must all have infinite global dimension.
On the geometric means of entire functions of several complex variables
A. K.
Agarwal
651-657
Abstract: Let $f({z_1}, \ldots ,{z_n})$ be an entire function of the $n( \geqq 2)$ complex variables ${z_1}, \ldots ,{z_n}$, holomorphic for $ \vert{z_t}\vert \leqq {r_t},t = 1, \ldots ,n$. We have considered the case of only two complex variables for simplicity. Recently many authors have defined the arithmetic means of the function $ \vert f({z_1},{z_2})\vert$ and have investigated their properties. In the present paper, the geometric means of the function $\vert f({z_1},{z_2})\vert$ have been defined and the asymptotic behavior of certain growth indicators for entire functions of several complex variables have been studied and the results are given in the form of theorems.
Applications of the Tumura-Clunie theorem
Chung-chun
Yang
659-662
Abstract: Some applications of the Tumura-Clunie theorem are given. Most of these concerned fixed points of compositions of entire functions.
Finite groups with quasi-dihedral and wreathed Sylow $2$-subgroups.
J. L.
Alperin;
Richard
Brauer;
Daniel
Gorenstein
1-261
Abstract: The primary purpose of this paper is to give a complete classification of all finite simple groups with quasi-dihedral Sylow 2-subgroups. We shall prove that any such group must be isomorphic to one of the groups ${L_3}(q)$ with $q \equiv - 1 \pmod 4,{U_3}(q)$ with $q \equiv 1 \pmod 4$, or ${M_{11}}$. We shall also carry out a major portion of the corresponding classification of simple groups with Sylow 2-subgroups isomorphic to the wreath product of $ {Z_{{2^n}}}$ and ${Z_2},n \geqq 2$.
Strong renewal theorems with infinite mean
K. Bruce
Erickson
263-291
Abstract: Let F be a nonarithmetic probability distribution on $(0,\infty )$ and suppose $1 - F(t)$ is regularly varying at $\infty$ with exponent $\alpha ,0 < \alpha \leqq 1$. Let $U(t) = \Sigma {F^{{n^ \ast }}}(t)$ be the renewal function. In this paper we first derive various asymptotic expressions for the quantity $U(t + h) - U(t)$ as $t \to \infty ,h > 0$ fixed. Next we derive asymptotic relations for the convolution ${U^ \ast }z(t),t \to \infty $, for a large class of integrable functions z. All of these asymptotic relations are expressed in terms of the truncated mean function $m(t) = \smallint _0^t[1 - F(x)]dx$, t large, and appear as the natural extension of the classical strong renewal theorem for distributions with finite mean. Finally in the last sections of the paper we apply the special case $ \alpha = 1$ to derive some limit theorems for the distributions of certain waiting times associated with a renewal process.
Meromorphic functions of elements of a commutative Banach algebra
Barnett W.
Glickfeld
293-307
Quadratic functionals of second order
Walter
Leighton
309-322
Abstract: In this paper we study the minimizing of the general second-order quadratic functional (1.3) in a class of admissible functions $y(x)$ with fixed endpoint conditions on $y(x)$ and its derivative at $x = a$ and at $x = b$. Necessary conditions and sufficient conditions are obtained. These lead, in turn, to natural extensions of the Sturm comparison theorem to fourth-order selfadjoint equations. These extensions include and are more general than previously stated comparison theorems. Finally, it is found that the present variational theory provides an orderly approach to second-order Wirtinger-like inequalities, and the results include as special cases many published results of this type.
Dual spaces of weighted spaces
W. H.
Summers
323-333
Abstract: The topological duals of a large class of weighted spaces of continuous functions are characterized as spaces of Radon measures which can be factored into a product of a weight function and a bounded Radon measure. We next obtain a representation for a base for the equicontinuous subsets of these dual spaces and for the extremal points of the members of this base. Finally, among other applications, these ideas make possible an extension of the representation theorem for biequicontinuous completed tensor products of weighted spaces obtained by the author in an earlier paper.
Global dimension of orders
Richard B.
Tarsy
335-340
Abstract: We prove that the finitistic global dimension (fGD) of an order in a quaternion algebra over the quotient field of a Dedekind domain is one. Examples are given of orders of global dimension $n - 1$ in $n \times n$ matrices over the quotient field of a discrete valuation ring.
Spaces of countable and point-countable type
J. E.
Vaughan
341-351
Abstract: These spaces were introduced by M. Henriksen and J. R. Isbell, and A. V. Arhangel'skiĭ, who proved results about the placement of such spaces in their compactifications. In the present paper, these results are consolidated using new terminology. In addition, further results concerning the heredity of these spaces are obtained. Generalizations of these spaces are introduced, and an analogous treatment is given for them. Finally, some examples are given of which one gives a solution to a problem raised by Arhangel'skiĭ by showing that the perfect image of a first countable space need not be of point-countable type.
On $2$-groups with no normal abelian subgroups of rank $3$, and their occurrence as Sylow $2$-subgroups of finite simple groups
Anne R.
MacWilliams
345-408
Abstract: We prove that in a finite $2$-group with no normal Abelian subgroup of rank $ \geqq 3$, every subgroup can be generated by four elements. This result is then used to determine which $2$-groups $T$ with no normal Abelian subgroup of rank $ \geqq 3$ can occur as $ {S_2}$'s of finite simple groups $G$, under certain assumptions on the embedding of $T$ in $G$.
Union extensions of semigroups
L. A. M.
Verbeek
409-423
The Hasse invariant of a vector bundle
Richard R.
Patterson
425-443
Abstract: The object of this work is to define, by analogy with algebra, the Witt group and the graded Brauer group of a topological space $X$. A homomorphism is defined between them analogous to the generalized Hasse invariant. Upon evaluation, the Witt group is seen to be $ \tilde KO(X)$, the graded Brauer group $1 + {H^1}(X;{Z_2}) + {H^2}(X;{Z_2})$ with truncated cup product multiplication, while the homomorphism is given by Stiefel-Whitney classes: $1 + {w_1} + {w_2}$.
Spectral mapping theorems and perturbation theorems for Browder's essential spectrum
Roger D.
Nussbaum
445-455
Abstract: If $T$ is a closed, densely defined linear operator in a Banach space, F. E. Browder has defined the essential spectrum of $T,\operatorname{ess} (T)$ [1]. We derive below spectral mapping theorems and perturbation theorems for Browder's essential spectrum. If $T$ is a bounded linear operator and $f$ is a function analytic on a neighborhood of the spectrum of $T$, we prove that $ f(\operatorname{ess} (T)) = \operatorname{ess} (f(T))$. If $T$ is a closed, densely defined linear operator with nonempty resolvent set and $ f$ is a polynomial, the same theorem holds. For a closed, densely defined linear operator $T$ and a bounded linear operator $B$ which commutes with $T$, we prove that $\operatorname{ess} (T + B) \subseteq \operatorname{ess} (T) + \operatorname{ess} (B) = \{ \mu + v:\mu \in \operatorname{ess} (T),v \in \operatorname{ess} (B)\}$. By making additional assumptions, we obtain an analogous theorem for $B$ unbounded.
Singly generated homogeneous $F$-algebras
Ronn
Carpenter
457-468
Abstract: With each point $ m$ in the spectrum of a singly generated $F$-algebra we associate an algebra ${A_m}$ of germs of functions. It is shown that if ${A_m}$ is isomorphic to the algebra of germs of analytic functions of a single complex variable, then the spectrum of $A$ contains an analytic disc about $m$. The algebra $A$ is called homogeneous if the algebras $ {A_m}$ are all isomorphic. If $A$ is homogeneous and none of the algebras $ {A_m}$ have zero divisors, we show that $A$ is the direct sum of its radical and either an algebra of analytic functions or countably many copies of the complex numbers. If $A$ is a uniform algebra which is homogeneous, then it is shown that $A$ is either the algebra of analytic functions on an open subset of the complex numbers or the algebra of all continuous functions on its spectrum.
An exponential limit formula for nonlinear semigroups
Joel L.
Mermin
469-476
Abstract: In recent papers, many writers have developed the theory of semigroups of operators generated by nonlinear accretive operators. In the present paper, we construct this semigroup by means of an exponential limit formula, and then use this means of obtaining the semigroup to prove an approximation theorem that is a direct generalization of the Kato-Trotter theorem for linear semigroups.
Topology and the duals of certain locally compact groups
I.
Schochetman
477-489
Abstract: We consider some topological questions concerning the dual space of a (separable) extension $G$ of a type I, regularly embedded subgroup $ N$. The dual $ \hat G$ is known to have a fibre-like structure. The fibres are in bijective correspondence with certain subsets of dual spaces of associated stability subgroups. These subsets in turn are in bijective correspondence with certain projective dual spaces. Under varying hypotheses, we give sufficient conditions for these bijections to be homeomorphisms, we determine the support of the induced representation $U^L$ (for $ L \in \hat N$) and we give necessary and sufficient conditions for a union of fibres in $\hat G$ to be closed. In a much more general context we study the Hausdorff and CCR separation properties of the dual of an extension. We then completely describe the dual space topology of the above extension $G$ in an interesting case. The preceding results are then applied to the case where $N$ is abelian and $G/N$ is compact.
A representation of the solutions of the Darboux equation in odd-dimensional spaces
H.
Rhee
491-498
Abstract: It is shown that determining a function from its averages over all spheres passing through the origin leads to an explicit representation of the even solutions of the Darboux equation in the exterior of the characteristic cones in terms of the hyperboloidal means of the boundary data on the cones.
A diophantine problem on groups. I
R. C.
Baker
499-506
Abstract: The following theorem of H. Weyl is generalised to the context of locally compact abelian groups. Theorem. Let ${\lambda _1} < {\lambda _2} < {\lambda _3} \cdots$ be a sequence such that, for some $c > 0,\varepsilon > 0,{\lambda _{n + k}} - {\lambda _n} \geqq c$ whenever $k \geqq n/{(\log n)^{1 + \varepsilon }}(n = 1,2, \ldots )$. Then for almost all real $u$ the sequence ${\lambda _1}u,{\lambda _2}u, \ldots ,{\lambda _n}u\pmod 1$ is uniformly distributed.
$G\sb{2n}$ spaces
Donald O.
Koehler
507-518
Abstract: A complex normed linear space $X$ will be called a ${G_{2n}}$ space if and only if there is a mapping $\left\langle { \cdot , \ldots , \cdot } \right\rangle$ from ${X^{2n}}$ into the complex numbers such that: ${x_k} \to \left\langle {{x_1}, \ldots ,{x_{2n}}} \right\rangle$ is linear for $k = 1, \ldots ,n;\left\langle {{x_1}, \ldots ,{x_{2n}}} \right\rangle = {\left\langle {{x_{2n}}, \ldots ,{x_1}} \right\rangle ^ - }$; and ${\left\langle {x, \ldots ,x} \right\rangle ^{1/2n}} = \vert\vert x\vert\vert$. The basic models are the $ {L^{2n}}$ spaces, but one also has that every inner product space is a $ {G_{2n}}$ space for every integer $n$. Hence ${G_{2n}}$ spaces of a given cardinality need not be isometrically isomorphic. It is shown that a complex normed linear space is a ${G_{2n}}$ space if and only if the norm satisfies a generalized parallelogram law. From the proof of this characterization it follows that a linear map $ U$ from $X$ to $X$ is an isometry if and only if $\left\langle {U({x_1}), \ldots ,U({x_{2n}})} \right\rangle = \left\langle {{x_1}, \ldots ,{x_{2n}}} \right\rangle$ for all ${x_1}, \ldots ,{x_{2n}}$. This then provides a way to construct all of the isometries of a finite dimensional ${G_{2n}}$ space. Of particular interest are the $\operatorname{CBS} {G_{2n}}$ spaces in which $\vert\left\langle {{x_1}, \ldots ,{x_{2n}}} \right\rangle \vert \leqq \vert\vert{x_1}\vert\vert \cdots \vert\vert{x_{2n}}\vert\vert$. These spaces have many properties similar to inner product spaces. An operator $A$ on a complete $\operatorname{CBS} {G_{2n}}$ space is said to be symmetric if and only if $\left\langle {{x_1}, \ldots ,A({x_i}), \ldots ,{x_{2n}}} \right\rangle = \left\langle {{x_1}, \ldots ,A({x_j}), \ldots ,{x_{2n}}} \right\rangle$ for all $ i$ and $j$. It is easy to show that these operators are scalar and that on ${L^{2n}},n > 1$, they characterize multiplication by a real $ {L^\infty }$ function. The interest in nontrivial symmetric operators is that they exist if and only if the space can be decomposed into the direct sum of nontrivial ${G_{2n}}$ spaces.
On the radius of convexity and boundary distortion of Schlicht functions
David E.
Tepper
519-528
Abstract: Let $w = f(z) = z + \sum\nolimits_{n = 2}^\infty {{a_n}{z^n}}$ be regular and univalent for $ \vert z\vert < 1$ and map $\vert z\vert < 1$ onto a region which is starlike with respect to $w = 0$. If ${r_0}$ denotes the radius of convexity of $w = f(z)$, $d_0 = \min \vert f(z)\vert$ for $\vert z\vert = {r_0}$, and ${d^ \ast } = \inf \vert\beta \vert$ for $f(z) \ne \beta$, then it has been conjectured that $ {d_0}/{d^ \ast } \geqq 2/3$. It is shown here that ${d_0}/{d^ \ast } \geqq 0.343 \ldots$, which improves the old estimate ${d_0}/{d^ \ast } \geqq 0.268 \ldots$. In addition, sharp estimates for ${r_0}$ are given which depend on the value of $ \vert{a_2}\vert$.
Bounded and compact vectorial Hankel operators
Lavon B.
Page
529-539
Abstract: Operators $ H$ satisfying ${S^ \ast }H = HS$ where $S$ is a unilateral shift on Hilbert space are called Hankel operators. For a fixed shift $S$ of arbitrary multiplicity the Banach spaces of bounded Hankel operators and of compact Hankel operators are described, and it is shown that the former is always the second dual of the latter. Representations for bounded and for compact Hankel operators are given in a standard function space model.
Existence and stability of a class of nonlinear Volterra integral equations
Stanley I.
Grossman
541-556
Abstract: In this paper we study the problem of existence and uniqueness to solutions of the nonlinear Volterra integral equation $x = f + {a_1}{g_1}(x) + \cdots + {a_n}{g_n}(x)$, where the ${a_i}$ are continuous linear operators mapping a Fréchet space $ \mathcal{F}$ into itself and the ${g_i}$ are nonlinear operators in that space. Solutions are sought which lie in a Banach subspace of $\mathcal{F}$ having a stronger topology. The equations are studied first when the ${g_i}$ are of the form ${g_i}(x) = x + {h_i}(x)$ where ${h_i}(x)$ is ``small", and then when the $ {g_i}$ are slope restricted. This generalizes certain results in recent papers by Miller, Nohel, Wong, Sandberg, and Beneš.
Automorphism groups on compact Riemann surfaces
W. T.
Kiley
557-563
Abstract: For $g \geqq 2$, let $N(g)$ be the order of the largest automorphism group on a Riemann surface of genus $ g$. In this paper, lower bounds for $N(g)$ for various sequences of $g$'s are obtained. Sequences of appropriate groups are constructed. Each of these groups is then realized as a group of cover transformations of a surface covering the Riemann sphere. The genus of the resulting surface is then found by using the Riemann-Hurwitz formula and the automorphism group of the surface contains the given group. Each lower bound which is found is also shown to be sharp. That is, there are infinitely many $g$'s in the sequence to which the bound applies for which $N(g)$ does not exceed the bound.
Packing and reflexivity in Banach spaces
Clifford A.
Kottman
565-576
Abstract: A measure of the ``massiveness'' of the unit ball of a Banach space is introduced in terms of an efficiency of the tightest packing of balls of equal size in the unit ball. This measure is computed for the ${l_p}$-spaces, and spaces with distinct measures are shown to be not nearly isometric. A new convexity condition, which is compared to $B$-convexity, uniform smoothness, and uniform convexity, is introduced in terms of this measure, and is shown to be a criterion of reflexivity. The property dual to this convexity condition is also exposed and examined.
Finite dimensional inseparable algebras
Shuen
Yuan
577-587
Abstract: We determine the structure of finite dimensional algebras which are differentiably simple with respect to a set of higher derivations.
An alternative proof that Bing's dogbone space is not topologically $E\sp{3}$
E. H.
Anderson
589-609
Harmonic analysis on nilmanifolds
Jonathan
Brezin
611-618
Abstract: We compute, using a device of A. Weil, an explicit decomposition of $ {L^2}$ of a nilmanifold into irreducible translation-invariant subspaces. The results refine previous work of C. C. Moore and L. Green.
Estimates for the number of real-valued continuous functions
W. W.
Comfort;
Anthony W.
Hager
619-631
Abstract: It is a familiar fact that $ \vert C(X)\vert \leqq {2^{\delta X}}$, where $\vert C(X)\vert$ is the cardinal number of the set of real-valued continuous functions on the infinite topological space $X$, and $\delta X$ is the least cardinal of a dense subset of $X$. While for metrizable spaces equality obtains, for some familiar spaces--e.g., the one-point compactification of the discrete space of cardinal $2\aleph 0$--the inequality can be strict, and the problem of more delicate estimates arises. It is hard to conceive of a general upper bound for $ \vert C(X)\vert$ which does not involve a cardinal property of $X$ as an exponent, and therefore we consider exponential combinations of certain natural cardinal numbers associated with $ X$. Among the numbers are $ wX$, the least cardinal of an open basis, and $wcX$, the least $ \mathfrak{m}$ for which each open cover of $X$ has a subfamily with $ \mathfrak{m}$ or fewer elements whose union is dense. We show that $\vert C(X)\vert \leqq {(wX)^{wcX}}$, and that this estimate is best possible among the numbers in question. (In particular, ${(wX)^{wcX}} \leqq {2^{\delta X}}$ always holds.) In fact, it is only with the use of a version of the generalized continuum hypothesis that we succeed in finding an $X$ for which $ \vert C(X)\vert < {(wX)^{wcX}}$.
The strict topology for double centralizer algebras
Donald Curtis
Taylor
633-643
Abstract: Sufficient conditions are given for a double centralizer algebra under the strict topology to be a Mackey space.
Compact riemannian manifolds with essential groups of conformorphisms.
A. J.
Ledger;
Morio
Obata
645-651
Abstract: A solution to the following conjecture: A compact connected riemannian $ n$-manifold $(n > 2)$ with an essential group of conformorphisms is conformorphic to a euclidean $ n$-sphere.
Infinite products which are Hilbert cubes
James E.
West
1-25
Abstract: Let $Q$ denote the Hilbert cube. It is shown that if $P$ and $P'$ are compact polyhedra of the same simple homotopy type then $P \times Q$ and
On the shrinkability of decompositions of $3$-manifolds
William L.
Voxman
27-39
Abstract: An upper semicontinuous decomposition $G$ of a metric space $M$ is said to be shrinkable in case for each covering $ \mathcal{U}$ of the union of the nondegenerate elements, for each $\varepsilon > 0$, and for an arbitrary homeomorphism $h$ from $M$ onto $M$, there exists a homeomorphism $f$ from $M$ onto itself such that (1) if $ x \in M - ( \cup \{ U:U \in \mathcal{U}\} )$, then $ f(x) = h(x)$, (2) for each $g \in G$, (a) $\operatorname{diam} f[g] < \varepsilon$ and (b) there exists $ D \in \mathcal{U}$ such that $ h[D] \supset h[g] \cup f[g]$. Our main result is that if $G$ is a cellular decomposition of a $ 3$-manifold $M$, then $M/G = M$ if and only if $G$ is shrinkable. We also define concepts of local and weak shrinkability, and we show the equivalence of the various types of shrinkability for certain cellular decompositions. Some applications of these notions are given, and extensions of theorems of Bing and Price are proved.
Polynomial approximation on compact manifolds and homogeneous spaces
David L.
Ragozin
41-53
Abstract: We prove several theorems which relate the smoothness of a function, $ f$, defined on a compact ${C^\infty }$-submanifold of a Euclidean space to the rate at which the error in the best uniform approximation to $f$ by polynomials of degree at most $ n$ tends to zero.
Commutators modulo the center in a properly infinite von Neumann algebra
Herbert
Halpern
55-68
Coefficient estimates for Dirichlet series
W. T.
Sledd
69-76
An abstract nonlinear Cauchy-Kovalevska theorem
François
Trèves
77-92
Abstract: A nonlinear version of Ovcyannikov's theorem is proved. If $F(u,t)$ is an analytic function of $ t$ real or complex and of $ u$ varying in a scale of Banach spaces, valued in a scale of Banach spaces, the Cauchy problem ${u_t} = F(u,t),u(0) = {u_0}$, has a unique analytic solution. This is an abstract version of the Cauchy-Kovalevska theorem which can be applied to equations other than partial-differential, e.g. to certain differential-convolution or, more generally, differential-pseudodifferential equations.
Twin-convergence regions for continued fractions $K(a\sb{n}/1)$
William B.
Jones;
W. J.
Thron
93-119
Subgroups and automorphisms of extended Schottky type groups
Vicki
Chuckrow
121-129
On expansive transformation groups
Ping-fun
Lam
131-138
Rings for which certain flat modules are projective
S. H.
Cox;
R. L.
Pendleton
139-156
On successive approximations in homological algebra
V. K. A. M.
Gugenheim;
R. J.
Milgram
157-182
Theorems of Krein-Milman type for certain convex sets of operators.
P. D.
Morris;
R. R.
Phelps
183-200
Abstract: Let $M$ be a real (or complex) Banach space and $C(Y)$ the space of continuous real (or complex) functions on the compact Hausdorff space $ Y$. The unit ball of the space of bounded operators from $M$ into $C(Y)$ is shown to be the weak operator (or equivalently, strong operator) closed convex hull of its extreme points, provided $Y$ is totally disconnected, or provided ${M^ \ast }$ is strictly convex. These assertions are corollaries to more general theorems, most of which have valid converses. In the case $M = C(X)$, similar results are obtained for the positive normalized operators. Analogous results are obtained for the unit ball of the space of compact operators (this time in the operator norm topology) from $ M$ into $C(Y)$.
Elements with trivial centralizer in wreath products
Wolfgang P.
Kappe;
Donald B.
Parker
201-212
Abstract: Groups with self-centralizing elements have been investigated in recent papers by Kappe, Konvisser and Seksenbaev. In particular, if $G = A$wr$B$ is a wreath product some necessary and some sufficient conditions have been given for the existence of self-centralizing elements and for $G = \left\langle {{S_G}} \right\rangle$, where ${S_G}$ is the set of self-centralizing elements. In this paper ${S_G}$ and the set ${R_G}$ of elements with trivial centralizer are determined both for restricted and unrestricted wreath products. Based on this the size of $\left\langle {{S_G}} \right\rangle$ and $ \left\langle {{R_G}} \right\rangle$ is found in some cases, in particular if $ A$ and $B$ are $p$-groups or if $B$ is not periodic.
Finitely generated ideals of differentiable functions
B.
Roth
213-225
Abstract: In some spaces of differentiable functions, the finitely generated ideals which are closed are characterized in terms of the zeros of the generators. Applications are made to problems of division for distributions.
The subgroups of a free product of two groups with an amalgamated subgroup
A.
Karrass;
D.
Solitar
227-255
Abstract: We prove that all subgroups $H$ of a free product $G$ of two groups $A,B$ with an amalgamated subgroup $U$ are obtained by two constructions from the intersection of $H$ and certain conjugates of $A,B$, and $U$. The constructions are those of a tree product, a special kind of generalized free product, and of a Higman-Neumann-Neumann group. The particular conjugates of $ A,B$, and $U$ involved are given by double coset representatives in a compatible regular extended Schreier system for $G$ modulo $H$. The structure of subgroups indecomposable with respect to amalgamated product, and of subgroups satisfying a nontrivial law is specified. Let $ A$ and $B$ have the property $P$ and $U$ have the property $Q$. Then it is proved that $G$ has the property $P$ in the following cases: $ P$ means every f.g. (finitely generated) subgroup is finitely presented, and $ Q$ means every subgroup is f.g.; $P$ means the intersection of two f.g. subgroups is f.g., and $Q$ means finite; $P$ means locally indicable, and $ Q$ means cyclic. It is also proved that if $N$ is a f.g. normal subgroup of $G$ not contained in $U$, then $NU$ has finite index in $G$.
A fundamental solution of the parabolic equation on Hilbert space. II. The semigroup property
M. Ann
Piech
257-286
Abstract: The existence of a family of solution operators $\{ {q_t}:t > 0\}$ corresponding to a fundamental solution of a second order infinite-dimensional differential equation of the form $\partial u/\partial t = Lu$ was previously established by the author. In the present paper, it is established that these operators are nonnegative, and satisfy the condition ${q_s}{q_t} = {q_{s + t}}$.
Embeddings in division rings
John
Dauns
287-299
Abstract: A method for embedding a certain class of integral domains in division rings is devised. Integral domains $A$ are constructed with a generalized valuation into a (noncommutative) totally ordered semigroup that need not be discrete. Then the multiplicative semigroup $ A\backslash \{ 0\}$ is expressed as an inverse limit of semigroups each of which is embeddable in a group. Thus $A\backslash \{ 0\}$ can be embedded in a group $ G$. The main problem is to introduce addition on $G$ in order that $G$ becomes a division ring by the use of eventually commuting maps of inverse limits.
A characterization of integral currents
John E.
Brothers
301-325
Two methods of integrating Monge-Amp\`ere's equations
Michihiko
Matsuda
327-343
Abstract: Modifying Monge's method and Laplace's one respectively, we shall give two methods of integration of Monge-Ampère's equations. Although they seem quite different, the equivalence of our two methods will be shown. The first method will be from a point of view different from that of Lewy. The second will present a solution to a problem of Goursat.
Extending monotone decompositions of 3-manifolds
R. H.
Bing
351-369
On the structure of certain idempotent semigroups
Ahmad
Shafaat
371-378
Abstract: Some general theorems concerning residual finiteness of algebras are given that are applied to show that every idempotent semigroup satisfying $ xyzx = xzyx$ identically is a subcartesian product of certain simple semigroups of order two and three.
On $n$-parameter discrete and continuous semigroups of operators
James A.
Deddens
379-390
Abstract: We prove that n commuting operators on a Hilbert space can be uniquely simultaneously extended to doubly commuting coisometric operators if and only if they satisfy certain positivity conditions, which for the case $n = 1$ state simply that the original operator is a contraction. Our proof establishes the connection between these positivity conditions and the backward translation semigroup on ${l^2}({Z^{ + n}},\mathcal{K})$. A semigroup of operators is unitarily equivalent to backward translation (or a part thereof) on ${l^2}({Z^{ + n}},\mathcal{K})$ if and only if the positivity conditions are satisfied and the individual operators are coisometries (or contractions) whose powers tend strongly to zero. Analogous results are proven in the continuous case ${R^{ + n}}$.
Some fixed point theorems for compact maps and flows in Banach spaces.
W. A.
Horn
391-404
Abstract: Let ${S_0} \subset {S_1} \subset {S_2}$ be convex subsets of the Banach space X, with $ {S_0}$ and ${S_2}$ closed and ${S_1}$ open in ${S_2}$. If f is a compact mapping of $ {S_2}$ into X such that $\cup _{j = 1}^m{f^j}({S_1}) \subset {S_2}$ and ${f^m}({S_1}) \cup {f^{m + 1}}({S_1}) \subset {S_0}$ for some $m > 0$, then f has a fixed point in $ {S_0}$. (This extends a result of F. E. Browder published in 1959.) Also, if $ \{ {T_t}:t \in {R^ + }\}$ is a continuous flow on the Banach space X, $ {S_0} \subset {S_1} \subset {S_2}$ are convex subsets of X with $ {S_0}$ and ${S_2}$ compact and ${S_1}$ open in ${S_2}$, and ${T_{{t_0}}}({S_1}) \subset {S_0}$ for some $ {t_0} > 0$, where ${T_t}({S_1}) \subset {S_2}$ for all $t \leqq {t_0}$, then there exists ${x_0} \in {S_0}$ such that ${T_t}({x_0}) = {x_0}$ for all $t \geqq 0$. Minor extensions of Browder's work on ``nonejective'' and ``nonrepulsive'' fixed points are also given, with similar results for flows.
On properties of subspaces of $l\sb{p},\,0<p<1$
W. J.
Stiles
405-415
Abstract: The material presented in this paper deals with some questions concerning projections, quotient spaces, and linear dimension in $ {l_p}$ spaces, and also includes a remark about weak Schauder bases in $ {l_p}$ spaces and an example of an infinite-dimensional closed subspace of ${l_p}$ which is not isomorphic to ${l_p}$.
A generalization of the Siegel-Walfisz theorem
Larry Joel
Goldstein
417-429
Abstract: The uniform prime number theorem for primes in arithmetic progressions is generalized to the setting of Hecke L-series.
Analogues of Artin's conjecture
Larry Joel
Goldstein
431-442
Abstract: Based on heuristic, a general conjecture is made, which contains Artin's primitive root conjecture as a special case. Special cases of the general conjecture are verified using the generalized Siegel-Walfisz theorem. It is shown that the general conjecture can be considered as an infinite-dimensional analogue of the Tchebotarev density theorem.
On a lemma of Milutin concerning averaging operators in continuous function spaces
Seymour Z.
Ditor
443-452
Abstract: We show that any infinite compact Hausdorff space S is the continuous image of a totally disconnected compact Hausdorff space $S'$, having the same topological weight as S, by a map $\varphi$ which admits a regular linear operator of averaging, i.e., a projection of norm one of $ C(S')$ onto ${\varphi ^ \circ }C(S)$, where $f \in C(S)$ into $f \circ \varphi$. A corollary of this theorem is that if S is an absolute extensor for totally disconnected spaces, the space $S'$ can be taken to be the Cantor space $ {\{ 0,1\} ^\mathfrak{m}}$, where $ \mathfrak{m}$ is the topological weight of S. This generalizes a result due to Milutin and Pełczyński. In addition, we show that for compact metric spaces S and T and any continuous surjection $\varphi :S \to T$, the operator $u:C(S) \to C(T)$ is a regular averaging operator for $\varphi$ if and only if u has a representation $uf(t) = \smallint _0^1f(\theta (t,x))$ for a suitable function $\theta :T \times [0,1] \to S$.
On entropy and generators of measure-preserving transformations
Wolfgang
Krieger
453-464
Abstract: Let T be an ergodic measure-preserving transformation of a Lebesgue measure space with entropy $h(T)$. We prove that T has a generator of size k where ${e^{h(T)}} \leqq k \leqq {e^{h(T)}} + 1$.
Systems of derivations
Frances
Gulick
465-488
Abstract: Let A and B be two complex algebras. A system of derivations of order m from A into B is a set of $m + 1$ linear operators ${D_k}:A \to B(k = 0,1, \ldots ,m)$ such that for $ x,y \in A$ and $k = 0,1,2, \ldots ,m$, $\displaystyle {D_k}(xy) = \sum\limits_{j = 0}^k {\left( {_j^k} \right)} ({D_j}x)({D_{k - j}}y).$ If A is a commutative, regular, semisimple F-algebra with an identity, B the algebra of continuous functions on the closed maximal ideal space of A and $({D_0},{D_1}, \ldots ,{D_m})$ a system of derivations from A into B with ${D_0}$ the Gelfand mapping, then each $ {D_k}$ is continuous. The continuity of the operators in a system of derivations from ${C^n}(U)$ into $C(U)(U \subset R \;{\text{open}})$ is used to obtain a formula for ${D_k}f,f \in {C^n}(U)$, in terms of the ordinary derivatives of f and functions in $C(U)$. Each system of derivations from A into B and each multiplicative seminorm on B determine a multiplicative seminorm on A. Let U be a subset of C and $({D_0},{D_1}, \ldots ,{D_m})$ a system of derivations from the algebra $P(x)$ of polynomials on U into $ C(U)$ with ${D_0}$ the identity operator. Then the system of derivations determines a Hausdorff topology on $P(x)$. If U is open in R and ${D_1}x(t) \ne 0$ for $t \in U(x(t) = t)$, then the completion of $ P(x)$ in this topology is $ {C^m}(U)$. If U is open in C, then the completion of $P(x)$ in this topology is the algebra of functions analytic on U.
A characterization of the Peano derivative
J. Marshall
Ash
489-501
Abstract: For each choice of parameters $\{ {a_i},{b_i}\} ,i = 0,1, \ldots ,n + e$, satisfying certain simple conditions, the expression $\displaystyle \mathop {\lim }\limits_{h \to 0} {h^{ - n}}\sum\limits_{i = 0}^{n + e} {{a_i}f(x + {b_i}h)}$ yields a generalized nth derivative. A function f has an nth Peano derivative at x if and only if all the members of a certain subfamily of these nth derivatives exist at x. The result holds for the corresponding $ {L^p}$ derivatives. A uniformity lemma in the proof (Lemma 2) may be of independent interest. Also, a new generalized second derivative is introduced which differentiates more functions than the ordinary second derivative but fewer than the second Peano derivative.
Representations of twisted group algebras
Robert C.
Busby;
Harvey A.
Smith
503-537
Abstract: We construct a general class of Banach algebras which include as special cases the group algebra of a locally compact group, the group algebra of a group extension (in terms of the subgroup and quotient group), and some other examples, special cases of which have been studied under the name of covariance algebras. We develop the general representation theory and generalize Mackey's theory of induced representations.
Mersions of topological manifolds
David
Gauld
539-560
Abstract: We here generalise the immersion and submersion theorems of Smale, Hirsch, Haefliger and Poenaru, Phillips, Lees, and Lashof, giving a relative version in the case of mersions of topological manifolds. A mersion is a map of manifolds ${M^m} \to {Q^q}$ which in the appropriate local coordinate systems has the form ${R^m} \to {R^q}$ of the standard inclusion or projection of one euclidean space in another. Such a mersion induces a map of tangent bundles satisfying certain properties. In this paper the problem of classifying mersions is reduced to that of classifying such bundle maps.
Exact dynamic systems are tree-like and vice-versa
Kuo-tsai
Chen
561-567
Abstract: This paper gives an analytic characterization of those dynamic systems whose graph of trajectories is a tree.
The tension field of the Gauss map
Ernst A.
Ruh;
Jaak
Vilms
569-573
Abstract: In this paper it is shown that the tension field of the Gauss map can be identified with the covariant derivative of the mean curvature vector field. Since a map with vanishing tension field is called harmonic the following theorem is obtained as a corollary. The Gauss map of a minimal submanifold is harmonic.
Some applications of Waldhausen's results on irreducible surfaces
C. D.
Feustel
575-583
Piecewise linear groups and transformation groups
Herman
Gluck
585-593
Structure spaces of semigroups of continuous functions
K. D.
Magill
595-600
Abstract: In a previous paper, we associated a topological space with each left ideal of a semigroup. Here, we determine this space when the semigroup under consideration is the semigroup of all continuous selfmaps of any space belonging to a fairly extensive class of topological spaces and the left ideal is taken to be the kernel of the semigroup.
Quasi-states on $C\sp{\ast} $-algebras
Johan F.
Aarnes
601-625
Amalgamation of polyadic algebras
James S.
Johnson
627-652
Abstract: The main result of the paper is that for I an infinite set, the class of polyadic I-algebras (with equality) has the strong amalgamation property; i.e., if two polyadic I-algebras have a given common subalgebra they can be embedded in another algebra in such a way that the intersection of the images of the two algebras is the given common subalgebra.
The homotopy type of Fredholm manifolds
Kalyan K.
Mukherjea
653-663
Abstract: Banach manifolds whose tangent bundles admit a reduction to the Fredholm group have been intensively studied in the last few years. Here we show that such a manifold (under appropriate smoothness and separability restrictions) is homotopy equivalent to the union of a nested sequence of closed finite-dimensional submanifolds.
The free envelope of a finitely generated commutative semigroup
Pierre Antoine
Grillet
665-682
Monotone approximation by algebraic polynomials
G. G.
Lorentz;
K. L.
Zeller
1-18
Abstract: A given real continuous function f on [a, b] is approximated by polynomials ${P_n}$ of degree n that are subject to certain restrictions. Let $1 \leqq {k_1} < \cdots < {k_p} \leqq n$ be given integers, ${\varepsilon _i} = \pm 1$, given signs. It is assumed that $ P_n^{({k_i})}(x)$ has the sign of ${\varepsilon _i},i = 1, \ldots ,p,a \leqq x \leqq b$. Theorems are obtained which describe the polynomials of best approximation, and (for $p = 1$) establish their uniqueness. Relations to Birkhoff interpolation problems are of importance. Another tool are the sets A, where $ \vert f(x) - {P_n}(x)\vert$ attains its maximum, and the sets ${B_i}$ with $P_n^{({k_i})}(x) = 0$. Conditions are discussed which these sets must satisfy for a polynomial $ {P_n}$ of best approximation for f. Numbers of the points of sets A, $ {B_i}$ are studied, the possibility of certain extreme situations established. For example, if $ p = 1,{k_1} = 1,n = 2q + 1$, it is possible that $ \vert A\vert = 3,\vert B\vert = n$.
Separability of metric spaces
Prabir
Roy
19-43
The pseudo-circle is unique
Lawrence
Fearnley
45-64
On the Mann iterative process
W. G.
Dotson
65-73
On the maximality of sums of nonlinear monotone operators
R. T.
Rockafellar
75-88
On Wiener process sample paths
G. J.
Foschini;
R. K.
Mueller
89-93
Abstract: Let $\{ {X_t}(\omega )\}$ represent a version of the Wiener process having almost surely continuous sample paths on $( - \infty ,\infty )$ that vanish at zero. We present a theorem concerning the local nature of the sample paths. Almost surely the local behavior at each t is of one of seven varieties thus inducing a partition of $( - \infty ,\infty )$ into seven disjoint Borel sets of the second class. The process $\{ {X_t}(\omega )\}$ can be modified so that almost surely the sample paths are everywhere locally recurrent.
Generic bifurcation of periodic points
K. R.
Meyer
95-107
Abstract: This paper discusses the bifurcation of periodic points of a generic symplectic diffeomorphism of a two manifold that depends on a parameter. A complete classification of the types of bifurcation that can occur in the generic case is given.
Inequalities satisfied by entire functions and their derivatives
Boo-sang
Lee;
S. M.
Shah
109-117
Abstract: For a class of entire functions with simple and positive zeros, it is shown that the maximum of the moduli of the first two Taylor coefficients at any point z, dominate all the remaining Taylor coefficients, provided $ \vert z\vert$ is sufficiently large. Further, there is a subclass for which this result holds at every point z.
Two point boundary problems for second order matrix differential systems
Garret J.
Etgen
119-132
Abstract: This paper is concerned with second order matrix differential systems involving a parameter together with boundary conditions specified at two points. The object of the paper is to establish sufficient conditions for the existence of eigenvalues for the system. Although such problems have been considered using the results of and techniques from the calculus of variations, the methods and results here are entirely in the context of ordinary differential equations. Use is made of the matrix generalization of the polar coordinate transformation introduced by J. H. Barrett and the unitary transformation suggested by F. V. Atkinson and V. A. Jakubovič. The sufficient conditions for the existence of eigenvalues obtained here represent certain extensions of W. M. Whyburn's work concerning linear and nonlinear boundary problems for second order differential systems.
A noncommutative Hilbert basis theorem and subrings of matrices
S. A.
Amitsur
133-142
Abstract: A finitely generated central extension $A[{u_1}, \ldots ,{u_k}]$ of a commutative noetherian ring A, satisfies the ascending chain condition for ideals P for which $A[{u_1}, \ldots ,{u_k}]/P$ can be embedded in matrix rings ${M_n}(K)$ over arbitrary commutative rings K and n bounded. The method of proof leads to an example of a ring R which satisfies the same identities of ${M_n}(K)$ but nevertheless cannot be embedded in any matrix ring over a commutative ring of arbitrary finite order.
Balanced rings and a problem of Thrall
Victor P.
Camillo
143-153
Abstract: Balanced ring is defined and related to Thrall's QF-1 rings. Several theorems are obtained which show that balanced rings enjoy strong homological and chain conditions. The structure of commutative balanced rings is determined. Also, the structure of commutative artinian QF-1 rings is gotten. This is a generalization of a theorem of Floyd.
Noncompact simplices
S.
Simons
155-161
Abstract: A bounded, but not necessarily closed, (Choquet) simplex in $ {R^n}$ with nonempty interior is the intersection of $n + 1$ half-spaces. There is no bounded simplex with nonempty interior in an infinite dimensional Hausdorff real linear topological space.
Inseparable Galois theory of exponent one
Shuen
Yuan
163-170
Abstract: An exponent one inseparable Galois theory for commutative ring extensions of prime characteristic $p \ne 0$ is given in this paper.
Markuschevich bases and duality theory
William B.
Johnson
171-177
Abstract: Several duality theorems concerning Schauder bases in locally convex spaces have analogues in the theory of Markuschevich bases. For example, a locally convex space with a Markuschevich basis is semireflexive iff the basis is shrinking and boundedly complete. The strong existence Theorem III.1 for Markuschevich bases allows us to show that a separable Banach space is isomorphic to a conjugate space iff it admits a boundedly complete Markuschevich basis, and that a separable Banach space has the metric approximation property iff it admits a Markuschevich basis which is a generalized summation basis in the sense of Kadec.
Identities involving the coefficients of a class of Dirichlet series. IV
Bruce C.
Berndt
179-185
Abstract: We consider a class of Dirichlet series satisfying a functional equation with gamma factors. We define a generalized Dirichlet series that is analogous to the generalized zeta-function of Riemann. An analytic continuation for these generalized series is derived, and a few simple properties are established. Secondly, we prove a theorem on the Abel summation of Dirichlet series that satisfy Hecke's functional equation.
The continuity of functions on Cartesian products
N.
Noble
187-198
Zero-one laws for Gaussian processes
G.
Kallianpur
199-211
Abstract: Some zero-one laws are proved for Gaussian processes defined on linear spaces of functions. They are generalizations of a result for Wiener measure due to R. H. Cameron and R. E. Graves. The proofs exploit an interesting relationship between a Gaussian process and its reproducing kernel Hilbert space. Applications are discussed.
Harmonic analysis on certain vector spaces
J.
Kuelbs;
V.
Mandrekar
213-231
The asymptotic manifolds of a perturbed linear system of differential equations
T. G.
Hallam;
J. W.
Heidel
233-241
Quadratic variation of potentials and harmonic functions
Gunnar A.
Brosamler
243-257
Abstract: We prove the existence of a finite quadratic variation for stochastic processes $u(Y)$, where Y is Brownian motion on a Green domain of ${R^n}$, stopped upon reaching the Martin boundary, and u is a positive superharmonic function on the domain. As by-products we have results which are also of interest from a non-probabilistic point of view.
An elementary theory of the category of topological spaces
Dana I.
Schlomiuk
259-278
Abstract: An elementary system of axioms was given by F. W. Lawvere for the category of sets and mappings. The purpose of this paper is to provide a finite number of elementary axioms for the category of topological spaces and continuous mappings and to prove that any model of these axioms is equivalent to ``the category of topological spaces'' constructed over some model of Lawvere's axioms. Furthermore, we prove that any complete category, model of the given axioms is equivalent to the category of topological spaces.
State spaces for Markov chains
J. L.
Doob
279-305
Abstract: If $p(t,i,j)$ is the transition probability (from i to j in time t) of a continuous parameter Markov chain, with $p(0 + ,i,i) = 1$, entrance and exit spaces for p are defined. If $L[{L^ \ast }]$ is an entrance [exit] space, the function $p( \cdot , \cdot ,j)[p( \cdot ,i, \cdot )/h( \cdot )]$ has a continuous extension to $ (0,\infty ) \times L[(0,\infty ) \times {L^ \ast }$, for a certain norming function h on $ {L^ \ast }$]. It is shown that there is always a space which is both an entrance and exit space. On this space one can define right continuous strong Markov processes, for the parameter interval [0, b], with the given transition function as conditioned by specification of the sample function limits at 0 and b.
Extension methods in cardinal arithmetic
Erik
Ellentuck
307-325
Abstract: Functions (relations) defined on the nonnegative integers are extended to the cardinal numbers by the method of Myhill (Nerode) respectively. We obtain various results relating these extensions and conclude with an analysis of AE Horn sentences interpreted in the cardinal numbers. Let $ \mathfrak{A}$ be the sentence $(\forall {x_1}) \cdots (\forall {x_n})(\exists !y)\mathfrak{b}$ where quantifiers are restricted to the Dedekind cardinals and $ \mathfrak{b}$ is an equation built up from functors for cardinal addition, multiplication, and integer constants. One of our principal results is that $ \mathfrak{A}$ is a theorem of set theory (with the axiom of choice replaced by the axiom of choice for sets of finite sets) if and only if we can prove that the uniquely determined Skolem function for $ \mathfrak{A}$ extends an almost combinatorial function.
Representations of certain compact semigroups by ${\rm HL}$-semigroups
J. H.
Carruth;
C. E.
Clark
327-337
Abstract: An HL-semigroup is defined to be a topological semigroup with the property that the Schützenberger group of each $ \mathcal{H}$-class is a Lie group. The following problem is considered: Does a compact semigroup admit enough homomorphisms into HL-semigroups to separate points of S; or equivalently, is S isomorphic to a strict projective limit of HL-semigroups? An affirmative answer is given in the case that S is an irreducible semigroup. If S is irreducible and separable, it is shown that S admits enough homomorphisms into finite dimensional HL-semigroups to separate points of S.
Two-sided semisimple maximal quotient rings
Vasily C.
Cateforis
339-349
Abstract: Let R be an associative ring with singular right ideal zero and finite right Goldie dimension; F. L. Sandomierski has shown that the (R. E. Johnson) maximal right quotient ring Q of R is then semisimple (artinian). In this paper necessary and sufficient conditions are sought that Q be also a left (necessarily the maximal) quotient ring of R. Flatness of Q as a right R-module is shown to be such a condition. The condition that R have singular left ideal zero and finite left Goldie dimension, though necessary, is shown to be not sufficient in general. Conditions of two-sidedness of Q are also obtained in terms of the homogeneous components (simple subrings) of Q and the subrings of R, they induce.
Higher products
Gerald J.
Porter
315-345
Analytic sheaves of local cohomology
Yum-tong
Siu
347-366
A density theorem with an application to gap power series
K. G.
Binmore
367-384
Abstract: Let N be a set of positive integers and let $\displaystyle F(z) = \sum {{A_n}{z^n}}$ be an entire function for which ${A_n} = 0(n \notin N)$. It is reasonable to expect that, if D denotes the density of the set N in some sense, then $F(z)$ will behave somewhat similarly in every angle of opening greater than $2\pi D$. For functions of finite order, the appropriate density seems to be the Pólya maximum density $ \mathcal{P}$. In this paper we introduce a new density $ \mathcal{D}$ which is perhaps the appropriate density for the consideration of functions of unrestricted growth. It is shown that, if $ \vert I\vert > 2\pi \mathcal{D}$, then $\displaystyle \log M(r) \sim \log M(r,I)$ outside a small exceptional set. Here $M(r)$ denotes the maximum modulus of $ F(z)$ on the circle $ \vert z\vert = r$ and $ M(r,I)$ that of $F(r{e^{i\theta }})$ for values of $\theta$ in the closed interval I. The method used is closely connected with the question of approximating to functions on an interval by means of linear combinations of the exponentials ${e^{ixn}}(n \in N)$.
Representation theorems for complemented algebras
Freda E.
Alexander
385-398
Strongly separable pairings of rings
Robert S.
Cunningham
399-416
Abstract: The theory of adjoint functors has been used by Morita to develop a theory of Frobenius and quasi-Frobenius extensions subsuming the work of Kasch, Müller, Nakayama, and others. We use adjoint functors to define a pairing of the two rings and develop a theory of relative projective and injective modules for pairings generalizing that of Hochschild for extensions. The main purpose of this paper is to define ``strongly separable pairings'' generalizing strongly separable (i.e. finitely generated projective separable) algebras. We show that such pairings have very close connections to category equivalences, so that it is natural to investigate those properties shared by two rings which admit a strongly separable pairing. We show that most ``categorical'' properties are so shared.
The pseudo-circle is not homogeneous
James T.
Rogers
417-428
Recursive functions modulo ${\rm CO}-r$-maximal sets
Manuel
Lerman
429-444
Abstract: Define the equivalence relation ${ \sim _A}$ on the set of recursive functions of one variable by $f\sim_A g$ if and only if $ f(x) = g(x)$ for all but finitely many $x \in \bar A$, where $ \bar A$ is an r-cohesive set, to obtain the structure $\mathcal{R}/\bar A$. Then the recursive functions modulo such an equivalence relation form a semiring with no zero divisors. It is shown that if A is r-maximal, then the structure obtained above is not a nonstandard model for arithmetic, a result due to Feferman, Scott, and Tennenbaum. Furthermore, if A and B are maximal sets, then a necessary and sufficient condition for $\mathcal{R}/\bar A$ and $\mathcal{R}/\bar B$ to be elementarily equivalent is obtained. It is also shown that many different elementary theories can be obtained for $\mathcal{R}/\bar A$ by proper choice of $ \bar A$.
Compact imbedding theorems for quasibounded domains
Robert A.
Adams
445-459
Convolutions with kernels having singularities on a sphere
Robert S.
Strichartz
461-471
Abstract: We prove that convolution with $(1 - \vert x{\vert^2})_ + ^{ - \alpha }$ and related convolutions are bounded from ${L^p}$ to ${L^q}$ for certain values of p and q. There is a unique choice of p which maximizes the measure of smoothing $1/p - 1/q$, in contrast with fractional integration where $1/p - 1/q$ is constant. We apply the results to obtain a priori estimates for solutions of the wave equation in which we sacrifice one derivative but gain more smoothing than in Sobolev's inequality.
Nice homology coalgebras
A. K.
Bousfield
473-489
Sur le rel\`evement des representations modulaires d'un groupe fini
François
Aribaud
491-499
Mesures associ\'ees aux fonctionnelles additives de Markov. I
D.
Revuz
501-531
Abstract: With each additive functional of Markov processes we associate a measure and characterize, under duality hypotheses, those which correspond to $\sigma$-finite measures. This enables us to weaken the hypotheses of Meyer's theorem on representation of potentials of measures as potentials of additive functional. We characterize also the measures which are associated with continuous additive functionals. This leads us to show that for each finite continuous additive functional of the process there exists a finite continuous additive functional of the dual process such that the corresponding time-changed processes are in duality. Similar results are also stated for subprocesses which generalize results by Hunt and Blumenthal and Getoor.
Finite nilpotent characteristic nonverbal groups
Orin
Chein
533-548
Abstract: In this paper, we study nilpotent groups which are quotient groups of finitely generated free groups with respect to characteristic but nonverbal subgroups. We show that there are no abelian groups of the type in question. We also show that all such groups of nilpotence class 2 or 3 are finite and have minimal sets of two generators. In fact, formal presentations for all such groups are given. The direct product of two finite CNV groups (as the groups in question will be called) which have minimal sets of generators of the same size is shown to again be a CNV group, provided that the orders of the original two groups are relatively prime. Conversely, if a finite CNV group is a direct product of groups of relatively prime orders, then at least one of these direct factors is a CNV group. Several other related results are also obtained.
Injective and projective Heyting algebras
Raymond
Balbes;
Alfred
Horn
549-559
A class of decompositions of $E\sp{n}$ which are factors of $E\sp{n+1}$
John L.
Bailey
561-575
Characterizations of $C\sp{\ast} $-algebras. II
T. W.
Palmer
577-588
Nest generated intersection rings in Tychonoff spaces
A. K.
Steiner;
E. F.
Steiner
589-601
A topologically strongly mixing symbolic minimal set
K. E.
Petersen
603-612
Abstract: Recent papers by the author, Keynes and Robertson, and others have shown that weakly mixing minimal flows are objects of considerable interest, but examples of such flows, other than the horocycle flows, have been scarce. We give here a ``machinal'' construction of a bilateral sequence with entries from 0, 1 whose orbit closure is topologically strongly mixing and minimal. We prove in addition that the flow we obtain has entropy zero, is uniquely ergodic, and fails to be measure-theoretically strongly mixing.
Some remarks on self-dual locally compact Abelian groups
Lawrence
Corwin
613-622
Abstract: The main results of this paper are the construction of some new self-dual locally compact Abelian groups and the proof of a structure theorem for a certain class of such groups. The construction is based on an investigation of when the extension of a compact Abelian group by its dual yields a self-dual group. It turns out that such extensions can be described algebraically ; the structure theorem follows from an analysis of the algebraic description.
Corrections to: ``On sequential convergence''
R. M.
Dudley
623-624
Errata to: ``Vector cross products on manifolds''
Alfred
Gray
625
Existence and uniqueness theorems for extensions of zero-dimensional compact metric spaces
R. S.
Pierce
1-21
A general theorem on the convergence of operator semigroups
Thomas G.
Kurtz
23-32
An infinite-dimensional Schoenflies theorem
D. E.
Sanderson
33-39
On Milnor's invariant for links. II. The Chen group
Kunio
Murasugi
41-61
Collaring and $(n-1)$-manifold in an $n$-manifold
C. L.
Seebeck
63-68
Topological semigroups with invariant means in the convex hull of multiplicative means
Anthony To-ming
Lau
69-84
Embedding as a double commutator in a type I $AW\sp{\ast} $-algebra
Herbert
Halpern
85-98
Boundaries of semilinear spaces and semialgebras
R. E.
Worth
99-119
Isotopisms of semigroups of functions
Kenneth D.
Magill
121-128
A characterization of unitary duality
David W.
Roeder
129-135
Abstract: The concept of unitary duality for topological groups was introduced by H. Chu. All mapping spaces are given the compact-open topology. Let G and H be locally compact groups. ${G^ \times }$ is the space of continuous finite-dimensional unitary representations of G. Let ${\operatorname{Hom}}({G^ \times },{H^ \times })$ denote the space of all continuous maps from ${G^ \times }$ to $ {H^ \times }$ which preserve degree, direct sum, tensor product and equivalence. We prove that if H satisfies unitary duality, then $ {\operatorname{Hom}}(G,H)$ and $ {\operatorname{Hom}}{\mkern 1mu} ({H^ \times },{G^ \times })$ are naturally homeomorphic. Conversely, if $ {\operatorname{Hom}}(Z,H)$ and $ {\operatorname{Hom}}{\mkern 1mu} ({H^ \times },{Z^ \times })$ are homeomorphic by the natural map, where Z denotes the integers, then H satisfies unitary duality. In different contexts, results similar to the first half of this theorem have been obtained by Suzuki and by Ernest. The proof relies heavily on another result in this paper which gives an explicit characterization of the topology on $ {\operatorname{Hom}}{\mkern 1mu} ({G^ \times },{H^ \times })$. In addition, we give another necessary condition for locally compact groups to satisfy unitary duality and use this condition to present an example of a maximally almost periodic discrete group which does not satisfy unitary duality.
Infinite deficiency in Fr\'echet manifolds
T. A.
Chapman
137-146
Abstract: Denote the countable infinite product of lines by s, let X be a separable metric manifold modeled on s, and let K be a closed subset of X having Property Z in X, i.e. for each nonnull, homotopically trivial, open subset U of X, it is true that $ U\backslash K$ is nonnull and homotopically trivial. We prove that there is a homeomorphism h of X onto $X \times s$ such that $h(K)$ projects onto a single point in each of infinitely many different coordinate directions in s. Using this we prove that there is an embedding of X as an open subset of s such that K is carried onto a closed subset of s having Property Z in s. We also establish stronger versions of these results.
Banach spaces of Lipschitz functions and vector-valued Lipschitz functions
J. A.
Johnson
147-169
The module index and invertible ideals
David W.
Ballew
171-184
Abstract: A. Fröhlich used the module index to classify the projective modules of an order in a finite dimensional commutative separable algebra over the quotient field of a Dedekind domain. This paper extends Fröhlich's results and classifies the invertible ideals of an order in a noncommutatives eparable algebra. Several properties of invertible ideals are considered, and examples are given.
Quasiconformal mappings and Schwarz's lemma
Peter J.
Kiernan
185-197
Abstract: In this paper, K quasiconformal maps of Riemann surfaces are investigated. A theorem, which is similar to Schwarz's lemma, is proved for a certain class of K quasiconformal maps. This result is then used to give elementary proofs of theorems concerning K quasiconformal maps. These include Schottky's lemma, Liouville's theorem, and the big Picard theorem. Some of Huber's results on analytic self-mappings of Riemann surfaces are also generalized to the K quasiconformal case. Finally, as an application of the Schwarz type theorem, a geometric proof of a special case of Moser's theorem is given.
Approximation by polynomials subordinate to a univalent function
Thomas H.
MacGregor
199-209
L\'evy measures for a class of Markov semigroups in one dimension
Ken iti
Sato
211-231
Abstract: Given a Markov semigroup of linear operators in the space of realvalued continuous functions on the line vanishing at infinity, we prove that the Lévy measure exists if the domain of the infinitesimal generator contains $ \mathcal{D}_K(D_m D_s^+)$, the domain of William Feller's generalized second order differential operator restricted to functions with compact supports. We give estimate of singularity of the Lévy measure and representation of the infinitesimal generator. Conversely, given Lévy measure or the form of infinitesimal generator, existence of the corresponding Markov semigroup is shown under some conditions. The case of circles is also discussed.
An asymptotic property of Gaussian processes. I
Hisao
Watanabe
233-248
The Blaschke condition for bounded holomorphic functions
Pak Soong
Chee
249-263
Countable paracompactness and weak normality properties
John
Mack
265-272
Nonlinear evolution equations and product integration in Banach spaces.
G. F.
Webb
273-282
Abstract: The method of product integration is used to obtain solutions to the nonlinear evolution equation $g' = Ag$ where A is a function from a Banach space S to itself and g is a continuously differentiable function from $ [0,\infty )$ to S. The conditions required on A are that A is dissipative on S, the range of $(e - \varepsilon A) = S$ for all $\varepsilon \geqq 0$, and A is continuous on S.
Stable maps and Schwartz maps
Andre
de Korvin
283-291
The Freudenthal-Springer-Tits constructions revisited
Kevin
McCrimmon
293-314
On the upper and lower class for stationary Gaussian processes
Tunekiti
Sirao;
Hisao
Watanabe
301-331
Semigroups of right quotients of topological semigroups
Hanns Joachim
Weinert
333-348
Approximate point spectrum of a weighted shift
William C.
Ridge
349-356
On a second order nonlinear oscillation problem
C. V.
Coffman;
J. S. W.
Wong
357-366
Structures determined by prime ideals of rings of functions
Richard G.
Montgomery
367-380
Slices in transformation groups
Larry M.
King
381-388
Concerning arcwise connectedness and the existence of simple closed curves in plane continua
Charles L.
Hagopian
389-402
Conjugate functions for Laguerre expansions
Benjamin
Muckenhoupt
403-418
Mean convergence of Hermite and Laguerre series. I
Benjamin
Muckenhoupt
419-431
Mean convergence of Hermite and Laguerre series. II
Benjamin
Muckenhoupt
433-460
Completely regular mappings with locally compact fiber
Stephen B.
Seidman
461-471
Generalized stochastic integrals and equations
Donald A.
Dawson
473-506
Positive harmonic functions on Lipschitz domains
Richard A.
Hunt;
Richard L.
Wheeden
507-527
On the isometric immersions in Euclidean space of manifolds with nonnegative sectional curvatures. II
Philip
Hartman
529-540
$P$-uniform convergence and a vector-valued strong law of large numbers
Anatole
Beck;
Daniel P.
Giesy
541-559
The decomposition of $3$-manifolds with several boundary components.
Jonathan L.
Gross
561-572
Completions in abstract homotopy theory
Alex
Heller
573-602
Semigroups on a half-space
J. G.
Horne
1-53
Lie isomorphisms of factors
C. Robert
Miers
55-63
A class of nonlinear evolution equations in a Banach space
J. R.
Dorroh
65-74
Espaces classifiants en $K$-th\'eorie
Max
Karoubi
75-115
Closed subalgebras of group algebras
Stephen
Friedberg
117-125
Dynamical systems with an invariant space of vector fields
Robert I.
Jewett;
Sol
Schwartzman
127-134
Some immersion theorems for projective spaces
A. Duane
Randall
135-151
A quantum dynamical, relativistically invariant rigid body system
Richard
Arens
153-201
On the counting function for the $a$-values of a meromorphic function.
Joseph
Miles
203-222
Deleted products with homotopy types of spheres
C. W.
Patty
223-240
Maximal closed preprimes in Banach algebras
P.
Civin;
C. C.
White
241-260
A $1$-linked link whose longitudes lie in the second commutator subgroup
H. W.
Lambert
261-269
Some restricted partition functions: Congruences modulo $2$
D. B.
Lahiri
271-278
Fibering $3$-manifolds that admit free $Z\sb{k}$ actions
Jeffrey L.
Tollefson
279-287
Characterization of taming sets on $2$-spheres
J. W.
Cannon
289-299
$2$-transitive symmetric designs
William M.
Kantor
1-28
The principle type-scheme of an object in combinatory logic
R.
Hindley
29-60
The $C\sp{k}$-classification of certain operators in $L\sb{p}$. II
Shmuel
Kantorovitz
61-67
Cyclic vectors and parts of the spectrum of a weighted shift
Ralph
Gellar
69-85
A nonlinear Tauberian theorem in function theory
Albert
Baernstein
87-105
An analogue of a problem of J. Bal\'azs and P. Tur\'an. III
A. K.
Varma
107-120
A representation theorem for biequicontinuous completed tensor products of weighted spaces
W. H.
Summers
121-131
Degree of symmetry of a product manifold
H. T.
Ku;
L. N.
Mann;
J. L.
Sicks;
J. C.
Su
133-149
Principal cofibrations in the category of simplicial groups
Norbert H.
Schlomiuk
151-165
Periodic solutions of hyperbolic partial differential equations in a strip
A. K.
Aziz;
A. M.
Meyers
167-178
On an ordinary differential equation involving a convex function
Philip
Hartman
179-202
Limit-operators and topological coreflections
Horst
Herrlich
203-210
On hypersingular integrals and certain spaces of locally differentiable functions
Richard L.
Wheeden
211-230
Partial prolongations and characteristics of differential equations.
H. H.
Johnson
231-240
On the row convergence of the Walsh array for meromorphic functions.
E. B.
Saff
241-257
Maximal abelian subalgebras of hyperfinite factors
Ole A.
Nielsen
259-272
Classifying spaces and infinite symmetric products
M. C.
McCord
273-298
The potential kernel and hitting probabilities for the general stable process in $R\sp{N}$
W. E.
Pruitt;
S. J.
Taylor
299-321
Identities involving the coefficients of a class of Dirichlet series. III
Bruce C.
Berndt
323-348
Two-parameter groups of formal power series
Abselom
Ran
349-368
Abstract: By ${\Omega ^F}$ we denote the group of the formal power series having the form $ F(z) = \Sigma _{q = 1}^\infty {f_q}{z^q},{f_1} \ne 0$, with respect to formal composition of power series. The problem of analytic iteration leads to the study of subgroups of ${\Omega ^F}$, having the form $\displaystyle F(z,s) = \sum\limits_{q = 1}^\infty {{f_q}(s){z^q}}$ where the coefficients ${f_q}(s)$ are analytic functions of the complex parameter s, such that for any two complex numbers s and t the formal law of composition $\displaystyle F[F(z,s),t] = F(z,s + t)$ is valid [6], [8]. The purpose of the present paper is to study similar two-parameter subgroups of ${\Omega ^F}$. In §1 r-parameter analytic subgroups of $ {\Omega ^F}$ are defined, as well as other concepts connected with the problem. In §2 the importance of two-parameter subgroups is emphasized. It is shown that the number of parameters of analytic subgroups of $ {\Omega ^F}$ can always be reduced to two at most. The existence of a countable number of classes of the two-parameter subgroups of ${\Omega ^F}$ is shown. §3 gives the explicit form of the coefficients ${f_q}({a^1},{a^2})$ of a two-parameter subgroup of ${\Omega ^F}$: $\displaystyle F(z,{a^1},{a^2}) = \sum\limits_{q = 1}^\infty {{f_q}({a^1},{a^2}){z^q}} .$ In §4 the existence of canonical representations for two-parameter analytic subgroups of ${\Omega ^F}$ is proven, and it is shown that every two-parameter analytic subgroup of ${\Omega ^F}$ is globally isomorphic to one of the groups $\displaystyle {H_n}(z,{a^1},{a^2}) = (1 + {a^1})z/{(1 + {a^2}{z^n})^{1/n}},\quad n = 1,2, \ldots $ (no two of which are globally isomorphic to each other).
Transitive semigroup actions
C. F.
Kelemen
369-375
Counterexamples in stable semigroups
L.
O’Carroll
377-386
The first exotic class of a manifold
David L.
Frank
387-395
Conjugate points and simple zeros for ordinary linear differential equations
Thomas L.
Sherman
397-411
Extension and some applications of the coincidence theorems
Zalman
Rubinstein;
J. L.
Walsh
413-427
Resolutions of singularities in prime characteristic for almost all primes
Paul C.
Eklof
429-438
On the univalence of a certain integral
Mamoru
Nunokawa
439-446
Embeddings of topological lattice-ordered groups
Robert L.
Madell
447-455
A lemma in transcendental number theory
Robert
Spira
457-464
Cohomology for ergodic groupoids
Joel J.
Westman
465-471
On the number of multipications on ${\rm SU}(3)$ and ${\rm Sp}(2)$
Mamoru
Mimura
473-492
The uniqueness of the Cauchy problem for partial differential equations which may have multiple characteristics
Peter M.
Goorjian
493-509
Antichains and products in partially ordered spaces
E. D.
Tymchatyn
511-520
$H\sp{p}$ spaces on bounded symmetric domains
Kyong T.
Hahn;
Josephine
Mitchell
521-531
Continuity of linear operators commuting with continuous linear operators. II
B. E.
Johnson;
A. M.
Sinclair
533-540
The cohomology of the complex projective Stiefel manifold
Carlos Alfredo
Ruiz
541-547
Errata to: ``Sturmian theorems and positive resolvents''
Kurt
Kreith
549
Manifolds of negative curvature
R. L.
Bishop;
B.
O’Neill
1-49
Upcrossing probabilities for stationary Gaussian processes
James
Pickands
51-73
Asymptotic properties of the maximum in a stationary Gaussian process.
James
Pickands
75-86
Taming embeddings of certain polyhedra in codimension three
T. B.
Rushing
87-103
Stacks, costacks and axiomatic homology
Yuh-ching
Chen
105-116
Anosov diffeomorphisms on tori
John
Franks
117-124
Multiplier rings and primitive ideals
John
Dauns
125-158
On the existence of normal subgroups containing their centralizer
Ulrich
Schoenwaelder
159-169
Asymptotic behavior of products $C\sp{p}=C+\cdots +C$ in locally compact abelian groups
W. R.
Emerson;
F. P.
Greenleaf
171-204
A generalization of the classical partition theorems
George E.
Andrews
205-221
A class of translation planes and a conjecture of D. R. Hughes
F. W.
Wilke
223-232
On extending semigroups of contractions
James A.
Deddens
233-239
Decomposition theories for modules
Joe W.
Fisher
241-269
Applications of Radon-Nikod\'ym theorems to martingale convergence
J. J.
Uhl
271-285
Covering theorems with applications to some questions on normal orders.
W. R.
Emerson
287-307
Certain cohomology rings of finite and formal group schemes
Gustave
Efroymson
309-322
A class of convex bodies
Ethan D.
Bolker
323-345
Phragm\`en-Lindel\"of theorems for elliptic equations in the plane
J. K.
Oddson
347-356
On free exterior powers
Harley
Flanders
357-367
An equivalence for the embeddings of cells in a $3$-manifold
R. J.
Daverman;
W. T.
Eaton
369-381
Lie isomorphisms of derived rings of simple rings
Richard A.
Howland
383-396
Some spectral sequences associated with fibrations
Eldon
Dyer;
Daniel S.
Kahn
397-437
On the differentiability of arbitrary real-valued set functions. I.
Harvel
Wright;
W. S.
Snyder
439-454
Free modules over free algebras and free group algebras: The Schreier technique
Jacques
Lewin
455-465
General Wiener-Hopf operators
Allen
Devinatz;
Marvin
Shinbrot
467-494
Maximal theorems for some orthogonal series. I
John E.
Gilbert
495-515
Ambient surgery and tangential homotopy quaternionic projective spaces.
Douglas N.
Hertz
517-545
Uniformly reflexive structures: On the nature of g\"odelizations and relative computability
Eric G.
Wagner
1-41
Abstract: In this paper we present an axiomatic theory within which much of the theory of computability can be developed in an abstract manner. The paper is based on the axiomatically defined concept of a Uniformly Reflexive Structure (U.R.S.). The axioms are chosen so as to capture what we view to be the essential properties of a ``gödelization'' of a set of functions on arbitrary infinite domain. It can be shown that (with a ``standard gödelization") both the partial recursive functions and the meta-recursive functions satisfy the axioms of U.R.S. In the first part of this paper, we define U.R.S. and develop the basic working theorems of the subject (e.g., analogues of the Kleene recursion theorems). The greater part of the paper is concerned with applying these basic results to (1) investigating the properties of gödelizations, and (2) developing an intrinsic theory of relative computability. The notion of relative computability which we develop is equivalent to Turing reducibility when applied to the partial recursive functions. Applied to appropriate U.R.S. on arbitrary domains, it provides an upper-semi-lattice ordering on the set of all functions (both total and partial) on that domain.
Henselization of a ring with respect to an ideal
Silvio
Greco
43-65
Smooth homology spheres and their fundamental groups
Michel A.
Kervaire
67-72
Local and global subordination theorems for vector-valued analytic functions
Mark
Finkelstein;
Robert
Whitley
73-76
Two notes on recursive functions and regressive isols
Joseph
Barback
77-94
Moduli for special Riemann surfaces of genus $2$
John
Schiller
95-113
On the closure of the bicyclic semigroup
Carl
Eberhart;
John
Selden
115-126
Nonsingular sections to Euclidean bundles
Kenneth C.
Millett
127-143
Deceptive convergence of Fourier series on ${\rm SU}(2)$
R. A.
Mayer
145-172
The Hauptvermutung for $3$-complexes
Edward M.
Brown
173-196
Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients
James S. W.
Wong
197-215
Lie algebras of characteristic $2$
Ronald C.
Hamelink
217-233
\v Ceby\v sev sets in Hilbert space
Edgar
Asplund
235-240
Bounded analytic functions on domains of infinite connectivity
Lawrence
Zalcman
241-269
The dual Poisson-Laguerre transform
Frank M.
Cholewinski;
Deborah Tepper
Haimo
271-300
Chain conditions in the distributive free product of lattices
G.
Grätzer;
H.
Lakser
301-312
Univalent functions with univalent derivatives. II
S. M.
Shah;
S. Y.
Trimble
313-320
The Poisson kernels and the Cauchy problem for elliptic equations with analytic coefficients
Yoshio
Katô
321-331
The spectrum of an operator on an interpolation space
James D.
Stafney
333-349
Topologically stationary locally compact groups and amenability
James C. S.
Wong
351-363
Heegaard splittings and splitting homomorphisms
William
Jaco
365-379
Two subspaces
P. R.
Halmos
381-389
Building Cartesian products of surfaces with $[0,\,1]$
Robert
Craggs
391-425
Families of valuations and semigroups of fractionary ideal classes.
Elbert M.
Pirtle
427-439
Homotopy invariants in differential geometry. I
Tadashi
Nagano
441-455
Induced and produced representations of Lie algebras
Robert J.
Blattner
457-474
The cohomology of transitive filtered modules. I. The first cohomology group
Charles
Freifeld
475-491
Systems of singular integral operators on spheres
Daniel A.
Levine
493-522
The motion of a large particle
Richard
Holley
523-534
The automorphism group of a homogeneous almost complex manifold
Joseph A.
Wolf
535-543
On the nullity and enclosure genus of wild knots
James M.
McPherson
545-555
Coherent sheaves on bordered Riemann surfaces
Martin
Jurchescu
557-563
Correction and addendum to: ``On algebras of finite representation type''
Spencer E.
Dickson
565-568
Addendum to: ``On some algebraic properties of the Bessel polynomials''
Emil
Grosswald
569-570
Correction to: ``Orthogonal representations of algebraic groups''
Frank
Grosshans
571
Solvable primitive permutation groups of low rank
David A.
Foulser
1-54
Analytic and entire vectors for representations of Lie groups
Roe W.
Goodman
55-76
Analytic sheaf cohomology groups of dimension $n$ of $n-dimensional$ $complex$ $spaces.$
Yum-tong
Siu
77-94
Analytic functions with quasi-analytic boundary values
William A.
Groening
95-107
Families of arcs in $E\sp{3}$
R. B.
Sher
109-116
Directed Banach spaces of affine functions
Leonard
Asimow
117-132
Semisimple bands
B. D.
Arendt
133-143
Piercing points of crumpled cubes
L. D.
Loveland
145-152
Some function-theoretic aspects of disconjugacy of linear-differential systems
Meira
Lavie
153-171
Brownian local times and taboo processes
Frank B.
Knight
173-185
Constructive techniques in rational approximation
T. W.
Gamelin;
J.
Garnett
187-200
Generic properties of polynomial vector fields at infinity
Enrique A.
González Velasco
201-222
Some tameness conditions involving singular disks
Warren
White
223-234
Diagonal nuclear operators on $l\sb{p}$ spaces
Alfred
Tong
235-247
On the minimal property of the Fourier projection
E. W.
Cheney;
C. R.
Hobby;
P. D.
Morris;
F.
Schurer;
D. E.
Wulbert
249-258
Mapping cylinder neighborhoods
Victor
Nicholson
259-268
Harmonic analysis of local times and sample functions of Gaussian processes
Simeon M.
Berman
269-281
Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries. III
Charles
Goldstein
283-301
Representations of semigroups and the translational hull of a regular Rees matrix semigroup
Mario
Petrich
303-318
Hypergeometric operator series and related partial differential equations
L. R.
Bragg
319-336
On Bazilevi\v c and convex functions
Mamoru
Nunokawa
337-341
Zeros of entire functions
Hans-J.
Runckel
343-362
Characterizations of functions with higher order derivatives in $\mathcal{L}_p$
J. W.
Jerome;
L. L.
Schumaker
363-371
On the Hauptvermutung for a class of open manifolds
A. J.
Machusko
373-392
Ascoli theorems and the exponential map
N.
Noble
393-411
Manifold phenomena in the theory of polyhedra
Ethan
Akin
413-473
On a coefficient problem in univalent functions
Dean
Phelps
475-485
${\rm UV}$ properties of compact sets
Steve
Armentrout
487-498
Homotopy properties of decomposition spaces
Steve
Armentrout
499-507
Polynomials of interpolation and approximation to meromorphic functions
E. B.
Saff
509-522
Self-equivalences of $S\sp{n}\times S\sp{k}$
Jerome
Levine
523-543
Uniform approximation on a real-analytic manifold
Michael
Freeman
545-553
A lower bound for the $\Delta $-Nielsen number
Robin B. S.
Brooks;
Robert F.
Brown
555-564
On the differentiability structure of real functions
A. M.
Bruckner;
J. G.
Ceder;
Max L.
Weiss
1-13
Approximation theorems on differentiable submanifolds of a complex manifold
Ricardo
Nirenberg;
R. O.
Wells
15-35
On a reflective subcategory of the category of all topological spaces.
Ladislav
Skula
37-41
Prime ideal structure in commutative rings
M.
Hochster
43-60
Cardinal algebras and measures invariant under equivalence relations.
Rolando
Chuaqui
61-79
A generalization of Bernstein's theorem and a differential inversion formula
W. J.
Studden
81-92
On the elementary ideals of link modules
R. H.
Crowell;
D.
Strauss
93-109
Hierarchies of effective descriptive set theory
Peter G.
Hinman
111-140
Factorization in Banach algebras and the general strict topology
F. Dennis
Sentilles;
Donald Curtis
Taylor
141-152
A variational problem related to an optimal filter problem with self-correlated noise
Leonard D.
Berkovitz;
Harry
Pollard
153-175
Manifolds in which the Poincar\'e conjecture is true
Jonathan L.
Gross
177-189
A unique decomposition theorem for $3$-manifolds with connected boundary
Jonathan L.
Gross
191-199
On detecting open collars
L. C.
Siebenmann
201-227
Relationships between reducibilities
Carl G.
Jockusch
229-237
Rearrangements of series of functions
Vladimir
Drobot
239-248
On the Marx conjecture for starlike functions of order $\alpha $
Renate
McLaughlin
249-256
On a class of stochastic processes with two states and continuous time parameter
Dudley Paul
Johnson
257-267
Homeomorphic continuous curves in $2$-space are isotopic in $3$-space.
W. K.
Mason
269-290
On the minimum computation time of functions
Stephen A.
Cook;
Stȧl O.
Aanderaa
291-314
Factors of infinite-dimensional manifolds
R. D.
Anderson;
R.
Schori
315-330
A generalization of the Ahlfors-Heins theorem
Matts
Essén
331-344
On the extension of partial orders on semigroups of right quotients.
Hanns Joachim
Weinert
345-353
Structure in simplexes. III. Composition series
Edward G.
Effros;
Alan
Gleit
355-379
Measure-theoretic uniformity in recursion theory and set theory
Gerald E.
Sacks
381-420
Reduction theorems for relative Grothendieck rings
T.-Y.
Lam;
I.
Reiner
421-435
Lie isomorphisms of prime rings
Wallace S.
Martindale
437-455
Maximal orders over regular local rings of dimension two
Mark
Ramras
457-479
Sur les th\'eories du premier ordre cat\'egoriques en un cardinal
J. P.
Ressayre
481-505
Groups with the same lower central sequence as a relatively free group. II. Properties
Gilbert
Baumslag
507-538
Indirect abelian theorems and a linear Volterra equation
Kenneth B.
Hannsgen
539-555
Correction to: ``The Stefan problem in several space variables''
Avner
Friedman
557
Decidability of second-order theories and automata on infinite trees.
Michael O.
Rabin
1-35
A recursion-theoretic characterization of the ramified analytical hierarchy
Richard
Boyd;
Gustav
Hensel;
Hilary
Putnam
37-62
Aligning functions defined on Cantor sets
Jo
Ford;
E. S.
Thomas
63-69
$m$-bounded uniformities between two given uniformities
Ellen E.
Reed;
W. J.
Thron
71-77
Approximation by rational and meromorphic functions having a bounded number of free poles
E. B.
Saff
79-92
On the distribution of elements belonging to certain subgroups of algebraic numbers
Carole
Sirovich
93-98
Endomorphism rings generated by units
Paul
Hill
99-105
Boundary functions and sets of curvilinear convergence for continuous functions
T. J.
Kaczynski
107-125
Sets of uniqueness for the vibrating string problem
Victor L.
Shapiro
127-146
Entire functions with prescribed values at discrete point sets
L. D.
Neidleman
147-157
Abstract evolution equations
Jerome A.
Goldstein
159-185
A uniqueness theorem for Haar and Walsh series
William R.
Wade
187-194
On theta functions and Weil's generalized Poisson summation formula.
Jun-ichi
Hano
195-210
Extensions of group representations over nonalgebraically closed fields
I. M.
Isaacs
211-228
Some theorems on hopficity
R.
Hirshon
229-244
Infinite primes and unique factorization in a principal right ideal domain
Raymond A.
Beauregard
245-253
Properties of $H\sp{p}$ $(0<p<1)$ and its continuing Banach space
P. L.
Duren;
A. L.
Shields
255-262
Free surfaces in $S\sp{3}$
John
Hempel
263-270
The folded ribbon theorem. A contribution to the study of immersed circles
George K.
Francis
271-303
The approximation of one-one measurable transformations by diffeomorphisms
H. E.
White
305-322
On the Frattini subgroup
Alice
Whittemore
323-333
Generalized cohomology operations and $H$-spaces of low rank
J. R.
Hubbuck
335-360
Absolute gap-sheaves and extensions of coherent analytic sheaves
Yum-tong
Siu
361-376
A commutative local ring with finite global dimension and zero divisors
B. L.
Osofsky
377-385
A new notion of semicharacters
Dennison R.
Brown;
Michael
Friedberg
387-401
On inflation-restriction exact sequences in group and Amitsur cohomology
A. J.
Berkson;
Alan
McConnell
403-413
A class of integer valued entire functions
Afton H.
Cayford
415-432
Decompositions into compact sets with $UV$ properties
Steve
Armentrout;
Thomas M.
Price
433-442
Some more results on rates of convergence in the law of large numbers for weighted sums of independent random variables
D. L.
Hanson;
F. T.
Wright
443-464
Vector cross products on manifolds
Alfred
Gray
465-504
Extensions of abstract valued set functions
James E.
Huneycutt
505-513
The Stone space of the saturated Boolean algebras
S.
Negrepontis
515-527
Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem ${\rm mod} 2$
William A.
Veech
1-33
On the convergence of Poisson integrals
E. M.
Stein;
N. J.
Weiss
35-54
On the deformation of commutative algebras
David W.
Knudson
55-70
A class of ``central limit theorems'' for convolution products of generalized functions
Reuben
Hersh
71-85
Projections of zero-sets (and the fine uniformity on a product)
Anthony W.
Hager
87-94
Topological conjugacy of affine transformations of compact abelian groups
Peter
Walters
95-107
An application of Holmgren's theorem and convexity with respect to differential operators with flat characteristic cones
E. C.
Zachmanoglou
109-115
A representation for a class of lattice ordered groups
F. D.
Pedersen
117-126
$n$-aposyndetic continua and cutting theorems
Eldon Jon
Vought
127-135
On an analytical decomposition of the Poisson law
R. G.
Laha
137-148
A very general theorem on systems of distinct representatives
Richard A.
Brualdi
149-160
Geometric groups and Whitehead torsion
E. H.
Connell;
John
Hollingsworth
161-181
Module homomorphisms of a von Neumann algebra into its center
Herbert
Halpern
183-193
Irreducible module homomorphisms of a von Neumann algebra into its center
Herbert
Halpern
195-221
On compact topological lattices of finite dimension
Tae Ho
Choe
223-237
Categorical homotopy and fibrations
F.-W.
Bauer;
J.
Dugundji
239-256
A generalization of Hall-complementation in finite supersolvable groups
Homer
Bechtell
257-270
Recursive theory and Dedekind cuts
Robert I.
Soare
271-294
Decompositions of $S\sp{3}$ and pseudo-isotopies
T. M.
Price
295-299
Multipliers of trigonometric series and pointwise convergence
N. M.
Rivière;
Y.
Sagher
301-308
Locally tauberian theorems for meromorphic functions of lower order less than one
Albert
Edrei
309-332
Asymptotic behavior of meromorphic functions with extremal deficiencies
Allen
Weitsman
333-352
Compactness of the Neumann-Poincar\'e operator
E. J.
Specht;
H. T.
Jones
353-366
Isomorphism and approximation of general state Markov processes
Richard
Isaac
367-380
Products with closed projections
N.
Noble
381-391
Boundary behavior of Poisson integrals on symmetric spaces
Adam
Korányi
393-409
Functions whose derivatives at one point form a finite set
L. D.
Neidleman;
E. G.
Straus
411-422
Pairs of inner functions on finite Riemann surfaces
Walter
Rudin
423-434
Modular representations of split $BN$ pairs
Forrest
Richen
435-460
On the coefficient problem for bounded univalent functions
Menahem
Schiffer;
Olli
Tammi
461-474
Some restricted partition functions: Congruences modulo $7$
D. B.
Lahiri
475-484
A commutativity theorem for unbounded operators in Hilbert space
A. E.
Nussbaum
485-491
Recursive density types. I
H.
Gonshor;
G.
Rice
493-503
Recursive density types. II
H.
Gonshor
505-509
Additions and corrections to ``On a convexity condition in normed linear spaces''
Daniel P.
Giesy
511-512
A characterization of the finite projective symplectic groups ${\rm PSp}\sb{4}(q)$
W. J.
Wong
1-35
On hypersingular integrals and Lebesgue spaces of differentiable functions. II
Richard L.
Wheeden
37-53
Commutators in properly infinite von Neumann algebras
Herbert
Halpern
55-73
A Radon-Nikodym theorm for Stone algebra valued measures
J. D. Maitland
Wright
75-94
A priori estimates for continuation problems for elliptic and principally normal differential equations
Paul E.
Saylor
95-108
On $ABA$-groups of finite order
Martin M.
Guterman
109-143
Locally compact rings having a topologically nilpotent unit
Seth
Warner
145-154
On homotopy spheres bounding highly connected manifolds
Douglas R.
Anderson
155-161
A note of correction to a theorem of W. E. Bonnice and R. J. Silverman.
Ting On
To
163-166
The problem of integration in finite terms
Robert H.
Risch
167-189
Higher homotopy-commutativity
Francis D.
Williams
191-206
Regular neighbourhoods and the stable range
W. B. R.
Lickorish;
L. C.
Siebenmann
207-230
Poisson integrals for Hermite and Laguerre expansions
Benjamin
Muckenhoupt
231-242
Hermite conjugate expansions
Benjamin
Muckenhoupt
243-260
A solution to the splitting mixed group problem of Baer
Phillip
Griffith
261-269
On the Fourier transform of the indicator function of a planar set.
Burton
Randol
271-278
On the asymptotic behavior of the Fourier transform of the indicator function of a convex set
Burton
Randol
279-285
The radical of the row-finite matrices over an arbitrary ring
N. E.
Sexauer;
J. E.
Warnock
287-295
A result on the Weil zeta function
Saul
Lubkin
297-300
Probabilistic treatment of the blowing up of solutions for a nonlinear integral equation
Masao
Nagasawa;
Tunekiti
Sirao
301-310
Differentials and matrices on Riemann surfaces
Joseph
Lewittes
311-318
Sturmian theorems and positive resolvents
Kurt
Kreith
319-327
A rate of convergence for the von Mises statistic
Walter A.
Rosenkrantz
329-337
Oscillation theorems for systems of linear differential equations
Zeev
Nehari
339-347
Some stability conditions for a nonlinear differential equation
Don
Hinton
349-358
Eigenvalue theorems in topological transformation groups
Harvey B.
Keynes;
James B.
Robertson
359-369
On common fixed points of commuting continuous functions on an interval
John Philip
Huneke
371-381
Mountain climbing
John Philip
Huneke
383-391
Irreducible matrix representations of finite semigroups
Gérard
Lallement;
Mario
Petrich
393-412
Smoothing derivatives of functions and applications
F. Wesley
Wilson
413-428
A compatibility condition between invariant riemannian metrics and Levi-Whitehead decompositions on a coset space
Joseph A.
Wolf
429-442
Gradients of convex functions
E.
Asplund;
R. T.
Rockafellar
443-467
Wave operators and similarity for generators of semigroups in Banach spaces
Sue-chin
Lin
469-494
The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras
Kevin
McCrimmon
495-510
Correction to ``Measures on product spaces''
E. O.
Elliott
511
Random walks. I
Donald S.
Ornstein
1-43
Random walks. II
Donald S.
Ornstein
45-60
Extensions of D. Jackson's theorem on best complex polynomial mean approximations
J. L.
Walsh;
E. B.
Saff
61-69
On the metamathematics of rings and integral domains
Bruno
Scarpellini
71-96
Vector bundles over orbit manifolds
Denis
Sjerve
97-106
A variational method for functions of bounded boundary rotation
Bernard
Pinchuk
107-113
Minimal first countable topologies
R. M.
Stephenson
115-127
Monotonicity of solutions of Volterra integral equations in Banach space
Avner
Friedman
129-148
Extreme positive operators on $C(X)$ which commute with given operators
George Arthur
Converse
149-158
On $H$-closed and minimal Hausdorff spaces
Jack
Porter;
John
Thomas
159-170
Bounded approximation by polynomials whose zeros lie on a circle
Charles Kam-tai
Chui
171-182
Local mapping relations and global implicit function theorems
Werner C.
Rheinboldt
183-198
Hermitian forms. I
H. P.
Allen
199-210
On quasiconformal mappings which keep the boundary points fixed
Edgar
Reich;
Kurt
Strebel
211-222
Free differentiable actions of $S\sp{1}$ and $S\sp{3}$ on homotopy $11$-spheres
Hsu-tung
Ku;
Mei-chin
Ku
223-228
On Schwarz's lemma and inner functions
Stephen D.
Fisher
229-240
Flat regular quotient rings
Vasily C.
Cateforis
241-249
The homology of symmetric products
R. James
Milgram
251-265
The tensor product of semigroups
Pierre-Antoine
Grillet
267-280
The tensor product of commutative semigroups
Pierre-Antoine
Grillet
281-293
Solving sequential conditions by finite-state strategies
J. Richard
Büchi;
Lawrence H.
Landweber
295-311
On the dimension theory of overrings of an integral domain
Jimmy T.
Arnold
313-326
Modules over polydisc algebras
Walter
Rudin;
E. L.
Stout
327-342
On the Wedderburn principal theorem
D. J.
Rodabaugh
343-361
Extremal structure of well-capped convex sets
L.
Asimow
363-375
$K\sb{1}$ of some abelian categories
Leslie G.
Roberts
377-382
Uniqueness theorems for convolution-type equations
J. S.
Byrnes;
D. J.
Newman
383-397
Extremal quasiconformal mappings with prescribed boundary values
Richard S.
Hamilton
399-406
Locally weakly flat spaces
O. G.
Harrold;
C. L.
Seebeck
407-414
Straightening-out and semirigidity in associative algebras
Francis J.
Flanigan
415-425
Abstract first order computability. I, II
Yiannis N.
Moschovakis
427-464
Abstract first order computability. II
Yiannis N.
Moschovakis
465-504
On finitely generated flat modules
Wolmer V.
Vasconcelos
505-512
On the theory of linear partial differential operators with analytic coefficients
François
Trèves
1-20
On handle decompositions and diffeomorphisms
Max K.
Agoston
21-26
Relative knots and tangential equivalences
Max K.
Agoston
27-52
Functional analytic properties of extremely amenable semigroups
Edmond E.
Granirer
53-75
Commuting functions with no common fixed point
William M.
Boyce
77-92
Transitivity of the automorphisms of certain geometric structures
William M.
Boothby
93-100
On algebraic extensions and order-preserving isomorphisms of certain partially ordered fields
Robert K.
Wright
101-114
Some applications of an inequality in locally convex spaces
C. W.
McArthur;
J. R.
Retherford
115-123
$A$-transforms and Hilbert functions in local lattices
E. W.
Johnson
125-139
Necessary conditions for embedding rings into fields
Abraham A.
Klein
141-151
Haar series
James R.
McLaughlin
153-176
Systems of difference equations with general homogeneous boundary conditions
Stanley
Osher
177-201
Boundary functions for bounded harmonic functions
T. J.
Kaczynski
203-209
The $C\sp{\ast} $-algebra generated by an isometry. II
L. A.
Coburn
211-217
The proof that a game may not have a solution
William F.
Lucas
219-229
Concerning cellular decompositions of $3$-manifolds with boundary
Steve
Armentrout
231-236
Weak wave operators for the nonlinear wave equation
A. Robert
Brodsky
237-244
Ideals in Chevalley algebras
James F.
Hurley
245-258
Lie algebras of characteristic three with nondegenerate Killing form.
Gordon
Brown
259-268
The automorphism groups of almost Hermitian manifolds
Shûkichi
Tanno
269-275
Local times and sample function properties of stationary Gaussian processes
Simeon M.
Berman
277-299
A note on the connectivity of components of Kleinian groups
Charles
Kalme
301-307
Complete models of differential fields
Peter
Blum
309-325
Centralizers in free associative algebras
George M.
Bergman
327-344
Identities involving the coefficients of a class of Dirichlet series. I
Bruce C.
Berndt
345-359
Identities involving the coefficients of a class of Dirichlet series. II
Bruce C.
Berndt
361-374
Extreme invariant positive operators
George
Converse;
Isaac
Namioka;
R. R.
Phelps
375-385
Harmonic functions and their conjugates on Hilbert space
Michael J.
Fisher
387-405
Pre-self-injective rings
G. B.
Klatt;
L. S.
Levy
407-419
Continuous characters of compact semigroups
Ronald
Fulp
421-430
On symmetric neighborhood systems in metric, strongly paracompact and some other types of spaces
Margaret Reames
Wiscamb
431-450
Triangulations of the $3$-ball with knotted spanning $1$-simplexes and collapsible $r$th derived subdivisions
W. B. R.
Lickorish;
J. M.
Martin
451-458
Complemented $B\sp{\ast} $-algebras
F. E.
Alexander;
B. J.
Tomiuk
459-480
A cohomological description of abelian Galois extensions
Morris
Orzech
481-499
Functions of exponential type not vanishing in a half-plane and related polynomials
N. K.
Govil;
Q. I.
Rahman
501-517
Othogonal representations of algebraic groups
Frank
Grosshans
519-531
Characteristic classes for modules over groups. I
L. S.
Charlap;
A. T.
Vasquez
533-549
On topological equivalence of $\aleph \sb{0}$-dimensional linear spaces
Raymond Y. T.
Wong
551-560
Random linear functionals
R. M.
Dudley
1-24
Compactification of strongly countable-dimensional spaces
Arlo W.
Schurle
25-32
$L\sp{p}$ estimates for integral transforms
Robert S.
Strichartz
33-50
Operator-stable probability distributions on vector groups
Michael
Sharpe
51-65
Problems involving diagonal products in nonnegative matrices
Richard
Sinkhorn;
Paul
Knopp
67-75
Multiplicities and Grothendieck groups
Marshall
Fraser
77-92
Picard-Lefschetz theorem for families of nonsingular algebraic varieties acquiring ordinary singularities
C. H.
Clemens
93-108
Convergence of Poisson integrals on generalized upper half-planes
Norman J.
Weiss
109-123
Prediction theory over discrete Abelian groups
D. M.
Eaves
125-137
Commutative group algebras
Warren
May
139-149
Chebyshev approximation by families with the betweeness property
Charles B.
Dunham
151-157
Existence theorems for optimal problems with vector-valued cost function
Czesław
Olech
159-180
Induced representations of Lie algebras and a theorem of Borel-Weil.
Nolan R.
Wallach
181-187
A general theory of relative regular neighborhoods
Marshall M.
Cohen
189-229
Completeness of $\alpha \sb{n}\,{\rm cos}\,nx+\beta \sb{n} {\rm sin} nx$
Robert P.
Feinerman;
Donald J.
Newman
231-245
Quantitative polynomial approximation on certain planar sets
D. J.
Newman;
L.
Raymon
247-259
A representation theory for prime and implicative semilattices
Raymond
Balbes
261-267
The theory of infinitely divisible matrices and kernels
Roger A.
Horn
269-286
Infinitely divisible positive definite sequences
Roger A.
Horn
287-303
The initial value problem for nonlinear wave equations in Hilbert space
Luiz Adauto
Medeiros
305-327
Self-intersections of some immersed manifolds
John G.
Miller
329-338
On the product of $F$-spaces
Stelios
Negrepontis
339-346
Irregularities of distribution. II
Wolfgang M.
Schmidt
347-360
Norm decreasing homomorphisms of measure algebras
Roger
Rigelhof
361-371
A characterization of $M(G)$
Roger
Rigelhof
373-379
A weak projection of $C$ onto a Euclidean subspace
Edward
Silverman
381-390
Secondary characteristic classes in $K$-theory
Connor
Lazarov
391-412
On the potential operator for one-dimensional recurrent random walks.
Charles J.
Stone
413-426
Asymptotic behavior of the convolution of a pair of measures
Charles J.
Stone
427-445
On two dimensional Markov processes with branching property
Shinzo
Watanabe
447-466
Tensor products of Banach algebras with involution
Kjeld B.
Laursen
467-487
The cohomology ring of a smooth manifold
James B.
Carrell
489-498
Reflectors as compositions of epi-reflectors
S.
Baron
499-508
Compact holomorphically convex subsets of a Stein manifold
Reese
Harvey;
R. O.
Wells
509-516
Uniqueness of the Cauchy problem for linear partial differential equations with variable coefficients
E. C.
Zachmanoglou
517-526
On the number of singularities of an analytic curve
Henry B.
Laufer
527-535
Permutational products and regular products of groups
R. B. J. T.
Allenby
537-543
Addendum to: ``On the strong law of large numbers and the central limit theorem for martingales''
Miklós
Csörgő
545
Addendum to ``On the distribution of values of meromorphic functions''
Fred
Gross
547
Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries. I
Charles Irwin
Goldstein
1-31
Eigenfunction expansions associated with the Laplacian for certain domains with infinite boundaries. II. Applications to scattering theory
Charles Irwin
Goldstein
33-50
Local triviality of Hurewicz fiber maps
Soon-kyu
Kim
51-67
On the regularity of generalized convex functions
J. H. B.
Kemperman
69-93
Quasiconformal structures and the metrization of $2$-manifolds
Raymond J.
Cannon
95-103
$L$-subalgebras of $M(G)$
Joseph L.
Taylor
105-113
The first hitting distribution of a sphere for symmetric stable processes
Sidney C.
Port
115-125
On algebras of finite representation type
Spencer E.
Dickson
127-141
Complex-valued stable measures and their domains of attraction
Ludwig
Arnold;
Johannes
Michalicek
143-158
The Poincar\'e-Bendixson theorem for the Klein bottle
Nelson G.
Markley
159-165
The uniqueness of Hermite series under Poisson-Abel summability
Lawrence S.
Kroll
167-192
A characterization of the Riesz space of measurable functions
J. J.
Masterson
193-197
An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite $w\sp{\ast} $-algebras
Donald
Bures
199-212
Symmetries of links
W. C.
Whitten
213-222
Invariant measures and Radon-Nikodym derivatives
Dorothy
Maharam
223-248
On the degrees of index sets. II
C. E. M.
Yates
249-266
Dependence in rings. II. The dependence number
P. M.
Cohn
267-279
Some inequalities concerning functions of exponential type
Q. I.
Rahman
281-293
Functions of exponential type
Q. I.
Rahman
295-309
Basic representations of arbitrary semigroups
D. B.
McAlister
311-342
On the functional equations involving nonlinear generalized $P$-compact operators
W. V.
Petryshyn;
T. S.
Tucker
343-373
Noether-Lasker decomposition of coherent analytic subsheaves
Yum-tong
Siu
375-385
A spectral theory for inner functions
Malcolm J.
Sherman
387-398
On the existence of good stationary strategies
William D.
Sudderth
399-414
Derivations and automorphisms of $L\sp{1}\,(0,\,1)$
Herbert
Kamowitz;
Stephen
Scheinberg
415-427
Convex and starlike univalent functions
S. D.
Bernardi
429-446
On the zeta function of biprojective complete intersections
Jacob
Barshay
447-458
On the set of subsequential limit points of successive approximations.
J. B.
Diaz;
F. T.
Metcalf
459-485
Disconjugacy of complex differential systems and equations
David
London;
Binyamin
Schwarz
487-505
Harmonic functions on Hermitian hyperbolic space
Adam
Korányi
507-516
Free $\Sigma $-structures
G.
Grätzer
517-542
On the Littlewood-Paley $g$-function and the Lusin $s$-function
George
Gasper
385-403
An almost everywhere direct power
Philip
Olin
405-420
On hypersingular integrals and Lebesgue spaces of differentiable functions
Richard L.
Wheeden
421-435
Pseudocompact spaces
R. M.
Stephenson
437-448
Convergent transfer schemes for $N$-person games
R. E.
Stearns
449-459
On the equation $z\sb{1}{}\sp{n}z\sb{2}{}\sp{n}\cdots z\sb{k}{}\sp{n}=y\sp{n}$ in a free semigroup
K. I.
Appel;
F. M.
Djorup
461-470
Measurable multivalued mappings and Lusin's theorem
Marc Q.
Jacobs
471-481
On Jacobi sums of certain composite orders
Joseph B.
Muskat
483-502
Spaces of affine continuous functions on simplexes
Aldo J.
Lazar
503-525
Analytic canonical forms for nonlinear difference equations
Jon W.
Tolle
527-544
Geometric dimension of vector bundles over lens spaces
Denis
Sjerve
545-557
On hamiltonian line-graphs
Gary
Chartrand
559-566
Algebraic models for measure preserving transformations
N.
Dinculeanu;
C.
Foiaş
215-237
Real Borel cohomology of locally compact groups
Arthur Mason
DuPre
239-260
The structure of group-like extensions of minimal sets
Robert
Ellis
261-287
The \v Silov boundary of $M(G)$
B. E.
Johnson
289-296
On polynomials in self-adjoint operators in spaces with an indefinite metric
C.-Y.
Lo
297-304
Measurable functions and spherical summability of multiple Fourier series
Roger
Cooke
305-313
The decomposability of torsion free modules of finite rank
Eben
Matlis
315-324
An identity for elliptic equations with applications
C. A.
Swanson
325-333
Continuity of metric projections
D. E.
Wulbert
335-341
The structure of ${\rm QF}-3$ rings
Kent R.
Fuller
343-354
Regular boundary value conditions for ordinary differential operators.
Stephen
Salaff
355-373
Homology comodules
Arunas
Liulevicius
375-382
Errata: ``The theory of operations on binary relations''
T.
Tamura
383
The Riemann sphere of a commutative Banach algebra
Barnett W.
Glickfeld
1-28
H\"older conditions for Gaussian processes with stationary increments
M. B.
Marcus
29-52
On disconjugate differential equations
Philip
Hartman
53-70
On primitive elements in differentially algebraic extension fields
A.
Babakhanian
71-83
On minimal blocks
Michael D.
Plummer
85-94
On the coefficient problem for univalent functions
Menahem
Schiffer
95-101
A hierarchy based on a type two object
J. R.
Shoenfield
103-108
Hankel forms, Toeplitz forms and meromorphic functions
Douglas N.
Clark
109-116
Nonequality of dimensions for metric spaces
Prabir
Roy
117-132
Maximal separation theorems for convex sets
Victor
Klee
133-147
Inductive ring topologies
John O.
Kiltinen
149-169
On a similarity invariant for compact operators
Reese T.
Posser
171-181
A functional characterization of Tor for noetherian rings of global dimension $1$
David
Newell
183-192
Gaussian processes and Hammerstein integral equations
Martin
Pincus
193-214
Concerning cellular decompositions of $3$-manifolds that yield $3$-manifolds
Steve
Armentrout
307-332
Tauberian theorems and slowly varying functions
David
Drasin
333-356
A method for the construction of reflection laws for a parabolic equation
C. Denson
Hill
357-372
On uniform spaces with quasi-nested base
Elias
Zakon
373-384
Unbounded normal operators on Banach spaces
Theodore W.
Palmer
385-414
\v Cech extensions of contravariant functors
C. N.
Lee;
Frank
Raymond
415-434
Uniform distribution in locally compact Abelian groups
I. D.
Berg;
M.
Rajagopalan;
L. A.
Rubel
435-446
Cohomology and the classification of liftings
James C.
Becker
447-475
The order-continuous operators on $L\sb{p}$-spaces. I. General theory
Milton P.
Olson
477-504
The spectra of operators having resolvents of first-order growth
C. R.
Putnam
505-510
On the existence of minimal ideals in a Banach algebra
Bruce A.
Barnes
511-517
Limit of lattices in a Lie group
S. P.
Wang
519-526
Systems of Toeplitz operators on $H\sp{2}$. II
H. R.
Pousson
527-536
Spaces for which the Stone-Weierstrass theorem holds
R. M.
Stephenson
537-546
Cycle length in a random function
P. W.
Purdom;
J. H.
Williams
547-551
The integral representation ring $a(R\sb{k}G)$
T. A.
Hannula
553-559
The abstract time-dependent Cauchy problem
Matthew
Hackman
1-50
The Stefan problem in several space variables
Avner
Friedman
51-87
One dimensional Stefan problems with nonmonotone free boundary
Avner
Friedman
89-114
On finite groupoids and $\mathcal{K}$-prime algebras
Ralph
McKenzie
115-129
On the residual finiteness of generalized free products
Joan Landman
Dyer
131-143
Martingale integrals
P. Warwick
Millar
145-166
Lower bounds for approximation by nonlinear manifolds
Hugh E.
Warren
167-178
Ratio properties in locally compact amenable groups
William R.
Emerson
179-204
Cohomology of algebras over Hopf algebras
Moss Eisenberg
Sweedler
205-239
On quasi-elliptic boundary problems
Tadato
Matsuzawa
241-265
Measures on $F$-spaces
G. L.
Seever
267-280
A spectral decomposition for self-adjoint elements in the maximum ${\rm GCR}$ ideal of a von Neumann algebra with applications to noncommutative integration theory
Herbert
Halpern
281-306
On a class of stochastic processes and its relationship to infinite particle gases
D. Paul
Johnson
275-295
Irreducibility of polynomials with low absolute values
R. J.
Levit
297-305
On the boundary values of harmonic functions
Richard A.
Hunt;
Richard L.
Wheeden
307-322
The $C\sp{k}$-classification of certain operators in $L\sb{p}$
Shmuel
Kantorovitz
323-333
Some ring theoretic Schr\"oder-Bernstein theorems
Ian G.
Connell
335-351
On Bazilevi\v c functions
D. K.
Thomas
353-361
Uniform limit theorems for the maximum cummulative sum in probability
S.
Sawyer
363-367
Some spectral properties of an operator associated with a pair of nonnegative matrices
M. V.
Menon
369-375
Abstract: An operator--in general nonlinear--associated with a pair of non-negative matrices, is defined and some of its spectral properties studied. If the pair of matrices are a square matrix A and the identity matrix of the same order, the operator reduces to the linear operator A. The results obtained include generalizations of one of the principal conclusions of the theorem of Perron-Frobenius.
Generalizations of weakly compact operators
Richard H.
Herman
377-386
On the existence of immersions and submersions
Emery
Thomas
387-394
Finite groups whose $n$-maximal subgroups are subnormal
Avino’am
Mann
395-409
The Plancherel transform on the nilpotent part of $G\sb{2}$ and some applocations to the representation theory of $G\sb{2}$
Kenneth I.
Gross
411-446
Ergodic theory and boundaries
M. A.
Akcoglu;
R. W.
Sharpe
447-460
Vladimirov-type approximation of Wiener integrals of suitably dominated functionals
Henry C.
Finlayson
461-469
An analysis of the group operation in universal Teichm\"uller space
Fred
Gardiner
471-486
On the fractional differentiation of a function of several variables
G. V.
Welland
487-500
The integral cohomology rings of groups of order $p\sp{3}$
Gene
Lewis
501-529
Free actions on $S\sp{n}\times S\sp{n}$
Gene
Lewis
531-540
Polynomial bases for compact sets in the plane
Victor
Manjarrez
541-551
On Schreier varieties of linear algebras
Jacques
Lewin
553-562
Errata to: ``Convergence in measure and related results in finite rings of operators''
A. R.
Padmanabhan
563
On dimension of support for stochastic processes with independent increments
Howard H.
Stratton
1-29
Convergent higher derivations on local rings
Nickolas
Heerema
31-44
Inertial automorphisms of a class of wildly ramified $v$-rings
Nickolas
Heerema
45-54
On the corona theorem and its application to spectral problems in Hilbert space
Paul A.
Fuhrmann
55-66
Seminorms and the Egoroff property in Riesz spaces
John A. R.
Holbrook
67-77
Double centralizers and extensions of $C\sp{\ast} $-algebras
Robert C.
Busby
79-99
Bisimple inverse semi-groups
N. R.
Reilly
101-114
Embeddability of discrete time simple branching processes into continuous time branching processes
Samuel
Karlin;
James
McGregor
115-136
Embedding iterates of analytic functions with two fixed points into continuous groups
Samuel
Karlin;
James
McGregor
137-145
Extensions of uniformly continuous pseudometrics
T. E.
Gantner
147-157
Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients
A. E.
Hurd;
D. H.
Sattinger
159-174
A description of the topology on the dual spaces of certain locally compact groups
Lawrence
Baggett
175-215
Homological dimension and the continuum hypothesis
B. L.
Osofsky
217-230
On the ring of holomorphic functions on an open Riemann surface
Irwin
Kra
231-244
Holomorphic hulls and holomorphic convexity of differentiable submanifolds
R. O.
Wells
245-262
An infinite subalgebra of ${\rm Ext}\sb{A}\,(Z\sb{2},\,Z\sb{2})$
Mark
Mahowald;
Martin
Tangora
263-274
The word problem and consequences for the braid groups and mapping class groups of the $2$-sphere
Richard
Gillette;
James
Van Buskirk
277-296
Harmonic differential with prescribed singularities
Mitsuru
Nakai
297-302
The minimal boundary of $C(X)$
S. L.
Gulick
303-314
Injective hulls of $C\sp*$ algebras
Harry
Gonshor
315-322
The $K$-theory of a class of homogeneous spaces
Bruno
Harris
323-332
Secondary characteristic classes for $k$-special bundles
Robert E.
Mosher
333-344
The monotone union property of manifolds
Lawrence S.
Husch
345-355
Changes of variables near a periodic surface or invariant manifold
Al
Kelley
356-364
Representation of bilinear forms in Hilbert space by linear operators
Alan
McIntosh
365-377
Some new finite translation planes
F. W.
Wilke;
J. L.
Zemmer
378-397
On the number of automorphisms of a closed Riemann surface
Robert D. M.
Accola
398-408
Closure theorems for some discrete subgroups of $R\sp{k}$
R. P.
Gosselin
409-419
Semirecursive sets and positive reducibility
Carl G.
Jockusch
420-436
Singular integral operators over a Hilbert space
Michael J.
Fisher
437-465
The Radon-Nikodym theorem for the Bochner integral
M. A.
Rieffel
466-487
The characters of the finite symplectic group ${\rm Sp}(4,\,q)$
Bhama
Srinivasan
488-525
Recursive pseudo-well-orderings
Joseph
Harrison
526-543
The structure and ideal theory of the predual of a Banach lattice
E. B.
Davies
544-555
Correction to: ``Higher obstructions to sectioning a special type of fibre bundle''
W.-C.
Hsiang
556
Equivalence of Markov processes
Donald A.
Dawson
1-31
On Klein's combination theorem. II
Bernard
Maskit
32-39
Topological conjugacy of affine transformations of tori
Peter
Walters
40-50
Classification of the actions of the circle on $3$-manifolds
Frank
Raymond
51-78
The branch point structure of extensions of interior boundaries
Morris L.
Marx
79-98
Functions starlike of order $\alpha $
Melvyn
Klein
99-106
On the Hewitt realcompactification of a product space
W. W.
Comfort
107-118
Point derivations in certain sup-norm algebras
S. J.
Sidney
119-127
Properties of the sequence of closed powers of a maximal ideal in a sup-norm algebra
S. J.
Sidney
128-148
Primary ideals and valuation ideals. II
Robert
Gilmer;
William
Heinzer
149-162
Some properties of Laplace transforms of measures
O. S.
Rothaus
163-169
Conditions implying that a $2$-sphere is almost tame
L. D.
Loveland
170-181
Some continuity properites of Brownian motion with the time parameter in Hilbert space
Simeon M.
Berman
182-198
On the distribution of values of meromorphic functions
Fred
Gross
199-214
On factorization of meromorphic functions
Fred
Gross
215-222
Families of nonnegative divisors
Theodore J.
Barth
223-245
A Riesz representation theorem in the setting of locally convex spaces
Robert K.
Goodrich
246-258
On the strong law of large numbers and the central limit theorem for martingales
Miklós
Csörgő
259-275
$I$-bisimple semigroups
R. J.
Warne
367-386
Some point-set properties and the edge path group of a generalized uniform space
Robert W.
Deming
387-405
Approximation by bounded analytic functions: Uniform convergence as implied by mean convergence
J. L.
Walsh
406-413
Localization for Fourier series on ${\rm SU}(2)$
R.
Mayer
414-424
Dominant representing measures and rational approximation
I.
Glicksberg
425-462
Locally compact vector spaces and algebras over discrete fields
Seth
Warner
463-493
On the equation $n=p+x\sp{2}$
R. J.
Miech
494-512
Simple systems of roots in $L\sp{\ast} $-algebras
V. K.
Balachandran
513-524
Conjugate locally convex spaces. II
V.
Krishnamurthy
525-531
An extension of P\'olya's theorem on power series with integer coefficients
Raphael M.
Robinson
532-543
Extremal properties of weakly starlike $p$-valent functions
James A.
Hummel
544-551
A description of all globally idempotent threads with zero
Charles R.
Storey
185-207
Functors of Artin rings
Michael
Schlessinger
208-222
A characterization of the double point structure of the projection of a polygonal knot in regular position
L. B.
Treybig
223-247
Prime mappings
L. B.
Treybig
248-253
A theorem on Riesz groups
J. Roger
Teller
254-264
Schauder bases and K\"othe sequence spaces
Ed
Dubinsky;
J. R.
Retherford
265-280
On constructing distribution functions: With applications to Lommel polynomials and Bessel functions
Daniel
Maki
281-297
Approximation and Schoenflies extension of $C\sp{m}$-diffeomorphisms $(m\geq 0)$
Jerome L.
Paul
298-321
On the classification of transitive effective actions on Stiefel manifolds
Wu-yi
Hsiang;
J. C.
Su
322-336
Isomorphisms of simple Lie rings
Eugene A.
Klotz
337-352
A fixed-point theorem for inward and outward maps
Benjamin R.
Halpern;
George M.
Bergman
353-358
Components of invertible elements in quotient algebras of operators
L. A.
Coburn;
A.
Lebow
359-365
On the lattice of recursively enumerable sets
A. H.
Lachlan
1-37
Computable algebraic structures and nonstandard arithmetic
Eugene W.
Madison
38-54
On a necessary condition for the validity of the Riemann hypothesis for functions that generalize the Riemann zeta function
Ronald
Alter
55-74
Families of finite character and essential valuations
Malcolm
Griffin
75-85
Absolutely closed spaces
Chen-tung
Liu
86-104
The formal linearization of a semisimple Lie algebra of vector fields about a singular point
Robert
Hermann
105-109
Abstract: A classical theorem by Poincaré gives conditions that a nonlinear ordinary differential equation $\displaystyle dx/dt = A(x),$ with $A(0) = 0$ in n variables $x = ({x_1}, \ldots ,{x_n})$ can be reduced to a linear form $\displaystyle \frac{{dx'}}{{dt}} = \frac{{\partial A}}{{\partial x}}(0)x'$ by a change of variables $x' = f(x)$. A generalization is given for a finite set of such differential equations, which form a semisimple Lie algebra.
Remarks on a paper of Hermann
Victor W.
Guillemin;
Shlomo
Sternberg
110-116
Function algebras, means, and fixed points
Theodore
Mitchell
117-126
Invariant means and fixed points: A sequel to Mitchell's paper
L. N.
Argabright
127-130
Addition and reduction theorems for medial properties
R. L.
Wilder
131-140
Infinite-product Markov processes
T. E.
Harris
141-152
Irreducible Jordan algebras of self-adjoint operators
Erling
Størmer
153-166
The convergence of sequences of rational functions of best approximation. III
J. L.
Walsh
167-183
Correction to ``Measurable gambling houses''
Ralph E.
Strauch
184
Representation theory of central topological groups
Siegfried
Grosser;
Martin
Moskowitz
361-390
Complex cubic splines
J. H.
Ahlberg;
E. N.
Nilson;
J. L.
Walsh
391-413
A study of metric-dependent dimension functions
Keiô
Nagami;
J. H.
Roberts
414-435
Classes of semi-inner-product spaces
J. R.
Giles
436-446
On a piece of hypersurface in a Riemannian manifold with mean curvature bounded away from zero
Yoshie
Katsurada
447-457
The numerical solutions of the eigenvalue problem for compact integral operators
Kendall E.
Atkinson
458-465
Extremal problems in the class of close-to-convex functions
Bernard
Pinchuk
466-478
Abstract: The class K of normalized close-to-convex functions in $D = \{ z:\vert z\vert < 1\}$ has a parametric representation involving two Stieltjes integrals. Using a variational method due to G. M. Goluzin [2] for classes of analytic functions defined by a Stieltjes integral, variational formulas are developed for K. With these variational formulas, two general extremal problems within K are solved. The first problem is to maximize the functional
Subharmonic functions in the half-plane
Jun-iti
Itô
479-499
Disconjugate linear differential operators
Zeev
Nehari
500-516
Real characters and the radical of an abelian group
J. E.
Diem;
F. B.
Wright
517-529
$M$-semiregular subalgebras in hyperfinite factors
Sister Rita Jean
Tauer
530-541
On rates of convergence for the invariance principle
Walter A.
Rosenkrantz
542-552
Some means of convex bodies
W. J.
Firey
181-217
Potential theoretic analysis of a certain integral equation
Gunnar A.
Brosamler
218-248
Hyperanalytic predicates
Yiannis N.
Moschovakis
249-282
Almost everywhere convergence of Poisson integrals on tube domains over cones
Norman J.
Weiss
283-307
Groups with the same lower central sequence as a relatively free group. I. The groups
Gilbert
Baumslag
308-321
An algebraic theory of local knottedness. I
Samuel J.
Lomonaco
322-343
Factoring pointlike simplicial mappings
C. L.
Wiginton
344-359
A rigidity theorem for discrete subgroups
Howard
Garland
1-25
Sets of constant relative width and constant relative brightness
G. D.
Chakerian
26-37
The forcing method and the upper semilattice of hyperdegrees
S. K.
Thomason
38-57
Homological algebra and the Eilenberg-Moore spectral sequence
Larry
Smith
58-93
Cobordism operations and Hopf algebras
P. S.
Landweber
94-110
The exit characteristics of Markov processes with applications to continuous martingales in R$\sp{n}$
Gene
Denzel
111-123
Gibbs phenomenon for functions of two variables
Fred
Ustina
124-129
Dissipative ordinary differential operators of even order
A.
Olubummo
130-139
The construction of certain $0-$dimensional transformation groups
R. F.
Williams
140-156
Markov processes whose hitting distributions are dominated by those of a given process
Chung-tuo
Shih
157-179
Convergence in measure and related results in finite rings of operators
A. R.
Padmanabhan
359-378
Measures on product spaces
E. O.
Elliott
379-388
Abstract: The theory of regular conditional probability is generalized by replacing a probability measure by a (perhaps non-$\sigma $-finite) outer measure and a resulting measure is obtained on the product space. A Fubini-like theorem is obtained for the integrable functions of this measure and a condition is given for this measure to impart the topological properties of being inner regular and almost Lindelöf to the product space when the component spaces also have these topological properties. Thus some theorems for the Morse-Bledsoe product measure [1] are generalized by methods very similar to those used in their paper on product measures [1].
A study of the proximal relation in coset transformation groups
Harvey B.
Keynes
389-402
An existence analysis for nonlinear equations in Hilbert space
John
Locker
403-413
On Serre duality and envelopes of holomorphy
Henry B.
Laufer
414-436
Nonlinear approximation. II. Curvature in Minkowski geometry and local uniqueness
John R.
Rice
437-459
On the integral representation of positive linear functionals
A. E.
Nussbaum
460-473
Prime associator-dependent rings with idempotent
Nicholas J.
Sterling
474-481
Connectivity, divisibility, and torsion
Lewis C.
Robertson
482-505
Generic splitting fields of composition algebras
J. C.
Ferrar
506-514
Dual spaces of a vector lattice and its cut-completion
J. J.
Masterson
515-522
A criterion for rings of analytic functions
Ian
Richards
523-530
Two theorems on hyperhypersimple sets
Robert W.
Robinson
531-538
On fixed point properties of plane continua
Harold
Bell
539-548
Differentiable monotone maps on manifolds
P. T.
Church
185-205
Doubly-connected minimal surfaces
Paul M.
Bailyn
206-220
Generalization of Schwarz-Pick lemma to invariant volume in a K\"ahler manifold
K. T.
Hahn;
Josephine
Mitchell
221-231
Isotopy classes of imbeddings
C. W.
Patty
232-247
On a constructive definition of the restricted Denjoy integral
Donald W.
Solomon
248-256
Convolution measure algebras with group maximal ideal spaces
Joseph L.
Taylor
257-263
Outer derivations of Lie algebras
Shigeaki
Tôgô
264-276
A weak containment theorem for groups with a quotient $R$-group
Lawrence
Baggett
277-290
Cohomology of local group extensions
S.
Świerczkowski
291-320
On multipliers of $p$-integrable functions
Liang-shin
Hahn
321-335
Upper and lower bounds of the radii of gyration of convex bodies
Tsuan Wu
Ting
336-357
Distal functions on groups
A. W.
Knapp
1-40
The cotangent complex of a morphism
S.
Lichtenbaum;
M.
Schlessinger
41-70
Some new Hilbert algebras
R.
Keown
71-87
Continuity of linear operators commuting with continuous linear operators
B. E.
Johnson
88-102
Darboux transformations
A. M.
Bruckner;
J. B.
Bruckner
103-111
Semisimple maximal quotient rings
Francis L.
Sandomierski
112-120
On the alternating projections theorem and bivariate stationary stochastic processes
Habib
Salehi
121-134
Abstract: In this paper we shall first use the theorem of von Neumann on alternating projections to obtain an algorithm for finding the projection of an element x in a Hilbert space $\mathcal{H}$ onto the subspace spanned by $\mathcal{H}$-valued orthogonally scattered measures ${\xi _1}$ and ${\xi _2}$. We then specialize this algorithm to the case that ${\xi _1}$ and ${\xi _2}$ are the canonical measures of the components of a bivariate stationary stochastic process (SP), and thereby get an algorithm for finding the best linear predictor in the time domain.
On some improperly posed problems for quasilinear equations of mixed type
L. E.
Payne;
D.
Sather
135-141
On homogeneous spaces and reductive subalgebras of simple Lie algebras
A.
Sagle;
D. J.
Winter
142-147
On homeomorphisms of certain infinite dimensional spaces
Raymond Y. T.
Wong
148-154
Diffeomorphisms of $1$-connected manifolds
William
Browder
155-163
Operators commuting with Boolean algebras of projections of infinite multiplicity
L.
Tzafriri
164-175
On a class of nodal noncommutative Jordan algebras
Jerry I.
Goldman
176-183
Homological algebra in locally compact abelian groups
Martin
Moskowitz
361-404
Conjugating representations and related results on semi-simple Lie groups
J. M. G.
Fell
405-426
Elementary differences between the isols and the co-simple isols
Louise
Hay
427-441
Induced fibrations and cofibrations
T.
Ganea
442-459
Invariant differential operators
W.
Smoke
460-494
The realization of a semisimplicial bundle map is a $k$-bundle map
S.
Weingram
495-514
Hopf algebras with one grouplike element
Moss Eisenberg
Sweedler
515-526
Generically algebraic algebras
Kevin
McCrimmon
527-551
Generalized interpolation in $H\sp{\infty }$
Donald
Sarason
179-203
Two characterizations of compact local trees
G. E.
Dimitroff
204-220
Amenable groups and groups with the fixed point property
Neil W.
Rickert
221-232
On quasi-pseudoconformality in several complex variables
Ricardo
Nirenberg
233-240
Nonautonomous differential equations and topological dynamics. I. The basic theory
George R.
Sell
241-262
Nonautonomous differential equations and topological dynamics. II. Limiting equations
George R.
Sell
263-283
Automorphisms of a class of metabelian groups
S.
Bachmuth
284-293
Automorphisms of a class of metabelian groups. II
S.
Bachmuth;
H. Y.
Mochizuki
294-301
Stable subalgebras of Lie algebras and associative algebras
Stanley
Page;
R. W.
Richardson
302-312
Jordan algebras of self-adjoint operators
Edward G.
Effros;
Erling
Størmer
313-316
On central topological groups
Siegfried
Grosser;
Martin
Moskowitz
317-340
A decomposition of mixed abelian groups
John A.
Oppelt
341-348
A nonzero determinant related to Schur's matrix
Stephen
Salaff
349-355
Corrections: ``A sufficient condition for total monotonicity''
B. E.
Rhoades
356-360
On parabolic and umbilic points of immersed hypersurfaces
E. A.
Feldman
1-28
A Galois theory for noncommutative rings
H. F.
Kreimer
29-41
Galois theory for noncommutative rings and normal bases
H. F.
Kreimer
42-49
A nonconstructible $\Delta \sb{3}{}\sp{1}$ set of integers
Robert M.
Solovay
50-75
The generalized Pellian equation
Leon
Bernstein
76-89
Unstable homotopy of ${\rm O}(n)$
Gade V.
Krishnarao
90-97
On localization and domains of uniqueness
Roe W.
Goodman
98-106
Elliptic systems of singular integral operators. I. The half-space case
Eliahu
Shamir
107-124
Group-like extensions of minimal sets
Robert
Ellis
125-135
Global dimension of valuation rings
B. L.
Osofsky
136-149
The subgroups of ${\rm PSL}(3,\,q)$ for odd $q$
David M.
Bloom
150-178
Topological modules. Banach algebras, tensor products, algebras of kernels
Jesús
Gil de Lamadrid
361-419
Oscillating sequences modulo one
Underwood
Dudley
420-426
Some characterizations of systems of linear differential equations having regular singular solutions
Donald A.
Lutz
427-441
Some properties of $p(n)$ and $c(n)$ modulo powers of $13$
A. O. L.
Atkin;
J. N.
O’Brien
442-459
Extremal length and conformal capacity
William P.
Ziemer
460-473
The strict topology and compactness in the space of measures. II
John B.
Conway
474-486
On the theory of exponential groups
Lajos
Pukánszky
487-507
Simple-connectivity and the Browder-Novikov theorem
M.
Kervaire;
A.
Vasquez
508-513
Invariants of Pfaffian systems
Robert B.
Gardner
514-533
On analytic functions which have algebraic values at a convergent sequence of points
D. L.
Hilliker
534-550
Generalizations of the Riemann derivative
J. Marshall
Ash
181-199
Topological properties of the Hilbert cube and the infinite product of open intervals
R. D.
Anderson
200-216
A criterion for cellularity in a manifold. II
D. R.
McMillan
217-224
Locally compact topologies for groups
Neil W.
Rickert
225-235
Topologies with the Stone-Weierstrass property
Paul R.
Meyer
236-243
Spaces determined by their homeomorphism groups
E. S.
Thomas
244-250
On the point spectrum of a Toeplitz operator
Douglas N.
Clark
251-266
Classification of normal subgroups of the modular group
Morris
Newman
267-277
Comparison theorems for elliptic equations on unbounded domains
C. A.
Swanson
278-285
The dual space of an operator algebra
Charles A.
Akemann
286-302
Capacities of sets and harmonic analysis on the group $2\sp{\omega }$
L. H.
Harper
303-315
Changes of variables near a periodic orbit
Al
Kelley
316-334
On the entropy of uniquely ergodic transformations
Frank
Hahn;
Yitzhak
Katznelson
335-360
Oscillation theorems of arithmetical functions
Emil
Grosswald
1-28
Gaussian-Markov processes and a boundary value problem
J. A.
Beekman
29-42
Remarks on fixed point theorems and their extensions
W. V.
Petryshyn
43-53
Reduced Teichm\"uller spaces
Clifford J.
Earle
54-63
Measurable gambling houses
Ralph E.
Strauch
64-72
Maximal $R$-sets, Grassmann spaces, and Stiefel spaces of a Hilbert space
Erhard
Luft
73-107
Descendingly incomplete ultrafilters
C. C.
Chang
108-118
Some conditions for manifolds to be locally flat
C.
Lacher
119-130
Volterra integral equations in Banach space
Avner
Friedman;
Marvin
Shinbrot
131-179
Higher-order indecomposable isols
Alfred B.
Manaster
363-383
On the range of an invariant mean
E. E.
Granirer
384-394
Locally compact simple rings having minimal left ideals
Seth
Warner
395-405
Complexification of real analytic groups
G.
Hochschild
406-413
Rings in which certain subsets satisfy polynomial identities
Thomas P.
Kezlan
414-421
A reciprocity law for maximal fields
Ladnor
Geissinger
422-431
$q$-complete spaces and cohomology
Giuliano
Sorani;
Vinicio
Villani
432-448
Construction of isology functors
Arthur H.
Copeland
449-460
A mixing condition for extreme left invariant means
S. P.
Lloyd
461-481
Disconjugacy of complex differential systems
Binyamin
Schwarz
482-496
The Kleene hierarchy classification of recursively random sequences
D. W.
Loveland
497-510
The Fredholm method in potential theory
Josef
Král
511-547
Geometry of immersions. II
E. A.
Feldman
181-215
$P\sp{p}$-conjecture for locally compact groups. I
M.
Rajagopalan
216-222
On maximal congruences and finite semisimple semigroups
Robert H.
Oehmke
223-237
A sufficient condition for conjugacy of metric automorphisms
N. F. G.
Martin
238-249
Operators and harmonic analysis on the sphere
Charles F.
Dunkl
250-263
Noncommutative Markov processes
John
de Pillis
264-279
Central automorphisms of a finite$p$-group
Albert D.
Otto
280-287
Indecomposable representations of groups with a cyclic Sylow subgroup
G. J.
Janusz
288-295
Pointwise ergodic theorems
M. A.
Akcoglu
296-309
Connexions in differential geometry of higher order
William F.
Pohl
310-325
The uniqueness of solutions of the heat equation in an infinite strip
Victor L.
Shapiro
326-361
A translation theorem for analytic Feynman integrals
R. H.
Cameron;
D. A.
Storvick
1-6
Analytic continuation for functions of several complex variables
R. H.
Cameron;
D. A.
Storvick
7-12
On the differentiability of generalized solutions of first order elliptic equations with discontinuous coefficients
Alfred
Schatz
13-31
An integral representation of a normal functional on a von Neumann algebra
Herbert
Halpern
32-46
Representation theory for denumerable Markov chains
John G.
Kemeny
47-62
Some asymptotic formulas for Wiener integrals
M.
Schilder
63-85
Ratio limit theorems for random walks on groups
Charles
Stone
86-100
A lattice-point problem. II
Burton
Randol
101-113
On a convexity condition in normed linear spaces
Daniel P.
Giesy
114-146
Decomposable modules
Eben
Matlis
147-179
Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. I
Lamberto
Cesari
369-412
Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. II. Existence theorems for weak solutions
Lamberto
Cesari
413-430
Massey higher products
David
Kraines
431-449
Lattices and the adjoint group of a Lie group
Howard
Garland;
Morikuni
Goto
450-460
Congruence-invariant measures in uniform spaces
David B.
Lewin
461-467
A factorization algorithm for $q\times q$ matrix-valued functions on the real line $R$
Habib
Salehi
468-479
Integral geometry in homogeneous spaces
John E.
Brothers
480-517
Piercing points of homeomorphisms of differentiable manifolds
Jerome L.
Paul
518-532
Inverse limits and homogeneity
Richard M.
Schori
533-539
Functions satisfying a weighted average property. II
Anil Kumar
Bose
540-551
A characterization of the cutpoint-order on a tree
Shwu-yeng T.
Lin
552-557
Some inclusion relations between matrices compounded from Cesaro matrices
A. J.
White
558-568
Regularity criteria for integral and meromorphic functions
J. M.
Anderson
185-200
On the Valiron deficiencies of meromorphic functions of finite order
D. F.
Shea
201-227
Martingale theory and pointwise convergence of certain orthogonal series
Richard F.
Gundy
228-248
When is a complex fibered by a subcomplex?
S. Y.
Husseini
249-291
Functions resembling quotients of measures
Ethan D.
Bolker
292-312
Classification of the finite nonlinear primitive Lie algebras
Takushiro
Ochiai
313-322
Limit theorems for Markov processes on topological groups
S. R.
Foguel
323-333
Abstract: Limit theorems for ${P^n}(x,A)$, as $n \to \infty$, are established, where $P(x,A)$ is the transition probability of a Markov process on a topological group. The transition probability is assumed to satisfy certain commutativity relations with translations. Thus special cases of our investigation are spatially homogenous processes and processes induced by automorphisms of the group.
On differential operators and automorphic forms
H. L.
Resnikoff
334-346
Some results giving rates of convergence in the law of large numbers for weighted sums of independent random variables
W. E.
Franck;
D. L.
Hanson
347-359
A partition theorem
J. D.
Halpern;
H.
Läuchli
360-367
Post's problem, admissible ordinals, and regularity
Gerald E.
Sacks
1-23
The cohomology of group extensions
L. S.
Charlap;
A. T.
Vasquez
24-40
Subgroups of the multiplicative group of a division ring
R. J.
Faudree
41-48
On primary groups with countable basic subgroups
Paul
Hill;
Charles
Megibben
49-59
Singular quadratic functionals of $n$ dependent variables
E. C.
Tomastik
60-76
Inertial subalgebras of algebras over commutative rings
Edward C.
Ingraham
77-93
On Milnor's invariant for links
Kunio
Marasugi
94-110
Integral represetation algebras
Irving
Reiner
111-121
The Hausdorff-Beiscovich dimension of the level sets of Perron's modular function
John R.
Kinney;
Tom S.
Pitcher
122-130
Products of nearly compact spaces
C. T.
Scarborough;
A. H.
Stone
131-147
Invariant integration over the infinite dimensional orthogonal group and related spaces
David
Shale
148-157
$H\sp{p}$ spaces and extremal functions in $H\sp{1}$
T. W.
Gamelin
158-167
Equilibrium processes
Sidney C.
Port
168-184
Hausdorff measures on abstract spaces
M.
Sion;
R. C.
Willmott
275-309
Compactification and duality of topological groups
Hsin
Chu
310-324
Finite topological spaces
R. E.
Stong
325-340
Bol loops
D. A.
Robinson
341-354
Tame surfaces and tame subsets of spheres in $E\sp{3}$
L. D.
Loveland
355-368
Elastic-plasitc torsion of a square bar
Tsuan Wu
Ting
369-401
Limit theorems for semi-Markov processes
James
Yackel
402-424
The level curves of harmonic functions
Leopold
Flatto;
Donald J.
Newman;
Harold S.
Shapiro
425-436
Dilations on invertible spaces
Ellard
Nunnally
437-448
Abstract: This paper primarily concerns certain groups of homeomorphisms which are associated in a natural way with a variety of spaces, which satisfy a set of axiomatic conditions put forth in §1. Let us suppose that $X$ is a space of the type in question and that $G$ is an appropriate group of homeomorphisms of $ X$ onto itself. In §2 we demonstrate the existence of a nonvoid subcollection $ \mathcal{D}$, the ``topological dilations,'' of $G$ which is characterized in Theorem 1 in the following fashion: suppose $f \in \mathcal{D}$ and $g \in G$, then $ g \in \mathcal{D}$ if and only if $f$ is a $G$-conjugate of $g$, that is if and only if there exists an element $ h$ of $G$ such that $f = hg{h^{ - 1}}$. We proceed then to show in §3 that if $f$ and $g$ are nonidentity elements of $G$, then we may find $\delta ,r \in G$ such that the product $(rg{r^{ - 1}})(\delta f{\delta ^{ - 1}}) \in \mathcal{D}$. We then combine this fact with the characterization of $ \mathcal{D}$ mentioned above to conclude that each element of $\mathcal{D}$ is a ``universal'' element of $ G$ in the sense that if $d \in \mathcal{D}$, then any element $g$ of $G$ may be represented as the product of two $ G$-conjugates of $ d$. Furthermore we conclude that if $g$ is not the identity element of $G$, then $g$ can be represented as the product of three $ G$-conjugates of any nonidentity element of $G$. Finally, we apply the conclusions to groups of homeomorphisms of certain spaces: for example spheres, cells, the Cantor set, etc.
Decay at infinity of solutions to partial differential equations with constant coefficients
Walter
Littman
449-459
Decomposable chainable continua
J. B.
Fugate
460-468
Regular minimal sets. I
Joseph
Auslander
469-479
Unknotting in $M\sp{2}\times I$
E. M.
Brown
480-505
Polynomial automorphic forms and nondiscontinuous groups
Marvin Isadore
Knopp
506-520
Singular perturbations on the infinite interval
Frank Charles
Hoppensteadt
521-535
On Witt's theorem in the denumerably infinite case
Herbert
Gross
536-547
Errata to ``Szeg\"o functions on a locally compact Abelian group with ordered dual''
I. I.
Hirschman
548
Quasi-norm spaces
Richard
Metzler;
Hidegoro
Nakano
1-31
Convergence of sequences of convex sets, cones and functions. II
R. A.
Wijsman
32-45
Level sets and continuity of conjugate convex functions
R. T.
Rockafellar
46-63
Some theorems on diophantine approximation
Charles F.
Osgood
64-87
Some normal subgroups of homomorphisms
James V.
Whittaker
88-98
Groups with normal solvable Hall $p'$-subgroups
D. S.
Passman
99-111
On product measures and Fubini's theorem in locally compact space
Roy A.
Johnson
112-129
Stationary measures for the flow of a linear differential equation driven by white noise
Harry
Dym
130-164
Decompostions of $E\sp{3}$ with a compact ${\rm O}$-dimensional set of nondegenerate elements
Steve
Armentrout
165-177
On badly approximable numbers and certain games
Wolfgang M.
Schmidt
178-199
On the classification of noncompact complex abelian Lie groups
Akihiko
Morimoto
200-228
Multipliers on $D\sb{\alpha }$
Gerald D.
Taylor
229-240
The semi-simplicity manifold of arbitrary operators
Shmuel
Kantorovitz
241-252
Simple and weakly almost periodic transformation groups
Robert N.
Rich
253-266
Graph topology for function spaces
Somashekhar Amrith
Naimpally
267-272
Errata to ``The theory of operations on binary relations''
T.
Tamura
273
Uniform approximation on noncompact spaces
R. E.
Edwards
249-276
Outer measures on a linear lattice
Leon
Brown;
Hidegoro
Nakano
277-288
Oscillatory properties of certain nonlinear matrix differential systems of second order
Garret J.
Etgen
289-310
Intersections of combinatorial balls and of Euclidean spaces
L. C.
Glaser
311-320
Some counterexamples related to integral closure in $D[[x]]$
Jack
Ohm
321-333
The cohomology of augmented algebras and generalized Massey products for ${\rm DGA}$-Algebras
J. Peter
May
334-340
Extensions of the Rauch comparison theorem to submanifolds
F. W.
Warner
341-356
Topology of quaternionic manifolds
Vivian Yoh
Kraines
357-367
Symbolic dynamics and transformations of the unit interval
William
Parry
368-378
The lattice of topologies: Structure and complementation
A. K.
Steiner
379-398
A duality between certain spheres and arcs in $S\sp{3}$
Carl D.
Sikkema
399-415
Cohomology as the derived functor of derivations
Michael
Barr;
George S.
Rinehart
416-426
A necessary condition that a cellular upper semi-continuous decomposition of $E\sp{n}$ yield $E\sp{n}$
T. M.
Price
427-435
The semi-simplicial free Lie ring
James W.
Schlesinger
436-442
Mean approximation on an interval for an exponent less than one
T. S.
Motzkin;
J. L.
Walsh
443-460
Separable algebras over commutative rings
G. J.
Janusz
461-479
Differentiable actions on homotopy seven spheres
D.
Montgomery;
C. T.
Yang
480-498
An extension problem for cancellative semigroups
Charles V.
Heuer;
Donald W.
Miller
499-515
Asymptotic distribution of eigenvalues and eigenfunctions for general linear elliptic boundary value problems
Bui An
Ton
516-546
Correction to the paper ``On the zeros of polynomials over division rings''
B.
Gordon;
T. S.
Motzkin
547
Induced automorphisms of free groups and free metabelian groups
S.
Bachmuth
1-17
A characterization of the Mathieu group $\mathfrak{M}_{12}$
Richard
Brauer;
Paul
Fong
18-47
Flexible partially stable algebras
Chester E.
Tsai
48-63
Decomposing pairs of modules
Lawrence S.
Levy
64-80
Braid groups of compact $2$-manifolds with elements of finite order
James
van Buskirk
81-97
Linear transformations of Gaussian measures
Dale E.
Varberg
98-111
On complex quadratic fields with class number equal to one
Harold
Stark
112-119
Semicontinuity of integrals
A. W. J.
Stoddart
120-135
Structures and operators on almost-Hermitian manifolds
Chuan-chih
Hsiung
136-152
Gauges and their densities
Gerald
Freilich
153-162
On limits of scalar operators
James E.
Simpson
163-176
Asymptotic behavior of the solutions of an $n{\rm th}$ order nonhomogeneous ordinary differential equation
Thomas G.
Hallam
177-194
Fixed points and multiplicative left invariant means
Theodore
Mitchell
195-202
Sums of solid horned spheres
Lawrence O.
Cannon
203-228
On the Radon-Nikod\'ym derivatives of measurable transformations
Dorothy
Maharam
229-248
Hausdorff means and the Gibbs phenomenon
Jonah
Mann
277-295
Some properties of pseudo-complements of recursively enumerable sets
Robert A.
Di Paola
296-308
On the degrees of index sets
C. E. M.
Yates
309-328
The conjugacy problem in wreath products and free metabelian groups
Jane
Matthews
329-339
Ordered cycle lengths in a random permutation
L. A.
Shepp;
S. P.
Lloyd
340-357
Commutators, generalized eigenfunction expansions and singular integral operators
Joel David
Pincus
358-377
On the Mills-Seligman axioms for Lie algebras of classical type
Richard E.
Block
378-392
Maximal subgroups of symmetric groups
Ralph W.
Ball
393-407
Closed ideals in the group algebra $L\sp{1}(G)\cap L\sp{2}(G)$
C. Robert
Warner
408-423
Continuous images of ordered compacta and a new dimension which neglects metric subcontinua
Sibe
Mardešić
424-433
Global structure in von Neumann algebras
Edward G.
Effros
434-454
Coefficient extremal problems for schlicht functions
J. T.
Poole
455-475
On function spaces which are Lindel\"of spaces
H. H.
Corson;
J.
Lindenstrauss
476-491
Continuous selections with nonmetrizable range
H. H.
Corson;
J.
Lindenstrauss
492-504
Deformations, related deformations and a universal subfamily
Kimyong
Kim
505-515
On the dimensions of certain spaces of homeomorphisms
Beverly L.
Brechner
516-548
Correction to ``The Wiener integral and the Schr\"odinger operator''
Donald
Babbitt
549-552
Hankel matrices
Harold
Widom
1-35
On the Wedderburn principal theorem for commutative power-associative algebras
Robert L.
Hemminger
36-51
A generalisation of Eberlein's integral over function space
V. L. N.
Sarma
52-61
On Ree's series of simple groups
Harold N.
Ward
62-89
Operator limit theorems
Norton
Starr
90-115
The Martin boundary for random walk
P.
Ney;
F.
Spitzer
116-132
Szeg\"o functions on a locally compact Abelian group with ordered dual
I. I.
Hirschman
133-159
Flat chains over a finite coefficient group
Wendell H.
Fleming
160-186
Structure and representations of noncommutative Jordan algebras
Kevin
McCrimmon
187-199
Limit theorems for Markov processes
S. R.
Foguel
200-209
The Hahn-Banach theorem for finite dimensional spaces
William E.
Bonnice;
Robert J.
Silverman
210-222
A torsion theory for Abelian categories
Spencer E.
Dickson
223-235
The entropy of Chebyshev polynomials
R. L.
Adler;
M. H.
McAndrew
236-241
K\"unneth formulas for bordism theories
Peter S.
Landweber
242-256
A lattice-point problem
Burton
Randol
257-268
Bounds for dimensions of odd order nonsingular immersions of $RP\sp{n}$
Haruo
Suzuki
269-275
Quasi-martingales
Donald L.
Fisk
369-389
On a theorem of Vorono\"\i
F. L.
Cleaver
390-400
On coverings of four-space by spheres
F. L.
Cleaver
401-416
Representations of a semigroup
Rebecca Ellen
Slover
417-427
Cohomology and homology theories for categories of principal $G$-bundles
Jerrold
Siegel
428-437
On the Jordan structure of $C\sp{\ast} $-algebras
Erling
Størmer
438-447
Extensions of Dehn's lemma and the loop theorem
David W.
Henderson
448-469
Self-unlinked simple closed curves
David W.
Henderson
470-480
On the homotopy groups of the exceptional Lie groups
P. G.
Kumpel
481-498
On Klein's combination theorem
Bernard
Maskit
499-509
On the existence of boundary values of a class of Beppo Levi functions
Hans
Wallin
510-525
A Jordan decomposition for operators in Banach space
Shmuel
Kantorovitz
526-550
The geometry of immersions. I
E. A.
Feldman
185-224
Normal operators and uniformly elliptic self-adjoint partial differential equations
Leo
Sario;
Georges
Weill
225-235
Prime $z$-filters on completely regular spaces
Carl W.
Kohls
236-246
Homotopically homogeneous spaces and manifolds
Peter
Rice
247-254
Bounded holomorphic functions on finite Reimann surfaces
E. L.
Stout
255-285
On the volume elements on a manifold
Jürgen
Moser
286-294
Structure of spectral measures on locally convex spaces
Bertram
Walsh
295-326
On characteristic functions and renewal theory
Charles
Stone
327-342
The theory of operations on binary relations
T.
Tamura
343-358
On creative sets and indices of partial recursive functions
Louise
Hay
359-367
Errata: Some properties of Pontryagin classes $\bmod\, 3$
Yasuro
Tomonaga
368
The principal semi-algebra in a Banach algebra
Edward J.
Barbeau
1-16
Representations of graded Lie algebras
Leonard E.
Ross
17-23
Generalized Fekete means
Paul
Schaefer
24-36
Homotopy properties of the space of homeomorphisms on $P\sp{2}$ and the Klein bottle
Mary-Elizabeth
Hamstrom
37-45
Normal curves arising from light open mappings of the annulus
Morris L.
Marx
46-56
On the representations of an abstract lattice as the family of closed sets of a topological space
David
Drake;
W. J.
Thron
57-71
Multivalent functions star-like in one direction
Teruo
Takatsuka
72-82
$\lambda $-continuous Markov chains. II
Shu-teh C.
Moy
83-107
Abstract: Continuing the investigation in [8] we study a $\lambda$-continuous Markov operator $ P$. It is shown that, if $ P$ is conservative and ergodic, $P$ is indeed ``periodic'' as is the case when the state space is discrete; there is a positive integer $ \delta$, called the period of $P$, such that the state space may be decomposed into $ \delta$ cyclically moving sets ${C_0}, \cdots ,{C_{\delta - 1}}$ and, for every positive integer $ n,{P^{n\delta }}$ acting on each ${C_i}$ alone is ergodic. It is also shown that $ P$ maps ${L_q}(\mu )$ into $ {L_q}(\mu )$ where $ \mu$ is the nontrivial invariant measure of $P$ and $ 1 \leqq q \leqq \infty$. If $\mu$ is finite and normalized then it is shown that (1) if $ f \in {L_\infty }(\lambda )$, then $ \{ {P^{n\delta + k}}f\}$ converges a.e. $ (\lambda )$ to ${g_k} = \sum\nolimits_{i = 0}^{\delta - 1} {{c_{i + k}}} {1_{{C_i}}}$ where ${c_j} = \delta {\smallint _{{C_j}}}fd\mu$ if $0 \leqq j \leqq \delta - 1$ and ${c_j} = {c_i}$ if $j = m\delta + i,0 \leqq i \leqq \delta - 1$, (2) $\{ {P^{n\delta + k}}f\}$ converges in ${L_q}(\mu )$ to ${g_k}$ if $ f \in {L_q}(\mu )$, and(3) $\lim {\inf _{n \to \infty }}{P^{n\delta + k}}f = {g_k}$ a.e. $(\lambda )$ if $f \in {L_1}(\mu )$ and $f \geqq 0$. If $\mu$ is infinite, then it is shown that (1) if $ f \geqq 0,f \in {L_q}(\mu )$ for some $ 1 \leqq q < \infty$, then $\lim {\inf _{n \to \infty }}{P^n}f = 0$ a.e. $(\lambda )$, (2) there exists a sequence $\{ {E_k}\}$ of sets such that $X = \cup _{k = 1}^\infty {E_k}$ and ${\lim _{n \to \infty }}{P^{n\delta + i}}{1_{{E_k}}} = 0$ a.e. $ (\lambda )$ for $i = 0,1, \cdots ,\delta - 1$ and $k = 1,2, \cdots$.
Convergence rates in the law of large numbers
Leonard E.
Baum;
Melvin
Katz
108-123
Generalized Taylor series and orders and types of entire functions of several complex variables
Fred
Gross
124-144
An eigenvalue problem for nonlinear elliptic partial differential equations
Melvyn S.
Berger
145-184
On the distribution of the supremum for stochastic processes with interchangeable increments
Lajos
Takács
367-379
Essential cluster sets
Ulysses
Hunter
380-388
A formal solution of certain dual integral equations
Charles
Fox
389-398
Two notes on locally Macaulay rings
Louis J.
Ratliff
399-406
On the behavior of solutions of quasi-linear elliptic equations
Howard
Jenkins
407-416
Continuous Stolz extensions and boundary functions
Larry E.
Snyder
417-427
Spectral methods for a generalized probability theory
Stanley P.
Gudder
428-442
On ``essentially metrizable'' spaces and on measurable functions with values in such spaces
Elias
Zakon
443-453
Correction to a paper of R. H. Bruck
Richard
Roth
454-456
Invariant eigendistributions on a semisimple Lie group
Harish-Chandra
457-508
On the spectra of semi-normal operators
C. R.
Putnam
509-523
Potent rings
R. E.
Johnson
524-534
An extension of the Denjoy-Carleman-Ahlfors theorem in subharmonic form
J. E.
Skeath
535-551
Errata to: ``Noncommutative unique factorization domains''
P. M.
Cohn
552
Fixed point theorems on infinite dimensional manifolds
Felix E.
Browder
179-194
On Bieberbach-Eilenberg functions. III
James A.
Jenkins
195-215
Sign-invariant random variables and stochastic processes with sign-invariant increments
Simeon M.
Berman
216-243
Constant functions and left invariant means on semigroups
Theodore
Mitchell
244-261
On sets of completeness for families of Haar functions
J. J.
Price;
Robert E.
Zink
262-269
The radial heat polynomials and related functions
L. R.
Bragg
270-290
On affine symmetric spaces
Sebastian S.
Koh
291-309
Probability and the $(C,\,r)$ summability of Fourier series
Walter A.
Rosenkrantz
310-332
A generalization of Brauer characters
W. F.
Reynolds
333-351
Vector-valued analytic functions
Daniel O.
Etter
352-366
Solvability of ordinary differential equations near singular points
Homer G.
Ellis
1-20
A reduction of the three body problem by means of transformation groups
James W.
Sauve
21-36
Liapunov functions and $L\sp{p}$ solutions of differential equations
Aaron
Strauss
37-50
On orbits under ergodic measure-preserving transformations
Dorothy
Maharam
51-66
On the fourth coefficient of bounded univalent functions
M.
Schiffer;
O.
Tammi
67-78
Abstract homotopy theory
Edgar H.
Brown
79-85
On minimal and almost-minimal systems of notations
David
Luckham;
Hilary
Putnam
86-100
Approximation in the metric of $L\sp{1}(X,\,\mu )$
B. R.
Kripke;
T. J.
Rivlin
101-122
Disks in $E\sp{3}$. II. Disks which ``almost'' lie on a $2$-sphere
Ralph J.
Bean
123-124
Quasi-convexity and lower semi-continuity of multiple variational integrals of any order
Norman G.
Meyers
125-149
The structure of convolution measure algebras
Joseph L.
Taylor
150-166
Meromorphic multivalent close-to-convex functions
A. E.
Livingston
167-177
A characterization of locally euclidean spaces
O. G.
Harrold
1-16
Classical expansions and their relation to conjugate harmonic functions
B.
Muckenhoupt;
E. M.
Stein
17-92
Automorphisms of free metabelian groups
S.
Bachmuth
93-104
Green's formula, linear continuity, and Hausdorff measure
Robert M.
Fesq
105-112
On the field extension by complex multiplication
Tomio
Kubota
113-122
On the ergodic mixing theorem
H. A.
Dye
123-130
Tensor products over Banach algebras
B. R.
Gelbaum
131-149
Rings with zero right and left singular ideals
R. E.
Johnson
150-157
On continuous rings and self injective rings
Yuzo
Utumi
158-173
On the Lefschetz number and the Euler class
Robert F.
Brown
174-179
Path fields on manifolds
Robert F.
Brown
180-191
Some ways of constructing a propositional calculus of any required degree of unsolvability
M. D.
Gladstone
192-210
On spherical Bessel functions
James M.
Horner
211-220
Elementary methods in the theory of primes
Patrick R.
Ahern
221-242
On cardinality, cohomology and a conjecture of Rosenberg and Zelinsky
Robert E.
MacRae
243-246
An extension of differential Galois theory
H. F.
Kreimer
247-256
Multiple K\"unneth formulas for abelian groups
Thomas W.
Hungerford
257-275
Fourier analysis of nonstationary stochastic processes
Tatsuo
Kawata
276-302
Reflective functors in general topology and elsewhere
J. F.
Kennison
303-315
On a necessary and sufficient condition that an infinitely divisible distribution be absolutely continuous
Howard G.
Tucker
316-330
On the Marx conjecture for starlike functions
P. L.
Duren
331-337
Multi-valued monotone nonlinear mappings and duality mappings in Banach spaces
Felix E.
Browder
338-351
Orientability of manifolds for generalised homology theories
W. C.
Hsiang;
C. T. C.
Wall
352-359
Fourier analysis of sub-stationary processes with a finite moment
R. M.
Dudley
360-375
Interval topology in subsets of totally orderable spaces
Mary Ellen
Rudin
376-389
On the polar equations for linear systems and related nonlinear matrix differential equations
Glen J.
Culler
390-405
On the central decomposition for positive functionals on $C\sp{\ast} $-algebras
Shôichirô
Sakai
406-419
Acyclic semigroups and multiplications on two-manifolds
Haskell
Cohen;
R. J.
Koch
420-427
Transformation groups on compact symmetric spaces
Tadashi
Nagano
428-453
The classification of singular points of algebraic curves
Sherwood
Ebey
454-471
Functions satisfying a weighted average property
Anil K.
Bose
472-487
On groups and graphs
Chong-yun
Chao
488-497
Rings of continuous integer-valued functions and nonstandard arithmetic
Norman L.
Alling
498-525
Lie loops with invariant uniformities. II
Sigmund N.
Hudson
526-533
Some results on crumpled cubes
Lloyd L.
Lininger
534-549
A correction of some theorems on partitions
Peter
Hagis
550
On semicontinuous fuctions and Baire functions
Robert E.
Zink
1-9
Large Abelian subgroups of $p$-groups
J. L.
Alperin
10-20
The dimension theory of certain cardinal algebras
Peter A.
Fillmore
21-36
Some properties of Pontryagin classes ${\rm mod} 3$
Yasuro
Tomonaga
37-42
Some theorems on integral currents
Herbert
Federer
43-67
$\lambda $-continuous Markov chains
Shu-teh C.
Moy
68-91
Orbits of $L\sp{1}$-functions under doubly stochastic transformations
John V.
Ryff
92-100
The distribution of irreducibles in ${\rm GF}[q,\,x]$
David R.
Hayes
101-127
Implicative semi-lattices
William C.
Nemitz
128-142
Eigenfunction expansions associated with an integral operator
Marvin
Shinbrot
143-156
Power series whose sections have zeros of large modulus
J. D.
Buckholtz
157-166
Integro-differential operators on vector bundles
R. T.
Seeley
167-204
On a certain class of limit distributions and their domain of attraction
D.
Mejzler
205-236
Primary ideals and valuation ideals
Robert
Gilmer;
Jack
Ohm
237-250
Complete Riemannian manifolds and some vector fields
Yoshihiro
Tashiro
251-275
A theorem on mappings into Riemann surfaces of infinite genus
Leo
Sario
276-284
On the computational complexity of algorithms
J.
Hartmanis;
R. E.
Stearns
285-306
Reduced products and Horn classes
H. Jerome
Keisler
307-328
Undefinability of addition from one unary operator
Robert
McNaughton
329-337
On the behavior of quasiconformal mappings at a point
Edgar
Reich;
Hubert R.
Walczak
338-351
A nonhomogeneous eigenfunction expansion
Allan M.
Krall
352-361
On random walks with a reflecting barrier
Sidney C.
Port
362-370
Iterates of measurable transformations and Markov operators
Arshag B.
Hajian;
Yuji
Ito
371-386
On a certain numerical invariant of link types
Kunio
Murasugi
387-422
A fixed-point formula for the classical groups over a finite field
Louis
Solomon
423-440
A characterization of $\Delta \sb{2}$-sets
B.
Scarpellini
441-450
Ergodic decompositions induced by certain Markov operators
Benton
Jamison
451-468
Hyponormal operators and spectral density
J. G.
Stampfli
469-476
Relative homological algebra and homological dimen- sion of Lie algebras
Byoung-song
Chwe
477-493
Theory of provable recursive functions
Patrick C.
Fischer
494-520
Some results on stability in semigroups
L. W.
Anderson;
R. P.
Hunter;
R. J.
Koch
521-529
Nonlinear elliptic boundary value problems. II
Felix E.
Browder
530-550
Some results in the location of the zeros of linear combinations of polynomials
Zalman
Rubinstein
1-8
Dimension theory in complete orthocomplemented weakly modular lattices
Arlan
Ramsay
9-31
A characterization of commutative group algebras and measure algebras
Marc A.
Rieffel
32-65
The Wiener integral and the Schr\"odinger operator
Donald
Babbitt
66-78
Uniformization of symmetric Riemann surfaces by Schottky groups
Robert J.
Sibner
79-85
On modular forms of dimension $-2$
John Roderick
Smart
86-107
Differentiability properties of solutions to higher order double integral variational problems
Julius
Smith
108-125
On regularity in Hurewicz fiber spaces
Patricia
Tulley
126-134
On the sum of two solid Alexander horned spheres
B. G.
Casler
135-150
Transitive permutation groups of degree $p=2q+1,\,p$ and $q$ being prime numbers. III
Noboru
Itô
151-166
Linear operators in $VH$-spaces
R. M.
Loynes
167-180
Volterra operators similar to $J:f(x)\to \int \sb{0}{}\sp{x}f(y)dy$
J. M.
Freeman
181-192
The essential spectrum of elliptic differential operators in $L\sp{p}(R\sb{n})$
Erik
Balslev
193-217
On the zeros of polynomials over division rings
B.
Gordon;
T. S.
Motzkin
218-226
The convergence of sequences of rational functions of best approximation. II
J. L.
Walsh
227-237
On the line graph of a symmetric balanced incomplete block design
A. J.
Hoffman;
D. K.
Ray-Chaudhuri
238-252
Products of automata and the problem of covering
Abraham
Ginzburg;
Michael
Yoeli
253-266
On the extreme eigenvalues of Toeplitz operators of the Hankel type II
J. R.
Davis
267-299
Generating functions for products of recursive sequences
David
Zeitlin
300-315
Generalized matrix functions
Marvin
Marcus;
Henryk
Minc
316-329
Integral equations associated with Hankel convolutions
Deborah Tepper
Haimo
330-375
Some types of Banach spaces, Hermitian operators, and Bade functionals
Earl
Berkson
376-385
The category of cofinal types. I
Seymour
Ginsburg;
J. R.
Isbell
386-393
The category of cofinal types. II
J. R.
Isbell
394-416
On Galois conditions and Galois groups of simple rings
Takasi
Nagahara
417-434
Purely inseparable extensions and higher derivations
Morris
Weisfeld
435-449
Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines
Kenneth
Krohn;
John
Rhodes
450-464
Integral equations with difference kernels on finite intervals
A.
Leonard;
T. W.
Mullikin
465-473
Limit distributions of the minimax of independent identically distributed random variables
Herman
Chernoff;
Henry
Teicher
474-491
Abstract: The class of limiting distributions of the normalized minimax (or maximin) of independent identically distributed random variables is obtained and the domains of attraction of the three limiting types are characterized. Asymptotic independence of the minimax and maximin is also demonstrated.
Expansions of distributions
Gilbert G.
Walter
492-510
Improving the side approximation theorem
R. H.
Bing
511-525
Outer product rings
David
Lissner
526-535
Function theory and multiplicative linear functionals
Kenneth
Hoffman;
Hugo
Rossi
536-543
The integrability problem for $G$-structures
Victor
Guillemin
544-560
Generalized commuting properties of measure-preserving transfomations
Roy L.
Adler
1-13
Vector lattices of self-adjoint operators
David M.
Topping
14-30
Generalized Rodriques formula solutions for certain linear differential equations
James M.
Horner
31-42
Some theorems on equiconnected and locally equiconnected spaces
Charles J.
Himmelberg
43-53
Oscillation properties of even-order linear differential equations
Robert W.
Hunt
54-61
General solution of nonlinear difference equations
W. A.
Harris;
Y.
Sibuya
62-75
Continuous transformations and stochastic differential equations
D. A.
Woodward
76-82
On properties of regressive sets
K. I.
Appel;
T. G.
McLaughlin
83-93
On isometric immersions in Euclidean space of manifolds with non-negative sectional curvatures
Philip
Hartman
94-109
The space of minimal prime ideals of a commutative ring
M.
Henriksen;
M.
Jerison
110-130
A duality in function spaces
I.
Namioka
131-144
On the degree of convergence of extremal polynomials and other extremal functions
J. L.
Walsh;
A.
Sinclair
145-160
$p$-valent close-to-convex functions
A. E.
Livingston
161-179
On the Chern classes of representations of finite groups
Leonard
Evens
180-193
Classification of operators by means of their operational calculus
Shmuel
Kantorovitz
194-224
Bockstein spectra
Thomas W.
Hungerford
225-241
The eigenvalue behavior of certain convolution equations
H. J.
Landau
242-256
On co-groups in the category of graded algebras
Israel
Berstein
257-269
Extremal problems and coefficient regions for analytic functions represented by a Stieltjes integral
J. A.
Pfaltzgraff
270-282
Chevalley bases for Lie modules
David A.
Smith
283-299
Fractional integration in Orlicz spaces. I
Richard
O’Neil
300-328
On semi-cylinders, splinters, and bounded-truth-table reducibility
Paul R.
Young
329-339
On the structure of continuous functions of several variables
David A.
Sprecher
340-355
A pair of unsymmetrical Fourier kernels
Roop N.
Kesarwani
356-369
Factorization of differentiable maps with branch set dimension at most $n-3$
P. T.
Church
370-387
Convergence of best rational Tchebycheff approximations
Barry
Boehm
388-399
Asymptotic behavior of solutions of nonlinear differential equations
Richard K.
Miller
400-416
Lie loops with invariant uniformities
Sigmund N.
Hudson
417-432
Spectral theory of generalized Toeplitz operators
R. G.
Douglas;
Carl
Pearcy
433-444
An extension of Ascoli's theorem and its applications to the theory of optimal control
S. S. L.
Chang
445-470
Twisting spun knots
E. C.
Zeeman
471-495
Derivations on $\bar p$-adic fields
Joseph
Neggers
496-504
Generalized eigenvectors and separation of variables
Maurice
Machover
505-524
On embedding highly connected manifolds in Euclidean space
J. B.
Minkus
525-540
A class of projective planes
Donald E.
Knuth
541-549
Errata: Hyponormal operators and spectral density
J. G.
Stampfli
550
Maximal abelian subalgebras in finite factors of type II
Rita Jean
Tauer
281-308
Topological entropy
R. L.
Adler;
A. G.
Konheim;
M. H.
McAndrew
309-319
On estimating the ``rates of convergence'' of iterative methods for elliptic difference equations
Seymour V.
Parter
320-354
Elementary groups
Homer
Bechtell
355-362
On properties of derivatives
Clifford E.
Weil
363-376
On the Minkowski unit of slice links
Kunio
Murasugi
377-383
Carath\'eodory measure of cylinders
Gerald
Freilich
384-400
Nearly modular orthocomplemented lattices
M. Donald
MacLaren
401-416
Some renewal theorems for non-negative independent random variables
John A.
Williamson
417-445
Translation-invariant functionals on functions defined in Euclidean spaces
Joel Alan
Smoller
446-467
A generalization of the flexible flaw
D.
Rodabaugh
468-487
Generalized normal bundles for locally-flat imbeddings
Edward
Fadell
488-513
Categoricity in power
Michael
Morley
514-538
Plane flows with closed orbits
Anatole
Beck
539-551
Retractions and other continuous maps from $\beta X$ onto $\beta X_X$
W. W.
Comfort
1-9
Abstract: Our two main theorems are stated below. The first is proved with the aid of the continuum hypothesis. Theorem 2.6. [CH] Suppose that there is a retraction from $ \beta X$ onto $\beta X\backslash X$. Then X is locally compact and pseudocompact. Theorem 4.2. Let D be a discrete space whose cardinal number m exceeds 1. In order that there exist a continuous function from $\beta D$ onto $\beta D\backslash D$, it is necessary and sufficient that $\mathfrak{m} = {\mathfrak{m}^{\aleph _0}}$. The proof of Theorem 2.6 rests on a result of Walter Rudin concerning P-points (see 1(d) and 1(e) below); Theorem 4.2 depends on the following simple result, which appears to be new. Theorem 4.1. Let D be the discrete space with cardinal number $ \mathfrak{m} (\geqq{\aleph _0})$. The smallest cardinal number which is the cardinal number of some dense subset of $\beta D\backslash D$ is ${\mathfrak{m}^{\aleph _0}}$.
Cellular subsets of the $3$-sphere
D. G.
Stewart
10-22
On a theorem concerning existence of interpolating functions
Richard F.
DeMar
23-29
Geometry of the zeros of the sums of linear fractions
J. L.
Walsh
30-39
Multiplication rings as rings in which ideals with prime radical are primary
Robert W.
Gilmer;
Joe Leonard
Mott
40-52
On Noetherian prime rings
Carl
Faith;
Yuzo
Utumi
53-60
Orders in simple artinian rings
Carl
Faith
61-64
Limits of polynomials whose zeros lie in a radial set
J. E.
Lange;
J.
Korevaar
65-79
Characterizations of tame surfaces in $E\sp{3}$
C. E.
Burgess
80-97
Measures and tensors
Jesús
Gil de Lamadrid
98-121
On algebras with a finite number of indecomposable modules
C. W.
Curtis;
J. P.
Jans
122-132
Spectral sequences and Frobenius groups
Ernst
Snapper
133-146
A difference property for polynomials and exponential polynomials on abelian locally compact groups
F. W.
Carroll
147-155
Two topological problems concerning infinite-dimensional normed linear spaces
Bor-luh
Lin
156-175
On close-to-convex analytic functions
Ch.
Pommerenke
176-186
Invariant means on spaces of continuous or measurable functions
Chivukula
Ramamohana Rao
187-196
Inverse limit sequences with covering maps
M. C.
McCord
197-209
Generalized polynomial identities and pivotal monomials
S. A.
Amitsur
210-226
The dual spaces of Banach algebras
J. M. G.
Fell
227-250
Perturbations of the shift operator
J. M.
Freeman
251-260
On the category of certain classes of transformations in ergodic theory
A.
Ionescu Tulcea
261-279
Errata: ``A topology for certain measure spaces''
N. F. G.
Martin
280
The $(\varphi ,\,s)$ regular subsets of $n$-space
John M.
Marstrand
369-392
Generators in $l\sb{1}$
D. J.
Newman
393-396
The existence of topologies on field extensions
Lowell A.
Hinrichs
397-405
Concentricity in $3$-manifolds
C. H.
Edwards
406-423
On algebraic equivalence between pairs of linear transformations
Uri
Fixman
424-453
Transitive permutation groups of degree $p=2q+1$, $p$ and $q$ being prime numbers. II
Noboru
Iţo
454-487
Unbounded normal operators in Hilbert space
S.
Kaniel
488-511
The second dual of the space of continuous functions. IV
Samuel
Kaplan
512-546
Errata to ``Holomorphic functions with values in locally convex spaces and applications to integral formulas''
L.
Bungart
547
Linearly orderable spaces
I. L.
Lynn
189-218
The order dual of the space of Radon measures
John
Mack
219-239
Topological divisors of zero and Tauberian theorems
Albert
Wilansky
240-251
Strictly singular operators and their conjugates
R. J.
Whitley
252-261
On Gaussian measures equivalent to Wiener measure
Dale E.
Varberg
262-273
Invariants of Euclidean reflection groups
Louis
Solomon
274-286
On a problem connected with Beta and Gamma distributions
R. G.
Laha
287-298
Self adjoint function spaces on Riemannian symmetric manifolds
Joseph A.
Wolf
299-315
Chandrasekhar's $X$ and $Y$ equations
T. W.
Mullikin
316-332
Bounded ${\rm ALGOL}$-like languages
Seymour
Ginsburg;
Edwin H.
Spanier
333-368
The existence of eigenvalues for integral operators
Samuel
Karlin
1-17
Orthogonal conjugacies in associative and Lie algebras
Earl J.
Taft
18-29
The Tomita decomposition of rings of operators
Joseph L.
Taylor
30-39
Extensions and low dimensional cohomology theory of locally compact groups. I, II
Calvin C.
Moore
40-63
Extensions and low dimensional cohomology theory of locally compact groups. II
Calvin C.
Moore
64-86
Spectral multiplicity theory for a class of singular integral operators
Walter
Koppelman
87-100
On the spectral theory of singular integral operators
Joel David
Pincus
101-128
Weakly compact sets
Robert C.
James
129-140
The automorphism group of a geometric structure
Hsin
Chu;
Shoshichi
Kobayashi
141-150
Locally compact semigroups with dense maximal subgroups
T. S.
Wu
151-168
Boundedness in linear topological spaces
S.
Simons
169-180
Normalizers of system normalizers
J. L.
Alperin
181-188
Periodic transformations on the product of two spheres
J. C.
Su
369-380
Spaces of vector valued real analytic functions
J.
Barros-Neto
381-391
Differential equations invariant under finite reflection groups
Robert
Steinberg
392-400
On a class of partitions with distinct summands
Peter
Hagis
401-415
Nodal noncommutative Jordan algebras
Robert H.
Oehmke
416-431
Minimizing variational curves restricted to a preassigned set
J.
Warga
432-455
Two new classes of simple Lie algebras
Marguerite
Frank
456-482
On sequential convergence
R. M.
Dudley
483-507
Some implications of the generalized Gauss-Bonnet theorem
R. L.
Bishop;
S. I.
Goldberg
508-535
The number of solutions of some congruences modulo a product of primes
L.
Carlitz;
A. L.
Whiteman
536-552
Singulary cylindric and polyadic equality algebras
Donald
Monk
185-205
Disks in $E\sp{3}$. I. Subsets of disks having neighborhoods lying on $2$-spheres
Ralph J.
Bean
206-213
Some asymptotic formulas in the theory of numbers
Eckford
Cohen
214-227
The topology of certain spaces of paths on a compact symmetric space
Lawrence
Conlon
228-248
On dominant dimensions of ${\rm QF}$-3 algebras
Hiroyuki
Tachikawa
249-266
On K\"othe spaces
Robert
Welland
267-277
Restrictions of subspaces of $C(X)$
T. W.
Gamelin
278-286
Groups which have a faithful representation of degree less than $p-1$
Walter
Feit
287-303
Toeplitz operators on $H\sp{2}$ spaces
Allen
Devinatz
304-317
Completely monotone sequences as invariant measures
Richard
Scoville
318-329
Distributivity and perspectivity in orthomodular lattices
Samuel S.
Holland
330-343
Grothendieck groups of orders in semisimple algebras
A.
Heller;
I.
Reiner
344-355
Factorization of polynomials over Banach algebras
John A.
Lindberg
356-368
A topology for certain measure spaces
N. F. G.
Martin
1-18
On Lie algebras of rank one
Richard E.
Block
19-31
A generalization of alternative rings
Frank
Kosier
32-42
On the Hopf-Toda invariant
K. A.
Hardie
43-54
Intrinsic Markov chains
William
Parry
55-66
Further geometric consequences of conformal structure
Tilla
Klotz
67-78
Errata and addenda to ``A subgroup theorem for free nilpotent groups''
S.
Moran
79-83
On automorphisms of polyadic algebras
Aubert
Daigneault
84-130
Additive functionals of Markov processes in duality
R. M.
Blumenthal;
R. K.
Getoor
131-163
Intertwining forms for summable induced representations
Adam
Kleppner
164-183
A theorem on amenable semigroups
Edmond
Granirer
367-379
Characterizations of the continuous images of the pseudo-arc
Lawrence
Fearnley
380-399
Distinguished rings of linear transformations
R. E.
Johnson
400-412
A class of nilstable algebras
William G.
Witthoft
413-422
Sur l'int\'egration des syst\`emes de Pfaff complets
Mendel
Haimovici
423-439
Topological spaces which admit unisolvent systems
John A.
Lutts
440-448
Sequentially $1-{\rm ULC}$ tori
David S.
Gillman
449-456
Hierarchies in recursive function theory
H. B.
Enderton
457-471
Algebraic rings
Marvin J.
Greenberg
472-481
The foundations for an extension of differential algebra
H. F.
Kreimer
482-492
Ideals and invariant subspaces of analytic functions
Michael
Voichick
493-512
Disjoint open subsets of $\beta X_X$
W. W.
Comfort;
Hugh
Gordon
513-520
General boundary conditions for an ordinary linear differential system
Randal H.
Cole
521-550
A generalized Koszul complex. I
David A.
Buchsbaum
183-196
A generalized Koszul complex. II. Depth and multiplicity
David A.
Buchsbaum;
Dock S.
Rim
197-224
The bilinear relation on open Riemann surfaces
Albert
Marden
225-239
The structure of ideals and point derivations in Banach algebras of Lipschitz functions
Donald R.
Sherbert
240-272
A surface in $S\sp{3}$ is tame if it can be deformed into each complementary domain
John
Hempel
273-287
Combinatorial equivalence versus topological equivalence
Barry
Mazur
288-316
Holomorphic functions with values in locally convex spaces and applications to integral formulas
Lutz
Bungart
317-344
Oka's Heftungslemma and the Levi problem for complex spaces
Aldo
Andreotti;
Raghavan
Narasimhan
345-366
Semi-translation planes
T. G.
Ostrom
1-18
Coordinate systems of some semi-translation planes
D. L.
Morgan;
T. G.
Ostrom
19-32
Total positivity, absorption probabilities and applications
Samuel
Karlin
33-107
Manifolds which are joins
Kyung Whan
Kwun;
Frank
Raymond
108-120
Some uncomplemented function algebras
I.
Glicksberg
121-137
Homotopy classification of maps by cohomology homomorphisms
Emery
Thomas
138-151
On the characterization of spectral operators
Shmuel
Kantorovitz
152-181
On sums of large sets of integers
J. H. B.
Kemperman
375-392
Higher obstructions to sectioning a special type of fibre bundles
Wu-chung
Hsiang
393-412
On points of Jacobian rank $k$
P. T.
Church
413-423
Weak containment and induced representations of groups. II
J. M. G.
Fell
424-447
Reproducing kernels and Beurling's theorem
Harold S.
Shapiro
448-458
Some uniqueness theorems for differential equations with operator coefficients
Robert
Carroll;
Jérome
Neuwirth
459-472
The structure of associator dependent rings
E.
Kleinfeld;
F.
Kosier;
J. Marshall
Osborn;
D.
Rodabaugh
473-483
Nonassociative algebras satisfying identities of degree three
Frank
Kosier;
J. Marshall
Osborn
484-492
Metrical theorems on fractional parts of sequences
Wolfgang M.
Schmidt
493-518
Operation calculus in Banach algebras for algebra-valued functions
S.
Kantorovitz
519-537
The Hurewicz homomorphism and finite homotopy invariants
M.
Arkowitz;
C. R.
Curjel
538-551
Instantaneous states of Markov processes
Gerald
Smith
185-195
Overrings of commutative rings. II. Integrally closed overrings
Edward D.
Davis
196-212
Integral representations of dihedral groups of order $2p$
Myrna Pike
Lee
213-231
Contributions to the theory of optimal control. A general procedure for the computation of switching manifolds
D. C.
Lewis;
P.
Mendelson
232-244
General product measures
E. O.
Elliott;
A. P.
Morse
245-283
The geometry of Banach spaces. Smoothness
Dennis F.
Cudia
284-314
On obstruction theory in orientable fiber bundles
Mark
Mahowald
315-349
Invariant and reducing subalgebras of measure preserving transformations
Roy L.
Adler
350-360
Analytic mean periodic functions
D. G.
Dickson
361-374
Differential equations and differential calculus in Montel spaces
E.
Dubinsky
1-21
Asymptotic behavior of solutions of ordinary difference equations
Charles V.
Coffman
22-51
Boundary images of meromorphic functions
P. T.
Church
52-78
Spectral representations for some unbounded normal operators
George
Maltese
79-87
On the existence and characterization of best nonlinear Tchebycheff approximations
John R.
Rice
88-97
Injective structures
J.-M.
Maranda
98-135
Weakly wandering sets and invariant measures
Arshag B.
Hajian;
Shizuo
Kakutani
136-151
Invariant measures for Markov processes
Yuji
Ito
152-184
Weak eigenvectors and the functional calculus
R. T.
Harris
367-384
Extending a disk to a sphere
Joseph M.
Martin
385-399
Transformations preserving the Grassmannian
William C.
Nemitz
400-410
Derivations on an arbitrary vector bundle
C. J.
Henrich
411-419
On obstructions in sphere bundles and immersion of manifolds
J.
Levine
420-429
On Amitsur's complex and restricted Lie algebras
Astrid J.
Berkson
430-443
Tchebycheff approximation in several variables
John R.
Rice
444-466
Substitution minimal sets
W. H.
Gottschalk
467-491
An application of valuation theory to rings of continuous real and complex-valued functions
Norman L.
Alling
492-508
Higher order operations
E.
Spanier
509-539
A theorem in homological algebra and stable homotopy of projective spaces
Arunas
Liulevicius
540-552
Groups of automorphisms of Borel spaces
V. S.
Varadarajan
191-220
Baer-invariants of algebras
A.
Fröhlich
221-244
Duality theorems for some commutative semigroups
C. W.
Austin
245-256
Poincar\'e duality and cobordism
R.
Lashof
257-277
Asymptotic behavior of the eigenvalues of certain integral equations
Harold
Widom
278-295
Some fundamental lemmas on projective schemes
Yoshikazu
Nakai
296-302
The isomorphism problem for some classes of multiplicative systems
Trevor
Evans
303-312
Noncommutative unique factorization domains
P. M.
Cohn
313-331
Rings with a weak algorithm
P. M.
Cohn
332-356
On the residual nilpotence of some varietal products
Gilbert
Baumslag
357-365
On the differential geometry of homogeneous vector bundles
Phillip A.
Griffiths
1-34
Stationary equations in continuous time Markov chains
Rupert G.
Miller
35-44
Partially bounded $J$-fractions
Lester G.
Riggs;
W. T.
Scott
45-55
Random walks and a sojourn density process of Brownian motion
F. B.
Knight
56-86
Differentiable open maps on manifolds
P. T.
Church
87-100
On the algebra of representative functions of a Lie algebra
William B.
Giles
101-120
The role of the Appell transformation in the theory of heat conduction
D. V.
Widder
121-134
Creativity and effective inseparability
Raymond M.
Smullyan
135-145
$\varepsilon $-mappings onto polyhedra
Sibe
Mardešić;
Jack
Segal
146-164
Local partial differential algebra
S.
Halfin;
A.
Robinson
165-180
Topological loops with invariant uniformities
Sigmund
Hudson
181-190
Noncommuting random products
Harry
Furstenberg
377-428
Two-element generation of the symplectic group
Peter
Stanek
429-436
On semi-parabolic Riemann surfaces
Robert D. M.
Accola
437-448
Holomorphic functions with positive real part on polycylinders
A.
Korányi;
L.
Pukánszky
449-456
Analytic iteration
Eri
Jabotinsky
457-477
Energy inequalities for hyperbolic equations in several variables with multiple characteristics and constant coefficients
Gideon
Peyser
478-490
Picard-Vessiot theory of linear homogeneous difference equations
Charles H.
Franke
491-515
Some subgroup theorems for free $\mathfrak{v}$-groups
Gilbert
Baumslag
516-525
Free bases for normal subgroups of free groups
D. E.
Cohen;
R. C.
Lyndon
526-537
The admissible mean values of a stochastic process
T. S.
Pitcher
538-546
Errata to ``Summability $C$ of series of surface spherical harmonics''
Aaron
Siegel
547-548
Separation of the $n$-sphere by an $(n-1)$-sphere
James C.
Cantrell
185-194
Differential forms on general commutative algebras
George S.
Rinehart
195-222
Recursive enumerability and the jump operator
Gerald E.
Sacks
223-239
On spherical characteristic cohomology
A. W.
Adler
240-250
On the Schr\"odinger and heat equations for nonnegative potentials
Jacob
Feldman
251-264
Extreme positive operators and homomorphisms
R. R.
Phelps
265-274
Homotopy associativity of $H$-spaces. I
James Dillon
Stasheff
275-292
Homotopy associativity of $H$-spaces. II
James Dillon
Stasheff
293-312
Combinatorial lemmas in higher dimensions
O.
Barndorff-Nielsen;
Glen
Baxter
313-325
Differential operators preserving relations of automorphy
R. C.
Gunning
326-352
Higher torsion in $H$-spaces
William
Browder
353-375
Simple algebras and derivations
Klaus
Hoechsmann
1-12
Scattering for hyperbolic equations
Walter A.
Strauss
13-37
Some geometric consequences of conformal structure
Tilla
Klotz
38-53
A generalization of the transfer map in the cohomology of groups
Leonard
Evens
54-65
A Radon-Nikodym theorem in dimension lattices
Samuel S.
Holland
66-87
Proximity relations in transformation groups
Jesse Paul
Clay
88-96
Minimal topological spaces
Manuel P.
Berri
97-105
Recursive functionals and quantifiers of finite types. II
S. C.
Kleene
106-142
The Hahn embedding theorem for abelian lattice-ordered groups
Paul
Conrad;
John
Harvey;
Charles
Holland
143-169
Existence in the large of parallelism homomorphisms
Robert
Hermann
170-183
The critical points of a linear combination of Green's functions
Morris
Marden
369-381
Limit ultrapowers
H. Jerome
Keisler
382-408
Cameron-Martin translation theorems in the Wiener space of functions of two variables
J.
Yeh
409-420
An isoperimetric inequality for moments of inertia of plane convex sets
Tsuan Wu
Ting
421-431
Induced maps for Postnikov systems
Donald W.
Kahn
432-450
Almost locally polyhedral curves in Euclidean $n$-space
J. C.
Cantrell;
C. H.
Edwards
451-457
Left valuation rings and simple radical rings
Edward C.
Posner
458-465
On generators of the Banach algebras $l\sb{1}$ and $L\sb{1}(0,\,\infty )$
D. J.
Newman;
J. T.
Schwartz;
H. S.
Shapiro
466-484
Reduction of ordinary differential equations to the Birkhoff canonical form
H. L.
Turrittin
485-507
Bilateral birth and death processes
William E.
Pruitt
508-525
Determination of $H\sp{\ast} ({\rm BO}(k,\cdots,\infty ),Z\sb{2})$ and $H\sp{\ast} ({\rm BU}(k,\cdots,\infty ),Z\sb{2})$
Robert E.
Stong
526-544
Some sample function properties of a process with stationary independent increments
Shashanka Shekhar
Mitra
545-558
Errata: Kernel constructions and Borel sets
A. H.
Stone
558
Principal fibrations
Jean-Pierre
Meyer
177-185
Invariant manifolds for some autonomous systems
F. S.
Van Vleck
186-196
Isology theory
Arthur H.
Copeland
197-216
Order isomorphisms of $B\sp{\ast} $ algebras
P. E.
Miles
217-236
On the dominance of partial differential operators
Martin
Schechter
237-260
Families of principal solutions of ordinary differential equations.
Erling William
Chamberlain
261-272
Hereditary orders
Manabu
Harada
273-290
Tchebycheff quadrature on the infinite interval
J. L.
Ullman
291-299
Meromorphic starlike multivalent functions
W. C.
Royster
300-308
A sufficient condition for total monotonicity
B. E.
Rhoades
309-319
Unbounded coverings of Riemann surfaces and extensions of rings of meromorphic functions
Helmut
Röhrl
320-346
Reducible von Neumann geometries
Israel
Halperin
347-359
Roots and canonical forms for circulant matrices
C. M.
Ablow;
J. L.
Brenner
360-376
Two theorems about relations
H.
Kenyon
1-9
Coerciveness in $L\sp{p}$
Martin
Schechter
10-29
Darboux property for functions of several variables
C. J.
Neugebauer
30-37
Projection constants and spaces of continuous functions
J. R.
Isbell;
Z.
Semadeni
38-48
Homomorphisms of filtered modules
Alex
Heller
49-57
Kernel constructions and Borel sets
A. H.
Stone
58-70
Extensions of ordered groups and sequence completion
Charles
Holland
71-82
A decomposition theory for representations of $C\sp{\ast} $-algebras.
Edward G.
Effros
83-106
Abstract ergodic theorems
Alexandra
Ionescu Tulcea;
Cassius
Ionescu Tulcea
107-124
An addendum to the general coefficient theorem
James A.
Jenkins
125-128
The collineation groups of free planes
Reuben
Sandler
129-139
The convergence of measures on parametric surfaces
J. H.
Michael
140-152
On differential Hopf algebras
William
Browder
153-176
The Marcel Riesz theorem on conjugate functions
Frank
Forelli
369-390
Extreme eigenvalues of $N$-dimensional convolution operators
Harold
Widom
391-414
Real functions coming from flows on compact spaces and concepts of almost periodicity
L.
Auslander;
F.
Hahn
415-426
Matrix representations of $d$-simple semigroups
R. J.
Warne
427-435
Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion
Daniel
Ray
436-444
Some Hilbert spaces of analytic functions. I
Louis
de Branges
445-468
Limiting distributions for critical multitype branching processes with discrete time
T. W.
Mullikin
469-494
Boundary theory for recurrent Markov chains
John G.
Kemeny;
J. Laurie
Snell
495-520
Islands and peninsulas on arbitrary Riemann surfaces
Leo
Sario
521-533
On certain representations of the measure algebra of a locally compact abelian group
Horst
Leptin
534-540
Errata: ``Homogeneous manifolds of zero curvature''
Joseph A.
Wolf
540
On the residual finiteness of generalised free products of nilpotent groups
Gilbert
Baumslag
193-209
Some multiplicative properties of complete ideals
H. T.
Muhly;
M.
Sakuma
210-221
Regularity and positional games
A. W.
Hales;
R. I.
Jewett
222-229
Abelian extensions of arbitrary fields
D. K.
Harrison
230-235
Extremal problems for analytic functions with positive real part and applications
M. S.
Robertson
236-253
Hausdorff summability methods, addendum
B. E.
Rhoades
254-258
On the classification of noncompact surfaces
Ian
Richards
259-269
A lower bound for the volume of strictly convex bodies with many boundary lattice points
George E.
Andrews
270-279
Solutions to cooperative games without side payments
Bezalel
Peleg
280-292
Homeotopy groups
G. S.
McCarty
293-304
Transformation groups on cohomology projective spaces
J. C.
Su
305-318
A useful functor and three famous examples in topology
R. F.
Williams
319-329
On finitely generated modules over Noetherian rings
J. P.
Jans
330-340
Correction to ``Construction of automorphic forms on $H$-groups and supplementary Fourier series''
Marvin Isadore
Knopp
341-345
Operational calculus and the finite part of divergent integrals
T. K.
Boehme
346-368
Semi-discrete analytic functions
G. J.
Kurowski
1-18
Locally compact semigroups in which a subgroup with compact complement is dense
Karl Heinrich
Hofmann
19-51
Homogeneous locally compact groups with compact boundary
Karl Heinrich
Hofmann
52-63
Unique subdirect sums of prime rings
Lawrence
Levy
64-76
On a special class of regular $p$-groups
J. L.
Alperin
77-99
Simplification of turning point problems for systems of linear differential equations
Wolfgang
Wasow
100-114
Actions of elementary $p$-groups on manifolds
L. N.
Mann;
J. C.
Su
115-126
The asymptotic distribution of the number of zero-free intervals of a stable process
R. K.
Getoor
127-138
Classes of predictably computable functions
Robert W.
Ritchie
139-173
Some calculations of homotopy groups of symmetric spaces
Bruno
Harris
174-184
On the orthogonality of measures induced by $L$-processes
Marek
Fisz
185-192
Existence of interpolating functions of exponential type
Richard F.
DeMar
359-371
Measurable functions on Hilbert space
Leonard
Gross
372-390
On sets of algebraic integers whose remaining conjugates lie in the unit circle
David G.
Cantor
391-406
The action of an algebraic torus on the affine plane
A.
Gutwirth
407-414
Measures orthogonal to algebras and sets of antisymmetry
Irving
Glicksberg
415-435
Topological objects and sheaves
Yeaton H.
Clifton;
J. Wolfgang
Smith
436-452
A generalization of K. T. Chen's invariants for paths under transformation groups
H. H.
Johnson
453-461
The intersection of norm groups
James
Ax
462-474
Singular parabolic partial differential equations with time dependent coefficients
Doris S.
Stockton
475-495
Integral currents ${\rm mod}$ $2$
William P.
Ziemer
496-524
Polynomial approximation of Bernstein type
Gilbert
Strang
525-535
Sums of four squares in a quadratic ring
Harvey
Cohn;
Gordon
Pall
536-556
Axiomatic primary and tertiary decomposition theory
John A.
Riley
177-201
Quadratic division algebras
J. Marshall
Osborn
202-221
Fixed point free involutions and equivariant maps. II
P. E.
Conner;
E. E.
Floyd
222-228
On the algebraic closure of certain partially ordered fields
Walter
Strodt
229-250
On ring extensions for completely primary noncommutative rings
E. H.
Feller;
E. W.
Swokowski
251-263
Vector bundles and projective modules
Richard G.
Swan
264-277
An existence theorem for fields with Krull valuations
P.
Ribenboim
278-294
On commutative algebras of degree two
Robert H.
Oehmke
295-313
The fundamental group of certain deleted product spaces
C. W.
Patty
314-321
On some extremal functions and their applications in the theory of analytic functions of several complex variables
Józef
Siciak
322-357
Finite groups with isomorphic group algebras
D. B.
Coleman
1-8
Semigroups on a half-plane
J. G.
Horne
9-20
Regular $\mathcal{D}$-classes in measure semigroups
H. S.
Collins;
R. J.
Koch
21-31
Summability of Fourier series in $L\sp{p}(d\mu )$
Marvin
Rosenblum
32-42
Some Hilbert spaces of entire functions. IV
Louis
de Branges
43-83
On a theorem of D. Ridout in the theory of Diophantine approximations.
Aviezri S.
Fraenkel
84-101
Fourier transforms of certain classes of integrable functions
Robert
Ryan
102-111
On solutions of Chandrasekhar's integral equation
I. W.
Busbridge
112-117
A closure property of sets of vectors
Robert
Steinberg
118-125
On extensions of $H$-spaces
James Dillon
Stasheff
126-135
Direct and iterative methods for the solution of linear operator equations in Hilbert space
W. V.
Petryshyn
136-175
On Riesz and Riemann summability
Dennis C.
Russell
383-391
The Liouville theorem for a quasi-linear elliptic partial differential equation
S. Elwood
Bohn;
Lloyd K.
Jackson
392-397
A conjecture on weak compactness
Victor
Klee
398-402
Uniqueness of the invariant mean on Abelian topological semigroups
Indar S.
Luthar
403-411
Continuous matrices and approximate similarity
James C.
Lillo
412-419
Weak structural synthesis for certain Banach algebras
Paul
Civin
420-424
A complete set of unitary invariants for $3\times 3$ complex matrices
Carl
Pearcy
425-429
On convergence of stochastic processes
John
Lamperti
430-435
$m$-pseudocompactness
J. F.
Kennison
436-442
The cohomology of a subalgebra of the Steenrod algebra
Arunas L.
Liulevicius
443-449
Line element fields on manifolds
W. S.
Massey;
R. H.
Szczarba
450-456
Local operators on trigonometric series
Gen-ichirô
Sunouchi
457-461
Homogeneous manifolds of zero curvature
Joseph A.
Wolf
462-469
On an integral inequality of Z. Opial
Paul R.
Beesack
470-475
On the differentiability of the solutions of quasilinear partial differential equations
J.
Peetre
476-482
A class of non-Desarguesian affine planes
T. G.
Ostrom
483-487
On the least positive eigenvalue of integral equations with equimeasurable kernels
Binyamin
Schwarz
488-494
On the proximate linear orders of entire Dirichlet series
A. G.
Azpeitia
495-501
An almost everywhere existence theorem for solutions of Volterra functional equations
J.
Yeh
502-509
The Pythagorean theorem in certain symmetry classes of tensors
Marvin
Marcus;
Henryk
Minc
510-515
Invariant factors and two criteria for projectivity of modules
Maurice
Auslander;
David A.
Buchsbaum
516-522
On $\pi (x+y)\leq \pi (x)+\pi (y)$
Sanford L.
Segal
523-527
A note on some new finite division ring planes
R.
Sandler
528-531
Functions whose derivative has a positive real part
T. H.
MacGregor
532-537
Commutative algebras and cohomology
D. K.
Harrison
191-204
Theory of covering spaces
Saul
Lubkin
205-238
A fiber homotopy extension theorem
Guy
Allaud;
Edward
Fadell
239-251
A decomposition theory for unitary representations of locally compact groups
John A.
Ernest
252-277
Dual et quasi-dual d'une alg\`ebre de Banach involutive
J.
Dixmier
278-283
Summability $C$ of series of surface spherical harmonics
Aaron
Siegel
284-307
The embedding of two-spheres in the four-sphere
Herman
Gluck
308-333
Higher degrees of distributivity in lattices of continuous functions
Elliot Carl
Weinberg
334-346
Constructive analogues of the group of permutations of the natural numbers
Clement F.
Kent
347-362
On unitary equivalence of arbitrary matrices
Heydar
Radjavi
363-373
Nonexistence of low dimension relations between Stiefel-Whitney classes
Edgar H.
Brown
374-382
Chain sequences and orthogonal polynomials
T. S.
Chihara
1-16
Real commutative semigroups on the plane. II
J. G.
Horne
17-23
The problem of locally-$A$ functions in a commutative Banach algebra $A$
Richard
Arens
24-36
Uniformly semi-primitive multiplicative processes
Garrett
Birkhoff
37-51
Overrings of commutative rings. I. Noetherian overrings
Edward D.
Davis
52-61
Semi-stable stochastic processes
John
Lamperti
62-78
On necessary and sufficient conditions for the convergence of the renewal density
Walter L.
Smith
79-100
Classifications of recursive functions by means of hierarchies
Solomon
Feferman
101-122
Semi-Gaussian subspaces
D. L.
Burkholder
123-131
Connected abelian groups in compact loops
Karl Heinrich
Hofmann
132-143
Symmetric random walk
L. A.
Shepp
144-153
On global asymptotic stability of solutions of differential equations.
Philip
Hartman;
Czesław
Olech
154-178
Iteration methods for nonlinear problems
Samuel
Schechter
179-189
On the existence of real-closed fields that are $\eta \sb{\alpha }$-sets of power $\aleph \sb{\alpha }$
Norman L.
Alling
341-352
Rings and quasiconformal mappings in space
F. W.
Gehring
353-393
A comparison of the work done by generalized sequential machines and Turing machines
Seymour
Ginsburg;
Gene F.
Rose
394-402
Some imbedding and nonimbedding theorems for $n$-manifolds
Beauregard
Stubblefield
403-420
Killing the middle homotopy groups of odd dimensional manifolds
C. T. C.
Wall
421-433
First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path
Z.
Ciesielski;
S. J.
Taylor
434-450
Category-isomorphisms and endomorphism rings of modules
Kiiti
Morita
451-469
The equation of evolution in a Banach space
Joanne
Elliott
470-483
Solvable groups and modular representation theory
Paul
Fong
484-494
A subgroup theorem for free nilpotent groups
S.
Moran
495-515
Locally affine spaces with nilpotent fundamental groups
J. C.
Sanwal
516-524
Tensor products and related questions
B. R.
Gelbaum
525-548
A decomposition for complete normed abelian groups with applications to spaces of additive set functions
Richard B.
Darst
549-558
Errata to ``Decision problems of finite automata design and related arithmetics''
Calvin C.
Elgot
558-559
Errata to ``Some new analytical techniques and their application to irregular cases for the third order ordinary linear boundary-value problem''
Nathaniel R.
Stanley
559
Some probabilistic theorems on Diophantine approximations
Harry
Kesten
189-217
On the random walk and Brownian motion
Frank B.
Knight
218-228
A superlinear Sturm-Liouville problem
George H.
Pimbley
229-248
Approximation in systems of real-valued continuous functions
Frank W.
Anderson
249-271
Representations of relatively complemented modular lattices
Bjarni
Jónsson
272-303
States and representations
Richard V.
Kadison
304-319
Plane semigroups
Paul S.
Mostert
320-328
The extension of interiority, with some applications
C. J.
Titus;
G. S.
Young
329-340
On the cohomology of exact sequences of compact groups
Sufian Y.
Husseini
1-29
Cocycle formulas for homotopy classification; maps into projective and lens spaces
Paul
Olum
30-44
Hermitian moment sequences
J. S.
MacNerney
45-81
Occupation times for Markov and semi-Markov chains
Harry
Kesten
82-112
The module type of a ring
W. G.
Leavitt
113-130
Positive bases for linear spaces
Richard L.
McKinney
131-148
Linear symmetries of free boson fields
David
Shale
149-167
Construction of automorphic forms on $H$-groups and supplementary Fourier series
Marvin Isadore
Knopp
168-188
Differential forms on regular affine algebras
G.
Hochschild;
Bertram
Kostant;
Alex
Rosenberg
383-408
Arithmetic properties of Bernoulli convolutions
Adriano M.
Garsia
409-432
Extremals on compact $E$-surfaces
Eugene M.
Zaustinsky
433-445
A class of characteristic-value problems
R. M.
Moroney
446-470
Linear connexions with zero torsion and recurrent curvature
Yung-chow
Wong
471-506
Analyticity in certain Banach spaces
Errett
Bishop
507-544
Erratum to volume 96; Errata to volume 101
545
Exponential convergence rates for the law of large numbers
Leonard E.
Baum;
Melvin
Katz;
Robert R.
Read
187-199
On the category of indecomposable distributions on topological groups
K. R.
Parthasarathy;
R.
Ranga Rao;
S. R. S.
Varadhan
200-217
A decomposition theory for finite groups with applications to $p$-groups
Paul M.
Weichsel
218-226
Generalized homology theories
George W.
Whitehead
227-283
Difference properties for continuity and Riemann integrability on locally compact groups
F. W.
Carroll
284-292
The real cohomology ring of a sphere bundle over a differentiable manifold
D. G.
Malm
293-298
On a class of nonflexible algebras
Frank
Kosier
299-318
Torsion free and projective modules
Hyman
Bass
319-327
Families of measures and representations of algebras of operators
E. J.
McShane
328-345
Derivations on $p$-adic fields
Nickolas
Heerema
346-351
Cohomology groups of commutative Banach algebras
Herbert
Kamowitz
352-372
Some contractible open $3$-manifolds
D. R.
McMillan
373-382
Concentric solid tori in the $3$-sphere
C. H.
Edwards
1-17
Injective dimension in Noetherian rings
Hyman
Bass
18-29
A problem on partitions with a prime modulus $p\geq 3$
Peter
Hagis
30-62
Finitely generated pathological extensions of difference fields
Albert E.
Babbitt
63-81
Variational methods for functions with positive real part
M. S.
Robertson
82-93
Quasi-additive set functions and the concept of integral over a variety
Lamberto
Cesari
94-113
Extension problem for quasi-additive set functions and Radon-Nikodym derivatives
Lamberto
Cesari
114-146
Bodies for which harmonic functions satisfy the mean value property
Avner
Friedman;
Walter
Littman
147-166
Functions satsifying the mean value property
Avner
Friedman;
Walter
Littman
167-180
A note on the mean value property
Adriano M.
Garsia
181-186
A characterization of the $n$-sphere
Kyung Whan
Kwun
377-383
Point transitive transformation groups
Robert
Ellis
384-395
Hausdorff summability methods
B. E.
Rhoades
396-425
Malcev algebras
Arthur A.
Sagle
426-458
Coincidence sets and transformations of function spaces
James V.
Whittaker
459-476
Best approximations and interpolating functions
John R.
Rice
477-498
Symmetrization of rings in space
F. W.
Gehring
499-519
On some classes of analytic functions of several variables
Adam
Korányi
520-554
Erratum to ``Reflective $N$-prime rings with the ascending chain condition''
E. H.
Feller;
E. W.
Swokowski
555
Maximal commutative algebras of linear transformations
R. C.
Courter
177-199
Bounds for certain sums; a remark on a conjecture of Mahler
Wolfgang M.
Schmidt
200-210
On quotient varieties and the affine embedding of certain homogeneous spaces
Maxwell
Rosenlicht
211-223
The cohomology ring of a finite group
Leonard
Evens
224-239
Function-theoretic characterization of Einstein spaces and harmonic spaces
Avner
Friedman
240-258
Ordinal factorization of finite relations
C. C.
Chang
259-293
A surface is tame if its complement is $1$-ULC
R. H.
Bing
294-305
Quadratic variational theory and linear elliptic partial differential equations
Magnus R.
Hestenes
306-350
Some new analytical techniques and their application to irregular cases for the third order ordinary linear boundary-value problem
Nathaniel R.
Stanley
351-376
Abstract: 1. For the operator $T_3^ - (D)$ defined by $- {d^3}/d{x^3}$ and a triple of boundary conditions irregular in the sense of Birkhoff, the reduction of this triple to canonical forms is implicit in the reduction made for a more general third order operator (Theorem 1.2). 2. A new technique is developed for calculating the Green's function for the nth order ordinary linear boundary-value problem (Theorem 2.4), and is applied to $T_3^ -$; a necessary and sufficient condition is given for the identification of degenerate sets of boundary conditions for $T_3^ -$ (Theorem 2.6). 3. A new technique is developed for calculating asymptotic expansions for large zeros of exponential sums, and the form of the expansion, which includes a logarithmic asymptotic series, is established by induction (Theorem 3.1); expansions for the cube roots of the eigenvalues of $ T_3^ -$ then follow as special cases. 4. A theorem of Dunford and Schwartz (Theorem 4.0) giving a sufficient condition for completeness of eigenfunctions in terms of growth of the norm of the resolvent operator, is applied to prove that, with a possible exception, the eigenfunctions of $T_3^ -$ span $ {L_2}(0,1)$ (Theorem 4.5).
The weak topology of a Banach space
H. H.
Corson
1-15
Commutators in the special and general linear groups
R. C.
Thompson
16-33
The second dual of the space of continuous functions. III
Samuel
Kaplan
34-51
Analytically uniform spaces and some applications
Leon
Ehrenpreis
52-74
First passage times for symmetric stable processes in space
R. K.
Getoor
75-90
Oscillation criteria for self-adjoint differential systems
William T.
Reid
91-106
On the characterization of linear and projective linear groups. II
John H.
Walter
107-123
Locally compact transformation groups
James
Glimm
124-138
On the definition and properties of certain variational integrals
James
Serrin
139-167
Likelihood ratios for diffusion processes with shifted mean values.
T. S.
Pitcher
168-176
Rings of integer-valued continuous functions
R. S.
Pierce
371-394
On difference methods for the solution of a Cauchy problem for a hyperbolic equation with data on a parabolic line
Hajimu
Ogawa
395-403
On difference methods for the solution of a Tricomi problem
Hajimu
Ogawa
404-424
Analytic distribution kernels
J.
Barros-Neto
425-438
Algebras of holomorphic functions on one-dimensional varieties
Hugo
Rossi
439-458
A general class of linear transformations of Wiener integrals
D. A.
Woodward
459-480
On the characterization of linear and projective linear groups. I
John H.
Walter
481-529
Filtered algebras and representations of Lie algebras
R.
Sridharan
530-550
The homology of twisted cartesian products
R. H.
Szczarba
197-216
Partition functions whose logarithms are slowly oscillating
S.
Parameswaran
217-240
Wedges in abelian groups and in linear topological spaces
J. G.
MacCarthy
241-251
Extreme eigenvalues of translation kernels
Harold
Widom
252-262
On the extreme eigenvalues of Toeplitz matrices
Seymour V.
Parter
263-276
Semi-similarity invariants for spectral operators on Hilbert space.
Alvin N.
Feldzamen
277-324
Constructive versions of ordinal number classes
Donald L.
Kreider;
Hartley
Rogers
325-369
Correction to ``Generalized means''
Fred B.
Wright
370
Transient behavior of single-server queueing processes with Erlang input
Lajos
Takács
1-28
Semi-inner-product spaces
G.
Lumer
29-43
The Eichler cohomology groups and automorphic forms
R. C.
Gunning
44-62
On differentiably simple algebras
Laurence
Harper
63-72
Some Hilbert spaces of entire functions. III
Louis
de Branges
73-115
On the Kronecker products of irreducible representations of the $2\times 2$ real unimodular group. I
Lajos
Pukánszky
116-152
Orthonormal sets with non-negative Dirichlet kernels. II
J. J.
Price
153-161
On the structure of orbit spaces of generalized manifolds
Glen E.
Bredon
162-196
Automorphisms of formal power series under substitution
Benjamin
Muckenhoupt
373-383
Change of time scale for Markov processes
Steven
Orey
384-397
Discrete uniform subgroups of solvable Lie groups
Louis
Auslander
398-402
H\"older conditions for realizations of Gaussian processes
Z.
Ciesielski
403-413
Almost periodicity and convergent trigonometric series
Robert G.
Uttley
414-424
Finite groups with nilpotent centralizers
Michio
Suzuki
425-470
A convergence equivalence related to polynomials orthogonal on the unit circle
Glen
Baxter
471-487
$p$-adic groups of transformations
G. E.
Bredon;
Frank
Raymond;
R. F.
Williams
488-498
Central intertwining numbers for representations of finite groups
John A.
Ernest
499-508
On differential operators and connections
H. K.
Nickerson
509-539
On the distribution of first hits for the symmetric stable processes.
R. M.
Blumenthal;
R. K.
Getoor;
D. B.
Ray
540-554
Nets and recurrence in transformation groups
F. J.
Hahn
193-200
An order relation among topological spaces
C. N.
Maxwell
201-204
Examples relating to normality in topological spaces
H. H.
Corson
205-211
Some structure theorems for lattice-ordered groups
Paul
Conrad
212-240
On certain character sums
B. W.
Brewer
241-245
Decomposition theory for nonsemimodular lattices
Peter
Crawley
246-254
Some semigroups on an $n$-cell
Anne Lester
Hudson
255-263
Reflective $N$-prime rings with the ascending chain condition
E. H.
Feller;
E. W.
Swokowski
264-271
An asymptotic expression for the number of solutions of a general class of Diophantine equations
George E.
Andrews
272-277
A limit theorem for a function of the increments of a decomposable process
Robert
Cogburn;
Howard G.
Tucker
278-284
A solution of Chandrasekhar's integral equation
Charles
Fox
285-291
A note on abstract integration
Richard B.
Darst
292-297
Tchebycheff approximations by functions unisolvent of variable degree
John R.
Rice
298-302
Simplifying the structure of second order partial differential equations
Avner
Friedman
303-307
On the restricted Ces\`aro summability of double Fourier series
A. J.
White
308-319
Extension of a result of Beurling on invariant subspaces
Peter L.
Duren
320-324
Recurrent tensors on a linearly connected differentiable manifold
Yung-chow
Wong
325-341
Examples of non-Gaussian quasi-invariant distributions in Hilbert space
Jacob
Feldman
342-349
A regression problem concerning stationary processes
M. P.
Heble
350-371
On canonical conformal maps of multiply connected domains
H. J.
Landau
1-20
Generalized archimedean groups
Elias
Zakon
21-40
Compact abelian transformation groups
L. N.
Mann
41-59
Note on a theorem of Grosswald
Joseph A.
Cima
60-61
Homogeneous extensions of positive linear operators
Dorothy
Maharam
62-82
Vitali's theorem for invariant measures
W. W.
Comfort;
Hugh
Gordon
83-90
Recurrence-partitions of finite measure spaces with applications to ergodic theory
Rudolph W.
Preisendorfer;
Bernard W.
Roos
91-101
On Diophantine approximations
L. C.
Eggan
102-117
Some Hilbert spaces of entire functions. II
Louis
de Branges
118-152
Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations
Seymour V.
Parter
153-192
Extending homeomorphisms on the pseudo-arc
G. R.
Lehner
369-394
The $G$ and $H$ functions as symmetrical Fourier kernels
Charles
Fox
395-429
Stable processes and integral equations
Harold
Widom
430-449
On the K\"unneth theorem
Alex
Heller
450-458
On projective class groups
Dock Sang
Rim
459-467
On automorphisms and derivations of simple rings with minimum conditions
Andrzej
Białynicki-Birula
468-484
Generalized eigenfunction expansions for operator algebras
R. T.
Harris
485-500
The \v Silov boundary induced by a certain Banach algebra
W. W.
Comfort
501-517
Banach algebras with scattered structure spaces
Irving
Glicksberg
518-526
Abelian torsion groups having a minimal system of generators
Samir A.
Khabbaz
527-538
The core of a cooperative game without side payments
Robert J.
Aumann
539-552
Generalized means
Fred B.
Wright
187-203
Currents and area
Hebert
Federer
204-233
Inscribed squares in plane curves
R. P.
Jerrard
234-241
Modules over a complete discrete valuation ring
Joseph
Rotman;
Ti
Yen
242-254
On a theorem of A. Borel
W. H.
Cockcroft
255-262
On the characters of $p$-solvable groups
P.
Fong
263-284
Matrices over polynomial rings
David
Lissner
285-305
On a decomposition theorem for measures in Euclidean $n$-space
William C.
Nemitz
306-333
Cartan decompositions for $L\sp{\ast} $ algebras
John R.
Schue
334-349
On homologically trivial $3$-manifolds
D. R.
McMillan
350-367
A new proof and generalizations of the Cauchy-Kowalewski theorem
Avner
Friedman
1-20
Decision problems of finite automata design and related arithmetics
Calvin C.
Elgot
21-51
Uniform distribution of sequences of integers
Ivan
Niven
52-61
On the integration of the diffusion equation with boundary conditions.
R. S.
Phillips
62-84
Viscous fluids, elasticity and function-theory. I
James
Sanders
85-147
Cohomology of Lie triple systems and Lie algebras with involution
Bruno
Harris
148-162
A fluxintegral theorem for functions which have harmonic support
Guy
Johnson
163-185
Correction on a previous paper
E. M.
Stein
186
The Brauer group of a commutative ring
Maurice
Auslander;
Oscar
Goldman
367-409
Imbedding of abelian categories
Saul
Lubkin
410-417
Tight Galois connections and complete distributivity
George N.
Raney
418-426
Local differential algebra
Abraham
Robinson
427-456
Direct products of modules
Stephen U.
Chase
457-473
On dimension subgroups
Gerald
Losey
474-486
Uniqueness of the topology in certain compact groups
T. E.
Stewart
487-494
Modules with descending chain condition
Eben
Matlis
495-508
On the passage to equilibrium of Gibbsian ensembles
Clifford C.
Brown
509-522
Some developments in the theory of numerations
A. H.
Kruse
523-553
On the group of all homeomorphisms of a manifold
Gordon M.
Fisher
193-212
Isotopies in $3$-manifolds
J. M.
Kister
213-224
Periodicity modulo $m$ and divisibility properties of the partition function
Morris
Newman
225-236
A construction for the normalizer of a ring with local unit with applications to the theory of $L$-algebras
Barron
Brainerd
237-253
Martingales in a $\sigma $-finite measure space indexed by directed sets
Y. S.
Chow
254-285
On automorphisms of Lie algebras of classical type. III
George B.
Seligman
286-316
On bounded matrices and kinematic similarity
C. E.
Langenhop
317-326
On Amitsur's complex
Alex
Rosenberg;
Daniel
Zelinsky
327-356
The asymptotic expansions for the odd periodic Mathieu functions
Gertrude
Blanch
357-366
Maximal orders
Maurice
Auslander;
Oscar
Goldman
1-24
Lower bounds for the degree of approximation
G. G.
Lorentz
25-34
On the spectral theory of singular integral operators
Walter
Koppelman
35-63
Representations of complemented modular lattices
Bjarni
Jónsson
64-94
Area as the integral of lengths of contours
Raymond
Rishel
95-119
Separation theorems for some plane-like spaces
Steve
Armentrout
120-130
Singular integral equations in $L\sb{p}$
Harold
Widom
131-160
Obstruction theory of principal fibre bundles
Albert T.
Lundell
161-192
A class of arithmetical functions in several variables with applications to congruences
Eckford
Cohen
355-381
The loop-space functor in homological algebra
Alex
Heller
382-394
Bounds for determinants with positive diagonals
Emilie V.
Haynsworth
395-399
Some remarks on abstract machines
Seymour
Ginsburg
400-444
Equations in free groups
Roger C.
Lyndon
445-457
On monosplines of least deviation
R. S.
Johnson
458-477
Remark on free products of groups
P. J.
Hilton
478-487
A class of graphs
Paul
Kelly;
David
Merriell
488-492
On boundary value problems for systems of ordinary, nonlinear, second order differential equations
Philip
Hartman
493-509
One-parameter semigroups in a semigroup
Paul S.
Mostert;
Allen L.
Shields
510-517
Groups with parametric exponents
Roger C.
Lyndon
518-533
On certain coefficients of univalent functions. II
James A.
Jenkins
534-545
Errata
546-546
Differentiation of set functions using Vitali coverings
W. E.
Hartnett;
A. H.
Kruse
185-209
A two-dimensional operator identity with application to the change of sign in sums of random variables
Glen
Baxter
210-221
Elementary properties of ordered abelian groups
Abraham
Robinson;
Elias
Zakon
222-236
On uniformization of sets in topological spaces
Maurice
Sion
237-245
Note on degree of approximation by bounded analytic functions: Problem $\beta $
J. L.
Walsh
246-258
Some Hilbert spaces of entire functions
Louis
de Branges
259-295
Oscillation criteria for fourth-order linear differential equations.
Henry
Howard
296-311
A new class of continued fraction expansions for the ratios of Heine functions. III
Evelyn
Frank
312-321
The characterization of best nonlinear Tchebycheff approximations
John R.
Rice
322-340
On analytic sets in topological spaces
Maurice
Sion
341-354
Decomposition theory for lattices without chain conditions
R. P.
Dilworth;
Peter
Crawley
1-22
Some problems concerning Kummer's congruences for the Euler numbers and polynomials
L.
Carlitz;
Jack
Levine
23-37
Sequences generated by iteration
W. J.
Thron
38-53
A class of differential-functional systems. I
Robert B.
Kelman
54-66
On the cohomology of the real Grassmann complexes and the characteristic classes of $n$-plane bundles
Emery
Thomas
67-89
On uniqueness theorems for ordinary differential equations and for partial differential equations of hyperbolic type
J. B.
Diaz;
W. L.
Walter
90-100
The stable suspension of an Eilenberg-MacLane space
W. D.
Barcus
101-114
On the Gauss curvature of minimal surfaces
Robert
Osserman
115-128
On the inverse limit of Euclidean $N$-spheres
Morton
Brown
129-134
On the dimensions of the irreducible modules of Lie algebras of classical type
Charles W.
Curtis
135-142
The bilinear relation on open Riemann surfaces
Robert D. M.
Accola
143-161
On absolutely convergent exponential sums
Leon
Brown;
Allen
Shields;
Karl
Zeller
162-183
An extension of the general coefficient theorem
James A.
Jenkins
387-407
Nonlinear Volterra functional equations and linear parabolic differential systems
J.
Yeh
408-432
Wiener measure in a space of functions of two variables
J.
Yeh
433-450
Projection constants
B.
Grünbaum
451-465
Finitistic dimension and a homological generalization of semi-primary rings
Hyman
Bass
466-488
Sur la repr\'esentation d'une fonction arbitraire par des fonctions jouissant de la propri\'et\'e de Darboux
Solomon
Marcus
489-494
Some equivalence conditions for the uniform convergence in distribution of sequences of stochastic processes
Ernest G.
Kimme
495-515
A metrical theorem in geometry of numbers
Wolfgang M.
Schmidt
516-529
Analytic group kernels and Lie algebra kernels
R. A.
Macauley
530-553
On the construction of sets of mutually orthogonal Latin squares and the falsity of a conjecture of Euler
R. C.
Bose;
S. S.
Shrikhande
191-209
Well-quasi-ordering, the Tree Theorem, and Vazsonyi's conjecture
J. B.
Kruskal
210-225
Lorentz structures on the plane
J. Wolfgang
Smith
226-237
Uniqueness of Hahn-Banach extensions and unique best approximation.
R. R.
Phelps
238-255
Orthonormal sets with non-negative Dirichlet kernels
J. J.
Price
256-262
Some theorems on stable processes
R. M.
Blumenthal;
R. K.
Getoor
263-273
$L$-structure in $L$-spaces
F.
Cunningham
274-299
The lattice theory of Post algebras
George
Epstein
300-317
On a certain class of operator algebras
James G.
Glimm
318-340
Computable algebra, general theory and theory of computable fields.
Michael O.
Rabin
341-360
Convergence of interpolatory polynomials, $(0,\,1,\,2,\,4)$ interpolation
R. B.
Saxena
361-385
Group extensions by primary abelian groups
Saunders
Mac Lane
1-16
A new class of continued fraction expansions for the ratios of Heine functions. II
Evelyn
Frank
17-26
The homology of cyclic products
Richard G.
Swan
27-68
Hilbert space methods in the theory of Lie algebras
John R.
Schue
69-80
Unique continuation for elliptic equations
M. H.
Protter
81-91
The meaning of the form calculus in classical ideal theory
Harley
Flanders
92-100
On a class of nonlinear second-order differential equations
Zeev
Nehari
101-123
Sublinear differential polynomials
D. G.
Mead
124-136
Hankel multiplier transformations and weighted $p$-norms
Douglas L.
Guy
137-189
Correction and addition to ``Some theorems concerning partitions''.
Emil
Grosswald
190
The boundary behavior and uniqueness of solutions of the heat equation.
F. W.
Gehring
337-364
The dual spaces of $C\sp{\ast} $-algebras
J. M. G.
Fell
365-403
Integration and nonlinear transformations in Hilbert space
Leonard
Gross
404-440
Functional composition patterns and power series reversion
George N.
Raney
441-451
On automorphisms of Lie algebras of classical type. II
George B.
Seligman
452-482
A property of the differential ideal $y\sp{p}$
Kathleen B.
O’Keefe
483-497
On ordered divisible groups
Norman L.
Alling
498-514
Studies in Faber polynomials. I
Joseph L.
Ullman
515-528
Topologies and projections on Riesz spaces
Hugh
Gordon
529-551
The spectral theory of bounded functions
Carl S.
Herz
181-232
A fibering of a class of homogeneous complex manifolds
Jun-ichi
Hano;
Shoshichi
Kobayashi
233-243
On the powers of a real number reduced modulo one
Fred
Supnick;
H. J.
Cohen;
J. F.
Keston
244-257
Homomorphisms of transformation groups
Robert
Ellis;
W. H.
Gottschalk
258-271
A semigroup associated with a transformation group
Robert
Ellis
272-281
Combinatory methods and stochastic Kolmogorov equations in the theory ofqueues with one server
V. E.
Beneš
282-294
Probabilities on a compact group
Karl
Stromberg
295-309
Nodal noncommutative Jordan algebras and simple Lie algebras of characteristic $p$
R. D.
Schafer
310-326
On the spectral theory of symmetric finite operators
Harro
Heuser
327-336
The cyclic Jacobi method for computing the principal values of a complex matrix
G. E.
Forsythe;
P.
Henrici
1-23
On the asymptotic solutions of a class of ordinary differential equations of the fourth order. I. Existence of regular formal solutions
C. C.
Lin;
A. L.
Rabenstein
24-57
Energy inequalities for the solution of differential equations
Milton
Lees
58-73
On some classes of noncontinuable analytic functions
F. W.
Carroll
74-85
A generalization of the Banach and Mazur game
Haim
Hanani
86-102
Brownian motion of rotation
C. D.
Gorman
103-117
Eigen operators of ergodic transformations
Anatole
Beck
118-129
On the frequency of small fractional parts in certain real sequences. II
W. J.
LeVeque
130-149
A Tauberian theorem and its probability interpretation
Frank
Spitzer
150-169
A theorem on translation kernels in $n$ dimensions
Harold
Widom
170-180
The stability of weighted Lebesgue spaces
R. E.
Edwards
369-394
Infinite Cartesian products and a problem concerning homology local connectedness
Sibe
Mardešić
395-417
Curvature measures
Herbert
Federer
418-491
A uniqueness theorem for Haar measure
E. J.
Mickle;
T.
Radó
492-508
Parabolic equations of the second order
Avner
Friedman
509-530
On a sojourn time problem in the theory of stochastic processes
L.
Takács
531-540
Some remarks on commutative algebras of operators on Banach spaces
D. A.
Edwards;
C. T.
Ionescu Tulcea
541-551
Graphs and subgraphs. II
Oystein
Ore
185-204
On a class of distribution functions where the quotient follows the Cauchy law
R. G.
Laha
205-215
Generalized resolvents of ordinary differential operators
Earl A.
Coddington;
Richard C.
Gilbert
216-241
Immersions of manifolds
Morris W.
Hirsch
242-276
Hyperbolic equations with multiple characteristics
R. Bruce
Kellogg
277-291
On the growth of meromorphic functions with several deficient values.
Albert
Edrei;
Wolfgang H. J.
Fuchs
292-328
The second dual of the space of continuous functions. II
Samuel
Kaplan
329-350
Embedding any semigroup in a $\mathcal{D}$-simple semigroup
G. B.
Preston
351-355
On the semigroup structure of continua
Robert P.
Hunter
356-368
Some theorems on lacunary Fourier series, with extensions to compact groups
Edwin
Hewitt;
H. S.
Zuckerman
1-19
On a homomorphism property of certain Jordan algebras
A. A.
Albert;
L. J.
Paige
20-29
Nonoscillation theorems for a class of nonlinear differential equations
Richard A.
Moore;
Zeev
Nehari
30-52
A formula for the multiplicity of a weight
Bertram
Kostant
53-73
A new proof of the completeness of the \L ukasiewicz axioms
C. C.
Chang
74-80
On canonical conformal maps of multiply connected regions
J. L.
Walsh;
H. J.
Landau
81-96
Affine semigroups
Haskell
Cohen;
H. S.
Collins
97-113
An analysis of the Wang algebra of networks
R. J.
Duffin
114-131
On the embedding of vector lattices in $F$-rings
Barron
Brainerd
132-144
Some operators on uniform spaces
Seymour
Ginsburg;
J. R.
Isbell
145-168
Int\'egration d'une forme diff\'erentielle le long de certaines courbes non rectifiables
Georges
Glaeser
169-183
Formal solutions and a related equation for a class of fourth order differential equations of a hydrodynamic type
Rudolph E.
Langer
371-410
Spaces of functions with values in a Banach algebra
G. Philip
Johnson
411-429
On automorphisms of Lie algebras of classical type
George B.
Seligman
430-448
Representation of modular lattices and of relation algebras
Bjarni
Jónsson
449-464
A strong maximum modulus theorem for maximal function algebras
H. S.
Bear
464-469
Stability theory and adjoint operators for linear differential-difference equations
Richard
Bellman;
Kenneth L.
Cooke
470-500
Interpolation in $H\sp{\infty }$
D. J.
Newman
501-507
A multiplicity theory for Boolean algebras of projections in Banach spaces
William G.
Bade
508-530
On the law of the iterated logarithm for uniformly bounded orthonormal systems
Mary
Weiss
531-553
On characterizations of linear groups. I
Michio
Suzuki
191-204
On characterizations of linear groups. II
Michio
Suzuki
205-219
Expansions in terms of heat polynomials and associated functions
P. C.
Rosenbloom;
D. V.
Widder
220-266
Geometry of bounded domains
Shoshichi
Kobayashi
267-290
Some theorems on $L\sp{p}$ Fourier series
Richard P.
Gosselin
291-301
Functions which have harmonic support
Guy
Johnson
302-321
On a decompostion theorem of Federer
Earl J.
Mickle
322-335
Symmetric random walks on groups
Harry
Kesten
336-354
Integrated continuity conditions and degree of approximation by polynomials or by bounded analytic functions
J. L.
Walsh;
H. G.
Russell
355-370
A sampling theorem for stationary (wide sense) stochastic processes.
S. P.
Lloyd
1-12
Implicit alternating direction methods
Garrett
Birkhoff;
Richard S.
Varga
13-24
The average first recurrence time
Bernard
Friedman;
Ivan
Niven
25-34
Free lattices with infinite operations
Peter
Crawley;
Richard A.
Dean
35-47
On the arithmetic of order types
Anne C.
Morel
48-71
Model theory for the higher order predicate calculus
Steven
Orey
72-84
On a subrecursive hierarchy and primitive recursive degrees
Paul
Axt
85-105
The rate of increase of real continuous solutions of certain algebraic functional equations
K. L.
Cooke
106-124
Natural operations on differential forms
Richard S.
Palais
125-141
A generalization of the Gauss-Bonnet theorem
James
Eells
142-153
Vanishing algebras
Arthur B.
Simon
154-167
Conformal connections and conformal transformations
Noboru
Tanaka
168-190
Inclusion theorems for congruence subgroups
M.
Newman;
I.
Reiner
369-379
Semigroups with identity on a manifold
Paul S.
Mostert;
Allen L.
Shields
380-389
Formes diff\'erentielles m\'eromorphes localement exactes. Compl\'ements
Pierre
Dolbeault
390-398
The finite convolution transform
Jerome
Blackman;
Harry
Pollard
399-409
Solution about a singular point of a linear differential equation involving a large parameter
Robert
McKelvey
410-424
On integration of quasi-linear parabolic equations by explicit difference methods
J. Wolfgang
Smith
425-443
The law of the iterated logarithm for lacunary trigonometric series.
Mary
Weiss
444-469
On Hardy-Littlewood series
Mary
Weiss
470-479
Crossed-inverse and related loops
R.
Artzy
480-492
Finite reflection groups
Robert
Steinberg
493-504
The cohomology groups of a fibre space with fibre a space of type $\mathcal{K}(\pi, n)$. II
W. H.
Cockcroft
505-524
Mosaics of compact metric spaces
Walter F.
Davison
525-546
Connected ordered topological semigroups with idempotent endpoints. II.
A. H.
Clifford
193-208
Partly transitive modules and modules with proper isomorphic submodules
R. A.
Beaumont;
R. S.
Pierce
209-219
Partly invariant submodules of a torsion module
R. A.
Beaumont;
R. S.
Pierce
220-230
Polynomials of best approximation on a real finite point set. I
T. S.
Motzkin;
J. L.
Walsh
231-245
On a generalized Dirichlet problem for plurisubharmonic functions and pseudo-convex domains. Characterization of \v Silov boundaries
H. J.
Bremermann
246-276
On exterior derivatives and solutions of ordinary differential equations
Philip
Hartman
277-293
Weighted quadratic norms and ultraspherical polynomials. I
Richard
Askey;
Isidore
Hirschman
294-313
Weighted quadratic norms and ultraspherical polynomials. II
I. I.
Hirschman
314-329
An operational calculus for infinitesimal generators of semigroups.
A. V.
Balakrishnan
330-353
On the zeros of the derivatives of some entire functions
Robert M.
McLeod
354-367
Recursive functionals and quantifiers of finite types. I
S. C.
Kleene
1-52
Pansions and the theory of Fourier transforms
Jacob
Korevaar
53-101
On the distributivity and simple connectivity of plane topological lattices
Lee W.
Anderson
102-112
Classical differential equations on manifolds
L.
Auslander;
L.
Markus
113-128
On an integral of Marcinkiewicz
Daniel
Waterman
129-138
Composita, equations, and freely generated algebras
A.
Nerode
139-151
The existence of certain types of manifolds
M. L.
Curtis;
R. L.
Wilder
152-160
On the representation of cardinal algebras by directed sums
A. Bruce
Clarke
161-192
Stone-\v Cech compactifications of products
Irving
Glicksberg
369-382
Complex function algebras
H. S.
Bear
383-393
Specializations in differential algebra
Azriel
Rosenfeld
394-407
On the ideal structure of certain semirings and compactification of topological spaces
J. G.
Horne
408-430
Some properties of the Fourier-transform on semisimple Lie Groups. III.
L.
Ehrenpreis;
F. I.
Mautner
431-484
A class of bilinear forms
Howard A.
Osborn
485-498
On Lebesgue area. II
Maurice R.
Demers;
Herbert
Federer
499-522
The Hahn-Banach theorem and the least upper bound property
R. J.
Silverman;
Ti
Yen
523-526
Matroids and graphs
W. T.
Tutte
527-552
Dissipative operators and hyperbolic systems of partial differential equations
R. S.
Phillips
193-254
Sufficient conditions that a Peano-interior function have an extension with constant boundary values
William C.
Fox
255-271
On spaces having the homotopy type of ${\rm CW}$-complex
John
Milnor
272-280
A classification of immersions of the two-sphere
Stephen
Smale
281-290
On conjugate spaces of Nakano spaces
Sadayuki
Yamamuro
291-311
Derivative manifolds and Taylor series in the mean
D. S.
Greenstein
312-322
Nondegenerate surfaces of finite topological type
W. H.
Fleming
323-335
Primary intersections for two sided ideals of a Noetherian matrix ring.
Edmund H.
Feller
336-339
Repr\'esentation asymptotique des fonctions de Mathieu et des fonctions sph\'eroidales. II
Robert
Sips
340-368
On fiber spaces
Edward
Fadell
1-14
Integration theorems for gages and duality for unimodular groups
W. Forrest
Stinespring
15-56
Mandelbrojt's inequality and Dirichlet series with complex exponents.
T. M.
Gallie
57-72
Simple algebras and cohomology groups of arbitrary fields
S. A.
Amitsur
73-112
The asymptotic solutions of a linear differential equation of the second order with two turning points
Rudolph E.
Langer
113-142
Flexure
L. M.
Milne-Thomson
143-160
Characterizations of generalized uniserial algebras
Drury W.
Wall
161-170
Another homogeneous plane continuum
R. H.
Bing;
F. B.
Jones
171-192
Variation of the branch points for an analytic function
A. W.
Goodman
277-284
On variation formulas for univalent functions
A. W.
Goodman
285-294
On the critical points of a multivalent function
A. W.
Goodman
295-309
On the ramification of algebraic functions. II. Unaffected equations for characteristic two
Shreeram
Abhyankar
310-324
On the oscillation of solutions of self-adjoint linear differential equations of the fourth order
Walter
Leighton;
Zeev
Nehari
325-377
Linear operators on quasi-continuous functions
Ralph E.
Lane
378-394
Integration of paths---a faithful representation of paths by non-commutative formal power series
Kuo-sai
Chen
395-407
Cohomology of equivariant maps
J.
Dugundji
408-420
New simple Lie algebras of prime characteristic
Richard
Block
421-449
Analytic functions and the Fourier transform of distributions. II
Leon
Ehrenpreis
450-483
The application of stability analysis in the numerical solution of quasi-linear parabolic differential equations
Jim
Douglas
484-518
$L\sb{p}$ Fourier transforms on locally compact unimodular groups
R. A.
Kunze
519-540
Dual symmetry of projective sets in a finite modular lattice
S. P.
Avann
541-558
Errata, volume 85; Errata, volume 88
559-559
On the H\"older continuity of quasi-conformal and elliptic mappings
Robert
Finn;
James
Serrin
1-15
Stable processes with an absorbing barrier
Daniel
Ray
16-24
Some theorems on classes of recursively enumerable sets
J. C. E.
Dekker;
J.
Myhill
25-59
The variation of the sign of the real part of a meromorphic function on the unit circle
Jun-iti
Itô
60-78
Properties of primary noncommutative rings
Edmund H.
Feller
79-91
On positive-definite integral kernels and a related quadratic form.
J.
Chover;
J.
Feldman
92-99
On the location of the zeros of the derivative of rational functions of distance polynomials
Augusta
Schurrer
100-112
Some theorems concerning partitions
Emil
Grosswald
113-128
Variation formulas for multivalent functions
A. W.
Goodman
129-148
Lie algebras of characteristic $p$
Irving
Kaplansky
149-183
Some properties of partitions. II
A. O. L.
Atkin;
S. M.
Hussain
184-200
Local topological invariants. II
H. B.
Griffiths
201-244
Prime ideals in nonassociative rings
Bailey
Brown;
Neal H.
McCoy
245-255
Multiplicateurs et classes caract\'eristiques
J.-L.
Koszul
256-266
Algebraic structure and conformal mapping
Maurice
Heins
267-276
Admissibility of semigroup structures on continua
R. J.
Koch;
A. D.
Wallace
277-287
A new class of continued fraction expansions for the ratios of Heine functions
Evelyn
Frank
288-300
Algebraic topological methods for the synthesis of switching systems. I
J. Paul
Roth
301-326
Addendum to ``Invariant means and cones with vector interiors''
R. J.
Silverman;
Ti
Yen
327-330
Singular self-adjoint boundary value problems for the differential equation $Lx=\lambda Mx$
Fred
Brauer
331-345
Weak asymptotic properties of partitions
Eugene E.
Kohlbecker
346-365
Continua and various types of homogeneity
C. E.
Burgess
366-374
Uniqueness and the force formulas for plane subsonic flows
Robert
Finn;
David
Gilbarg
375-379
An occupation time theorem for a class of stochastic processes
John
Lamperti
380-387
A characterization of the everywhere regular solution of the reduced wave equation
O. G.
Owens
388-399
A functional calculus using singular Laplace integrals
Edward
Nelson
400-413
An existence theorem for second order parabolic equations
Edward
Nelson
414-429
On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz
E. M.
Stein
430-466
Algebraic analysis of many valued logics
C. C.
Chang
467-490
On the eigenvalues of certain Hermitian operators
Harold
Widom
491-522
Clans with zero on an interval
Haskell
Cohen;
L. I.
Wade
523-535
Inverse-producing extensions of normed algebras
Richard
Arens
536-548
An atomic decomposition of continua into aposyndetic continua
Louis F.
McAuley
1-11
Distributions in Hilbert space and canonical systems of operators
I. E.
Segal
12-41
On the prolongation of valuations
Peter
Roquette
42-56
On the homotopy classification of the extensions of a fixed map
W. D.
Barcus;
M. G.
Barratt
57-74
Invariant means and cones with vector interiors
R. J.
Silverman
75-79
Connected ordered topological semigroups with idempotent endpoints. I.
A. H.
Clifford
80-98
Lie algebras of types ${\rm A}$, ${\rm B}$, ${\rm C}$, ${\rm D}$, and ${\rm F}$
Marvin L.
Tomber
99-106
Remarks on lattice ordered groups and vector lattices. I. Carath\'eodory functions
Casper
Goffman
107-120
$B$-sets and fine-cyclic elements
Christoph J.
Neugebauer
121-136
Compact homogeneous hypersurfaces
Shoshichi
Kobayashi
137-143
A homotopy theorem for matroids. I
W. T.
Tutte
144-160
A homotopy theorem for matroids. II
W. T.
Tutte
161-174
The strain-energy function for anisotropic elastic materials
G. F.
Smith;
R. S.
Rivlin
175-193
Homological dimension in noetherian rings. II
Maurice
Auslander;
David A.
Buchsbaum
194-206
On a canonical conformal mapping of J. L. Walsh
James A.
Jenkins
207-213
A finite recursion formula for the coefficients in asymptotic expansions
T. D.
Riney
214-226
The geometry of numbers over algebraic number fields
K.
Rogers;
H. P. F.
Swinnerton-Dyer
227-242
Harmonic integrals on almost product manifolds
Bruce L.
Reinhart
243-276
Transformation groups on a homological manifold
C. T.
Yang
261-283
Orbits of highest dimension
D.
Montgomery;
C. T.
Yang
284-293
Adjoint functors
Daniel M.
Kan
294-329
Functors involving c.s.s. complexes
Daniel M.
Kan
330-346
On subgroups of the orthogonal group
Morio
Obata
347-358
Semigroup of endomorphisms of a locally compact group
Morikuni
Goto;
Naoki
Kimura
359-371
Homogeneous algebras on compact abelian groups
K.
de Leeuw
372-386
On integrating factors and on conformal mappings
Philip
Hartman
387-406
Construction and properties of some $6$-dimensional almost complex manifolds
Eugenio
Calabi
407-438
Mean convergence of martingales
Lester L.
Helms
439-446
Boundary behavior of generalized analytic functions
Kenneth
Hoffman
447-466
On approximation by bounded analytic functions
J. L.
Walsh
467-484
Lattices with involution
J. A.
Kalman
485-491
Regular curves on Riemannian manifolds
Stephen
Smale
492-512
Commutators and absolutely continuous operators
C. R.
Putnam
513-525
Generalized powers of the difference operator
D. B.
Sumner
526-540
On the realization of homology classes by submanifolds
Haruo
Suzuki
541-550
Fragments of many-valued statement calculi
Alan
Rose;
J. Barkley
Rosser
1-53
The dependence of an axiom of \L ukasiewicz
C. A.
Meredith
54
Proof of an axiom of \L ukasiewicz
C. C.
Chang
55-56
Addendum to the paper on partially stable algebras
A. A.
Albert
57-62
Lie simplicity of a special class of associative rings. II
Willard E.
Baxter
63-75
Automorphisms of the Gaussian unimodular group
Joseph
Landin;
Irving
Reiner
76-89
The Fourier coefficients of the invariants $j(2\sp{{1\over 2}};\,\tau )$ and $j(3\sp{{1\over 2}};\,\tau )$
John
Raleigh
90-107
A functional calculus for general operators in Hilbert space
Morris
Schreiber
108-118
A maximum principle for hyperbolic equations in a neighborhood of an initial line
M. H.
Protter
119-129
Partial fraction decompositions and expansions of zero
Joseph
Lehner
130-143
A device for studying Hausdorff moments
Chandler
Davis
144-158
Interpolation of operators with change of measures
E. M.
Stein;
G.
Weiss
159-172
On the deviations of the empiric distribution function of vector chance variables
J.
Kiefer;
J.
Wolfowitz
173-186
Some theorems concerning $2$-dimensional Brownian motion
Frank
Spitzer
187-197
On the analytic continuation of mapping functions
Philip
Davis;
Henry
Pollak
198-225
A class of noncommutative power-associative algebras
Robert H.
Oehmke
226-236
On the frequency of small fractional parts in certain real sequences.
William J.
LeVeque
237-260
Topologies of group algebras and a theorem of Littlewood
Sigurđur
Helgason
269-283
Collineations and generalized incidence matrices
D. R.
Hughes
284-296
Peano spaces which are either strongly cyclic or two-cyclic
G. Ralph
Strohl
297-308
On the fundamental theorems of the calculus
István S.
Gál
309-320
On the continuity and limiting values of functions
István S.
Gál
321-334
L'\'etude de l'\'equation $du/d\tau=A(\tau)u$ pour certaines classes d'op\'erateurs non born\'es de l'espace de Hilbert
C.
Foiaş;
Gh.
Gussi;
V.
Poenaru
335-347
Characteristic classes of homogeneous spaces
Alfred
Adler
348-365
The classification of birth and death processes
Samuel
Karlin;
James
McGregor
366-400
The cyclotomic numbers of order sixteen
Albert Leon
Whiteman
401-413
Spectral theory for operators on a Banach space
Errett
Bishop
414-445
The approximate functional equation of Hecke's Dirichlet series
T. M.
Apostol;
Abe
Sklar
446-462
On order-preserving integration
R. R.
Christian
463-488
Some generalizations of full normality
M. J.
Mansfield
489-505
A note on summability methods and spectral analysis
Carl S.
Herz
506-510
Some limit theorems for nonhomogeneous Markoff processes
A.
Fuchs
511-531
Abstract: We intend to study some problems related to the asymptotic behaviour of a physical system the evolution of which is markovian. The typical example of such an evolution is furnished by an homogeneous discrete chain with a finite number of possible states considered first by A. A. Markoff. In §1 we recall briefly the main results of this theory and in §2 we treat its obvious generalization to the continuous parameter case. In §3 we pass to the proper object of this paper and we establish a limit theorem for time-homogeneous Markoff processes. This limit theorem is then extended to the nonhomogeneous case under some supplementary conditions (§4). Finally we give an application of this theory to random functions connected with a Markoff process (§5).
Algebraic logic. IV. Equality in polyadic algebras
Paul R.
Halmos
1-27
A general functional equation
J. H. B.
Kemperman
28-56
The boundary integral of ${\rm log}\vert \varphi \vert $ for generalized analytic functions
Richard
Arens
57-69
On the second dual of the space of continuous functions
Samuel
Kaplan
70-90
Modular Lie algebras. II
Charles W.
Curtis
91-108
Dissipative hyperbolic systems
R. S.
Phillips
109-173
The cohomology ring of product complexes
F. P.
Palermo
174-196
Functional cohomology operations
Franklin P.
Peterson
197-211
Localization on spheres
Victor L.
Shapiro
212-219
Syntactical transforms
A. H.
Lightstone;
A.
Robinson
220-245
Fourier-Stieltjes series of Walsh functions
N. J.
Fine
246-255
Continuous mappings into nonsimple spaces
J.
Dugundji
256-268
Acyclic models and fibre spaces
V. K. A. M.
Gugenheim;
J. C.
Moore
265-306
Acyclic models and de Rham's theorem
Amasa
Forrester
307-326
Specialization and Picard-Vessiot theory
Lawrence
Goldman
327-356
The asymptotic behavior of the stable initial manifolds of a system of nonlinear differential equations
J. J.
Levin
357-368
Densities of sets of integers and transforms of sequences of zeros and ones
Ralph Palmer
Agnew
369-389
Homological dimension in local rings
Maurice
Auslander;
David A.
Buchsbaum
390-405
Extensions of normal bases and completely basic fields
Carl C.
Faith
406-427
Oscillation criteria for second-order linear differential equations
Zeev
Nehari
428-445
Adjoint linear differential operators
William T.
Reid
446-461
On singularities of solutions of certain differential equations in three variables
Stefan
Bergman
462-488
The differential equations of birth-and-death processes, and the Stieltjes moment problem
S.
Karlin;
J. L.
McGregor
489-546
A scalar transport equation
Z. A.
Melzak
547-560
The automorphisms of the holomorph of a finite abelian group
W. H.
Mills
1-34
On supercomplexes
V. K. A. M.
Gugenheim
35-51
An extended Markov property
R. M.
Blumenthal
52-72
On the distribution of the supremum functional for processes with stationary independent increments
Glen
Baxter;
M. D.
Donsker
73-87
Asymptotic perturbation series
Vernon A.
Kramer
88-105
The Fatou theorem and its converse
F. W.
Gehring
106-121
On the second theorem of consistency in the theory of absolute Riesz summability
B. N.
Prasad;
T.
Pati
122-133
Borel structure in groups and their duals
George W.
Mackey
134-165
Positivity in $H$-systems and sufficient statistics
T. S.
Pitcher
166-173
Representation as a Poisson transform
Harry
Pollard
174-180
Complex analytic connections in fibre bundles
M. F.
Atiyah
181-207
On the representation of $\alpha$-complete Boolean algebras
C. C.
Chang
208-218
Extensions of Jentzsch's theorem
Garrett
Birkhoff
219-227
On the foundations of quasigroups
Sherman K.
Stein
228-256
Some growth theorems for linear ordinary differential equations
W. S.
Loud
257-264
Generalized random variables
L. E.
Dubins
273-309
On univalent mappings by solutions of linear elliptic partial differential equations
Paul W.
Berg
310-318
Closed coverings in \v Cech homology theory
E. E.
Floyd
319-337
An asymptotic formula in the theory of numbers
H.
Halberstam
338-351
On free products of groups
Daniel H.
Wagner
352-378
Four dimensional compact locally hermitian manifolds
Louis
Auslander
379-391
On some simple groups defined by C. Chevalley
Rimhak
Ree
392-400
Semi-direct products with ample homomorphisms
Franklin
Haimo
401-425
Jordan homomorphisms onto prime rings
M. F.
Smiley
426-429
On partially stable algebras
A. A.
Albert
430-443
On occupation times for Markoff processes
D. A.
Darling;
M.
Kac
444-458
A class of limit theorems
M.
Kac
459-471
On Walsh-Fourier series
George W.
Morgenthaler
472-507
Structure theory of faithful rings. I. Closure operations on lattices
R. E.
Johnson
508-522
Structure theory of faithful rings. II. Restricted rings
R. E.
Johnson
523-544
Multiplication on spheres. II
I. M.
James
545-558
Addenda to the paper ``On a problem of Mihlin''
A. P.
Calderon;
A.
Zygmund
559-560
Some properties of the Fourier transform on semi-simple Lie groups. II
L.
Ehrenpreis;
F. I.
Mautner
1-55
On quasi-conformality and pseudo-analyticity
Akira
Mori
56-77
On a theorem of Mori and the definition of quasiconformality
Lipman
Bers
78-84
Factor sets and differentials on abelian varieties
Iacopo
Barsotti
85-108
Graphs and subgraphs
Oystein
Ore
109-136
Sums of normal endomorphisms
Nickolas
Heerema
137-143
On the asymptotic solutions of a class of ordinary differential equations of the fourth order, with special reference to an equation of hydrodynamics
Rudolph E.
Langer
144-191
On a family of Lie algebras of characteristic $p$
S. A.
Jennings;
Rimhak
Ree
192-207
On the convergence of sequences of stochastic processes
Ernest G.
Kimme
208-229
Higher degrees of distributivity and completeness in Boolean algebras
E. C.
Smith;
Alfred
Tarski
230-257
The independence of certain distributive laws in Boolean algebras
Dana
Scott
258-261
Spherical fractional integrals
Nicolaas
du Plessis
262-272
Rings of analytic and meromorphic functions
H. L.
Royden
269-276
Recursive and recursively enumerable orders
H. G.
Rice
277-300
Duality in products of groups with operators
J. W.
Ellis
301-312
Convergence of martingales with a directed index set
K.
Krickeberg
313-337
The critical points of Peano-interior functions defined on $2$-manifolds
William C.
Fox
338-370
Least $p$th power polynomials on a finite point set
T. S.
Motzkin;
J. L.
Walsh
371-396
Characterization of a continuous geometry within the unit group
Gertrude
Ehrlich
397-416
Necessary and sufficient conditions for Carlson's theorem on entire functions
L. A.
Rubel
417-429
Algebraic logic. III. Predicates, terms, and operations in polyadic algebras
Paul R.
Halmos
430-470
Compact homogeneous almost complex spaces of positive characteristic
Robert
Hermann
471-481
Interpolation of linear operators
Elias M.
Stein
482-492
On the projection geometry of a finite $AW\sp *$-algebra
S. K.
Berberian
493-509
On generalized Witt algebras
Rimhak
Ree
510-546
Congruences in algebraic number fields involving sums of similar powers
Eckford
Cohen
547-566
An inversion of the Laplace and Stieltjes transforms utilizing difference operators
R. S.
Pinkham
1-18
Boundaries induced by non-negative matrices
William
Feller
19-54
Homotopy groups of joins and unions
George W.
Whitehead
55-69
The $l\sb 1$-algebra of a commutative semigroup
Edwin
Hewitt;
Herbert S.
Zuckerman
70-97
The characters of semisimple Lie groups
Harish-Chandra
98-163
The maximum of sums of stable random variables
D. A.
Darling
164-169
Approximately finite algebras
Harold
Widom
170-178
On the determination of the phase of a Fourier integral. I
Edwin J.
Akutowicz
179-192
A type of convexity in the space of $n$-complex variables
K.
deLeeuw
193-204
Variational measure
Maurice
Sion
205-221
Means on semigroups and the Hahn-Banach extension property
Robert J.
Silverman
222-237
Completely free lattices generated by partially ordered sets
R.
Dean
238-249
The invariance principle for dependent random variables
Patrick
Billingsley
250-268
Degrees of computability
Norman
Shapiro
281-299
On the distortion in conformal mapping of variable domains
S. E.
Warschawski
300-322
A combinatorial lemma and its application to probability theory
Frank
Spitzer
323-339
A study of function spaces by spectral sequences
Herbert
Federer
340-361
Some remarks about elementary divisor rings
Leonard
Gillman;
Melvin
Henriksen
362-365
Rings of continuous functions in which every finitely generated ideal is principal
Leonard
Gillman;
Melvin
Henriksen
366-391
Stochastic processes connected with harmonic functions
Joanne
Elliott;
William
Feller
392-420
On the numerical solution of heat conduction problems in two and three space variables
Jim
Douglas;
H. H.
Rachford
421-439
On the definition of significant multiplicity for continuous transformations
Earl J.
Mickle
440-451
Stationary Markov processes with continuous paths
Daniel
Ray
452-493
Formes diff\'erentielles m\'eromorphes localement exactes
Pierre
Dolbeault
494-518
Elementary solutions for certain parabolic partial differential equations
Henry P.
McKean
519-548
Primal ideals and isolated components in noncommutative rings
Wilfred E.
Barnes
1-16
Complex convexity
H. J.
Bremermann
17-51
Singular integral operators and spherical harmonics
J.
Horváth
52-63
The moduli of hyperelliptic curves
Irwin
Fischer
64-84
Cohomology of infinite algebras
Alex
Rosenberg;
Daniel
Zelinsky
85-98
A theorem on planar graphs
W. T.
Tutte
99-116
Variational property of cusp forms
Harvey
Cohn
117-127
On the conformal mapping of multiply connected regions
J. L.
Walsh
128-146
On functions subharmonic in a half-space
Alfred
Huber
147-159
Modular Lie algebras. I
Charles W.
Curtis
160-179
On products in minimal complexes
J. F.
Adams
180-189
Inductive limits of normed algebras
Seth
Warner
190-216
Riemann surfaces of class ${\rm A}$
Robert
Osserman
217-245
Relative homological algebra
G.
Hochschild
246-269
Regular $\mathcal{D}$-classes in semigroups
D. D.
Miller;
A. H.
Clifford
270-280
Spectral type of the shift transformation of differential processes with stationary increments
Kiyosi
Itô
253-263
Semi-groups of measures on Lie groups
G. A.
Hunt
264-293
Some theorems concerning Brownian motion
G. A.
Hunt
294-319
Intrinsic algebras on the torus
K.
de Leeuw;
H.
Mirkil
320-330
Jordan homomorphisms
I. N.
Herstein
331-341
The lattice of submodules of a module over a noncommutative ring
Edmund H.
Feller
342-357
Characteristic and order of differentiable points in the conformal plane
N. D.
Lane;
P.
Scherk
358-378
Generalized analytic functions
Richard
Arens;
I. M.
Singer
379-393
On the construction of related differential equations
Rudolph E.
Langer
394-410
Invariant linear functions
Robert J.
Silverman
411-424
On associative products of groups
Ruth Rebekka
Struik
425-452
A new class of continued fraction expansions for the ratios of hypergeometric functions
Evelyn
Franik
453-476
Some theorems on boundary distortion
James A.
Jenkins
477-500
A Banach algebra generalization of conformal mappings of the disc
Richard
Arens
501-513
The symmetric derivative on the $(k-1)$-dimensional hypersphere
Victor L.
Shapiro
514-524
On linear and nonlinear perturbations of linear systems of ordinary differential equations with constant coefficients
Philip
Hartman;
Aurel
Wintner
1-24
The enclosing of cells in three space by simple closed surfaces
H. C.
Griffith
25-48
On the boundary behavior of a conformal map
J. L.
Walsh;
L.
Rosenfeld
49-73
On decomposition of continua into aposyndetic continua
Louis F.
McAuley
74-91
Functions typically-real and meromorphic in the unit circle
A. W.
Goodman
92-105
Tensor algebras over Hilbert spaces. I
I. E.
Segal
106-134
On strong bounds for sums of independent random variables which tend to a stable distribution
Miriam
Lipschutz
135-154
On the logarithm of a uniformly bounded operator
G. L.
Krabbe
155-166
On functions analytic in a half-plane
Daniel
Waterman
167-194
Some new developments in Markov chains
K. L.
Chung
195-210
Nonoscillation and disconjugacy in the complex domain
Paul R.
Beesack
211-242
On differential geometry of hypersurfaces in the large
Chuan-Chih
Hsiung
243-252
The linear difference-differential equation with linear coefficients
Barbara G.
Yates
281-298
Homotopy resolutions of semi-simplicial complexes
Alex
Heller
299-344
On Boolean algebras of projections and algebras of operators
William G.
Bade
345-360
Finite subgroups of division rings
S. A.
Amitsur
361-386
Errata, volume 78; Errata, volume 79
386
Pairs of matrices with property $L$. II
T. S.
Motzkin;
Olga
Taussky
387-401
The characters of the finite general linear groups
J. A.
Green
402-447
On Galois groups of local fields
Kenkichi
Iwasawa
448-469
Symmetric measures on Cartesian products
Edwin
Hewitt;
Leonard J.
Savage
470-501
Planar division neo-rings
D. R.
Hughes
502-527
Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold
Bertram
Kostant
528-542
Exact categories and duality
D. A.
Buchsbaum
1-34
Automorphisms of the symplectic modular group
Irving
Reiner
35-50
Some functional equations in the theory of dynamic programming. I. Functions of points and point transformations
R.
Bellman
51-71
A class of affinely connected spaces
Hsien-Chung
Wang;
Kentaro
Yano
72-92
On the asymptotic forms of the solutions of ordinary linear differential equations of the third order in a region containing a turning point
Rudolph E.
Langer
93-123
Some theorems about the Riesz fractional integral
Nicolaas
du Plessis
124-134
Restricted Lie algebras and simple associative algebras of characteristic $p$
G.
Hochschild
135-147
Some theorems on holonomy groups of Riemannian manifolds
Shigeo
Sasaki;
Morikuni
Goto
148-158
Complex nonoscillation theorems and criteria of univalence
Binyamin
Schwarz
159-186
Monomial groups
Ralph B.
Crousch
187-215
A probability approach to the heat equation
J. L.
Doob
216-280
A central limit theorem for uniformly bounded orthonormal systems
George W.
Morgenthaler
281-311
Arithmetical predicates and function quantifiers
S. C.
Kleene
312-340
Order types and similarity transformations
Seymour
Ginsburg
341-361
Extremal problems and harmonic interpolation on open Riemann surfaces
Leo
Sario
362-377
On curvature in Finsler geometry
Louis
Auslander
378-388
Some contributions to the theory of rings of operators. II
Ernest L.
Griffin
389-400
Potential theory in the geometry of matrices
Josephine
Mitchell
401-422
On circumferentially mean $p$-valent functions
James A.
Jenkins
423-428
Galois theory of continuous transformation rings
Alex
Rosenberg;
Daniel
Zelinsky
429-452
Extension of derivations in continuous transformation rings
Alex
Rosenberg;
Daniel
Zelinsky
453-458
Maximal sets of involutions
Irving
Reiner
459-476
Simple algebras with purely inseparable splitting fields of exponent $1$
G.
Hochschild
477-489
Compound group extensions. III
Robert L.
Taylor
490-520
Finite extensions of Abelian groups with minimum condition
Reinhold
Baer
521-540
Some contributions to the theory of denumerable Markov chains
Cyrus
Derman
541-555
Abstract: §1 deals with the statistical regularity properties of a denumerable number of particles, all moving about the states of a Markov chain according to the same transition probabilities. §2 deals with the problem of obtaining a sharper version of a strong limit theorem proved independently by Harris and Lévy.
Automorphisms of function fields
Maxwell
Rosenlicht
1-11
A characterization of tame curves in three-space
O. G.
Harrold;
H. C.
Griffith;
E. E.
Posey
12-34
On compactness in functional analysis
Robert G.
Bartle
35-57
Monotone and convex operator functions
Julius
Bendat;
Seymour
Sherman
58-71
The laws of apparition and repetition of primes in a cubic recurrence
Morgan
Ward
72-90
Partitions of the plane into sets having positive measure in every non-null measurable product set
Paul
Erdős;
John C.
Oxtoby
91-102
The solutions of second order linear ordinary differential equations about a turning point of order two
Robert W.
McKelvey
103-123
A contribution to the theory of Toeplitz matrices
Ulf
Grenander
124-140
A generalization of the Riesz theory of completely continuous transformations
Lawrence M.
Graves
141-149
Remarks on singular points of functional equations
Lawrence M.
Graves
150-157
On interpolation to a given analytic function by analytic functions of minimum norm
J. P.
Evans;
J. L.
Walsh
158-172
Product measures
W. W.
Bledsoe;
A. P.
Morse
173-215
Simply connected spaces
Robert S.
Novosad
216-228
On kernel representation of linear operators
Dorothy
Maharam
229-255
Completeness and compactness in linear topological spaces
Heron Sherwood
Collins
256-280
Abelian varieties over function fields
Wei-Liang
Chow
253-275
Meromorphic functions with three radially distributed values
Albert
Edrei
276-293
On asymptotic values of functions analytic in a circle
Makoto
Ohtsuka
294-304
Asymptotic expansions for the Whittaker functions of large complex order $m$
Nicholas D.
Kazarinoff
305-328
On Burnside's problem. II
R. C.
Lyndon
329-332
Some theorems on bounded analytic functions
Walter
Rudin
333-342
A theory of analytic functions in Banach algebras
E. K.
Blum
343-370
Note on the Fourier inversion formula on groups
F. I.
Mautner
371-384
Causality and analyticity
Y.
Fourès;
I. E.
Segal
385-405
Eigenfunction expansions associated with singular differential operators
Joanne
Elliott
406-425
Hilbert space methods in the theory of harmonic integrals
Matthew P.
Gaffney
426-444
The number of linear, directed, rooted, and connected graphs
Frank
Harary
445-463
Difference sets in a finite group
R. H.
Bruck
464-481
On the application of the individual ergodic theorem to discrete stochastic processes
H. D.
Brunk
482-491
On linear, second order differential equations in the unit circle
Philip
Hartman;
Aurel
Wintner
492-500
Summation of bounded divergent sequences, topological methods
Albert
Wilansky;
Karl
Zeller
501-509
On Bieberbach-Eilenberg functions. II
James A.
Jenkins
510-515
Strict convexity and smoothness of normed spaces
Mahlon M.
Day
516-528
Some analytical properties of continuous stationary Markov transition functions
David G.
Kendall
529-540
Abstract: A systematic treatment of Markov processes with Euclidean state-spaces has recently been presented by Doob [1], the restriction on the nature of the state-space being associated with the very illuminating probabilistic method which he uses throughout. At about the same time a new step was taken by Kolmogorov [4] who established for countable state-spaces the existence and finiteness of the derivative of the transition-function $ {p_{ij}}(t)$ at $t = 0 +$ when $i \ne j$. In this paper some of Doob's and Kolmogorov's results are combined and shown to be valid (when suitably formulated) for an arbitrary state-space. For the sake of a generality which proves useful in the discussion of existence theorems the transition-function $ {P_t}(x,\;A)$ is not assumed to be ``honest"; i.e., if $X$ is the state-space then it is supposed that $ {P_t}(x,\;X) \leqq 1$.
The Poisson transform
Harry
Pollard
541-550
On the theory of queues with many servers
J.
Kiefer;
J.
Wolfowitz
1-18
The inversion of the generalized Fourier transform by Abelian summability
Jerome
Blackman
19-29
Some topological properties of convex sets
V. L.
Klee
30-45
Analytic functions of class $H\sb p$
Walter
Rudin
46-66
Least $p$th power polynomials on a real finite point set
T. S.
Motzkin;
J. L.
Walsh
67-81
Almost bounded functions
A. W.
Goodman
82-97
Quadratic functionals with a singular end point
Walter
Leighton;
Allan D.
Martin
98-128
Productive sets
J. C. E.
Dekker
129-149
Normal automorphisms and their fixed points
Franklin
Haimo
150-167
Martingales and one-dimensional diffusion
J. L.
Doob
168-208
On a problem of Mihlin
A. P.
Calderón;
A.
Zygmund
209-224
Locally uniformly convex Banach spaces
A. R.
Lovaglia
225-238
Additive polynomials. II
T. H. M.
Crampton;
G.
Whaples
239-252
New results on power-associative algebras
Louis A.
Kokoris
363-373
Areas of $k$-dimensional nonparametric surfaces in $k+1$ space
Robert N.
Tompson
374-407
Note on the Bessel polynomials
M.
Nassif
408-412
On zero and pole surfaces of functions of two complex variables
Stefan
Bergman
413-454
Integral representations of positive definite functions. II
A.
Devinatz
455-480
Sets of ``positive'' functions in $H$-systems
Tom
Pitcher
481-489
Homeomorphism groups and coset spaces
Lester R.
Ford
490-497
Symplectic modular complements
Irving
Reiner
498-505
Asymptoticity in topological dynamics
John D.
Baum
506-519
On the topology of cyclic products of spheres
S. D.
Liao
520-551
Errata to Volume 76
552-552
On the existence of functions having given partial derivatives on a curve
Maurice
Sion
179-201
On Burnside's problem
R. C.
Lyndon
202-215
On $P$-soluble rings
Jakob
Levitzki
216-237
Some integralgeometric theorems
Herbert
Federer
238-261
A general coefficient theorem
James A.
Jenkins
262-280
Coverings of a topological space
R. S.
Pierce
281-298
On spectra of second-order differential operators
Daniel
Ray
299-321
Summability and uniqueness of double trigonometric integrals
Victor L.
Shapiro
322-339
Concerning rings of continuous functions
Leonard
Gillman;
Melvin
Henriksen
340-362
Diffusion processes in one dimension
William
Feller
1-31
On even entire functions with zeros having a density
R. M.
Redheffer
32-61
The space of point homotopic maps into the circle
B. J.
Eisenstadt
62-85
Semimartingales and subharmonic functions
J. L.
Doob
86-121
Further results on order types and decompositions of sets
Seymour
Ginsburg
122-150
Mapping for elliptic equations
J. J.
Gergen;
F. G.
Dressel
151-178
On the absolute Riesz summability of Fourier series and its conjugate series
T.
Pati
351-374
Semi-prime rings
R. E.
Johnson
375-388
On Bieberbach-Eilenberg functions
James A.
Jenkins
389-396
Contributions to the theory of Markov chains. II
K. L.
Chung
397-419
A study of $\alpha$-variation. I
F. W.
Gehring
420-443
The exact Peano derivative
H. William
Oliver
444-456
A generalized notion of boundary
W. H.
Fleming;
L. C.
Young
457-484
The Plancherel formula for complex semisimple Lie groups
Harish-Chandra
485-528
Surfaces without conjugate points
L. W.
Green
529-546
On the representations of an integer as the sum of products of integers
S. M.
Johnson
177-189
On representations and extensions of bounded linear functionals defined on classes of analytic functions
Philip
Davis;
J. L.
Walsh
190-206
Branch points of solutions of equations in Banach space. II
Jane
Cronin
207-222
Circular summability $C$ of double trigonometric series
Victor L.
Shapiro
223-233
Representations of semisimple Lie groups. III
Harish-Chandra
234-253
Schlicht solutions of $W''+pW=0$
M. S.
Robertson
254-274
A group ring method for finitely generated groups
Kuo-Tsai
Chen
275-287
On solution of partial differential equations by the Hahn-Banach theorem
P. R.
Garabedian;
Max
Shiffman
288-299
The boundary value problems and semigroups associated with certain integro-differential operators
Joanne
Elliott
300-331
$q$-Bernoulli and Eulerian numbers
L.
Carlitz
332-350
Interpolation series for integral functions of exponential type
A. J.
Macintyre
1-13
The conformal rigidity of certain subdomains on a Riemann surface
H. L.
Royden
14-25
Representations of semisimple Lie groups. II
Harish-Chandra
26-65
Infinite general linear groups
Richard V.
Kadison
66-91
Iterative methods for solving partial difference equations of elliptic type
David
Young
92-111
An extension of a theorem of G. Szeg\"o and its application to the study of stochastic processes
Ulf
Grenander;
Murray
Rosenblatt
112-126
Global structure of ordinary differential equations in the plane
L.
Markus
127-148
Generalized $n$th primitives
R. D.
James
149-176
Isolated singularities of solutions of non-linear partial differential equations
Robert
Finn
385-404
Invariantive theory of equations in a finite field
L.
Carlitz
405-427
A theorem on holonomy
W.
Ambrose;
I. M.
Singer
428-443
Congruence representations in algebraic number fields
Eckford
Cohen
444-470
Some contributions to the theory of rings of operators
Ernest L.
Griffin
471-504
On a certain class of ideals in the $L\sp 1$-algebra of a locally compact abelian group
H. J.
Reiter
505-509
Positive temperatures on a semi-infinite rod
D. V.
Widder
510-525
Functions of potential type
Maynard G.
Arsove
526-551
Nonlinear transformations of Volterra type in Wiener space
R. H.
Cameron;
R. E.
Fagen
552-575
Representations of a semisimple Lie group on a Banach space. I
Harish-Chandra
185-243
On residues of polygenic functions
Vincent C.
Poor
244-255
Some inequalities in the theory of functions
Zeev
Nehari
256-286
Summability of Fourier series on the quaternions of norm one
Gustave
Rabson
287-303
Compound group extensions. II
Robert L.
Taylor
304-310
Development of an extended exterior differential calculus
Harley
Flanders
311-326
Functions representable as differences of subharmonic functions
Maynard G.
Arsove
327-365
Singular points of functional equations
Robert G.
Bartle
366-384
Propositional calculus and realizability
Gene F.
Rose
1-19
Group elements of prime power index
Reinhold
Baer
20-47
Complex Tauberian theorems for power series
Dieter
Gaier
48-68
Simultaneous partitionings of two sets
G. H. M.
Thomas
69-79
The asymptotic behavior of the solutions of linear and nonlinear differential-difference equations
K. L.
Cooke
80-105
Compound group extensions. I. Continuations of normal homomorphisms
Robert L.
Taylor
106-135
Ideal lattices and the structure of rings
Robert L.
Blair
136-153
The representation of abstract integrals
Dorothy
Maharam
154-184
On the generation function of a doubly infinite, totally positive sequence
Albert
Edrei
367-383
On the structure of algebraic algebras and related rings
Jakob
Levitzki
384-409
Some interconnections between modern algebra and mathematical logic
Leon
Henkin
410-427
On the non-isomorphism of certain holomorphs
W. H.
Mills
428-443
Complex variable Tauberians
Shmuel
Agmon
444-481
On Rademacher's extension of the Goldbach-Vinogradoff theorem
Raymond
Ayoub
482-491
Finite projective plane geometries and difference sets
Gerald
Berman
492-499
Concerning webs in the plane
Mary-Elizabeth
Hamstrom
500-513
Some remarks on order types and decompositions of sets
Seymour
Ginsburg
514-535
An indirect sufficiency proof for the problem of Lagrange with differential inequalities as added side conditions
Louis L.
Pennisi
177-198
Perturbation theory for semi-groups of linear operators
R. S.
Phillips
199-221
The mathematics of second quantization
J. M.
Cook
222-245
On P\'olya frequence functions. III. The positivity of translation determinants with an application to the interpolation problem by spline curves
I. J.
Schoenberg;
Anne
Whitney
246-259
On $n$-dimensional Riemannian spaces admitting a group of motions of order $n(n-1)/2+1$
Kentaro
Yano
260-279
Continuity properties of sample functions of Markov processes
J. R.
Kinney
280-302
Linear functions on almost periodic functions
Edwin
Hewitt
303-322
On commutative power-associative algebras of degree two
A. A.
Albert
323-343
A unified space for mappings
G. T.
Whyburn
344-350
Representations of prime rings
R. E.
Johnson
351-357
Classes of recursively enumerable sets and their decision problems
H. G.
Rice
358-366
An interpolation series expansion for a meromorphic function
J. L.
Walsh
1-9
Convex bodies and periodic homeomorphisms in Hilbert space
Victor L.
Klee
10-43
The positive cone in Banach algebras
J. L.
Kelley;
R. L.
Vaught
44-55
Integral representations of positive definite functions
A.
Devinatz
56-77
Finite Fourier series and equations in finite fields
Albert Leon
Whiteman
78-98
On the algebra of networks
R.
Bott;
R. J.
Duffin
99-109
Cohomology of group extensions
G.
Hochschild;
J.-P.
Serre
110-134
Some properties of the double Laplace transformation
Geraldine A.
Coon;
Dorothy L.
Bernstein
135-176
Primitive roots in a finite field
L.
Carlitz
373-382
On local structure of finite-dimensional groups
Sze-Tsen
Hu
383-400
Investigations in harmonic analysis
H. J.
Reiter
401-427
Approximation and variance
Arthur
Sard
428-446
Degree of approximation to functions on a Jordan curve
J. L.
Walsh
447-458
An extremal method on arbitrary Riemann surfaces
Leo
Sario
459-470
First passage and recurrence distributions
T. E.
Harris
471-486
Jacobi sums as ``Gr\"ossencharaktere''
André
Weil
487-495
A theory of spherical functions. I
Roger
Godement
496-556
Integrability along a line for a class of entire functions
R. P.
Boas
191-197
The integral geometry definition of arc length for two-dimensional Finsler spaces
O. G.
Owens
198-210
On monotone interior mappings in the plane
R. D.
Anderson
211-222
Complex boundary value problems
P. R.
Garabedian;
D. C.
Spencer
223-242
Truth definitions and consistency proofs
Hao
Wang
243-275
Galois theory of simple rings
Tadasi
Nakayama
276-292
Applications of martingale system theorems
J. L.
Snell
293-312
Remarks on some modular identities
Morris
Newman
313-320
On the symmetry and bounded closure of locally convex spaces
William F.
Donoghue;
Kennan T.
Smith
321-344
On double trigonometric integrals
Leonard D.
Berkovitz
345-372
The coefficient problem in the theory of linear partial differential equations
Stefan
Bergman
1-34
Adjunction of subfield closures to ordered division rings
Arno
Jaeger
35-39
Harmonic functions on open Riemann surfaces
H. L.
Royden
40-94
The influence of the maximum term in the addition of independent random variables
D. A.
Darling
95-107
Pairs of matrices with property ${\rm L}$
T. S.
Motzkin;
Olga
Taussky
108-114
Subrings of simple rings with minimal ideals
Alex
Rosenberg
115-138
On certain minimum problems in the theory of convex curves
Marlow
Sholander
139-173
Some basic theorems in differential algebra (characteristic $p$, arbitrary)
A.
Seidenberg
174-190
On the structure of unitary groups
Jean
Dieudonné
367-385
Infinite unitary groups
Richard V.
Kadison
386-399
Mappings between function spaces
Robert G.
Bartle;
Lawrence M.
Graves
400-413
An arithmetic theory of adjoint plane curves
Daniel
Gorenstein
414-436
Existential definability in arithmetic
Julia
Robinson
437-449
On the primary subgroups of a group
Christine Williams
Ayoub
450-466
Automorphisms of the projective unimodular group
L. K.
Hua;
I.
Reiner
467-473
Some asymptotic formulae for the mock theta series of Ramanujan
Leila A.
Dragonette
474-500
Integral transformations and Tauberian constants
Ralph Palmer
Agnew
501-518
On the relation between Green's functions and covariances of certain stochastic processes and its application to unbiased linear prediction
C. L.
Dolph;
M. A.
Woodbury
519-550
Diophantine approximation in fields of characteristic $p$
L.
Carlitz
187-208
The automorphism group of Lie group
G.
Hochschild
209-216
A structure theory of Lie triple systems
William G.
Lister
217-242
The Radon-Nikod\'ym theorem for finite rings of operators
H. A.
Dye
243-280
A linear operator method on arbitrary Riemann surfaces
Leo
Sario
281-295
On nonassociative division algebras
A. A.
Albert
296-309
Homomorphisms of Jordan rings of self-adjoint elements
N.
Jacobson;
C. E.
Rickart
310-322
Banach spaces with the extension property
J. L.
Kelley
323-326
Modules over Dedekind rings and valuation rings
Irving
Kaplansky
327-340
A class of nonharmonic Fourier series
R. J.
Duffin;
A. C.
Schaeffer
341-366
Representations of alternative algebras
R. D.
Schafer
1-17
On hyperdirichlet series and on related questions of the general theory of functions
Benjamin
Lepson
18-45
Finitely additive measures
Kôsaku
Yosida;
Edwin
Hewitt
46-66
On Fermat's last theorem (thirteenth paper)
Taro
Morishima
67-81
On the zeros of total sets of polynomials
Philip
Davis;
Henry
Pollak
82-103
An application of doubly orthogonal functions to a problem of approximation in two regions
Philip
Davis
104-137
On periodic maps and the Euler characteristics of associated spaces
E. E.
Floyd
138-147
Generalization of Runge's theorem to approximation by analytic functions
Annette
Sinclair
148-164
On the oscillation of sums of random variables
M.
Rosenblatt
165-178
On the application of the Borel-Cantelli lemma
K. L.
Chung;
P.
Erdös
179-186
Automorphisms of the unimodular group
L. K.
Hua;
I.
Reiner
331-348
Local properties of algebraic correspondences
I.
Barsotti
349-378
Multiple holomorphs of finitely generated abelian groups
W. H.
Mills
379-392
Spectral theory for semi-groups of linear operators
R. S.
Phillips
393-415
A boundary value problem for an equation of mixed type
M. H.
Protter
416-429
Necessary and sufficient conditions for the representation of a function by a Weierstrass transform
D. V.
Widder
430-439
Distance sets in metric spaces
L. M.
Kelly;
E. A.
Nordhaus
440-456
Identities in two-valued calculi
R. C.
Lyndon
457-465
Certain linear combinations of Hausdorff summability methods
B.
Vanderburg
466-477
Estimates on the mapping function and its derivatives in conformal mapping of nearly circular regions
E. J.
Specht
183-196
On some algebraic properties of the Bessel polynomials
Emil
Grosswald
197-210
Curvature of closed hypersurfaces and non-existence of closed minimal hypersurfaces
S. B.
Myers
211-217
On the foundations of inversion geometry
Alan J.
Hoffman
218-242
Differentiable inequalities and the theory of convex bodies
E. R.
Lorch
243-266
Higher-dimensional hereditarily indecomposable continua
R. H.
Bing
267-273
A geometric approach to the representations of the full linear group over a Galois field
R.
Steinberg
274-282
Arithmetic translations of axiom systems
Hao
Wang
283-293
Homology theories for multiplicative systems
Samuel
Eilenberg;
Saunders
MacLane
294-330
On approximation to functions satisfying a generalized continuity condition
H. Margaret
Elliott
1-23
The fundamental solution of a degenerate partial differential equation of parabolic type
Maria
Weber
24-37
Random Fourier transforms
G. A.
Hunt
38-69
On the representations of a number as the sum of three squares
Paul T.
Bateman
70-101
The structure of valuations of the rational function field $K(x)$
B. N.
Moyls
102-112
An open mapping approach to Hurwitz's theorem
G. T.
Whyburn
113-119
On a class of Markov processes
M.
Rosenblatt
120-135
On the spherical summability of multiple orthogonal series
Josephine
Mitchell
136-151
Topologies on spaces of subsets
Ernest
Michael
152-182
Uniqueness theory of Hermite series
Walter
Rudin
387-403
The Wedderburn principal theorem for Jordan algebras
A. J.
Penico
404-420
The coefficient problem for schlicht mappings of the exterior of the unit circle
G.
Springer
421-450
The lines and planes connecting the points of a finite set
Th.
Motzkin
451-464
Boundary value problems for minimal surfaces with singularities at infinity
Lipman
Bers
465-491
Functions of exponential type in an angle and singularities of Taylor series
Shmuel
Agmon
492-508
General representation theory of Jordan algebras
N.
Jacobson
509-530
The regular representation of a restricted direct product of finite groups
F. I.
Mautner
531-548
Fundamental properties of Hamiltonian operators of Schr\"odinger type
Tosio
Kato
195-211
On the existence of solutions of the helium wave equation
Tosio
Kato
212-218
The structure of certain operator algebras
Irving
Kaplansky
219-255
Exponential limiting products in Banach algebras
B. O.
Koopman
256-276
A law of small numbers in Markoff chains
B. O.
Koopman
277-290
Jordan division algebras and the algebras $A(\lambda)$
Charles M.
Price
291-300
Universal spaces under strong homeomorphisms
S. W.
Hahn
301-311
A general theory of conjugate nets in projective hyperspace
Chuan-Chih
Hsiung
312-322
Circular summation and localization of double trigonometric series
Leonard D.
Berkovitz
323-344
On the lattice of subgroups of finite groups
Michio
Suzuki
345-371
On the $L$-homomorphisms of finite groups
Michio
Suzuki
372-386
The behavior at infinity of certain convolution transforms
I. I.
Hirschman
1-14
A characterization of $3$-space by partitionings
R. H.
Bing
15-27
On some applications of the universal enveloping algebra of a semisimple Lie algebra
Harish-Chandra
28-96
Extensional invariance
A. D.
Wallace
97-102
Characterizations of certain spaces of continuous functions
Meyer
Jerison
103-113
Some theorems on integral functions with negative zeros
N. A.
Bowen;
A. J.
Macintyre
114-126
A class of multivalent functions
A. W.
Goodman;
M. S.
Robertson
127-136
Undecidable rings
Raphael M.
Robinson
137-159
Additive functionals on a space of continuous functions. I
R. H.
Cameron;
Ross E.
Graves
160-176
Convergence- and sum-factors for series of complex numbers
E.
Calabi;
A.
Dvoretzky
177-194
Sur la premi\`ere d\'eriv\'ee
Z.
Zahorski
1-54
Neumann series of Bessel functions. II
J. Ernest
Wilkins
55-65
The generalized Walsh functions
N. J.
Fine
66-77
Hilbert's characteristic function and the arithmetic genus of an algebraic variety
H. T.
Muhly;
O.
Zariski
78-88
Projections in normed linear spaces
Dwight B.
Goodner
89-108
On the asymptotic distribution of the elementary symmetric functions $\pmod p$
N. J.
Fine
109-129
On the strong differentiation of the indefinite integral
Athanasios
Papoulis
130-141
Decompositions of measure algebras and spaces
Dorothy
Maharam
142-160
A class of domain functions and some allied extremal problems
Zeev
Nehari
161-178
On unicoherent continua
Harlan C.
Miller
179-194
On iterative methods in linear differential equations
Kenneth S.
Miller
195-207
Branch points of solutions of equations in Banach space
Jane
Cronin
208-231
On the measure of Cartesian product sets
Gerald
Freilich
232-275
Means for the bounded functions and ergodicity of the bounded representations of semi-groups
Mahlon M.
Day
276-291
Automorphisms of simple algebras
G.
Hochschild
292-301
Some theorems on groups with applications to ring theory
Bailey
Brown;
Neal H.
McCoy
302-311
On Fourier-Stieltjes integrals
R. S.
Phillips
312-323
Density ratios and $(\phi,1)$ rectifiability in $n$-space
Edward F.
Moore
324-334
On the degree of variation in conformal mapping of variable regions
S. E.
Warschawski
335-356
The hyperplane sections of normal varieties
A.
Seidenberg
357-386
Coverings with connected intersections
R. H.
Bing;
E. E.
Floyd
387-391
The classes $L\sb p$ and conformal mapping
P. R.
Garabedian
392-415
On simultaneous interpolation and approximation by functions analytic in a given region
J. L.
Walsh;
H. G.
Russel
416-439
Gibbs' phenomenon for Hausdorff means
Otto
Szász
440-456
On $n$-parameter families of functions and associated convex functions
Leonard
Tornheim
457-467
On some high indices theorems
Daniel
Waterman
468-478
Jordan homomorphisms of rings
N.
Jacobson;
C. E.
Rickart
479-502
A theory of power-associative commutative algebras
A. A.
Albert
503-527
On mapping of a uniform space onto itself
Albert
Edrei
528-536
Theory of reproducing kernels
N.
Aronszajn
337-404
On generalized Jacobians
T.
Rado;
P. V.
Reichelderfer
405-419
Abelian group algebras of finite order
Sam
Perlis;
Gordon L.
Walker
420-426
The normal completion of the lattice of continuous functions
R. P.
Dilworth
427-438
Free mobility and orthogonality
Reinhold
Baer
439-460
On solutions with algebraic character of linear partial differential equations
Stefan
Bergman
461-507
Euclid's algorithm in certain quartic fields
H.
Davenport
508-532
Extensions of semigroups
A. H.
Clifford
165-173
The formal structure of a denumerable system
R. L.
Goodstein
174-182
Polynomial approximation to harmonic and analytic functions: generalized continuity conditions
J. L.
Walsh;
H. Margaret
Elliott
183-203
On the Schwarz-Christoffel transformation and $p$-valent functions
A. W.
Goodman
204-223
Uniform asymptotic formulae for functions with transition points
T. M.
Cherry
224-257
Conformal mapping of open Riemann surfaces
Zeev
Nehari
258-277
Integral representation of continuous functions
Walter
Rudin
278-286
Uniqueness theory for Laplace series
Walter
Rudin
287-303
Stokes multipliers for asymptotic solutions of a certain differential equation
H. L.
Turrittin
304-329
On Ge\"ocze's problem for non-parametric surfaces
H. P.
Mulholland
330-336
Some properties of free groups
Herbert
Federer;
Bjarni
Jónsson
1-27
A theorem of the Hahn-Banach type for linear transformations
Leopoldo
Nachbin
28-46
On the behaviour of harmonic functions at the boundary
A. P.
Calderón
47-54
On a theorem of Marcinkiewicz and Zygmund
A. P.
Calderón
55-61
Topological representation of algebras. II
Irving
Kaplansky
62-75
Integration in noncommutative systems
Frank M.
Stewart
76-104
The existence of mulitple solutions of elliptic differential equations
Jane
Cronin
105-131
On a class of nonlinear difference equations in the complex domain
Walter
Strodt
132-164
Principal quadratic functionals
Walter
Leighton
253-274
On the semi-continuity of double integrals
Paul V.
Reichelderfer
275-285
The imbedding theorem for fibre bundles
Shiing-Shen
Chern;
Yi-Fone
Sun
286-303
Parallelisability of principal fibre bundles
Shiing-Shen
Chern;
Sze-Tsen
Hu
304-309
The topology of ordered Abelian groups
L. W.
Cohen;
Casper
Goffman
310-319
On the growth of derivatives of functions omitting two values
A. S.
Galbraith;
W.
Seidel;
J. L.
Walsh
320-326
Some problems in conformal mapping
J. A.
Jenkins
327-350
On the distribution of the characteristic values and singular values of linear integral equations
Shih-Hsun
Chang
351-367
Continued fractions with absolutely convergent even and odd parts
R. E.
Lane;
H. S.
Wall
368-380
On the degree of polynomial approximation to harmonic and analytic functions
J. L.
Walsh;
W. E.
Sewell;
H. M.
Elliott
381-420
Coset representations in free groups
Marshall
Hall
421-432
On a monotone singular function and on the approximation of analytic functions by nearly analytic functions in the complex domain
H.
Kober
433-450
Concerning upper semi-continuous collections of continua
R. D.
Anderson
451-460
The asymptotic solutions of ordinary linear differential equations of the second order, with special reference to a turning point
Rudolph E.
Langer
461-490
Abel's theorem and a generalization of one-parameter groups
J. F.
Ritt
491-497
Schwarz's lemma and the Szeg\"o kernel function
P. R.
Garabedian
1-35
An estimate concerning the Kolmogoroff limit distribution
Kai-Lai
Chung
36-50
On the strong law of large numbers
P.
Erdös
51-56
A note on the pseudo-arc
Edwin E.
Moise
57-58
An application of Banach linear functionals to summability
Albert
Wilansky
59-68
A representation theory for a general class of convolution transforms
I. I.
Hirschman;
D. V.
Widder
69-97
Fluctuation theory of recurrent events
William
Feller
98-119
Euclidean domains with uniformly Abelian local fundamental groups
O. G.
Harrold
120-129
Virtual mass and polarization
M.
Schiffer;
G.
Szegö
130-205
The Charlier $B$-series
R. P.
Boas
206-216
Abstract ergodic theorems and weak almost periodic functions
W. F.
Eberlein
217-240
On the zeros of successive derivatives of integral functions
Sheila Scott
Macintyre
241-251
The transformation of Wiener integrals by nonlinear transformations
R. H.
Cameron;
W. T.
Martin
253-283
A nonhomogeneous ergodic theorem
Paul R.
Halmos
284-288
Nonlinear integral equations of the Hammerstein type
C. L.
Dolph
289-307
Exceptional values of meromorphic functions
E. F.
Collingwood
308-346
On cyclic additivity theorems
E. J.
Mickle;
T.
Rado
347-365
On the permutational representation of general sets of operations by partition lattices
Alfred L.
Foster
366-388
Incidence relations in multicoherent spaces. I
A. H.
Stone
389-406
On the zeros of rational functions having prescribed poles, with applications to the derivative of an entire function of finite genre
Morris
Marden
407-418
Rectilinear congruences
Chuan-Chih
Hsiung
419-439
On the size of certain number-theoretic functions
Wm. J.
LeVeque
440-463
Elementary divisors and modules
Irving
Kaplansky
464-491
Determination of a certain family of finite metabelian groups
G.
Szekeres
1-43
Orthogonal properties of independent functions
Samuel
Karlin
44-64
A theory of transfinite convergence
L. W.
Cohen;
Casper
Goffman
65-74
Weak topology and nonlinear integral equations
E. H.
Rothe
75-92
Repr\'esentation asymptotique des fonctions de Mathieu et des fonctions d'onde sph\'eroidales
Robert
Sips
93-134
The inversion of a general class of convolution transforms
I. I.
Hirschman;
D. V.
Widder
135-201
On ordered division rings
B. H.
Neumann
202-252
Existence of minimal surfaces with a simple pole at infinity and condition of transversality on the surface of a cylinder
Yu Why
Chen
331-347
The dynamics of transformation groups
W. H.
Gottschalk;
G. A.
Hedlund
348-359
Some theorems on rectilinear congruences and transformations of surfaces
Chenkuo
Pa
360-371
On the Walsh functions
N. J.
Fine
372-414
On the generators of the symplectic modular group
L. K.
Hua;
I.
Reiner
415-426
Incidence relations in unicoherent spaces
A. H.
Stone
427-447
On the order of $\zeta(1/2+it)$
Szu-Hoa
Min
448-472
Correction to ``Projective planes.''
Marshall
Hall
473-474
A generalization of Meyer's theorem
Irving
Reiner
170-186
Identities in the theory of conformal mapping
P. R.
Garabedian;
M.
Schiffer
187-238
On functions analytic in a region: approximation in the sense of least $p$th powers
J. L.
Walsh;
E. N.
Nilson
239-258
On the differential equation $u\sb {xx}+u\sb {yy}+N(x)u=0$
Martin M. E.
Eichler
259-278
The representation of abstract measure functions
Dorothy
Maharam
279-330
On distributions of certain Wiener functionals
M.
Kac
1-13
On the dense packing of spheres
Fred
Supnick
14-26
The $L\sb 2$-system of a unimodular group. I
W.
Ambrose
27-48
Homology of spaces with operators. II
Samuel
Eilenberg;
Saunders
MacLane
49-99
A new class of orthogonal polynomials: The Bessel polynomials
H. L.
Krall;
Orrin
Frink
100-115
Representation of Jordan and Lie algebras
Garrett
Birkhoff;
Phillip M.
Whitman
116-136
On the genus of curves of an algebraic system
Wei-Liang
Chow
137-140
Classification and representation of semi-simple Jordan algebras
F. D.
Jacobson;
N.
Jacobson
141-169
The asymptotic expansion of integral functions and of the coefficients in their Taylor series
E. M.
Wright
409-438
Linear difference equations and exponential polynomials
Walter
Strodt
439-466
Measures in Boolean algebras
Alfred
Horn;
Alfred
Tarski
467-497
The unions of trajectorial series of lineal elements generated by the plane motion of a rigid body
Don
Mittleman
498-518
Direct decompositions into infinitely many summands
Reinhold
Baer
519-551
Power-associative rings
A. A.
Albert
552-593
On Maharam's conditions for measure
J. L.
Hodges;
Alfred
Horn
594-595
On the maximum partial sums of sequences of independent random variables
Kai Lai
Chung
205-233
Non-oscillation theorems
Einar
Hille
234-252
A study of analytic surfaces by means of a projective theory of envelopes
E.
Greer;
P. O.
Bell
253-267
Nonlinear difference equations analytic in a parameter
C. F.
Stephens
268-282
Interpolation series
R. Creighton
Buck
283-298
Representations of lattices by sets
Garrett
Birkhoff;
Orrin
Frink
299-316
Some applications of the Dirichlet integral to the theory of surfaces
L. C.
Young
317-335
Mappings of a normal space into an absolute neighborhood retract
Sze-Tsen
Hu
336-358
Neumann series of Bessel functions
J. Ernest
Wilkins
359-385
On Schreier systems in free groups
Marshall
Hall;
Tibor
Radó
386-408
Cohomology and Galois theory. I. Normality of algebras and Teichm\"uller's cocycle
Samuel
Eilenberg;
Saunders
MacLane
1-20
On the existence and boundedness of solutions of nonlinear partial differential equations of parabolic type
Richard
Bellman
21-44
Rings of real-valued continuous functions. I
Edwin
Hewitt
45-99
Ultrafilters and compactification of uniform spaces
Pierre
Samuel
100-132
Manifolds of difference polynomials
Richard M.
Cohn
133-172
Some generalization of quasi-Frobenius algebras
R. M.
Thrall
173-183
Coloring six-rings
D. W.
Hall;
D. C.
Lewis
184-191
A sufficiency theorem for the Plateau problem
J. W.
Lawson
192-204
Asymptotic properties of Markoff transition prababilities
J. L.
Doob
393-421
Renewal theory from the point of view of the theory of probability
J. L.
Doob
422-438
A semi-strong minimum for a multiple integral problem in the calculus of variations
William
Karush
439-451
On $n$-arc connectedness
G. T.
Whyburn
452-456
Topological representation of algebras
Richard F.
Arens;
Irving
Kaplansky
457-481
On the theory of clusters
R. A.
Good
482-513
Topology of level curves of harmonic functions
Wilfred
Kaplan
514-522
Functions of extended class in the theory of functions of several complex variables
Stefan
Bergman
523-547
Generalized power series fields
Philip W.
Carruth
548-559
Unrestricted solution fields of almost-separable differential equations
Philip
Hartman
560-580
An indecomposable plane continuum which is homeomorphic to each of its nondegenerate subcontinua
Edwin E.
Moise
581-594
Rectifications to the papers: Sets of uniqueness and sets of multiplicity, I and II
R.
Salem
595-598
Errata, volume 61; Errata, volume 62; Errata, volume 63
599-600
The boundedness of orthonormal polynomials on certain curves of the fourth degree
Dunham
Jackson
193-206
T\"oplerian $(L\sp 2)$-bases
Philip
Hartman;
Aurel
Wintner
207-225
Tauberian theorems and Tauberian conditions
G. G.
Lorentz
226-234
A gap theorem
M.
Kac;
R.
Salem;
A.
Zygmund
235-243
On $k$-periodic systems of linear equations
I. M.
Sheffer
244-313
Generalized spherical harmonics
M. H.
Protter
314-341
Discontinuous integrals and generalized potential theory
Alexander
Weinstein
342-354
The mean convergence of orthogonal series. II
Harry
Pollard
355-367
Metric foundations of continuous transformations
Earl J.
Mickle
368-391
Les fonctions de type positif et la th\'eorie des groupes
Roger
Godement
1-84
Cohomology theory of Lie groups and Lie algebras
Claude
Chevalley;
Samuel
Eilenberg
85-124
Mapping by analytic functions. I. Conformal mapping of multiply-connected domains
Leonard
Greenstone
125-143
The asymptotic behavior of solutions of systems of Volterra integral equations
Alfred
Horn
144-174
On some determinants related to $p$-valent functions
A. W.
Goodman
175-192
On the boundedness of solutions of nonlinear differential and difference equations
Richard
Bellman
357-386
The mean convergence of orthogonal series. I
Harry
Pollard
387-403
The endomorphisms of the total operator domain of an infinite module
R. E.
Johnson;
F.
Kiokemeister
404-430
Congruence and superposability in elliptic space
Leonard M.
Blumenthal
431-451
Two-dimensional subsonic flows of a compressible fluid and their singularities
Stefan
Bergman
452-498
Characterization of the space of continuous functions over a compact Hausdorff space
R. F.
Arens;
J. L.
Kelley
499-508
An indirect sufficiency proof for the problem of Bolza in nonparametric form
Magnus R.
Hestenes
509-535
Dimension and measure
Herbert
Federer
536-547
The $(\varphi,k)$ rectifiable subsets of $n$-space
Herbert
Federer
114-192
The propagation of error in linear problems
A. T.
Lonseth
193-212
On M\"obius' inversion formula and closed sets of functions
Otto
Szász
213-239
Maximal sums of ordinals
Ben
Dushnik
240-247
Homology properties of arbitrary subsets of Euclidean spaces
Samuel
Kaplan
248-271
On the real parts of the zeros of complex polynomials and applications to continued fraction expansions of analytic functions
Evelyn
Frank
272-283
On the definition of the essential multiplicity for continuous transformations in the plane
P. V.
Reichelderfer
284-314
Polygons circumscribed about closed convex curves
Mahlon M.
Day
315-319
Some characterizations of inner-product spaces
Mahlon M.
Day
320-337
Interpolation to some classes of analytic functions by functions with pre-assigned poles
Yu-Cheng
Shen
338-356
The concept of a simple point of an abstract algebraic variety
Oscar
Zariski
1-52
On the connection between gaps in power series and the roots of their partial sums
P.
Erdös;
H.
Fried
53-61
Direct decompositions
Reinhold
Baer
62-98
Concerning the uniformization of certain Riemann surfaces allied to the inverse-cosine and inverse-gamma surfaces
Gerald R.
MacLane
99-113
Contributions to projective theory of singular points of space curves
Su-Cheng
Chang
369-377
Homology of spaces with operators. I
Samuel
Eilenberg
378-417
Perfect blankets
Anthony P.
Morse
418-442
A theorem on surface area
R. G.
Helsel
443-453
On functions holomorphic in a strip region, and an extension of Watson's problem
S.
Mandelbrojt;
G. R.
MacLane
454-467
Note on a paper by Mandelbrojt and MacLane
Jacqueline
Ferrand
468
Meromorphic functions with simultaneous multiplication and addition theorems
Irving
Gerst
469-481
On special Jordan algebras
G. K.
Kalisch
482-494
Descriptive collineations in spaces of $K$-spreads
Buchin
Su
495-507
Splitting endomorphisms
Reinhold
Baer
508-516
Endomorphism rings of operator loops
Reinhold
Baer
517-529
The isoperimetric inequality and the Lebesgue definition of surface area
Tibor
Radó
530-555
Geometrices of matrices. II. Study of involutions in the geometry of symmetric matrices
Loo-Keng
Hua
193-228
Geometries of matrices. III. Fundamental theorems in the geometries of symmetric matrices
Loo-Keng
Hua
229-255
An alternate sufficiency proof for the normal problem of Bolza
Magnus R.
Hestenes
256-264
Orthogonality and linear functionals in normed linear spaces
Robert C.
James
265-292
Approximation of continuous functions by integral functions of finite order
H.
Kober
293-306
Recursive functions and intuitionistic number theory
David
Nelson
307-368
Univalent majorants
Raphael M.
Robinson
1-35
On absolute convergence of multiple Fourier series
S.
Minakshisundaram;
Otto
Szász
36-53
Density theorems for power series and complete sets
R. P.
Boas
54-68
The group algebra of a locally compact group
I. E.
Segal
69-105
Measures on locally compact topological groups
W.
Ambrose
106-121
Direct sum theorem for Haar measures
W.
Ambrose
122-127
On Sokolovsky's ``momentless shells.''
C.
Truesdell
128-133
On the extension of interval functions
Lawrence A.
Ringenberg
134-146
Multiplicative Riemann integration in normed rings
P. R.
Masani
147-192
Obituary: George David Birkhoff
R. E.
Langer
1-2
Distributive postulates for systems like Boolean algebras
George D.
Birkhoff;
Garrett
Birkhoff
3-11
Interpolation to certain analytic functions by rational functions
Yu-Cheng
Shen
12-21
A study of the projective differential geometry of surfaces by means of a modified tensor analysis
P. O.
Bell
22-50
The Weierstrass $E$-function in the calculus of variations
Magnus R.
Hestenes
51-71
Theorem of Lindeberg in the calculus of variations
Magnus R.
Hestenes
72-92
Sufficient conditions for the isoperimetric problem of Bolza in the calculus of variations
Magnus R.
Hestenes
93-118
Geometry of the Fourier heat equation
Edward
Kasner;
John
DeCicco
119-132
Concerning simple plane webs
R. H.
Bing
133-148
Geometrical characterizations of some families of dynamical trajectories
L. A.
MacColl
149-166
Singular integral equations with Cauchy kernels
W. J.
Trjitzinsky
167-214
Invariant measures in groups which are not locally compact
John C.
Oxtoby
215-237
On the Phragm\'en-Lindel\"of principle
Maurice
Heins
238-244
Contributions to the theory of loops
R. H.
Bruck
245-354
Chromatic polynomials
G. D.
Birkhoff;
D. C.
Lewis
355-451
Complemented modular lattices and projective spaces of infinite dimension
Orrin
Frink
452-467
Rings with a finite number of primes. I
I. S.
Cohen;
Irving
Kaplansky
468-477
A class of sequences of functions
D. G.
Bourgin
478-518
On convex topological linear spaces
George W.
Mackey
519-537
On the ergodic theorem
Nelson
Dunford;
D. S.
Miller
538-549
Errata, volume 58; Errata, volume 59
550
Metric characterization of elliptic space
Leonard M.
Blumenthal
381-400
On the average of a certain Wiener functional and a related limit theorem in calculus of probability
M.
Kac
401-414
Geodesics on surfaces of revolution
Donald P.
Ling
415-429
On the associate and conjugate space for the direct product of Banach spaces
Nelson
Dunford;
Robert
Schatten
430-436
On linear expansions. I
Leopoldo
Nachbin
437-440
Coincidence functions and their integrals
Herbert
Federer
441-466
Contributions to the theory of surfaces in a 4-space of constant curvature
Yung-Chow
Wong
467-507
Orthogonal classification of Hermitian matrices
Loo-Keng
Hua
508-523
On Jordan algebras of linear transformations
A. A.
Albert
524-555
Further contribution to the geometry of numbers for non-convex regions
L. J.
Mordell
189-215
A class of harmonic functions in three variables and their properties
Stefan
Bergman
216-247
Application of analytic functions to two-dimensional biharmonic analysis
Hillel
Poritsky
248-279
On generalized quaternions
Gordon
Pall
280-332
Descriptive geometries as multigroups
Walter
Prenowitz
333-380
Extension sets
A. D.
Wallace
1-13
The approximation by partial sums of Fourier series
R.
Salem;
A.
Zygmund
14-22
Capacity of sets and Fourier series
R.
Salem;
A.
Zygmund
23-41
Equilong maps of the $\infty\sp 3$ circles
John
DeCicco
42-53
On the structure and ideal theory of complete local rings
I. S.
Cohen
54-106
Contour integration in the theory of the spherical pendulum and the heavy symmetrical top
Walter
Kohn
107-131
A new foundation of the projective differential theory of curves in five-dimensional space
Su-Cheng
Chang
132-165
The theory of Boolean-like rings
Alfred L.
Foster
166-187
Arithmetic of ordered systems
Mahlon M.
Day
1-43
The Gauss-Green theorem
Herbert
Federer
44-76
Quasi-complements and closed projections in reflexive Banach spaces
F. J.
Murray
77-95
The membrane theory of shells of revolution
C.
Truesdell
96-166
The boundedness of certain sets of orthonormal polynomials in one, two, and three variables
Dunham
Jackson
167-183
Transformations of Wiener integrals under a general class of linear transformations
R. H.
Cameron;
W. T.
Martin
184-219
On generalized convex functions
E. F.
Beckenbach;
R. H.
Bing
220-230
General theory of natural equivalences
Samuel
Eilenberg;
Saunders
MacLane
231-294
Representations of groups as quotient groups. I
Reinhold
Baer
295-347
Representations of groups as quotient groups. II. Minimal central chains of a group
Reinhold
Baer
348-389
Representations of groups as quotient groups. III. Invariants of classes of related representations
Reinhold
Baer
390-419
On continuous mappings of Peano spaces
Tibor
Radó
420-454
Markoff chains---denumerable case
J. L.
Doob
455-473
Errata, volume 48; Errata, volume 57
474
Certain classes of analytic functions of two real variables and their properties
Stefan
Bergman
299-331
Orthonormal sets of periodic functions of the type $\{f(nx)\}$
D. G.
Bourgin;
C. W.
Mendel
332-363
Structure theorems for a special class of Banach algebras
Warren
Ambrose
364-386
Valuation ideals in polynomial rings
A.
Seidenberg
387-425
Fields in terms of a single operation
Robert J.
Levit
426-440
Geometries of matrices. I. Generalizations of von Staudt's theorem
Loo-Keng
Hua
441-481
Geometries of matrices. $I\sb 1$. Arithmetical construction
Loo-Keng
Hua
482-490
On infinite-dimensional linear spaces
George W.
Mackey
155-207
On ordered groups
C. J.
Everett;
S.
Ulam
208-216
Arcs and geodesics in metric spaces
S. B.
Myers
217-227
Structure theory of simple rings without finiteness assumptions
N.
Jacobson
228-245
Remarks on ergodic theorems
Einar
Hille
246-269
Dynamical and curvature trajectories in space
John
DeCicco
270-286
Coherent and saturated collections
G. T.
Whyburn
287-298
Intersections of algebraic and algebroid varieties
Claude
Chevalley
1-85
Separable Diophantine equations
E. T.
Bell
86-101
Note on the converse of Fabry's gap theorem
P.
Erdös
102-104
A general theory of surfaces and conjugate nets
V. G.
Grove
105-122
Lattices with unique complements
R. P.
Dilworth
123-154
The degree of a linear homogeneous group
William Benjamin
Fite
1-6
Approximation by integral functions in the complex domain
H.
Kober
7-31
Sets of uniqueness and sets of multiplicity. II
R.
Salem
32-49
An abstract Radon-Nikodym theorem
C. E.
Rickart
50-66
On a class of functions defined by partial differential equations
Lipman
Bers;
Abe
Gelbart
67-93
The fundamental theorems of elementary geometry. An axiomatic analysis
Reinhold
Baer
94-129
The theorem of Bertini on the variable singular points of a linear system of varieties
Oscar
Zariski
130-140
Some results in the theory of linear non-associative algebras
Richard H.
Bruck
141-199
Local metric geometry
Herbert
Busemann
200-274
Net homotopy for compacta
D. E.
Christie
275-308
Conformal differential geometry of a subspace
Aaron
Fialkow
309-433
An extension of metric distributive lattices with an application in general analysis
Malcolm
Smiley
435-447
On highly composite and similar numbers
L.
Alaoglu;
P.
Erdös
448-469
Some applications of Fourier analysis and calculus of probability to the study of real roots of algebraic equations
Alfred M.
Peiser
470-493
Cleft rings
Bernard
Vinograde
494-507
Two Tauberian theorems in the theory of Fourier series
Ching-Tsün
Loo
508-518
Note on the strong summability of Fourier series
Ching-Tsün
Loo
519-527
The free Lie ring and Lie representations of the full linear group
Angeline
Brandt
528-536
On the distribution of quadratic nonresidues and the Euclidean algorithm in real quadratic fields. I
Loo-Keng
Hua
537-546
On the distribution of quadratic non-residues and the Euclidean algorithm in real quadratic fields. II
Loo-Keng
Hua;
Szu-Hoa
Min
547-569
Errata, volume 55; Errata, volume 56
570
Approximation theories for measure preserving transformations
Paul R.
Halmos
1-18
Some results in the theory of quasigroups
Richard H.
Bruck
19-52
Interpolation and approximation by functions analytic and bounded in a given region
E. N.
Nilson;
J. L.
Walsh
53-67
Some properties of ideals in rings of power series
Claude
Chevalley
68-84
Positive temperatures on an infinite rod
D. V.
Widder
85-95
Quasi-analyticity and analytic continuation---a general principle
S.
Mandelbrojt
96-131
The identity of weak and strong extensions of differential operators
K. O.
Friedrichs
132-151
On the extension of linear transformations
Andrew
Sobczyk
153-169
On certain integrals
A.
Zygmund
170-204
A theory of covering and differentiation
Anthony P.
Morse
205-235
The $\phi$ rectifiable subsets of the plane
A. P.
Morse;
John F.
Randolph
236-305
Tensor theory of invariants for the projective differential geometry of a curved surface
Thomas C.
Doyle
306-348
On a problem of Walsh concerning the Hadamard three circles theorem
Maurice H.
Heins
349-372
Contributions to the analytic theory of $J$-fractions
H. S.
Wall;
Marion
Wetzel
373-392
Postulate-sets for Boolean rings
B. A.
Bernstein
393-400
Quasigroups. II
A. A.
Albert
401-419
Surface area. I
Herbert
Federer
420-437
Surface area. II
Herbert
Federer
438-456
A new criterion for completely monotonic functions
Harry
Pollard
457-464
Metric foundations of geometry. I
Garrett
Birkhoff
465-492
Galois connexions
Oystein
Ore
493-513
Closure operators and Galois theory in lattices
C. J.
Everett
514-525
Generalization of a probability limit theorem of Cram\'er
W.
Feller
361-372
The general form of the so-called law of the iterated logarithm
W.
Feller
373-402
Regularity of label-sequences under configuration transformations
T. N. E.
Greville
403-413
The theory of analytic functions in normed Abelian vector rings
Edgar R.
Lorch
414-425
Tests of statistical hypotheses concerning several parameters when the number of observations is large
Abraham
Wald
426-482
On some trigonometric summability methods and Gibbs' phenomenon
Otto
Szász
483-497
On reflexive norms for the direct product of Banach spaces
Robert
Schatten
498-506
Quasigroups. I
A. A.
Albert
507-519
Permanent configurations in the $n$-body problem
Carl
Holtom
520-543
On spaces in which two points determine a geodesic
Herbert
Busemann
171-184
Spectral theory. I. Convergence to projections
Nelson
Dunford
185-217
Sets of uniqueness and sets of multiplicity
R.
Salem
218-228
Projective planes
Marshall
Hall
229-277
On the oscillation of differential transforms. III. Oscillations of the derivative of a function
A. C.
Schaeffer
278-285
On homological situation properties of complexes and closed sets
Paul
Alexandroff
286-339
Independent integral bases and a characterization of regular surfaces
H. T.
Muhly
340-360
Infinite linear difference equations with arbitrary real spans and first degree coefficients
R. H.
Cameron;
W. T.
Martin
1-22
A generalized theory of dynamical trajectories
Edward
Kasner;
John
DeCicco
23-38
A property of the zeros of Legendre polynomials
A.
Zygmund
39-56
Flat space congruences of order one in $[n]$
Edwin J.
Purcell
57-69
On the approximation to integrable functions by integral functions
H.
Kober
70-82
The transformation of double integrals
R. G.
Helsel;
T.
Radó
83-102
A theory of crossed characters
Reinhold
Baer
103-170
On approximating certain integrals by sums
C. Raymond
Adams;
Anthony P.
Morse
363-426
On some singular monotonic functions which are strictly increasing
R.
Salem
427-439
On the partial sums of Fourier series at points of discontinuity
Otto
Szász
440-453
The characteristic of a quadratic form for an arbitrary field
Rufus
Oldenburger
454-462
On the oscillation of differential transforms. IV. Jacobi polynomials
G.
Szegö
463-468
On structures of infinite modules
R. E.
Johnson
469-489
Foundations of a general theory of birational correspondences
Oscar
Zariski
490-542
Some Einstein spaces with conformally separable fundamental tensors
Yung-Chow
Wong
157-194
On the direct product of Banach spaces
Robert
Schatten
195-217
Direct methods in geometrical optics
M.
Herzberger
218-229
Mean-values and harmonic polynomials
E. F.
Beckenbach;
Maxwell
Reade
230-238
The converse of the Fatou theorem for positive harmonic functions
Lynn H.
Loomis
239-250
On bounded variation and absolute continuity for parametric representations of continuous surfaces
Paul V.
Reichelderfer
251-291
A theory for ordinary differential boundary problems of the second order and of the highly irregular type
Rudolph E.
Langer
292-361
Torsion of hollow cylinders
R. C. F.
Bartels
1-13
Heat conduction in an infinite composite solid with an interface resistance
W. A.
Mersman
14-24
Congruences in unitary space
N.
Coburn
25-40
Recursive predicates and quantifiers
S. C.
Kleene
41-73
B\'ezout's theorem and algebraic differential equations
J. F.
Ritt
74-82
The continuity of functions of many variables
Richard
Kershner
83-100
The Gauss-Bonnet theorem for Riemannian polyhedra
Carl B.
Allendoerfer;
André
Weil
101-129
Linear operators in the theory of partial differential equations
Stefan
Bergman
130-155
A new class of self-adjoint boundary value problems
William T.
Reid
381-425
Bounded univalent functions
Raphael M.
Robinson
426-449
On the oscillation of differential transforms. I
G.
Szegö
450-462
On the oscillation of differential transforms. II. Characteristic series of boundary value problems
Einar
Hille
463-497
Integration in a convex linear topological space
C. E.
Rickart
498-521
The restricted problem of three bodies
Monroe H.
Martin
522-538
Analytic extension by Hausdorff methods
Ralph Palmer
Agnew
217-237
The spectrum of linear transformations
Edgar R.
Lorch
238-248
On the oscillation of the derivatives of a periodic function
George
Pólya;
Norbert
Wiener
249-256
Structure of linear sets
Ernst
Snapper
257-264
On a generalization of the problem of quasi-analyticity
S.
Mandelbrojt;
F. E.
Ulrich
265-282
A unified theory of projective spaces and finite abelian groups
Reinhold
Baer
283-343
Sufficient conditions for a weak relative minimum in the problem of Bolza
E. J.
McShane
344-379
Quadratic Diophantine equations in the rational and quadratic fields
Ivan
Niven
1-11
On the partial sums of harmonic developments and of power series
Otto
Szász
12-21
Hyperspaces of a continuum
J. L.
Kelley
22-36
Topics in the theory of Markoff chains
J. L.
Doob
37-64
On converse gap theorems
George
Pólya
65-71
N\"orlund summability of double Fourier series
John G.
Herriot
72-94
Transitives of betweenness
Everett
Pitcher;
M. F.
Smiley
95-114
On the basis theorem for differential systems
E. R.
Kolchin
115-127
On the derivatives of functions analytic in the unit circle and their radii of univalence and of $p$-valence
W.
Seidel;
J. L.
Walsh
128-216
Segments of ordered sets
W. D.
Duthie
1-14
Theory of monomial groups
Oystein
Ore
15-64
The finite displacement of thin rods
G. E.
Hay
65-102
Homomorphisms and modular functionals
Saul
Gorn
103-116
Complete sets of logical functions
William
Wernick
117-132
On distortion in pseudo-conformal mapping
Stefan
Bergman;
D. C.
Spencer
133-163
Lebesgue theory on a Boolean algebra
John M. H.
Olmsted
164-193
On the number of partitions of a number into unequal parts
Loo-keng
Hua
194-201
Theory of reduction for arithmetical equivalence. II
Hermann
Weyl
203-231
The general invariant theory of irregular analytic arcs or elements
Edward
Kasner;
John
DeCicco
232-254
The transformation of series and sequences
W. T.
Scott;
H. S.
Wall
255-279
On conformal mapping of infinite strips
S. E.
Warschawski
280-335
On the semi-continuity of double integrals in parametric form
Tibor
Radó
336-361
Manifolds without conjugate points
Marston
Morse;
Gustav A.
Hedlund
362-386
Completely convex functions and Lidstone series
D. V.
Widder
387-398
Ergodic theorems for Abelian semigroups
Mahlon
Day
399-412
On measure in abstract sets
Dorothy
Maharam
413-433
The conformal theory of curves
Aaron
Fialkow
435-501
An arithmetical theory of the Bernoulli numbers
H. S.
Vandiver
502-531
On the structure of differential polynomials and on their theory of ideals
Howard
Levi
532-568
Topology in lattices
Orrin
Frink
569-582
Operation in Banach spaces
Mahlon M.
Day
583-608
The Ramanujan identities under modular substitutions
Hans
Rademacher
609-636
The structure of locally connected topological spaces
G. E.
Albert;
J. W. T.
Youngs
637-654
Factor-sets of a group in its abstract unit group
A. H.
Clifford;
Saunders
Mac Lane
385-406
On the behavior of trigonometric series and power series
J.
Marcinkiewicz;
A.
Zygmund
407-453
The paradox of Kleene and Rosser
Haskell B.
Curry
454-516
A refinement of an inequality of the brothers Markoff
R. J.
Duffin;
A. C.
Schaeffer
517-528
On differential geometry intrinsically connected with a surface element of projective arc length
P. O.
Bell
529-547
Inequalities for harmonic polynomials in two and three dimensions
A. C.
Schaeffer;
G.
Szegö
187-225
Fourier integrals and metric geometry
J.
von Neumann;
I. J.
Schoenberg
226-251
Properties of growth for solutions of differential equations of dynamical type
W. J.
Trjitzinsky
252-294
Normal algebraic number fields
Saunders
MacLane;
O. F. G.
Schilling
295-384
The decomposition of meromorphic functions into rational functions of univalent functions
Lynn H.
Loomis
1-14
Restricted Lie algebras of characteristic $p$
N.
Jacobson
15-25
Conformal unitary spaces
N.
Coburn
26-39
On a generalized form of Plateau's problem
R.
Courant
40-47
Pencils on an algebraic variety and a new proof of a theorem of Bertini
Oscar
Zariski
48-70
Solution of the inverse problem of the calculus of variations
Jesse
Douglas
71-128
Sur l'existence de fonctions enti\`eres satisfaisant \`a certaines conditions lin\'eaires
G.
Pólya
129-139
The parastrophic criterion for the factorization of primes
Fred
Kiokemeister
140-159
On a theorem of Schur and on fractional integrals on purely imaginary order
H.
Kober
160-174
The structure of the group ring of a $p$-group over a modular field
S. A.
Jennings
175-185
Ideals in Birkhoff lattices
R. P.
Dilworth
325-353
Generalizations to space of the Cauchy and Morera theorems
Maxwell
Reade;
E. F.
Beckenbach
354-377
Families of curves conformally equivalent to circles
Edward
Kasner;
John
De Cicco
378-391
Structure of abelian quasi-groups
D. C.
Murdoch
392-409
Concerning the decomposition and amalgamation of points, upper semi-continuous collections, and topological extensions
R. G.
Lubben
410-466
On the Jacobi series
J. H.
Curtiss
467-501
On sets of matrices with coefficients in a division ring
Richard
Brauer
502-548
On the foundations of calculus of variations
Herbert
Busemann;
Walther
Mayer
173-198
On the growth properties of a function of two complex variables given by its power series expansion
Abe
Gelbart
199-210
On the possible rate of growth of an analytic function
P. W.
Ketchum
211-228
On the degree of polynomial approximation to analytic functions: problem $\beta$
J. L.
Walsh;
W. E.
Sewell
229-257
A theory of absolutely continuous transformations in the plane
T.
Radó;
P.
Reichelderfer
258-307
On the classification of the mappings of a 2-complex
Herbert
Robbins
308-324
Two-to-one transformations on 2-manifolds
Venable
Martin;
J. H.
Roberts
1-17
The integral representation of weakly almost-periodic transformations in reflexive vector spaces
Edgar R.
Lorch
18-40
General combinatorial topology
Paul
Alexandroff
41-105
Products of normal semi-fields
Albert
Neuhaus
106-121
The accuracy of the Gaussian approximation to the sum of independent variates
Andrew C.
Berry
122-136
Minimal positive harmonic functions
Robert S.
Martin
137-172
Analytic systems of central conics in space
J. L.
Coolidge
359-376
On circavariant matrices and circa-equivalent networks
Richard Stevens
Burington
377-390
The position of the radical in an algebra
Marshall
Hall
391-404
Integers of quadratic fields as sums of squares
Ivan
Niven
405-417
On finitely mean valent functions. II
D. C.
Spencer
418-435
Integral sets of quaternion algebras over a function field
436-450
Order types and structure of orders
André
Gleyzal
451-466
Expansions of analytic functions
R. P.
Boas
467-487
On the integro-differential equations of purely discontinuous Markoff processes
Willy
Feller
488-515
On linear transformations
R. S.
Phillips
516-541
On a type of algebraic differential manifold
J. F.
Ritt
542-552
Continued fractions and totally monotone sequences
H. S.
Wall
165-184
Hausdorff methods of summation and continued fractions
H. L.
Garabedian;
H. S.
Wall
185-207
Polyadic groups
Emil L.
Post
208-350
On a minimum problem in the theory of analytic functions of several variables
W. T.
Martin
351-357
On kernels of faltung transformations
Ralph Palmer
Agnew
1-20
Topological group foundations of rigid space geometry
Deane
Montgomery;
Leo
Zippin
21-49
Conformality in connection with functions of two complex variables
Edward
Kasner
50-62
Arc- and tree-preserving transformations
D. W.
Hall;
G. T.
Whyburn
63-71
Orthogonal polynomials with auxiliary conditions
Dunham
Jackson
72-81
Continuous additive functionals on the space $(BV)$ and certain subspaces
C. Raymond
Adams;
Anthony P.
Morse
82-100
A new special form of the linear element of a surface
Jesse
Douglas
101-116
On strong summability of Fourier series
Otto
Szász
117-125
Theory of reduction for arithmetical equivalence
Hermann
Weyl
126-164
Linear operations on summable functions
Nelson
Dunford;
B. J.
Pettis
323-392
Nilpotent groups and their generalizations
Reinhold
Baer
393-434
Nets and groups. II
Reinhold
Baer
435-439
Regular normal extensions over complete fields
O. F. G.
Schilling
440-454
Regularity properties of certain families of chance variables
J. L.
Doob
455-486
On the arithmetic of quaternions
Gordon
Pall
487-500
A theorem on quadratic forms and its application in the calculus of variations
M. R.
Hestenes;
E. J.
McShane
501-512
Harmonic minimal surfaces
W. C.
Graustein
173-206
The geometry of fields of lineal elements
John
De Cicco
207-229
The role of the mean curvature in the immersion theory of surfaces
H. W.
Alexander
230-253
On the degree of convergence of sequences of rational functions
J. L.
Walsh
254-292
Note on the degree of convergence of sequences of analytic functions
J. L.
Walsh
293-304
On a decomposition of true cyclic elements
Dick Wick
Hall
305-321
Correction to ``The asymptotic forms of the Hermite and Weber functions''
Nathan
Schwid
322
The theory of integration
G. Baley
Price
1-50
On the gradient of solid harmonic polynomials
G.
Szegö
51-65
On measurable stochastic processes
Warren
Ambrose
66-79
Contributions to the theory of Hermitian series. II. The representation problem
Einar
Hille
80-94
On $0$-regular surface transformations
W. T.
Puckett
95-113
Integration in a convex linear topological space
R. S.
Phillips
114-145
Harmonic surfaces in Riemann metric
S.
Bochner
146-154
A convergence theorem for continued fractions
W. T.
Scott;
H. S.
Wall
155-172
Continuous groups and Schwarz' lemma
Max
Zorn
1-22
Steinitz field towers for modular fields
Saunders
MacLane
23-45
On interpolation by functions analytic and bounded in a given region
J. L.
Walsh
46-65
Mean motions and almost periodic functions
Philip
Hartman
66-81
Maximal orders in rational cyclic algebras of composite degree
Sam
Perlis
82-96
Convergence properties of analytic functions of Fourier-Stieltjes transforms
R. H.
Cameron;
Norbert
Wiener
97-109
Nets and groups
Reinhold
Baer
110-141
On a generalization of the Stieltjes moment problem
R. P.
Boas
142-150
The boundary problem of an ordinary linear differential system in the complex domain
Rudolph E.
Langer
151-190
On the representation of a function by certain Fourier integrals
Harald
Cramér
191-201
General theory of singular integral equations with real kernels
W. J.
Trjitzinsky
202-279
Limits of a distribution function determined by absolute moments and inequalities satisfied by absolute moments
Abraham
Wald
280-306
An interpretation of the index of inertia of the discriminant matrices of a linear associative algebra
R. F.
Rinehart
307-327
Geometric aspects of relativistic dynamics
L. A.
MacColl
328-347
The differential geometry of series of lineal elements
John
De Cicco
348-361
On the remainders and convergence of the series for the partition function
D. H.
Lehmer
362-373
Contributions to the transformation theory of dynamics
Daniel C.
Lewis
374-388
A study of curved surfaces by means of certain associated ruled surfaces
P. O.
Bell
389-409
Invariance of the admissibility of numbers under certain general types of transformations
T. N. E.
Greville
410-425
Non-commutative residuated lattices
R. P.
Dilworth
426-444
Ideal theory and algebraic difference equations
J. F.
Ritt;
H. W.
Raudenbush
445-452
Exponent trajectories in symbolic dynamics
Rufus
Oldenburger
453-466
Correction to ``The boundary problem of an ordinary linear differential system in the complex domain''
Rudolph E.
Langer
467
A correction to ``Properties of functions $f(x,y)$ of bounded variation.''
C. R.
Adams;
J. A.
Clarkson
468
Residuated lattices
Morgan
Ward;
R. P.
Dilworth
335-354
Kakeya's problem on the zeros of the derivative of a polynomial
Morris
Marden
355-368
Abstract symmetric boundary conditions
J. W.
Calkin
369-442
Conformal geodesics
Aaron
Fialkow
443-473
Bilinear transformations in Hilbert space
Francis J.
Murray
474-507
Some existence theorems in the calculus of variations. IV. Isoperimetric problems in non-parametric form
E. J.
McShane
173-196
Some existence theorems in the calculus of variations. V. The isoperimetric problem in parametric form
E. J.
McShane
197-216
On a calculus of operators in reflexive vector spaces
Edgar R.
Lorch
217-234
Thermal stresses in elastic plates
I. S.
Sokolnikoff;
E. S.
Sokolnikoff
235-255
Geometry of a surface in the neighborhood of a spine
V. G.
Grove
256-264
Conditions on $u(x,y)$ and $v(x,y)$ necessary and sufficient for the regularity of $u+iv$
Philip T.
Maker
265-275
Irreducible systems of algebraic differential equations
Walter Charles
Strodt
276-297
On the coefficients of certain modular forms belonging to subgroups of the modular group
Herbert S.
Zuckerman
298-321
The type of certain Borel sets in several Banach spaces
C. Raymond
Adams;
James A.
Clarkson
322-334
The iterated Stieltjes transform
R. P.
Boas;
D. V.
Widder
1-72
The abstract groups $G\sp {m,n,p}$
H. S. M.
Coxeter
73-150
Some existence theorems in the calculus of variations. III. Existence theorems for nonregular problems
E. J.
McShane
151-171
Groups with abelian central quotient group
Reinhold
Baer
357-386
Groups with preassigned central and central quotient group
Reinhold
Baer
387-412
Definitely self-adjoint boundary value problems
Gilbert A.
Bliss
413-428
Some existence theorems in the calculus of variations. I. The Dresden corner condition
E. J.
McShane
429-438
Some existence theorems in the calculus of variations. II. Existence theorems for isoperimetric problems in the plane
E. J.
McShane
439-453
Contributions to the theory of multivariate statistical analysis
William G.
Madow
454-495
A continuous function with no unilateral derivatives
Anthony P.
Morse
496-507
A system of ordinary linear differential equations with two-point boundary conditions
William T.
Reid
508-521
Metric spaces and positive definite functions
I. J.
Schoenberg
522-536
On the transitivity of perspectivity in continuous geometries
Israel
Halperin
537-562
Permanent configurations in the problem of five bodies
W. L.
Williams
563-579
Geometry in an $n$-dimensional space with the arc length $s=\int\{A_i(x,x') x''^i + B(x,x')\}^{1/p} dt$
Akitsugu
Kawaguchi
153-167
On the reduction of dynamical systems by means of parametrized invariant relations
E. R.
van Kampen;
Aurel
Wintner
168-195
An isomorphism between linear recurring sequences and algebraic rings
Marshall
Hall
196-218
Rational equivalence of a form to a sum of $p$th powers
Rufus
Oldenburger
219-249
Two-dimensional spaces in which there exist contiguous points
E. C.
Klipple
250-276
On integration in vector spaces
B. J.
Pettis
277-304
Uniformity in linear spaces
Nelson
Dunford
305-356
Analytic continuation of diagonals and Hadamard compositions of multiple power series
R. H.
Cameron;
W. T.
Martin
1-7
Equivalence of pairs of matrices
Merrill M.
Flood
8-17
Fixed points under transformations of continua which are not connected im Kleinen
O. H.
Hamilton
18-24
The two conformal invariants of fifth order
Edward
Kasner
25-31
The absolute optical instrument
J. L.
Synge
32-46
Polynomial approximations for elliptic functions
E. T.
Bell
47-57
Plane Peanian continua with unique maps on the sphere and in the plane
V. W.
Adkisson
58-67
The law of apparition of primes in a Lucasian sequence
Morgan
Ward
68-86
Stochastic processes with an integral-valued parameter
J. L.
Doob
87-150
Correction to the paper: ``The multinomial solid and the chi test'' [Trans. Amer. Math. Soc. {\bf 31} (1929), no. 1, 133--144; 1501472]
Burton H.
Camp
151
Comparison of products of methods of summability
Ralph Palmer
Agnew
327-343
The geometry of whirl series
John
De Cicco
344-358
An extension of Schwarz's lemma
Lars V.
Ahlfors
359-364
Normality and abnormality in the calculus of variations
G. A.
Bliss
365-376
A theorem in finite projective geometry and some applications to number theory
James
Singer
377-385
Symmetric and alternate matrices in an arbitrary field. I
A. Adrian
Albert
386-436
Generalized integrals and differential equations
Hans
Lewy
437-464
Decompositions and dimension of closed sets in ${\bf R}\sp n$
Arthur N.
Milgram
465-481
Concerning limiting sets in abstract spaces. II
R. G.
Lubben
482-493
A correction to the paper: ``On effective sets of points in relation to integral functions'' [Trans. Amer. Math. Soc. {\bf 42} (1937), no. 3, 358--365; 1501926]
V. Ganapathy
Iyer
494
A class of polynomials
Leonard
Carlitz
167-182
Analyticity of equilibrium figures of rotation
Bernard
Friedman
183-198
Generalizations of the Gauss law of the spherical mean
Hillel
Poritsky
199-225
The Sch\"onemann-Eisenstein irreducibility criteria in terms of prime ideals
Saunders
MacLane
226-239
On the growth of analytic functions
Norman
Levinson
240-257
On differential geometry in the large. I. Minkowski's problem
Hans
Lewy
258-270
On the series for the partition function
D. H.
Lehmer
271-295
A problem in additive number theory
R. D.
James
296-302
Transformations of a surface bearing a family of asymptotic curves
G. D.
Gore
303-320
On functions with bounded derivatives
Oystein
Ore
321-326
Postulates for abelian groups and fields in terms of non-associative operations
B. A.
Bernstein
1-6
The Stieltjes transform
D. V.
Widder
7-60
Analytical groups
Garrett
Birkhoff
61-101
On an approximate functional equation of Paley
W. C.
Randels
102-125
On the solutions of quasi-linear elliptic partial differential equations
Charles B.
Morrey
126-166
Consistency of the conditions determining Kollektivs
Arthur H.
Copeland
333-357
On effective sets of points in relation to integral functions
V. Ganapathy
Iyer
358-365
Fourier series and mean moduli of continuity
Otto
Szász
366-395
The characterization of the closed $n$-cell
D. W.
Woodard
396-415
Partially ordered sets
H. M.
MacNeille
416-460
On mechanical quadratures, in particular, with positive coefficients
J.
Shohat
461-496
Existence theorem for the flow of an ideal incompressible fluid in two dimensions
A. C.
Schaeffer
497-513
A new class of transcendental numbers
Philip
Franklin
155-182
A direct expansion proof of sufficient conditions for the non-parametric problem of Bolza
William T.
Reid
183-190
On a classification of integral functions by means of certain invariant point properties. A supplement
M. T.
Bird
191-193
On the space $(BV)$
C. Raymond
Adams;
Anthony P.
Morse
194-205
Abstract derivation and Lie algebras
Nathan
Jacobson
206-224
Theory of non-linear singular differential systems
W. J.
Trjitzinsky
225-321
Almost periodic transformation groups
Deane
Montgomery
322-332
The involution curve determined from a special pencil of $n$-ics
Josephine H.
Chanler
1-15
Relative linear sets and similarity of matrices whose elements belong to a division algebra
M. H.
Ingraham;
M. C.
Wolf
16-31
The Binet of quadrics in $S\sb 3$
Temple R.
Hollcroft
32-40
Concerning essential continua of condensation
R. L.
Moore
41-52
Concerning certain topologically flat spaces
F. Burton
Jones
53-93
The geometry of isogonal and equi-tangential series
Edward
Kasner
94-106
Stochastic processes depending on a continuous parameter
J. L.
Doob
107-140
A direct sufficiency proof for the problem of Bolza in the calculus of variations
Magnus R.
Hestenes
141-154
On certain inequalities and characteristic value problems for analytic functions and for functions of two variables
Kurt
Friedrichs
321-364
A priori limitations for solutions of Monge-Amp\`ere equations. II
Hans
Lewy
365-374
Applications of the theory of Boolean rings to general topology
M. H.
Stone
375-481
Errata: ``Convergence in variation and related topics'' [Trans. Amer. Math. Soc. {\bf 41} (1937), no. 1, 48--83; 1501891]
Anthony P.
Morse
482
Errata: ``On the theorem of Jordan-H\"older'' [Trans. Amer. Math. Soc. {\bf 41} (1937), no. 2, 266--275; 1501901]
Oystein
Ore
482
Functions defined by sequences of integrals and the inversion of approximate derived numbers
R. L.
Jeffery
171-192
Tangent lines and planes in topological spaces
Charles C.
Torrance
193-207
On rings of operators. II
F. J.
Murray;
J.
von Neumann
208-248
Generalized weight properties of the resultant of $n+1$ polynomials in $n$ indeterminates
Oscar
Zariski
249-265
On the theorem of Jordan-H\"older
Oystein
Ore
266-275
Linear divisibility sequences
Morgan
Ward
276-286
Concerning uniqueness-bases of finite groups with applications to $p$-groups of class $2$
Charles
Hopkins
287-313
Geometry of dynamical trajectories at a point of equilibrium
Edward
Kasner;
Aaron
Fialkow
314-320
On Phragm\'en-Lindel\"of's principle
Lars V.
Ahlfors
1-8
Extensions of the four-vertex theorem
W. C.
Graustein
9-23
On the summability of Fourier series
W. C.
Randels
24-47
Convergence in variation and related topics
Anthony P.
Morse
48-83
Generalized derivatives and approximation by polynomials
W. E.
Sewell
84-123
Stereographic parameters and pseudo-minimal hypersurfaces. II
Otto
Laporte
124-137
On complementary manifolds and projections in spaces $L\sb p$ and $l\sb p$
F. J.
Murray
138-152
A correction: ``A local solution of the difference equation $\Delta y(x)=F(x)$ and of related equations'' [Trans. Amer. Math. Soc. {\bf 39} (1936), no. 3, 345--379; 1501852]
I. M.
Sheffer
153-159
On some functionals. II
S.
Saks
160-170
On Boolean functions of many variables
J. C. C.
McKinsey
343-362
A construction for absolute values in polynomial rings
Saunders
MacLane
363-395
Uniformly convex spaces
James A.
Clarkson
396-414
Remarks on the preceding paper of James A. Clarkson: ``Uniformly convex spaces'' [Trans. Amer. Math. Soc. {\bf 40} (1936), no. 3; MR1501880]
Nelson
Dunford;
Anthony P.
Morse
415-420
The space of functions of bounded variation and certain general spaces
C. Raymond
Adams
421-438
On ideals in a quaternion algebra and the representation of integers by Hermitian forms
Claiborne G.
Latimer
439-449
On some Hermitian forms associated with two given curves of the complex plane
Gabriel
Szegö
450-461
On a classification of integral functions by means of certain invariant point properties
R. D.
Carmichael;
W. T.
Martin;
M. T.
Bird
462-473
Integration and linear operations
Nelson
Dunford
474-494
Initial motion in fields of force
Aaron
Fialkow
495-501
Summation of multiple Fourier series by spherical means
Salomon
Bochner
175-207
On certain generalizations of the Cauchy-Taylor expansion theory
Gertrude Stith
Ketchum
208-224
Bernstein's theorem and trigonometric approximation
Dunham
Jackson
225-251
Singular quadratic functionals
Marston
Morse;
Walter
Leighton
252-286
Some theorems on Fourier transforms and conjugate trigonometric integrals
R. P.
Boas
287-308
Differentiable functions defined in arbitrary subsets of Euclidean space
Hassler
Whitney
309-317
On the class number of a quaternion algebra with a negative fundamental number
Claiborne G.
Latimer
318-323
On simply transitive groups with transitive abelian subgroups of the same degree
Dorothy
Manning
324-342
The maximal orders of generalized quaternion division algebras
Ralph
Hull
1-11
On the zeros of successive derivatives of integral functions
I. J.
Schoenberg
12-23
Periodic one-parameter groups in three-space
Deane
Montgomery;
Leo
Zippin
24-36
The theory of representations for Boolean algebras
M. H.
Stone
37-111
Simple algebras of degree $p\sp e$ over a centrum of characteristic $p$
A. Adrian
Albert
112-126
Asymptotic solutions of certain linear differential equations in which the coefficient of the parameter may have a zero
Henry
Scheffé
127-154
Differential geometry of a surface at a planar point
V. G.
Grove
155-166
The generalized theorem of Stokes
Stewart S.
Cairns
167-174
A functional equation in arithmetic
E. T.
Bell
341-344
A local solution of the difference equation $\Delta y(x)=F(x)$ and of related equations
I. M.
Sheffer
345-379
On prime numbers of real quadratic fields in rectangles
Hans
Rademacher
380-398
On distributions admitting a sufficient statistic
B. O.
Koopman
399-409
Statistical estimation
J. L.
Doob
410-421
Non-singular multilinear forms and certain $p$-way matrix factorizations
Rufus
Oldenburger
422-455
The existence of an extremum in problems of Mayer
Lawrence M.
Graves
456-471
Some properties of conversion
Alonzo
Church;
J. B.
Rosser
472-482
On $k$-commutative matrices
William E.
Roth
483-495
On the order of groups of automorphisms
Garrett
Birkhoff;
Philip
Hall
496-499
Correction to the paper: ``A problem concerning orthogonal polynomials'' [Trans. Amer. Math. Soc. {\bf 37} (1935), no. 1, 196--206; 1501782]
Gabriel
Szegö
500
Some arithmetic means connected with Fourier series
L. S.
Bosanquet
189-204
A new method for Waring theorems with polynomial summands. II
L. E.
Dickson
205-208
The volume of the fundamental domain for some infinite groups
Carl Ludwig
Siegel
209-218
Mittelwerte arithmetischer Funktionen in Zahlk\"orpern
Carl Ludwig
Siegel
219-224
\"Uber die algebraischen Integrale des restringierten Dreik\"orperproblems
Carl Ludwig
Siegel
225-233
On the characteristic values of the matrix $f(A,B)$
William E.
Roth
234-243
A classification of generating functions
D. V.
Widder
244-298
Fuchsian groups and ergodic theory
Eberhard
Hopf
299-314
On asymptotic distributions of arithmetical functions
I. J.
Schoenberg
315-330
The integral representation of unbounded self-adjoint transformations in Hilbert space
Frederick
Riesz;
E. R.
Lorch
331-340
Inequalities for the zeros of Legendre polynomials and related functions
Gabriel
Szegö
1-17
Trigonometrische Reihen und Potenzreihen mit mehrfach monotoner Koeffizientenfolge
Leopold
Fejér
18-59
Differential geometry of a certain type of surface in $S\sb 4$
V. G.
Grove
60-70
Covariants of $r$-parameter groups
C. C.
MacDuffee
71-82
Linear transformations in $\mathfrak{L}_p, p>1$
F. J.
Murray
83-100
Quasi-commutative rings and differential ideals
Neal H.
McCoy
101-116
Converse theorems of summability for Dirichlet's series
Otto
Szász
117-130
Notes on linear transformations. I
Einar
Hille
131-153
Stereographic parameters and pseudo-minimal hypersurfaces
Otto
Laporte;
G. Y.
Rainich
154-182
Normal division algebras of degree $p\sp e$ over $F$ of characteristic $p$
A. Adrian
Albert
183-188
Continuity and summability for double Fourier series
J. J.
Gergen;
S. B.
Littauer
401-435
On ideals in generalized quaternion algebras and Hermitian forms
Claiborne G.
Latimer
436-446
The Diophantine equation $X\sp 2-DY\sp 2=Z\sp M$
Morgan
Ward
447-457
Interpolation in regularly distributed points
John
Curtiss
458-473
Abstract theory of inversion of finite series
Louis
Weisner
474-484
Some properties of prime-power groups
Louis
Weisner
485-492
General relations between Bernoulli, Euler, and allied polynomials
E. T.
Bell
493-500
Algebraic characterizations in complex differential geometry
T. Y.
Thomas
501-514
Maximal orders in rational cyclic algebras of odd prime degree
Ralph
Hull
515-530
Carath\'eodory measure and a generalization of the Gauss-Green lemma
John F.
Randolph
531-548
Existence theorems for double integral problems of the calculus of variations
E. J.
McShane
549-563
The characterization of plane collineations in terms of homologous families of lines
Walter
Prenowitz
564-599
Corrections to the paper: ``Integration in general analysis'' [Trans. Amer. Math. Soc. {\bf 37} (1935), no. 3, 441--453; 1501796]
Nelson
Dunford
600-601
Cyclotomy when $e$ is composite
L. E.
Dickson
187-200
Potentials of positive mass. II
Griffith C.
Evans
201-236
Some generalizations of Paley's theorems on Fourier series with positive coefficients
Michael
Fekete
237-249
On Fourier transforms. III
A. C.
Offord
250-266
Die Differentialgeometrie der Untermannigfaltigkeiten des ${\bf R}\sb n$ konstanter Kr\"ummung
Walther
Mayer
267-309
On the higher derivatives at the boundary in conformal mapping
Stefan E.
Warschawski
310-340
Simply connected sets
R. E.
Basye
341-356
Integration of functions with values in a Banach space
Garrett
Birkhoff
357-378
Functional dependence
Arthur B.
Brown
379-394
Note on irreducible quartic congruences
H. R.
Brahana
395-400
Inter-relations among the four principal types of order
Edward V.
Huntington
1-9
Contribution \`a l'\'etude du saut d'une fonction donn\'ee par son d\'eveloppement en s\'erie d'Hermite ou de Laguerre
Ervand
Kogbetliantz
10-47
Distribution functions and the Riemann zeta function
Børge
Jessen;
Aurel
Wintner
48-88
Trajectories and lines of force
Aaron
Fialkow
89-105
The Dirichlet problem for domains with multiple boundary points
F. W.
Perkins
106-144
The equivalence of pairs of Hermitian matrices
M. H.
Ingraham;
K. W.
Wegner
145-162
On the equivalence of quadrics in $m$-affine $n$-space and its relation to the equivalence of $2m$-pole networks
Richard Stevens
Burington
163-176
The potential function method for the solution of two-dimensional stress problems
C. W.
MacGregor
177-186
Cyclotomy and trinomial congruences
L. E.
Dickson
363-380
On the expansion of the remainder in the Newton-Cotes formula
J. V.
Uspensky
381-396
On the asymptotic solutions of ordinary differential equations, with reference to the Stokes' phenomenon about a singular point
Rudolph E.
Langer
397-416
A priori limitations for solutions of Monge-Amp\`ere equations
Hans
Lewy
417-434
An enumerative problem in the arithmetic of linear recurring series
Morgan
Ward
435-440
Integration in general analysis
Nelson
Dunford
441-453
On cyclic fields
A. Adrian
Albert
454-462
A canonical power series expansion for a surface
Ernest P.
Lane
463-482
Convergence properties of Fourier series
Otto
Szász
483-500
Certain contact properties of linear systems of hypersurfaces
B. C.
Wong
207-215
A basis for residual polynomials in $n$ variables
Marie
Litzinger
216-225
On potentials of positive mass. I
Griffith C.
Evans
226-253
An elliptic system of integral equations on summable functions
J. H.
Binney
254-265
On convex functions
Tibor
Radó
266-285
Systems of algebraic mixed difference equations
Fritz
Herzog
286-300
Linear transformations between Hilbert spaces and the application of this theory to linear partial differential equations
F. J.
Murray
301-338
The asymptotic forms of the Hermite and Weber functions
Nathan
Schwid
339-362
On complete topological spaces
John
von Neumann
1-20
Almost periodic functions in groups. II
S.
Bochner;
J.
von Neumann
21-50
On reversible dynamical systems
G. Baley
Price
51-79
Laplace integrals and factorial series in the theory of linear differential and linear difference equations
W. J.
Trjitzinsky
80-146
Sufficient conditions in the problem of Lagrange without assumptions of normalcy
Marston
Morse
147-160
Metabelian groups of order $p\sp m, p>2$
Charles
Hopkins
161-195
A problem concerning orthogonal polynomials
G.
Szegö
196-206
Properties of functions $f(x,y)$ of bounded variation
C. Raymond
Adams;
James A.
Clarkson
711-730
A new method for Waring theorems with polynomial summands
L. E.
Dickson
731-748
Derived numbers with respect to functions of bounded variation
R. L.
Jeffery
749-758
Probability and statistics
J. L.
Doob
759-775
Metabelian groups of order $p\sp {n+m}$ with commutator subgroups of order $p\sp m$
H. R.
Brahana
776-792
Sufficient conditions for the problem of Bolza in the calculus of variations
Magnus R.
Hestenes
793-818
Groups in which the squares of the elements are a dihedral subgroup
G. A.
Miller
819-825
A projective generalization of metrically defined associate surfaces
M. L.
MacQueen
826-840
On the power series for elliptic functions
E. T.
Bell
841-852
Some inequalities for non-uniformly bounded ortho-normal polynomials
M. F.
Rosskopf
853-867
On bounded linear functional operations
T. H.
Hildebrandt
868-875
A set of four postulates for Boolean algebra in terms of the ``implicative'' operation
B. A.
Bernstein
876-884
On normal Kummer fields over a non-modular field
A. Adrian
Albert
885-892
Almost periodic functions in a group. I
J.
v. Neumann
445-492
Waring's problem for cubic functions
G. Cuthbert
Webber
493-510
The application of the theory of admissible numbers to time series with constant probability
Francis
Regan
511-529
Inscribed sequences of surfaces associated with generalized sequences of Laplace
G. D.
Gore
530-541
The geometry of Riemannian spaces
W. C.
Graustein
542-585
Some points in the theory of trigonometric and power series
Antoni
Zygmund
586-617
The Riemann multiple-space and algebroid functions
B. O.
Koopman;
A. B.
Brown
618-626
On a certain correspondence between surfaces in hyperspace
V. G.
Grove
627-636
The solutions of the Mathieu equation with a complex variable and at least one parameter large
Rudolph E.
Langer
637-710
The moving trihedron
E. P.
Lane
696-710
Converses of Gauss' theorem on the arithmetic mean
Oliver D.
Kellogg
227-242
Contributions to the theory of finite fields
Oystein
Ore
243-274
Errata in my paper: ``On a special class of polynomials'' [Trans. Amer. Math. Soc. {\bf 35} (1933), no. 3, 559--584; 1501703]
Oystein
Ore
275
Almost periodic transformations
R. H.
Cameron
276-291
The Bertini transformation in space
F. R.
Sharpe;
L. A.
Dye
292-305
On the problem of $n$ bodies
J. J. L.
Hinrichsen
306-314
The ruled $V\sp 4\sb 4$ in $S\sb 5$ associated with a Schl\"afli hexad
John
Eiesland
315-326
On quasi-commutative matrices
Neal H.
McCoy
327-340
On the distributions of the zeros of sums of exponentials of polynomials
L. A.
MacColl
341-360
Ideal theory and algebraic differential equations
H. W.
Raudenbush
361-368
Differentiable functions defined in closed sets. I
Hassler
Whitney
369-387
Normal division algebras over a modular field
A. Adrian
Albert
388-394
The value of the number $g(k)$ in Waring's problem
R. D.
James
395-444
Waring's problem for cubic functions
L. E.
Dickson
1-12
On the convergence and overconvergence of sequences of polynomials of best simultaneous approximation to several functions analytic in distinct regions
J. L.
Walsh;
Helen G.
Russell
13-28
The first and second variations of an $n$-tuple integral in the case of variable limits
H. A.
Simmons
29-43
Consecutive covariant configurations at a point of a space curve
Abba V.
Newton
44-62
Analytic extensions of differentiable functions defined in closed sets
Hassler
Whitney
63-89
The asymptotic solutions of certain linear ordinary differential equations of the second order
Rudolph E.
Langer
90-106
The inversion of the Laplace integral and the related moment problem
D. V.
Widder
107-200
On the distribution of values of bounded analytic functions
Wladimir
Seidel
201-226
Notes on the theory and application of Fourier transforms. III, IV, V, VI, VII
R. E. A. C.
Paley;
N.
Wiener
761-791
Infinite systems of ordinary differential equations with applications to certain second-order partial differential equations
Daniel C.
Lewis
792-823
On definitions of bounded variation for functions of two variables
James A.
Clarkson;
C. Raymond
Adams
824-854
The general web of algebraic surfaces of order $n$ and the involution defined by it
Temple R.
Hollcroft
855-868
An iterative process in the problem of plateau
T.
Radó
869-887
Effects of linear transformations on the divergence of bounded sequences and functions
Joseph
Lev
888-896
Groups in which every operator has at most a prime number of conjugates
G. A.
Miller
897-902
Polynomial Diophantine systems
E. T.
Bell
903-914
Sections of point sets
Deane
Montgomery
915-928
Invariants of Pfaffian systems
Mabel
Griffin
929-939
The complete existential theory of the Whitehead-Huntington set of postulates for the algebra of logic
A. H.
Diamond
940-948
Cyclic fields of degree eight
A. Adrian
Albert
949-964
Addition to the note: ``On some functionals'' [Trans. Amer. Math. Soc. {\bf 35} (1933), no. 2, 549--556; 1501701]
Stanislaw
Saks
965-970
Correction to a paper: ``The Moore-Kline problem'' [Trans. Amer. Math. Soc. {\bf 34} (1932), no. 3, 705--721; 1501658]
Leo
Zippin
972
On a special class of polynomials
Oystein
Ore
559-584
The degree and class of multiply transitive groups. III
W. A.
Manning
585-599
The arithmetical theory of linear recurring series
Morgan
Ward
600-628
Sets of $k$-extent in $n$-dimensional space
R. L.
Jeffery
629-647
Subharmonic functions and minimal surfaces
E. F.
Beckenbach;
Tibor
Radó
648-661
Subharmonic functions and surfaces of negative curvature
E. F.
Beckenbach;
T.
Radó
662-674
A transformation of the problem of Lagrange in the calculus of variations
Lawrence M.
Graves
675-682
Contributions to the theory of transformations of nets in a space $S\sb n$
V. G.
Grove
683-688
On the equation $P(A,X)=0$ in matrices
William E.
Roth
689-708
A special integral function
R. E. A. C.
Paley
709-715
Parametrizations of saddle surfaces, with application to the problem of plateau
E. J.
McShane
716-733
The Latin square, or cyclic, functions
E. T.
Bell
734-745
Sufficient conditions in the problem of the calculus of variations in $n$-space in parametric form and under general end conditions
Sumner Byron
Myers
746-760
On a series of involutorial Cremona transformations of space defined by a pencil of rules surfaces
Virgil
Snyder
341-347
Notes on the theory and application of Fourier transforms. I, II
R. E. A. C.
Paley;
N.
Wiener
348-355
Pfaffian systems of species one
Joseph Miller
Thomas
356-371
Families of groups generated by two operators of the same order
Abraham
Sinkov
372-385
Groups $\{S,T\}$ whose commutator subgroups are abelian
H. R.
Brahana
386-396
On the representation of a polynomial in a Galois field as the sum of an even number of squares
Leonard
Carlitz
397-410
On the class number of a cyclic field
Claiborne G.
Latimer
411-417
The boundary values of analytic functions. II
Joseph L.
Doob
418-451
On finite-rowed systems of linear inequalities in infinitely many variables. II
I. J.
Schoenberg
452-478
Sufficient conditions for the general problem of Mayer with variable end points
M. R.
Hestenes
479-490
The structure of the number of representations function in a binary quadratic form
Gordon
Pall
491-509
On analytical complexes
S.
Lefschetz;
J. H. C.
Whitehead
510-517
Non-conjugate osculating quadrics of a curve on a surface
R. C.
Bullock
518-531
On Riesz and Ces\`aro methods of summability
Ralph Palmer
Agnew
532-548
On some functionals
Stanislaw
Saks
549-556
Boolean algebra. A correction to: ``New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell's {\it Principia mathematica}'' [Trans. Amer. Math. Soc. {\bf 35} (1933), no. 1, 274--304; 1501684]
Edward V.
Huntington
557-558
Systems of linear difference equations and expansions in series of exponential functions
R. D.
Carmichael
1-28
Convergence criteria for double Fourier series
J. J.
Gergen
29-63
On the numerators of the convergents of the Stieltjes continued fractions
Jacob
Sherman
64-87
Three-dimensional manifolds and their Heegaard diagrams
James
Singer
88-111
Non-cyclic algebras of degree and exponent four
A. Adrian
Albert
112-121
On abelian fields
Leonard
Carlitz
122-136
On the derivatives of Newtonian and logarithmic potentials near the acting masses
Mildred M.
Sullivan
137-171
The degree of convergence of a series of Bessel functions
M. G.
Scherberg
172-183
On the properties of polynomials satisfying a linear differential equation. I
I. M.
Sheffer
184-214
On the resultant of a system of forms homogeneous in each of several sets of variables
Neal H.
McCoy
215-233
An axiomatic basis for plane geometry
Stewart S.
Cairns
234-244
Proof of the fundamental theorems on second-order cross partial derivatives
A. E.
Currier
245-253
The cancellation law in the theory of congruences to a double modulus
Morgan
Ward
254-260
A characterization of the closed $2$-cell
Hassler
Whitney
261-273
New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell's {\it Principia mathematica}
Edward V.
Huntington
274-304
Sufficient conditions for a problem of Mayer in the calculus of variations
G. A.
Bliss;
M. R.
Hestenes
305-326
The total variation of $g(x+h)-g(x)$
Norbert
Wiener;
R. C.
Young
327-340
One-sided minimal surfaces with a given boundary
Jesse
Douglas
731-756
On the summability of Fourier series. I
Einar
Hille;
J. D.
Tamarkin
757-783
Canonical forms for symmetric linear vector functions in pseudo-Euclidean space
R. V.
Churchill
784-794
Types of involutorial space transformations associated with certain rational curves
Amos Hale
Black
795-810
Exterior motion in the restricted problem of three bodies
Carl Jenness
Coe
811-837
Permanent configurations in the problem of four bodies
W. D.
MacMillan;
Walter
Bartky
838-875
Classes of maximum numbers and minimum numbers that are associated with certain symmetric equations in $n$ reciprocals
H. A.
Simmons
876-907
The numbers of solutions of congruences involving only $k$th powers
Ralph
Hull
908-937
On lacunary trigonometric series
Antoni
Zygmund
435-446
On the asymptotic solutions of differential equations, with an application to the Bessel functions of large complex order
Rudolph E.
Langer
447-480
The apsides of general dynamical systems
J. L.
Synge
481-522
Interpolation and functions analytic interior to the unit circle
J. L.
Walsh
523-556
Parallelism and equidistance in classical differential geometry
W. C.
Graustein
557-593
On finite-rowed systems of linear inequalities in infinitely many variables
I. J.
Schoenberg
594-619
On normal simple algebras
A. Adrian
Albert
620-625
A complete system of tensors of linear homogeneous second-order differential equations
Clyde M.
Cramlet
626-644
Non-absolutely convergent integrals with respect to functions of bounded variation
R. L.
Jeffery
645-675
Surfaces and curvilinear congruences
Ernest P.
Lane
676-688
The variable end point problem of the calculus of variations including a generalization of the classical Jacobi conditions
A. E.
Currier
689-704
The Moore-Kline problem
Leo
Zippin
705-721
A determination of all normal division algebras over an algebraic number field
A. Adrian
Albert;
Helmut
Hasse
722-726
Additional note to the author's: ``Theory of cyclic algebras over an algebraic number field'' [Trans. Amer. Math. Soc. {\bf 34} (1932), no. 1, 171--214; 1501634]
Helmut
Hasse
727-730
On summability of double series
C. Raymond
Adams
215-230
On the covering of analytic loci by complexes
B. O.
Koopman;
A. B.
Brown
231-251
Concerning topological transformations in $E\sb n$
J. H.
Roberts
252-262
On cyclic numbers of one-dimensional compact sets
W. A.
Wilson
263-273
On certain polynomial and other approximations to analytic functions
Hillel
Poritsky
274-331
The condition for an orthonomic differential system
Joseph Miller
Thomas
332-338
Non-separable and planar graphs
Hassler
Whitney
339-362
Normal division algebras of degree four over an algebraic field
A. Adrian
Albert
363-372
Theory of measure and invariant integrals
Eberhard
Hopf
373-393
Continuous transformations of abstract spaces
Rothwell
Stephens
394-408
General theorems on the convergence of sequences of Pad\'e approximants
H. S.
Wall
409-416
A third-order irregular boundary value problem and the associated series
Lewis E.
Ward
417-434
On the cluster values of analytic functions
Wladimir
Seidel
1-21
On interpolation and approximation by rational functions with preassigned poles
J. L.
Walsh
22-74
Poincar\'e's rotation number and Morse's type number
Gustav A.
Hedlund
75-97
On sums of two or four values of a quadratic function of $x$
Gordon
Pall
98-125
On the geometry of the Riemann tensor
R. V.
Churchill
126-152
The boundary values of analytic functions
Joseph L.
Doob
153-170
Theory of cyclic algebras over an algebraic number field
Helmut
Hasse
171-214
The $k$-function, a particular case of the confluent hypergeometric function
H.
Bateman
817-831
On the curvatures of a curve in Riemann space
E. H.
Cutler
832-838
Frenet formulas for a general subspace of a Riemann space
E. H.
Cutler
839-850
Necessary and sufficient conditions for the representation of a function as a Laplace integral
D. V.
Widder
851-892
On a theorem of S. Bernstein-Widder
J. D.
Tamarkin
893-896
Arithmetical composition and inversion of functions over classes
E. T.
Bell
897-933
On Stieltjes polynomials
Morris
Marden
934-944
Arithmetic of double series
D. H.
Lehmer
945-957
On the existence of acyclic curves satisfying certain conditions with respect to a given continuous curve
C. M.
Cleveland
958-978
Problems of closest approximation connected with the solution of linear differential equations
W. H.
McEwen
979-997
Errata: ``On Fermat's last theorem'' [Trans. Amer. Math. Soc. {\bf 31} (1929), no. 4, 613--642; 1501503]
H. S.
Vandiver
998
Errata: ``Elementary properties of the $t\sb \nu$ functions'' [Trans. Amer. Math. Soc. {\bf 32} (1930), no. 4, 905--911; 1501570]
Wilhelm
Maier
998
Errata: ``On direct products, cyclic division algebras, and pure Riemann matrices'' [Trans. Amer. Math. Soc. {\bf 33} (1931), no. 1, 219--234; 1510586]
A. Adrian
Albert
999
Errata: ``A proof of the generalized second-limit theorem in the theory of probability'' [Trans. Amer. Math. Soc. {\bf 33} (1931), no. 2, 533--543; 1501604]
M.
Fréchet;
J.
Shohat
999
Remarks concerning the paper of W. L. Ayres on the regular points of a continuum
Karl
Menger
663-667
The existence of rational functions of best approximation
J. L.
Walsh
668-689
On direct products
A. Adrian
Albert
690-711
Conditions for the solubility of the Diophantine equation $x\sp 2-My\sp 2=-1$
Morgan
Ward
712-718
On a solution of Laplace's equation with an application to the torsion problem for a polygon with reentrant angles
I. S.
Sokolnikoff
719-732
The transformation $C$ of nets in hyperspace
V. G.
Grove
733-741
On universal quadratic null forms in five variables
R. S.
Underwood
742-758
On the approximate representation of a function of two variables
E. L.
Mickelson
759-781
A geometric theory of solution of linear inequalities
Ruth Wyckliffe
Stokes
782-805
On certain types of plane continua
N. E.
Rutt
806-816
Moderately thick circular plates with plane faces
H. W.
Sibert
329-369
Note on the overconvergence of sequences of polynomials of best approximation
J. L.
Walsh
370-388
Matrices of integers ordering derivatives
Joseph Miller
Thomas
389-410
The effects of general regular transformations on oscillations of sequences of functions
Ralph Palmer
Agnew
411-424
The discriminant matrix of a semi-simple algebra
C. C.
MacDuffee
425-432
The integrability of a sequence of functions
R. L.
Jeffery
433-440
On the interchange of limit and Lebesgue integral for a sequence of functions
T. H.
Hildebrandt
441-443
Non-separated cuttings of connected point sets
G. T.
Whyburn
444-454
Expansion theory associated with linear differential equations and their regular singular points
Leonard
Bristow
455-474
Expansion problems associated with a system of integral equations
William T.
Reid
475-485
On the derivatives of harmonic functions on the boundary
Oliver D.
Kellogg
486-510
Abstract: L \phi (x,y)$ and the boundary values of U on E have continuous derivatives of order n which satisfy a Dini condition, then the partial derivatives of U of order n exist, as limits, on E, and are continuous in R at any interior point of E. Hölder conditions on the boundary values of U, or on their derivatives of order n, imply Hölder conditions on U, or the corresponding derivatives, in R, in the neighborhood of the interior points of E.
On the Pad\'e approximants associated with a positive definite power series
H. S.
Wall
511-532
A proof of the generalized second-limit theorem in the theory of probability
M.
Fréchet;
J.
Shohat
533-543
A certain type of continuous curve and related point sets
P. M.
Swingle
544-556
The geometric configuration defined by a special algebraic relation of genus four
Frances
Harshbarger
557-578
The theory of multiplicative arithmetic functions
R.
Vaidyanathaswamy
579-662
A method for finding units in cubic orders of a negative discriminant
J. V.
Uspensky
1-22
On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order
Rudolph E.
Langer
23-64
Singular relations between certain arithmetical functions
E. T.
Bell
65-71
The critical points of a function of $n$ variables
Marston
Morse
72-91
A new definition of genus for ternary quadratic forms
Burton W.
Jones
92-110
The regularity of a genus of positive ternary quadratic forms
Burton W.
Jones
111-124
On parallel displacement in a non-Finsler space
Homer Vincent
Craig
125-142
On the maximum absolute value of the derivative of $e\sp {-x\sp 2}p\sb n(x)$
W. E.
Milne
143-146
The transformation $E$ of nets
V. G.
Grove
147-152
The characteristic number of a sequence of integers satisfying a linear recursion relation
Morgan
Ward
153-165
The distribution of residues in a sequence satisfying a linear recursion relation
Morgan
Ward
166-190
On linear connections
J. H. C.
Whitehead
191-209
On sequences defined by linear recurrence relations
H. T.
Engstrom
210-218
On direct products, cyclic division algebras, and pure Riemann matrices
A. Adrian
Albert
219-234
On normal division algebras of type $R$ in thirty-six units
A. Adrian
Albert
235-243
Integrals whose extremals are a given $2n$-parameter family of curves
David R.
Davis
244-251
On the regular points of a continuum
W. L.
Ayres
252-262
Solution of the problem of Plateau
Jesse
Douglas
263-321
On orthogonal polynomials
J.
Geronimus
322-328
Manifolds of functions defined by systems of algebraic differential equations
J. F.
Ritt
569-598
The foundations of a theory of the calculus of variations in the large in $m$-space. II
Marston
Morse
599-631
A converse of the Jordan-Brouwer separation theorem in three dimensions
R. L.
Wilder
632-657
On the zeros of certain rational functions
Morris
Marden
658-668
The behavior of bounds and oscillations of sequences of functions under regular transformations
Ralph Palmer
Agnew
669-708
Some differential and algebraic consequences of the Einstein field equations
K. W.
Lamson
709-722
Generalized differentiation
Emil L.
Post
723-781
Integral surfaces of pairs of partial differential equations of the third order
Ernest P.
Lane
782-793
On the overconvergence of sequences of polynomials of best approximation
J. L.
Walsh
794-816
On Birkhoff's Pfaffian systems
Lucien
Feraud
817-831
On certain identities in theta functions
Claiborne G.
Latimer
832-846
The consistency and ultimate distribution of optimum statistics
Harold
Hotelling
847-859
The notion of the Green's function in the theory of integro-differential equations. II
J. D.
Tamarkin
860-868
Analytic functions in the irregular field of all numerical functions
E. T.
Bell
869-878
The theorem of Dedekind in the ideal theory of Zolotarev
H. T.
Engstrom
879-887
Polynomials of best approximation on an infinite interval
James M.
Earl
888-904
Elementary properties of the $t\sb \nu$-functions
W. E.
Maier
905-911
On division algebras
O. C.
Hazlett
912-925
On the structure of connected and connected IM Kleinen point sets
G. T.
Whyburn
926-943
Errata: ``Algebras which do not possess a finite basis'' [Trans. Amer. Math. Soc. {\bf 26} (1924), no. 4, 395--426; 1501287]
J. H. M.
Wedderburn
944
Errata: ``Algebraic combinations of exponentials'' [Trans. Amer. Math. Soc. {\bf 31} (1929), no. 4, 654--679; 1501505]
J. F.
Ritt
944
Boundary values of an analytic function and the Tchebycheff method of approximation
J. L.
Walsh
335-390
On power characters of singular integers in a properly irregular cyclotomic field
H. S.
Vandiver
391-408
On Gibbs's phenomenon for the developments in Bessel's functions
Charles N.
Moore
409-416
Surface transformations
F. R.
Bamforth
417-450
On Galois fields of certain types
Leonard
Carlitz
451-472
A canonical form of Green's projective analogue of the Gauss differential equations
V. G.
Grove
473-484
On irreducible polynomials in several variables which become reducible when the variables are replaced by powers of themselves
Eli
Gourin
485-501
Functional properties of the solutions of differential systems
William M.
Whyburn
502-508
An inverse problem of the calculus of variations for multiple integrals
Lincoln
La Paz
509-519
Postulates for the inverse operations in a group
Morgan
Ward
520-526
On sequences of continuous functions having continuous limits
D. C.
Gillespie;
W. A.
Hurwitz
527-543
Series associated with certain irregular third-order boundary value problems
Lewis E.
Ward
544-557
Hypergeodesic mapping of a surface on a plane
Ernest P.
Lane
558-568
On a certain ring of functions of two variables
J. F.
Ritt
155-170
New results in the theory of normal division algebras
A. Adrian
Albert
171-195
On composition of singularities
W. J.
Trjitzinsky
196-215
Residues of polygenic functions
Vincent C.
Poor
216-222
On the common index divisors of an algebraic field
H. T.
Engstrom
223-237
The boundary problem associated with a differential system rational in the parameter
Rudolph E.
Langer
238-250
Involutorial transformations in $S\sb 3$ of order $n$ with an $(n-1)$-fold line
Leaman A.
Dye
251-261
The limit of transitivity of a substitution group
Marie J.
Weiss
262-283
Properties of solutions of an infinite system of ordinary linear differential equations of the first order with auxiliary boundary conditions
William T.
Reid
284-318
Construction of division algebras
L. E.
Dickson
319-334
Numbers of representations of integers in a certain triad of ternary quadratic forms
E. T.
Bell
1-5
Concerning non-dense plane continua
J. H.
Roberts
6-30
Pairs of rectilinear congruences with generators in one-to-one correspondence
Alexander J.
Cook
31-46
On the solution of the equations of elastic equilibrium suitable for elliptic boundaries
Suddhodan
Ghosh
47-60
On the unilateral equation in matrices
William E.
Roth
61-80
On the zeros of linear partial fractions
Morris
Marden
81-109
On differential equations admitting polygenic integrals
Georges
Calugaréano
110-113
Divergent series and singular points of ordinary differential equations
George D.
Birkhoff;
Frederic R.
Bamforth
114-146
Cut points of connected sets and of continua
G. T.
Whyburn
147-154
On Fermat's last theorem
H. S.
Vandiver
613-642
The degree and class of multiply transitive groups. II
W. A.
Manning
643-653
Algebraic combinations of exponentials
J. F.
Ritt
654-679
On the zeros of exponential polynomials
J. F.
Ritt
680-686
On the singularities of linear partial differential equations
Bernard Osgood
Koopman
687-693
A generalization of Dirichlet's series and of Laplace's integrals by means of a Stieltjes integral
D. V.
Widder
694-743
A study of continuous curves and their relation to the Janiszewski-Mullikin theorem
Leo
Zippin
744-770
On extended Stieltjes series
Hubert Stanley
Wall
771-781
The asymptotic solution of an operational equation
John R.
Carson
782-792
On commutation formulas in the algebra of quantum mechanics
Neal H.
McCoy
793-806
The basic power series of interpolation
George
Rutledge
807-820
The limit points of a group
Lester R.
Ford
821-828
On linear upper semi-continuous collections of bounded continua
Wallace Alvin
Wilson
829-836
The asymptotic location of the roots of a certain transcendental equation
Rudolph E.
Langer
837-844
Nets with equal $W$ invariants
V. G.
Grove
845-852
Contributions to the general theory of transformations of nets
V. G.
Grove
853-860
Note on integro-$q$-difference equations
C. Raymond
Adams
861-867
The expansion problem in the theory of ordinary linear differential systems of the second order
Rudolph E.
Langer
868-906
On the degree of convergence of expansions in an infinite interval
W. E.
Milne
907-918
A kinematical treatment of some theorems on normal rectilinear congruences
Charles H.
Rowe
919-930
Errata: ``The singular points of analytic space-curves'' [Trans. Amer. Math. Soc. {\bf 31} (1929), no. 1, 145--163; 1501473]
Arthur
Ranum
931
The foundations of the calculus of variations in the large in $m$-space. I
Marston
Morse
379-404
Certain invariant sequences of polynomials
E. T.
Bell
405-421
On the degree of convergence of the Gram-Charlier series
W. E.
Milne
422-443
Foucault's pendulum in elliptic space
James
Pierpont
444-447
On a certain formula of mechanical quadratures with non-equidistant ordinates
J. A.
Shohat
448-463
Generalized Vandermonde determinants
E. R.
Heineman
464-476
On approximation by rational functions to an arbitrary function of a complex variable
J. L.
Walsh
477-502
Separation theorems with applications to questions concerning accessibility and plane continua
R. G.
Lubben
503-522
Tetrads of ruled surfaces
A. F.
Carpenter
523-528
On the congruences connected with certain magic squares
D. N.
Lehmer
529-551
Certain problems related to the cutting of a simply connected plane region by a continuum
Wallace Alvin
Wilson
552-562
An application of the calculus of variations to boundary value problems
A. O.
Hickson
563-579
An extension of Pascal's theorem
Charles A.
Rupp
580-594
Concerning the arc-curves and basic sets of a continuous curve. II
W. Leake
Ayres
595-612
A note on closest approximation
Dunham
Jackson
215-222
Homogeneous polynomials with a multiplication theorem
O. C.
Hazlett
223-232
Generalized factorial series
Tomlinson
Fort
233-240
On extending a continuous ($1$-$1$) correspondence. II
Harry Merrill
Gehman
241-252
A determination of all normal division algebras in sixteen units
A. Adrian
Albert
253-260
Expansions in generalized Appell polynomials, and a class of related linear functional equations
I. M.
Sheffer
261-280
Systems of infinitely many linear differential equations of infinite order, with constant coefficients
I. M.
Sheffer
281-289
On general topology and the relation of the properties of the class of all continuous functions to the properties of space
E. W.
Chittenden
290-321
Properties of functions represented by the Dirichlet series $\sum(a\nu+b)\sp {-s}$, or by linear combinations of such series
J. I.
Hutchinson
322-344
Concerning zero-dimensional sets in Euclidean space
R. L.
Wilder
345-359
On the linear partial $q$-difference equation of general type
C. Raymond
Adams
360-371
Les fonctions polyg\`enes comme int\'egrales d'\'equations diff\'erentielles
G.
Calugaréano
372-378
The boundary problem associated with a differential equation in which the coefficient of the parameter changes sign
Rudolph E.
Langer
1-24
Triple and multiple systems, their geometric configurations and groups
Arnold
Emch
25-42
On the expansion of analytic functions of the complex variable in generalized Taylor's series
D. V.
Widder
43-52
Note on the expansion of analytic functions in series of polynomials and in series of other analytic functions
J. L.
Walsh
53-57
A general problem of minimizing an integral with discontinuous integrand
C. F.
Roos
58-70
An introduction to the theory of ideals in linear associative algebras
C. C.
MacDuffee
71-90
On the Pad\'e approximants associated with the continued fraction and series of Stieltjes
Hubert S.
Wall
91-116
On curvilinear congruences
C. E.
Weatherburn
117-132
The multinomial solid and the chi test
B. H.
Camp
133-144
The singular points of analytic space-curves
Arthur
Ranum
145-163
Universal quadratic forms
L. E.
Dickson
164-189
Boundary value problems for potentials of a single layer
E. R. C.
Miles
190-203
On a generalization of the associative law
Anton
Suschkewitsch
204-214
Some non-linear problems in approximation
Dunham
Jackson
621-629
Second-order differential systems with integral and $k$-point boundary conditions
William M.
Whyburn
630-640
Analytic functions of hypercomplex variables
P. W.
Ketchum
641-667
Concerning limiting sets in abstract spaces
R. G.
Lubben
668-685
Riesz summability for double series
Florence M.
Mears
686-709
The inverse problem of the calculus of variations in higher space
David R.
Davis
710-736
Types of motion of the gyroscope
A. H.
Copeland
737-764
The invariant integral and the inverse problem in the calculus of variations
Thomas H.
Rawles
765-784
The projective differential geometry of systems of linear homogeneous differential equations of the first order
Ernest P.
Lane
785-796
The behavior of a boundary value problem as the interval becomes infinite
W. E.
Milne
797-802
The second derivative of a polygenic function
Edward
Kasner
803-818
Diophantine equations in division algebras
Ralph G.
Archibald
819-837
On the degree of approximation to an analytic function by means of rational functions
J. L.
Walsh
838-847
Existence and oscillation theorems for non-linear differential systems of the second order
William M.
Whyburn
848-854
Errata: ``A symbolic theory of formal modular covariants'' [Trans. Amer. Math. Soc. {\bf 24} (1922), no. 4, 286--311; 1501227]
Olive C.
Hazlett
855
Errata: ``On the irregular cases of the linear ordinary difference equation'' [Trans. Amer. Math. Soc. {\bf 30} (1928), no. 3, 507--541; 1501443]
C. Raymond
Adams
855
Transversality in space of three dimensions
Edward
Kasner
447-452
On integral equations with discontinuous kernels
J. D.
Tamarkin;
Rudolph E.
Langer
453-471
On approximation to an arbitrary function of a complex variable by polynomials
J. L.
Walsh
472-482
Transformations of nets
V. G.
Grove
483-497
Allgemeine Eigenschaften der Cantorschen Koh\"arenzen
Miron
Zarycki
498-506
On the irregular cases of the linear ordinary difference equation
C. Raymond
Adams
507-541
On the convergence of quadrature formulas related to an infinite interval
J. V.
Uspensky
542-559
Second-order linear systems with summable coefficients
J. H.
Sturdivant
560-566
Concerning the arc-curves and basic sets of a continuous curve
W. L.
Ayres
567-578
A solution of the matric equation $P(X)=A$
William E.
Roth
579-596
Concerning the cut points of continua
Gordon T.
Whyburn
597-609
Some theorems on the connection between ideals and group of a Galois field
Oystein
Ore
610-620
The foundations of a theory in the calculus of variations in the large
Marston
Morse
213-274
Topological invariants of knots and links
J. W.
Alexander
275-306
On the expansion of analytic functions in series of polynomials and in series of other analytic functions
J. L.
Walsh
307-332
Primitive groups which contain substitutions of prime order $p$ and of degree $6p$ or $7p$
Marie J.
Weiss
333-359
Generalized Lagrange problems in the calculus of variations
C. F.
Roos
360-384
On Jacobi's arithmetical theorems concerning the simultaneous representation of numbers by two different quadratic forms
J. V.
Uspensky
385-404
On relative content and Green's lemma
H. L.
Smith
405-419
On Bell's arithmetic of Boolean algebra
Wallie Abraham
Hurwitz
420-424
A theorem on orthogonal functions with an application to integral inequalities
Lloyd L.
Dines
425-438
A theorem on orthogonal sequences
L. L.
Dines
439-446
Simpler proofs of Waring's theorem on cubes, with various generalizations
L. E.
Dickson
1-18
Cubic curves and desmic surfaces. II
R. M.
Mathews
19-23
Possible orders of two generators of the alternating and of the symmetric group
G. A.
Miller
24-32
Optics in hyperbolic space
James
Pierpont
33-48
Geodesics on surfaces of genus zero with knobs
Donald Everett
Richmond
49-62
Concerning end points of continuous curves and other continua
Harry Merrill
Gehman
63-84
The apportionment of representatives in Congress
E. V.
Huntington
85-110
Conditions for associativity of division algebras connected with non-abelian groups
John
Williamson
111-125
A generalization of Taylor's series
D. V.
Widder
126-154
A problem in the calculus of variations with an infinite number of auxiliary conditions
R. G. D.
Richardson
155-189
A contribution to the theory of fundamental transformations of surfaces
M. M.
Slotnick
190-212
The differential equation of the elliptic cylinder
J. H.
McDonald
647-682
On a class of integral equations with discontinuous kernels
Rudolph E.
Langer;
Eleanor P.
Brown
683-715
Some third-order irregular boundary value problems
Lewis E.
Ward
716-745
Concerning point sets which can be made connected by the addition of a simple continuous arc
Gordon T.
Whyburn
746-754
The notion of the Green's function in the theory of integro-differential equations
J. D.
Tamarkin
755-800
A general theory of nets on a surface
Vernon G.
Grove
801-814
Simply transitive primitive groups
W. A.
Manning
815-825
The expansion problems associated with regular differential systems of the second order
M. H.
Stone
826-844
Errata: ``On the representation of a number as the sum of any number of squares, and in particular of five'' [Trans. Amer. Math. Soc. {\bf 21} (1920), no. 3, 255--284; 1501144]
G. H.
Hardy
845-847
Errata: ``Manifolds with a boundary and their transformations'' [Trans. Amer. Math. Soc. {\bf 29} (1927), no. 2, 429--462; 1501397]
Solomon
Lefschetz
848
The contact of a cubic surface with an analytic surface
Ernest P.
Lane
471-480
Dynamical space-times which contain a conformal Euclidean $3$-space
H. P.
Robertson
481-496
Reduction of the ordinary linear differential equation of the $n$th order whose coefficients are certain polynomials in a parameter to a system of $n$ first-order equations which are linear in the parameter
Charles E.
Wilder
497-506
On the ``third axiom of metric space''
V. W.
Niemytzki
507-513
Implicit functions and differential equations in general analysis
Lawrence M.
Graves
514-552
Concerning acyclic continuous curves
Harry Merrill
Gehman
553-568
On a general formula in the theory of Tchebycheff polynomials and its applications
J.
Shohat
569-583
A factorization theory for functions $\sum\sb {i=1}\sp na\sb ie\sp {\alpha\sb ix}$
J. F.
Ritt
584-596
Arithmetic of logic
E. T.
Bell
597-611
Functionals of $r$-dimensional manifolds admitting continuous groups of point transformations
Aristotle D.
Michal
612-646
Singular case of pairs of bilinear, quadratic, or Hermitian forms
L. E.
Dickson
239-253
Triads of ruled surfaces
A. F.
Carpenter
254-275
Certain uniform functions of rational functions
Emory P.
Starke
276-286
On rejection to infinity and exterior motion in the restricted problem of three bodies
Bernard Osgood
Koopman
287-331
A connected and regular point set which has no subcontinuum
R. L.
Wilder
332-340
Meromorphic functions with addition or multiplication theorems
J. F.
Ritt
341-360
Real functions with algebraic addition theorems
J. F.
Ritt
361-368
Concerning continua in the plane
Gordon T.
Whyburn
369-400
Extremals and transversality of the general calculus of variations problem of the first order in space
Jesse
Douglas
401-420
A figuratrix for double integrals
Paul R.
Rider
421-428
Manifolds with a boundary and their transformations
Solomon
Lefschetz
429-462
On sets of functions of a general variable
Lloyd L.
Dines
463-470
Sets of independent postulates for the arithmetic mean, the geometric mean, the harmonic mean, and the root-mean-square
Edward V.
Huntington
1-22
Irregular differential systems of order two and the related expansion problems
M. H.
Stone
23-53
Some problems in the theory of interpolation by Sturm-Liouville functions
Carey M.
Jensen
54-79
Singular ruled surfaces in space of five dimensions
E. B.
Stouffer
80-95
On a type of completeness characterizing the general laws for separation of point-pairs
C. H.
Langford
96-110
Integers and basis of a number field
N. R.
Wilson
111-126
Implicit functions and their differentials in general analysis
T. H.
Hildebrandt;
Lawrence M.
Graves
127-153
Application of the theory of relative cyclic fields to both cases of Fermat's last theorem. II
H. S.
Vandiver
154-162
Riemann integration and Taylor's theorem in general analysis
Lawrence M.
Graves
163-177
Alternatives to Zermelo's assumption
Alonzo
Church
178-208
On a general theorem concerning the distribution of the residues and non-residues of powers
J. M.
Vinogradov
209-217
On the bound of the least non-residue of $n$th powers
J. M.
Vinogradov
218-226
On convergence factors in multiple series
Charles N.
Moore
227-238
A boundary value problem for a system of ordinary linear differential equations of the first order
Gilbert Ames
Bliss
561-584
On the theory of integral equations with discontinuous kernels
Rudolph E.
Langer
585-639
The figuratrix in the calculus of variations
Paul R.
Rider
640-653
On the existence of fields in Boolean algebras
B. A.
Bernstein
654-657
Asymmetric displacement of a vector
Joseph Miller
Thomas
658-670
Tensors determined by a hypersurface in Riemann space
Harry
Levy
671-694
A comparison of the series of Fourier and Birkhoff
M. H.
Stone
695-761
Analytic approximations to topological transformations
Philip
Franklin;
Norbert
Wiener
762-785
Errata: ``Solution of certain functional equations relative to a general linear set'' [Trans. Amer. Math. Soc. {\bf 28} (1926), no. 2, 287--300; 1501345]
Mark H.
Ingraham
786
Errata: ``On the zeros of the function $\beta(z)$ associated with the gamma function'' [Trans. Amer. Math. Soc. {\bf 28} (1926), no. 3, 391--399; 1501352]
T. H.
Gronwall
786
Double binary forms with the closure property
Arthur B.
Coble
357-383
On a class of polynomials in the theory of Bessel's functions
J. H.
McDonald
384-390
On the zeros of the function $\beta(z)$ associated with the gamma function
T. H.
Gronwall
391-399
Osculating curves and surfaces
Philip
Franklin
400-416
On Laplace's integral equations
J. D.
Tamarkin
417-425
On Volterra's integro-functional equation
J. D.
Tamarkin
426-431
On the discriminant of ternary forms and a certain class of surfaces
Arnold
Emch
432-434
On the convergence of certain methods of closest approximation
Elizabeth
Carlson
435-447
Functions of plurisegments
Alfred J.
Maria
448-471
Sets of postulates for the logic of propositions
B. A.
Bernstein
472-478
Multiple-sheeted spaces and manifolds of states of motion
Harold
Hotelling
479-490
A theory of a general net on a surface
Vernon G.
Grove
491-501
Cubic curves and desmic surfaces
R. M.
Mathews
502-522
Expansion problems in connection with homogeneous linear $q$-difference equations
M. G.
Carman
523-535
Some properties of limited continua, irreducible between two points
Wallace Alvin
Wilson
536-553
Application of the theory of relative cyclic fields to both cases of Fermat's last theorem
H. S.
Vandiver
554-560
New division algebras
L. E.
Dickson
207-234
The first and second variations of a double integral for the case of variable limits
H. A.
Simmons
235-251
On extending a continuous $(1\text{-}1)$ correspondence of two plane continuous curves to a correspondence of their planes
Harry Merrill
Gehman
252-265
Systems of equations in an infinity of unknowns, whose solution involves an arbitrary parameter
I. M.
Sheffer
266-286
Solution of certain functional equations relative to a general linear set
Mark H.
Ingraham
287-300
Combinatorial analysis situs
J. W.
Alexander
301-329
Geometries of paths for which the equations of the paths admit $n(n+1)/2$ independent linear first integrals
L. P.
Eisenhart
330-338
Multiply transitive substitution groups
G. A.
Miller
339-345
An extension of Lagrange's expansion
H.
Bateman
346-356
Intersections and transformations of complexes and manifolds
Solomon
Lefschetz
1-49
Divergent double sequences and series
G. M.
Robison
50-73
On certain families of orbits with arbitrary masses in the problem of three bodies
F. H.
Murray
74-108
On certain families of orbits with arbitrary masses in the problem of three bodies. II
F. H.
Murray
109-118
Existence theorems for a linear partial difference equation of the intermediate type
C. Raymond
Adams
119-128
An algebra of sequences of functions, with an application to the Bernoullian functions
E. T.
Bell
129-148
Bundles and pencils of nets on a surface
Ernest P.
Lane
149-167
On the momental constants of a summable function
R. E.
Langer
168-182
Fundamental systems of formal modular protomorphs of binary forms
W. L. G.
Williams
183-197
Non-synchronized relative invariant integrals
K. P.
Williams
198-206
Cone cubic configurations of a ruled surface
A. F.
Carpenter
397-415
Concerning upper semi-continuous collections of continua
R. L.
Moore
416-428
On the oscillation of a continuum at a point
Wallace Alvin
Wilson
429-440
On the conditions of integrability of covariant differential equations
J. A.
Schouten
441-473
Complete groups of points on a plane cubic curve of genus one
M. I.
Logsdon
474-490
On the existence of the Stieltjes integral
H. L.
Smith
491-515
The mutual inductance of two square coils
T. H.
Gronwall
516-536
On the development of continuous functions in series of Tchebycheff polynomials
J. A.
Shohat
537-550
Analytic functions in three dimensions
E. R.
Hedrick;
Louis
Ingold
551-555
The Beltrami equations in three dimensions
E. R.
Hedrick;
Louis
Ingold
556-562
Fields of parallel vectors in a Riemannian geometry
Luther Pfahler
Eisenhart
563-573
A symbolic treatment of the geometry of hyperspace
Louis
Ingold
574-599
Errata: ``Operations with respect to which the elements of a Boolean algebra form a group'' [Trans. Amer. Math. Soc. {\bf 26} (1924), no. 2, 171--175; 1501271]
B. A.
Bernstein
600
Errata: ``Relative extrema of pairs of quadratic and Hermitian forms'' [Trans. Amer. Math. Soc. {\bf 26} (1924), no. 4, 479--494; 1501293]
R. G. D.
Richardson
600
The linear complex of conics
Earl E.
Libman
265-269
On the Weddle surface and analogous loci
Arnold
Emch
270-278
Algebraic surfaces with reducible bitangent and osculating hyperplanar sections
Maria
Castellani
279-286
Polynomials of several variables and their residue systems
Aubrey J.
Kempner
287-298
A criterion for the conformal equivalence of a Riemann space to a Euclidean space
Jesse
Douglas
299-306
The deflection of a rectangular plate fixed at the edges
H. W.
March
307-317
On irredundant sets of postulates
Alonzo
Church
318-328
Three-dimensional manifolds of states of motion
Harold
Hotelling
329-344
Relations between the critical points of a real function of $n$ independent variables
Marston
Morse
345-396
The subgroup composed of the substitutions which omit a letter of a transitive group
G. A.
Miller
137-145
On the closeness of approach of complex rational fractions to a complex irrational number
Lester R.
Ford
146-154
Solutions of the Einstein equations involving functions of only one variable
Edward
Kasner
155-162
A general theory of linear sets
Mark H.
Ingraham
163-196
On the representation of a certain fundamental law of probability
H. L.
Rietz
197-212
The group of motions of an Einstein space
John
Eiesland
213-245
A generalization of Levi-Civita's parallelism and the Frenet formulas
James Henry
Taylor
246-264
On normal forms of differential equations
William F.
Osgood
1-14
Congruences with constant absolute invariants
H. L.
Olson
15-42
On the prime divisors of the cyclotomic functions
C. M.
Huber
43-48
On the roots of the Riemann zeta function
J. I.
Hutchinson
49-60
A generalisation of the Riemannian line-element
J. L.
Synge
61-67
Elementary functions and their inverses
J. F.
Ritt
68-90
Analytic transformations of everywhere dense point sets
Philip
Franklin
91-100
An algebraic solution of the Einstein equations
Edward
Kasner
101-105
Electrodynamics in the general relativity theory
G. Y.
Rainich
106-136
Algebras which do not possess a finite basis
J. H. M.
Wedderburn
395-426
Determination of all the prime power groups containing only one invariant subgroup of every index which exceeds this prime number
H. A.
Bender
427-434
Invariants of the linear group modulo $\pi=p\sb 1\sp {\lambda\sb 1}p\sb 2\sp {\lambda\sb 2}\cdots p\sb n\sp {\lambda\sb n}$
Cornelius
Gouwens
435-440
A uniqueness theorem for the Legendre and Hermite polynomials
K. P.
Williams
441-445
A new type of class number relations
E. T.
Bell
446-450
A new method in the equivalence of pairs of bilinear forms
R. G. D.
Richardson
451-478
Relative extrema of pairs of quadratic and Hermitian forms
R. G. D.
Richardson
479-494
Errata: ``Permutable rational functions'' [Trans. Amer. Math. Soc. {\bf 25} (1923), no. 3, 399--448; 1501252]
J. F.
Ritt
494
Errata: ``Discontinuous boundary conditions and the Dirichlet problem'' [Trans. Amer. Math. Soc. {\bf 25} (1923), no. 3, 307--314; 1501246]
Norbert
Wiener
494
The general theory of a class of linear partial $q$-difference equations
C. Raymond
Adams
283-312
The summability of the triple Fourier series at points of discontinuity of the function developed
Bess M.
Eversull
313-334
An unusual type of expansion problem
M. H.
Stone
335-355
On the independence of principal minors of determinants
E. B.
Stouffer
356-368
A necessary and sufficient condition that two surfaces be applicable
W. C.
Graustein;
B. O.
Koopman
369-372
Extensions of relative tensors
Oswald
Veblen;
Tracy Yerkes
Thomas
373-377
Geometries of paths for which the equations of the paths admit a quadratic first integral
Luther Pfahler
Eisenhart
378-384
A general mean-value theorem
D. V.
Widder
385-394
On the expansion of analytic functions in series of polynomials
J. L.
Walsh
155-170
Operations with respect to which the elements of a Boolean algebra form a group
B. A.
Bernstein
171-175
Isometric $W$-surfaces
W. C.
Graustein
176-204
Space-time continua of perfect fluids in general relativity
Luther Pfahler
Eisenhart
205-220
Equivalent rational substitutions
J. F.
Ritt
221-229
Extension of Bernstein's theorem to Sturm-Liouville sums
Elizabeth
Carlson
230-240
An existence theorem
Einar
Hille
241-248
On the complete independence of the postulates for betweenness
W. E.
Van De Walle
249-256
A new set of postulates for betweenness, with proof of complete independence
Edward V.
Huntington
257-282
An existence theorem for the characteristic numbers of a certain boundary value problem
H. T.
Davis
1-16
A theorem on the factorization of polynomials of a certain type
Lloyd L.
Dines
17-24
A fundamental class of geodesics on any closed surface of genus greater than one
Harold Marston
Morse
25-60
The Hilbert integral and Mayer fields for the problem of Mayer in the calculus of variations
Gillie A.
Larew
61-67
Normal congruences and quadruply infinite systems of curves in space
Jesse
Douglas
68-100
The equivalence of certain regular transformations
L. L.
Silverman
101-112
Maclaurin expansion of the interpolation polynomial determined by $2n+1$ evenly spaced points
George
Rutledge
113-123
On covariants of linear algebras
C. C.
Macduffee
124-132
A generalized problem in weighted approximation
Dunham
Jackson
133-154
Generalized limits in general analysis. II
Charles N.
Moore
459-468
The equilong transformations of Euclidean space
B. H.
Brown
469-484
Invariant sets of equations in Riemann space
Philip
Franklin
485-500
Some properties of spherical curves, with applications to the gyroscope
O. D.
Kellogg
501-524
The greatest and the least variate under general laws of error
Edward Lewis
Dodd
525-539
The intersection numbers
Oswald
Veblen
540-550
The geometry of paths
Oswald
Veblen;
Tracy Yerkes
Thomas
551-608
Discontinuous boundary conditions and the Dirichlet problem
Norbert
Wiener
307-314
A type of differential system containing a parameter
F. H.
Murray
315-324
On a remarkable class of entire functions
J. I.
Hutchinson
325-332
Note on an ambiguous case of approximation
Dunham
Jackson
333-337
Expansions in terms of solutions of partial differential equations. II. Multiple Birkhoff series
Chester C.
Camp
338-342
Circular plates of constant or variable thickness
Carl A.
Garabedian
343-398
Permutable rational functions
J. F.
Ritt
399-448
On approximation by functions of given continuity
Dunham
Jackson
449-458
Developments associated with a boundary problem not linear in the parameter
Rudolph E.
Langer
155-172
Invariant points of a surface transformation of given class
J. W.
Alexander
173-184
Applications of analysis to the arithmetic of higher forms
E. T.
Bell
185-189
On the second derivatives of an extremal-integral with an application to a problem with variable end points (supplementary paper)
Arnold
Dresden
190-192
Abstract group definitions and applications
W. E.
Edington
193-210
On the integrals of elementary functions
J. F.
Ritt
211-222
Invariants of the linear group modulo $p\sp k$
M. M.
Feldstein
223-238
On certain polar curves with their application to the location of the roots of the derivatives of a rational function
B. Z.
Linfield
239-258
Orthogonal systems of hypersurfaces in a general Riemann space
Luther Pfahler
Eisenhart
259-280
Ruled surfaces with generators in one-to-one correspondence
Ernest P.
Lane
281-296
Symmetric tensors of the second order whose first covariant derivatives are zero
Luther Pfahler
Eisenhart
297-306
The $(1,2)$ correspondence associated with the cubic space involution of order two
F. R.
Sharpe;
Virgil
Snyder
1-12
Sur certaines \'equations aux diff\'erences finies
N. E.
Nörlund
13-98
Differential geometry of an $m$-dimensional manifold in a Euclidean space of $n$ dimensions
Charles E.
Wilder
99-122
Expansions in terms of solutions of partial differential equations. I. Multiple Fourier series expansions
Chester C.
Camp
123-134
Euler algebra
E. T.
Bell
135-154
Representations of a complex point by pairs of ordered real points
W. C.
Graustein
245-254
Non-loxodromic substitutions and groups in $n$ dimensions
Edward B.
Van Vleck
255-273
Birational transformations simplifying singularities of algebraic curves
Gilbert Ames
Bliss
274-285
A symbolic theory of formal modular covariants
Olive C.
Hazlett
286-311
On the mean-value theorem corresponding to a given linear homogeneous differential equation
G.
Pólya
312-324
On the location of the roots of certain types of polynomials
J. L.
Walsh
163-180
A note on the preceding paper: ``On the location of the roots of certain types of polynomials'' [Trans. Amer. Math. Soc. {\bf 24} (1922), no. 3, 163--180; 1501220] by J. L. Walsh
D. R.
Curtiss
181-184
Determination of all general homogeneous polynomials expressible as determinants whose elements are homogeneous polynomials
H. S.
Everett
185-194
General vector calculus
James Byrnie
Shaw
195-244
Generalized limits in general analysis. I
Charles N.
Moore
79-88
Anharmonic polynomial generalizations of the numbers of Bernoulli and Euler
E. T.
Bell
89-112
New properties of all real functions
Henry
Blumberg
113-128
A fundamental system of invariants of a modular group of transformations
John Sidney
Turner
129-134
The Gaussian law of error for any number of variables
J. L.
Coolidge
135-143
Certain theorems relating to plane connected point sets
Anna M.
Mullikin
144-162
Associated sets of points
Arthur B.
Coble
1-20
On algebraic functions which can be expressed in terms of radicals
J. F.
Ritt
21-30
On the location of the roots of the Jacobian of two binary forms, and of the derivative of a rational function
J. L.
Walsh
31-69
$I$-conjugate operators of an abelian group
G. A.
Miller
70-78
A proof and extension of the Jordan-Brouwer separation theorem
J. W.
Alexander
333-349
Oscillation theorems in the complex domain
Einar
Hille
350-385
On certain relations between the projective theory of surfaces and the projective theory of congruences
Frank Edwin
Wood
386-408
Asymptotic planetoids
Daniel
Buchanan
409-431
Errata: ``Prime and composite polynomials'' [Trans. Amer. Math. Soc. {\bf 23} (1922), no. 1, 51--66; 1501189]
J. F.
Ritt
431
Some generalizations of geodesics
E. J.
Wilczynski
223-239
On the gyroscope
William F.
Osgood
240-264
The relative distribution of the real roots of a system of polynomials
C. F.
Gummer
265-282
A general theory of conjugate nets
Ernest P.
Lane
283-297
Parallel maps of surfaces
W. C.
Graustein
298-332
Differential geometry of the complex plane
Julian Lowell
Coolidge
117-134
Invariantive characterizations of linear algebras with the associative law not assumed
Cyrus Colton
Macduffee
135-150
Curves invariant under point-transformations of special type
Mary F.
Curtis
151-172
Die Zerlegung von Primzahlen in algebraischen Zahl-K\"orpern
Andreas
Speiser
173-178
The elliptic modular functions associated with the elliptic norm curve $E\sp 7$
Roscoe
Woods
179-197
Linear equations with two parameters
Anna J.
Pell
198-211
The theory of functions of one Boolean variable
Karl
Schmidt
212-222
Relatively uniform convergence and the classification of functions
E. W.
Chittenden
1-15
Periodic functions with a multiplication theorem
J. F.
Ritt
16-25
Note on Dirichlet and factorial series
Tomlinson
Fort
26-29
Functions of infinitely many variables in Hilbert space
W. L.
Hart
30-50
Prime and composite polynomials
J. F.
Ritt
51-66
Some two-dimensional loci connected with cross ratios
J. L.
Walsh
67-88
On transformations with invariant points
J. W.
Alexander
89-95
Invariant points in function space
G. D.
Birkhoff;
O. D.
Kellogg
96-115
On certain numerical invariants of algebraic varieties with application to abelian varieties (cont.)
Solomon
Lefschetz
407-482
On restricted systems of higher indeterminate equations
E. T.
Bell
483-488
Maximum modulus of some expressions of limited analytic functions
Sōichi
Kakeya
489-504
Differential variations in ballistics, with applications to the qualitative properties of the trajectory
T. H.
Gronwall
505-525
An expansion theorem for a system of linear differential equations of the first order
Wallie Abraham
Hurwitz
526-543
Errata: ``A memoir upon formal invariancy with regard to binary modular transformations. Invariants of relativity'' [Trans. Amer. Math. Soc. {\bf 21} (1920), no. 3, 285--312; 1501145]
Oliver Edmunds
Glenn
543
Polynomials and their residue systems (cont.)
Aubrey J.
Kempner
267-288
On certain two-point properties of general families of curves
Jesse
Douglas
289-310
Properties of the solutions of certain functional-differential equations
William Benjamin
Fite
311-319
Note on a class of polynomials of approximation
Dunham
Jackson
320-326
On certain numerical invariants of algebraic varieties with application to abelian varieties
Solomon
Lefschetz
327-406
On division algebras
J. H. M.
Wedderburn
129-135
Oscillation theorems for the real, self-adjoint linear system of the second order
H. J.
Ettlinger
136-143
New proofs of certain finiteness theorems in the theory of modular covariants
Olive C.
Hazlett
144-157
On the convergence of certain trigonometric and polynomial approximations
Dunham
Jackson
158-166
Determination of all general homogeneous polynomials expressible as determinants with linear elements
Leonard Eugene
Dickson
167-179
Pseudo-canonical forms and invariants of systems of partial differential equations
Alfred L.
Nelson
180-197
Arithmetical paraphrases. II
E. T.
Bell
198-219
On the zeros of solutions of homogeneous linear differential equations
Clarence N.
Reynolds
220-229
A generalization of the Fourier cosine series
J. L.
Walsh
230-239
Polynomials and their residue systems
Aubrey J.
Kempner
240-266
Arithmetical paraphrases
E. T.
Bell
1-30
The construction of algebraic correspondences between two algebraic curves
Virgil
Snyder;
F. R.
Sharpe
31-40
Concerning certain equicontinuous systems of curves
Robert L.
Moore
41-55
Fundamental systems of formal modular seminvariants [semi-invariants] of the binary cubic
W. L. G.
Williams
56-79
A property of two $(n+1)$-gons inscribed in a norm-curve in $n$-space
H. S.
White
80-83
Recurrent geodesics on a surface of negative curvature
Harold Marston
Morse
84-100
On the location of the roots of the Jacobian of two binary forms, and of the derivative of a rational function
J. L.
Walsh
101-116
On functions of closest approximation
Dunham
Jackson
117-128
Minima of functions of lines
Elizabeth
LeStourgeon
357-383
Invariants of infinite groups in the plane
Eugene Francis
Simonds
384-390
On triply orthogonal congruences
James Byrnie
Shaw
391-408
A set of properties characteristic of a class of congruences connected with the theory of functions
E. J.
Wilczynski
409-445
On the equilibrium of a fluid mass at rest
James W.
Alexander
446-450
Concerning approachability of simple closed and open curves
John Robert
Kline
451-458
On the representation of a number as the sum of any number of squares, and in particular of five
G. H.
Hardy
255-284
A memoir upon formal invariancy with regard to binary modular transformations. Invariants of relativity
Oliver Edmunds
Glenn
285-312
Properties of the subgroups of an abelian prime power group which are conjugate under its group of isomorphisms
G. A.
Miller
313-320
On the order of magnitude of the coefficients in trigonometric interpolation
Dunham
Jackson
321-332
Concerning simple continuous curves
Robert L.
Moore
333-347
On the iteration of rational functions
J. F.
Ritt
348-356
Differential equations containing arbitrary functions
Gilbert Ames
Bliss
79-92
Functions of lines in ballistics
Gilbert Ames
Bliss
93-106
On the summability of the developments in Bessel's functions
Charles N.
Moore
107-156
One parameter families and nets of ruled surfaces and a new theory of congruences
E. J.
Wilczynski
157-206
Nets of space curves
G. M.
Green
207-236
A set of postulates for fields
Norbert
Wiener
237-246
A theorem on modular covariants
Olive C.
Hazlett
247-254
The strain of a gravitating sphere of variable density and elasticity
L. M.
Hoskins
1-43
The geometry of Hermitian forms
Julian Lowell
Coolidge
44-51
Certain types of involutorial space transformations
F. R.
Sharpe;
Virgil
Snyder
52-78
Line-geometric representations for functions of a complex variable
E. J.
Wilczynski
271-298
On the combination of non-loxodromic substitutions
Edward B.
Van Vleck
299-312
On a general class of integrals of the form $\int\sp \infty\sb 0\phi(t)g(x+t)dt$
R. D.
Carmichael
313-322
Transformations of surfaces applicable to a quadric
Luther Pfahler
Eisenhart
323-338
Note on two three-dimensional manifolds with the same group
J. W.
Alexander
339-342
A general system of linear equations
A. J.
Pell
343-355
Errata: ``On a new treatment of theorems of finiteness'' [Trans. Amer. Math. Soc. {\bf 20} (1919), no. 3, 203--212; 1501122]
Oliver Edmunds
Glenn
356
Certain types of involutorial space transformations
F. R.
Sharpe;
Virgil
Snyder
185-202
On a new treatment of theorems of finiteness
Oliver Edmunds
Glenn
203-212
On the theory of developments of an abstract class in relation to the calcul fonctionnel
E. W.
Chittenden;
A. D.
Pitcher
213-233
On the influence of keyways on the stress distribution in cylindrical shafts
T. H.
Gronwall
234-244
Some convergent developments associated with irregular boundary conditions
James W.
Hopkins
245-259
Groups possessing a small number of sets of conjugate operators
G. A.
Miller
260-270
Memoir on the general theory of surfaces and rectilinear congruences
Gabriel M.
Green
79-153
Modular concomitant scales, with a fundamental system of formal covariants, modulo $3$, of the binary quadratic
Oliver Edmunds
Glenn
154-168
Concerning a set of postulates for plane analysis situs
Robert L.
Moore
169-178
On the limit functions of sequences of continuous functions converging relatively uniformly
E. W.
Chittenden
179-184
Necessary conditions in the problems of Mayer in the calculus of variations
Gillie A.
Larew
1-22
[Obituary for Maxime B\^ocher (d. 1918)]
i-ii
Linear equations with unsymmetric systems of coefficients
A. J.
Pell
23-39
On convex functions
Henry
Blumberg
40-44
Projective transformations in function space
L. L.
Dines
45-65
On the order of primitive groups. IV
W. A.
Manning
66-78
Spiral minimal surfaces
J. K.
Whittemore
315-330
On the group of isomorphisms of a certain extension of an abelian group
Louis C.
Mathewson
331-340
Concerning the zeros of the solutions of certain differential equations
William Benjamin
Fite
341-352
Differentiation with respect to a function of limited variation
P. J.
Daniell
353-362
Linear integro-differential equations with a boundary condition
Minfu Tah
Hu
363-407
On scalar and vector covariants of linear algebras
Olive C.
Hazlett
408-420
Singular points of analytic transformations
William F.
Osgood
251-274
Space involutions defined by a web of quadrics
Virgil
Snyder;
F. R.
Sharpe
275-290
On the location of the roots of the Jacobian of two binary forms, and of the derivative of a rational function
J. L.
Walsh
291-298
Sets of independent generators of a substitution group
G. A.
Miller
299-304
The problem of Mayer with variable end points
Gilbert Ames
Bliss
305-314
Proof that certain ideals in a cyclotomic realm are principal ideals
Howard H.
Mitchell
119-126
The order of primitive groups. III
W. A.
Manning
127-142
On the degree of convergence of Birkhoff's series
W. E.
Milne
143-156
Problems in the theory of ordinary linear differential equations with auxiliary conditions at more than two points
Charles E.
Wilder
157-166
Transformations of applicable conjugate nets of curves on surfaces
Luther Pfahler
Eisenhart
167-185
A new integral test for the convergence and divergence of infinite series
Raymond W.
Brink
186-204
On the reduction of certain differential equations of the second order
William Duncan
Macmillan
205-222
Invariants of differential configurations in the plane
Eugene Francis
Simonds
223-250
Rational approximations to irrational complex numbers
Lester R.
Ford
1-42
On the imaginary roots of a polynomial and the real roots of its derivative
Henry Bedinger
Mitchell
43-52
Relations entre les notions de limite et de distance
Maurice
Fréchet
53-65
On the foundations of the calcul fonctionnel of Fr\'echet
A. D.
Pitcher;
E. W.
Chittenden
66-78
Existence theorems for the general real self-adjoint linear system of the second order
H. J.
Ettlinger
79-96
On boundary value problems in linear differential equations in general analysis
T. H.
Hildebrandt
97-108
A fundamental system of formal covariants modulo $2$ of the binary cubic
Oliver Edmunds
Glenn
109-118
Expansion problems of ordinary linear differential equations with auxiliary conditions at more than two points
Charles E.
Wilder
415-442
A memoir on the doctrine of associated forms
O. E.
Glenn
443-462
The degree and class of multiply transitive groups
W. A.
Manning
463-479
Some geometric characterizations of isothermal nets on a curved surface
Gabriel M.
Green
480-488
A new method in boundary problems for differential equations
R. G. D.
Richardson
489-518
Note supplementary to the paper ``On certain pairs of transcendental functions whose roots separate each other''
Maxime
Bôcher
519-521
A theorem for space analogous to Ces\`aro's theorem for plane isogonal systems
Percey F.
Smith
522-540
Errata: ``On a theory of linear differential equations in general analysis'' [Trans. Amer. Math. Soc. {\bf 18} (1917), no. 1, 73--96; 1501063]
T. H.
Hildebrandt
540
Sets of independent postulates for betweenness
Edward V.
Huntington;
J. Robert
Kline
301-325
Haskins's momental theorem and its connection with Stieltjes's problem of moments
Edward B.
Van Vleck
326-330
Point sets and allied Cremona groups. III
Arthur B.
Coble
331-372
On the second derivatives of the extremal-integral for the integral $\int F(y;y')dt$
Arnold
Dresden
373-378
Concerning singular transformations $B\sb k$ of surfaces applicable to quadrics
Luigi
Bianchi
379-401
Types of $(2,2)$ point correspondences between two planes
F. R.
Sharpe;
Virgil
Snyder
402-414
Differential equations and implicit functions in infinitely many variables
William Leroy
Hart
125-160
On the equivalence of \'Ecart and voisinage
E. W.
Chittenden
161-166
On the theory of associative division algebras
Olive C.
Hazlett
167-176
The converse of the theorem concerning the division of a plane by an open curve
J. R.
Kline
177-184
On the conformal mapping of curvilinear angles. The functional equation $\phi[f(x)]=a\sb 1\phi(x)$
G. A.
Pfeiffer
185-198
Dynamical systems with two degrees of freedom
George D.
Birkhoff
199-300
On the consistency and equivalence of certain definitions of summability
Wallie Abraham
Hurwitz;
L. L.
Silverman
1-20
The resolution into partial fractions of the reciprocal of an entire function of genus zero
J. F.
Ritt
21-26
On a general class of linear homogeneous differential equations of infinite order with constant coefficients
J. F.
Ritt
27-49
On the expressibility of a uniform function of several complex variables as the quotient of two functions of entire character
T. H.
Gronwall
50-64
Certain formal invariances in Boolean algebras
Norbert
Wiener
65-72
On a theory of linear differential equations in general analysis
T. H.
Hildebrandt
73-96
Transformations $T$ of conjugate systems of curves on a surface
Luther Pfahler
Eisenhart
97-124
On the invariant system of a pair of connexes
O. E.
Glenn
405-417
Algebraic properties of self-adjoint systems
Dunham
Jackson
418-424
On the second derivatives of an extremal-integral with an application to a problem with variable end points
Arnold
Dresden
425-436
Deformable transformations of Ribaucour
Luther Pfahler
Eisenhart
437-458
Jacobi's condition for the problem of Lagrange in the calculus of variations
David M.
Smith
459-475
On transcendental numbers
Aubrey J.
Kempner
476-482
The linear dependence of functions of several variables, and completely integrable systems of homogeneous linear partial differential equations
Gabriel M.
Green
483-516
On the factorization of expressions of various types
Henry
Blumberg
517-544
The formal modular invariant theory of binary quantics
O. E.
Glenn
545-556
On a general class of series of the form $\sum c\sb ng(x+n)$
R. D.
Carmichael
207-232
The geometries associated with a certain system of Cremona groups
John Wesley
Young;
Frank Millett
Morgan
233-244
A reduction of certain analytic differential equations to differential equations of an algebraic type
William Duncan
Macmillan
245-258
A new canonical form of the elliptic integral
Bessie I.
Miller
259-283
On the notion of summability for the limit of a function of a continuous variable
L. L.
Silverman
284-294
On the factorization of Cremona plane transformations
James W.
Alexander
295-300
Weierstrass's non-differentiable function
G. H.
Hardy
301-325
Finite groups represented by special matrices
G. A.
Miller
326-332
On infinite regions
William F.
Osgood
333-344
Point sets and allied Cremona groups. II
Arthur B.
Coble
345-385
Infinite products of analytic matrices
George D.
Birkhoff
386-404
On multiform solutions of linear differential equations having elliptic function coefficients
Wilson Lee
Miser
109-130
On the foundations of plane analysis situs
Robert L.
Moore
131-164
On the generalized Jacobi-Kummer cyclotomic function
Howard H.
Mitchell
165-177
Proof of a theorem of Haskins
Dunham
Jackson
178-180
On the measurable bounds and the distribution of functional values of summable functions
Charles N.
Haskins
181-194
Jacobi's condition for problems of the calculus of variations in parametric form
Gilbert Ames
Bliss
195-206
On functions of several complex variables
William F.
Osgood
1-8
A study of certain functional equations for the $\theta$-functions
E. B.
Van Vleck;
F.
H’Doubler
9-49
A set of four independent postulates for Boolean algebras
B. A.
Bernstein
50-52
Transformations of surfaces $\Omega$. II
Luther Pfahler
Eisenhart
53-99
On figures of equilibrium of a rotating compressible fluid mass; certain negative results
E. J.
Moulton
100-108
A type of singular points for a transformation of three variables
W. V.
Lovitt
371-386
The reduction of multiple $L$-integrals of separated functions to iterated $L$-integrals
J. K.
Lamond
387-398
Independent generators of a group of finite order
G. A.
Miller
399-404
On the zeros of the function, $P(x)$, complementary to the incomplete gamma function
Charles N.
Haskins
405-412
Group properties of the residue classes of certain Kronecker modular systems and some related generalizations in number theory
Edward
Kircher
413-434
Sur l'int\'egrale de Lebesgue
C.
De La Vallée Poussin
435-501
A new development of the theory of algebraic numbers
G. E.
Wahlin
502-508
Ruled surfaces whose flecnode curves have plane branches
A. F.
Carpenter
509-532
Errata: ``Lebesgue integrals containing a parameter, with applications'' [Trans. Amer. Math. Soc. {\bf 15} (1914), no. 1, 87--106; 1500967]
Burton H.
Camp
532
Sur les fonctionnelles bilin\'eaires
Maurice
Fréchet
215-234
Oriented circles in space
David F.
Barrow
235-258
A new isosceles-triangle solution of the three body problem
Daniel
Buchanan
259-274
Surfaces $\Omega$ and their transformations
Luther Pfahler
Eisenhart
275-310
The general theory of congruences
E. J.
Wilczynski
311-327
On matrices whose coefficients are functions of a single variable
J. H. M.
Wedderburn
328-332
Conformal classification of analytic arcs or elements: Poincar\'e's local problem of conformal geometry
Edward
Kasner
333-349
Extensions of Descartes' rule of signs connected with a problem suggested by Laguerre
D. R.
Curtiss
350-360
On parastrophic algebras
James Byrnie
Shaw
361-370
Quartic curves modulo $2$
L. E.
Dickson
111-120
Mixed linear integral equations of the first order
Wallie Abraham
Hurwitz
121-133
Prime power groups in which every commutator of prime order is invariant
William Benjamin
Fite
134-138
On the order of primitive groups. II
W. A.
Manning
139-147
A proof of the invariance of certain constants of analysis situs
J. W.
Alexander
148-154
Point sets and allied Cremona groups. I
Arthur B.
Coble
155-198
Scroll directrix curves
Chas. T.
Sullivan
199-214
On the theory of curved surfaces, and canonical systems in projective differential geometry
G. M.
Green
1-12
The multitude of triad systems on $31$ letters
Henry S.
White
13-19
The $\phi$-subgroup of a group
G. A.
Miller
20-26
On a set of postulates which suffice to define a number-plane
Robert L.
Moore
27-32
The equivalence of complex points, planes, lines with respect to real motions and certain other groups of real transformations
William Caspar
Graustein
33-44
Invariants of the rational plane quintic curve and of any rational curve of odd order
Joseph Eugene
Rowe
45-50
A set of postulates for general projective geometry
Meyer G.
Gaba
51-61
Certain quartic surfaces belonging to infinite discontinuous Cremonian groups
Virgil
Snyder;
F. R.
Sharpe
62-70
The functions of a complex variable defined by an ordinary differential equation of the first order and the first degree
Joseph
Slepian
71-88
On the differential geometry of ruled surfaces in $4$-space and cyclic surfaces in $3$-space
Arthur
Ranum
89-110
The conic as a space element
Roger A.
Johnson
335-368
The Weierstrass $F$-function for problems of the calculus of variations in space
Gilbert Ames
Bliss
369-378
The subgroups of the quaternary abelian linear group
Howard H.
Mitchell
379-396
Transformations of conjugate systems with equal point invariants
Luther Pfahler
Eisenhart
397-430
Proof of the finiteness of the modular covariants of a system of binary forms and cogredient points
Forbes Bagley
Wiley
431-438
On the degree of convergence of Sturm-Liouville series
Dunham
Jackson
439-466
Singular integral equations of the Volterra type
Clyde E.
Love
467-476
On the reduction of integro-differential equations
Griffith C.
Evans
477-496
Invariants in the theory of numbers
L. E.
Dickson
497-503
Addenda and errata: ``Proper multiple integrals over iterable fields'' [Trans. Amer. Math. Soc. {\bf 11} (1910), no. 1, 25--36; 1500854]
Ernest B.
Lytle
504-505
Errata: ``Decomposition of an $n$-space by a polyhedron'' [Trans. Amer. Math. Soc. {\bf 14} (1913), no. 1, 65--72; 1500936]
Oswald
Veblen
506
A new principle in the geometry of numbers, with some applications
H. F.
Blichfeldt
227-235
An application of Severi's theory of a basis to the Kummer and Weddle surfaces
F. R.
Sharpe;
C. F.
Craig
236-244
Transformations of surfaces of Voss
Luther Pfahler
Eisenhart
245-265
Birational transformations of certain quartic surfaces
F. R.
Sharpe;
Virgil
Snyder
266-276
One-parameter families of curves in the plane
Gabriel M.
Green
277-290
The minimum of a definite integral for unilateral variations in space
G. A.
Bliss;
A. L.
Underhill
291-310
On a method of comparison for triple-systems
Louise D.
Cummings
311-327
An existence theorem for a certain differential equation of the $n$th order
William Raymond
Longley
328-334
Sur la notion de diff\'erentielle d'une fonction de ligne
Maurice
Fréchet
135-161
A type of primitive algebra
J. H. M.
Wedderburn
162-166
Properties of surfaces whose asymptotic curves belong to linear complexes
Charles T.
Sullivan
167-196
Relatively uniform convergence of sequences of functions
E. W.
Chittenden
197-201
Note on Fermat's last theorem
H. S.
Vandiver
202-204
A set of axioms for line geometry
E. R.
Hedrick;
Louis
Ingold
205-214
The Cauchy problem for integro-differential equations
Griffith C.
Evans
215-226
On the degree of convergence of Laplace's series
T. H.
Gronwall
1-30
Linear associative algebras and abelian equations
L. E.
Dickson
31-46
Some theorems concerning groups whose orders are powers of a prime
William Benjamin
Fite
47-50
Limits in terms of order, with example of limiting element not approachable by a sequence
Ralph E.
Root
51-71
The symbolical theory of finite expansions
O. E.
Glenn
72-86
Lebesgue integrals containing a parameter, with applications
Burton H.
Camp
87-106
Congruences and complexes of circles
J. L.
Coolidge
107-134
Applications and generalizations of the conception of adjoint systems
Maxime
Bôcher
403-420
On a certain class of self-projective surfaces
E. J.
Wilczynski
421-443
On the representation groups of given abstract groups
G. A.
Miller
444-452
On the accuracy of trigonometric interpolation
Dunham
Jackson
453-461
On a simple type of irregular singular point
George D.
Birkhoff
462-476
On quadratic residues
J.
McDonnell
477-480
A set of five independent postulates for Boolean algebras, with application to logical constants
Henry Maurice
Sheffer
481-488
Formal modular invariants with application to binary modular covariants
Mildred
Sanderson
489-500
Proof of the finiteness of modular covariants
Leonard Eugene
Dickson
299-310
On transcendentally transcendental functions
R. D.
Carmichael
311-319
Sur les classes $V$ normales
Maurice
Fréchet
320-324
Implicit functions defined by equations with vanishing Jacobian
Guy Roger
Clements
325-342
On the approximate representation of an indefinite integral and the degree of convergence of related Fourier's series
Dunham
Jackson
343-364
Certain continuous deformations of surfaces applicable to the quadrics
Luther Pfahler
Eisenhart
365-402
A study of the circle cross
J. L.
Coolidge
149-174
Projective differential geometry of developable surfaces
William Wells
Denton
175-208
The solutions of non-homogeneous linear difference equations and their asymptotic form
K. P.
Williams
209-240
An application of finite geometry to the characteristic theory of the odd and even theta functions
Arthur B.
Coble
241-276
Conformal transformations on the boundaries of their regions of definitions
William F.
Osgood;
Edson H.
Taylor
277-298
The triad systems of thirteen letters
F. N.
Cole
1-5
Triple-systems as transformations, and their paths among triads
H. S.
White
6-13
Proof of Poincar\'e's geometric theorem
George D.
Birkhoff
14-22
On the existence of loci with given singularities
Solomon
Lefschetz
23-41
Singular multiple integrals, with applications to series
Burton H.
Camp
42-64
Decomposition of an $n$-space by a polyhedron
Oswald
Veblen
65-72
On convergence factors in double series and the double Fourier's series
Charles N.
Moore
73-104
Algebraic surfaces invariant under an infinite discontinuous group of birational transformations. I
Virgil
Snyder
105-108
Concerning Van Vleck's non-measurable set
N. J.
Lennes
109-112
Some asymptotic expressions in the theory of numbers
T. H.
Gronwall
113-122
Determination of the finite quaternary linear groups
Howard H.
Mitchell
123-142
On the character of a transformation in the neighborhood of a point where its Jacobian vanishes
L. S.
Dederick
143-148
On the pseudo-resolvent to the kernel of an integral equation
Wallie Abraham
Hurwitz
405-418
Infinite systems of indivisible groups
G. A.
Miller
419-433
Improper multiple integrals over iterable fields
J. K.
Lamond
434-444
On a theorem of Fej\'er's and an analogon to Gibbs' phenomenon
T. H.
Gronwall
445-468
The southerly and easterly deviations of falling bodies for an unsymmetrical gravitational field of force
William H.
Roever
469-490
On approximation by trigonometric sums and polynomials
Dunham
Jackson
491-515
Errata: ``On the extension of a theorem of Poincar\'e for difference-equations'' [Trans. Amer. Math. Soc. {\bf 13} (1912), no. 3, 342--352; 1500922]
Edward B.
Van Vleck
517
Errata: ``One-parameter projective groups and the classification of collineations'' [Trans. Amer. Math. Soc. {\bf 13} (1912), no. 3, 353--386; 1500923]
Edward B.
Van Vleck
517
Quaternion developments with applications
James Byrnie
Shaw
279-292
Theory of finite algebras
H. S.
Vandiver
293-304
On the degree of convergence of the development of a continuous function according to Legendre's polynomials
Dunham
Jackson
305-318
Functional-differential geometry
Louis
Ingold
319-341
On the extension of a theorem of Poincar\'e for difference-equations
Edward B.
Van Vleck
342-352
One-parameter projective groups and the classification of collineations
Edward B.
Van Vleck
353-386
Bicombinants of the rational plane quartic and combinant curves of the rational plane quintic
J. E.
Rowe
387-404
A generalization of Weierstrass' preparation theorem for a power series in several variables
Gilbert Ames
Bliss
133-145
An existence theorem for periodic solutions
William Duncan
Macmillan
146-158
A condition that a function in a projective space be rational
William F.
Osgood
159-163
A method of proving certain theorems relating to rational functions which are adjoint to an algebraic equation for a given value of the independent variable
J. C.
Fields
164-174
The dependence of focal points upon curvature for problems of the calculus of variations in space
Marion Ballantyne
White
175-198
Orthocentric properties of the plane directed $n$-line
Joseph Ellis
Hodgson
199-231
Certain singularities of point transformations in space of three dimensions
S. E.
Urner
232-264
Multiple correspondences determined by the rational plane quintic curve
J. R.
Conner
265-278
\"Uber eine idealtheoretische Funktion
Edmund
Landau
1-21
Theorems of oscillation for two linear differential equations of the second order with two parameters
R. G. D.
Richardson
22-34
The absolute minimum of a definite integral in a special field
E. J.
Miles
35-49
An existence theorem for a problem of the calculus of variations in space
E. Gordon
Bill
50-58
Linear algebras
Leonard Eugene
Dickson
59-73
A note concerning Veblen's axioms for geometry
Robert L.
Moore
74-76
Natural families of curves in a general curved space of $n$-dimensions
Joseph
Lipke
77-95
A class of periodic orbits of superior planets
F. R.
Moulton
96-108
Harmonic functions and Green's integral
O. D.
Kellogg
109-132
On the limit of the degree of primitive groups
W. A.
Manning
375-386
Isomorphisms of a group whose order is a power of a prime
G. A.
Miller
387-402
On minimal lines and congruences in four-dimensional space
John
Eiesland
403-428
Volterra's integral equation of the second kind, with discontinuous kernel. II
Griffith C.
Evans
429-472
One-parameter families and nets of plane curves
E. J.
Wilczynski
473-510
Errata: ``Irreducible homogeneous linear groups in an arbitrary domain'' [Trans. Amer. Math. Soc. {\bf 10} (1909), no. 3, 315--318; 1500842]
William Benjamin
Fite
511
Errata: ``On the base of a relative number-field, with an application to the composition of fields'' [Trans. Amer. Math. Soc. {\bf 11} (1910), no. 4, 487--493; 1500875]
G. E.
Wahlin
511-512
On properties of a domain for which any derived set is closed
Earle Raymond
Hedrick
285-294
Important covariant curves and a complete system of invariants of the rational quartic curve
J. E.
Rowe
295-310
An application of Moore's cross-ratio group to the solution of the sextic equation
Arthur B.
Coble
311-325
On the use of the co-sets of a group
G. A.
Miller
326-334
The southerly derivation of falling bodies
William H.
Roever
335-353
An application of a $(1,2)$ quaternary correspondence to the Weddle and Kummer surfaces
Virgil
Snyder
354-366
On semi-discriminants of ternary forms
O. E.
Glenn
367-374
Biorthogonal systems of functions
Anna Johnson
Pell
135-164
Applications of biorthogonal systems of functions to the theory of integral equations
Anna Johnson
Pell
165-180
On the uniform convergence of the developments in Bessel functions
Charles N.
Moore
181-206
Determination of the ordinary and modular ternary linear groups
Howard H.
Mitchell
207-242
General theory of linear difference equations
George D.
Birkhoff
243-284
An invariantive investigation of irreducible binary modular forms
Leonard Eugene
Dickson
1-18
An application of symbolic methods to the treatment of mean curvatures in hyperspace
William Hunt
Bates
19-38
On the order of linear homogeneous groups. IV
H. F.
Blichfeldt
39-42
The metrical aspect of the line-sphere transformation
Julian Lowell
Coolidge
43-69
Natural systems of trajectories generating families of Lam\'e
Edward
Kasner
70-74
A fundamental system of invariants of the general modular linear group with a solution of the form problem
Leonard Eugene
Dickson
75-98
Linear difference equations and their analytic solutions
R. D.
Carmichael
99-134
Conjugate line congruences contained in a bundle of quadric surfaces
Virgil
Snyder
371-387
On the fundamental number of the algebraic number-field $k(\root p\of m)$
Jacob
Westlund
388-392
Volterra's integral equation of the second kind, with discontinuous kernel
Griffith C.
Evans
393-413
Ein Seitenst\"uck zur Moebius'schen Geometrie der Kreisverwandtschaften
H.
Beck
414-448
Vector interpretation of symbolic differential parameters
Louis
Ingold
449-474
Surfaces with isothermal representation of their lines of curvature and their transformations. II
Luther Pfahler
Eisenhart
475-486
On the base of a relative number field, with an application to the composition of fields
G. E.
Wahlin
487-493
The strain of a non-gravitating sphere of variable density
L. M.
Hoskins
494-504
Die nat\"urlichen Gleichungen der analytischen Curven im Euklidischen Raume
E.
Study
249-279
Two-dimensional chains and the associated collineations in a complex plane
John Wesley
Young
280-293
Groups of rational transformations in a general field
Lewis Irving
Neikirk
294-300
On osculating element-bands associated with loci of surface-elements
Percey F.
Smith
301-324
Fields of extremals in space
Gilbert Ames
Bliss;
Max
Mason
325-340
Groups generated by two operators $(s\sb 1,s\sb 2)$ satisfying the equation $s\sb 1s\sp 2\sb 2=s\sb 2s\sp 2\sb 1$
G. A.
Miller
341-350
Congruences of the elliptic type
Luther Pfahler
Eisenhart
351-370
The theorem of Thomson and Tait and natural families of trajectories
Edward
Kasner
121-140
The introduction of ideal elements and a new definition of projective $n$-space
Frederick William
Owens
141-171
The group of classes of congruent quadratic integers with respect to a composite ideal modulus
Arthur
Ranum
172-198
A simplified treatment of the regular singular point
George D.
Birkhoff
199-202
The strain of a gravitating, compressible elastic sphere
L. M.
Hoskins
203-248
Theorems on simple groups
H. F.
Blichfeldt
1-14
Infinite discontinuous groups of birational transformations which leave certain surfaces invariant
Virgil
Snyder
15-24
Proper multiple integrals over iterable fields
Ernest B.
Lytle
25-36
On a class of hyper-Fuchsian functions
Clyde Firman
Craig
37-54
Periodic orbits about an oblate spheroid
William Duncan
Macmillan
55-120
The summability of the developments in Bessel functions, with applications
Charles N.
Moore
391-435
Singular points of ordinary linear differential equations
George D.
Birkhoff
436-470
Automorphisms of order two
G. A.
Miller
471-478
Resolution into involutory substitutions of the transformations of a non-singular bilinear form into itself
Dunham
Jackson
479-484
On singular points in the approximate development of the perturbative function
Frank Walker
Reed
485-509
Errata: ``Generalization of the groups of genus zero'' [Trans. Amer. Math. Soc. {\bf 8} (1907), no. 1, 1--13; 1500770]
G. A.
Miller
510
Errata: ``Natural families of trajectories: conservative fields of force'' [Trans. Amer. Math. Soc. {\bf 10} (1909), no. 2, 201--219; 1500834]
Edward
Kasner
510
Indices, volumes 1-10: 1900-1909
511-545
Projective differential geometry of curved surfaces. V
E. J.
Wilczynski
279-296
On the osculating quartic of a plane curve
William Wells
Denton
297-308
Note on a system of axioms for geometry
Arthur Richard
Schweitzer
309-314
Irreducible homogeneous linear groups in an arbitrary domain
William Benjamin
Fite
315-318
On the integration of the homogeneous linear difference equation of second order
Walter B.
Ford
319-336
On Cantor's theorem concerning the coefficients of a convergent trigonometric series, with generalizations
William F.
Osgood
337-346
Equivalence of pairs of bilinear or quadratic forms under rational transformation
Leonard Eugene
Dickson
347-360
On a complete system of invariants of two triangles
David D.
Leib
361-390
General theory of modular invariants
Leonard Eugene
Dickson
123-158
Beitr\"age zur Theorie der Gruppen linearer homogener Substitutionen
I.
Schur
159-175
Projective differential geometry of curved surfaces. IV
E. J.
Wilczynski
176-200
Natural families of trajectories: conservative fields of force
Edward
Kasner
201-219
Plane fields of force whose trajectories are invariant under a projective group
George Wilber
Hartwell
220-246
On the order of primitive groups
W. A.
Manning
247-258
Existence and oscillation theorem for a certain boundary value problem
George D.
Birkhoff
259-270
On the regions of convergence of power-series which represent two-dimensional harmonic functions
Maxime
Bôcher
271-278
Zur Differentialgeometrie der analytischen Curven
Eduard
Study
1-49
The central of a group
G. A.
Miller
50-60
The hypergeometric functions of $N$ variables
J. I.
Hutchinson
61-70
Surfaces derived from the cubic variety having nine double points in four-dimensional space
Virgil
Snyder
71-78
On a certain class of isothermic surfaces
Archer E.
Young
79-94
A geometrical application of binary syzygies
Aubrey E.
Landry
95-108
Definite forms in a finite field
Leonard Eugene
Dickson
109-122
Boundary value and expansion problems of ordinary linear differential equations
George D.
Birkhoff
373-395
An application of the form-problems associated with certain Cremona groups to the solution of equations of higher degree
Arthur B.
Coble
396-424
On the differential equations of the equilibrium of an inextensible string
Edwin Bidwell
Wilson
425-439
The properties of curves in space which minimize a definite integral
Max
Mason;
Gilbert Ames
Bliss
440-466
The second derivatives of the extremal-integral
Arnold
Dresden
467-486
Sets of metrical hypotheses for geometry
Robert L.
Moore
487-512
Brilliant points of curves and surfaces
William H.
Roever
245-279
Continuous increasing functions of finite and transfinite ordinals
Oswald
Veblen
280-292
Projective differential geometry of curved surfaces. III
E. J.
Wilczynski
293-315
Invariants of the function $F(x,y,x',y')$ in the calculus of variations
Anthony Lispenard
Underhill
316-338
The integration of a sequence of functions and its application to iterated integrals
R. G. D.
Richardson
339-372
Representations of the general symmetric group as linear groups in finite and infinite fields
Leonard Eugene
Dickson
121-148
Surfaces with isothermal representation of their lines of curvature and their transformations
Luther Pfahler
Eisenhart
149-177
The equilong transformations of space
Julian Lowell
Coolidge
178-182
Concerning linear substitutions of finite period with rational coefficients
Arthur
Ranum
183-202
On hypercomplex number systems belonging to an arbitrary domain of rationality
R. B.
Allen
203-218
On the asymptotic character of the solutions of certain linear differential equations containing a parameter
George D.
Birkhoff
219-231
On the holomorph of the cyclic group of order $p\sp m$
G. A.
Miller
232-236
On non-measurable sets of points, with an example
Edward B.
Van Vleck
237-244
Certain periodic orbits of $k$ finite bodies revolving about a relatively large central mass
Frank Loxley
Griffin
1-33
Further note on Maclaurin's spheroid
George Howard
Darwin
34-38
Potential functions on the boundary of their regions of definition
O. D.
Kellogg
39-50
Double distributions and the Dirichlet problem
O. D.
Kellogg
51-66
Groups defined by the orders of two generators and the order of their commutator
G. A.
Miller
67-78
Projective differential geometry of curved surfaces. II
E. J.
Wilczynski
79-120
The expansion of a function in terms of normal functions
Max
Mason
427-432
Sur les op\'erations lin\'eaires. III
Maurice
Fréchet
433-446
The elliptic integral in electromagnetic theory
Alfred George
Greenhill
447-534
Errata: ``Applicable surfaces with asymptotic lines of one surface corresponding to a conjugate system of another'' [Trans. Amer. Math. Soc. {\bf 8} (1907), no. 1, 113--134; 1500778]
Luther Pfahler
Eisenhart
535
Errata: ``A class of periodic orbits of an infinitesimal body subject to the attraction of $n$ finite bodies'' [Trans. Amer. Math. Soc. {\bf 8} (1907), no. 2, 159--188; 1500780]
William Raymond
Longley
535
Errata: ``Invariants of binary forms under modular transformations'' [Trans. Amer. Math. Soc. {\bf 8} (1907), no. 2, 205--232; 1500782]
Leonard Eugene
Dickson
535
Errata: ``On the introduction of convergence factors into summable series and summable integrals'' [Trans. Amer. Math. Soc. {\bf 8} (1907), no. 2, 299--330; 1500786]
Charles N.
Moore
535-536
Errata: ``A new form of the simplest problem of the calculus of variations'' [Trans. Amer. Math. Soc. {\bf 8} (1907), no. 3, 405--414; 1500795]
Gilbert Ames
Bliss
536
General theory of approximation by functions involving a given number of arbitrary parameters
John Wesley
Young
331-344
On derivatives over assemblages
E. R.
Hedrick
345-353
Projective geometries of congruence and finite projective geometries
Beppo
Levi
354-365
Collineations in a finite projective geometry
Oswald
Veblen
366-368
Geometry in which the sum of the angles of every triangle is two right angles
R. L.
Moore
369-378
Non-Desarguesian and non-Pascalian geometries
O.
Veblen;
J. H.
Maclagan-Wedderburn
379-388
Modular theory of group-matrices
Leonard Eugene
Dickson
389-398
Existence proof for a field of extremals tangent to a given curve
Oskar
Bolza
399-404
A new form of the simplest problem of the calculus of variations
Gilbert Ames
Bliss
405-414
On certain isothermic surfaces
Archer Everett
Young
415-426
Dynamical trajectories: the motion of a particle in an arbitrary field of force
Edward
Kasner
135-158
A class of periodic orbits of an infinitesimal body subject to the attraction of $n$ finite bodies
William Raymond
Longley
159-188
A proof of some theorems on pointwise discontinuous functions
Edward B.
Van Vleck
189-204
Invariants of binary forms under modular transformations
Leonard Eugene
Dickson
205-232
Projective differential geometry of curved surfaces
E. J.
Wilczynski
233-260
A method for constructing the fundamental region of a discontinuous group of linear transformations
J. I.
Hutchinson
261-269
Oblique reflections and unimodular strains
Edwin Bidwell
Wilson
270-298
On the introduction of convergence factors into summable series and summable integrals
Charles N.
Moore
299-330
Generalization of the groups of genus zero
G. A.
Miller
1-13
On reflexive geometry
F.
Morley
14-24
The groups in which every subgroup is either abelian or Hamiltonian
G. A.
Miller
25-29
On modular groups isomorphic with a given linear group
H. F.
Blichfeldt
30-32
Denumerants of double differentiants
William Edward
Story
33-70
The group of classes of congruent matrices with application to the group of isomorphisms of any abelian group
Arthur
Ranum
71-91
A theorem of Abel and its application to the development of a function in terms of Bessel's functions
Clara E.
Smith
92-106
Irreducible linear homogeneous groups whose orders are powers of a prime
William Benjamin
Fite
107-112
Applicable surfaces with asymptotic lines of one surface corresponding to a conjugate system of another
Luther Pfahler
Eisenhart
113-134
Weierstrass' theorem and Kneser's theorem on transversals for the most general case of an extremum of a simple definite integral
Oskar
Bolza
459-488
Area of curved surfaces
James
Pierpont
489-498
On multiply transitive groups
W. A.
Manning
499-508
Zur Theorie der vollst\"andig reduciblen Gruppen, die zu einer Gruppe linearer homogener Substitutionen geh\"oren. (Aus einem Briefe an Herrn Alfred Loewy)
Ludwig
Stickelberger
509-513
On commutative linear algebras in which division is always uniquely possible
L. E.
Dickson
514-522
On the order of linear homogeneous groups (supplement)
H. F.
Blichfeldt
523-529
On automorphic groups whose coefficients are integers in a quadratic field
J. I.
Hutchinson
530-536
A class of periodic solutions of the problem of three bodies with application to the lunar theory
Forest Ray
Moulton
537-577
A problem in the reduction of hyperelliptic integrals
John Hector
McDonald
578-587
On the differential invariants of a plane
Charles Nelson
Haskins
588-590
Errata: ``Note on the definitions of abstract groups and fields by sets of independent postulates'' [Trans. Amer. Math. Soc. {\bf 6} (1905), no. 2, 181--197; 1500705]
Edward V.
Huntington
591
Errata: ``Differential parameters of the first order'' [Trans. Amer. Math. Soc. {\bf 7} (1906), no. 1, 69--80; 1500733]
Heinrich
Maschke
591
On the boundary value problems of linear ordinary differential equations of second order
Max
Mason
337-360
The resolution of any collineation into perspective reflections
M. W.
Haskell
361-369
Linear algebras in which division is always uniquely possible
Leonard Eugene
Dickson
370-390
Correspondences and the theory of groups
Joseph Edmund
Wright
391-400
The trajectories of dynamics
Edward
Kasner
401-424
On the automorphic functions of the group $(0,3;l\sb 1,l\sb 2,l\sb 3)$
Richard
Morris
425-448
Improper multiple integrals
R. G. D.
Richardson
449-458
On geometries in which circles are the shortest lines
Carl Eben
Stromquist
175-183
A generalization of the notion of angle
Gilbert Ames
Bliss
184-196
The square root and the relations of order
Oswald
Veblen
197-199
The problem of partial geodesic representation
Edward
Kasner
200-206
On the pentadeltoid
R. P.
Stephens
207-227
The groups of order $p\sp m$ which contain exactly $p$ cyclic subgroups of order $p\sp \alpha$
G. A.
Miller
228-232
Groups in which a large number of operators may correspond to their inverses
W. A.
Manning
233-240
Finite projective geometries
Oswald
Veblen;
W. H.
Bussey
241-259
On the analytic extension of functions defined by double power series
W. B.
Ford
260-274
On quadratic, Hermitian and bilinear forms
Leonard Eugene
Dickson
275-292
Die kinematische Erzeugung von Minimalfl\"achen. I
Paul
Stackel
293-313
A fifth necessary condition for a strong extremum of the integral $\int\sp {x\sb 1}\sb {x\sb 0} F(x,y,y')dx$
Oskar
Bolza
314-324
A problem of the calculus of variations in which the integrand is discontinuous
Gilbert Ames
Bliss;
Max
Mason
325-336
On the relation between the three-parameter groups of a cubic space curve and a quadric surface
A. B.
Coble
1-20
On certain hyperabelian functions which are expressible by theta series
J. I.
Hutchinson
21-25
On the form of a plane quintic curve with five cusps
Peter
Field
26-32
The symbolic treatment of differential geometry
Arthur Whipple
Smith
33-60
Groups whose orders are powers of a prime
William Benjamin
Fite
61-68
Differential parameters of the first order
Heinrich
Maschke
69-80
The Kronecker-Gaussian curvature of hyperspace
Heinrich
Maschke
81-93
Groups containing only three operators which are squares
G. A.
Miller
94-98
Theorems converse to Riemann's on linear differential equations
David Raymond
Curtiss
99-106
General mean value and remainder theorems with applications to mechanical differentiation and quadrature
George David
Birkhoff
107-136
Determination of the abstract groups of order $p\sp 2qr$; $p$, $q$, $r$ being distinct primes
Oliver E.
Glenn
137-151
Note on the differential invariants of a surface and of space
Charles Nelson
Haskins
152-154
On improper multiple integrals
James
Pierpont
155-174
Sur l'\'ecart de deux courbes et sur les courbes limites
Maurice
Fréchet
435-449
On a certain system of conjugate lines on a surface connected with Euler's transformation
John
Eiesland
450-471
Surfaces of constant curvature and their transformations
Luther Pfahler
Eisenhart
472-485
Volumes and areas
N. J.
Lennes
486-490
On a problem including that of several bodies and admitting of an additional integral
Edgar Odell
Lovett
491-495
On the stability of the motion of a viscous liquid
Francis Robert
Sharpe
496-503
\"Uber die vollst\"andig reduciblen Gruppen, die zu einer Gruppe linearer homogener Substitutionen geh\"oren
Alfred
Loewy
504-533
On the Cayley-Veronese class of configurations
Walter B.
Carver
534-545
Errata: ``A theorem concerning the invariants of linear homogeneous groups with some applications to substitution-groups'' [Trans. Amer. Math. Soc. {\bf 5} (1904), no. 2, 461--466; 1500684]
H. F.
Blichfeldt
546
Errata: ``A set of postulates for real algebra, comprising postulates for a one-dimensional continuum and for the theory of groups'' [Trans. Amer. Math. Soc. {\bf 6} (1905), no. 1, 17--41; 1500691]
Edward V.
Huntington
546
Errata: ``On the coefficients in the quotient of two alternants'' [Trans. Amer. Math. Soc. {\bf 6} (1905), no. 1, 63--74; 1500695]
E. D.
Roe
546-547
Errata: ``Definitions of a group and a field by independent postulates'' [Trans. Amer. Math. Soc. {\bf 6} (1905), no. 2, 198--204; 1500706]
Leonard Eugene
Dickson
547
Sur les lignes g\'eod\'esiques des surfaces convexes
Henri
Poincaré
237-274
The classification of quadrics
T. J. I’A.
Bromwich
275-285
On differential invariants
Joseph Edmund
Wright
286-315
Groups of order $p\sp m$, which contain cyclic subgroups of order $p\sp {m-3}$
Lewis Irving
Neikirk
316-325
On the invariant subgroups of prime index
G. A.
Miller
326-331
On a general method for treating transmitted motions and its application to indirect perturbations
Ernest William
Brown
332-343
On hypercomplex number systems
L. E.
Dickson
344-348
A theorem on finite algebras
J. H.
Maclagan-Wedderburn
349-352
The relation of the principles of logic to the foundations of geometry
Josiah
Royce
353-415
On multiple integrals
James
Pierpont
416-434
General projective theory of space curves
E. J.
Wilczynski
99-133
Sur les op\'erations lin\'eaires. II
Maurice
Fréchet
134-140
Surfaces whose geodesics may be represented in the plane by parabolas
Edward
Kasner
141-158
The doubly periodic solutions of Poisson's equation in two independent variables
Max
Mason
159-164
Definition in terms of order alone in the linear continuum and in well-ordered sets
Oswald
Veblen
165-171
On the structure of hypercomplex number systems
Saul
Epsteen;
J. H.
Maclagan-Wedderburn
172-178
On a definition of abstract groups
Eliakim Hastings
Moore
179-180
Note on the definitions of abstract groups and fields by sets of independent postulates
Edward V.
Huntington
181-197
Definitions of a group and a field by independent postulates
Leonard Eugene
Dickson
198-204
On semi-groups and the general isomorphism between infinite groups
Leonard Eugene
Dickson
205-208
A set of postulates for ordinary complex algebra
Edward V.
Huntington
209-229
On imprimitive linear homogeneous groups
H. F.
Blichfeldt
230-236
On the linear transformations of a quadratic form into itself
Percey F.
Smith
1-16
A set of postulates for real algebra, comprising postulates for a one-dimensional continuum and for the theory of groups
Edward V.
Huntington
17-41
On the primitive groups of class $3p$
W. A.
Manning
42-47
The minimum degree $\tau$ of resolvents for the $p$-section of the periods of hyperelliptic functions of four periods
Leonard Eugene
Dickson
48-57
Determination of all the groups of order $2\sp m$ which contain an odd number of cyclic subgroups of composite order
G. A.
Miller
58-62
On the coefficients in the quotient of two alternants
E. D.
Roe
63-74
General theory of curves on ruled surfaces
E. J.
Wilczynski
75-82
Theory on plane curves in non-metrical analysis situs
Oswald
Veblen
83-98
On a method of dealing with the intersections of plane curves
F. S.
Macaulay
385-410
On certain complete systems of quaternion expressions, and on the removal of metric limitations from the calculus of quaternions
James Mills
Peirce
411-420
Three particular systems of lines on a surface
Luther Pfahler
Eisenhart
421-437
On ruled surfaces whose flecnode curve intersects every generator in two coincident points
E. J.
Wilczynski
438-446
On the automorphic functions of the group $(0,3;2,6,6)$
J. I.
Hutchinson
447-460
A theorem concerning the invariants of linear homogeneous groups, with some applications to substitution-groups
H. F.
Blichfeldt
461-466
On the geometry whose element is the $3$-point of a plane
F.
Morley
467-476
Sufficient condition for a minimum with respect to one-sided variations
Gilbert Ames
Bliss
477-492
Sur les op\'erations lin\'eaires
Maurice
Fréchet
493-499
On groups in which certain commutative operations are conjugate
H. L.
Rietz
500-508
On hypercomplex number systems. I
Henry
Taber
509-548
Errata: ``On a class of particular solutions of the problem of four bodies'' [Trans. Amer. Math. Soc. {\bf 1} (1900), no. 1, 17--29; 1500520]
Forest Ray
Moulton
549
Errata: ``A complete set of postulates for the theory of absolute continuous magnitude'' [Trans. Amer. Math. Soc. {\bf 3} (1902), no. 2, 264--279; 1500598]
Edward V.
Huntington
549
Notes and errata: ``A definition of abstract groups'' [Trans. Amer. Math. Soc. {\bf 3} (1902), no. 4, 485--492; 1500616]
Eliakim Hastings
Moore
549
Errata: ``Definitions of a field by independent postulates'' [Trans. Amer. Math. Soc. {\bf 4} (1903), no. 1, 13--20; 1500619]
Leonard Eugene
Dickson
549-550
Errata: ``The generalized Beltrami problem concerning geodesic representation'' [Trans. Amer. Math. Soc. {\bf 4} (1903), no. 2, 149--152; 1500632]
Edward
Kasner
550
Errata: ``On a certain congruence associated with a given ruled surface'' [Trans. Amer. Math. Soc. {\bf 4} (1903), no. 2, 185--200; 1500637]
E. J.
Wilczynski
550
Errata: ``On the point-line as element of space: a study of the corresponding bilinear connex'' [Trans. Amer. Math. Soc. {\bf 4} (1903), no. 3, 213--233; 1500639]
Edward
Kasner
550
Errata: ``On the subgroups of order a power of $p$ in the quaternary abelian group in the Galois field of order $p\sp n$'' [Trans. Amer. Math. Soc. {\bf 4} (1903), no. 4, 371--386; 1500648]
Leonard Eugene
Dickson
550-551
Errata: ``On solutions of differential equations which possess an oscillation theorem'' [Trans. Amer. Math. Soc. {\bf 4} (1903), no. 4, 423--433; 1500652]
Helen A.
Merrill
551
Errata: ``Semireducible hypercomplex number systems'' [Trans. Amer. Math. Soc. {\bf 4} (1903), no. 4, 437--444; 1500654]
Saul
Epsteen
551
Errata: ``The subgroups of order a power of $2$ of the simple quinary orthogonal group in the Galois field of order $p\sp n=8l\pm3$'' [Trans. Amer. Math. Soc. {\bf 5} (1904), no. 1, 1--38; 1500658]
Leonard Eugene
Dickson
551
Errata: ``Determination of all the subgroups of the known simple group of order $25\,920$'' [Trans. Amer. Math. Soc. {\bf 5} (1904), no. 2, 126--166; 1500666]
Leonard Eugene
Dickson
551
Errata: ``On the smaller perturbations of the lunar arguments'' [Trans. Amer. Math. Soc. {\bf 5} (1904), no. 3, 279--287; 1500674]
Ernest William
Brown
551
Errata: ``Sets of independent postulates for the algebra of logic'' [Trans. Amer. Math. Soc. {\bf 5} (1904), no. 3, 288--309; 1500675]
Edward V.
Huntington
552
Errata: ``Algebras defined by finite groups'' [Trans. Amer. Math. Soc. {\bf 5} (1904), no. 3, 326--342; 1500677]
James Byrnie
Shaw
552
On the convergence of algebraic continued fractions whose coefficients have limiting values
Edward B.
Van Vleck
253-262
The Sylow subgroups of the symmetric group
William
Findlay
263-278
On the smaller perturbations of the lunar arguments
Ernest William
Brown
279-287
Sets of independent postulates for the algebra of logic
Edward V.
Huntington
288-309
On the order of linear homogeneous groups. II
H. F.
Blichfeldt
310-325
Algebras defined by finite groups
James Byrnie
Shaw
326-342
A system of axioms for geometry
Oswald
Veblen
343-384
An existence theorem for a differential equation of the second order, with an application to the calculus of variations
Gilbert Ames
Bliss
113-125
Determination of all the subgroups of the known simple group of order $25920$
Leonard Eugene
Dickson
126-166
On the invariants of quadratic differential forms. II
Charles Nelson
Haskins
167-192
On the coefficients in the product of an alternant and a symmetric function
E. D.
Roe
193-213
The groups of order $p\sp 3q\sp \beta$
F. N.
Cole
214-219
Green's theorem and Green's functions for certain systems of differential equations
Max
Mason
220-225
Studies in the general theory of ruled surfaces
E. J.
Wilczynski
226-252
The subgroups of order a power of $2$ of the simple quinary orthogonal group in the Galois field of order $p\sp n=8l\pm 3$
Leonard Eugene
Dickson
1-38
On certain invariants of two triangles
John Gale
Hun
39-55
Isothermal systems of geodesics
Edward
Kasner
56-60
Zur Gruppentheorie mit Anwendungen auf die Theorie der linearen homogenen Differentialgleichungen
Alfred
Loewy
61-80
On the group of sign $(0,3;2,4,\infty)$ and the functions belonging to it
John Wesley
Young
81-104
On the definition of reducible hypercomplex number systems
Saul
Epsteen
105-109
A simple proof of a theorem in the calculus of variations (extract from a letter to Mr. W. F. Osgood)
E.
Goursat
110-112
On the subgroups of order a power of $p$ in the quaternary abelian group in the Galois field of order $p\sp n$
Leonard Eugene
Dickson
371-386
On the order of linear homogeneous groups
H. F.
Blichfeldt
387-397
Non-abelian groups in which every subgroup is abelian
G. A.
Miller;
H. C.
Moreno
398-404
On nilpotent algebras
James Byrnie
Shaw
405-422
On solutions of differential equations which possess an oscillation theorem
Helen A.
Merrill
423-433
On the reducibility of linear groups
Leonard Eugene
Dickson
434-436
Semireducible hypercomplex number systems
Saul
Epsteen
437-444
A symbolic treatment of the theory of invariants of quadratic differential quantics of $n$ variables
Heinrich
Maschke
445-469
Congruences of curves
Luther Pfahler
Eisenhart
470-488
Similar conics through three points
T. J. I’A.
Bromwich
489-492
On the point-line as element of space: a study of the corresponding bilinear connex
Edward
Kasner
213-233
On the formation of the derivatives of the lunar co\"ordinates with respect to the elements
Ernest William
Brown
234-248
On reducible groups
Saul
Epsteen
249-250
Theory of linear associative algebra
James Byrnie
Shaw
251-287
Projective co\"ordinates
F.
Morley
288-296
On an extension of the 1894 memoir of Stieltjes
Edward Burr
Van Vleck
297-332
On the variation of the arbitrary and given constants in dynamical equations
Ernest William
Brown
333-350
The primitive groups of class $2p$ which contain a substitution of order $p$ and degree $2p$
W. A.
Manning
351-357
Complete sets of postulates for the theory of real quantities
Edward V.
Huntington
358-370
The approximate determination of the form of Maclaurin's spheroid
George Howard
Darwin
113-133
On twisted cubic curves that have a directrix
H. S.
White
134-141
Ueber Curvenintegrale im $m$-dimensionalen Raum
Lothar
Heffter
142-148
The generalized Beltrami problem concerning geodesic representation
Edward
Kasner
149-152
On the holomorph of a cyclic group
G. A.
Miller
153-160
Quadric surfaces in hyperbolic space
Julian Lowell
Coolidge
161-170
Ueber die Reducibilit\"at der Reellen Gruppen linearer homogener Substitutionen
Alfred
Loewy
171-177
On the possibility of differentiating term by term the developments for an arbitrary function of one real variable in terms of Bessel functions
Walter B.
Ford
178-184
On a certain congruence associated with a given ruled surface
E. J.
Wilczynski
185-200
On the class number of the cyclotomic number field $k(e\sp {2\pi i/p\sp n})$
Jacob
Westlund
201-212
Orthocentric properties of the plane $n$-line
F.
Morley
1-12
Definitions of a field by independent postulates
Leonard Eugene
Dickson
13-20
Definitions of a linear associative algebra by independent postulates
Leonard Eugene
Dickson
21-26
Two definitions of an abelian group by sets of independent postulates
Edward V.
Huntington
27-30
Definitions of a field by sets of independent postulates
Edward V.
Huntington
31-37
On the invariants of differential forms of degree higher than two
Charles Nelson
Haskins
38-43
\"Uber die Reducibilit\"at der Gruppen linearer homogener Substitutionen
Alfred
Loewy
44-64
The quartic curve as related to conics
A. B.
Coble
65-85
The cogredient and digredient theories of multiple binary forms
Edward
Kasner
86-102
On the envelope of the axes of a system of conics passing through three fixed points
R. E.
Allardice
103-106
A Jordan curve of positive area
William F.
Osgood
107-112
On the groups of order $p\sp m$ which contain operators of order $p\sp {m-2}$
G. A.
Miller
383-387
On the circuits of plane curves
Charlotte Angas
Scott
388-398
Note on the real inflexions of plane curves
Charlotte Angas
Scott
399-400
La th\'eorie des plaques \'elastiques planes
Jacques
Hadamard
401-422
Covariants of systems of linear differential equations and applications to the theory of ruled surfaces
E. J.
Wilczynski
423-450
On the rank, order and class of algebraic minimum curves
Arthur Sullivan
Gale
451-466
On the determination of the distance between two points in space of $n$ dimensions
H. F.
Blichfeldt
467-481
On superosculating quadric surfaces
Heinrich
Maschke
482-484
A definition of abstract groups
Eliakim Hastings
Moore
485-492
Algebraic transformations of a complex variable realized by linkages
Arnold
Emch
493-498
Errata: ``On a class of particular solutions of the problem of four bodies'' [Trans. Amer. Math. Soc. {\bf 1} (1900), no. 1, 17--29; 1500520]
Forest Ray
Moulton
499
Errata: ``Canonical forms of quaternary abelian substitutions in an arbitrary Galois field'' [Trans. Amer. Math. Soc. {\bf 2} (1901), no. 2, 103--138; 1500559]
Leonard Eugene
Dickson
499
Errata: ``Geometry of a simultaneous system of two linear homogeneous differential equations of the second order'' [Trans. Amer. Math. Soc. {\bf 2} (1901), no. 4, 343--362; 1500572]
E. J.
Wilczynski
500
Errata: ``Theory of linear groups in an arbitrary field'' [Trans. Amer. Math. Soc. {\bf 2} (1901), no. 4, 363--394; 1500573]
Leonard Eugene
Dickson
500
Errata: ``The groups of Steiner in problems of contact'' [Trans. Amer. Math. Soc. {\bf 3} (1902), no. 1, 38--45; 1500585]
Leonard Eugene
Dickson
500
Errata: ``On the invariants of quadratic differential forms'' [Trans. Amer. Math. Soc. {\bf 3} (1902), no. 1, 71--91; 1500588]
Charles Nelson
Haskins
501
Errata: ``A determination of the number of real and imaginary roots of the hypergeometric series'' [Trans. Amer. Math. Soc. {\bf 3} (1902), no. 1, 110--131; 1500590]
Edward B.
Van Vleck
501
Errata: ``On the projective axioms of geometry'' [Trans. Amer. Math. Soc. {\bf 3} (1902), no. 1, 142--158; 1500592]
Eliakim Hastings
Moore
501
On the group defined for any given field by the multiplication table of any given finite group
Leonard Eugene
Dickson
285-301
Proof of the sufficiency of Jacobi's condition for a permanent sign of the second variation in the so-called isoperimetric problems
Oskar
Bolza
305-311
On hypercomplex number systems
Herbert Edwin
Hawkes
312-330
On metabelian groups
William Benjamin
Fite
331-353
Conjugate rectilinear congruences
Luther Pfahler
Eisenhart
354-371
Constructive theory of the unicursal cubic by synthetic methods
D. N.
Lehmer
372-376
The groups of Steiner in problems of contact. II
Leonard Eugene
Dickson
377-382
On the small divisors in the lunar theory
Ernest William
Brown
159-185
On the holomorphisms of a group
John Wesley
Young
186-191
A simple non-Desarguesian plane geometry
Forest Ray
Moulton
192-195
On the real solutions of systems of two homogeneous linear differential equations of the first order
Maxime
Bôcher
196-215
On a recent method for dealing with the intersections of plane curves
Charlotte Angas
Scott
216-263
A complete set of postulates for the theory of absolute continuous magnitude
Edward V.
Huntington
264-279
Complete sets of postulates for the theories of positive integral and positive rational numbers
Edward V.
Huntington
280-284
On a class of automorphic functions
J. I.
Hutchinson
1-11
Concerning the existence of surfaces capable of conformal representation upon the plane in such a manner that geodetic lines are represented by a prescribed system of curves
Henry Freeman
Stecker
12-22
Zur Erkl\"arung der Bogenl\"ange und des Inhaltes einer krummen Fl\"ache
O.
Stolz
23-37
The groups of Steiner in problems of contact
Leonard Eugene
Dickson
38-45
Quaternion space
Arthur S.
Hathaway
46-59
Reciprocal systems of linear differential equations
E. J.
Wilczynski
60-70
On the invariants of quadratic differential forms
Charles Nelson
Haskins
71-91
On the nature and use of the functions employed in the recognition of quadratic residues
Emory
McClintock
92-109
A determination of the number of real and imaginary roots of the hypergeometric series
Edward B.
Van Vleck
110-131
The second variation of a definite integral when one end-point is variable
Gilbert Ames
Bliss
132-141
On the projective axioms of geometry
Eliakim Hastings
Moore
142-158
Geometry of a simultaneous system of two linear homogeneous differential equations of the second order
E. J.
Wilczynski
343-362
Theory of linear groups in an arbitrary field
Leonard Eugene
Dickson
363-394
On certain aggregates of determinant minors
W. H.
Metzler
395-403
Ueber die Anwendung der Cauchy'schen Multiplicationsregel auf bedingt convergente oder divergente Reihen
Alfred
Pringsheim
404-412
Ueber den Goursat'schen Beweis des Cauchy'schen Integralsatzes
Alfred
Pringsheim
413-421
New proof of a theorem of Osgood's in the calculus of variations
Oskar
Bolza
422-427
On certain pairs of transcendental functions whose roots separate each other
Maxime
Bôcher
428-436
On the system of a binary cubic and quadratic and the reduction of hyperelliptic integrals of genus two to elliptic integrals by a transformation of the fourth order
John Hector
McDonald
437-458
On the theory of improper definite integrals
Eliakim Hastings
Moore
459-475
On the convergence and character of the continued fraction $\frac{a\sb 1z}1{{}\atop +}\frac{a\sb 2z}1{{}\atop +}\frac{a\sb 3z}1{{}\atop +}\cdots$
Edward B.
Van Vleck
476-483
Errata: ``On the existence of the Green's function for the most general simply connected plane region'' [Trans. Amer. Math. Soc. {\bf 1} (1900), no. 3, 310--314; 1500539]
W. F.
Osgood
484-485
Errata: ``The invariant theory of the inversion group: geometry upon a quadric surface'' [Trans. Amer. Math. Soc. {\bf 1} (1900), no. 4, 430--498; 1500550]
Edward
Kasner
485
Errata: ``A simple proof of the fundamental Cauchy-Goursat theorem'' [Trans. Amer. Math. Soc. {\bf 1} (1900), no. 4, 499--506; 1500551]
Eliakim Hastings
Moore
486
Errata: ``Invariants of systems of linear differential equations'' [Trans. Amer. Math. Soc. {\bf 2} (1901), 1--24; 1500552]
E. J.
Wilczynski
486
Errata: ``On the reduction of the general Abelian integral'' [Trans. Amer. Math. Soc. {\bf 2} (1901), no. 1, 49--86; 1500556]
J. C.
Fields
486
Errata: ``On the convergence of continued fractions with complex elements'' [Trans. Amer. Math. Soc. {\bf 2} (1901), no. 3, 215--233; 1500565]
E. B.
Van Vleck
486
Errata: ``Concerning Harnack's theory of improper definite integrals'' [Trans. Amer. Math. Soc. {\bf 2} (1901), no. 3, 296--330; 1500570]
Eliakim Hastings
Moore
487
On the convergence of continued fractions with complex elements
E. B.
Van Vleck
215-233
Geometry within a linear spherical complex
Percey F.
Smith
234-248
A new determination of the primitive continuous groups in two variables
H. F.
Blichfeldt
249-258
Determination of all the groups of order $p\sp m$ which contain the abelian group of type $(m-2,1), p$ being any prime
G. A.
Miller
259-272
On a fundamental property of a minimum in the calculus of variations and the proof of a theorem of Weierstrass's
W. F.
Osgood
273-295
Concerning Harnack's theory of improper definite integrals
Eliakim Hastings
Moore
296-330
Zur linearen Transformation der $\theta$-Reihen
F.
Mertens
331-342
Canonical forms of quaternary abelian substitutions in an arbitrary Galois field
Leonard Eugene
Dickson
103-138
Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence
Maxime
Bôcher
139-149
An elementary proof of a theorem of Sturm
Maxime
Bôcher
150-151
On the determination of surfaces capable of conformal representation upon the plane in such a manner that geodetic lines are represented by algebraic curves
Henry Freeman
Stecker
152-165
On the geometry of planes in a parabolic space of four dimensions
Irving
Stringham
183-214
Invariants of systems of linear differential equations
E. J.
Wilczynski
1-24
Divergent and conditionally convergent series whose product is absolutely convergent
Florian
Cajori
25-36
Sets of coincidence points on the non-singular cubics of a syzygetic sheaf
M. B.
Porter
37-42
Note on non-quaternion number systems
Wendell M.
Strong
43-48
On the reduction of the general Abelian integral
J. C.
Fields
49-86
Ueber Fl\"achen von constanter Gaussscher Kr\"ummung
David
Hilbert
87-99
Note on the functions of the form $f(x)\equiv\phi(x)+a\sb 1x\sp {n-1}+a\sb 2x\sp {n-2}+\cdots+a\sb n$ which in a given interval differ the least possible from zero
H. F.
Blichfeldt
100-102
On surfaces enveloped by spheres belonging to a linear spherical complex
Percey F.
Smith
371-390
On certain relations among the theta constants
J. I.
Hutchinson
391-394
On the groups which have the same group of isomorphisms
G. A.
Miller
395-401
Die Hessische und die Cayley'sche Curve
Paul
Gordan
402-413
Application of a method of d'Alembert to the proof of Sturm's theorems of comparison
Maxime
Bôcher
414-420
Two plane movements generating quartic scrolls
E. M.
Blake
421-429
The invariant theory of the inversion group: geometry upon a quadric surface
Edward
Kasner
430-498
A simple proof of the fundamental Cauchy-Goursat theorem
Eliakim Hastings
Moore
499-506
Errata: ``On certain crinkly curves'' [Trans. Amer. Math. Soc. {\bf 1} (1900), no. 1, 72--90; 1500526]
Eliakim Hastings
Moore
507
Errata: ``Plane cubics and irrational covariant cubics'' [Trans. Amer. Math. Soc. {\bf 1} (1900), no. 2, 170--181; 1500530]
Henry S.
White
508
Errata: ``A new method of determining the differential parameters and invariants of quadratic differential quantics'' [Trans. Amer. Math. Soc. {\bf 1} (1900), no. 2, 197--204; 1500533]
Heinrich
Maschke
508
Errata: ``Wave propagation over non-uniform electrical conductors'' [Trans. Amer. Math. Soc. {\bf 1} (1900), no. 3, 259--286; 1500536]
M. I.
Pupin
509
Errata: ``On linear criteria for the determination of the radius of convergence of a power series'' [Trans. Amer. Math. Soc. {\bf 1} (1900), no. 3, 293--309; 1500538]
Edward B.
Van Vleck
509
Errata: ``An application of group theory to hydrodynamics'' [Trans. Amer. Math. Soc. {\bf 1} (1900), no. 3, 339--352; 1500542]
E. J.
Wilczynski
509
Wave propagation over non-uniform electrical conductors
M. I.
Pupin
259-286
Ueber Systeme von Differentialgleichungen denen vierfach periodische Functionen Gen\"uge leisten
Martin
Krause
287-292
On linear criteria for the determination of the radius of convergence of a power series
Edward B.
Van Vleck
293-309
On the existence of the Green's function for the most general simply connected plane region
W. F.
Osgood
310-314
``$D$'' lines on quadrics
Alexander
Pell
315-322
Sundry metric theorems concerning $n$ lines in a plane
Frank H.
Loud
323-338
An application of group theory to hydrodynamics
E. J.
Wilczynski
339-352
Determination of an abstract simple group of order $2\sp 7\cdot3\sp 6\cdot 5\cdot 7$ holoedrically isomorphic with a certain orthogonal group and with a certain hyperabelian group
Leonard Eugene
Dickson
353-370
On the metric geometry of the plane $n$-line
F.
Morley
97-115
On relative motion
Alexander S.
Chessin
116-169
Plane cubics and irrational covariant cubics
Henry S.
White
170-181
A purely geometric representation of all points in the projective plane
Julian Lowell
Coolidge
182-192
The decomposition of the general collineation of space into three skew reflections
Edwin B.
Wilson
193-196
A new method of determining the differential parameters and invariants of quadratic differential quantics
Heinrich
Maschke
197-204
On the extension of Delaunay's method in the lunar theory to the general problem of planetary motion
G. W.
Hill
205-242
On the types of linear partial differential equations of the second order in three independent variables which are unaltered by the transformations of a continuous group
J. E.
Campbell
243-258
Conics and cubics connected with a plane cubic by certain covariant relations
Henry S.
White
1-8
Formentheoretische Entwickelung der in Herrn White's Abhandlung \"uber Curven dritter Ordnung enthaltenen S\"atze
Paul
Gordan
9-13
Sur la d\'efinition g\'en\'erale des fonctions analytiques, d'apr\`es Cauchy
E.
Goursat
14-16
On a class of particular solutions of the problem of four bodies
Forest Ray
Moulton
17-29
Definition of the abelian, the two hypoabelian, and related linear groups as quotient-groups of the groups of isomorphisms of certain elementary groups
Leonard Eugene
Dickson
30-38
Note on the unilateral surface of Moebius
Heinrich
Maschke
39
On regular singular points of linear differential equations of the second order whose coefficients are not necessarily analytic
Maxime
Bôcher
40-52
The elliptic $\sigma$-functions considered as a special case of the hyperelliptic $\sigma$-functions
Oskar
Bolza
53-65
On the groups which are the direct products of two subgroups
G. A.
Miller
66-71
On certain crinkly curves
Eliakim Hastings
Moore
72-90
A new definition of the general abelian linear group
Leonard Eugene
Dickson
91-96